P W i i *m£4J*+Sd< -.'::■ :-■■ 0; erm a ' •' ■' ' - ■ jB5B Proceedings of lh Seventh Conference l^Si nffii nraK IS ■■■■HbI SSXaS Hi ■■■■■1 ' rt ftg n£fiK£i£tta| ' IS ' HHHi - ■■■■■I ffiBEHH mi ■nH United bta MI|l!HfflllMHn»«J^»l«JM)< National Bureau of Standards Special Publication 302 CMP Y'** TrA-' ShcsESopwk BBS B flfr^ofra J sHfej C I 3. i o ". 5cs-j_ (\" cy" Digitized by the Inte in 2013 http://archive.org/details/thermalconductivOOflyn UNITED STATES DEPARTMENT OF COMMERCE C. R. Smith, Secretary NATIONAL BUREAU OF STANDARDS • A. V. Astin, Director Thermal Conductivity Proceedings of the Seventh Conference Held at the National Bureau of Standards, Gaithersburg, Maryland November 13-16, 1967 Edited by Daniel R. Flynn and Bradley A. Peavy, Jr. Institute for Applied Technology- National Bureau of Standards Washington, D.C. 20234 National Bureau of Standards Special Publication 302 Issued September 1968 For sale by the Superintendent of Documents U.S. Government Printing Oflleo Washington, D.C. 20402 - Price $8.26 Library of Congress Catalog Card Number: 68-61552 FOREWORD The determination of the thermal and physical properties of materials and the development of methods of making such determinations are continuing responsibilities of the National Bureau of Standards. The Bureau carries out these responsibilities through its own research program, by pro- viding services that support and stimulate research in other laboratories, and by encouraging and promoting the dissemination of relevant information. To provide a forum for exchange of information and ideas among specialists in the field of heat conduction, the Bureau sponsored the "Seventh Conference on Thermal Conductivity", held November 13 - 16, 1967 in the new NBS Laboratories at Gaithersburg, Maryland. Over 180 scientists and engineers, representing industry, government, and university laboratories in the United States and ten other countries, participated. This volume contains the text or abstracts of the ninety papers contributed at this Conference. A. V. ASTIN, DIRECTOR NATIONAL BUREAU OF STANDARDS HI Abstract The Seventh Conference on Thermal Conductivity was held at the National Bureau of Standards on November 13 - 16, 1967. This volume contains the texts of the papers presented. Topics covered include surveys of the present state of knowledge regarding the thermal conductivity of different materials, descriptions of different apparatuses for measuring thermal conductivity, new experimental data on the thermal conductivity or diffusivity of a variety of materials, and correlations between experimental results and theoretical predictions. Key Words: Conductance, conductivity, contact conductance, contact resistance, electrical conductivity, electrical resistivity, heat transfer, Lorenz function, resistivity, temperature, thermal conductivity, thermal diffusivity, thermal resistivity, thermophysical properties. IV Preface This conference was the seventh in a series of annual conferences which began in 1961. The first conference stemmed from an informal discussion of thermal conductivity measurement methods by Messrs. D. L. McElroy, Oak Ridge National Laboratory, C. F. Lucks and H. W. Deem, Battelle Memorial Institute, D. R. Flynn, National Bureau of Standards, and A. I. Dahl, General Electrical Company, while attending the Symposium on Temperature, Its Measurement and Control in Science and Industry, in Columbus, Ohio. This group believed it would be very worthwhile to expand the exchange of ideas and experiences to an informal conference on thermal conductivity measurement methods applicable to solid materials at ele- vated temperatures. Battelle Memorial Institute agreed to develop and sponsor such a conference. During the first conference, which was held in October, 1961, M. J. Laubitz of the National Research Council of Canada agreed to sponsor a second conference the following year. Each subsequent year a different laboratory has volunteered to host the next conference. The sponsor and time of each of these conferences are listed below: Sponsoring Organization Chairman Date Battelle Memorial Institute C. F. Lucks October , 1961 National Research Council (Canada) M. J. Laubitz October , 1962 Oak Ridge National Laboratory D. L. McElroy October , 1963 U.S. Naval Radiological Defense Laborat ory R. L. Rudkin October , 1964 University of Denver J. D. Plunkett October , 1965 Air Force Materials Laboratory M. L. Minges and G. L Denman October , 1966 National Bureau of Standards D. R. Flynn and B. A. Peavy November , 1967 The keen interest in and active enthusiasm for these conferences is evidenced by the fact that they have grown larger in attendance and scope, and moreover, have been self-perpetuating in that there has been no single sponsoring organization or society pressing for their continuance. The first six thermal conductivity conferences were informal in the sense that the proceedings of these conferences were not formally published. A limited number of copies of the proceedings were printed for distribution to active workers in the field. These previous proceedings were not supposed to be referenced in the open literature. The informal proceedings of these first six conferences contained approximately 200 technical papers plus a number of shorter progress reports; these proceed- ings occupy over 1 foot of shelf space and weigh almost 28 pounds. Many of the papers in these V proceedings have never been published formally. These considerations led us to the conclusion that a great deal of valuable information was not reaching the open literature and led us to the decision that the proceedings of the Seventh Conference on Thermal Conductivity would be published formally. The Eighth Conference on Thermal Conductivity will be held in October, 1968, under the sponsorship of the Thermophysical Properties Research Center and the chairmanship of C. Y. Ho and R. E. Taylor. DANIEL R. FLYNN BRADLEY A. PEAVY, JR. April 1, 1968 VI Introduction The last two decades have witnessed a very large growth in the amount of activity in the various fields of thermophysical property measurement. Thermal conductivity is no exception. Engineering data on thermal conductivity are needed for a wide variety of heat transfer applications in such diverse industries as nuclear, aerospace, housing, cryogenics, electronics, and food processing. Since the thermal conductivity of many materials is extremely sensitive to trace impurities or small imperfec- tions, the measurement of thermal conductivity has also become a useful tool for the study of other properties of matter. This book represents the formal proceedings of the Seventh Conference on Thermal Conductivity, which was held at the new site of the National Bureau of Standards in Gaithersburg, Maryland, from November 13 through November 16, 1967. This Conference brought together over 180 leading authorities, both from the United States and abroad, in the fields which are concerned with conductive heat transport in solids, liquids, gases, and phase mixtures. In this book are included the text or abstract of nearly all of the papers which were orally presented at the conference. In addition, a few papers are included which were no.t pre- sented orally because of travel problems. We thank the session chairmen for an excellent job of conducting their sessions and for their assistance in reviewing and editing the manuscripts. Particular thanks go to W. L. Carroll for his able assistance in handling much of the pre-conference arrangements and correspondence and in editing the manuscripts. The NBS Office of Technical Information and Publications, under the direction of W. R. Tilley, gave invaluable assistance in different areas ranging from the excellent handling of the logistics of the Conference by R. T. Cook, G. T. Leighty, and coworkers to the final publication of this book under the direction of J. E. Carpenter and his coworkers. DANIEL R. FLYNN BRADLEY A. PEAVY, JR. April 1, 1968 VII Contents Page Foreword Preface Introduction III V VII INTRODUCTORY SESSION Chairman: D. L. McElroy Oak Ridge National Laboratory Oak Ridge, Tennessee 1. The State of Knowledge Regarding the Thermal Conductivity of the Metallic Elements R. W. Powell and C. Y. Ho Thermophysical Properties Research Center Purdue University West Lafayette, Indiana 2. The State of Knowledge Regarding the Thermal Conductivity of the Non-Metallic Elements--Those Solid at Normal Temperature C. Y. Ho and R. W. Powell Thermophysical Properties Research Center Purdue University West Lafayette, Indiana 3. Some Aspects Concerning Thermal Conductivity Data of Liquids and Proposals for New Standard Reference Materials H. Poltz Physikalisch-Technische Bundesanstalt Braunschweig, Germany 4. The Thermal Conductivity of Non-Metallic Elements (In Liquid or Gaseous State at NTP) In Solid, Saturated and Atmospheric Pressure States P. E. Liley Thermophysical Properties Research Center Purdue University West Lafayette, Indiana 5. Thermal Conductivity of Vitreous Silica: Selected Values Lois C. K. Carwile and Harold J. Hoge Pioneering Research Laboratory U. S. Army Natick Laboratories Natick, Massachusetts 6. Development of High Temperature Thermal Conductivity Standards 33 47 57 59 77 Alfred E. Wechsler Arthur D. Little, Inc. Cambridge, Massachusetts Merrill L. Minges Air Force Materials Laboratory Wright-Patterson Air Force Base, Ohio VHI Page 7. Glass Beads--A Standard for the Low Thermal Conductivity Range? 89 Alfred E. Wechsler Arthur D. Little, Inc. Cambridge, Massachusetts THEORY -MAINLY PHONONS Chairman: P. G. Klemens University of Connecticut Storrs, Connecticut 8. Lattice Thermal Conductivities of Semiconductor Alloys 97 I . Kudman RCA Laboratories Princeton, New Jersey 9. Thermal Conductivity of N-type Germanium from 0.3°K to 4.2°K 103 Bruce L. Bird and Norman Pearlman Department of Physics Purdue University Lafayette, Indiana 10. The Lattice Thermal Conductivity of Bismuth Telluride and 111 Some Bismuth Telluride - Bismuth Selenide Alloys D. H. Damon and R. W. Ure , Jr. J. Gersi Westinghouse Research Laboratories Westinghouse Atomic Equipment Division Pittsburgh, Pennsylvania Cheswick, Pennsylvania 11. Thermal Conductivity of Cadmium Arsenide from 60 to 400°K 123 D. P. Spitzer American Cyanamid Company Stamford, Connecticut 12. Measurement of Dislocation Phonon Scattering in Alloys 131 P. Charsley, A. D. W. Leaver and J. A. M. Salter Department of Physics, University of Surrey Battersea Park Road London, England 1J. The Thermal Resistivity Resulting From the Combined Scattering 139 of Phonons by Edge Dislocation Dipoles and by Normal Processes P. P. Gruner Boeing Scientific Research Laboratories Seattle, Washington METHODS AND MATHEMATICAL ANALYSIS Chairman: M. J. Laubitz National Research Council Ottawa, Ontario, Canada 14. Thermal Diffusivity Measurements With Stationary Waves Method 151 A. Sacchi, V. Ferro and C. Codegone Instituto di Fisica Tecnica del Politecnico di Torino (Italy) IX Page 15. Rotating Thermal Field 163 C. Codegone, V. Ferro and A. Sacchi Istituto di Fisica Tecnica, Politecnico Turin, (Italy) 16. Thermal Attenuation through Homogeneous and Multilayer Slabs 177 in Steady Periodic Conditions - Theory and Experiments V. Ferro, A. Sacchi and C. Codegone Istituto di Fisica Tecnica, Politecnico Turin, (Italy) 17. Heat Losses in a Cut-Bar Apparatus: Experimental-Analytical 197 Comparisons Merrill L. Minges U. S. Air Force Materials Laboratory Wright-Patterson Air Force Base Dayton, Ohio 18. Application of Neutron Diffusion Theory Techniques to 207 Solution of the Conduction Equation I. Catton and J. R. Lilley Douglas Aircraft Company Santa Monica, California 19. Conduction from a Finite-Size Moving Heat Source Applied to 209 Radioisotope Capsule Self-Burial C. R. Easton Nuclear Technology and Subsystems Research and Development Missile and Space Systems Division Douglas Aircraft Company Santa Monica, California 20. Evaluation of Steady-Periodic Heat Flow Method for Measuring the 219 Thermal Diffusivity of Materials with Temperature-Dependent Properties C. J. Shirtliffe and D. G. Stephenson Davision of Building Research National Research Council of Canada Ottawa, Canada 21. Comparison of Modes of Operation for Guarded Hot Plate Apparatus 229 with Emphasis on Transient Characteristics C. J. Shirtliffe and H. W. Orr Division of Building Research National Research Council of Canada Ottawa, Canada 22. Method for Measuring Total Hemispheric Emissivity of Plane Surfaces 241 with Conventional Thermal Conductivity Apparatus Nathaniel E. Hager, Jr. Research and Development Center Armstrong Cork Company Lancaster, Pennsylvania X Page METALS: VERY LOW TEMPERATURES Chairman: R. L. Powell National Bureau of Standards Boulder, Colorado 23. Heat Pulse Experiments and the Study of Thermal Transport 247 at Low Temperatures R. J. von Gutfeld IBM Watson Research Center Yorktown Heights, New York 24. Deviations from Matthiessen' s Rule in the Low Temperature 249 Thermal and Electrical Resistivities of Very Pure Copper J. T. Schriempf Metallurgy Division Naval Research Laboratory Washington, D. C. 25. Thermal Conduction in Bismuth at Liquid Helium Temperatures- 253 Effects at Intermediate Fields S. M. Bhagat and B. Winer University of Maryland College Park, Maryland 26. Thermal Conductivity of Pure and Impure Tin in the Normal 257 and Superconducting States C. A. Reynolds, J. E. Gueths, F. V. Burckbuchler , N. N. Clark, D. Markowitz, G. J. Pearson, R. H. Bartram and C. W. Ulbrich Physics Department and Institute of Materials Science University of Connecticut Storrs, Connecticut 27. Temperature Dependence of the Lattice Thermal Conductivity 259 of Copper-Nickel Alloys at Low Temperatures J. C. Erdmann and J. A. Jahoda Boeing Scientific Research Laboratories P. 0. Box 3981 Seattle, Washington 28. Thermal Conductivity of Aerospace Alloys at Cryogenic Temperatures 271 J. G. Hust, D. H. Weitzel and Robert L. Powell Cryogenics Division Institute for Materials Research National Bureau of Standards Boulder, Colorado XI Page METALS: INTERMEDIATE TEMPERATURES Chairman: R. W. Powell Thermophysical Properties Research Center Purdue, University West Lafayette, Indiana 29. Physical Properties of Indium from 77 to 350°K 279 M. Barisoni, R. K. Williams and D. L. McElroy Metals and Ceramics Division Oak Ridge National Laboratory Oak Ridge, Tennessee 30. Thermal Conductivity of Aluminum Between About 78 and 373°K 293 K. E. Wilkes and R. W. Powell Thermophysical Properties Research Center Purdue University West Lafayette, Indiana Jl. Physical Properties of Chromium from 77 to 400"K 297 J. P. Moore, R. K. Williams and D. L. McElroy Metals and Ceramics Division Oak Ridge National Laboratory Oak Ridge, Tennessee 32. The Thermal Conductivity of Chromium Above and Below the 311 Neel Temperature - An Analysis J. F. Goff U. S. Naval Ordnance Laboratory White Oak, Silver Spring, Maryland 33> Thermal Conductivity of a Round-Robin Armco Iron Sample 323 D. C. Larsen and R. W. Powell Thermophysical Properties Research Center Purdue University West Lafayette, Indiana 34. High Temperature Transport Properties of Some Platinel Alloys 325 M. J. Laubitz and M. P. van der Meer Division of Applied Physics National Research Council Ottawa, Canada 35. The Thermal Diffusivity of Gold 331 H. R. Shanks, M. M. Burns and G. C. Danielson Institute for Atomic Research & Department of Physics Iowa State University, Ames, Iowa 36. Experimental Determination of the Thermal Conductivity of 337 Solids Between 90 and 200°K J. G. Androulakis and R. L. Kosson Thermodynamics Section Grumman Aircraft Engineering Corporation Bethpage, New York XII Page METALS: INTERMEDIATE AND HIGH TEMPERATURES Chairman: M. L. Minges Air Force Materials Laboratory Wright-Patterson Air Force Base Dayton, Ohio 37. The Thermal Conductivities of Several Metals: An Evaluation of 349 A Method Employed By the National Bureau of Standards David R. Williams and Harold A. Blum SMU Institute of Technology Dallas, Texas 38. A Longitudinal Symmetrical Heat Flow Apparatus for the Determination 355 of the Thermal Conductivity of Metals and their Alloys H. Chang and M. G. Blair Energy Conversion Division Nuclear Materials and Equipment Corporation Leechburg, Pennsylvania 39. The Intrinsic Electronic Thermal Resistivity of Iron 365 M. S. R. Chari National Physical Laboratory of India New Delhi , India 40. The Thermal Conductivity and Electrical Resistivity of 371 Tungsten-26 Weight Percent Rhenium A. D. Feith Nuclear Materials and Propulsion Operation General Electric Company Cincinnati, Ohio 41. Thermal Conductivity of Magnesium Stannide 381 J. J. Martin, H. R. Shanks and G. C. Danielson Institute for Atomic Research and Department of Physics Iowa State University, Ames, Iowa NON-METALLICS AND GRAPHITES Chairman: J. Kaspar Aerospace Corporation Los Angeles, California 42. Thermal Properties of SNAP Fuels 387 C. C. Weeks, M.M. Nakata and C. A. Smith Atomics International A Division of North American Rockwell Corporation Canoga Park, California 43. Thermal Conductivity, Diffusivity, and Specific Heat of 399 Calcium Tungstate from 77° to 300°K Phillipp H. Klein Materials Branch NASA Electronics Research Center Cambridge, Massachusetts xni Page 44. Measurement of the Thermal Conductivity of Self-Supporting 405 Thin Films of Aluminum Oxide 325 to 2000 Angstroms Thick G. A. Shirfin and Robert W. Gammon Hughes Research Laboratory Malibu, California 45. Impulse Thermal Breakdown in Silicon Oxide Films 407 N. Klein and E. Burstein Department of Electrical Engineering Technion, Israel Institute of Technology Haifa, Israel 46. High Temperature Radial Heat Flow Measurements 415 J. P. Brazel K. H. Styhr General Electric Company National Beryllia Corporation Re-Entry Systems Department Haskell, New Jersey Philadelphia, Pennsylvania 47. An Investigation of the Mechanisms of Heat Transfer in 425 Low-Density Phenolic-Nylon Chars E. D. Smyly and C. M. Pyron, Jr. Southern Research Institute Birmingham, Alabama 48. Biological Specimen Holder for Thermal Diffusivity Determinations 455 Floyd V. Matthews, Jr. Carl W. Hall Agricultural Engineering Department Agricultural Engineering Department University of Maryland Michigan State University College Park, Maryland East Lansing, Michigan NUCLEAR MATERIALS Chairman: A. D. Feith General Electric Company Cincinnati, Ohio 49. Thermal Diffusivity of Actinide Compounds 461 J. B. Moser and 0. L. Kruger Argonne National Laboratory Argonne, Illinois 50. The Thermal Diffusivity and Thermal Conductivity of 467 Sintered Uranium Dioxide J. B. Ainscough and M. J. Wheeler The General Electric Company Limited Central Research Laboratories Hirst Researcb Centre Wembley, England 51. Suitability of the Flash Method' for Measurements of the Thermal 469 Diffusivity of Uranium Dioxide Specimens Containing Cracks D. Shaw, L. A. Goldsmith and A. J. Little Reactor Development Laboratories U. K. Atomic Energy Authority, Windscale Works Sellafield, Seascale Cumberland, England XIV Page 52. Thermal Conductivity of Uranium Oxycarbides 477 J. Lambert Bates Battelle Memorial Institute Pacific Northwest Laboratory Richland, Washington 53- The Effect of Irradiation on the Thermal Conductivity of Some 491 Fissile Ceramics in the Range 150-1600°C Up To a Dose of 10 21 Fissions cm~5 D. J. Clough United Kingdom Atomic Energy Authority Metallurgy Division Harwell, England 54. Phase Studies on Fueled Zirconium Hydride 493 M. M. Nakata, C. A. Smith and C. C. Weeks Atomics International A Division of North American Rockwell Corporation Canoga Park, California BUILDING ELEMENTS, ROCKS AND SOILS Chairman: H. E. Robinson National Bureau of Standards Washington, D. C. 55. On the Development of Methods for Measuring Heat Leakage 495 of Insulated Walls with Internal Convection G. Lorentzen, E. Brendeng and P. Frivik Norges Tekniske Hogskole Trondheim, Norway 56. Thermal Diffusivity Measurements from a Step Function Change 507 in Flux Into a Double Layer Infinite Slab E. K. Halteman and R. W. Gerrish, Jr. Pittsburgh Corning Corporation Pittsburgh, Pennsylvania 57. A Guarded Hot Plate Apparatus for Measuring Thermal Conductivity 513 From -80 +100 °C Jean-Claude Rousselle Laboratorire National d'Essais (L.N.E.) Paris, France 58. A Study of the Effects of Edge Insulation and Ambient Temperatures 521 On Errors in Guarded Hot-Plate Measurements H. W. Orr Division of Building Research National Research Council of Canada Ottawa, Canada 59. Ring Heat Source Probes for Rapid Determination of 527 Thermal Conductivity of Rocks Wilbur H. Somerton and Mohammad Mossahebi University of California, Berkeley National Iranian Oil Company Tehran, Iran XV Page 60. Thermal Conductivity and Thermal Diffusivity of Green 529 River Oil Shale S. S. Tihen, H. C. Carpenter and H. W. Sohns Laramie Petroleum Research Center Bureau of Mines, U. S. Department of the Interior Laramie, Wyoming 61. Experimental Study Relating Thermal Conductivity to 537 Thermal Piercing of Rocks V. V. Mirkovich Mineral Processing Division, Mines Branch Department of Energy, Mines and Resources Ottawa, Canada GASES Chairman: C. F. Bonilla Columbia University New York, New York 62. Measurement of Thermal Conductivity of Gases 539 S. C. Saxena Thermophysical Properties Research Center Purdue University West Lafayette, Indiana 63. Thermal Conductivity of Gases: Hydrocarbons at Normal Pressures 547 Dipak Roy and George Thodos Northwestern University Evanston, Illinois 64. The Thermal Conductivity of 46 Gases at Atmospheric Pressure 549 P. E. Liley Thermophysical Properties Research Center Purdue University West Lafayette, Indiana 65. Calculation of Total Thermal Conductivity of Ionized Gases 551 Warren F. Ahtye Vehicle Environment Division Ames Research Center, NASA Moffett Field, California 66. Thermal Conductivity of the Alkali Metal Vapors and Argon 561 Chai-sung Lee and Charles F. Bonilla Liquid Metals Research Laboratory Department of Chemical Engineering Columbia University New York, New York 67. Recent Developments at Bellevue on Thermal Conductivity 579 Measurements of Compressed Gases B. Le Neindre, P. Bury, R. Tufeu, P. Johannin and B. Vodar Laboratoire des Hautes Pressions, C.N.R-S. 1, Place A. Briand, Bellevue, France XVI Page 68. Theoretical Developments on the Density Dependence of the 595 Thermal Conductivity of Compressed Gases J. V. Sengers National Bureau of Standards Washington, D. C. 69. Discrepancies Between Viscosity Data For Simple Gases 597 H. J. M. Hanley and G. E. Childs Cryogenic Data Center Institute for Materials Research National Bureau of Standards Boulder, Colorado 70. Thermal Conductivity of Binary, Ternary and Quaternary 605 Mixtures of Polyatomic Gases S. C. Saxena G. P. Gupta Thermophysical Properties Research Center Department of Physics Purdue University University of Raj as than West Lafayette, Indiana Jaipur, Rajasthan, India 71. Thermal Conductivities of Gaseous Mixtures of Diethyl Ether 615 with Inert Gases P. Gray, S. Holland and A. 0. S. Maczek School of Chemistry, The University Leeds 2, England 72. Determination of the Eucken Factor for Parahydrogen at 77°K 627 Lee B. Harris Research Laboratories Xerox Corporation Rochester, New York LIQUIDS Chairman: E. McLaughlin Louisiana State University Baton Rouge, Louisiana 73- Prediction of Minor Heat Losses in a Thermal Conductivity Cell 633 and Other Calorimeter Type Cells D. R. Tree and W. Leidenfrost Mechanical Engineering Purdue University Lafayette, Indiana 74. The Thermal Conductivity of Pure Organic Liquids 659 J. E. S. Venart and C. Rrishnamurthy Faculty of Engineering The University of Calgary Calgary, Alberta, Canada 75. Pressures and Temperature Dependence of the Thermal Conductivity of Liquids 671 E. McLaughlin Chemical Engineering Department Louisiana State University Baton Rouge, Louisiana XVII Page 76. Thermal Conductivity of Several Dielectric Liquids 677 of Low Boiling Point R. C. Chu, 0. Gupta, J. H. Seely International Business Machines Corporation Systems Development Division Poughkeepsie , New York TWO-PHASE SYSTEMS Chairman: A. E. Wechsler Arthur D. Little, Inc. Cambridge, Massachusetts 77. Thermal Conductivity of Two-Phase Systems 685 Alfred L. Baxley, Nicholas C. Nahas and James R. Couper Department of Chemical Engineering University of Arkansas Fayetteville , Arkansas 78. Measurements of the Thermal Conductivity of Granular Carbon and 695 The Thermal Contact Resistance at the Container Walls C. A. Fritsch and P. E. Prettyman Bell Telephone Laboratories, Incorporated Whippany, New Jersey 79. The Thermal Conductivity of Thoria Powder From 400° to 1200°C 703 In Various Gases at Atmospheric Pressure A. D. Feith Nuclear Materials and Propulsion Operation General Electric Company Cincinnati, Ohio 80. Thermal Conductivity of a 58% Dense MgO Powder in Nitrogen 711 J. P. Moore, D. L. McElroy and R. S. Graves Oak Ridge National Laboratory Oak Ridge, Tennessee 81. Measurements of the Thermal Conductivity of Glass Beads in 713 a Vacuum at Temperatures from 100° to 500°K R. B. Merrill Space Sciences Laboratory The George C. Marshall Space Flight Center Huntsville, Alabama 82. The Measurement of the Effective Thermal Conductivities of Well-Mixed 721 Porous Beds of Dissimilar Solid Particles by use of The Thermal Conductivity Probe C. S'. Beroes and H. D. Hatters Chemical and Petroleum Engineering Department University of Pittsburgh Pittsburgh, Pennsylvania 83. Thermal Conductivity Measurements on a Fibrous Insulation Material 729 R. R. Pettyjohn General Dynamics Corporation Convair Division San Diego, California XVIII Page 84. The Effect of Thickness and Temperature on Heat Transfer 737 Through Foamed Polymers T. T. Jones Rideal Research Laboratories Monsanto Chemicals Limited Corporation Road Newport, Monmouthshire, England 85. Measurements of In-Vivo Thermal Diffusivity of Cat Brain 749 G. J. Trezek, D. L. Jewett and T. E. Cooper University of California Berkeley, California THERMAL CONTACT CONDUCTANCE Chairman: E. Fried General Electric Company Philadelphia, Pennsylvania 86. A Correlation for Thermal Contact Conductance of 755 Nominally-Flat Surfaces in a Vacuum C. L. Tien University of California Berkeley, California 87. Thermal Conductance of Imperfect Contacts 761 Evan Charles Brown, Jr., and Vernon Emerson Holt North Carolina State University Raleigh, North Carolina 88. Ultrasonic Measurement of the Thermal Conductance of Joints in Vacuum 769 Ludwig Wolf, Jr., and Constantine Kostenko IIT Research Institute Chicago, Illinois 89. Thermal Resistance of Sapphire-Sapphire Contact 777 Y. Baer Laboratorium fur Festkorperphysik ETH, Zurich, Switzerland 90. Heat Transfer Across Surfaces In Contact: Effects of Thermal 787 Transients on One-Dimensional Composite Slabs C. J. Moore, Jr. H. A. Blum North Carolina State University Southern Methodist University Raleigh, North Carolina Dallas, Texas XIX The State of Knowledge Regarding the Thermal Conductivity of the Metallic Elements R. W. Powell and C. Y. Ho Thermophysical Properties Research Center, Purdue University West Lafayette, Indiana 47906 The program of the Thermophysical Properties Research Center has since its inception included the extraction, evaluation, and publication of all thermal conductivity data that have been combed from a major portion of the world literature. The values so de- rived for many metallic elements have now been critically assessed. Some examples are presented which indicate the problems involved in arriving at probable thermal conductivity values from the limited information. Much more work is still required, and the elements and temperature ranges are clearly indicated for which the investigation shows the need for additional measurements, in- deed for 19 metals a first measurement has yet to be reported. The provisional recommendations are presented as logarithmic plots of thermal conductivity against temperature. Key Words: Conductivity, critical evaluation, elements, heat conductivity, liquid metals, metallic elements, metals, most probable values, provisional recommendations, solids, state of knowledge, thermal conductivity. 2 1. Introduction In many fields of scientific investigation information centers have been estab- lished and at these centers groups of workers are engaged in combing and storing the ever-increasing output of literature relating to certain specified topics so that others may readily receive answers to relevant aspects of their current problems with- out making costly independent searches. The Thermophysical Properties Research Center (TPRC) was one of the earliest of such centers, and, for rather more than a decade, thermal conductivity, the property that forms the subject of this Conference, has been one of several properties investigated by that Center. A more recent development has been the work promoted by the National Standard Reference Data System which seeks to make readily available the quantitative data of physical science after it has been critically evaluated. The National Bureau of Standards has the responsibility for administering this effort, and, on their behalf, the TPRC is endeavouring to assess critically the available data for the thermal con- ductivity of certain substances. Not only does the progress of this work relate closely to the programme of this Conference, but there is a more important reason why the authors welcome this opportunity to present these papers at this time. They seek your help and that of others who will read these Proceedings. This help may be of the immediate or short-term type that could lend support to a tentative suggested value, or, what is more important, could disagree with it so convincingly as to lead to re- vision and a more satisfactory final set of data. Maybe relevant information has been overlooked or new values are known to be in course of publication. Or the help may be long term. TPRC would like to receive reprints of subsequent publications in this field. Senior Researcher and Head of Reference Data Tables Division, respectively. 2 Much of this Introduction also applies to the accompanying paper by C. Y. Ho and R. W. Powell and to that of P. E„ Liley. .. The following outlines the position reached at the time of writing. The report [l] for the first years' work dealt with the thermal conductivity of the metallic elements, aluminum, copper, gold, iron (also Armco iron) , mercury, platinum, silver and tungsten, the oxides of aluminum, beryllium, magnesium, silicon, thorium and titanium, two alloys, diamond, two glasses, liquid and gaseous argon, helium, nitrogen, carbon tetrachloride, toluene, diphenyl and water and of two other liquids. The report for the second year [2] dealt with the metallic elements, cadmium, chromium, lead, magnesium, molybdenum, nickel, niobium, tantalum, tin, titanium, zinc and zirconium, several grades of graphite and gaseous acetone, ammonia and methane. In the case of gases and liquids (not liquid metals) most probable values are tabulated usually at 10 deg intervals and departure plots of the percent difference between available and recommended values are included. For each of the other materials each report contains complete updated specifica- tion tables of the type used in the TPRC data sheets[3] (An example is included later in Table 1) and graphical plots, on linear thermal conductivity and temperature scales. These include all of the existing data. The number assigned to each curve is included in the accompanying specification table and serves to indicate the authorship, year, method used and reference. This last relates to the number as listed in the relevant TPRC Data Book. On the basis of this available information, recommended curves have been inserted in each figure and tables of the recommended values are included. This assessment of available data is proving a formidable task, possibly an un- enviable one; it is clearly an essential service, but its value depends on its relia- bility. In order to give a greater degree of confidence to the recommendations made, the first drafts of these two reports have been internationally distributed to fifty or so workers who are active in this field, for comment and criticism. The response has proved most helpful and many changes have been made in the published versions, for which, it should be noted, the authors assume complete responsibility. Work is currently on hand on the rest of the elements for which thermal conduc- tivity information is available, and early in 1968 it is proposed to issue a first draft, covering not less than 59 additional elements, for similar reviewing. The new data now presented should therefore be regarded as provisional. It is hoped subsequent- ly to prepare for publication by the NBS-NSRDS a book on the thermal conductivity of the elements. Hence, information on those elements for which no or incomplete values are given, and constructive comments bearing on the sections already published will also be welcomed to assist in the final updating and revision. In the present account graphs are used to illustrate the amount and range of the information available, the data for some specific elements and the final recommended values. Tables of recommended values will become available in the final report to NSRDS. One of the primary purposes of this presentation is to direct attention to the areas in which further work would be helpful. 2. Existing Information In connection with this investigation the many specification tables that had been prepared and published [3] have been carefully checked against the original papers and, in many cases expanded by the inclusion of pertinent information regarding the electrical resistivity, or conductivity of a specimen. A typical example of such an updated table is reproduced as table 1. This relates to zirconium for which the corresponding linear and logarithmic plots of thermal conductivity against temperature appear as figures 1 and 2. So far, experimental thermal conductivity information is available for 62 of the 81 metallic elements, and, since each of these elements cannot be treated in detail in a paper of the present type, the number of available measurements for each element is shown in figure 3 whilst figure 4 indicates the temperature or temperature range covered by these measurements. Figure 4 brings to light a few surprises. Among the 19 metals listed in Table 2 for which no experimental information has been found are the three related metals calcium, strontium and barium. That no information is available -for calcium is truly surprising since calcium rates thiiri after iron and magnesium in the scale of abundance 'Figures in brackets indicate the literature references at the end of this paper. 00 I a •a a 2 H K' o i CO H N N N §S 9 P H 00 IO I I I «! s « n t- • • oi *■ e» *■ 09 « ►■" J O K d*. d*. d ft fs •*l --"1 a | X as a d§: d.% "3 a a" oo* * 3 M 1 I l i 00 1 s s IN CM o IO 1> -V rH U9 l-l A •* IO CO rt •* a m tri oa 1 I 1 t* t* tB fci «' ►J 11 i- J M « HV § 6 d • a d § s i £ s * * AH to H rH fj * O 9 ? 09 ■♦ IO CO I "O o o S|i id -a fi * m Z is* H K S3 £ £ £ in Ik 1 3; - a - (3 K ■ o • O O 2 a to a a 5 5 8 2 "3 1 m h m h a = s of the elements now under consideration. A value W m~l deg~l) at 20C has been listed for calcium i Handbook" [6] but the source of this value has not would appear to be low since with P 2 q^k as ^.6 x has the low value of 1.53xlO~ 8 V 2 K- 2 '. Cobalt has such high melting point metals as iridium, ruthen been measured above 493 to 592K or only for about for which they are solid. These stand out as def others will appear as the attempt is made to asse of 0.3 cal cm -1 s~l deg -1 (125.5 n several editions of the ASM"Metals been traced, furthermore, this value 10~ 8 ohm m [7] the Lorenz function not been measured above 430K and ium, rhodium and palladium have not one quarter of the temperature range inite gaps in knowledge, and several ss the available data. Table 2. Metallic Elements that Lack Experimental Thermal Conductivity Values Name Atomic No. Name Atomic No. Name Atomic No Actinium 89 Einsteinium 99 Nobel ium 102 Americium 95 Europium (a) 63 Polonium 84 Barium 56 Fermium 100 Promethium (b) 61 Berkelium 97 Prancium 87 Protactinium 91 Calcium 20 Lawrencium 103 Radium 88 Californium 98 Mendelevium 101 Strontium 38 Curium 96 (a) For estimated value at 291K see [4]. (b) For estimated values range 300-2250K see [5], This paper also contains estimated values for neodymium up to its melting point. 3„ Assessment of Information - Some Guides Earlier analytical work at the TPRC [8,9,10] has shown that the low-temperature thermal conductivities of many metals can be correlated to a single curve given by X* = [l/3 (T*) 2 + Ir*]" 1 (1) where X* = X and T* = T , X* being the ratio of thermal conductivity X at temperature *~m T m T to the maximum conductivity X at temperature T . It was further shown that in the low temperature region below 1.5T , X could be derived from the equation X = [a'T n + /8T- 1 ]" 1 (2) where a' = a" (Mm" +~T In this equation a, m, n and a" are constants for a metal whereas a' and /3 depend on the purity and perfection of the particular sample. Theoretically, ' " <>o L o -1 (3) where L is the theoretical Lorenz function and p is the residual electrical resistiv- ity and this parameter, as derived through equation (2) or (3) gives a measure of the purity and/or perfection of the sample. For those metals which conform to this eq- uation its use enables the full curve to be derived up to 1 5 T m even if only a few observations are available. This course has usually been adopted to complete the curve through the highest points and thus to obtain a more complete curve for the sample of highest apparent purity. If p Q is known, use of the equation X = L o Tp Q -l = 2.443xi0 -8 Tp o - 1 (4) also serves to give the lower end of the curve, or can serve as a check in this region. The low-temperature thermal conductivity values to which equation (2) applies depend most strongly on the sample characteristics of a particular metal. At higher temperatures, whilst the purest sample usually has the highest thermal conductivity, the differences tend to be smaller. Furthermore, for any particular metal, even although the Wiedemann-Franz-Lorenz law does not always hold, a sample with a higher electrical conductivity should generally have a higher thermal conductivity. Liquid metals appear to conform reasonably well with the WFL correlation [ll], although for some metals the Lorenz function has been found to be lower than the theoretical value by as much as 15 per cent. The foregoing are the main guidelines in considering this group of elements. They are seen to rely strongly on parallel electrical conductivity information being available for the same sample. This fact is now well recognized and it is mainly for older determinations that the corresponding electrical measurements are found to be lacking. 4. Some Typical Sets of Data In this section it is proposed to present and discuss the thermal conductivity data that are available for the four metals zirconium, zinc, tungsten and lithium. Tungsten is included because of its potential use as a thermal conductivity standard at high temperatures and the others as illustrating behaviour or problems of particular interest. 4.1 Zirconium Table 1 lists the experimental investigations into the thermal conductivity of zirconium and contains details of each specimen and of the range covered. The letters L,C,R and T in the column headed 'method used' are the initial letters of methods of the following types; longitudinal, comparative, radial and thermoelectrical. The corresponding thermal conductivity values are plotted against temperature in figures 1 and 2. Using the 0-value calculated from data due to White and Woods [12] the low- temperature section of the proposed curve was derived to about 20K and this continues smoothly to pass through their values near 90K. Above 90K uncertainties enter. From 94 to 297K only one measured value is available. This is the 121K limiting point of White and Woods which may be uncertain due to increased radiation transfer. From 298K to about 900K a + 20 per cent band of values are available with minima of from 17 to 24. 5 W m'^deg-l at around 600K. The highest curve is by Mikryukov [13] for a 99.9 per cent sample of iodide zirconium. Zirconium readily combines with oxygen and it seems likely that differences in the pre-treatment and test conditions could help to account for some of the observed differences in thermal conductivity. Treco [14] had made a study of the influence of oxygen content on the electrical resistivity of oxygen-free high-purity zirconium at 273K, the value increasing to 0.577 \i ohm m for 2.5 atomic per cent of oxygen. In view of these measurements it is surprising to note that Mikryukov gives 0.361 \x ohm m as the resistivity of his higher conductivity sample at 331K, which would probably extrapolate to about 0.26 jxohm m at 273K. This seems to be an excessive difference if both samples are polycrystalline zirconium. But a-zirconium has a hexagonal crystal structure so varying degrees of preferred orientation may be another factor associated with the observed differences in con- ductivity. No documented information is available regarding the anisotropy of zir- conium for either heat or electrical conduction. When attention is directed to the reported mean values for the Lorenz function in this temperature range it is seen that the values for Mikryukov 's two samples decrease from 3.42 x 10~ 8 V 2 K~ 2 at 323K to 3.11 x 10~ 8 V 2 K -2 at 833K; those of Powell and Tye [15] from 3.10 x 10~ 8 V 2 K~ 2 for three samples at 323K to 2.73 x 10~ 8 V 2 K -2 for two samples at 823K whilst those of Bing et al. [16] for three samples decrease from 3.14 x 10 _8 V 2 k~ 2 at 323K to 2.84 x 10~ 8 V 2 K~ 2 at 523K. On the basis of these data an extrapolated average Lorenz function at 273K has been obtained, and used with Treco ' s electrical resistivity value to give a thermal conductivity of 22.9Wm~l deg~l at 273K. This value is 3 per cent above the value obtained by Moss[17] for nominally high purity zirconium. The curve now proposed for the thermal conductivity of pure zirconium continues smoothly from 35.0W m"-'- deg -1 at 90K to 22.9 Wm~l deg~l at 273K, runs about 3 percent above Moss's curve, to cross the curve of Fieldhouse and Lang [18] and to continue some 3 percent below this curve to its upper temperature limit of 1925K. The other data of Timrot and Peletskii [19] in this high temperature region which cross the recommended curve yield a Lorenz func- tion of only about 2 x 10 -8 V 2 K~ 2 at its lower end, which seems unduly low for this element, despite there being a phase transformation at 1135K. A drop of about 14 per cent in the electrical resistivity has been reported in this region. The thermal con- ductivity measurements of Fieldhouse and Lang are the only ones to span the transition region and these indicate no corresponding change. Hence, the proposed curve has been drawn with no discontinuity. Clearly zirconium is a metal to which much more attention should be given. Once single crystals become available, information on their anisotropy is certainly recruired for both thermal and electrical conductivities. The phase transition region requires study, and, pending the production of the further required high temperature thermal conductivity values, it would be useful to extend electrical resistivity determinations above their present limit of 1280K, even into the liquid phase. The effects of high temperature working conditions on the samples might also be investigated through a preliminary series of electrical resistivity measurements. Thermal conductivity data are also required for the 100 to 300K region. 4.2 Zinc Zinc has been included so as to be able to mention a difficulty that had not appeared previously. Zinc has a hexagonal crystal structure and both thermal and electrical conductivities show definite anisotropy. The unusual feature is that the low-temperature curves for X^ and X y/ appear to cross both above and below T m - In figure 5, curves 22 and 23, both due to Mendelssohn and Rosenberg[20] behave in this way and their findings agree above T m with some earlier data due to Goens and Gruneisen[2l] and below T m with subsequent data of X by Zavaritskii[22] . incidentally, electrical resistivity data of Goens and Gruneisen showed no corresponding cross over, p, exceeding p.. at 21.2, 83.2 and 293. 2K, which suggests that the explanation might require a markedly different temperature dependence of the anisotropy of the lattice component of the thermal conductivity. The subject is clearly one that calls for further investigation. The type of thermal conductivity behaviour shown by the above three sets of data is quite incompatible with the usual treatment of Cezairliyan and Touloukian[8] [9] [10] , which has formed the basis of the evaluation of thermal conductivity values in this low temperature region, since different values of j8 yield a family of non-intersecting curves. It has also been thought undesirable to give recommended curves that cross over in this way at the present time, and so the recommendations given for zinc relate only to the polycrystalline form. The highest low-temperature value is a single observation of X ± by Zavaritskii[22] at 0.825K. The corresponding X (polycrystal) is derived through his ratio X. ± /k H for two other samples at this temperature. This has then been used to evaluate )3 and to derive the curve for polycrystalline zinc in the normal (non-superconducting) state up to about 8K. It is then continued through the 83K value for polycrystalline zinc and about 2 per cent below the 293K value, both being derived from the single crystal data of Goens and Gruneisen. The available data at higher temperatures are shown more clearly in figure s . There have been no measurements reported on zinc at normal temperatures and above within the past twenty years, with the exception of a single low value of Parker, Jenkins, Butler and Abbott [23]. Such early measurements as those of Shelton and Swanger[24] were probably reliable but to allow for the possible re- duced purity of these specimens the proposed curve lies about mid-way between that of Shelton and Swanger and the most recent determinations, those reported by Mikryukov and Rabotnov[25] in 1944. Clearly the high purity melting-point zinc now available should be used for new measurements on polycrystalline and on single crystal samples. There is also a strong case for the new determinations to be extended into the liquid phase. The two early measurements by Konno[26] and Bidwell[27] happen to agree well, but they yield a solid/liquid ratio X s /X^,,of 1.67 whereas Roll and Motz[28] report 2 2 for the ratio pi/p„. Also both Konno and Bidwell obtain a negative temperature coefficient where consideration of the Lorenz function would qive a strong positive coefficient, at least near the melting point. Dutchak and Panasyuk [29] have recently reported a positive coefficient but their values seem too high as they would yield a ratio of 1.72 for X Ai. The recommended curve has accordingly been drawn in a lower position. Of course it may be that the proposed value of X g is too low. Zinc certainly calls for some new determinations just above and below the melting point and well into the liquid phase. 4.3 Tungsten The thermal conductivity data available for tungsten to high temperatures is shown in a linear plot against temperature in figure 7. Some widely-scattered values are evident but the recommended curve lies within about 5 per cent of several recent determinations and is thought to be satisfactory to within about these limits. The corresponding Lorenz functions are approaching 20 per cent above the theoretical value. 4.4 Lithium The experimental data for lithium are shown in figures 8 and 9. The recommend- ed curve has a maximum value of 743 W m~ deg -1 at about 16. 5K and above 60K it follows an intermediate course between the values of Rosenberg[30] and of MacDonald, White and Woods[3l], These last stop at 121K and from this temperature to the melting point (453. 7K) further information is required. Rather tentatively the proposed curve is shown to continue with a gradual decrease to the melting point passing close to Bidwell's values[32] for the solid near this region. Much more attention has been given to the liquid state and at about 500K the five available sets of experimental thermal conductivity measurements lie within about 10 per cent of 43 w m~ldeg~l. These measurements, however, show serious differ- ences regarding the variation of thermal conductivity with temperature. Two[33,34] indicate liquid lithium to have a large initial negative coefficient, one a small positive coef f icient&5] and two indicate large positive coef f icients[36, 37] . These last receive support from estimates based on the Lorenz function and knowledge of the electrical resistivity. Noteworthy among such estimates is that of Grosse[38], who includes lithium among the several metals for which he has made thermal con- ductivity predictions from the melting point to the critical point. He does admit that more uncertainty is associated with his estimate of 4150K for the critical temperature of lithium; it also seems that the empirical method used to extrapolate the essential electrical resistivity data to the critical point is less certain for lithium than for some of the other liquid metals that he has treated in this way. Mercury is the only metal for which the electrical conductivity, a, has been determined to the critical point[39]. From these results and others covering less complete, temperature ranges, Grosse[40] has derived for molten metals an equation of the form: (a* + b) (T + + b) = a (5) where cr* = ofi phase transformation. (Similar suggestions hold for titanium,) In planning any future work it is important to include electrical resistivity determinations whenever possible. Indeed, within the liquid helium temperature range and at high temperatures, particularly for the liquid phase of a metallic conductor, such measurements can often provide a basis for the estimation of thermal conductivity values. 8. References [l] Powell, R.W., Ho., C.Y., and Liley, P.E., Thermal Conductivity of Selected Materials, National Standard Reference Data System, NSRDS-NBS 8 (Nov. 1966). [2] Ho, C Y„, Powell, R.W., and Liley, P.E., Thermal Conductivity of Selected Materials, Part 2, National Standard Reference Data System, NSRDS- NBS-16, in press. [3] Touloukian, Y.S., Thermophysical Properties Research Center Data Book, Volume 1, Metallic Elements and Their Alloys (In Solid, Liquid or Gaseous State) , Chapter 1, Thermal Conductivity, 849 pages in Chapter 1 as of December 1966 [4] Jolliffe, B.W., Tye, R.P., and Powell, R. W., The thermal and electrical con- ductivities of scandium, yttrium, and manganese and twelve rare-earth metals, at normal temperatures, J. Less-Common Metals 11, 388-394 (1966) . [5] Williams, R.K. and McElroy, D.L„, Estimated Thermal Conductivity Values for Solid and Liquid Promethium, ORNL-TM-1424 (March 1966). [6] Metals Handbook, 8th Edition, Editor T. Lyman, Published by American Society for Metals, Vol. 1, p. 48 (1961) . [7] Meaden, G.T., Electrical Resistance of Metals , Plenum Press, New York, p. 'Jf (1965). [8] Cezairliyan, A., Prediction of Thermal Conductivity of Metallic Elements and Their Dilute Alloys at Cryogenic Temperatures, Purdue University, Thermophysical Properties Research Center, TPRC Rept 14, 1-140 (1962). [9] Cezairliyan, A., and Touloukian, Y.S., Generalization and calculation of the thermal conductivity of metals by means of the law of corresponding states, High Temperature 3_, 63-75 (1965) ; A translation of Teplofizika Vysokikh Temperatur 3_# 75-85 (1965) . [10] Cezairliyan, A a and Touloukian, Y.S., Correlation and prediction of thermal conductivity of metals through the application of the principle of cor- responding states, in Advances in Thermophysical Properties at Extreme Temperatures and Pressures, 3rd Symposium on Thermophysical Properties, A.S.M.E., 301-303 ( 1965) . [11] Powell, R„W., Correlation of metallic thermal and electrical conductivities for both solid and liquid phases, Intnl. J. Heat Mass Trans. 8_, 1033- 1045 (1965) . 11 [12] White, G.K. and Woods, S B., Electrical and thermal resistivity of the transition elements at low temperature, Phil. Trans. Roy. Soc. A251 (995), 273-302 (1959). [13] Mikryukov, V.E., Temperature depend- ence of the heat conductivity and electrical resistance of titanium, zirconium and of zirconium alloys, Vest. Moskov Univ 0/ Ser. Mat.Mekh. Astron. Fiz.Khim . 12 (5). 73-80 (1957). [14] Treco, R.M., Some properties of high purity zirconium and dilute alloys with oxygen ,Trans. Amer. Soc. Metals 45 , 872-892 (1953) . [15] Powell, R.W. and Tye, R„P., The thermal and electrical conductivities of zirconium and of some zirconium alloys, J. Less-Common Metals 3_, 202-215 (1961). [16] Bing, G. , Fink, F.W., and Thompson, H.B., The Thermal and Electrical Conductivities of Zirconium and Its Alloys, USAEC Rept. BMI-65, 1-19 (1951) . [17 J Moss, M., An apparatus for measuring the thermal conductivity of metals in vacuum at high temperatures, Rev. Sci. Instrum. .26, 276-280 (1955). [18] Fieldhouse, I.B. and Lang, J. I., Measurement of Thermal Properties, WADD TR 60-904, 1-119 (1961) . [AD268304] [19] Timrot, D.L. and Peletskii, V.E., The thermal conductivity and integral blackness of zirconium, High Temperature _3 (2) , 199-202 (1965) ; Translation of Teplofiz Vysokikh Temperatur 2(2), 223-227 (1965). [20] Mendelssohn, K. and Rosenberg, H.M , The thermal conductivity of metals at low temperatures, I: The elements of groups 1,2, and 3, II: The transition elements, Proc. Phys. Soc. (London) A65 . 385-394 (1952). [2l] Goens, E.and Gruneisen, E., Electrical and thermal conductivities of single crystals of zinc and cadmium, Ann. Physik 14, 164-180 (1932) . [22] Zavaritskii, N.V„, Measurement of heat conductive anisotropy of zinc and cadmium in the superconducting state, Zhur. Eksptl. i Teoret. Fiz. 39, 1193-7 (1960); Soviet Phys. JETP 12(5), 831-3 (1961). [23] Parker, W J., Jenkins, R.J., Butler. C.P , and Abbott, G.L„ , Flash method of determining thermal diffusivity, heat capacity, and thermal conduc- tivity, J. Appl. Phys. 32.(9), 1679- 84 (1961) . [24] Shelton, S.M and Swanger, W.H., Thermal conductivities of irons and steels and some other metals in the temperature range 0-600°, Trans. Amer. Soc. Steel Treating 21., 1061- 77 (1933). [25] Mikryukov, V.E. and Rabotnov, S N., Thermal and electrical conductivities of mono-and polycrystalline sub- stances from 100 degrees to the melting point, Uchenye Zapiski Moskov Ordena Lenina Gosudarst Univ. M.V. Lomonosova Fizika 24, 167-79 (1944) . [26 J Konno, S., On the variation of ther- mal conductivity during the fusion of metals, Sci. Repts. TShoku Imp. Univ. 8, 169-79 (1919) . [27] Bidwell, C.C., A precise method of measuring thermal conductivity applicable to either molten or solid metals. Thermal conductivity of zinc, Phys. Rev. .56, 594-598 (1939). [28] Roll, A. and Motz, H., The electrical resistance of metals, I. Measuring method and electrical resistance of molten pure metals, Z. Metallkunde 48(5), 272-80 (1957). [29] Dutchak, Ya.I. and Panasyuk, P.V., Investigation of the heat conduc- tivity of certain metals on transi- tion from the solid to the liquid state, Fiz. Tverdogo Tela 8,(9) > 2805- 2808 (1966); for Transl. see Soviet Phys. -Solid State 8.(9), 2244-2246 (1967) a [30] Rosenberg, H.M., The thermal and electrical conductivity of lithium at low temperatures, Phil. Mag. (Eighth Ser.) jL, 738-746 (1956). [31] MacDonald, D.K.C., White, G.K,,, and Woods, S.B., Thermal and electrical conductivities of the alkali metals at low temperatures, Proc. Roy. Soc. (London) A235 . 358-374 (1956) . [32] Bidwell, C.C., Thermal conductivity of lithium and sodium by a modifi- cation of the Forbes bar method, Phys. Rev. 28/ 584-597 (1926). 12 [33] Yaggee, F.L. and Untermyer, A., The Relative Thermal Conductivities of Liquid Lithium, Sodium and Eutectic NaiK / and the Specific Heat of Liquid Lithium, USAEC Rept. ANL-4458, 1-27 (1950). [34] Webber, H.A., Goldstein, D. , and Fellinger, R„C., Determination of the thermal conductivity of molten lithium, Trans. Amer. Soc. Mech. Engrs, 77. 97-102 (1955). [35] Nickol'skii, N.A., Kalakutskaya,N.A. , Pchelkin,I.M. , Klassen, T.V., and Vel 'tishcheva, V.A., Thermophysical properties of some metals and alloys in the molten state, Voprosy Teploobmena, Akad. Nauk S.S.S.R., Energet. Inst. im. G.M. Krzhizharovo- skogo, 11-45 (1959). [36] Cooke, J.W. , Experimental determin- ation of the thermal conductivity of molten lithium from 320° to 830°C, J. Chem. Phys. 40(7), 1902-1909(1964). [37] Rudnev, I.I., Lyashenko, V.S., and Abramovich, M.D., Diffusivity of sodium and lithium, At. Energy (USSR) 11(3), 230-232(1961); for English trans, see At. Energy (USSR) 11(3), 877-880 (1962). [38] Grosse, A.V. , Electrical and thermal conductivities of metals over their r- entire liquid range, Rev. Hautes Temper- et Refract. 3., 115-146 (1966) (in English) . [39] Birch, F., The electrical resistance and the critical point of mercury, r Phys. Rev. 41, 641-648 (1932). L [40] Grosse, A.V., An empirical relation- ship between the electrical conduc- tivity of mercury and temperature over its entire liquid range also its thermal conductivity and the latter' s regular behaviour, J. Inorg. Nucl. Chem. 28* 803-811 (1966) . [41] Freedman, J.F. and Robertson, W.D., Electrical resistivity of liquid sodium, liquid lithium, and dilute liquid sodium solutions, J. Chem. Phys. 34* 769-780 (1961) . [42] Rigney, D.V. , Kapelner, S.M., and Cleary, R.E., The Electrical Resistivity of Lithium and Columbium (Niobium) -1 Zirconium Alloy to 1430° Pratt and Whitney Aircraft-Canel, ' Middletown, Conn., USAEC Rept. TIM- 854, 1-14(1965). [43] Tepper, F., Zelenak, J., Rochlich, F., and May, V. , Thermophysical and Transport Properties of Liquid Metals, AFML-TR-65-99, 1-103.(1965). [AD 464138; X65-18506] [44] Kapelner, S.M„, The Electrical Resistivity of Lithium and Sodium- Potassium Alloy, Pratt and Whitney Aircraft Division, USAEC Rept. PWAC -349, 1-33 (1961) . [45] Gates, D.S. and Thodos, G., The critical constants of the elements, A.I. Ch.E.J. 6.(1), 50-54 (1960). [46] Weatherford, W.D.,Jr., Tyler, J.C., and Ku, P.M., Properties of Inor- ganic Energy-Conversion and Heat- Transfer Fluids for Space Applications, Southwest Research Inst., San Antonio, Texas, WADD TR- 61-96, 1-470 (1961). [AD 267 541] [47] Grosse, A.V., Thermal Conductivity of Liquid Metals over Their Entire Liquid Range, i.e., from Melting Point to Critical Point, in Relation to Their Electrical Conductivities, and the Fallacy of Dividing Metals into "Normal" and "Abnormal" Thermal Conducting Ones, (Temple University Philadelphia, Pa., Research Institute) TID-21737, 1-76 (Dec. 7, 1964). 48]Flynn D. R. and 0'Hagan, M. E., Mea- surements of the thermal conductivity and electrical resistivity of platinum from 100 to 900 °C J J. Research NBS 71C , 255 - 284 (1967). 49]Plynn, D. R. and 0'Hagan, M. E. , Mea- surements of the thermal conductivity and electrical resistivity of platinum from 100 to 900 °C. Engelhard Tech. Bull. (March, 1968). [50] Martin, J.J., Sidles, P.H., and Danielson, G.C., Thermal diffusivity of platinum from 300 to 1200K, J. Appl. Phys. 38 (8), 3075-3078 (1967). [51] Laubitz, M.J. and van der Meer, M.P., The thermal conductivity of platinum between 300 and 1000K. Canadian J. Phys. 44(12), 3173-3184 (1966). [52] Fieldhouse, I.B., Hedge, J.C., Lang, J.I., and Waterman, T.E., Measurements of Thermal Properties, WADC-TR-55-495 Part II, 1-18 (1956). [AD110 510] 13 [53] McClelland, J.D. , Rasor,N.S . , Dahleen, R.C., and Zehms, E.H., Thermal Conductivity of Liquid Copper, WADC-TR-56-400 Part 11,1-6 (1957) . [AD 118243] [54] Lucks, C.F. and Deem, H.W., Thermal properties of Thirteen Metals, American Society for Testing and Materials, Special Technical Publication No»227, 1-29(1958). [55] Ewing, C.T., Grand, J. A., and Miller, R.R., Thermal conductivity of liquid sodium and potassium, J. Am. Chem. Soc. 74,11-14(1952). [56] Deem, H.W. and Matolich, J., Jr., The Thermal Conductivity and Electrical Resistivity of Liquid Potassium and the Alloy Niobium-1 Zirconium, NASA Rept. BATT-4673-T6, 1-26 (1963). [57] Arajs, S. and Colvin, R.V>_v Thermal conductivity of scandium between 5 and 320°K, in 4th Rare Earth Re- search Conference, Phoenix, Ariz., Conf. -405-23, 1-10 (1964). [58] Aliyev, N.G. and volkenshteyn, N.V., Low-temperature thermal conductivity of scandium and yttrium, Fiz. Metal, i Metalloved 19(5), 793 (1965); English Translation: Phys . Metals Metallog. .19(5), 141 (1965). 14 T 1 I i I i i 1 1 1 1 I THERMAL CONDUCTIVITY OF ZIRCONIUM i - - ) i _ 3 - 4 v JL s - 5 - (6) -J 3 ji — k 3 \ U_JS H raiBk " - $ [©© - (=> <•) l! a | , | U i [ ! i ! 1 - i 1 /> i i ® ^ / _ 1 r P i \ J / \ - J_ 1 I i I 1 1 I 1 1 1 1 co (D ■H m u CO o ■H }-l en o e •H c o o u ■A N M-l O >i -P ■A > ■rH -P U 3 T3 fi O U (0 £ a> <0 u 3 ,JI ,J« 4»om 'XllAllOnONOO 1VWH3H1 15 ,_X , w UDM 'AllAllOnONOO ~IVWy3Hl 16 i i i r i i r i r i i i r j I L j i i i I I i i L J I I L J I I L o^ U < « •H 3 (0 .o K +J 1 ■P > 0. ■H £ ■P o Q. 3 « Q. T> 8 „ C w o .o z U z _ z iH a. (0 :i e u 5 o> V I ,C Eh a. 5 Q) 3 +J _l • * C _i o n _l c Ol ^ ■rH c 4-> , f X ■H .« •P (0 !* 0) > O G E 1 H UJ <4-l O w » u X^ ai ^ ■^ 3 p ° O z 3 SN0UVDI1S3ANI JO «38WnN 17 1 1 1 1 II II I l I I 1 Nl 1 II II | 1 1 1 1 | 1 1 1 1 | 1 1 1 1 | | | 1 1 1 1 1 1 | 1 1 1 1 1 1 1 1 ? : ^^^ l_ 1 jL ? i> f t i i — ?_ ™ ? L- ss 1 I f 1 1 I 1 1 % ? 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Thermal Conductivity of Zinc (Linear Scales) 20 a i -P ■H > •H +J u C o u (0 s M 0) En 0) p ,_»l_UJ HOM ' AilAllOnQNOO "IWVH3H1 21 T 1 1 1 1 n~ 1 1 1 1 1 THERMAL CONDUCTIVITY OF LITHIUM — - — — — \ I hd) — — (wj s — — — — — a> r-\ (0 O w u •H E -P -rH 10 CP o e 3 •P •H M-l O 4J U TJ C O u .13 e u 0) "Td co to m * ro cj ^ eo ijo ,.» ,.UJ »»DM 'AllAllOnQNOO 1VHM3H1 22 e ■H +J M-l O >i •P > •r-l +J u 3 TJ C O U to e u o Q Z o 1 ' 1 1 \ I 1 i i ! 1 T 1 1 1 L - i - i 1 THERMAL CONDUCTIVITY OF METALLIC ELEMENTS OF GROUPS IA AND IB (Provisional Recommendations) - - i i i 1 : - i - / i 1 i 1 I — — ' 1 — i ■ ! — ; i ; / / i i i j / / / ^ — — / y 1 i i : ; - / / / / \ \&- \ \ \\\ j j ' - / / / y^ N / i / is»-^^ = . 1 1 | 4T L vk \ \ """^-i - \ \N j Ag OiqJ - - i-r^ ; — - / ^~ ^ \ \ j -// 1 v^ i i i i "***• "1 L% %l — V V " '/ Rubidiurr ^T~i "~^ \jj^— -^ ft m i 1 1 - / — —. n X^ \ - Cs (liq) \\ \ J 1 \ \ \\ \ \ \ \ I in i ir ni ii 1 1 : 1 1 Vl \ - V \\ \V \ \ \ 1 1 1 1 1 1 1 1 1 | 1 | 1 li i 11 1 1 ' 1 ' ! 1 1 1 1 1 1 1 i 1 1 1 1 1 - I : 1 1 1 1 1 I I — ii 1 * 1 1 1- I I I i! ii! - 1 1 i i i 1 i 1 _L 1 1 3 4 5 6 8 10 2 3 4 5 6 8 I0 2 TEMPERATURE. K 5 6 8 I0 3 3 4 5 6 8 10* Figure 10. Thermal Conductivity of Metallic Elements of Groups IA and IB (Provisional Recommendations) 24 m» _ i i 1 1 1 1 1 1 1 1 i i i i - a - b THERMAL CONDUCTIVITY OF ELEMENTS OF GROUPS HA AND HB (Provisional Recommendations) b 4 3 - - 2 ii-/* - 10 r yy 1 - 8 0/'-' /2\\ - 6 r ^ X UN J 4 3 -x\o A A > - c I0 3 8 6 5 V \ \ ^v« \ \ \\ A \ N a\ ■/ AT W - .^ v / /\V \\ \t - J^ X L\\\ - A/j \\v \% \ \€ / *r 3 Nr N \ z' \ - 4\ k^L \ ^^ - in 2 B ^ 10 - c ^ - 8 - & c admium (poly 7 •'\ - 5 4 \\ ^-~.___Mer E ur y GTfefcu. " ! \yr o--»__ ~— --I'y^^ox/s, \M i **"■-*■ — ~- = ~3E^Zr^Z^. — i & 2 10 ^-Mercury (poly.) - .*' & --, \ - - \ \ - - \ \ - \ 4 3 2 t \ i - - i 1 1 J_ I 1 1 I I 1 i i i - 3 4 5 6 8 10 2 3 4 5 6 8 I0 2 TEMPERATURE, K — 3 4 5 6 8 10 3 4 5 Figure 11. Thermal Conductivity of Elements of Groups IIA and IIB (Provisional Recommendations) 25 10" > 8 '£ 6 t 5 |4 >* 3 > P 2 O 3 Q Z ° in 2 _J 8 < % 6 _ 1 1 1 i 1 i mm i i i 1 i 1 i | i ii II | i | _ - THERMAL CONDUCTIVITY OF METALLIC ELEMENTS OF GROUPS DIA AND MB 4- (Provisinnnl RsrnmmsnHMInncl - 1 ! 1 III' - y 1 1 - / - ! i - / ~ /\^^TX \ \% \\\ % \ i - \ \ \ \^- \ \ \ \ \^ i / \\ \ X % \\ \ \ \ - \ \\ \c \ \ \ \ \ \Q, \ \ - A\ \ \ ! - \ \ \ \— \ \ i \ \ \ \^ \ \ \& \ \% \ > i - \ \^k \\ \ i i ! \ \ y \\ \ - \ v^ \ \°\ \\% \ N Y" — — .j^^w i i _ - ! N - - £ \T ,- \ ... ^—-^^^LiPoW 7+- \ \ \/- f \ \ - \ - f y ^ ^ *C^. \ - S> "— ""■" |- - 4 jf X / J - / 4 v/$ — '- / " , r s 1- 1 / 'A i i 1 1 l 1 . i i i i 1 .]_ 4 5 6 8 10 3 4 5 6 8 I0 2 TEMPERATURE. K- 4 5 6 4 5 6 Figure 12. Thermal Conductivity of Metallic Elements of Groups IIIA and IIIB (Provisional Recommendations) 26 T E 6 i : > 3 i- o o or x 1 1 i I 1 1 1 | 1 i 1 1 1 1 1 - 1 THERMAL CONDUCTIVITY OF _ METALLIC ELEMENTS OF GROUPS J EA AND IEB (Provisional Recommendations) >X/^~ > \Vv / \ \N - - - - \\\ v\ \\\ - \\\ \\\ - - - \\> 1, - - l\ a\YL - - \ v\v \ %w - \ \v \ - \ W \ \ \\ - ' / \ \ v\ \ V > *X ^^_ V - yf \ "\ ^\ ^"■^r*~-— J - - / / N v ^^^ijsr-^^ - .< ^ — ~-^^ \H // v^ *\ * / / ^""^5^ BC-i v^b«*^^> - / / / / / ~ / / / £ - - J, ?/ - - V / - / &/ / < / / / / / - / <- / / / /I / / 1 1 1 1 1 I J_ 1 1 1 1 1 - 3 4 5 6 8 10 2 3 4 5 6 8 10" TEMPERATURE. K — 2 3 4 5 6 8 IC 3 4 5 Figure 13. Thermal Conductivity of Metallic Elements of Groups IVA and IVB (Provisional Recommendations) 27 •E 1 >-" > H <_> Q Z o o a: u 10 8 6 5 4 3 _ 1 i 1 1 1 1 1 | | | i | i | " j ' | | i | ' | ' | '| | - THERMAL CONDUCTIVITY OF IETALLIC ELEMENTS OF GROUPS VA AND VB (Provisional Recommendations) hi 2 IC? 1 - - o - 6 D /\ '\ 3 - I0 2 - ^^ - o ^^ "la - - / 7 & lV^ D y / j> <\0& 2 10 8 6 \ ^V ^^T sr - / # 4 j^O^N. j - - 4 x Vv x^ - - - 4 3 2 / - - 8 6 - - - - 4 3 2 - - - i i I 1 i 1 1 I 1 1 i 1 1 1 - I 2 3 4 5 6 8 10 2 34568 I0 2 2 34568 IC? TEMPERATURE, K - 2 3 4 5 Figure 14. Thermal Conductivity of Metallic Elements of Groups VA and VB (Provisional Recommendations) 28 10 8 6 5 4 3 _ 1 1 L 1 J__ 1 1 1 ~r 1 1 1 1 1 1 1 i - HERMAL CONDUCTIVITY OF MENTS OF GROUPS EIB AND WB (Provisional Recommendations) 1 ELE / / — / - / 4 I0 3 8 \ 1 . & y 4 / 6 y* y 5 3 2 io 2 V 8 r E 6 C 5 > / / t / - / / t *s / f s F 2 o 3 Q Z o 10 ^ 8 1 6 Ul 5 X H 4 3 \/ / - / / / - / / ^■^""^ ~— ■ <* ^ ^ 2 i - 8 6 5 4 3 2 in 1 ' 1 | 1 1 _L 1 I 1 1 1 1 10 1 2 : 4 R € £ 1 l( ) 2 TFM PFF \ £ (AT 5 ( UF i E *E. i l( K- f c 1 \ A f > i > £ 1 l( f i i 4 ; > e > e i k ) 4 Figure 15. Thermal Conductivity of Elements of Groups VTB and VIIB (Provisional Recommendations) 29 d 2 '£ 1 >-" H > H O Q o o _) < s cr UJ I I- 1 ! i — i — 1 ~r 1 1 1 1 ' 1 ' 1 II! 1 ' ' 1 ' 1 II ~r - - 1 1 1 1 1 1 i ill IERMAL CONDUCTIVITY OF ELEMENTS OF GROUP WL - — Th - - - - - -- .. 1 - - 1 1 — — - - vC? y, ~o 6 /■ ~\ \ \ - - o } - - ///w /A\\\\' /T /%> // \\ WW "— _ - - | - / '/Y&y v. : // Vv — \\\\\ ] ///W v& x^ v7 ; ' AS 4** "9\ N - '•^ ?> /'^Sv *^. k ---Si - // 0£ - 7 ,*y 1 ^*r--^ 'Z^^ i - V / \ *■ 1 : / / / \ JS,- J i \<2. \ \ - - \ \ - - - - li i \ - 1 i i 1 1 1 J_ 1 1 1 1 1 3 4 5 6 8 10 2 3 4 5 6 8 I0 2 TEMPERATURE, K 4 5 6 8 10- 5 6 8 I0 4 Figure 16. Thermal Conductivity of Elements of Group VTII (Provisional Recommendations) 30 -"k ° - o Q Z o o 6 5 4 3 2 Id 1 _ 1 1 I I 1 1 I i 1 1 1 i iii | | | i 1 1 1 i ii - - THERMAL CONDUCTIVITY OF ELEMENTS OF THE RARE EARTH AND THE ACTINIDE GROUPS (Provisional Recommendations) - ' - - - - - - - - - - - - - - - - ~"^— --2>'or/ujn • -CV - / VJtO r^utf JgS^^ fnnlv) OJS*^^^ - - / J^P^^Z^^r — ^* Promeiniui^^-.^^ ,- - ^; -"bi '-N eodymium (p olyj - J //"W/fyf/ / ^v — ^^^^—^^^^ a / &/y /// // / . .«wy'Ss — Holmlum (nolv) ^tf'vX ^--t j T> - - -J ^«/ L^^^^^^^f, \/ - f/SM'LJ^ Vy/w// 7 s / /J$y /j/ ^^~ \^r 1 / Neptuni um t / V - '// //i i A # R dy - */ 1/ > i'/ 1 / 1 /— / ' / / / ■ - - i / // - / / - /, 1 1 1 1 1 i I 1 1 1 1 1 - 3 4 5 6 8 10 4 5 6 8 10^ 3 4 5 6 8 10- 3 4 5 TEMPERATURE. K- Figure 17. Thermal Conductivity of Elements of the Rare Earth and the Actinide Groups (Provisional Recommendations) 31 The State of Knowledge Regarding the Thermal Conductivity of the Non-Metallic Elements — Those Solid at Normal Temperature 1 C.Y. HoandR.W. Powell 2 Thermophysical Properties Research Center, Purdue University, West Lafayette, Indiana 47906 Of the 103 chemical elements, experimental thermal conductivity data are available for 62 metallic elements, 9 non-metallic solids and 11 non-metallic fluids. In this paper, the available thermal conductivity data for the 9 non-metal- lic solid elements are reviewed. These are arsenic, boron, carbon (in the forms of diamond and graphite), iodine, phosphorous, selenium, silicon, sulfur, and tellurium. Key words: Amorphous solid, arsenic, boron, carbon, conductivity, diamond, graphite, iodine, liquid phase, non -metallic elements, phosphorus, polycrystal, selenium, single crystal, silicon, sulfur, tellurium, thermal conductivity. 1 . Introduction 3 The present paper reviews the information available for the thermal conductivity of the nine non-metallic elements that are solid at normal temperature . It is complementary to the accompanying paper by Powell and Ho [l] 4 , and, the two papers seek to give the complete information for the normal solid elements. As before attention is directed to the areas that need to be studied. In the course of this work, earlier TPRC publications [2] have been considerably up-dated and attempts have been made to assess the available results so as to yield most probable values. Similar assessments were previously made for diamond [3] and for graphite [4] but those for amorphous carbon, arsenic, boron, iodine, phosphorus, selenium, silicon, sulfur, and tellurium are new. No information has been located for astatine. As was explained in [l], the present assessments regarding the most probable thermal conductivities should be treated as provisional values for which constructive comments would be welcomed, together with notes of any further relevant information. Because of the provisional nature of these values it has not seemed desirable to tabulate them at this stage. Graphical presentation of these interim values should suffice for those having expert knowledge of these materials to judge whether the present proposals are reasonable ones, and at the same time the curves will give clear indications of the temperature ranges for which values are available. The forthcoming work on the thermal con- ductivities of the elements referred to in [l] will of course contain tabulated recommended values. 2. Notes on the Existing Information and Its Assessment Table 1 is included as an example of the manner in which the experimental information available for the thermal conductivity of a particular material has been tabulated. This is for sulfur, an element for which only 21 sets of experimental data have been located and the relevant information is contained in a table of reasonable size. For graphite there are 582 entries. ^his paper covers a more restricted field than was originally indicated. Some of the omitted matter has appeared in "Thermal Conductivity of Selected Material," Powell, R.W. , Ho, C.Y., and Liley, P. E. , National Standard Reference Data System NSRDS-NBS 8 (November 1966). 2 Head of Reference Data Tables Division and Senior Researcher, respectively. 3 Much of the "Introduction" of the accompanying paper by R.W. Powell and C.Y. Ho applies equally to this paper but is not repeated. 4 Figures in brackets indicate the literature references at the end of this paper. 33 ■BOO 1 IE 0) ri * 0) n S3 o 4 «z 6 d S 8 j j j j j CT> O iH i-H Ifl OJ W i-t ^H CD Oi OS CTi 0} 0} in to be M he bn 2 2 1 1 d 6 . o ^ ^ W W TO io m co lo oi © © in 30 -t- CD C7> CO CO IH >H .8 8 .&« ■a S o -S m ° -s i f a § 3 1 B H CD CO CO CO J hJ J _l N N N N rH IO 10 IO OS CT> OS OS fe fe fe < < < Sfc. M< 11 w > ft m c h tlu O Ol O f-H CM CO © i-H CM © a o> a> 5 « « £ O hX -h t-H O « « fl . > > > u . <■ < < a - • - s* <£ «G S3 g o o o 34 In table 1 the numbered references have been altered to correspond with those of this paper. The last column contains the summarized specimen details which were thought to be significant and it seems desirable to comment on the sixth and seventh entries. These relate to measurements by Slack [7] for two single crystals of sulfur, exceeding 99 percent purity; each a rod-shaped specimen about 0.7 cm long and 0.2 cm in diameter, cut from a crystal of the stable orthorhombic modification to give the heat flow perpendicular to the c-axis in No. 6 and in No. 7 parallel to this direction. Slack noted that a few internal growth flaws were visible in the samples studied. Since these samples were tested to temperatures below that of the thermal conductivity maximum, the most important of these facts is the sample size, since the thermal conductivity of a non-metallic single crystal near absolute zero is mainly governed by the boundary scattering. An otherwise similar specimen, cut from the same crystal, but of larger diameter will have a proportionately greater thermal conductivity over the very low temperature region. The crystallite size will behave similarly. The corresponding logarithmic plot of the temperature variation of the thermal conductivity of sulfur is given in figure 1, and it is of interest to comment first on the two low temperature curves numbered 6 and 7. Over about the range 12 to 90 K the two curves lie within 10 percent crossing over at 25 K. These differences are prob- ably consistent within the experimental uncertainty, but at the maxima and at lower temperatures the differences increase to over 300 percent, the thermal conductivity for the direction parallel to the c-axis appearing greater. However, Slack did not regard these differences as due to true crystal anisotropy, but thought them to result from differences in crystal perfection. Slack reported the confirmation of little if any anisotropy in X for sulfur by some room temperature measurements on other samples by a comparative measurement of the de Senarmont type [19], the values so obtained all agreeing to within 20 percent. The foregoing rather incidental remarks are important in that they introduce some of the precautions involved at low temperatures with this type of work. Slack's measurements , as shown in curves 6 and 7 of figure 1 stop at about 90 K. He considered radiation transfer to have become a limiting factor in the attainable experimental accuracy, and stated that the measurements of Eucken [ 6~\ (curve 4), which link on well with the data of several other workers near room temperature, probably represent the behavior at higher temperatures. This would leave the measurements for sulfur in a fairly satisfactory state except for two points at 300 K due to A. V. Ioffe and A. F. Ioffe [18], These are numbered 20 and 21 and are for single crystals having the heat flow along two mutually perpendicular axes. Slack only mentioned this Russian work as a 'Note added in proof and it is unfortunate that this work came to his notice after his own measurements were completed,for several uncer- tainties now arise. The ratio for these two axial directions is 1.27 and indicates an anisotropy which exceeds the 20 percent regarded as negligible by Slack. He deduced that the higher value would be for the c-axial direction, and, if so this would agree with his own results at low temperatures but not at his upper temperature limit. As the Russian values are much higher, the question remains, are these on curves which would remain higher to low temperatures, or should Slack's curves flatten out towards these values? These questions now remain for future work to answer. For the present it has been decided to ignore these Russian values until more evidence is available and to follow the course suggested by Slack. This gives a drop of about 33 percent at the a-*/8 phase transformation, a constant value for the monoclinic form, which could be sub- ject to further investigation, and another drop of about 18 percent on passing into the liquid phase. Three out of four observers agree with the curve chosen for the liquid phase, and all four, that the phase has a small positive coefficient in this range near the melting point (See [3] or fig. 7). Much more could be said about individual measurements and other uncertainties would probably arise for each element considered. In this short account , further detailed treatment is impossible and an attempt has been made to convey an overall picture of the present position through figures 2 and 3. Figure 2 il- lustrates for the three main types of carbon and for the other elements under consideration the number of sets of measurements for which TPRC has information, whilst figure 3 shows the total ranges of temperature that have been covered for each material. It will be noted that no information is available for astatine and but little for arsenic and iodine. Indeed, knowledge of the thermal conductivity of arsenic is limited to one single measurement at 293 K by Little [20] in 1926, and that of iodine for the solid phase rests on measurements published in 1923 [21] for a narrow range of temperature close to normal. The measurements shown for the range 447 to 693 K relate to the vapor phase of iodine, there being none for the liquid phase, although estimated curves extending to the critical point (785 K), have been proposed [22] for both liquid and vapor. No other measurements for the vapor phase occur, so any other instance where the vertical line of figure 3 exceeds the melting point relates to the liquid phase. The available data for silicon are shown by means of logarithmic and linear plots in figures 4 and 5. Both figures are included because silicon is an element that has attracted a lot of interest and it seems desirable to comment on the data available at both low and high temperatures. Much of the low temperature data included in figure 4 relate to samples that have been doped to give silicon of either n- or p-type. The location of the thermal conductivity maximum reveals no well-defined dependence on the crystal type. Holland [23] reports the highest maximum value of 5230 Wm -1 deg -1 . This occurs at 22. 5 K and is for an n-type single crystal of silicon doped with phosphorus. The same worker is also responsible for the highest value for a p-type crystal, one doped with boron for which his value is 4700 W m~i deg _ l at 24 K. 35 This figure has been included to illustrate an unusual type of variation in the location of these maxima. For most materials, increase of purity leads to a higher maximum which occurs at a proportionately lower temperature. This type of variation is in fact seen for many of the silicon results where T m decreases from about 200 to 22. 5 K as X m increases from 42 to 5230 Wm -1 deg -1 . On the other hand there are maxima which remain in the temperature range 18 to 28 K as X m increases from 116 to 5230 Wm^deg" 1 . In each group there are examples of both n- and p-type silicon, though in the latter group the majority of the samples are of n-type and in the former group the majority are of p-type. In this low temperature range the most probable curve for pure silicon has been based on Holland's highest curve. Below the lowest temperature his curve has been extrapolated parallel to sets of data by Carruthers, Geballe, Rosenberg, and Ziman [24] as shown by curves 4 and 5 and in this range X ~ T 2- 9 , in fair agreement with the T 3 dependence of specific heat. At 120 K, the upper limit of Holland's measurements on the chosen sample, the curve merges into No. 31 of Glassbrenner and Slack [25]. Their measurements extend almost to the melting point and over much of the high temperature range, shown more clearly in figure 5. They agree with the values of curve 82 due to Fulkerson et al. [26] to within about 4 percent, the latter values being lower. The proposed curve has been drawn intermediately between these two sets of data and has been continued smoothly to the melting point. No determinations appear to have been reported for the thermal conductivity of liquid silicon, and, in view of the electronic contribution to the thermal conductivity of the solid near the melting point, which is believed to be about 5 percent [25], any estimation for the liquid phase from electrical resistivity data would seem to present considerable uncertainties . The treatment given the remaining elements has been similar but can only be briefly indicated. The curves that have been previously recommended for diamond and graphite and those which are now provisionally recom- mended for the other elements are reproduced in figures 6 and 7. Arsenic: — The one available measurement is for room temperature only, but should the thermal conduc- tivity of arsenic be required over a wider range, examination of figure 7 suggests that extrapolation in keeping with the form of the other curves could produce some very provisional values. Boron: — Values and predictions by Slack [7] have been adopted. His measurements are only for the range 3 to 300 K, but he gave extrapolated values to about 1473 K, thea->/J transformation temperature. At 630 K however some values reported by Morris [27] are some 60 percent greater. Carbon; — In the amorphous form the values are low, have a positive coefficient and vary considerably, presumably due to such variables as density, particle size, manufacturing process, etc. No assessment has been attempted. Thermal conductivity of diamond has been reviewed [3] . The very limited data have all been at normal and low temperatures, 320 K being the maximum measurement temperature. Graphite with nearly 600 sets of measurements on its many different forms and grades, and its unusual temperature dependence of thermal conductivity at low and high temperatures, and its higher electrical conductivity is a complex special subject for which reference should be made to an earlier publication [4] in which the general features of the thermal conduc- tivity of graphite have been discussed and separate treatments have been given to pyrolytic graphite and to four commercial grades. Figure 6 shows the tremendous anisotropy of pyrolytic graphite. For a limited range of temperatures above 100 K the thermal conductivity parallel to the layer planes exceeds all known solid materials except diamond, whereas that of pyrolytic graphite in the direction perpendicular to the layer planes is compar- able with that of insulators. Between these two curves lie all other grades of graphite. The precise curve at low temperatures for the highly-oriented and well -annealed pyrolytic graphite in the direction perpendicular to the layer planes is largely conjectural since no measurements have been made. Phosphorus: — The data of Slack [7] have been accepted for black phosphorus for the range 2. 7 to 300 K and those of Turnbull [28] for white phosphorus for temperatures a little above and below the melting point. The latter's data in the liquid phase are preferred over those of Kramer and Schmeiser [29] as his method had appeared to yield satisfactory values for liquid sulfur. Also his values are thought to be in better agreement with simple ' quasi-lattice' theories. Selenium: — In the case of selenium the recently published measurements of Adams, Baumann, and Stuke [30] have provided curves for the two main crystal directions. For temperatures above 20 K the thermal conductivity for the direction 'parallel to the c-axis is about 4 times that for the perpendicular direction. This difference persists to about 100 K, the upper limit of their experiments, and in this temperature range the de- rived values for polycrystalline selenium show reasonable agreement with a curve due to White, Woods, and Elford [31]. The recommended curves for the two crystal directions are based on the anisotropy ratio and the highest conducting crystal values of [30] but at low temperatures the values are clearly very dependent on crystal per- fection and specimen size. No dimensions are given in [30] . Above 100 K the two curves have been extrapolated in line with other data, turning up at about 350 K. The two sets of data for the conductivity of liquid selenium differ considerably and a somewhat tentative intermediate curve has been indicated in figure 7. 36 Tellurium: — Available data for tellurium also indicate considerable thermal conductivity anisotropy and the two curves shown in figure 7 for the parallel and perpendicular directions have a ratio of about two. The thermal conductivity of liquid tellurium is far from certain. The proposed curve follows recent values by Benguigui [32] , which show liquid tellurium to behave as a semi-metal with a Lorenz function decreasing from 4.4 x 10 -8 to 3.2 x 10" 8 V 2 K~ 2 over the range 743 to 923 K, whereas two earlier sets of measurements [33,34] had indicated much higher values. 3. Summary of Position, with Suggestions for Future Work The attemptto assess the thermal conductivity data for the non-metallic elements has served to emphasize the value of the additional knowledge that is available for use with metallic elements. (a) (b) (c) (d) The differences as they apply at present are compared below: Metallic Elements At moderate and high temperatures (including liquid phase) X = LTp" 1 approx. , where L = Lorenz function p = electrical resistivity At low temperatures (1) Close to absolute zero X = LqTPq 1 approx. , where L = theoretical Lorenz number p = residual electrical resistivity (2) Over range < T < 1 . 5 T m X = [a'T n + 0T- 1 ] -1 where a' =a"(/?/na") a/(m + l) See references [3] and [4] for details. Purity and/or perfection increases as: — (1) P295/P0 increases (2) j3 decreases j3 can be derived from equation of (b) (2) or from ft = Ljj^Po Dependence of X on sample size None Non-metallic Elements Nothing similar Nothing similar Nothing similar Nothing similar Nothing similar At low temperatures (below Tjj,) X of a pure crystal increases with size of crystal. Any suggestions or investigations leading to an acceptable guiding parameter for non-metals would be most useful. At the time when Cezairliyan [35] applied the generalized principle of corresponding states to metallic elements and found the equation given above for the range to 1. 5 T m to hold, he also considered some semi- metals and non-metals. Whereas the various sets of data for bismuth, germanium, silicon, selenium, tellurium, and carbon (graphite) could be similarly correlated in separate curves by plotting the reduced thermal conduc- tivity ( XfAm) as a function of the reduced temperature (T/T m ), these curves differed amongst themselves as well as from that for the metals. It was concluded that it would be necessary to include some additional parameters to bring these elements collectively or in groups to one or more common curves. This further work is still re- quired. Until a satisfactory criteria for the net purity and perfection of a non-metallic sample becomes available an element of doubt will exist as to whether a set of measurements made on one sample will be applicable to another sample. The above are regarded as directions for further work that are common to all non-metallic elements. Some of the thermal conductivity requirements for the individual elements will now be summarized. 37 Arsenic: — Measurements for the full temperature range are required to support the single data point at room temperature and any prediction that might be based on this. Astatine: — The longest lived isotope of astatine has a half -life of only 8. 3 hours, so the thermal conduc- tivity would probably be of academic interest only, and the lack of information is likely to continue. Boron: — Boron should be measured above 300 K, the present upper limit, for which no values have been reported. This includes the liquid phase Carbon: — Diamond with its high thermal conductivity could be an important material. The proposed values for the three types are based on only a few measurements, all at or below room temperature. More measurements are required and these should be extended to higher temperatures. Graphite in the pyrolytic form requires study to low temperatures for the direction normal to the layer planes. As new forms and distinct grades of graphite are produced new measurements will be required for the full temperature range. Above 3000 K is still a region where some doubt exists. Iodine: — Iodine requires measurements over a wide temperature range for both solid and liquid phases. The one available set of values for the solid, reported in 1923, has a negative coefficient that appears unduly large; also measurements have yet to be made for the liquid phase. Phosphorus: — Black phosphorus requires further study above 300 K, and white phosphorus below 250 K and check measurements should be included near the solid to liquid transition and well into the liquid phase. Selenium: — The present curves for the two main single crystal directions have been extrapolated to tem- peratures above about 100 K. These require confirmation particularly above about 300 K and well into the liquid phase. Silicon: liquid phase. The solid phase appears to have been adequately covered. No values have been reported for the Sulfur: — The data appear to be sufficient for polycrystalline sulfur but new determinations on good single crystals are needed to resolve the differences between the Russian values at room temperature and the extrapolated data of Slack. Should the former be confirmed the measurements on these crystals will be required over the full temperature range . Tellurium: — The values adopted for the liquid phase need confirmation. 4. References Cl] Powell, R.W. and Ho, C.Y. , The state of know- ledge regarding the thermal conductivity of the metallic elements, the preceding paper of these Proceedings. [2] Touloukian, Y.S. , Thermophysical Properties Research Center Data Book, Volume 3, Non- metallic Elements, Compounds and Mixtures (in Solid State at Normal Temperature and Pressure), Chapter 1, Thermal Conductivity, 457 pages in Chapter 1 as of December 1966. [3] PoweU, R.W. , Ho, C.Y. , and Liley, P.E., Thermal Conductivity of Selected Materials , National Standard Reference Data System NSRDS-NBS 8 (Nov. 1966). [4] Ho, C.Y., Powell, R.W., and Liley, P.E., Thermal Conductivity of Selected Materials, Part 2, National Standard Reference Data System NSRDS-NBS 16, in press. [5] Kaye, G.W.C. and Higgins, W. F. , Thermal conductivity of solid and liquid sulfur, Proc. Roy. Soc. (London) A122 , 633-46 (1929). [6] Eucken, A. , On the temperature dependence of the thermal conductivity of solid non-metals, Ann. Physik34(2), 185-221 (l91l). [7] Slack, G.A. , Thermal conductivity of elements with complex lattices: B, P, S, Phys. Rev. 139 (2A), A507-15 (1965). [8] Yoshizawa, Y. , Sugawara, A., and Yamada, E. , Thermal conductivity of a -sulfur, J. Appl. Phys. 35(4), 1354-5 (1964). [ 9] Sugawara, A. , Thermal conductivity of sulfur accompanying crystal transition and phase change, J. Appl. Phys. 36(8), 2375-7 (1965). [10] Lees, C.H. , On the thermal conductivities of single and mixed solids and liquids and their variation with temperature, Phil. Trans. Roy. Soc. (London) 191A , 399-440 (1898). [ll] Mogilevskii, B. M. and Chudnovskii, A. F. , Measurement of the thermal conductivity of semi- conductors by the transient probe method, Inzh. - Fiz. Zh. , Akad. Nauk Belorussk. SSR, 7 (12), 23-31 (1964) (Russian). [12] Mogilevskii, B.M. and Chudnovskii, A. F. , Measurement of semiconductor thermal conduc- tivity by a nonsteady probe method, English trans- lation of above, Foreign Technology Div. , Wright- Patterson AFB, Ohio, DDCandCFSTI, AD 627 112, FTD-TT -65-1257, TT 66-60371, 1-15 (1966). 38 [13] Green, S. E. , The spherical shell method of deter- mining the thermal conductivity of a thermal in- sulator, Proc. Phys. Soc. (London) 44, 295-313 (1932). [14] Turnbull, A. G. , Thermal conductivity of molten salts and other liquids , Ph.D. Thesis, Imperial College of Science and Technology. London, 146 pp. (1959). [15] Hecht, H. , Method to determine the thermal conductivity of poorly conducting bodies in sphere and cube form and its performance on marble, glass, sandstone, gypsum and ser- pentine, basalt, sulfur and coal, Ann. Physik 14, 1008-30 (1904). [16] Lees, C. H. , The thermal conductivity of crystals and other bad conductors, Phil. Trans. Roy. Soc. (London) 183, 481-509 (1892). [17] Niven, C. and Geddes, A. E.M., On a method of finding the conductivity for heat, Proc. Roy. Soc. (London) 87A , 535-9 (1912). [18] Ioffe, A.V. andloffe, A. F. , Simple method of measuring thermal conductivities, Zh. tekh. Fiz. 22 (l2), 2005-13(1952). [19] de Senarmont, M. H. , Memoir on the conduc- tivity of crystalline substances for heat, Ann. Chim. Phys. , 3, 21, 457-70 (1847) and 22 , 179-211 (1848); also Compt. Rend., 2, 25, 459-61 (1847). [20] Little, N. C. , Thermomagnetic and galvano- magnetic effects in arsenic, Phys. Rev. 28, 418-22 (1926). [21] Pochettino, A. and Fulcheris, G. , On the electrical and thermal properties of iodine, Atti. Reale Accad. Sci. Turino 58 (l4a), 311-20 (1923). [22] Schaefer, C.A. andThodos, G. , Thermal conductivity of diatomic gases: liquid and gaseous states, A. I. Ch. E. Journal 5_(3), 367-372 (1959). [23] Holland; M. G. , Effect of oxygen on the thermal conductivity of silicon, Proc. Intern. Conf. Low Temp. Phys. , 7th Toronto, Canada, 1960, 280-4 (1961). [24] Carruthers, J. R. , Geballe, T.H. , Rosen- berg, H.M. , and Ziman, J.M., The thermal conductivity of germanium and silicon be- tween 2 and 300 K, Proc. Roy. Soc. (London) 238A , 502-14 (1957). [25] Glassbrenner, C.J. and Slack, G.A. , Thermal conductivity of silicon and germanium from 3 K to the melting point, Phys. Rev. 134 (4 A), A1058-69 (1964). [26] Fulkerson, W. , Moore, J. P. , Williams, R. K. , Graves, R.S. , and McElroy, D.L. , Thermal conductivity, electrical resistivity and See- beck coefficient of silicon from 100 to 1300 K, Phys. Rev. , in press. [27] Morris, R.G., High-temperature thermal conductivity measurements in semiconduc- tors, South Dakota School of Mines and Tech. , Status Rept. for 1 Oct. 64 to 30 Sept. 65 under Contract Nonr296401, AD 430 154 (September 1965). [28] Turnbull, A. G. , Thermal conductivity of liquid and solid a -phosphorus, Z. Physik. Chem.( Frankfurt) 42 (3/4), 243-6 (1964). [29] Kraemer, H. and Schmeiser, K. , Thermal conductivity of yellow phosphorus in the temperature range +16 to +80°, Z. Physik. Chem. (Frankfurt)35, 1-9 (1962). [30] Adams, A.R., Baumann, F. , and Stuke, J., Thermal conductivity of selenium and tel- lurium single crystals and phonon drag of tellurium, Physica Status Solidi 23 (l), K99-K104 (1967). [31] White, G.K. , Woods, S.B. , and Elford, M.T. , Thermal conductivity of selenium at low temperatures, Phys. Rev. 112_(2), 111-13, (1958). [32] Benguigui, L. , Thermal conductivity of liquid tellurium between 740 and 900 K, Physik Kondensierten Materie 5_ (3), 171-77 (1966). [33] Amirkhanov, Kh. I., Bagdnev, G.B. , and Kazhlayev, M. A. , Anisotropy of thermal conductivity in single crystals of tellurium, Akad. Nauk. SSSR Doklady 124(3), 554-6, (1959). [34] Cutler, M. and Mallon, C.E., Thermo- electric study of liquid semiconductor solutions of tellurium and selenium, J. Chem. Phys. 37, 2677-83 (1962). [35] Cezairliyan, A. , Predition of Thermal Conductivity of Metallic Elements and Their Dilute Alloys at Cryogenic Temperatures, Purdue Univ. , Thermophysical Properties Research Center, TPRC Rept. 14 (1962); also USAF Rept. ASD-TDR-63-291 (1963). 39 .X ,_«« W°M 'AllAllOnQNOO 1VWH3H1 40 250 225 200 175 150 S2 K UJ > Z 125 U. O tr. LJ CD Z> 100 75 50 25 I -Estimations -Experiments 1 £ e O J__l As At B Se Figure 2. Number of Investigations on the Thermal Conductivity of Non-metallic Solid Elements. 41 2500 M.P2573 M.P>3773 Subl. 3925-3970 3786 2250 I Estimated values -Experimental data 2000 1750 1500 1 '250 St! UJ -I-M.P 1000 750 500 250 -i-HP -rM.P -MP MP As At B I Se Si Te Figure 3. Temperature Range of Available Thermal Conductivity Data for Non-metallic Solid Elements. 42 -a o bD O a o o > •rH -4-> O TJ C O O 13 5 H 2 ,.X ,J« H°M 'AilAllOfiaNOO ivHuam 43 I0O20O3OO4OO3OO6OO70OBO09OO I0O0 IIOO I2O0 1300 1400 1600 ITOO TEMPERATURE, K Figure 5. Thermal Conductivity of Silicon (Linear Scales). 44 _ 1 1 1 \ i i i i 1 1 1 1 1 I - „ 7j \ THERMAL CONDUCTIVITY OF / V \ \ GRAPHITE (Provisional Recommendations) / / \ y l> JSffi-m V | \ - j 1 / / / / \ \ \ \ \ v \ - I \ - / / / / / \ \ S. \ V / V / / / /-(a to mSVmtmimi \ \ \ \ - / / / / f | yS 10 «Jtof wrrutton) - / / ~-\ \ / •■trustor i - 1 / 1 i i t i \ \ /Y//// / ,. 1 "OC^S/ - \ /////7V y/ / /// s > 1 ATJ X \ ^ t t f — i // M/ / ' i 1 J / / / If;// f/fff\ I//K nj \ \ \ \ \ PL a to aw o< otrutton) i i - j 1 f i / / ll Ifi/ 1 \/ a to kqw pkml I - 1 / t Iff Iff \ \ / / / \ X / / / j / / Iff, \ v / 7 / / // ft '/ i ill / i \ \ / / / / / / / / / / / / f li, V \ V s / / / / J t 1 1 1 \ \ / 1 t I, >' / V ' 1 li I ' 1 ! w ' \ / I i It El * I) nj i HI a i ~ j 1 Hi y i 1 1 1 ff' I i l 1 1 1 1 1 1 > « I • • PO t 1 4 S • • K>* TEMPERATURE, K t I 4 9 • • t I 4 t Figure 6. Thermal Conductivity of Graphite: Recommended Curves. 45 ] i i i i i i 1 1 1 1 i r~n — r= THERMAL CONDUCTIVITY OF NONMETALLIC SOLID ELEMENTS (Provisional Recommendations) Figure 7. Thermal Conductivity of Non-metallic Solid Elements: Provisional Recommendations. 46 Some Aspects concerning Thermal Conductivity Data of Liquids and Proposals for New Standard Reference Materials H. Poltz Physikalisch-Technische Bundesanstalt , Braunschweig, Germany The thermal conductivity of organic liquids poorly absorbing iR-radiation contains a term of measurable magnitude arising from radiative heat transfer. This term can be, depending on the apparative arrangements, between about one and seven per cent of the total thermal conductivity at room temperature and increases like T*. The influence of the apparative conditions on the temper- ature coefficient of thermal conductivity is in general considerably greater. Therefore one should, if possible, choose liquids strongly absorbing IR-radiation for standard reference purposes because their thermal conduc- tivity shows a smaller radiative term. We present thermal conductivity values of some alco- hols which have a high IR-absorption. As it was to be expected, the measured values show a small dependence on the thickness of the liquid layer. Therefore we think some of these alcohols well-suited as standard reference materials at moderate temperatures. Their hygroscopic reaction is moderate and very pure products are obtainable in trade. As reference materials for a higher temperature range phthalic acid esters should be taken into consider- ation. Key words: Alcohols, conductivity, heat transfer, hydrocarbons, IR-absorption, liquids, organic liquids, phthalic acid esters, radiation, radiative heat transfer, reference materials, standard, temperature coefficient, thermal conductivity. 1 . The Radiative Term of Thermal Conductivity Of the different physical processes contributing simultaneously to the heat transport in a liquid, the radiative heat transfer depends on the thickness of the liquid layer and on the optical qualities of the liquid and the surfaces bordering it. This was shown by measuring thermal conductivities of organic liquids using a guarded parallel plate apparatus by varying the plate distances D» 2 mm. Hence a perceptible influence of convection seems to exist only for plate distances of more than 2 mm. Fig. 2 and 3 show the increase of the thermal conductivity and its temperature coefficient respectively within Figures in brackets indicate the literature references at the end of the text of this paper. 47 the range from s" = 0.4 to 2 mm each for three liquids which have an IR-absorption of different magnitude. The slopes of the curves Increase with decreasing IR-absorptlon coefficients for the liquid. Calculating the radiative term R oi the total thermal conductivity k e ff gives the following expression valid for small temperature differences between the plates (T-|-T2« T2 o s o O O *- vo »- O ON ON KMACM fvl I I I I <* onvo O ON CO CO VD 1- CM on ON *- T- ON t— T- CM ON ON *• T" on\o -3- m ON ON e © T m T* CO •* TfrUN ONC^ o T- T* [— O t— CO on on t- *" [— O t«- CO ON ON o *" T 0>CtM UN oo co on as CT\OMJ\0\ o e r* • f* T- •f T- OVO KN UN 00 CO ON ON ON ON ON ON o • • • T» T- T- T~ vo o o T- o o o CM CM ir\ »• o o O o CM CM f- VO IT- IfN T m *- CM CM o o O O e o CM CM CM CM CM \0 CM ■«*• T- T- CM CM o o O O o o • CM CM CM CM s UN CM ON T- CM CM VO VO M3 UN KN ON T- CM CM VO KN KN ^3- UN CO "Sf UN UN e. 00 CO ITNCT\ UN UN in CM •<4- ON -"^ ON CO KN VO NO MD CM M3 f- 00 CO VO VO VO VO CMVO I— 00 00 \o \o V£> CO CO ON UN UN CM ■«*• ON ON KN h CD 0) -P H a) o En £ to CD T3 to CD o a UN O o a CO co CO fi • •** M X) -H U 04> 3 • •H o< >> Ja «h -H af S OHH UN KN UN ON KN CM CM O I I I I oo f— cm r— KN "3" UN UN O O O O CO CO CM M3 KN *d- UN UN o o o o UN -*3- ^ UN O O vo "tf- "3" UN O O CM O UN CO UNVO ^O ^O o o o o 1- ONVO CO UN UNVO VO o o o o CO UN O ON UN O UN CM 00 CO "-O C— f- t— o o o o UN CM c— ON VO t— f— t— O O O O & 00 UN ON UN UN CM ON ^ ON 51 Table 4 Purity of alcohols us ed for the measurements Substances purity guar ante 2d real, purity max. water percentage real water percentage other impurities Methanol R.G. Ethanol, absolute R 99.0 .G. 99.1 0.05 0.2 Methanol 0.1 Propanol - (1) R.G. iso-Propanol R.G. 99.0 99.7 99.8 - 99.9 0.05 0.1 t-Butanol Butanol - (1) R.G. 99.0 99.5 - 99.9 0.2 0.1 Butanol - (2) for chromatography 98.5 99.8 - 99.9 0.2 0.05 iso-Butanol R. G. 99.0 99.2 - 99.9 0.05 t-Butanol for chromatography 99.0 99.8 - 99.9 0.1 Butanol - (2) R.G. = Reagent Grade 4. Alcohols as Standard Reference Materials at Moderate Temperatures As a conclusion from our experiences in measuring alcohols, we can say that the propanols and the butanols with exception of tertiary butanol are well fitted for reference purposes. In order to show how much the radiative term of them is reduced by their high IR-absorption we have presented in the same scale the measured depen- dence of thermal conductivity upon the plate distances for an alcohol and a hydrocarbon each in the figures 7 and 8. Unfortunately there are not enough thermal conductivity measurements of high accuracy available on alcohols. Table 5 shows a comparison of some values measured by several authors [6 to 15] . We think it necessary to gain more reliable mean values by further measurements. 52 Table 5 Thermal conductivity of alcohols at 25 measured by several authors Author Ref. Meth- Eth- Propanol Butanol anol anol n iso n sec. iso tert. Thermal conductivity, W cm" 1 deg' 1 x 10" 3 Riedel w 2.01 1.66 1.55 1.40 1.52 Mason w 2.01 1.61 Filippov EJ 2.05 1.70 1.52 1.41 1.44 1.34 Tsederberg M 1.63 Challoner [10] 1.68 Scheffy [H] 1.07 Grassmann [12] 1.64 Jamieson [13] 2.01 Jobst [14] 1.64 1.62 1.52 1.47 Tufeu [15] 2.09 1.73 1.58 This work 1.97 1.64 1.52 1.34 1.48 1.35 1.31 1.08 w T -3.1 emperature coefficient, W cm" 1 deg" 2 x 10~ 6 Riedel -3.1 -1.1 -1.2 -2.0 Mason CO -1.3 -5.5 Filippov [81 -2.5 -2.4 -2.1 -1.6 -2.0 -1.3 Tsederberg KJ -2.6 Challoner [10] -3.0 t Scheffy [11J -0.3 Grassmann [12] -3.3 Jamieson [13] - Jobst [H] -3.7 -2.8 -2.0 -1.7 Tufeu [15] -3.8 -3.1 -2.4 This work -3.0 -3.0 -2.3 -2.0 -2.1 -2.0 -1.6 -1.3 5. Standard Reference Materials for Elevated Temperatures Because of the radiative term of thermal conductivity rising rapidly with tempe- rature, the necessity is evident to use liquids of high -absorption for reference purposes at elevated temperatures. But there seem to be no measured values of suit- able materials sufficiently reliable. We believe that phthalic acid esters are well fitted for these purposes. They are to be obtained in a very high degree of purity. They are nonhygroscopic, nontoxic, liquid at room temperature and have relatively high boiling points. Besides this, they are nonaggressive and resistant to chemical and thermal influences. We hope to present results about their thermal conductivity and its radiative term in some time. 53 6. References [l] Fritz, W. and Poltz, H., Absolutbe- stimmung der Warmeleitfahigkeit von Plttssigkeiten I - Kritische Versuche an einer neuen Plattenapparatur, Int. J. Heat Mass Transfer 5, 307 (1962). [2] Poltz, H., Me Warmeleitfahigkeit von Plttssigkeiten III. Abhangigkeit der Warmeleitfahigkeit von der Schicht- dicke bei organischen Plttssigkeiten, Int. J. Heat Mass Transfer 8, 609 (1965). [3] Poltz, H., Die Warmeleitfahigkeit von Plttssigkeiten II. Der Strahlungsan- teil der effektiven Warmeleitfahig- keit, Int. J. Heat Mass Transfer 8, 515 (1965). [A] Kohler, M. , EinfluB der Strahlung auf den Warmetransport durch sine Plttssig- keitsschicht, Z. Angew. Phys. 18, 356 (1965). [5] Poltz, H. and Jugel, R. , The thermal conductivity of liquids IV - Tempera- ture dependence of thermal conductivi- ty, In,t. J. Heat Mass Transfer 10, 1075 (1967). [6] Riedel, L. , Neue Warmeleitfahigkeits- messungen an organischen Plttssigkei- ten. Chemie Ingr.-Tech. 23, 321, (1951); - Warmeleitf ahigke it smes sun- gen an Mlschungen verschiedener orga- nischer Yerbindungen mit Wasser. Che- mie Ingr. Tech. 23, 465 (1951). [7] Mason, H. 1., Thermal conductivity of some industrial liquids from to 100 deg. C. Trans. Amer. Soc. Mech. Engrs. 76, 817 (1954). [8*] Pilippov, L. P. , Thermal conductivity of fifty organic liquids. Vestnik Moskov, Univ. Ser. Piz.-Mat. i Estestven. Hauk, £, 45 (1954). [9] Tsederberg, N.V. and Timrot, D.L., Experimental determination of the coefficient of thermal conductivity for a 94 per cent volume solution of ethanol for the temperature range -75 to +200 °C. Zh. tekh. Piz., 25, 2458 (1955). [10] Challoner, A.R. and Powell R.W. , Thermal conductivities of liquids: new determinations for seven liquids and appraisal of existing values, Proc. R. Soc. A 238 , 90 (1956). [11J Scheffy, W.J. and Johnson, E.P., Thermal conductivities of liquids at high temperatures. J. Chem. and Engng. Data 6, 245 (1961). [12] Grassmann, P., Straumann, W. , Widmer, P. and Jobst, W. , Measurement of thermal conductivity of liquids by an unsteady state method. In Amer. Soc. Mech. Engrs. Progress in Intern. Res. on Thermodyn. and Transport Prop., pp. 447-453. New York and London: Academic Press (1962). [13J Jamieson, D.T. and Tudhope, J.S., A simple device for measuring the thermal conductivity of liquids with moderate accuracy. NEL Report No. 81. East Kilbride, Glasgow (1963). [14] Jobst, W. , Messung der Warmeleitfa- higkeit von organischen, aliphati- schen Plttssigkeiten und von Gasen nach einem instationaren Abeolutver- fahren, Dissertation EidgenSssische Technische Hochschule Zurich, Switzer- land (1965). [15J Tufeu, R. , Le Neindre, B. and Johannin, P. , Conductlbilite thermique de quelques liquides, C.R.Acad. Sc. Paris. 262, 229 (1966). 54 7.50 6 1.35 1.25 2 3 4 5 Layer thickness s, mm Figure 1. Thermal conductivity k e £ f as a function of the layer thickness at 25 C, 1.0 1.5 2.0 Layer thickness s, mm Figure 2. Thermal conductivity k ft ff a» a function of the layer thickness at 25 °C. -020 -1.50 -1.60 ■1.70 10 1.5 2.0 Layer thickness s, mm Figure 3. Temperature coefficients of thermal conductivity as functions of the layer thickness at 25 °C. 1.0 0.8 0.6 0.4 0.2 0.0 - ^£=7.0 ^0.< "^0.2 ~--0.0 - gp^\ - . It^ 0.2 0.5 1 2 5 Optical thickness T 20 Figure 4. Y as a function of the optical thickness T 55 1.34 ? 1.33 S 132 1.31 V » * /.2S Iheor. _ <^- -'"' P xP-- " Iheor. y Jl^, .--'"" ^y Toluene AT 4r -2.'! -g -2.5 o -27 ; 1 -2.S 0.5 W 15 2.0 Layer thickness s, mm Figure 5. The thermal conductivity of toluene and its temperature Coefficient measured and calculated for 25 °C. 200 300 Temperature, °C Figure 6. Thermal conductivity of liquid paraffin for several layer thicknesses extrapolated at 300 °C. 40 50 60 70 Temperature , °C Figure 7. Thermal conductivities of iso- butanol and liquid paraffin at three layer thicknesses. 140 s = 1.93 mm °^ < 0.96 mm 0.46 mm 20 30 40 50 60 Temperature , °C 56 Figure 8. Thermal conductivities of iso- propanol and toluene at three layer thicknesses. The Thermal Conductivity of Nonmetallic Elements [in liquid or gaseous state at NTP] In Solid, Saturated and Atmospheric Pressure States P. E. Liley Thermophysical Properties Research Center Purdue University 2595 Yeager Road West Lafayette, Indiana 47906 Information in the literature on the thermal conductivity of argon, bromine, chlorine, deuterium, fluorine, helium, hydrogen, iodine, krypton, neon, nitrogen, oxygen, radon, tritium and xenon for solid, saturated liquid and vapor and gas at atmospheric pressure has been collected and analyzed. Where data were lacking, a variety of estimation methods have been employed to obtain values. Most probable values are suggested and departure plots given where appropriate to show the concordance between different data sets. Keywords: Conductivity, elements, gases, liquids, recsmimen-ded values, solids, thermal conductivity, vapors. 57 Thermal Conductivity of Vitreous Silica: Selected Values' o Lois C. K. Carwile and Harold J. Hoge Pioneering Research Laboratory U. S. Array Natick Laboratories Natick, Massachusetts OI76O The published literature on the thermal conductivity of vitreous silica has been assembled and the results critically evaluated. Best values of thermal conductivity as a function of temperature have been selected. These are presented in both graphical and tabular form. They cover the range 80 to 660° K and published data extending from 3° to 2100° K are shown in the graphs. An attempt was made to con- sult all work that could significantly affect the choice of best values. Published papers were located with the aid of Chemical Abstracts , Physics Abstracts , the Thermophysical Properties Retrieval Guide , and some other general sources. In addition, relevant references in the papers themselves were followed up until a substantially "closed system" had been generated, as shown by the fact that no new references were being turned up. Key Words: Fused quartz, fused silica, quartz glass, radiative heat transfer, reference material, silica, thermal conductivity, vitreous silica. 1. Introduction Vitreous silica is an important material in research, and is used to some extent in industry. It is easily obtained in a state of high purity. It is chemically inert to most substances, including many strong acids; hence it is valued as a storage container, and as a reaction vessel for chemicals of the highest purity. It is much used in optical work because of its wide range of transmission. It has excellent elastic properties: it transmits ultrasonic waves with very little absorption of energy (except at low temperatures); and fibers of it, being relatively immune to permanent set, make excellent electrometer suspensions and delicate spring balances. Vitreous silica has a very small coefficient of thermal expansion, hence is highly resistant to thermal shock. As a refractory, it can withstand temperatures of 1300°K (l027°C) for long periods, and higher temperatures for short periods. It is an excellent electrical insulator. This paper contains the material presented in Technical Report 67-7-PR (July 1966) of the U. S. Army Natick Laboratories, with no significant change. This paper reports research undertaken at the U. S. Army Natick (Mass.) Laboratories and is approved for publication. The findings in this report are not to be construed as an official Department of the Army position. 2 Member and Head, respectively, of Thermodynamics Group. 59 Vitreous silica is a noncrystalline material (glass) of the composition SiC^. It is usually made either from large selected quartz crystals or from very pure quartz sand. It is available in two basic types: clear (transparent) and translucent (satiny, silky). Clear vitreous silica is made by melting large crystals of quartz, and for this reason it is often called fused quartz or quartz glass; however, we prefer to reserve the word quartz for the commonest of the crystalline forms of silica, and to refer to the glass as vitreous silica, fused silica, or silica glass. Sosman [751, in his book on silica published in 1927* accepted the name "vitreous silica", and Laufer [76] in a more recent paper recommends it. When vitreous silica is made from sand, the translucent or satiny type is obtained. The satiny appearance is caused by tiny air bubbles that are trapped when the material is fused, and then are elongated into striations when the material is drawn or worked into useful shapes. The present study is almost entirely concerned with clear, vitreous silica, which contains no trapped bubbles of air or other gas. Vitreous silica is marketed under the names of fused silica, quartz glass, vitreous silica, and also under a number of trade-mark names. Among these are Amersil, Homosil, Infrasil, and Vitreosil; some of these names refer to special varieties having, for example, high optical transmission in a certain region of the spectrum. Because vitreous silica is a highly reproducible material with suitable mechanical, chemical, and refractory properties, it has been suggested as a standard material for thermal- conductivity work. How- ever, we believe the large radiative transfer of heat through vitreous silica makes it less desirable than an opaque substance as a thermal- conductivity standard, at least at high temperatures. We will return to this point later. Characterization . Vitreous silica consists of a network of tetrahedrons of SiO £ , loosely packed. Each silicon atom is tetrahedrally surrounded by k oxygen atoms, and each oxygen is shared by 2 silicons; but long-range order is lacking. In a crystal, all corresponding Si-O-Si bond angles are equal, but in the glass this angle varies somewhat; also, according to Warren and Biscoe [77] > the orientation of adjacent tetrahedra about these bonds is random in the glass. The X-ray analysis of vitreous silica shows the short-range, near-neighbor order that is characteristic of liquids and glasses, but no order extending over a distance as large as 3 or k times a unit- cell dimension. The density of vitreous silica has a slight dependence on the rate at which the material is cooled from the molten state. Sosman [75] selected 2.203 g cm 3 as the best value for the density of clear vitreous silica, and stated that different samples were likely to differ in density by as much as 0.001 percent. Subsequent work gives little reason for differing from Sosman' s conclusions. Crawford and Cohen [13] observed both the density and the thermal conductivity of vitreous silica before and after neutron bombardment. Using a hydrostatic method, they found the density at room temperature to be 2.2002 g cm before bombardment. This value was increased by 2.7 percent during bombardment, but subsequent annealing substantially restored the original value. The lack of long-range order in vitreous silica is associated with an open molecular structure with numerous vacant sites. These vacant sites reduce the density of vitreous silica to a value about 17 percent less than that of crystalline quartz. Vitreous silica does not have a sharp melting point. The crystalline form of silica that is stable at the highest temperature is cristobalite [75] J it melts at about 1710°C (l983°K), and becomes a liquid of high viscosity. Devitrification . The high-temperature limit of usefulness of vitreous silica is generally the temperature at which devitrification (crystallization) occurs, rather than that at which the material begins to soften or melt. Below about 1000°C (l273°K) it shows no tendency to crystallize, but at 1100°C and above, minute crystals gradually form. The crystals become visible when the material is cooled, as they undergo a transformation that gives them a white, chalky appearance. Devitrification is accompanied by a decrease in mechanical strength and an increase in tendency to shatter. It proceeds more rapidly as the temperature is raised; it may be complete in k to 8 hours, or it may be a matter of days. Devitri- fication begins at a surface; it is accelerated by water vapor, by alkalis, and by many other substances. It is retarded when the amount of oxygen in the vitreous silica is slightly less than stoichiometric [78]. High-quality transparent vitreous silica, in the absence of catalysts, has been held at 1300°C (l573°K) 3 Figures in brackets indicate the literature references at the end of this paper. 60 for several hours with only a slight cloudiness resulting. The product of devitrification is usually cristobalite, even at temperatures where cristobalite is not the most stable form of silica. Effects of OH. Although clear vitreous silica made by melting crystals is ordinarily very pure, there are usually traces of metals in it. For some optical work, these traces of metals must be avoided; for this purpose, silica is made by reacting SiCLi with water and depositing the Si0 2 in the glassy state. Vitreous silica made from SiCl 4 is metal-free, but it contains [79] a relatively large amount of OH, which comes from the water formed in the reaction. This OH can amount to more than 0.001 percent by weight, whereas vitreous silica made by melting natural crystals contains only about 3 parts of OH per million. This difference in OH-content causes small changes in density, thermal expansion, index of refraction, and other properties, but perhaps its greatest importance is in spectroscopic experiments that require transmission of the OH frequencies; the presence of OH in the glass causes these frequencies to be absorbed. The effect of OH-content on the thermal conductivity of vitreous silica has not been specifically studied, but presumably it is small in comparison with the scattering of even the better data now available. Satiny vitreous silica , \-values . As discussed earlier, satiny vitreous silica contains elongated air bubbles that form striations in the material. Since the selected values in the present paper refer to clear, rather than satiny vitreous silica, we will present here the small amount of information that is available on the difference in thermal conductivity of the clear and satiny varieties. One would expect the striations to reduce the conductivity, especially when the direction of heat flow is perpen- dicular to the long axes of the striations. However, the effect appears to be rather small. Koenig and co-workers [jh] reported measurements on clear vitreous silica, and on satiny material with the heat flow parallel to the striations. These measurements show the X-values of the satiny type to be about 1 percent below those of the clear material in the range kO to 120°C. The authors did not consider this difference significant, but we note that it is in the direction to be expected. Later the same group [35] made similar measurements on two new satiny samples, one having the striations parallel, the other per- pendicular, to the direction of heat flow. Again, in approximately the same temperature range, one sample had a conductivity about 3 percent lower than the other, but unfortunately the report does not tell us which sample was which. Both of the sets of measurements just referred to were made rather early in an extensive series of investigations. More accurate results were obtained later in the work, but the comparison between clear and satiny vitreous silica does not appear to have been repeated. 2. Selection of the Values The methods used in evaluating the experimental data, drawing a master curve, and preparing a table of selected values of the thermal conductivity, A., will now be described. As indicated earlier, the variety of material to which the curve and tables refer is clear (transparent) vitreous silica. In some cases the same set of experimental data has been published in several different papers. In such cases we have selected one of the papers as the primary reference and have listed the others in the group of references "for which another reference is preferred. " The governing factors considered in the selection of the primary reference will be found in the annotations to the various references. Because of the large amount of data on the thermal conductivity of vitreous silica and the wide range of temperature covered, three separate graphs have been used to present the data. All three graphs start at 0°K; one extends upward to 100 °K, another to 1000 °K, and the third to 2100 °K. The results of a particular research are given in one graph but are not duplicated in the others; thus over- crowding of the graphs is avoided. The values of X selected and recommended for use are given in Tables 1 and 2, and are represented in Figs. 1, 2, and 3 by heavy solid lines (the master curve). The values were selected by plotting the available data on large-scale graphs similar to Figs. 1, 2, and 3- A master curve was drawn, studied, revised, and tentatively acdepted. Values were read from this master curve; they were differenced, smoothed, and checked for consistency with the master curve. When table and curve were smooth and mutually consistent, they were accepted. Table 1 resulted from this process. Table 2 was derived from Table 1 and is consistent with it. The master curve and the tables extend only from about 80° to about 660°K (-193° to 38t°C). At higher temperatures there is a dashed line joining with the master curve and extending it. This exten- sion was derived by using heat-capacity data, and it is believed to represent roughly the thermal con- ductivity of vitreous silica in the absence of radiative heat transfer. The method of deriving the dashed curve is described later in this report. 61 Thermal conductivity of vitreous silica Table 1 T X T A. millical millical °K cm sec °C °K cm sec °C 80 1.31 38O 3-56 ; 30 5 100 1.61 1+00 3.61 27 5 120 1.88 1+20 3.66 2k 6 i4o 2.12 1+1+0 3.72 21 5 160 2-33 1+60 3-77 19 6 180 2.52 1+80 3.83 17 200 2.69 500 3.88 15 5 220 2.81+ 520 3-93 13 6 240 2.97 5I+O 3-99 12 5 260 3.09 560 1+.01+ 10 6 280 3-19 580 1+.10 9 5 300 3.28 600 I+.15 8 5 320 3.36 620 1+.20 7 1+ 3I+O 3.1+3 61+0 1+.21+ 7 1+ 36O 3.50 6 660 1+.28 Table 2 T X Btu in. °R °R ft 2 hr 150 3-95 117 200 5-12 100 250 6.12 83 300 6.95 73 350 7.68 61 1+00 8.29 51 1+50 8.80 ^3 500 9.23 36 550 9-59 30 600 9.89 31 650 10.2 2 700 10.1+ 2 750 10.6 2 800 10.8 3 850 ll.l 2 900 11.3 2 950 11.5 2 1000 11.7 2 1050 11.9 2 1100 12.1 2 1150 12.3 2 1200 12.5 62 Data Shown in Fig. 1 . Figure 1 extends from 0° to 1000°K. It contains the data from references [1] to [11] j these references include much but not all of the best data for intermediate temperatures. Although the master curve is based on the data in all three figures, the data of Birch and Clark [1], shown in Fig. 1, are perhaps the most valuable single set of data. These authors used a guarded hot- plate apparatus and a steady-state absolute method. The temperature difference across the sample was usually about 5°K. Each test was run in a nitrogen atmosphere and again in helium, and between the two results the thermal- contact resistances were eliminated. Considerable weight has been given to the data obtained by Ratcliffe [2], which cover the range 123° to 322°K. Steady-state measurements were made in 3 different apparatuses; each apparatus had the hot plate sandwiched between 2 specimens. Care was taken to reduce or eliminate contact resistance, and corrections were made for lateral heat losses. At least 12 different samples were measured. Valuable data were obtained by Kamilov [3] from about 89° to about 500°K. The apparatus was con- tained in a Dewar flask within another flask, with provision for regulating the pressure between the walls of the inner flask. One unusual feature of the apparatus was the type of "thermocouple" used to register temperature equality between the top shield and the hot plate. This " thermocouple" consisted of a copper sheet coated with cuprous oxide; it actuated an electronic temperature controller. The published data represent the averaged results for k specimens with different thicknesses. Kurepin and Platunov [h] described an apparatus for measuring either thermal conductivity or thermal diffusivity over a wide range of temperature. In measuring X, heat was supplied so as to get a steady rate of temperature rise in the entire sample, for example, 0.10 deg/sec, and the temperature drop across the sample was measured as a function of time. The amount of heat flowing was determined from the known heat capacity and temperature rise of a heat sink, which was a core of Armco iron. Their data appear to be of good accuracy. Eucken [5] used a steady-state hot-plate method, and obtained surprisingly good results considering that his apparatus appears to have been unguarded. Vasil'ev [6] used a qua si stationary (steady-rise) method for 39 of his plotted points, and a transient method for 2 points. He used radial heat flow in a guarded cylindrical apparatus that did not require a heat sink. Automatic temperature control and recording were used, and the data appear to be good. We note that the data in his Fig. 3 taken from Ratcliffe appear to be incomplete; and those taken from Khapp, to be misplotted. Kaye and Higgins [7] reported careful measurements by a comparison method, using aluminum bars as the reference material. Data were obtained for 3 samples of vitreous silica; they are 6 to 8 percent lower than our accepted curve. Kingery and Norton [8] used 2 different methods. The first method employed a prolate spheroidal envelope; it was an absolute, steady-state method. The second method was a comparative one. Their data are about k percent higher than our accepted curve; however, the data are valuable because they extend to relatively high temperatures. Bil 1 and Avtokratova [9] used vitreous silica as a standard material for testing their apparatus. The values found are about 8 to 15 percent lower than our selected values. Their samples were stated to be of unsatisfactory purity and to contain air pockets. The results obtained by Scholes [10] and by Koenig [11] were obtained by a comparison method. These values are higher and rise more rapidly with temperature than those we have selected. Data Shown in Fig. 2 . Figure 2 extends from 0° to 100°K. In this range, the work of Berman [12] appears to be the most reliable. He measured 3 samples by a steady-state method, with a radiation- shielded apparatus surrounded by a vacuum jacket. The entire apparatus could be cooled by liquid helium or other coolant. The \-values of Crawford and Cohen [13] increase very rapidly with temperature, and do not extrapo- late toward the accepted values at higher temperatures as Berman 's do. Crawford and Cohen attributed the difference between their values and those of Berman to a difference in the materials tested, but the difference seems rather large for such an explanation. Data Shown in Fig. 3 - Figure 3 extends from 0° to 2100 °K. Very good data were reported by Devyatkova, Petrov, Smirnov, and Moizhes [lb], over a wide range of temperature. Their work was under- taken to determine the potential usefulness of vitreous silica as a standard of reference for measurements 63 of thermal conductivity. They used 3 different apparatuses. Temperature gradients were measured by means of thermocouples fastened to metal pins inserted into holes drilled in the samples. Corrections were made for lateral heat losses. Their results have significantly influenced our selection of best values throughout the range for which a master curve and tables have been given. As will be seen below, we believe that radiant energy transfer has caused the results at higher temperatures to represent an apparent rather than true thermal conductivity. Seemann [15] reported careful work by a radial- flow method with 2 different apparatuses and a single test sample. During measurements, both surfaces of the specimen were in contact with liquid metal: mercury at ordinary temperatures, lead-tin eutectic at higher temperatures. Seemann' s results appear to be about 10 percent low in the range of temperatures for which we have drawn a master curve. However, at higher temperatures Seemann' s results appear to be among the most successful in avoiding radiant-heat transfer . Ito [16] also used the radial- flow method. His results are lower than those of most other workers; at the lower end of his range, he is about 20 percent below our master curve. Ito's results as reported by Birch and Clark [1] and by Ratcliffe [2] in the form of straight lines do not agree with the plotted points in Ito's paper; the discrepancy can be explained if we assume that these authors used Ito's equation with temperatures in °K, whereas it is actually for temperatures in °C. Whatever the cause, the discrepancy is substantial. Vishnevskii and Dzyubenko [17] built a guarded apparatus for measuring X by a transient method. They report results obtained with this apparatus and also by a steady- state method; the two sets of results show good agreement and are represented by a single curve. However, these values are about 8 percent lower than those of our accepted curve. An unusual method, involving steep temperature gradients, was used by Wray and Connolly [18] to measure X. of vitreous silica at high temperatures "without radiative effects". Each sample was in the form of a cylindrical rod with a tungsten wire along the axis, so prepared as to have continuous bonding between wire and sample as shown by microscopic examination. The wires were of very small diameter (0.008 cm or less), and served as both heater and thermometer. From the graphs of observed power vs. wire temperature, the slopes were determined; and these slopes were used to find X by a formula which Wray and Connolly derived theoretically. Their results were presented as smooth curves, and give values of X at higher temperatures than any other papers that we have located. The curves of Wray and Connolly remain substantially horizontal at higher temperatures, where those of others rise steeply. Observations at high temperatures were also reported by Lucks, Matolich, and Van Velzor [19]; Kingery and Norton [20]; and Pustovalov [21]. In all these data the steepening rise shows the presence of radiative heat transfer. Most observers made some effort either to reduce or to correct for radiative heat transfer. For example, Lucks, Matolich, and Van Velzor made some experiments with thin low emissivity foil "adjacent to the surface of the transparent solid specimens" (they used a comparison method, with Armco iron as the accepted standard). They found that the \-values obtained without foil showed a more rapid increase than those obtained with foil, above about *i50°C (723°K). Kingery and Norton used 3 different apparatuses; all employed the comparison method. The paper of Lee and Kingery [64] reports the same work with a somewhat better description of experimental methods. Pustovalov used an apparatus of cylindrical symmetry, with an axial heater; the positions' of the thermocouples in the samples were determined by using X-rays [80]. The \-values reported are about 25 percent lower than our master curve. It is possible that the porosity of the sample is responsible; Pustovalov gives [69] the apparent porosity of his sample as 1.0 percent; but from his measured density (2.06 g cm ) and our accepted density (2.20^), we calculate a porosity of 6.5 percent. Also shown in Fig. 3 are the results of Colosky [22], and those of Knapp [23]; these appear to be too high and to rise too steeply. At the higher temperatures, the data reported by almost all the experimenters lie above the master curve and the dashed curve that extends it upward. As mentioned earlier, this dashed curve represents our estimate of the thermal conductivity of vitreous silica in the absence of radiative heat transfer. The method of obtaining this curve will be described in the next section. 3- Nonradiative Heat Transfer at High Temperatures Thermal conductivity as defined does not include the transfer of heat by either convection or radiation. However, in many investigations of thermal conductivity some heat transfer by convection or 64 radiation takes place in the sample. When the amount is small, this transfer is often included with the transfer by conduction, and an apparent thermal conductivity is calculated. In the case of vitreous silica such a procedure is permissible up to about 600°K, but as the temperature increases the radiative heat transfer becomes very large, and it is not desirable to lump it with thermal conductivity. The problem of radiative heat transfer is of special importance in vitreous silica because the substance is highly transparent. Vitreous silica transmits a wider range of wave lengths than most glasses; the range extends from 0.2 to k.O microns, approximately, and in addition there are other bands of transmission or partial transmission at longer wave lengths. In estimating the ordinary (nonradiative ) thermal conductivity of vitreous silica at high tempera- tures we can follow either of two paths. We can estimate the effect of radiation and substract it from the total observed heat transfer, or, we can assume a relation between thermal conductivity and some other property of the material such as the specific heat c , and use this to give us X. P It is easy to make rough calculations of radiative heat transfer in vitreous silica, but much harder to make accurate ones using the absorption coefficient appropriate for each wave length. We have avoided such difficult and perhaps uncertain calculations by assuming that X/c is a constant. Accepting c , we have calculated the nonradiative value of X, and have checked the results so obtained by some rough cal- culations of radiative heat transfer that are described in the next section. It is well known that heat capacity and thermal conductivity are closely related. According to a very simple theory X is proportional to the product c vl, where v is the mean velocity of phonons and 1 their mean free path. Our assumption that iV/c is constant is equivalent to assuming that the product vl is constant. This assumption is used simply because no better one is at hand, and no great accuracy is claimed for it. If we were to go beyond the assumption X/c = a constant, we would assume that this quantity decreases somewhat with temperature, because increasing lattice vibrations tend to reduce the mean free path 1. Sosrnan [reference 75, p. 31^] has published a table giving c of vitreous silica. Although there are more recent data in some temperature regions than those that Sosrnan used, his table is convenient for our present purpose because it covers a wide range, extending from 18° to 19T3°K. With c -values from this table and \-values from our master curve, the quantity X/c_ was calculated and plotted vs. T. Throughout the interval kfj) to 623 °K, this quantity had the value 16.6 mg cm sec . At both ends of the interval the curve turned upward. The value 16.6 was accepted. Multiplying Sosman's c -values by 16.6 gave the values of X that are represented by the dashed extension of the master curve in Fig. 3. This dashed curve we believe to be a rough but reasonable estimate of the thermal conductivity of vitreous silica in the absence of radiative heat transfer. It will be seen that the results of Wray and Connolly [18], whose experiments were designed to minimize radiative heat transfer, are in much better agreement with the dashed curve than those of any other investigator. Sosman's c values have a change in slope between 1^00° and 1500°K, which shows up in our dashed curve. Whether this change in slope is real or is due to experimental error we do not know. The uncertainty in assuming iv/c to be constant is so large that the irregularity in the dashed curve is probably comparatively unimportant. k. Radiative Heat Transfer As a check on he values of nonradiative heat transfer obtained as described above, a few calcula- tions of the radiative heat transfer to be expected in vacuum and in our idealized model of vitreous silica have been made. First the "thermal conductivity" of a "slab of vacuum" 1 cm thick, with black-body walls, was calculated. Suoh a model would account for the radiative heat transfer in a perfectly trans- parent material and might be useful for a material that was a uniformly gray absorber at all wave lengths. It did not account satisfactorily for the sharp increase in slope shown by the experimental data for vitreous silica, principally because it caused the rise to occur at temperatures that were too low. Next we tried a slightly more sophisticated model, one in which the sample of 1 cm thickness was considered to be a perfect transmitter for all radiation of wave lengths lying between 0.2 and k.O microns, and to be perfectly opaque to all other wave lengths. (Actually, the energy at wave lengths shorter than 0.2 u was negligible, so a "window" extending from to k.O \i would have given the same result. ) This model was quite satisfactory. When the radiative "conductivity" of the model was added to the values of the master curve and the dashed curve, the dot-dash curve shown in Fig. 3 vas obtained. 65 This curve is a reasonably good representation of the experimental data in which radiation was important. Note that it comes close to the data of Kingery and Norton, and to those of Lucks, et al. "without foil". Since the more sophisticated model of radiative transfer and the dashed X-curve based on c are mutually consistent they tend to confirm each other, but even so the breakdown of X into radiative and nonradiative parts can only be considered a rough approximation. A few remarks pointing out the weakness of the model used above in the calculation of radiative transmission are in order. Perfect transmission of radiation gives a conductance that is independent of the sample thickness and a conductivity that is proportional to sample thickness. For many wave lengths, vitreous silica is neither a perfect transmitter nor a perfect absorber, but is intermediate, with a mean free path for the radiation that may be of the order of a few millimeters. An individual quantum will, on the average, travel this distance before it is absorbed, but new quanta to replace those absorbed will be generated throughout the glass. Gardon [8l] has summarized the theory of heat transfer within glass by absorption and re-emission of radiation. A radiative thermal conductivity proportional to T 3 and inversely proportional to the absorption coefficient of the glass is found. However, the concept of a radiative thermal conductivity is not very useful because it is applicable only in the interior of large samples and not near the heat source and heat sink (hot plate and cold plate). The boundary conditions when radiative heat transfer is present are normally such that the temperature gradient within the sample is nonuniform, being steeper near the hot and cold plates than elsewhere. We see from this that when the absorption coefficient has any small non-zero value (as well as when it is actually zero) the "radiation conductiv- ity" depends on sample thickness. Some workers have reduced the effect of radiative heat transfer by using highly reflective metal foil such as aluminum or platinum, placed on either side of the sample, between it and the hot and cold plates. Although this arrangement substantially reduces the amount of radiative heat transfer, it does not eliminate it. Even if the foil had zero emissivity so that heat entered and left the sample only by conduction, a portion of the heat would be converted to radiation within the sample and would travel through the sample in the form of radiation. 5- Vitreous Silica as a Standard of Thermal Conductivity Clear vitreous silica exhibits many of the characteristics desirable in a standard material for thermal- conductivity measurements. Among these characteristics are its high purity, uniformity, low thermal expansion, and chemical inertness. Vitreous silica has been used as a standard by a number of investigators; among them Benfleld [27], Bullard and Niblett [82], Kamilov [3], and R. W. Powell [67]. Noting the wide use of vitreous silica as a calibrating standard, Devyatkova, Petrov, Smirnov, and Moizhes [14] made experiments to examine its suitability for this purpose. Using three different apparatuses in overlapping ranges, they made measurements from 82° to ll66°K. Comparing their data with those of Berman [12], Ratcliffe [2], and Kingery [60], they found good agreement, and concluded that vitreous silica is suitable for use as a standard substance in thermal- conductivity measurements. Although we agree in part with this conclusion, we believe that the usefulness of clear vitreous silica as a standard is limited to low and moderate temperatures, where radiation is not important. As was brought out in the two preceding sections of this paper, radiative heat transfer in vitreous silica undoubtedly exceeds nonradiative transfer at the higher temperatures. The agreement of the high-tempera- ture data of Devyatkova, et al., with those of Kingery is better than we would expect. The sample used by the first group had a thickness of 1 cm. The thickness of Kingery' s sample was probably 2.5^ cm. The relative importance of radiation in the two samples would be different and should lead to different apparent values of X.. It is possible that some compensating effect was present that kept the two sets of results in agreement. Methods of making vitreous silica opaque have been devised, but so far as we are aware there are no measurements of the thermal conductivity of opaque samples. If vitreous silica could be rendered opaque without changing its thermal conductivity, it would make an excellent standard material for use at any temperature up to the point where significant devitrification may occur. 6. Reliability of the Tables The tabulated values of X between 150° and U5O K are believed to be accurate to ± 7 percent; near room temperature they may be somewhat better. Below 150° K the uncertainty increases and may reach ± 12 percent. Likewise above U5O K the uncertainty increases and may be as large as ± 15 percent at the upper limit of the table. 66 7. Data for Conversion of Units T (°R) = T (°K) x 1.8. T (°K) = t (°C) + 273-15- T (°R) = t (°F) + ^59-67- Watt cm K _1 cm" 2 = cal cm °K~ 1 cm" 2 sec" 1 x l+.l8^0. Btu in. °R" 1 ft" 2 hr" 1 = cal cm °K" 1 cm" 2 sec" 1 x 2902. 9. Btu ft. R _1 ft" 2 hr" 1 = cal cm "k" 1 cm" 2 sec" 1 x 2^1. 91. 8. Acknowledgment We are indebted to Mr. Ronald A. Segars of this laboratory for help on various phases of this survey. 9- Annotated References 9.1. Containing Data Plotted in Fig. 1 [1] Birch, Francis, and Clark, Harry, The thermal conductivity of rocks and its dependence upon temperature and composition, Part 1. Am. J. Sci. 238 , 529-58 (19^0). Values were read from their Fig. 6 and also from their Fig. 7, and agreed within the error of read- ing. Corresponding values from the two graphs were averaged before being plotted in our Fig. 1. [2] Ratcliffe, E. H., Thermal conductivities of fused and crystalline quartz. Brit. J. Appl. Phys. 10, 22-5 (1959)- In addition to the plotted points of Ratcliffe 's "present measurements" in his Fig. 3> we have also accepted the point taken by Ratcliffe from D. W. Butler, "Unpublished data (National Physical Laboratory, 1950 )•" [3] Kamilov, I. K. , Investigation of the thermal conductivity of solids in the interval from 80 to 500°K. Instr. Exptl. Tech. Wo. 3, 583-7 (1962). This is a translation of reference [58]. Figure 3 of this paper con- tains vitreous-silica data in the form of plotted points. We found that the x's repre- sent Kamilov 's data; the circles were taken by him from reference [It]. [h] Kurepin, V. V., and Platunov, E. S., Apparatus for rapid wide- temperature thermophysical tests of thermally insulating and semi-conducting materials (dynamic a \- calorimeter ), Izv. Vysshikh Uchebn, Zavendenii, Priborostr. No. 5, 119-26 (1961). Their Fig. k contains values of thermal conductivity (A.) and thermal diffusivity (0C); we have used only the former. [5] Eucken, A., The temperature dependence of thermal conductivity of solid nonmetals. Ann. Physik {h) 3k, 185-221 (1911). The value for 100 °C (373 °K) is followed by a question mark in the paper; also it does not agree with the value given by Eucken for the same tempera- ture in a later paper (reference 56). Hence we have omitted this value from our Fig. 1. [6] Vasil'ev, L. L., Method and apparatus for the determination of the thermophysical properties of heat-insulating materials in the tempera- ture range 80 - 500°K. Inzh.-Fiz. Zhur. J, No. 5, 76-8*1 (1964). Thermal-conductivity values are given in his Fig. 3j thermal- diffusivity and specific-heat values are given in his Fig. h. We have used only the thermal- conductivity data as given in his Fig. 3- [7] Kaye, G.W.C., and Higgins, W.F., The thermal conductivity of vitreous silica, with a note on crystalline quartz. Proc. Roy. Soc. (London) A 113 , 335-51 (1926). We have plot-, ted their results from Table 5- [8] Kingery, W. D., and Norton, F. H., The Measure- ment of Thermal Conductivity of Refractory Materials. U. S. At. Energy Comm. Rept. NY0- 6kkk (June 30, 195*0, 16 p.; AD-J+0873. We have taken the experimental data from Fig. 1 of this report. This figure has been pub- lished in several different papers; we have selected this one as our primary reference because the figure in it was the largest. Reference [60], which contains a smaller version of the same figure, contains a fuller description of the work than the present reference. 67 [9] Bil', V. S. and Avtokratova, N.D., Temperature dependence of the thermal and temperature con- ductivities of some unfilled polymers. High Temp. 2, 169-75 (1964). This is a translation of reference [4-9]. The data given in their Fig. 1 have been read off and are plotted in our Fig. 1. [10] Scholes, William A., Thermal conductivity of bodies of high BeO content. J. Am. Ceram. Soc. 23, 111-7 (1950). The points for vitreous silica given in his Fig. 3 have been read off and are plotted in our Fig. 1. [11] Koenig, John H., Progress Report No. 4 from. September 1 to December 1, 1953, N. J. Ceram. Research Sta., Rutgers Univ., 159 p.; AD-29335- We have selected the data given in engineering units in Table II, p. 6 of this report, for conversion and plotting in our Fig. 1. 9.2. Containing Data Plotted in Fig. 2 [12] Berman, R., The thermal conductivities of some dielectric solids at low temperatures. Proc. Roy. Soc. (London) A 208 , 90-108 (1951). We have read off and used the plotted points in the upper section of Fig. 2 of Berman 's paper. [13] Crawford, J. H., Jr. and Cohen, A. F., The effect of fast neutron bombardment on the thermal conductivity of silica glass at low temperature. Bull. Inst. Intern. Froid, Annexe 1958-I (1958), p. 165-72. These authors give data obtained before irradiation, after irradiation, and after the effects of irradia- tion have been removed by annealing. The lowest curve in their Fig. 1 represents the data obtained in the unirradiated state and the data obtained after annealing; we have used both of these sets of data. 9.3. Containing Data Plotted in Fig. 3 [Ik] Devyatkova, E. D., Petrov, A.V., Smirnov, I. A. and Mbizhes, B. Ya., Fused quartz as a standard material in heat conductivity measurements. Fiz. Tverdogo Tela 2, 738-46 (i960). Figure 3 of this paper contains 3 sets of data obtained by the authors and 4 sets of data taken from other workers. The data of the authors have been read from this figure. An English trans- lation of the paper exists (reference 55 )> hut it was considered preferable to take the data from the Russian original. [15] Seemann, Herman E., The thermal and electrical conductivity of fused quartz as a function of temperature. Phys. Rev. £1, 119-29 (1928). The data were read from Fig. 3 of this paper. [16] Ito, Shuyo, Thermal conductivity of a trans- parent vitreous silica tube. J. Mazda Co. (Japan) Vol. 4, No. 2, 185-8 (1929). A copy of this paper was obtained for us from The Diet Library, Tokyo, by a Japanese colleague. The experimental points plotted in Fig. 3 were read from the graph in this paper. [17] Vishnevskii, I.I. and Dzyubenko, M.I., Measure- ment of the coefficients of thermal conducti- vity and thermal diffusivity of refractory materials by a transient method. Inzh.-Fiz. Zhur., Akad. Nauk Belorussk. S.S.R. 7, No. 10, 45-8 (1964). [18] Wray, Kurt L. and Connolly, Thomas J., Thermal conductivity of clear fused silica at high temperatures. J. Appl. Phys. 30, 1702-5 (1959). These authors give \-values only as the smooth curves plotted in their Fig. 3 '> values have been read from these curves. Fig. 5 of reference [74] could have been used instead, but there are slight differences in the two figures, and the present paper is presumably the later one. In our Fig. 3> the data for Wray and Connolly have been shown as short sections of smooth curves at 100°-inter- vals. There is no correspondence between these short sections and any plotted points of Wray and Connolly, since these authors gave their \-data only as smooth curves. [19] Lucks, C. F., Matolich, J. and Van Velzor, J. A. The Experimental Measurement of Thermal Conductivities, Specific Heats and Densities of Metallic, Transparent, and Protective Materials, Part III. Wright Air Development Center AF Tech. Rept. 6l45, Part III, March 1954. 71 p.; AD-95406. We have used the data for vitreous silica from Tables 7 and 8. [20] Kingery, W. D. and Norton, F. H., The measure- ment of thermal conductivity of refractory materials. Quarterly Progress Report for the Period Ending January 1, 1955- U. S. At. Energy Comm. Rept. NY0-6447 (1955), 16 p.; AD- 55595. Data are given in Fig. 4 of this paper. They extend from 95 to 12l4 c C (368 to l487°K), but not all have been shown in our Fig. 3. Those below 700°C are uniformly spaced, and we have omitted them on the assumption that they represent the same data that are presented in reference [8]. The data at 700 °C and above are less regular and we have accepted them as new data. The 3 highest- temperature points in the present reference are omitted because they have \-values beyond the range of Fig. 3. 68 [21] Pustovalov, V. V., Change of thermal conduc- tivity of quartz glass in the process of crystallization. Steklo i Keram. 17, No. 5, 28-50 (i960). The plotted points from the top curve in Fig. 3 of this paper have been read off and used in our report. The paper has been translated into English (reference 68). [22] Colosky, Benjamin P., Thermal conductivity measurements on silica. Bull. Am. Ceram. Soc. 31, 465-6 (1952). Figure 2 contains the experimental data. The two upper curves refer to crystalline quartz. The bottom curve represents data obtained with 3 different samples of vitreous silica. Two of these were clear and one was smoky, but there is no significant difference between the X-values of the 3 samples. The points for all three have been read from the graph and plotted in our Fig. 3. [23] Knapp, W. J., Thermal conductivity of nonmetallic single crystals. J. Am. Ceram. Soc. 26, 1+8-55 (19^3). The vitreous- silica data from Table II of this paper are plotted in our Fig. 3, except for the point of highest T and X, which comes beyond the range of the figure. 9.4. Containing Data Not Plotted in the Figures These references contain the data judged to be less important than those in references [1] to [23]. The arrangement is alphabetical. Each of the references in this group is followed by a brief annotation, in which the X-values reported in the paper are included. Numerical values given below are in cal cm sec °C [24] Ballard, Stanley S., McCarthy, KathrynA., and Davis, William C, A method for measuring the thermal conductivity of small samples of poorly conducting materials such as optical crystals. Rev. Sci. Instr. 21) 905-7 (1950). A relative method was used, but the specific reference material used in each experiment is not stated. For vitreous silica they found X = 0.00282 at 4l°C, using a "Homosil" sample from the W. C. Heraeus Co. [25] Barratt, Thomas, Thermal conductivity. Part II: Thermal conductivity of badly- conducting solids. Proc. Phys. Soc. (London) 27, 81-93 (1914). For vitreous silica of density 2.17 g cm , the thermal conductivity was found to be 0.00237 at 20°C, and 0.00255 at 100°C. [26] Beck, A., A steady state method for the rapid measurement of the thermal conductivity of rocks. J. Sci. Instr. 3_4, 186-9 (1957)- A relative method was used, with brass (cali- brated versus crystalline quartz ) as the reference material. For vitreous silica at 28°C, the thermal conductivity obtained was O.OO323. [27] Benfield, A. E., Terrestrial heat flow in Great Britain. Proc. Roy. Soc. (London) A 173, 428-50 (1939)- A comparison method was used, with brass bars as the reference material. The brass was calibrated versus crystalline quartz, and the effect of resist- ance at contact surfaces was eliminated by using 4 samples of different thickness. The value of X found for vitreous silica (Vitreosil from the Thermal Syndicate) was O.OO3I7 at 21.1°C. [28] Bullard, E. C, Heat flow in South Africa. Proc. Roy. Soc. (London) A 173. 474-502 (1939)- The thermal conductivity of vitreous silica was measured as a check on the accur- acy of the apparatus, which had been cali- brated against a crystalline quartz sample. For vitreous silica at 25°C, Bullard found x = 0.00307. [29] Crawford, J. H., Jr., and Wittels, M. C, Radiation stability of nonmetals and ceramics. Proc. 2d U.N. Int. Conf. Peaceful Uses Atomic Energy, Vol. 5, p. 300-10, 1958. They report X = O.OO35 at 30°C; there is also a graph (Fig. 7) of low- temperature values. This graph appears to be identical with one in reference [53] > reference [13] is preferred over both. [30] Eucken, A., The heat conductivity of ceramic refractory materials. Its calculation from the heat conductivity of the constituents. Forschungsh. 353 (Suppl. Forsch. Gebiete Ingenieurw. B 3, Mar. - Apr. 1932 ), 16 p. The data in this paper appear to be a selection of values based on previously published work of Eucken [5], Kaye and Higgins [7], and Seemann [15]- However, if this is the case, the values ought to be about 20 percent higher. We cannot be sure that these values are not based in part on new unpublished data obtained by Eucken. [31] Gafner, G., The application of a transient method to the measurement of the thermal conductivity of rocks and building materials. Brit. J. Appl. Phys. 8, 393-7 (1957). A method of "steady rise" was employed; three samples of vitreous silica were measured in one apparatus and three others in another. The mean of the 6 measurements was X = O.OO323 with a total spread of 6 percent; the temperature of all the measurements was 30°C. 69 [32] Jakob, M. , The material properties most im- portant in heat transfer. Per Chemie - Ingenieur , A. Eucken and M. Jakob, editors, Akad. Verlag, Leipzig, 1933- Vol. 1, Pt. 1, p. 308-79- The information on vitreous silica, p. 3^0-l> in- cludes a smooth curve of thermal conductivity vs. temperature, but no plotted points. It seems likely that this curve is based on previously- published work of other authors, but we are not sure that this is the case. [33] Kaganov, M. A., Lisker, I. S. and Chudnovskii, A. F., Rapid method for determining thermal conductivity of semiconductor materials. Inzh.-Fiz. Zhur., Akad. Nauk B. S. S. R. h, No. 3, 110-2 (1961). By a transient method they found X of vitreous silica to be 0.00331? at a temperature that we take to be approxi- mately 12.5°C. [3M Koenig, John H., Progress Report No. VI from March 1 to June 1, 1952. N.J. Ceram. Research Sta., Rutgers Univ., 98 p.; ATI-I63519. This paper is of interest because it contains a comparison of thermal conductivities of clear and silky vitreous silica. For quantitative data on clear vitreous silica we prefer reference [11], in which the measurements were made with improved equipment. [35] Koenig, John H., Progress Report No. 2 from March 1 to June 1, 1953- N.J. Ceramic Re- search Sta., Rutgers Univ., 117 p.; AD- 1315*1-. This paper contains a comparison of thermal conductivities of silky vitreous silica measured parallel and perpendicular to the striations. Although it is not clear which set of data refers to each form, the data show that there was not much difference between the two. [36] Luikov, A. V., Vasiliev, L. L. and Shashkov, A. G., A method for the simultaneous deter- mination of all thermal properties of poor heat conductors over the temperature range 80 to 500 °K, in Advances in Thermophysical Prop - erties at Extreme Temperatures , and Pressures , Third Symposium on Thermophysical Properties. ASME, Purdue Univ., Mar. 22-25, 1965, Serge Gratch, editor (The American Society of Mechanical Engineers, New York, 1965), p. J>lk-9. This paper contains some ambiguities or errors that have caused us not to include the data in any of our graphs. The density given for the vitreous silica is 2.650 g cm 3 , which is 20 percent higher than accepted values (it is roughly the accepted density for crystalline quartz). The data of other observers, plotted in Fig. 6 of the present reference, appear to be incorrectly given, at least in part. Also the data for vitreous silica in the present reference are from 8 to 17 percent lower than those given by Vasiliev [6] one year earlier. [37] Milnes, M. V., Low- temperature thermal con- ductivity apparatus. North American Aviation Inc., Rept. SDL ^08, 9 Sept. 1963, 1^ p. A sample of vitreous silica was measured, pre- sumably to check out the apparatus. The X-value found was .0033? and the sample mean temperature, which we computed from the given thermocouple data, was 23.1°C. [38] Nukiyama, Shiro, The thermal conductivity of glass, chilled glass, quartz, fused quartz (transparent), Bakelite, India-rubber, coal, Isolite, porcelain, slate, granite and marble. Trans. Soc. Mech. Engrs. (Japan) 2, 3UU-5 (1936). English abstract, ibid., p. S-96, S-97- Three plotted A-values for vitreous silica are given, covering the range 15 to 33° C. An equation is given that appears to re- present the points satisfactorily. The equation gives, at 300°K (26.85°C): A. = 0.00294. [39] Phillips, L. S., The measurement of thermo- electric properties at high temperatures. J. Sci. Instr. k2_, 209-11 (1965). The appa- ratus was so designed that the Seebeck coef- ficient, the electrical resistivity, and the thermal conductivity of a specimen could each be measured. The thermal conductivity of vitreous silica was measured in the range l60 to 595 °C "to assess the behavior of the appa- ratus." The results are given in the form of a smooth curve. In view of this, and in view of the fact that the apparatus emphasized rapid operation rather than high accuracy, we have not plotted the results in any of our figures. At the lowest temperature, Phillips' curve is 5 percent below our master curve; at the highest temperature it is 18$ above it. [40] Powell, R. W., Thermal conductivity as a non- destructive testing technique, in Progress in Non-destructive Testing , E. G. Stanford and J. H. Fear on, editors (Heywood and Company, London, 1958), Vol. 1, p. 199-226. The X- value of vitreous silica at "about 28°C" is given as O.OO33O in Table k and again in Table 5- This same value of X was given earlier by Powell in reference [67]. However, the earlier paper does not give the tempera- ture of the measurement. [Ul] Sisman, 0., Bopp, C. D. and Towns, R. L., Radiation effects on the thermal conductivity of ceramics. U. S. At. Energy Comm., Oak Ridge Natl. Lab. Rept. ORNL-I852, 33-5 (1955)- The value reported for unirradiated vitreous silica is O.OO35, "at about 40°C." 70 [42] Weeks, James L. and Seifert, Ralph L., Note on the thermal conductivity of synthetic sapphire. J. Am. Ceram. Soc. _35, 15 (1952). Measurements made on a 6-mm rod of vitreous silica from Hanovia Chemical Company yielded 0.0041 at 73°C and 0.0057 at 135°C. 9.5. Containing Data for Which Another Reference is Preferred. The arrangement is alphabetical. [43] Barratt, T., and Winter, R. M., The thermal conductivity of wires and rods. Ann. Physik 77, 1-15 (l925)' Condensed German translation of 2 papers of Barratt, one of which was reference [25]. [44] Berman, R., Thermal conductivity of glasses at low temperatures. Phys. Rev. j6j 315-6 (l9^9)- This appears to he the place of first publication of Berman' s data on vitreous silica. However, we prefer to use the fuller account, reference [12]. [45] Berman, R., The thermal conductivity of dis- ordered solids at low temperatures. Bull. Inst. Intern. Froid. Annexe 1952-1 (1952), p. 13-20. This paper contains a smooth curve for vitreous silica; we prefer to take the data from the plotted points of reference [12]. [46] Berman, R., Thermal conductivity of some polycrystalline solids at low temperatures. Proc. Phys. Soc. (London) A 6£, 1029-40 (1952). This paper contains 7 tabulated values of \ of vitreous silica. Six are at low temperatures, and agree with those in reference [12], which is preferred. The re- maining value is for room temperature, and we have not found it elsewhere in Berman 's published papers; however, it is given to only one significant figure, and we have not used it. [47] Berman, R., The thermal conductivity of dielectric solids at low temperatures. Advances Phys. (London) 2, 103-4-0 (1953). This paper contains a smooth curve for vitreous silica. Reference [12] is preferred. [48] Berman, R., Klemens, P. G., Simon, F. E., and Fry, T. M. , Effect of neutron irradiation on the thermal conductivity of a quartz crystal at low temperature. Nature 166, 864-6 (1950). This paper contains a smooth curve of \ versus temperature, taken from an earlier paper of Berman [44], Reference [12] is preferred. [49] Bil', V. S. and Avtokratova, N. D., Tempera- ture dependence of thermal conductivity and thermal diffusivity of some unfilled polymers. Teplofiz. Vysokikh Tempera tur, Akad. Nauk S. S. S. R. 2, 192-8 (1964). We have not yet obtained a copy of this paper; it is the Russian original of [9]. [50] Bopp, C. D., Sisman, 0. and Towns, R. L., Radiation stability of ceramics. U. S. At. Energy Comm. Oak Ridge Natl. Lab. Rept. 0RNL- 1945 (Aug. 30, 1955), 35-6. Reference [4l] is preferred. [51] Cohen, A. Foner, Low- temperature thermal con- ductivity in neutron- irradiated vitreous sili- ca. U. S. At. Energy Comm. Oak Ridge Natl. Lab. Rept. ORNL-2413, 70-1 (Aug. 31, 1957). This appears to be the same work as that re- ported in reference [13]> which we prefer. [52] Cohen, Anna Foner, Low temperature thermal conductivity of nonmetals including radiation effects, in Low Temperature Physics and Chemistry , Proc . Fifth Intern . Conf ■ on Low Temperature Phys . and Chem . . . . 1957 ; Joseph R. Dillinger, editor (Univ. of Wisconsin Press, Madison 1958), p. 385-8. Reference [13] is preferred. [53] Cohen, Anna Foner, Low- temperature thermal conductivity in neutron irradiated vitreous silica. J. Appl. Phys. 2£, 591-3 (1958). This paper contains less than half as many plotted points as [13 ], although it covers a greater temperature range. We prefer refer- ence [13] , which appears to be a later and better report of the work. [54] Crawford, J. H.,Jr., and Cohen, A. F., Effect of fast neutron bombardment on the thermal conductivity of silica glass at low tempera- ture. U. S. At. Energy Comm. Oak Ridge Natl. Lab. Rept. ORNL-26l4, 43-6 (31 Aug. 1958). This article closely resembles reference [13]; however, the latter is preferred. [55] Devyamkova, E. D., Pemrov, A. V., Smirnov, I. A., and Moizhes, B. Ya. Fused quartz as a model material in thermal conductivity measurements. Soviet Phys. Solid State 2j 681-8 (i960). English translation of reference [14]. [56] Eucken, A. The heat conductivity of solid materials at low temperatures. Z. Tech. Physik 6, 689-94 (1925). Reference [5] is preferred. The present paper contains a value at 21°K that is not given in reference [5] and another at 373°K that differs from the value given in reference [5]. We have omitted the 21° value of the present report on the assumption that it is not an original deter- mination but an extrapolation, and we have omitted both of the 373° values given by Eucken because they do not agree. The remain- ing values in the present reference agree with 71 those in reference [5]- [57] Eucken, A., Thermal conductivity of nonmetals and metals. Physik. Z. 29, 563-6 (1928). We prefer reference [5]. [58] Kamilov, I. K., Investigation of the thermal conductivity of solids in the interval from 80 to 500°K. Pribory i Tekhn. Eksperim. No. 3, 176-9 (May-June 1962). This is the Russian original of reference [3]. [59] Kingery, W. D., Factors affecting thermal stress resistance of ceramic materials. J. Am. Ceram. Soc. ^8, 3-15 (1955). This paper contains two numerical \-values for vitreous silica; however, they appear to be only in- cidental to the subject of the paper. We prefer reference [8]. [60] Kingery, W. D., Thermal conductivity: XII, Temperature dependence of conductivity for single-phase ceramics. J. Am. Ceram. Soc. 38, 251-5 (1955). This paper is similar to refer- ence [8]j it has a better description of experimental techniques than reference [8]. [6l] Kingery, W. D. , Thermal conductivity: XIV, Conductivity of multicomponent systems. J. Am. Ceram. Soc. 42, 617-27 (1959)- Reference [8] is preferred. [62] Klemens, P. G., The thermal conductivity of dielectric solids at low temperatures (Theoretical). Proc. Roy. Soc. (London) A 208, IO8-33 (1951)- The X-data on vitreous silica in Fig. 2 of this paper are from Berman [12]. However, the data in the present refer- ence are less numerous than those in Berman 's paper, which we prefer. In addition, the data in the present reference have been "adjusted". [63] Koenig, John H., Progress Report No. 1 from December 1, 1952 to March 1, 1953 . N. J. Ceram. Research Sta., Rutgers Univ., 100 p; AD-5552. This paper contains X-data for vitreous silica. However, we consider that they are superseded by the data reported in reference [11]. [64] Lee, D. W. and Kingery, W. D., Radiation energy transfer and thermal conductivity of ceramic oxides. J. Am. Ceram. Soc. 43, 59^-607 (i960). Reference [20] is preferred. However, the present reference contains a somewhat fuller description of the experiments than reference [20]. [65] Lucks, C. F., Deem, H. W. and Wood, W. D. Thermal properties of six glasses and two graphites. Bull. Am. Ceram. Soc. 39, 313-9 (i960). After studying this paper and plot- ting the \-values given for vitreous silica we have concluded that the data are smoothed values taken from the original data of reference [19]. [66] Norton, F. H., Kingery, W. D. "et al.", The measurement of thermal conductivity of refractory materials. U. S. At. Energy Comm. Rept. NYO-3646, 1-8 (July 1, 1953). Reference [8] is preferred. Although the present refer- ence appears to be the earliest report of this work, its graph of the data is smaller than the graph in reference [8]. [67] Powell, R. W., Experiments using a simple thermal comparator for measurement of thermal conductivity, surface roughness and thickness of foils or of surface deposits. J. Sci. Instr. 34, 485-92 (1957)- Vitreous silica was one of the materials used in standardizing the apparatus. The X- value given here is also given in reference [40]. Reference [40] is preferred because it contains the temperature of the measurement, whereas no temperature is given in the present reference. [68] Pustovalov, V. V., Change in thermal conduct- ivity of quartz glass in the process of crystallization. Translation by A. J. Peat, General Electric Research Lab. Rept. 6O-RL- (2525M), September i960. 8 p. English trans- lation of reference [21] . Graph showing data is smaller than the one in reference [21]. [69] Pustovalov, V. V., Thermal conductivity of some refractory materials. Ogneupory 26_, 302-5 (1961). The unsmoothed values of reference [21] are preferred. However, the present paper gives the density of the vitreous- silica sample, from which we computed a greater porosity than that given by Pustovalov. [70] Ratcliffe, E. H., Preliminary measurements to determine the effect of composition on the thermal conductivity of glass. Phys. Chem. Glasses 1, 103-4 (i960). Reference [2] is preferred. The smooth curve in the present reference agrees with the data in reference [2]. It goes to a higher temperature, but we have assumed that this is an extrapolation of the previous results. [71] Ratcliffe, E. H., A survey of most probable values for the thermal conductivities of glasses between -150 and 100°C, including new data on twenty-two glasses and a working formula for the calculation of conductivity from composition. Glass Technol. 4, 113-28 (1963). Reference [2] is preferred. The present reference contains a short table of "assessed best" values for vitreous silica. These values average 1.0 percent above our master curve. 72 [72] Sisman, 0., Bopp, C. D. and Towns, R. L., Radiation stability of ceramic materials. U. S. At. Energy Comm. Oak Ridge Natl. Lab. Rept. 0RNL-21H3 (Aug. 31, 1957), 80-2. Reference [4l] is preferred. [73] Smoke, Edward J. and Koenig, John H., Thermal properties of ceramics. Rutgers Univ. Eng. Research Bull. kO (Jan. 1958). 53 P- This is a summary of results previously issued in their progress reports. We prefer reference [11] as a source of \-data on vitreous silica. [fh] Wray, Kurt L., and Connolly, Thomas J., Thermal conductivity of clear fused silica at high temperatures. Avco Research Lab. Res. Rept. kh (February 1959). 13 P- We prefer reference [18]. Containing No Original Thermal- Conductivity Data on Fused Silica [75] Sosman, Robert B., The Properties of Silica . (The Chemical Catalog Company. Reinhold Pub. Corp., New York, 1927). 856 p. [76] Laufer, Jerome S., High silica glass, quartz, and vitreous silica. J. Opt. Soc. Am. 55> U58-6O (1965). [77] Warren, B. E. and Biscoe, J., The structure of silica glass by X-ray diffraction studies. J. Am. Ceram. Soc. 21, ^9-5^ (1938). [78] Wagstaff, F. E., Brown, S. D. and Cutler, I.B. The influence of H 2 and 2 atmospheres on the crystallization of vitreous silica. Phys. Chem. Glasses 5, 76-81 (196k). [79] Hetherington, G. and Jack, K. H., Water in vitreous silica. Part I: Influence of 'water' content on the properties of vitreous silica. Phys. Chem. Glasses 3, 129-33 (1962). [80] Pustovalov, V. V., Determination of thermal conductivity of refractories to 1200° by the method of stationary heat flow. Ogneupory 2k, No. k, 180-5 (1959)- Gives a description of the experimental method used by Pustovalov in reference [21] . [8l] Gardon, Robert, A review of radiant heat transfer in glass. J. Am. Ceram. Soc. hk, 305-12 (1961). [82] Bullard, E. C. and Niblett, E. R., Terrestrial heat flow in England. Monthly Notices Roy. Astron. Soc. Geophys. Suppl. 6, 222-38 (1951). 73 t o * 09S z ujo /ujo idojimuj * y 74 o CM £ o O o £ o "o o £ /C Figure 2. Thermal conductivity data of references [l2J and [l3j . The master curve begins at 80 °K. 75 -« 09S 2 UJO o /UJD IDOjII.IUJ'y 76 Development of High Temperature Thermal Conductivity Standards Arthur D. Little, Inc. Cambridge, Massachusetts 02140 and Merrill L . Minges 2 Air Force Materials Laboratory Wright-Patterson Air Force Base, Ohio 45433 The continued use of high-temperature materials has led to the requirements for accurate thermal conductivity and thermal diffusivity standards. The program described herein encompasses an analytical and experimental study to establish high-temperature thermal conductivity and thermal diffusivity standards and to test these standards at several property measurement laboratories. Candidate ceramics, intermetallics , metals, and graphites were selected and used in a preliminary laboratory evaluation consisting of heating tests, visual and microscopic examination, and room-temperature thermal conductivity and electrical resistivity measurements. On the basis of test results, alumina, thoria, tungsten, and RVD and AXM-5Q1 graphites were chosen for the labora- tory measurement program. Specifications written for these materials and the results of receiving inspection and density measurements of the samples obtained arc presented. The status of measurements at participating laboratories is briefly described. Key Words: Alumina, conductivity standards, diffusivity standards, graphite, high temperature thermophysical properties, thermal conductivity, thermal diffusivity, thoria, tungsten. 1. Introduction The increased importance of high-temperature materials has led to the requirements for accurate thermal property data for design purposes. Measurements of thermal conductivity and diffusivity have been limited in their applicability by the unavailability of basic reference standards to evaluate and calibrate thermal property measurement apparatus. The selection of calibration standards for use at temperatures up to 1200°C (2200°F) has been proceeding for several years; laboratories have participated in round-robin tests evaluating tentative standard materials, such as Armco iron, Pyrex glass, and Pyroceram Code 9606. The objectives of this program are to establish high-temperature thermal conductivity and thermal diffusivity standards and to test these standards at several thermal property measurement laboratories. The program is being carried out in two phases: Phase I consisted of the identification, selection, characterization and preliminary evaluation of candidate standard materials. Phase II consists of specification of sample materials, a test program at participating laboratories, and detailed examination of the thermal property data obtained. This paper summarizes the results of the Phase I program, presents materials specifications and the results of visual examination and density measurements of the samples, and reviews the status of the Phase II laboratory test program. 2. Phase I — Materials Selection and Evaluation 2.1. Preliminary Materials Selection To initiate the program, we chose potential candidate materials from those for which thermal prop- erty measurements had been reported in the literature. Materials which were obviously unsuitable because Group Leader, Engineering Sciences, Research and Experiments Group. 2 Technical Manager, Structural Materials 77 of their low melting points were not considered further. Other materials were rated on a qualitative scale according to criteria such as phase change, melting point, sample reproducibility, availability, isotropy, homogeneity, microstructure stability, vapor pressure, mechanical strength, fabricability , shelf life, and cost. Materials were categorized as ceramics, intermetallics , metals, and graphites to aid their comparison, to provide for selection of potential standards which would cover a broad range of thermal conductivities (0.1 to 100 watt m °C ) and to select candidates for use in vacuum, oxidizing, reducing or graphitic environments. The materials chosen for further experimental evaluation were: Ceramics Intermetallics Metals Graphites A1 2 3 , BeO, CaZr0 3 , Th0 2 , and Zr0 2 (Y 2 3 stabilized) B^C, SiC, TiC, ZrB 2 Mo, Mo-30W, Ta, Ta-lOW, W Union Carbide Corp. Pure Carbon Co. Poco Graphite, Inc. Spear Carbon Co. Ultra Carbon Co. - AT J, CEQ, and RVD - PO-3 and DS-13 - 1924-H - 7890 - XUT-31 and UF4S Figures 1 and 2 show the approximate thermal conductivity of these materials as a function of temperature. 2.2. Sample Procurement Specifications for materials which were not commercial products were established, and samples in sizes and shapes similar to those required for thermal conductivity and diffusivity apparatus were ob- tained from qualified suppliers. Table 1 shows the materials used, the suppliers and the general sample characteristics. The samples were examined visually upon receipt and compared with the purchase speci- fications. All samples were accepted for preliminary experimental evaluation, even though some of them differed from the purchase specification. Table 1. Materials Used in Preliminary Laboratory Evaluation Sample Supplier Sample Size Remarks Ceramics Alumina Coors Porcelain Co. (1) AD-995 composition, poly- Golden, Colorado crystalline, sintered alumina Beryllia National Beryllia Corp. (1) Berlox composition, 99.5% Haskell, N. J. BeO, polycrystalline, sintered beryllia Brush Beryllium Co. (1) Thermolox 99.5 composi- Elmore, Ohio tion, polycrystalline, sintered beryllia Calcium Zirconate Zirconium Corp. of America (1) 99.5% pure, polycrystal- Solon, Ohio line, sintered material Zirconia Zirconium Corp. of America (1) Polycrystalline, sintered Solon, Ohio material, stabilized with 12 mole % yttria Thoria National Beryllia Corp. (1) Thorox composition, 99.5% Haskell, N. J. pure, polycrystalline, sintered material Intermetallics Boron Carbide Carborundum Co. (2) 90+% dense, 76 wt% Boron, Niagara Falls, N. Y. pressed samples Silicon Carbide Norton Co. Worcester, Mass. (2) Crystalon R grade Carborundum Co. (2) KT grade Niagara Falls, N. Y. Titanium Carbide Carborundum Co. Niagara Falls, N. Y. (2) 95% pure 78 Titanium Carbide Norton Co. Worcester, Mass. (2) 96-98% pure, 79.7% Ti Zirconium Diboride Carborundum Co. Niagara Falls, N. Y. (2) 95% pure, 79% Zr Norton Co. Worcester, Mass. (2) 98% pure, 78% Zr Metals Molybdenum Climax Molybdenum New York, N. Y. (1) Arc melted, wrought bar Molybdenum- 30% Tungsten Climax Molybdenum New York, N. Y. (1) Arc melted, wrought bar Tantalum National Research Corp. Cambridge, Mass. (1) Arc melted, wrought bar Tantalum-10% Tungsten National Research Corp. Cambridge, Mass. (1) Arc melted, wrought bar Tungsten Fansteel Metallurgical Corp. Chicago, Illinois 3/4" dia bar Pressed and sintered, wrought bar Universal-Cyclops Bridgeville, Penn. 9/16" dia bar Arc melted, wrought bar Graphites Grade RVD Union Carbide Corp. Parma, Ohio 5" x 5" x 3" Density 1.88 gm/cm 3 CEQ Union Carbide Corp. Parma, Ohio 5" x 5" x 3" Density 1.53 gm/cm 3 ATJ Union Carbide Corp. Parma, Ohio 5" x 5" x 3" Density 1.77 gm/cm 3 7890 Spear Carbon Co. St. Marys, Penn. 5" x 2" x 10" Density 1.47 gm/cm 3 1924-H Poco Graphite, Inc. Garland , Texas 1 3/4" dia x 6" Density 1.84 gm/cm 3 PO-3 Pure Carbon Co. St. Marys, Penn. 1" dia x 2" Density 1.82 gm/cm 3 XUT-31 Ultra Carbon Co. Bay City, Michigan 1" dia x 12" Density 1.67 gm/cm 3 UF4S Ultra Carbon Co. Bay City, Michigan 1" dia x 12" Density 1.72 gm/cm 3 Sample Size: (1) 2 th 1/2" dia x 1/2" thick disc witl ick disc l 1/2" dia center hole and 5/8" dia x 1/8" (2) 2 1/2" dia x 1/2" thick disc 2.3. Experimental Evaluation The experimental evaluation program consisted of visual examination of the samples and comparison with the specifications, density measurements, microstructural examination, heating tests, preliminary thermal conductivity measurements, and electrical conductivity measurements. Ceramics, metals, and intermetallics were heated in a vacuum environment at temperatures up to 2200°C (4000°F) for up to 24 hours per test cycle. Graphites were heated in a graphite and atmospheric pressure argon environment at temperatures up to 2300°C (4200°F) for similar durations. Preliminary thermal conductivity measure- ments were made at room temperature in a cut bar apparatus; measurement precision ranged between + 2 and + 3%, depending upon the samples used. Electrical resistivity was measured at room temperature and at liquid nitrogen temperature using the d.c. four-point method with an apparatus and instrumentation capable of + 1 to + 2% precision. The objective of these thermal conductivity and electrical resistivity measurements was to assess changes in the samples caused by heat treatment. Details of the apparatus and test methods are given in the report on the Phase I program [1] . 3 Figures in brackets indicate the literature references at the end of this paper. 79 Among the ceramics, aluminum oxide and thorium oxide were found to be suitable for further evalu- ation in the standards program because tests indicated that any changes in their structure or thermal properties induced by heat treatment were small and would not affect the application of these materials as standards. Beryllium oxide was rejected from further consideration because of its high vaporization rates in vacuum and because its unpredictable high-temperature reactions could result in toxicity prob- lems. Calcium zirconate was rejected because of its poor sample reproducibility, its significant va- porization at moderate temperatures, and its reaction with refractory metals and ceramics used in typical measurement apparatus. Yttria stabilized zirconia was rejected because of its reaction with refractory metals, its apparent change in physical and chemical composition (resulting in a change in thermal properties) on heating, and its high vaporization rates at high temperatures. None of the intermetallic candidate materials was found suitable for continued use in the program. Boron carbide was unsatisfactory because the high rate of vaporization at low temperatures causes the preferential vaporization of boron and thus changes its physical and chemical properties. Commercially available silicon carbide was unsatisfactory because of the high rate of vaporization of silicon in vacuum and the accompanying change in chemical composition. Titanium carbide was rejected because of the difficulties in obtaining reproducible samples, lack of control of stoichiometry , and rapid volatilization at moderate temperatures. Zirconium diboride appeared to be the most promising of the intermetallic materials. However, high vaporization rates resulting in a preferential loss of boron, changes in thermal conductivity on heating, and difficulties in obtaining reproducible samples of con- trolled stoichiometry indicate that this material should not be used as a standard in its present state of development. Arc cast tungsten was selected as the most suitable candidate metal because heat treatment up to 2200°C (4000°F) did not significantly change its thermal and electrical properties or produce effects known to detract from its use as a standard. Changes in the thermal and electrical properties of arc- melted tantalum after heating were also minimal; however, tungsten was preferred to tantalum because it has greater physical and chemical stability at high temperatures. Molybdenum and molybdenum- 30% tungsten were rejected because of their high vaporization rates and their lower maximum use temperatures compared to the other refractory metals. Tantalum-10% tungsten was rejected because of difficulties in obtaining samples of reproducible composition and changes in thermal and electrical properties after heat treat- ment. Pressed and sintered tungsten was not considered for further use because of intergranular crack- ing at high temperatures and resultant changes in thermal and electrical properties. On the basis of our preliminary experimental examination, it was difficult to distinguish between acceptable and non-acceptable graphites. However, available manufacturers' data, the data in the literature, and our experiments indicate that Union Carbide Corporation's grade RVD and Poco Graphite, Inc.'s grade 1924-H are the most suitable materials. Grade RVD is recommended by its homogeneity, reproducibility, and the long experience in its production; its principal disadvantage is anisotropy. Grade 1924-H is very isotropic compared to other graphites. 3. Phase II — Sample Specification, Procurement and Preparation 3.1. Sample Specification and Procurement The alumina, thoria, and graphite samples for Phase II were to be obtained from the suppliers who provided Phase I samples. The specifications for these materials, except for dimensions, were the same as those used in Phase I. Because of the sample sizes required and the limited availability of large- diameter arc-cast tungsten, samples were obtained from Climax Molybdenum Corp. rather than from Universal-Cyclops Steel Corp. The material specification was almost identical to that used previously. In Phase I we used a Poco Graphite, Inc., sample designated 1924-H. During the program the desig- nation was changed to AXF-5Q1. (The symbols designate the type and condition of the material: e.g., A indicates fine grain, X indicates graphitization, F indicates the high density range, 5 indicates graphitization at 2500°C, and 1 indicates a purified product.) The information we received at the completion of Phase I indicated the desirability of using grade AXM-5Q1, to improve sample reproducibility and to make the material more valuable as a standard. Grade AXF-5Q1 is of very high density; the material is available only in small sample sizes and there may be more variation in density across individual sample billets or bars. This grade is produced on a limited basis; but, because of its high density and high mechanical strength, it has been of considerable in- terest to aerospace companies. Grade AXM-5Q1 is similar in all respects to grade AXF-5Q1 except that the molding pressure is less, the density is lower, and the density-related properties (strength and conductivity) are correspondingly lower. Because grade AXM-5Q1 is available in larger sizes than AXF-5Q1 and should have less variation of properties in a particular billet, it was selected for Phase II. Materials specifications for Phase II are given below: a. Aluminum Oxide — Coors Porcelain Co. Aluminum oxide to be used as a standard for determination of thermal conductivity and thermal dif- fusivity. To ensure homogeneous density and reduce internal stresses the material is to be fabricated 80 by isostatic pressing. Fired pieces must be free of cracks, visible discolorations or inclusions, and be uniform in color. The microstructure must be uniform within the specification and free of second phase. Fired pieces must conform to the following specification: Grade Chemical Purity Chromium Content Sintered Density Water Absorption Maximum Average Grain Size Surface Finish AD 995 99.5% Alpha A1 2 3 (nominal) 0.20 + 0.05% 3.84 gm/cm typical, 3.80 gm/cm minimum Zero 25 microns 55 micro inches maximum Edges to be left sharp as ground. A representative analysis of the finished product must be supplied; lot identification and sintering temperature must be given. All samples are to be packed individually to eliminate damage in shipment . b. Thorium Oxide — National Beryllia Corp. Thorium oxide to be used as a standard for determination of thermal conductivity and thermal diffu- sivity. To ensure homogeneous density and reduce internal stresses the material is to be fabricated by isostatic pressing then sintered. The fired pieces must be free of cracks, visible discolorations or inclusions and must be uniform in color. The microstructure must be uniform within the specification and free of second phase. Fired parts must conform to the following specification: Grade Chemical Purity Sintered Density Water Absorption Thorox 99.9% cubic Th02 (minimum) 9.2 + 0.1 gm/cc 1.0% maximum, 0.1% typical Maximum Average Grain Size 30 microns Surface Finish 55 micro inches (maximum) The chemical composition of the starting powder and a representative analysis of the finished product must be supplied; lot identification, sintering time and temperature must be given. All samples are to be packed individually to eliminate damage in shipment. c. Tungsten — Climax Molybdenum Co. Arc-cast tungsten — extruded and stress relieved — to be used as a standard for determination of thermal conductivity and thermal diffusivity. General Composition: w 99.87% min Sn less than 20 ppm Mn less than 10 ppm Mo 0.1% max Pb less than 20 ppm Si less than 20 ppm Ni 0.001% max 02 less than 30 ppm Cr less than 10 ppm Fe 0.003% max N2 less than 30 ppm V less than 10 ppm C 0.005% max H 2 less than 5 I >pm Ti less than 10 ppm Co less than 10 ppm Al less than 20 ppm Cu less than 10 ppm Mg less than 10 ppm Identification: The material is to be identified according to heat number, the identifying symbol or number to be marked on 'each rod or bar. The chemical composition of the heat and the conditions of stress relieving are to be given. Inspection: All samples are to be inspected using a dye-penetrant technique and an ultrasonic tech- nique to assure freedom from defects; the material is to be free of defects such as slivers, cracks, and seams . d. RVD Graphite — Union Carbide Corp. RVD graphite to be used as a standard for determination of thermal conductivity and thermal diffu- sivity. Pieces must be free of cracks and inclusions and conform to the following chemical and 81 mechanical specifications. The direction of the grain is to be marked on each piece, and the heat or batch number is to be indicated. RVD Two-fold anisotropic High density Medium 0.016" (maximum) 1.86-1.92 gm/cm 3 (average density 1.89 gm/cm 3 ) 1260 + 20% micro ohm-cm variation within the lot + 10% (measured with grain) ; 2000 + 20% micro ohm-cm variation within the lot + 10% (measured across grain) 3.0 x 10~ 6 (+ 20%) in/in°C variation within the lot + 10% (measured across grain) ; 1.7 x 10~ 6 (+ 20%) in/in°C variation within the lot + 10% (measured with grain) 2800°C e. AXM-5Q1 Graphite — Poco Graphite, Inc. AXM-5Q1 graphite to be used as a standard for determination of thermal conductivity and thermal diffusivity. Pieces must be free of cracks and inclusions and must conform to the following chemical and mechanical specification. The direction of the grain is to be indicated. Grade Characteristic Structural' Class Grain Size Range Particle Size Density Electrical Resistivity Coefficient of Thermal Expansion Process Temperature Grade Characteristic Structural Class Grain Size Range Particle Size Density Compressive Strength Electrical Resistivity Coefficient of Thermal Expansion Process Temperature AXM-5Q1 (special density) Isotropic Medium i dens ity Fine 0.001 inch (maximum) 1.74-1.76 gm/cm 3 14,000 psi (minimum) 1750 + 20% micro ohm-cm 6.3 x 10" in/in°C (room temperature to 100°C) 2500°C Alumina and thoria samples were obtained in the sizes required for the apparatus to be used in the participating laboratories (see Section 3.3). Tungsten was ordered as bar stock and machined to the sizes required. Graphites were ordered in billet form and machined to size after non-destructive tests had been completed. 3.2. Sample Inspection and Preparation The sample materials were visually inspected upon receipt and the densities of finished samples and bulk material were measured. Microstructural examination and chemical analysis of samples are in pro- gress. Significant observations are noted below; Table 2 summarizes density measurements on samples. The samples distributed to test laboratories have been further selected to reduce the density variation. a. Alumina Finished alumina samples appeared to be of good quality with minimum surface defects. A slight color variation was noted on disc samples (2" dia x 1/2" thick); more pronounced variation was noted on rod samples (1/4" and 3/8" dia). Dimension tolerances were within the + 0.003" specified, and sample flatness was within the specification of 0.002" TIR. 82 Table 2. Densities of Samples Obtained for Use in Phase II Size Number Average Density Average Deviation Density Range Tested gm/cm 3 gm/ cm 3 gm/cm 3 Alumina 2" dia disc 48 3.807 0.005 3.786-3.824 3/8" dia rod 11 3.782 0.006 3.770-3.794 1/4" dia rod 11 3.801 0.008 3.779-3.823 Thoria 2" dia disc 55 9.256 0.045 9.137-9.399 3/8" dia rod 10 9.293 0.026 9.24 -9.37 1/4" dia rod 4 9.288 0.033 9.25 -9.34 AXM-5Q1 Graphite 2" x 4" x 6" 20 1.757 0.006 1.747-1.766 4" dia x 12" 3 1.759 0.007 1.748-1.766 As shown in Table 2, the density of the disc samples is at the low end of the specification range, and the rod samples are in many cases outside the specification range. We have selected specific samples for use in conductivity and diffusivity measurements based upon these values and have attempted to use materials of constant density for different measurement methods, where possible. It should be possible to correct the data obtained for these density variations. Note that within any given type of specimen, good uniformity is obtained. b. Thoria Thoria samples were of apparent good quality with minimum surface defects. A color variation from light to dark yellow was observed. Sample edges tfere sometimes dark and grayish. Principal sample de- fects were chips on the edges and color centers in several samples. Samples were within the dimension tolerance specified. As shown in Table 2, the average density values are within the specified limits although several specific samples were outside the range. The density range and average deviations are significantly larger for the thoria samples than for alumina samples. c. Tungsten Visual examination of tungsten samples showed no significant defects. Samples were generally within the tolerances specified and were machined to tolerances similar to the alumina and thoria specimens. The measured density of all mgchined samples was approximately 19.23 gm/cm 3 . d. AXM-5Q1 Graphite Cursory visual inspection of AXM-5Q1 graphite showed no obvious flaws or defects. As indicated in Table 2, there was little variation in the bulk density of the billets and bars of this material. How- ever, NDT radiometric density determinations carried out at Avco Corporation on these samples showed local variations of up to + 4% in density over a single block. To reduce the effects of these variations, we have prepared density maps of the individual graphite billets and bars and carefully selected certain portions for the preparation of samples. Material having the least density variation and a good average value was chosen for sample preparation. Figures 3 and 4 show representative results of NDT tests on cylindrical AXM-5Q1 samples. The variations in density appear to be random on both billets and bars although there are slight trends toward greater density variations near the edges of the samples. e. RVD Graphite RVD graphite was obtained as a large billet, approximately 25 x 25 x 45 cm. The billet was cut in three slabs, each 8 cm thick. Visual observation showed no obvious flaws or deficiencies. The bulk density of the three slabs was measured to be 1.84, 1.87 and 1.89 gm/cm 3 . NDT radiometric density mea- surements of the slabs showed good uniformity in the center portion of the slab (i.e., a variation of about + 1% in density). The peripheral areas of the sample had higher densities than the center. Samples were selected from the central portions of the slabs to minimize density variations and to ob- tain a uniform average density. 3.3. Thermal Conductivity and Diffusivity Measurements Measurements of thermal conductivity and diffusivity, sponsored under this program, are in progress at the laboratories indicated in Table 3. The samples to be measured and the methods used are also 83 given in the table. Measurements of thermal conductivity of alumina and thoria by a comparative method and of tungsten by a longitudinal heat flow method are planned by the National Bureau of Standards. Measurements of thermal conductivity (and electrical resistivity, where applicable) of the five candidate standards by longitudinal and radial heat flow methods are planned by Oak Ridge National Laboratories. Microscopic and metallurgical examination of samples, combined with chemical analysis will be used both before and after testing to aid the interpretation of the thermal property data. Table 3. Thermal Property Measurements in Progress in Phase II Organization Method Samples Tested Temperature Range , ° C Remarks Thermal Conductivity Atomics International Radial inward RVD graphite 1000-2500 Duplicate samples heat flow AXM-5Q1 graphite 1000-2500 each material General Electric Radial outward Alumina 1000-1800 Duplicate data, (Cincinnati) heat flow Thoria 1000-2200 duplicate samples Tungsten 1000-2500 each material Illinois Institute of Radial outward Alumina 1000-1800 Duplicate samples Technology Research heat flow Tungsten 1000-2200 each material Institute National Beryllia Radial outward Thoria 1000-2200. Duplicate samples Corporation heat flow Southern Research Radial inward RVD graphite ' 800-2700 Duplicate samples Institute heat flow and longitudinal heat flow (800- 1000°C) AXM-5Q1 graphite 800-2700 each material; measurements made in 2 orientations with AXM-5qi graphite Thermal Diffusivity Atomics International Laser flash Alumina 1000-1800 Duplicate samples Thoria 1000-2200 each material; Tungsten 1000-2200 graphite measure- RVD graphite 1000-2200 ments to be made AXM-5Q1 graphite 1000-2200 in 2 perpendicular orientations Battelle Memorial Laser flash Alumina 1000-1800 Duplicate samples Institute Thoria 1000-2200 each material; Tungsten 1000-2200 graphite measure- RVD graphite 1000-2200 ments to be made AXM-5Q1 graphite 1000-2200 in 2 perpendicular orientations 4. Acknowledgments This work is supported by the Air Force Materials Laboratory, Wright-Patterson Air Force Base, Ohio, under Contract AF 33(615)-2874. The authors gratefully acknowledge the assistance of the many co-workers in the field of thermal property measurements who contributed to the program. 5. References [1] Wechsler, A. E. , Development of high tempera- ture thermal conductivity standards, AFML-TR- 66-415, January 1967. [4] Loser, J. B., Moeller, C. E., and Thompson, M. B. , Thermophysical properties of thermal in- sulating materials, ML-TDR-64-5, April 1964. [2] Powell, R. W. , Ho, C. Y. , and Liley, P. E. , Thermal conductivity of selected materials, NSRDS-NBS 8, November 25, 1966. [3] Goldsmith, A., Hirschhorn, H. J., and Waterman, T. E. , Thermophysical properties of solid materials, Vol. Ill, WADS Technical Report 58- 476, Revised Edit., November 1960. [5] Jun, C. K. , and Hoch, M. , Research on thermal conductivity of tantalum, tungsten, rhenium, Ta-lOW, Tni» T 222» and w_25 Re in the tempera- ture range 1500-2800°K, Contract AF 33(615)- 1759, 1966. 84 [6] Taylor, R. E. , and Nakata, M. M. , Thermal properties of refractory materials, WADD-TR- o0-581, Part IV, November 1963. [7] Wood, W. D. , and Deem, H. W. , Thermal proper- ties of high temperature materials, RSIC-202, June 1964. [8] Neel, D. S., Pears, C. D. , and Oglesby, S., The thermal properties of thirteen solid materials to 5000°F or their destruction temperature, WADD-TR-60-924, February 1962. 85 TEMPERATURE l°C) 500 400 - B 300 - 200 100 1000 2000 Data From Reference 1. 3000 4000 TEMPERATURE -°F 5000 6000 7000 Figure 1 Approximate Thermal Conductivity of Various Grades of Graphite Versus Temperature Temperature (°F) 500 1000 2000 3000 1000 1400 „ 1800 Temperature (°K) 2200 1.6 1.4 1.2 5« 200 500 1000 Temperature (°F) 2000 3000 4000 T 1000 800 600 % E 0) 400 £ 200 \ * Boron Carbide (7) 600 1000 1400 1800 2200 Temperature (°KI 2600 Figure 2 Thermal Conductivity of Candidate Standards 86 CO I E o c o 1 00 1. 7U CO 60 J y\ _^_J ^Detector Source \\f J~ 1 85 l.oP i an 1. oU -^mk'-mTf^- ^5^ SSsr^ ^^5 1 75 1. /-> 1 7(1 - fl» 45° — «: 135 1. f U 1 f, | 1 Geometric Density 1.770 gm cm 1 1 1 1 1 "3 12 3 4 5 Position Figure 3 Axial Density Distribution in AXM-5Q1 Graphite 1. 7-» 1.90 1 ffi 1 2 1 1 1 4 % — I.S* Co 60 1 Source j" tjt-| Detector ^ i tin 1 Ti \ \ s V *•* >53 r ""^- V sr-x — J/.- •y s.. ,40 / ^ 1 1.70 1 — •• — «: D5° Geometric Density 1.770 gm cm J r 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 Position Figure £ Radial Density Distribution in AXM-5Q1 Graphite 87 Glass Beads — A Standard for the Low Thermal Conductivity Range? Alfred E. Wechsler 1 Arthur D. Little, Inc. Cambridge, Massachusetts 021A0 A measurement standard for low thermal conductivity, suitable for a broad tem- perature range, is required in the many applications of thermal insulation materials. Glass beads have been suggested as a candidate standard because of their potential reproducibility, thermal stability, and amenability to analysis of the heat transfer mechanisms. Literature thermal conductivity data for several types of evacuated and gas-filled glass bead systems are summarized. Recent measurements using the line heat source and guarded cold plate apparatus are presented. The effects of particle size, temperature, and gas pressure on thermal conductivity are discussed. Consider- able discrepancy exists in the conductivity data for evacuated glass bead systems ob- tained by different methods at different laboratories. The results suggest that variations in the sample preparation and cleanliness, particle size and distribution, as well as apparatus limitations may be the cause of these discrepancies. Key Words: Comparative method, evacuated powders, glass beads, guarded cold plate method, line heat source method, thermal conductivity, thermal conduc- tivity probe. Introduction The increased use of powder, fibrous, foam, and multilayer insulations as components of thermal pro- tection systems for low and intermediate temperature applications has led to the requirements for a suitable standard of low thermal conductivity. The present standard used by the National Bureau of Standards and others is a semi-rigid resin bonded fiberglass insulating board with a thermal conduc- tivity of about 0.035 W m" x deg" The need for a standard with a lower conductivity value is shown in Table 1 which summarizes the range of thermal conductivity values of common insulation materials. Table 1. Thermal Conductivities of Various Insulations Material Thermal Conductivity W m" 1 deg" 1 Multilayer Insulations Opacified Evacuated Powders Evacuated Glass Fibers Evacuated "Powders Unevacuated Foams, Powders, Fibers 1.0 x 10" 5 to 1.0 x 10" 1 * 2.8 x lO" 1 * to 1.0 x 10-3 5.8 x 10" 4 to 2.8 x lO" 3 8.5 x 10" 1 * to 5.8 x 10 3 6.0 x 10" 3 to 3.6 x 10 2 The usual criteria of homogeneity, isotropy, reproducibility, and availability apply to the selec- tion of a low thermal conductivity standard. In addition, it is desirable to (1) have a knowledge of the mechanisms of heat transfer in the candidate standard, (2) select a material suitable for use in many types of apparatus over a broad temperature range, and (3) have a single standard that will be ap- plicable to the large conductivity range of many types of insulating materials. Group Leader, Engineering Sciences, Research and Experiments Group. 89 Glass beads have been suggested as a candidate standard for this application. They are readily available; their inherent form should lead to homogeneity, isotropy, and reproducibility. They can be used in many types of apparatus because of their small particle size and their fluidity. A large range in thermal conductivity can be obtained by using gas-filled and evacuated beads. However, the reproduci- bility of thermal conductivity data is questionable, and the effects of particle size, mechanical load- ing, temperature, and sample preparation procedures on thermal conductivity have not yet been fully established. The objectives of this paper are to summarize available thermal conductivity data on glass beads, to discuss the factors which cduse variations in the data, and to suggest procedures for further work to improve the applicability of glass beads as a thermal conductivity standard. 2. Materials Used Solid beads or spheres are commercially prepared from soda-lime-silica glass, borosilicate glass, and fused quartz. The first of these has been used most often in thermal conductivity measurements and the data' presented in this paper will be restricted to this readily available material. The principal commercial suppliers of soda-lime-silica glass beads are the 3M Company and Microbeads, Inc. The beads are available in grades with average diameters from about 0.025 mm to 0.5 mm, graded with respect to shape as well as size and size distribution. Manufacturers may designate the beads according to the percentage of "true spheres", typically 65% to 95%. When obtaining glass beads, the material is usually specified by the fraction retained by or passing through standard sieves. Thus, the upper and lower values for bead diameters are established. In practice, however, a small percentage of fines is often found in most samples. For example, Table 2 below gives both manufacturers 1 size distribution data on two typical bead sizes. Table 2. Glass Beads Size Distribution — Manufacturers' Data Bead Type Percent Retained on U. S. Sieve No. 60 (0.25 mm) 70 (0.21 mm) 80 (0.177 mm) 100 (0.149 mm) 120 (0.125 mm) 140 (0.105 mm) 170 (0.088 mm) 710A 100B (200 micron) 1217A 120B (120 micron) 10 0-2 40 40 92-100 10 0-2 35 60 92-100 5 Properties of the bulk glass which are significant to the use of the glass beads in thermal conduc- tivity measurements are listed in Table 3. Glass beads are supplied in quantities of one to several thousand pounds. Special orders with more narrow size distributions can also be obtained. The beads are relatively clean and free from foreign material when received. Table 3. Properties of Bulk Glass from Which Glass Beads are Prepared' Density 2.5 x 10 3 kg/m 3 Hardness , 6 (Mohs' scale) Coefficient of Thermal Expansion 9.5 x 10~ 6 deg -1 (20 to 300°C) Thermal Conductivity 1.04 W m -1 deg -1 (room temperature) Specific Heat 1080 J kg l deg -1 7.5 x 10* ° N/m 2 Modulus of Elasticity Refractive Index (6200A) 1.53 From manufacturers' data sheets. Density of the beads depends upon the packing and size distribution but is usually in the range o f 1.45 to 1.55 x 10 3 kg/m 3 . 90 3. Measurement Methods Table 4 summarizes the methods used by various investigators to measure the thermal conductivity of glass beads, as well as the types and sizes of beads, temperature ranges, and environmental conditions. The thermal conductivity probe, line heat source, and guarded cold plate methods are commonly used tech- niques well known to most investigators. Table 4. Methods and Environmental Conditions Used in Measuring Thermal Conductivity of Glass Beads Investigators Method Bead Properties Temperature Environmental Conditions Type Diameter Density mm kg /m 3 °K Masamune & Smith[l] 2 Comparative, Lucite disc 3M 0.029-0.470 1500 315 Air and vacuum Watson[2] One-dimensional radiative flux Microbeads <0. 037-0, 84 1170-1820 150-350 Vacuum Messmer [3] Probe Microbeads 0.177-0.84 1560-1660 300 Air , vacuum and mechani- cal load Wechsler & Simon [4] Line heat source Microbeads 0.044-0.062 1420 190-350 Vacuum Wechsler & Glaser[5] Probe and guarded cold plate 3M 0.029-0.150 1460 300 Air and vacuum Salisbury & Glaser[6] Line heat source 3M 0.050-0.200 1500 300 Vacuum Arthur D. Little [7] Guarded cold plate Microbeads 0.59 -0.84 1590 190-270 Vacuum, N2, He The comparative apparatus used by Masamune and Smith [1] consisted of a Lucite disc 3.8 cm in diam- eter and 1.25 cm thick adjacent to a similar sized sample. One-dimensional heat flow was assumed between hot and cold plates which were in contact with the Lucite disc and sample, respectively. Temperature measurements were made with thermocouples at the Lucite-heat source, Lucite-sample, and sample-heat sink interfaces. The conductivity of the Lucite was taken as 0.205 W m * deg -1 In the method used by Watson [2] , a sample was placed on a warm temperature bath plate; the top of the sample radiated to a cavity maintained at 77°K. The heat flux emitted from the sample was measured using an 8-l4u filtered infrared photoconductor photometer and the temperature of the sample surface was determined from this emitted energy. From the measured heat flux, bath temperature, sample thickness, and the calculated temperature of the free surface of the sample, the thermal conductivity was then evaluated. Errors in measurement are discussed by each of the investigators cited in Table 4. Generally, this method should have accuracies of from 5 to 20%, although greater accuracy is claimed by several investigators . The method of sample preparation is not adequately described by most investigators. Messmer [3] resleved, washed, and dried (at 110°C) the beads used in his work. The beads used by Wechsler and Simon [4] were sieved, washed, and baked at 200 C C in high vacuum for a prolonged period. Watson [2] washed some of the samples in acetone prior to use. Other investigators provide only limited indications of how the samples were handled, cleaned, and used. The only common measurement control used in most investigations is bulk density. The measurements are usually gravimetric and are often made during preparing the sample. A desirable characteristic of the glass beads is that they will pack under random filling methods to densities between 1450 and 1550 kg/m . Values differing significantly from this range are indicative of a broad size range or errors in measurement . Figures in brackets indicate the literature references at the end of this paper. 91 4. Measurement Results In this section, the effects of gas pressure, particle size, temperature, and mechanical loading will be discussed. The results of different investigators will be compared; the discussion will empha- size data obtained at atmospheric pressure and at high vacuum. 4.1. Effect of Gas Pressure The dependence of thermal conductivity of a powder on gas pressure has been studied for many years [8]. Theoretical and empirical correlations have been developed to predict the contribution of the gas phase to thermal conductivity of the heterogeneous composite [9,10], The correlations are generally satisfactory in predicting the variation of thermal conductivity with gas type at atmospheric pressure, and in predicting the rate of decrease in thermal conductivity with decreasing gas pressure. The prin- cipal discrepancy between available theories and current experiments is in establishing the residual thermal conductivity of the solid when the interstitial gas pressure is reduced to a very low value. The effects of physically or chemically adsorbed gases on this residual conductivity are not well known. Figure 1 shows data of three investigators on the effect of gas pressure. Unfortunately, the mea- surements were made with different particle size beads; however, all measurements were near 300 C K. The data of Messmer [3] and Wechsler and Glaser [5] are in reasonable agreement. The larger the particle size, the greater the reduction in gas pressure that is required to achieve the equivalent reduction in thermal conductivity. Note that these data are in fair agreement at atmospheric pressure and at low pressures. The variation in conductivity with particle size is in agreement with other data described later. The data of Masamune and Smith [1] agree with the data of Messmer [3] and Wechsler and Glaser [5] at atmospheric pressure but differ greatly at low pressure. This may be a result of the limitations imposed by the design of the comparative apparatus using a relatively high conductivity standard. These data point out two ranges where glass beads may be useful as a possible standard — at atmospheric pressure and at high vacuum. It would be difficult to use the beads as a standard at intermediate gas pressures because of the large slope of the conductivity-pressure curve and the difficulty in accurate pressure measurements within powder systems. 4.2. Effect of Particle Size Figure 2 shows the data obtained by several investigators at atmospheric pressure and at gas pressures below 10 -2 torr, where the effects of gas pressure are small. Although there appears to be a wide variation in the data, several definitive conclusions can be drawn. (1) At atmospheric pressure, the data of all investigators are relatively consistent with respect to the absolute value of thermal conductivity. This is attributed to the large gas conduction contribu- tion at atmospheric pressure. There is only a small increase of thermal conductivity with an increase in particle size. (2) Under high vacuum conditions, there is considerable variation of thermal conductivity with particle size. The results of Watson [2], Wechsler and Glaser [4], Salisbury and Glaser [6], and Arthur D. Little, Inc. [7 J are consistent in both the absolute value of thermal conductivity and the increase of thermal conductivity with increased particle size. Note that the thermal conductivity of beads with very small particle size appears to be higher than that of beads with an intermediate particle size. This can be explained by the different magnitudes of the conduction and radiation contributions to ther- mal conductivity. (3) Examination of Figure 2 and Table 4 shows that there is no consistent effect of the source of materials. The comparative method of Masamune and Smith [1] gives higher thermal conductivity values and less effect of particle size on thermal conductivity than any other method. The thermal conductivity probe methods used by Messmer [3] and Wechsler and Glaser [5] give higher thermal conductivity values than the line heat source or guarded flat plate methods. These data suggest that the probe and compara- tive method may have lower limits of applicability or may yield inaccurate data at low conductivity values. The high value of conductivity for beads with a 193u diameter obtained by Messmer [3] at 10 torr does not agree with the other observed trends. (4) The beads which have been thoroughly cleaned and outgassed at low pressures (Watson [2] and Wechsler and Simon [4]) seem to have lower thermal conductivity values than those for which the prepara- tion method is not specified. This may be a coincidence or it could result from removal of conducting films on the beads by cleaning or prolonged outgassing. The results suggest that careful sample preparation is necessary to obtain good data and each appa- ratus and method should be examined to determine the errors involved in measurements of low thermal con- ductivity materials. 4.3. Effect of Temperature The effect of temperature on the thermal conductivity of glass beads has been investigated by Watson [2] and Wechsler and Simon [4]. These investigators interpret the conductivity data as the sum 92 of a term with a constant value, representing the contribution of solid conduction, and a term which has a cubic temperature dependence, representing the contribution of thermal radiation. This approach seems to give consistent results for glass beads as well as other powders. Figure 3 shows typical results of these investigators. Each curve represents the equation: B + AT 3 (1) where B is the solid conduction contribution, A is a radiation term — a constant for a given material — k is the effective thermal conductivity, and T is the absolute temperature. Table 5 shows values of these constants as a function of particle size. Table 5. Radiative and Solid Conduction Contributions to the Thermal Conductivity of Glass Beads Investigator Particle Size Radiative Term (A) Solid Conduction Term (B) microns W m _1 deg -1 * W m -1 deg -1 Watson [2] 840-590 26.6 x 10 -11 -6.6 x 10~ 5 350-250 13.4 x 10" 11 9.5 x 10 5 125- 88 8.5 x 10" 11 32.2 x 10~ 5 74- 53 3.4 x 10 n 70.0 x 10~ 5 <37 6.3 x 10 -11 95.4 x 10~ 5 Wechsler & Simon [4] 44- 62 3.0 x 10 n 46.6 x 10" 5 The general trends of increased solid conduction contribution with increased particle size and de- creased radiation contribution with decreased particle size are evident. Relatively good agreement is obtained between the work of these investigators. The two opposing trends can result in apparent anoma- lies of the effects of particle size on thermal conductivity at very low temperatures. It would be de- sirable to extend the temperature range of measurement as well as the quantity of data to improve the characterization of the solid conduction and radiation mechanisms. Little published data are available on the effect of temperature on the thermal conductivity of gas filled glass beads. However, because of the large contribution of gas conduction, the effect of tem- perature on conductivity of gas filled glass beads should parallel the effect of temperature on gas conductivity. 4.4. Effect of Mechanical Loading Messmer [3] has investigated the effect of compressive loading on the thermal conductivity of glass beads. Evacuated glass beads of 710u and 193u diameter were subjected to external pressures of from 10 5 to 7 x 10 6 N/m 2 at 300°K. The thermal conductivity of both materials increased with the 1/4 power of the applied external pressure. This differs slightly from the 1/3 power of the externa] pres- sure obtained from Hertzian loading analysis. Data obtained on gas filled beads by Messmer [3] show only a small effect of mechanical load because of the large contribution of the gas to the system's thermal conductivity. Preliminary data obtained at Arthur D. Little, Inc. [7] using 710u diameter evacu- ated glass beads at 225°K in a guarded cold plate apparatus show that the thermal conductivity increases with about the 1/10 power of the load over the pressure range from 5 x 10 3 to 3 x 10 1 * N/m 2 . The radi- ation component of thermal conductivity should be almost invariant with loading, whereas the conduction component should increase with the 1/3 power of the load. Depending upon the ratio of solid conduction to radiation, the dependence of thermal conductivity on external load can vary significantly. In most thermal conductivity apparatus, natural mechanical loading of the sample will not be important. 5. Recommendations Evacuated and gas filled glass beads are a candidate standard for the low thermal conductivity range; however, additional carefully controlled data are required before this material can be accepted as a standard. The following procedures are suggested to assist the further evaluation of glass beads. (1) Select two or three particle size ranges and distributions for use in subsequent measurements. The size ranges should be chosen so that sieving can be used to obtain the required size distribution rather than more complex separation or classification methods. 93 (2) Establish a controlled method for cleaning, outgassing, and preparing samples. (3) Perform repeated thermal conductivity measurements using different methods and apparatus under carefully specified environmental conditions of temperature, temperature gradient, and gas pressure. A temperature range of 100 to 500°K, with temperature gradients of less than 10°K/cm is recommended. Gas pressures should be kept below 10 torr or at atmospheric pressure. Nitrogen or an inert gas of high purity should be used. Measurements should be made under carefully controlled mechanical load conditions. Density data should be obtained as a control parameter. (A) Analyze the random and systematic errors in each thermal conductivity method used and report the precision and estimated accuracy of the measurements with the data. Through the cooperative efforts of the participants of this conference, it may be possible to estab- lish glass beads as a low conductivity standard within several years. 6. Acknowledgments The author gratefully acknowledges the assistance of Mr. F. Ruccia, Mr. I. Black, Mr. E. and Mr. R. Carroll in obtaining the published and unpublished data at Arthur D. Little, Inc. of Professor R. C. Reid of MIT in compiling and examining the data was greatly appreciated. Boudreau, The help References [1] Masamune, S. , and Smith, J. M. , Thermal con- ductivity of beds of spherical particles, Ind. & Eng. Chem. Fundamentals, 2_, No. 2, p 136-142 (1963). [6] Salisbury, J. W. , and Glaser, P. E. , editors, Studies of the characteristics of probable lunar surface materials, AFCRL-64-970 (Jan. 1964) . [2] Watson, K. , The thermal conductivity measure- ments of selected silicate powders in vacuum from 150°-350°K, Ph.D. Thesis, California Inst, of Tech. , 1964. [3] Messmer, J., personal communication (1964). [7] Arthur D. Little, Inc., unpublished work in progress. [8] Smoluchowski, M. , Sur la conductibilite calorifique des corps pulverisis, Bull. L'Acad. Sci. de Cracovie, A129 (1910). Int. [4] Wechsler, A. E. , and Simon, I., Thermal con- ductivity and dielectric constant of silicate materials, Final Report prepared by Arthur D. Little, Inc., for NASA George C. Marshall Space Flight Center, under Contract NAS8-20076, August 1966. [5] Wechsler, A. E. , and Glaser, P. E., Thermal conductivity of non-metallic materials, Final Report prepared by Arthur D. Little, Inc., for NASA George C. Marshall Space Flight Center, under Contract NAS8-1567, June 1964. [9] Schotte, W. , Thermal conductivity of packed beds, AIChE Journal, ji, 63 (1960). [10] Everest, A., Glaser, P. E. , and Wechsler, A. E.. Thermal conductivity of non-metallic materials, Summary Report prepared by Arthur D. Little, Inc., for NASA George C. Marshall Space Flight Center, under Contract NAS8-1567, 27 April 1962. 94 Gas Pressure (Torr) 10" 10 -3 10" 10 -1 1 10 10< 760 20 10 2 5 o E £ re c o o Measurements at approximately 300°K .5 o .1 LEGEND: Thermal Conductivity Probe A 360 /a Size (Messmer) □ 50/i Size (Wechsler & Glaser) O 150/a Size (Wechsler & Glaser) Comparative Method • 200/a Size (Masamune & Smith) ■ 30/i Size (Masamune & Smith) I I 10" 10" 10 10' 10 3 10 4 10 5 Gas Pressure (Newton/Meter') Figure 1 Effect of Gas Pressure on Thermal Conductivity of Glass Beads (63) (M) (f0§) 7 Masamune & Smith Watson *-» u 1 20 o o 15 44 - 62/t Microbeads (Wechsler & Simm) 150 1 200 250 300 Temperature (°K) 350 400 Figure 3 Effect of Temperature on the Thermal Conductivity of Glass Beads 96 Lattice Thermal Conductivities of Semiconductor Alloys I . Kudman RCA Laboratories Princeton, New Jersey The lattice thermal conductivities of disordered semiconductor alloys are treated using the phenomenological model derived by Abeles. The thermal conductivity is expressed in terms of the lattice parameters and mean atomic weights of the alloy and its constituents. This treatment is applied to both III-V and IV-VI semi- conductor alloy systems. Comparison of the theoretically derived and experimentally determined lattice thermal conductivities are discussed. Key words: Lattice thermal conductivity, semiconductor alloys. 1. Introduction Solid solution alloys are ideally suited for studying the effects of defects on thermal transport. The small concentrations of intentionally and unintentionally added impurities have a negligible effect on the thermal conductivity, as compared to that of alloying. Above the Debye temperature, the thermal conductivity of a semiconductor can be determined, to a fair degree of accuracy, by use of the 3-phonon scattering equation of Leibfried and Schlomann [1]1. For the case of combined scattering, the relaxation time approximation has been applied with considerable success. The relaxation time approximation of Klemens [2] and Callaway [31 has been used by Abeles T41 to derive a simple phenomenological model to describe the variation of thermal conductivity of semi- conductor alloys as functions of both composition and temperature. It is the purpose of this paper to compare the thermal conductivities derived by the Abeles model to those determined experimentally. The experimental results are given for III-V and IV-VI semi- conductor alloy systems. 2. Experimental The III-V alloy systems; GaAs-InAs, GaSb-GaAs, and GaSb-InSb, were prepared by the zone leveling technique. This procedure consists of moving a molten zone having the liquidus composition through charge material having the isothermally corresponding solidus composition. The IV-VI alloy system, PbTe-PbSe, was prepared by the vertical Bridgman method. The thermal conductivities were determined from high temperature diffusivity measurements [51. The specific heats required to convert the thermal diffusivity, K/Cp, into K were obtained from measurements of th III-V compounds [61 and the Dulong-Petit value was used for the PbTe-PbSe alloy system [7]. 3. Discussion 3.1 Theory For temperatures above the Debye temperature, the thermal conductivity of a semiconductor alloy results from the scattering of phonons by mass fluctuation and strain scattering due to alloying and anharmonlc phonon-phonon scattering. A phenomenological theory describing the variation of the thermal resistance on alloy composition and temperature has been derived by Abeles [4]. It was assumed that the reciprocal relaxation times depend on frequency uo as uo for strain and mass point defects and as u)2 for normal and umklapp three-phonon anharmonic processes. The lattice thermal resistance of a disordered 1 Figures in brackets indicate the literature references at the end of this paper. 97 semiconductor alloy is given by the expression (reader is referred to reference 4 for a detailed dis- cussion) : 1/k = TAl (TMr)* 6 3 + [B] M5 6 7/2 T, where [M = 9.67xl0 5 (1 + 5/9 a)* y : P~ 2 (1) [Bl = 7.08xl0 -2 (1 + 5/9 a) y 2 p~ 3 Here a is the ratio of normal to umklapp scattering rates: y, is the anharmonicity parameter; p is defined by p = 9MS 6^/2, where 9 is the Debye temperature, M is the mass, and 6 = a/2 for diamond and zinc-blende lattices, where a is the lattice constant. The disorder parameter T = x(l-x) [£M/M + e A6/6], where x is the composition, /\M is the mass difference, A6 is the difference in lattice constants, and f is the strain term. The model of Abeles includes two major modifications which have not, in general, been considered in theories of point defect scattering: (1) the inclusion of normal processes and (2) the introduction of a strain term due to atomic misfit. A similar expression was derived by Parrott [8], however, the contribution of strain scattering was not included in the derivation. The introduction of normal processes was based on the suggestion of Klemens [9] that these pro- cesses contribute significantly to the thermal resistance even at high temperatures. While normal processes do not contribute directly to the thermal resistance, they scatter phonons into modes which can then undergo scattering by another mechanism e.g. point defect scattering. The strain term introduced by Abeles is that due to Klemens [2] in which an impurity is introduced in a simple cubic lattice with only nearest neighbors acting. Abeles then proceeds to treat the scattering in terms of the elastic continum "sphere-in-hole" model and arrives at the definition of r given above. The strain term e is assigned a value of 39 based on the impurity model of Klemens. Although p is readily calculated, e assigned a value of 39, and y^ can be determined from the thermal conductivity of the components, it is not obvious how to choose a value for the adjustable parameter a. The thermal resistivity of the Ge-Si alloy system was computed by Abeles and agreement obtained between the theoretical values at 300°K and the experimental data for y± = 1.77 and a = 2.5. 3.2 III-V Compound Alloys The calculations of the lattice thermal resistivities of disordered semiconductor alloys will now be applied to the III-V compound alloy systems. The strain term e remains unchanged and the adjustable parameter a will be assumed equal to 2.5 as in the base of the Ge-Si alloys. (It must be emphasized that a is an adjustable parameter and may vary from one class of semiconductors to the next, it is used only for the initial calculations since it was successfully applied to the Ge-Si alloys). Abeles computed the thermal resistivities for two III-V alloys systems assuming y\ to be constant, as was found for the Ge-Si alloys. For one system, InAs-InP, the agreement was poor, especially as the InP content increased. While the assumption of a constant y^ for Ge-Si alloys is justified on the basis of experimental data, this is not the case for the III-V compounds. In the III-V compounds it was found that the anharmonicity parameter varied significantly from compound to compound [1C]. yi depends on the mass ratio of the constituent elements in such a manner that compounds with a mass ratio close to unity, e.g., InSb, have large values of yy ; and those with a large mass ratio, e.g. AlSb have smaller values of y. . The variation of y. as a function of the mass ratio is shown in figure 1.3 The variation of the room temperature lattice thermal resistivities of the III-V alloys were computed using the values of a and e given above. The anharmonicity parameter was determined from figure 1 using the average mass ratio of the alloy. The lattice thermal conductivities of the alloys were obtained by subtracting the electronic contribution from the total thermal conductivity. The electronic thermal conductivity was computed for the case of combined polar plus ionized impurity scattering [11].' o The variation of the anharmonicity parameter given in reference 10 was determined by fitting the experimental data to the 3-phonon equation of Leibfried and Schlomann [1]. The only unknown in this expression was y, which was found to vary from compound to compound. It was not intended to imply, however, that the parameter y would depend on the mass ratio rather than the thermal conductivity. Since no theoretical expression is available to explain this behavior, it is convenient to use. a varying y to describe the thermal conductivity of the III-V compounds and their alloys. o 2 9 3 Note that y, eq ( 1) = [y + |~| (reference 10). 98 The theoretical curve (dashed line) for the alloy system InAs-GaAs is shown in figure 2 together with the experimental data (crosses), the agreement being better than 20% at all compositions. In view of the assumption made for the value of a, the agreement between theory and experiment is quite good. The value of a necessary to fit the experimental data in figure 2 was determined prior to the compu- tation of the thermal resistivities of the remaining III-V alloy systems. It was pointed out, previously, that a is an adjustable parameter and may vary from one class of semiconductors to another. It is expected, however, that a will not vary greatly from system to system within a particular covalent crystal system such that the value of a determined for the InAs-GaAs system should be applicable to other III-V alloy systems. The best fit was obtained for a = 1.5 and shown in figure 2 by the solid line, the agreement being excellent at all compositions. The thermal resistivities of the alloy systems GaAs-GaSb, GaSb-InSb, and InAs-IriP [12] were computed using a = 1.5, e = 39, and y\ determined from figure 1. The computed curves and experimental results are shown in figures 3,4 and 5 for III-V alloy systems, the agreement being excellent in all cases. 3.3 IV -VI Alloys The calculation of the thermal resistivity was extended to the IV-VI alloy system PbTe-PbSe which has been investigated in the past for thermoelectric applications [13]. The anharmonicity parameter was determined for the components and then assumed to vary linearly as a function of composition. The parameters a and e were assumed to be the same as the III-V compounds. The electronic contribution to the thermal conductivity was computed for the case of acoustical scattering and subtracted from the total thermal conductivity. The computed curve and experimental results are shown in figure 6, the agreement, as for the other alloy systems, being excellent at all compositions. 4. Conclusion It has been quite clearly demonstrated that the theory of Abeles is applicable to a wide range of cubic semiconductor alloys systems. It is apparent that information obtained from one alloy system can be used to obtain greater accuracy in predicting the thermal conductivities of other alloy systems within a class of semiconductors. The introduction of a strain contribution to the disorder parameter must be considered for all alloy systems as well as the effects of normal processes. For many of the alloy systems discussed above, the strain contribution was considerably greater than the mass fluctuation contribution to the disorder parameter F. Li] [2] [3] [4 J [5] [61 [7] Leibfried, G. and Schlomann, E. , Nachr. Akad. Wiss. Gottingen, Math. Physik Kl. Ila 4, 71 (1954). References [8] Klemens, P. G. , Proc. Roy. Soc. (London) A208 , 108 (1951); Proc. Roy. Soc. (London) A68 , 1113 (1955); Phys. Rev. 119, 507 (1960). Callaway, J., Phys. Rev. 113, 1046 (1959); Callaway, J, and von Baeyer, H. C, Phys. Rev. 120 , 1149 (1960). Abeles, B., Phys. Rev. 131 , 1906 (1963). Abeles, B. , Cody, G. D. , and Beers, D. S., J. Appl. Phys. 31, 1585 (1960). Miller, R. , unpublished. Parkinson, D. H. , and Quarrington, J. E. , Proc. Phys. Soc. A67 , 569 (1954). [9] [10] [11] [12] [13] Parrott, J. E., Proc. Phys. Soc. 81, 726 (1963). Klemens, P. G. Westinghouse Research Report, 929-8904-R3 (1961); Klemens, P. G. , White, G. , and Tainsh, R. J., Phil. Mag. 7, 1323 (1962). Steigmeier, E. F. and Kudman, I. Phys. Rev., 141 , 767 (1966). Amith, A., Kudman, I., and Steigmeier, E.F., Phys. Rev. 138, A1270 (1965). Bowers, R. , Bauerle, J. E., and Cornish, A. J., J. Appl. Phys. 30, 1050 (1959). Joffe, A. F., Semiconductor Thermoelements and Thermoelectric Cooling (Infosearch Ltd., London 1957). 99 ^m (M x /M n ) Figure 1 The variation of the square of the anharmonicity parameter y. for the III-V compounds as a function of the mass ratio M^/M^^ of the constituent elements. The values shown by crosses are derived experimentally. CALCULATED EXPERIMENTAL - X 10 20 30 40 50 60 MOLE % GaAs 70 80 90 100 Figure 2 The lattice thermal resistivity at 300°K as a function of composition for GaAs-InAs alloys. Dashed curve (a = 2.5) and solid curve (a = 1.5) computed from eq. 1. 100 20 15 - ?I0 ■o E 5 5 1 1 1 1 1 1 1 1 GoSb — Go As T« 300°K /% X — CALCULATED X MEASURED - 1 1 1 1 1 1 1 1 V — 10 20 30 70 80 90 100 40 50 60 MOLE % GoAs Figure 3 The lattice thermal resistivity at 300°K as a function of composition for GaAs-GaSb alloys. Solid curve computed from eq. 1. 40 50 60 MOLE % GoSb Figure 4 The lattice thermal resistivity at 300°K as a function of composition for GaSb-InSb alloys. Solid curve computed from eq. 1. 101 10 •o E o 5- 50: t- Figure 6 1 1 1 1 II i I x v /* / x / 1 1 InAs- In P T=300°K i i i i i l 1 10 20 30 40 50 60 MOLE % InP 70 80 90 100 Figure 5 The lattice thermal resistivity at 300°K as a function of composition for InAs-InP alloys. Solid curve computed from eq. 1. 600 °K CALCULATED X MEASURED 10 J L 20 30 40 SO 60 MOLE % PbSe J L 70 80 90 100 Lattice thermal resistivity at 300°K as a function of composition for PbTe-PbSe alloys. Solid curves at 300 and 600°K computed from eq. 1. 102 Thermal Conductivity of N-type Germanium from 0.3°K to 4.2°k1 Bruce L. Bird and Norman Pearlman Department of Physics Purdue University Lafayette, Indiana The thermal conductivity of N-type germanium has been measured between 0.3°K and 4.2°K. Excellent agreement with the theoretical model of Griffin and Carruthers is found for an Sb doped sample (N.p= 3.6x 10 cm' J ), The model pro- posed by Keyes, however, is also able to give a reasonable fit for different values of the parameters. Experimental results for an As doped sample (N_= 5.3x 10 cm ) and a P doped sample (N_= 2.6x lO^ cm - ^) are in complete disagreement with both theoretical models. Both samples show a decreased ther- mal conductivity which varies as T^ down to the lowest temperature measured with no indication of any resonance scattering. Key Words: Electron-phonon scattering, low temperature, N-type ger- manium, point defect, semiconductor, thermal conductivity. 1. Introduction Previous measurements of the low temperature thermal conductivity of N and P-type germanium [1, 2, 3, 4, 5] , have shown that shallow donors or acceptors are very strong scatterers of phonons. For ex- ample, a concentration of as little as 4x10 Sb donors per cm" . reduces the thermal conductivity at 2 K by a factor of eight. Measurements by Goff and Pearlman [2] on highly doped, highly compensated N-type material as com- pared with singly doped material having the same concentration of neutral donors indicated that the ex- tra thermal resistance was correlated with the number of neutral donors and not with the total impurity concentration. Thus the phonon scattering occurs because of an interaction with the electron bound to the donor ion not with the donor ion itself. An impurity species effect was also found with Sb being a more effective scatterer of phonons than either As or P at 2°K for concentrations less than 6xKr'cm . For concentrations greater than this the extra thermal resistance becomes species independent. The scattering process then can be separated into at least two types, one for concentrations in which there is little overlap of the electronic wave functions between neighboring impurities and the other in which the concentration is so large that the interaction of donor electrons leads to the for- mation of an impurity band. In this article we will only consider the case of low concentration, non- compensated samples for which the donor electrons are assumed to be isolated. The problem of isolated shallow donors in germanium has been reviewed by Kohn [6], The electronic energy levels of the bound donor electron were calculated using the effective mass approximation. There is good agreement between theory and experiment for the excited states but not for the ground state. The effective mass approximation predicts that all shallow donors should have an ionization energy of 9.2 milli-electron volts and their ground state should be 4-fold degenerate, neglecting spin. Experi- mentally it has been found that the donor ground state is split by an amount 4A producing a singlet ground state and a triply degenerate excited state. The type of chemical impurity determines 4A , thus it is referred to as the "chemical shift" or "valley-orbit" splitting. Table 1 lists the ionization energy (e.) and chemical shift for some of the shallow donors in germanium, as determined from infrared absorbtion measurements by Reuszer and Fisher [7], Work supported in part by ARPA. 2 Figures in brackets indicate the literature references at the end of this paper. 103 Table 1 Effective Sb P As Mass 10.19 12.76 14.04 9.2 0.32 2.83 4.23 0.0 3.71 32.8 49.1 44.2 35.3 32.0 65.0 2.2xl0 16 4 .4xl0 16 5.8 xlO 16 7.0 x 10 15 e . (milli-e.v.) 4A (milli-e.v.) c T R (°K)=4A c /k r Q (A) N^cm" 3 ) Also listed in Table 1 is the effective Bohr radius r for an electron in the ground state. It was calculated from the measured ionization energy by using the equations applicable to the hydrogen atom immersed in a medium with dielectric constant K, that is, e . = R(m*/m) (1/K ) and r = a (m/m*)K where R= 13.6 e.v. and a = 0.53Ai The dielectric constant for germanium is 16. The value of r listed un- der the effective mass value was obtained by Kohn [6] as a result of a variational calculation to deter- mine e . . l Fritzsche [8], on the basis of impurity conduction measurements, has suggested that the condition <5, where a is an average distance i the impurity atoms can be considerec inequality holds, N5J, is shown in Table 1, a/r <5, where a is an average distance between impurity atoms, a= (3/4ttN ) ' , defines the region in which the impurity atoms can be considered isolated. The largest concentration of donors for which this 2. Models Used for Electron- Phonon Scattering In order to explain the thermal conductivity results Keyes [9] proposed a model in which the scat- tering arises from the large effect of strain on the energy of the bound donor electron. The electronic energy states have a quadratic dependence on the strain produced by the phonons. This results in scat- tering of the point defect type T* a ll) . A further dependence on the phonon frequency arises when the phonon wavelength becomes shorter then the diameter of the electronic orbitcausing the average strain to approach zero, so that the phonon scattering rapidly decreases as the phonon frequency rises past u)~v/r . This results in a scattering rate which Keyes expresses in the form t"^(w, P ) K = M(u)) V (v J F (w,p) (1) ep p E 4 D uo 4 with MC) = -rS [ 6X " I 3 TTd Z (4A ) 2 c v(v p ) = fc^7+- i 7 ] and the cutoff factor F(uo,p) given by F(u ) ,p)=[l+(r o u ) /2v p ) 2 ]" 8 The anistropy factor D has the values P D 1 = 4/5 D 2 = D 3 = 3/5 where p=l, 2, 3 indicates the phonon polarization. 104 In eq. (1) d is the density of germanium, n is the concentration of neutral donors, E is the shear deformation potential constant, v. and v„ are the velocity of longitudinal and transverse sound waves, and D is an anisotropy factor averaged overangle. Keyes expression for the inverse relaxation time is small at low frequencies because of the tu dependence, reaches a maximum at about uu=v/r then rapidly declines due to the cutoff factor F(u),p). It is species dependent through its dependence on 4A and r . c o In a more detailed calculation, Griffin and Carruthers [10] argued that whereas Keyes treated the electron as moving in the static strain field created by the phonons, the electron actually adjusts to the phonon perturbation with a characteristic frequency 4A fh which lies in the range of phonon fre- quencies important for heat conduction. Griffin and Carruthers thus treat the problem in analogy with the resonance fluorescence scattering of photons. During the elastic scattering process the electron undergoes a virtual transition between the singlet and triplet states while the phonons' crystal momen- tum is changed. • Using this approach Griffin and Carruthers obtained the following expression for the relaxation time t" 1 (u), P ) GC = M , (u))R(u)) W(a),v p) B(T) (2) ep p 4 4 . n E H ud n.i. ™i/ \ ' r ex u p -i with M'(u))= —7 \ [ 5 ] 3TTd Z ( 4A C ) R(ti)) = (4A c ) 4 /[(M 2 -(4A c ) 2 ] 2 W(oo,v p) = {-\- (-^-5 + — ^-5)] V p S p S 1 V 1 2S 2 V 2 2 where S is related to the cutoff factor F(uo,p) by S = 1/F(uu,p). H is an anistropy factor with the values H l = 43/225 H 2 = 32/225 H 3 = 40/225 and B(T) = {B s (T) + B T dx < 3 > 6tt ft J P v (e X -l) C P where the total relaxation time t is given by 105 T~ (U),p) = T~ (U),p) + T~ (U)) + T~ (u)) + T~ (u),p). The relaxation times included in the total relaxation time t are t.,, boundary scattering; t t , isotope scattering; r , umklapp and normal processes; and t , electron-phonon scattering. Casimir's [12] relation Will be used for boundary scattering but modified to take into account the difference in velocity of the differently polarized waves, i.e., tI = v /L . The Casimir length, L =1.13(A ) ' , defines an effective length associated with a sample whose cross-sectional area is A . Klemens [13] expression for the isotopic relaxation time is given by t~ = Auu with the constant A = V r/(4Trvv in which Vo is the volume per atom, v is some average sound velocity, and r is a constant determined by the relative abundance of the isotopes and their masses. For germanium r= 5.89x10 . This results in the theoretical value of A being 2.35 x lCT^sec . , assuming v = 3.56 x lCrcm./sec.as calculated from the Debye temperature at 0°K of 375°K for germanium. It should be noted that it would be more consistent to take the polarization dependence of the sound velocity into account when using t^ *- n tne thermal conductivity as was done for boundary scat- tering. However, in the temperature range of interest, since the isotopic contribution to the thermal re- sistance is small, the usual approximation of taking A to be equal for all three branches will be fol- lowed. Callaway [14], in his phenomenological development of an appropriate method of combining various scattering processes in the thermal conductivity integral, suggested that scattering due to momentum con- serving normal processes could be allowed for by including their relaxation time as part of the total relaxation time t . Using Herring's [15] suggested form for the relaxation time for low frequency phonon-phonon scattering (t~ « uu ^T ), Callaway was able to fit the thermal conductivity of normal and isotope enriched germanium reasonably well over the range from 2 to 100°K. With T~ = V S A< C + Aid + (B +B 2 )uTT 3 he obtained the best fit with A = 2.57 x 10~ 44 sec 3 and (B + B 2 ) = 2.77xlCT 23 sec. deg" 3 . 4. Description of samples and method of measurement All germanium samples were cut from the ingot in the form of rectangular prisms with their long axis in the (100) direction perpendicular to the growth axis of the crystal in order to minimize im- purity concentration gradients. The impurity concentration listed in Table 2 was determined from the room temperature Hall coefficient. The samples as cut from the ingot with a diamond saw had approximate dimensions of 5x5x 25mm. All surfaces were then ground with #280 carborundum powder reducing the sample dimensions to about 4x4x24mm. and ensuring diffuse boundary scattering. Table 2 _3 Sample n (cm ) Ge - 422A 5.4 xlO 13 Ge(Sb) - 172AI 3.6xl0 15 Ge(As) - 200A 5.3 xlO 15 Ge(P) - 229A (2) 2.6 xlO 16 3 Measurements were made in a recirculating He cryostat capable of covering the temperature range from 0.27 K to above 4.2 K. The thermal conductivity was measured using the absolute method with the temperature difference along the sample determined by two germanium, thermometers. The germanium ther- mometers were calibrated each run against the vapour pressure of He and He from 0.7 to 4.2 K. Below 0.7°K a paramagnetic salt, chromic methylammonium alum, was used for calibration. For each sample forty or more data points were taken between 0.3 K and 4.2°K and about 26 calibration points. The estimated absolute accuracy of X is ±6% while comparison of X values for a given run is within ±4% for T>2.2°K and ±2% for T< 2.2°K with the exception of the data points for sample Ge(As)-200A above 2.2°K which are uncertain to ±10% because the measured AT was too small. 106 5. Results and Discussion Representative data for the pure germanium sample, Ge - 422A, are shown in figure 1. GC1 is the curve calculated from eq (3) but setting t =0. The values for normal germanium found by Callaway, i.e., A= 2.57 x 10" 44 sec. 3 and (B +B„) = ep 2 . 77 x 10" 23 sec deg" 2 were used along with v 1 = 5. 37x 10 5 cm/sec. and v, = v. = 3.28 x 10 cm. /sec. as calculated by Hasegawa [16] from the measured elastic constants. The Debye temperature was taken to be 9 = 375 K. A least mean square fit of the seven experimental points between 0.37°K and 0.56°K to the equation \ = CT n gave C= 0.1314 Watt cm deg" 4 and n= 2.99. Agree- ment between values for \ calculated from this equation and the experimental values was better than 0.5% for each point. This value of C gives L =0.438 cm. as compared to L =0.443 cm. calculated from Casimir's relation, that is, L /L =98.9%. ' exp c Representative data for the Sb doped sample, Ge(Sb) - 172AI, are also shown in figure 1. Two dif- ferent runs were made on this sample, as indicated by the squares and triangles, in order to ensure no effect on the measurements due to strains introduced by the method of mounting the sample. The data points indicated by triangles were taken with the sample attached to the He cryostat and to the sample heater with copper clamps. The thermometers were glued to thin, pure germanium slabs which were soldered to the sample. The squares indicate data points taken with the sample attached to the He cryostat and to the sample heater by soldering to a thin slab of tungsten. The mounting of the thermometers was also changed by gluing them directly to the sample. Comparison of the results between 0.3 K and 1.2 K of the two mounting methods gave agreement to within 1%. In all the calculated curves the values for A, (B.+B„), v.. , v„, and 6 used for GC1 will be used and the value for L will be taken as L= .989 L p in order to be consistent with the results obtained for the pure sample. Also all electrons will be assumed to be in the ground singlet state, i.e., B(T)= 1.0 in eq (2). Curve GC2 in figure 1 was calculated using the Griff in-Carruthers expression for t v.^u E =16.0 e.v., 4A =0.32x10" e.v., and r = 44A. An excellent fit, within 10%, over tne entire tem- perature range is obtained. Since E appears in the expressions for t raised to the fourth power, any uncertainty in its value will be important when comparing theory with experiment. This is shown in curve GC2A which has the same values for all parameters except E is changed from 16.0 e.v. to 19.0 e.v. Unfortunately values of E determined by different experiments [17, 18, 19] cover the range from 15 to 19 e.v. A mean value appears to be around 17 e.v. In figure 2 the data for Ge(Sb) - 172A(2) is reproduced along with curves Kl and K2 which were obtained by using Keyes 1 expression for t . The same values of E , 4A , and r used in calculating GC2 were used to calculate K2. If the true resonance factor R(u>) is replaced by its low frequency li- mit in GC ' s expression (eq 2) the calculated curve lies within a few percent of K2 . Thus the true re- sonance factor is what causes the excellent fit of GC2 to the data above 1°K, suggesting that the me- chanism of resonance fluorescence does occur. Unfortunately the strength of this conclusion is weakened somewhat by considering curve Kl which uses the same values for the parameters as GC2 and K2 except that r is set equal to the effective mass value of 65A. This curve agrees with the data down to about 1.4 K but then deviates to higher values at the lower temperatures. Thus by treating E , which is most ef- fective near the maximum in the resonance, and r as adjustable parameters one could also obtain a reasonable fit to the experimental results using Keyes expression for t • Figure 3 shows representative data for an As doped sample, Ge(As)-200A, and a phosphorous doped sample, Ge(P)-229A(2) . Curve GC3 is calculated with 4A = 4.23xl0" 3 e.v. and r =32. 0A while curve GC4 used 4A = 2.83x 10 e.v. and r =35.3A. The surprising result is that neither As or P doped samples exhibit any resonance scattering at all but only a constant added thermal resistance implying a frequency independent scattering mechanism. Both GC3 and GC4 lie somewhat below the experimental points at the high temperature side but then gradually return to the pure sample limit as the temperature is re- duced due to the oi dependence of t -• ep In view of these results it is of interest to compare the measurements down to 0.2 K reported by Carruthers et al.f50 on indium doped samples (10 , 1.9x10 , and 10 cm" 3 ). The energy level struc- ture for acceptors in germanium is very different from that of donors, consisting of a 4- fold degenerate ground state with the next excited state 6 to 7 milli-e.v. higher. However these samples also gave a de- creased thermal conductivity varying as T 3 but the extra thermal resistance was not well correlated with the amount of impurity added. Klemens' £13} expression for the scattering of phonons by grain boundaries gives a frequency independent relaxation time but when applied to germanium leads t a a mean free path 10 times larger than observed. They therefore surmised that the added impurities may locate preferen- tially near low angle grain boundaries in germanium and give an enhanced grain boundary scattering. Ultrasonic attenuation measurements made by Pomerantz C20} on N-type germanium (Sb, As, and P doped) in the temperature range from 5°K to 50°K were explained by him as due to the electronic relaxation be- tween the triplet states which had been split in energy because of the strain field created by the ultra- sonic wave. Kwok C21J has developed the theory for this process in more detail. 107 The mechanism proposed by Pomerantz, since it is proportional to the number of electrons occupying the triplet states, would predict negligible phonon scattering for As and P doped germanium in the tem- perature range from 0.3°K to 4.2°K. (See table 1 for the temperature equivalent, T , of 4A .) In addition to the different effect Sb donors have on the thermal conductivity of germanium as com- pared with As and P donors, Mathur and Pearlman [3] have found a difference in their behavior on compen- sation. They found that highly compensated P and As doped samples have an extra thermal resistance which , while correlated with the number of neutral donors, shows an increased thermal resistance as compared to the uncompensated material. This is in contrast to the behavior of Sb doped samples which show a similar or slightly smaller value of the extra thermal resistance upon compensation. Further measurements are in progress on samples with different concentrations, both compensated and uncompensated, to see if these two differences in behavior between Sb doped as compared to As or P doped germanium are correlated. References [1] J.F. Goff and N. Pearlman, in Proceedings of the 7th International Conference on Low Tem- perature Physics, edited by G.M. Graham and A.C. Hollis Hullet (University of Toronto Press, Toronto, 1961), p. 284-288. [2] J.F. Goff and N. Pearlman, Phys. Rev. 140 , No. 6A, A2151 (1965). [3] M.P. Mathur and N. Pearlman, to be published. [4] J. A. Carruthers, T.H. Geballe, H.M. Rosenberg, and J.M. Ziman, Proc. Royal Soc. (London) 238A, 502 (1957). [5] J. A. Carruthers, J.F. Cochran, and K. Mendelssohn, Cryogenics, 2, No. 3, 1 (1962). [6] W . Kohn , in Advances in Solid State Physics , edited by F. Seitz and D. Turnbull (Academic Press Inc., New York, 1957), Vol. 5, p. 258-321. [7] J.H. Reuszer and P. Fisher, Phys. Rev. 135 , No. 4A, A1125 (1964). [10] A. Griffin and P. Carruthers, Phys. Rev. 131 , No. 5, 1976 (1963). [11] P. Carruthers, Rev. Mod. Physics 33, 92 (1961). [12] H.B.G. Casimir, Physica 5, 495 (1938). [13] P.G. Klemens, Advances in Solid State Physics , edited by F. Seitz and D. Turnbull (Academic Press Inc., New York, 1958), Vol. 7, p. 1. [14] J. Callaway, Phys. Rev. JL13, 1046 (1959). [15] C. Herring, Phys. Rev. 95, 954 (1954). [16] H. Hasegawa, Phys. Rev. _U8, 1513 (1960). [17] B. Tell and G. Weinreich, Phys. Rev. 143 , No. 2, 584 (1966). [18] R.E. Pontinen and T.M. Sanders, Jr., Phys. Rev. 152, No. 2, 850 (1966). [19] S. Riskaer, Phys. Rev. _15_2, No. 2, 845 (1966). [8] H. Fritzsche, J. Phys. Chem. Solids j>, 69(1958). [20] M. Pomerantz, Proc. IEEE, 53, No. 10, 1438 (1965). [9] R.W. Keyes, Phys. Rev. 222, 1171 (1961). [21] P.C. Kwock, Phys. Rev. Ih9_, No. 2, 666 (1966). 108 1 \— o GE-42ZA a,oGE(SBM72AI Figure 1. Thermal conductivity of pure and Sb doped germanium. .0 2.0 4.0 T(°K) 10 - Figure 2. Thermal conductivity of Sb doped germanium. o T 2I0 cr> i— I. . cI72AI — A/ 7 ' / /A ' / — / / / / J / GClV / / 7 / ' / A / / ' / / / / / f t _ / /tl / - / ki ^d^K2 - _ / •'x-A- _ /a 1 1 1 1 1 1 1 1 u o GE(AS)-200A io 2 - a GE(P)-229A(2) y//f y — r^ A/^ GCI-y^ a/ C GC-3 T 2I0' ^-GC-4 - in t JOy/£ 5g Jo/* o ,~0 /?r " C Te, and a slug 12.5 cm long of stoichiometric Bi2Te*. This procedure gives a melt composition of 63 atomic $ Te which remains approximately constant as the molten zone is moved through the ingot. With a melt of this composition, the solid contains slight excess Te which makes the crystal n-type [9]. Table 1. Description of the samples used in this investigation. The compositions given are those of the starting material. For all samples, K was measured with the heat flow parallel to the cleavage plane. Sample Number Code Number R(77°K) 3„-l cm C Extrinsic Carrier Concentration ~ -3 p(77°K) x lo 6 a(100°K) ohm-m nvA Preparation and Composition 1 6jEk-l 2 62AW3-1 3 67F-3 +0.24 2.6 x 10' .19 -0.79 7.9 x 10 18 +0.192 3.3 x 10 19 'JJ-13 -0.30 2.1 x 10 19 I.56 1.1+8 1.92 + 84 9h 3.6 Single zone, BigTe^ melt, sample Bi rich, single crystal, p-type Single zone, Bi2Te^ + excess Te, single crystal, n-type + 81 Single zone, BigTe* + 2 mole $ BigSe^, nearly single crystal, no grains revealed by etching, cleavage not perfect. -115 Bridgman, BigTe^ + 10 mole # BiaSe^ + I dop- ing, etching revealed 6 nearly single crys- tal grains 5 B-23-B4 ■N-47 N-VfZ -1.46 4.3 x 10 18 3.01 3.42 11.1 -118 -124 -241 Bridgman, Bi2Tej + 10 mole $ BigSe^ + I dop- ing, many small crystallites Bridgman, Bi2Te5 + 10 mole $ Bi 2 Sej + I dop- ing, many small crystallites Single zone pass on ingot from which sam- ple 6 was cut, single crystal 113 Samples were cut from the ingots with a spark cutter. Some of the characteristics of the samples are listed in Table 1. The samples were cut from the ingots and then etched in either a 1:1 HClrHNOj etch or a Br etch consisting of a few drops of Br in ~50 cc of methanol. These etches are effective in revealing grain boundaries. They do not reveal small distortions of the crystal within the grains. For example, the etching showed Sample k to be composed of four large and two small grains. Upon completion of the thermal conductivity measurements, one grain was cut from the sample and attempts were made to cleave a bar (~1 x 3 x 12 ram) suitable for measurement of the Hall coefficient from this grain. This cleaving showed that the c axis was not in the same direction throughout the grain. However, the c axis did not deviate by more than 5° from the normal to the long dimension of the crystal. This was found to be true for all the samples. For the Bridgraan ingots, we were successful in obtaining a good single crystal Hall bar for Sample k only. The single zone ingots presented no such problems. These samples were prepared with cleavage planes as two faces of the sample. The other four planes completing the right rectangular parallelepiped were cut with a spark cutter. 3. Experimental Results Figures 1, 2 and 3 show the measured values of K, a and p plotted against temperature, T. Where comparisons can be made, these results are in good agreement with previously published results. The procedure [17] used for separating the measured thermal conductivity into an electronic thermal conductivity, Ke> a nd a lattice thermal conductivity, K^, is well known and only a few brief remarks are required here. One assumes that (1) the scattering of the carriers can be described by a relaxa- tion time that is proportional to the carrier energy raised to some power, i.e., t = TqE 3 , where E is the carrier energy, and (2) the bands are of standard form, i.e., the density of states varies as EV 2 . One then uses the measured values of a to calculate values of the Fermi energy £. The electronic thermal conductivity is then given by Ke = LT/p where the Lorenz number, L, is a known function of £ and s. This procedure is probably reasonably good so long as there is electrical conduction in one band only. At high temperatures there is conduction in both the valence and conduction bands and the electronic thermal conductivity is enhanced by ambipolar diffusion. The calculation of this contribu- tion requires knowledge of several parameters whose values are not well known for the alloys. The results shown in figure 1 clearly show ambipolar contributions to the thermal conductivities of Sample k at 38o°K and Sample 7 at 300°K and suggest that there may be a small contribution in Sample 2 at 300°K. Accordingly, we have restricted our analysis to results at temperatures below 2lK)°K. At 2 i K)°K the value of £ calculated from a is greater than -0.1 kT for all our samples with the exception of Sample 7. As usual, (; is measured in hole energy from the valence band edge for the p-type specimens and in electron energy from the conduction band edge for the n-type specimens. Since the energy gaps for these samples are about 8 kT at this temperature, it seems safe to ignore two-band conduction below 2U0°K. As shown in figure 3, the electrical resistivity of all the samples increases at least as fast as T between 8o°K and 2 i K) K. This surely indicates that phonon scattering predominates. The simplest theory of electron-phonon scattering in semiconductors would therefore suggest s = -1/2. This value, however, does not provide a good account of the temperature dependence of the mobility of the carriers in Bi2Te5, particularly for p-type material. Moreover, for the alloys the resistivity is less sensi- tive to temperature than for the Bi 2 Te* samples. This suggests that carrier scattering by the solute atoms is not negligible in the alloys. The relaxation time for combined impurity and phonon scattering is not proportional to E s , but must be represented by a more complicated function of E. For small or moderate charge carrier concentrations, t is a decreasing function of E for phonon scattering, an increasing function of E for charged impurity scattering, and independent of E for neutral impurity scattering. Qualitatively then, one can study the effect of impurity scattering by calculating Ke using t = TqE 8 with values of s in the range -1/2 to 3/2. It will be appreciated that at present we cannot calculate very accurate values of Kg. Figure k shows the temperature dependence of W L = 1/Kl = l/fK-Kg) calculated from the measured values of K and the calculated values of Ke* With the exception of Sample k, we have used s = -1/2. For Sample k two results are shown, one for s = -1/2 and one for s = +1/2. Within experimental error and considering the uncertainty in the values of Ke, the experimental results may be represented by W = PT + r* If Ke is calculated using s » -1/2, then p has the following values: Samples 1 and 2 (Bi2Te3) p = 2.0 x 10" 3; Sample 3 (Bi2Tej + 2& Big^) B = 2.2 x 10" 5; Samples h, 5, 6 (Bi2Te3 + 10# Bi2Se3 + I) p = 2.7 x 10" 3; Sample 7 = 2.6 x 10-3 mW - - 1 -. We observe that this difference cannot be attributed to improper values of Ke» If there is any significant difference between the energy depen- dence of the t's for the Bi2Tej samples and for the alloys containing 10$ Bi2Se:j, then the t for the latter samples should be a less rapidly decreasing function of E than the t for the Bi2Tej samples. As shown by the results with s = +1/2, this difference would result in an even larger difference between the values of p. k. Discussion For the case of phonon scattering by three phonon U-processes combined with scattering by a random distribution of point defects, Klemens [l] obtained the formula 114 O -1 f T) K, = K -H tan" x f -£ ) (2) for the lattice thermal conductivity. In this expression up is the Debye frequency, w is the phonon frequency at which the scattering rates for the two processes are equal and K u is the lattice thermal conductivity of the defect-free crystal. If we have K u => KqT -1 with Kq independent of T, then u> is proportional to T-w 2 N"-V 2 where N is the concentration of defects as shown by KLemens [l]. Figure 5 shows a plot of Wl = l/Kj, calculated from this formula for various values of N. Both T, Wl and N are in arbitrary units. The lowest line W = W u = T/Kq shows the thermal resistivity of the defect- free material. For small defect concentrations, i.e., w » up, eq (2) may be written W^ = W u + V.' with W = (W u /3)(wp/w ) 2 which is independent of T and proportional to N. This behavior is shown by the curve labeled N = 1. As shown by the curves labeled N = U and N = 9, Wl cannot be approximated by eq (1) with W u independent of N for larger values of N. However, this figure clearly shows that over a small range of temperatures the curvature in a plot of measured values of Wl vs. T would not be observed. One would represent all the results between say T = 10 and T = 50 by equations of the form Wl = PT + v with both p and y increasing with N. This, we suggest, is the explanation of our experi- mental results shown in figure k. Moreover, the results shown in figure k do not exactly fit the straight lines. Because of experimental error and the uncertainty in Kg, we do not claim that the curvature indicated by our results is well established but it certainly is of the form predicted by eq (2). An attempt to give a quantitative account of the experimental results is beset with many diffi- culties. The uncertainty in the values of Kg has already been described. The use of K u = KqT"^- is certainly not completely correct. Ioffe has pointed out that the equation W u = (l/Ko)(T - 0/3) might better represent the high temperature (T £ 0) lattice thermal resistivity of a defect-free material. Goldsmid [17J has reported that neither formula is perfectly satisfactory for Bi2Te5 # The value of Kq calculated from the results shown in figure h for Samples 1 and 2 is 5«0 x 10 2 Wm"l, in good agreement with the value reported by Satherthwaite and Ure [9] but about lk*f> larger than the value found by Walker [l8]. The data for Samples 1 and 2 do not answer the question of how to represent W u as a function of T since these are not completely defect-free samples. No foreign atoms have been added to these samples but they contain defects, such as vacancies, interstitials, atoms on wrong sites, and dislocations, which are effective phonon scatterers. The value of K u should vary with Se concentration. However, if it is permissible to use a virtual crystal model, this is a very small effect, since the lattice thermal conductivity of Bi2Se^ is only about 1.3 times that of Bi2Te^ [10]. The compositions given in Table 1 are those of the starting material and are not necessarily those of the samples measured. Samples h, 5 and 6 contain I added to maintain a high carrier concentration. A quantitative account of the measured values of Wl would necessitate the separation of the scattering due to Se from that due to I since iodine is an effective phonon- scatterer in BigTej. The addition of I to the 10 mole $ Bi2Se5 samples was necessary for the following reason. Bi2Te* crystallized from a stoichiometric melt is p-type (Bi rich). As stoichiometric amounts of Bi2Se* are added, the extrinsic hole concentration decreases until, for a composition of ~20 mole $ Bi2Se*, the materials become n-type [19]. If the iodine had not been added to the samples containing ~10 mole $ Bi2Se3, then their extrinsic hole concentrations would have been small and the uncertainty in Kg due to ambipolar diffu- sion would have been encountered at lower temperatures. This is seen in the results for Sample J. The values of R, p and a given in Table 1 show that a large fraction of the iodine was lost in the process of growing Sample 7 from the material from which Sample 6 was cut. The value of W^ for Sample 7 at 2'K) K (figure h) is almost certainly incorrect because an ambipolar contribution is an important part of Kg at this temperature. At lower temperatures the temperature dependence of W^ for Sample 7 is not qualitatively different from that for Samples h, 5 and 6 (we know that at least part of this reduction in defect concentration is due to the loss of iodine). We can then conclude that there is no difference between the lattice thermal conductivities of the Bridgman and single-zone grown alloys for T > 8o° and Se concentrations below 10 mole $ other than that caused by different point defect concentrations . Despite these uncertainties, it nevertheless seems worthwhile to try to fit the KLemens formula to the data. For Ge-Si alloys, it is known that this formula does not give a good account of the tempera- ture dependence of K^. As shown by Parrot [2] and Abeles [3], the theory can be brought into reason- able agreement with experimental results only when the effect of N-processes are also included. Therefore, it is of interest to see if the formula can do somewhat more than predict the qualitative changes in the temperature dependence of the thermal resistivity of the alloys. By taking Wo = ( w u/3)(wd/w ) , eq (2) becomes "W - . _ /3W u . -1 w o K W «± tan"" 1 " \J—2 . (3) u V 3W » W K " 3W o u We assume that W u = 2 x 10"5 T °K mW -1 is the same for all specimens, ignoring the small change in W u 115 with the addition of Bi 2 Se* predicted by the virtual crystal model. With the choice of W = 0.18 and 0.59 ra °K W"-*- for Samples 3 and k respectively, figure 6 shows that the formula gives an excellent account of the experimental results. At first thought, the ratio 59/18 ~ 3«3 does not seem large enough. Sample h should contain roughly 5 times as much Se as Sample 3> as well as some iodine that is absent from Sample 3« However, we do know that there must also be a difference in the vacancy /inter- stitial concentrations between the two samples and until the effect of this difference is understood, a final judgement must be reserved. This fit of the Klemens formula suggests that the N-processes may be of relatively less importance to the high temperature thermal conductivity of BigTe^ alloys than to the Ge-Si alloys. This would seem to be a reasonable result. It is known [20] that the Leibfried- Schloemann formula overestimates the lattice thermal conductivity of both elements and compounds. However, the discrepancy is larger for compounds than for elements (in the case of Ge, Si and I^Te^, the Leibfried-Schloemann formula overestimates the lattice thermal conductivity by factors of 2.1, 1.5 and 5.1 respectively). Klemens suggested [20 ] that this is the result of additional U-processes arising from the extra zone boundaries in the compounds. If this is correct, then we might expect that the ratio of the strengths of U to N processes would be larger in the Bi 2 Te-z alloys than in the Ge-Si alloys. However, N-processes are most effective when the point defect scattering is dominant and this is not the case even for Samples 4, 5 and 6 at 8o°K. For Sample 4 at 8o°K, wd/w ~ 3«3 whereas Ge-Si alloys have been measured for which wp/w ~ 13« It must, therefore, be recognized that we have not put the Klemens formula to an exacting test. In figure 4 we have included a line labeled "Lenningrad" . This represents the results found by the group at the Lenningrad Semiconductor Institute [12-15 ] for a sample of 97$ BigTej + yf> Sb2S3. Moreover, they find dWL/dT to be independent of defect concentrations for samples containing as much as 9 mole $ St^Sj so that their results can all be represented by W = W u + W Q even when W > W u . We do not understand their results but it seems quite clear that point defect scattering alone cannot account for the changes in K^ that they have produced by alloying Bi2Te3 with Eft^Sj. They also claim the same type of results for Bi2Tej - Bi2Se^ alloys for samples containing as much as 8$ Bi2Sej. Our results clearly disagree with theirs. It does not seem possible to understand their results in terms of the current theory of point defect scattering. In summary, the lattice thermal conductivity of Bi2Te-* alloys containing as much as 10 mole $ Bi2Se3 is determined only by the chemical composition of the sample. The experimental results are in excellent qualitative agreement with the theory that treats the phonons as being scattered both by three phonon processes and by a random array of point defects. Quantitative comparison of theory and experiment must await the solution of a number of problems. However, our preliminary attempts to fit the theory to the experimental results suggests that good quantitative agreement between theory and experiment will be found. 5. Acknowledgment The authors thank P. Piotrowski and D. Zupon for assistance with the sample preparation and the measurements . 6. References [1] P.G. Klemens, Phys. Rev. 119, 507 (I960). [8] H.J. Goldsmid, Proc. Phys. Soc. B69, 203 (1956). [2] J.E. Parrot, Proc. Phys. Soc. 8l, 726 (1963). [3] B. Abeles, Phys. Rev. 131 , 1906 (1963). [4] For example see, Thermoelectricity; Science [lo] H.J. Goldsmid in Materials Used in Semi [9] C.B. Satherthwaite and R.W. Ure, Jr., Phys. [3] B. Abeles, Phys. Rev. 131, 1906 (1963). Rev. 108, 1164 (1957). and Engineering , eds. R.R. Heikes and R.W. conductor Devices , ed. C.A. Hogarth Ure, Jr. (Interscience. New York, 1961). (Interscience Publishers, New York, I965), p. I65ff. [5] A.F. Ioffe, S.V. Airapetiants, A.V. Ioffe, N.V. Kolomoets and L.S. Stilbans, Doklady [ll] See Ref. 10 for a discussion of these Akad. Nauk SSSR I06 , 981 (1956); English results and references to the original work. translation: Russian Literature Survey: SEM-l-56 (infosearch, London, 1956). For a [12 ] E.A. Gurieva, V.A. Kutasov and I.A. Smirnov, discussion of this argument, see Ref. 4, Sov. Phys. - Solid State 6, 1945 (1965). p. 383. [13] V.A. Kutasov, B.Y. Moizhes and I.A. Smirnov, [6] H.J. Goldsmid, Thermoelectric Refrigeration Sov. Phys. - Solid State 7, 81*5 (1965). (Plenum Press, New York, 1964), p. 130. [14] E. A. Gurieva, A.I. Zaslavski, V.A. Kutasov [7J R.W. Ure, Jr., Proc. of the IEEE/AIAA Thermo- and I.A. Smirnov, Sov. Phys. - Solid State electric Specialists Conf., Washington, D.C., 7, 978 (I965). May 17, 1966 (Inst, of Electrical & Electronic Engineers, New York, 1966), Paper No. 11; Adv. Energy Conversion, to be published. 116 [151 V.A. Kutasov and I.A. Smirnov, Sov. Phys. - tl8] P.A. Walker, Proc. Phys. Soc. 76, 113 Solid State 8, 2153 (1967). (i960). [16] D.H. Damon, J. Appl. Phys. 37, 3l8l (1966). [19 ] U. Birkholtz, Z. Naturforsch 13a, 780 (1958). [17] H.J. Goldsmid, Proc. Phys. Soc. 72, 17 (1958). [20] P.G. KLemens in Solid State Physics , eds. F. Seitz and D. Turnbull (Academic Press, New York, 1958), Vol. 7, p. 46ff. 117 Fig. 1 Measured values of the thermal conductivity, K, plotted against absolute temperature, T. 118 a -40 Fig. 2 Measured values of the Seebeck coefficient, a, plotted against absolute temperature, T. 100 200 Temperature, °K Fig. 5 Measured values of the electrical resistivity, p, plotted against absolute temperature, T. E 8- 100 Temperature, °K 119 o CM O 120 160 T, °K 280 Fig» k Values of the lattice thermal resistivity, W T , plotted against absolute temperature, T. W T is calculated from W = 1/lC. and K_ is calculated from K_ = K-K where K is the electronic thermal conductivity computed from the measured values of p and a Lorenz number deduced from the measured values of a. All values of the Lorenz number were calculated assuming s = -1/2 with the exception of the curve labeled s = +1/2 for Sample k (see text). The line labeled "Lenningrad" shows the results given in reference [13 ] for a sample of 97$ Bi 2 Te 5 + % Sb^. 120 CO *C 10 20 30 40 T, arbitrary units 50 60 Fig. 5 The variation of the lattice thermal resistivity, W , with temperature, T, and defect concentration, n, predicted by the KLeraens formula (eq 2). W T , T and n are in arbitrary units. The lowest curve shows the variation of W , the lattice thermal resistivity of the defect-free crystal; the upper curves show the effects of adding point defects. 121 E T, °K Fig. 6 The fit of the KLemens formula (eq 3) to the measured values of W for Samples 3 and k. The points are the same as those shown in figure k. The curves have been calculated from equation 3 with the assumption W = 0.002 T °K m W - for pure BipTe.,. 122 Thermal Conductivity of Cadmium Arsenide from 60 to 1+C0°K D. P. Spitzer American Cyanamid Company Stamford, Connecticut 0690U Measurements of the thermal conductivity of Cd 3 As 2 have been extended to lower temperatures than previously reported [l] . It now appears, contrary to an earlier interpretation, that the lattice thermal conductivity of either undoped, Cu-doped, or Se-doped Cd 3 As 2 is l.k t 0.2 W m -1 deg"" 1 ( Ik ± 2 mW cm -1 deg -1 ) over the whole stated range of temperature, and total thermal conductivities are anomalously low because of electronic components that are much less than given by the Wiedemann- Franz law. Similar results have been reported for Ag 2 Se, HgSe, and PbTe at low temperatures, and may be explained by assuming inelastic scattering mechanisms, probably of electrons by electrons. Inelastic scattering processes probably occur in many semiconductors, but the effects are clearly noticeable only in materials with high carrier mobility and low lattice thermal conductivity. Key Words: Cadmium arsenide, electron-electron scattering, electronic thermal conductivity, inelastic electron scattering, lattice thermal conductivity, Lorenz number, thermal conductivity, thermal resistivity. 1. Introduction An earlier publication from our laboratories presented thermal and electrical conductivities for over thirty variously -doped samples of Cd 3 As 2 at room temperature and for three representative samples from 2U0 to ^00°K [lj. At that time, low thermal conductivities of undoped samples (and some doped ones) were ascribed to low lattice components, due to unusually effective phonon scattering by a lattice defect (As vacancy). Cd 3 As 2 doped with Se, which increased the charge carrier concentration but decreased carrier mobility, had normal thermal conductivities in accordance with the Wiedemann-Franz law. It appeared that more information on the cause of the low thermal conductivities might be obtained from thermal conductivity measurements at lower temperatures, on both undoped and Se-doped Cd 3 As 2 . The small differences in lattice thermal conductivity found in [l] should be much greater at lower tempera- tures if the postulated explanation were correct (namely, the Se was assumed to fill up anion vacancies thought to be normally present in undoped Cd 3 As 2 so that the usual l/T law is found for lattice thermal conductivity instead of the temperature independent conductivity that results from impurity scattering) . 2. Experimental The samples used were polycrystalline pieces, a few millimeters on a side, cut from ingots made by fusion of the elements in vacuum, as described previously [l]. In addition to the samples described in [l], we attempted to make CdsAs 2 with a smaller number of carriers by zone-refining, as reported by Ugai and Zyubina [2]. CdsAs 2 was completely sealed in a graphite tube, except for a small hole through which it could be evacuated, and this in turn was sealed in a Vycor tube. A molten zone about two centimeters wide was passed through the 20 cm long sample at the rate of 2.5 cm/hr. After forty such passes, no significant variation in carrier concentration along the length of the sample was found. Swiggard [3] recently reported that zohe-refining of Cd 3 As 2 under conditions where vaporization was eliminated reduced the number of carriers by only a factor of two. Appreciable sublimation had taken place in our case, and this might have evened out otherwise small differences in concentration. Measurements of thermal conductivities at room temperature and below were made in an apparatus designed to fit into a helium dewar with only a 1.27 cm (0.5 inch) diameter sample chamber. The arrange- ment used is illustrated in figure 1. The steady heat applied at the bottom end of the sample was cal- Figures in brackets indicate the literature references at the end of this paper. 123 culated directly from the current and voltage across the resistor which was used as a heater. A correc- tion was made for radiation losses from the heater block surface (this surface was intentionally black- ened so that its approximate emissivity would be known). This correction ranged from about 10$ at room temperature to less than 0.5$ at 100°K. Another correction was made for conduction through the leads connected to the heat source and through the thermocouple wires; this was 2-5$ of the total heat and essentially independent of temperature. The thermal conductivity of the sample is simply the corrected heat input divided by the resultant temperature difference across the sample, and multiplied by the ratio of the sample's length to its cross-sectional area. Measurements on samples of Inconel 702 of known thermal conductivity gave results that were correct within t 5$, at least down to about 8o°K. The measurements on the Cd 3 As 2 samples, however, were gener- ally reproducible only to within 10$. At lower temperatures, the temperature of the heat sink drifted too much to permit accurate measurements of the temperature difference across the sample (the heat capacity of the copper sink decreases by more than a factor of ten between 8o and 20°K) . It is likely that this difficulty could be corrected by using metal instead of polytetrafluoroethylene rings around the sink. Electrical conductivity and Seebeck coefficient were measured on the same set-up, after moving the position of only one lead (from the end of the heater resistor to the copper block around it). 3. Results Total and lattice thermal conductivities for one undoped, one Cu-doped, and two Se-doped samples of Cd 3 As 2 are given in figures 2 and 3. For the undoped and Cu-doped samples, the two lattice conductivi- ties given result from two different estimates of the electronic thermal conductivity, as described below. The lattice conductivity is simply the difference between the measured total thermal conducti- vity and the calculated electronic thermal conductivity. Electrical conductivities for the four samples are given in figure h. Seebeck coefficients varied smoothly from about 15 (iV/°K at 6o°K to 70-80 uV/°K at U00°K for undoped or Cu-doped Cd 3 As 2 . At these same temperatures, the Seebeck coefficients for the Se-doped samples were about 10 and Uo liV/ c K, respectively. Measurements which were obtained down to about 20°K on the present samples did indicate that total thermal conductivities continued to decrease with decreasing temperature, as expected from X « LOT and the decreasing lattice heat capacity term. The scatter in these data became quite bad at these lower temperatures, however, and such data are not included here. More accurate measurements at lower tempera- tures are not expected to affect the interpretation of the results to be presented. From [l] it is known that there is very little change in total or lattice thermal conductivity between 300 and lK)0°K, for all Cd 3 As 2 samples. (Unfortunately, the measurements reported here were not made on the identical samples used in [l]; the Cu-doped sample was from the same ingot, however, and the other samples were from different ingots, but prepared under identical conditions.) In figure 2 of [l], Xl for a Se-doped sample is shown to be increasing with decreasing temperature, but it is now thought that a greater uncertainty should be admitted for this X T , because of the large electronic component. It is significant that the points for this Xy fall in the range l.k ± O.k W m -1 deg -1 , where the mean value is the same as that ->und here. Thermal conductivity of Cd 3 As 2 in the temperature range 80-380°K has also been recently reported by Giraudier [!»■]. The thermal and electrical conductivities of his sample are about twice those of our undoped sample. Giraudier 1 s thermal conductivities seem generally too high, especially above 300°K, where the thermal conductivity is shown to be increasing considerably with temperature. The lattice conductivity can hardly be increasing, and the electronic conductivity should be nearly independent of temperature. Below 300°K, good agreement with our X_ of l.k W m _1 deg -1 is obtained only if an L as high as 3.l(k/e) 2 is assumed (k is Boltzmann's constant and e is the electronic charge). This is reasonable only if the carrier concentration is much higher than for our samples, but this is not known. A description of the calculation of the electronic thermal conductivity has been given in the earlier publication [l]. This is calculated as usual from the electrical conductivity and the Lorenz constant. L is determined from a knowledge of the carrier scattering mechanism, as given by temperature dependence of mobility, and from the position of the Fermi level, calculated from the Seebeck coeffi- cient. According to our calculations and those of others [5], even undoped Cd 3 As 2 is quite degenerate, with a Fermi level about 0.12 eV (~ 5kT at 300°K) above the bottom of the conduction band and not much dependent on temperature. Consequently, L calculated as above for undoped Cd 3 As 2 is near the Lq = (fi /3)(k/e) value that applies to a completely degenerate conductor, namely, L~ 3-0(k/e) at 300°K [l]. Similarly, L ~ 2.9(k/e) 2 was found for the Cu-doped Cd 3 As 2 and L ~ 3.1(k/e) 2 for Se-doped Cd 3 As 2 . Using these values for L gives the X't curves for samples 1 and 2 (fig. 2) and the X L values for samples 3 &n& k (fig. 3). The X L curves for samples 1 and 2 result from taking L = 2.0(k/e)2 instead. It is thought to be just coincidental that this is the L for the completely non-degenerate case. 124 h. Discussion There is now considerably more evidence for anomalously low electronic thermal conductivities than for low lattice thermal conductivities: ( l) The anomaly decreases as electrical conductivity decreases. (2) It is now found the X L is independent of temperature for Se-doped as well as undoped Cd 3 As 2 . This means that our explanation as to why Se-doping apparently increased lattice thermal conductivity (i.e., Se atoms were thought to fill up As vacancies) has no support. (3) The X-r for all of our samples are identical over the entire temperature range if we assume that the Lorenz number for undoped ( or Cu-doped) Cd 3 As 2 is a low 2(k/e) 2 . (k) X L = 1.2 W m -1 deg -1 at 8o°K has been found by Goldsmid [6] from measure- ments utilizing a magnetic field to directly eliminate the electronic thermal conductivity. This is in good agreement with our value of l.k t 0.2 W m -1 deg -1 found when a low electronic thermal conductivity is assumed. Such measurements using magnetic fields will ultimately provide the most direct separation of lattice and electronic thermal conductivity. As more data become available on thermal conductivities of materials that have comparable lattice and electronic thermal conductivities, it appears that adherence to the Wiedemann-Franz law, with Lorenz numbers calculated as indicated above, is the exception rather than the rule. There are presently only a handful of materials for which measurements of thermal conductivity have been made on samples of vary- ing electrical conductivity. Besides Cd 3 As 2 , this list includes Bi 2 Te 3 [7], AfeTe [8], Ag 2 Se [9, 10] , SnTe [ll], InAs [l2], HgSe [13], and PbTe [li, 15, 16] (some such data are also available for Bi 2 Se 3 [17] but are too limited to permit definite conclusions regarding scattering mechanism). Most of these materials clearly show considerable deviations from the electronic thermal conductivity when calculated as previously mentioned. Perhaps Bi 2 Te 3 is the only compound on the above list that closely obeys the Wiedemann-Franz law. Low values for total thermal conductivity have been found for some samples of Bi 2 Te 3 [7] , but this has quite conclusively been shown to be due to a low lattice component ( caused by a large phonon scattering cross-section for the halogen ion dopants used) . Available data for Ag 2 Te [8] do not extend to very high electrical conductivities and could be fit equally well with a lower Lorenz number if it is assumed that the samples with the lowest electrical conductivities have a lattice component that is about 0.2 W m -1 deg -1 less than for the other samples (this seems reasonable because some of these samples are stated to be defective in silver and the low electrical conductivity of the others suggests that they are more than usually disordered). SnTe is different from the other materials in the list above in that it has a rather low carrier mobility and is not a good thermoelectric material. The carrier concentrations in the SnTe samples [ll] were at least a factor of ten greater than for the other materials in the list. Here the Lorenz numbers were found to be generally greater than Lq. It was suggested that there was a connection between the behavior of the Lorenz number and a band structure involving two valence bands, but the physical pro- cesses involved could not be identified. The anomaly in the electronic thermal conductivity of SnTe is almost certainly quite different from that in Cd 3 As 2 in any event. Cd 3 As 2 , PbTe, Ag 2 Se, HgSe, and InAs all definitely exhibit anomalously low Lorenz numbers, at least over some range of temperature and carrier concentration. It seems likely that some type of inelastic electron scattering is responsible for the low Lorenz numbers in all of these materials. In some cases the inelastic electron scattering may be caused by optical phonons and in other cases may be caused by other electrons. At room temperature, Lorenz numbers are markedly low only for Cd 3 As 2 and Ag 2 Se. No explanation has been given for Ag 2 Se, but its behavior is generally more similar to Cd 3 As 2 than to any of the other materials mentioned. InAs has such a high lattice thermal conductivity that accurate experimental determination of the electronic component has not been possible. Thermal conductivity measurements on highly doped InAs have been reported [12] only for the temperature range 300-700°K, and even at these temperatures the lattice conductivity is fairly high and the electronic contribution is rather uncertain. The electronic com- ponent is definitely lower than expected and can be at least partly accounted for by scattering of electrons by optical phonons; a reduction in L to 0.6oL was calculated for scattering by optical pho- nons. The most unequivocal separation of electronic and lattice thermal conductivities is obtained by measurement of thermal conductivity with and without a strong magnetic field. The latter measurement gives the lattice conductivity directly, and the difference of the two measurements is the electronic component. Such measurements have been made for PbTe and HgSe at low temperatures (below 300° K) . The HgSe samples [13] were all very degenerate, with the Fermi level being 7-30 kT above the bottom of the conduction band. The Lorenz number was found to be only about 70$ and h5 of Lq at 95 and 205°K, respectively. Since there was also previous evidence that optical phonons were involved in the scattering processes, it was suggested that the low Lorenz numbers were associated with the inelastic nature of the electron scattering by the optical phonons (as was done for InAs above). Ho quantitative analysis was attempted. It is generally agreed that scattering of electrons in Cd 3 As 2 is predominantly by acoustic phonons [l8, 19, 5], and it is unlikely that the explanation given for HgSe applies to Cd 3 Aa 2 . 125 Measurements on FbTe [l5» l6] were made on somewhat less degenerate samples, with Fermi levels about 2-12 kT above the bottom of the conduction band. Low Lorenz numbers were found below room tempera- ture, with a minimum of about 0.6L in the vicinity of 100°K, for samples with carrier concentration in the range 10 24 to 1£> 25 m -3 ( 10 18 to lO 1 * 1 cm -3 ). The low Lorenz numbers were explained on the basis of electron-electron collisions, which had also been satisfactorily used to explain thermoelectric power data for FbTe [20]. Equation ( l) is given by Moizhes and Ravich [20] for the thermal resistivity due to electron- electron scattering: W ee b ^ 4 e 3 (kT) g r s unN = 273 4 I * ' Wo 56^ * v F kj, Here W is the thermal resistivity associated with elastic scattering mechanisms, as given by the Wiedemann-Franz law, k™ is the average crystal momentum at the Fermi level, kj. = (3n 2 n/N) l/3 (2) vp is the velocity corresponding to this crystal momentum, N is the number of energy ellipsoids, r is the screening radius, r s - (ejkve z p) 1 / 2 (3) p is the density of states, u is the electron mobility, and £„ is the high frequency dielectric con- stant. Using eq ( 1) , Moizhes and Ravich find W /W = 1.0 at 77°K, in excellent agreement with the value of 0.7 required to explain their experimental data. In repeating this calculation, we obtain a value of 9-3 instead of 1.0 for W ee /w . Moizhes and Ravich, however, did not explicitly state what value they used for the effective mass (which enters into v™) , and it may be that they used something other than the 0.12 m that we used. One cannot really expect such good agreement as is claimed, any- way, and it is significant enough that the calculation gives the right order of magnitude. The same calculation may readily be made for Cd 3 As 2 . Using values of n = 2-10 m , p = 1.7" 10 25 (eV) -1 m -3 , e m = l6.3 [21], u = 3.0 m 2 V -1 sec" 1 , m = 0^05 mo, eq ( l) gives W ee /w = 2.2 at 77°K, which is of the same order of magnitude as for FbTe. The form of eq (1) also agrees with the temperature dependence of the electron-electron resistance in FbTe below about 100°K, i.e., W ge decreases with decreasing temperature. Our data on Cd 3 As 2 do not go to low enough temperatures to make a comparison. For FbTe above 100°K, however, the experimental W ee decreases with increasing temperature, contrary to eq ( l) . This is ascribed in [l6] to the sample becoming non-degenerate, but this seems doubtful. For Cd 3 As 2 , the extra thermal resistivity is practi- cally independent of temperature. If the vy and kjp are expressed in terms of n in eq ( l) , it is seen that W should decrease with increasing n (W ee does not, on the contrary, increase indefinitely as n decreases, since n must always be large enough for degenerate statistics to apply). This dependence on n was actually observed for PbTe, and this agrees qualitatively with our results on Cd 3 As 2 . Namely, the Se-doped Cd 3 As 2 samples with about six times the number of carriers as the undoped samples do not show the extra thermal resistivity that the undoped samples do. (However, the fact that the mobility dependence on temperature is less for Se-doped Cd 3 As 2 (u«T-i/2) than for undoped Cd 3 As 2 (u a T _1 ) suggests that impurity scat- tering is greater in the Se-doped samples, and this may also lead to a higher Lorenz number.) As pointed out in [2l], an equation similar to ( l) is given by Ziman [22], for W only: W ee = ( 9 A0)(TT 2 )(e 4 T/v F E F 3 )(R/q) (k) where Ej, is the Fermi energy, R is the same as kj. with N = 1, and q is the same as l/r s with €„, = 1. Using the same numbers for Cd 3 As 2 as above, eq (£) yields W = 5 m deg W _1 , approximately, at room temperature. The extra thermal resistance found experimentally is about 0.3 m deg W -1 over our whole temperature range. Ziman recognizes that eq (k) overestimates the electron-electron thermal resistance. 126 This may be due in part to the neglect of the dielectric constant of the material; a high dielectric con- stant would be expected to decrease the strength of the interaction between electrons and result in a lower W . It is most significant, however, that the W ee calculated for Cd3As 2 is a factor of 10 4 larger than that quoted by Ziman for a typical metal ( sodium) . This is seen to be primarily connected with the much lower Fermi energy of the semiconductor. According to Ziman [22], the mean free path for electron- electron collisions is proportional to (E-/kT) 2 , i.e., collisions are much more likely in the material with the lower Fermi energy. 5. Conclusions It appears that inelastic scattering processes occur in many semiconductors, but the effects are clearly noticeable only in materials with high carrier mobility and low lattice thermal conductivity. Inelastic scattering of electrons may be caused by either optical phonons or by other electrons. Scattering by optical phonons has been invoked to explain the electronic thermal conductivity of HgSe and InAs, while electron-electron scattering qualitatively explains the behavior of the electronic thermal conductivity in PbTe below 100°K and gives at least the right order of magnitude for the extra thermal resistance in Cd3As 2 . The band structure of Cd 3 As 2 is thought not to be simple, and it has been proposed [l8, 25] that there are two conduction bands with different effective masses. Such details may be important for the electronic thermal conductivity. It is hoped that the approximations presently available for calculating inelastic scattering processes can be improved and that more direct measure- ments of electronic thermal resistance in semiconductors will be made. If one expresses all parameters in eq ( l) in terms of effective mass and carrier concentration (taking u proportional to T" 3 / 2 m* -5 / and p proportional to m n 1 / 3 ), then W ee /Wo«(TV= m *)/(c 3/ V /e ) (5) The ratio m /e varies from one semiconductor to another by, at most, a factor of about ten; for thermoelectric materials only, the variation is just a factor of two. Hence W ee may be significant in any semiconductor that is degenerate at a relatively low carrier concentration, i.e., one that has a low effective mass. The lattice thermal conductivity of Cd 3 As 2 has been rather ignored here. In a way, the interpreta- tion of our data is straightforward since the temperature independence of the lattice thermal conductivi- ty clearly indicates that the phonon scattering is predominantly by impurities, both for undoped and doped Cd 3 As 2 . The impurities must be lattice defects, and are still presumed to be arsenic vacancies; there is still no direct evidence on this point, however. 6. Acknowledgements We wish to thank Mr. S. B. Hladky for his part in development of the experimental apparatus and Mr. D. L. Wiegand for preparation of the samples and assistance with measurements. 7. References [l] Spitzer, D. P., Castellion, G..A., and [9] Conn, J. B. and Taylor, R. C, J. Electrochem. Haacke, G., J. Appl. Phys. 37, 3795 (1966). Soc. _307, 977 (i960). [2] Ugai, Ya. A. and Zyubina, T. A., Inorganic [lo] Busch, G., Hilti, B., and Steigmeier, E. , Materials 1, 790 (1965). Z. Naturforsch. l6a, 627 (1961). [3] Swiggard, E., J. Electrochem. Soc. 11^, 976 [ll] Damon, D. H., J. Appl. Phys. 37, 3l8l ( 1966) . (1967). . ' ~~ [12] Sheard, F. W., Phil. Mag. J?, 887 (i960). [k~\ Giraudier, L., J. Phys. (Paris) 26, 129A (1965). [13] Aliev, S. A., Korenblit, L. L. and Shalyt, S. S., Soviet Phys. - Solid State 8, 565 ( 1966) . [5] For example: Zdanowicz, W. , Acta Phys. Polon. 20, 6*4-7 (1961). [lk] Devyatkova, E. D., Soviet Phys. - Tech. Phys. [6] Goldsmid, H. J., private communication. 2, klk (1957). [15] Muzhdaba, V. M. and Shalyt, S. S., Soviet Phys. [7] Goldsmid. H. J., Proc. Roy. Soc. (London) 72, - Solid State 8, 2997 (1966). 17 (19595. [l6] Ravich, Yu. I., Smirnov, I. A., and Tikhonov, [8] Taylor, P. F. and Wood, C, J. Appl. Phys. 32, V. V.. Soviet Phys. - Semiconductors 1, 163 1(1961). (1967). 127 [17] Hashimoto, K., Mem. Fac. Sci. Kyushu Univ., [2l] Haidemenakis, E. D., Balkanski, M., Palik, Ser. B, 2_, 187 (1958) (Quoted hy Drabble, E. D., and Tavernier, J., Bull. Am. Phys. Soc. J. R. and Goldsmid, H. J., Thermal Conduc- 11, 1+01 (1966). tion in Semiconductors (Pergamon Press, (I961). [22] Ziman, J. M. , Electron and Phonons, p. 1+17 (Clarendon Press, i960). [18] Sexer, N., Phys. Status Solid! lU, KU3 (1966). [23] Sexer, N., Phys. Status Solidi 21, 225 ( 1967) . [19] Sexer, N., J. Phys. Radium 22, 807 ( 196l) . [20] Moizhes, B. Ya. and Ravich, Yu. I., Soviet Phys. - Semiconductors 1, 1U9 (1967). 128 TO VACUUM FEED-THROUGH Figure 1. Thermal conductivity apparatus. A - main heater; B - copper heat sink; C - polytetrafluoroethylene spacers; D - sample ; E - copper-constantan thermo- couples; F - copper heater block; G - heater resistor (Corning glass) ; H - brass shield. The diameter of the body of the apparatus is 1.20 cm, and the overall length is about 18 cm. < •=0 I 3.6 3.2 2.8 L 2 -4 L ■2.0 1.6 1.2 I 0.8 0.4 I O TIT Jt$^l? * 2-Ai I-Al 2-A L > I-Al 1 J. _L X J_ Figure 2. Total and lattice thermal conduc- tivities of (1) Cd 3 As 2 (2'10 24 e lectrons/m 3 ) and (2) Cd 3 As 2 + 0.1 wt. 7. Cu (1.6-10 electrons/nH) as a function of temperature. 40 80 120 160 200 240 280 320 129 << -a Figure 3. Total and lattice thermal conductivities of two different samples of Cd 3 As 2 + 1.0 wt. 7. Se 25 "\ (about 1.5*10 electrons/nT for both samples) as a function of temperature. 40 80 120 160 200 240 280 320 °K 14 12 10 Figure 4. Electrical conductivities versus temperature for the samples of figures 2 and 3. J L J L 40 80 120 160 200 240 280 320 130 Measurement of Dislocation Phonon Scattering in Alloys P. Charsley, A.D.W. Leaver and J. A.M. Salter Department of Physics, University of Surrey Battersea Park Road, London, S.W. 11. England Measurement of thermal conductivity between 1.5 and 4.2 K combined with the determination of dislocation density by transmission electron microscopy yielded values for dislocation phonon scattering in three copper aluminium alloys, and one copper zinc alloy. The results indicated that dislocation phonon scattering in copper alloys depends on both the type and amount of solute, and that the effects of dislocation arrangement are small. More careful experiments to detect any effect due to dislocation arrangements have been carried out. An experiment to measure the thermal conductivities of a deformed 12 at.% copper -aluminium alloy while held under tension near its flow stress, and while released, indicated no difference between the two cases. In an experiment on a deformed single crystal of a 12 at.% copper -aluminium alloy, a small effect ascribed to an anisotropy in the scattering of phonons by screw dislocations has been found. In both these experiments the A/T against T graphs showed a kink at about 3 K. This may be explained by considering the scattering of phonons from edge dislocation dipoles. Key Words: anisotropy, copper alloys, dislocations, lattice conductivity, phonon scattering. 1. Introduction At liquid helium temperatures the thermal conductivity of annealed copper alloys with residual resistivities less than about 0.1 uOm can be well represented by A = AT + BT 2 (1) where the quadratic term is entirely due to lattice conduction, its magnitude is limited by electron- phonon scattering which has the same dependence on phonon wave number q as the expected scattering due to individual dislocations. So the effect of plastic deformation is to reduce the value of B in eq (1) . Changes in B -1 give values of W,T 2 which is a measure of the extra thermal resistivity due to the dislocations introduced. From measurements of W,T 2 combined with the determination of dislocation densities N, using transmission electron microscopy we have obtained values for W,T 2 for some oc phase copper aluminium alloys. The results, which have been submitted for publication else- where (1) , were that o varied from 1.0 * .3 x 10 -13 W -1 m 3 deg 3 for a 2 at.% alloy to 2.1 ± .7 x 10 -13 W -1 m 3 deg 3 for a 12 at.% alloy. We thought that this variation was due to the marked changes in dislocation structures with aluminium content which is observed in these alloys (2) . The 12 at.% alloy has a low stacking fault energy and the dislocation structure is quite regular with the frequent occurrence of rows of dislocations of the same sign. In the 2 at.% alloy the dislocations are very tangled. It seemed possible that the higher value of o in the 12 at.% alloy was due to the effect of dislocation pile ups. This was suggested previously by Klemens (3) as a possible explana- tion of the discrepancies which exist between theoretical and experimental values of o. However, 70-30 a brass has a dislocation structure in the deformed state very similar to that found in our 12 at.% copper aluminium alloy, and the value of o obtained by Lomer and Rosenberg was about 0.5 x 10 W~ m 3 deg 3 , (4). Believing this might be in error we made a similar measurement and Figures in brackets indicate the literature references at the end of this paper. 131 obtained a value for a brass of 0.4 * .2 x 10 W m deg . We therefore suggested that the values of a obtained in copper alloys depend upon both the type and amount of solute present, and that they appear to be substantially independent of dislocation arrangement. Because this conclusion seems at variance with the strain field model for dislocation-phonon scattering we have carried out some further experiments to investigate the effect of pile ups and dislocation arrangements on the lattice thermal conductivity of a 12 at.% copper-aluminium alloy. 2. The Specimens The alloy of nominal composition 12 at.% was prepared and analysed by the International Research and Development Co. as described previously (5). The mean composition of the ingot from which all the specimens were made was 11.3 at.% aluminium in copper. Single crystals were grown in graphite moulds using the Bridgman technique. The polycrystalline specimens used in the under stress experiment described below were turned out from as received material and annealed for about 15 hours at 750 C before use. 3. The Experiment to Detect Pile-Ups 3.1 The Idea The strain field of a dislocation pile-up depends upon the spacing of the dislocations in it. This can be altered by applying an external stress. The phonons are scattered by the strain field, so that if pile-ups in any number are present one might expect to find a difference between the lattice thermal conductivities of a deformed specimen while in a relaxed state and while held under tension. Since the effect of a stress is to decrease the average spacing between dislocations in a pile-up it is to be expected that a specimen under stress will have a lower conductivity than a relaxed specimen. To perform such an experiment the specimen must be constrained by a jig, and at the same time the thermal conductivity measured. 3.2 The Techniques Used in the Under-Stress Experiments It was decided to use a jig of very high thermal conductance and to supply the heat to the specimen from the centre. Thus the two ends of the specimen are maintained at equal temperatures. The jig is shown in figure 1. Stainless steel threaded end pieces E were brazed on to the speci- men S, these served both to take the nuts N, and to enable the specimen to be fitted into x tensile machine. The thermal contact from the ends of the specimen to the copper jig J were made by soldering copper wires w between them and the jig corners. The connection of the jig to the heat sink was made by a 3 mm diameter copper wire W from the centre of the cross piece B. The specimen, with a heater H. and thermometers Tj and T 2 attached as shown, was fitted in the jig and then mounted in a tensile machine. After a suitable amount of plastic deformation, and with the stress still applied, the nuts N were tightened until the stress registered on the tensile machine had dropped to zero. The specimen is then held in the jig under tension near the flow stress, and when mounted in the cryostat and cooled to 4.2 K remains in this condition. We believe this to be the case because; (a) Separate experiments in the tensile machine indicated that after an initial load drop of about 2% of the flow stress no load relaxation in the specimen and end pieces occurred over periods of up to 12 hours. (b) The differential contraction between specimen and jig which occurs on cooling is of such a sign as to cause further plastic deformation. The cryostat was similar to one previously described (5) . Measurements of Q and AT, the temperature difference down one half of the specimen, were made between 1.5 and 4.2 K. The system was then warmed to room temperature, the nuts N loosened and removed, and the measurement of Q and AT repeated. For a symmetrical arrangement A is easily calculated since half the heat produced in the heater goes in each direction; but in any case A may be calculated knowing the size factor of the gauge length and the relative size factors of the two halves of the specimen between the heater tag and the points where the wires w are attached. A comparison of the values of X determined in this way for a relaxed specimen in the jig with the values obtained by the standard method for the same specimen removed from the jig gave results which agreed to within 1% of A. 132 4. The Experiment to Detect Anisotropy 4.1 The Idea This is similar to the experiment carried out by Moss (6) on non-metallic crystals. We tried to prepare from a deformed single crystal specimens in which the angle between the heat current and the dislocation lines differed. This experiment is very dif f icu 1 t in a face centred cubic material because one must limit plastic deformation as much as possible to one slip system. This in turn limits the dislocation densities which can be introduced, and one is trying to detect small changes in W,T 2 against a relatively large background of electron-phonon scattering. 4.2 Specimen Preparation Figure 2 shows diagrammatically a single crystal of 12 at.% copper-aluminium of dimensions .2 x 10 x 2.5 cm. The operative slip plane containing the primary burgers vector b_ is shown. _b made an angle of 10 with the trace of the slip plane on the thin face of the crystal. This crystal was deformed in tension as much as possible consistent with only one slip system operating. This was judged from the absence of a significant number of slip lines on any but the primary system. (In this connection it is worth noting that Steeds and Hazzledine (7) measured the relative densities of primary and secondary dislocations in a 10 at.% copper-aluminium alloy after 45% shear strain. They found that the density of the primary dislocations was about four times that of the secondary dislocations.) Then the directions in the crystal of the primary edge and screw dislocations may be determined. Two specimens of the shape shown in figure 2 complete with thermometer tags T were obtained by cutting perpendicular to the large face ABCD of the crystal. This was done by spark machining. In one specimen, referred to here-after as cross 1, the orientation of the cross was chosen such that the primary edge dislocations made equal angles with both arms Aj A2 and 1i\ T>2- The angle between the screw dislocations and these two directions however differed. In cross 2 the situation with regard to the edge and screw dislocations was reversed. Thus a comparison of the thermal conductivities of the two arms of cross 1 or of the two arms of cross 2 will show any aniso- tropy in the scattering of phonons by screw and edge dislocations respectively. One arm of each cross made an angle of about 30 with the tensile axis so that it is reasonable to assume equal mean dislocation densities and the same dislocation structures for both arms of any one cross. 5. Results 5.1 The Jig Experiment The results for the measurements done in the jig are shown in figure 3. The upper set of points is for the single crystal. This had been deformed about 20% in tension to a stress of 7 kg mm -2 . The lower set of points is for the polycrystal which was deformed by about 5% at a stress of 20 kg mm -2 For both specimens the open circles refer to the relaxed condition. Note that the ordinates for the two specimens are shifted relative to one another. The vertical arrows on the graph indicate * 1% — and within this range, (a) there is no signifi- cant difference between the measurements under stress or relaxed for either the poly or the single crystal, and (b) straight lines may be drawn through the points. However, the scatter on the individual curves is considerably less than * 1%, and in the case of the polycrystal the relaxed measurements give lower values throughout the temperature range. This is the opposite of what one would expect for pile-ups, and any change in size factor caused by the elastic deformation of the specimen under tension can only increase this difference. Moreover, if one joins up the points the X/T against T plots appear as a curve for the single crystal under stress, but more like a kinked straight line for the remaining three sets of points. 5.2 The Anisotropy Experiment The flat single crystal was deformed about 25% in tension at a flow stress of 6 kg mm 2 . This took the crystal through stage I of the work hardening curve and into the beginning of stage II. The table below gives the angles between the heat current and the primary edge and screw dislocations for each cross. The edges in cross 1 were not at exactly the same angle due to the rotation of the cry- stal axis during deformation. 133 Table 1. The angles between the primary dislocations and the heat flow for the directions measured. The graphical symbols used in figure 4 are also given. Cross 2 Aj A 2 Bx B 2 80° 46° Cross 1 Al A 2 Bl B 2 Angle to Edges 55° 63° Angle to Screws 35° 73° Graphical Symbol O □ X The results of the thermal conductivity measurements are shown in figure 4. The graphical symbols used are indicated in table 1, and the points for cross 2 are shown displaced to the right by 0.6 K. Almost all the points can be made to lie within * 1% X/T of the same straight line. Taken at their face value each set of points is better represented by a kinked straight line as drawn in figure 4. The results for cross 2 indicate that there is no detectable anisotropy due to edge dislocations. The vertical difference between the results for the two arms of cross 1 cannot be considered significant since each direction involves the measurement of a different size factor. Nevertheless, for cross 1 in the region above about 3 K, the points for arm AiA 2 lie on a line of greater slope than those for arm BjB 2 . This would be consistent with an anisotropy in the scattering of phonons by screw dislocations. 5.3 Evidence that the Kink is Real The drawing of other than straight lines through the points for the X/T against T plots needs justification. The effect looks like a systematic error. It cannot be due to the super fluid transi- tion in liquid helium because it occurs well above this. We measured cross 1 in the direction AjA 2 in a different cryostat using different thermometers and heater. The two sets of measurements are shown in figure 5. They agree extremely well. Recent measurements in this latter cryostat of the thermal conductivity of an annealed and deformed 2 at.% alloy gave very good straight lines in both cases with all the points within * 0.2% X/T of the lines. We are at present carrying out further test experiments . 6. Discussion No effect corresponding to the expected behaviour of dislocation pile-ups was observed in the jig experiments. For the polycrystal the difference between the two measurements, under stress and relaxed, could be significant but it is in the opposite direction to that expected. The limitations on the dislocation density which can be introduced in the anisotropy experiments together with the results obtained indicate perhaps the unsuitability of a face centred cubic alloy. The experiment might be better done using an alloy with a hexagonal close packed structure in which slip only occurred on the basal planes. Nevertheless, considering the results for cross 1 in the temperature region above the kink there is a difference in the slopes of the X/T against T plots for the two arms. The smaller slope occurs for direction BiB 2 in which the screw dislocations make a greater angle with the heat flow in agreement with the expected behaviour (6) . There still remains the kink in the X/T against T graphs which occurs in all the specimens. This kink is not present in annealed and deformed alloys of 2 at.% copper- aluminium. We believe that it may be due to the presence of large numbers of dislocation dipoles in the deformed 12 at.% alloy. The deformation of annealed single crystals of this alloy in the early stages proceeds by the formation of localized bands of slip, separated by regions in which comparatively little or no slip has occurred. As the deformation proceeds the bands get wider and fresh bands appear until at the end of stage I the crystal is uniformly covered with slip. Recently Mitchell et al (8) studied the dislocation struc- ture of these bands in an 8 at.% copper- aluminium alloy in the first few percent of plastic deformation. They found that the dislocations in the bands of deformation were present in the form of interleaved pile-ups of parallel positive and negative edge dislocations. The distance between the glide planes on which the adjacent groups of dislocations of opposite sign occurred varied from about 50 angstroms to a few microns. Steeds and Hazzledine (7) also found large numbers of dipole groups in their 10 at.% alloy. 134 The effect of dipoles on lattice thermal conductivity has been discussed by Moss (6); it depends upon the relative sizes of the dipole spacing d and the dominant phonon wave length given by For I << d the phonons see a dipole as separate dislocations giving the usual q wave number dependence for the dislocation phonon scattering relaxation time. For I >> d L the phonons see the equivalent of a line of point defects, the relaxation time is proportional to q , and the scattering is less effective than that from two isolated dislocations for the same I. 0.6 a— where a is the lattice spacing and 9 the Debye temperature, ■1 For a fixed value of d, as the temperature is lowered, there will come a point where the phonons will begin to see the dipole aspect of the dislocation arrangement. The phonon scattering will then begin to decrease from the value expected for a random arrangement of dislocations sufficient to account for the phonon scattering at higher temperatures. Correspondingly the values of X/T will begin to lie above a line extrapolated from the higher temperatures. Moss (6) has suggested that this change will begin when I - 2d. In our alloys the kink occurs at about 3 K where the dominant phonon wavelength is about 200 angstroms. So to account for the if feet observed, the majority of the dipoles must have a spacing of about 100 angstroms. This is consistent with the available electron microscope evidence. What one might expect to happen at lower temperatures is that the X/T against T plots would turn again, but in the opposite direction to a line whose slope would correspond to the scattering from electrons and the residual density of randomly distributed dislocations. The effect of an external stress on a sheet of dislocation dipoles will be to move one set of dislocations relative to the other but not to change the relative spacing of dislocations of a given sign. Thus a null effect in the under-stress experiment is readily explained on the assumption that the majority of the piled-up groups of dislocations exist in dipole sheets. The effect in the 'wrong' direction might be explainable in terms of an overall decrease in dipole spacings under the action of the applied stress. With regard to our values for a(l). Neglecting the existence of the kink in deformed copper- aluminium alloys of the higher compositions could lead to an over estimate of the values of W,T 2 obtained. At the worst, the values we obtained would be one-and-a-half times too big. 7. References (1) Charsley, P., Salter, J. A.M. and Leaver, A.D.W. An experimental investigation of dislocation- phonon scattering in some copper alloys . Sub- mitted to phys. stat. sol for publication. (5) Charsley, P. and Salter, J. A.M. The lattice thermal conductivities of annealed copper- aluminium alloys. phys. stat. sol. W 575 (1965) (2) Swann, P.R. and Nutting, J. The influence of stacking-fault energy on the modes of deforma- tion of polycrystalline copper alloys. J.Inst. Met. 90 133 (1961) (6) Moss, M. Effect of plastic deformation on the thermal conductivity of calcium fluoride and lithium fluoride. J. Appl. Phys. 36 , 3308 (1965) (3) P.G. Klemens. Thermal conductivity and lattice (7) vibrational modes. Solid State Physics 1_ 1 (1958) (4) Lomer, J.N. and Rosenberg, H.M. The detection of dislocations by low temperature heat conduc- (8) tivity measurements. Phil. Mag. k_ 467 (1959) Steeds, J.W. and Hazzledine, P.M. Disloca- tion configurations in deformed copper and copper 10 at.% aluminium alloy. Disc, of Faraday Soc. 38 103 (1964) Mitchell, J.W., Chevrier, J.C., Hockey, B.J. and Monaghan, J. P. Jr. The nature and formation of bands of deformation in single crystals of a-phase copper aluminium alloys. Can. J. Phys. 45 453 (1967) 135 w N CF 3 i T ' $ B=oJ w I Of w •B W w 70 end top Figure 1 The specimen and jig. See section 3.2 of the text for explanation. 35' B. A/ 3 cm A, TkU B A, specimen shape Figure 2 The single crystal and specimen shape. The trace of the primary slip plane on the crystal faces is shown. _b is the primary burgers vector. 136 CT> T3 'E O 40 / / / / / / 9 / /y V 40 3-5 T K- Figure 3 — against T for the specimens measured in the jig. The open circles are for the relaxed condition. The lower set of points and the right hand — axis refer to the polycrystal. 50 en JO 'e o 4-5 40 / / /° V a / a' V />> / V 7 / T U K Figure 4 — against T for the cross shaped specimens. The lower temperature scale refers to the right hand set of points which are for the two directions of cross 2. For a further indi- cation of which points refer to which directions see table 1 in section 5.2. 137 T°K Figure 5 — against T for the direction A1A2 of cross 1 measured in two different cryostats. The triangles refer to the second cryostat. 138 The Thermal Resistivity Resulting From the Combined Scattering of Phonons by Edge Dislocation Dipoles and by Normal Processes P. P. Gruner Boeing Scientific Research Laboratories P.O. Box 3981 Seattle, Washington 98124 The thermal resistivity due to the scattering of phonons by edge dislocation dipoles and by normal three-phonon processes has been calculated. Deviations from the normal T - ^ - dependence of the resistivity (T:absolute temperature), which is due to the scattering of phonons by isolated dislocations, occur at temperatures where the dominant wavelength of the phonons is larger than the distance between the two dislocations of the dipole. It is found that the resistivity becomes independent of the temperature in the neighborhood of T~ and remains finite at T=0. Towards high temperatures the resistivity due to dipoles approaches asymptoti- cally the resistivity which is due to isolated dislocations. Numerical results were obtained with the material constants of copper. The temperature above which the resistivity differs less than 10 % from the asymptotic value is given by T=370 ( |d"| /R) °K (D*:Burgers vector, R:dislocatlon distance). The interactions of the phonons with the dipoles were treated by non-linear elasticity theory which includes also the normal three-phonon interaction. The Boltzmann equation was solved with the variational method. Key Words: Boltzmann equation, edge dislocation dipole, lattice thermal conductivity, non-linear elasticity theory, normal processes, variational method. 1. Introduction The possibility to distinguish between different kinds of lattice defects by measurements of the lattice thermal conductivity as a function of the temperature is based on two facts. Firstly, the scattering cross-sections of various kinds of defects are characteristic functions of the phonon wave- length. Secondly, the phonon spectrum depends at low temperatures strongly on the temperature. On account of these features the thermal resistivities due to various kinds of defects show characteristic temperature dependencies. Yet, the thermal resistivity does not only depend on the type of the scatterers, it also depends on the arrangement of the defects. If the defects are far apart from each other and are distributed in a random manner then the scattering is incoherent. Then, the total scattering cross-section of a particu- lar kind of defect is simply the sum of the single scattering cross-sections. In case of deviations from a random distribution of defects, the various phases of the scattered waves must be taken into account; the scattering cross-section contains a coherent part as well as an incoherent part. The contribution due to the coherent scattering will be significant if the distance between the defects is comparable with the dominant phonon wavelength. Investigations of the thermal resistivity due to pile-ups of screw dislocations [1] and of edge dislocations [2] show that coherent scattering is not negligible below a certain temperature. The coherent part of the scattering gives rise to an increase in the resistivity. It also leads to devia- tions from the normal quadratic temperature dependence of the thermal conductivity, which arises from scattering of phonons by randomly distributed dislocations. In order to give a value for the temperature'' below which the arrangement of the dislocations becomes significant, we mention the result which was obtained for a group of dislocations piled up under a stress against an obstacle. In case of screw dislocations, the arrangement becomes significant below 2.5°K. Tht corresponding value for edge disloca- tions is 13.5°K. These data were calculated with the material constants of copper. The resolved shear stress was assumed to be 10~3 G (G:shear modulus). The number of dislocations within the group was 15. 1 Figures in brackets indicate the literature references at the end of this paper. ' The limiting temperature is defined as that temperature above which the relative difference between the resistivity due to randomly distributed dislocations and due to dislocation groups is smaller than 10%. 139 Another important example where the resistivity strongly depends on the arrangement of the dislocations is the dipole configuration. In this case two parallel edge dislocations of opposite signs are separated by a distance R which is small compared with other dislocation distances. A few qualitative conclusions on the thermal resistivity due to this kind of defect can be drawn without calculation. The lattice thermal conductivity which is limited by a dislocation dipole must be one half of the conductivity which is limited by an isolated dislocation if the distance between the two dis- locations is sufficiently large. At high temperatures, where most of the energy is transported by phonons with wavelengths shorter than the distance R , one expects also the resistivity due to a dipole to be twice that which is due to a single dislocation. If the distance R is small, however, parts of the strain fields cancel since the two dislocations have opposite signs. Hence, decreasing distances between the dislocations of a dipole lead to a reduction of the resistivity. Thus, whereas in case of phonon scattering by dislocation pile-ups the resistivity increases with decreasing separation between the dislocations, the contrary is true in case of the phonon-dislocation dipole interaction. Before presenting an outline of the theory which was used for the calculation of the thermal resistivity, we anticipate a result of the following calculations concerning the convergence of the conductivity integral. It turns out that the scattering by dipoles alone is not strong enough in order to produce a non-zero resistivity. The partial cancellation of the strain fields gives rise to a rapid drop of the scattering strength for long wavelength phonons. On account of this decrease of scattering strength the energy current carried by the low frequency phonons will not reach a stationary state. A natural way out of this situation is given by taking into account the normal three-phonon interactions. These latter interactions contain processes which diminish the rapid drop of the scattering strength for long wavelength phonons [3] . It must be noted that the normal processes themselves cannot produce a non-zero resistance, as was first shown by Peierls [4]. The conservation of the quasi-momentum of the phonon gas, which is a characteristic feature of these interactions, prevents the phonons from reaching a stationary state. But, in connection with the phonon scattering by lattice defects, which destroys the conservation of the momentum, the normal processes are important. 2. Outline of the Theory 2.1 The Classical Hamiltonian of the Lattice Vibrations We are interested in the scattering of phonons at low temperatures. In this temperature region the wavelengths of the excited phonons are large compared with the lattice constant. We can therefore treat the dynamics of the lattice waves with continuum theory. For convenience we shall restrict ourselves to an isotropic continuum. The interactions between the lattice waves and the defects are in this model treated by deviations from Hooke's law. We shall take into account deviations from Hooke's law up to terms which are quadratic in the strains. Bross [5] showed that the potential energy density of the lattice vibrations in an isotropic continuum which contains lattice defects is of the form f = * {a.; z r e lz ; Y ' r Y ir Y m ; Uy)^, CeeY> Iir , < E YY) m } (2 which describes the state of the phonon gas-*: a + (k) <|i (N£) = (N£+l) 1/2 <|i (NjH-1) , (2.3a) a(k) * (N£) = (N£) 1/2 <|i (N+-1) . (2.3b) N£ is the number of phonons in the mode k . The Hamilton operator which corresponds to eq (2.2) can be written as H = H Q + V D + V N (2.4a) 4 with the Hamilton operator of the undisturbed phonon gas H Q = I nui(£) |a + (k")a(k") + 1/2 } . (2.4b) k The frequency u(k) of the mode k is in the continuum approximation given by u)(k) = v|k| . (2.4c) The sound velocity, denoted by v, can have two different values, one for the longitudinal waves and another for the transversal waves. V and V., are the operators for the phonon-def ect interaction and the normal three-phonon interaction. The phonon-defect operator can be represented in the form V = I V(k,k') a (k)a(k') + other terms quadratic in the operators a (2.4d) k,k' with V(k,k') = A [e(k)-e(k')] (k-k') T (g) + ••• . (2.4e) i L e(k) is a polarization vector and T is the Fourier transform of e . The coefficients V(k,k') vanish, unless k + P + g = 0. (2.4f) Equation (2.4f) shows that the wave vector is not conserved in collisions of the phonons with defects. The consequence is that the quasi-momentum of the phonon gas will be changed in the phonon-defect interaction. The phonon-phonon interaction operator can be written as V = £* V(k,1c' ,^")a + (tc)a + (1c')a(1c") + other terms cubic in the operators a. (2.4g) W k.k'.k" In the following equations we denote the mode (k,j) by k. 4 , n is Planck's constant divided by 2ir 141 The coefficient V(k,k ,k ) is a function of products between polarization vectors and wave vectors and does not contain the strain field of the defects. V(k,k',K'') vanishes, unless £ + k' + k" = . (2.4h) Both coefficients, V(k,k') and V(k,k',k''), are explicitly represented in Reference [5]. Scattering processes of the type (2. Ah) are called normal processes. As already mentioned, the quasi-momentum of a phonon gas in which the phonons interact only by normal processes is conserved. This conservation has the consequence that the thermal conductivity is infinite if the phonons were not in addition scattered by those processes (eq(2.4f)) which destroy the conservation of the quasi-momentum. The operators V and V„ , if acting on the wave function i|> , cause transitions from one phonon state to another. The probability rate for those transitions can be calculated with the help of time dependent perturbation theory. Having the probability rate, one calculates the total rate of change of the number of phonons in mode t.. The result can be written in the following form: (dN+/dt) „. . = I U(X,-P)(.^,-N?) k collision + . k k k + I WCk.k'.-k'^tCR^flXN^.+DN^,, - N£N£,(N£,,+1)] k*',k" + two similar expressions . (2.5a) — -* ish* is tne mean number of phonons in the mode k and is defined by % = I \ P(...N+...,...N+...,t) , (2.5b) where the summation runs over all phonon states ( . . .Nr>-. . . ) . P obeys the master equation and is the probability of finding N+ phonons at a time t in mode t. if there were Nj* phonons in the same mode at t=0. The quantities W (k,-k') are, except for the mean phonon numbers N^t, equal to the probability per unit time for a transition from mode k to mode k' which is caused by the phonon- defect interaction. It is W(k,-k*)=2TT(27T) 6 (V p )" 2 |v(k,-k*)| 2 [o J (k)o J (k')]" 1 6 {aKkVu^k')}, (2.5c) where V. and P are the crystal volume and the mass density in the undeformed state, respectively. Equation (2.5c) shows that the scattering by static defects is elastic, i.e., the energy of the incident phonon must be equal to the energy of the scattered phonon: noj(k) = nu(k') . (2.6) The transitions due to normal processes between three phonon states are governed by two selection rules which follow from the corresponding transition probabilities [5] . One of the selection rules we have already met in eq (2.4h). For transitions in which, for example, one phonon of mode £' ' is annihilated and one phonon in each of the modes Ic and £' is created, eq (2.4h) reads k" = k + k' . (2.7a) On the other hand, the transition probability per unit time, W(k,k' ,-k' ' ) , for the processes under consideration vanishes, unless co(k") = w(k) + w(k') . (2.7b) The consequences of the selection rules (2.7) will be discussed later on. 142 2.3 The Transport Equation A temperature gradient changes the distribution of the phonons in thermal equilibrium , N£ = [exp(lWKT) - l]" 1 , (2.8) until, on account of the various scattering mechanisms, a stationary state is reached. The change of the mean phonon numbers, Nh* , per unit time which is caused by the field of the temperature gradient is ^1 field = " [v(k).gradT]ON£/3T) , (2.9) where v(k) is the group velocity. In the stationary state the total rate of change of the mean phonon numbers must vanish. This yields %t,. , , + % | nl . . = . (2.10) k| field k | collision The Boltzmann equation (2.10) is an integral equation for the mean phonon numbers N-J . It is con- venient to subtract from the occupation numbers N£ of the stationary state their thermal equilibrium value and to write Ng - H£ = = I nu)(3N/3no))v i (k)Z.(k) = j> Z . (k) L (2.16b) k k k Condition (2.16a) is chosen since only the odd part of Z(k) contributes to the conductivity (2.12b). On account of the condition (2.16b) and eqs (2.15a) and (2.15b), it follows that among all vector functions Z(k) the exact solution X(k) leads to a maximum of the expression - £ Z . L-JZ . I . One finds - I Z.(k)L |z i (k)l < - I X ± (fc)L |x.(k)l . (2.17) k k From this inequality it follows that among all arbitrary functions Z;(k) those which make the left hand side of (2.17) a maximum lead together with eq (2.12b) to the exact diagonal components of the conductivity tensor. We choose as a trail function the expansion co oo n v y y 2& = £i ~t IT n za.ni.n.r)^)^^,.),) . (2.18) r=U n=l m=-n nm Since Z(k) depends, according to our convention that k stands for k and j , on the polarization branch, the expansion coefficients Z(j,m,n,r) depend a^lso on the polarization. The spherical angles S and determine the direction of the wave vector k . Y are normalized spherical harmonics. Substitution of the trial function (2.18) into eq (2.17) and variation of the coefficients Z(j,m,n,r) under the conditions (2.16a) and (2.16b), yields the following system of equations I M(v,v 1 )Z + (v') = N(v) , (2.19) The sum^ ...is in the usual way replaced by (V_/(2'") J ...d k 144 where v stands for (j,m,n,r). The matrix M contains all information on the scattering mechanisms. If more than one scattering mechanism is present then M is equal to the sum of the matrices which belong to each scattering mechanism. In our present case, M contains two parts. One part is due to the phonon-dipole interaction, _£he other part belongs to the normal processes . The matrix M for defects and the vector matrix N are represented in the Appendix. From eqs (2.12b) and (2.19) and the expansion (2.18) we find for the thermal conductivity: (1/TV Q ) I N(v) Z(v) . (2.20) 3. Application of the Theory to the Scattering of Phonons by Edge Dislocation Dipoles We shall not present the calculations in detail and shall rather restrict ourselves to a few remarks which will be needed in the following discussion of the results. We consider two parallel edge dislocations of opposite signs, as shown in Figure 1. The dislocation lines are parallel to the z-axis and have Burgers vectors in the direction of the positive x-axis. The vector R and the x^-axis enclose the angle n . As already mentioned, the strains e in the energy density (2.1a) are those which are given by linear elasticity theory. The composed strain field of the two dislocations is then simply the sum of the single strain fields. Taking the Fourier transform of the composed strain field, one finds t (g) = T 5 and 0', <}> ' of the wave vectors k and k' and the elastic constants. The structure factor in eq (3.2a), resulting from the different positions of the scattering centers, may be omitted if the distance R between the dislocations is very large. Then W(k,-k') = 2 W^(k,-k' ) and noj' belong to transversal branches and if the phonon no)'' belongs to a longitudinal branch, then oj' depends on io. The studies of the possible normal processes in a continuum [3] showed that processes for which to' does not depend on io- energetically unlimited processes-yield a rate of change of the mean phonon numbers (2.5a) which is proportional, to co for urH). The other kind of interaction, for which cj' is a function of id, yields 3N/3t<*io . According to the last section, the rate of change of the phonon numbers due to the phonon-dipole interaction is proportional to to , for oj-H) j and becomes insignificant at oj-v.0 if compared with 3N/3t which results from energetically unlimited normal processes. Hence, at small frequencies, the frequency dependence of the deviations X(k) of the phonon numbers from their thermal equilibrium values is governed by energet- ically unlimited processes. Then, as follows from the Boltzmann equation, the X(k) are independent of the frequency. The consequence is that the thermal conductivity will no longer have a singularity at oj=0 . We note, that we already used in eq (2.18) the result that X(k) is independent of ui, for ur>0 (the power series starts with r=0) . We shall not go into details of the integrations which must be carried out in order to get numeri- cal values for the matrices M . We only point out that for the normal processes the matrix elements of M depend on each other. One finds, as a consequence of the two selection rules (2.7a) and (2.7b), that M(j,m,l,r;L,-m,l,l) + (v L /v T )M(j ,m,l,r ;T,-m,l,l) (4.1) Here L and T stand for longitudinal and transversal branches, v and v are the corresponding sound velocities. On account of eq (4.1) the determinant of M vanishes identically. processes are the only scattering mechanisms then the vector thermal resistance is zero, as mentioned earlier. Z(v) in eq If normal (2.19) is infinite and the 4.2 The Thermal Conductivity Limited by Dipoles and by Normal Processes We consider the dislocation arrangement of Figure 1. Let be equal to tt/4 . The resistivity for configurations with other angles has also been calculated. Since the main features of the tempera- ture behaviour of the thermal conductivity do not change very much with varying angles n , we shall discuss here only the 45 degree case, due to the other dislocation vanishes Note, that the shear stress in the glide plane of a dislocation Hence, this configuration is stable with respect to gliding. The conductivity has been calculated with the elastic constants of copper. Numerical values of the elastic constants can be found in Reference [11]. In Figures 2 through 4, we have plotted the components of the reduced thermal conductivity ~ , which is defined by :[K2p v*/Qib\)] k , (4.2) versus the reduced temperature T = KT(p Q h 3 vj b 2 N D )' (4.3) N is the number of dislocation pairs per cm perpendicular to the x, , x„-plane; constant and b is the absolute value of the Burgers vector. The parameter h is Planck's 146 p = (b 2 N D )^(R/b) (4.4) is a measure for the distance between the two dislocations of a dipole. Q Figures 2,3, and 4 show the components "k , ~ , and -~ , respectively. The component Tc is infinite since the z-component of the wave vectors is conserved (see eq (3.2b)). We note that the xy-component is negative. This means that a temperature gradient in positive y-direction produces a positive heat current component in x-direction. The component Tc appears since in this configuration the x and y axes are not principal axes of the conductivity "ellipsoid". In the case of isolated edge dislocations parallel to the z-axis and with Burgers vectors parallel to the x-axis only 1< and "k do not vanish, except for k yy r zz Curve 1 in Figures 2 and 3 shows the components ~ and k , respectively, of the conductivity which is limited by isolated dislocations and normal processes. These components are at low and high temperatures proportional to T . The magnitude of the conductivity at low temperatures, however, is larger than that at high temperatures. This difference is due to a change in slope of the conduc- tivity at intermediate temperatures, which results from the three-phonon interactions [3]. In case of phonon scattering by dipoles and by normal processes the conductivity approaches asymptotically curve 1 for increasing temperatures or increasing distances R. This follows immediately from the transition probability (3.2a) which approaches for large products u> R the transition probability due to two isolated dislocations. At very low temperatures, the conductivity limited by dipoles and normal processes is independent of the temperature and has a finite value at T=0. The magnitude of the conductivity in this temperature region is inversely proportional to the square of the distance R . As expected, the conductivity increases on account of the reduction in scattering strength which is due to the partial cancellation of the strain fields. At intermediate temperatures, we find a minimum in the conductivity if the distance R is sufficiently small (p small) . The reason for the occurrence of this minimum follows from the transition probability (3.2a) which has a main maximum for values of kR~Tr/2. If the temperature is such that the maximum of the phonon distribution coincides with the main maximum of the transition probability then the resistivity will have a maximum. An increase in R will shift the main maximum of the transition probability towards smaller k values. Hence, the minimum of the conductivity occurs at lower tempera- tures. The minimum disappears if the distance R becomes too large. Increasing R leads to a reduc- tion of the width of the main maximum. The consequence is that the number of phonon states which are affected by an enhanced scattering is reduced. The -k" component, shown in Figure 4, is at very low temperatures also independent of the temperature and proportional to 1/R and drops at high temperatures very rapidly to zero. Note, that in case of phonon scattering by isolated dislocations, which are parallel to the z-axis and which have Burgersvectors parallel to the x-axis, the component 1? is zero. The temperature above which the difference between the thermal conductivity limited by dipoles , k , and the conductivity limited by isolated dislocations, K g , is smaller than < q /10 has been calculated for the xx-component of k. It is found that this limit lies at a temperature T - . . = 370 b/R°K. llmlt 5. Conclusion The influence of edge dislocation dipoles on the lattice thermal conductivity has been studied in an isotropic continuum. The scattering of phonons by normal processes was included in the calculations in order to get finite results for the thermal conductivity. Deviations from the normal T -dependence of the conductivity, which is found if the phonons are scattered by isolated dislocations, appear below a certain temperature. This temperature depends on the distance between the two dislocations of a dipole. The conductivity is independent of the temperature at T~0 and reached a finite value at T=0 . It is x = x. , y = x» and z = x„. 147 6- Refei [1] Gruner, P., Zeitschr. Naturforsch. 20a , Heft 12, 1626 (1965). [2] Bross, H., Zeitschr. Phys. 189, 33 (1966). [3] Bross, H. , Gruner, P., and Kirschenmann, P. Zeitschr. Naturforsch, 20a, Heft 12, 1612 (1965). [4] Peierls, R.E., Quantum Theory of Solids, Oxford Clarendon (1955). [5] Bross, H., Phys. Stat. Sol. 2, 481 (1962). [6] Murnaghan, F. D. , Finite Deformation of an Elastic Solid, John Wiley and Sons, Inc., New York (1951). [7] Klemens, P.G., Solid State Physics, Vol. 7, Academic Press, New York (1958). [8] Callaway.J., Phys. Rev. 113, 1046 (1958) [9] Krumhausl, J. A., Proc. Phys. Soc. 85, 921 (1965) [10] Bross, H., Seeger, A., and Haberkorn, R., Phys. Stat. Sol. 3^ 1126 (1963) [11] Seeger, A., Buck, 0., Zeitschr. Naturforsch. 15a , Heft 12, 1056 (1960) 7 . Appendix The matrix M for static defects is given by -KT[ (2tt) 3 /V ] 2 M(j ,m,n,r;j ' ,m' ,n' ,r' ) = I //w(j ,k; j ' ' ,-P) [N(j ,k")+l]N(j ,k") [lU(j ,k) ] r Y (0,< i^' J J [naj(j,k)] r 'Y , ,(0,) -6 j ' j " [nco(j ' ,k' ) ] r *Y , , (0' ,' ) \ d 3 kd\' . n m n m • The vector matrix N is given by -KT[(2tt) 3 /V ]N(j,m,n,r) = / [N(j ,k)+l]N(j ,k) [tfu>(j ,k) ] r+1 v(j ,k)Y (6,40 d 3 k 148 m - CN - ^x^^fe \ vcss NSSN. \ ^C^vs^ ^ \ \\v\ \ \^^n 10 — cn — m — ^s. ,_ cn - CN O CN o CO CN 6 CN 6 CN CO CN 6 ■t' •0 CN \ 21 00 cn\ o\ ■d \ «o \ (N I *n — II II If II II n" 11 1 a. a. a. a. a. a. a. 1 CN — 1 1 1 10 CN 1 I 7 T 1 •0 1 CN lo I 1 1 CN 1 *Jt 3 -a a o CJ u a crt B -J z 0) j= !* 4-1 -O 4-1 -rl O C m •u C m <1J (U c rH O a a. ft ■H a 1 c X ■x ■H •u T) nt HI CJ 3 iH T) CO 0) ■H Pi a 60 u < CD O a) 00 ai M 3 00 149 a R u o •H o p, ►J u t= Ph o M u (1) rH 1 a CO a. o a O o *o a CD c o •H s»> 3 ^ ■o >> 0) u T-l Pi •H C > cfl 4J tn •H » o 01) C D. 01 •H B o O Xx X- 6 t > •H >, U „o O 3 -O T3 C ra .H >-l cn •H OJ Q ,c iJ where H is a real constant. Equation (3) is equivalent to the following: ©„ = H © (4) i o The system obtained in this manner (figures la and lb) is a feedback system. It is unstable when the open circuit gain is: G (p) " H = -1 (5) Equation (5) is both modulus and phase valid. It can therefore be divided into the following equalities : I G (p) * H I = 1 ; arg [g (p) * H ] = (2n + l) n (6) in which n is an arbitrary whole number. Equations (6) and (2) lead to the following: H = I cosh (qx) I ; arg [ cosh (qi)J = (2n + l)n (7) Persistent oscillations are introduced into the system, if u) = 2rrf = 2n/T (f being the oscillation frequency and T the oscillation period) when p = jo). It follows that: / p / jq). r* qZ = Z J~T = i, V~~ a = (1 + 3)1 V2a~" , from which: CK1 (8) R (qx) = 1 (/2tt, and each of them has a tt/2 phase difference from the preceding wave when examined successively in one direction (e.e. clockwise) around the specimen. A prism sufficiently elongated in the direction perpendicular to the lines of thermal flux is assumed (figure 1). Therefore the effects of thermal dispersions on the extreme unheated sides (B. ) and (B.) may be neglected in the middle square section (A) (these sides are well insulated anyway). In the middle section there is a two dimensional temperature field which is a superposition of com- ponents H and H , where H and H are the two perpendicular components due to the four externally applied thermal waves. A stabilized state of oscillations can now be considered in which the amplitudes of the imposed temperatures are sufficiently limited (10 - 15°C), so that in such a range, the thermal properties of the medium, seat of the studied phenomenon, may be held constant. Under such conditions the unidimensional problem then consists of a sinusoidal thermal field de- fined by the gradient H = S9 /9s (s = x,y) along any of the tw t (5) y Sy l v ' which are perpendicular and out of phase with one another. In the middle section (A) , the thermal field resulting from the vectorial sum of H and H will xv rotate with angular velocity uu and have magnitude 2 / l , and in the complex plane can be represented by the rotating vector (fig. 3) • 2©„ + i H = - y H = H + j H = — -£— e jU)t . (6) In fact, considering (4) and (5) as parametric equations of the resulting field and eliminating (Dt, we obtain the ordinary equation of the circular trajectory of the point of the resultant rotating vector: H x + H y = 4 ®t ' ^ ■ (7) In the general case where the thermal field components H and H have different amplitudes, the parametric equations of these components are [3] 1 2® H = — — — sin w t = a sin ou t (8) 2© y 1 cos (out + cp) = b cos (u)t + cp) (9) where and are the amplitudes of the externally applied thermal waves in the x and y direction respectively, and cp is the phase angle between the components H and H . The equation of the resulting trajectory is an ellipse: 2 Figures in brackets indicate the literature references at the end of this paper. 164 In this manner v", (and consequently 9.) has the same time dependence as v 1 ,. For zone 2, which is opposed to zone 4 (fig. 5b), the regulation unit maintains constant v" + v' at the value 3 mV, so that v'_ = v', , v"_ (and consequently 9 ) oscillates out of phase with respect to v'_ (and to 9,), but always in the neighborhood of the constant of 2.5 mV (the equivalent of the tempera- ture on the axis of the specimen). Temperatures 9 and 9 , starting from a signal on the oscillator having a phase lag of tt/2 with respect to v 1 ^, present a phase lag of rr/2 in comparison to 92 an d 64 respectively. If v' lags behind v', the thermal field rotates clockwise, if v' anticipates v', the thermal field rotates counterclockwise. 3. Results of Experimental Observations and Measurements In figure 6, recordings are given of temperature waves imposed on the four sides 1,2,3 an d 4 of the prism specimen in the case when all the waves have the same amplitude (® = @ ®„)> an d waves on adjacent sides have a rr/2 phase difference. The mean temperature 9^ of the oscillations is constant and equal to +60°C, the value imposed on the axis of the prism. o Figure 7 illustrates the position of thermocouples a, b, and o, used for detecting thermal fields, in the interior and on the middle plane of the specimen. Couples a, b are positioned on the diagonals 15 mm from the axis, i.e. the position where the couple o is placed. This distance of 15 mm must be sufficiently small when compared to the thermal wavelength K used in the experim ents , for previously mentioned reasons. Operating with a period T = 6 hours, the wavelength is X = 2 \fn a T = 520 mm, assuming for the present specimen of expanded phenolic resin, according to preceding measurements [1], that a = 12 • 10" 7 m 2 /s. In figure 8 the connections of the thermocouples for detecting the field are indicated together with the connections between them and the XY plotter. Due to the particular symmetry in positioning of sensitive elements, it follows, that the pair a - o of thermocouples, connected in series, gives the thermal gradient S9/SX which oscillates sinusoidally with time along the X axis, while the second pair b - o gives the thermal gradient 39/dY , perpendicular to the first and a sinusoidal function of time in the direction of Y axis. These gradients, analogous to the thermal fields H and H mentioned in paragraph 1, are fed into a twin axial XY plotter, in which the motion of the writing p¥n is directed by one of the thermal fields and the motion of the paper sheet by the other one. In this manner (fig. 9) the recording of the field resulting from the superposition of the two per- pendicular fields oscillating tt/2 out of phase is obtained. From the recording we can observe a field rotating with an angular velocity uu "^ 2.9 x 10 rad/s. The trajectory of the point of the vector representative of the field is practically circular and has the following radius: A® A® A 1,? = A ' = 2.2°C/cm . Ax Ay With an imposed wave amplitude of 12°C, the amplitude of the thermal field should become, accord- ing to eqs. (4), (5), and the assumption of a linear temperature variation within the specimen and there- fore of propagation without attenuation, 2® 1 1 = 2.4°C/cm. This value is nearly 10% greater than the experimental value. Figure 10 shows a recording of the unidimensional thermal field A9, ,/Ax alone; the temperature on sides 2 and 4 (constant at the value 9 = +60°C) and the amplitude and period of temperature waves imposed on faces 1 and 3 (ffg« 11) ar e the same as those already mentioned regarding figure 6. In this recording, which was linear as expected, the thermocouple pair a - o was connected to the X axis and the pair b - o to the Y axis of the XY plotter. Figures 12a and 12b show respectively the general view of the apparatus and the prism mounted in place. Further experiments could be executed using different amplitudes and frequencies of the imposed temperature waves and choosing different phases for the perpendicular thermal fields. This would enable evaluation of thermal properties of materials, such as thermal diffusivity and the attenuation constant, as in order to detect eventual anisotropics in the same materials. In other experiments we should also be able to verify with thermal fields the general properties of oscillatory phenomena. For example, thermo- and galvano-magnetic effects could be investigated with a superposition of magnetic and thermal fields in electrical conductors and semiconductors. Since the rotation of the thermal field is actuated by means of fixed apparatuses which are external to the medium being studied, we may foresee other applications, for instance to the processes of zone fusion for the purification of metals. 165 H 2H H x y . 2 + — b sin cp = cos cp (10) The resulting field still rotates with angular velocity tu , and the ratio of the semiaxes is; tan i|r (11) where t is; n , -2ab sin 2 \|r = — ^ r cos cp aT + b^ (12) and the azimuth X is: 1 - /• -2ab . . X = -g - arctan ( — — sin cp ) a + b (13) 2. Description of the Experimental Apparatus The experiments were made on a prismatic specimen of expanded phenolic resin of dimensions 100 x 100 x 45 mm, having eleven 0.2 mm diameter Cu-Constantan thermocouples^ , according to the schemes of figures 4a and 4b. Four of these couples, each placed on one of the sides of the specimen, are used for the automatic regulation; four others, side by side with the ones above, are connected to a tem- perature recorder for controlling the imposed oscillations; the remaining three, positioned inside the specimen, are used for detecting the rotating thermal field and are connected to an XY recorder. The sides of the specimen are covered by four externally blackened aluminum plates of dimensions 95 x 400 x 4 mm, which are used for the purpose of making the surface temperatures uniform. The com- plete specimen assembly was positioned inside a support structure bearing polished aluminum shields to direct radiating energy fluxes, emitted by each group of infrared radiation lamps L, onto the cor- responding surface of the specimen. Four automatic regulators R feed the groups of lamps in such a manner as to cause the temperature in the center of all four sides of the prism to vary sinusoidally with the same amplitude, and such that there is a tt/2 phase difference between the variations of adjacent sides in a clockwise sense around the specimen. The programming apparatus G of the thermal oscillations is a function generator producing two sinusoidal signals of variable amplitude and phase lag, respectively to 10 V and to 2rr. Each signal, used as a program by two opposed sides, was suitably reduced in amplitude through a dividing resistance P (fig. 4a), so as to reduce the amplitude of the e.m. f. to ± 0.5 mV around the set value of 0.5 mV. This e.m.f. amplitude is equivalent to 12°C of amplitude of temperature oscil- lation, for the given type of thermocouples. In this temperature range the linearization of the characteristic curve of the thermocouples results in an error of less than 0.1°C. The e.m.f. extracted from the dividing resistance is alternatively added to and subtracted from the signal given by the couples of the opposing faces, and the resulting signals are sent to the regulation units which feed the related infrared radiation heaters. On the basis of figures 4a and 4b, figure 5a presents the evolution in time of the e.m.f. to the regulation of zones 4 and 2, which are connected to the same signal from generator G. related v' extracted from dividing resistance P , i zone 4, and v". - v'. is sent into regulatio The e.m.f. thermocouple of *. -. , ;J / heaters of the relative zone in order to maintain the e. constant value 2 mV. s subtracted from v", , given by the on unit R4, which suitably feeds the f. corresponding to the said zone at the Systematic experiments, executed with various diameters of thermocouple joints and with various frequencies, indicated that in the foresaid conditions, the influence of the thermal capacity of the joints on the results of the measurements is not detectable by the registering apparatus. 166 References [1] Codegone.C, Ferro, V., and Sacchi, A., Prove di risonanza di oscillazioni termiche, - Atti Accademia delle Scienze di Torino 100 , 377-388, (1965/66). Ancora sulle oscillazioni termiche in piane, - Atti Accademia Scienze di Torino 100 , 627-634, (1965/66). [2] Grober, H. , and Erk, S. , Die Grundgesetze der Wanneubertragung, J. Springer, Berlin 81, (1933). [3] Perucca, E., Fisica generale e sperimentale , U.T.E.T. , Vol. 1, Torino, 128, (1949). & A = - 6 r cos coT = ©pcoscoT Figure 1 Scheme of the principle for the actuation of a bidimensional thermal field in the prismatic specimen of the considered material. 167 coT= 17/2,-11/2 COT- TT/4 / 3n/4 coT = o,tt -x Figure 2 Thermal unidimendional field in a slab with linear distribution of temperature along the thickness of the same. fix Figure 3 Rotating thermal field in a transversal section of the prism. 168 R4 R2 Figure 4a R3 R1 L2 -tr3 ! cm I a H N in. If ^ 2 If — ! L4 ;(I o°c^j £j 13 l>°c Figure 4b i-a3 - ^P \\//A \i V Jl l_3 :a iL 2 Li ^wr L 3 dcd 1 - Ixi 3 d> Figure 4a Functional scheme of the experimental apparatus. = wave generator L = infrared radiation lamps P = partition resistance R = P.I.D. action regulators with magnetic power amplifiers. Figure 4b Transversal section of the specimen with schematic indication of thermp couple used for: O regulation A recording • XT recording S = polished aluminum shields filled by insulating material. 169 C ;1 mV Figure 5a Bearing of e.m.f. relative to the regulation of zone 4. Figure 5b Bearing of e.m.f. relative to the regulation of zone 2. 170 1.7 1.9 2.1 2,3 2,5 27 2,9 3,1 3,3 I i i i ■ ■ i i I i f ■ f ■ f ■ i PC] 45 5C i ) ' 55 i 60 65 70 Hl'i | 75 80 i 1 l||]|| III lii \\\ ! : \H i[|j| Pj ; i! |||l|!||||||| fej 1 ' 1 ; 01 ! J ' 1 1 | liiji H " II 1 1 1 1 •ii mU 1 ' | ! If] j! II iff - II I II i t V Imii' : 1 I ill 1 1 II i Item ill Ml III ill 1 7 jir j .■ ' I | 4 [ j 1 J [Ftm 1 M 1 ' i . fl II [RmJ 1 JFM 11 I I' "" II 1 Ml | ! II 1 1 1 IjJII iii : M i Unl "m" iJJfff ImT frit : i ll/i rmi 1 Ii In J !■: H Fl m\\ IE il ill 1 j IJiUfmU lUllllIBi Jflflfll 1 I '111 - II ' ; - ||:i I i J fit' 1 ! i ■ ! jfl lii : ' ' ' il 1 1 - \i\ 1 ■ ' II i J|C- ■ l ; Mi M J .ill TMi 1 -■■ 1- i i Jfl IJlftD] TTTtprr J] 1 111 ^ l| i ' 1 |. ; II ! j ' III ! | I j; || : | - •1 i •\\w il 1 II | i - 1 ■ ' ■ ml Ni- 1 '1 j Mm il j I I; | • I | 3 : | r , lll-'Pil ii Li i J ' jj .1 1 f 1 i - inn ! h f 1 JK1 :!■ j 1 11 JiP'r i|tt z II | \ %\ m 11141111 ■ I T*6I II lillr ! | M | i j ||f j » II tjjjjjf II ' 'ill III r JJli j i 1 3 II \ III imlurl i II | [ ||| I 1 I II 1 1 1 f 1 1 1 II JJ] j ID 1 -1 II 1 II i 111 Ml 1 - II 111 'lm 1 11 Iff li II llllJIIII ill II 1 ■■% HI ! II 11 1 1 llrrrl 11 il Iflfft - II i N 1 ; - II jjjf Ml] lii 1 | lliU Hi ; li - II II IJJml JJJfii 1 - II 1 | , 1 1 Hfnil lily 1 Hi ' i! Ihmiii i 1 ^ II 1 i Hi ! WB i ' [Ltfffljl ' ill 1 - || ! J M M 1 i 1 1 i :jl| i| |Bf M JHHtti II |jj[l ! iiiiii!iyfin fcii Ml 1 lUlffllj . I lIlM 11 <7 |r| Mill! Mi 1 1 ii'ifii i 'j ■ K 'il 'lllilllllllllllllli m j] fI 1 i i || : ■ 1 1 i 1 illllllllllll il ij > in * r 1 hi 1 III i!llllli:i]||||i!lllllllllll!iJ|iL ml HldlUIIUll will 1 •_) Figure 6 Recording in time of the waves of temperature ^t>^?>^>^< imposed on the four sides of the prism (see fig. l) for a period T = 6 hs. 171 Figure 7 100 1 t+x 11 -y 1 % ■— ... J j! /' — \a b/ / ! x # 2 J — *»o / i li 3i i-x Figure 8 L.J-- Figure 7 Position of the thermocouples in the transversal section of the spe cimen for the detection of the rotating thermal field H. Figure 8 Scheme of the connections of the couples in figure 7 to the XY plotter for the field H. 172 o" CM o I CM i <*H \ 00- CM- O-i i EcM-f U 1 ' t > ' r o 1 !°C/cmj -3-2-10123456 I I I I I I I I i i ImVi -0,2 -0.1 0.1 0,2 0.3 0,4 Figure 9 Recording of the thermal rotating field H in the section of figure 7, imposing on the external sides the waves of figure 6 . CM o' 00 CM O- fj | * ^V 5 « 1 i 1 » 1 \ [°C/cmj -3-2-10123456 l I I i I I l l l I [mV] -0,2 -0,1 0,1 0.2 0,3 0.4 ! I I I I I I Figure 10 Recording of the unidimensional thermal field H , with connections to XY plotter as in figure 8. 173 [mV] 1.7 1.9 2.1 2.3 2.5 2.7 2.9 3.1 3.3 l I I l l l I I I I I I I I I I I Figure 11 Recording in time of temperature waves # and ®, imposed 6n sides 1 and 3 of the prism, for detecting unidimensional field H , the value of temperature on sides 2 and 4 is equal 0_ = 9, = 9 = +60°C. ^ 2 4 o 174 Figure 12a Photographic view of the experimental apparatus. 1 - Wave generator 2 - "Regulators of imposed thermal waves 3 - Recorder of imposed thermal waves 4 - XY plotter 5 - Lamps 6 - Vertical shields 7 - Horizontal shields 8 - Prism, seat of rotating thermal field. 175 Figure 12b Photographic view of the prismatic specimen, seat of rotating thermal field with basal and side shields. 176 Thermal Attenuation through Homogeneous and Multilayer Slabs in Steady Periodic Conditions - Theory and Experiments V. Ferro, A. Sacchi and C. Codegone 1 Istituto di Fisica Tecnica - Politecnico 10129 - Turin - (Italy) Tests of thermal oscillation through homogeneous or multilayer slabs have been carried out on a device consisting of two equal and plane samples with and without thermal capacity interposed. By a suitable control system identical, sinusoidal temperature oscillations have been imposed onto both external surfaces of the device and recorded, and the temperature oscillations received at the internal surfaces have been recorded. With a proper test arrangement, reduction factors of temperature waves at various frequencies have been found by this method, and the characteristic parameters of the thermal propagation through the homogeneous or composite slabs have been experimentally determined. The experimental results have been compared with those one can calculate from the theory of periodic thermal conduction through slabs. Key Words: Attenuation through slabs, characteristic parameters of periodic conduction, periodic thermal waves. 1. Introduction The study of the transmission of thermal steady waves through homogeneous or multilayer slabs has been carried out by several authors in the past[l - 6]^ and more recently by others[7 - 11]. In this work an experimental apparatus is examined to evaluate the attenuation and the phase dif- ference of thermal sinusoidal waves through a simple or a composite slab with and without thermal capacity in series to the same slab. Parameters are obtained from these tests which fully characterize the behaviour of the slabs in periodic thermal conditions, and moreover the field of agreement of theory with the experimental re- sults is determined. 2. Theory of the Phenomenon 2,1. Homogeneous Slab The Fourier equation for the one-dimensional heat flow through a flat and homogeneous slab infinitely extended in the direction normal to its thickness and which is free of heat sources or sinks, is expressed through the Laplace transform © of the temperature 9, by the well-known relation: £f = ±p© (i) dx^ a ^Assistants and Professor of Technical Physics, respectively. ^Figures in brackets indicate the literature references at the end of this paper. 177 where x is the abscissa of the generic isothermal plane surface, a is the thermal diffusivity and p the auxiliary variable introduced by the transformation. If a thermal capacity 2C. , referred to the unit surface and expressed in kJm~ 2 °C -1 is put in the middle of the slab, eq (1) is still valid with the following boundary conditions transformed according to Laplace, (fig. 1): at x = + i © = ©„=©.(, (2) at x = \CjLJ) =c,p© =-$ . (3) dx x=0 where I is the thickness of the slab and \ is the thermal conductivity of the material, and $ repre- sents the Laplace transform of the flux cp. When the temperature changes on the external surfaces of the slab are sinusoidal with pulsation U) = 2rrf , the steady state solution of eq (1) with p = jtu (j = /-l) and with the boundary conditions (2) and (3), becomes: © = O cos h {(1 + j) x [uj/2a] 1 / 2 } + j*C 1 4 ) ® sin h {(1 + j) x |>/(2a)] 1/2 } = A© + BjiuC^ (4) where the expressions for A and B are obvious from eq (4). The heat flux in the slab is given by cp = -XdG/dx (5) and its Laplace transform for this case is: * = [$ /jc U C i Z ro )]{sinh (1 + j) x [u)/(2a)] 1/2 + i Q cosh {(1 + j) x [u ) /(2a)] 1 / 2 } *o ■ C — + D$ *o < 6 > jiuC. r O' where Zx = (1 - j)/ /2u)Xpc (p = density [kg m"3] and c = specific heat [ Jkg" lo C"l] is the character- istic thermal impedance of the medium which is assumed to be infinitely extended in the x-direction, and through which the thermal wave propagates. The expression for the parameters A and b are easily deduced from eq (4). The expressions for C and D result from eq (6) and for a homogeneous slab are: A = D and B = C^ 2 . Developing eqs (4) and (6), and separating the real part from the imaginary part yield the re- duction factors a of temperature and heat flux oxcillations through the slab between the surfaces x = + I and x = 0. We obtain for the moduli of a: I 2 + < N i + ReN 2 ) 2 ]" 1/2 (7) I <7q> I = %l®i =

l , eqs. (9), (11) and (12) can be written: q, = 8.686e - 6.02 [db] (17) i|r . ~ e dt = (1 - tan|eV(l + tane) (18), (19) °>9 °>cp and, if e = TT, the eqs. (17), (18) and (19) are simply: Of =21.082 t = tt * = tt/4 (20), (21), (22) 0,9 O,0 O ,m which represent half-wave stationary conditions through the slab and in this case the pulsation uj =■ 2-rr 3 a/£ 2 results [12]. In figures 2,3 and 4 eqs (7)-(19) are plotted versus the non-dimensional parameters e and R. 2.2. Multilayer Slabs Theoretical analysis, performed by several authors [5], [6], [ 131 > has established that the com- posite slabs, made of several sublayers of homogeneous isotropic materials arranged normally to flow lines, may be treated as a cascade of four- terminal networks if the boundary effects are negligible. Of course, each four-terminal network of the cascade is supposed to be passive, linear, and bilateral, and moreover the wave amplitudes are assumed relatively small so that the thermal diffusivity may be considered a constant and hence the superposition of the effects due to different frequencies is appli- able. The discussion is carried out generally using the theory of complex variable functions. We indi- cate a generical frequency f = uu /(2tt) = 1/t ( E F G H v v Measurement of D v Both slab elements are reversed (fig. 5c) , so that the surfaces S are in mutual contact and we then proceed according to section 3.1. The temperature oscillation 9 is applied to the outer surfaces S and the corresponding oscillation q is now recorded at the facing surfaces S . From eq (24) , remembering that with this arrangement m = , we obtain: ^ov ov If the slab is symmetric, it follows A = D and the test shown in this section is not necessary. This is also true for the case of a homogeneous slab, where a = d . The parameter C can now be evaluated from eq (25). For a homogeneous slab it is simply c = b /Z 2 . 4. Description of the Experimental Equipment The equipment used to perform the experiments on the multilayers slabs is shown schematically in figure 6. Slab samples M 1 and M , placed according to one of the arrangements described in the pre- ceding section, are fastened between two metal plates 6a and 6b with guard rings 3 a an d 3b. The plates and the rings are equipped with nickel-chrome heating resistances 2a, 2b, 5a and 5b, and face the outer surfaces of the sample layers M. , which are in good thermal contact with them. Two cooling tanks, la and lb, are located next to the heating plates. Each one of them faces the heating elements and has a copper plate (size 500 x 500 x 10 mm) which is kept isothermal by circulating water or methyl alcohol at a temperature about 25-40°C lower than the mean temperature of the heating elements. The temperature value in the range 25-40°C is chosen as a function of the wave frequency. The temperature of the cooling tanks is kept at a value much lower than the oscillating temperature minimum, while the control systems deliver the power required to impose the programmed temperature oscillation on the slab surfaces. Two asbestos sheets, 4a and 4b, separate the cooling tanks from the heating plates. The temperature oscillation amplitudes must have limited values (=s5-10°C) , so that the transport parameters can be considered constant and therefore the heat flux equations linear, as it was assumed in the previous sections. A sinusoidal wave generator G, with pre-arrangeable amplitude and frequency, causes the temperature to oscillate at the central heating unit (5a and 5b) through two control systems R 1 and R,-, whereas the two guard sections 2a and 2b follow the temperature of the corresponding heating unit by means of two differential thermocouples, which are connected to control units R fi and R respectively. In order to avoid errors due to heat losses at the edges when the composite slab includes some good conductor layer, or in some instances a layer with a thermal diffusivity higher than others in the sample, the influence of the conducting layer is compensated for on an average by means of the peripheral heating resistances r and r , controlled by the systems R and R . These control systems operate so as to equalize the temperatures of the center and of the periphery of the conducting layer, which has a thermal capacity of 2C. . An additional control element R, eliminates the heat losses at the edges of the ther- mal capacity 2C. by means of a peripheral heating resistance r, and a cooling pipe 7. -2 -1 The thermal capacity 2C . is built of an aluminum plate 1 mm thick [C. = 1195 Jm °C J or 2 mm -2 -1 1 -2 -1 :L thick [C. = 2390 Jm °C ] or 4 mm thick [C. = 4780 Jm °C ], which is suitably divided in concentric squares separated by air annuli to further reduce the peripheral heat losses. All control systems used are of the P.I.D. (proportional, integral and differential) type, with final power amplifiers. The whole assembly, including samples, heating units, and cooling tanks, is tightened in the thermal flux direction and is thermally insulated on the peripheral surfaces with fiberglass. It is then placed in a usual room for oscillating tests at mean temperature in the range +40 to +60°C, or in a refrigerated box with dried atmosphere for tests at a mean temperature of about 0°C or lower. In this second case the box is kept at the average temperature of the programmed thermal oscillation by means of a suitable control system. Chromel-Alumel thermocouples (0.32 mm dia.) are located on outer and inner surfaces of the slab samples and in the middle of the system within suitable grooves. They follow in part an isothermal path and are connected to a multipoint potentiometric recorder T. This experimental apparatus is obviously also employed for tests on homogeneous slabs. A modifi- cation of this device uses a radiant heating source [12], [14] instead of a conducting heating source like plates 6a, b and 3 a > b of figure 6. 182 5. Experimental Results 5.1. Homogeneous Slabs Figures 7a and 7b reproduce the recordings of the imposed temperature wave 9 and of the trans- mitted wave 9 t obtained for a sample of rigid polyurethane foam, with and without thermal capacity C. in series. Tnree types of synthetic expanded materials were tested: polystyrene, phenolic resin and polyurethane. Through the experimental measurements of the equivalent thermal conductivity \, the apparent den- sity , and the equivalent specific heat c 3 , the diffusivity value _a and therefore the value of the dimensionless parameter e were deduced for the samples tested. From this value of e and through the theoretical relationships (7), (9), (11), (13), (15), (17), (18), we have plotted the curves of the attenuation (or reduction factor) and the phase difference, which in figures 8-11 are compared with the experimental values obtained by means of the assembly described in the previous paragraph. The results also confirm the validity of the previous relationships and the admissibility of the expression a = \/ (pc) of the thermal diffusivity for expanded insulating materials, in which temperature oscillations of limited amplitudes (~5-10°C) occur. These materials are microscopically non-homogeneous, insofar as they are composed of small cells, where the conduction phenomenon is coupled with radiation and con- vection phenomena. The latter indeed is rather limited, being questionable for materials of low gas permeability. Tables 1, 2 and 3 show the physical properties of tested materials and the wave periods, which have been used to calculate the theoretical curves. Further results are given in reference [15]. The wave periods in these tables are also the same as those of the tests. Table 1. Expanded Phenolic Resin. Mean temperature of the slab Slab thickness Equivalent thermal conductivity at +45°C Equivalent specific heat at +45°C Apparent density Diffusivity Thermal capacity R = C./( p cl) = Period Table 2. Expanded Polystyrene. Mean temperature of the slab 9 = +45°C Slab thickness 2Z =0.1 Equivalent thermal conductivity at +45°C \ = O.O38 Equivalent specific heat at +45°C c = 1.215 Apparent density p = 14.6 Diffusivity a = \/( p c) = 21.2-10" 7 Thermal capacity C. = 0, 1.195, 2.390, 4.780 R = C./( p cl) = =0, 1.35, 2.70, 5.40 Period T = 0.5, 1,2,3 hours 9 = +45 m °C 2£ =0.1 m \ = 0.042 Wm" 1 ^" 1 c = 1.17 kJ kg" lo C _1 p = 42.6 -3 kgm ' a = X/( p c) = 8 .43' 10" ■7 2 -1 m s C. = 0, 1.195 l 2.390, 4. 780 -P -1 kJm °C = 0, 0.48, .96 1 92 t = 0.5, 1, 3 - > 3 hours Wm •lo c -l kJ kg^'c- 1 kg m-3 2 m -1 s kJ m L We speak of equivalent or apparent transport property because we have an expanded, porous material as a sample. 183 W m" 1 ^" 1 kJ kg-^c" 1 kg .-3 2 m -l s kJ m C Table 3. Rigid Polyurethane Foam (Expanded with F12) . Mean temperature of the slab 9 = +63°C Slab thickness 2£ = 0.102 Equivalent thermal conductivity at +63°C \ = 0.102 Equivalent specific heat at +63°C c = 1.385 Apparent density p = 36.9 Diffusivity a = \/( p c) = 4.64-10" Thermal capacity C. = 0, 2.390, 4.780 R = C./( p cl) = =0, 0.92, 1.84 Period _ = 1, 2.5, 6 hours 5.2. Multilayer Slabs Some types of multilayer slabs have been examined, and their characteristics are listed in table 4. The experiments outlined in sections 3.1-. and 3-2. have been performed on symmetric samples, where as those explained in sections 3-l-> 3«2. and 3»3- were performed on asymmetric samples. The imposed thermal wave has a mean temperature value of about +40°C and an amplitude in the range ± 5-7°C . The moduli and the phase differences of the reduction factor , which have been found by the experiments, are given in table 5. According to eqs (28), (30) , (32) and (25) the characteristic para- meters A , B , C , D of multilayer slabs have been determined. The value of B has been taken as the y ' V ' V * V J V average between the values of B_ and B, relative to the two different thermal capacities fB_ with -2„ -1 2v 4v _p„ -1 v L 2v C. = 2390 J m °C and B, with C = 4780 J m °C ]. The results are listed in table 6; in figure 12 are given some recordings of the temperature waves. Afterwards numerical calculations have been made to check the agreement between theoretical and experimental values of the oversaid parameters. The theoretical values for the various tested slabs have been calculated following the theory given in a preceding paper [16] and assuming the quantities given in table 4. The calculated values are listed in table 7. Further results are given in reference [17 J'. In the range of temperature amplitudes chosen the agreement between theoretical and experimental values is satisfactory, considering that the slabs examined are made of usual building material. 4 To evaluate the influence of the thermal joint capacities on the attenuation and on the phase lag, some experiments have been made with wire of diameters O.32, 0.5 and 1.02 mm, at frequencies in the range 1x10"^ to 1x10"^ Hz. In this range and with the wire of diameter O.32 mm the influence of the said capacity is not detectable by the recorders. 184 Table 4. Properties of the Tested MultiLayer Slabs, Slab Layer Material I A C c q (a) IP - - - m Wm- l0 C _1 kg m Jkg-^C" 1 w ffl - 2o c"' 1 1 Eternit 0.02 0.55 1850 930 2 Expanded polystyrene 0.018 0.041 12.3 1215 2.11 £ 2 1 Expanded polystyrene 0.018 0.041 12.3 1215 2.11 2 Eternit 0.02 0.55 1850 930 13 3 1 Eternit 0.005 0.55 1880 930 o 2 Expanded polystyrene 0.019 0.041 13 1215 2.08 e £ Slab tesr j.i. f\ C; = 2390 J m" 2 °C _1 Ci = 4780 J m" 2 °C _1 i es i o . o . Uy, ft /ft 1 k /ft 1 ov r\v ft /ft ov nv k /ft 1 ov nv ft /ft 1 ov nv| Iff /ft 1 ov x\v k At 1 nv ov Iff /ft 1 nv ov - hours - rad - rad - rad - rad 1 2 3 4 2.5 6 15 2.5 6 15 1 2.5 6 1 2.5 6 11.30 5.41 2.26 1.17 1.03 1.02 7.63 3.03 1.56 3.70 1.73 1.22 1.49 1.35 1.11 0.49 0.20 0.11 1.49 1.13 0.84 1.76 1.16 0.61 12.03 5.71 2.37 1.49 1.10 1.024 9.6 3.78 1.82 4.63 1.98 1.29 1.52 1.37 1.14 1.00 0.51 0.25 1.54 1.24 0.98 1.89 1.26 0.70 12.78 6.03 2.48 2.07 1.27 1.04 11.5 4.50 2.13 5.4 2.24 1.36 1.55 1.39 1.15 1.37 0.77 0.38 1.58 1.30 1.07 2.05 1.35 0.78 1.17 1.03 1.02 11.30 5.41 2.26 0.49 0.20 0.11 1.49 1.35 1.11 185 Table 6. Experimental Characteristic Parameters of Multilayer Slabs (a) Slab TV &2v Hy bv C V mod arg mod arg mod | arg mod arg - hours m C W rad 2o_...-l m C W rad 2o_ ,.,-1 m C W rad ... -2o_-l Wm C rad 1 2.5 0.485 0.380 0.479 0.416 0.482 0.398 28.3 1 .66 6 0.460 0.143 0.475 0.153 0.468 0.148 12.1 1.56 15 0.467 0.116 0.471 0.143 0.469 0.130 4.63 1.53 2 2.5 0.443 0.313 0.484 0.393 0.464 0.353 29.4 1.70 6 0.483 0.148 0.454 0.156 0.469 0.152 12.0 1.55 15 0.514 0.153 0.500 0.173 0.507 0.163 4.28 1.50 3 1 0.485 0.183 0.478 0.185 0.482 0.184 123 2.79 2.5 0.500 0.078 0.479 0.046 0.490 0.062 20.1 2.28 6 0.505 0.078 0.508 0.026 0.507 0.052 5.40 2.01 4 1 0.258 0.785 0.238 0.842 0.248 0.814 58.7 2.71 2.5 0.190 0.292 0.191 0.328 0.191 0.310 19.6 2.21 6 0.191 0.101 0.186 0.127 0.189 0.114 7.83 1.79 (a) The values of A,, and D,, are listed in table V Table 7. Theoretical Characteristic Parameters of Multilayer Slabs, Slab \ Av B, C v »v mod arg mod arg mod arg mod arg - hours - rad 2o_ . . .-1 m C W rad Wm" 2 °C rad - rad 1 2 3 4 2.5 6 15 2.5 6 15 1 2.5 6 1 2.5 6 12.35 4.967 2.074 1.067 1.012 1.002 7.33 3.09 1.68 3.683 1.709 1.150 1.596 1.393 1.052 0.473 0.206 0.099 1.547 1.254 0.91 1.760 1.205 0.613 0.492 0.480 0.472 0.492 0.480 0.472 0.484 0.495 0.497 0.214 0.193 0.183 0.405 0.150 0.134 0.405 0.150 0.134 0.180 0.061 0.044 0.710 0.340 0.130 27.9 10.8 4.03 27.9 10.8 4.03 113 21.0 6.30 68.2 19.2 7.37 1.718 1.639 1.524 1.718 1.639 1.524 2.909 2.497 2.037 2.793 2.237 1.839 1.067 1.012 1.002 12.35 4.967 2.074 7.33 3.09 1.68 3.683 1.709 1.150 0.473 0.206 0.099 1.596 1.393 1.052 1.547 1.254 0.910 1.760 1.205 0.613 186 References [1] Broeber H. , Erk, S. , Die Grundgesetze der Warmeubertragung , Springer Verlag, Berlin, 70, (19J3). [2] Alford, J. S., Ryan, J.E., Urban P.O., Effect of Heat Storage and Variation in Outdoor Temperature and Solar Intensity on Heat Trans- [10] fer Through Walls, Ashve Transaction, 4_5, 369, (1939). [31 Marmet Ph. , Etude du regime thermique sinusoidal des murs et des locaux par voie ' analogie elec- trique, Compte Rendu du Septieme Congres [H] International du Chauffage, Ventilation et du Conditionnement, Paris, 236-302, (1947). [9] Warsi, Z. U. A., Choudhury, N. K. D. , Weighing Function and Transient Thermal Response of Buildings, Part II, Composite Structure, Int. Journal of Heat and Mass Transfer, 7_, 1322-1334. (1964). [4] Mackey, C. 0., Wright, L. T. , Periodic Heat Flow. Homogeneous Walls, Heating, Piping, 16 , 546-555, (1944). Periodic Heat Flow. Composite Walls or Roofs, Heating, Piping, 18, 107-110, (1946). [5] Carslaw, H. S., Jaeger, J. C. , Conduction of Heat in Solids, Clarendon Press, Oxford, 88, (1948). Columba, M. , Lo Giudice , G. , Trasmissione del calore in regime periodico negli edifici, Calcolo Numerico, Atti Accademia di Scienze, Lettere ed Arti di Palermo, Serie IV, Vol. XXV, Parte 1, 267-322, (1964-65). Sacchi, A., Sul Calcolo della attenuazione termica di pareti in regime periodico, La Termotecnica, Vol. XX, 60-65, (1966). [12] Ferro, V., Sacchi, A., Oscillazioni termiche in pareti semplici anche in condizioni di risonanza, La Termotecnica, Ricerche n. 16, Vol. XX, 8, (1966). [13] Vodicka, V., Conduction of Fluctuating Heat Flow in a Wall Consisting of Many Layers, Applied Scientific Research, Vol. A-V, 108- 114, (1956). [6] Mattarolo, L., Analogia elettrica e metodo dei [1^] quadripoli in problemi di trasmissione termica, La Termotecnica, Vol. VI, 355-363, (1952). n Ferro, V., Sacchi, A., Oscillazioni termiche in pareti composete, La Termotecnica, Ricerche 16, Vol. XX, 22, (1966). [7] Barducci, I., Calcolo approssimato dell'attenu- [15] azione termica di pareti in regime periodico, sotto condizioni piu generali di quelle di Alford, Ryan e Urban, Quaderni di Fisica Tecnica n. 2, Universita di Roma, Facolta di Ingegneria, (1964). [16] [8] Choudhury, N. K. D. , Warsi, Z. U. A., Weighing Function and Transient Thermal Response of Buildings. Part I, Homogeneous Structure, Int. Journal of Heat and Mass Transfer, ]_, 1309-1321, (1964). [17] Codegone , C. , Ferro, V., Sacchi, A., Experi- ments on Thermal Oscillations through Insulat- ing Materials, Supplement au Bull. del ' Inst. Int. du Froid, Annexe, 175, (1966-2). Boffa, C. , Ferro, V., Sacchi, A., Tabelle numeriche per il calcolo rapido dei parametri caratteristici di pareti composte, La Termotecnica, Ricerche n. 16, Vol. XX, 52, (1966). Codegone, C. , Ferro, V., Sacchi, A., Thermal Oscillations through Walls , Comm. 2, XII e Congres International du Froid, Madrid, (1967), 187 Fig. 1 Thermal test diagram with interposed capacity 2 Cj «J db ] 0,001 2.5 e=hcj/<2a) Fig. 2 Reduction factor -nodulus O^n anc ' attenuation d. « of temperature oscillations vs. adimensional parameters 5 and R. 188 Ye.Yt l\ Trail 4- 2- i I farad i sessagl 200 150 2.5 c oj 3_ Fig. 3 Phase difference >!/„ and •v^% vs. adimensional parameters £ and R. 189 Fig. 4 3 e=e\rzGA2^ Reduction factor modulus \0'\f and attenuation -y^ of thermal flux oscil lations vs. adimensional parameters £ and R. Oo ^n^n ^o Figure 5a. Thermal test diagram to measure the parameter A v . bn ^o^>o 5n Figure 5b. Thermal test diagram to measure the parameter B v . Figure 5c. Thermal test diagram to measure the parameter Dj, . 190 ■o c o D Q. QL D E o b) D "5 c o Q. O 00 •rH 191 50 55 60 65 70 75 r;|iiii['||i|;|[|[||[|||[ iinii in ii in ii iiiiHiiiiiii lllll lllll Irtish mil! 1 1 ii iiiiii'iPiiiiin mil ii jjUfflfM ' I 1 !', 1 i Utt 1 .'! i;|| Ii; 1 1 1 1 ill i, ! 1 II]] j;| ' | / i ' , -| I 1 1 1 1 Fif 1 ' hl+Jil f III Hill 1 '! il II (a) H| 1 ! "jJIJjjii | I, | :| If i IH '11 "I 1 i i' 1 'ill >\ P llllP'l '!! ! i ' ' r ! '' J It 50 55 60 65 70 75 Fig. 7-a Recording of the imposed temperature waves 'ft « = -6' « and of the recorded wave $ tested material: rigid urethane foam expanded with F 12 - y& / 'ftp I = 0.49; arg ( -9 Q /S^ ) = 1.40 rad; Period £=2.5h. Fig. 7-b Recording of the imposed temperature waves ~& a = ty n and of the received waves *& -2-1 ° with interposed thermal capacity C. =4.780 kJ m C -tested material: the same of Fig. 7-a -\^ Q / ^ ^\ =0.108; arg ( W / # , ) = 1 .92 rad; Period 1 =2.5h. 192 XI ■a CD o o CM I o "3- E c (-»- o ^ a> O C\J « ■a -o i- C o c o a. Q. E X z _o c o o c % M- JS o _y "5 u> *o -*- c -* £ w> ii c^^ a. 0) X _c — ■* T3 o c o, C o 14- ac >. so "to > -a >v c 3 o o O Q- ~o U) o "O L. C o a. _c X i— E m o* -o 8 193 Fig. 9 Theoretical curves and experimental values of "U/"q,vs. adimensional parameter 8 and R for the same slabs and mean temperature of fig. 8. 194 200 175 150- 125 100- 75 50 25 fgradi 1 [sessag.J To [rad.; 0,5 1,5 2,5 £ Fig. 10 Like fig. 8 but the slab tested is rigid urethane foam expanded with F 12 {P = 36.9 kg/n and the mean temperature is >& = + 63 C , 69 1.00- 0,01 0,5 O e [db] -0 20 40 1,0 1.5 2,0 2,5 3,0 £ Fig. 11 Like fig. 9 for the same material of fig. 10. 195 °C40-[ 30- _ 1 Ci 2390 J/(m»°C) p~ 2 S r || ^ jj= gig t=g= u__ r „ !gj 3s! &i ifet-— F=p -r ^ pr. ic- pS |I nine ^ = OD^ _ -.--p ===== I— i — | — | — i — |- *5 m" Ifcj — [■ =3 fe (=< 51= *$ =R^ =s g^ : 1 *L — L - Jj|| °C40- -: - = :■" - - -| = : --H - -^ i j = = =3 =&-= = = - 1 J ^ = E= - | ~H= = == j^5 ^0 r i z ^V ^ = ^ & = == : : i - \^E — r = = ~ =: ' -- = 1 - t 1-- -J ' . -~ -. - ZP'-l = 45- ^i ^ R ri- :- = =- = _ -- ^ i 7 : ~%~ = ~ 3 ~ t =-"-- r- i=- 3 = - - 7 : ^ f ; --■- = = - j -: - - i - = °C40 Fig. 12 Recordings of imposed temperature waves '©' and received 4^ in the case of multilayer slab N° 3 and for the period £ = 2.5h. The intermediate wave recorded between 'O' and (5) where , P (J!pt), E . I- (i!pL) + F . Ko (i!p) G n(n. rJ )=Io(^)K (^)-I (^)K (ip) Substituting B.C. 1 and 2 in the above equation and developing the Fourier coefficient integral conditions in the usual way for this orthogonal family (3) leads to the following general solution of the differential equation meeting boundary conditions 1 and 2, co . / n7rz \ _ 2 » sin(-j— ) T(r ' z) = f Z r I I (Gn(r,r )/ f,(z ) sin (-Ofi-) dz + G n (r„r) jf f (z)sin (^) dz} (6) Considering now the solution involving boundary condition 4, separation of variables leads to the following general solution of Laplace's equation, T(r,z) = [AJ (/cr) + BY (/cr)] [c cosh (/cz) + D sin h(/cz)] (7) where , A, B, C, D, = CONSTANTS The T = boundary condition along the surface z = leads to the requirement that Applying the boundary conditions C=0 r = r , Z, T=0 r = r,, Z, T = and eliminating the constants A and B from the equations that result leads to the following condition which must be fulfilled: 200 Y (/cr ) JofArrJ-YoUr.JJotArro) = (8) The above condition is met by an infinite number of eigenvalues, K n , which are the positive roots of this equation. Introducing cylindrical function notation, U (/c n ; r ;r,) = J (^n r o) Yo(K" n r^) - J (K n r,) Y (K n r ) (9; the eigenvalues, K n , are defined by u o(*ni r oi r,) = 0, n = l,2,3, (10) Thus a particular solution of the differential equation becomes T(r,z) n = [A n J (/c n r) + B n J (/c n r)] [d„ sin h (/c n z)] (11) where again the constants are subscripted to reflect the family of solutions represented by the eignevalues, K n Writing the corresponding infinite series solution and recasting the form of the constants leads to the following representation for T(r,z), R(/c n r|)LUo(/c n ;r ;r)J + R(/c n ro)[u (K- ni r ; r,)J T(r,z) = V — sinh(/ and r = r respectively, then it can be f 2 (r)=f (L)+[f l (L)-f (L)]in(-f o )/ln(i.) (14) Based on the experimental temperature distribution measurements f Q (z) and f-^(z) can be quite accurately represented in the form, fo(2)'=T o (l-0C)+CCT, LO (15) where , f,(Z)=T,(l — CC) + CCT L1 CC=2/l To=f (0), T L ,= f,(L) (16) Tlo = Ul) Using eqs (14)-(16) for explicit evaluation of the various Fourier coefficient integrals leads to the following equation for the temperature distribution in the insulation segment. . . y y r Jo(*nro)Uo(*n;r;r )SIN h(* n z) "I ^ ln (fe) wi L [ J o 2 Kr )- J 2 (* n r,)]siN h(*c n L)J [{T LO ln(-^)(l + lnr )-T u ln(r )}{j (' jo(fnn) ( r m(Ji)£r n,7nj o(*nro) .' COSh(/c n L)-l (18) Where , ^n=J (' c nro)/[{jo 2 (^nro)-J 2 (K n r l )}siNh(/c n L)] 7 7n = [T LO ln(-^-){l + lnr }-T L1 lnr ][j (/c n r )-J (/c n r 1 )] + ( T Li-T L0 ){j (' ,: nr )lnr l -J (/fnn)lnr } r'^.r.i-T/IML / n77 " r J\, /nirri \ /nTrrjx Alternate terms of the first summation are zero due to the factor[|-(-| ) Jhence only odd values of n need be considered. In the second summation determination of the eigenvalues, K n , as defined by eq (10) is necessary. An infinite series approximation given by Gray and Mathews (4) was applied; limited tabulations are also found in Jahnke and Emde (5). The first few roots are given in Table 1 for the case where r^/^ = 6.0. ^ or n > 6, the roots are given to within 0.1% by eq (19), ^n lo T\1T/ (19) Table 1 - Positive Rofrts of U (Knj r jr. ) = When r,/r = 6.0 •o ^n 1 n K. 1 0.226 2 1.233 3 1.874 4 2.506 5 3.134 6 3.764 7 4.393 8 5.023 203 IBM 7094 evaluation of eq ( 18 ) , discussion of convergence characteristics and comparison with ex- perimental heat loss measurements is given in the next section. 3. Experimental-Analytical Comparisons In evaluating the series in eq (18) term by term print out of the computer results was em- ployed to study convergence characteristics. Hand calculation of several terms was also run to check the computer results. The first series was carried to n = 40. For n > 20 the terms con- tributed no more than 0.3% to the sum. In the case of the second series the first term in the summation, represented by K t t varied between 70% and 93% of the series sum. Subsequent terms alternated sign with decreasing magnitude, as might be expected for a series having linear J (K n r) factors. Terms through k 9 were evaluated to obtain a sum varying less than 1% with in- clusion of higher terms. Equation (18) requires values of the thermal insulation conductivity. This conductivity was not measured in the investigation, thus it was necessary to estimate values from data in the literature. Literature data on alumina powders are reported for a range of densities and parti- cle sizes. Extrapolation of the data to other ranges of density or particle size is not straightforward because of the high radiative contribution to the total thermal conductivity. The data of Laubitz (7) was found to be particularly useful since the solid conduction and radiation contributions to the effective conductivity were presented separately as a function of temperature. Further, the data were found to be consistent with other alumina powder data in the same density and particle size range (8,9). The results of Laubitz, measured in vacuum, on a powder with density of 1.77 gms/cm^ and particle sizes in the range 300-420 /J, is given as curve (1) in figure 2. This particle size range encompasses the average particle diameter of the alumina bubbles used here, d = 396 /x. The alumina bubbles have a lower density, however, the average value being 1.01 gms/cm^. The bubbles are hollow but since the radiative contribution is essentially size dependent, (see eq (2), ref. 7), it was assumed that this contribution was the same as that inferred by Laubitz for material having the same particle size range as the alumina bubbles (curve (2), fig 2). The solid conduction contribution for the alumina bubbles was calculated by reducing that measured by Laubitz in the ratio 1.01/1.77. The estimated effective conductivity for the alumina bubbles, found by adding the conductive and radiative contributions, is given as curve (3) in figure 2. That this curve is of the right order of magnitude is demonstrated by the dotted curve in the figure. This curve gives vacuum environment conductivity results by Wechsler and Glaser (10)for an alumina bubble/graphite fiber composite. Using curve (3), figure 2, and the experimental temperature distribution data given in Table 2, eq (18) was evaluated. Table 2. Insulation Segment Temperature Distribution Data Data Point T L l(°C) Tl(°C) Tlo(°C) T (°C) T i (oc) w """ i(r deg 105.2 44.6 74.0 28.1 74.9 0.0115 145.0 55.3 120.5 43.9 100.2 0.0144 363.9 181.9 397.7 73.9 272.9 0.0259 437.4 216.2 502.2 80.6 326.8 0.0302 479.0 237.2 510.0 86.7 358.1 0.0332 531.3 257.2 522.2 90.0 394.3 0.0374 571.1 264.7 657.2 86.7 418.0 0.0418 651.3 278.5 836.5 45.6 461.8 0.0475 723.0 282.9 1042.5 15.6 503.2 0.0548 The comparison between the experimental heat loss measurements and those predicted by eq (18) are presented in Table 3 as percentages of the average heat flux in the specimen column. Positive values represent heat flow from the specimens into the insulation. If (T„ ) m is the mid-point temperature of the beryllium oxide specimen and T quarc j is the guard heater temperature at the same axial position, then (Tg uar< 3/ (^ggQ ) m -l )100 is the percentage by which the guard temperature exceeds that of the BeO specimen; this quantity will be termed the guard mismatch. For data points 4-6, the mean temperature of the BeO was held approximately constant while the guard mismatch varied from 8 to 23%. As seen in Table 3 the heat exchange between specimen column and insulation varied only 1.3% as observed experimentally and 2.9% according to the 204 analytical prediction. Thus thermal guarding of the specimen column does have an effect on the heat flowing in the column but the effect is not of high sensitivity. The experimental con- ductivity data was obtained with guard mismatching ranging from +23% to -11%. Across this wide range the average heat exchange between insulation and specimen column was 4.7%. Table 3. Experimental-Analytical Heat Exchange Values Data Point Average Heat Flux in Predicted Value Experimental Value the Specimen Column(W) 1 11.0 2.6% 5.4% 2 18.4 2.8 6.4 3 56.6 7.1 5.5 4 67.6 9.0 5.4 5 67.5 10.0 5.4 6 67.2 11.9 4.1 7 81.0 14.0 4.0 8 103.8 16.8 -1.1 9 130.0 19.7 -5.5 It is important to note that there is some uncertainty in the experimental heat exchange values. For data points. 4 and above these exchange values are obtained by taking the difference between large heat flux values, hence uncertainty in these fluxes greatly accentuates un- certainties in the heat exchange values. For low temperature data points the principal un- certainty in the heat exchange values is the error in the temperature gradient determinations in the meter bars above and below the BeO specimen. This uncertainty is particularly important here where the overall temperature drops in the meter bars are relatively low. The change in sign of the experimental heat exchange values for the higher temperature data points are internally consistent. That is, in those cases where the exchange is positive (heat flow into the insulation), the thermocouples on the specimen surface give lower readings than those in the same plane at the specimen centerline. When the experimentally observed heat ex- change values change sign, the surface thermocouples are found to read higher than the corres- ponding centerline couples. Now comparing the experimental heat exchange values with the analytical predictions, based on the uncertainties in the experimental values discussed above, the predictions are consistent at least through data point 7. The last two data point comparisons in Table 3 require further elaboration. Examining the experimental thermocouple data, there is an uncertainty of the gradient in the meter bars of 3.1%. In the cases where the measured heat exchange is near zero and going negative this un- certainty in the meter bar gradient introduces a particularly large uncertainty in the heat ex- change value. The range for data point 8 is from -1.1% to +2.8% while for data point 9 it is from -5.5% to -2.2%. Further, the magnitude of the heat exchange is large here compared with the low temperature cases. For data points 8 and 9 this radial flux is from 10 to 12 times that for data point 1. For these higher temperature data points, the comparatively high radial fluxes introduce the possibility of significant temperature drops at the specimen-insulation interfaces and the insulation-guard heater interface. These temperature discontinuities result from the contact resistance associated with the interfaces and become significant when the radial flux is large. Assuming a conservatively high interface thermal contact conductance of 20 BTU/hr-f t 2 -°F, the interfacial temperature drop is at least 20°F for the last two data points in Table 3. This factor is significant with respect to the analytical predictions because the temperature distributions used in evaluating eq (18) were not measured directly in the insulation but in the specimen column and the guard heater ring. This is certainly one of the largest factors in the disagreement between the heat exchange values for the high temperature data points. In general the heat exchange predictions are of the right magnitude. However, it is apparent that to provide a more rigorous test of heat exchange predictions it would be necessary to conduct extensive measurements with that specific aim in mind. Such measurements would be most difficult to perform due to the necessity of accurately measuring temperatures directly in the low density insulation material. 205 100 1.2 3 00 0.8 > h- 0.6 o O o o 0.4 UJ I i- 0.2 200 200 300 TEMPERATURE, °C 400 500 600 700 800 900 DENSITY PARTICLE SIZE CURVE (gm/cm 3 ) (/*) REF. © 1.77 300-420 7 ® 1.77 300-420 7 (D 1.01 396 - — 0.98 250-300 10 400 600 1200 1400 800 1000 TEMPERATURE, °F FIGURE 2 - Thermal Insulation Conductivity Comparisons 1600 0.16 o. T3 I E •v 0.12 >* o 3 Q 08 o tr u X 0.04 H 4. Nomenclature Because the final write-up for this paper was completed before the information on units and nomenclature was available, the symbols are not entirely consistent with those of the Proceed- ings. The following listing gives symbol designations unique to this paper. q Thermal Conductivity Eigenvalue Parameters Heat Flux References (1) Minges, M.L. , "Comparative Conductivity and Thermal Contact Resistance Measurements" , p. IV-B-20 to IV-B-31, Proc 4th Thermal CondConf, (1964). (2) Minges, M.L. and Denman, G.L. , "Progress Report, AFML Thermal Properties Group", Proc Fifth Thermal Cond Conf , II, (1965). (3) Kreyszig, E. , Advanced Engineering Mathe - matics , John Wiley and Sons, New York, N. Y. , (1962). (4) Gray, A. and Mathews, G.B., A Treatise on Bessel Functions , p. 260, MacMillan and Co. , Ltd., London, (1931). (5) Jahnke, E. , Emde, F. , Funktionentafeln , p. 205, Dover Publications, New York, N.Y. , (1945). (6) Norton Company, Worcester, Mass., Data Sheets, Aluminum Oxide Bubbles, Product E-163. (7) Laubitz, M.J., "A Note on the Thermal Con- ductivity of Powders in Vacuum", Proc Third Conf Thermal Cond. II, p. 628-635, (1963). (8) Godbee, H.W. , Ziegler, W.T. , "The Thermal Conductivity of Mg0,Al203, and Zr02 Pow- ders to 850°C", Proc Third Conf Thermal Cond , II, p. 557-612, (1963). (9) Flynn, D.R., J. Res NBS-C , Engrg and Inst., 67C(2), p. 129-137, (1963). (10) Wechsler, A.E., Glaser, P.E. , "Investi- gation of the Thermal Properties of High Temperature Insulating Materials", ASD- TDR-63-574, 91, (1963) 206 Application of Neutron Diffusion Theory Techniques to Solution of the Conduction Equation I. Catton and J. R. Lilley Douglas Aircraft Company Santa Monica, California Computer codes for neutronic analysis of nuclear reactor cores have developed in parallel with, and sometimes spurred the development of, improved electronic computers. While this large effort has yielded spectacular increases in the effi- ciency with which the neutron diffusion equation can be solved, the diffusion of these techniques into heat conduction codes has not continued apace. This paper reports on the application to the conduction equation of techniques commonly used to solve Poisson's equation for neutron diffusion in one and two dimensions. Specifically, a non-iterative solution to the one-dimensional steady state equa- tion is described, as well as comparisons of the Liebman, Aitken and Young-Frankel methods as applied to two-dimensional problems. 207 Conduction from a Finite- Size Moving Heat Source Applied to Radioisotope Capsule Self-Burial 1 2 C. R. Easton Nuclear Technology and Subsystems Research and Development Missile and Space Systems Division Douglas Aircraft Company Santa Monica, California 90^05 The re-entry of a radioisotope heat source and its impact on the Earth's surface may result in burial of the source in soil. Internal heat generation may increase the temperature of the capsule and the surrounding soil enough to melt the soil and initiate capsule self -burial. Analytical studies of heat conduction from a finite, moving heat source were conducted to determine the conditions under which self -burial occurs, the magnitude of self -burial velocity, and the parameters important to self- burial. Configurations studied were a point source, a constant-temperature spherical- shell source, and a finite-line source. For one typical case, self -burial was initiated at a capsule power density of 2.2 x 10" 6 W/m3. The self -burial velocities were on the order of 10"? m/s or less. The dimensionless groups found to control self-burial are / Velocity * capsule radius \ I 2 * thermal diffusivity / Heat source strengt h (lMT * radius * thermal conductivity * soil-melting temperature/ and, for the cylindrical capsules, the length-to-diameter ratio. Key Words: Analytical, cylindrical, finite, heat conduction, heat source, moving source, radioisotope, self -burial, spherical. 1. Problem Statement The application of radioisotope heat sources to electric-power systems used in space vehicles poses a safety problem because random re-entry and impact on the Earth's surface nay occur. A heat source designed to re-enter intact possesses terminal velocities high enough to result in Impact burial for some soil conditions and capsule designs. The power-systems of many capsules have a sufficiently high power density to melt the surrounding soil if they are partially or totally buried. If the capsule is denser than the soil, which is usually the case, it. will sink through the molten soil and bury itself. The answers to questions concerning recovery, possible contamination of underground water, and self- burial as a passive disposal mechanism depend on an understanding of the initiation and rate of self- burial. Finally, to design for or against self -burial, one needs to know what parameters and dimensionless groupings affect self -burial and how. An analytical solution to self -burial will clarify these problems . The solution involves the treat- ment of a finite-size, moving heat source. The assumptions necessary for this solution degrade its accuracy, but the speed and ease of obtaining answers and the understanding achieved more than compensate Credit: This work was conducted by the Douglas Missiles and Space Systems Division under Independent Research and Development Account No. 81311-006. 2 Mechanical Engineer 209 for this shortcoming. One necessary assumption is that the soil has constant thermal properties. This assumption linearizes the governing differential equations to make a solution possible. It is also assumed that the temperature field surrounding the capsule as seen from the capsule is steady. This assumption implies an infinite, homogeneous medium with constant self -burial velocity. Temperature falls rapidly to the front and sides of the capsule, making it reasonable to assume a homogeneous medium. Data of Jeff ers and Baker [1] 3 show transient times of about it- hours for a constant-power electric heater buried in soil. The self -burial velocities reported here are of the order of 10~5 m/s or less. The capsule can move through the region of important temperature effects in about the same time it takes to initiate motion. Hence, the portion of the transient during which motion of the capsule occurs should be small, compared to the portion which initiates motion. It is then reasonable to consider only steady motion of the capsule. Observing the capsule from a distance, one would see a monopole (caused by the internal heat genera- tion) and superimposed dipoles (caused by the melting and refreezing of the soil and by the distributed source). The strength of the melting-freezing dipole is sharply reduced because the high thermal con- ductivity of the capsule makes it nearly a thermal short circuit for the dipole. The opposite effect applies to the distributed source dipole. One can normally neglect dipole effects in the presence of a monopole. Such an assumption will be shown invalid for the present case because of the inhomogeneity caused by the capsule in the soil medium. It will be assumed instead that the high thermal conductivity of the capsule dominates all multipole effects. Additional discussion of this assumption is presented with the analysis of a monopole heat source. The problem then becomes that of solving for the motion of a finite heat source in an infinite, homogeneous medium of constant properties. The motion is to be along a straight line and such that the temperature at a fixed distance in front of the capsule center is a specified constant, i.e., the soil melting temperature. 2. Basic Equations The development of the basic differential equation for a moving heat source is a textbook problem discussed by Eckert and Drake [2] and others. The transient heat conduction equation, 3T 2 (1) is the starting point. In eq (l), T is the temperature excess over the surroundings, t is the time, a is the thermal diffusivity, and V 2 is the Laplacian operator. The motion is assumed to be in the x direction. Placing 4 = x - Ut (2) transforms eq (l) to coordinates moving with the source, if U is the speed of the source. Considering steady state only, the transformed equation is v 2 _ _U 8T (3) where Vt is the Laplacian operator in the new coordinates. Equation (3) has a solution of the form T = e 2 *f(£, y, z) , 00 where f satisfies n 2 ' - &) ' Equation (5) is the basic differential equation for this study. (5) Figures in brackets indicate the literature references at the end of this paper. 210 2.1. Solution for a Monopole For a monopole, f has no angular dependence. The solution follows the Eckert and Urake concept. In spherical coordinates, eq (5) becomes, 2 2 d f 2, df , 2 r dr dr fe) (6) This equation has a general solution of the form Ur Ur" ] \ e + C 2 S j (7) Substituting from eq (7) into eq (k) gives T = ■£ r HI 2a| -S^ - [ Ur Ur_" (8) Boundary conditions are r— oo, T— _ 9T r-0, — - Imr k , (9) where Q = Heat source strength k = Soil thermal conductivity The first boundary condition gives C = 0, while the second gives Wk (10) since r = implies % = 0. The solution for temperature is T = -&-+0 Wkr (11) For a capsule of radius, r , the temperature at £, = r = r is T m , the soil melting temperature. The self -burial speed is found by solving eq (ll) for U with the above conditions applied r £n (w kT ) o \ o m/ U = > Urvr kT o m < kvr kT o m (12) 211 The maximum heat-source strength that will not cause self -burial is given by Sn = W o kT m • («) A value of self -burial speed can be calculated by assuming the following parameters: Soil density = 1,600 kg/m 3 , , Soil specific heat = 81+0 J kg deg Soil conductivity = 0.35 W m"l deg"-'- Source strength = ^00 W Soil melting temperature = 1,330 deg K above ambient Capsule radius = 0.025 m The self -burial speed for these values is U = 1.1 x 10 r5 m/s . (1*0 The maximum value of heat-source strength, Q ra , which will not cause self -burial is 150 W. The tem- perature difference between the bottom and the top of the capsule, T, is approximately equal to 2,200 deg. Such a large temperature difference could clearly not be tolerated by a real capsule. However, conduc- tion through the capsule should substantially reduce this temperature difference. An estimate of the true temperature difference can be easily formed using the following idealiza- tions: (l) conduction takes place in the cladding only, (2) all internally generated heat is delivered uniformly to the cladding, (3) all heat transferred to the soil is transferred uniformly over the lower hemisphere, and (k) all heat liberated by freezing is transferred uniformly to the upper hemisphere to be conducted to the lower hemisphere. These assumptions result in the differential equation, fe ( sin ° ^ : >) - : — " • ^ ls ) where q ■ - 1 ■■ . ! . C-6) and G is the polar angle, b is the cladding thickness, k c is the cladding thermal conductivity, P s is the soil density, and Qf is the soil heat-of -fusion. Equation (15) is integrated separately for the upper and lower regions with boundary and matching conditions . dT\ de/ 2 • a qr sin 9 b k c « , P s UQ f ij-iTr 2 JT ae )=0, IT Ml ) = \ (i> T„ (0) = T i m 212 Integrating and applying conditions of eq (17) gives, 2 r o :-'- jq In (2) + q^ [in (sin 9) - C in (esc 6 - cot 6)]} , (l8) ' c where C = +1 for the lower half and -1 for the upper half. Solving for the temperature difference gives, 2 !n(2)r 2 / Q P U(L\ AT = T (,r) - T„ (0) = r=— 2" /— S_ + - 1 , q < 1 (27) ™ /Zk ^(Z) \ I (- 1 )" ^ + H%r - p) i < z > p=o \ P / (28) These equations are shown graphically in figure 1. The two equations show different magnitudes and functional forms. Also shown is an estimate of the self -burial speed of a real capsule. To form this estimate, it is assumed that the difference between the monopole and the isothermal sphere is caused by the unrestrained temperature away from the forward stagnation point of the monopole. The heat carried away by the excess of temperature can be estimated by Za , nZ ,v —q- (e - 1) (29) 214 where a is a multiplier on the capsule radius representing the effective radius of excess heating, and n is a number representing the average excess temperature. At the capsule surface, n has a range from 1 to 2. Assuming n = 1.5? a can be calculated from the data of figure 1. The variation of a with Z is shown in figure 2. The dashed curve in figure 2 is from the family a = 1 + -| , (30) which we might expect a to follow, analogous to Mach waves in supersonic flow. It is appropriate to choose n = 0.2*4- to estimate the self -burial speed of a real capsule, as this value gives the appropriate temperature difference across the capsule. This value of n should be an upper limit since the temperature excess is a high estimate. The deviation from the isothermal sphere is not negligible, but the error in estimating the self -burial velocity of the real spherical capsule is felt to be small. There is no need to pursue spherical models further. It is more desirable to consider cylindrical models because these better approximate real capsules. 2.3. Solution of Finite Cylinder Capsules The model chosen for analysis of cylindrical capsules is a finite-line integral of monopole heat sources. The capsule is assumed to have its axis horizontal. Tigure 3 shows the manner in which eq (ll) is used as an integrating kernel for the model. In dimensionless form, the line integral is _ ^ kT _ f e- 2 * 1 + JT77 ^ dn , (3D w/O where H is the capsule L/D and the other variables are as previously defined. The relationship between heat flux and self -burial velocity is Q/l Z e wfkT~ - ft / 2 " (32) O m I _-i vl + r| A + rf dn Equation (32) can be integrated for the limiting case of zero self -burial velocity. The result is v = : — l, 1 a,»2 ] • (33) log e U + v/l + 2 2 J The limiting case of infinite L/D is found from eq (5) solved in cylindrical coordinates. The solu- tion is similar, except that hyperbolic Bessel's functions appear in lieu of exponentials. The result is Z e o There is no minimum value of q for the infinite cylinder, analogous to 0,^. The fallacy of assuming an infinite cylinder is thus underscored. The general case of eq (32) must be integrated numerically. The result is given in figure h, with the zero line calculated from eq (33) and the asymptotic limit from eq (3*0 . Only for large self -burial speed does the heat-source strength approach the asymptotic limit by an L/D of 20. The finite length of a real capsule will be an important consideration. The solution for the case of a finite, isothermal cylinder is very difficult. It would probably be as easy to develop a generalized computer program to solve the problem with capsule internal conduction, 215 melting and freezing, and variable thermal properties as it would be to solve the case of the isothermal cylinder. This course is recommended should further pursuit of this subject prove warranted. The correlation between the monopole and the isothermal sphere can be us.ed to estimate the behavior of an isothermal cylinder. The value of n should be somewhat larger for a cylinder than for a sphere. Assuming n = 1.6, the equation for the excess source is ' Za / nZ . \ /„,-\ Figure 5 shows the line source and estimated isothermal cylinder for a capsule L/D of 12. The deviation of the line-source model from the isothermal -cylinder model is estimated to be much less important than for the comparable spherical cases . This result can be attributed to the smaller relative area encompassed by the dimension r a in the case of the cylinder. Although the excess of temperature is the same for both cases, the mass of soil involved is much less for the cylinder, therefore the excess of heat removed is much less. 3. Conclusions Self-burial is a real possibility for a radioisotope capsule on the ground. The maximum heat-source strength which will not bury itself after an initial burial is easily shown to be % " W o kT m (36) , 1+TrLk T \ - r /— gq <3T) iog e [* +v/i + r\ for the spherical and cylindrical capsules, respectively. Expressions for the relationship between heat- source strength and self -burial speed were derived for idealized conditions. The solutions were demonstrated to be reasonably accurate within the limitations of steady-state, homogeneous medium and constant properties. It is recommended that the capsule designer study the effect of self -burial on capsule design from the standpoint of passive disposal, potential hazards, and recovery. If designing for self -burial is desirable and practical, a computer program should be developed to solve the general equations and provide design data. k. References [l] Jeffers, S. L. and Baker, F. L., Private [2] Eckert, E. R. G. and Drake, R. M., Heat and Communication, Sandia Corporation, Aerospace Mass Transfer, (McGraw-Hill, 1959)- Nuclear Safety Group. 216 0.1 0.2 0.3 0.4 0.5 0.6 DIMENSIONLESS SELF-BURIAL VELOCITY (Z) Figure 1. Dimensionless self-burial velocity. 6.0 5.0 CO 5 4.0 o 3.0 2.0 1.0 5C V a = 1 + c . ' ^ z ' *—•—-< 0.1 0.2 0.3 0.4 0.5 DIMENSIONLESS SELF-BURIAL VELOCITY (Z) Figure 2. Effective radius of excess heating. ► X Figure 3. Continuous-line-source model of cylindrical capsule. 217 Figure k. Effect of capsule length-to-diameter ratio on heat-source strength per unit. o LU a: l— CO LU o o co CO CO a o CO 4.0 8.0 12.0 16.0 CAPSULE L/D 20.0 00 0.9 0.8 cr H I— O 0.7 2: LU re i— CO 0.6 LU <_> o; o 0.5 CO ll LU 3: 0.4 CO CO LU 1 0.3 o CO LU 0.2 S: Q 0.1 I .INE-S( 5URCE l/IODEL ISOTH CYLIN IRMAL DER Figure 5. Estimate of self-burial velocity of an isothermal cylinder with L/D = 12. 0.04 0.08 0.12 0.16 0.20 0.24 0.28 DIMENSIONLESS SELF-BURIAL VELOCITY (Z) 218 Evaluation of Steady-Periodic Heat Flow Method for Measuring the Thermal Diffusivity of Materials with Temperature-Dependent Properties C.J. Shirtliffe and D. G. Stephenson Division of Building Research National Research Council of Canada Ottawa Temperatures in a material with temperature-dependent properties were cal- culated by finite difference methods. The calculated values were analyzed for harmonic content and used with the solution to the linear form of the heat conduction equations to determine the mean value of thermal diffusivity. Errors in determining the mean pro- perties were calculated and the distortion produced by the temperature dependence was tabulated. Distortion of the temperature waves in the material places limitations on the use of the method for moist materials. The main harmonic content was found to have little distortion so long as only odd harmonics were present in the boundary conditions. Accuracy of the method was established for the configuration and range of parameters anticipated for moist building materials. Key Words: Dry materials, infinite slabs, moist materials, periodic heat flow, temperature dependence, thermal conductivity, thermal diffusivity. 1. Introduction Water migrates when a temperature gradient is applied to a moist porous material. This fact has to be borne in mind when designing experiments to measure the thermal properties of moist materials. The movement of moisture contributes to the heat transfer and it makes the apparent thermal properties quite dependent on the temperature and moisture content of the sample. It also causes the moisture distribution through the sample to change during the course of an experiment making it difficult to say what moisture content and temperature correspond to the measured con- ductivity or diffusivity. This problem is alleviated by using steady periodic boundary conditions because this makes the time average condition of the sample constant. The Angstrom method has been used, therefore, to measure the thermal diffusivity of moist materials (1, 2), but it has never been established that the method gives true mean values, even for "dry" materials, when the thermal properties are temperature dependent. The objective of this study was to determine the magnitude of the errors that might occur when the Angstrom method was used to measure the thermal diffusivity of materials that contain moisture. 2. Method of Determining Errors The error inherent in the Angstrom method was checked in the following way: (i) the tempera- tures atdifferent locations through a sample were calculated for steady periodic boundary conditions assuming that the thermal properties of the sample varied in a particular way with variations of temperature; (ii) the calculated temperatures were used to compute a mean value of thermal dif- fusivity just as if they had been obtained by experiment; and (iii) the resulting "experimental" value of diffusivity was compared with the "correct" mean value, i. e. the mean of the values used in the Figures in parentheses indicate the literature references at the end of this paper. 219 original calculations; the difference was the error inherent in the method of calculating a mean value from experimental results. The initial calculations were made using the finite difference equations given in Appendix I. This numerical approach was used because no exact analytical solution was known that would permit the simulation of an experimental set-up: quasi-linear approximate methods were tried but there was no sure way of knowing when the calculated results began to have a non-negligible error. The purely numerical results could be made as accurate as required by using finer increments in space and time and the accuracy could be checked by comparing results for successively smaller increments. This study indicated that there is an optimum value of the inverse Fourier number, 1/F Q - (Ax) /aAt, at about 6 where "a" is the thermal diffusivity. Values of Ax and At were selected, therefore, so as to make this parameter close to six when "a" was at its mean value. This also ensured that the finite difference expressions remained stable since the stability limit is 1/F > 2; in no case did "a" vary by as much as a factor of 3 from its mean value. The results of the finite difference calculations were first analyzed to determine the mean value and the amplitude and phase of the main harmonic of the temperature at those locations where temper- atures would normally be measured. The main harmonic component of each of these temperatures was then used to calculate the experimental value of "a. " The formulas used for this computation are given in Appendix II. These equations are based on the premise that the thermal properties of the sample are independent of temperature; it is this simplifying assumption that causes an error in the calculated value of diffusivity. The errors associated with all the numerical calculations were determined by comparing the "experimental" results with the "correct" values for a case where there was no variation of the pro- perties with temperature. This showed that the error was less than 0.003 per cent. 3. Scope of Study The initial study was limited to problems subject to the three following conditions: (i) two identical, two-dimensional slabs (ii) properties varying linearly with temperature X = X (1 +0.6); dc = dc (1 + 39) o * o l r ' where d is density, a and 3 are the temperature coefficients, X and dc Q are the mean values; (iii) boundary conditions 1.8, = E, sin wt and fi = E„ sin fwt + A ) 11 2 2 o where w = 2TT/p, circular frequency, A is a phase shift, p is the period 2. Square waves; + E and ± E ; A = or 180°. The range of the variables investigated was: a = 2.6 x 10" 8 m 2 /s and 2. 6 x 10" 7 m 2 /s (0. 001 and 0.01 ft 2 /hr) E and E = 2. 8 to 28 C (5 to 50 F) p = 7.2 x 10 2 to 3. 6 x 10 4 sec (0.2 to 10 hr) a and g - -0. 03 6 to +0. 03 6 deg c" (-0. 02 to +0. 02 deg F" ) 220 The boundary conditions were selected so as to have no even harmonics. A quasi-linear analysis, to be published later, showed that this type of boundary condition should produce the best results. 4. Results 4. 1 Distortion of Wave Form The temperature dependence of the properties caused a distortion of the temperature wave form in the material. Tables 1 and 2 summarize some of the results for one set of boundary conditions. Distortion occurs for both sine-wave and square-wave boundary conditions. Table 1. Effects of Temperature Dependence of Properties on the Temperature Variation in a Slab Tabulated results for: - centerline of slab - 13.9 deg C amplitude - 1 hr period -l/F o =5.9 - 5th cycle results -thickness = 0.061 m square-waves in phase 1000 steps/cycle -7 2 a = 2.58 x 10 m /s o zero mean temp, at surfaces Average Error in Temp. Coefficients Temp. Main Harmonic Second Harmonic $4 a o a x 10 3 3 x 10 3 °C Amplitude, °C Phase Amplitude, °C Phase % 0. 0* 0. 0* 0. 000* 6. 172* -102.22* 0.000* _ _ 0.0 0.0 0. 002 6. 172 -102.22 0.00005 10.88 +0. 002 0. -3.6 -0. 00050 6. 172 -102.22 0. 077 101.58 -0. 005 0. -7.2 -0.0011 6. 172 -102. 20 0. 156 101.64 + 0.018 0.0 7.2 0.0011 6. 172 -102. 20 0. 156 -78.28 + 0.018 3.6 0.0 0. 307 6. 172 -102. 17 0. 040 143.48 -0.020 -3.6 0.0 -0. 307 6. 172 -102. 17 0. 040 -36.38 -0. 020 -7.2 0.0 -0. 612 6. 167 -102. 07 0.080 -34.20 -0.040 3.6 -3.6 0. 301 6. 178 -102. 12 0. 091 118.60 -0. 080 3.6 3.6 0.312 6. 167 -102.22 0. 038 122. 10 + 0. 011 -3.6 -3.6 -0.312 6. 167 -102.22 0. 038 57.74 -0.011 -3.6 -7.2 -0.317 6. 161 -102.26 0.089 84. 18 -0. 182 3.6 7.2 -0.317 6.161 -102. 26 0.089 -95.73 -0. 182 7.2 -3.6 0. 604 6. 183 -101.92 0.546 -175.47 -0. 112 7.2 3.6 0. 618 6. 156 -102. 12 0.053 170.28 -0.268 7.2 7.2 0. 623 6. 150 -102. 12 0.076 122.15 +0.444 -7.2 -7.2 -0. 623 6. 144 -102. 22 0. 077 57.77 + 0.444 36.0 -36.0 2.797 6. 644 -92.07 1.006 132.24 -7.55 36.0 36.0 2.963 5. 617 -102. 14 0.283 -123.63 + 10.2 * Solutions for constant properties ** from real part of eq (3), Appendix II The temperature dependence of the properties gives rise to a non-zero average temperature at the internal points, even though the average boundary temperatures are zero. Typical profiles are given in figure 1. Table 1 shows that, for the conditions studied, this effect is less than 20 per cent of the amplitude of the main harmonic. The distortion in the mean temperature depends primarily on the coefficient a and varies linearly with it. 221 Table 2. Effects of Frequency on the Temperature Variation in a Slab with Temperature Dependent Properties (non-metal; 5 cm thickness) a = 0.0036/°C L = 0. 061 m 13.9 deg C sine waves, zero mean 18 slices, 1/F = 6.2, 5th cycle results Boundaries at mean temperature = 0.0072 /deg C a = 2. 58 x 10-' m 2 /s 120° phase shift Results for the Center line Period hr Mean Temp, °C Main Harmonic Second Harmonic Ampl, ° C Phase Ampl, ° C Phase 1. 0. 5 0.2 0. 171 0. 179 0. 193 2.42 1. 12 0. 25 -162.24 156.43 75. 35 0. 0218 0. 0050 0. 0012 131. 84 36.91 -2.34 Distortion of the main harmonic is primarily affected by the coefficient a as shown by Tables 1 and 2. These are relatively small and it can be inferred that the fundamental frequency can be used to obtain good approximations to the mean thermal diffusivity. The error in the main harmonic is also affected by 8, 9 max, and the period. The temperature dependence of properties produced even harmonics at internal points. The second harmonic content (Tables 1 and 2) depends on both the temperature coefficients. The amplitude of the second harmonic is more than 15 per cent of the main harmonic for the worst case shown. Results not shown indicate that the odd harmonics of square-wave boundary conditions have dis- tortion similar to the main harmonic, but because of the fewer number of points used in the harmonic analysis, they have more inherent errors. The frequency has a marked effect on the distortion (Table 2). The distortion of the average temperature increases with frequency, but the relative amplitude of the second harmonic decreases. No exact explanation can be offered for this at present. The relative phase of the boundary temperatures affects the results. The effect is second order for mean temperature as shown in figure 1. The second harmonic distortion is worse when the boundary temperatures are in phase than when they are 180° out of phase. 4.2 Errors in Determination of Thermal Diffusivity The results of a quasi-linear analysis, to be published later, and the basic form of the finite dif- ference equation, given in Appendix I, indicated that the errors are functions of a, 3, a - 3 and d + 8 and sums and products of these parameters. These relationships were checked for the results of the finite difference calculations by plotting error in determining thermal diffusivity against the same combinations of a and S. A summary of the most important characteristics of the error in the determination of the thermal diffusivity is given in figures 1 and 2 for a = 2.58 x 10 -7 m 2 /s (0.01 ft 2 / hr), and, + 13.9 C (25 F) square-wave boundary conditions. The error for a thermal diffusivity of 2. 58 x 10"°m /sec was similar. Figure 2 gives the errors in determining the thermal diffusivity from the real part of eq (3) in Ap- pendix II while figure 3 gives the error when they are determined from the imaginary part of the equation. The errors are smaller using the real part, except when (X = 0, i, e. when the thermal dif- fusivity is a constant. The slope of the curves indicates that the error terms are essentially quadratic functions of a and 6. Figures 2 and 3 are not exact, expecially at the lower values, because of the 222 finite difference error in the calculations. Typical values for the temperature coefficients, a and |3, for dry materials are given in Table 3, The range in a is from -0. 01 to +0. 01 per deg K and 3 from -0. 003 to +0. 004 per deg K. Values for moist materials are unknown, but a rough estimate would be, -0. 02 to +0. 02 per deg K for a and -0.005 to +0. 005 for 0. Table 3. Typical Values for the Temperature Coefficients of Thermal Conductivity and Thermal Diffusivity of Dry Solids at Temperatures Above 100°K Type of Material Temperature Coefficient Temperature Coefficient a 3 per deg. C per deg. C metals -0. 010 to +0. 010 (rare 0. 020) -0. 003 to +0. 004 non-metals to 0. 004 (rare 0. 020) -0. 0003 to +0. 003 s emi - c onduc tor s -0. 006 to +0. 006 -0. 0003 to +0. 003 (Data from American Institute of Physics Handbook, 2nd Edition) The errors in determining thermal diffusivity in the range considered can be estimated from figures 2 and 3, as being less than 1 per cent for dry materials and 3 per cent for moist materials using the real part of the linear solution, and less than 2 and 6 per cent, respectively, using the imaginary part of the linear solutions. 5. Conclusions Distortion produced by the temperature dependence of thermal properties places limitations on the use of steady-periodic methods for measuring the thermal properties of moist materials. The steady mean temperature gradient can be expected to produce a nonuniform mean moisture content in the material. The method has advantages, however, over other methods since the mean gradient is only a small fraction of the maximum temperature variation in the material. The main harmonic must be found by accurate harmonic analysis because of the harmonic dis- tortion that is introduced by the temperature dependence of the thermal properties. The mean thermal diffusivity of the material determined from the main harmonic has some error but the accuracy of the method is adequate for most purposes even for large temperature dependence. Steady periodic methods can, therefore, be used to determine the thermal diffusivity of normal dry materials. Errors due to the temperature dependence can be estimated from the curves in figures 2 and 3. 6. References (1) Higashi, A. On the thermal conductivity of soil, with special reference to that of frozen soil, Trans. Amer. Geophys. Union, 34, 5, (1963). (2) Jackson, R. D. and Kirkham, D. Method of measurement of the real diffusivity of moist soil. Proc. Soil Sci. Amer., 22, 6, (1958). (3) Car slaw, H. S. and Jaeger, J. C. Conduction of heat in solids. 2nd Edition, Oxford Press. (4) Shirtliffe, C. J. and Stephenson, D. G. The determination of the thermal diffusivity and con- ductivity of materials subjected to periodic temper- ature variations. NRC/DBR, Computer Program No. 12 (1962). 223 Appendix I 1. Finite Difference Solution 1. 1 Basic Equation The basic equation approximated by the finite differences was a_8 _ _i_ a_ at " cd 9: Mi) (i) with: zero initial conditions, periodic boundary temperature, cd and X functions of temperature. 1.2 Finite Differences Central difference approximations were used to approximate the derivatives, yielding 9 (n,t + At) = 6(n,t) + -f— ^(n-D + 4X (n)-X(n+l) j, e(n . lt) . 2X (n) e ( n , t) Ax dc(n) X (n+1) + 4X (n) - X (n- 2^1) -9 (n+l,t)l ' t (2) where: dc and X are arbitrary functions of temperature, n indicates the node point, At indicates the time step. If X = X (1 +0.9 ) and cd = c d (1 + 89 ), eq (2) reduces to 9 (n, t + A t) = 6 (n,t) + a *VH -I f "> VV ) H - a(e( T 1 ' ^ " 9 (n '^' ^^ ] ° Ax L VI + 09 (n, t) I 4(1 + 09 (n, t)) J where H = (9 (n+1, t) - 29 (n, t) + 9 (n-1, t)J and is the form for constant properties and Ax /a At is the normal non-dimensional inverse Fourier number, 1/F . o The second term is maximum when a = -0. Appendix II 2. Linear Solution for Two Slabs 2. 1 General The solution for the temperature variation in an infinite slab of a material of constant properties with periodic boundary temperatures is given in Car slaw and Jaeger (3). In terms of the temperature at three points in the material, and in matrix notation the solution is A B D A (1) and D* * " 9 3 * . q 3. (2) where 9 , 9 , and 9 are periodic temperatures at points 1, 2, and 3; q , q , and q are the periodic heat fluxes at points 1, 2, and 3, and 224 , . , . . R sinh { 1 + i ) $ _ ( 1 + i) $ sinh ( 1 + i) j A = cosh (l + i)$ B = (1 ^ iH D = -i ' — * 1 — 2 I where § = (ttL /aP) i , R = L/X, L = distance between respective points, X = thermal conductivity, a = thermal diffusivity = . X/dc, P - period, d = density, c = specific heat per unit mass, and * refers to functions for the slab between points 2 and 3. Equations (1) and (2) hold for all pure harmonics of the boundary conditions when the properties are constants. 2. 2 Similar Slabs For the case where the three temperatures are measured at evenly spaced points in a homogeneous material, the equations can be combined to give the following expression for the thermal diffusivity (Shirtliffe and Stephenson (4)): _! / 9 1 + 9 3 \ (1 + i) $= cosh ( 2 e ) 0) Values of $ can be obtained for both the real and imaginary part of the equation and the two values should be equal. This configuration can be used as an absolute method for finding the thermal diffusi- vity of a material. 2.3 Dissimilar Slabs For two slabs of dissimilar material in contact at plane 2, the properties of the slab of material between points 1 and 2 can be found if the properties of the material between points 2 and 3 are known. Again from Ref (4) we obtain '1 ' q = A + B ^2 A* where = — + 8, B* ? Cr) ■ The second equation shows that the reference slab is used as a "heat meter. " When eq (5) is sub- stituted in eq (4), the complex equation for 9i /9-> contains the tv/o unknowns R and §, or X and a. These can be found from the real and imaginary parts of the equation. The solution has been done on a digital computer using standard numerical techniques. 2.4 Application of Equations Apparatus based on these equations has been built and used at intervals over the past eight years. The results from the experimental investigation including considerable error analysis will be published elsewhere. Computer analysis was found necessary to adequately handle the harmonic analysis of the temper- ature waves to 3olve the equations, and for error analysis. This paper is a contribution from the Division of Building Research, National Research Council of Canada, and is published with the approval of the Director of the Division. 225 o o CC < X/L Figure 1 Mean Temperature Profile Produced by Temperature Dependence of Properties for Conditions Given. 226 20 I I I I I I 1 1 I I I I I 10 * l 2 0.5 OH 0.2 0. 1 0.05 0.02 USING REAL PART OF LINEAR S a =2.5 p = 1 h L = 0.0 SQUARE WA Figure 2 0.001 0.01 a, and /3 per °C Error in Determining a Using Real Part of Linear Solution and for Particular Combinations of a and £ . 227 0.02 0.001 0.01 oe, and {2> per °C Figure 3 Error in Determining a Using Imaginary Part of Linear Solution o and for Particular Combinations of a and p. 228 Comparison of Modes of Operation for Guarded Hot Plate Apparatus with Emphasis on Transient Characteristics C. J. Shirtliffe and H. W. Orr Division of Building Research National Research Council of Canada Ottawa, Canada The constant heat flux and constant temperature modes of operation of the guarded hot plate apparatus are compared and the existence of an optimum heat flux boundary condition is examined. The step response equations for the different modes and for various preconditioning temperatures are tabulated and presented in graphical form. Optimum preconditioning temperatures are determined. Effects of disturbances and plate capacity are discussed and control and measurement problems compared. A hybrid mode of operation is suggested. The extension of the results to the use of heat meters and transient heat flow apparatus is discussed. Constant temperature or a hybrid mode of operation is found to be the most practical. The response times can be an order of magnitude less than constant heat flux operation. Use of the guarded hot plate apparatus in the constant temperature mode is suggested for other measurements. Key Words: Guarded hot plate, heat meters, modes, optimum boundary conditions, response time, thermal conductivity, thermal diffusivity, transient errors, transient heat flow. 1. Introduction Historically, the guarded hot plate apparatus (1) is a "Standards Lab" type instrument where the emphasis is on accuracy rather than on minimizing the duration of the test. The transient heat flow characteristics and control problems have therefore been neglected. Other types of apparatus have been evolved for short duration tests of lesser accuracy. It was believed that the standard guarded hot plate apparatus was much more versatile than assumed and that with the proper control system and measurement techniques its use could be extended to rapid, high accuracy testing. The apparatus was thought to offer advantages in the transient heat flow area and for use in conjunction with heat meters. Also, since the guarded hot plate apparatus is being proposed for testing "super" insulations with extremely long response times, it was believed that the control mode that is optimum in terms of speed and accuracy should be determined and the response functions tabulated to allow estimation of the time to reach an adequate steady state condition. The guard balance is assumed to be controlled automatically in this analysis so that it does not introduce additional disturbances into the system. This is believed to be a reasonable assumption as most laboratories have provided this type of control for their apparatus. The temperature of the cold plate is assumed to be constant. Figures in parentheses indicate the literature references at the end of this paper. 229 2. Modes of Operation and Transient Characteristics Two modes of control of the hot plate in the guarded hot plate apparatus are in general use. For the simplest and most widely used the plate is supplied with constant power, thereby producing a constant heat flux at the surface of the plate. The second, less widely used, method is to keep the temperature of the hot plate constant by controlling the power with a temperature controller. The latter method was used by Sommers (2) for "high speed" tests, but has not been widely used because of the extra equipment involved and the lack of knowledge of the relative advantages and problems of this mode compared to the constant heat flux mode. 2. 1. Constant Heat Flux Operation The usual sequence of events in performing a test with the guarded hot plate apparatus with constant heat flux from the hot plate is as follows: (i) The sample is conditioned to the mean temperature of the test (ii) The cold plates are at the required temperature (iii) No power is being supplied to the hot plate and it is not at the required test temperature (iv) The thermal conductivity and, therefore, the power to the test area are estimated (v) The sample is placed in the plates, the power to the test area increased for an unspecified period of time, to heat up the plate and the sample, then reduced to the estimated power. The conditions then are allowed to reach steady state (vi) Additional adjustments of the power are usually necessary and disturbances must settle out. There has been no easy way to estimate the time required to reach the steady state condition using this procedure. 2. 2. Constant Temperature Operation The usual sequence of operations for use with the hot plate at a controlled temperature is as follows: (i) The sample is conditioned to the mean temperature of the test (ii) Both the hot and cold plates are at the required temperatures (iii) The sample is placed in the apparatus and the test area power increases automatically to supply the required heat and then drops to the steady state value. It should be noted that no guessing or second adjustments are required. 2. 3. Optimum Boundary Conditions Constant temperature operation is easier than constant heat flux operation, and probably faster, but the question remains as to whether there is a boundary condition that will give even faster response. Assuming there is a better, nonconstant, heat flux boundary condition, F(t), where t is time, and the appropriate unit step response of the sample is V(x, t) , the surface temperature of the hot plate can be written as given in ref. (3). 9(x,t) x I"* F(t-t") V (x,t») d V oj t 230 where t* is the dummy time variable, and subscript t indicates the time derivative. By normal response factor methods, F(t) can be found for any desired response. Since the answer depends, however on the step response of the sample, V(x, t), which is a function of the unknown thermal properties, there is no practical way to find this optimum condition for thermal conductivity apparatus. The most practical boundary conditions would appear to be the constant temperature boundary condition, since it is the only boundary condition that can be specified without regard to sample properties. 2. 4. Response Times The theoretical solutions to the heat transfer problems corresponding to the constant temperature and constant heat flux modes of operation are given in Appendix I. The solutions have been derived in nondimensional form to make them general. The temperature is nondimensionalized as shown in Appendix I to be zero at the cold plate and 1 at the hot plate under steady state conditions. The non- dimensional heat flux is also 1 under steady conditions. Time zero is referred to as the time at which the boundary conditions are applied, that is, the test is started, and is nondimensionalized by the thermal diffusivity "a" and the thickness, L. The solutions are given in general form and also for specific locations in the sample at which the measurements are to be made for determining the thermal conductivity, since it is these values that must approach steady state before the measurements can be made accurately. The response time of a sample is defined here to mean the time required for the uncontrolled, or unspecified, variable at the hot plate to reach an adequate steady state so as to give an error in the measurement of thermal conductivity of less than the specified value. A curve of error in the measurement versus time will result for each mode. The equations for error are given in Appendix II. Response times for the constant temperature and constant heat flux modes with the initial temperature equal to the steady mean temperature, W = 0. 5, are given in figure 1. The response time for constant temperature mode is seen to be almost an order of magnitude less than for the constant heat flux mode when the errors are small. The response times for specimens at various initial temperatures, W, are shown in figures 2 and 3, for values of W » 0, the cold plate temperature, to W = 1, the hot plate temperature. In each mode there is an optimum value of W at which the response time is least. The error reverses sign, that is, overshoots, when the initial temperature is greater than 0. 5 for the constant temperature mode and 0. 635 for the constant heat flux mode, and the negative portion of the curve can govern the response time of a test. The optimum values of the initial temperature, W, are therefore approximately 0. 52 and 0. 64, respectively, for the two modes. The response times for the two modes at the optimum W are approximately the same. The response time for the constant flux mode at optimum conditioning is an order of magnitude less than that for conditioning to the mean temperature; this can be useful in cases where the thermal conductivity of a specimen is being checked and the steady state heat flux can be determined beforehand. If the thermal conductivity is not known then the final hot plate temperature and therefore the optimum W cannot be determined. The equations and figures do not take into account the heat storage capacity of the plates, the three-dimensional nature of the heat flow, or the temperature dependence of the properties. It should therefore be recognized that the equations and figures are only a guide to the actual response time and that checks should be made on the applicability of the figures to any particular apparatus by testing samples of known thermal diffusivity. A test was made on a foam polystyrene sample, 2. 5 cm thick with a 20- by 20 -cm guarded hot plate to compare the theoretical response times to the actual values. For constant heat flux operation, the response time to \ per cent error was less than 15 sec, when the hot plate was preheated to the steady state temperature, and over 10 hr when the plate was not preheated. In each case the exact constant heat flux was used to obtain the desired hot plate temperature. The theoretical value was approximately 40 sec for the specimen. In constant temperature mode the measured response time was approximately 70 sec. The response curves are, therefore, better estimations of the actual response times for the constant temperature mode than the constant heat flux mode. 231 2, 5. Measurement Problems Each mode of operation has inherent measuring problems. The constant heat flux mode of operation has such long response times that the true steady state value can be difficult to recognize. Measurements must be made over an extended period to ensure that it has been reached. The constant temperature mode of operation requires that the temperature of the hot plate be controlled, by controlling the power to the heater. The steady state power to the heater will, by definition, vary a small amount with time. It is necessary to average the power over a sufficient length of time to obtain the mean values. The averaging time will depend on the controller characteristics and the temperature sensor. The problem of averaging can be eased considerably by using a strip chart recorder, automatic data logger, or a signal averaging circuit. 2. 6. Hybrid Mode The best of both methods can be obtained by starting the test in the constant temperature mode and switching later to the constant heat flux mode. This hybrid mode of operation requires a signal averager and is accomplished by starting in the constant temperature mode, monitoring the average power supply voltage, and when it becomes sufficiently constant, setting the output of the controller to that value, and switching the controller to manual mode, thereby supplying constant heat flux to the hot plate. Depending on the success of the transfer to constant heat flux operation, the measure- ment can be made with a short settling time, and with normal measuring instruments. 2. 7. Disturbances and Readjustments If the temperature or the heat flux of the hot plate must be readjusted so as to conform to a standard test condition, or if it is disturbed, the time for the transient to settle can be estimated from the response time curves. A change in a parameter at one surface corresponds to the case of a change at the surface and zero initial temperature. The error that the disturbance produces in the steady state value should be used in determining the response time. For example, a 10 per cent change need only settle to 10 per cent of its final value to produce a 1 per cent error in the measure- ment. 2. 8. Plate Capacitance The hot plate assembly can have more than 100 times the thermal capacitance of the sample and can therefore be the major factor in determining response time (3). Failure to preheat the hot-plate assembly to the desired temperature can greatly extend the test, especially for the constant heat flux mode where the small source of heat must raise the temperature of the large thermal mass. The increase in response time, or lag, decreases as the conductance of samples increases, and can vary from a few seconds to many hours. For constant temperature operation, the hot plate is normally at the required temperature and the thermal mass can be used to advantage to supplement the heater capacity in heating the specimen. In the hybrid mode, the thermal capacitance is helpful during the constant temperature interval and detrimental in the constant heat flux interval. 2. 9. Characteristics of Control Systems for Hot Plates It has been found that the response time for the normal design of hot plates and normal samples is so great that reset is required, i. e. integral action with "anti -windup" feature along with the proportional action in the controller. The sudden changes in power output required when high capacitance samples are placed in the apparatus make it desirable to have rate, i. e. derivative action, as well. Proper design of the hot plate and placement of the temperature sensor can decrease the lags in the control system and ease these requirements. 2. 10. Further Refinement If the shortest possible test time is required, there is no reason why the samples cannot be preconditioned to the proper temperature gradient in a large, simplified pair of temperature 232 controlled plates. The samples could then be tested rapidly in an apparatus operated in the constant temperature mode. 3. Application to Heat Meter Apparatus The guarded hot plate apparatus, when operated in the constant temperature mode^ forms the basic configuration required for measuring thermal conductivity with a heat meter; the advantage of this heat meter method is the simplicity of the measurements. The solutions in Appendix I can be applied to the heat meter apparatus as well, as long as the heat meter is thin and has a low thermal capacitance. Equations (5), (6), and (7) of Appendix I show that there are three locations in a sample where optimum response without overshoot can be attained. The locations and relevant conditions are listed below. Test Conditions (i) the sample is conditioned to a uniform temperature (ii) the temperature of the hot and cold plates are controlled. Locations and Initial Temperatures (i) at the hot surface - sample at the mean temperature (ii) at the cold surface - sample at the mean temperature (iii) at the center of the sample - any uniform temperature. The third location, at the center of the sample, is not only ideal in terms of response but also provides some isolation from any small independent disturbances at the plates and, ideally, does not require the sample to be conditioned to any specific temperature. The initial temperature must be uniform or at least symmetric about the center point. If there is the possibility that thermal conductivity is a function of temperature, the center position affords the proper location to indicate if the temperature has deviated from the mean value. The center position requires two identical samples. Lang (4) has developed an apparatus with the heat meter in this location but it is not clear if all the advantages were recognized. Norris and Fitzroy (5) gave the theoretical justification for the method. 4. Extension to Transient Measurement Conditions Beck (6) has determined the optimum configuration for transient heat flow apparatus for simul- taneous determination of thermal conductivity and specific heat. When the thermal conductivity is low, the optimum configuration is a large thermal mass "cold plate, " preconditioned but not continuously cooled, and a constant temperature warm side with heat flux metered. The cold plate is insulated on all but the act've surface. The thermal properties are inferred from the cold plate temperature and the hot side heat flux, using a "nonlinear estimation" procedure. The guarded hot plate apparatus with temperature controlled hot plate can be modified to attain the required configuration; this allows the extension of the apparatus to measurement of both thermal conductivity and specific heat. With samples of high thermal conductivity, the optimum configuration at the cold plate remains the same while that of the hot side condition is zero heat flux. Thermal properties are inferred from temperatures measured on the two sides by the same procedure. By removing the hot plate from the guarded hot plate apparatus, this condition can be accomplished. Further work should be considered on the adaptation of the guarded hot plate apparatus to this type of measurement, and a study of the sources of error. Modification in design of the hot and cold plates might be required to make them "universal, " or interchangeable plates might be considered. 233 5. Conclusion The constant temperature and hybrid modes of operation have been shown to be the most practical for the guarded hot plate apparatus. The response times can be an order of magnitude better than for constant heat flux operation. The figures presented can be used as a guide to estimate the time required to reach an adequate steady state. Guarded hot plate apparatus with temperature-controlled plates can be used for rapid measure- ments, either alone or in conjunction with a heat meter. The apparatus also might be used for the simultaneous determination of thermal conductivity and thermal diffusivity. This report is a contribution from the Division of Building Research, National Research Council of Canada, and is published with the approval of the Director of the Division. References (l ) Method of test for thermal conductivity of materials by means of the guarded hot plate (C177-63. Book of ASTM Standards, 1963, Part 3. (2) Somers, E. V., High-speed guarded hot plate apparatus for thermal conductivity of thermal insulation. Trans. , ASME, 78, August 1956, p. 1191-6. (3) Churchill, R. V. , Operational mathematics, Second Edition, McGraw-Hill Book Co. Inc. New York, 1958. (1 ) Lang, D. L. , A quick thermal conductivity test on insulating materials, ASTM Bulletin No. 216, September 1965, p. 58-60. (5) Norris, R. H. and Fitzroy, (Mr s. ) Nancy D. A quick thermal conductivity test on insulating materials, Materials Research and Standards, VI, September 1961, p. 727-729. (6) Beck, J. V. , The optimum analytical design of transient experiments for simul- taneous determinations of thermal conduc- tivity and specific heat. Michigan State University, Ph.D., 1964 Engineering, Mechanical. Appendix I Response Times for an Infinite Slab 1. Basic Equation The nondimensionalized heat equation for an infinite slab with uniform boundary conditions is e>x 2 ae 3t where 6 = [t (x, t) - T(0,=>)1 / Ft (L, °°) - T(0,°°)1 X = x/L, t = a t/L , L s= the thickness of the slab, x = the space variable, running from to L, t s time, a = the thermal diffusivity. The nondimensionalized heat flux is written as a_ _ ajL q, " dX where q = heat flux, q 1 = the steady state heat flux ( -X [~T(L, <*>) - T(0,c°)l /L), X «e the thermal conductivity, » = indicates the steady state value. 234 1.1. Boundary Conditions (B.C.)^ Constant temperature at X = 9(0, t) = 0, Two separate boundary conditions at X = 1 will be considered: (i) Constant temperature: 9(1,t) = 1, and (ii) Constant heat flux: — = 1. '1, T In each case the steady state temperature difference and steady state heat flux are equal to unity. 1.2. Initial Conditions The following uniform initial condition is assumed: 9(X,0) * W where W is a finite constant. 1. 3. Solutions to Basic Problem Solutions have been determined using basic solutions and the "Superposition Principle" as given by Churchill (3). The solutions are given in terms of the variable not specified at X * 1 , that is, in terms of heat flux for constant temperature boundary condition, and in terms of temperature for the constant heat flux boundary condition. The solutions are first given in general form, then for special cases with X, or both X and W, specified. Tables of the functions may be obtained by writing the authors directly. (1) General Form of Solutions (a) Constant Temperature B. C. \ CO 00 |4) = 1 + 2 Y (-l) n . cos (A X) . exp(-A 2 t) + 4W V cos (B X) . exp ( -B 2 T ) (1) oX/, L>, n n £j, n n '1,T n=l n=l where A = nrr, B = (2n-l)rr, exp ( ) is the exponential function, e n n (b) Constant Heat Flux B.C. GO 9(X,t) = X - 8 Y r(-l) n_1 /B 2 ] . sin (B X /Z) . exp(-B 2 T/4) Z_i , L nj n n n*l CO + 4W y (l/B ) . sin(B X/2) . exp(-B 2 T/4) (2) Li, n n n n*l 2 Boundary conditions will be abbreviated as B. C. 235 (2) Special Cases (a) Constant Temperature B. C. , with X = 1 aj. dX = 1 + 2 Y exp(-A 2 T) - 4W Y exp(-B 2 T) (3) 1,T n=l n*l (b) Constant Heat Flux B. C. , with X = 1 CO CO 9(1, t) = l -8 y (l/B 2 ) . exp(-B 2 T /4) + 4W T U-l) n ' l /B i . exp(-B 2 T/4) (4) i-i, n n i-j, L nJ n n=l ■ n=l (c) Constant Temperature B.C., with X = 0. 5, W finite ||) ■ l + 2 I «p(-4AT) (5) '0. 5,T n=l (d) Constant Temperature B.C., with X = 1, W = 0. 5 ae * right-hand side of eq (5) (6) 9X/ 0. 5,T (e) Constant Temperature B.C. , with X = 0, W = 0. 5 as r^ ' * right-hand side of eq (5) (7) SX /0,T (f) Constant Temperature, with X = 1, W = 1 || = 1 + 2 y (-l) n . exp (-A 2 i) (8) aX >l,T n=l 2. Calculation of Errors Errors in determination of thermal conductivity, if measurements are made before steady state conditions are attained, are derived below. 2.1. Constant Temperature B.C. Error at time T and position X. _ q( X,T) - q(X , -) Error = a -rri — ^ - q(X,co) . 51) -1 ax /x, T a&JEl = ii\ and 11 ! ° o x /x,t' ax /x , 236 - 1 2. 2. Constant Heat Flux B. C. Error at time T and at position X » L6(X,c°) - 9(0, » )J Error = — * f [e(x, T j - 9(0, t)] 1 = 9(X,t) 1 9(1, t) 9(0, t) = 0, all t, and 9(X, ») = X. 1 at X = 1 237 100 TT TT "P 50 INITIAL TEMPERATURE = 0.5 20 10 < o CONSTANT HEAT FLUX 0.5 0.2 0. 1 -CONSTANT TEMPERATURE B 1.0 2.0 3.0 T - at W 1 Figure 1 Response Times for Measuring Thermal Conductivity - Error at Time T After Start of Test - Constant Temp, and Constant Heat Flux Modes - Initial Temp. , W = 0. 5 238 100 T- o-t \V Figure 2 Response Times for Measuring Thermal Conductivity - Error at Time T After Start of Test - Constant Heat Flux Mode - Initial Temp. , W - to 1. 239 100 10 DC o 0. 1 0.2 0.4 0.6 0.8 1.0 1.2 T- at \V L Figure 3 Response Times for Measuring Thermal Conductivity - Error at Time T After Start of Test - Constant Temp, Mode - Initial Temp. , W * to 1.0 240 METHOD FOR MEASURING TOTAL HEMISPHERIC EMISSIVITY OF PLANE SURFACES WITH CONVENTIONAL ■THERMAL CONDUCTIVITY APPARATUS Nathaniel E. Hager, Jr. Research & Development Center Armstrong Cork Company Lancaster, Pennsylvania The conductance of an air space bounded by two large horizontal parallel plane surfaces is measured in conventional thermal conductivity apparatus. Convection is kept unimportant by having the hotter surface on top, and conduction of heat by the air is subtracted from the measurement to obtain a value for the net radiation exchange. When the two surfaces are identical, this value is used to compute their total hemispheric emissivity. A high-emissivity surface so measured is retained as a standard, and single samples of other surfaces are then tested by placing them in the opposite position. It is estimated that use of this technique with the Northrup thermal conductivity apparatus yields emissivity values accurate to within 0.015. Key words: Emissivity, heat transfer, surface properties, thermal conductivity, thermal properties, thermal radiation. 1. Introduction Infrared radiation plays an important and sometimes dominant role in practical heat transfer problems. To calculate the rate at which thermal energy is emitted, one usually needs to know the total hemispheric emissivity of surfaces in the temperature range found in the normal environment. The literature describes apparatus for producing detailed data on the directional and spectral characteristics of emitting surfaces (1,2,3,^)^-. Such data are ideal for checking emission theories, but they must be integrated to get values useful for engineering and apparatus design purposes. There remains a practical need for simpler apparatus which directly produces integrated values. A number of techniques have been described for direct measurement of total hemispheric emissivities of wires (5), spheres (6), and cylinders (7); the samples are heated electrically in evacuated chambers, and emissivity determinations are made by observing the rate of energy loss to the enclos- ing surfaces. Worthing has used a parallel-plate-ln- vacuum method (8); one plate is held at a fixed temperature, and the heating rate and heat capacity of a second plate are used to compute the radiant exchange between the enclosed surfaces. The method described here also employs parallel plates, but it does not require evacuation. The thermal conductance of an air space between two parallel plane surfaces is measured in conventional thermal conductivity apparatus of the type described by Northrup (U). The apparatus produces total hemispheric emissivity values of coatings, sheets, and surfaces of thin slabs with sufficient accuracy for most design purposes. Calibrated black body sources and delicate radiometric devices are not needed, and temperature-controlled shielding devices are not required for eliminating background interference even when the sample is approximately at room temperature. 2 . Theory Elementary heat transfer theory suggests an idealized experiment for determining emissivity values for infinite plane samples. This experiment is described in the following discussion. Later, real apparatus is described which puts the experiment on a practical basis; i.e., steps are taken to permit measurements to be made with samples of limited size. 1 Figures in brackets indicate the literature references at the end of this paper. 241 2.1. Idealized Experiment A volume of air is enclosed, as shown in figure 1, between two infinite horizontal isothermal planes. A calibrated slab of thermal insulating board is placed immediately below the air space. The top surface of the assembly is heated uniformly to temperature T^ while the bottom surface is held at a lower temperature T Q - Under steady-state conditions the interface between the air space and the board assumes a temperature T c . The heat flux q through the air space is the same as that through the board and is given by the equation q = k AT , (1) o o ' x ' D o where k and D Q are known values of the thermal conductivity and thickness of the board, and AT = T c " To- Since the upper plane is at a higher temperature than the lower, heat is transferred downward through the air space by gaseous conduction and thermal radiation, and convection is negligible. Assuming that the separation between planes is so small that no appreciable amount of radiation is absorbed by the air in passing from the one plane to the other, the conduction and the radiation processes are independent, and the total heat flux through the air space may be broken down into two components as indicated by the equation q = % + 1 r ) (2) where qa is the heat conducted by the air, and q r is the net thermal radiation exchange between surfaces. If the thermal conductivity of the air is k a , and the distance between the enclosing surfaces is D, then the gas conducted heat is calculated from the equation q a = k a A T ' (3) where AT = T. - T . h c According to standard texts (9), the net radiation exchange is given by the equation Wt 3 1 ± + i where 6^ and £ c are respectively the emissivities of the "hot" and "cold" surfaces bounding the air space, c is the Stefan radiation constant, and T is the mean absolute temperature of the air space. For this equation to be valid, the emissivity of each surface must equal its absorptivity for that radiation emitted by the opposite surface. This means that the surfaces must be "gray", or AT must be small. Combining the above equations, and solving for £ h yields the equation Va 1 h of 3 C. k AT k c c o o a D AT D o If the emissivity of the one surface £ is known, then, since all other factors on the right-hand side are known or can be measured, the emissivity of the other surface £^ can be calculated. If the emissivity of neither surface is known, but both surfaces are known to have the same emissivity, so that €" c = 6h> then the equation can be re-solved to get an expression for the emissivity common to the two surfaces. 242 2.2 Practical Experiment Experience with conventional thermal conductivity apparatus suggests that the idealized experiment is closely approximated by the configuration sketched in figure 2. Heavy horizontal high-conductivity plates, one electrically heated and the other kept at a cooler temperature, are used to form nearly isothermal surfaces. The plates are square with edge length L. A calibrated slab of insulating board having thickness D rests on the lower plate, and the upper plate is supported at a distance D above the upper surface of the slab. The choice of design depends on selecting values for D and D, and then sizing L, in view of the magnitude of D and D, so that the heat flow in the central region of the system is essentially the same as it would be if L were infinite. The reasoning involved in making these choices is included in a paper being prepared for later publication, and only the results are stated here. Experience suggests that about 2.5 cm is a good value to take for D and D. It is estimated from published data (10) that, as long as D does not exceed this value, the absorption of radiation by normal room air does not affect the practical experiment. Both analysis and experiment show that the infinite plane theory is approximately valid within the central portions of the finite plate system when L is at least twelve times as large as D, and when at least one of the surfaces enclosing the air space emits with characteristics approximating those of an ideal black surface. 3. Apparatus and Procedure Apparatus constructed along the lines discussed, as shown in figure 3(a), consists of a simple Northrup thermal conductivity apparatus, and a special emissivity cell. The latter unit is positioned between an electrically-heated plate and the calibrated insulation slab; the hot and cold plates are about 30.5 cm square and are made of 6.35-mm and 12.7-mm aluminum respectively. 3.1. Emissivity Cell Construction The emissivity cell detailed in figure 3(b) nominally measures 30.5 x 30-5 x 2.5^-cm, and consists of a plywood frame covered on one side with a tightly stretched sheet of 0.15-mm paper. At the center of the paper sheet is mounted a 12 x 12 x 0.05-mm platelet made of edge-welded copper-Constantan thermocouple ribbon. The paper and the central platelet are sprayed with flat black enamel, and the lower surface of the hot plate is treated with the same material. In the latest version of the cell, the interior surfaces of the sides of the frame are lined with 5-M, aluminum foil. 3.2. Standard Slab Selection The standard slab is 0.295-g/cm3 corkboard (ll) which is sanded to obtain plane surfaces and a uniform thickness of about 2.5^ cm. This board has maintained its thermal conductivity to within 2$ of the original value measured five years ago, and according to the experience of workers elsewhere in this laboratory, similar slabs have been found to be at least this stable over even longer periods of time (12). 3.3 Standard Surface Calibration To calibrate the black surface in the emissivity cell, the apparatus is run as described with the two surfaces having identical coatings. To make sure that the emissivities of the two surfaces are identical, it is necessary to make sure that the emissivity is characteristic of the coating and is not affected by the base. Assurance is obtained that this is the case by building up the coating on one surface at a time with several applications until successive applications lead to no further change in emissivity measurements. The heater power is adjusted so that the temperature difference between hot and cold plates is between 10 and 20 deg. After the apparatus has come to thermal equilibrium, which usually takes 6 to 8 hours, the temperature differences across the standard slab and the emissivity cell are read, and the temperature of the hot plate is measured. These data, along with known parameters associated with the standard slab and the test cell, are substituted into eq (5) to obtain the emissivity value shown in the first row of table 1. Note that this value is obtained before the cell is lined with aluminum foil. 243 Table 1. Emlssivity values measured for standard surface. 6 Emissivity Cell Number Condition of Tests Unlined 11 0.91U ±0.003 Lined with Foil h 0. 928 + 0.005 3.^. Performance Tests Several tests are run for the purpose of establishing the extent to which the finite plate system approximates the ideal infinite plate system. The first test is to add the aluminum foil lining. This produces an effect comparable to increasing the extent of the surfaces facing the air space. The value shown in the table indicates that a 1.5% increase in measured emissivity values results from using the foil. Because increasing the effective size of the surfaces makes such a small difference, it is concluded that, at least at the center of the cell where the measurement is made, the radiant flux is close to the infinite plane ideal. Other tests demonstrate the degree of uniformity of the temperature of the median plane, or the plane defined by the paper surface in the emissivity cell. At a distance of 6 cm from the center, the temperature is only 0.02 deg lower than at the center, and at a distance of 11.5 cm, it is 0.15 deg lower. These data can be used to show that, in the central region, lateral heat flow has little effect on the over-all transfer of heat from the heated plate to the median plane; i.e., the conducted heat flows uniformly perpendicular to the plates in the central region. These tests have led to the belief that heat flows through the center of the emissivity cell nearly as it would in the infinite plate case. 3 . 5 • Accuracy In view of the accuracy with which the parameters of the system are known, it is estimated that the probable absolute error in the measurements does not exceed O.OlU when the foil-lined cell is used. As noted in the table, the spread in measured values is somewhat less than the estimated error. 3.6. Test Surface Techniques After the standard surface in the emissivity cell is calibrated, other surfaces are measured by mounting them on the hot plate, measuring the conductance of the air space, and using eq (5) to obtain the emissivity. When measuring the emissivity of a surface of a thin sheet, the sheet is stretched over the hot plate or mounted with thin double-faced pressure-sensitive tape. Coatings are put on a thin metal foil or other thin base material, which is in turn mounted to the hot plate. The surface of sheets which do not have negligible thermal resistance can also be tested, provided appropriate corrections are made in the evaluation. 3-7. Results Compared with Published Measurements Some verification of the correctness of the described method is obtained by comparing measured values with values tabulated in the literature. It is difficult to do this with a high degree of exactness because surface condition is so important in most cases, and because it is difficult to reproduce surfaces measured by others. Nevertheless certain surfaces may be expected to have emissi- vity values falling within limited ranges, and some degree of checking is possible. Table 2 shows the comparison. Especially important is the fact that appropriately low emissivity values are obtained with reflective surfaces. This helps to establish the fact that convection heat transfer has indeed been made unimportant by using the horizontal configuration with the hotter surface on top. 244 Table 2. Experimental emlssivity values compared with published values, Surface T("C) New Apparatus Published (a,b) Commercial aluminum kk 0.28 - paint Various aluminum 2U-38 0.20-0.60 O.27-O.67 paints Flat black alkyd 38 O.928 . enamel Black lacquer 38 - O.87-O.98 Commercial aluminum 38 O.065 foil Commercial aluminum 38 - 0.055-0.11 foil and sheet goods (a) W. H. MacAdams, "Heat Transmission". (b) G. B. Wilkes, "Heat Insulation". k. (1) Maki, A. G., Stair, R., and Johnston, R. G. J. Research Natl. Bur. Standards 6**C, 99 (I960). (2) Maki, A. G., and Plyler, E. K., J. Research Natl. Bur. Standards 66c , 283 (1962). (3) McDermott, P. F., Rev. Sci. Instr. 8, 185 (1937). (h) Wilkes, G. B., "Heat Insulation", (John Wiley and Sons, Inc., New York, 1950). (5) Davisson, C. and Weeks, J. R., J. Opt. Soc. Am. and Rev. Sci. Instr. 8, 581 (192U). (6) Gest, G., J. Opt. Soc. Am. 32, 1009 (19^9)- References (7) (8) (9) (10) (11) (12) Fulk, M. M., and Reynolds, M. M., J. Appl. Phys. 28, 1'46U (1957). Worthing, A. G., J. Opt. Soc. Am. 30, 91 (19^0). Jakob, Max, "Heat Transfer", (John Wiley and Sons, Inc., New York, 1957), Vol. II, p. 3- McAdams, W. H., "Heat Transmission", (McGraw- Hill Book Company, Inc., New York 195*0, PP- 85-88. Vibracork, Armstrong Cork Company, Lancaster, Pa. Private Communication, Z. Zabawsky, Physical Standards Department, Research and Develop- ment Center, Armstrong Cork Company. 245 \ r HEATED SURFACE Do/ / / L COOLED SURFACE AIR SPACE kg, T -T h / / / CALIBRATED SLAB /k / / HEAT FLOW DIRECTION Tn Figure 1. Schematic view of idealized experiment. Figure 2. Schematic view of practical apparatus. HEATED PLATE -^ \ \ V \ \ \ \ ,\ AIR t D I / / / /CALIBRATED / / SLAB / / f \ \ \ \ \ \ \ \ \ COOLED PLATE — ^ (a) GENERAL VIEW Yt ELECTRICALLY HEATED PLATE i4 wi -THERMAL INSULATION /EMISSIVITY CELL /STANDARD SLAB (b) EMISSIVITY CELL CONSTANTAN V f COPPER FOIL LINING FLAT BLACK' ENAMEL ON PLATELET 0.25- IN. PLYWOOD FLAT BLACK ENAMEL ON PAPER Figure 3. Sketches of apparatus, 246 Heat Pulse Experiments and the Study of Thermal Transport at Low Temperatures R. J. von Gutfeld IBM Watson Research Center Yorktown Heights, New York Heat pulse propagation experiments in solids offer a relatively simple and direct method for determining (1) the thermal carrier mean free path from the shape of the detected pulse and (2) the carrier velocity from the first arrival time of the thermal signal. In dielectric samples this gives rise to the phonon mean free path and the phonon energy velocities. In metal samples, the electron mean free path and the Fermi velocity are obtained. Recent heat pulse experiments on sapphire and gallium will be descriV-ed to illustrate the technique and its useful- ness in the study of thermal transport properties. 1 Key Words: Electron and phonon mean free path, electrons-scattering, Fermi velocity, gallium, heat conduction, sapphire. 1 Full test to appear in W. P. Mason, Physical Acoustics Vol. V, (Adademic Press, New York, 1968). Preliminary results were reported by the author and A. H. Nethercot, Jr., in Phys. Rev. Letters 12 , 641 (1964) and .18, 855 (1967). 247 Deviations from Matthiessen's Rule in the Low Temperature Thermal and Electrical Resistivities of Very Pure Copper J. T. Schriempf Metallurgy Division Naval Research Laboratory Washington, D. C. 20390 Measurements have been made of the electrical and thermal resistivities of copper at temperatures from 2 to 20°K. The ratios of room temperature to helium temperature electrical resistivities of the three specimens studied ranged from 1000 to 3000. In both the electrical and thermal resistivities, deviations from Matthiessen's rule of the order of 100% were observed. These results are consistent with recent work by Dugdale and Basinskifl] 1 which indicates that the deviations from Matthiessen's rule can be very large when metallic impurities are the major source of the residual resistivities. Key Words: Copper, electrical conductivity, Lorenz number, Matthiessen's rule, thermal conductivity. 1. Introduction Matthiessen's rule assumes that the experimentally observed electrical resistivity (p ex p) of a metallic specimen can be written as: Pexp = Po + f>i> (1) where p is that portion of the electrical resistivity which is due to electrons scattering from impurities and imperfections, and p± is the portion of the electrical resistivity which is characteristic of the ideally pure metal. In pure metals it is a good assumption that the thermal transport is due entirely to the motion of the electrons, and in this case, Matthiessen's rule can be extended to the experimentally observed thermal resistivity (w ex p) , or w = w +w . , (2) exp o i' where w and w. are the thermal analogues of p and pj^. Since changes in atomic volume are small at tne temperatures of interest here (below 20°K) , p is independent of tem- perature. Furthermore, p and w are related according to the Wiedemann-Franz law, V w^T < 3 > o 1 Figures in brackets indicate the literature references at the end of this paper. 249 where L is the Lorenz number and has the theoretical value, L,,= 2.443xl0~ 8 V 2 /deg: 2 . o tn D Although the Wiedemann-Franz law as used in eq (3) has been well established in pure metals, Matthiessen 's rule is seldom strictly obeyed [2]. Furthermore, if one takes A = p -p-p (4) p *exp r i r o v ' and A = w -w . -w (5) w exp l o v ' as the deviations from Matthiessen's rule for the electrical and thermal resistivities, respectively, it can be shown that the deviations are always positive [2]. Although several workers [1,3,4] have studied these deviations in copper, a careful comparison of the thermal and electrical deviations has not heretofore been done. The data pre- sented below permit a comparison of these deviations at temperatures below 20°K for copper specimens with ratios of room temperature to helium temperature resistivities of the order of 1000 to 3000. 2. Experimental The apparatus used to measure the electrical and thermal resistivities of the copper specimens from ~ 2.5° to 20°K has already been described [5]. In this apparatus both thermal and electrical contact is made at each end of a specimen by a single copper clamp, so that a single geometric factor determines the net thermal, as well as the net electrical, resistance. Thus the apparatus permits the measurement of both the electrical and thermal resistivities on exactly the same specimen. Three copper specimens were prepared from American Smelting and Refining Company copper rod of spectrographic purity (99.999%). Specimen Cu 1 was swaged from the stock diameter of 3/8 inch to 0.125 inch diameter, annealed for 12 hr at 1000°C in air at a pressure of about 10~ 3 torr, and etched to a final diameter of 0.119 inch. Specimen Cu 2 was made from the same stock by swaging to a diameter of about 0.080 inch. This specimen was etched heavily in 50% nitric acid between steps in the swaging process to reduce surface contamination. Finally, the specimen was etched to a diameter of about 0.076 inch and annealed for 3 hr at 530°C in a vacuum of about 1x10" s torr. After measurements were completed on specimen Cu 2, it was annealed for about 22 hours at 1000°C in an atmosphere achieved by using a continuous air leak to reduce the lxlO -6 torr vacuum to 5xl0~ 4 torr. This "oxidized" specimen, labeled Cu 2-0, was found to have a slightly larger diameter than Cu 2. Metallographic examination of Cu 2-0 revealed the presence of "holes", presumably pockets of copper oxide formed during the oxidation annealing process. No indication of oxide pockets was found in Cu 1 or in a test piece prepared by the same technique used for Cu 2. Spectrographic examination of the three specimens verified the 99.999% purity of specimens Cu 2 and Cu 2-0, but indicated about 0.01% Mn in Cu 1. The presence of 0.01% Mn in Cu 1 was confirmed by colorimetric chemical analysis. Geometric shape factors for Cu 1 and Cu 2 were obtained from optical comparator measurements of diameters and distances between contacts. The shape factor for Cu 2-0 was obtained by assuming its electrical resistivity at 24°C was identical to that of Cu 2. Errors introduced in the determination of specimen geometry are estimated to be no greater than 2%. Apart from errors in the shape factors, errors in the measurement are estimated to be no greater than ± 1%. 3. Results and Discussion The values of p and w Q T obtained by extrapolation of the data to 0°K are shown in Table 1, along with other pertinent characteristics of the three specimens. In the case of p , the extrapolation ignored a small minimum (of the order of 1/2%) in the p-T curve which occurred at about 5°K for specimens Cu 1 and Cu 2. It is noteworthy that the values of L Q for all three specimens are within 2% of the theoretical value. This is in agreement with results of Tainsh and White [6] on a specimen of comparable purity, and is in marked contrast with earlier results by Powell, Roder, and Hall [4], In view of the sound theoretical and extensive experimental evidence which indicates Lo = ^th * n good metallic conductors of high purity [2], it is felt that the value of L obtained 250 by Powell, Roder , and Hall must not be correct for high purity copper, Table 1. Characteristics of copper specimens. Specimen p xlO 11 r o On P24°Cxl0 8 On w TxlO* o m deg 2 /w L xlO 8 o VVdeg 2 Annealing Metallic impurities ppm temp. °C pressure torr time hr Cu 1 1.73 1.77 7.20 2.40 1000 -10" 3 12 100 (Mn) Cu 2 .579 1.69 2.34 2.47 530 10" 6 3 <10 Cu 2-0(a) 1.12 1.69 4.52 2.48 1000 5x10" 4 22 <10 (a) Shape factor obtained by assuming p 9 .o r to be equal to that of Cu 2, In figure 1 are shown in lo cated in figure a specimen of p is in quite good The lack of agre Hall's observati resistivity is a 5° to 20°K. For roughly T s near for Cu 2 and T 1 the thermal and electrical resistiv arithmic plots of w ex p-w and Pexp~Po 1 by a dotted line are the thermal da ■ 1.2xl0 -11 Qm. Our data on specimen agreement with the Powell, Roder, an ement at lower temperatures is cons is }f L < L-tb. The observed temperat pproximately T 4 ' 6 for all three spec the thermal resistivity, the observe 20°K, but becomes weaker with decreas for Cu 2-0. ity data for all three specimens versus temperature. Also indi- ta of Powell, Roder, and Hall for Cu 2-0, which has p ~ 1. 1x10" 11 f^n, d Hall results above, say, 15°K. tent with Powell, Roder, and ure dependence of the electrical imens at temperatures from about d temperature dependence is ing temperature, approaching T 1 ' 7 In order to establish the deviations from Matthiessen's rule, as defined in eqs (4) and (5), a knowledge of p^ and w^ is necessary. The usual practice is to assume that Ap and A w are essentially zero in the most pure specimen; thus in this work P± = ^exp-Po^u 2 (6) and w.= (w -w )„ „, l exp o Cu 2 (7) With assumptions (6) and (7) then, the data in figure 1 indicate that A w and A„ are much larger in specimen Cu 1 than in specimen Cu 2-0. Recent work by Dugdale and Basinski [4] with dilute alloys of Au, Ge, and Sn in Cu indicates that the deviations from Matthiessen's rule might be much larger in the case where impurities are the major source of p and w than in the case where dislocations are the source of p and w . Thus it is tempting, although by no means conclusive, to claim that the p and w of specimen Cu 1 are due primarily to the Mn impurity, while Cu 2-0 exhibits a p and w due chiefly to dislocations. In both the present work and in Dugdale and Basinski's studies, the ideal resis- tivities were found by assuming A_ and A w to be negligible in the purest specimen. However, all the specimens used in the present study were quite pure by current stand- ards, and yet significant deviations from Matthiessen's rule were found among them. Therefore, any conclusions from studies of this nature are strongly influenced by the characteristics of the specimen chosen as the "ideal" specimen. Nevertheless, the present investigation, in agreement with Dugdale and Basinski's studies, indicates that very small amounts of metallic impurities can cause extremely large deviations from Matthiessen's rule in copper at low temperatures. 251 4. Acknowledgments The author is indebted to G. Picklo for performing the spectrographs and chemical analyses, and to R. Meussner and the Metallurgy Division metallography section for the metallographic examinations of the specimens. References [l] Dugdale, J. S. , and Basinski, Z. S. , Phys. Rev. 157 , 552 (1967). [2] Klemens, P. G. , Handbuch der Physik \A, 198 (1956). [3] White, G. K. , Australian J. Phys. 6, 397 (1953). [4] Powell, R. L. , Roder, H. M. , and Hall, W. J., Phys. Rev. 115 , 314 (1959) [5] Schriempf, J. T. , Sixth Conference on Thermal Conductivity, Dayton, Ohio, p. 615 (1966). [6] White, G. K. , and Tainsh, R. J., Phys. Rev. 119, 1869 (1960). a> -a in O x c I 20 1 1 1 Mill II ' 1 o o II 1 1 1 II ' l o o — 10 8 I i ° J- o o = 5 — o o° f II ii ° & + o o i + — 3 o ii o & 2 o o o° + • + ' + / II o + o - 1.0 o° ii ii + °% — 0.5 — o ii II ° + — + II 0.3 o + + + ii o •+■ 0.2 0.1 1 +o 1 Mill Mill II n + II 1 1 - 1.0 0.5 E 0.3 C3 0.2 ~o X * — » 0.10 o I a. 0.05 o X . o o x • o X 1 * • X . x • X . X . x . x . nj u H in •d <0 i— ( C o 4) S-i 1) H •H Ph ph m n) •H X 0) cd in h X 01 ■rf > h 01 +-> C o e« m H V) +J •H X) rt M e c •H o X CO +-> 3 Ul e ■H / — , « +J •H T3 X X> .— 1 in •H ■H § O X h 3 rt T3 0) O fc CJ cd h c at •H Ph ns -O CD e in B bo CD o .c in 4-> •M ■H h 14-1 3 CD o O in *-J nl CD cd 0) in •H J5 U) ^1 o n! O o 255 4> o o o o X • K • o o 00 X €*3 X • K • x • • X • X « X • o o (0 43 fn 3 •H O TJ tu 3 OJ w C C o crt u Sn •H X • O o §8 * o x X • x » x • x • x • o o X • X • x« x • X • x e x# x • x • X • x« x • x • (VJ o i ^ K |« X-X I X o o * K * x # "I x* X 8 3 W E= ■H , — \ vi 4-> •H T3 X 43 t— t 14-1 ■H O 4-1 T3 « +J C •H cd > Ul •H i/> (3 4J ■H 3 O x S 3 cd T5 cd u nj h e cd 1— 1 ■H ft nj 43 a> F in in M cd 4-> 4-" •H Sh 4h 3 tu O O 4-1 <0 C 4n U O •H 4-> l/> 4-> 0) in ■H 43 in h o nj O o 256 Thermal Conductivity of Pure and Impure Tin in the Normal and Superconducting States' C. A. Reynolds, J. E. Gueths. 2 F. V. Burckbuchler N. N. Clark, 3 D. Markowitz, G. J. Pearson, ^ R. H. Bartram, and C. W. Ulbrich 5 Physics Department and Institute of Materials Science University of Connecticut Storrs, Connecticut 06268 Measurements of the normal- and superconducting-state thermal conductivities (K n and K s ) were made on two pure (99.999$) and twenty lightly doped tin specimens. Nine polycrystalline samples contained mercury, lead or bismuth impurity up to 1 at. % . Eleven single crystal samples containing up to 0.3 at. % cadmium were measured with the heat current nearly perpendicu- lar to the tin tetrad axis. The lattice component in the normal state, K n , is fractionally very small, but is enhanced in the superconducting state, K»; the data admit to analysis in terms of the "universal curve" formalism of Lindenfeld and Pennebaker for, KjJ , and the theory of Bardeen, Rickayzen and Tewordt for K|/K n • The electronic component in the normal state, K n , is assumed to be of the conventional foftn f (3 /T + «T 2 ) -1 . The ratio, /° / /Q , where /2 is the residual resistivity, is found to be independent of impurity content and in agreement with the theoretical Lorenz number. The anisotropy of /3 is found to be similar to that of /o B . The variation of <* was studied as a function of impurity concentration and found to increase monotonically with concentration, as reported by previous investigators. Comparison of some of our K|,/Kg data with the theory of Kadanoff and Martin produced an apparent contradiction to Anderson's theorem. This prompted us to formulate a theory, based on features taken from the work of Markowitz and Kadanoff and of Hohenborg, of the effect of energy gap anisotropy on K 3 . An expression was developed for K 3 /K n as a function of crystal orientation, temperature, and mean free path. The gap edge, which is central to thermal conductivity, is rendered isotropic by impurities well before their effect is felt on quantities such as the average gap or transition temperature. The theory is found to reproduce the qualitative features of our data, as well as that of Guenault for pure tin, but it under-estiraates the magnitude of the anisotropy effect by about a factor of two. From the isotropic theory of Kadanoff and Martin and Guenault 's data we have estimated the effective gaps parallel and perpendicular to the tin tetrad axis. Using the pic- ture of the Fermi surface proposed by Klemens, we have calculated the average gap after anisotropy has been washed out and found that our data is in good agreement with theory when the gap varia- tion is determined in this semi- empirical way. Key Words: Electrical conductivity, Lorenz number, superconducting energy gap, superconductor, thermal conductivity, tin, tin alloys. 1. Full text has been published as follows: J. E. Gueths, N. N. Clark, D. Markowitz, F. V. Burckbuchler and C. A. Reynolds, Phys. Rev. 165(2) 364 (1967). G. J. Pearson, C. W. Ulbrich, J. E. Gueths, M. A. Mitchell and C. A. Reynolds, Phys. Rev. 1U, 329 (1967). C. W. Ulbrich, D. Markowitz, R. H. Bartram, and C. A. Reynolds, Phys. Rev. 1^, 338 (1967). Supported by U. S. Air Force Office of Scientific Research grant AF-AFOSR-^7A-67 and Office of Naval Research Contract N0NR2967 (00) 2 Wisconsin State University, Oahkosh, Wisconsin. (Present address) ^Present address: Universldad de Costa Rica, San Jose, Costa Rica. ^Present address: Eastman Kodak, Rochester, New York. 'Present address: Clemson University, Clemson, South Carolina 257 2. References 1. P. Lindenfeld and W. B. Pennebaker , Phys. Rev. 127 , 1881 (1962). 2. J. Bardeen, G. Rickayzen, and L. Tewordt, Phys. Rev. 115 . 982 (1959). 3. Leo P. Kadanoff and Paul C. Martin, Phys. Rev. 124 , 670 (1961). 4. P. W. Anderson, J. Phys. Chem. Solids 11, 26 (1959). 5. D. Markowitz and L. P. Kadanoff, Phys. Rev. 151 . 56J (1963). 6. P. Hohenberg, Zh. Eksperim. i Teor. Fiz. 45 , 1208 (1963) (English transl. : Soviet Phys.-' JETP 18, 834 (1964)). 7. A. M. Guenault, Proc. Roy. Soc. (London) A262 420 (1961). 8. P. G. Klemens, C. Van Baarle and F. W. Gorter, Physica, 3_0, 1470 (1964). 258 Temperature Dependence of the Lattice Thermal Conductivity of Copper-Nickel Alloys at Low Temperatures J. C. Erdmann and J. A. Jahoda Boeing Scientific Research Laboratories P.O. Box 3981 Seattle, Washington 98124 The lattice thermal conductivity of coRper-nickel alloys at low temperatures deviates systematically from the ordinary T - dependence. This deviation is small in the alloys containing only very little nickel. For higher concentrations of nickel, however, the temperature dependence approaches T in the liquid helium temperature range. Only for very nickel-rich alloys a temperature dependence of the lattice conductivity like T is found. These findings result from experiments with polycrystals as well as single crystals. The behavior of the alloys with more than 50 percent copper can be understood in terms of Pippard's simple relaxation model for the ultrasonic attenu- ation by s-electrons. Zimmerman and Lindenfeld and Pennebaker have already in this way interpreted similar results in other alloy systems. To understand the behavior of the nickel-rich alloys, however, Pippard's model has to be extended to include d-electrons. Such an extended model has been applied. We find that consistency with experimental results exists, if the mean free path of the d-electrons is up to one order of magnitude smaller than the mean free path of the s-electrons. Other conclusions concern the ratios of the effective masses and the relaxation rates. Key Words: Copper-nickel alloys, electrical conductivity, electron mean free path, thermal conductivity. 1. Introduction The lattice thermal conductivity of copper-nickel alloys at low temperatures is interesting mainly for the following reason: The mean free path SL of the electrons can become small enough so that its product with the predominant phonon wave number q is smaller than unity. This situation is usually met in studies of the ultrasonic attenuation by electrons. Pippard [1] has given a theory of this effect, also applicable for the case q i>l. The value of Pippard's theory for investigations of the lattice thermal conductivity lies in the fact, that the coefficient of ultrasonic attenuation can be regarded as the reciprocal of the phonon mean free path. It is then possible to study the transition of the lattice thermal conductivity from the regime q 1>1 (ordinarily valid for thermal phonons) to the regime q 1<1 (often characteristic for ultrasonic experiments). This transition means a breakdown of the adiabatic principle, as discussed by Ziman [2], but the arising theoretical difficulties are avoided by Pippard's approach, unlike the usual treatment of electron-phonon scattering in terms of well- defined quantum states. Zimmerman [3] and Lindenfeld and Pennebaker [4] have shown by investigations of various alloys that the low temperature lattice thermal conductivity, if limited by the electron-phonon interaction, varies (a) as the square of the temperature, if q 5. >>1 (longitudinal modes); (b) as the cube of the temperature, if q5. >> 1 (transversal modes); (c) proportional to the temperature, if qJ. << 1 (all modes). At least the cases (a) and (c) seem to be well verified. Staff Members 2 Figures in brackets indicate the literature references at the end of this paper. 259 As in the case of palladium-silver alloys, recently investigated by Fletcher and Greig [5], the interest in the thermal conductivity of the copper-nickel system is enhanced by considerations of the important role of d-electrons. We shall concentrate on this topic and show that certain specific features of our results can be best explained in terms of the contributions by d-electrons. In partic- ular we can conclude that the mean free path of the d-electrons is substantially smaller than that of the s-electrons, so that the d-electrons remain in the "ultrasonic" regime ql, < 1 for most of the nickel-rich alloys. The chosen alloys lend themselves well to this type of investigation, because they form a continuous series of solid solutions without miscibility gap. Furthermore, the Fermi-surfaces of copper and nickel, have been investigated extensively [6 - 9]. 2. Experiments 2.1. Sample Preparation Eleven specimens have been investigated, three of them single crystals. Their composition and residual resistivity are listed in Table 1. The single crystals were purchased from Materials Research Corp. They were prepared by electron beam float zoning, 12 cm long, 6.0 - 7.5 mm diameter. The polycrystalline alloys, together with their chemical analysis and spectrographic surveys were provided by the courtesy of the International Nickel Company, Inc. The vacuum cast ingots were hammer forged, hot rolled to 18 mm diameter and rough turned. The rough turned bars were swaged to 10 mm diameter. Intermediate annealing was performed at 930°C when necessary. These specimens were finally machined to size, 10 cm long, 5.0 mm diameter. They were annealed for 24 hrs . in the argon furnace and allowed to cool slowly. Table 1. Composition in weight percent and electrical residual resistivity p in uficm of the investigated specimens. s c. - single crystal, p.c. - polycrystal Notation Type Composition p uficm o Cu 98 s . c. 2.29 Ni, bal. Cu 2.17 Cu 96 s .c. 4.05 Ni, bal. Cu 4.95 Cu 91 p.c. 9.30 Ni, 0.025 Al, bal. Cu 11.22 Cu 72 p.c. 27.96 Ni, 0.023 Al, bal. Cu 33.38 Cu 49 p.c. 50.50 Ni, 0.030 Al, bal. Cu 46.10 Ni 65 p.c. 64.87 Ni, 0.051 Al, bal. Cu 27.62 Ni 85 p.c. 84.70 Ni, 0.054 Al, bal. Cu 11.14 Ni 90 p.c. 90.24 Ni, 0.060 Al, bal. Cu 8.24 Ni 91(a) p.c. 91.05 Ni, 0.046 Al, bal. Cu 15.88 Ni 96 p.c. 95.60 Ni, bal. Cu 3.91 Ni 98 s . c. 99.35 Ni, bal. Cu 0.907 Ni 98(b) p.c. 97.97 Ni, bal. Cu .1.66 (a) the large value of appr. 80 percent Ni (b) no thermal conductivity data suggests a different composition, 2.: Measurements All measurements were performed in the apparatus described previously [10]. One end of the cylindrical specimen was soldered to the heat sink, variable in temperature. To the other end a heater was connected, and the temperature difference along the rod was measured by means of a differential gas thermometer. The electrical resistance also was measured. The vacuum of the specimen chamber was 10 -6 mm Hg. Results The thermal conductivity of the eleven specimens listed in Table 1 was measured at temperatures between 4.2 8 K and approximately 40°K. Below 20-30°K all alloys are well within the range of residual electrical resistivity [10], so that the Wiedemann-Franz-law can be regarded as valid (L = 2.45 -10 V 2 /deg 2 ) . The total thermal conductivity of the alloys and its electronic and lattice component found by using the Wiedemann-Franz-law are listed in Tables 2-12. The lattice thermal conductivity of the 260 alloys is plotted in figures 1-4. Tables 2-12. Ambient temperature in deg K, total thermal conductivity k, its electronic part k e) and its lattice part k p ^ in 10"3 Wcm - J-deg~l for tha various speci- mens. The data for k are entered as measured, k is determined from p (see Table 1) using the Wiedemann-Franz law with L = 2.45 • 10~ 8 V 2 deg-2 . The limit of the validity range of this separation method is indicated where necessary. The exponent n expresses the temperature dependence of the lattice part between 4.2° and 20°K. Table 2 : Cu 98 Table 3: Cu 36 Table 4: Cu 91 n=2 00 n= =2.00 n= =1.97 T k k e kph T k k s k ph T k k e kph 4.23 11.52 47.92 23,60 4.24 36.68 ""Z0Y98" 15.70 ~4T2~4~ 20.86 9.27 11.59 5.92 109.9 66.95 42.92 4.24 38.15 20.97 17.18 4.52 23.36 9.88 13.49 6.59 120.4 74.60 45.81 4.21 35.93 20.83 15.10 4.67 23.80 10.20 13.60 7.41 154.5 83.87 70.58 5.19 48.91 25.65 23.26 4.87 26.87 10.63 16.24 8.21 172.8 93.43 79.41 5.42 53.96 26.83 27.13 5.12 28.65 11.17 17.46 9.31 217.2 105.4 111.8 5.82 60.68 28.80 31.88 5.32 31.50 11.72 19.77 10.34 243.6 117.0 126.6 6.12 66.59 30.28 36.31 5.54 32.77 12.09 20.68 11.41 296.2 129.1 167.1 6.41 69.22 31.70 37.52 6.62 42.32 14.46 27.86 12.59 338.1 142.4 195.7 6.73 71.27 33.30 37.97 7.08 49.28 15.45 33.83 14.01 391.8 158.5 233.3 7.35 85.36 36.35 49.01 7.63 53.62 16.67 36.94 15.83 458.5 179.2 279.3 7.78 93.44 38.50 54.94 8.01 57.64 17.50 40.14 18.85 630.9 213.3 417.6 8.67 110.6 42,90 67.73 8.54 66.06 18.66 47.40 19.34 604.7 218.9 385.8 9.92 133.0 49. C8 83.94 9.20 69.58 20.10 49.48 22.03 679.0 249.3 429.7 10.44 151.7 51.66 100.0 9.69 80.15 21.17 58.98 24.34 781.3 275.4 505.9 10.93 160.0 54.06 105.9 10.89 98.52 23.79 74.73 29.54 878.2 334.3 543.9 11.99 182.4 59.29 123.1 11.76 110.2 25.69 84.46 36.04 1051. 407.8 643.1 12.63 211.7 62.45 149.2 13,25 147.5 28.94 118.6 39.74 1157. 449.8 707.1 13.53 220.7 66.94 153.8 15.02 170.4 32.81 137.6 44.93 1227. 508.5 718.8 14.52 258.7 71.83 186.8 16.36 183.4 35.74 147.7 49.18 1243 . 556.6 687.2 16.02 293.0 79.24 213.8 18.21 216.5 39.78 176.7 55.33 1423. 626.1 797.1 17.30 318.9 85.58 233.3 19.88 250.2 43.43 206.8 62.73 1411. 709.9 701.8 17.25 322.8 85.33 237.5 22.07 265.5 48.20 217.3 70.02 1503. 792.4 710.1 20.02 393.3 99.00 294.3 23.78 286.7 51.93 234.8 21.52 436.3 106.5 329.8 25.30 334.0 55.26 278.8 23.53 489.3 116.4 372.9 26.99 345.7 58.95 286.8 25.02 488.5 123.8 364.7 32.11 375.5 70.15 305.3 27.99 603.8 138.5 465.3 36.22 449.8 79.11 370.7 31.02 590.3 153,4 436.9 41.07 431.2 89.71 341.5 39.03 701.9 193.0 508.8 49.92 510 . 2 109.0 401.2 . . 51.02 848.4 252.4 596.0 60.23 70.92 534.3 566.4 131.5 154.9 402.7 411.5 Table 5: Cu 72 Table 6: Cu 49 Table 7: Ni 65 n=l. 50 n=l 15 n= 1.26 T k k e kph T k k e k ph T k k e k ph 4.24 16.05 3 . 11 12.94 4.23 15.30 2.24 13.06 4.24 15.74 3.76 11.98 4.59 19.00 3.37 15.63 4.68 17.20 2.48 14.72 4.24 15.74 3.76 11.98 4.91 20.88 3.60 17.28 4,68 16.80 2.48 14.32 4.43 16.46 3.93 12.53 5.09 20.53 3.73 16.79 5.25 16.79 2.79 14.00 4.49 16.86 3.98 12.86 5.43 25.33 3.99 21.34 5.60 18.86 2.97 15.89 4.74 18.26 4.20 14.06 5.83 27.71 4.28 23.43 6.30 22.79 3.35 19.44 4.89 18.51 4.34 14.17 6.16 27.75 4.52 23.23 6.30 24.59 3.35 21.24 5.23 20.61 4.64 15.96 6.27 30.86 4.61 26.25 7.00 26.30 3.72 22.58 5.44 22.27 4.83 17.44 6.66 33.57 4.89 28.68 7.00 26.90 3.72 23.18 5.57 23.49 4.94 18.55 7.01 36.26 5.15 31.12 8.05 30.13 4.28 25.85 5.82 23.66 5.16 18.49 7.43 37.54 5.45 32.09 9.05 36.09 4.81 31.28 6.10 22.07 5.41 16.66 7.99 43.52 5.86 37.66 10.50 39.01 5.53 33.48 6.31 26.18 5.60 20.59 8.51 43.79 6.24 37.55 10.50 43.30 5.53 37.77 6.41 27.26 5.69 21.57 9.06 50.21 6.65 43.56 10.50 42.20 5.53 36.67 6.42 26.97 5.69 21.27 10.03 61.43 7.36 54.05 10.50 40.80 5.53 35.27 6.67 27.25 5.92 21.34 10.68 67.42 7.84 59.59 12.10 50.82 6.43 44.39 7.27 30.29 6.45 23.84 11.78 84.89 8.64 76.25 16.80 74.80 8.92, 65.88 7.37 27.86 6.54 21.31 261 Table 5 (continued) Table 6 (continue d) Table 7 (continued) 12.38 82.90 9.08 73.81 20.00 90.25 10.62 79.63 7.37 30.70 6.54 24.16 12.85 87.26 9.43 77.83 24.00 111,9 12.74 99.16 7.67 31.15 6.81 24.35 13.01 83.80 9.55 74.26 30.40 135.0 16.14 118.8 8.00 31.69 7.10 24.59 14.25 108.6 10.46 98.16 35.00 142.5 18.58 123.9 8.26 36.57 7.32 29.25 14.28 99.80 10.48 89.32 45.00 155.0 23.90 131.1 8.55 38.30 7.58 30.71 15.50 116.6 11.38 105.2 8.86 37.21 7.86 29.35 16.52 115.6 12.12 103.5 9.32 37.74 8.27 29.48 17.54 128.7 12.87 115.8 9.34 43.25 8.28 34.97 18.40 137.1 13.50 123.6 9.64 41.25 8.55 32.70 19.95 161.9 14.65 147.3 10.46 43.26 9.28 33.98 19.96 142.1 14.65 127.5 10.81 52.79 9.59 43.20 21.53 152.1 15.80 136.3 11.48 52.54 10.19 42.35 23.24 177.9 17.05 160.9 12.51 53.15 11.10 42.04 25.08 195.1 18.41 176.7 12.54 60.14 11.12 49.02 25.13 196.4 18.44 177.9 13.76 59.26 12.20 47.06 26.99 222.5 19.81 202.7 15.27 76.94 13.54 63.40 26.99 217.9 19.81 198.1 16.98 84.52 15.06 69.46 28.65 249.2 21.03 228.2 18.54 95.62 16.45 79.17 34.95 283.5 25.65 257.8 20.18 102.7 17.91 84.83 35.02 306.5 25.71 280.8 21.49 100.5 19.06 81.46 38.22 291.0 28.06 262.9 24.07 114.6 21.36 93.28 42.05 306.2 30.86 275.3 26.57 124.7 23.57 101.1 47.37 322.8 34.77 288.0 29.13 137.0 25.84 111.2 54.04 359.3 39.66 319.6 32.97 44.23 54.16 65.34 139.9 188.9 197.7 219.58 29.25 39.23 48.05 57.96 110.65 149.7 149.7 161.6 Tab] e 8: Ni 85 Table 9: Ni 90 Table 10: Ni 91 n=1.33 n =1.34 n= =1.30 T k K. kph T k k e kph T k k e kph 4.22 17.54 9.27 8.27 4.22 19.75 12.56 7.19 4.23 12.14 6.53 5.61 4.24 17.57 9.32 8.25 4.23 19.44 12.58 6.85 4.24 12.47 6.54 5.93 4.39 17.90 9.64 8.26 4.32 20.31 12.85 7.46 4.28 12.13 6.60 5.53 4.52 19.79 9.95 9.84 4.42 20.20 13.14 7.05 4.55 13.73 7.01 6.72 5.03 20.63 11.07 9.56 4.56 21.01 13.56 7.45 4.82 14.49 7.44 7.04 5.31 22.86 11.68 11.18 4.71 21.89 13.99 7.91 5.00 15.52 7.72 7.80 5.83 25.57 12.81 12.75 4.87 22.39 14.46 7.92 5.01 16.05 7.73 8.32 6.21 26.16 13.66 12.50 5.37 26.47 15.97 10.50 5.49 17.76 8.48 9.28 6.29 27.76 13.84 13.92 5.61 27.10 16.68 10.42 5.87 19.18 9.06 10.12 6.39 26.94 14.05 12.89 5.82 29.56 17.31 12.25 6.16 19.92 9.51 10.41 6.67 29.90 14.66 15.24 6.22 29.29 18.50 10.78 6.74 23.80 10.39 13.40 6.99 30.97 15.37 15.60 6.60 32.42 19.61 12.85 6.92 23.59 10.68 12.91 7.34 33.96 16.13 17.82 6.93 34.04 20.60 13.44 7.69 25.34 11.86 13.47 7.69 34.67 16.90 17.77 7.40 36.81 21.99 14.86 8.16 28.34 12.59 15.75 7.94 35.24 17.47 17.77 7.54 38.66 22.42 16.23 8.85 32.38 13.66 18.73 8.63 40.01 18.98 21.03 7.97 40.42 23.73 16.69 9.17 31.24 14.15 17.09 9.01 41.08 19.81 21.27 8.51 42.77 25.31 17.46 9.87 34.36 15.23 19.12 9.93 43.98 21.84 22.13 9.78 49.52 29.06 20.45 10.57 36.25 16.30 19.95 10.00 45.94 21. '98 23.95 10.29 53.82 30.58 23.23 10.94 41.72 16.88 24.84 11.04 53.96 24.28 29.68 10.91 58.78 32.44 26.34 12.00 44.42 18.51 25.91 12.11 1 56.98 26.64 30.34 11.96 63.74 35.55 28.19 13.02 47.64 20.09 27.55 12.51 61.79 27.54 34.25 12.55 73.14 37.31 35.83 14.58 55.88 22.49 33.40 13.97 72.31 30.73 41.59 13.93 85.51 41.41 44.10 15.78 61.51 24.35 37.16 15.72 81.92 34.57 47.35 13.54 73.86 40.25 33.61 16.63 65.61 25.65 39.96 17.51 89. 1C 38.51 50.59 15.29 89.23 45.46 43.77 18.77 68.10 28.96 39.15 19.42 105.8 42.70 63.07 17.41 103.1 51.75 51.30 21.20 86.18 32.71 53.47 21.03 114.8 46.25 68.60 18.91 113.9 56.23 57.67 23.20 95.09 35.80 59.29 22.50 123.3 49.48 73.82 20.49 125.6 60.92 64.65 24.88 106.4 38.38 68.05 24.97 146.3 54.91 91.36 24.06 147.8 71.54 76.28 25.11 105.1 38.74 66.38 27.24 157.2 59.91 97.27 26.07 170.1 77.51 92.55 26.35 115.9 40.66 75.20 262 Table 8 (continued) Table 9 (continued} Tabl e 10 (continued) 29.51 175.9 64.89 111.0 28.14 180.0 83.66 96.32 29.07 142.3 44.84 97.47 31.75 176.2 69.83 106.4 30.55 181.6 90.83 90.81 31.14 135.0 48.05 86.91 33.65 194.6 74.00 120.6 39.00 262.7 115.9 146.7 35.05 136.7 54.07 82.57 36.88 203.7 81.10 122.6 40.70 185.1 62.79 122.3 39.54 222.2 86.95 135.3 50.74 217.8 78.28 139.5 43.99 245.6 96.73 148.9 60.58 205.4 93.47 111.9 48.31 254.8 106.2 148.6 49.43 239.6 108.7 130.9 53.18 278.9 116.9 162.0 Table 11: Ni 96 Table 12: Ni 98 n= =2.0 n=2.0 T k k e kph T k k e V 4.31 31.81 27.04 4.77 4.23 117.76 114.28 3.47 4.44 31.38 27.83 3.55 4.24 118.56 114.62 3.93 4.58 31.76 28.70 3.05 4.25 117.90 114.80 3.10 4.72 33.07 29.57 3.50 4.51 126.77 121.88 4.89 5.42 38.43 34.00 4.43 4.68 129.63 126.43 3.20 6.47 46.48 40.57 5.91 5.23 146.68 141.40 5.27 7.03 57.28 44.07 13.21 5.73 159.70 154.88 4.82 7.03 55.06 44.07 10.99 6.32 178.82 170.72 8.10 8.15 63.17 51.07 12.10 7.59 219.94 205.03 14.9 9.58 64.79 60.04 14.75 8.48 246.74 229.05 17.7 11.64 95.97 72.94 23.03 8.95 254.04 241.79 12.2 13.82 118.2 86.65 31.55 9.50 274.17 256.57 17.6 19.70 174.4 123.5 50.89 10.49 308.26 283.47 24.8 24.95 230.3 156.4 73.94 10.82 314.79 292.37 22.4 32.08 284.3 201.1 83.17 11.39 342.95 307.72 35.2 41.96 372.6 263.0 109.6 11.69 11.87 13.39 13.78 15.83 16.65 17.94 18.87 20.66 21.86 24.25 26.75 28.09 352.36 356.84 401.89 418.33 472.75 509.85 557.49 574.12 615.05 655.45 754.88 844.70 845.98 315.87 320.75 361.79 372.11 427.69 449.66 484.73 509.83 558.11 590.48 654.93 722.66 758.76 36.5 36.1 40.1 46.2 45.1 60.2 72.8 64.3 56.9 65.0 100. 122. 87.2 The temperature dependence of the lattice thermal conductivity in the temperature range between 4.2°K and 20°K can be well described as T n by a single power n of the temperature T. The value of n varies from alloy to alloy and is included in tables 2-12. It is also plotted vs. the composition in figure 5. The same figure also contains the value of the lattice thermal conductivity at 4.2°K as a function of the composition. The following features of the two curves are conspicuous: (a) The exponent n decreases from the value 2.0 for pure copper to about 1.1 for Cu50-Ni50. (b) For very nickel-rich alloys again a value n~2 is found. (c) In the composition range Cu50-Ni50 to Cul0-Ni90 the exponent n varies only from 1.1 to about 1.3. The lack of symmetry with composition about Cu50-Ni50 is obvious. (d) The lattice thermal conductivity of copper is about 7.5 times larger than that of nickel. (e) The lattice thermal conductivity of copper decreases rapidly with the addition of nickel, while the lattice thermal conductivity of the two most nickel-rich alloys is approximately constant . (f) For intermediate compositions, the lattice thermal conductivity is largest at the composition Cu40-Ni60. It drops continuously on the nickel-rich side of the alloy system. On the copper- 263 rich side this drop is also observable, but it interferes with the effect described in (e) . 4. Discussion The ratio of the lattice thermal conductivity to the total measured thermal conductivity at 4.2°K is plotted vs. the alloy composition in figure 6. One notices that for most of the alloys this ratio is larger than 60 percent. We think that this extraordinary enhancement of the lattice component is basically caused by the incomplete electron-phonon interaction which is typical for a situation where the electron mean free path I is smaller than the wave length of the phonons with the predominant wave number q, i.e. where qJl <1. The theory of this interaction has been given by Pippard [1] for the coefficient of ultrasonic attenuation. It is, however, also relevant to the thermal conductivity, because the attenuation coefficient can be regarded as the reciprocal phonon mean free path. Some of the consequences of Pippard 's theory have already been summarized in the Introduction. Indeed, the product ql is smaller than unity for most of the copper-nickel alloys. We find this result, shown in figure 7, if we calculate qS, from the known values of the electrical conductivity o. The predominant phonon wave number is approximately [11] q = 2 KT/(nv s ) (1) (K - Boltzmann constant, T - abs . temperature, n=h/2TT - Planck's constant, v s - sound velocity). We shall assume that the electrical conductivity arises from contributions by s- as well as d-electrons, but most of the current is always carried by the s-electrons [12], thus a ss o , where 2 a = e r n /m (2) s s s s (t - relaxation time of the s-electrons, n - number of s-electrons per atom, m - effective mass of s s s the s-electrons). If we assume spherical energy surfaces, we can rewrite eq (2) as '•-^■» • where S is the area of the Fermi-surface occupied by the s-electrons (e - electronic charge) . From eqs (l)"and (3) there follows ql = 24tt 3 (KT/e 2 v ) • (a /S_ ) . (4) s s s Fs The number of s-electrons per atom n for a particular copper-nickel alloy is fairly well known from magnetic measurements [13]. It drops from 1.0 in copper to about 0.6 in Cu60-Ni40 and retains the value 0.6 approximately for the rest of the system. The area of the Fermi-surface S is approximately proportional to n 2/3, F 0r pure copper its value is 23.3 fi~2 in the approximation of spherical energy surfaces [14], and 23.0 8~2 if we use Garcia Moliner's nonspherical representation [15], calculated to fit Pippard's experimental results [6]. The number of d-electrons, on the other hand, increases from 0.0 in Cu60-Ni40 to 0.6 per atom in nickel [13]. Hence, in nickel their number is equal to the number of s-electrons. But since the mass ratio of d-- and s- electrons may be of order 10 or larger, as one can conclude from specific heat measurements [16], the contribution of the d-electrons to the electrical conductivity is quite small, so that we cannot derive their mean free path from a. The situation is different in the case of the ultrasonic attenuation a by electrons, which is given by the expression [1] s 2 2 -J q I tan '(qjQ a long dv o T [ 'i[qi-taTi" l {ql)] for the longitudinal waves and by -] (5) 'trans = iS^ H (q*) 2 +l] tan" V) - q,}" 1 (5a) 264 for transversal waves. It is important to note that these expressions for the attenuation result as integrals over a spherical Fermi-surface. They have been derived for s-electrons, but it can be shown that the same expressions are obtained for a spherical pocket for d-electrons. Provided t ~ x , , it is then obvious from eqs (5) and (5a) that d-electrons can contribute significantly to a because of their large effective mass. Let us write, using an obvious notation, a = a (qJ, ) + a (qd ) . (6) s s d d The bar indicates some modification of a due to s-d or d-s - scattering, respectively. Other modifica- tions may arise, because the d-electrons may not occupy separate pockets, but rather deep depressions of the main sheet of the Fermi-surface, as it seems to be the case in nickel [7-9]. For a first approximation we assume a. as a. . If we now take a as the reciprocal phonon mean free path, we can calculate the lattice thermal conductivity (compare with Ref. [4]): 3 2 °° k. = -^ *=■ 7— I / c /a. (c ,c.) • f(x) dx , (7) long , 2 u 2 I v . ' s long s d 6 6tt H s slong & = -^- ^- T °r k = _ 2 SL ' J c /a (c ,c.) ■ f(x) dx , (8) trans 3it n i. v o s trans s' d s strans where and ql , c = ql , f(x) = e x (e -1) , nqv -2 s s ' d n d' v ' ' KT v , = 5* 10 cm s , v = 2.5 10' cm s slong strans In eqs (7) and (8) we use the calculated values for & (fig. 7) The appearance of I in the denominator in the expressions (7) and (8) explains in principle the observation (f) mentioned under "Results". We do not expect that such a crude approach results in good absolute values of the lattice conductivity. The exponent n expressing the temperature dependence, however, should be less affected by wrong scaling constants. Following the Pippard theory consequently, we should add the two components (7) and (8), in order to obtain the total lattice conductivity. This amounts to the assumption of equally strong interaction of the conduction electrons with both longitudinal and transversal vibrational modes (Makinson scheme) . The other case often discussed in the literature [17] , is restricted to the interaction of electrons with longitudinal modes only (Bloch scheme) . The trans- versal modes are coupled to the longitudinal modes by at least the normal phonon-phonon processes. If we calculate the temperature exponent n from eqs. (7) and (8) for both the Makinson and the Bloch scheme, we find that the former appears to represent the results for the most copper-rich alloys better, while the latter seems to apply better to the rest of the system, i.e. to most of the alloys. This is particularly obvious for the alloy Cu49-Ni51. The calculated exponent n is plotted in figure 8 vs. the alloy composition. This curve should be compared with the corresponding curve in figure 5. It seems that with approximately equal relaxation times for s- and d-electrons, a large mass ratio, and a small ratio of the Fermi-velocities, the experimental data for the alloys containing more than 40 percent nickel can be fairly well fitted by the theoretical values of n , if only longitudinal modes are considered. Since q a V„ T n s Fs s (9) where v and v denote the Fermi-velocities of the s- and d-electrons, respectively, we can conclude tor confirm) thit the mean free path of the d-electrons must be significantly smaller than the mean free path of the s-electrons. It also means that below 10°K the lattice conductivity of the nickel-rich alloys, including those containing only a few percent copper, belongs to the regime ql <1, although there may be q£ >1. This situation has been indicated in figure 7. 265 The results are not so satisfactory for the copper-rich alloys, although inclusion of the transversal modes brings some improvement. It is interesting that in this composition range where it is difficult to fit the exponent n, the lattice conductivity drops rather rapidly for even small additions of solute (Fig. 5). This effect is typical for copper- and silver-rich alloys and has been given particular attention by Tainsh and White [19]. The true causes of this effect, however, are still rather uncertain, although the most likely cause might have to be found in variations of the electron-phonon interaction itself. The fact that the lattice conductivity of nickel is so much smaller than that of copper [observa- tion (d) ] can be explained by the arguments used by Fletcher and Greig [5] who made a similar observation in the silver-palladium system. They explain the low value of the lattice conductivity in transition metals by the effectiveness of s-d and d-d electron transitions. We think that the copper-nickel system should behave correspondingly. 5. Summary The temperature dependence of the lattice conductivity of well-annealed copper-nickel alloys can be understood through Pippard's theory of the ultrasonic attenuation by electrons. When extended to include d-electrons, this model explains several basic observations. The most important result, however, is the sensitivity of the lattice conductivity to properties of the d-electrons (quite unlike the electrical conductivity), particularly, if their effective mass is large. The experiments confirm that the ratio of the mean free paths of d- and s-electrons in nickel-rich alloys is appr. 0.1, and that the mass ratio is appr. 10. 6. References [1] Pippard, A. B. , Ultrasonic attenuation in metals, Phil. Mag. 46, 1104 (1955). [2] Ziman, J. M. , Electrons and Phonons , Ch. 5.12 (Clarendon Press, Oxford, 1960). [3] Zimmerman, J. E. , Low-temperature lattice heat conduction in high-resistivity alloys, J. Phys. Chem. Solids 11, 299 (1959). [4] Lindenfeld, P. and Pennebaker, W. B., Lattice conductivity of copper alloys, Phys. Rev. 127 , 1881 (1962). [11] See Ref. 2, p. 307. [12] Jones, H., Theory of electrical and thermal conductivity in metals, in Handbuch der Physik, edited by S. Flugge(Springer - Verlag, Berlin, 1956), Vol. 19, p. 265. [13] Van Elst, H. C, Labach, B., and Van Den Berg, G. J., The magnetization of some nickel alloys in magnetic fields up to 15 kOe between 0°K and 300°K, Physica 28, 1297 (1962), [14] See Ref. 2, p. 114. [5] Fletcher, R. and Greig, D. , The lattice thermal conductivity of some palladium and platinum alloys, Phil Mag. 16, 303 (1967). [6] Pippard, A. B., An experimental determination of the Fermi surface in copper, Phil. Trans. Roy. Soc. A250 , 325 (1957). [7] Ehrenreich, H. , Phillipp, H.R. , and Olechna, D. J., Optical properties and Eermi surface of nickel, Phys. Rev. 131, 2469 (1963). [8] Tsui, D. C. and Stark, R. W. , De Haas-Van Alphen effect in ferromagnetic nickel, Phys. Rev. Letters JL7, 871 (1966). [15] Garcia Moliner, F. , On the Fermi surface of copper, Phil. Mag. 3, 207 (1957). [16] Keesom, W.H. and Kurrelmeyer, B., Specific heats of alloys of nickel with copper and with iron from 1.2° to 20°K, Physica ]_, 1003 (1940). [17] Klemens, P. G., Thermal conductivity of solids at low tempertures, in Handbuch der Physik, edited by S. Flugge (Springer-Verlag. Berlin, 1956), Vol. 14, p. 198. [18] Rosenberg, H. M. , Low Temperature Solid State Physics, p. 123 (Clarendon Press, Oxford, 1963) . [9] Hanus, J., Feinleib, J., and Scouler, W. J., Low-energy interband transitions and band structure in nickel, Phys. Rev. Letters 19 , 16 (1967). [19] Tainsh, R. J. and White, G. K., Lattice thermal conductivity of copper- and silver-alloys at low temperatures, J. Phys. Chem. Solids 23 , 1329 (1962) . [10] Schroeder, P. A., Wolf, R. , and Woollam, J. A. , Thermopowers and resistivities of silver- palladium and copper-nickel alloys, Phys. Rev. 138A, 105 (1965). 266 1.0 E u u D o 0.1 u ~o E i_ (D U 0.01 Cu 98 S.C.(2.00) Temperature [°K I L-l —l 100 Figure 1: The lattice thermal conductivity vs. the abs . temperature of the alloys Cu98, Cu96, Cu91. The broken line refers to pure copper and has been taken from the literature [18]. Single crystals are denoted by s. c. E u D C O u D E CD 0) '£ 0.01 o - / f ^^^— • // _ V* y- j\ - • J* - V* - m •/• - /• - . 7 Cu 72 (1.60) 1 I 1 J 1 _ 1 /Cu 49 /. (1.15) • III II r* Ni 65 /- 0.26) 6 8 10 15 20 30 40 60 4 6 8 10 15 20 4 6 8 10 15 20 Temperature [°Kj Figure 2: The lattice conductivity vs. temperature of the alloys Cu72, Cu49, Ni65. 267 ^ E u u D c o U "a E i_ Jj 4 6 8 10 15 20 4 6 8 10 15 20 30 Temperature (°K) Figure 3: The lattice conductivity vs. temperature of the alloys Ni85, Ni90. 0.1 E */ u 1 >* - i /• 7 > -*- u */ z> -a c O u %/ "a 0.01 E 7 Ni 91 (1.30) i— • 02 u — "_l a i i i i i 1 Ni96 (2.00) Ni 98 (2.00) S.C. I L- I I L _J I I L 4 6 8 10 15 20 4 6 8 10 15 20 4 6 8 10 15 20 Temperature [°K] Figure 4: The lattice conductivity vs. temperature of the alloys Ni91, Ni96, Ni98. 268 0.03 20 40 60 80 Weight Percent Nickel 100 Figure 5: The lattice thermal conductivity at 4.2°K and its temperature exponent n vs. alloy composition of the copper-nickel system. -1 100 20 40 60 80 Weight Percent Nickel 100 Figure 6: The ratio of the lattice thermal conductivity to the total measured thermal conductivity at 4.2°K vs. alloy composition. 269 E 60 o _=3. 'co " h- > h- O ID Q ■z. o 10 o _l < 0.5 en LlI X 03 H — PRELIMINARY THERMAL CONDUCTIVITY Ti A- 110 AT 4 5 10 50 100 TEMPERATURE, °K Figure 3. Thermal Conductivity of titanium alloy A-110AT. 300 £200 £100 >-" t > 50 O Z) Q 8 a io rr UJ x _ h- 5 — PRELIMINARY THERMAL CONDUCTIVITY Al 7039 4 5 10 50 100 TEMPERATURE, °K Figure 4. Thermal Conductivity of aluminum alloy 7039. 276 300 e a. > t- > UJ en < o UJ 1.7 1.6 — 1.3 ~i — r "i — i — r T i — i — r PRELIMINARY ELECTRICAL RESISTIVITY Ti A-IIOAT J I I L J I L 100 200 TEMPERATURE,°K 0.05 300 e "I 1 i r "i — i — i — r PRELIMINARY ELECTRICAL RESISTIVITY Al 7039 100 200 TEMPERATURE,°K 300 Figure 5. Electrical resistivity of titanium alloy A- 11 OAT. Figure 6. Electrical resistivity of aluminum alloy 7039. > 6 =4. of UJ |4 0- O i — i — r n — i — r PRELIMINARY THERM0PCWER Ti A-IIOAT vs. n-Ag J I L _l ! I L J I I L. 100 200 TEMPERATURE, °K 300 PRELIMINARY THERM0P0WER A I -7039 vs n-Ag J I I I I I L 100 200 TEMPERATURE,°K 300 Figure 7. Thermopower of titanium alloy A- 110 AT. (vs. normal silver). Figure 8. Thermopower of aluminum alloy 7039 (vs. normal silver). 277 CD < tr N ■z. UJ o 15 Ph CO ra > 3. £ o a lf< i la hD (U H = 13 Cd CQ \ ft > •H r3 O C"— ■< •H H H h O 1 Ph O Ml QJ 0) J -P o ■H r< a O P CJ P O 0) £ H ^ CD -H H P H cd ^H a cd o Ph tfl CO > ft i ■H o C^ H ■H H l Ph O T3 0) H A a 1" p o r< B ■H CJ O P P o & 0) H h cd -H r O S 1 . u p a) — . u W l~ 01 r^vD^DC\jr\CNitOO[>tOtC s -C^-<.OC\|tr\l>vD -J-OHsfr-OCM^^DtOOHCNJl^ CO^DlAC^HOOO^ONtCMrHO^Oin ONOoa\otot)ototototooc--r~ oooooooooooooo CD cd mr-lc^toC^CMOmmtiO-xt-Nf^Or^ l>oatONl'OI>< y iOl>N!-CMOtt)t> cd OONOONOHO^vDO-srOstOt>H Of T if>HvOO-Ni-tOr r \I>iHvDHi-'^tO CT\C^OONC>COtOt)OtOt»tOI>l>I>r- ooooooooooooooo cd HsfC~--60CiOI>QI>vO>tl>Nf HO PlHtOlTir'inKMSmflOOttflO t>C^OOCMvDrvJOH-^0«)tX3CMm l>cnc5NmHl>sti-ltomcMotot>H vDmo\i>tONj-tor-it>oHir\o^c> CAJO(^^DCT.CM'Nf«0t0OOt0t>Nf o o o o + cd i>CMr^HvDl>mOOtMC\jnO-NrC3> vDlfistMHOMMnsl C\J H O L> O OC7>C^OOC^tX)tOtOtOt)000!>I>l> ooooooooooooooo NTl>H^ooc^f^c^vorviriCMr^s)-st !>O^0-vDr-OmCMOH t>c-\«ONt'0'-o< v> \c^voc v '\r-itox>xttx) n»OvDsf(00>invD(M(\!tOTlO^ nvDOC\JsfvDt!OC5NOOONtOI>Nf ? OOHHHHHCMCNJHHHH cd i>toHtol>tonoor\Ji>iAvoONt> OCMml>o< Y ^[>Hmo>c"\tor , i.corH t^-moHoto^Omo-iHOtoOmm ooooooooooooooo cd i-ia\iHOv£>C\JOC\JOmO!>C-CMiSO ntMOmrKOM^vDfflW^iMOO Osl-r^cn\OOI>vDtX)CNJC3NC^C\i60C\J M^OMTlHtOstrKOiDnHOWCM HCMCMrONf^tn^^DC^-tOC^OOi-l OOOOQOOOO „ HHHrHrH(MOJC\liM(MClC\rin o o o o o J 285 The results of this study are compared to resistance-temperature characteristics of indium ther- mometers in figure 5. Resistance ratio values calculated for an ideal, isotropic polycrystal form the base line for this plot and the resistance ratios derived from the two principal resistivities define the outer limits of variations which could be expected from anisotropy effects. None of the data shown were corrected for the residual resistivities. Based on the thermometer sensitivity, d(R/R )/dT, the width of the anisotropy band decreases from about 3.1° at 100 °K to 1.9° at 250 °K, and illustrates the necessity for thermometer stabilization and calibration. The resistivity values of Powell et al. [8], which fell outside the anisotropy envelope, were not included in the plot. The temperature dependences found by White and Woods [10], Yates [ll], and values for our polycrystalline specimen are seen to be in good agreement and to be consistent with only a small amount of anisotropy. This plot also offers a possible explanation for the results of Swenson [12], which had been thought to be anomalous [ll]. Swenson's results are in good agreement with the curve for an [001] oriented wire, and these results thus indicate the importance of anisotropy effects for indium resistance thermometers. 4.2. Thermal Conductivity Data Treatment of the thermal conductivity data for the three specimens followed the procedure outlined for the electrical resistivity results. The empirical equation chosen was: A = A - BT ± CT 2 . (2) The fit of the uncorrected data to this equation is shown in figure 3. The equation works rather well at temperatures above about 150 °K, but small systematic deviations appear at lower temperatures. Equations for the actual samples, principal conductivities and conductivity of isotropic indium, both corrected and uncorrected for volume changes are shown in table 4. The temperature and orientation dependence of the smoothed, corrected thermal conductivity values are shown in figure 6. The curves for the two principal conductivities are nearly parallel at high temperatures and converge at about 80 °K. The maxim-am anisotropy effect is only about 2.5%, which is within the combined uncertainties of the two measurements. The anisotropy of the thermal conductivity is thus not well defined, but the magnitude and direction of the observed effect is consistent with the anisotropy of the electrical resistivity. The curve for the polycrystalline specimen follows the trend shown by the principal conductivities but lies slightly above the latter values at the lowest temper- atures. This difference is also well within the combined uncertainties of the two measurements and does not seem to be due to difficulties with the volume change correction because the electrical resistivity does not show this behavior. Powell et al. [8] have reported values for the thermal conductivity of a polycrystalline indium specimen from 73 to 393 °K and these values are compared to the results of this investigation in figure 6. This comparison shows that the values of Powell et al. are consistently larger than all of the results of this study, the difference decreasing from about 10.5% at 78°K to A. 3% at room temperature. The curves for the principal conductivities show that the difference cannot be attributed to preferred orientation effects, and the resistivity values of Powell et al. , which are consistently higher than the data from this study, show that the difference is inconsistent with the difference in sample purity. By elimination, then, the discrepancy must be attributed to experimental difficulties. The results of the present study are supported by an intercomparison of results on platinum specimens which has been summarized by Flynn [13]. This summary shows that the data due to Powell [l<4] are also consistently higher than the values from other laboratories in this temperature range and the measurements of Moore [15] on Powell's platinum specimen indicate that the difference cannot be attributed to sample variations. A-.3. Seebeck Coefficient Values for the absolute Seebeck coefficient of indium are shown in figure 7 and table 5. These data were obtained from the measured Seebeck coefficients (Sin^c^s^) , values for the Seebeck coefficient of constantan with respect to' platinum [15] and the absolute Seebeck coefficient of platinum [16] . The absolute Seebeck coefficient of indium appears to be small and positive throughout the range of the measurements, but the total uncertainties are so large that even the sign of the coefficient is not well established. Including estimated uncertainties for SQons.t-Pt and Spt, the total maximum error for the data would vary from ±130% at 100 °K to ±50% at 300 °K. The anisotropy effect (Sj-S,,) found from these data is, of course, better defined at a given temperature because most of the errors must have the same effect on both coefficients. Analysis of the errors associated with the difference, S,— S||, indicates this quantity may be significant. It should be noted that the Seebeck coefficient apparently becomes more isotropic as the temperature increases, which is opposite to the trend shown by the electrical resistivity and thermal conductivity. 286 5. Discussion of Results Although no theory of the transport properties of anisotropic metals has been developed [17], it is interesting to see what sort of conclusions can be drawn from using the simple theories for isotropic metals. Applying the modified Bloch-Griineisen [3] equation to the electrical resistivity results offers one such test. This equation, which gives a good description of the electrical resistivity-temperature behavior of many metals [6], reduces to a simple polynomial in the temperature range studied (0.72e D to 3.186): T + A 2 T + A 3 T< (3) The A3 term, which arises from the temperature variation of the Debye temperature, should be related to the dominant linear term: A3 = 2a v .yA 2 (4) where a = volumetric expansion coefficient, and Y = Gruneisen constant. Also, the Aj_ term should amount to only a few percent of the total over the reduced temperature range of interest. As shown by table 6, the smoothed, corrected resistivity values from table 5 fit this equation rather well. The Ai term is seen to contribute a maximum of about 5$ of the total resistivity at 100 °K and the maximum variation of the linear coefficient (A 2 ) is only about <+$. Table 6. Least-squares values for modified Bloch-Grtineisen equation.' Resistivity Data A 2 Maximum Deviation at Temperature (*) (°K) Parallel to [001] -8.716 2.117 X 10" 2 2.805 x 10 -5 Perpendicular to [001] -10.214- 2.131 x 10 " 2 3.160 x 10 " 5 Theoretical polycrystal -9.720 2.126 x 10 ~ 2 3.041 x 10 " 5 Experimental polycrystal -5.671 2.051 x TO" 2 3.244 x TO" 5 0.5 0.3 0.06 1.08 80 80 140 80 ^ata from Table 5. Examination of the coefficients obtained for the two principal directions shows that the anisotropy arises almost exclusively from the A 3 term. The ratio of the first two terms (Aj/T + A 2 T)_ L /(A 1 /T + A 2 T) .. varies from 0.995 at 80°K to 1.006 at 350°K while the anisotropy ratio varies from 1.000 to 1.045. The theoretical expression for the A3 term [eq (4) ] does not allow for an anisotropy effect, but if one takes the liberty of replacing a v with the two principal expansion coefficients, the term can at least correlate the signs of the resistivity and linear thermal expansion anisotropies [4]. This correlation, which is not valid for all metals, has of course been recognized for some time [18]. This is about as far as the point can be pressed, the eq (4) could not be expected to hold for indium which has a very anisotropic thermal expansion coefficient and the magnitude of the A3 term cannot be quantitatively predicted from the measured GrUneisen constant and average volumetric expansion coefficient. Adopting the A 2 and A3 values for the theoretical polycrystal (table 6) and taking <* v = 90 X 10" 6 "K" 1 , y = 2.4 [5] yields: Al. 2a y a 2 v' 3.31. (4a) A reduced Wiedemann-Franz ratio plot for indium is shown in figure 8. The ±2$ error band associated with this ratio makes it impossible to reach any conclusions about the isotropy of the electronic Lorenz 287 number for indium. Values for copper [3,19] are included for comparison, and the striking similarity- gives one indication that indium, like copper, is a dominant electronic conductor. Estimates of the phonon conductivity from the Leibfried and Schl'omann equation [20] are also consistent with this hypothesis. Strictly speaking, this treatment should not be applied to a material with an anisotropic lattice conductivity, but since the values required were only estimates, this difficulty was ignored. The equation has been shown to consistently over estimate the lattice conductivity of insulators [21], and this tendency was further enhanced by choosing a Debye temperature and Griinelsen constant from the extremes of the reported ranges [5]. This leads to an expression for the maximum lattice thermal conductivity: ^Vmax = ¥ • . (« The maximum phonon conductivity estimate's correspond to 6 and 1.5$ of the total conductivities at 80 and 350 °K. The lattice conductivity of copper has been determined at temperatures up to about 100 °K [21], and extrapolation of these results to higher temperatures shows that the relative impor- tance of the lattice conductivity is about equal in the two metals. For copper, the extrapolated lattice conductivity amounts to 4 and 1.3$ of the total at reduced temperatures (T/9 ) corresponding to 80 and 350 °K for indium. The W-F ratio behavior for indium thus seems to be predominately due to the electronic thermal conductivity. The decrease in this ratio at lower temperatures would be attributed to inelastic electron-phonon scattering and the fact that the ratio approaches, but does not equal, the Sommerfeld value would indicate that the Lorenz number deviations, which are being found in many transition metals [22], are not present. The resistivity and thermal conductivity data for indium fit the Sondheimer-Wilson theory [23] quite well if an effective free electron concentration of about 1 electron per 4 atoms is chosen. This result is quite similar to the value for copper [19 ] but is not quite so disturbing because indium is not a monovalent metal. Also, Pimentel and Sheline [24] have derived a concentration of 1 electron per 2 atoms for indium from low temperature specific heat and thermoelectric power measurements. 6. Conclusions Analysis of the data obtained during this study has led to several conclusions about the behavior of the physical properties of indium: 1. The thermal conductivity is principally due to electronic transport and the values obtained are somewhat lower than previous literature values. 2. The effects of crystal anisotropy on the electrical resistivity and thermal conductivity are rather small and the anisotropy ratios observed for the two properties are internally consistent 3. The anisotropy ratios for the thermal conductivity and electrical resistivity increase with increasing temperature, a trend which is also shown by the thermal expansion coefficients. 4. Measurements of the electrical resistivity anisotropy are useful for interpreting the variability of indium resistance thermometers. 5. The absolute Seebeck coefficient is small over the whole temperature range studied, probably positive, and, the anisotropy effect observed, if real, has a temperature dependence opposite that of the electrical resistivity and thermal conductivity. 7. References [l] Nye, J. F., Physical Properties of Crystals, [4] Smith, J. F. and Schneider, V. L., J. Less- p. 195, Oxford University Press, London, 1960. Common Metals 7, 17 (1964). [2] American Institute of Physics Handbook, [5] Gschneidner, K. A., Solid State Phys. 16, 2nd ed., p. 2-20, McGraw-Hill, New York, 1963. 314 (1964). [3] Moore, J. P., McElroy, D. L. , and [6] Meaden, G. T. , Electrical Resistance of Graves, R. S., Canadian Journal of Physics Metals, p. 8, Plenum Press, New York, 1965. (to be published in December 1967) . 288 [7] Powell, R. W., Phys. Letters 24A, 445 (1967). [8] Powell, R. W., Woodman, M. J., Tye, R. P., Phil. Mag. 7, 1183 (1962) . [9] Olsen, J. L. , Electron Transport in Metals, Interscience Publishers, p. 107, Wiley, New York, 1962. [10] White, G. K. and Woods, S. B. , Rev. Sci. Instr. 28, 638 (1957) . [11] Yates, B. and Panter, C. H. , J. Sci. Instr. 38, 196 (1961) . [12] Swenson, C. H. , Phys. Rev. 100(6), 1607 (1955) . [12a] Aleksandrov, B. N. , Zh. Eksperim. I. Teor. Fiz. 43, 399 (1962). [13] Flynn, D. R. and O'Hagan, M. E., J. Researeih NBS 21C, 255 (1967). [14] Powell, R. W. and Tye, R. P., Brit. J. Appl. Phys. 14, 662 (1963) . [15] Moore, J. P., private communication. [16] Cusack, N. and Kendall P., Proc. Phys. Soc. (London) 72, 893 (1958) . [17] Ziman, J . M. , Electrons and Phonons, p. 396, Oxford University Press, London, 1963. [l8] Boas, W. and Mackenzie, J. K. , Progr. Metal Phys. 2, 90 (1950) . [19] Laubitz, M. J., Canadian Journal of Physics (to be published in December 1967) . [20] Fulkerson, W. , private communication. [21] Klemens, P. G. , Solid State Phys. 7, 47 (1958) . [22] Fulkerson, W. and Williams, R. K. , Proceedings of the Sixth Conference on Thermal Conductivity, p. 807, Materials Application Division, Air Force Materials Laboratory, Wright-Patterson Air Force Base, Ohio, 1966. [23] Wilson, A. H. , The Theory of Metals, p. 290 University Press, Cambridge, 1958. [24] Pimentel, G. C. and Sheline, R. K. , J. Chem. Phys. 17(7) , 644 (1949) . [25J* Roizin, N. M. , Mostovlyanskii , N.S., and Strod, R. K. , Trans. Solid State Phys. USSR 5, 887 (1963). [26]* Roizin, N. M. , Fiz. Metall. Metalloved L5, 800 (1963). *These are two additional references on the thermal conductivity of indium. BASE HEATER Figure 1. Indium sample PC. 10X. Figure 2. Schematic of ORNL longi- tudinal heat flow apparatus. 289 200 250 TEMPERATURE CK) Figure 3. Deviation of the Experimental Electrical Resistivity and Thermal Conductivity Data from the Empirical eqs (l and 2) . ®PC Run 1; x PC Run 2; H SC-A; O SC-C. - X^X XX > 7.0 X XX XX 5 XX XX ui XX 6.0 . xyX _i < u s 1- O 5.0 u ;j 40 3.0 2.0 1 1 1 200 250 TEMPERATURE ("HI Figure 4-. Effects of Temperature and Crystal Anisotropy on the Electrical Resistivity of Indium. 290 O 0.2 100 150 200 TEMPERATURE (°K) 250 300 Figure 5. Deviation of R/Rrj Values for Indium Resistance Thermometers from Values Calculated for an Isotropic Polycrystal. Ro is the ice point resistance. Upper bold curve is for R/Ro parallel to [001]; lower bold curve is for R/Ro perpendicular to [00l]. A Sample PC; Q White and Woods; Yates [11]; Swenson [12]. 200 250 TEMPERATURE C Figure 6. Effects of Temperature and Crystal Anisotropy on the Thermal Conductivity of Indiurr Lower curves from present study. - - - - Powell, Woodman, and Tye [8]. 291 - - - — ""^Z--"''^^ - ! -I?^ \ ^\ y^^x \ yy- // - %*.// / - yr/ / _ // / " // Spc s - / . /$. 1 1 1 1 1 - Figure 7. Effects of temperature and crystal anisotropy on the absolute Seebeck coefficient of indium. zoo TEMPERATURE I °K ) Figure 8. Reduced Wiedmann-Franr ratio plot for indium and copper. Indium n = 110 °K; copper 8 Q 300 °K. O copper [3,19]; D polycrystalline indium, Powell, Woodman, and Tye [8]; ® indium perpendicular to [00l] , present study; X indium parallel to [lOOj , present study. 292 Thermal Conductivity of Aluminum Between About 78 and 373°K K. E. Wilkes and R. W. Powell Thermophysical Properties Research Center Purdue University West Lafayette, Indiana An apparatus is described which employs the absolute longi- tudinal method and permits the measurement of the thermal conduc- tivity of metals in the intermediate low temperature range between the boiling point of nitrogen and the steam point. Results are given for reasonably pure samples of iron and of aluminum. Key Words: Aluminum, conductivity, electrical resistivity, iron, resistivity, thermal conductivity. 1. Introduction Recently, considerable attention has been focused on the possibility of a minimum in the thermal conductivity vs. temperature relationship for aluminum in the inter- mediate low temperature range. The purpose of this investigation was to explore this minimum by measurements on a very pure specimen of aluminum. The results presented here are for an unannealed specimen. Later work will be directed at measuring this effect in an annealed specimen and in specimens with known amounts of impurities. 2. Apparatus The apparatus used for these measurements is very similar to the one used by Powell, et.al.Cl] . A schematic of the apparatus is shown in Figure 1. it consists of an internally polished cylindrical chamber connected to a vacuum system capable of producing vacua of the order of 5 x 10~^torr. The specimen is fitted in a heavy brass base and the chamber is immersed in liquid nitrogen, dry ice and acetone, ice and water, or water. For measurements at intermediate temperatures below the ice point, an aux- iliary heated container filled with either isopentane or acetone surrounds the chamber and this in turn is surrounded by either liquid nitrogen or dry ice and acetone. Heat is supplied electrically at the free end of the rod. For the preliminary measurements on iron, the heater consisted of nichrome wire wound on a thin layer of mica around the end of the rod and then covered with aluminum foil. This proved to be unsatisfactory in that the temperature of the aluminum foil was much higher than that of the specimen, thus introducing large radiation errors. For the measurements on aluminum, the heater consisted of helical coils of 38 gauge nichrome wire insulated by alumina tubing and inserted in a small cylinder of aluminum which was then cemented to the end of the rod. The entire heater was then covered with aluminum foil to minimize radiative losses. Current leads were made of 24 gauge nichrome and two sets of 38 gauge constantan potential leads were spot welded to them. The two constantan leads on each current lead could then be read differentially to determine the amount of heat flowing along the current leads. This type of heater is much like that described by Laubitz[2], and later used by Flynn and 0'Hagan[3-l. Chromel-contantan thermocouples were pinned by a pencil point into small holes,, located at points about 1.9, 5.0, and 8.1 cm. from the free end of the aluminum rod. Other thermocouples were attached to the heater, one of the support rods, the inside wall of the can, and several places on the outside of the can. All thermocouples and leads from the specimen were securely tied to one of the support rods in order to minimize heat losses. 1 2 Graduate .Research Assistant and Senior Researcher,, respectively. "Figures in brackets indicate the literature references at the end of this paper. 293 3. Procedure The specimen was first suspended freely and heat was supplied in order to de- termine the amount of heat lost by radiation from the specimen. Next, the specimen was fitted into the brass base and an equilibrium gradient was measured with no heat sup- plied to the specimen. Then heat was supplied in order to create a temperature dif- ference of about 5°C between the two outside thermocouples. Measurements were made after equilibrium was established, usually after several hours. All temperature and voltage measurements were made with a Leeds and Northrup six-dial potentiometer. The thermal conductivity was determined by using the difference of that gradient with heat supplied and that without heat supplied to the specimen. Corrections were made for radiation losses, heat either gained or lost through the current leads, and heat lost by conduction through the other leads. The relative amount of these heat losses are summarized in Table 1. Table 1. Summary of Minor Heat Losses. Specimen Rad 373°K Lation Heat 273°K Losses 195°K 88°K Fn^ufie'n't Leaas gn^StEe-r™ Leads Aluminum 2 . 5% 1.0% 0.4% .04% .5% .05% Iron — 1 . 9% — — 1% .09% 4. Preliminary Measurements Since these were the first measurements made with this apparatus, a preliminary check on its accuracy was made by measuring the thermal conductivity of a specimen of pure iron near 273°K. The specimen was a rod of 99.96% iron, 1.25 cm in diameter and 10.44 cm. long,. At 25°C, its density was 7871 kg m~3. The electrical resistivity was found to be ,0900/i-^-m. at 273°K, and . 1044/1 A-m at 302°K. At 280°K, its thermal con- ductivity was found to be 81.5 Wm~ldeg~ . This compares well with literature values of from 80.5 to 82.5 Wm- 1 deg -1 [4]. 5. Measurements on Aluminum Next, a series of measurements were made on a polycrystalline aluminum rod which was 1.225 cm. in diameter and 10.16 cm. long. The specimen was supplied by Advanced Research Materials with a stated analysis of .5ppm. Cu, ,5ppm. Si. and . lppm Mg, in- dicating 99.99989°/ Al by difference. It was measured in the unannealed state as received from the supplier. Its density was 2698 kgm -3 at 23°C and its electrical re- sistivity was .02425 /i-^-m at 273°K and .00221 /iO.-m at 77°K. The results of this series of measurements are shown in Figures 2 and 3. In Figure 2, the calculations were based on the table of emf vs. temperature for chromel-con- stantan thermocouples which was generated by the Oak Ridge National LaboratoryC5] and was based on NBS Circular 561. This curve shows a minimum ©f a little over 2% at about 180°K, with a maximum at about 330°K. However, at the lowest temperatures measured, liquid nitrogen temperatures, the output of the thermocouples was around 8710Mv. This corresponds to a deviation of about 107/xv, from the above table by usinq the boilina point of liquid nitrogen, corrected for atmospheric pressure. Later, it was called to the authors' attention by Dr. R. L. Powell that newer thermocouple tables exist£6J. at the lowest temperatures, the thermocouples deviated from these tables by about 57/iv, the actual temperature being read with a platinum resistance thermometer. The thermocouples were then calibrated in the region below the ice point by using the resistance thermometer and the above tables. In this way, a temperature vs. emf table was established for our particular thermocouples. The results of the calculations based on this table are shown in Figure 3. The calculations above the ice point are based on the ORNL tables with a calibration against a platinum-rhodium 294 thermocouple. This curve shows a minimum of about 2% at about 220°K and a maximum at about 330 K. By comparing Figures 2 and 3, it is seen that the influence of the proper calibration is to shift the minimum to a higher temperature and to make the calculated thermal conductivity higher as one goes to lower temperatures. At the lowest tem- perature, the thermal conductivity is changed from 328.4 at 84.6°K to 348.0 at 88.2°K, a change of around 6%. Both the above curves, along with the results of previous workers are plotted on an enlarged scale in Figure 4. Curve 1 is the results obtained by NBS[7-1. No specimen characterization is available for this data. This curve shows a minimum of a little over 2% at about 240 K. Curve 2 shows the results of measurements by Powell, Tye and Woodman [8], on a specimen of 99.993% Al. Here, the minimum is about 8% and occurs at about 220°K. Curve 4 shows the results of measurements by Moore, McElroy, and Barisoni [9] on a specimen of 99.999% Al. It indicates a minimum of about 10% at 160°K. Curves 4 and 5 are the results of the present work both before and after account was taken of the thermocouple calibration. From these curves, although all agree in indicating a thermal conductivity min- imum below room temperature and a maximum above, it is seen that serious discrepancies exist in the individual data. The present work agrees most closely to the results obtained by NBS. Some indication of the importance of the thermocouple calibration has been shown and it is hoped that the proposed further measurements will help to provide more definite information on the influence of sample purity. 6 . Acknowledgment The above work has been carried out at the Thermophysical Properties Research Center and forms part of a program supported by the National Science Foundation to whom grateful acknowledgment is made. 7. References [l] Powell, R.W. , Tye, R.P., and Woodman, M.J. The Thermal Conductivity and Electrical Resistivity of Rhenium, J. Less-Common Metals, 5., pp. 49-55 (1963) . [2] Laubitz, M.J., The Unmatched Guard Method of Measuring Thermal Conductivity, II, The Guardless Method, Can. J. Phys., 43, pp. 232 (1965). T3l Flynn, D.R., and O'Hagan, M.E., Measurements of the Thermal Conductivity and Electrical Resistivity of Platinum from 100 to 900°C, J. Res. NBS TIC, 255 (1967), [4] Powell, R.W., Ho, C.Y., and Liley, P.E., Thermal Conductivity of Selected Materials, NSRDS-NBS 8 (Nov., 1966). [5] Adams, R.K., and Davisson, E.G., "Smoothed Thermocouple Tables of Extended Significance (°C) , Chrome 1-Constantan Thermocouples, " ORNL-3649, vol. 2, Sec. 2.6. [6] Powell, R.L., and Sparks, LL., Final Report on Thermometry Project, NBS Report 9249 (June, 1966). [7] Flynn, D.R., private communication. [8] Powell, R.W. , Tye, R.P., and Woodman, M.J., The Thermal Conductivity of Pure and Alloyed Aluminum, I. Solid Aluminum as a Reference Material, Advances in Thermophysical Prop- erties at Extreme Temperature s and Pressures, ASME, pp. 277-288 (1965). [9] Moore, J. P., McElroy, D.L., and Barisoni, M., Thermal Conductivity Measurements between 78 and 340°K on Aluminum, Iron, Platinum and Tungsten, Proceedings of the Sixth Conference on Thermal Conductivity, pp. 737-778 (1966). 295 To vacuum pump* >' fe i a= 31" o n o o rs— 7T- n ft ri Heating Wire Double -Walled Vessel Lead O-Ring Cotton Radiation Shield Support Rod - Thermocouples Heater Covered With Aluminum Foil Specimen Outer Chamber Iso-pentane or Acetone THERMAL CONDUCTIVITY OF ALUMINUM BEFORE THERMOCOUPLE CORRECTIONS 200 250 TEMPERATURE, 350 400 Figure 1. TPRC low temperature apparatus. Figure 2. Thermal conductivity of aluminum before thermocouple corrections. 360 340 THERMAL CONDUCTIVITY OF ALUMINUM AFTER THERMOCOUPLE CORRECTIONS 320 I 300 \ C7> \ X) \ J 280 \ •< \ 260 " V 240 * . r , g 6__- rr ^ ° — --~-^, 220 200 250 300 350 400 TEMPERATURE, °K 280 THERMAL CONDUCTIVITY OF HIGH PURITY ALUMINUM 270 % 260 i 250 \ \\ @ ©@© w •o ~ 240 230 220 ■d ^JIT \j£ y 1 NBS 11958) 2 POWELL, TYE AND WOODMAN (1965) 3 MOORE, M-.ELROY AND BARIS0NKI966) 210 4 PRESENT WORK BEFORE THERMO- COUPLE CORRECTIONS 200 5 PRESENT WORK AFTER THERMO- COUPLE CORRECTIONS 200 250 300 TEMPERATURE, °K 350 400 Figure 3. Thermal conductivity of aluminum after thermocouple corrections. Figure 4. Thermal conductivity of high purity aluminum. 296 Physical Properties of Chromium from 77 to U00 "K 1 J. P. Moore, R. K. Williams, and D. L. McElroy Metals and Ceramics Division, Oak Ridge National Laboratory Oak Ridge, Tennessee 3733O The electrical resistivity, thermal conductivity, and absolute Seebeck coeffi- cient of two high-purity chromium specimens were measured from 77 to ij-00 °K. No anomaly was observed in any of these properties in the region of 122 °K, where a local minimum occurs in the elastic constants. At the Neel temperature the Seebeck coefficient and the electrical resistivity drop sharply and the thermal conductivity has a broad shallow minimum. In addition, the electrical resistivity between 310 and 317 °K was also found to be sensitive to temperature gradients. Within experimental accuracy limits, these properties do not show hysteresis near the N^el temperature on thermal cycling from 77 to ^-00 C K . Key Words; Electrical resistivity, thermal conductivity, Seebeck coefficient, chromium, Lorenz function, Neel temperature. 1 . Introduction Many properties of chromium have been investigated in the vicinity of the ant if erromagne tic- paramagnetic transition at approximately 312 °K . Bolef and De Klerk [1] have observed local minima in the elastic constants at 120 and 310 C K as well as a deep local minimum in the coefficient of thermal expansion at 310 °K . Magnetic susceptibility measurements by Collings et a l. [2] indicate a local maximum at 312.2 ± 0.5 °K and Beaumont et al. [3] report a peak of about 2.5$ in the specific heat at 311.7 °K. Arajs et al. [k] have measured the electrical resistivity of a chromium single crystal from k to 33O °K. Their results do not indicate an anomaly in p around 120 °K but do show a distinct, local minimum at the Neel temperature, TW. Their results show some hysteresis at the Neel temperature presumably caused by thermal cycling between k.2 and 370 °K . The thermal conductivity (\) of chromium has been measured above 323 °K by Powell and Tye [5] and by Lucks and Deem [6] and below 150 °K by several investigators including Harper et al . [7]. There has not, however, been a careful study of the \ behavior at the Neel temperature nor do any measure- ments exist between 150 and 300 °K to connect smoothly with high- and low- temperature data. The experiment described in this paper was initiated to measure the electrical resistivity, the thermal conductivity, and the absolute Seebeck coefficient(S) of chromium simultaneously from 77 to 1+00 °K . We were especially interested in the exact behavior of \ at the Neel temperature since the p data of Arajs et a l. [k] indicate a local minimum might occur in \ if the electronic component of \ were dominant . 2. Specimen Characteristics Two samples, designated A and B, were used for this study. Iodide chromium crystals, purchased from the Chromalloy Corporation, West Nyak, New York, were used as the starting material for both Research sponsored by the U.S. Atomic Energy Commission under contract with the Union Carbide Corporation , 2 Figures in brackets indicate the literature references at the end of this paper. 297 samples. The samples were produced by different techniques and different lots of crystals were used. Sample A was produced by compacting the cleaned crystals, sealing them in an evacuated steel jacket and hot extruding the assembly. The extrusion, which was carried out at the Battelle Memorial Institute, Columbus, Ohio, was recommended as the best available method for producing dense, high-purity, poly- crystalline chromium. The extruded rod produced by BMI was 60 era long and 1.6 cm in diameter. The second sample (B) was produced in our laboratory by inert gas tungsten electrode arc melting. In this method the crystals are first consolidated into a disc-shaped ingot and this ingot is drop-cast into a rod. The rod produced by this method was approximately 15 cm long and 1.6 cm in diameter. Both samples, which were 7.7 cm long by 0.96-cm diameter, were machined from the two large diameter rods that yielded extra samples for chemical and metallographic evaluation. A summary of the chemical analysis is shown in table 1. These results indicate that both samples were 99.98+$ pure and the Table 1. Chemical analysis of chromium specimens Element Sample A (Extruded) Sample B (Cast) Manufacturing Analysis Crystals for Samp le B (Lot 613) Ag 1 Al x(b) As 1 Au X B 3 Ba 2 Be X Bi <0.1+ Ca 6 Nb 0.1+ Cd <0.8 Co <0.3 Cu 2 Fe 30 Ga <20 Ge <0.2 Hg <1 In <0.2 K 0.1+ Li <0.1 Mg <20 Mn 10 Mo <0.5 Na ) small inclusions were present in the samples. There appeared to be some tendency for the inclusions to favor grain-boundaries and, in sample A, a few longitudinal stringers of inclusions were observed. Etching revealed that the grain sizes of the two specimens were quite different. Specimen A had a fairly uniform, fine-grained structure and this observation, along with some evidences of grain growth phenomena indicate that no residual plastic deformation was present. Sample B had very coarse grains and some evidence of subgrain formation could be detected near the centers of the sections. Radial and axial grain size variations were present in this sample but these variations were much smaller than the difference between the grain sizes of samples A and B. Hardness and grain-size measurements were also obtained and are summarized in table 2. The hardness measurements give another indication of sample quality and the values for sample A, which agree well with reported values for recrystallized iodide chromium (see Edwards et al. [9]), also indicate that the samples were well annealed. The densities of the specimens were obtained by the immersion method, using distilled, degassed water. Samples A and B were found to have densities of 7.19 and 7.15 g/cm 3 , and it is presumed that the 0.6$> difference was due to the presence of voids in the cast sample (B). The value for Table 2. Summary of Hardness and Grain Size Results for Chromium Samples Hardness Number kg/ mm (Diamond pyramid, 1 kg load) Average Grain Diameter (microns) Sample A — Longitudinal Transverse Sample B — Transverse — Top — Bottom 121 120 123 123 63 8U0 kko sample A is less than the x-ray density (7.22 g/cm 3 ) [10], but is in agreement with a value tabulated for polycrystalline chromium [ll]. No voids were observed during the microscopic examination of the samples, but a single void was present at one end of the cast sample. The specimen heater was wrapped on this end of the specimen so that it would not affect the results. 299 3. Apparatus Description A guarded-longitudinal technique described by Moore et al . [l2]was used to determine p, S, and \ over the temperature range from 77 to 1*00 Tf . This technique has most probable errors of ±0.36% ±2.0$, and ±1.2$ for p, S, and X, respectively. The absolute Seebeck coefficient of Cr-A was determined by measuring the emf between the Constantan legs of the two Chromel-Constantan thermocouples and using a calibration curve relating the Constantan to platinum. The absolute Seebeck of platinum as determined by Casack and Kendall [13 ] was then used to calculate the absolute Seebeck coefficient of Cr-A. This method was also used on Cr-B; but, in addition, two platinum wires were attached to the sample near the thermocouple junctions to enable measurements of the Seebeck coefficient with respect to platinum directly. 1*. Presentation of Results l*.l Electrical Resistivity Smoothed values of p, X, and S are given in table 3 for both chromium samples. The electrical resistivity from 1*.2 to 375 °K is shown in figure 3. The electrical resistivity of both samples increases with increasing temperature up to the antiferromagnetic-paramagnetic transition at the Nisei temperature of about 3H °K • The p of each specimen has a distinct local maximum and minimum in the region of the N^el temperature. Values obtained on Cr-A in the vicinity of 122 °K indicate that the pos- sible spin flip which occurs around 122 °K does not have a noticeable effect on p. Table 3. Smoothed values of p, S, and X for chromium Chromium A (a) Chromium B (8.) -1 rtO \ x 10" 2 „ ^ 8 ~2 X X 10 p X 10 pxi0 T(°K) (fi-m) W m _1 deg -1 S( U V deg X ) (I)-m) (W m deg ) S(uV deg 1 ) 80 0.860 _ _ 1.060 _ _ 90 1.225 1.697 3.^8 1.1*1*5 1.522 5.27 100 1.630 1.593 3-95 1.860 1.1+53 5.1*5 120 2.605 1.^39 4.97 2.860 1.31+3 6.01* 11*0 3.760 I.327 6.25 1*.050 1.259 7.06 160 5.000 1.241 7.85 5.295 1.187 8.55 180 6.315 1.171 9.92 6.575 1.122 10.62 200 7-5^5 1.111 12.1*2 7.830 1.066 12.92 220 8.79O 1.061 15.16 9.100 1.020 15.70 2l*0 10.015 1.019 18.13 10.300 O.983 I8.30 260 11.095 O.98I* 20.52 11.385 0.951+ 20.58 280 I2.O3O 0.957 22.00 12.270 0.930 21.90 300 12.710 0.935 21.63 12.880 0.915 21.65 30U 12.792 0.93 2 * 21.40 12.9'*6 0.915 21.25 306 12.813 0.93'-+ 21.20 12.961 0.916 20.95 308 12.816 0.934 20.80 12.958 O.9I6 20.50 310 12.791 0.934 20.10 12.931 0.916 19.83 312 12.772 0.93*+ 19.1*0 12.900 0.916 19.OO 31A 12.780 0.931* 18.75 12.898 0.917 18.60 316 12.808 0.934 18.50 12.91*0 O.9I8 18.1*0 320 12.925 0.93 1 ! 18.20 13.080 0.918 18.20 31*0 13.605 0.932 17.65 13.765 0.917 17.60 360 1U.3U0 O.928 17.20 14.1*70 0.913 17.16 360 15.085 0.920 16.80 15.200 0.906 16.77 1*00 15.8>*5 0.910 16.42 15.935 0.900 16.1*1 (a) The smoothed p and X data have not been corrected for thermal expansion. Over the range from 80 to 1*00°K the correction would amount to less than 0.1$. 300 The p of Cr-A Is lower than that of Cr-B over the entire temperature range . The difference amounts to about 13$ at 100°K and decreases to 0.6$ at 1+00°K. This difference cannot be explained in terms of normal specimen characteristics such as impurity content or density since the two specimens are quite similar in those respects. If anything, Cr-A has a slightly higher total impurity concen- tration than Cr-B and this is in the wrong direction to explain the p difference. The relative grain sizes of the two samples is also in the wro ng direction to have a pertinent effect. The region around the Neel temperature is shown in greater detail in the inset of Fig. 3. In contrast to the results obtained by Arajs et al. [k ] on a single crystal of chromium, our results show no hysteresis outside our maximum imprecision (±0.1$). During one heating cycle, Arajs et al . observed a 3$ decrease in p at the Neel temperature and a 1.3$ decrease on a subsequent heating cycle and this compares to the 0.5$ decrease observed in our samples when the specimen had a 0.014 °K m - 1 gradient and about 1$ when the specimen was isothermal. From 310 to 317 °K the electrical resistivity is sensitive to the presence of a temperature gradient. The results indicated by the solid line were obtained simultaneously with the \ and S measurements when a 0.014 C K m gradient existed along the rod. 3 The dashed line for each specimen denotes measurements obtained when the sample was isothermal. The difference between the two curves for each specimen was a maximum of about 0.4$ at 312. 9 °K. The p data obtained with a gradient can be explained to within 0.2$ based on an averaging of the isothermal p curve. Possibly, if we could reduce the maximum imprecision on each curve (about ±0.1$ for this data), the difference might be totally attributable to a gradient averaging. At all other temperatures, that is, below 310 °K and above 317 °K, the results were not sensitive to the presence of a thermal gradient. Since the thermal gradient was generated with a radially wound heater on one end of the specimen, a small longitudinal magnetic field was unavoidably applied to the specimen .while the gradient was present. To insure that the small magnetic field was not causing the apparent resistivity difference, we applied an external magnetic field to Cr-B and took p data while the sample was isothermal. Although the external field was some 25 times the field from the specimen heater, the isothermal p data was unchanged. We feel, therefore, that the small magnetic field present when there was a gradient along the rod had a negligible effect on the results. The electrical resistivity of our Cr-A is compared to the results of previous investigators in figure h. This figure shows the difference in p between the other measurements and our results on Cr-A as a function of temperature . Many of the existing p data on chromium are very high due to low sample density and/ or impurity content and we have not considered these data in making this comparison plot. The p of Cr-A is lower than any chromium reported-to-date except for one run by Arajs et al . [ k ] which crosses the base line above 300 °K. The p of chromium definitely does not obey Matthieson's rule since none of the curves are parallel. The relative behavior of the various results is rather amazing if we generously assume that the results are all accurate. The two sets of data which behave most remarkably are the results of Powell and Tye [ 5 ] and Arajs et al . [ h ]. Powell and Tye's results on their high density electrodeposited sample (annealed at 1410 °C) are in good agreement with our results at low temperatures but are about 8$ higher than ours above 300 °K. Conversely, the data of Arajs et al. are extremely high at low temperatures and almost converge with ours above 300 C K . Data reported by White and Woods [14] on an annealed, recrystallized sample are in very good agreement with our results on Cr-A below 100 °K but are higher above 100 °K. The data reported by White and Woods were obtained from sample 5 studied by Harper et al . [ 7 ] . There are two curves on figure 4 representing data obtained by De Morton [15 ] on a heavily cold-worked chromium wire and the same wire after annealing at 700°C in a vacuum of 2 x 10~~ ^ mm of mercury. De Morton's data indicated that p of the annealed chromium was higher than p of the cold worked chromium below the Neel temperature. Above the Neel temperature the cold-worked chromium had the higher electrical resistivity. This behavior appears anomalous especially considering the results which Harper et al .[ 7 ] obtained on cold-worked chromium specimens. Their data showed that p at 4.2 °K decreased from 0.255 X 10~ ° Qm for a cold-worked sample to 0.055 X 10 fim for an annealed fully recrystallized sample. There is a possibility that De Morton's wire sample became contaminated during the annealing in the relatively poor vacuum which could possibly have caused an upward shift of the annealed sample p over the entire temperature range. A sample diameter decrease due to chromium evaporation could also cause the same effect. 4.2 Seebeck Coefficient The absolute Seebeck coefficient for both chromium samples is shown in figure 5. The coefficient for both specimens shows a rapid increase with increasing temperature up to about 287 °K where the The smoothed values of p in table 3 represent results obtained with a thermal gradient, 301 values reach a maximum. A continuous decrease of about 20$ occurs in the region of the- Neel tempera- ture. There is a large difference between the two samples at low temperatures but this difference disappears above 250 °K . The difference in S at low temperatures is consistent with the p difference in reflecting an impurity-like effect. The lower curve in figure 5 represents data obtained by Potter [16] on a chromium of unspecified character. it. 3 Thermal Conductivity The thermal conductivity results for Cr-A and Cr-B are shown in figure 6. The thermal conductivity of both specimens decreases with increasing temperature to 300 °K where a broad shallow minimum is observed. Above 3U0 °K \ decreases slowly with increasing temperature. The many data points taken on Cr-A in the region of 122 °K indicate that any anomaly in \ was less than the maximum imprecision (« tl/kfi) of the measurements. The \ results from the two specimens differ by about 11$ at 90 °K and 1.1$ at U00 °K with Cr-A being higher over the entire temperature range. The large difference between the two samples at 90 C K and the convergence at higher temperatures appears to reflect an impurity concentration effect . We attempted to measure with a very small gradient to determine if \ was sensitive to the thermal gradient in the vicinity of the Keel temperature. With the markedly reduced gradient, however, the X. results were too imprecise to resolve a .k$> difference. Our results are compared to those of previous investigators with a percentage deviation plot in figure 7. Our X. results on Cr-A, which are used for a basis, are higher than any reported data on chromium. The only high-temperature data which overlaps our measurements are those of Powell and Tye [ 5 ] . At 323 °K their results on a high-density electrodeposited sample of chromium are about 8$ below \ of Cr-A. If the electronic component of \ were dominant, this difference would be expected since their reported p on the same sample is about 8$ higher than the p of Cr-A. The comparison plot includes three sets of data obtained by Harper et al. [7] on chromium specimens with various amounts of cold working. Although their low-temperature \ results (<50 °K) relate well with the sample residual resistivities (p = electrical resistivity at k .2 °K), the \ results which overlap our measurement range do not. Their specimen with the lowest p has the lowest measured values of \ from 90 to 120 C K . 5. Discussion of Results 5.1 Seebeck Coefficient The various aspects of the Seebeck coefficient results can be discussed with respect to the equation^ 2 . 2 3 e 1 Me) I , . given by Ziman [17] where k is Boltzmann's constant, e is the electronic charge, T is the absolute temperature and a(e) is the electrical conductivity if the Fermi energy were e and £ is the Fermi energy. Ziman [17] and Fulkerson et al . [18] have shown that this equation can be extended to explain the deviations between the S values of the samples at low temperatures and their convergence with increasing temperature in terms of the p differences . They obtain AS * _I^( S*nAp ) Ap_ 3e V de J £ = ^ P r ' where, in this case, Ap is the resistivity difference between Cr-A and Cr-B and p is the resistivity of Cr-A. Since — decreases with increasing temperature, the convergent behavior of the S values r with increasing temperature is not surprising. Fulkerson et al . [18 ] present S data on two iron samples of different purity which behave similarly . Actually eq (l) is not valid at low temperatures (T < S D where D is the Debye temperature). Ziman [IV ] gives a low-temperature expression for S which depends on the Lorenz function and approaches eq (1) only when L approaches L Q (2 .kk^ x 10-8 V 2 "XT 2 ). In view of this we shall limit our use of eq (1) to a comparison of the two chromium samples. 302 Fine et al. [19] obtained S results by taking the first derivative of a curve for the total emf of chromium vs platinum and they indicate a discontinuity in S at Tjj. Our S data are not sufficient to indicate whether or not S is discontinuous at the Neel temperature. The measured values indicate that S is continuous but this may only be apparent since the thermal gradient necessary for measuring S dictates the measurement of an average S over a small temperature range. With a gradient as small as 0.01i+ °K m—1 (about 5 °K between the two measuring thermocouples) S still appeared to be continuous; however, the decrease was as much as 5$> in one degree of temperature change. In addition, there may be an unresolvable effect of the thermal gradient on S analogous to the p results . 5.2 Thermal Conductivity The total thermal conductivity of a metal can be expressed as * = \ + \ (3) where \ is the component carried by the electrons and therefore related to the electrical conductivity, 1 e a = — , and \ is the component associated with the lattice vibrations. The electronic component can be related to p with the Lorenz function L to yield Several values for the Lorenz function (L) have been theoretically predicted. The Sommerfeld value (L Q = 2.V+3 x 10~ °) should apply to a completely degenerate electron gas at high temperatures and seems to be reasonably well attained in the alkali and noble metals. In transition metals, however, the valid- ity of L is in doubt and several possible reasons for high-temperature deviations have been suggested. These have been discussed by Fulkerson and Williams [20]. Since the expression for L obtained from solution of the Boltzmann equation is an infinite series, higher order terms above the first term (L ) may be nonnegligible . Keeping the first three terms, L can be written for high temperatures as .2 8ir k kV 1 a 2 a(e) L n - b + 7^ o - T77T — o O) o W e 2 o^IT ae 2 a 2 q(e ' Thus a value for L - . . .. . . . . , . If \ R is assumed to be zero and an experimental L calculated using experimental data de 'e = £ for p and \ and eq (h) , we find that the experimental Lis 3.95 X 10~~ ° at 300 °K. This is much greater than the value of L and indicates that L > L Q and/ or \ is not negligible. Values of \„ have been calculated by Fulkerson [21] using a modified Leibfried -Schlomarui [22] equation to account for the three phonon Umklapp thermal resistivity and an equation from Ziman [17] to account for the electron-phonon thermal resistivity. The most significant aspect of the calculations is that \ g can vary from almost zero to values higher than \. Thus, this approach seems futile for determining X and X . & g e Fulkerson and Williams [20] outline an approach which involves fitting experimental p and \ lata to eq (h) with the assumptions that L is constant and K - I (6) 303 Equation (6) is of the form predicted by the modified Leibfried-Schlomann equation and eq (7) is of the form experimentally observed on many electrical insulators. If we assume that \ is given by eq (6), then eq (h) becomes ° ™ = ~ + A . (8) p This equation indicates that a plot of \T vs T /p would be useful in determining an experimental value for L and A. This type plot is shown in figure 8 for chromium. The high-temperature data of Powell and Tye on their sample annealed at li+10 °C is also plotted. From 200 to 1300 °K all points are within ±3$ of the straight line which yields L ~ 2.80 X 10~° and A = 8000 W m- 1 . These values are similar to ones obtained on W by Fulkerson and Williams [20], The above result for A indicates that at 300 °K, \g ~ 66.8 W m -1 and \ ~ 26.7 W m -1 . To look at the behavior of L at T N (where the actual data shows a small deviation from the line in figure 8) we have calculated L for Cr-A using eq (8) with an A of 8000 W m -1 and an A of zero corresponding to a \ of zero. The results are shown in figure 9. Although the value of L is definitely in severe doubt because of the uncertainty in X , a distinct slope change occurs in L when we assume a smooth behavior for X . We could just as well assume a smooth or constant value for L and calculate a X which would show a large slope change at Tjj. This implies that the Lorenz function and/or \_ must be irregular at Tjy. To obtain a better idea of the location of Tjj of Cr-A we have calculated the first derivative of each property with respect to temperature and plotted the results in figure 10. Hall coefficient data from De Vries and Ratherau [23] were used to calculate ARt/aT. The slopes of all the properties have extreme values between 309 a nd 3H °K with the exception of the isothermal a data which appears to peak at 311-5 °K. This indicates that the Nlel temperature location can be determined most easily by observing the derivative of the property with respect to T as a function of T. The range of temperature for the maximum slope change for S, L, and the a data obtained during gradient conditions may be caused by the presence of the thermal gradient and this somewhat complicates location of T™. The large difference in the properties of the two chromium samples at low temperatures is not understood. All the p and \ data indicate that Cr-A is superior to Cr-B in being closer to an ideal, defect-free material. The chemical analysis, however, indicates that Cr-B has lower total impurity concentrations than Cr-A so that this does not appear to be a factor. The difference in the properties cannot be attributed to cold working effects since Cr-B was obtained from a casting and Cr-A was hot extruded and the hardness of both specimens was consistent with values for well annealed chromium. There are two possible reasons for the discrepancy. There might be a particular impurity present in Cr-B (Np possibly) which has a large effect on the transport properties, or the impurities might be in solution in Cr-B and in fine precipitates in Cr-A. We plan to investigate this problem further with a series of heat treatment experiments on the two samples. Based on these differences we are not convinced that our results, even on Cr-A, are representative of those of ideal high-purity chromium, although the results do indicate that Cr-A is the best sample studied to date. 6. Conclusions The results of p, \, and S measurements on the two chromium specimens permit the following conclusions ; 1. Although the two chromium samples used for this study were almost identical chemically, their properties differ significantly at low temperatures. 2. The electrical resistivity has a local maximum and minimum at Tjj. The p of both samples of chromium was sensitive to the presence of a thermal gradient from 310 to 3I7 °K. The p of the chromium samples does not obey Matthieson's rule. 3. There was no hysteresis observed in p, \, or S around Tjj on thermal cycling between 77 an< ^ ^00 °K . h. The Seebeck coefficients of both chromium samples are large and positive and both exhibit a sharp decrease at T„. 5. The thermal conductivity of chromium decreases with increasing temperature and has a broad shallow minimum at T^. The lattice component of \ is probably significant in magnitude compared to \g . 6. The first derivatives of S, — , and L with respect to temperature have extreme values at the Neel temperature. This appears to be the most sensitive approach for determining % from transport property data . 304 7. The behavior of \ at T N cannot be explained in terms of the p behavior assuming a uniform behavior for L and \ at T, T . Thus L and/ or \ must be irregular at T, T . g W g H 7. References [1] Bolef, D. I. and De Klerk, J., Phys. Rev. 129(3), 1063 (1963). [2] Collings, E. W., Hedgcock, F. T. and Siddiqi, A., Phil. Mag. 6, 155 (1961) . [3] Beaumont, R. H., Chihara, H. and Morrison, J. A., Phil. Mag. 5, 188 (i960). [h] Arajs, Sigurds, Colvin, R. V. and Marinkowski, M. J., J. Less Common Metals k, 1+6 (1962). [5] Powell, R. W. and Tye, R. P., J. Inst. Metals 85, 185 (1957). [6] Lu^ks, C. F. and Deem, H. W., WADC TR 55-U96 [AD 97 185]. [7] Harper, A. F. A., Kemp, W. R. G., Klemens, P. G., Tainsh, R ; J. and White, G. K., Phil. Mag, 2, 577 (1957). [8] Schindler, A. I. and La Roy, B. C, J. Appl. Phys. 9(37), 3610 (I966). [9] Edwards, A. R., Nish, J. I. and Wain, H. L., Met. Rev. h(lG) , U03 (1959). [10] American Institute of Physics Handbook, 2nd Ed., 1963, p. 2-20. [11] Metals Handbook, 8th Ed., 1961, p. 44. [12] Moore, J. P., McElroy, D. L. and Graves, R. S., Can. J. Phys., Dec. 1967. [13] Cusack, N. and Kendall, P., Proc. Phys. Soc. (London) 72, 898 (1958). [14] White, G. K. and Woods, S. B., Phil. Trans. Roy. Soc. London, Ser. A, 25_1, 273 (1959). [15] De Morton, M. E., Nature 181, I+77 (I958) . [16] Potter, H. H., Proc. Phys. Soc. 53, 6 95 (I9UI). [17] Ziman, J. M., Electrons and Phonons, (Oxford University Press, i960) . [18] Fulkerson, W., Moore, J. P. and McElroy, D. L., J. Appl. Phys. 22(7), 2636 (1966). [19] Fine, M. E., Greiner, E. S. and Ellis, W.C., J. Metals, Trans. Met. Soc. AIME 191, 56 (1951). [20] Fulkerson, W. and Williams, R. K., Proc. Sixth Conf . on Thermal Conductivity, Dayton, Ohio, October 18-22, I966, p. 807, I966. [21] Fulkerson, W., private communication, 1967. [22] Leibfried, G. and SchlSmann, E., Akad. Wiss. Gottingen Mat.-Physik. Kl.IIA, 71 (195^) (translation obtainable from Technical Library Research Service, Purchase Order Bl+B-60150, Letter Release S-70). [23] De Vries, G. and Ratherau, G. W., Phys. Chem. Solids 2, 339 (1957). 305 c -■ I s( iV/2 Figure 1. Photomicrograph of Cr-A, 38X, longitudinal section, etchant: 80 ml H 2 0, 10 g KOH, 10 g K 3 Fe(CN) 6 . Figure 2. Photomicrograph of Cr-B, 38X, transverse section, etchant: 80 ml H20, 10 g KOH, 10 g K 3 Fe(CN) 6 . 306 13.0 - ^^'H / 4 ' / m -12.8 V"* JHT / ^^ j/- oo 2 12 X 1 1 3 05 310 315 320 "e £'0.0 // ~ *S 8.0 Cr-B — 7/ 6 4 20 -=-r~t i i,i, Figure 3. Electrical resistivity of Cr-A, and Cr-B versus temperature. O: Cr-B with 0.014 °K m gradient. A: Cr-B isothermal; X: Cr-A with 0.014 °K m" 1 gradient; •: Cr-A isothermal. 100 200 TEMPERATURE (°K) 300 50 100 150 200 250 TEMPERATURE (°K) 300 350 400 Figure it. Deviation Plot Showing Difference Between p Results of Other Investigators and Cr-A., X:Cr-B; Q :Arajs et al . [U]j A : Powell and Tye [5]; O :White and Woods [14]; H :De Morton [15] Cold Worked Sample; V :De Morton [15] Sample Annealed at 973 °K for 2 X 10 ^ mm of Hg. 307 23 21 19 A 17 15 13 M-l F 3 o CO CD < 1 r^ / It // / •^ // / V V. II 1 •9 / 1 1 1 1 1 // / // / // / 1 / / >> ^/, / / / / / / * / / 50 100 150 200 250 300 350 400 TEMPERATURE (°K) Figure 5. Absolute Seebeck coefficient of chromium. X: Cr-B using constantan leads; + : Cr-B using platinum leads; O : Cr-A using constantan leads; — --: Potter [l6J . Figure 6. Thermal, conductivity of chromium from 90 to 400 °K. X: Cr-A; o : Cr-B. 150 200 250 300 TEMPERATURE (°K) 350 400 308 o o >- o t- co o 450 200 TEMPERATURE 250 K) 400 Figure 7. Percentage Deviation of Other \ Results from Results on Cr-A. X:Cr-B; TJ :Harper et al . [7] on Cold Worked Sample with p = 0.255 x 10""^ ftm; Q :Harper et al . [7] on Cold Worked o Sample with p = 0.181 x 10 Qm; A :Harper et al . [7] on Annealed Sample with _8 _ p = 0.055 X 10 ftm; P :Powell and Tye [5] °n Annealed Eleetrodeposited Sample. 8.0 7.0 ^% - 6.0 ^X O :/* X ~ 5.0 - 'E 5 - 4.0 X^^ — 1- 3.0 — 2.0 <"tP 1.0 1 1 1 1 1 0.5 1.0 1.5 T«/ , " , (°K*flr l m') x I0 -4 2.0 2.5 Figure 8. \T versus Tp for Chromium; O :0RNL Results on Cr-A; X:Powell and Tye [5] 309 O 4-0 K £~ > - / -I _l < H z 30 til 2 111 a. X Ul — / Lo= 2 443 x 10"' 1 1 1 1 1 1 100 200 TEMPERATURE (°K) 300 Figure 9. Experimental Lorenz Function for Chromium. X:Calculated using eq (8) with A = 0; O'.calculated using eq (8) with A = 8000 W m -1 . 270 300 TEMPERATURE (°K) 330 Figure 10. First Derivative of a,S, L, and Hall Coefficient with Respect to Temperature. D :Hall coefficient data from De Vries and Ratherau [23]; X: a of Cr-A when sample under a gradient of 0.01^ °K m -1 : cr of Cr-A when sample was isothermal; -1 O :Lorenz function from eq (8) with A = 8000 W m ; A :Seebeck coefficient of Cr-A. 310 The Thermal Conductivity of Chromium Above and Below the N^el Temperature - An Analysis J. F. Goff 1 U. S. Naval Ordnance Laboratory White Oak, Silver Spring, Maryland 20910 Measurements of the thermal conductivity of polycrystalline Cr between 2°K and 290°K indicate that at the higher temperatures it increases. As a result the measured Lorenz No. L shows a T departure from the Sommerfeld value L in the temperature range between 120°K and 280°K. The Method of Moments promulgated by Klemens predicts that this departure is due to a second order term in the expansion of the specific conductivity a(E) about the Fermi level. Previous measurements made by Powell and Tye between 323°K and 1273°K found that L was greater than L over the whole range of temperature but decreased as the temperature was raised. Consequently AL has been calculated for a model in which a(E) has a parabolic well symmetrically arranged about the Fermi level but with a finite width E . The calculations give reasonable agree- ment with the data between 120°K and 1000°K if E is allowed to decrease during the antif erromagnetic-paramagnetic change at the N^el temperature. A surprising result is that a parameter of the model - which is related to the constant of the parabolic well - is independent of magnetic changes taking place at the N4el temperature. Key Words: chromium, electrical resistivity, Lorenz No., N^el temperature, thermal conductivity. 1. Introduction Both of the most recently reported measurements of the thermal conductivity X of Cr [1,2] indicate that it is anomalously large above 90°K; that is, the Lorenz No. L found from these data is much larger than the expected Sommerfeld value [3] L = 2.4453 x 10~8( v/deg ) . The low temperature measurements were made on a series of polycrystals between about 2°K and 150°K by Harper et al [1]. Above 90°K the Lorenz No. constructed from their data becomes rapidly greater than L . The high temperature measurements were made on a well-annealed, electrolytically deposited sample between 323°K and 1273°K by Powell and Tye [2], They found that L was anomalously larger over the whole temperature range and attempted to ascribe this anomaly to a lattice component of the thermal conductivity. They concluded that this lattice component would itself have to be anomalously large. The present measurement [4] of X and the electrical resistivity p over the range 2°K to 291°K confirms this anomaly. At the higher temperatures X begins to increase, the concomitant L becomes larger than L , and this extra aL = L - L varies as T^ between 120°K and 280°K. This temperature dependence is exactly the prediction of the second order extension of the simple theory of electrical conduction which has been made by Klemens [5]. Thus, the purpose of this paper is to show that it is possible to combine these measurements with those of Powell and Tye [2] to develop a model which explains the anomalous thermal conduction without the assumption of a lattice component. Physicist. 2 Figures in brackets indicate the literature references at the end of this paper, 311 2. Apparatus The apparatus has been described previously [6]. The salient features are: 1. A gas thermometer which is used as a calibration standard in the temperature ranges where liquid gas standards are not readily available, 2. a radiation shield which is thermally anchored to the base of the sample so that the thermal environment is nearly the same as the temperature gradient in the sample itself, and 3. electrical connections made of two meter lengths of #40 copper wire thermally anchored to the sample base to minimize conduction losses. Since the previous description [6], there have been four changes made in the apparatus. The sample heater is now a 2000 ohm, !•£ watt, metal film resistor which is sealed into a copper block with glyptal enamel. The thermocouple which is used to measure the temperature gradient between 60°K and 300°K has been changed to a Au-2.11 at % Co: Manganin couple to minimize thermal conduction losses. The temperature gradient in the 4°K to 60°K range is measured by two Allen-Bradley 100 ohm, 1/10 watt carbon resistors that are calibrated at 4.2°K and 77°K connected by a straight line on a modified Clement and Quinell plot \T\. Finally, pressure contacts are used in the sample mounting. The justification of the calibration procedure in the 4°K to 60°K range is three- fold. Further calibration of the carbon resistors at lower temperatures against the vapor pressure of liquid He and at higher ones against the gas thermometer indicate a linear Clement and Quinell plot at temperatures up to 120°K. Secondly the data taken with these thermometers at the lower temperatures agree well with the data taken at the higher ones with the thermocouple. Finally, the data taken in this way agree with theoretical predictions. The data are taken in several ways to obviate systematic errors. No attempt is made to maintain the same size temperature gradient. Some of the data are taken by heating above the liquid He bath and others are taken by allowing the apparatus to drift up in temperature after the bath has boiled away. These data agree with each other within a few percent. The same temperature gradient is used in determining the thermoelectric power of the sample. Since this quantity has a different temper- ature dependence than X, it would be difficult for a systematic error to hide in both sets of data. At the very highest temperatures the thermoelectric power measured with this temperature difference agrees with the published data of Miller et al [8] within 1%. An estimate of radiation losses at the higher temperatures has been made by assuming a gradient between the sample and the radiation shield that is about 10 times the measured gradient. The calculated error in that case is 10% at 300°K and 3% at 200°K. Therefore it is assumed that radiation losses can be ignored. 3. Sample The sample was cast in an oxygen-free copper boat by melting with an argon arc. It was polycrystalline with a resistivity ratio p(297°K)/p(4°K) = 72. It was annealed in a vacuum at 900°C for 24 hours and then ground to size. The dimensions between thermometers were approximately 4mm x 4 mm x 35mm. Electrolytic Cr was used. It has been analysed as follows (in weight percent): Cr - 99.92%, Mn - 0.004%, Fe - 0.005%, Mg - 0.002%, Cu - 0.003%, and the balance mostly S, P, Ni, and Mn. 4. Data 4.1 Thermal Conductivity The thermal conductivity data for this measurement are shown plotted in figure 1 together with those data that have been previously reported for similar samples [1,2]. The lower temperature data were obtained by Harper et al [1] from an as-cast sample with a partial anneal (sample #2). It has a residual resistivity p = 1.81 x 10" y ohm-m which is almost the same as that of the present sample for which p Q = 1.83 x 10~9 ohm-m. The high temperature data were obtained by Powell and Tye [2] from an electro- lytically deposited sample that had been annealed at 1410°C. Both X ma x an d Po depend on the quality of the sample. These two quantities are shown plotted against each other in figure 2 where it is seen that the present measure- ment agrees well in this respect with the data of Harper et al [1]. The reason for the different temperature dependences of the two samples shown in figure 1 is not understood. However, it will be shown that the present data agree with the theoretical 312 prediction of the temperature dependence of the thermal resistivity [9] due to phonon scattering, w^. In an elemental metal of reasonable purity, one expects X to be due to electrons; that is X At low temperatures where the electron distribution is surely degenerate the thermal analog to Matthiessen ' s Rule should hold; the thermal resistance is the sum of the residual and ideal thermal resistivities: w = X, = w + w. O 1 can be calculated from L Q and the measured value of p : w Q = (p o A o )T ■1 (4-1) (4-2) The temperature dependence of w^ separated from w in this way is plotted in figure 3. Between 60°K and 90°K the measured values of w^ « T2, the dependence predicted by theory. Both of these measurements indicate that below 30°K w^ is greater than would be expected, although the temperature dependences differ in the two cases. The source of this extra resistivity is not known and will not be pursued in this paper. From the point of view of this paper the important phenomenon is the increase of X above 200°K. It will be shown that this effect is electronic. It is unfortunate that it was not possible to measure through the N^el temperature [10] Tj\j = 313. 0°K with the present apparatus because one would like to know if the data would join those of Powell and Tye. It would seem reasonable to expect a discontinuity in X since one is observed in p [11]. 4.2. Electrical Resistivity The electrical resistivity p of the sample was measured in the range from 1.2°K to 297°K. The residual resistivity p was computed to have an average value of 1.834 x 10~9 ohm-m on the basis of nine measurements made between 1.2°K and 12.9°K. If it is assumed that Matthiessen ' s Rule holds: P = P + H o (4-3) where p^ is the ideal component, it is possible to compute pj_ from the data. These data are shown in figure 4 as the circles. The solid line comes from the much more accurate data of Arajs and Dunmyre [10], The agreement between the two sets of data supports the contention that method used to calibrate the carbon resistors at low temperatures is accurate to a fraction of a degree because these same calibrated resistors were used to determine the temperature of the electrical resistivity data. The temperature dependence of pi shown in figure 4 is completely anomalous. At low temperatures where one expects [12] pi « j5 f a T^»2 dependence is actually found. This latter dependence has been taken to indicate a predominant proportion of inter- band scattering [1]. At the higher temperatures where a T dependence is expected [12], Pi a T±'°. It is not the purpose of the paper to discuss this anomaly, but it seems reasonable to suppose that it indicates a large proportion of electron-»electron scattering, which is expected to give a T behavior [13]. The small anomaly occurring at the N^el temperature itself has been well studied [10,11], 4.3, Lorenz Number The Lorenz No. L = Xp/T has been computed from values of X and p taken from the best smooth lines drawn through the data. This value is shown in figure 5 along with values for the other two comparable samples [1,2]. The circles are the actual values given by Powell and Tye in their paper [2], while the dashed line was constructed from Harper et al's data [1] for X and p and the values of pj_ shown in figure 4. In normal metals L is expected to equal L whenever there are no other thermal conduction mechanisms and the electronic scattering processes are elastic. The effect of other conduction mechanisms such as lattice conduction is to increase L because then X = Xe + ^q« n the other hand the effect of inelasticity is to make L < L , a condition that obtains generally at intermediate temperatures. In the 313 present case the behavior of L below 90°K is qualitatively similar to normal metallic behavior with \ = X e only. Now if there is no appreciable lattice thermal conductivity \q at these low temperatures, it is unlikely that there would be any at the higher ones because the maximum value of \„ is expected at approximately T = 9/20 where the Debye temperature [14] 6(Cr) = 402°K. On the other hand Klemens [5] has pointed out that semimetals depart from normal metallic behavior as the temperature increases because the energy KT embraces more of the overlapping band structure. Therefore it is the purpose of this paper to show that the abnormally large values of L above 90°K can be reasonably ascribed to the effects of a semimetallic - or more simply - a complicated band structure. For this purpose it is more instructive to treat AL shown in Figure 6. The striking feature is that between about 120°K and 280°K AL « T 2 . 5. Analysis 5.1. Method of Moments The Method of Moments (hereafter MM) as promulgated by Klemens [5] is a way of treating the electronic Boltzmann equations so that the resulting transport coefficients become expressed in forms that are tractable when applied to substances with complicated Fermi surfaces. The essence of the method is to focus the effect of such complicated surfaces on the integrations over energy that occur in all of the transport integrals. It becomes possible to expand the integrands of these integrals about the Fermi energy and then to allow the data to determine the first derivatives of the expansions. The MM represents the integration over a constant energy surface by the specific conductivity a(e) where e = E/KT is the reduced energy. Then if it is assumed that a relaxation time exists which is the same for both thermal and electrical processes and that the phonon system is in equilibrium, the transport coefficients of interest to this paper become: electrical conductivity a = M~ (5-la) 2 thermal conductivity X e = f^j T (M 2 - jjpj (5-lb) ft)" Lorenz No. L = fp /tt^V S 2 .. ('..-Ic M is the n ' th moment of a(e 3f. M n = - e n a(e) jf de (5-2) ^ — CO 2 where f is the Fermi-Dirac distribution function. It is possible to neglect the S term in eq (5-lc) (and similarly the Mi term in (5-lb)) because M2/MQ > 1, (K/e) 86.3 u.V/deg, and S max (Cr) = 21 uV/deg at 300°K [8]. This neglection is about a 6% error. In the following formulae these terms will no longer be included. In the transport properties of metals one is concerned with the distribution of electrons about the Fermi surface. Therefore this surface is chosen as the energy zero and a(e) is expanded in a McLauran series about it. o(e) = o fl + o, £ +T£ 2 + -7? e 3 + '*• (5-3) 2 fc T 6 where a "= (a c^de ) Q . If only the first a which makes M non-vanishing is retained these coefficients nave normal metallic form. If the second non-vanishing term is retained, these coefficients become 2 a = °0 + "6" °2 (5-4al 314 5)* 1 L M -V £ 7tt 30 M2)' M?J (5-4b) (5-4c) These formulae are expressed in terms of the reduced energy e; the a depend implicitly on temperature. The temperature dependence of these second order corrections becomes explicit if they are rewritten in terms of the energy E. It is easy to see by comparing expansions of a(E) and a(e) that (KT! 'mE (5-5: Properly speaking L is not a transport coefficient, but since L = Xp/T it has the advantage that it has the constant value L = (tt /3)(K/e)^ for a normal metal whenever the assumptions made at the beginning of this section apply. Therefore the second order effects are most easily seen in AL = L - L n : L_ V-V ( 2 'oe) !ttK) 2 T 2 / 1 + 4 « 2 m ^ (5-6) which has a saturation value of 7.815 10 :v/deg)- a value which is 3.2 times L. The assumptions made in MM can be expected to apply to metals at temperatures of the order of 100°K or more, and it is seen in figure 6 that the temperature dependence of AL is exactly the initial temperature dependence of eq(5-6): AL /16\ / °2E \ L o " W l 2 W (ttK) 2 T 2 '.5-7: The quantity C = (o"2e/2ctoe) has been determined from the data to be 195 (electron volts )~2. This is a very interesting quantity not only because it is temperature independent below T n , but it will be shown in the next section that the same number is found from the data above T n . 5.2 Model Since second order terms in the ct(e) expansion cause AL > 0, the non-zero values of AL above T found by Powell and Tye [2] and shown in figure 6 suggest that these terms still exist at these high temperatures but in a vestigial manner. That is, these second order terms extend only over a limited range of energy E ; and as the temperature increases the normal portion of a(E) outside this range becomes more important. Therefore the model shown in figure 7 by the solid line is suggested by the data, and AL will be calculated for it. For even values of n the moment integrals of eq (5-2) become "lO^'o 5 + a 20^ F n (ro) F n<«o>3 + W e o> (5-8: The F (e ) are the Fermi-Dirac integrals in the finite interval e , c I \ F 2 n 3t o , (5-9) It is convenient to define a new function G(e ) ^C ) = M n /a 1Q (5-10) because then Oiq, which yields the normal metallic electrical conductivity at low temperatures and so contains unknown temperature dependences implicitly cancels out of L. The transport coefficients are 315 a = o 10 (T) G o (e o ) (5-lla) X e = o 10 (T) T (A G 2 (e o ) (5-llb) (5-llc) K\ 2 G 2 (e o These coefficients contain two constants which must be determined from the data. The first, a = 020/010, is the ratio of the zero'th derivitive outside the parabolic well to that inside. The second is the energy width of the well E . These constants are related to the quantity C defined by the previous section; C = 4(a - l)/E *, The model was calculated on the CEIR computor by first computing the F _(e ) by a three point Gaussian integration with 100 intervals and then storing these in a list to compute Gn(e ) for various values of a. It is somewhat surprising that the F n (e ) do not approach their saturation values Fn(°°) until the interval e /2 attains rather high values: n = 0, e /2 = 16; n = 2, e /2 = 20; n = 4, e /2 = 26. As a result the effect of E is seen in the coefficients 5-11 at temperatures for which KT « E . The results of these calculations are shown in figure 8 where the reduced AL/(K/e) = AL r are plotted against KT /E for a series of a. The shape of the high temperature portion of the curve is sensitive to the form of aoo» anc ' the agreement of Powell and Tye's data with the model implies that 020 1S indeed rather independent of KT/Eq up to 1000°K. At low temperatures |E| > E /2 do not affect the transport coefficient, and it is possible to state only that the data must fall somewhere in the cross-hatched region of the figure. The very interesting result of the model is that the value of C computed from the parameters needed to fit Powell and Tye's data is in excellent agreement with the value obtained from the low temperature data. E was found to be 0.188 electron- volts and a to be 2.7. These parameters give a value of C of 192 (electron-volts) -2 . 6. Conclusion Since the thermal conductivity data were taken on different samples below and above T n , there must be some reservations in any conclusions reached from the analysis, However, it is well known that there is an anomaly in p( 10,11) at T n , and therefore one would expect an anomaly in either X or L. Calculations based on the high temperature model shown in figure 7 predict that AL r vs KT/E should be a continuous curve like those shown in figure 8 by the family of curves. It is not possible for the data to fit on one such line, and so in the vicinity of T n the two parameters a and E must decrease as T increases. It is a surprising result of the calculations that C = (a ?F /2ap,p) - which is sort of the parabolic constant of the well - is independent of the changes occurring at the N^el temperature. As is shown in figure 7 the change in a(E) at T n amounts to an enormous decrease in the height of the parabolic well, which is itself unchanged. What can be the meaning of such a result? Since ct(e) can be considered as a product of the density-of-states (hereafter DS ) and a carrier mobility [5], there are two possible sources of explaination : scattering or band structure. However the very fact that the parameters of the model are temperature independent suggests that they represent band parameters. The energy E = 0.188 electron-volts corresponds to 0.0120 rydbergs , an energy which is comparable to parabolic type wells in the computed DS of transition metals [15,16], Then if such an interpretation be correct, one would expect to find aL > in other transition metals at the higher temperatures. It has been pointed out by the author [17] that considerations of the transport coefficients from the multiband point of view indicate that they represent different portions of the Fermi surface. The conductivities are weighted toward those portions of the surface which have large carrier occupations and rather pedestrian mobilities when compared with the Hall coefficient and Thermoelectric power. Thus, the DS of interest is not the total DS but more likely the DS in some principal direction of the Brillouin zone. One can then guess that in Cr the principal electron conduction is occurring in those portions of the Fermi surface which are relatively unaffected by the disappearance of the magnetic Brillouin zone planes at T n . Therefore C is unchanged, but there is some reshuf f leling of DS which corresponds to a large decrease of a with a concomitant small change in p. 316 This interpretation is quite speculative. However, the very real possibility exists that measurements of thermal conductivity at high temperatures in pure metals can give some insight into the Fermi surface at these temperatures. 7. Acknowledgements The author would like to thank Dr. P. G. Klemens for a very helpful discussion. Also, he appreciates the help given by A. C. Verbalis in the computerized analysis and W. J. Buehler in preparing the sample. 8. References [1] A. F. A. Harper, W. R. G. Kemp, P. G. [9] P. G. Klemens, Solid State Phys 7 Klemens, R. J. Tainsh, and G. K. White; 1 (1958). * ~' Phil. Mag. 2, 577 (1957). [2] R. W. Powell and R. P. Tye . J. Inst, of Metals 85, 185 (1956-57). [3] J. M. Ziman, Electrons and Phonons Oxford: Clarendon Press, 1960. [4] J. F. Goff, Bull. Am. Phys. Soc. 12, 348 (1967). [5] P. G. Klemens, to be published. [6] J. F. Goff, J. Appl. Phys. 35, 2919 (L964). [7] J. P. Clement in "Temperature - Its Measurement and Control in Science and Industry" edited by H. C. Wolfe, Rheinhold: Hew York, 1955, p. 382. [8] H. B. Miller, A. L. Trego, and A. R. Mackintosh, Sol. State Comm. _3, 137 (1965). [10] S. Arajs and G. R. Dunmyre , J. Appl. Phys. 36, 3555 (1965). [11] S. Arajs, R. V. Colvin, M. J. Marcinkowski; J. Less-Comm. Metals 4, 46 (1962). [12] J. M. Ziman op. cit. p. 334. [13] N.F. Mott, Adv. in Phys. 13, 325 (1964 [14] I. Estermann, S. A. Friedberg, and J. E. Goldman; Phys. Rev. 87, 582 (1952). [15] M. Asdente and J. Friedel, Phys. Rev. 124 . 384 (1961). [16] L. F. Mattheiss, Phys. Rev. 139, A1893 (1965). [17] J. F. Goff, submitted to J. Appl. Phys. 317 10 3 00 0) ^ 1()2 10 J - HARPER ET AL^ x 4/* "fc^. •> POWELL & TYE y^GOFF 1 1 1 1 10 100 T(°K) 1000 Figure 1. The thermal conductivity X of chromium vs temperatures. The solid line is the present measurement; the dashed line is for other measurements of comparable samples. At low temperatures these data are for sample #2 of Harper et al [1], At high temperatures the data are for Powell and Tye's sample [2] which was annealed at 1410°C. 500 • GOFF a HARPER ET AL -5 400 - \^ E ~ 300 re 1 200 — ^"-^^ /< 100 — 1 1 0.001 0.002 0.003 P (iifl-m) Figure 2. The maximum value of the thermal conductivity compared with the residual resistivity . 318 Figure 3. The thermal resistivity vs temperature. The circlas are the values of the ideal thermal resistivity computed from the actual data. 10- Figure 4. The ideal electrical resistivity. The circles are the data taken for this measurement. The solid line is taken from the data of Arajs and Dunmyre [lo] . E 2 i to the Lorenz number, L Q , as a function or temperature. Figure 6. The ratio of the Lorenz function, L e , to (L Q -S ) , as a function of temperature. 330 The Thermal Diffusivity of Gold H. R. Shanks, M. M. Burns and G. C. Danielson Institute for Atomic Research and Department of Physics Iowa State University, Ames, Iowa 50010 The thermal diffusivities of four gold samples have been measured from 325 to 1225°K. The sample purities varied from about 95 to 99. 999% and the resistivity ratios from 3 to 600. The thermal diffusivity of gold was found to be sensitive to the impurity concentration. The thermal dif- fusivity data were combined with published specific heat data to obtain thermal conductivity values. Electrical resistivity and Seebeck coefficient measurements were also made on the samples. Key Words: Gold, Lorenz number, Seebeck coefficient, thermal conductivity, thermal diffusivity. 1. Introduction Recently there has been increasing interest in understanding the high temperature thermal con- ductivity of metals. In order to do so, much additional accurate data is needed on the simpler metals such as Cu, Ag, and Au. Thermal diffusivity data on a series of gold samples of different purities are reported here. Gold was chosen because of the availability of reasonably pure samples and the almost total lack of published thermal conductivity data above 300°K. 2. Measurements The thermal diffusivity measurements were made by a modified Angstrom method. The tech- . niques and equipment which were used for these measurements have been described previously [l, 2] . The three sample thermocouples consisted of 0. 025 cm diam Pt and Pt/10% Rh wires spot welded to the gold sample. Current for electrical resistivity measurements was applied to the sample through 0. 061 cm diam Pt wires spot welded to the sample ends. The apparent electrical resistivity was measured after each measurement of thermal diffusivity by a standard 4 -wire technique. Two of the Pt thermocouple wires were used as voltage probes. The thermal diffusivity measurements were made with the samples in one atmosphere of helium and with a 2. 5°K amplitude 30 sec period tempera- ture sine wave. The absolute Seebeck coefficients of the samples were measured by a method described by Heller and Danielson [3], Two of the thermocouples welded to the sample for thermal diffusivity measurements were used for the Seebeck coefficient measurements. The wires were reconnected ex- ternally so that the instantaneous voltage developed across the Pt - Au - Pt junctions could be plotted as a function of the voltage across the Pt/l0% Rh - Au - Pt/10% Rh junctions on an x-y recorder. The 30 sec period sine wave used for thermal diffusivity measurements was also used to supply the varying temperature gradient for the Seebeck coefficient measurements. The absolute Seebeck coefficient of gold could then be calculated from the absolute Seebeck coefficient of platinum reported by Cusack and Kendall [4] and the Pt - Pt/10% Rh thermocouple tables published by Adams and Davisson [5]. 3. Samples Four gold samples of different purity were obtained from four different sources in the form of rods 0. 35 cm in diam and 30 cm long. Table 1 lists the samples, the manufacturer, the purity stated by the manufacturer and our measured values for the ratio of the electrical resistivity at 300°K to the electrical resistivity at 4. 2°K after the samples had been annealed at 1225°K for 1 hr. Figures in brackets indicate the literature references at the end of this paper. 331 Table 1 . Sample Characterization. Sample Manufacturer Purity L 30C/ R 4. 2 Au 1 Au 2 Au 3 Au 4 Johnson, Matthey and Co. Sigmund Cohn Aremco Mint Gold 99. 999% 99. 99% 99. 9999% 600 310 110 3 4. Results The thermal diffusivity results for the four samples are shown in fig. 1. Each point represents the average of two diffusivity values obtained from the two thermocouple combinations at a given tem- perature. The deviation of the points for each sample from a smooth curve drawn through them was less than 1. 5%. The thermal diffusivity was dependent on the purity of the sample and decreased with increasing impurity concentration. For the most impure sample, Mint gold, the thermal diffusivity was found to go through a maximum value at about 725°K. The electrical resistivities of the four samples are shown in fig. 2. The electrical resistivities of the two samples of highest purity were the same, and were lower than the values for the other two samples. The electrical resistivity versus temperature curve for Au 4 is higher than, and not paral- lel to, the curves for the other three samples. This difference between the curves implies a high impurity concentration and also a violation of Matthiessen's rule. The Seebeck coefficients for the four gold samples are shown in fig. 3. The data for the gold samples are based on the absolute Seebeck coefficient of platinum reported by Cusack and Kendall [4]. They report an accuracy of ±5% for their platinum data. The absolute Seebeck coefficients for sam- ples Au 1, Au 2, and Au 3 agreed within ± 2% over the entire temperature range; and since the dif- ferences in the data for the three samples were less than the total errors for the measurements, the values were averaged to give one curve of Seebeck coefficient versus temperature. As expected the results for Mint gold were considerably lower and approached the values observed for the other three samples only at the highest temperatures. 5. Discussion The thermal diffusivities (fig. 1) of the two highest purity samples, Au 1 and Au 2, which have resistivity ratios of 600 and 310, respectively, differ by only 0. 7% at 350°K and by 3% at 1225°K. Since the errors in the measurement are about 2%, it is difficult to report any real difference in the thermal diffusivities of the two samples, although the higher purity sample appears to have a higher thermal diffusivity. Au 3, with a resistivity ratio of 110, has a thermal diffusivity 7% lower than Au 1 at 350°K. This difference represents a real change in thermal diffusivity with purity. For the most impure sample, Au 4, the thermal diffusivity at 350°K is 36% below the value observed for Au 1, the most pure sample. The shape of the thermal diffusivity versus temperature curve for Au 4 is different from the shape of the other sample curves and represents the effect of large amount of impurities on the thermal diffusivity. The thermal conductivities of the four samples were obtained from the product of the measured values of thermal diffusivity and the published values of density and specific heat. A constant density [6] of 19. 30 gm/cm^ was used. The specific heat results of Jaeger, Rosenbohm, and Bottema [7] were used in the calculation since their data is the most recent published and, if extrapolated to low temperatures, connects smoothly to the data of Geballe and Giauque [8]. Figure 4 shows the thermal conductivities calculated .for each of the four samples and also shows data reported by Mikryukov [9]. The data reported by Mikryukov [9] is on a polycrystalline sample cf gold with a reported purity of 99. 99%. The thermal conductivity curves reflect the same differences in purity observed in the thermal diffusivity data because the same specific heat values were used for all samples. This is probably justified for the three samples of highest purity since the specific heat is usually rather insensitive to small amounts of impurities. It may not be valid to use the specific heat for pure gold to calculate the thermal conductivity of Au 4 which has a high impurity concentration. The smoothed thermal dif- fusivity, the thermal conductivity, the electrical resistivity and Seebeck coefficient for Au 1 are given in table 2. These results represent our best values for the properties of pure gold. 332 Table 2. Transport properties of pure gold Temperature °K Thermal Diffusivity cm /sec Thermal Electrical Seebeck Conductivity Resistivity Coefficient watts/cm-deg ohm-cm /iV/deg 2. 97 2.43 1.93 2. 96 3. 05 2. 37 2. 94 3.93 2. 83 2. 92 4. 84 3. 15 2. 90 5.76 3. 35 2. 86 6.75 3. 51 2. 82 7. 76 3. 67 2. 78 8. 89 3. 85 2.73 10. 06 4. 03 2.66 11. 35 4. 29 325 400 500 600 700 800 900 1000 1100 1200 1. 140 1. 121 1. 093 1. 062 1. 028 0. 994 0. 954 0. 920 0. 874 0. 826 The apparent Lorenz numbers were calculated from the thermal conductivity and electrical resistivity data. The resulting curves are shown in fig. 5. The curves for all four samples are in good agreement and, at high temperatures, are all within ± 7% of the theoretical value of 2. 45 X 10"° V 2 /°K?. 6. Conclusions The thermal diffusivities of four gold samples with resistivity ratios ranging from 600 down to 3 have been measured from 325 to 1225°K. The thermal diffusivity values and published specific heat values were used to calculate the thermal conductivities. The thermal conductivities for the samples of higher purity are in good agreement with data on gold reported by Mikryukov [9]. The calculated Lorenz numbers agree well with theory. References [l] Sidles, P. H. and Danielson, G. C. , Thermal diffusivity measurements at high temperatures, In Thermoelectricity, P. H. Egli, Ed., Chap. 16, p. 270 (John Wiley and Sons, 1960). [2] Martin, J. J., Sidles, P. H. , and Danielson, G. C. , J. Appl. Phys. 38 3075 (1967). [3] Heller, M. W. and Danielson, G. C. , J. Phys. Chem. Solids 23^ 601 (1962). [4] Cusack, N. and Kendall, P., Proc. Phys. Soc. London 72_ 898(1958). [5] Adams, R. K. and Davisson, E. C. , ORNL-3649 Vol. 2, Section 2. 1 (1965). [6] Seybold, A. U. , USAEC Publ. AECD 2434 (1943). [7] Jaeger, F. M. , Rosenbohm, E. , and Bottema, J. A. , Pro. Acad. Sci. (Amster- dam) 35 772 (1932). [8] Geballe, T. H. and Giauque, W. F. , J. Am. Chem. Soc. 74 2368(1952). [9] Mikryukov, V. E. , Vestnik Moskov. Univ. Ser. Mat. Mekh, Astron. Fiz, i Khim. , L2 (6) 57 (1957). [10] Rudnitskii, A. A., AEC-tr-3724 (July, 1959), translated from a publication of the publish- ing house of the Academy of Sciences, U.S.S.R. (Moscow, 1956). 333 300 500 700 900 TEMPERATURE (°K) 1100 1300 Figure 1. The thermal diffusivity of gold as a function of temperature. The data show the effect of different amounts of impurities with Au 1 the most pure sample and Au 4 the most impure sample. a a. Figure 2. The electrical resis- tivity of gold as a function of temperature above 300 °K. 1 1 1 " / 12 10 GOLD & _ 8 O Aul D Au2 X Au3 A Au4 / x&i v'tcr — A # 6 * j^ 4 i Y 2 i i l 1 300 500 700 900 TEMPERATURE <°K) 334 GOLD Aul, Au2, Au3 Au4 RUDNITSKII [10] CUSACK and KENDALL [9] '300 500 700 900 TEMPERATURE (°K) 1100 1300 Figure 3. The Seebeck coefficient of gold as a function of temperature. The effect of large amounts of impurities on the Seebeck coefficient can be seen in the data for Au 4. 335 E o V. (/> \- < > I- o o z o o < tr UJ 3.5 — 3.0 2.5 2.0 1.5 1.0 300 GOLD — THIS INVESTIGATION V MIKRYUKOV 1 500 700 900 TEMPERATURE (°K) 1100 1300 Figure 4. The thermal conductivity of gold as a function of temperature. 2.8XI0" 8 300 O Aul Au2 X Au3 A Au4 500 700 900 TEMPERATURE (°K) 1100 1300 Figure 5. The apparent Lorenz number as a function of temperature for gold. The Wiedemann-Franz _ Q T ? law predicts a constant Lorenz number of 2.45 X 10 V /K . 336 Experimental Determination of the Thermal Conductivity of Solids Between 90 and 200 ok 1 p J. G. Androulakis and R. L. Kosson Thermodynamics Section Grumman Aircraft Engineering Corporation Bethpage, New York An apparatus to measure the thermal conductivity of solids from 90 to 200 °K is described. This apparatus is based on the axial flow technique. The samples are bars of circular, square, or rectangular cross section. By varying sample geometry, the thermal conductivity of poorly conductive materials and very highly conductive materials can be determined. Experimental data, curves of conductivity vs. temperature, are presented for some metals and plastics. A detailed error analysis identifying important parameters is included. Key Words: Conductivity, thermal conductivity, measurement apparatus, stainless steels, Teflon, Nylon 1. Introduction The apparatus developed to measure the thermal conductivity of solids in the temperature range from 90 to 200 °K is based on the conventional axial flow bar method. With this technique the sample, electrically heated at one end, conducts to a liquid nitrogen reservoir at the other end. It is suspended inside a radiation shield guard heater assembly with the radiation shield having approximately the same temperature distribution as the sample. The conductivity is determined from the difference in temperature between any two points, the heat input to the sample heater, and the known geometry of the sample. The apparatus is similar in concept to many others previously used for low-temper- ature conductivity determination (1) (2)3 but differs significantly in detail design and instrumentation . Applicability of this method to the determination of the thermal conductivity of both poorly conductive materials and highly conductive materials was evaluated and an optimum sample geometry was defined in each case. 2. Experimental Arrangement The apparatus is shown schematically in Fig. 1 (see also Figs. 2 through 4). It consists of a central assembly, the conductivity rig proper surrounded by a liquid nitrogen cold wall jacket. Both are suspended coaxially from the top cover of an external vacuum chamber by means of the inlet and outlet flow pipes. The central assembly consists of an upper liquid nitrogen reservoir to which is attached,' through a copper block, a radiation shield and guard heater assembly. The specimen is suspended vertically inside the radiation shield by means of a copper plate to which it is attached by one central screw. This plate is housed inside the shield upper flange which is screwed to the copper sink. An upper heater is housed inside a groove in the copper sink. This effort was performed under the sponsorship of the Grumman Aircraft Engineering Corp. Advanced Development Program, Project No. AD 04-03. 2 Thermodynamics Engineer and Thermophysics Group Leader, respectively. 5 Figures in brackets indicate literature references cited at the end of this paper. 337 Various radiation shields are used to fit particular sample geometries. They are of stainless steel or tantalum sheet with a highly polished inner surface to minimize radiation exchange with the sample. A typical stainless steel construction is shown schematically in Fig. 2a. The miniature heater supplying heat to the specimen is illustrated in Fig. 2b. After preparation, the sample is inserted inside the radiation shield through the top flange and the whole assembly is attached to the copper block. All mating surfaces are covered with a thin layer of silicone grease to provide better thermal contact. The sample is from O.O76 to 0.23 m long and from 0.0064 to 0.0190 m in diameter (or of equivalent square or rectangular section.) For measurement of voltage and temperature, fine-diameter insulated wires are used (0.51 x 10 - 4 to 1.27 x 10~4 m ) i n order to minimize the conduction loss. The temperature is measured by copper-constantan thermocouples. They are either soldered directly to the sample or, when soldering is not possible, the junction is bur- ied in the sample and secured in place by a low temperature cement or some mechanical process. The thermocouple output voltage is measured with a Leeds and Northrup type K-3 potentiometer. The power input to the sample heater is supplied by a variable transformer, and is determined from measurements of the current and the voltage drop across the heater. The voltage terminals are near the heater to avoid introducing the I 2 R losses of the power leads into the power measurement. The sample heater current is measured from the voltage drop across a standard resistor. The voltage drop across the sample heater and the standard resistor are measured with the Fluke differential dc-ac voltmeter model 887AB. The power inputs to the guard heaters are monitored manually by variable transformers. In some cases, in order to avoid frequent inter- vention by the operator, the differential temperature between the sample heater and the bottom guard heater is controlled by a closed feedback control loop using a three-mode control action. This control loop continuously adjusts the power input to the guard heater and keeps the temperature difference less than 0.1 °K. 3. Experimental Procedure The temperature along the sample is measured at equal distances ranging from 0.0064 to 0.051 m, depending on the thermal conductivity of the material. For high-conductivity materials a small ratio A/Z of the cross-sectional area A to the distance Z between thermocouples usually will be adequate for accurate determination of the thermal conductivity. For low-conductivity materials a large value of A/Z is used consistent with minimum measurable power dissipation and maximum temperature in- terval within which an average value of the thermal conductivity is to be determined. The sample temperature is measured at three or more points depending on the length of the sample. At low energy levels, the choice of the sample length is greatly in- fluenced by uncertainty in the heat measurement due to heat losses or gains by conduc- tion through the thermocouple wires and by radiation exchange with the radiation shield. This uncertainty, however, becomes insignificant with increased power applied to the sample. Also, the time required to reach a stable thermal balance has some in- fluence on the choice of the sample length. For plastics, in general, a sample length of O.O76 m is chosen with temperatures red at two or three points. For metals, a length of 0.13 to 0.23 m is chosen, thermocouples spaced 0.025 to 0.051 m apart. measured at two or three points. For m with thermocouples spaced 0.025 to 0.05 The temperature of the radiation shield is measured at the midpoint and is kept within 6 °K of the temperature of the corresponding point at the sample by the use of two guard heaters. In some cases it is necessary to minimize the radiation losses to the cold wall jacket. This is done by loosely wrapping a number of superinsulation foils around the radiation shield. The conduction loss or gain through the thermocouple wires is minimized by. leaving a loose 0.25 m length of wire between the sample and the radiation shield, and by thermally grounding another 0.30 m of wire at the corresponding temperature on the shield, Similarly, the conduction loss or gain through the power leads is minimized by wrap- ping enough length of wire around the bottom guard heater. During operation, after the vacuum chamber is evacuated, the two reservoirs are filled with liquid nitrogen. Dry helium is then Introduced in the chamber to rapidly cool the sample to a uniform temperature near that of liquid nitrogen. The chamber is 338 re-evacuated and, after a stable condition is reached, the specimen heater is turned on, with subsequent operation of the bottom guard heater. The upper guard heater is used occasionally to minimize the temperature gradient in the copper block at high heat fluxes but is mainly provided to raise the sink temperature when measurements are made at higher sample temperatures. Unless this is done, for a given thermocouple spacing the temperature gradient between two measured points becomes very large at high power inputs. Measurements are made in a vacuum between 1.33 x 10-5 and 1.33 x 10-6 N/m 2. 4. Conductivity Determination For each experiment, the conductivity is determined at one or more average tempera- tures depending on the number of thermocouples. The simplified one-dimensional steady state heat balance equation used to determine the thermal conductivity from the measured values is: P = VI = X -|-AT U) where: P = power applied to the sample heater, V = voltage drop across the sample heater, I = current,AT = temperature difference between two points, A = average value of the thermal conductivity in the temperature interval AT. Corrections due to inherent errors of the apparatus are not applied to the calculated conductivity value from eq. (l). An error evaluation (presented below) shows that experimental conditions and sample geometry can be chosen to bring the uncertainty of the experimental value down to a value of about Q% at 90 °K and 5^ at 200 °K. 5. Error Evaluation 5.1 Analysis The following analysis considers the errors involved in using equation (l) to compute the value of the thermal conductivity, as well as the errors involved in measuring the basic physical quantities of a thermal conductivity sample. The most general heat balance expression for the sample is: P = Jk x ^ fl dA + Q i + WC P T (2 ) dT where: ~^r = temperature gradient in the axial direction, Qa = energy loss through the lead wires, residual gas and by radiation heat transfer, w = mass of measured portion, c p = specific heat, T = rate of change of sample temperature with time at the moment of measurement. The integral can be replaced with the one-dimensional equiva- lent expression A(T)A dT/dZ since the temperature variation over any cross-section where measurements are made is kept negligible (less than 0.1 °K) by the radiation shielding. Furthermore, in order that AfT)A dT/dZ = AA AT/Z, A T should be kept rela- tively small so that errors due to non-linear terms in X(T) will be small. Temperature differences from 8 to 28 deg. are established between measured distances, depending on the thermal conductivity of the material and the temperature. Eq (2) becomes: P = AA ^|- + Qj + wc p T (3) Equation (3) gives after differentiation and rearrangement: « 8Z 5A |*I + * P - **J - ™ P »T - 6(wc p )T a z - a - St aa z-i at [ ' For the evaluation of A , eq (3) is reduced to eq (l), setting Q/ = T =0. Equation (4) then becomes: 339 6\ _5P_ JA JZ SAT fj/ WC P * T X P AZAT P P 5) Since the errors in P, A, Z, and AT are random while errors due to Qn and T are sys- tematic, the total error can be better approximated by "* W?-wr ♦(*-■)♦ (%r] i/2 + ^ + ^ii p Each linear dimension is measured with an accuracy of +0.3$. This gives a cumulative error of the measured portion of the sample of: (^+( i !-) 2 -(^) 2 + ( i |) 2 =(^-3) 2 + (.3) 2 .(.67« 2 (7) The power measurement error contains the contribution to the total error due to the accuracy of the readout instruments for voltage and current. The accuracy of the Fluke high input impedance vacuum tube voltmeter is ±0.05$ and the overall accuracy of the precision portable shunt (used with the Fluke) is ±0.25$. In addition, there is an error of the order of 0.3$ in the current and the voltage measurements because of waveform deformation. Thus (¥) 2 = (^) 2 + {v I ~) 2 = (0 - 25 + °- 30)2 + (0 - 05 + °- 30)2 = (o - 6 ^ )2 (8) Equation (5) shows the uncertainty in the differential temperature measurement to increase rapidly with decreasing temperature interval AT. The limit of error of the copper-constantan thermocouples using premium grade wire is ±1$ within the temper- ature range 89 to 215 °K. Therefore, the absolute error 5AT increases with decreasing temperatures. The expression for the differential temperature measurement error is (m) 2 - <^> 2 +(' t i + 1 > 2 o) Vi ' < T i - T i + 1)2 where i and i+1 refer to successive thermocouples on the sample. This error is plotted in Fig. 5. The importance of keeping the temperature intervals small enough so that the thermal conductivity may be considered linear in T within the interval AT has been previously noted. In order to keep a temperature interval small with an acceptable uncertainty on the differential temperature measurement at the same time, an average value of the thermal conductivity is determined from measurements on two or more spans, in successive tests, arranged p so that each span is evaluated at the same mean temper- ature and with approximately the same differential temperature. This is accomplished by using the upper guard heater or increasing sample heater power. The error shown in Fig. 5 would then be divided by the number of spans used. The thermocouple readings are checked at ambient and at near liquid nitrogen temperatures (before heat is applied to the sample). The thermocouples respond nearly uniformly at both temperatures, indicating that errors are probably in the same direc- tion. Therefore the differential temperature error is smaller than the values in- dicated in Fig. 5- Accuracy can be further improved by using calibrated thermocouples. Heat flow through the thermocouple wires towards or away from the junction causes a temperature measurement error. This is minimized by thermally grounding the thermo- couple wires to the radiation shield. The term bQig = 5Q g + $0^ + 5Q rj where Qg is heat leakage through the residual gas is heat leakage through lead wires and Q r is heat leakage by radiation. 340 The heat loss by convection is generally neglected for vacuum pressures ^1.33 x 10 -2 N/m 2 . Thus, Qg contains only heat loss by gas conduction. Using the procedure indicated in Ref. [3~]i at systems pressures below 0.13 N/m , this gas conduction is given by: = 2.426 x 10-^ 7+ 1 v — (T 1 Vmt s T ) o' (10) with accommodation coefficients a g = a = 1, where: 5Q is heat transfer in watts, A is surface area in cm 2 , 7 is ratio of specific heats, P is pressure in microns of mercury, T is temperature in degrees Kelvin, and M is molecular weight of residual gas. Subscripts "s" and "o" refer to sample and cold wall respectively. The temperature T without subscript is measured at the gauge that measures the pressure p. The heat leaks by gas conduction are greatest at the upper limit of sample temper- ature. However 6q„/P depends also on the power input to the sample, which is a min- imum at the lowest temperature. For T c = 77°K, T = 300 U K, _p = 0.001 microns, , A c = 68.5 cm 2 (maximum sample surface), 7= 1.41 and M = 28.8, #Qg/P is evaluated for min- imum sample heat inputs of P = 0.0293W at 89°K and P = 0.293W at 200°K, yielding c 10-4' m 2 , €S = °- 6 > € r = 0.2 c > Average Sample Temperature deg K 5Q r - Watts Minimum Power Input to the Sample P - Watts 6% P t 89 200 0.0434 x 10-2 0.510 x 10 _ 2 0.0293 0.293 1.48 1.74 This maximum error occurs for the measured portion nearest the sink since the effect of heat leaks by radiation is cumulative proceeding away from the sample heater. It is important to note that when the upper heater is used to bring the temper- ature of the sink to a higher value (instead of increasing the heat input to the sample) for measurements of the conductivity at higher temperatures, the percentage 342 error ^Q r /p increases, since P remains small. In such cases, the heat leak $Q r must be kept small by decreasing the temperature difference between the radiation shield and the sample, or finding some other compromise solution towards a minimum error. The error due to the changing of internal energy may be written: wc p 61 Z c p d'6T P " AAT where d is the density of the material. Tables 4 and 5 show this error for two test conditions. 12 Table 4. Stainless steel rod of D = 0.95 x 10- 2 m, measured portion length: Z = 5.1 x 10-2 m, d = 7850 kg/m3, specific heat [3} cp = 4l4 JKg-1 deg- 1 at 200 °K and cp = 209 Jkg-1 deg-1 at 89 °K, Assume that at the moment of measurement the temperature at all points of the measured portion changes uniformly at the rate of T = 0.11 deg/hr Average Temperature of the measured portion deg K wc p 6T Watts Minimum power input to the sample P Watts wc p 6T/P * 89 200 0.182 x 10~3 O.361 x 10" 3 0.073 0.73 0.25 0.049 Table 5. Teflon rod of D = 0.019 m. Z = 0.013 m , d = 2130 Kg/m . 3 , specific heat[3]c p = 695 Jkg-1 deg" 1 at 200 °K.and cp = 356 Jkg-1 deg" 1 at 89 °K. Assume at the moment of measurement <$T = 0.11 deg/hr Average Temperature of the measured portion deg K wcp 6T Watts Minimum power input to the sample P - Watts wcp St % p 89 200 O.O85 x 10 -3 0.167 x 10~ 3 0.0293 0.293 0.29 0.057 5.2 Summary of Errors An estimate of the total error is summarized below, as calculated using eq (6). The largest single source of error is that due to the measurement of the differ- ential temperature which is assumed not to exceed 5$ at 89°K and 3$ at 200°K (either by keeping large temperature intervals or by using more than one span as noted pre- viously) . Table 6. Estimate of Total Error Average Temperature, deg K «5A X Average or High Conductivity Samples Low Conductivity Samples 89 200 8.06 fo 4.68 8.09 % 5.07 343 6 tc 6. Experimental Results Some of the experimental results for steels and plastics are presented in Figures For all steels, except the 410 stainless steel, the thermal conductivity deter- mined in this apparatus is in close agreement with values reported in [3] and [4] . For the 410 S.S. the variation of the conductivity with temperature follows the same trend, however, the values are 10$ lower than those reported in [4] . The thermal conductivity of the Teflon rod near liquid nitrogen temperature com- pares satisfactorily with the value reported in [5] . 7. References 1 Ziegler, W. T., Mullins, J. C. and Hwa S.C.P., "Specific Heat and Ther- mal Conductivity of Four Commercial Titanium Alloys', Advances in Cryo- genic Engineering, Volume 8. 2 Tyler, W. W. , Nesbitt, L. B. , and Wilson A. C. Jr., J. Metals 1104 (Sept. 1953) 3 Scott, R. B. , Cryogenic Engineering, D. Van Nostrand Co., N.Y., 1959. Powell, R. L. and Blanpied, W. A., "Thermal Conductivity of Metals and Alloys at Low Temperatures", National Bureau of Standards Circular 556. Chelton, D. B. and Mann D. B. , "Cryogenic Data Book", National Bureau of Standards Cryogenic Engi- neering Laboratory, Boulder, Colorado, May 15, 1956. Nitrogen Gas Outlet Inert Gas Inlet Power Feedthru Liquid Nitrogen Inlet Liquid Nitrogen Coldwall Superinsulation Radiation Sh Differential Thermocouple Bellows Liquid Nitrogen Inlet High Vacuum Coupling Thermocouple Feedthru Nitrogen Gas .::. Outlet Liquid Nitrogen Reservoir Copper Block Upper Heater Sample Thermocouple Sample Heater Guard Heater Removable Copper Disk Pumping Station Fig. 1. Schematic drawing of the apparatus 344 Copper Block Shield Upper Flange (Copper) Sample End Plate Radiation Shield (Stainless Steel) Sample Sample Heater Copper Superinsulation Blanket Guard Heater Coil (2a) Pig. 2 2.4x10" -J To Fit Sample Cross Section 8.9x10 4 m Copper Leads (1.3 x 10-4m DIA; Nichrome, Wire 2 x 10- K DIA Aluminized Mylar (Aluminum Out) Low Temp . Epoxy Cement Radiation shield and guard heater assembly (2a) and sample heater (2b) On the top are shown various radiation shields and bottom guard heaters assemblies. On the bottom are wired samples. The three samples on the right are shown with heaters in place. Fig. 3- Wired shields and samples 345 Shown on the right is the main power supply, and on the left the pumping station. The apparatus is mounted on a Jnst^menSSn^ *" ^^ P °^ ^^ ^ "^r Fig. 4. The conductivity apparatus in operation 346 OJ I OJ Eh En l EH EH •H ■P o 73 C O O 14 - 12 - 10 - 303 Stainless Steel A 321 Stainless Steel D 410 Stainless Steel D D - D - D - A <£ A O O A O A 1 I 1 1 1 100 120 140 160 Temperature, deg K 180 200 220 Fig. 6. Thermal conductivity of three stainless steels 347 80 bO O) 73 70 -p •H 60 -p O C O o 50 60 100 120 140 Temperature, deg K 160 200 Fig. 7- Thermal conductivity of SAE 1020 steel 0.30 H M CD H Teflon ( Polytetraf luorethylene I \ AMS 3651 j A Nylon 101 (Polypenco) | 0.28 * A A 1>> ■P •H .5 0.24 O -p 73 C O O O A n ?n 1 1 1 1 1 80 100 120 140 Temperature, deg K 160 180 Fig. 8 Thermal conductivity of nylon and teflon 348 The Thermal Conductivities of Several Metals: An Evaluation Of A Method Employed By The National Bureau of Standards David R. Williams and Harold A. Blum SMU Institute of Technology Dallas , Texas An absolute method, developed by Watson and Robinson at the National Bureau of Standards, for the determination of thermal conductivity as a function of temperature was tested with four metals whose thermal conductivities ranged from 16 to 120 W m °C~ 1 from 38 to 3^5°C. In addition, a comparative method and a low heat loss absolute method were also developed with this ap- paratus. Some remarks about costs and operations are also in- cluded . Key Words : thermal . Aluminum, conductivity, iron, metals, steel, Intro duct ion An absolute method for determination of thermal conductivity of metals was de- veloped at the National Bureau of Standards by Watson and Robinson [l] • We con- structed the apparatus and measured the thermal conductivities of four metals, namely AISI 316, AISI 303MA, ARMCO iron, and 2024-T 351. func are usin tive cond the spe c in o to r sibi by W The -advantag tion of tempe conducted. I g this appara procedure in uctivity is w temperatures imen to guard rder to obtai eport the res lities of thi illiams [ 3 ] . e of ratu n ad tus whi ell of t was n th ult s s sy the re wi ditio whi ch ch th known he gu negl e the of t stem . meth thin n to als e ap 2] ard igib rmal his Th od is that a wide ran this appro o has merit paratus is and in the and specime le . These conducti vi study and t e complete the r ge o ach , in cali sec n in latt ty d o di de s c mal conductiv f temperature we tested tw some cas es . brated with A ond case we f such a manne er methods on ata . It i s t s cus s s ome of ription of th ity s pr o ot One rmco ound r th ly r he p the e st can be o ovided t her proc of these iron wh it poss at the h equire o urpose o limit at udy has btai wo e edur is ose ible eat ne e f th ions been ned as a xperiments es for a compara- thermal to adjust loss from xperiment is paper and pos- reported 2. Description of Methods Three methods of analysis are applied to the experimental data obtained in tests using the previously described apparatus. These methods are: 1. An absolute method devised by Watson and Robinson [l] which requires the data obtained from a two-run experiment. In one run the temperature of the specimen is higher than the guard while the opposite holds for the second run. A heat balance equation is written for the specimen in each run. The heat input to the specimen is equated to the thermal conductivity and the radial heat loss - both expressed as functions of temperature. The two equations are solved simultaneously to yield the thermal conductivity and heat loss. 2. A comparative method, wherein by a series of experiments, using Armco iron, the apparatus is calibrated for heat transfer between specimen and guard. There- after, thermal conductivities for other specimens are obtainable on a relative basis from the results of a one-run experiment. 3. A "no-loss" absolute method for which the temperature differences between specimen and guard are reduced to the extent that heat transfer between the two 1 Figures in brackets indicate the literature references at the end of this paper. 349 becomes negligible. On this basis, thermal conductivities are determinable on an absolute basis from the results of a one-run experiment. 3 . Res ult s 3.1 The Experiments In this investigation, the first seven experiments included two runs each. As explained previously, the data obtained from a run (a) and a run (b) are required for the solution of X values by the absolute method of Watson and Robinson [l]. These seven experiments are the following: Experiment No. Spe cimen Mat eri al 1 2 3 AISI 316 AISI 303MA AISI 303MA h Armco iron 5 6 7 Armco iron 202U-T351 2021+-T351 Average Specimen Temperatures Span VI Span I 72 °C 231 ' 50 137 73 23U 55 1^7 77 2^5 55 111 It 78 216 Remarks Tested by N.B.S In this tabulation, span VI and span I refer to segments of the specimen near the sink and heater respectively. The average specimen temperature in each span designates the arithmetic average of the temperatures in that span from runs (a) and (b). Further experiments consisted of nine individual runs using the Armco iron speci- men. The object of these nine runs was to calibrate the apparatus for heat transfer between specimen and guard as a basis for determining X values by a comparative method. These nine calibration runs were divided into three subsets of three runs each. In each subset, the specimen temperature was maintained essentially constant, and the guard temperature was changed for each of the three runs. Each subset of three runs provided bonus results in that any two runs of a subset can be paired and analyzed by the Watson and Robinson [2] absolute method. This pair- ing of runs was, in fact, done to the extent that six additional analyses were made on Armco iron by this absolute method. For consistency with the above table, these six pairs of runs are identified as six additional two-run experiments, numbered 8 through 13 in the continuing tabulation below: Experiment Spe cimen No. Mate] Armco -ial 8 iron 9 Armco iron 10 Armco iron 11 Armco iron 12 Armco i ron 13 Armco iron Average Specimen Temperatures Span VI Span I 1+8 U8 67 67 87 86 °c 127 128 212 211 311 302 Calibration Runs 1 and 2 2 and 3 k and 5 5 and 6 7 and 8 8 and 9 3.2 Thermal Conductivities of AISI 3l6 Stainless Steel The particular AISI 3l6 specimen used in Experiment 1 was earlier tested by the National Bureau of Standards, and the results were reported by Watson and Robinson [ 1+ ] . These investigators determined thermal conductivities of AISI 3l6 , in the range from 90°C to 81+0°C, to be: 2 3 X = 13.33 + 17.27 [ 1000 ]- U.333U[ 1000 3.32 [ 1000 (4-1) 350 where \ is thermal conductivity, W m" 1 "C -1 , and T is temperature, °C. The data from Experiment 1 performed here were analyzed by the Watson and Robinson [l] method, and the relationship of A and temperature was found by the least-squares method for a temperature range from 72°C to 231°C to be: A = 13. 478 + 9.47 x 10 3 T + 5.60 x 10 5 T 2 (fc-2) Values of A, obtained, are presented in Table 1. Comparisons are also made for values of A computed by the comparative method and the "no-loss" absolute method. Table 1. Thermal conductivity values of AISI 3l6 stainless steel Span No. Specimen Mid-Span Temperature (°C) Run ( a) Run (b ) Thermal Conductivities (W m °C ) N.B.S. W. & R. Method % Diff. Comparat Method i ve % Diff. "No-Loss" Method % Diff. 17.12 18.27 + 6.7 17.78 + 3.8 17.04 -0.5 16.65 17.47 + 4 .9 17.26 + 3.6 16. 64 -0.1 16.16 16.57 + 2.6 16.78 + 3.9 16.61 + 2.8 15.66 15.71 + 0.3 15 • hi -1.5 15.83 + 1.1 15.10 14.96 -0.9 lit. 03 -7.1 lit. 83 -1.8 lit. 55 Ik.kk -0.7 13.63 -6.3 lit. 1*1 -0.9 I 231 3 230 7 II 201 7 201 III 170 .9 170 3 IV 139 1 139 1 V 105 8 105 5 VI 71 6 71 7 3.3 Thermal Conductivities of AISI 303MA Stainless Steel Experiments 2 and 3 were performed with the AISI 303MA specimen. From the data in Experiment 2, thermal conductivities were determined for a temperature range from 50°C to 137°C; from Experiment 3, A values were found in the temperature range from 73°C to 23 1 t°C. Application of the Watson and Robinson [l] absolute method to Experiment 2 data yields the following relationship for thermal conductivity: 13.257 + 2.46 x 10 2 (50 < T < 137C) (4-3) For Experiment 3, the Watson and Robinson absolute method provides the following equation for A as a function of temperature: 14. 189 + 1.99 x 10 T (73 < T < 234C (4-4) In the overlapping temperature range, the values of A determined by equations (4-3) and (4-4) agree within approximately three percent. Published values of thermal conductivity could not be found for this particular stainless steel. The alloying elements and percentages thereof in AISI 303MA and AISI 303 are identical with two exceptions: AISI 303MA contains 0.60$ molybdenum, maximum, and 0.50/0.90% aluminum; whereas, AISI 303 does not contain these elements. The influence which these added elements might have on thermal conductivity can only be surmised to be small. Table 2 includes thermal conductivities for AISI 303 as given by McAdams [3]. In Table 2, values of thermal conductivity of AISI 303MA are presented for the Watson and Robinson method applied to Experiment 2 data; in addition, values are given for the comparative and "no-loss" methods as applied to run (a) of Experiment 2. Arbitrarily, the values of A obtained by the "no-loss" method are used as a basis 351 l6 62 16 30 -2 Q 16 75 16 28 + 1 15 69 -3 h 16 52 15 79 -3 8 15 1*1 -6 1 16 30 15 29 -5 .6 Ik 76 -9 16 07 lit 81* -6 6 Ik 15 -10 9 15 85 lU 58 -6 .2 13 80 -11 1 15 60 for comparison of the values derived by the other methods. Comparison is expressed in percent of difference from the "no-loss" value in each span. Table 2. Thermal conductivities of AISI 303MA stainless steel. Span Specimen Mid-Span _, , _ , . . . . ... -1 „ -i f. _ , / on \ Ther mal Conductivities Wm °C ) Mo. Temperature ( C) " No _ Loss „ w . & R . Comparative AISI 303 Run (a) Run (b) Method Method % Method % (Information I 136.7 137.8 16.62 II 119. h 120.6 16. 21* III 102.2 103.1+ 16.1*2 IV 85.0 86.0 16.19 V 67.7 68.3 15.90 VI 50.0 50.1 15.55 3.1* Thermal Conductivities of Armco Iron Equation U-5 represents the application of the method of least-squares to the values reported by thirteen investigators [2] of Armco iron and is valid over a temperature range from -18°C to 538°C. X = 7^.32 - 6.22 x 10" 2 T (U-5) Consequently, knowing the thermal conductivity for Armco iron permits further evaluation of the apparatus and the method of Watson and Robinson [l]. Also, it provides the basis for the comparative method and a means of evaluating the "no-loss" method . To this end, eight experiments were performed using the Armco iron specimen. These experiments are numbers 1* , 5, 8, 9, 10, 11, 12, and 13. The eight experiments were analyzed by the Watson and Robinson method and by the "no-loss" method. In addition, the comparative method was applied to run (a) of experiment 1* and run (a) of experiment 5. Thermal conductivities are presented for Experiment 12 in Table 3 and Experiment 1* in Table h as typical of the data obtained. The basis for presenting the errors is to assume the Goldsmith [2] data is correct. The application of the Watson and Robinson [2] absolute method produced poor results for Experiments 1* and 5 [Data in Table 3]. "The reason is not known. In another six experiments, however, the Watson and Robinson [2] analysis technique provided overall results within 1.5% of the reference values from equation 1+-5. Validity of the "no-loss" method is seen to be related to the specimen-to-guard temperature difference. For example, in Experiment 9, the specimen temperature is higher than guard temperature by approximately 6°C over the length of the bar. Thus, the cumulative loss of heat from the specimen results in progressively poorer results from span I to span VI for the "no-loss" method. In experiment 10, where the "no-loss" method affords very favorable values for A the average radial temperature difference between specimen and guard is approximately 2°C. 352 Table 3. Thermal conductivity values of Armco iron from experiment 12 Span No . Specimen Mid-Span Temperature (°C) Run (a) Run (b) Th srmal Conductivities ( W m °C ) Ref (2) W. & R. Method % Diff. Comparative Method % Diff. "No-Los s " Method % Diff. I 315 .6 306.0 5^ 67 55.08 +0.8 Not Used 56. 1U + 2.7 II 265 .6 258.2 57 77 58. 30 +0.9 60.20 + 1+ .2 III 218.1 211. 7 60 n 61.57 +1. 1 * 6U.8U + 6.8 IV 172. 8 167.8 63 58 65.05 +2.3 68.70 + 8.0 V 129. k 125 .6 66 26 68.80 +3.8 68.87 + 3.9 VI 88.1+ 86.1 68 82 72. 88 +5.9 71.66 + 1+.1 Table k. Thermal conductivity values of Armco iron from experiment h Span Spe cimen Mid-Span No. Temperat Run ( a ) are (°c) Run (b) Th ;rmal Conductivities (W m "c" 1 ) Ref (2) W. & R. Method % Comparat Method i ve % "No-Los s Method II % Diff. Diff. D iff. I 1U7.2 1^6.1 65 17 62 . 1+0 -1+ . 3 62.83 -3.9 63.30 -2.9 II 128. 3 127.2 66 3U 63.68 -1+.0 62.1+3 -5.9 62.9I+ -5.2 III 109 . k 109.0 67 50 63.92 -5.3 61+. 1+7 -1+.5 65 .20 -3.1+ IV 91.0 90.5 68 61+ 63.78 -7.1 61+. 53 -6.0 65 .60 -1+.1+ V 72.8 72.2 69 77 63.59 -8.9 65 .22 -6.5 66.62 -1+.5 VI 55.1 55.0 70 88 63.58-10.1+ 66.97 -5.5 68.75 -3.0 3.5 Thermal Conductivities of 2021+-T351 Aluminum Alloy Two experiments, numbers 6 and 7» were performed using the 2021+-T351 aluminum specimen. The temperature range in each of these experiments overlapped to permit a comparison of X values in the common temperature range. No reference was found for the thermal conductivity of this particular aluminum. Table 5 summarizes the results from Experiment 6. The three methods of analysis provide generally comparable values for \. Values calculated by the "no-loss" method are higher than the mean and are progressively departing therefrom in the successive spans on the specimen. Nominally, a 5°C radial temperature difference existed over all the spans. Therefore, the departure of \ values by the "no-loss" method is reasonable . In the overlapping temperature range of experiments 6 and 7, the application of the Watson and Robinson method to each experiment gives values of \ which agree within three pe r cent . 353 I 1U3 9 II 127 III 109 It IV 91 7 V 73 8 VI 55 k Table 5. Thermal conductivity values of 2024-T351 aluminum from experiment 6 Span Specimen Mid-Span m , , - , . . , . , -l Oo -i. „, * . /ooi Thermal Conductivities (W tn °C _) No. Temperature ( C) ; — — m„ T n W. & R. Comparative No-Loss Run (a) Run (b) Method Method % Method % Diff ■ Dif f . 143.4 138.90 139.63 +0.5 142.40 +2.5 126.7 135.46 133.59 -1.4 137. 34 + 1.4 109.3 131.96 129.35 -2.0 134.33 +1.8 91.7 128. 54 127.26 -1.0 133.86 +3.8 73.9 125.27 122.29 -2.1+ 129.43 +3.3 55.5 122.24 120.08 -1.8 127.90 + 4.6 4 . Dis cus s ion The method of Watson and Robinson [l] has been tested in a limited range (38-345C) for four metals whose thermal conductivities range from about 16 to 120 watts/M-°C. Agreement within 3% was obtained on one common sample. Two other procedures were found to be successful in many cases. One method depended upon the calibration of the system with Armco iron while the second method depended on obtaining an average temperature difference between the specimen and guard of less than 2°C. This latter method assumes no radial heat losses and is treated as if it were an absolute method. The advantage of these latter methods, when applicable, is that only one experimental run is required to obtain thermal conductivity as a function of temperature compared with the two runs required for the Watson and Robinson absolute method. This apparatus does provide the advantage of obtaining thermal conductivity as a function of temperature in one experimental setup and in one or two experimental runs. It requires a relatively large specimen of specific size (about 0.37 meters long and about 0.4 cm. diameter) which can be machined in order to provide a place for the source and for the sink. The experiment takes about one day per run, about a day for machining and only a short time for analysis with a computer program. The safe mini- mum elapsed time from the receipt of the sample to the reporting of data would be about three days. There are possibilities for further simplification and greater flexibility with this apparatus. 5. Acknowledgements We appreciate the help of Daniel R. Flynn of the National Bureau of Standards for furnishing us with the details of the apparatus and a sample for comparison. This study was part of a project supported by the National Aeronautics and Space Administration (NsG 711/004 007 004 ) . 6 . Ref e rences [l] Watson, T.W. and Robinson, H.E., [3] Williams, David R., Thermal conduc- Thermal conductivity of some com- tivity of some metals, M.S. Thesis, mercial iron-nickel alloys, Journal November, 1966 , SMU Institute of of Heat Transfer, November, 1961. Technology, Dallas, Tex. [2] Goldsmith, A., Waterman, T., [4] Watson, T.W., and Robinson, H.E., Hirschhorn, H., Handbook of thermo- Thermal conductivity of a sample of physical properties of solid materi- "type 3l6 stainless steel, NBS Rept. als , Volume 1 - Elements, revised ed., 78l8, March 22, 1963. Armour Research Found., Pergamon Press, 196l. [5J McAdams, W. H. , Heat transmission, Third ed. , McGraw-Hill Book Co. , 1954. 354 A Longitudinal Symmetrical Heat Flow Apparatus For The Determination Of The Thermal Conductivity Of Metals And Their Alloys H. Chang 1 and M. G. Blair 2 Energy Conversion Division Nuclear Materials and Equipment Corporation Leechburg, Pennsylvania 15656 The present paper describes an apparatus used for determining the thermal con- ductivities of metals and their alloys in the temperature range 50° to 1000°C. The measurements are made on a small-diameter cylindrical rod sample subjected to one- dimensional longitudinal heat transfer. The colorimetric heater, welded into a cavity half-way between the ends of the sample, provides, because of its location and construction, equal and symmetrical heat transfer from the center of the sample to either of the two ends, and allows input heat to be calculated precisely. Be- cause of the presence of the guard, which is radially isothermal with the sample, no heat loss is expected through the heater leads or by radiation from any portion of the sample. The thermal conductivities of Tophel Special (approximate composition 90% Ni 10% Cr) up to 500°C has been measured in this apparatus and reported. Tophel Special is an excellent metallic thermoelectric material. The precision and accuracy of the apparatus is discussed, and modifications may be easily made to measure the thermal conductivities of high melting point metals and their alloys. The reported data are preliminary, and current measurements on standard materials are designed to confirm the accuracy of the apparatus. Keywords: Electrical resistivity, figure of merit, nickel-base alloy, Seebeck coefficient, thermal conductivity. 1 . Introduction The electrical power output of a radioisotope thermoelectric generator is directly proportional to the figure of merit, Z, of the thermoelectric materials used to convert thermal energy directly into electrical energy. Z is defined as S 2 /pA, where S, p, and A are respectively the Seebeck Coefficient, electrical resistivity, and thermal conductivity of the material. Thermoelectric materials with high figures of merit are, therefore, excellent candidates for use in thermoelectric generators. In order to compute the figure of merit of a given material, therefore, it is essential to know its thermal conduct- ivity. Although the figure of merit of metallic thermoelectric materials is often an order of magnitude smaller than those of semi-conductors, metallic thermoelectric materials may be preferred for practical reasons. Such is the case for Tophel Special (manufactured by Wilbur B. Driver Company) , an alloy of Ni and Cr, the thermal conductivity of which is unknown. The largest Tophel Special samples available for testing, which have the same physical properties as those of the usual fabricated small diameter thermo- electric elements, are cylindrical rods with a diameter of approximately 3.2 mm. A precise and an accur- ate method for measuring the thermal conductivity of metals and their alloys which are available only in small diameter samples is, therefore, required. 1 Manager, Energy Conversion Physics Department 2 Physicist 355 Although the present absolute method of measuring thermal conductivity is reminiscent of those employed by Schofield (1) and R. W. Powell (2) , the present method utilizes flexibility of design and temperature control in order to accommodate cylindrical samples of small diameter. The accuracy of the data depends on the degree to which the thermal response of the sample can be made to approach the ideal conditions assumed in theory and on the magnitude of the errors involved in the various experimental measurements. The unique feature of the apparatus reported in this paper is the heater built into a cavity half-way between the ends of the cylindrical rod sample. This construct- ion minimizes heat loss and allows input heat to be calculated precisely. The apparatus is also capable of measuring materials with high melting points, such as refractory metals, which may not be available in large sizes. 2. Description of the Method The absolute method for measuring the thermal conductivities of metals and alloys described here utilizes a small heater placed in a cavity midway between the two ends of a long cylindrical sample. Radial radiative heat transfer from the surface of the sample is eliminated by a coaxial tube sample guard which may be heated so that the difference between the sample surface temperature and the guard surface temperature, measured at any location equidistant from the longitudinal center of the sample and guard, is zero. This arrangement insures bidirectional conductive heat transfer from the center of the sample to either of the two ends. Considering the axis of the cylindrical sample to be coincidental with the y axis of a rectangular coordinate system, with the longitudinal center coincidental with the origin, the one-dimensional steady-state heat transfer equation (assuming the thermal conductivity is independent of temperature) governing heat conduction from the center of the sample to the top end is simply given as Q i = " A A l TT ) 1 ' (1) where Q^ = heat transferred from the center of the sample to the top end A = area perpendicular to the direction of heat transfer d 8 (-5 ) = temperature gradient established in the positive y direction Similarly, the one-dimensional steady-state heat transfer equation governing heat conduction from the center of the sample to the lower end is Q 2 = - AA 4f->; 2 ' (2) where Q2 = heat transfer from the center of the sample to lower end (-= ) = temperature gradient established in the negative y direction. The total power input Q t to the sample heater is the sum Q t = Ql + Q2 , (3) or Qt = - X A (— ) + (— ) (4) 356 The thermal conductivity X is calculated from Eq. (4) to be (Qt/A) X = v— ) + (— ) Ldy \ + ( d y ; 2 . -1 (5) ,d 9 d e Equation (5) is used to calculate X if (-5 ) and (3——) are not equal, which indicates that the heat a Y 1 Y 2 transfer from the center of the sample to either of the two ends is unequal. However, when the heat transfer from the center of the sample to both of the two ends is made to be equal, ,de > equals A 6 >, ( d y \ l dy \ and Equation (5) reduces to X = - (Q t /2A) l dy' (6) Either Equation (5) or Equation (6) may be used to calculate the thermal conductivity, A, the choice depending upon the symmetry of the experimentally measured gradients. Moreover' in the case where all the heat input is made to flow in only one direction by creating an isotherm at either the top or bottom half of the cylindrical sample, then one of the temperature grad- ients in (5) is zero. If the heat is made to flow to the top end, then. d 9 . in Equation (5) is made ( dy \ ■zero, and Equation (5) reduces to X = - (Qt/A) (7) Deliberate creation of such an isotherm in one direction provides essentially a second experiment. With its experimental findings, the results of the bidirectional experiment may be correlated, thereby giv- ing a double check on the accuracy of both results. 3 . Apparatus The schematic diagram and the details of the apparatus used to measure the thermal conductivity of Tophel Special (a thermoelectric material, of approximate composition 90% Ni and 10% Cr, supplied by Wilbur B. Driver Company) are shown in Figure 1 and Figure 2 respectively. The high - temperature sample heater (detail A of Figure 2) is made by threading a continuous length of .178 mm (.007") diameter tan- talum wire through all of the 7 bores of a multibore alumina tube 2.5 cm in length (Degussa catalog No. 60-077175, outside diameter = 1.75 mm and bore diameter = .20 mm) . The two free ends of tantalum wire, which form the power leads to the heater, are at opposite ends of the alumina tube, and are brought out through the sample wall through two alumina-insulated holes discussed below. The cylindrical sample rod is fabricated from a single 3.18 mm (.1274") diameter 200 mm long Tophel Special rod. To place the heater in a cavity within the sample, located at the longitudinal center of the sample rod, the rod was cut into two pieces of equal length, and flat bottomed 2.38 mm (.0935") holes were bored concentric with the rod in either direction at the break. The cavity holes were bored deep enough to allow for the symmetric spacing of the heater in the sample and to include 1 mm alumina insul- ators at both ends. A .838 mm diameter hole was drilled radially into the cavity 1.5 mm toward the break from the bottom of each cavity hole for passage of the heater power leads and their alumina insulators. After insertion of the heater and alumina insulators, the sample was rejoined by TIG (Tungsten Inert Gas) welding at the break. To facilitate the welding, the ends at the break were machined to a close- toler- ance fit, mating a relief entubulation on one side to a counter bore on the other. This joint was sub- sequently fused by the weld. 357 The dual encapsulation of the sample heater within the ceramic and within the sample provides, aside from the necessary electrical insulation, the dual benefit of more uniform heat transfer to the sample and the elimination of heat leakage from the sample, for which accounting would otherwise be difficult. A .127 mm diameter (.005") platinum 10% rhodium wire is tack welded to each of the heater leads at the point of their emergence from the sample to serve as voltage leads. The locations of the voltage leads are so chosen that they effect the most accurate reading of the voltage drop across the sample heater possible. Precise measurement of this voltage and of the current through the heater gives the most accurate possible measurement of the power input. The two-bore oval cross-section alumina insulator carries each pair of leads radially through the guard, and away from the apparatus, effecting complete electrical insulation from each other and from other parts of the system. When the sample guard, which will be discussed later, is raised to the temper- ature of the sample at this location, the power lead is in an isothermal environment and no heat loss from the sample through it or the voltage leads is expected. At the top and bottom ends of the sample, holes are drilled to receive alumina rods which extend to serve as centering and insulating supports. Heaters, which are made by threading a length of .178 mm tantalum wire through 23 uniformly spaced, alumina- insulated holes in a copper cylinder 27 rim in diameter and 12.7 mm in height are attached at top and bottom of the sample. A center hole of the proper diameter allows the heaters to be fitted over the sample, and a positive mechanical contact is made by slotting the toroidal heater and drawing the open side together with screws. The thermal gradient is obtained by taking temperature differences between the seven pairs of Pt- Pt 10% Rh thermocouples strategically located along the length of the sample. The .127 mm (.005") thermo- couple wire pairs were flame fused at the end into a small bead, which was spot welded into a small posi- tioning indentation in the sample at the desired location. One thermocouple is located at the longitudinal mid-point of the sample, which is also the mid-point of the sample heater. The remaining six pairs of thermocouples are symmetrically located with respect to the center pair (See Figure 2): three pairs attached to the sample above the heater and three below. The thermocouples are evenly spaced on both sides of the center, respectively 19.05 mm (.75"), 31.75 mm (1.25") and 44.45 mm (1.75") from the sample center. The two nearest thermocouples to the sample heater are 6.35 mm (.25") from the end of the center sample heater nearest them. The sample guard is a tube with an inside diameter of 6. 35mm (.25") and a wall thickness of .381 mm (.015"). Since the purpose of the guard is to provide an isothermal radial environment for the sample at all points, a guard is required which will have the same thermal gradient as the sample. This condit- ion can be satisfied ideally if the thermal conductivity and the cross-sectional areas of both tube and sample are the same. This wall thickness was chosen to allow the cross-sectional area of the guard per- pendicular to the direction of heat flow to equal that of the sample. Due to the fact that the Tophel Special sample material is not available in rods with diameter greater than 3.175 mm in diameter (.125") ; Tophel Special with slightly different heat treatment was selected for the guard. The tube is machined from a solid rod 7.97 mm in diameter (.314") which was also supplied by Wilbur B. Driver Company. Since the total length of the guard tube required was 143 mm (5.625") , it was difficult to machine such a tube and maintain tolerances on walls of .381 mm thickness. The guard tube was, therefore, machined in two shorter pieces, and subsequently E.B. welded together. (See detail A of Figure 2) . Three heaters are attached to the guard in the same fashion as the sample heaters, one in the center and one on each end of the tube. Six pairs of the Pt-10% Rh thermocouples are attached to the surface of the guard in the same manner as those attached to the surface of the sample. They are located at positions radially opposite those attached on the samples when the guard is properly positioned around the sample. (See Figure 2) A .254 mm diameter wire (.010") Pt-Pt-10 Rh thermcouple is attached to each of the five copper heaters. These thermocouples are placed in the slots of the heater and mechanically affixed when the heaters are tightened on the guard and sample. In the final assembly, the lower alumina sample support rod fits into a hole drilled into the base of the support framework for this purpose. The top alumina centering rod passes through a hole drilled in the center of the top plate of the framework, and is free to move up and down to allow for the thermal expansion of the cylindrical sample. The top and bottom guard heaters are supported by the top and bottom sample heaters through three alumina support rods, which not only center the guard with respect to the sample but also insulate the components electrically and thermally from each other. The entire structure is thermally insulated by three 1 mil titanium radiation shields. The guard and sample heaters are similarly isolated by sets of radiation shields, and the entire system is operated in 1 x 10~° mm Hg vacuum. Power is supplied to the sample heater by a Sorensen QRC 40-30A DC power supply, and both the 358 voltage across the heater and the current through it are measured with a Fluke model 895A DC voltmeter, which has a 1 yv sensitivity. The latter measurement is made across a 1ft (.05% tolerance) 10 watt pre- cision resistor in series with the heater. The five peripheral copper heaters are all powered in the same manner (Insert AA of Figure 1) by Powerstat bridges indicated schematically, which are connected directly to the output of a Sorensen 2501 AC voltage regulator. The inputs to these heaters are regulated and controllable to within 10 pv as measured by a Fluke 931 PB RMS differential voltmeter. The thermocouple emfs are measured by a Leeds & Northrup K-5 potentiometer against a zero-degree reference temperature bath. The distance between the thermocouples was measured by a Nikon measuring microscope. 4. Experimental Procedure and Errors In conducting the bidirectional heat flow experiment, a symmetrical temperature gradient from the center of the sample to either of the two ends of approximately .8 °C/mm (20 °C/inch) is established under steady-state heat transfer conditions. The top and bottom sample heaters were used to set the temper- ature at which the thermal conductivity was to be measured; and the center sample heater was used to establish the heat flux and thermal gradient. Top and bottom sample heaters were adjusted by reading the seven sample thermocouples to balance the thermal gradients in both direction to the desired degree. Simultaneously, the three guard heaters were adjusted to achieve a steady-state condition in which the temperatures measured by each of the thermocouples on the sample were matched by the temperature readings of the thermocouples radially opposite each of them on the guard. Careful manipulation of the guard heaters could achieve simultaneous matching of the thermocouple pairs to within + 2.5°C; and in most cases the matching was considerably better. Thermal gradients established in this experiment were usually 9° - 10°C between the sample thermo- couples spaced at 1.29 cm intervals. Since indications were that the thermal conductivity was a slowly varying function of the temperature, the assumption that there is a linear gradient between thermocouple pairs is realistic. The calculated thermal conductivity was, therefore, postulated to be that at the average temperature of the two thermocouples. When the gradients were matched to within the 2.5°C tolerance mentioned above, the thermal conduct- ivities measured were consistent within 2%. The single largest source of experimental error inherent in the system is the measurement of the length between thermocouples, affecting the gradient calculation. Couples were made from J.27 mm diameter wires by flame-fusing a bead at their juncture. This bead formed a sphere with diameter in excess of 0.5 mm, which was then spot-welded to the sample. Determination of the exact distance between contact points was, therefore, indetermirtant by at least 0.5 nm; and since the length between couples was 12.7 mm (.50") , the 2% indeterminacy could account for the 2% spread in the data. The precision of the measurements themselves was .1%. The confirming procedure for the conductivity data, in the form of the unidirectional heat flow experiment, utilizes the ability of the apparatus to control both temperature and thermal gradient in different portions of the apparatus independently. To make this second measurement, the sample heaters at center and one end are set to establish an isotherm, with resultant net zero heat flux in that direct- ion from the center. The remaining sample heater is set at a reduced temperature to produce the desired gradient in that direction. Meanwhile, the guard heaters are manipulated to make the guard temperature conform to the temperature profile of the sample. Experience with the apparatus operating in this mode indicated that the measured thermal conductivity was relatively insensitive to variation between sample and guard temperatures, on the isothermal side, up to approximately 5°C; but that this method was very sensitive to small differences in sample and guard gradients on the heat transfer side. Confirmation of the bidirectional data, however, was within 2.5% for guard-sample gradients matched to less than 1°C on the gradient side, and guard-sample matched isotherms of 5°C or less. 359 5. Results and Discussion of Results The thermal conductivity of Tophel Special has been measured in the temperature range between 70° and 500°C. The values obtained have been tabulated in Table 1 and plotted in Figure 3 (Ref . 3 has been consulted in the numerical reduction of the data) . Table 1. Measured Thermal Conductivity of Tophel Special Mean Temperature Thermal Conductivity Mean Temperature Thermal Conductivity T (°C) Xx 10-2 (watt/M°C) SS T (°C) X x 1(T 2 (watt/M°C) .2276 17 340.2 .254 .2258 18 341.0 .251 .2236 19 341.1 .249 .2218 20 341.4 .248 .2224 21 x 335.3 .251 .2254 22 x 336 .256 .2230 23 x 335 .257 .268 24 x 337.5 .240 .267 25 x 337.9 .253 .264 26 506.3 .197 .261 27 506.1 .197 .259 28 506.1 .197 .268 29 506.0 .198 .249 30 505.7 .199 .250 31 505.5 .200 .248 32 505.3 .199 x Indicates unidirectional confirming runs The measurements have been made by the bidirectional heat transfer method and verified by the uni- directional method at approximately 350°C. The agreement is excellent, as shown in Figure 3. The thermal conductivity of Chromel P (approximate composition Ni plus Co 90%, Chromium 10%) was measured by Shelton (4) , and the thermal conductivity of this material, which is closest to Tophel Special in composition, has also been plotted in Figure 3 for comparison. Although the thermal conductivities of Tophel Special and Chromel P are of the same order of magnitude, there appears to be a difference in their behaviors as a function of temperature. The thermal conductivity of pure nickel (99.999 pure) was most recently measured by Kirichenko (5) in the temperature range 67-647°C,and by Powell and Tye (6) between 50 and 1050°C. The thermal conduct- ivity of nickel, interestingly enough, exhibits a negative temperature coefficient up to the temperature of the magnetic transformation (ranging from 340 to 350°C) , beyond which a positive temperature coeffic- ient was observed. The thermal conductivity of chromium, as measured by Powell and Tye (7) after a variety of heat treat- ments shows a negative temperature coefficient for heat treatment above 500°C, but positive temperature coefficients for samples treated at lower temperatures. Thermal conductivity of the Tophel Special sample is much the same as that of the low-temperature chromium at relatively low temperatures, for which data is available. The low temperature data of Sochtig (9) corroborates the order of magnitude of the Powell and Tye data, although their sample had been heat treated at 1000°C. The heat-treated sample of Lucks and Deem (8) , on the other hand, corroborates the high temperature annealed sample data observed by Powell and Tye. 1 73.9 2 73.9 3 73.9 4 73.8 5 73.8 6 73.5 7 74.0 8 188.2 9 187.0 10 186.5 11 188.2 12 188.2 13 187.6 14 338.0 15 337.9 16 339.7 360 6 . Acknowledgement The work described in this paper was conducted as part of the Radioisotope Powered Cardiac Pace- maker Program under the terms of the Contract AT (30-1) -3731 with the Atomic Energy Ccmnission in cooper- ation with the National Heart Institute. Thanks are due to Mr. David L. Purdy and Mr. Thomas F. Hursen of Energy Conversion Division for their encouragement and to Mr. Stanley L. Stein, Mr. Ray G. Burkett, and Mrs. Carol B. Duffman for assembly, drafting and typing respectively. 7 . References (1) F. N. Schofield, "The Thermal and Electrical (5) P. I. Kirichenko, V. E. Mikryvkov, "High Conductivities of Some Pure Metals, " Proc. Temperature," 2_ (2), 176-180 (1964). Roy. Soc. London, 107A, 206-227 (1925) . (6) R. W. Powell, R. P. Tye, M. J. Hockman, Int. (2) R. W. Powell, "The Thermal and Electrical J. Heat Mass Transfer, 8^ 679-88 (1965). Conductivities of Metals and Alloys: Part I, Iron from 0° to 800°C," Proc. Phys. Soc., 46, (7) R. W. Powell, R. P. Tye, J. Inst. Metals, 85, 659-674 (1934) . 185-95 (1957) . (3) R. K. Adams, E. K. Davis, "Smooth Thermocouple (8) C. F. Lucks, H. W. Deem, WADC TR55-496, 1-65, Tables of Extended Significance," VoL II, (1956), (AD97185) . Section 2.1, (Platinum versus Platinum - 10% Rhodium Thermocouple) , ORNL 3649. (9) H. Sochtig, Ann Physik, 5, 38, 97-120 (1940). (4) S. M. Shelton, W. H. Swanger, "Thermal Con- ductivity of Irons and Steels and Some Other Metals in the Temperature Range to 600°C," Trans. Am. Steel Treating, 21, 1061-78 (1933) . 361 LEEDS8N0RTHRUP K-5 POTENTIOMETER VACUUM SYSTEM lAJl^-AJLAAJUUUUlA/ STEP DOWN TRANSFORMER SORENSEN 2501 AC VOLTAGE REGULATOR POWER STAT PRECISION RESISTOR SORENSEN QRC40-30A DC POWER SUPPLY DIFFUSION i PUMP — r-r— 1 I J MECHANICAL PUMP ._! Figure 1 Schematic Diagram of the Apparatus For Measuring Thermal Conductivity of Metals and Alloys 362 CONAX FITTINGS THERMOCOUPLE LEADS ALUMINA CENTERING ROD (FOR EXPANSION) VACUUM SYSTEM RADIATION SHIELDS TOP SAMPLE HEATER TOP GUARD HEATER THERMOCOUPLE SEE DETAIL "A" CENTER GUARD HEATER HEATER BOTTOM GUARD HEATER SUPPORT STRUCTURE ALUMINA SUPPORT ROD BOTTOM SAMPLE HEATER HEATER LEADS DETAIL A SAMPLE CERAMIC SPACER SAMPLE HEATER VOLTAGE TAP HEATER WIRE MULTIBORE CERAMIC SAMPLE GUARD Figure 2 Details of Apparatus For Measuring Thermal Conductivity of Metals and Alloys 363 < o o CM O. X I- O O O X 0.5 0.4 0.3 0.2 0.1 THERMAL CONDUCTIVITY OF T0PHEL SPECIAL vs TEMPERATURE O-UNIDIRECTI0NAL O-BIDIRECTI0NAL A-CHR0MEL P(4) 100 200 300 400 500 600 700 800 TEMPERATURE, °C Figure 3 Thermal Conductivity of Tophel Special Vs Temperature 364 The Intrinsic Electronic Thermal Resistivity of Iron National Physical Laboratory of India New Delhi, India Starting from Linde's empirical observation that, at temperatures where the electrical resistivity of a (normal, non- transitional) metal has a linear tempera- ture dependence, its electronic thermal resistivity is independent of temperature, Backlund has employed a modified Wiedemann-Franz-Lorenz relation to evaluate the intrinsic component (w e „) of the electronic thermal resistivity of iron. This, in effect, uses a Lorenz parameter smaller than the normal value L n . Chari has recently obtained values for the w _ of iron, in the temperature region 600 - 1200 °K, from the usual Wiedemann-Franz-Lorenz relation, using the value of the electronic Lorenz parameter L e , as given by Makinson for a pure monovalent metal having the Debye parameter of iron. Here also, w„ e comes out independent of temperature. In the present paper, a more detailed comparison is made of the two methods. It is shown that, if iron is assumed to contribute one electron per atom to the conduction band, the fractional drops (AL/Ljj) in the effective electronic Lorenz number, used in these two methods are respectively given by 0. 179/1. 17T and (b - a)/ (1 + b - a) , where b = 3£9 2 /(2tt 2 DT 2 ) and a = J 7 /(2tt 2 J 5 ), the symbols £, D, J^ and J_ having the same meaning as in Makinson 's paper. At the Debye temperature 9, the two methods give the same AL/I^. At higher temperatures, the Backlund method gives an effective AL/L proportional to 1/T while Makinson 's method leads to a 1/T 2 variation and an ideal electronic thermal resistivity w which is constant in the temperature range 49 to 29 but increases by about 3% in going from 29 to 39/2 and by about 67„ in going from 39/2 to 9 . At lower temperature, w e „ rises rapidly reaching, at 9/4, a value about twice the constant high-temperature value. This rising trend of w„ e corresponds to the thermal con- ductivity minimum at low temperatures (& 9/5), which is inherent in the standard theory and is presumably due to an over- estimation of the strength of the electron-phonon interaction. Key Words: Electronic thermal resistivity, iron, Lorenz number, resistivity, thermal resistivity. 1. Introduction The component of the electrical resistivity due to the magnetic scattering of free electrons in metals of half-integral spin has been evaluated by Friedal and De Gennes[l] , using the Born approxi- mation and assuming an exchange interaction between the atomic spin and that of the conduction electron. In the paramagnetic state of such a metal, where it is permissible to consider the electron scattering as elastic, they find that the magnetic contribution (p m ) to the electrical resistivity is propor- tional to s(s + 1), where s is the spin of the magnetic atom. Senior Scientist. 9 Figures in brackets indicate the literature references at the end of this paper. 365 Weiss and Marotta[2] have demonstrated this proportionality even in the case of metals having a non-integral spin, using the Van Vleck configuration, replacing s(s + 1) by E p.s ( (sj + 1), where p. is the fraction of atoms in the spin state s ± . They find a surprisingly linear variation of p m with respect to this sum over a wide range of spins, indicating a very narrow range of values for the ex- change coupling constant G in the various transitional metals and their alloys. Assuming the ideal (intrinsic) electrical resistivity p g to be given by the Bloch-Gruneisen expression £1 e/T ( e z - 1)(1 - e" z ) (1) (where 9 is the characteristic Debye temperature and A is a constant of the metal), and the magnetic resistivity p m to be proportional to s(s + 1), they analyzed the electrical resistivity data of Pallister [3] on iron. At the temperatures large with respect to T c and 9, the Bloch-Gruneison term reduces to AT/9 (T being the Curie temperature). Since Pallister 's p - T curve for iron, at such high temperatures, shows the electrical resistivity to have a linear variation with T, they equated the slope of this curve to A/9 and evaluated the ideal electrical resistivity p„. Since the residual electrical resistivity (p ) of iron is negligibly small (compared to p and p m ) , subtracting p„ from the experimental value of p gave p m . The corresponding components of the thermal resistivity of iron were obtained from Powell's data [4] by Backlund[5], using a modified Wiedemann-Franz relation and, by the present author[6], using the appropriate values of the electronic Lorenz number. A detailed comparison is made here between these two methods. Recently, careful measurements on the electrical and thermal resistivities of high-purity iron [7, 8, 9] and Armco iron [8, 9] have been reported, which show that the resistivities at the alpha- gamma transition temperature (1180 °K) present a stem-like singularity. It is pointed out in the present paper that these new data may not radically affect the above mentioned discussions (which were based on the old data of Pallister[3] and of Powell[4]). 2 . The Lorenz Number The early theories of metallic conduction were based on a classical concept of the collision pro- cess between the conduction electrons and the atomic lattice and predicted a Lorenz number L having a universal constancy. The modern Quantum theory of metals, on the other hand, recognizes the inelastic nature of the electron-phonon interaction, requiring an exchange of the phonon energy in the scattering process. Thus, at temperatures T, which are much higher than the Debye temperature 9, where the maxi- mum phonon energy k9 would be small compared to kT, the electronic Lorenz number L e has the temperature- independent value L^. This applies also at low temperatures where elastic impurity scattering is predominant. At intermediate temperatures where the energy of the phonon absorbed or emitted is not negligible compared to kT, the electronic Lorenz parameter falls below the value 1^. Starting from Wilson's expression[ 10] for the ideal (intrinsic) electrical resistivity and the ideal electronic thermal resistivity of a metal, Makinson[ll] obtained the following expression for L e of a metal, assuming the lattice waves to be those of an isotropic continuum having an upper frequency limit: L f> 4A V9; 2 (2) n 4A + V where ■(?•?)- (D h {• ♦*&)'}-* '7]. and 9/T J z n dz ... (e z - 1)(1 - e z )- W 366 Here the symbols Q, D, X, Jy and J~ have the same meaning as in Makinson's paper: Q is the Fermi energy, £/D - (2N a )1'3, where N a is the number of conduction electrons per atom, p /4A is the impurity parameter, p being the residual electrical resistivity, and A/9 is the slope of the ideal electrical resistivity p g at temperatures T/9 > 0.6. For an ideally pure metal, for which p /4A = 0, eq (1) reduces to L e /[ 1+ ^dS) 2 -F g]=d + b-a) (5) The drop in the value of L below the value L at the temperature T is given by L = L n " L e = (b _ a) ^i/ (1 + b " a) < 6) 3. The Two Methods for Deriving the Ideal Electronic Thermal Resistivity 3.1. The Bases of the Two Methods The component w of the electronic thermal resistivity of iron, due to phonon scattering has been obtained by two methods, both starting from the value of the corresponding component p„ in the electrical resistivity of iron. One method, due to Backlund[5] is based on the fact that the Block-Gruneison expression (eq 1) leads to a reduced resistivity r (= p^/pg) of a metal being a universal function of the reduced temperature t (= T/9). At temperatures above, say 9/3, this function is a straight line given by [12]: r = -0.17 + 1.17t (7) so that dp/dT is 1.17 Pg/9. Following Linde's observation[ 13] for normal (non-transitional) metals that w e _, the phonon-scattering component of the electronic thermal resistivity, is independent of temperature as long as the electrical resistivity has a linear temperature dependence, Backlund[5] has evolved a modified Wiedemann-Franz relation w eg - 9 , the T~ variation of AL/Ljj is maintained, but below 9, the variation is less rapid, as in the case of N a = 1. This latter behaviour is in accordance with the partial filling up, at low N a , of the theoretically predicted electronic thermal conductivity minimum. 3.3. The Instrinsic Electronic Thermal Resistivity Backlund's analysis[5] starts with the assumption of a constant value for w eg in the temperature region of a linear p - T curve. His modification of the Wiedemann-Franz relation makes sure of this down to about 120 °K for iron. The value of w e „ obtained by us, following Makinson's theoretical evaluation of Lorenz number, is sensibly constant in the temperature region 49 to 29 . For N a = 1, w eg increases by about 3% for the temperature drop from 29 to 39/2 and by about 6/i between 39/2 and 9. Below 9 it rises rapidly, reaching at about 9/4 a value about twice the high temperature value. This rising trend of w eg cor- responds to the thermal conductivity minimum at low temperatures (?» 9/5) which is inherent in the standard theory of Wilson and Makinson and is presumably due to an overestimation of the strength of the electron-phonon interaction. For N a = 0.7, (it is immaterial whether or not the impurity parameter p /4A is taken into account) the value of w e „ at high temperatures is not affected, but now the rise is only about 50% for the temperature drop to 9/4. This again corresponds to the filling up, at small N„, of the theoretically predicted minimum in the thermal conductivity. 4. The Alpha-Gamma Transformation More recent measurements on the conductivities of iron[7, 8, 9] show a step-like anomaly at the temperature Tq^, of the transformation of iron from the magnetically disordered body-centered cubic (b.c.c.) to a face-centered cubic (f.c.c.) structure. The slope of the p - T curve above T c is changing and is different above and below Tq^ . This renders questionable the validity of the original analysis of Weiss and Marotta, who made use of the linear p - T curve above T c . In this connection, we wish to made the following comments. The slope of the p - T curve changes rapidly between the Curie temperature T c and Tq^ , getting smaller as we approach Tq^ from T c . Fulkerson et aj.[8] derive slopes of 3.95, 4.05, and 3.73 X 10" 10 Qm/ c K for the ORNL Armco iron, ORNL high purity iron and USS high purity iron respectively in the alpha region, and 3.74, 3.69, and 4.00 X 10"-'-^Om/ o K in the gamma region. Actually, since the temperature interval T to T^y is rather small, we have to visualize a true slope somewhat smaller than these values. Since the b.c.c. structure of ironT8j again becomes stable above 1680 °K, Coles[l6] has deduced a slope of 3 X 10"10Qm/°K from the resistivity values just above T c and those in the 6 - iron region. This seems reasonable and agrees with the value obtained by Weiss and Marotta from the old data of Pallister[3] , which does not re- cord the resistance drop at L^y. Whereas the use of dp/dT = 3 X 10"-'-"Qm/ c K results in a p s (which is the constant value of magne- tic resistivity above T c ) of 0.8u,Qm, higher values of dp/dT of say 4 or 5 X 10" 10 Qm/°K would lead to ss 0.65 and 0.54p,Qm respectively. These latter values for p s would be somewhat smaller than the value 0,75 which latter would conform to a linear variation of $ s with S p i s.(si + 1). The trend of the rest of the above discussions is not disturbed by such higher values of dd/dT, though the numerical values certainly change. We might add that Koveskiy and Samsonov[l7] have given a constant dp/dT of 4.79 x 10" 10 Qm/°K, with no step at T^. 368 To which component of the electrical resistivity is the step at T^y to be attributed? It was shown by Weiss and Tauer[l8] from the specific heat data on iron, that the Debye temperature is about 420 °K in the alpha phase and about 335 °K in the gamma phase. Since we have accepted for dp/dT, the limiting value, away from the temperatures of transition, it would imply our assuming a value for p s which is the same for the alpha and gamma phases. If this argument is correct and the Debye tempera- tures of the two phases are taken into account, we can use the Bloch relation p « RT/N a 9 (where R is the Fermi radius) and derive the ratio of the conduction electrons per atom in the two phases as Ny/1^2 = 1.6. Such an estimation has been indicated by Fulkerson et a\ who also discuss in detail the step in the thermal conductivity of iron at Tq.. 5. References [1] Friedel, J. and De Gennes, P, , Anomalies de resistivite dans certains metaux magnetigues, J. Phys. Chem. Solids. 4, 71 (1958). [2] Weiss, R. J. and Marotta, A. S., Spin-depen- dence of the resistivity of magnetic metals J. Phys. Chem. Solid. 9, 302 (1959). [3] Pallister, P. R. , The specific heat and resistivity of high purity iron up to 1250 °G. J. Iron Steel Inst. .161, 87 (1949). [4] Powell, R. W. , Further measurements of the thermal and electrical conductivity of iron at high temperatures, Proc. Phys. Soc. (Lond), 51, 407 (1939). [5] Backlund, N. G., An experimental investiga- tion of the electrical and thermal conduc- tivity of iron and some dilue iron alloys at temperatures above 100 °K, J. Phys. Chem. Solids. 20, 1, (1961). [6] Chari, M. S. R. , The electrical and thermal resistivities of iron. Phys. Stat. Sol. _19, 169 (1967). [7] Arajs, S. and Colvin, R. V., Electrical resistivity of high purity iron from 300 to 1300 °K. Phys. Stat. Sol. 6, 797 (1964). [8] Fulkerson, W. , Moore, J. P. and McElroy, D. L. , Comparison of the thermal conductivity, elec- trical resistivity, and Seebeck coefficient of a high-purity iron and an Armco iron to 1000 °C., J. Appl. Phys. 37, 2639 (1966). [9] Laubitz, M. L. , Thermal and electrical properties of iron at high temperatures. Can. J. Phys., 38, 887 (1960). T10] Wilson, A. H, , Second-order electrical effects in metals., Proc. Camb. Phil. Soc. 33, 371, (1937). [11] Makinson, R. E. B. , The thermal conducti- vity of metals., Proc. Camb. Phil. Soc. 34, 474, (1938). [12] Borelius, G. , Temperaturabhangigkeit des- widerstandes ferromagnetischer Metalle. Ann. Phys. LpZ. , 8, 261 (1931). [13] Linde, J. 0., An investigation of the valid- ity of the Wiedemann-Franz-Lorenz Law., Ark. Fys. 4, 541 (1952). [14] Borelius, G., On the connection between electrical resistivity and potential energy in pure metals., Ark. Fys., 11, 291 (1956). [15] White, G. K. and Woods, S. B. , Electrical and thermal resistivity of the transition elements at low temperatures. Phil. Trans. Roy. Soc. (Lond). A251 , 273 (1958). [16] Coles, B. R. , Spin-disorder effects in the electrical resistivities of metals and al- loys. Adv. Phys. 2. 4° (1958). [17] Kovenskiy, I. I. and Samsonov, G. V., Electrical resistivity of certain transi- tion metals at high temperatures., Phys. Met. Metallegr., 15, 124 (1963). [18] Weiss, R. J. and Tauer, K. J., Components of the thermodynamic functions of iron. Phys. Rev. H)2, 1490 (1956). 369 The Thermal Conductivity and Electrical Resistivity of Tungsten-26 Weight Percent Rhenium A. D. Feith 1 Nuclear Materials and 'Propulsion Operation General Electric Company Cincinnati, Ohio 1+5215 Radial heat flow measurements of the thermal conductivity of W-26Re (w/o) have been made over the temperature range from 300 to 2200 C. These data identify a small positive temperature coefficient on conductivity with conductivity values ranging from 56.0 WnT 1 deg -1 at 300 °C to about 69.0 Win -1 deg -1 at 2200 °C . Electrical resistivity measurements are also reported for both W-26Re and for unalloyed tungsten. This information was employed to calculate the electronic con- tribution to the thermal conductivity of the W-26Re alloy. At 300 C the electronic contribution was calculated to be about 655» of the thermal conductivity, whereas at 2200 C the conductivity is about 90% electronic. Key Words: Conductivity, electrical resistivity, electronic conduction, phonon conduction, thermal conductivity, tungsten, tungsten-26 rhenium. 1. Introduction Alloys of tungsten and rhenium have been found to be of considerable interest for potential structural applications at extremely high temperatures. In addition, these alloys have been used extensively as thermocouples with pure tungsten for high temperature measurements. Although much has been reported on the thermal electric output of W-26Re with pure tungsten as a thermocouple, little or no work has been reported on either the thermal conductivity or electrical resistivity of this alloy in the high temperature range (> 1200 C). The data presented in this report, therefore, fill a vital gap in the knowledge of the thermal properties of this important alloy in the tungsten-rhenium system. They also serve to verify data presented by Jun and Hoch (l) 2 on a similar alloy (W-25Re) but measured by a different method. The general agreement with the predicted values of McElroy (2) suggests that some confidence can be placed in the estimation of fairly reliable conductivity data for an alloy over a broad temperature range when electrical resistivities over this range are known. 2. Description of Specimens 2.1. Thermal Conductivity The thermal conductivity specimens (W-26Re) and the guard discs (tungsten) were obtained commer- cially. A powder-metallurgy process was used to fabricate discs which were then machined to the con- figurations shown in figure 1. The composition of the as-received material is given in table 1. Measured densities were 98.96% of theoretical for the specimen and 98.3$ (on the average) of theoreti- cal for the guards. 2.2. Electrical Resistivity Electrical resistivity specimens were made from the same material used in making the thermal con- ductivity specimens and also from powder-metallurgy tungsten. For these specimens though, circular rods, 0.32 cm in diameter by 11.1*3 cm in length were employed. Small holes which were required to accommodate voltage probes were generated by an EDM process. 'Principal Engineer 2 Figures in parentheses indicate the literature references at the end of this paper. 371 Table 1. Composition of specimens in percent by weight Tungsten-Rhenium Tungsten % rhenium 26.1*8 % tungsten 73.39 (by difference) % impurities % impurities Mo 0.01 - 0.10 Mo 0.01 - 0.10 Fe <0.01 Fe <0.01 Na <0.01 Na <0.01 Ti <0.01 Maximum impurities <0.12# Ni <0.01 Maximum impurities <0.13# 2.3. Microstructures of Specimens Typical microstructures of the materials used are shown in figure 2. X-ray diffraction studies have shown the alloy material to be a single-phase solid solution with a theoretical density of 19,575 kg/m3; also, a density of 19,199 kg/nP was determined for the unalloyed tungsten. 3 . Apparatus 3.1. Thermal Conductivity Thermal conductivity measurements were obtained in the radial heat flow apparatus which has been described previously (3,^,5). However, one important difference is noteworthy. Because of the high cost of material containing significant amounts of rhenium, some thought was given to the use of lower cost guard materials. In this study the guards were made from tungsten and three guard discs were employed above and below the W-26Re specimen disc. In addition, two discs of tungsten with flanges at the I.D. and O.D. were placed at each interface between the specimen and adjacent guard disc as shown in figure 3. This thermal isolation of the specimen was sufficient to restrict the heat flow to the radial direction in keeping with the boundary conditions of the experiment. In general, this procedure was proven to be satisfactory as indicated by the reproducibility of the data obtained. Temperatures were measured with a probe-type thermocouple (Pt/Pt + lORh) up to 1200 C and an optical pyrometer from 1000 C to the maximum temperature of the measurement, 2200 C. Again, as in the case of the molybdenum measurements (5), the radial thermal gradient was small (3 C to 20 C) and great care had to be exercised in the temperature measurement to obtain reasonable precision in the results. 3.2. Electrical Resistivity Electrical resistivity measurements were made on rod specimens (0.32 cm dia. x 11.1*3 cm long) by a four-probe method as described previously (5) except that two specimens were placed in series as shown schematically in figure h. One W-26Re specimen was used in conjunction with an unalloyed tungsten specimen. In this manner, electrical resistivity was measured on the two materials simul- taneously. A probe thermocouple was used to determine the temperature of the specimens up to 1200 C; above this temperature, an optical pyrometer was employed. Voltage probes were of W-26Re for the alloy specimen and tungsten for the unalloyed tungsten specimen. Verification of the temperature level and possible gradient along the specimens was obtained by measuring the thermal e.m.f. between the W and W-Re voltage probes when no external power was supplied to the specimens. Measurements at liquid nitrogen temperature (-196 C) were also made on as-received and tested specimens. h. Results k.l. Thermal Conductivity The results of the thermal conductivity measurements are shown in figure 5 and in table 2. The initial measurements (series 1, table 2) were made from 300 to 1600 C. After an interruption of about two months, during which time the specimen and guards were removed from the apparatus, the second series of measurements was made on the same specimen from 1300 to 2350 C and then down to 1550 C on final cooling. In both series, reproducibility was demonstrated by repeated measurements over the temperature ranges studied. It can be seen in figure 5 that the conductivity increases with temper- ature over the entire range although the slope decreases at higher temperatures. The agreement with the values reported by Jun and Hoch (l) is quite good considering the different methods employed. The predicted values of McElroy (2) up to 1500 C also appear to be verified by these results. McElroy (2) 372 shows two sets of values in his prediction based on two different assumptions for the lattice component. Both values are shown on the curve to demonstrate the agreement between these predicted values and the experimental results of this study. Table 2. Thermal conductivity data for W-26Pe Wm -1 deg -1 Series 1 Series 2 Run 1 Run 2 Run 3 Run 1 Run 2 T, U C X T, °C X T, "C X T, "C X T, U C X 308 kk.6 551 57.0 1268 67.1 1335 63.5 1685 66.6 UOU kk.l 578 51*. u 1360 58.6 1522 62.2 1708 65.9 507 58.3 781 68. k 1381 57-5 165U 63.5 1779 67.7 673 62.8 8U8 66.0 1392 61.8 161*1* + 63.7 1802 67.8 7^9 59.1 869 63.6 1566 58.1 1527+ 65.3 1999 62.7 81*5 61.7 932 65.3 1593 55.5 2008 66.7 759+ 62.3 9fc8 65.8 201*0 66.1* 728+ 60.lt 998 66.0 2021+ 61*. 65U+ 58.7 1021* 69.0 1910+ 67.1* 535+ 55.3 1082 69.7 2008 67.9 1*13+ 56.0 IIU7 11U8 1210 11U6+ 1120+ 69.2 63.3 63.6 63.U 55.8 2156 2185 2251* 231*9 1816+ 1657+ 1675 + 1679+ 15l*2+ 70.0 69.3 68.1+ 73.1* 68.8 68.6 71.9 66.1* 67.1* + Indicates measurements during cool down. Jun and Hoch (l) have reported two distinct values for the thermal conductivity of W-25Re both of which are independent of temperature and which depend upon the method of fabrication; i.e., 62.6 Wm~l deg -1 for powder-metallurgy material and 68.0 Wm -1 deg -1 for arc-cast material in the 1500 - 2700 K range. 1*.2. Electrical Resistivity Results The measured values of the electrical resistivity of W-26Re and unalloyed tungsten are presented in table 3 by individual runs. -These data are combined to give the curves in figure 6. Second degree polynomial equations were developed based on these data. The equation for tungsten-rhenium is con- tinuous and fits well within ±1.5% up to about 2370 K with few exceptions; however, the precision is about ±6? for the few data points above 2370 is: K. The equation for the electrical resistivity of W-26Re 7 io — 1 1* ? p = 1.861*30 x 10"' + 3.37090 x 10 T - 1.291*57 x 10 T = 2.1*3036 x 10" 9 (2m (1) where T is in K, p is in Dm and o is the standard deviation. Results of this study are somewhat higher than those reported by McElroy (2). This shift is probably due to differences in the materials used in the two experiments. In the ORNL study small diameter tungsten-26 rhenium thermocouple wire (6) was used which may have had a slightly different composition, whereas the material used in this study was a 0.32 cm diameter rod and is characterized by the composition given in table 1. The values for tungsten were fitted more precisely when two second degree polynomial expressions were used rather than one. This is similar to the method proposed by Moore, et al. (7), and is mainly due to the desire to restrict the expression to a second degree polynomial rather than to suggest a physical change in temperature dependence of resistivity. The equations are: 373 Table 3. Electrical resistivity of tungsten-26 rhenium and tungsten T, °C W-26Ren fim x 10 W to x 10 30 28.691 5.525 216 3U.705 9.893 225 35.023 10.391 680 1*9. 610 23.213 32 28.292 5.302 23*+ 35.297 10.321+ 573 1+6.309 19.722 1062 59.982 35.1+20 30 27.290 5.253 907 56.654 31.282 1308 67.056 1+3.312 1615 76.598 52.891 31+ 29.019 5.605 28 28.955 5.551+ 299 37.679 12.002 7*+2 51.378 25.159 752 51.818 25.157 982 58.990 33.111 762+ 52.61*2 25.81+5 307 + 37.937 12.235 1213 66.416 I+O.562 1968 88.356 65.928 1978 88.559 66.608 2196 9U.U9U 69.626 2120+ 91.U92 68.276 1955+ 86.360 61*. 990 1209+ 65.725 40.368 852+ 55.097 28.901 29 28.692 5.539 803 53.616 27.281 1122 63.027 37.562 1503 73.79 1 * 1+9.51*3 185I* 8U.312 61.1+1*8 2097 90.827 68.551 21 5.327 A.R. 2k 5.1+52 P.T. 25 28.239 A.R. 25 28.275 P.T. -196 LN„ -196 d 19.1+86 A.R. 0.7029 A.R. 19.686 P.T. 0.6276 P.T. + Indicates measurements during cool down. A.R. - As-received P.T. - Post-test 1*. 33471 x 10 1 T 2 + 2.19691 x 10~ 10 T - 1.61*011 x 10 (300°K < T < 12l+0°K) (2) = 3.03318 x 10 9 Qm and p = -1*. 06012 x 10 -lli T 2 + 1*. 67093 x 10 10 T - 1.97071 x 10 7 (I2l+0°K < T < 2570°K) (3) c = 1.56328 x 10 Qm 374 where p is in fim, T is in K and a is the standard deviation. Numerous resistivity studies have been made on unalloyed tungsten over the past four decades. Moore, et al. (7) , have shown that the values of most other investigators are within ±h% of their work; however, each investigator is consistently above or below their results up to lUOO C. The results of this study are within ±1% of the Moore data over the 200 to ll+00 C range. 5. Discussion of Results 5.1. Thermal Conductivity The increase in thermal conductivity with temperature for refractory alloys has been shown to be typical (2) even though the constituents of the alloy, when considered alone, have a decreasing value with temperature. When separating the total thermal conductivity into two components, phonon conduc- tion (A.) and electronic conduction (A ), it can be shown by the Wiedemann-Franz-Lorenz law that the flow of heat by electrons is the major component and, therefore, has the greatest influence on the temperature dependence of the thermal conductivity, tivity is: An equation for total (measured) thermal conduc- X T = X e + h LT A+BT w where A = thermal conductivity due to electrons A = thermal conductivity due to phonons total conductivity Lorenz number = constant and T = P = 2.1+1*3 x 10" 8 V 2 /deg 2 absolute temperature , U K electrical resistivity in Qm A and B are constants. Since L is virtually constant within a small range of values, it is generally considered constant for the purpose of discussion. An analysis of the various components of conductivity is shown in table h and figure 7. The A e curve has about the same characteristic shape as the total conductivity curve while the A^ curve has an inverse temperature dependence (l/T) similar to the behavior of typical insulators in which the bulk of the conductivity is by phonons. Table k. Components of thermal conductivity of W-26Re Wiri - -*- deg - -*- Temp., A A» *m C e I T 300 37.3 20.2 57.5 1*00 1*0.1+ 17.6 57.9 500 U3. 15.6 58.6 600 U5. 3 ll+.O 59.3 700 1*7.3 12.7 60.0 800 1+9.2 11.6 60.8 900 50.8 10.7 61.5 1000 52.3 9.9 62.2 1100 53.7 9.2 62.9 1200 55.0 8.6 63.6 1300 56.1 8.1 61*. 3 ii»oo 57.2 7.7 61+. 9 1500 58.3 7.3 65.5 1600 59.2 6.9 66.1 1700 60.2 6.6 66.7 1800 61.0 6.3 67.3 1900 61.9 6.0 67.9 2000 62.7 5.7 68.1+ 2100 63.5 5.5 69.O 2200 61+. 2 5.3 69.5 2300 61+. 9 5.1 70.0 375 A computer program (see Appendix) was used to fit the data to an equation of the form above, result obtained is: The 2.1*1*3 x 10 -8 T 7.1*925 x 10 3 + 7.31+269 x 10" >T J (5) where X m = Wm dee To T = K p = electrical resistivity eq (l) c = 1*. 58183 Wm -1 deg -1 5.2. Electrical Resistivity The electrical resistivity of the W-26He measured in this program is approximately 8% above the values reported by McElroy (2) at ll*00 C. Since McElroy did not report the composition of the material, an explanation for this difference cannot be made. However, the data presented herein have been shown to be reproducible and hence have been used in the section above as an aid in interpreting the thermal conductivity results and to show that the electronic heat conduction is the major portion of the total conductivity. Resistivity data obtained for unalloyed tungsten in this study fall in a band of values along with those of other investigators as shown in figure 6. The high temperature portion defined by eq (3) is concave downward since the second derivative is negative. This appears to contradict the work reported by Gumenyuk and Lebedev (8) and also that of Worthing, et al. (9). It is not known, at this time whether this is real or whether it is caused by the increased scatter in the high temperature range. R. P. Tye (10) reports resistivity data on pure tungsten to ll*50 C which has a considerably stronger temperature dependence and therefore greater departure from the work of this study and others (8,9). work reported by Moore, et al. (7), to ll*00°C gives two second degree polynomials with positive srivatives. Their values are within ±1% of this study in the 200 to 11+00 C temperature range. The second derivati However, Moore corrected his measured values for thermal expansion using an average coefficient of ex- pansion of 5 x 10~6 °C -1 . This correction amounts to an increase of about 0.7$ at ll+00 C which does not alter the relative values significantly. At 2200 C the increase in resistivity amounts to 1.5% for all three curves shown when correcting for the thermal expansion. 6. Conclusions The thermal conductivity of tungsten-26 w/o rhenium has been shown to be primarily due to thermal transport by electrons. The lattice or phonon portion of the thermal conductivity varies from approxi- mately 35$ at 300 °C to 10$ at 2200 °C. Predictions of thermal conductivity by McElroy (2) appear to have been verified. Electrical resistivity values for W-26Re and tungsten have been shown to be reproducible; compari- son of the tungsten values with those of Moore, et al. (7), and others shows good agreement. 7. References (1) Jun, C. K. and Hoch, M. , "Thermal Conductiv- ity of Tantalum, Tungsten, Rhenium, Ta-lOW, T-lll, T-222, W-25Re in the Temperature Range 1500° - 2800 °K," Third International Symposium on High Temperature Technology, Stanford Research Institute, Sept. 17-20, 1967, to be published. (2) McElroy, D. L. , "High Temperature Materials Program Quarterly Report for Period Ending April 30, 1966," ORNL-TM-1520 , Part 1, PP. 53-58. (3) Feith, A. D., "A Radial Heat Flow Apparatus for High Temperature Thermal Conductivity Measurements," GE-NMP0, GEMP-296 , August 1963. (1*) Feith, A. D., "Thermal Conductivity of Several Ceramic Materials to 2500 C," Advances in Thermal Physical Properties at Extreme Temperatures and Pressures, ASME, 1965 (5) Feith, A. D., "Measurements of the Thermal Conductivity and Electrical Resistivity of Molybdenum," GE-NMPO, GE-TM 65-10-1, October 1965. (6) Fulkerson, W. , private communication, September 15, 1965. 376 (7) Moore, J. P., Graves, P. S., Fulkerson, W. and McElroy, D. L. , "The Physical Properties of Tungsten," 1°65 Conference on Thermal Conductivity, Denver, Colorado, October 1965, by permission. (8) Gumenyuk, V. S. and Lebedev, V. V., "Inves- tigating Heat and Electro-Conduction of Tungsten and Graphite at High Temperatures," Fizika Metallov and Metalovedeniye (2 Jan. 1961), Nr. 1, pp. 29-33, Translation USAEC-NP-TRT33. (9) Worthing, A. G. and Watson, E. M. , "Resist- ance and Radiation of Tungsten as a Function of Temperature," J. Opt. Soc . A., Vol. 2k (111*), 193U. (10) Tye, R. P., "Preliminary Measurements on the Thermal and Electrical Conductivities of Mo, Nb, Ta, and W," from "Niobium, Tantalum, Molybdenum, and Tungsten," A. G. Quarell, Editor, Elsevier Pub. Co., Amsterdam, 196l. 8 . Appendix Procedure used to derive the thermal conductivity equation for W-Re. The desire to fit the thermal conductivity to an equation relating it to the electrical resistivity and to include a term with the proper temperature dependence for phonon conduction required that special data reduction procedures be developed. The fact that the precision in the data appeared to vary as a function of temperature suggested that a method employing weighting functions be used. The electrical resistivity data were treated first for an analysis of variance. This showed the variance to have a temperature dependence which could be correlated with the experiment . The characteristic of the furnace and power controller is that temperatures are maintained very well at high temperatures (> 1000 C) but relatively poorly at lower temperatures. However, the un- certainties in the measurement of temperature increase at high temperatures. At very high tem- peratures (> 2000 C) the voltage drop measurements were not as steady as at lower temperatures. This statistical approach is used to obtain a weighting function which takes into account the experimental confidence in the data. In this manner the second degree polynomial for resistivity as a function of temperature was derived . The analysis of variance of the thermal conductivity data showed it to be independent of temperature and, therefore, all data were weighted evenly. This may appear inconsistent with the statement above on electrical resistivity; however, when considering the difference in the magnitude of the error in the measurements in the two properties, ±1.5% for resistivity compared to ±8% to ±10$ for conductivity, one can see that the precision in any one property would be sensitive to test condi- tions whereas the other may not. Using the equation for resistivity of W-Re as determined above and starting with the classical Lorenz constant, the constants A and B were determined from the thermal conductivity data. Then L was increased in steps and new values of A and B were obtained. A final analysis to minimize the standard deviation while forcing L back to the classical constant yielded the reported equation. 4 hol.i. 0.065 • 0.00S din., 0.38 do.p GE-NMPO 1.686 • 0.010 dio. 0.686 • 0.010 dio All dimension, in inches Upper guard 1.985 • 0.015 dio., typ.' 0.375 • 0.00S dio., lyp.' Specimen Lower guard Figure 1. Design of thermal conductivity specimen and guards. 377 Tungsten W-26Re Figure 2. Microstructures of W-26Re and unalloyed tungsten. Specimen Figure 3. Thermal conductivity specimen and guard assembly. 378 Reversing switch 1.5 rin L i T T Selector switch Standard cell K-2 batteries r — i r-1'IH'M'h L & N I ight beam L & N K-2 potentiometer galvanometer Figure k. Schematic of electrical resistivity measurement system. 0> E V 3 -o C O *$±jrjj:£r § This study -- Jun and Koch-powder metallurgy — Jun and Hoch-arc cast V ORNL new estimate A ORNL old estimate I I I 400 600 800 1000 1200 1400 1600 1800 2000 2200 2400 Tempero»-TS, °C Figure 5. Thermal conductivity versus temperature for W-26Re alloy. 379 100 90 80 70 x E C 60 > '•5 50 40 30 20 10 A tk i y W-26Re ' / & // / / / // /.' // /** A '/ O / // // / ' // // ft // // // k n-ion« < This work ORNL (2) w — This work O ORNL (7) O TYE (10) A Worthing (9) V Gumenyuk (8) 1 1 1 1 1 400 800 1200 1600 2000 2400 2800 Temperature, °C Figure 6. Electrical resistivity versus temperature for W-26Re alloy and unalloyed W. 70 60 7 50 E ~ 40 Z 30 | 20 10 ' Xt r x. X T= X e +X^ I43xl0 e T , 1 P + 7 4 92S«!0" 3 + 7.34269 «io- s i ^t 400 600 800 1000 1200 1400 1600 1800 2000 2200 Temperature, °C Figure "J. Components of thermal conductivity vs temperature for W-26Re. 380 Thermal Conductivity of Magnesium Stannide J. J. Martin, H. R. Shanks and G. C. Danielson Institute for Atomic Research and Department of Physics Iowa State University, Ames, Iowa 50010 The thermal conductivity of the semiconductor Mg^Sn has been de- termined in the temperature range 80 to 700°K from measurements of the thermal diffusivity. At low temperatures the thermal conduction is entirely lattice conduction, as shown by our earlier measurements from 4 to 300°K. At higher temperatures, however, our results show that bipolar -electronic conduction (diffusion and recombination of electron-hole pairs) becomes im- portant and actually equals the lattice conduction at 700°K. The difference between our measured thermal conductivity and the lattice thermal conduc- tivity is in good quantitative agreement with the calculated bipolar -electronic thermal conductivity, and this agreement gives strong support to the theory of bipolar conduction in semiconductors. Key Words: Bipolar conductivity, lattice conductivity, magnesium stannide, semiconductor, thermal conductivity, thermal diffusivity. 1. Introduction MgpSn is a II-IV compound semiconductor with the antifluorite structure and is a member of the Mg£X family of compounds where X can be Si, Ge, Sn or Pb. Elastic constants and calculated lattice vibration frequencies have been reported by Davis et al. [l] . The electrical properties of Mg2Sn have been reported by Winkler [2] and others [3-517 Umeda [6] and Crossman L7] report that the electronic structure of Mg£Sn consists of ellipsoids along the [lOO] directions. Busch and Schneider [8] have reported thermal conductivity data on MgpSn from 77 to 400°K. Recently, Martin and Danielson [9] measured the thermal conductivity of several Mg^Sn samples from 4. 2 to 300°K. These results indicated that below 300°K lattice conduction is completely dominant in Mg^Sn; the thermal conductivity is proportional to T"' from 150 to 27 5°K. Above room temperature Mg£Sn is completely intrinsic and the lattice conductivity is small (0. 075 W/cm-deg at 300°K); hence, the elec- tronic contribution should be a large part of the total thermal conductivity. Therefore, we have ex- tended our measurements by a thermal diffusivity technique to 700°K in order to determine the elec- tronic thermal conductivity of Mg2Sn. 2. Experimental Procedure The single crystal sample was grown by a modified Bridgman technique. The sample was grown from 99. 999% purity tin and from magnesium distilled under high vacuum in this Laboratory from 99. 99% purity starting material. The sample was n-type. Figure 1 shows the electrical re- sistivity data taken on the diffusivity sample before and after the thermal diffusivity measurements. Figure 1 also shows the Hall coefficient of the sample that was measured prior to the thermal diffu- sivity measurements. The Hall data indicates that the sample contained approximately 2. 5 X 10*-" cm"3 uncompensated donors before the thermal diffusivity measurements were made. The change in the electrical resistivity data after the diffusivity measurements were completed indicates that the donor concentration increased to about 3 X 10*" cm" . The sample dimensions were 4. 30 X 4. 24 X 20mm. We have developed new instrumentation so that thermal diffusivity data may be taken on short samples by our finite difference method. The new data recording apparatus is similar to our old gear [10] except that Astrodata model 120R amplifiers are used for D. C. amplification and a four trace oscilloscope is used to record the temperature versus time curves. Typical pulse times are 5 sec. Figures in brackets indicate the literature references at the end of this paper. 381 The sample was mounted in a sample holder which contained Pt versus Pt-10% Rh and Cu versus constantan thermocouples for temperature measurements above and below room temperature. Three Cu wires were spark welded to the sample at distances of 5, 7. 5 and 10mm from the resistance wire heater wound on one end of the sample. A fourth Cu wire was spark welded to the opposite end of the sample for use as a reference lead. The three Cu versus Mg2Sn versus Cu reference differential thermocouples were used to measure the three temperature versus time curves in the sample. The data were taken in a cryostat-furnace which could cover the temperature range 80 to 700°K. 3. Results Our thermal diffusivity results are shown in fig. 2. The curve represents the best fit to the data and was used to calculate thermal conductivity values. The thermal conductivity was calculated from the smoothed thermal diffusivity, the x-ray density [ll] of 3. 592 gm/cm and the specific heat. The specific heat values of Jelinek et al. Ll2] were used between 80 and 300°K, and the specific heat values of Gschneider [13] were used Between 300 and 700°K. The thermal conductivity curve is shown in fig. 3 along with our low temperature thermal conductivity data and the earlier data of Busch and Schneider [8]. Between 80 and 300°K our new data is in good agreement with our old thermal con- ductivity data and with the data of Busch and Schneider. 4. Discussion In a semiconductor, heat may be carried by the lattice (phonons) and by the charge carriers (electrons and holes). The total thermal conductivity, K, can be expressed as a sum of the lattice conductivity, Kr, and the electronic thermal conductivity, K e ; K=K e+ K L . (1) In order to determine K we must subtract K-^from the total thermal conductivity. From 150 to 275°K the thermal conductivity of Mg^Sn goes as T"-*- which is characteristic of three phonon scattering. Since the Debye temperature is about 200°K and three phonon scattering processes will continue to dominate the lattice conductivity at higher temperatures, we expect the T temperature dependence to continue. Mg?Sn becomes completely intrinsic near 300°K so we expect the electronic thermal con- ductivity to become important above 300°K. Figure 4 shows a plot of the thermal resistivity of our sample (K~l) versus T. The line represents the K~* ~ T extrapolation of our data. We will use this line as our high temperature extrapolation of the lattice conductivity. The electronic thermal conductivity, K was obtained by subtracting the lattice conductivity, K L , from the total thermal conductivity. At 700°K, K w K.. If we assume acoustic mode scattering of both the electrons and holes, the electronic thermal conductivity can be expressed as K e ( k )2 | 2+ (4+AE/kT) npb| gT- (2) 6 L (bn+p^ } Ineq(2) k is Boltzmann's constant, e is the electronic charge, AE is the energy gap at temperature T, b is the mobility ratio ^ e //i n , n and p are the electron and hole concentrations, and O is the electrical conductivity. The first term expresses the ordinary Wiedemann-Franz law for a non-degenerate material. The second term represents the transport of the recombination energy of the electrons and holes and is called the bipolar contribution to the electronic thermal conductivity. The bipolar term is usually much larger than the Wiedemann-Franz term because AE/kT >> 1. Winkler [2] found an energy gap of (0. 36-3 X 10" T)eV for Mg2Sn. The ratio, b, of the electron mobility to the hole mobili>ty in the intrinsic range was found to be 1. 20-1. 25 by Lipson and Kahan [14] and 1. 23 by Blunt et al. [3]. For the calculation of K e we shall use b = 1. 23. In the intrinsic region (T > 300°K for Mg2SnT~n=p. The solid line in figure 5 shows the electronic thermal conductivity cal- culated from eq 2. The electrical conductivity measured on our sample was used in the calculation along with the energy gap of Winkler [2] and b = 1. 23. The calculated and measured electronic ther- mal conductivities agree to within 3mW/cm-deg which is about 5% of the total thermal conductivity. 382 5. Summary UU CLL L11C1 111(11 LUllUULLiVlLy X ^ O HZ L3 . AZ X U1II 1, ^ W — £j | _^ J. ^ , Ulb U1C.1 lliai V,UltUuC 11 VI IV gO^O CL © X W11LLJ1 characteristic of high temperature lattice conductivity. The electronic thermal conductivity which was determined by subtracting the extrapolated lattice conductivity from the measured thermal con- ductivity is in good agreement with the calculated electronic thermal conductivity. At 700°K the elec> tronic thermal conductivity was equal to the lattice thermal conductivity. 6. References [l] Davis, L. C. , Whitten, W. B. , and Danielson, G. C. , J. Phys. Chem. Solids, 28^439 (1967). [2] Winkler, U. , Helv. Phys. Acta. 28 633 (1955). [3] Blunt, R. F. , Frederikse, H. P. R. , and Hosier, W. R. , Phys. Rev. 100 633(1955). [4] Lawson, W. D. , Nielsen, S. , Putley, E. H. and Roberts, V. , J. Electronics 1 203 (1955). [5] Lichter, B. D. , J. Electrochem. Soc. 109 819 (1962). [6] Umeda, J., J. Phys. Soc. Japan 19 2052 (1964). [7] Crossman, L. D. , Ph.D. thesis, Iowa State University, 1967 (unpublished). [8] Busch, G. and Schneider, M. , Physica, 20^ 1084 (1959). [9] Martin, J. J. and Danielson, G. C. , Phys. Rev. to be published. ClO] Kennedy, W. L. , Sidles, P. H. , and Danielson, G. C. , Advanced energy con- version Vol. 2, p. 53 (Pergamon Press, Oxford, 1962). [ll] Swanson, H. E. , Gilfrich, N. T. , and Uriginic, G. M. , X-ray diffraction pow- der patterns. National Bureau of Stan- dards Circular 539, Vol. 5: 41 (1955). [12] Jelinek, F. J., Shickell, W. D. , Gerstein, B. C. , J. Phys. Chem. Solids, 28 267 (1967). [13] Gschneider, K. , private communication (1967). [14] Lipson, H. G. and Kahan, A., Phys. Rev. 133 A800 (1964). TPK) E ICT 500 300 200 "i r Mg^n T~ ~r 100 80 -1 1 10 -CD - 10' BEFORE k MEASUREMENTS A /> D R AFTER k MEASUREMENTS O P 3 I 1 1 1 I I I I I I I I I I I 2 3 4 5 6 7 8 9 10 II 12 13 |V I0 3 /T («K) Figure 1. The Hall coefficient and electrical resistivity of our thermal diffusivity sample that were measured before the thermal diffusivity data were taken and the electrical resis- tivity that was measured after the thermal diffusivity was measured as shown. 383 0.50 0.40 - 0.30 0.20 - 0.10 Figure 2. The thermal diffusivity of Mg Sn. 100 200 300 400 500 600 700 T(°K) 1.00 0.50 0.20 0.10 0.05 0.03 A4a & # O SAMPLE I a SAMPLE 2a A SAMPLE 2b + SAMPLE 2c V SAMPLE 3 — FROM THERMAL DIFFUSIVITY x BUSCH S SCHNEIDER rf> J L Mg 2 Sn 10 20 50 100 200 TCK) 500 1000 Figure 3. The smoothed thermal conductivity calculated from the thermal diffusivity is shown with the results of our earlier thermal conductivity measurements. 384 Figure 4. The points show the smoothed thermal resistivity of our sample. The straight line represents our estimate of the lattice conductivity. 300 400 TCK) 700 30 20 - 10 Figure 5. The curve represents the cal- culated electronic thermal conductivity. The points represeut the difference between our total thermal conductivity and the estimated lattice conductivity. 1 1 Mgg Sn 1 1 1 / X CALCULATED / X X K-K L x / / x 1 1 x / X / X 1 1 1 100 200 300 400 500 600 700 T(°K) 385 Thermal Properties of SNAP Fuels C. C. Weeks, M. M. Nakata, and C. A. Smith Atomics International A Division of North American Rockwell Corporation P. 0. Box 309 Canoga Park, California The thermal conductivity of uranium-fueled zirconium hydride was obtained from thermal diffusivity, density, and specific heat over the composition range of H/Zr - 1.58 to 1.81 (atomic ratio). The thermal properties measurements of SNAP fuel materials was complicated by the fact that the hydride decomposes at elevated temperatures and also undergoes phase changes in the temperature range of interest. Uncer- tainties in the phase diagram of the zirconium hydride system also contributed to the difficulties. Diffusivity was measured by the pulsed laser technique. Enthalpy and specific heat were determined with a high pressure hydrogen drop calorimeter. The calorimeter employs a variable temperature receiver to permit measurements to temperatures above and below the numerous phase boundaries in the Zr-U-H system. The data on the various properties measured were analyzed and plotted with the aid of a computer program and are presented as functions of temper- ature and composition. Key Words: Calorimeter, density, diffusivity, SNAP reactor fuel, specific heat, thermal conductivity, thermal diffu- sivity, thermal properties. 1. Introduction The thermal properties of zirconium hydride as a function of composition and temperature are of particular interest to the SNAP reactor program since these reactors employ zirconium hydride and uran- ium as a homogeneous fuel-moderator. Thus, the thermal properties of this material, and in particular the thermal conductivity, are essential to predict the behavior of the moderator during reactor opera- tion. Thermal conductivity is most readily obtained from the relation X = adc p , (1) where a is the thermal diffusivity, d is the density and c p is the specific heat. Since zirconium hydride decomposes at elevated temperatures, studies of this material are compli- cated by the need to maintain composition during measurements by providing a balancing hydrogen over- pressure which is a strong function of temperature. To complicate the matter further, the phase diagram for zirconium hydride is complex and was not well established at atomic ratios of hydrogen to zirconium of 1.6 and greater. Fig. 1 shows a phase 'diagram for the zirconium hydride system. The boundaries of the 6 + e region were not well known at the beginning of the properties program, as indicated by the shaded bands. Data from another program, concurrent with this one, to determine phase boundaries, indicate the 6 and e boundaries curve toward each other, joining in the vicinity of 450°C. The 6 boundary continues its characteristic curvature to higher H/Zr ratios beyond the intersection. Preliminary results indicate Hifork done under AEC Contract AT (04-3) -701 387 a 6 to e boundary near 700 °C, for H/Zr = 1.81. The dashed curves in the upper portion of the figure delineate the equilibrium hydrogen pressures. The required hydrogen overpressure sets a practical limit on the maximum temperature at which the material can be used or studied. In addition to the usual requirements of instrumentation for experi- mental determination of thermophysical properties, the systems must be capable of operation at hydrogen pressures up to about ten atmospheres. The samples for specific heat and thermal diffusivity were machined from adjacent cuts of the same fuel rod which, in turn, was from the same extrusion of fuel alloy and contained the reference concen- tration of modifier. Each sample was fully analyzed before measurements were made. The hydrogen pressure was maintained at equilibrium during temperature changes by following the isochores [l]^ shown in Fig. 2. Specific heat was measured using a drop calorimeter specifically designed for use with hydride materials. The copper receiver, into which the samples were dropped, can be heated to any predeter- mined temperature up to 650°C to avoid crossing phase boundaries during sample cooling. Also, hydrogen pressure can be maintained in equilibrium with the sample up to 150 psi. The hydrogen content of each sample was adjusted to a selected H/Zr ratio by maintaining the sample at temperature and equilibrium pressure in the drop calorimeter prior to each series of drops. The method and its limitations are described further in Section 2. Thermal diffusivity was measured by the flash method using a laser beam as the source of energy. Samples for these measurements were cut from material adjacent to the corresponding specific heat sam- ples. The hydrogen content of each sample was adjusted within the equipment to conform with the same isochores used for specific heat. The flash method is described in more detail in Section 3. Thermal conductivity can be calculated from specific heat, thermal diffusivity, and density by the relationship X = ado p . (2) The experimental program described in this paper yielded specific heat and thermal diffusivity. The thermal conductivity was then computed from these data and from the known density of SNAP fuels as a function of temperature. 2. Specific Heat 2.1 Technique Drop calorimetry is probably the simplest and most generally applicable method of determining high temperature specific heats. In principle, the method consists of determining the enthalpy change in a material by measuring the heat absorbed in a cool receiver after it receives a heated sample. The spe- cific heat, c , of the sample is then obtained from the relation y-f • « The most serious limitation of this method is that cooling of the sample may be too rapid to permit phase changes so that the sample is quenched in a non-equilibrium state and evolves less heat than equilibrium conditions would yield. To eliminate this problem, the calorimeter was designed so that the receiver could be maintained up to 650°C, thus eliminating the necessity for dropping through phase boundaries. In the experimental procedure, the samples were heated to a predetermined temperature, T, in an atmosphere of hydrogen maintained at the equilibrium dissociation pressure. The samples were then dropped (under gravity with electromagnetic braking) into the massive copper receiver maintained at a suitable lower temperature, T r ; the hydrogen overpressure was also re-adjusted to maintain a constant fuel composition. The loss of heat, Q(T), by the sample results in a small temperature increase, 6(T), in the receiver which was measured by a thermopile. The specific enthalpy, H, of the sample at tempera- ture, T, relative to T r , is then given by H(T) =Sl2i = c 6(T) M ( j x m r x m ' 2 Figures in brackets indicate the literature references at the end of this paper. 388 when m is the mass of the sample, C r is the specific heat of the receiver, and M is the mass of the receiver. By making drops from several sample temperatures, the specific enthalpy was determined as a function of temperature H = ao + a^T + agf + aa T 3 + . (5) The specific heat, c , was then determined from the derivative of eq (5) with respect to temperature, c = % + 2aa T + 3aa T 8 + . (6) The values of H and c were determined by the use of computer techniques. 2.2 Apparatus a. System The drop calorimeter used for obtaining specific heat during this program was designed and con- structed by the Dynatech Corporation for Atomics International. Fig. 3 is a cross section of the calorimeter, simplified for clarity. The three basic parts of the apparatus are the furnace, the transition zone, and the receiver. b. Furnace The furnace is molybdenum wound with aluminum oxide insulation and molybdenum radiation shields adjacent to the furnace winding. The central portion of the furnace is lined with a thick-walled iron tube to flatten the temperature profile. The temperature is controlled and monitored by Geminol ther- mocouples, which are relatively insensitive to the hydrogen environment. c. Transition Zone The transition zone is heated to reduce heat loss from the receiver along its supporting column. It also contains a pair of movable heat shields to reduce convection losses from the receiver and to prevent furnace radiation from reaching the receiver. These heat shields are actuated by rotary sole- noids and are opened and closed quickly during a drop to allow the sample to pass through the transi- tion zone. d. Receiver The receiver, which is a copper cylinder weighing about 100 pounds, is vacuum insulated and is thermally shielded by a surrounding copper shell or receiver guard maintained at the receiver tempera- ture. The transition zone heaters and the receiver guard heaters are differentially controlled with reference to the receiver temperature so that they follow it and maintain receiver losses at a very low value. By proper adjustment of the guard and the transition zone temperatures, it is possible to ob- tain receiver time constants of several hours. The receiver contains a dual-purpose heater. It is used to heat the receiver during electrical calibrations and, when it is desirable, to raise the receiver temperature to avoid dropping through phase boundaries. The temperature rise of the receiver following a sample drop is determined by means of a thermopile consisting of ten chromel-alumel couples. e. Temperature Control Temperature sensing for control purposes is provided by chromel-alumel thermocouples throughout the calorimeter except in the furnace. Geminol thermocouples were used in the furnace for control and monitoring because they are relatively insensitive to hydrogen throughout their temperature range. Furnace control is provided by a null-balance type controller with a controlled rectifier output. The temperature profile in the furnace is controlled manually by adjustment of a secondary heater to correct for end losses. The control thermocouples for both the transition zone heater and the receiver guard are connected differentially to thermocouples mounted on the receiver. These controllers are also the null-balance type with band width control to minimize temperature overshoot. 389 f . Drop Mechanism The drop mechanism permits multiple drops of a sample without opening the furnace. When in place, the sample is supported by a molybdenum wire attached to a magnetic armature which travels inside a nonmagnetic thimble located above the furnace. The armature is supported by a matching set of pole pieces and permanent magnets that fit the outside of the thimble. The magnet assembly is driven up or down the thimble by means of a linear AC motor. g. Safety Several safety precautions were taken to minimize the possibility of a hydrogen conflagration. The entire outside vessel was traced with cooling water tubes to maintain the seals near room temperature and to cool any gas which might leak out. A large hood was installed above the calorimeter and exhausted continuously to prevent the chance of hydrogen buildup in the laboratory. A second exhaust system, vented to the outside of the building, was installed on the calorimeter. During operation, it was continuously purged with argon. Hydrogen from the calorimeter was exhausted through this system. System pressures of one atmosphere and less were pumped into the vent with a mechanical vacuum pump, but gases in the calorimeter at pressures above one atmosphere were exhausted by a quick release mechanism. This mechanism bypasses the vacuum system and consists primarily of a solenoid valve for quick release and a check valve to prevent impurities from backstreaming. 2.3 Calibration Before the calorimeter could be used, it had to be calibrated to determine the response of the receiver to an addition of heat. The relationship for the electrical calibration is h = jEIdt =JE(e)l(e) -p. de f (7) where h = heat gained by the receiver E = potential drop across the calibration heater I = calibration heater current e = thermopile emf t = time. From eq (7) we obtain: i = E(e)l(e)f| , (8) where db/de is the receiver characteristic which must be determined to calibrate the receiver since it gives the heat input for an observed change in thermopile emf. To determine dh/de, power is applied continuously to the receiver, and E, I, e, and t are measured at regular intervals over the desired range of receiver temperatures. Values of dh/de are then computed from eq (8) preferably by a com- puter program. To perform the electrical calibration, power was supplied to the receiver heater from a voltage-regulated, solid-state power supply. The heater current was determined by reading the potential drop across a 0.01$ shunt with a precision potentiometer. The voltage across the heater was measured with a potentiometer using probes connected directly to the heater terminations. The receiver response was obtained by a thermopile consisting of ten chromel-alumel thermocouples. The thermopile junctions were imbedded in the outer surface of the receiver so that they would not be in hydrogen. The heater voltage was held constant at 10 volts; current was approximately 4.46 amperes when the receiver was near room temperature. The receiver heating rate was about 10°C per hour. The hydrogen pressure in the receiver was kept at one atmosphere during the calibration. Originally an electrical calibration was made for the receiver temperature range of room tempera- ture to 350°C, using a single thermocouple on the receiver. These results were compared to a predicted calibration curve obtained by multiplying the mass of the receiver by the specific heat of copper, assuming a linear thermocouple response. The electrical calibration exhibited deviation of about 3% 390 from the predicted curve. Short runs at various receiver temperatures duplicated the original data quite well, so that it was concluded that the general shape of the experimental curve was real. It showed a direct relationship to deviations from linearity of the thermocouple output as a function of tempera- ture, amounting to about ± 30 microvolts. This effect is, of course, a property of the thermocouple materials and is repeatable. In addition to deviations caused by the characteristics of the thermocouple materials, there was scattering of data, but to a much lesser degree. It was determined that an error of one microvolt, in a thermocouple reading, resulted in about a 1$ change in the value dh/de. Since the potentiometer used to measure the thermocouple output could be read only to one-half microvolt, it was concluded that a single thermocouple was not sensitive enough to maintain the desired accuracy during drop tests at the lower end of the temperature range. In order to increase the receiver thermocouple output to an accept- able value, it was decided to replace the single couple by a ten-element chromel-alumel thermopile. Because the phase boundaries for the 6 + e region had not been established, as indicated by the shaded regions in Fig. 1, all drops were made to room temperature. Thus, the ten-element thermopile was calibrated only for a short temperature range of 34° to 50°C. The dh/de values for the receiver are relatively constant for these short spans and are given as average values in Table 1. Table 1. Receiver Calibration dh/de (Joules/millivolt) Receiver Electrical Copper Drops Calculated Temperature dh/de 34°-50°C 4.1883 4.2522 4.22 Five calibration drops were made to room temperature values from temperatures between 200° and 600°C, using a 135.6 gm sample of copper. The receiver was at a slightly higher temperature for each successive drop, but remained within the 34° to 50°C range selected for electrical calibration. The average dh/de value for the copper drops is shown in Table 1. The specific heat data for the copper sample was taken from literature values [2], Calculated values were obtained by multiplying the mass of the receiver by the specific heat of copper, assuming a linear thermocouple response. This was not performed as a precise calibration but to predict the approximate results from the experimental calibration. 2.4 Measurement Technique Each zirconium hydride sample, before a series of drops, was placed in the calorimeter and the system pumped to a vacuum of approximately 1.3 x 10~3 Nm at room temperature. Then, after several purges of ultrapure hydrogen, sample temperatures were raised to values which insured relatively short equilibration periods. The samples were then maintained at the correct temperature and hydrogen pres- sure for adequate periods of time to insure no more than small fractional percentage deviations from the preselected H/Zr ratios. Once the sample had been conditioned in the calorimeter, it was subjected to a series of drops from furnace temperatures ranging from about 200°C to 900°C, except for H/Zr 1.81 in which case the upper temperature was restricted by the ten-atmosphere hydrogen pressure limitation placed on the calorimeter. When the sample was dropped from the furnace, the heat shields in the transition zone were opened long enough to permit passage into the receiver. The emf output from the receiver thermopile was then read with a potentiometer to determine the receiver temperature and the temperature rise. The thermo- pile emf was recorded at regular increments of time to establish when the sample and receiver reached thermal equilibrium and to check the time constant of the receiver. Normally the system was adjusted to give the receiver a thermal relaxation period of several hours. In most cases the sample cooled rapidly enough in the receiver following a drop to maintain com- position without adjusting the hydrogen pressure. At several of the higher temperatures, the hydro- gen pressure was reduced during the sample cooling to maintain composition. 2.5 Data Reduction To determine specific heat for each composition of zirconium hydride, each series of drops was analyzed to obtain the enthalpy change for each drop. These data were then machine-fitted to a 391 polynominal expression of enthalpy. The expression for enthalpy was then differentiated to obtain an expression for specific heat. The enthalpy change for each drop is given by the expression H(T,9 a ) -i C ^de -ijjk (%-e,) , (9) v ' 3 ' m J e, de m de " ^ ' w ' where H(T,6 8 ) = enthalpy per unit mass from T to 6 8 Qg = the receiver temperature after cooling of the sample ej = thermopile emf before the drop eg = thermopile emf after cooling of the sample Since the initial receiver temperature changes with each drop, a correction must be made to nor- malize all droos in a particular series to the same receiver temperature, 6 . A further correction is required due to the molybdenum wire suspension system. The mass of the wire, n^, is small compared to that of the sample; therefore, its average specific heat, 1^, may be used to determine its heat contri- bution over the temperature range T to 9 a . Upon applying these corrections, the expression for the enthalpy change of the sample becomes H ( T > 9 °> = m" S <*-* ) - - w C w (T - 8 S ) + § | 9o g ( %Hb ) , (10) where dH/d8 |6 is the specific heat of the sample at % and % is the thermopile emf at 6 . Since the average initial receiver temperature was approximately 34°C, all data were adjusted to 6 = 34°C The specific heat, dH/d9, was obtained from earlier work [33 in which pulse heating was used. Since specific heat is the derivative of enthalpy, with respect to temperature, the use of 34°C rather than 0° as the reference temperature does not affect its value. 2.6 Enthalpy and Specific Heat Enthalpy for each of the four compositions^ 1.58, 1.65, 1.70, and 1.81, was determined by a least squares fit of the polynomial expression, eq (5), to the experimental data. The curves for the enthalpy change, with reference to 34°C, are shown in Fig. 4. The specific heat is the first derivative of the enthalpy; thus, it is expressed as a second degree polynomial. The curves of the specific heat for the four compositions are shown in Fig. 5. The boundaries between the 6 and e regions had not been well defined before these measurements were made. Therefore, some of the curves may include heat of transformation from one phase to the other. It is believed that any boundary that was crossed in this work had a very small heat of transformation, since it was not apparent from the enthalpy data, and could be neglected in a second degree expression for specific heat. The enthalpy data were also fitted to a fourth degree polynomial to determine if any subtle effects could be detected as a result of crossing phase boundaries. In comparing the probable error of the coefficients, a midtemperature range effect for the 1.65 and 1.70 samples and a higher temperature range effect for the 1.81 sample were noticed where the probable error became greater than the coefficient. Derivatives of these expressions, giving a third degree equation for specific heat, gave S curve con- figurations for the specific heat of 1.65 and 1.70. Even though these results are not conclusive, they do have a correlation with preliminary results from the phase study program indicating curvature of the 6 to e phase boundary and, thus, there may be a small effect on the specific heat values determined in this program. 3. Thermal Diffusivity Thermal diffusivity measurements were made on 1.58, 1.65, and 1.70 H/Zr materials using the pulsed 392 laser method [4]. In this method, thermal diffusivity is determined from the temperature history of the rear face of a wafer, following a pulse of energy impinging on the front face. For valid results, the following conditions must hold: the energy impulse must be uniformly distributed over the face of the wafer; the duration of the pulse must be negligibly small compared to the characteristic rise time of the specimen; and heat losses must be sufficiently small so that the energy transfer is axially directed through the wafer and the specimen temperature is independent of heat losses for a period long compared to the transient time. When the above conditions are satisfied, thermal diffusivity, a, can be obtained from the simple expression: a _ 0.139 £ H (11) where t is the specimen thickness and ti is the half-time, or the time required to reach half the maxi- mum temperature rise, following the pulse of energy. Where heat losses from the specimen are signifi- cant or where the duration of the pulsed energy is not sufficiently short, techniques have been devel- oped [5>6] for applying the necessary corrections to the coefficient given in the above expression. 3.1 Apparatus and Method The apparatus and method used in determining the thermal diffusivity of hydrides have been reported previously l3>73> s0 that only a brief description will be given here. Figure 6 is a schematic diagram of the apparatus. The specimens were in the form of small discs nominally 6.4 x 10 J m diameter and with thicknesses as shown in Table 2. The specimen was mounted in a stainless steel holder by means of three pins located equidistant around the circumferential edge. Geminol thermocouple wires, 7.6 x 10 - **tn diameter (3-mil), were spot-welded to the center of the back face, approximately 1.5 x 10~^m apart. The speci- men holder was then placed within a stainless steel tube furnace, which was designed to contain hydro- gen gas atmospheres to pressures in excess of 8 x 10-* N/m*- (6000 torr) at temperatures up to about 1000°C. The system could be evacuated to < 3 x 10"^ N/m 2 (< 2 x 10~° torr) and back-filled with ultra- pure hydrogen gas. Gas pressure was measured with Wallace and Tiernan - 6.7 x \Qr N/m 2 (0 - 50 torr) and U. S. Gauge Co. - 8 x 10^ N/m 2 (0 - 6000 torr) precision dial gauges. Table 2. Thermal Diffusivity Specimen Data H/Zr Specimen No. Thickness (m) x 10"-3 1.59 ' ' "~"" 1 " 1.920 2 1.895 1.65 1 1.869 2 1.874 3 2 .581 1.70 1 1.816 2 1.849 3 1.984 In order to maintain the composition of the material being measured, the specimen was first heated to temperatures greater than 600°C and allowed to equilibrate with the proper hydrogen overpressure in the system. Diffusivity measurements were then made at selected temperature intervals, generally during the cooling cycle. When the specimen temperature was stabilized at a given level, the front face was heated with a short pulse from the laser and the output of the thermocouple was displayed on an oscil- loscope and recorded on photographic film. 3.2 Results and Discussion The data obtained for the three H/Zr compositions were fitted to least squares polynominal equa- tions of the form: a = Aq + \B + AgB 3 + AaS 3 + , (12) where a is in cm /sec, using a digital computer. In each case, a third degree polynominal was found to fit the data best. Graphs were also obtained by means of the computer, as shown in Fig. 7. The 1.81 393 H/Zr material was not measured in the present work, since the data obtained previously [3>7] were con- sidered accurate. The 1.81 data, however, are presented here since they were used, along with the specific heat data, in calculating the thermal conductivity of the 1.81 H/Zr material. Two 1.58 H/Zr specimens were measured. One specimen only, however, was measured over the entire temperature range shown. The second specimen gave values which were in good agreement with those of the first specimen, at the high temperature end, i.e., > 860°C. During the cooling cycle, however, the back face temperature transients were observed to be anomalous. Examination of the specimen after the run indicated the presence of hair-line cracks, which could account for the anomalous behavior. The remain- ing 1.58 H/Zr specimens available for testing were also found to contain fine cracks, so that no fur- ther measurements could be made on the 1.58 H/Zr composition. Since previous examination of the speci- mens, using gamma graph, showed no apparent flaws, it seems possible that the material is susceptible to structural degradation during storage. The apparent inflection in the curve at about 600°C was not expected, since no phase boundary is known in the temperature range covered. Fine cracks forming at high temperatures might account for the anomaly. The 1.65 H/Zr curve in Fig. 7 represents measurements made on three specimens in the range of room temperature to 900°C. The individual curves, for each of the three specimens (not shown here) were within a ± 2% band over the entire temperature range. There is an inflection in the curve near 300°C; this would be compatible with a 6 + e •» 6 phase boundary at about 300°C for the 1.65 H/Zr composition. The results of measurements in the room temperature to 870°C range, on three 1.70 H/Zr specimens, are shown in Fig. 7. The curves for each specimen (not shown here) fell within a ± 1.5$ band over the entire temperature range. The inflection in the curve at about 500°C suggests a phase change; the 6 + e ♦* 6 boundary at approximately this temperature for the 1.70 H/Zr composition would be in accord with this implication. The thermal diffusivity of 1.81 H/Zr material, as reported previously [3>7l, is included in Fig. 7- The values represent the "best data" from measurements made in a joint program by Atomics International and Battelle Memorial Institute. 3.3 Summary and Conclusions The 1.65 and 1.70 H/Zr materials gave nearly identical thermal diffusivity values over most of the temperature range covered. The diffusivity versus temperature curve for 1.70 H/Zr, however, deviates from the curve for 1.65 H/Zr at the lower temperatures, the values for 1.70 H/Zr being about 1.5$ lower at room temperature. The lower values for 1.70 H/Zr in the lower temperature region is not unexpected since two phases, 6 and e, are presumed to be present, whereas the 1.65 H/Zr is expected to be single- phased down to room temperature. The diffusivity of the 1.58 H/Zr material is lower than those of the 1.65 and 1.70 H/Zr materials, by approximately 11$, over most of the temperature range covered. Near room temperature, however, the values for 1.70 and 1.58 are nearly equal. The best data for the 1.58 H/Zr material in the region above 700°C would probably be an extrapolation of the curve obtained in the room temperature to 700 °C interval. 4. Thermal Conductivity Since the thermal conductivity, X, of hydrides cannot be measured directly with meaningful results, it was calculated from the relationship X = adc p , (13) where a, c , and d are the thermal diffusivity, specific heat, and density. The variation of the densities of SNAP fuel with temperature were determined from earlier thermal expansion measurements [8] obtained on compositions ranging from 1.70 H/Zr to 1.81 H/Zr. Isotropic expansion was assumed since all specimens used in both studies were polycrystalline and were subjected to high temperatures before measurements. Values of the ratio, R, of the density at temperature, 6, to the density at 0°C were obtained from linear expansion measurements on each composition. The ratios obtained (more than 144 values in each case) were fitted to polynomials by a least squares method. The polynomials obtained for the density ratios of 1.70 H/Zr and 1.81 H/Zr were found to be: Rj_ ?0 = 1 - 1.78 i io" 5 e - 3.31 • io" 8 e 2 + 2.3 • io" i:L e 3 , (14) h.i .81 with standard deviations of 0.00015 and 0.00009, respectively. 394 = 1 - 2.6 • 10" 5 e - 1.64 • io~ 8 e 2 , (15) The ratio at 900°C for 1.70 H/Zr was found to be only 0.015? more than that of 1.73 H/Zr and 0.7? less than that of zirconium, so that it was considered reasonable to use the 1.70 H/Zr expression for 1,58 H/Zr and 1.65 H/Zr as well. The error introduced into the conductivity values for the two lower compositions will thus be considerably less than 0.7? and probably near 0.02?. In general, data points of enthalpy and thermal diffusivity were obtained at different temperatures. Therefore, values of specific heat, thermal diffusivity, and density were calculated at 10°C intervals for the required temperature ranges from the polynomial expressions for these quantities. Thermal con- ductivity was then obtained at each of the temperatures as the product of the three quantities. The resulting values were then subjected to a polynomial least squares treatment which yielded fourth-degree polynomials as the best fit in each case. The thermal conductivities obtained in this study are given in Fig. 8. The unusual shape of the curve obtained for the 1.58 H/Zr composition, as compared with the curves for the other compositions, is due to two factors. First, the specific heat for 1.58 H/Zr (Fig. 5) appears to be almost a para- bolic function of temperature, whereas, the specific heats for the other compositions vary almost linearly with temperature. Secondly, the diffusivity decreases more rapidly in the low temperature region for 1.58 H/Zr fuel than it does for the other compositions. 5. References [1] H. E. Johnson, "Hydrogen Dissociation Pres- sures of Modified SNAP Fuels", NAA-SR-9295, March 15, 1964. [2] T. S. Touloukian, "Recommended Values of the Thermophysical Properties of Eight Alloys, Major Constituents and Their Oxides", Ther- mophysical Properties Research Center, Purdue University, February, 1966. [3] M. M. Nakata, C. J. Ambrose, and R. A. Finch, "Thermophysical Properties of Zirconium- Uranium Hydrides", Proceedings of the 6th Conf. on Thermal Conductivity, Dayton, Ohio, pp. 478-507 (1966). [4] W. J. Parker, et al., "Flash Method of Deter- mining Thermal Diffusivity, Heat Capacity and Thermal Conductivity", J. Appl. Phys. 3.2 (9) 1679-84 (1961). [5] J. A. Cape and G. W. Lehman, "Temperature and Finite Pulse-Time Effects in the Flash Method for Measuring Thermal Diffusivity", J. Appl. Phys. 3A (7) 1909-13 (1963). [6] R. D. Cowan, "Pulse Method of Measuring Ther- mal Diffusivity at High Temperatures", J. Appl. Phys. 2k (4) 926-7 (1963). [7] C J. Ambrose, R. E. Taylor, and R. A. Finch, "Thermophysical Properties of Zirconium- Uranium Hydride", Symp. of the 4th Conf. on Thermal Conductivity, San Francisco, Calif. (October, 1964). [8] T. L. Lliff, "SNAP 10A Composition Fuel Alloy for Thermophysical Properties Reference Program", NAA-SR-Memo-11533. Figure 1. Phase diagram for the rirconiui hydride system. 395 io- MAGNET SAMPLE- SOLENOID- RECEIVER RECEIVER GUARD Figure 3. Cross section drawing of the drop calorimeter „ Figure 2. Dissociation pressure of H/Zr fuel. LINEAR MOTOR vzzzzzzzm FURNACE -VACUUM LINE -TRANSITION ZONE 3 96 1 1 H/Zr 1.58 — **y A~-\.bb - 1.31 / - - 1.70 "^T^ - 1 1 1 1 - Figure 4. Enthalpy for third degree fit for H/Zr fuel. 400 600 TEMPERATURE ( C) 1000 Figure 5. Specific heat for H/Zr fuel from third degree fit of enthalpy. 200 400 600 TEMPERATURE CO 1000 DUAL BEAM OSCILLOSCOPE-^ AMPLIFIER (xlOOOK \ TANTALUM \ \ TUBEFURNACEn Mvigffinrr \ \ SAMPLE^ LASER V-V-" -1 ^> o ^3 \ \\ II t" OT( DC( ~~ r VACUUM URNACE POWE V 1 LASER POWER SUPPLY i / \ L THERM R- 1 )CELL )UPLE Figure 6. Schematic drawing of system for measuring diffusivity. 397 400 600 800 TEMPERATURE ( C) 1000 Figure 7. Thermal diffusivity of H/Zr fuel. 200 400 600 TEMPERATURE (=C) 800 Figure 8. Thermal conductivity of H/Zr fuel. 398 Thermal Conductivity, Diffusivity, and Specific Heat of Calcium Tungstate from 77° to 300°K Fhilipp H. Klein Materials Branch NASA Electronics Research Center Cambridge, Mass. 02139 Thermal conductivity and diffusivity have been measured by a simultaneous method on a single crystal of calcium tungstate containing approximately 0.5 mole per cent of neodymium. These data were used for calculation of c throughout the temperature range 80-300°K. Thermal conductivity was found to obey ?he relationship X = 822/T from 80° to 180°, and remains nearly constant at 4.4 Wnr 1 dee -1 from l80° to 300°. Thermal diffusivity decreases smoothly from 17 X 10-6m 2 /s at 80° to 2.5 X lO-Om^/s at 200° and 1.4 X 10-6m 2 /s at 300°. The Debye temperature, estimated from specific heat values below 200°, is approximately 570°K. At 300°K Op is about 15$ greater than c V) which is consistent with the known large coefficient of thermal expansion of calcium tungstate. Key Words: Calcium tungstate, conductivity, Debye temperature, diffusivity, heat conductivity, laser material, neodymium, specific heat capacity, thermal conductivity, thermal diffusivity, thermal properties. 1. Introduction Phase changes and other optical distortions which result from transient thermal phenomena in lasers have been discussed in a number of experimental and analytical studies [1-5] . For rigorous treatment of the experimental data, and for predictive use of the analytical results, it is necessary that a detailed temperature distribution within the laser rod be available. Once the thermal diffusi- vity, a, is known, this distribution can readily be determined, using standard techniques of heat-flow analysis . Unfortunately, thermal diffusivity has been measured for only a few laser materials. Comnutation of a (a = X/c d, where X is the thermal conductivity, c is the specific heat, and d is the density) is also frequently Impossible because some of the necessary data are also not available. Computation also involves a risk of error, since measurements of X, c , d have most often been performed on different samples and under differing conditions. p We present here the results of simultaneous measurements of the thermal conductivity and diffusivi- ty of a specimen of neodymlum-doped calcium tungstate. From these data, we have calculated the specific heat capacity in the range from 80°K to 300°K, and estimated the Debye temperature. It is hoped that publication of these results will make easier the prediction and analysis of transient phenomena in lasers made from this material. 2 . Experimental 2.1 Method and Apparatus Details of the experimental method have been published [6] and are only summarized here. The appa- ratus was of the type used for conventional steady-heat-flow measurements of thermal conductivity. A rod-shaped specimen was mounted vertically atop a steel plaoue, which was firmly secured to a heavy copper heat sink. A fine-wire heater was wound directly on the sample at its upoer end, and two 80-um 1 Figures in brackets Indicate the leterature references at the end of this paper. 399 (0.002") copper-constantan thermocouples were cemented along Its length. A third thermocouple Junction was placed directly within the heat sink, and temperature differences were read with a potentiometer between three different pairs of couples. The absolute heat-sink temperature was determined separately, using a platinum resistance-thermometer. Radiation errors were kept to a minimum by surrounding the sample with two concentric radiation shields, the inner one of which was mounted directly on the heat sink. The outer shield served also to suspend the heat sink from the top of the calorimeter can, in which the pressure was kept below 0.1 torn As an additional precaution, power to the sample heater was adjusted to maintain the temperature differ- ence along the specimen at less than one per cent of the absolute temperature of the heat sink. The steady-state difference in temperature between the points of attachment of the sample thermo- couples was determined with constant input to the sample heater. This value was used for calculation of the thermal conductivity. Power was then removed from the heater, and the time required for the temnera- ture difference along the sample to decay to one-half its steady-state value was noted for computation of the thermal diffusivity.2 For recording the decay of temperature, the output of the thermocouples was amplified and applied to the vertical axis of a strip-chart recorder. Typical half-decay times were about 20 s at 80°K and about 95 s at 300°K. Chart speeds were selected so that the half-decay time rep- resented a deflection of at least 20-25 cm along the time axis. °-2. Sample The experimental sample was cut from a single crystal of calcium tungstate, grown by the Czochral- ski technique from a melt containing 0.5 mole percent each of neodymium and sodium (one atom of each for every 200 calcium atoms). Sylvania Crystal Grade calcium and sodium tungstates were used in prepar- ing the melt, with neodymium oxide and tungstic oxide of purities exceeding 99.99$ added in stoichio- metric quantities. The crystal was pulled at a rate of approximately 1 cm/h from an iridium crucible in 1 atm of oxygen, then annealed for 2k h at 1200°C in air. Following annealing, the crystal was oriented by X-ray methods and the sample was cut with its length parallel to the c_ axis. Final dimensions of the rectangular bar were 2.3*1 X 2.92 X 23.30 mm. 3. Results and Discussion 3.1. Thermal Conductivity Figure 1 is a logarithmic plot of the thermal conductivity as a function of temperature. The data are nearly identical to those of Holland [7] for a similar sample. In the region below l80°K, the equation X = 822/T fits the data quite well. The 1/T dependence of X in this region indicates that scattering by impurities is the dominant mechanism for scattering of phonons. This is consistent with the fact that our sample contained about 6 X 10^5 m~3 (6 X 10^9 cm - ') neodymium ions and an approximately equal number of sodium ions. Above l80°K, X remains almost constant up to 300°K. Ordinarily, the slopes of (log X)-vs-(log T) curves become more negative at temperatures above the impurity-scattering region. This results from phonon-phonon scattering (Umklapp processes), which yield a dependence of X proportional to exp(0_yctT), where 9- is the Debye temperature and a is a constant. However, the five data points between 180* and 300° in figure 1 represent a total of seven determinations of X, all within the range 4. 26-4.7'J Wrrl deg~l. There is good evidence that this apparent constancy is real, and not due to experimental error, as we explain below. Both Holland's [7] and our samples, although prepared and measured entirely independently, showed little change in thermal conductivity between 200°K and 300°K. This fact might be ascribable to increased thermal radiation near room temperature. Our two data points at 297° actually represent results of three determinations, in which the power input to the sample heater was varied by a factor of eight. Despite this, all three measurements yield X values between ^.^2 and 4.^5 W m-1 deg-1. If radiation from the heated sample had a significant influence at this temperature, greater divergence of the data would be expected. Accepting as real the invariance of X in the range 200-300°K does not, however, explain the anomaly That the phenomenon is a consequence of the presence of impurities is suggested by the single datum of Nassau and Broyer [8], who reported X = 3.3 W m - ^ deg -1 at 290°K for an undoped, c-axis-oriented speci- men. If plotted in figure 3, this value would lie close to the line 'x = 822/T. 2 The computation Involved solution of eqs (6) and (9) of reference [6], which are derived and explained therein. 400 One might be tempted to relate the apparent constancy of the thermal conductivity to some ordering of the inpurity ions. For example, Slack [9] ascribed the temperature-invariant thermal behavior of magnetite to reradiation (scattering) of phonons absorbed in the process of ordering of the octahedral ions. Ihis ordering is a consequence of the proximity of ferrous and ferric ions on neighboring sites, and of the easy exchange of electrons between them. Even if an analogous ordering process could be con- ceived in our case, it would be extremely unlikely to have an observable effect in our sample. Granting replacement of 1/200 of the calcium ions by neodyimium ions, and replacement of another 1/200 by sodium ions, only 1% of the Ca ++ sites would be affected. With a random distribution of imDurities and four molecules of calcium tungstate per tetragonal unit cell, there results a mean distance of at least three lattice spacings between pairs of impurity ions. The distance is far greater than that over which order- ing phenomena can be expected, particularly at temperatures as high as 300°K. 3.2. Thermal Diffusivity Thermal diffusivity results are shown in figure 2, a conventional plot of log a as a function of log T. The uniformity of curvature is notable, in marked contrast with the situation in figure 1. This is a significant point, for it is added evidence for the absence of radiation- induced errors. These results may be compared with estimates [6] of the thermal diffusivity of calcium tungstate containing 0.5 mole per cent neodymium. The estimated values, based on fragmentary and extrapolated data from the literature (and therefore representing, at best, measurements of different specimens) were a = 20.7 X 10-° m2/s at 80°K and a = 1.53 X 10-6 ra 2/ s a t 300°K. The corresponding data from figure 2 are each about 10 per cent smaller: 18.0 X 10"° m2/s and 1.12 X 10-° m 2 /s, respectively. 3.3. Specific Heat The data of figures 1 and 2 were used to compute the specific heat capacity at each temperature from the relationship a = X/c d. No correction was made for the change of density with temperature, a constant value of 6.06 X 103 Rg/m3 (6.06 g/cm3) [8] being used throughout. The error so introduced is estimated to be of the order of one per cent. Results are shown in figure 3. Two other room-temperature values of c are indicated on figure 3. That shown as a solid square is due to Kopp [10], and was determined more tnan a century ago. The sample was probably a mineral speci- men of scheelite. Kopp found Cp to be invariant from 292°K to 322°K. The solid circle represents more recent work [11] on a (presumably undoped) sample of high-purity calcium tungstate. Although this point lies remarkably close to the curve calculated from the Debye temperature (see section 3.1), it is impossible to compare its accuracy with that of our data, since no experimental details are given. It is unlikely, however, that our deliberate addition of a total of approximately one mole per cent of impurities is entirely responsible for the difference of 15 per cent between our room-temperature value and this one. 3.1. Debye Temperature The right-hand ordinate of figure 3 shows the mean heat capacity per gram-atom. This scale was obtained bv dividing the molar heat capacity (C p ) by 6, the number of atoms in a molecule of calcium tungstate. Its use in estimating the Debye temperature, 9 D , is described below. Heat-capacity values read from the righthand ordinate were compared with a graph showing 9n/T as a function of Cy for a solid conforming to the Debye specific-heat approximation^. A value of 0ryT was read from this curve, and Op was obtained by multiplication by the temperature of measurement. A "best" Qq was then found by trial and error, the criterion being the degree of congruency of the experi- mental curve and the C v -vs-T curve of a solid with a Debye temperature of 9n. The broken curve of figure 3 is a graph of the molar specific heat (at constant volume) of a monatcmic solid which obeys the Debye approximation and which has 0n = 570°K. This curve is seen to be a reasonable fit of the data at all temperatures up to about 190°K (Od/T = 3). The divergence of the experimental (C ) curve from the estimated (C v ) curve above this temperature is to be expected, in view of the approximations made in the estimate. In addition, the relatively large coefficients of thermal expansion (7.9 X 10~° dee -1 in the a direction and 12.7 X Iff" 6 deg -1 in the c direction throughout the temperature ranee of figure 3 [8] suggest a perceptible difference between (L and C v at the higher temperatures. ^or this purpose, Cy and CL were assumed to be equal. The validity of this assumption improves at the lower temperatures. 401 4. Summary and Conclusions 1. The thermal conductivity of neodymium-doped calcium tungstate, as determined in this study, has been found to agree well with previous measurements [7], although its apparent invariance between 200°K and 300°K remains unexplained. 2. Results for thermal diffusivity, obtained simultaneously with the thermal conductivity, repre- sent the first actual measurements of the property for this material. Published estimates [6], based on composites of data obtained with different samples, are consistently about 10? higher than the measured values. 3. The specific hear capacity at room temperature was found to be about 15% higher than observed by previous authors [11]. At temperatures below 190°K, the heat capacity corresponds to a Debye temp- erature of 570°K. 4. All results cover the temperature range within which laser oscillators and amplifiers are commonly studied, and are sufficiently accurate to permit refined analyses and predictions [5] of the transient behavior of laser rods made from this material. 5 . Acknowledgement s We wish to express our gratitude to Dr. J. D. Childress and Dr. D. H. Ridgley for beneficial discus- sions, and to Richard Andersson and Donald P. Ellingwood for assistance with portions of the experiment- al work. 6. References [1] Cabezas, A. Y., Komai, L. G., and Treat, R. P., [6] Klein, P. H. , Thermal conductivity, thermal Dynamic measurements of phase shifts in laser diffusivity, and specific heat of solids from amplifiers, Appl. Opt. 5, 647 (1966). a single experiment, with application to Y, Q o Nd„ 02 v J. Appl. Phys. 38, 1598 (1967). ,y [2] Sims, S. D. , Stein, A., and Roth, C. , Dynamic ' 3 optical path distortions in laser rods, Appl. [7] Holland, M. 0., Thermal conductivity of several Cpt. 5., 621 (1966). optical maser materials, J. Appl. Phys. 33, 2910 (1962). [3] Welling, H. and Bickart, C. J., Spatial and temporal variation of the optical path length [8] Slack, G. A., Thermal conductivity of MgO, in flash-pumped laser rods, J. Opt. Sci. Am. A1„0_, MgAlJDj., and Fe^O^. crystals from 3° to 56, 611 (1966). 300°K, Physf Rev. 126/427 (1962). [4] Blume, A. E. and Tittel, K. P., Thermal effects [9] Nassau, K. and Broyer, A. M. , Calcium tungstate: in laser amplifiers and oscillators, Appl. Opt. Czochralski growth, perfection, and substitu- 3, 527 (1964). tion, J. Appl. Phys. 33, 3064 (1962). [5] Quelle, P. W. , Jr., Thermal distortion of diff- [10] Kopp, H. , Investigations of the specific heat raction-limited optical elements, Appl. Opt. 5, of solid bodies, Phil. Trans. Roy. Soc. (London) 633 (1966). 155^.71 (1965). [11] Plournoy, P. A. and Brixner, L. H. , Laser characteristics of niobium compensated CaMoOn, J. Electrochem. Soc. 112, 779 (1965). 402 THERMAL 8 CONDUCTIVITY IN W m" ' deg - ' 10 I 1 1 1 1 1 1 ' I 1 1 l_ X= 822/T - 8 — ^W - - \? / - 6 — <^W - i 1 1 1 1 1 rh 4 V, o 1J/ . . . r 100 200 TEMPERATURE IN °K 300 +3 Figure 1. Thermal conductivity of a single crystal of CaWO^.: 0.5% Nd , measured parallel to the c axis THERMAL DIFFUSIVITY 5 IN 2 /Am /s 80 100 140 200 TEMPERATURE IN °K 300 Figure 2. Thermal dlffusivity of a single crystal of CaWO,: 0.5% Nd' +3 403 SPECIFIC HEAT CAPACITY IN J kg"' deg" 100 -6 «**-5 i 1 ' i i i | 1 I i l | 500 — ^^^ ° - 400 - /r' e D = 570 °K 300 — y4> 200 - - §/ 100 I 1 l i i i 1 i i i i 1 200 TEMPERATURE IN °K MEAN HEAT CAPACITY Per Gram-atom 300 Figure 3. Specific heat capacity, c , (lefthand ordinate) and mean heat capacity per gram-atcm ( = 1/6 molar heat capacity) (righthand ordinate) computed from figures 2 and 3. The solid curve is a fit of the experimental data, which the broken curve is the molar heat capacity, C , of a monatomic solid with o = 570°K. Solid circle: datum from reference 11; solid square: datum from reference 12. 404 Measurement of the Thermal Conductivity of Self-Supporting Thin Films of Aluminum Oxide and Evaporated Films of Silicon Monoxide * 2 G. A. Shifrin and Robert W. Gammon Hughes Research Laboratories Malibu, California 90265 A technique has been developed for measuring the thermal conductivity of self- supporting dielectric thin films for directions in the plane of the film. Two evaporated metal film bolometers on the sample are used both to supply the heat conducted across the narrow gap between them and to measure the temperature difference across the gap. We have applied the technique to anodic aluminum oxide films. The thermal conductivity has been found to be 1. 68 ± . 07 W m - ' deg"' (16. 8 mW cm-1 deg- 1 ) at 37°C for films 325 to 1500 R thick with no detectable thickness dependence. For comparison bulk polycrystalline aluminum oxide of the same density as the films shows a thermal conductivity of 30. 2 W m" 1 deg"- 1 - The lower value of conductivity is believed to be due to amorphous structure in the anodic films. The ± 4% uncertainty given on the value of conductivity represents mainly the geometrical uncertainties of the measurement. The precision on any one sample was much better (± 0. 3%). By an extension of the method we have also measured the thermal conductivity of films evaporated onto previously calibrated self-supporting films. To perform this measurement it is necessary to observe the additional conductance due to the evaporated film and to establish its thickness independently. This type of measurement is potentially applicable for studying the thermal conductivity of a great variety of evaporated thin films. We measured the thermal conductivity of evaporated silicon monoxide films to be 0. 3 ± 0. 1 W m _1 deg -1 (3 mW cm"' deg"') for thicknesses from 340 to 560 A. Since the measurement accuracy was ± 10%, the indicated 33% uncertainty is believed to represent a spread in conductivity values. Key Words: Aluminum oxide, silicon monoxide, thermal conductivity, thin films. A full report of the experiment and the results will be submitted to the Journal of Applied Physics. 2 Present address: The Catholic University of America, Washington, D. C. 20017. 405 Impulse Thermal Breakdown in Silicon Oxide Films N. Klein and E. Burstein 2 Dept. of Electrical Engineering Technion, Israel Institute of Technology Haifa, Israel Silicon oxide capacitors with aluminum electrodes were deposited on glass substrates by evaporation in vacuum. The dielectric was about 5000 A thick and weak spots were eliminated by self-healing breakdowns. On application of voltage pulses, the breakdown voltage was found to increase with pulse duration decreasing from 100 to 10"^ s. It was assumed that the breakdown was thermal and caused by Joule heat in the dielectric. The breakdown voltage was calcu- lated by solving the nonlinear equation of conduction of heat with the aid of a digital computer and by approximate analysis. The two solutions agreed with each other and with the experimental results. In principle the thermal conduc- tivity of the substrate can be calculated with the measured breakdown voltages. Key Words: Electrical breakdown, thermal breakdown, silicon oxide, thermal conductivity, equation of conduction of heat. 1. Introduction Although the physical process of thermal breakdown in dielectrics has been understood for a long time, difficulties have been encountered in the calculation and observation of thermal breakdown fields, especially when the effect of voltage pulses was considered. Copple, Hartree, Porter and Tyson[l] investigated thermal breakdown of thick dielectrics, which have the form of an infinite slab. They made the simplifying assumption that the electric field is uniform in the dielectric and obtained the thermal breakdown field, F , as a function of pulse dura- tion by numerical methods. Vermeer[2] and 0'Dwyer[3] derived relations for F , disregarding the effect of heat conduction during the pulse. Hanscomb 's[4] observations of thermal breakdown in sodium chloride at 350°C ambient temperature support the validity of O'Dwyer's relation, when the pulses are short, but not when they are of long duration. Vermeer's experimental results of F in glass are 10-20% larger than calculated. The discrepancies of calculations and experiment are in both cases ascribed to the influence of heat conduction. The present work is concerned with impulse thermal breakdown in thin film capacitors of silicon oxide. The temperature of the dielectric of these capacitors is uniform and this circumstance simpli- fies the analytical treatment considerably relative to the case of thick dielectrics. Account is taken of the effect of conduction of heat and F is calculated with the aid of a digital computer and also by approximate analysis. Results of the two methods agree very well, also with the experimental obser- vations for pulses shorter than 0.1 s, for pulses of a few seconds duration the differences of calcu- lated and experimental values is less than 3.5%. It is believed that the relations derived here for F are applicable for thicker dielectrics also, when the internal thermal resistance of the dielectric is less than one tenth of the external thermal resistance. The observations of breakdown field as a function of pulse duration make possible in principle the determination of the thermal conductivity of the capacitor substrate. ^The work reported herein was performed under the sponsorship of the National Bureau of Standards, The work reported herein forms part of the M.Sc. Thesis of E. Burstein, submitted to the Senate c the Technion in July, 1967. 3 Figures in brackets indicate the literature references at the end of this paper. 407 2. The Breakdown Experiments The capacitors were produced by vapor deposition at 10"^ to 10"^ Torr pressure on g.l cm thick soda glass substrates. The Jhickness of the silicon oxide dielectric was 3000 to 6000 A and that of the aluminum electrodes 500 A. Breakdowns were self-healing and nonshorting, producing a hole through the three layers of the capacitor. The self-healing property of the capacitor made possible hundreds of breakdown experiments on a single sample. Figure 1 shows a sketch of the tested capacitors. Their area was 4xl0" 3 cm 2 , where the dielectric was thinnest. The dielectric was reinforced at the capacitor edges adjacent to the electrode contacts, preventing preferential breakdown at these edges. Prior to the pulse tests the weak spots in the capacitor were eliminated by single hole breakdowns, by applying increasing dc voltages to the capacitor .[5] The remaining capacitor area was quasi-uniform and its electrical conductivity, a could be fitted well to a relation simple for analysis, a = CToe gV e a(T-T ) (1) Here a , a and g are constants, V is the voltage and T and T are temperatures of the dielectric and of ambient respectively. The maximum voltage for thermal breakdown, V.. was observed in the capacitors and this voltage could be calculated to an accuracy better than 1% with the relation derived by Klein and Gafni[5] v = A log __ Ql_ (2) dm S eaa AV 2 o dm Here h and A are the dielectric thickness and are respectively, T is the thermal conductance of the capacitor and e the natural logarithm base. The pulse experiments were carried out with step voltages V > V, . The pulse rise-time was shorter by at least three orders of magnitude than the time to breakdown t and the voltage Vp m was constant during the pulse to better than 0.17 o . On application of a pulse the leakage current i through the dielectric increased first slowly with time t, followed by a very fast increase. This is shown in Fig. 2 by oscillograms obtained on one capacitor for pulse voltages of increasing magnitudes. When the current increased after time t to a very large magnitude, the capacitor broke down approxi- mately over the whole area by evaporation. This descruction was prevented by diverting the current from the capacitor with a controlled rectifier at a time t', when the current became 5 to 8 times larger than at t = 0. The time t' is only a few percent smaller than t, and t' is quoted as the measured time to breakdown. The effect of this difference on V is negligible because it will be seen that V a - log t^. As destructive breakdowns were avoided, a large number of experiments were carried out on a single sample and V as a function of pulse duration could be ascertained in the whole range of interest. This is shown for a typical capacitor for the range 0.01 s to nearly 10 s in Fig. 3, where measured values are represented by dots. The breakdown voltage increases to a maximum for a pulse duration of about lO'^s. The breakdown mechanism radically changes at this voltage, becoming electrical in nature. These breakdowns are not of interest for the present treatment and will be discussed elsewhere. 3. Calculation of the Thermal Breakdown Voltage The breakdown of the silicon oxide capacitors on dc was found to be thermal. Thus, the nature of the current increase shown in Fig. 2 and the increase of V with decreasing t indicate that the break- down is thermal for pulses of duration longer than a certain minimum value. This assumption was investigated quantitatively. The source of heat in the dielectric is Joule heat in the pulse experiments and the power P per unit area of capacitor is, using eq (1) a(T-T Q ) a(T-T Q ) P = 0- o V 2 eSVe /h = P Q -e (3) As the capacitor thickness is three orders of magnitude smaller than its lateral dimensions and the substrate thickness, the capacitor is idealized by the following model: The effect of the capacitor is represented by an infinite plane heat source of power P, situated on an infinite substrate bounded by two parallel planes. Heat is conducted through the substrate only, and the substrate surface opposite the capacitor surface is a ambient temperature. The capacitor temperature is assumed to be uniform. To obtain the times t = t as function of applied voltage V , when the capacitor temperature 4 08 T - oo and breakdown arises, the equation of conduction of heat is solved with the appropriate boundary and initial conditions. The coordinates perpendicular to the substrate surfaces is denoted by x and x = H at the capacitor and x = at the opposite surface. The temperature is denoted by T g 3 at oT s ox a(T-T Q ) (4) (5) x=H with T s = T Q for t £ and T s = T *....». - of the substrate, k is replaced by K /c, Normalizing with (T g -T ) = a(T s - T Q ) , T = Kt/cH 2 t.o/i x = O.k is the thermal diffusivity and K the thermal conductivity c being the specific heat per unit volume of substrate. x/H, eq (4) and (5) are replaced by STs 3t o z T , aP ■1 and fig, ox Be (T-T Q ) (4a) and (5a) x=0 with T S -T Q = for t 5 and for x = 0; here B = hK (6) The analytical solution of this nonlinear problem of heat conduction does not seem t o be k nown and solution was obtained with the aid of a digital computer. The relevant result, when T-T -» » In the composite is shown in Fig. 4, where B, a function of the breakdown voltage V pm , is plotted against the normalized time to breakdown Kt/cH 2 . It should be remarked that the computer handled the problem only for a maximum value of 10 in T-T Q . The B values in Fig. 4 are therefore consistently smaller than for T-T -*», The difference should be less than 1%. The pulse breakdown voltage V_ m can now be calculated with eq. (6), v = 1 x hKB(K T /cH2) Pm * ^o^pm (7) with B = B(Kt/cH 2 ) obtained from Fig. 3. Equation (7) can be connected with the relation for the dc maximum thermal breakdown voltage V, , eq (2). For the capacitor model discussed T = KA/h and 'pm = 1 log - 8 eaa hK HV 2 , (2a) dm hence V pm - V dm + i l°gCeB(KT/cH 2 )] - 1 log(V pm /V dm ) (7a) g g The oscillograms of Fig. 2 show that the leakage current and the power input change relatively little before the current runaway. This property of the formation of the breakdown process leads one to attempt an approximate solution of the heat conduction equation for the case of breakdown, des- cribed in the following lines. A mentioned, the temperature of the thin capacitor is assumed to have a uniform value T. equation for the capacitor hedt balance can then be written simply for unit capacitor area as The P dt = chdT + K' dT B x=H-h (8) 409 The thermal conductivity and diffusivity are denoted by K' and k' for reasons that will be apparent later. The capacitor top and bottom surfaces are at x = H and x = H - h respectively. The influence of electrodes is disregarded in eq (8), because their thermal resistance is negligible and their heat capacitance is much smaller than that of the dielectric. For c the specific heat of the sub- strate is taken and this assumption is justified by later results. dT s An approximate expression is obtained for the conduction term K' ,jx by making use of the x=H-h solution of the heat conduction problem discussed here, when the power is not an exponential function of T, as in eq (3), but constant, P=P . This solution is denoted by P cp(x,t) and the assumption is made for the conduction term in eq (8) that K 1 ST As shown in Carslaw and Jaeger[6] = K'P e a(T - T ° ) frpfr.t) Je= . H_h Sx x=H-h (9) P x V P cp(x,t) = T fl " T = p-2, k'(2n + l) a n*t ,„ , ,. n ~- — 7^3 . (2n + l)nx_ (10) sm 2H (2n + l) 8 P x y (1) e *tf_ sin 2H Substituting for cp(x,t) in eq (9) and for the derivative in eq (8) and integrating from t = to t and T = T to T, o ' GO 16H 2 gV^Y (-l)n r e -k'(2n+l) 2 n 2 t/4H 2 1 cos (2n+l)n(H-h) _ c[ -a(T-T n {] ^V^2 °o* S -V L q -^p- L J 2H at J (11) Breakdown arises at the voltage V = V—, when at time t = t, T - T -* <*> and the relation obtained from eq (11) for V is ^ pm V - I loe ck'TT 3 h 2 pm g 16aH2 CTo V^ n X(x) with CO X(T) . I (-1)% [l - e-k'C^D^tMH 2 ] (2n n )n(H -h) * (T) £ (2n+l)J L J 2H (12) Since H - h « H, this is approximately KM -Ar, 8 ^ e -k-(2n + l)Vt/4H 2 n X(T) "TeiL 1 -JzZ, ( 2n+ l) 2 J (13) n=0 with ck' = K V pm = I log hf — (12a) pm g aa HV 2 pm Y( T ) with 2. r V e -k'(2n+l) 2 n 2 T/4H 2 Y(K' T /cH 2 ) = 1 - 8 ) £ ^ '-^ n 2 L> (2n+l) 2 n=0 410 Testing the validity of eq (12a) for the limiting case, when t -* m V = V and ¥(t) = 1, the rela- tion (12a) is found to differ from eq (2a) by the factor e in the denominarBr of the logarithmic expression. This discrepance can be avoided by replacing K/e for K 1 . Comparing eq (12a) with the earlier relation (7) for V it appears that the condition for equality of results is kb(Kt/ch2) = TvwJZry <14) Replacing in eq (14) numerical values for B and for Y as function of the normalized pulse duration, equality of the two sides is found, within 1% accuracy, when K' = K/e (15) The approximate relation for the pulse breakdown is thus V -I log ^ with P m g eaa HV 2 m , l'(KT/ecH) (12b) k(2n+l) 2 rr 2 T/4ecH 2 t (Kr/ecH) = l - 8 V n L (2n+l) 2 n=0 Equations (7) and (12b) were applied to calculate V = f(i-) for the capacitor, the experimental breakdown results, of which are presented in Fig. 3. The data of this capacitor and of the substrate were: h = 5000 A, g = p. 054 V, c= 2.06 J/cm 3 , °C, a = 0.03/°C, a Q = 1.3xl0" 12 /n-cm and H = 0.1 cm. The value of the thermal conductivity was determined by fitting eq (7) to the experimental results for t < 0.1 s and K = 0.031 W/°C cm was obtained. The results of the calculations with both eq (7) and eq (12b) are represented in Fig. 3 by the curve. When the approximate calculation of V is carried out for the case that the substrate is a semi- infinite solid, the relation for V™, is V m = T log I t-- h(TTcK t*v1 P™ 8 2*7^%, (16) Breakdown voltages calculated with eq (12b) and (16) agree for t < 0.1. 4. Discussion Comparison of experimental and calculated results show very good agreement for t < 0.1 s, and that V is practically proportional to log 1/t2. On the other hand, the experimentally determined thermal conductivity 0.031 W deg cm -1 is much larger than the highest values found in handbooks for soda glass, 0.012 W dep cm" . The reason seems to be connected with the idealizations of the model. Brestechko[7] showed that heat from the capacitor is conducted not only through the substrate, but also by convection in the ambient air and by conduction through the electrodes to the contact pads (see Fig. 1). The convection accounts for about a 507„ increase in thermal conductance. Rough calcu- lations indicate that the difference between the true and the effective value of K is plausibly explained by the influence of the two additional paths of thermal conduction. It is interesting to note that while this influence increases conduction, it does not effect the form of the relations of V = f("r). It is believed that when the ef feet i of convection and of thermal conduction by the electrodes can be eliminated, the thermal conductivity of the substrate can be determined with short time pulse breakdown measurements. 411 For pulse durations longer than 0.1 s, measured voltages become higher than calculated and the largest discrepancy is about 3.5% at t = 2s. These discrepancies are due to further departures from the idealized model, which can effect the functional relationship also. One difference can be ascertained in the side view of Fig. 1: The lateral dimensions of the capacitor are smaller than the substrate thickness. When on pulses of longer duration heat flux penetrates deep into the substrate, the cross section for the flux does not remain constant, as assumed by the model, but increases. This is equivalent to an increase in thermal conductance and in V . An opposing effect arises at the bottom surface of the substrate. Measurements shown that the temperature there is not T as assumed, but slightly larger, and this decreases the thermal conductance and V . The results in Fig. 3 indi- cate that the first effect is larger than the second one. It appears that the differences between the idealized model and the test capacitor effect little the validity of the derived relations, and the measure of agreement of observed and calcu- lated breakdown voltages indicate that the breakdowns are thermal. 8. References [l] Copple, C, Hartree, D. R. , Porter, A., and [5] Klein, N. and Gafni, H. , The maximum di- Tyson, H. , The evaluation of transient electric strength of thin silicon oxide temperature distributions in a dielectric films, IEEE Trans. Electr. Dev. ED- 13 , 281 in an alternating field, J. IEE, London, (1966). 85, 56, (1939). [6] Carslaw, H. S. and Jaeger, J. C. , Conduction [2] Vermeer, J., On the relation between ionic of heat in solids, (book) 2nd ed., Ch. Ill, conductivity and breakdown strength of Oxford Univ. Press, London (1959). glass, Physica, 22, 1269 (1956). [7] Brestechco, M. , DC electrical breakdown in [3] O'Dwyer, J. J., High temperature dielectric thin silicon oxide films in vacuum, M.Sc. breakdown of alkali halides, Australian J. thesis, Technion, Israel Inst, of Techn. , Phys. 13, 270 (1960). (1967). [4] Hanscomb, J. R. , High temperature electrical breakdown in sodium chloride, Australian J. Phys. L5, 504, (1962). 412 Fig. 1 Sketch of tested capacitor. ELECTRODE BOTTOM ELECTRODE TOP y> AREA A AFFECTED A BY y BREAKDOWN t REINFORCED DIELECTRIC TOP VIEW BOTTOM ELECTRODE r CAPACITOR SUBSTRATE SIDE VIEW y- (Z z LU uj Q. or 2 cc < Z> o 1.6X10' 0.8 Oo CVJCVJ > J in o C\J CJ — o ^ ooo £J CVJCVJCJ 2 00 0> O.I 0.2 0.3 0.4 TIME IN SECONDS Fig. 2 Oscillogram of thermal breakdown event. 413 CO 5 O > 210 UJ < 200 O > z o o < UJ cr m UJ CO 190 180 170 Q. 160 ^k. ftl A V 1 ft J I 1 ft **v MAAIMUm ruLoc, onLHiM/v/nn »ulihwl Si D.C. ^3 BREAKDOWN VOLTAGE 1 1 1 1 1 111 i i i I i i I i ^^^ "' O ' 1 ^°°\^ i i— T-rrn io- 2 io-' PULSE WiOTH IN SECONDS JO Fig. 3 Pulse breakdown voltage versus pulse duration. Dots measured values, curve calculated. NORMALIZED PULSE DURATION Kt/cH' Fig. 4 B as function of normalized time. 414 High Temperature Radial Heat Flow Measurements J. P. Brazel General Electric Company Re-Entry Systems Department Philadelphia, Pa. 19101 K. H. Styhr National Beryllia Corporation Haskell, New Jersey 07420 Steady state thermal conductivity measurements on three molded graphite structures and one phenolic Refrasil Char com- posite are presented. Electrical resistivity data and struct- ural characterization of the specimens are included for corre- lation with other literature values . Measurements were made in flowing argon at mean Specimen temperatures in the range 125°c to 2000°C. Thermal conductivity ( ^) of ATJ grade graphite can be reduced almost 50% at 1000°c from about 6 to 3 mW/m°c by a "purification" process of leaching and outgassing. Reduction in \ is less at higher temperatures, about 30% at 1800OC from 2.6 to 1.8 mW/m°C. Specimen hot face temperatures exceeding 2200°c in most cases could not be reach- ed due to the high thermal conductivity of selected specimens and a power supply limitation. A modified apparatus with a larger power supply and equipped for high vacuum as well as inert atmosphere operation is described. Key words: Apparatus, conductivity, graphites, high temp- erature, radial heat flow, steady state, thermal conducti- vity. 1. Introduction The original intention of presenting a paper at the Seventh Conference on Thermal Conductivity was to compare the thermal conductivity of selected, well- characterized commercial grade graphites as measured in a variety of atmospheres. Un- fortunately, a problem developed in the electronic wattmeter of the apparatus equipped with the high vacuum system. Only data acquired in flowing argon were completed in time for the program. Since the basic apparatus has been described in previous con- ference proceedings (1,2,3) , only a brief outline of the equipment is included herein. This stresses the difference between the instruments used in the various atmospheres concerned and specific instrumentation problems arising from the sample materials studied. It is the authors' intention to report complete results and stress interpretation and analysis of the data in a subsequent paper. 1 Figures in parentheses indicate the literature references at the end of this paper. 415 2. Specimen Geometry The specimen stack is cylindrical, 12.7 cm (5") tall and 5 cm (2") in diameter. It contains a 1.9 cm (0.75") concentric axial cavity to accomodate the inner heater which supplies the radial outward temperature gradient. The stack may be composed of a number of thin discs (1.2 cm or greater) or preferably 1 to 3 larger cylinders. In this study all specimen stacks are composed of 3 cylinders, a 5 cm (2") tall speci- men and two guard cylinders (upper and lower) of equal height sufficient to total 12.7 cm. A typical specimen, without upper and lower guards, is shown in Figure 1. Since the power dissipated in the 2.54 cm (1") central section of the 12.7 cm (5") tall assembly is separately metered, in addition to metering the total inner heater power, all sensing wells are located within the 2.54 cm central section. Both radial and axial temperature sensing wells were used in this study. Axial wells have the disadvantage of a finite width in the direction of the temperature gradient; in the examples of Figure 1, 1.57 mm (0.063"). The precise radial location (isotherm) of the average temperature measured by a thermocouple or optical pyrometer is uncertain to some factor dependent upon the temperature gradient across the specific axial well considered. However, since conduction along thermocouple leads in a radial gradient can be significant, axial wells are used exclusively for thermocouple measure- ments, but only at lower temperatures (below 1500°C) and in situations where the gradient across the well is not large. Radial sensing wells have the advantage of having a flat bottom plane essentially on an isothermal plane within the specimen. Although this isotherm can be precisely located (+ 0.03 mm, 0.001"), temperature measurement by disappearing filament optical pyrometer or automatic optical pyrometer is inherently less precise (± 2°C and 0.5°C), than good thermocouple thermometry (0.01°C) . Although it has been determined that black body optical temperature assumptions are valid in most non- isothermal cavities in materials of high emmittance (4\ errors increase with decreasing emittance, de- creasing L/D cavity ratios, and increasing thermal gradient in the radial cavity. These factors must be considered in each individual case. Thin opaque discs can be used to increase emittance in specimens which are unusually porous or translucent. All specimens in this study were of sufficiently high emittance and contained cavities of sufficiently high L/D ratio and small thermal gradient so that no corrections from these considerations were necessary. 3 . Apparatus The photograph of Figure 2 shows a typical specimen stack assembled on a zirconia pedestal. Although the newer, vacuum- equipped, apparatus is illustrated, specimen assembly, furnace geometry and console are identical to the argon apparatus used except as specifically mentioned here. Both radial sight holes and axial thermocouples are visible. This assembly is inserted into the furnace from underneath as shown in Figure 3. The entire high vacuum system, normally attached to the furnace by four wing bolts, has been lowered about 10 cm and pivoted out of the way to facilitate sample loading. The lower high current (600 amps) inner heater power terminal, consisting of a liquid metal contact assembly contained within the liquid nitrogen cold trap region, is clearly visible in the right foreground. Figure 4 illustrates how the oil diffusion pump system may be readily pivoted back and raised to accomplish the 0-ring seal to the furnace. The seven position, self cleaning, radial sight window assembly, as well as a top axial sight window, is also visible in the photograph. Thermocouple wires leading from a multi-contact vacuum feed thru behind the furnace are lead up an external exhaust port behind the vacuum control box to a remote multi-point recorder. Console controls for inner heater, outer heater, and fail-safe circuits have been previously described (2,3). In Figure 4, the center meter indicates power in watts dissipated from the middle 2.54 cm (1") region of a 13.3 cm (5 1/4") tungsten mesh inner heater. In the vacuum apparatus, a Hall multiplier circuit in a specially designed electronic wattmeter was added to decrease possible AC meter error from +4% to j+0.5% and to compensate for the small (±1%) power factor correction uncertainty in a VA determination of gauge watts . 416 A schematic of the furnace interior is shown in Figure 5. It is only slightly different from the previously described arrangement (2). The carbon atmosphere from the samples of this study attacks the tantalum wire voltage probes. A thin (0.13 mm, 0.005") tantalum sheet placed at the I.D. of the specimen within the anti-contamination space serves as a "getter" to retard attack on the probes. Alternately, tungsten may be used for probe wire when the higher impedance electronic wattmeter or a digital voltmeter are employed since I 2 R losses in the probe wire are then not significant. 4. Data and Analysis The formula for calculating X in a radial geometry has appeared in many papers and texts and is: rl A = q-- ln r2 2 7T 1AT when: q = power in watts, dissipated in the central gauge length 1 = central gauge length in meters ^T = temperature gradient in °C between two radial isotherms r^,r2 = radii of the two radial isotherms then A is the apparent thermal conductivity in W/m°C directly without need for comparative measurements. For convenience, a geometric factor may be calculated for any pair of tempera- ture sensing wells. The procedure is illustrated in Figure 1. The form of data accumu- lation is shown in Table 1 which lists one set of raw data for the As-Received ATJ graphite specimen No. 3. To conserve space, raw data is not presented for all specimens here but may be obtained from the authors. Similarly, calculations of A from the raw data, using the geometric factors are presented in Table 2. It may be seen from these examples that several types of thermocouples (Chrome 1-Constantan and Platinum-Rhodium alloys) are used dependent upon the temperature difference to be measured, the tempera- ture range of interest and the reactivity of the specimen or its vapors with the ther- moelements. TypicalZVT's for various sensing wells are shown. It is also apparent that the product of "sense volts" (an AC meter reading of the 2.54 cm central gauge section) and total inner heater current was used to calculate gauge watts. Curves of thermal conductivity vs . temperature in SI units for ATJ graphite both in the As-Received condition and "purified" state are presented in Figure 6. They indicated a marked reduction in thermal conductivity due to the "purification" process of leaching and degassing. Specimens were aligned with the "C" axis parallel to the inner heater. Therefore, the radial outward temperature gradient and measured conducti- vity is in the AB with-grain plane. Measurements of electrical resistivity as a function of orientation were made on the lower guard section of each of the graphite samples. These are presented along with density measurements in Table 3. Measurements on the center specimen were not made since this requires cutting the specimen and more conductivity values in other atmospheres are desired. Note that the lower guard of the "purified" ATJ graphite #2 appears to have been cut incorrectly by 90°. Thermal conductivity was to have been measured in the AB plane, and some uncertainty exists now as to whether the AB plane was the radial plane in the specimen. This may account for the converging conductivi- ties at higher temperatures in the two ATJ graphites and will be confirmed by subsequent tests. The value of electrical resistivity measurements, even in the absence of a Wiedemann-Franz law for graphite, is evident. Measurements were also made on a commercial POCO AXM grade "unpurified" graphite graphitized at a temperature of approximately 4000°F . Thermal conductivity is plotted as a function of temperature in Figure 7, and is similar both in magnitude and tempera- ture dependence to "purified" ATJ graphite. Density and electrical resistivity data, listed in Table 3, are similar but not precisely in the range listed for the standard 417 TABLE 1 DATA RECORDED AT STEADY STATE CONDITIONS As-Received ATJ Graphite RUN CENTRAL RECORDED CONDITION HEATER DATA THERMOCOUPLES OUTER HEATER OPTICAL TEMPERATURES Sense Gunge Volts Amps Volts Watts TC-1 TC-2 Volts Amps Tl T2 T3 Rl-1 3.0 300 .290 87 887 883 __ Rl-2 3.3 315 .330 104 957 952 — — — — — Rl-3 5.1 395 .610 243 1310 1296 — — — — — Rl-5 6.4 450 .770 347 — — 9 80 1295 1280 1258 Rl-6 6.9 450 .820 370 — — 12 135 1788 1769 1753 R2-1 1.2 165 .090 14.9 303.37 302. 47* — __ R2-2 1.8 245 .160 39.2 525.69 524. 44* — — — — — R3-1 1.2 186 .090 16.7 380.61 380. ,11* — R3-2 2.0 245 .180 44.1 662.33 660. 66* — — — — — R3-3 5.0 390 .620 234.0 — — — — 1243 1225 — R3-4 6.6 450 .790 355.0 — — 8 100 1714 1670 — * Thermocouples Chroma l-Con£tantan Traceable to NBS Calibration TABLE 2 CALCULATIONS FOR AS -RECEIVED ATJ GRAPHITE STEADY STATE GAUGE MEAN CONSTANT RECORDED THERMAL CONDITION SECTION TEMPERATURE FACTOR DATA CONDUCTIVITY °C Watts AT mW/m°C Rl-1 TC 885 0.0298 87.00 4 6.50 Rl-2 TC 955 0.0298 104.00 5 6.20 Rl-3 TC 1300 0.0298 243.00 14 5.17 Rl-5 1-2 1668 0.0135 347.00 15 3.12 Rl-6 1-2 1780 0.0135 370.00 19 2.63 R2-1 TC 303 0.0283 14.85 0. 90 4.669 R2-2 TC 525 0.0283 39.20 1. 25 8.875 R3-1 TC 380 0.0283 16.74 0. 50 9.46 R3-2 TC 661 0.0283 44.10 1. 67 7.50 R3-3 1-2 1241 0.0354 234.00 18 4.60 R3-4 1-2 1700 0.0354 355.00 44 2.79 418 in the AFML-ADL "Thermal Conductivity Standards Program" (5). This suggests specific specimens will have to be well characterized in order that meaningful, well correlated comparative measurements are possible. A plot of with-lamina thermal conductivity versus temperature of a commercial phenolic impregnated phenolic Refrasil Char composite is shown in Figure 8. This is included to show that the ablative plastic impregnation causes more than an order of magnitude decrease in thermal conductivity (note the scale change) . Moreover, the temperature dependence has changed radically. Top measurement temperature is limited by the temperature stability of the sample and its attack on both furnace components and thermocouples at higher temperatures . 5. Conclusions The description of apparatus, techniques and specimens and the results of initial tests on selected graphite structures presented here summarizes the first part of a continuing program designed to define more closely the high temperature characteristics of graphite materials. Equipment is available which can determine changes in X due to materials processing and which is sufficiently sensitive to differentiate between specimens of slightly different microstructure, composition or thermal history. The effects of these differences and of various atmospheres in the "pores" of porous graphite structures is being studied with the hope of presenting additional data at next year's conference. Table 3. Specimen Characterization. Electrical Resistivity Measurements. Graphite Material (direction) Resistivity (ohm-cm x 10~3) Poco AXM #1 Poco AXM #1 Purified ATJ #2 Purified ATJ #2 Purified ATJ #2 As Rec'd ATJ #3 As Rec'd ATJ #3 Material POCO AXM #1 Purified ATJ #2 As Rec'd ATJ #? c 1.72 AB 1.69 C 1.11 AB 1.83 AB (90°) 1.15 C 1.93 AB 1.12 Density Measurements Density (g/cc) Density (lbs /ft 3 ) 1.73 108.1 1.67 104.4 1.72 107.5 8. References (1) The third conference on Thermal Con- (4) ductivity, hosted by Oak Ridge Nation- al Laboratory, Gat linburg, Tenn. October 16-18,1963, Page 428. (2) The fifth conference on Thermal Con- ductivity, hosted by The University of Denver, Denver, Colorado, October 20-22, 1965, Page VI-J-1. (3) The sixth conference on Thermal Con- ductivity, hosted by AFML, Dayton, Ohio. October 19-21,1966, Page 405. (5) E.M. Sparrow "Radiant Emission Char- acteristics of Nonisothermal Cylindri- cal Cavities" Applied Optics (4) 1 (1965) Page 41. A. E. Wechsler and M.L. Minges "Devel- opment of High Temperature Thermal Conductivity Standards" The Seventh Conference on Thermal Conductivity, hosted by the National Bureau of Standards, Washington, D.C., November 13-16, 1967. 419 0.990" Deep by 0.063 dia. Factor Calculations Factor= ln ri 2 If 1 Where: 1-Voltage Probe Separation in cm. ri-r2=Appropriate Radii 1) . Thermocouple wells (Ftc) ■ 622 F TC= In .492 = In 1.498 = .0251 2T2.54 15.95 2) . Radial T;l-T2 .622 Fi-?= In .492 2 77*2.54 3) Radial T2-T3 In 1.265 = .0146 15.95 '2-3 ■ 867 In .622 = In 1.333 Dimensions in Inches 2 7T2.54 15.95 4) . Radial T-L-T3 .867 F l-3 = In -492 = In 1.568 277'2.54 15.95 .0208 .0355 Figure 1 - Specimen Measurements and Geometric Factor Calculations - Purified ATJ Graphite Figure 2. Specimen stack, with thermocouples, assembled on Zirconia Pedestal. 420 Figure 3. Sample installation into furnace. Lower Vacuum Bell Assembly Pivoted Aside. Figure 4. Front View of Apparatus in Operation. (All photographs courtesy of the General Electric Co.) 421 Tungsten or Tantalum Wire Voltage Probes Water Out 23 Water In — ""Cj s^gkjSfeg Axial Temperature Sensing Wells 90° to Radial Wells Water Cooled Steel Shell Al20 3 Spacers and Shields AI2O Coated Tungsten Heater Radial Temperati Sensing Wells Tungsten Mesh Main Heater Vacuum Exhaust- Ports Qyick -Disconnect Vacuum Bell and Lower Terminal Water Cooled Copper Cup with Liquid Metal Contact (1) Water Cooled ickel Power Terminals (3) -Water In BeO Plate Molybdenum Radiation Shields Inert Gas Inlets (3) -Fused Quartz Windows (3) Specimens ^nti -Contamina- tion Spacing O-Ring Seals ^~"iL*_ Angle Iron ■ Support To Liquid N2 Cold Trap Baffles and High Vacuum System Figure 5 422 10 9 Reported by- K. H. Styhr Material - AT J graphite Organization - NATIONAL BERYLLIA CORR Property- Thermal Conductivity \ A as received "purified" 8 7 E E_ -<* 3 2 aN A Thermocouple points • Optical pyrometer points A ' < A j > ^ > K w ^j» *^» ■^ "T* ■ *V_> -- 10 13 t,°c X io 2 16 19 22 Figure 6 10 9 8 7 ,o 6 E \ 5 5 E < 4 Ref Org >orted by- K. H. Styhr Material- POCO AXM graphite anization - NATIONAL BERYLLIA CORR Property- Thermal Conductivity • Thermocouple points A Optical pyrometer points n^_ • • ^Vj A ft" A 3 2 10 13 t, *CxlO z 16 19 22 Pigure 7 423 Reported by- K. H.Styhr Organization - NATIONAL BERYLLIA CORR Material- Phenolic Refrasil Char Property- Thermal Conductivity O o.3 £ >< Charring Schedul e Time Rate Temp. Atm. (hrs.) CF/hr.) (°F) 48 325 N? 3 12 500 N ? 20 12 1000 N? 30 12 1600 N 2 Virgin Density = 1.68 g/cc Phenolic Refrasil Char ? = 1.39 g/cc Measured in the with -lamina direction • Thermocouple points A Optical pyrometer points Figure 8 424 An Investigation of the Mechanisms of Heat Transfer in Low- Density Phenolic- Nylon Chars E. D. Smyly and C. M. Pyron, Jr. 1 Southern Research Institute Birmingham, Alabama 35205 An investigation is being made of the mechanisms of heat transfer during ablation in low-density phenolic- nylon chars. The total objective of the study is to develop a thermal model which can be correlated with characterization studies and property measurements made in the labora- tory to analytically extrapolate and predict the overall effective thermal conductivity of the material system under reentry conditions. The objective of the work to date has been to determine the apparent thermal conductivity of the char independent of the act of ablation. Ultimately, the influence of the ablation process will be included. The chars used in this phase were prepared in an induction plasma torch from three densities of a phenolic- ny Ion ablative material. The effective thermal conductivities of these chars were measured in vacuum (<0. 01 torr), in nitrogen and in helium between 200°P and 1000°F. A thermal model was developed from characterization studies of the char structure and correlated with the experimental data to develop an equation which would predict the solid conduction, gas conduction and radiation. Values predicted by this equation agreed closely with the experimental data over the temperature range covered by the measure- ments, but fell considerably below experimental values obtained on a similar char at elevated temperatures. It was concluded that the con- ductivity of flight chars is increased at the higher temperatures by an increase in the conductivity of the matrix resulting from graphitization, and by radiant transmission either through voids or through the matrix membranes. Key Words: Carbon, char, graphite, phenolic- nylon, porous material, thermal conductivity. 1. Introduction Efforts to describe the mechanisms of heat transfer through ablating materials have met with limited success. The application which concerns us here is the behavior of the char layer on an abla- tive material during reentry independent of the ablating process. There are several reasons why one desires to be able to predict the response of the char. For earth entry, the actual conditions cannot be duplicated in the laboratory, so that extrapolation is necessary. The heat flux in reentry is partially blocked by the evolution of pyrolysis products through the char. If one attempts to measure the thermal conductivity of a fully degraded char under steady state conditions, this blocking effect is absent and must be accounted for by analysis. On the other hand, if the measurements are made under transient conditions which are less severe than reentry, the data also require analytical corrections. There are arguments pro and con for both steady state and transient techniques. However, in brief, the arguments ^Associate Engineer and Head, Thermodynamics Section, respectively. 425 against the steady state technique are that fairly long times are required for the measurements during which the apparent thermal conductivity of the char may be altered by graphitization. On the other hand, the transient measurements are subject to errors in defining temperature gradients in the degrading material whose properties are changing with depth, while the heat blocking effect, previously mentioned, takes place. It is not the purpose of this paper to argue the merits of test techniques but to propose one approach to analytically predict flight performance from laboratory data. The method used was to produce chars in a plasma jet, and to develop a thermal model for this char which would provide the basis for analytical expressions for the apparent thermal conductivity of this char. These expressions are compared with experimental data obtained on these chars under various gas environments at tempera- tures ranging from ambient to 1000°F (540°C) and with data obtained in a helium environment at ele- vated temperatures. In subsequent programs, the ablation process will be injected into the solution. 2. Materials Chars were made from virgin material of three different densities (19,30 and 42 lb/ft ). The virgin material was a low-density phenolic- nylon which consisted of 40 wt% nylon (Du pont Zytel 103), 25 wt% phenolic resin (Union Carbide BRP-5549), 35 wt% phenolic Microballoons (Union Carbide BJO- 0930). Densities were varied through variation of the molding pressures. The starting materials were post cured at about 300°F. A more detailed description of composition and processing conditions is presented in Reference flj. 2 3. Preparation and Characterization of Samples The specimens were charred in an induction plasma jet consisting of about 70 percent argon- 30 percent nitrogen by volume. Cold wall heat flux densities, as measured by a slug calorimeter, averaged about 170 Btu/ft 2 -sec at the specimen location about one inch from the end of the quartz tube housing the plasma. Observed front surface temperatures of the specimens, measured with an optical pyrometer, averaged about 4100°F. The charring times were varied between 125 seconds and 180 seconds, depending on the density of the sample, to obtain chars ■% inch thick. The chars were machined into specimens 1 inch diameter x -§ inch thick. Since the chars were quite friable, it was necessary to impregnate them before machining to provide sufficient mechanical strength. The impregnant used was polyalphamethylstyrene, a material which could be completely volatilized by baking the specimens at about 700°F. Bulk densities were determined from weight and volume calculations. True densities were determined by pulverizing the samples, weighing them, and measuring their volume in a pycnometer. At least 75 percent of the pulverized sample consisted of particles less than two microns in diameter while none of the particles were larger than 12 microns. Porosity was calculated from the relation P=l- Pb PT (1) where P, p ■□, and p rp are the porosity, bulk density and true density, respectively. Results are given in Table 1. Note that the bulk densities of the chars ranged from 12 to 20 lb/ft 3 , whereas the respec- tive porosities ranged from 88 percent to 79 percent. Pore size distributions in the chars were obtained by visual observation under the microscope with the aid of a calibrated eye piece which reads in filar units. The specimens used for the photo- micrographic evaluations were not impregnated. The specimens were viewed at 100X magnification in a plane parallel to the charring direction. The pore diameter measurements were made in a central zone midway between the front and back surfaces of the sample. Two traverses a few mils apart were made across the char at this location. When grossly irregular pores were encountered, the diameter was approximated. Cracks which ran parallel to the charring direction were not included in the mea- surements, and only openings which were nearly enclosed were counted as pores. Some areas appeared to represent locations where several pores had been blown out. Since these areas were enclosed by pores on all sides, they were counted as large pores since they probably were created during the charring process. Shown in Figure 1 are photomicrographs of the char produced from the three *Figures in brackets indicate the literature references at the end of this paper. 426 materials of different virgin densities. Pictures 1(a) and 1(c) are typical, whereas pictures 1(b) and 1(d) are not typical (as many large openings as are shown do not appear in the typical photomicrograph). The porous areas shown in figure 1(b) (away from the large openings) are typical. The samples shown in figures 1 (a), 1(b), and 1(c) were used for the pore size measurements. Table 1. Results of bulk and true density measurements and porosity calculations for low-density phenolic- nylon chars. Nominal 1 Density Bulk Average of Virgin Densities Porosity Material of Chars True Density for Each (gm/cm ) Specimen (gm/cm 3 ) (gm/cm 3 ) Porosity (b) Virgin Density Remarks 0.305 19-4 0.196 1.526(a) 0.872 Before run 19-4 0.188 1.526(a) 0.877 0.876 After run 19-5 0.186 0.875/0.811 After run 0.481 30-3 0.282 0.813/0.823 Not evaluated 30-4 0.271 1.501(a) 0.820 0.824 After run 30-6 0.257 0.829/0.839 After run 0.673 42-3 0.322 0.783/0.786 After run 42-4 0.319 1.479(a) 0.785 0.786 After run 42-5 0.315 0.788/0.790 After run (a) Char was ground using mortar and pestle. Particle sizes of powder used in measurements were: 2 microns or less - 75 percent 3-5 microns or less - 20 percent 6-12 microns or less - 5 percent (b) A porosity range is given for those samples on which true density measurements were not made. Range is based on the two different true density measurements made on impregnated and unim- pregnated chars. Histograms of the distribution of pore diameters were plotted for each char porosity. The average pore diameters of the materials with porosities of 0. 88, 0. 82 and 0. 79 were 69, 18, and 35 microns, respectively. There was no distinct correlation between the pore diameter distribution and the bulk density of either the virgin material or the char. However, the histograms gave reasonable agree- ment and indicated the approximate range of pore diameters. A typical histogram is shown in Figure 2. Apparently, the average pore diameter decreases with increasing bulk density up to a point and then increases. Perhaps the porosity of the virgin structure permits outgassing through paths following the existing voids up to a point where the existing voids cannot handle the gas evolution and then larger paths (and pores) are formed to provide gas release. 4. Experimental Apparatus and Procedure The effective thermal conductivity measurements were made in one direction (the charring direction) in a comparative rod apparatus. Since this type of apparatus is fairly standard and has been described numerous times in the literature, only a brief description is included here. Basically, this apparatus consisted of a stacked column with the specimen sandwiched between two references of known thermal conductivity. Heat was made to flow through the column, and the thermal conductivity was measured by comparing the temperature drop across the specimen material to the temperature drop through the references. Guard heaters were employed to minimize radial heat losses and axial heat bypass. The guard section was concentric with the specimen column, and the annulus between the column and the guard was filled with diatomaceous earth as an insulation. A schematic of the apparatus is shown in figure 3. A typical temperature profile through the apparatus is given in figure 4. The apparatus was placed inside a vacuum chamber which was connected to a 15 cfm mechanical pump and a four inch diffusion pump. With this system, it was possible to achieve a vacuum level of 427 0. 002 torr. Pressures were measured with a Hastings thermocouple vacuum gage below 0. 1 torr, a McLeod vacuum gage between 0. 1 torr and 5 torr, a Dubrovin vacuum gage between 1 torr and 20 torr, and a bourdon tube gage above 20 torr. The specimens were machined after impregnating the chars with polyalphamethylstyrene to pro- vide sufficient mechanical strength. Machining was performed by first cleaning up the heated surface until it was flat (this usually required the removal of only a small amount of material since this was the best surface), then reducing the thickness a minimum amount until the back (substrate) surface was flat and parallel with the front surface. The specimen was then turned to 1 inch outside diameter, and 0. 040 inch thermocouple holes were drilled 0. 093 inch from either surface. The gage length between thermocouples generally averaged about 0. 25 inch. After being machined, the specimens were baked at 700°F under a nitrogen purge to remove the im pregnant. The basic uncertainty in the comparative rod apparatus is + 5 percent for a normal specimen. The uncertainty was increased for these evaluations because of the shorter specimens, and the low thermal conductivity of the specimens which resulted in a mismatch between the specimen conductivity and the reference conductivity. Two reference materials were available; one was Code 9606 Pyroceram, which has a conductivity of about 50 x 10" 5 Btu/sec-ft-°F, and the other was Pyrex, which has a conductivity of about 20 x 10~ 5 Btu/sec-ft-°F. The Pyrex gave a better match; however, because of the transparency of Pyrex to radiation at temperatures above 500°F, Pyroceram was used for most of the measurements. When a mismatch occurs, it sets up conditions for some of the heat flow through the upper reference to bypass the specimen, thus resulting in the measured value being higher than the true value (if the speci- men has a lower conductivity than the reference). Further, analyses by Flynn [2 J and Robinson [3j have shown that the error introduced is a function of the conductivity of the insulation material. Based on these analyses errors introduced by the mismatch were estimated to be negligible in vacuum, about + 5 percent in nitrogen and about + 15 percent in helium. However, the error calculations were based on conditions which did not conform exactly to the experimental conditions and, hence, are subject to considerable uncertainty themselves. For this reason, the corrections were not applied to the data. A consideration of all of the errors led to the conclusion that the random precision in the system was about + 10 percent with a possible bias error of +15 percent. 5. Experimental Data The apparent thermal conductivity was measured on two or more specimens each of chars made from 19, 30, and 42 lb/ft 3 virgin phenolic- nylon. The measurements were made in the charring direc- tion with the temperature gradient maintained in the same direction as it occurred during charring. Measurements were made in vacuum (less than 0. 01 torr), in nitrogen and in helium over a tempera- ture range from 400°F to 1000°F. At about 500°F data were obtained at various pressures ranging from less than 0. 01 torr to one atmosphere to determine the pressure dependence of the conductivity. The apparent thermal conductivity versus temperature is plotted in figures 5, 6, and 7 for 19,30 and 42 lb/ft 3 chars. The low data for specimen 42-4 probably resulted from deterioration of the lower or substrate surface of this specimen during the run. The gage length of specimen 42-3 was only 0. 133 inch (about half that of the other specimens) which may have created some additional errors in its eval- uation. A composite plot of the apparent thermal conductivities of the three chars is presented in figure 8. No distinct correlation is seen between the char porosity or density of the virgin material. In vacuum and in nitrogen the conductivity of the 0. 79 porosity char fell between that of the 0. 82 and 0. 88 porosity chars, but was the lowest of the three in helium. Measurements of the apparent conductivities as a function of ambient gas pressure were made for all three chars in nitrogen and in helium at an average mean temperature of 500°F. A typical plot show- ing the apparent thermal conductivity of the 0. 82 porosity char versus ambient gas pressure in nitro- gen is shown in figure 9. The 88 and 79 percent chars behaved similarly and for brevity these plots are not reproduced here, but may be found in Reference [4] . Observe that the apparent thermal conduc- tivity of the char was constant at pressures between about 300 torr and 760 torr. Below 300 torr the conductivity decreased, finally reaching a second plateau at about 1 torr. This behavior can be explained as follows: at higher pressures the mean free path of the gas molecules is shorter than the separation distance, or average pore size, of the char and energy is transferred through the gas almost entirely by interroolecular collisions. As the pressure is reduced, the mean free path becomes on the 428 order of the separation distance and the gas molecules transfer energy both by intermolecular collisions and by collisions with the cell walls. This range of pressures (or mean free paths) corresponds to the knee of the curve. As the pressure is further reduced, the mean free path becomes longer than the separa- tion distance, and the gas molecules give up energy almost entirely by collisions with the cell walls so that a further reduction in pressure has no effect on the conductivity. Comparing figure 3 with figure 9, it can be seen that the range of measured pore sizes corresponds rather closely with the cal- culated range of mean free paths over which the apparent thermal conductivity decreases. It is also important to note that the change in the apparent conductivity of the char as pressure is increased from vacuum to one atmosphere is greater than the thermal conductivity of the gas. This indicates either strong dependence on the gas thermal conductivity or convection effects. Radiation, if it were present, would be a constant component. Recall from the prior discussion that the theoretical analysis of the bias error introduced because of the conductivity mismatch between the specimens and the references predicted positive errors of 5 and 15 percent in nitrogen and helium, respectively. This is being recalled because there have been several analyses of porous materials which would predict nearly the total gas thermal conductivity as the contribution of the gas to the measured thermal conductivity for this range of porosities, and it is important to determine if this low prediction is probable within the experimental error. The last two columns in table 2 give the maximum possible percentage change in the measured apparent thermal conductivity which can be attributed to the gas and the difference between this value and the measured value, respectively. In the last column, observe that with one exception the difference in the measured increase in conductivity and the increase obtained by adding the thermal conductivity of the gas exceeded the 15 percent bias error in the measurement. This is important in that it indicates that the thermal conductivity of the gas can have more effect on the "apparent" thermal conductivity of the char than its own value. This effect has appeared in the literature before. Young, Hartwig and Norton [5] presented data for firebrick of 0.82 porosity. The apparent thermal conductivity increased in air (nitrogen)from 1.46 x 10" 5 Btu/sec-ft-°F at 0. 05 torr to 2. 52 x 10" 5 Btu/sec-ft-°F at 760 torr for a change of 73 percent. By assuming that the air contributed its total conductivity, the change would have been only 40 percent (based on a gas conductivity of 0.48 x 10" 5 Btu/sec-ft-°F). One might argue that convection effects are present. Evidence will be presented later which indicates that this effect is small. For the range of porosities which have been evaluated, there is small possibility that the matrix is continuous, especially considering the violent birth of the material. Therefore, delaminations and voids must exist which offer infinite resistance to heat flow at vacuum but which become conducting paths when a continuum gas is introduced. Table 2. Comparison of the experimental data at approximately 500°F for the three char densities Specimen Number Average Bulk Density (gm/cm3) Average apparent thermal conductivity in Absolute change in apparent thermal conductivity going from vacuum. to Average Porosity Vacuum in 10" 5 Btu sec ft °F Nitrogen in 10 -5 Btu Helium in 10" 5 Btu sec ft °F Nitrogen in 10 -5 Btu sec ft °F Helium in 10" 5 Btu sec ft °F sec ft °F 19-4 and 19-5 0.187 0.88 9.0 12.5 18.8 3.5 9.8 30-4 and 30-6 0.264 0.82 14.5 17.5 22.0 3.0 7.5 42-3 and 42-5 0.318 0.79 13.0 17.5 22.0 4.5 9.0 Specimen Number Percent Change in Going from Vacuum to (a) Maximum Possible % Change due to Gas Conduction Difference between measured % Change and Maximum Possible % Change due to Gas Conduction Nitrogen Helium Nitrogen Helium Nitrogen Helium 19-4 and 19-5 38.8 108.8 6.5 39 32.3 69.8 30-4 and 30-6 20.6 51.7 4.1 25 16.5 26.7 42-3 and 42-5 34.6 69.2 4.5 27 30. 1 32.2 (a) Percent change Apparent thermal conductivity in gas - apparent thermal conductivity in vacuum Apparent thermal conductivity in vacuum 429 6. Thermal Analysis In order to separate the effects of radiation, solid conduction, gas conduction and convection on the apparent thermal conductivity of the char, a thermal analysis was performed. This analysis also allowed the isolation of one of the important intrinsic properties of the char, the thermal conductivity of the matrix. In the thermal analysis, each of the aforementioned modes of heat transfer was considered. Before proceeding to the thermal model some work by other authors should be discussed. 6. 1. Analyses of Porous Materials Russell [6], Goring and Churchill [7 J , Euken (presented in Reference [8j), Loeb [9j, and many other authors have presented equations for predicting the apparent thermal conductivity of porous materials. In addition, an equation derived by Bruggeman (presented in a paper by Powers |_10] ) for dispersions may be applied to porous materials. This may be done by treating the pores as the dis- persed phase and using the conductivity of the gas within the pores as the conductivity of this phase. The char solid would be treated as the continuous phase. In all of these analyses the authors have assumed that the matrix was continuous (contained no delaminations or irregularities), and most of them give nearly identical results. For completeness, Russell's equation, which has found general acceptance, was applied to the data to ascertain how closely the models for continuous matrices apply. Russell's equation may be written [6 ] M* /3+ k™ (I" P 2/3 ) k^P 2 k g P + k m (1 P2 /3 + p) (2) where k_ = apparent thermal conductivity k m = thermal conductivity of matrix k„ = thermal conductivity of void P = porosity Russell's equation was applied to the data by reducing k (matrix conductivity) from the vacuum data and then predicting the values for nitrogen and helium. Also, k m was reduced from the nitrogen data, and the vacuum and helium values were predicted. A temperature of 500°F was assumed for these calculations and radiation was neglected. The vacuum data probably contain the least uncertainty; there- fore, for a comparison with the measured data, the predictions based on the vacuum data should be most meaningful. The results of these calculations are shown in table 3. Note that with one exception the values predicted by Russell's equation fell below the measured values by a greater percentage than can be attributed to uncertainty in the measurements (maximum of 25 percent), indicating that a non- continuous model should be assumed for the highly porous chars. Table 3. Results obtained by applying Russell's equation to experimental data at 500°F Apparent Thermal Conductivity Predicted by Rus sell's Equation Using Value of Matrix Conductivity Reduced from Vacuum Data in 10 -5 Btu/sec-ft-°F In vacuum Specimen Number Reduced at 500°F (532°K) In Nitrogen at 500°F (532°K) In Helium at 500°F (532°K) Predicted Measured Percent(a) Difference Predicted Measured Percent (a) Difference Measured 19-4 103 9.0 9.5 14.4 51.2 12.3 20^1 64.1 19-5 92 8.1 8.6 11.3 32.4 11.3 16.2 42.8 30-4,30-6 116 14.8 15.7 17.3 10.2 18.1 21.9 21.7 42-3 88 ia. 9 14.8 18.5 25.0 17.1 24.3 41.8 42-4 51 8.1 8.6 11.6 35.1 11.3 18.5 63.2 42-5 76 12.0 12.5 16.4 31.4 15.3 18.9 24.2 (a) Percent difference = measured conductivity - predicted conductivity _ ner cent predicted conductivity x 1UU P ercent 430 6. 2. Development of the Thermal Model a. Physical and Thermal Model A physical model was assumed which is a first approximation to a noncontinuous matrix. This model is shown in figure 10(a). It was assumed that the material contained cracks which were parallel to the direction of heat flow. These cracks were assumed to occupy some fraction of the total area and extend across the thickness of the material. The char does contain these long cracks but obviously they do not extend through the full thickness; some interconnections exist. However, the number of inter- connections appeared to be small. It was also assumed that a large number of one- dimensional heat flow channels existed. Note that a heat flow channel consisted of porous material. Some fraction of the total area, (l-f)F, was assumed to consist of continuous (undelaminated) flow channels and the re- maining area, (l-f)(l-F), was assumed to contain delaminations (cracks, breaks, or large voids, nor- mal to the heat flow). That these separations do exist is evident from photomicrographs of the char. Observe in figures 1(b) and 1(d) that voids which separate porous areas are readily visible. The voids shown are perhaps 50 to 100 microns in thickness. Also observe in figure 1(a) that areas exist where the pores are not continuous, and delaminations several pore diameters long are apparent. It is obvious that the char does not consist of well defined delaminated and undelaminated heat flow channels. Further the heat flow will tend to concentrate in the solid areas adjacent to a delamination and thus somewhat bypass the delamination. However, for purposes of this analysis, bypassing effects were neglected, and the heat flow was assumed to be one dimensional. The thermal conductance network which describes the heat flow through the assumed physical model is shown in Figure 10(b). The separate conductances are outlined below: K R = radiant conductance through cracks parallel to the heat flow Kq = gas conductance through cracks parallel to the heat flow K ' = solid conductance through undelaminated area K ' = radiant conductance through undelaminated area K ' = gas conductance through undelaminated area K r = radiant conductance through porous area in the delaminated area K = solid conductance through porous area in the delaminated area m K = gas conductance through porous area in the delaminated area K , = radiant conductance across delamination dr Kj. = gas conductance across delamination Now that the physical model and the thermal conductance network have been defined the equations for the conductances will be explored. Each of the modes of heat transfer, radiation, solid conduction, gas conduction and convection will be discussed under separate headings. 431 (1). Radiation. There are three radiation components of concern in porous chars. The first is radiation between the walls of the pores; the second is radiation through the cracks which are parallel to the heat flow; and the third is radiation across any delamination (separations normal to the heat flow) In his studies of porous materials (not chars), Russell [6] considered radiation across the pores and used the pore diameter as the effective radiation length. His expression has been modified slightly to account for a radiant length which may be longer than one pore diameter because of the somewhat open pore structure. The radiant conductance obtained in this manner is based on the heat transfer between two parallel plates a finite distance apart. The equation for the radiant conductance through the porous area in a delaminated flow channel may be written 3 / . 4a A eT x Ax / Radiation through \ K = ^~ r c A m A *V r (2 p + e )x ax I channe i / delaminated flow I (3) where 2p+ e a X 4o7r^ A r = (l-f)(l-F)p 2/3 A and a = Stefan- Boltzmann constant e = emittance p = reflectance x = characteristic radiant length T = absolute temperature f = fraction of total cross- sectional area, normal to the heat flow, occupied by the cracks which are parallel to the heat flow F = fraction of heat flow channels which are undelaminated A = total area P = porosity 4 4 3 Note that the radiation term a(T x - T 2 ) has been written as 4aT m AT. By algebraic manipulation it can be shown that these expressions are equivalent within three percent error if AT ^ 0. 346T m . This will be true for a reasonable heat flux and char thickness. Note in eq 3 that the term x/x has been introduced where x is some characteristic radiation length which is an unknown for phenolic- nylon chars in general. This term was introduced because K r is a conductance term and in order to obtain an expressionfor the radiant conductivity from eq 3, a length term is needed, i. e. , k r = K r x/A r where k r is the radiant conductivity. Thus, an expression for the radiant conductivity would contain x in the numerator only. If x is thought of as the spacing between parallel plates, it is obvious that the larger the number of plates (in a given length) the smaller x will be and consequently the radiant conductivity will be less. The term (2p + e ) in eq 3 arises from the use of the graybody shape factor for the heat transfer between the two surfaces with the shape factor being equal to one. As it is written eq 3 applies to radiation through the porous area in a delaminated flow channel. A similar expression also applies to the radiant conductance of an undelaminated flow channel. This equation may be obtained as follows: Take the reciprocal of the thermal conductance of an element of length x,(A r 'x/ax). This gives the thermal resistance, ax/A r 'x, of one element. Then sum all of the resistances over the total length L. In this manner the radiant thermal resistance aL/A 'x is obtained. Take the reciprocal of the thermal resistance to obtain the thermal conductance v = A r 'x /Radiation through aL undelaminated flow channel / (4) where 432 A r - = (l-f)FP 2/3 A L = thickness across which heat is being transferred The radiant conductance through the cracks parallel to the heat flow may be written 3 Kj^ = irz m R (Radiation through cracks) (5) L where A R = fA In eq 5, L represents the length of the crack and F^ r is a reradiation shape factor. Values of the shape factor may be estimated from curves developed by Jakob [11] for radiation through openings. All that is needed for the approximation is the ratio of the minimum crack dimension to the crack length. The term L/Lwas introduced for use later in summing conductances. The radiant conductance across the delaminations may be written 3 dr " d m d (Radiation across delaminations) (6) (2p + e)t at where t = delamination thickness A d = (l-f)(l-F)A Equations(3)through fc)should adequately describe the radiant heat transfer. The two major un- knowns in these equations are (1) the radiant length through the porous area, and (2) the delamination thickness. Transparency of the solid was neglected; this aspect requires further study. (2) Solid Conduction. Solid conduction through the char accounts for a considerable portion of the heat transfer. The matrix conductance for a delaminated flow channel may be written K = m^-m / Solid conduction through m I delaminated flow channel/ (7) where k = thermal conductivity of the matrix A™ = (l-f)(l-F)(l-P2/s)A In eq(7)the solid conduction length, x, is taken to be equal to the radiant length. This is of no conse- quence since any length can be used as long as the summing of lengths is properly performed. The determination of the effective cross- sectional area is most important. Some authors have used (l-P)A as the effective cross-sectional area for isometric pores, where P is the volume pore fraction [8] . Russell [6] and Ribaud (presented in Reference [8]) used (1-P 2 ' 3 )A as the effective conduction cross section. The latter expression appears to best describe the effective cross- sectional area. This rela- tion needs experimental verification for these highly porous chars as it has some bearing on reducing the matrix conductivity from experimental measurements of the "apparent" thermal conductivity. For an undelaminated flow channel the equation for the solid conductance is 433 k A ' m S3 /Solid conduction through L ( undelaminated flow channel) (8) where A m ' = (l-f)F(l-P 2/3 )A L = total thickness across which heat is being transferred (3). Gas Conduction. There are three gas conduction components in the chars: (1) gas conduction through the porous areas, (2) gas conduction through the cracks parallel to the direction of heat flow, and (3) gas conduction across any delaminations. These conductances may be written as follows: K = g g [Gas conduction through \ iq\ x \delaminated flow channel / K e ' = kg^g' /Gas conduction through \ /-~x L { undelaminated flow channel/ k A Kq = g R [Gas conduction through \ /--» L \c racks Kde = kg^d /Gas conduction across , ^de lamination where kg = thermal conductivity of the gas A g = (l-f)(l-F) P 2 ^ A A ' = (1-f) F P2'3A Ar = fA A d = (l-f)(l-F)A x = characteristic radiant length L = total thickness across which heat is being transferred t = delamination thickness (4). Convection. It was concluded that convection heat transfer was negligible in these experiments. This conclusion was based on the following observations: (1) The structure of the char - the miniscule pore sizes of the chars would tend to retard convection. (2) Experimental conditions - the heat flow was vertically downward through the chars, tending to negate bouyant forces. (3) Experimental observations - heat was induced to flow vertically downward, thus tending to negate bouyant forces. Also, since con- vection heat transfer is dependent on the temperature difference throughthe specimens, the apparent thermal conductivity should vary with AT if convection were significant. The experimental data showed no dependence on this parameter. (4) Results reported in the literature. Kreith [l2j stated that con- vection in horizontal air spaces, heat from below, does not begin until the Grashof number exceeds 1600. Grashof numbers for the porous chars were much lower. Vershoor and Greebler [13] deduced from measurements on glass wool insulation (having higher porosities and lower apparent conductivities than the porous chars discussed here) that convection accounted for only 16 percent of the total heat transfer. Thus, the evidence seems fairly conclusive that convection effects were negligible in these experiments. However, it would probably be worthwhile to pursue this effect further through experi- ments designed to induce convection in order to evaluate the effect of this mode on the flight performance. b. Equation for Apparent Thermal Conductivity of Char The conductances which are shown in figure 13 and presented in eqs (3) through (12) were com- bined using the relations for series and parallel electrical conductances to obtain an expression for the apparent conductance, K, of the char. In combining the conductances use was made of the fact that the total thickness of the delaminations in a delaminated flow channel is nt, where n is the total number of 434 delaminations and t is the average thickness of one delamination. Therefore, the length of the porous area in a delaminated flow channel is (L-nt). The expression for the apparent conductance was reduced to an expression for the apparent thermal conductivity of the char through the relation Q = KAT = k a AAT L (13) where K = apparent conductance k = apparent thermal conductivity A = total area L = total length AT = temperature difference The expression obtained for the apparent thermal conductivity of the char was (l-f)(l-F)(t/a + k g ) [ -^-^ + P 2 /3 k„ + (1 - P 2/3 ) k ] a ° * 5 ™ (14) (l-c)(t/a + kj + c ^£2ii + p2/ 3 k + (I-P2/3 ) k ] k m + 4 f a F ir2 T m 3 L + f k g + (l-f)F [ P 2/3 k g + d-p 2/3 )k m + -*£_!] where c = ill L The details of the derivation of eq (14) are presented in Reference [4] . The unknowns in eq (14) are F, x, t, c, and k . Continuum values taken from the literature were used for the thermal conductivities of nitrogen and helium at 760 torr. c. Correlation of Thermal Model with Measurements The first step taken in order to apply eq (14) to the experimental data was to estimate the radia- tion terms, xP 2 ' 3 /a, t/a, and 4f crF^ T m L. For a radiant length, x, of 508 microns (approximately 10 pore diameters), a reflectance, p , of 0. 2, an emittance, e , of 0. 8, and a reradiation shape factor, P. , of 0. 10 (estimated from approximate measurements of the crack size), the radiation terms were negligible at 500°F. Since the radiation terms were small, the value of t had little influence on the results; therefore, this term was ignored in the low temperature calculations. Only the total delamina- tion thickness, nt, was considered in accounting for gas conduction across the delaminations. There- fore, at the 500°F mean temperature only three unknowns remain, namely, c, F, and k . The frac- tion of the total area which was occupied by the .rackj, f, was estimated to be 0. 1. The experimental data at 500°F were used with eq (14) to determine the values of c, F, and k for each specimen. Experimental data were obtained for three different environmental conditions (vacuum, nitrogen and helium). Thus, the measurements represent the apparent thermal conductivities for three values of k (thermal conductivity of gas). One equation in c, F, and k m was written from eq (14) for each environmental condition. Values taken from the literature were used for the thermal conductivities of nitrogen and helium at 760 torr. In writing these equations the radiation terms were assumed to be zero since calculations had shown that the values of these terms were negligible at 500°F. 435 Also, the proper values for the char thickness, temperature and porosity, for a given specimen, were substituted into eq (14) along with the values of k a and k g for a given environmental condition. Hence, three simultaneous equations were obtained and these equations were solved for c, F, and k m . Only one set of values for c, F, and k m was obtained for each specimen at the given temperature level since there were three experimental conditions and three equations were required for a solution. The results of these calculations are shown in table 4. Also shown in table 4 are the data used in this analysis and the char porosities. The parameter c represents the ratio of the delamination length to the total length; note that the values ranged from 0. 6 percent to 7. 1 percent of the total length with a mean value of approximately 3 percent. This appears to be a reasonable value and does not represent an excessive amount of cracking. The factor which represented the fraction of the flow channels which were undelaminated, F, ranged from 0.48 to 0. 72, which again indicates a reasonable range of values. There was considerable scatter in the matrix conductivity from 127 x 10" 5 Btu/sec-ft-°F to 248 x 10" 5 Btu/sec-ft-°F. These values are higher than the range of values, 69 to 115 x 10" Btu/sec-ft-°F, which is generally accepted for carbon. The data reductions suggest that those specimens having the lowest densities, Specimens 19-4 and 19-5, had the highest matrix conductivity. Note in table 4 that the matrix conductivity was also reduced using P rather than P 2 ' 3 in eq (14) and yielded lower values which were still higher than the values normally reported for carbon. Table 4. Results of reducing the parameters c, F, and k in eq (14) from the experimental data Specimen Number Porosity Apparent thermal conductivity measured at 500°F(532°K) 10" 5 Btu/sec-ft-°F Vacuum Nitrogen Helium 19-4 19-5 30-4, 30-6 42-3 42-4 42-5 0.877 0.878 0.827 0.785 0.785 0.789 9.02 8.10 14.81 13.88 8.10 12.03 14.35 11.34 17.36 18.51 11.57 16.43 20.13 16.20 21.98 24.30 18.51 18.97 i Value of c, F , and k m in eq (14) Values obtained for k m by using P rather than P 2 ' 3 in eq (14) obtained by fitting data at 500°F (532°K) k m k m 10 Btu sec ft °F Specimen Number Porosity c F 10" 5 Btu sec ft °F 19-4 0.877 0.022 0.477 248 169 19-5 0.878 0.047 0.545 194 132 30-4,30-6 0.827 0.036 0.710 199 137 42-3 0.785 0.027 0.590 185 125 42-4 0.785 0.071 0.400 155 106 42-5 0.789 0.006 0.720 127 88 Since the range of porosities was small and there was considerable scatter in the parameters, it is hard to evaluate the model for its applicability. The fact that c, F, and k m varied from specimen to specimen prohibited using one set of values to attempt a correlation of all of the data. One additional check on the model is to see how the predicted variation in the "apparent" thermal conductivity with temperature agrees with the data. The apparent conductivity was calculated for each specimen over the temperature range from 250°F to 1000°F using the reduced values of c, F, and k m given in table 4. Typical plots showing a comparison of the measured and calculated apparent conduc- tivities for the 0. 79 porosity char, are presented in figure 11. Note that the values predicted at the 436 higher temperatures usually agreed with the measured values within the experimental uncertainty with the exception of the vacuum data at the higher temperatures for Specimen 42-3. The data for Specimen 42-3 at 500°F in vacuum appear erroneous. There is no reasonable explanation for this large increase of thermal conductivity with temperature. Similar plots for the other two chars may be found in Refer- ence [43. d. Correlation of Predicted Values with Prior High Temperature Data In use applications the char layer of an ablator is at temperatures on the order of 5000°F or higher. Thus the critical test of the foregoing analysis is its ability to predict performance at near 5000°F. Figure 12 shows the apparent conductivity obtained experimentally from measurements on a phenolic- nylon char of the same composition as those reported in this paper. These data were previous- ly reported in Reference [lj • Also plotted are values predicted by eq (14) for various combinations of radiant lengths x and matrix conductivities, kj^. The lowest value of k m of 231 x 10~ 5 Btu/sec-ft-°F corresponds approximately to the values calculated from eq (14) for the low temperature data. The highest value of k m corresponds to the conductivity of polycrystalline graphite. The radiant path lengths of 50 and 500/1 correspond to approximately 1 and 10 pore diameters, respectively. Values of c and F were those reduced from eq (14) for an 88 percent porous char. The delamination thickness t of 50ju corresponds to between 2 and 3 delaminations. There is some uncertainty as to the gas that should be used in the calculations. The experimen- tal measurements were obtained in helium but without evacuating the air beforehand. Thus, pores may have contained nitrogen. The calculations were based on nitrogen. Values predicted for helium would range from 20 to 40 percent higher. From the plots in figure 12 it can be seen that the lowest values of x and k give the best agreement with the experimental data at low temperatures, while at high temperatures the reverse is true. Two possible explanations can be offered for this behavior. The first is that the conductivity of the matrix increases due to graphitization. The second is that the ther- mal model does not adequately account for radiant heat transfer through the char. There is evidence to support both arguments. First, let us inspect the argument that the high apparent conductivity at high temperatures results solely from an increase in the conductivity of the matrix. In experimental measurements, it has been found that the apparent thermal conductivity of the char is higher on cooling than during heating (see fig. 12). Also, values obtained on reruns of the same specimens are usually higher at low temperatures and converge to about the same values as before at elevated temperatures. This behavior is illustrated in figure 13 which shows data for phenolic-car- bon chars. The characters of the curves for chars made at successively higher temperatures led to the proposal, presented in a previous paper [14J , that the apparent conductivity of "flight" chars could be represented by the curve drawn through the measured values for the chars at their charring temperatures. These are the "Boxed Values" represented by the heavy curves in figure 13. Similar effects of temperature on the apparent conductivity of chars have been reported by others. Nagler U53 presented data for a nylon char formed at about 1800°F, which after being heated to about 3800°F during conductivity measurements exhibited a conductivity at about 600°F which was about three times higher than the original value. Similar data (presented in Reference [l6j ) have been presented by Neubert, Royal and Van Dyken for heat treated pitch coke mixtures. There are two arguments which suggest additional graphitization with time at temperature exposure and subsequently higher matrix thermal conductivity values. The first is measurements on a plasma char made in this work which indicated that the side of the specimen which was exposed to the highest temperature for the longest period of time had a considerably higher thermal conductivity. The second argument pertains to some experimental data taken by Neubert, Royal and Van Dyken, which is presented in Reference [l6j . The material for their experiments consisted of Whiting coke and Barrett No. 30 pitch extruded in long bars and then reimpregnated with pitch. Room temperature measurements were made on heat treated specimens from this material in an axial heat flow apparatus. Separate samples were heated to 2812°F, 4352°F, and 5432°F. The hold times were not specified. The room temperature thermal conductivities of these samples were 485 x 10~ 5 , 1250 x 10~ 5 , and 2330 x 10" 3 Btu/ sec-ft-°F for the heating temperatures of 3812°F, 4352, and 5432°F, respectively. Further, they stated that the lowest conductivity specimen was typical of a product for which the graphitization had barely begun. This statement means little quantitatively, and the applicability of these data to phenolic- nylon char is questionable. However, these results correlate somewhat with the results of the thermal analy- 437 sis and indicate that "barely begun" graphitization can give high matrix conductivities. If barely begun graphitization means only a slight change in X-ray diffraction patterns from the carbon structure, these results may be significant. From the foregoing discussion it seems reasonable to conclude that the effective conductivity of the char might increase by a factor of 2 or more during the course of the measurements due to a change in the conductivity of the matrix. This behavior would account for a significant part of the discrepancy between the experimental and theoretical curves in figure 17. Exactly where in-flight values for the matrix conductivity would fall depends on the effects of time at temperature. An in-flight char would most likely not have as high a thermal conductivity as measured in the steady- state apparatus unless graphitization takes place very rapidly once a given temperature level is reached. Our measurements of the variation of conductivity across a char indicated that this effect may be rapid, occurring over a 120 second time interval. A knowledge of the effect of time at temperature is vital to predictions of the in-flight conductivity. If the time required for graphi- tization to occur is extremely small, then the thermal conductivity of the char is essentially dependent on the temperature alone. In this event, the thermal conductivity measured in the steady- state appara- tus represents the thermal conductivity of the in-flight char. It was mentioned that the discrepancy might also be due to the failure of the thermal model to account for radiant heat transfer in the char. This might result from the use of too short a radiant length x in the calculations or from transparency of the char matrix. Literature data as well as the glassy appearance of the char structure support this latter possibility. If the radiant energy trans- mitted through the char is assumed to be the difference between the experimental and theoretical curves in figure 12, (the theoretical curve used was x = 50, k m = 231, and it was shifted down to agree with the experimental curve at low temperatures), then a psuedo-transmittance for the char can be computed from the equation K = y4aT (15) where K r 7 CT T L the apparent radiant conductivity psuedo-transmittance Stefan- Boltzmann constant mean temperature of the char specimen gage length Values of y computed in this manner ranged from 0. 03 at 1500°F to 0. 13 at 5000°F for a radiant length, x, of 50/i. For x = 500/i, y ranged from 0. 02 at 2000°F to 0. 07 at 5000°F (table 5). Note that this pseudo transmittance probably is composed of both direct radiation through pores and trans- parency through the matrix membranes. The character of the portion resulting from transparency can be compared to the behavior of glass, which is practically opaque to radiation of long wave lengths (> 4. 5u) and is fairly transparent to radiation in the wavelength range from 1 to 2. 75/-t [)- r f\ • Thus, following this parallel, the psuedo-transmittance of the char might be expected to increase with increasing temperature not only as direct pore radiation becomes predominant but also as the spectral distribution of radiant energy shifts from longer to shorter wavelengths. Table 5. Psuedo-transmittance Temperature °F Radiant length, x, microns 50 500 1000 2000 3000 4000 5000 0.00 0.05 0.09 0.11 0.13 0.00 0.02 0.04 0.06 0.07 438 One other parameter which might affect the conductivity is time at temperature. The elevated temperature data presented in this paper were obtained in steady- state measurements during runs of several hours. "Flight" chars on the other hand are formed within periods of a few seconds. If the matrix conductivity remains constant during the short flight times, then the effects of temperature observed in the laboratory measurements are inapplicable. Some measurements on chars exposed to varying times at temperature are needed here. Data found here plus that reported by Nagler (15] indicate that soak times of several hours exert much less influence than does temperature in changing the conductivity. 7. Conclusions At low temperatures heat transfer through phenolic- nylon chars occurs primarily by solid and gas conduction. These two modes are inseparable in that the gas thermally shorts delaminations in the chars and effectively increases the conductive area. At temperatures below 1000°F, the effective conductivity in a nitrogen or helium environment may be twice that in vacuum. The difference in the two values ranges from two to three times the thermal conductivity of the gas. Equation (14) appears to adequately explain the effect of gas conductivity. At elevated temperatures eq (14) predicts values which are lower than experimental data on labor- atory chars by a factor of two or more. This discrepancy is probably due to (1) an increase in the ther- mal conductivity of the matrix with increasing temperature, (2) difficulty in proper definition of direct radiation transfer, and (3) transparency of the char at higher temperatures. Work on the higher tem- perature behavior is underway. Further investigation is needed to determine the effects of time at temperature on the conductivity of the matrix, and the nature of radiant heat transfer mechanisms in the char. After resolving these problems, methods of accounting for the process of ablation must be incor- porated in the mathematical model. 8. Acknowledgement This work was sponsored by the National Aeronautics and Space Administration, Langley Research Center. 9. References [1] Engelke W. T. , Pyron, C. M. , and Pears, C. D. , Thermophysical properties of a low-density phenolic- nylon ablation material, NASA CR- 809 prepared by Southern Research Institute (1967). [2] Flynn, D. R. , Thermal guarding of cut-bar apparatus, Conference on thermal conductivity methods (1961). (Referenced with permission of author. ) [3] Robinson, H. E. , Thermal conductivity refer- ence standards, proceedings of The Second Conference on Thermal Conductivity (1962). (Referenced with permission of the author. ) [4] Smyly, E. D. , Pyron, C. M. , and Pears, C. D. , An investigation of the mechanisms of heat transfer in low-density phenolic- nylon chars, NASA CR-966 prepared by Southern Research Institute (1967). [5] Young, R. C. , Hartwig, F. J. , and Norton, C. L. , Effect of various atmospheres on thermal conductance of refractories, J. Am.Cer. Soc. 47, No. 5, pp. 205-10(1964). [6] Russell, H. W. , Principles of heat flow in porous insulations, J. Am. Cer. Soc. 18 , No. 1, pp. 1-5 (1935). [7] Gorring, Robert L. , and Churchill, Stuart W. , Thermal conductivity of heterogeneous materials, Chem. Engr. Progress, 57, No. 7, pp. 53-9 (1961). [8] Francl, J. , and Kingery, W. D. , Thermal conductivity: IX, Experimental investigation of the effect of porosity on thermal conduc- tivity, J. Am. Cer. Soc, 37, No. 2, pp. 99- 151 (1954). 439 [9] Loeb, Arthur L. , Thermal conductivity: VIII, A theory of thermal conductivity of porous materials, J. Am. Cer. Soc. , 3J , No. 2 pp. 96-9 (1954). [10] Powers, A. E. , Conductivity in aggregates, proceedings of the Second Conference on Thermal Conductivity, Ottawa, Ontario, pp. 280-309 (1962). [11] Jakob, Max, Heat Transfer, John Wiley and Sons, Inc. , New York (1957). [12] Kreith, Frank, Principles of Heat Transfer, International Textbook Co. , Scranton, Pa. , pp. 318-19 (1962). [13] Verschoor, J. D. and Greebler, Paul, Heat transfer by gas conduction and radiation in fibrous insulations, Transactions of the ASME, No. 74, pp. 961-68 (1952). [14] Pears, C. D. and Pyron, CM., The thermal conductivity of ablative materials by the "boxing" analysis, presented at the Fifth Thermal Conductivity Conference, Denver, Colorado (1965). [15] Nagler, Robert G. , The thermal conduction process in carbonaceous chars, NASA TR 32-1010 prepared by Jet Propulsion Lab. , (1967). [16] Goldsmith, Alexander, Waterman, Thomas E. and Hirschhorn, Harry J. , Thermophysi- cal properties of solid material, Vol. I - Elements WADC TR 58-476, pp. I-C-3-c (1960). [17] Gardon, J. Am. Cer. Soc. , 44 pp. 305-12 (1961). No. 7, 440 (a) Char of 0. 88 porosity (19 lb/ft 3 virgin density) (b) Char of 0. 82 porosity (30 lb /ft 3 virgin density) Charring Direction t (c) Char of 0. 79 porosity (42 lb/ft? virgin density) (d) Char of 0. 79 porosity (42 lb/ft virgin density) Figure 1. Photomicrographs at 75X magnification of the three different densities of phenolic- nylon char 441 r ■ - ■ ■ *p 1— j iin r<- Median pore diameter = 12 m cronS' T ■ TT T it T'TT'"' ■ T " r - . L . X 1 . i 1 - . _ A ^^ 100 Also within structure the . . . . ■following pore diameters on , m i were observed 194. 1Rfi i 168 206 no, 242 uu _, o_ ;. .. ___. 250 . \ \ 1 <~ r j> 1 re 1 1 i 1 IT 11 TTTTTT 1 II 1 II 1 1 1 1 1 1 1 1 1 1 1 1 w ! , ' J- _j _, .J ::t „_i . g __. i , . i ._ .._. -J- i7n i i ; c ' (U | C . i co x.._. ...Sxzt - ip - . . -T 4 4: --X-X. __.4\_|^_. _r ! " I H U) j 1 n rn 1 i ! ! <" 2 u0 | c ■a :::.3 : x. . -3d tt x ttt ■ - <-• ' 11 S X-. - 4 - £ 4_ X X -0 it T + n i t a ' 1 1 2 ' C I 1 ! 1 Q. 1 | | 3 ! | ,-, | | | S 50 ! ni ! m x.§ it _ - - ,.- . - .- J_ 1 ; i 40 j:± ttr ._. __ it _ 1 4-h 1 ! j ' J_ L . on lit t it 30 ! 1 J 1 :: ~ :::px - -Xt ~ 4 " '--F ' -h ._ :tt 3::Lx __ iE ! 1 , 20 r ■"'■ . r_ ; ; 1 ' ' ' ; i 1 j 1 i 1 10 -■■-■■ Tt " . il it 1 X. L a. • -1 " .. .. i_. -t .. it it _„ .!_, _.._t _^ ±_ „_ t^™ n* 1 1 1 --< r ; _U- 1- n _._-. _L — ———H — :^— — 1: :____. ::i__.„..::i_--_-_.^^;- - 1 - 20 40 60 80 Pore diameter - microns 100 120 Figure 2. Histogram of pore diameters in the phenolic nylon char of 0. 82 porosity (30 lb /ft 3 virgin density) 442 tiHtMttiMP -X ■x- -l ,J -> (0.0254 m) Upper column heater ■Zirconia spacers Upper reference thermocouple Pyroceram reference Upper reference thermocouple Specimen thermocouple. Fiberfra ecimen thermocouple Lower reference thermocouple Pyroceram reference X Lower reference thermocouple Zirconia spacers Lower column heater * Upper reference thermocouple Upper reference * thermocouple ^Specimen "hot" v, face thermocouple ^Specimen thermocouple ^-Specimen thermocouple Specimen''cold" face thermocouple X^Lower reference thermocouple X Lower reference thermocouple Heat flow (a) Standard buildup for thermal conductivity evaluations Heat flow (b) Buildup for specimen 42-3 (second run) ■ attempt to measure variation in thermal conductivity across thickness Figure Z. Schematic diagram of comparative rod apparatus specimen configuration 443 750 650 600 550 500 £450 a 400 350 300 250 200 1 -- ■ t > \ \ i s Temperature gradient through upper Pyroceram reference ^, ■ i i i 1 j ; I i 1 J - 1 : ~ ^ \ 1 s ki ■ 1 Temperature gradient \ > \ through Fiberfrax pt 1U V " 1 1 1 1 1 1 1 1 1 1 1 1 ii n i > k V L i Temperature gradient through upper 0. 236 cm of specimen 1 I i i . Temperature gradient through . iwj. *-xxj v/i o^v,V4.mv>u ; 1 V Temperature gradient through lower 0. 236 cm of specimen \ , * * s\ V rhr-nncrh TPi'hprfrav nari ■ T V Temperature gradient through Lower Pyroceram reference \ ^ ~x_ \ I \ jz V !S 81 1 J. r 1 — i Guard profile -Note: This temperature profile ^is typical except for temperature I drops through Fiberfrax pads, "~ which were not used on other runs 1 1 1 Profile for specimen 42-3, - 760 torr, helium purge at — 500°F mean temperature \Z Distance - cm - t i i 3 75 p ? 54 5. 01 6. 25 ■- L. 27.: ! 645 589 533 as > S > ■r-i +-> o 3 fc TJ o c 1 o o «M r— i o cti 0) B UJ u 1 a> si m +-> c i cu o h r-i nj a 0, < .s >^ +J •r-l > -W CJ 3 Tj- T) a 1 o 1 I-H o cd (1) s «) h 3 (!) -4-> .13 PQ m a i (i) O Fh i-1 td a a «! O n o o O ^H si PQ -^ 10 c o •r4 fe o i Ti T, «-. a o o o cu W 1—i a 3 m (D r. ilS c I-l u c Oi tt a, > •d fe 3 ^2 ', a •+-< o • o u i— t a> ■, +-» •r-t > •|-t In a 1 T) a o o 0) tn ai s 3 in (1) CE CI +J ID O a> .— 1 ^ ri nt -r-l a a in jd V-/ OU 1 1 In helium - - A ^fl-fi 1 ■ U 40 ■ ' ... . _i-* 2± ; m , — -- •r-l C) fe 3 T3 ^_, C <4-l O 1 o (1) nJ ai E 3 PQ x! 115 +-> 1 ti o (J .— 1 a; h a rt ■r-l a a < c ■r-J >"i > +j fH U fl 1 T3 tr-l c o c; o 0) r-H to nl - s 3 h DQ u S3 -f-> o c i— i (1) fn C cti •i-H a a < •S >, +-» -t-i > -r-i fe -f-> 9 ' o • 1 T5 i o D C o u 1— 1 a! 3 B in n Olio XI ••-o ■MtH C (1) C h nl a a 40 20 I I I I.I I I I Symbols O A a I I I I I I.I . I. J. I I IIUJJ..I I I I I I I I I 1. 1 I I I I Specimens 42-3 (Second run) (Fiberfrax pads) 42-4 42-5 256 In helium 2.50 1.25 40 Symbols O d A 200 Specimens 400 600 800 Temperature - °F 1000 922 1200 42-3 (First run) 42-3 (Second run) (Fiberfrax pads) 42-3 (Third run) 42-4 42-5 In nitrogen 200 400 600 800 Temperature - °F 1000 1200 40 .Symbols O 6 A D ■• ' : — ' ■ tt± Specimens M. T _ 42-3 (Second run) (Fiberfrax pads) 42-3 (Third run) 42-4 42-5 In vacuum H (0. 01 torr or less) H-f +B-B 1.25 O I £ 2.50 o I S 1.25 £ 2.50 200 400 600 800 Temperature - °F 1000 1200 Figure 7. The apparent thermal conductivity of phenolic- nylon char of 0. 79 porosity (42 lb/ft 3 virgin density) in vacuum, nitrogen, and helium 447 ity (l9 1b/^ §.fe ■o°. c £ ° V ° o f-H (J) cd co u ^ CO 31 a d h- +j <» W ■B o -** O C! >-« <1> ^ at a Q. < 600 800 Temperature - °F 1000 1200 :±:::::q::::q:::: :: ±± ± ._ 4 20 ' I 1 1 -f-'o 82 Porosity (30Jb/fg)_: - „.±.±±.±— =fe | 4- B W4- 1 . /w j v - "1 1 " TTT"' i — 1""""f "--=— -f "Tn79 Porosity (42 lb/it ,i._ ' i ' "^i 1 1 1 — ---• ii 1 +f -+ ' riq lh/ft 3 ) RR Porosity U» id/" ; " 10 .____ __ __ __ __ _L__ __ i_ In nitrogen Temperature - °K 1 *. J ill ' 256 367lt ^78 589 700 811 "92 nil III! II III l,l,|, JJJ J 1.25 0.625 200 400 600 800 Temperature - °F 1000 1200 i on , . . , , £\l 1 — o 82 Porosity (30 lb /ft 3 ) i i i i i I l | i 'I i i,| i-H-HH -■ ' 1--- Mill M |l II II M j 1 1 J J ! l I- 1 ' ' _ I 0. 79 Poj "osity (42 lb/it i &pe<-. ■*« " ::::::::::: : - :: 4±- = i ---i^p-T" - "" in ^M^ _ _ — — H "|" 10 .-1-1 **' I-, a lv, /ft 3 ) „-!+\r liy ID/ *■ L ' 'J ' ' 0.88 Poi . oo»j Ml" "in vacuum i i l I M ~ ~ ' ""' " ' Temp erature - is. i 1 ' 1 ' 1 hi I *>l "7 A 70 •^RQ nr\r\ a • Q9' 25Ci 367 47o| oca ] 700 tsii "^' 1 1 i "TT TIM n oL J ,.„!,,! .±±_±___±LL±_ . J 1.25 0.625 200 400 600 800 Temperature - °F 1000 1200 Figure 8. Composite plot of the apparent thermal conductivities of the three phenolic- nylon chars in vacuum, nitrogen, and helium 448 Mo-^/AV do-U-oas^njg 9-OT U T ^TAijonpuoa lEtuaaqx 449 . — i cfl X ai 41 «t-* X fl a 4) RJ ■o a> rt fl s 5 m xT Is c e 0) u rt a; In •i-i QJ r-t y •5 -a o £2 to a! (U to a *§sg X 4> rt a 3 to o TJ P 01 £ o 3 W ni £ ••* a C 3. "a u -H O r-l 0) ■* c 2 *- to B C cd ™ a u to O 3 X! p 3 to O a C r« <" "^ o -c tm fl c 3 J3 «<-£ « 5 tfl w> 3 bo o 5P x: x: o S ^ S 3 O s S 5 bD u xi m oj o o u o o 3 ft JC *■" .P x: J3 O U ao -t-> o hO ■" Jp £ Sh 3 P 3 a -4 2" p rt a> o o n O 3 o P O XI o § C £ * 3 S Ja 3 .H o s flow channel [one •action of flow cha: ow channel [one of of flow channels w •!-■ O 3 i3 J j3 a '^_. co d 3X^3 •2 P .2 fl -<-■ O O " o § p ■§ u 3 X P 3 O U o T3 tO Ti m X CO rt -2 o P rt O y to c a P fl rt ^ a t O Sh /O cfl f 5 S 7 ^ P T5 3 r^ S o to rt o o •r-l 4-9 rt C •t4 i — i 4> X i i to to s rt O s o tO X ,« J- 1 p t; r "a 2 -* SS fa • W X s 11/ L 3 X O E ■? TY I J - X 1+ ^ fl o o CO X fj&8i rt u • rH > XI a. J i > si O C iS CJ 3 X X! O + n ► 6 < tiD< ► UJ < 1^ < cu 1 §s$8 < ^ IffiB t— i c T ark CD o O l raw X o T-( i, f 1 r e CU 3 I M'"7/\!| j\ rt Ih W) TT^ T •*4 v • 4> X! s / \ , *7 to T3 Eh A » 2 X! OJ 0*1 I P-, rt 3" >-^ fl **^ in -C c at ■fH 3 O a} ' — rn S 5 P =. -P 5 ti p o co 2 ■a 5 a C u o c i 450 ft o I £> u "£ m > a 2 a, o < 5 C O O •S 20 10 I I i I I I I I I I I I I I ll I i I I I I I I I i I - Specimen 42-4 - Values used in equation (L4 ) : P = 0.79 c = 0.071 ; F = 0.400 k m = 155 x 10 -5 Btu/sec- ft- TT 256 op D Measured in helium ■ A Measured in nitrogen O Measured in vacuum fe The solid lines represent the values predicted by equation (14) for the labeled environment 1.875 1.25 0.625 o I E 200 400 600 800 Temperature - °F 1000 922 1200 o 0) w — — p id a pq u m <\> i a o ■*-» i-H c 01 s-< ;>i a •"^ u. > LL < o 3 -o o 30 20 £ io I I I I I I I I I I I I I f'l I I I I I I I I I I I ! I Specimen 42-3 Values used in equation (14^ P =0.79 c = 0.027 F = 0.590 km = 185xl0" 5 Btu/sec-ft mrm^ O Measured in helium ■A Measured in nitrogen- O Measured in vacuum - The solid lines represent the values predicted by equation (14) for the labeled environment 256 367 i . 478 rr f- -^Temperature - °K ^-- ^ j - J I 1 — I — i — ; — | — I — I — : — I — | — 1 — I — I — I — F— 589 700 i r 811 i i i 1.875 1.25 0.625 o f B - 922 —Jo 200 400 600 800 Temperature - °F 1000 1200 Figure 11. Comparison of the predicted variation in apparent thermal conductivity with temperature with the measured values for specimens 42-3 and 42-4 451 fe PQ 256 810 Temperature - °K 1367 1920 2480 m 3035 3591 TTT I I I I I I I II I I I I I I I I I I I I I I I I I i i i i i r i i i Values of x and k shown on curves are in microns and 10" 5 Btu/sec-ft-°F, respectively. 80 70 60 b 50 c +-> > o 3 -o a o o I— I S U x; c a 40 30 20 10 x = 500 ( 10 pore dia) k m = 694 "J 5, 00 ° (polycrystalline gra phite; 4.375 625 - 3.75 3. 125 O 2.50 x = 50 (1 pore dia) 1. 875 k m= 231 (Corresponds approximately with values reduced from eq (14). ) ±t£ 1.25 0.625 3000 Temperature 600t Figure 12. Comparison of the high temperature apparent thermal conductivity predicted by equation (14) for a nitrogen environment with prior data 452 200 ' ! i | 1 1 | | 1 1 1 | 1 1 1 1 1 | 1 1 1 I I || 1 I 1 | 1 1 1 1 1 I 1 . M | i .|.l I 1 | 1 _ .. |! | < | 1 1 | . m | : . Mil | 1 | | | | | | II 1 1 1 II 1 i 1 II ! ! 1 ! II 1 II II 1 -- — 1-|— r -n '- 4 — h4"— t — 1 — i — ' — 1 1 1 With Lamina - solid curves } f " : "M Hi T " '" "^ ' --- "IX- .."' TiJ ITT ± X ~ X X jx Mi " " " ' O 1 000°F Char 10U i-f i.~ 4 4000°F Char 1st Exposure 1 ' ^ 4000°F Char, 2nd Exposure i 1 ! Tl i i 1 M\ 1 | H^Hial Inflow Apparatus SLJTiA 1 D 2000°F Char, With Lamina t ; ..□ ± . -h-til- -\JdA/S J ' i ' irt? / i / A 3000°F Char, With Lamina ■-JM\W±~ --/r- lou ! ! ■ j v 4000°F Char, specimen 1, nun i. _^ > jr ! /' JV'""y\ -f±.. 4| ' *£-*/< 'Jfi-i-4-i f a' 1 'J>r y/\ \ /Tiiy \\\J . l-i-L J-- ' -y ' - >*' '/*.*'. )/ i 1/ , J/jf '>?*' ' x aI*'Y\ ' /'• •U^'V^' ' ® 4000°F Char (Comparative Rod) ._* X 4000°F Char (Radial Inflow) + Z* A ^yrrp \'/ £f\\ /'■/ -+" " ill 1 1 V* / „« 1 *>n , ' j'Vff'y ' <•'/ /[J'Ti.i/ LiSf" "w ' " "" "" ~"" ya> , i : .. _j _ + t^-K^'T/ / M /Q 11./ -»* / \A / * X \\j' L -i-s^ / w ^^ j* / Lt* - i l/ /l^t h j* | 1 | -J_ - - ,\-Z&r*^7 / 4000°F Char >T/ ^ L -?'--- -2§ = = ---:"Boxed Values" ' - tpT- - +i:^ s -~y yr: 7 7'j^grj^r' +t i / (with lamina) - ^r so ' " j '' A "\V\ [ v\\'/ LW--fiil 1 ! I L$r // j / 4 y \\X\ \Y ,-'T i 1 1 1' ' w*' 1 i ii« ? ^ 1 '/! / / ,'>^^ ?fl00°F Char'^ iT ji 7 Jll"' T ' '\ { Jstf fa t i i j -.--4 — *•• aJCty J* i **/ A *H JJS^V ./^ !/| Jt'^f'r*^ a ' (in 1 j^ W ( | - r + L -|_ J I 1 J Aj' — + ± ___ -j-ni 1 ^ s ^jj jjji' * :i 2000°F Char ;,..,.--■ J 2* .-^? ---■ *^-* ' «iJj.--rrHlSFr j»* ! ' 3oxed Values " . ! i i l.i, LL.j L_ ± j 1 1000 2000 3000 Mean Temperature - °F 4000 50©0 Figure 13. Apparent thermal conductivity of MX4926 (Carbon Filled 30 Phenolic- Carbon Fabric- see AFML-TR-65-133) Precharred at Progressively Higher Temperatures 453 Biological Specimen Holder for Thermal Diffusivity Determinations Floyd V. Matthews, Jr. Agricultural Engineering Department University of Maryland College Park, Maryland 20742 and Carl W„ Hall 1 Agricultural Engineering Department Michigan State University East Lansing, Michigan 48823 A specimen holder was designed to hold a cylindrical heat source and a cyl- indrical potato section in intimate and uniform contact with each other during heat transfer experiments. Appropriate boundary conditions were maintained and thermocouples were placed in desired locations. A finite difference computer program and the recorded temperature history were used to calculate the thermal diffusivity of the potato section. The heat transfer methods combined with the finite difference method of data analysis had the advantages of speed, simplicity, ease of application to other products, and accuracy. Depending upon test duration, approximately four heat treatments could be completed in an hour. No difficult laboratory prepa- rations or techniques were involved. Other biological products could be a- dapted to the specimen holder, and large masses were not required. Random errors were cancelled by the finite difference method, but a continuous error of +1°F in temperature measurement was found to cause an error of -0.5% or less in the value of thermal diffusivity. Key Words: Biological specimen holder, computer program, finite difference, potato heat treatment, thermal diffusivity. 1. Introduction Commercial heat processing of vegetables is usually a transient process during which the phys- ical and thermal properties are changing (3, 7, 11). If an engineer is to calculate the net energy requirements of the process, he must know the values of the thermal properties over the working range. In the past, he has found it necessary to make vague determinations of the thermal properties and also to assume that these properties remain constant during the processing. With the present development of the science of heat transfer,, an engineer should not have to rely on these estimates and assumptions. Because some biological materials tend to undergo physical and/or chemical changes as heat is applied or removed (4, 9), a determination of the thermal properties should be fast enough to precede such changes. Water is evaporated when heat is applied to most biological materials. If much heat is applied, the final material is not the same as the initial material; the loss of moisture may cause a change in the thermal properties. Some researchers have used a guarded hot plate, a steady state device, to determine the thermal conductivity of meat (5)'. Because the guarded hot plate method is of long time duration, at normal processing temperatures a biological product would probably be cooked before the thermal conductivity could be determined. Others have used a modified Cenco-Fitch apparatus, a transient state device, to determine the thermal conductivity of meats, fruits, and vegetables (2, 10). The modified Cenco-Fitch method duplicates the transient process but is subject to errors of approximately 5% (2). ^-Associate Professor and Professor and Chairman, respectively. o Figures in brackets indicate the literature references at the end of this paper. 455 Therefore, when the thermal and/or physical properties of a vegetable vary with the heat absorbed, one must use other means to determine them. 2. Biological Specimen Holder 2.1 Description Beck (1) has developed a method for simultaneously determining the thermal conductivity and specific heat of a material. A heated piece of material with known thermal characteristics is held in uniform contact with material of unknown thermal properties. The temperature histories of the two materials are used to calculate the thermal properties of the unknown material. Based on the idea above, a biological specimen holder was designed to hold a cylindrical heat source and a cylindrical potato (solanum tuberosum) section in intimate and uniform contact with each other for the duration of an experiment, figure 1. Each half of the specimen holder was made from a 0.076m (3 in.) length of standard 0.1524m (6 in.) diameter steel pipe. The stationary half of the holder was welded to channel iron,, A flat base plate and angle iron rails were also welded to the channel iron. V-grooved wheels mated with the angle iron rails maintained accurate alignment in all planes between the movable and stationary halves. The longitudinal displacement of the V-grooved wheels from each other facilitated the rapid installation of the movable half onto the rails. During the ex- periments a constant pressure was maintained at the heat source-potato section interface by rotating the specimen holder 90° and supporting the holder on the channel iron, figure 2. A microswitch and its trip lever were fastened to the stationary half and movable half respec- tively, figure 1. When the halves of the specimen holder were brought together for an experiment, the microswitch completed a circuit for an electric interval timer and for an indexing pen on each recording potentiometer. The microswitch was adjusted to close the circuit when the heat source and potato section made contact. The insulation for the specimen holder was a polyurethane type with a low thermal conductivity. A 0.1016m (4 in.) length of insulation was machined to a snug fit in the bore of each half of the holder. Mating grooves were machined into the halves of insulation to minimize any air current and convection heat losses. A hole, 0.0246m (31/32 in.) diameter and 0.0254m (1 in.) deep, was machined into the face of the insulation of the movable half of the specimen holder; the potato sections were installed in the hole for the experiments. A hole of the same dimensions was machined into the face of the insulation for the stationary half of the specimen holder; a silicone rubber heat source was installed in the hole for the experiments, figure 1. When an electric heating element was used for the heat source, the temperatures were too high for the stability of the polyurethane insulation; a glass foam insert was made to go between the heating element and the polyurethane insulation, figure 3. A thermocouple silver soldered to the face of the metal cylinder indicated the temperature at X = when the cylinder was used for the heat source. To minimize thermocouple conduction errors, the lead wires were placed across the face of the heat source and wound partially around the circum- ference of the heat source before entering the insulation of the biological specimen holder. Thermo- couples were also located in the body of the specimen at X = X and at the end of the specimen noted as X = infinity in the boundary conditions. 3. Determination of Thermal Diffusivity 3.1 Finite Difference Equation Considering a body with the following characteristics, 1. X independent of position and temperature, 2. one dimensional heat flow in the X-direction, and 3. no internal heat generation, the general heat conduction equation reduces to: dVl 30 ax z a>zf CD A body subjected to a heat transfer process may be divided into nodal points that are a distance Ax apart. About any node an energy balance can be written and expressed in finite difference form (8). Temperatures in the body can be expressed with respect to position and time by means of sub- scripts and superscripts n and t respectively. For one dimensional heat flow in the X-direction in a finite body, nodes are located at X = 0, X = X, and X = L. Temperature boundary conditions may be 456 specified at X = and at X = L. In finite difference form, eq (1) becomes: "/?-/ fr/7 + "n*i &» (AX) Z Ai (2) Equation (2) can be solved for the temperature at node n for time t = t+1. A t+l n (3) where : (Axy 4 At a dimensionless number (4) A computer program using finite differences and based on the method of Beck (1) was used to calculate thermal diffusivity from temperature measurements. The basic operations of the program were: 1. The magnitude of A X was determined. 2. At the first and last node, a programmed time step was used to calculate temperatures with respect to time: t, t+l,...t+i where: (t+l) - t = programmed time step (t+i) - t = time interval of data 3. The finite difference eq (3) was used with an assumed a to calculate temperatures at each node. Calculations were made in two "passes"; an "odd pass" proceeding from first to last node and an "even pass" proceeding from last to first node. 4. The method of least squares was used to minimize with respect to a the difference between the calculated and measured temperatures. 5. The change in diffusivity was calculated and added to the assumed a to obtain a new value of a. 6. The new value of a was used in an iteration procedure to determine a more accurate value for a. The above program was based on the following assumptions: 1. Initial and boundary conditions of the body were: a. (X,0) = b. (0,t) = 8i c e (oo ,t) = o 2. One dimensional heat transfer. 3. Constant a with respect to time and temperature. 4. The last node was at X =eo. 3.2. Discussion of Results The biological specimen holder was used to obtain heat transfer data from approximately 150 tests on potato sections. The equipment has also been used to obtain similar data on other agricultural products. The thermal diffusivities of the potatoes, as calculated by the computer program, were within the range of values of other biological materials (6). Thermal diffusivity data could be obtained during the first 50 seconds of heat treatment; before moisture changes and physical reactions became a problem. This overcomes one of the objections to the guarded hot plate when used at cooking temperatures for biological materials. Because the temperature boundary conditions at X = did not have to be constant, heat loss from the sides of the heat source to the insulation was not critical. Many of the root mean square values of the difference between the actual temperatures and the calculated temperatures were less than 0.28°C(0.5°F). The recording potentiometers had a permissible error of 0.33°C(0.6 F). Random errors were cancelled by the finite difference method, but a continuous error of + 0.55 C (1 F) temperature was found to cause an error of -0.5% or less in the value of the thermal diffusivity. In the literature reviewed, errors of 5% were not uncommon in the determination of thermal properties of biological materials. 457 The heat transfer methods used in the experiments combined with the finite difference method of data analysis had the advantages of speed, simplicity, ease of application to other products, and accuracy. Depending upon test duration, approximately four heat treatments could be completed in an hour. No difficult laboratory preparations or techniques were involved. Other biological products could be adapted to the specimen holder and large masses were not required. 4. References (1) Beck, James V., Calculation of thermal diffusivity from temperature measurements. Transactions of the ASME, Journal of Heat Transfer, Series C, 85, 181-182 (1963)„ (7) Personius, Catherine J. and Sharp, Paul F., Adhesion of potato-tuber cells as influ- enced by temperature. Food Research 3_j_ 513-524 (1938). (2) Bennett, A. H., Chace, Jr., W. G. and Cubbedge, R. H. , Estimating thermal con- ductivity of fruit and vegetable components — the Fitch method. ASHRAE Journal, 4, No. 9 80-85 (1962). (3) Kaczmarzyk, Leonard M. , Fennema, 0. and Powrie, William D. , Changes produced in Wisconsin green snap beans by blanching. Food Technology 17, 943-946 (1963). (4) Matthews, Floyd V., Jr., Thermal diffu- sivity by finite differences and correla- tions with physical properties of heated potato. Thesis for degree of Ph D, Michigan State University, East Lansing (unpublished) (1966). (5) Miller, Herbert L. and Sunderland, J. Edward, Thermal conductivity of beef. Technology 17, 490-492 (1963). Food (8) Schneider, P. J., Conduction Heat Transfer. Addison- Wesley, Inc., Reading, Mass. 395 pp. (1955). (9) Simpson, Jean I. and Halliday, Evelyn G. , Chemical and histological studies of the disintegration of cell-membrane materials in vegetables during cooking. Food Research 6j_ 189-206 (1941). (10) Walters, Roger E. and May, K. N., Thermal conductivity and density of chicken breast muscle and skin. Food Technology 17, 808- 811 (1963). (11) Weiser, T. Elliot and Stocking, C. Ralph, Histological changes induced in fruits and vegetables by processing. Advances in Food Research 2. 297-342 (1949). (6) Parker, Rayburn E. and Stout, B. A., Thermal properties of tart cherries. American Society of Agricultural Engineers Paper No. 66-337. Saint Joseph, Michigan. 22 pp. (1966). Figure 1. Biological specimen holder with heat source mounted in the center of the white insulation, and with a microswitch mounted on the cylindrical frame, 458 Figure 2. From left to right; recording pen potentiometer, biological specimen holder in testing position, variable transformer, and a recording pen potentiometer, Figure 3- Biological specimen holder just prior to closing for a heat transfer test 459 Thermal Diffusivity of Actinide Compounds 1 J. B. Moser and 0. L. Kruger 2 Argonne National Laboratory Argonne, Illinois 60439 The thermal diffusivity of the carbides of uranium and plutonium was measured from 500 to 1500°C by means of the USNKDL flash technique. A ruby laser was used as the pulsing source while the disk-shaped specimens were heated in a high vacuum resistance furnace. The rear surface temperature of the specimen was monitored by an optical detector which contained a dual lead sulfide photoconductor in a temperature compensating circuit. The variations in thermal diffusivity as a function of effective carbon content and uranium- plutonium ratio are discussed and compared with results of other investigations. Key Words: Flash technique, heat conductivity, plutonium carbide, thermal diffusivity, uranium carbide, uranium-plutonium carbide. 1. Introduction The monocarbides of uranium and plutonium are of considerable interest as fuels for fast reactors, therefore the thermal diffusivity of these materials was measured to high temperatures using the well- known technique developed by Parker and his associates at USNRDL [1] . This method was found to be ideally suited for glove box adaptation necessary in plutonium research. In previous work by the authors [2,3] at temperatures from 25 to 700°C, the heat capacity was also measured and the thermal conductivity was calculated. Heat capacity measurements by this technique at high temperatures have not been successful to date, however data for UC and PuC are available [4,5]. The reactor designers are particularly interested in the diffusivity parameter per se at high temperatures, because nuclear fuels are subjected to transient heating in reactor excursions and their thermal performance is then governed by the thermal diffusivity. In order to evaluate the amplitude and phase response of the center temperature to these transient disturbances, computer programs using the diffusivity parameter are useful [6] . The thermal diffusivity of UC of variable density, and PuC, as well as the (Uq # 8^ u 0. 2^ s °lid solu- tion was measured. The sintered specimens wer'e of approximately 80% theoretical density and contained small amounts of oxygen and traces of nitrogen. They were prepared by crushing, cold-pressing and sintering arc-cast material. All samples were characterized by chemical analyses, geometrical density measurements and micrographic examinations. 2. Experimental 2.1 Apparatus The technique and apparatus used in this work have been described adequately [1,7] but a few of the experimental details are given below. A block diagram of the apparatus is shown in figure 1. The specimen was placed in the high temperature vacuum furnace and the front-face was illuminated evenly by the ruby laser mounted on the glove box directly above the furnace. No light from the laser flash was allowed to pass to the infrared detector which was located below the furnace outside the glove box. The water-cooled laser head contained a 1.43 cm ruby rod pumped by a helical flashtube. The system was capable of 30 joules output and was usually fired at 10 to 20 joules with a pulse duration of 500 microseconds. The delay circuit of the oscilloscope was used to trigger the laser after the CRT trace had started. The surface of the disk was coated with colloidal graphite to maintain a uniform emis- sivity. The lead sulfide detector contained two cells but only one was exposed to the radiation emanating from the specimen. The detection system contained filters which limited its spectral range from 2.0 to 2.6 microns. Collimating lenses made it possible to focus on an area of 0.16 cm diameter on the back of the sample, excluding all radiation coming from other portions of the furnace. Figure 2 Work performed under the auspices of the U. S. Atomic Energy Commission. 2 Associate Ceramist and Associate Metallurgist, respectively. Figures in brackets Indicate the literature references at the end of this paper. 461 shows a circuit diagram of the dual cell system. The temperature-compensating circuit kept the cold voltage constant in spite of ambient temperature changes. The output of the circuit was about 10 volts which was bucked with a null circuit and reduced to less than 0.1 volts. Figure 3 shows the calibra- tion curve of the instrument determined from a black-body calibration furnace of standard design. Filters of 75% and 95% absorption extended the low temperature range of the unfiltered unit. The sample temperature was measured by means of a digital voltmeter of 10 megohm impedance to the nearest millivolt. A signal strength of 3 to 5 mV per °C was obtained from the PbS detector which made further amplification unnecessary both for the temperature and the diffusivity measurements. The oscilloscope was used for all diffusivity measurements in the floating differential mode of essentially infinite impedance. The near linear range of the calibration curves was usually used for the diffusivity mea- surements in order to prevent distortion of the oscilloscope traces. Many measurements were also taken where the output of the ranges of the lead sulfide detector overlapped. Temperatures above 800°C were also measured by means of a disappearing filament optical pyrometer which was sighted on a black-body hole in the sample holder located adjacent to the specimen. The temperatures reported were those of the pyrometer reading above 1000° C and those of the infrared cell below this temperature. When the readings were taken at the same time, the maximum difference between the two devices was 10°C. A standard diffusivity disk taken from an alumina thermal conductivity specimen obtained from Dynatech Inc. was used to calibrate the apparatus. Figure 4 shows the results of these diffusivity determinations com- pared to the values calculated from the Dynatech thermal conductivity curve. A maximum error of + 7-1/2% and an average accuracy of + 5% was obtained; a calculation of the experimental precision by summing the errors in the various measurements also yielded + 5% error. 2.2 Specimen Characteristics All the compounds investigated were in the form of disk-shaped specimens 1.83 cm diameter and approximately 0.5 cm thickness which were subsequently lapped down to 0.2 to 0.3 cm thickness for the measurement. The carbides were prepared by fusion of 99.5 wt.% pure metal with spectrographically pure carbon in an arc furnace [8] . The oxygen and nitrogen content of the arc melted material was usually below 0.02 wt.%. The sintered specimens gained in oxygen and nitrogen content during the fabrication process as shown in Table 1. Table 1. Specimen characteristics. Material UC Arc-Cast UC Sintered 1800° C U 0.8 Pu 0.2 C Sintered 1800°C PuC Sintered 1400°C Density % Theoretical Carbon wt.% Oxygen wt.% Nitrogen wt.% Effective Carbon wt.% Effective Carbon at.% 99.0 4.93 0.0117 0.0021 4.94 50.8 78.8 4.69 0.106 0.0228 4.79 49.9 74.8 4.66 0.202 0.0206 4.83 50.2 83.8 4.12 0.567 0.0440 4.58 48.9 In addition to carbon, the major impurities of oxygen and nitrogen must be considered since they dissolve in the monocarbide to a considerable extent. A useful quantitative concept here is the equi- valent carbon content obtained by summing the oxygen and nitrogen contents as follows: Equivalent wt.% C = wt.% C + 12/16 wt.% + 12/14 wt.% N Single phase uranium monocarbide and mixed monocarbide of U/Pu ratio 4:1 can only be obtained if the nonmetal atom content does not exceed 50 at.%; for PuC the upper limit is 46.5 at.%. The arc-cast uranium carbide had an equivalent carbon content of 4.94 wt.% C (equal to 50.8 at.% C) . A small amount of second phase consisting of UC2 was found in the grains of this specimen (see figure 5) . The sintered uranium carbide specimen contained more oxygen but less carbon than the arc-cast material. Whereas the effective carbon content of the arc-cast specimen was only 0.01 wt.% higher than the actual carbon content, this difference in the sintered UC amounted to 0.1 wt.%. Thus an effective carbon con- tent of 4.79 wt.% is obtained on the sintered material which is equivalent to 49.9 at.% C. This results in a formula ratio of U0.50C0 49 < - ) 0.01 if the oxygen is considered separately. A photomicrograph of this sample is shown in figure 6 which gives a good indication of the pore configuration. No second phase was observed which is in good agreement with the calculated composition. Compounds containing plutonium generally have a greater affinity for oxygen, and the increase in oxygen pick-up with increasing plutonium content is shown in Table 1. The mixed carbide of nominal U/Pu ratio of 4:1 contained 0.2 wt.% oxygen. The calculated effective carbon content of this material was 4.83 wt.% which is considerably greater than the actual carbon content of 4.66 wt.%. The effective 462 atomic percentage of carbon is 50.2 resulting in stoichiometric (Uo.80P u O. 20)C]..00 or (U,Pu)q.50Co. 48500.015 if tne oxygen is taken into account separately. The pore configuration of this specimen is shown in figure 7. Since the nonmetal to metal ratio does not exceed one, no signi- ficant amounts of second phase would be expected and are in fact not observed in the photomicrograph. Since the plutonium monocarbide was expected to pick up oxygen during specimen fabrication, the carbon content of the arc-melt was purposely kept low. In spite of this precaution, the oxygen pick-up brought the effective carbon above 46.5 at.%. The effective carbon content of this material was 4.58 wt.% which gives an effective atomic percentage of carbon of 48.9 at.%. The atomic ratios are indicated by the formula Pug. 5lCo.4450o.045 • A photomicrograph of this structure seen in figure 8 shows approximately 10% plutonium sesquicarbide (PU2C3) phase. 3. Results and Discussion All of the present materials except the arc-cast uranium monocarbide, contained enough oxygen and nitrogen to make a calculation of porosity and heat capacity unrealistic, since the lack of stoi- chiometry and second phase content of the material, would influence both of these properties. For this reason it was decided to analyze the data directly in terms of diffusivity rather than convert them to conductivity as is often done. The radiation heat loss correction developed by Cowan [9] was applied to all the data. The PuC specimen was first run in vacuum and it was noted that vaporization occured above 1200°C. This speci- men was subsequently reduced from an original thickness of 0.31 to 0.26 cm and tested in argon at 1.3 atm pressure up to 1350°C. Even though the thickness and heat loss parameters were now different, the data points fell on the earlier curve. As has been discussed by the authors before [3] the porosity correction for diffusivity is con- siderably less than that for thermal conductivity. For the present work, the Maxwell correction was modified to read 2 + P where a is diffusivity, P the porosity, and the subscripts T and M denote theoretical and measured values, respectively. This correction was checked for UC where the arc-cast specimen of 99% theoretical density was compared with the sintered one of 79%. At temperatures above 700°C, this correction was found to be in good agreement with the data and the corrected 79% UC values fall on the curve of the arc-cast UC. At lower temperatures this correction did not account for the increase in thermal dif- fusivity of the arc-cast UC; this is at present thought to be due to the presence of impurities. In the case of the mixed carbide and the PuC the porosity correction would increase the experimental values about 10%. Since the rank of the curves is not changed by these corrections the data shown in figure 9 were not corrected to theoretical density. The order of increase seen here has been found previously by Leary and coworkers [10] who measured the thermal conductivity of UC, (Uo.sP u O. 2)C an ^ PuC from 200 to 400°C. Considerable work has been done in electrical conductivity measurements [10-12] on well characterized specimens of various purity levels. Comparison of this data shows that the same rank is followed in electrical conductivity as has been observed here in heat transfer. The lower electron transport capability of PuC is probably due to the fact that it is a defect structure, whereas the UC has a much higher electrical conductivity and a higher thermal conductivity as well. PuC could also have a lower lattice component than UC for the same reason. A comparison in the lower temperature region can be made between the present research and the diffusivity work at the Mound Laboratory [13]. A radial heat flow method was used with the specimen being heated at a constant rate. Some bonding problems were encountered with this method due to the encapsulation of the specimens in tantalum. The UC work was carried out on 96% dense carbide, with a sintering aid consisting of 0.1 wt.% Ni. A carbon analysis of 4.78 wt.% indicated the specimens were stoichiometric; however, the X-ray analysis showed some faint U2C3, U02 and UC2 lines. The data agreed quite well with our work on the high density carbide around 400°C, but the thermal diffusivity decreases rapidly to a minimum at 700°C and then rises gradually. The 91% dense (Uo. 8P U 0. 2) < -'0.95 va l ue s fall on the same curve as our work when the data are corrected to 100% theoretical density. Both the Mound material and our specimen contained about 0.2 wt.% oxygen and both were substoichiometric in carbon so that good agreement was to be expected. A comparison with thermal conductivity work on arc-cast high-purity material at LASL [10] by a standard axial heat flow method can be made at 400°C, the highest temperature of their research. The UC data is the same as ours but the (Uo.8^ u 0.2)C and the PuC values are much higher than found in our work. The calculated diffusivity of (Uo.8 Pu 0.2)C and PuC measured at LASL was 5.3 x 10 -6 and 4.3 x 10~ 6 M 2 sec -1 , respectively. Our dif fusivities for these materials were 3.5 for (Uq.8P u 0.2)C and 2.4 for PuC. Chemically stoichiometric UC of a carbon content of 4.8 wt.% but with varying impurity contents was investigated by Hayes and DeCrescente [11] who used a radial heat flow method. The results of this work generally fall within the range of our UC data and it covers the temperature span from 1000 to 1500°C. The thermal diffusivity of their highest purity material containing 0.086 wt.% and 0.052 wt.% 463 N was 6.5 x 10~ 6 M 2 sec -1 at 1000°C compared to our value of 5.3 x 10~ 5 . At 1175°C their curve went below ours and leveled out at around 1300°C, staying slightly below our results. Above 1250°C there was very little difference between this high purity material and two specimens of combined oxygen and nitrogen content of 0.22 and 0.40 wt.%, respectively. 4. Conclusions Thermal diffusivity measurements were made over a 1000° temperature range (500 to 1500°C) by means of the same apparatus on materials of similar microstructure and density. The relative magnitudes and trends shown are realistic and have been confirmed in part by other work. The study on the PuC was carried to within 200°C of the melting point. 5. Acknowledgments Thanks are due to J. W. Thompson, R. E. Mailhiot and L. J. Kocenko for assistance in the various technical aspects of this research. The authors also wish to express their appreciation to H. T. Goodspeed and J. H. Marsh, Jr., of the ANL Chemistry Division, for performing the chemical analyses. 6. References II] Parker, W. J., Jenkins, R. J., Butler, C. P., and Abbott, G. L. , Flash method of determining thermal diffusivity, heat capacity, and thermal conductivity, J. Appl. Phys. 32., 1679 (1961). [2] Moser, J. B. , and Kruger, 0. L., Heat pulse measurements on uranium compounds, J. Nucl. Mat. 17, 153 (1965). [3] Moser, J. B. , and Kruger, 0. L., Thermal con- ductivity and heat capacity of the monocar- bide, monophosphide and monosulfide of uranium, J. Appl. Phys. 38, 3215 (1967). [4] Harrington, L. C. , and Rowe, G. H. , The en- thalpy and heat capacity of uranium monocarbide to 1200°C, PWAC-426, Pratt & Whitney (Jan. 1964). [5] Kruger, 0. L. , and Savage, H. , Heat capacity of plutonium monocarbide from 400 to 1300°K, J. Chem. Phys. 40, 3324 (1964). [6] Sanathanan, C. K. , Carter, J. C, and Brittan, R. 0. , Transient temperature distributions in fast reactor fuels with widely varying thermal diffusivity, ANL-7120, Argonne National Laboratory (October, 1965). [8] Kruger, 0. L., Preparation and some properties of arc-cast plutonium monocarbide, J. Nucl. Mat. 7, 142 (1962). [9] Cowan, R. D. , Pulse method of measuring thermal diffusivity at high temperatures, J. Appl. Phys. 34, 926 (1963). [10] Leary, J. A., Thomas, R. L., Ogard, A. E. , and Wonn, G. C. , Thermal conductivity and electrical resistivity of UC, (U,Pu)C and PuC, symposium on Carbides in Nuclear Energy, Vol. 1, Macmillan Co., London (1964). [11] Hayes, B. A., and DeCrescente, M. A., Thermal conductivity and electrical resistivity of uranium monocarbide, PWAC-480, Pratt & Whitney (1965). [12] Kruger, 0. L., and Moser, J. B. , Electrical properties and electron-configuration of the monocarbide, monophosphide and monosulfide of plutonium, J. Chem. Phys. 46^, 891 (1967). [13] Wittenberg, L. J., and Grove, G. R. , Reactor fuels and materials development plutonium research: 1964 annual report, MLM-1244 (1965). [7] Deem, H. W. , and Wood, W. D. , High temperature thermal diffusivity using a ruby laser as a heat pulse source, Rev. Sci.'Instr. 13» 1107 (1962). Figure 1. Schematic of thermal diffusivity apparatus. GLOVEBOX I LASER HEAD 0® o oo LASER POWER SUPPLY "I VACUUM FURNACE I OSCILLOSCOPE INFRARED DETECTOR 464 22gV 470 K 330K X" 2K lPbS COMPENSATING ~J7 .ooTTfif jPbS SENSITIVE -I M 2K 10 ^aaaw — DIFFERENTIAL AMPLIFIER o Figure 2. Lead sulfide detector circuit. Figure 3. Output of lead sulfide detector as a function of temperature 1200 co 1000 O > H 800- 600 400 o X g 200 i 1 1 r FILTER II FILTER I (95% ABSORPTION)/ (75% ABSORPTION)/ NOTE- CALIBRATION THROUGH TWO 1/4" THICK QUARTZ WINDOWS 200 400 600 800 1000 1200 BLACK BODY TEMPERATURE (°C) 1400 1600 30 "a CO >- CO z> 20 1.0 < o: ui x \ \ \ .90 1 — i 1 1 1 r DETERMINATION OF a ON DYNATECH STANDARD SPECIMEN CALCULATED FROM DYNATECH X -o— — . ■- — -, O I 400 500 600 700 800 900 1000 TEMPERATURE (°C) Figure 4. Thermal diffusivity of AI2O3 standard. 1100 1200 465 Figure 5. Microstructure of high purity arc-cast UC. Samll amount of UC2 - precipitate within grains, (Nitric etch; 56X) . Figure 6. Microstruc ture of sintered UC showing pore-configuration (150), Figure 7. Microstructure of sintered (Uq qPuq 2)^ showing pore-configu- ration (i50). Figure 8. Microstructure of sintered PuC. Shows presence of PU2C3; possibly 107. (150). 8.0 "b 70 X 'a 6 05 1 1 r "1 ' 1 r I 2.0 a. a f 1.0 1 1 r o| 300 O ARC CAST UC O 79% DENSE UC o 75% DENSE U 08 Pu 02 C V 84% DENSE PuC J_ _L 900 1100 TEMPERATURE (°C) 1300 1500 Figure 9. Thermal diffusivity of uranium and plutonium monocarbides « 466 The Thermal Diffusivity and Thermal Conductivity of Sintered Uranium Dioxide J.B. Ainscough and M.J. Wheeler The General Electric Company Limited, Central Research Laboratories, Hirst Research Centre, Wembley, England. Thermal conductivity values are calculated from 700 thermal diffusivity measure- ments between 970°K and 2020°K on 25 samples of sintered 2.7% enriched U0 2 . 00 , 977° of theoretical density. The measurements were made using the modulated electron-beam technique and the thermal conductivities of all the samples, derived by using Moore and Kelley's specific heat values, are represented by k = .0227T 1 + 3.38 + 6 ' 6 * 10 ' 13t3 w cnf1 ^S^' (970-2020°K) the 35% confidence limits being + Vfrjifo. This representation of the thermal conduct- ivity is preferred because the values of the constants agree both with the theory and the results of other investigators. Key Words: Conductivity, diffusivity, heat conductivity, heat diffusivity, modulated electron beam, sintered uranium dioxide, thermal conductivity, thermal diffusivity, uranium dioxide. This paper will be published in the British Journal of Applied Physics, volume 19, 1968. 1 Dr. Ainscough is with the U.K.A.E.A., Springfield Works, Salwick, Preston, Lancashire, England. 467 Suitability of the Flash Method for Measurements of the Thermal Diffusivity of Uranium Dioxide specimens Containing Cracks D. Shaw, L. A. Goldsmith and A. J. Little Reactor Development Laboratories, U.K. Atomic Energy Authority, Windscale Works, Sellafield, Seascale, Cumberland, England. The accuracy of thermal diffusivity measurements by the flash-method on reactor fuel specimens containing cracks is discussed. A theoretical treatment for a speci- man containing a "model" crack showed that the thermal diffusivity as measured by the flash-method will probably be within 5% of the value derived from a steady-state heat flow method. Measurements on Armco Iron specimens containing restricted heat-flow paths confirmed this conclusion. Key Words: Armco Iron, cracks, flash-method, model -crack, reactor fuel, steady-state heat flow, thermal diffusivity. 1. Introduction r n 1 The flash-method, devised by Parker et al. I.lj is now an established technique in most laboratories interested in thermal property measurements. In particular, it is of great value in measuring the thermal diffusivities of relatively small specimens on which it would not have been possible to use a steady-state heat flow method. This is especially true of measurements on ceramic or ceramic-based reactor fuel specimens which are rarely manufactured with lengths or diameters in excess of 0.75 in. A possible limitation to bhe use of the flash-method is in cases where a crack or cracks form across the path of heat-flow, such as may occur in UO pellets during irradiation. The Parker method is, of course, based on heat flow through a continuous medium. It is clearly of interest to determine how well the method applies when used with specimens containing cracks. This note describes a theoretical and an experimental approach to the problem. 2. Thermal Conductivity Measurements on UO^ Specimens The flash-method apparatus of Shaw and Goldsmith, which has been described elsewhere [2] was used to carry out a series of measurements at temperatures between 400 and 1000 C on unirradiated and irradiated UO specimens, cut from UO- fuel pellets. Measurements were made on each specimen at four temperatures, nominally 400, 600, 800 and 1000 C. At any given temperature, six traces of time versus back-face temperature were taken. Values of t x (i.e. the time taken for the temperature of the back face to reach half its maximum rise from the time of delivery of the heat pulse to the opposite face of the specimen) were worked out for each-trace and the mean value was used in the formula q , =<<>1 /(t-,) where a is the thermal diffusivity (cm /s),»is a number (see below) and 1 is the thickness of trie specimen (cm). Provided heat losses do not occur, to has a value of 0.139. At temperatures of about 600 C and higher, significant heat losses occur and it becomes necessary to work out a revised value f or u>, using either the method of Cape and Lehmann [3] or of Cowan [4], Conductivity values were calculated from the product of diffusivity, density and specific heat, using the data of Moore and Kelley [5], A plot of thermal conductivity values for the unirradiated and the irradiated specimens is shown in figure 1. Irradiatiort conditions were such i.e. irradiation temperature between 500 and 1000 C and irradiation levels below 10,000 MWD/te, that, according to Clough and Sayers [6] it is most improbable that any of the observed decrease in conductivity was caused by the usual irradiation effects i.e. displaced atoms, defect clusters and porosity. Metallography revealed the presence of cracks (figs. 2 and 3) and it was concluded that the lower values were almost certainly due to the presence of cracks. Clearly, since the accuracy of fuel temperature calculations depends, amongst other things, on the accuracy to which the thermal conductivity is known, it becomes important to assess how accurately the diffusivity (and hence the conductivity) may be measured in a cracked specimen using the flash method. 1 Figures in brackets indicate the literature references at the end of this paper. 469 3. Theoretical Treatment Using an idealised model for the crack, the apparent conductivities for both transient and steady- state heat-flow conditions have been calculated and the results compared with the experimentally deter- mined values. 3.1 Model for the Crack In devising the model for the crack, the following assumptions were made: (1) The crack lies on a plane transverse to the direction of the heat flow and has a thickness which is small relative to the specimen thickness. (2) The temperature distribution in the crack is linear. (3) Solid contact exists between the two faces of the crack over a fraction of its surface area. This is demonstrably true, since heat transfer occurred readily through the specimens during tests, in a vacuum of better than 10~ mm Hg and at temperatures where heat transfer by radiation would be negli- gible. Metallography of several discs showed that all the major cracks were connected to the outer surfaces of the disc so that any helium filler gas originally present in the fuel pin or any of the gases formed during fission would have been pumped away thus eliminating the possibility of heat transfer by gas conduction. The equation to be solved is the heat conduction equation c-3l! -22 (i) _ 2 - at K ' ftx 2 4 o a = thermal diffusivity (m /10 s), T=temperature rise ( C) The boundary conditions for the transient problem are: (i) No loss of heat from the back surface, - ik^ (ii) The heat pulse from the laser beam is of the form *>x (iii) The flow of heat across the crack is continuous (2) k 2— = H for < t <>", x=o (3) = o otherwise fcx /x=a \ ^x /x=c T - T 1 2 (4) T , T are the temperatures of the material before and after the crack respectively. a is the distance of the crack from the front surface (cm), 1 is the length of the specimen (cm), k is the thermal conductivity (kWm deg ) , H is the amount of heat absorbed by the specimen from the laser pulse (J), n is a parameter which is a measure of the effectiveness of the crack and is in effect crack thickness 1 ,_. x — (5) ■conductivity of UO p p is the percentage of the crack surface which is in solid contact, (p, can also be interpreted in terms of a crack containing gas), T is the duration of the pulse. After working through the mathematics of the problem, a formula was obtained for the temperature on the rear face 470 T„(l t) - «o--i« Y f( . -X -et )e n (6) where fa_) (1 - e A n » T ) (7) n X fl cos A 1-kjA (b sin X a cos X b + a sin X b cos I a) - L sin X a sin X bl nj_ n^n n n n n "• n n I where a + b = 1 and ' satisfies n sin X 1 = ki.. X sin X a sin ; b n n n n (8) The first term in the series for T_ (1, t) is the dominant one and using this fact the following formula is obtained. log 41f(X ) «r \ C ' fc i (9) ti is the time taken to reach half the maximum temperature rise. Parker et al [2] have obtained a similar formula for the diffusivity in terms of t. a (.1391)' (10) Hence by eliminating ti we can obtain a formula relating a /a to the other parameters in the problem where a is the diffusivity of cracked UO and a is the diffusivity of uncracked UO . Assuming that the density and specific heat of the material do not change, the ratios o /a and k /k will be equivalent. a a 3.2 Steady State In the steady state case the ratio of the cracked to uncracked conductivities is given by k 1 k = 2R and 1 » 6. This latter condition was not fulfilled, but would probably not introduce an error of more than a few percent. The ratio of solid to 'effective' density was easily calculated, and as the specific heats would be the same, the ratio of 'effective' to solid thermal diffusivity a '' a . was found. This latter ratio was in fact virtually independent of 6, the width of the slot, as can be seen in Table 2, below. 472 Table 2. Thermal diffusivity ratios for the grooved-cylinder specimens Specimen No. Specimen size Groove dimensions Steady State Transient Diffusivity Ratio a , e/a s Total Length Dia. (2R) Length Dia. (2r) Thermal Conductivity ratio k e/k s Density ratio p^^ Diffusivity ratio a . e/cv s 1 A 1 B 1 C 1 D 0.750 0.759 0.732 0.759 0.764 0.764 0.760 0.768 0.258 0.156 0.258 0.144 0.536 0.528 0.266 0.256 0.61 0.66 0.20 0.24 1.21 1.12 1.45 1.20 0.74 0.73 0.28 0.29 (378°C) (577°C) 0.71 0.71 0.72 0.69 0.27 0.28 0.28 0.28 The steady-state conductivity ratios of the second set of specimens were calculated by splitting up the slab into longitudinal sections, as shown in figure 4, calculating the resistance of each section, and summing these resistances, using the resistance in parallel rule, to obtain the total resistance of the slab. The resistance of each elemental constricted slab (R ) was calculated in a similar manner to that of the first set of specimens, i.e. as the sum of the simple resistance of the constricted slab and the resistance due to the convergences of the heat flow lines. This latter resistance was calculated using the results from reference [7] and converting from cylindrical to linear co-ordinates. The circular holes in the slab were assumed, for ease of calculation, to be rectangular, but of the same cross-sectional area. The whole slab is, clearly made up of a parallel combination of n of these elemental constricted slabs, where n is the number of holes, and the two plain end sections E and E , whose resistances are R and R respectively. Thus the reciprocal of the total resistance R is: (R T ) 1 = n(R g ) 1 + (R a ) a -t- (R 2 ) 1 (14) where R 1 6 + x 3 yk s x 4 yk s TTyk '' Cosh 1 X 3 X 4 , (15) R„ (1+6) , D (1+6) and R„ X i yk s 2 " x 2 yk s (16) (See figure 4 for the meaning of symbols). Writing the effective resistance of the whole slab: 1+6 (17) and equating eq (15) to the reciprocal of eq (17), the ratio of the 'steady-state' effective to solid thermal diffusivity is then easily calculated, as in the first set of specimens. Unfortunately the steady-state conductivity values proved to be particularly sensitive both to the shape of the holes and to the specimen dimensions (Table 3). Consequently the steady-state conductivity ratios could not be calculated with an accuracy of better than about 20%. 473 Table 3. Thermal diffusivity ratios for drilled specimens Specimen Specimen size Hole Dimension Distance of hole centres from upper surface Steady-state rat ios Transient Method Diffusivity Ratio j No. Length Dia. No. Dia. Separation k e/k s P s/p e e/a s a i e/a s MAX 391°C 577°C 2 A 1.039 0.511 1? 7 .079 .034 .112 .399 0.78 II 1.07 0.83 0.80 " 0.82 0.79 0.81 0.79 0.81 2 B 1.024 It 0.513 7 .080 It .033 .262 .251 0.76 fl 1.07 0.81 0.78 " 0.70 0.70 0.69 0.68 0.69 2 C 1.029 0.514 9 0.052 .025 .258 .256 0.89 1.04 0.92 0.88 0.88 0.85 0.84 0.87 2 D 1.036 0.512 7 0.086 0.030 II .258 .254 0.75 1.08 II 0.81 0.74 0.69 0.69 0.69 0.69 2 E 1.055 0.512 7 0.080 0.034 .142 0.77 1.07 0.82 0.79 0.74 0.73 4.4 Results and Discussion Details of both sets of specimens, together with a comparison of their 'steady-state' and thermal diffusivity ratios are given in Tables 2 and 3. ' transient' With the grooved specimens, there is excellent agreement between the 'steady-state' and 'transient' conductivity ratios. The latter values are always less than or nearly equal to the former ones. The drilled-slab specimens show differences between the 'steady-state' and' transient' conductivity ratios of between 2 and 15%. This may, at first sight, appear exceptionally great, but because the 'steady- state' ratio is particularly sensitive to the specimen dimensions this degree of difference is not sur- prising. For example, a 15% change in the 'steady-state' ratio may be made by assuming the holes to be square instead of rectangular in shape, maintaining the same cross-sectional area, and a difference of only 0.001 in. in hole size and separation gives a 2-J% change in this ratio. In addition, practical difficulties made it impossible either to drill the holes accurately orthogonal to the direction of heat flow or parallel to one another. Thus the calculated steady state value could well be in error, and the discrepancies between the steady state and transient ratios are not necessarily due to erroneous diffusivity values. However, the results for these specimens do show that the value of the thermal diffusivity does not markedly depend on the direction of heat flow, as can be seen most clearly for specimen 2A. 5. Conclusion The theoretical analysis on a specimen containing a "model" crack and the experimental work on specimens containing restricted heat-flow paths, have both shown that the thermal diffusivity of a specimen containing a crack or impediment to heat flow may be obtained to an accuracy of better than about 5% using the flash-method with the equation of Parker et al. 6. References [l] Parker, W.T. et al. J. Appl. Phys., 32, pp 1979-84 (1961). [5] Moore, G. E. and Kelley, K.K. J. Amer. Ceram. Sec. 69, p2105 (1947). [2] Shaw, D. and Goldsmith, L.A. Journ. Sci. Instr., [6] Clough, D. J. and Sayers, J. B. The measure- Vol. 43 pp 594-596 (August 1966). ment of thermal conductivity of UO. under irradiation in the temperature range 150 - [3] Cape, J. A. and Lehmann, J.W. J. Appl. Phys. 1600°C. AERE-R4690. (1964). 34, No. 7 .(July 1963). [7] Cetinkale, T.N. and Fishenden, M. Thermal con- [4] Cowan, R.D. J. Appl. Phys. 34, No. 4, Pt. 1 ductance of metal surfaces in contact. Am. Soc. (April 1963). Mech. Eng. Proc. General discussion on heat transfer 271-5 (11-13 Sept., 1951). 474 0.05 0.04 > I- o o z o o _l < cc Id X 0.03 UNIRRADIATED SPECIMENS \© _l 100 300 500 700 TEMPERATURE °C 900 1100 Figure 1. Plot of thermal conductivity versus temperature for unirradiated and irradiated specimens. [ Figure 2. E f ? 3 -fTtSfcl — I Figure 3. Sections from irradiated UO2 pellets showing moderate and severe amounts of cracking respectively. 475 ASSUMED HOLE SHAPE Figure 4. Diagram of drilled-slab specimen showing elemental slab S and end sections Ei and Eo. 476 Thermal Conductivity of Uranium Oxycarbides J. Lambert Bates Battelle Memorial Institute Pacific Northwest Laboratory Richland, Washington 99352 The thermal diffusivity of uranium oxycarbide was measured from 100 to 1500°C as a function of oxygen concentration from 2 to 17 at.% of the anions. The thermal diffusivity (a) of the oxycarbides decreased and the shape of the a-versus-T curve changed as the oxy- gen content increased. Oxycarbides with the lowest oxygen content showed a decrease in a as the temperature increased, whereas the oxycarbides with the highest oxygen content exhibited a slight in- crease in a with an increase in temperature. The thermal dif- fusivities of all specimens approach a common value near 5 x 10"" m /s at the higher temperatures . The calculated thermal conductivities near 1500°C of the uranium oxycarbides are approximately 18 W m"l deg"-*-, very near the reported values for nearly stoichiometric and slightly hypo- stoichiometric UC. The thermal conductivities of the oxycarbides can be qualitatively separated into an electronic component, obeying the Wiedemann-Franz-Lorentz law, and a lattice component, varying inversely with temperature. The decrease in thermal conductivity at the lower temperatures with increasing oxygen content results primarily from changes in the lattice component. Small decreases also occur in the electronic component over the entire temperature range due to increases in oxygen content. Key Words: Armco iron, conductivity, electrical resistivity, pulse method thermal diffusivity, thermal diffusivity, thermal conductivity, uranium carbide, uranium oxycarbide. 1. Introduction Uranium monocarbide has received considerable attention as a high temperature fuel for nuclear reactors. Unfortunately, it has been difficult to prepare uranium mono- carbide entirely free of oxygen, the oxygen often being found in solid solution with the monocarbide structure. The difficulty in producing single-phase specimens has con- tributed significantly to the- wide variation in reported values for thermal conductivity and other high- temperature properties. Recent extensive investigations have been made of the phase relations in the uranium carbide region of the uranium - carbon - oxygen system [1-4]. Single-phase solid-solution uranium oxycarbides can be prepared with oxygen contents approaching 17 atomic percent. This paper is based 6n work performed under United States Atomic Energy Commission Contract AT(45- 1) - 1830 . 2 3 Research Associate, Physical Ceramics Unit, Materials Department, Pacific Northwest Laboratory, operated by Battelle Memorial Institute for the United States Atomic Energy Commission, Richland, Washington. Figures in brackets indicate the literature references at the end of this paper. 477 The purpose of this study was to determine the thermal dif fusivities of some uran- ium oxycarbides as a function of oxygen content (to 17 at.%) in the temperature range from 100 to 1500°C, to calculate thermal conductivity values, and to evaluate the data in terms of the electronic and lattice condition processes. 2. Thermal Diffusivity Measurements The uranium oxycarbide samples were prepared by reaction sintering of calculated quantities of uranium, graphite, and UO? powders. The compacts were pulverized and milled to a fine powder in an all-nickel mill under highly purified argon. Pellets were cold pressed and sintered at 1700°C for two hours in an atmosphere of carbon monoxide at or very near the decomposition pressure for each oxycarbide specimen. Details of the fabrication process have been published by Henry et al. [1]. Ceramographic specimens were mounted in resin and rough ground on dry 3/0 and 2/0 emery papers. The specimens were first polished for one hour on a vibratory polisher with 0.3 urn AI2O3 in water on Metcloth. A final polish was obtained using 20 pm MgO powder in water on Gamal cloth for 1/2 to 1 hour. The polished surfaces on some speci- mens were improved by five-second etching followed by hand polishing with MgO in water. This was repeated if necessary to remove final scratch marks. Specimens were etched in 1:1:1 nitric acid, acetic acid, and water at room temperatures. The compositions and densities of the table 1. The compositional relationships o carbon - oxygen phase diagram at 1700°C are ( U Q . 495 c . 485°0. 02? is single phase with mi (Widmanstatten precipitate) (figure 2) . The diffusivity measurements resulted in the in free uranium indicates inhomogeniety on a m occurred during measurements. UCON- 36 5 (U n hyperstoichiometric resulting in a higher W A precipitate in the grain boundaries in th sivity measurements is probably secondary s uranium oxycarbid f each specimen t illustrated in f nor traces of fre relative slow co crease in UC? pre icroscale. Some .48 C 0.49 O Q.03^ wa ldmanstatten prec e etched specimen esquicarbide (U2C e specimens are shown in o the reported uranium - igure 1. UCON- 283 e uranium and UC2 oling rate during the thermal cipitates. The presence of grain growth appears to have s deliberately made more ipitate of UC2 (figure 3) . after the thermal diffu- 3)- Table 1. Description of uranium oxycarbide specimens Designation Composition, Atom % Density kg/m 3 (10 ) Microstructure U C UCON-283 UCON- 2 88 UCON-289 UCON-365 49.5 49.5 49.5 48.0 48.5 35.5 33.5 49.0 2 15 17 3 12.7 12.5 12.3 13.1 Single phase UC x O y with trace-free U + UC2 Two-phase UC x O y and U0 2 with trace-free U Two-phase UC x and U0 2 with trace-free U Two-phase UC x O y + UC 2 with trace UO^. UCON- 2 88 (U C n ) and UCON-289 (U Q 4q ,C O n ) are two phase UC and U0 2 with traHe§ y df u f ill ufinium (figure 4) . Thg y 6o^-ig >b the light grey phase and y the free uranium is the white phase in the unetched micrographs. UO2 appears white in the etched specimens. The as-fabricated cylinders were cut into thin disks (~ 0.100 cm thick and 0.635 cm in diameter) for the thermal diffusivity measurements. The uranium oxycarbide cylinders for this study were fabricated at the Albany Research Laboratory, United States Bureau of Mines, Albany, Oregon. 478 The thermal diffusivity of uranium oxycarbide was measured from 100 to 1500°C using a laser-pulse technique [5,6]. The pulse was provided by a 1.27 cm diameter ruby laser with energy output of 3 - 7 joules. The pulse width was 0.54 ms . The temperature transients on the back surface of the specimens were measured optically using a liquid nitrogen cooled, indium antimonide infrared detector. The signal from the detector was displayed on an oscilloscope and recorded on polaroid film. Corrections were made for heat losses using the method proposed by Cowan [7,8]. No pulse time corrections were required. The specimen was heated in a vertical tungsten tube furnace which was heated by direct current resistance. The sample was held in a UO2 holder with small tungsten pins . Temperatures were measured using Pt versus Pt-131 Rh thermocouples positioned in the holder. This thermocouple was calibrated using another Pt/Pt-13! Rh thermocouple welded to a thin platinum disk set in the sample position. Temperatures were controlled manually. Measurements were made in a purified argon atmosphere; the inlet argon contained < 1 ppm oxygen and < 5 ppm H2O. The pressure in the furnace was one atmosphere. Each specimen was heated initially to 1000°C, and thermal diffusivity measurements made during the increase and decrease in temperature. The samples were removed from the furnace to check for possible reaction with the tungsten or UG^. The specimen was reinserted in the furnace and thermal diffusivity measurements were made to approxi- mately 1500 C. No reactions between the sample and the UO2 or W of the holder were detected. The thermal diffusivity apparatus was calibrated using Armco iron. The thermal diffusivity data (figure 5) for the iron are in satisfactory agreement with the data reported by Cody et al.[9], Godfrey et al.[10], and Shanks et al . [11]. 3. Results and Discussion Thermal diffusivity data for uranium oxycarbide from 100 to 1500°C are shown in figure 6. The thermal diffusivity (a) of the oxycarbides decreased with increasing oxygen content. This change was most pronounced at the lower temperatures. The diffusivity of the samples containing 15 to 17 at.% oxygen is approximately 601 of that of the samples containing 2 at.% oxygen. This difference decreases as the temperature increases . The temperature dependence of the , oxycarbides varied significantly with oxygen content. The oxycarbides with the lowest oxygen concentration showed a decrease in a as the temperature increased. In contrast, the oxycarbides with the highest oxygen concentration exhibited a very slight increase in a with an increase in temperature. The overall result is that the dif fusivities of all the specimens approach a common value near 5 x 10~6 m^/s at the higher temperatures. Since there are no reported thermal diffusivity or conductivity data for uranium oxycarbides, comparisons can best be made using data for uranium monocarbides . The magnitude of the thermal diffusivity values at high temperatures above 1100°C fit closely between the thermal diffusivity values of hyper- and hypostoichiometric uranium monocarbide reported by Wheeler [12] (figure 6). However, the temperature dependence of the thermal dif fusivities , with the exception of the 51 at.% C reported by Wheeler [12], are somewhat different, with a decrease at lower temperatures. It is more difficult to make a comparison between the thermal conductivities of UC x O y and UC because of the wide variations in reported UC thermal conductivity values [12-16] . Thermal conductivities of the uranium oxycarbides were calculated using heat capacity data for uranium monocarbide. Since the preliminary heat content data for UC x Oy do not differ significantly from that for uranium monocarbide [17] , the heat capacity data reported by Godfrey et al. [18] were used for these calculations. These results are shown in figure 7. The thermal conductivity values for the uranium oxycarbides above 1000°C agree best with nearly stoichiometric UC, (figure 8). 479 The temperature dependence of the thermal conductivity for UC reported by Dayton and Tipton [13] is most consistent with the temperature dependence of the UC x O data, decreasing with temperature at the lower temperatures. It is also consistent with the decrease in conductivity resulting from an increase in the oxygen content. The decrease in thermal conductivity due to an increase in oxygen content is attributed to a decrease in both the lattice and electronic components, with the larger decreases at lower temperatures due primarily to changes in lattice conduction. The small decreases in conductivity at the higher temperatures are attributed to decreases in the electronic component of the thermal conductivity. Since uranium carbide and uranium oxycarbide are relatively good electrical conductors in the temperature range of this study, the thermal conductivity will undoubtedly be comprised of an electronic component as well as a lattice component. The electronic component might be expected to obey the Wiedemann-Franz-Lorentz law, and the lattice component will vary inversely with temperature. The following analysis was made to determine qualitatively the changes in A^ and A which contribute to the decrease in the total thermal conductivity. The lattice component of the thermal conductivity can be derived by *i - * " * e (1) where \, = lattice conductivity, A = total conductivity and X Q = electrical conduction. A e can be estimated from the electrical conductivity assuming the validity of the Wiedemann-Franz-Lorentz law A e = L T a (2) Data on the electrical properties of uranium oxycarbide are very limited. Ideally, measurements should be made on the same specimens used for thermal diffusivity studies. However, because of the low resistance of the small oxycarbide specimens and the dif- ficulty in fabricating long specimens of high quality, only one reliable data point has been obtained to date. This result indicated that resistivity at room temperature for the oxycarbide was significantly higher than for uranium monocarbide (see figure 9) . Some recent preliminary electrical resistivities for some uranium oxycarbides have been measured by Warren and Lacis [19]. These results show that the electrical resis- tivity increases with the addition of oxygen, and that the temperature dependence is approximately the same for the oxycarbides as for UC but with higher resistivity values. However, only one set of data could be compared to the specimens used in this study since only electrical resistivity data for oxycarbides with the same stoichiometry ( ~50 at. I C) and same oxygen content could be used. The electrical resistivities for UC (~ 50 at.% C) and for stoichiometric uranium oxycarbide with - 15 at.% oxygen used in this analysis are illustrated in figure 9. The electrical resistivity of UC reported by Carneglia [20] is approximately 10% higher than the resistivity reported by Mustacchi et al. [21]. An average of the two sets of data for each temperature was used in this analysis. The resistivity curve for uranium oxycarbide combines the data of Warren and Lacis [19] for Uq # 50C0 • 38^0 . 15 at tne higher temperatures with the room temperature data point obtained fof Uq . 495C0 . 355OQ 15. The curve is estimated for the intermediate temperatures. The electrical resistivity curves for the intermediate oxycarbide compositions of 2 and 3 at.% oxygen were estimated from the resistivity curves for UC and u 0. 495^0. 355^0 15 (fig ure 9)- It was assumed that the predominant conducting mechanism at'l400°C for UC and the oxycarbides is A e and that the differences in the thermal conductivity at 1400°C reflect the differences in the electronic components between specimens. The thermal conductivity data of Dayton and Tipton [13] were used in this analysis since data over the entire temperature range was available. The ratio of the difference between the thermal conductivity of each oxycarbide and UC at 1400°C was the ratio used to estimate the electronic resistivity of the intermediate oxycarbides Since the thermal conductivity values for Un. 495CQ 355O0 15 and Uq .495C0 , 335O0 . 17 are very close, no attempt was made to separate ; only the former is used in the analysis. 480 (^0 . 495^0 . 485 ( - l 02 anc ^ ^0,48 ( -'Q 49^0.03^ at a ^ temperatures. This assumes that the change in lattice conduction due to'oxygen content at 1400°C is negligible. change The electron eq (2) assuming a (figure 10) . The thermal conductiv function of tempe inverse temperatu relatively good. was anticipated, conductivity data UC. (It has been to stoichiometry . ic component for UC and for the oxycarbides theoretical Lorentz number of 2.45 x 10"° lattice component was then determined usin ity values (figure 10) . The thermal resist rature to determine how closely the lattice re relation (figure 11) . The linearity of The thermal conductivity for UC had a lowe This could have resulted from the inapprop (Dayton and Tipton [13]) since the data re demonstrated that the thermal conductivity ) wa V 2 / s calculated, using Hp(t2 7watts-ohm\ g I fo K )2 ) g eq (1) and the measured ance is also plotted as a component agrees with an the oxycarbides and UC is r lattice conductivity than riate choice of thermal late to 50. 3 at.% C in the varies significantly due The results of this analysis of the thermal and electrical conductivity of UC x O v demonstrates qualitatively that both the lattice and electronic conduction of uranium oxycarbide decrease with increasing oxygen content. It also shows that the larger changes occur in the lattice component at the lower temperatures, whereas the thermal conductivity decrease above 1400°C results primarily from changes in the electronic component . Further study will be necessary to obtain a quantitative understanding of the effects of oxygen composition on the thermal and electrical properties of UC x O v . 4 . Acknowledgement The author is indebted to Jack Henry and his associates at the U. Mines, Albany, Oregon, for providing the specimens for this study. Bureau of References [1] Henry, J. L., Paulson, D. L., Blickensdorfer , R. and Kelley, H. J., Phase Relations in the Uranium Mono- carbide Region of the System Uranium - Carbon - Oxygen at 1700°C, US Bureau of Mines, No. 6968 (July 1967). [2] Stoops, R. F. and Hamme , J. V., Phase Relations in the System Uranium - Carbon - Oxygen, J. Am. Cer. Soc. 47_ No. 2, 59-62 (1964). [3] Morlevat, J. P., Contribution a L'-Etuce du Systeme Uranium - Carbone - Oxygene, CEA-R-2857 (1965). [4] Chiotte, P., Robinson, W." C. and Kanno, M. , Thermodynamic Properties of Uranium Oxycarbides, J. Less Common Metals 10 273-289 (1966). [5] Parker, W. J., Jenkins, R. J., Butler, C. P. and Abbott, G. L., Flash Method of Determining Thermal Diffusivity, Heat Capacity, and Thermal Conduc- tivity, J. Appl. Phys. 32^ 1679-1684 (1961) . [6] Rudkins, R. L., Jenkins, R. J. and Parker, W. J., Thermal Diffusivity Measurements on Metals at High Tem- peratures, Rev. Sci. Inst. 33 21-24 (1962). [7] Cowan, R. D. , Proposed Method of Measuring Thermal Diffusivity at High Temperatures, J . Appl. Phys. 32^ 1363- 1370 (1961). [8] Cowan, R. D. , Pulse Method of Measuring Thermal Diffusivity at High Tempera- tures, J. Appl. Phys. 34 976-977 (1963) . [9] Cody, G. D.,Abeles, B. and Beers, D. S., Thermal Diffusivity of Armco Iron, Trans. Met. Soc. AIME 221 (2) 25 (1961). [10] Godfrey, T. G., Fulkerson, W. , Kollie, T. G., Moore, J. P. and McElroy, D. L. , Thermal Conductivity of Uranium Dioxide and Armco Iron by an Improved Radial Heat Flow Technique, ORNL-3556 (June 1964) . [11] Shanks, H. R. , Klein, A. H. and Danielson, G. C, Thermal Properties of Armco Iron, J. Appl. Phys. 38 , No. 7, 2885-2892 (1967). [12] Wheeler, M. J., Thermal Conductivity of Uranium Monocarbide, Carbides in Nuclear Energy V-l (L. E. Russell, Editor) McMillan and Co., Ltd., London, 358-365 (1964). 481 [13] Dayton, R. W. , and Tipton, C. R. , Jr., [17] Henry, J. L. and Bates. J. L., Progress Relating to Civilian Private Communication - to be Applications during August 1959, published. Battelle Memorial Institute, Columbus, Ohio, BMI-1377 102 (1959). [18] Godfrey, T. G., Wooley, J. A. and Leitnaker, J. M., Thermodynamic [14] Grossman, L. N., High-Temperature Properties of Uranium Carbides, Thermophysical Properties of Uranium J. Nuclear Materials 2_1 175-189 Monocarbide, J. Am. Cer. Soc. 4_6, (1967). No. 6, 264-267 (1963). [19] Warren, I. H. and Lacis , J., Uni- [15] DeCrescenta and Meller, Uranium versity of British Columbia, Carbide at High Temperatures, Carbides Vancouver, Canada, Private Com- in Nuclear Energy V-l (L. E. Russell, munication (1967). Editor) McMillan and Co., Ltd., London, 344-349 (1964). [20] Carneglia, S. C. , Single Crystal and Dense Polycrystalline Uranium [16] Leary, J. A., Thomas, R. L., Ogard, Carbide: Thermal, Mechanical, and A. E. and Wonn, G. C. , Thermal Con- Chemical Properties, Carbides in ductivity and Electrical Resistivity Nuclear Energy V-l (L. E. Russell, of UC, (U,Pu)C, and PuC, Carbides in Editor) McMillan and Co., Ltd., Nuclear Energy V-l (L. E. Russell, London 407-412 (1964). Editor) McMillan and Co., Ltd., London, 365-372 (1964). [21] Mustacchi, C. and Guiliani, S., Development of Methods for the Determination of the High Temperature Thermal Diffusivity of UC, EUR-337.e (1963) . 482 5 10 15 Oxygen, Atomic Percent 20 FIGURE 1 Phases and Phase Relations in the U-C-0 System at 1700°C Showing Composition of Uranium Oxycarbide Samples [1]. • / '.: "\ > • \' • t .•••• ->w • *• t* •. .. v^ /• \^ . a i. <- • ^^^_ V ^^ • •. 4Jk •' • \\ \ ♦ • » .' ' • 4- ?S . ? • V; ft •/: • * V • i . • m * y As Polished As Etched of Uq 49 5 C 0t485 O 02 (UCON-283) Before lal Diffusivity Measurements to 1500 C. FIGURE 2 Microstructure and After Thermal Uittusivity The Oxycarbide is Single Phase with a Trace of Free Uranium (White Phase) . 281X 483 U O <4-l 0) PQ .s /_ •• • \ ' * • • * * •*'.r x '. • *• ' **. ,•'•*•* •. * / * .• ■, «.• • • • * • . / • * . • <* ..-.'•» • • • ■ *> - A \' • '9' . • • " ' £L* ''«**k • ^r^i«ij' u v • ''- 2ft •* ! ! - fc ,fc! ^ .' V. 8 T3 Ort CD O Ml) LD 3 .H LO<-H ,£> 1-OM-l H CO U >> COrH X oin) O • U CD 3 ^C H H ch O H • CD U CD -MO U CM O 3 <; o 4-> LO UflH 3 fi h rt O +-> 4-> tn CD O H i/l h O +-> o m e •H CD CD 2 m 6 CD CD U CO CTj ^ M Ph 3 CO CD T3 U CD CJ .£ ^ o o =1 a •H 4-1 O CD CD -H CO IS c h CO o C -3 ■ H CD S -C C/) H O S a, to • rt c/> x CD r-l C H °° •H O «M Ph CD V) CD to (-. ,d cO a, in CD •H (-. M n) u ^cO r-l em 3 •H c • C0^ U CD 3 U n5 CD 4-1 CD U i-< 3 a, co CHTd o CD x: cd in O-H cOt-t h o ha, tir 3 09 03 CD § CD U O pq t 7*,'' #•/ I LiirSSfcSR! i N .-A' ^ •a- « • • • • * • r-\ CD LO ^1 03 vo 3 C to to V) -H I OS d co 2 (D •H >- TD o s co 1 ^ X Co U 4-> M -C 3 ^ d + J °° CJ ^— ' -M O (UN W • H u to> CD (0 OH 4h o >s"-l O LJ rH «jS COOS OS CD «*e v^ O •H CD CO OCD ^3 1* 3 Ji H U H H r-g CM 3 CO o u CD uhht) CD -PO O S H 4-( o cO 3 < O CD ■M in C/l^ U id i— 1 OS c/) 3 a ,d CD U nj O Ph-h ■P ■M> ^ X) CD CD 13 CO O H t/> CO x: H O pop; co U CM C O 3 ■.H ■H CD CD CD O o 2 m S co pq CU < pq OS 484 0.14 (OT E 0.10 >■ 0.06 UJ 11 0.02 200 11) • SHANKS ET AL. CODY ET AL. (9) o THIS WORK (0.140 CM) • THIS WORK (0.071 CM) 400 600 TEMPERATURE iOO 1000 0.05 E >- > " 0.04 < 0.03 SHANKS ET AL. CODY ET AL. (9) THIS WORK (11) 680 720 760 800 TEMPERATURE - °C 840 FIGURE 5 Thermal Diffusivity of Armco Iron. 485 0.08 E >- < s 0.06 0.04 uc URANIUM OXYCARBIDE o U 0.495 C 0.485°0.02 ° N 2 A/0 0^ 3 A/0 15 A/0 ■^TSj^jRftl - 17 A/0 u 0.48Co.49°0.03 u 0.495 c 0.355°0.15 U 0.495 C 0.335°0.17 °—.< o-Q^. o 49.6 A/0 C '^>--_ 51.0 A/0 C 52.0 A/0 C 400 800 TEMPERATURE - °C 1200 1600 FIGURE 6 Thermal Diffusivity of Uranium Oxycarbide in Relation to the Thermal Diffusivity of Uranium Monocarbide. • U 0.495 C 0.485 °0.02 (UCON-283) A U 0.48 « '0.49 Q 03 (UCON-365) 22 X U 0.495 c n 0.355 u 0.15 (UCON-288) 1 en o U 0.495 C ^0.335 u 0.17 (UCON-289) CD T3 20 • E • • . * A S 18 • • 1 / • •a ft * A >- • £ A A (— A A — £ A A A A o 8 > A A A A A i— 16 °X « o X 4-* °x ^ O X o 14 X SP o 1 < X *X X ^ o x „x „x <*o dx o S x u O ac 12 o U_l i n 1 1 1 1 i i 1 1 200 400 600 800 TEMPERATURE 1000 0, 1200 1400 1600 FIGURE 7 Thermal Conductivity of Uranium Oxycarbides. (Not Corrected for Density) 486 I— > I— o Q O o —J < S 25 20 15 10 UC (50.3 A/0 C) IDAYTON-T I PTON 13 ) T UC (52.5 A/0 C) (GROSSMAN 14 ) 400 800 1200 1600 TEMPERATURE - C FIGURE 8 Thermal Conductivity of Uranium Oxycarbides in Relation to the Thermal Conductivity of Uranium Monocarbides . oo o >- > IS) UJ < CCL I— o 300 200 100 W \ BNW U 0.495 C 0.355°0.15 MUSTACCHI + GIULIANI UC (21] \^\ \ (19) WARREN-LACIS U _ 50 C 0<38 O _ 15 (.ARNIGI iA (18) UC 400 800 1200 1600 TEMPERATURE - C FIGURE 9 The Electrical Resistivity of Uranium Oxycarbide and Uranium Monocarbide. 487 ELECTR ICAL COMPONENT 20 01 CD I— > I— o o o < 12 - 04 - UC U 0.495 C 0.485°0.02 U 0.48 C 0.49°0.03 V \ U 0.495 C 0.355°0.15 16 LATTICE COMPONENT \ U 0.495 C 0.485°0.02 ^. 0.48"0.49 u *0.03 08 ^ ^^^^ / *^.^ uc 1 "~~ -* ^ ^ ^^^^rr^a* U 0.495 C 0.355°0.15 ""*" ""~ ^ "-^S^^S^B^ss*^, n 1,1.1. 400 800 TEMPERATURE 1200 FIGURE 10 The Electronic and Lattice Components of the Thermal Conductivity for Uranium Oxycarbides. 1— / ^ /'..- UJ U 0.48 C 0.49°0.03 ^ "S^**/" O a. 2 o <_> 20 LU urn ** •••' ^^ v U 0.495°0.355 U 0.15 ^ •''l^r UC — 1 *- en 1— a) < -a _j 1 S E ^****~ ^^^^^^ ^ 5 "* •jj****"*''^ «•""" 1— 10 — ....j^*^ *^'^' (•1 .••""""" *«^* " ~" ~~""^ \ co LU 0£ U 0.495 C 0.485°0.02 < 2 0; LU X 1 1 1 1 — n 1 1 1 1 1 1 1 400 800 1200 1600 TEMPERATURE - C FIGURE 11 The Thermal Resistance of the Lattice Component for Uranium Oxycarbide as a Function of Temperature. 489 The Effect of Irradiation on the Thermal Conductivity of some Fi3sile Ceramics in the Range 150-1600°C Up To a Dose of 10 21 Fissions cm**3 D. J. Clough United Kingdom Atomic Energy Authority, Metallurgy Division, Harwell, ENGLAND A study has been made of the effects of irradiation on the thermal conductivity of polycrystalline sintered UOp and arc cast and sintered UC. Measurements have been made continuously during irradiation in a series of » experiments, the irradiation capsule being designed to act as a calorimeter. Temperatures have been measured both at the centre of the specimen and its peri- phery to eliminate the contribution of the specimen/cladding interface to the measured temperature drop. 19 -3 It has been shown that up to 4 x 10 fissions cm measurable defect damage effects are restricted to continuous irradiation temperatures below 500 C both in UCv, and UC. Below 500°C in U0 9 reductions of up to 30$ in thermal conductivity 2 and UC. Below 500 C xn UO, reductions of up ive been measured up to 4 x 10*9 fissions cm***. have been measured up to 4 x 10 1 " fissions cm"*-'. The nature of the damage is both dose and temperature dependent as evidenced by anneals performed during irradiation. Values obtained for UCv, between 500 C and 1600 C lie close to presently accepted values for the unirradiated material as do those for UC between 500 C and 700°C. 21 -3 Extension of work to 10 fissions cm has shown that the effects of high burn-up on thermal conductivity in both U0„ and UC irradiated under conditions of high restraint between 500 and 700 C are small (<10$). Low temperature damage phenomena and factors contributing to the possible modification to the thermal conductivity under irradiation to high fission depletions are discussed. Key Words : Annealing behaviour, arc-cast, conductivity, damage, dose, high burn-up irradiation, restraint, sintered, thermal conductivity, uranium oxide, uranium carbide. 491 Phase Studies on Fueled Zirconium Hydride M. M. Nakata, C. A. Smith, and C. C. Weeks Atomics International A Division of North American Rockwell Corporation P. 0. Box 309 Canoga Park, California 91304 The phase boundaries of the zirconium-uranium-hydrogen system for atomic ratios from H/Zr - 1.60 to 1.81 were determined by electrical resis- tivity and thermal expansion measurements. The standard potential drop technique was employed for electrical resistivity measurements and a quartz dilatometer was used for the expansion measurements. Because the hydride dissociates at elevated temperatures, each apparatus was designed to oper- ate with hydrogen gas in the system to maintain the composition of the material. Resistivity measurements were made in the range of 20 to 900°C and with 1.33 x 10~ 4 to 9.33 x 10 5 N/m 2 (10 -6 to 7 x 10 J torr) of hydrogen gas in the system. Expansion measurements were made between.room temper- ature and 700°C and with 1.33 x 10"^ to 1.87 x 10 5 N/m 2 (10~ 6 to 10^ torr) of hydrogen. In addition to phase boundary determinations, the thermal expansion coefficients were determined for the various H/Zr ratios. The results of the electrical resistivity and thermal expansion measurements were found to correlate well with data of Moore & Young obtained by x-ray diffraction and hot-stage metallographic techniques. Key Words: Electrical resistivity, expansion, hydride, phase studies, resistivity, thermal expansion, zirconium hydride, zirconium-uranium-hydrogen system. (Publication of the complete phase studies work, including the x-ray diffraction and hot-stage metallographic work, in a scientific journal is contemplated; the journal is not identified at this time. ) Work done under AEC Contract AT(04-3)-701 493 On the Development of Methods for Measuring Heat Leakage of insulated Walls with Internal Convection G. Lorentzen, E. Brendeng and P. Frivik Norges Tekniske H0gskole Trondheim Norway The combined effect of conduction and convection has great practical importance in the design of insulated structures. Much more data is needed to permit an exact precalculation of the heat leakage. The ordinary types of apparatus for conductance measurements are inadequate when convection occurs as a result of its influence on the edge compensation and the scale effect. The performance of insulation with internal natural convection has been studied on full size structures for many years at the refrigeration laboratory of the Norwegian Institute of Technology. An account is given of the methods used and their limitations. Heat flowmeters can give quite accurate data for walls with fibrous or granular insulation with uniform flow resistance, while the convection effect in slab type insulations with slits is much more difficult to measure. A special large "hot box apparatus" has been designed for the testing of such structures. Computer programmes offer the possibility of calculating the convection influence for any wall design with known characteristics and operating conditions. Good agreement of calculated and measured results has been achieved. Key Words: Computer calculation of internal convection, convection effect, heat flowmeters, hot box apparatus, natural convection, thermal insulation, wall insulation testing. 1. Introduction It has been found that natural and forced convection can add significantly to the heat leakage of insulated structures (3)1. Convection may occur in fibrous or granular materials, which are not completely air tight, as well as in the channel system formed by slits and cracks between insulation slabs, created during erection or as a result of subsequent shrinkage. Data on the effect of convection under actual operating conditions are needed for accurate calculation of heating or cooling loads and for setting up standards of suitable insulation design. This report is concerned with natural convection only. In a horizontal plate apparatus with the heat flow going downwards no convection can occur, and even with upwards flow the effect will ordinarily be negligible because the level differences are so small. Published conductivity data for insulation materials can therefore usually be taken as valid for convection free conditions. For real structures the convection will depend on their physical dimensions, extension in the vertical direction and the temperature differences, and its effect can only be determined by measurement on life-size samples. A fair number of such investigations have been carried out in our laboratory since 1956 and the programme is still in progress. A record of the methods used will be given in the following. 1 Figures in parentheses indicate the literature references at the end of this paper. 495 During the first years the work was mainly concentrated on mineral fibre insulations, using a heat flowmeter technique (1). This method yields accurate results only when the air flow resistance is evenly distributed throughout the insulation material. For cases of non-uniform flow resistance, and particularly for slab type insulations with slits, a special guarded hot box method was developed (2). 2. Heat Flowmeter Method Extreme care is necessary in using heat flowmeters if reliable results shall be hoped for. Some commonly made errors are illustrated in figure 1 A. The flowmeter adds to the heat resistance of the wall in the area where it is applied, and the heat flow measured is too low. This is particularly so where the meter is placed against a surface with relatively high conductivity, changing the heat flow pattern as schematically indicated by arrows. Another common fault is that the instrument is not sealed completely air tight to the wall. Even very small air currents are sufficient to change the reading drastically. In order to avoid these difficulties the heat flowmeters can be fitted as schematically shown in figure 1 B, giving essentially the same additional insulation over the entire wall surface, unchanged surface film coefficient and complete air seal. A sufficient number of meters must be installed on both sides of the wall to accurately measure the variation of the unit heat flow and check the heat balance (1). If the difference between the measurements of total heat flowing into and out of the wall is too large, this is a clear indication that the method is inadequate. It is obvious that the heat flowmeter method can only be used when the heat flow varies continuously and fairly evenly over the wall area. This is the case with a homogeneous and uniformly packed mineral fibre insulation, figure 2 A. Even for this case it is necessary to use a fair number of meters to get a sufficiently accurate indication of the heat flow distribution. When the convection flow is less regular, it is impossible to get a sufficiently accurate record of the flow distribution on the basis of a reasonable number of meter readings. Figure 2 B shows conditions for a wall with slab insulation and 1.7 mm wide slits. The heat flow distribution in a vertical cross section is charted approximately by estimation on the basis of surface temperature measurements and should be taken only as an indication of principle. It is very uneven due to the local flow of air when the slit system emanates to the surface. Heat flowmeters placed for instance as indicated in figure 2 C would give a completely distorted indication and an unreliable average. Measurements on fibre insulations with uneven packing or structural subdivisions can give similar results. In all these cases conditions are complicated by the fact that the air flow follows a three-dimensional pattern extending over the width of the wall, and the areas of the heat flow diagrams in any given section are not necessarily equal. For this reason the heat flowmeter method has been used mainly for measurements of uniform mineral fibre insulations, and some results will be presented in section 6. The test arrangement is shown schematically in figure 3. The test wall was 2.0 m high and each section 1.1 m wide. Five heat flowmeters, each with 3 separate measuring sections, were used on both sides, giving altogether 30 individual measurements for each wall. The temperature field in the wall and air temperatures at both sides were measured with a total of 70 thermocouples per section. In order to limit the fluctuations of the heat flow measurements the temperatures on either side of the test wall were maintained constant within +_ 1/10°C. 3 . Guarded hot Box Method For measuring the heat leakage of a wall with irregular convection pattern, such as occurs in slab type insulations with slits or various loose fill insulations, a special guarded hot box apparatus was constructed, figure 4. The test wall area is 1.7 x 2.2 m 2 and the standard insulation thickness 200 mm. The required temperature on the cold side is produced by a refrigeration system which is running continuously with constant capacity, and an electric heater with low thermal inertia is used for accurate temperature adjustment. An electronic thermostat with a thermistor bead sensor controls the heater and maintains the temperature with a fluctuation less than + 0.15OC. The electric heat supply to the hot box is taken from a stabilized voltage source and is set constant for each test. It is measured by a precision type wattmeter. The guard box temperature is maintained equal to that of the hot box by a low inertia 496 electric heater and alcohol cooler, governed by a 230 point thermopile in the separating wall through an electronic null amplifier. This again alternately actuates relays for heating and cooling, and the necessary temperature difference to switch from one to the other is in the order of 0.003°C. The electric circuit diagram is reproduced in figure 5. The insulation to be tested is fitted between 10 mm panels of chipboard, stiffened on the external side by a framework made from 20 mm thin-wall steel tubing. The guard box is faced on the inside with aluminium sheet in order to limit the radiation heat exchange with the hot box. The entire apparatus is pivoted in a way to permit testing in a horizontal, convection free position or with the wall vertical, or at any intermediate angle. Temperature differences up to about 70°C can be attained at a mean temperature in the wall of 0OC, depending on the efficiency of the insulation tested. The minimum temperature difference is decided by the fan power dissipation in the hot box and is in the order of 10°C. 3.1. Design of the Edge Area The most critical design problem of a guarded hot box is the control of heat leakage in the edge area. The best solution necessarily depends on the nature and purpose of the test, and a number of different geometries have been used (2). One of the earlier solutions is schematically shown in figure 6 A, where the test wall extends to the inside perimeter of the hot box and has an area of 2 x 2.5 m^. A total of 66 small electric heaters along the rim supplied the necessary heat to the part of the wall which was covered by the gasket. The heaters were fed from a constant voltage source and could be regulated by individual rheostats to give isothermal conditions at the wall surface. This could be controlled by means of a number of fine thermocouples in the gasket area. A thin strip of insulation between compensation heaters and wall surface was used to compensate for the film coefficient of the free surface, in order to avoid any temperature difference between air and hot box. In this way all heat leakage through the hot box could be eliminated, and this could also be verified by means of a number of small heat flowmeters installed around the perifery. This method gave excellent results with regard to accuracy for reasonably convection free materials, but the extensive adjustments were rather time-consuming. When convection occurs right up to the edge region, the situation is different. In this case the adjustment of the compensation heaters to give isothermal surface constitutes a deviation from normal conditions and means that the convection air flow picks up extra heat from this source in the lower part of the test wall. Particularly in testing slab insulations with slits, where the convection current is concentrated right up to the edge, this could induce a sizable error. For such conditions another design was chosen, incorporating a convection free belt, figure 6 B. The complete hot box and rim section of the wall were made in one piece as a monolithic structure of polyurethane foam with glassf ibre-polyester covering. The slits in the test wall are displaced from the critical zone, and the width of the belt is chosen sufficiently large to minimize the effect on the temperature field in the actual edge area. The measurements are simplified by the elimination of the compensation heaters, but in all other respects the principle of the heating, cooling and control systems is unchanged. The actual test wall area was reduced to 1.7 x 2.2 m2 by the new design. Completely air tight enclosure of the test wall is essential in order to avoid extra heat leakage by air exchange. The hot side is effectively sealed by the plastic foam cast directly against the chipboard panel. The cold side seal is produced by an inflatable tube gasket as shown on the figure. The tightness was checked by pressure test. It is evident that the effect of convection in slits along the edge of the test section will influence the temperature field in the belt area to some extent, increasing the heat leakage in the lower part and causing a similar decrease in the upper part. The overall influence will be small and can be considered a natural part of the effect to be studied. The old design had the advantage that all heat produced in the hot box actually passed through the test section. With the new arrangement some of it is lost through the convection free belt and must be deducted from the reading. It was therefore necessary to calibrate the apparatus and find this heat flow as a function of the temperature difference. 497 3.2. Calibration of Belt Leakage The calibration was done with the test section carefully insulated with a mineral wool material for which the conductivity was known with great precision from a number of' previous measurements. The measured belt heat flow for some values of the temperature difference is indicated by the observation points in figure 7 B. It is also possible to calculate the temperature field and belt heat flow theoretically by means of an electronic computer. For this purpose a 10 mm square mesh was used, figure 7 A, and the isotherms and heat flow were calculated by iteration on Univac 1107. The necessary heat conductivity data for the polyurethane foam were measured in a normal plate apparatus while the surface temperatures were taken from actual observations on the structure itself. The calibration curve thus calculated is drawn in figure 7 B. It is seen that the deviation of the experimental points is quite small. The temperature field and heat leakage of the belt is influenced to some extent by the conductivity of the tested material. This effect was calculated on the computer for a wide variation of conductivities, and the result is given by the curve figure 7 C. Within the actual range, covering all common insulation materials, the influence is negligibly small, figure 7 D. The necessity to correct the measured heat leakage for the belt heat flow obviously means a somewhat reduced accuracy of the absolute value of the apparent conductivity found. However, it does not affect the accuracy of the convection heat flow, which is found as the difference of the figures measured in the vertical and the horizontal position. Any error in the belt heat flow calibration is thus automatically cancelled out. 3.3. Temperature Measurements The surface temperatures of the test insulation are measured by 20 thermocouples on each side on the inside of the chipboard panels. Each thermocouple is attached to a 20 x 20 mm thin copper plate which is imbedded in the board, flush with the surface. The temperature pattern in the test insulation is measured similarly by means of 80 thermocouples. Another 144 are used to ascertain the temperature distribution in the edge area. All the readings are recorded to an accuracy of 3-4 [J. V or about 0.1°C by an automatic datalogger system. The output is in the form of punched tape, which can be directly fed into the electronic computer for processing. 4. Computer Calculation of the natural Convection Effect The measurement of the convection effect on full size samples is very time- consuming and expensive, and it is hardly possible to really test all possible combinations of wall dimensions, insulation materials, slit geometries and temperature differences. It was therefore decided to extend the experimental results by computer calculation. As a first step a programme was designed for calculating the convection effect in rigid board type insulations with slits (4). It is then necessary to find the air flow pattern and its influence on the temperature distribution in the wall. The heat leakage can subsequently be found from the temperature field. As the mass and heat flow processes are closely interconnected they cannot be calculated separately, and a systematic approach to the correct solution by numerical iteration becomes necessary. The air flow in the slits is found from pressure and mass balance, figure 8. The coefficient of pressure drop in laminar flow depends on temperature and slit opening. Similarly the temperature distribution is determined by calculation of the heat balance for a large number of points, figure 9, and the accuracy depends on how fine a mesh is used. In this way we get two sets of numerous simultaneous equations, which can be solved on the computer. The programme thus developed can be used to find the convection effect for any insulation constructed from rigid board material, which is in itself air tight, with a given slit geometry. A similar method for calculation of fibrous or granular insulations with known resistance parameters is being designed, and offers no particular difficulties. 498 5. Measurements of Air Flow Resistance The computer calculations outlined require knowledge of resistance data, which can be measured in special tests. As the flow is always laminar, the pressure drop gradient is directly proportional to the velocity or mass flow. For a slit between parallel walls the pressure drop will be AP = 12 I) V • ^r = A • G a ^ where 77 = dynamic viscosity, Ns/m 2 v = mean velocity, m/s L = length of the slit, m a = width of the slit, m A = pressure drop constant, Ns/m 2 kg G = mass flow, kg/s The results of the measurements can be given directly as pressure drop constants A or in terms of equivalent slit opening a. It was found that the pressure drop in plane- parallel slits between polystyrene foam or "onazote" slabs was about 4% higher than corresponding to the opening measured by mechanical gauging. This is presumably due to the different influence of surface roughness. For fibrous or granular materials the flow resistance can be characterized by a "hydraulic diameter" D^. The laminar flow pressure drop is given by the equation dP/dL - 32 ^ ' V As an example figure 10 A gives the measured effective hydraulic diameter as function of packing density for two typical mineral wool materials with fibre spectra as shown in figure 10 B. 6. Some Results Altogether 21 different walls have been measured with the heat flowmeter method, most of them with mineral wool insulation of different density as follows: A. Loose fill fibreglass B. Fibreglass bats C . Rockwool bats 17.1 - 33.4 - 47.9 - 60.6 - 77.7 - 103.7 kg/m 3 47.0 - 63.7 - 80.0 kg/m3 22.6 - 27.3 - 36.0 - 41.3 - 52.0 - 66.1 - 84.4 kg/m 3 The fibre characteristics of these materials were given in figure 10 B. The insulation thickness was 200 mm and mean temperature about 0°C. Some results in the form of apparent heat conductivities are recorded in figure 11. So far 8 different walls have been tested in the guarded hot box apparatus, one of them with fibreglass bat insulation of 20.2 kg/m 3 density for calibration purposes, the other with rigid polystyrene or ebonite foam board and different slit openings. Measurements have been made in the horizontal, convection free position and with the wall vertical at temperature differences ranging from 8 to 76°C. A schematic of the typical slit geometry is shown in figure 12, while figure 13 records some of the results. It is obvious that important improvements can be achieved by the simple measure of sealing the surface seams. The actual effect of this operation for two different sealing methods is shown in figure 13 B. Figure 13 D illustrates the very rapid increase of convection for slits larger than 2-3 mm, particularly with high temperature difference. These curves were derived by computation. 7. Conclusion It is possible by a suitable computer technique to extend the results of laboratory measurements of convection influence to cover all actual wall designs with accurately known characteristics. Work is in progress to provide data in a practical form for a number of commonly used types of construction. Such information will be useful in calculating cooling or heating loads and in setting up quality requirements for insulation materials and workmanship. 499 8, References (1) Lorentzen, G. and Brendeng, E., On the influence of free convection in insulated, vertical walls, Proc. X Int. Congr. Refrig. Copenhagen 1959, I p. 254. (2) Lorentzen, G. and Brendeng, E., The design and performance of a large scale guarded hot box, Bull. Int. Inst. Refrig. Annexe 1962-1, p. 29. (3) Lorentzen, G., The influence of convection in cold store insulation, The Journal of Refrig., 1965, No. 2, p. 41. (4) Lorentzen, G. and Nesje, R. , Experimental and theoretical investigation of the influence of natural convection in walls with slab type insulation, Bull. Int. Inst. Refrig. Annexe 1966-2, p. 115. O V O 00 o ■*- o V o° °«°o % c <>o o oyo\v< ° o o o o 1 r D ° "0 ° O O o ° ^ "0 o °k« ««"0 o o < o -o o o o -D-D -0 T> -0 O O O o , o „ • • o ° o c o 6 «> *> O , o • ° O-o '.'% -0 O -n ,° ° o ' o° o V T> K <' o ft o r> „ D O o ,°%V ° o " o o 'oil 6 " °_ ° » ° ° « Cod * ' A u w « 0™ " o 00o 6c D * ° o 0° ° o 000 t> r, _ u ° » 2 O ° o p e. o o e o p c o °_ I ft m $ -n. M. '■ ":■ ■ ' o o p ' o " V ° ° ° o p ° ° B Figure 1 . A: Faulty installation of heat flowmeter may lead to large errors due to conduction in wall surface layer and air leaks, as indicated by arrows. B: Principle of installation used in the tests. a: Fibreboard cover plate, b: Soft "porolon" gasket, d: Plate with equal heat resistance. Heat flowmeter, 500 v> 1 1 1 J r n i i 1 i L J i I i I 1 1 1 1 1 1 i 1 1 1 1 1 Figure 2 A: B: C: B Air flow pattern and heat flow distribution on the warm and cold side of a wall with homogeneous fibre insulation. Similar indication for a wall with rigid board insulation and irregular slits, Typical location of heat flowmeters. 3S SajSS &. - Figure 3 , O I I 01 I I I I f-£ i i i .j i Arrangement of test wall. c: Heat flowmeters. f: Wall thermocouples. g: Air thermocouples, $01 :■'•.' :-'.';'i : ; . ; ::;^'.-;>:;':'.:'^f:':-S/:i ; ^;\Li: Figure 4, Design of the "guarded hot box" apparatus in its present form. a: Guard box. b: Hot box. c: Cold box. d: Alcohol cooling unit. e: Refrigeration unit. f: Surface thermocouples. ■j J J i, i j j J e J >e (i ■ i) c 1 j3 d i a n \S 3o e f Y 1 1 J ) 1 e ^ 1 1 Gu 3rd - box Ho t-i DO X Cc >ld -box Figure 5 , Electric circuit diagram for the "guarded hot box" apparatus. a: Voltage regulator, b: Null amplifier/thermostat. c: Voltage stabilizer. d: Precision wattmeter. e: Heater. H: Fan motor. 502 Figure 6. Previous design of edge area and compensation system, a: compensation heaters, b: Thermo- couples for control of isothermal conditions, c: Heat flowmeters for leakage control, d: Insulation strip with thickness S = X/a. New design with "convection free belt." e: Inflatable gasket seal on cold side. Figure 7. A: Geometry of edge area with calculated isotherms and heat flow lines. B: Calculated edge heat leakage as function of temperature difference (curve) . Points from calibration tests are indicated. C: Heat transfer from edge to test area in percent as a function of ratio of heat conductivity. D: Same correction for the actual range of conduc- tivities. O.06 0,02 -0.02 04 D 10' QS I 15 2 £5 , . 3 503 Figure 8. Pressure and mass balance for the convection air flow, a: Insulation slab, b: Paneling, c: Slits between individual slabs of insulation. Pi, P 2 . ? 3 •••• Air pressure in the slit junctions. G l> G 2> G 3 .... Mass flow in slits. A, B, C ..... Coefficient of pressure drop in laminar flow. H' H 2' H 3 "••• Level difference between slit junctions. p : Density of the air. Pressure balance P 1 =P 2 -a * P ' Hi-A Gi P2=P3+B • G2 etc. etc. Mass balance Gj-G 2 -G3=0 etc. etc. Similar equations can be written for the whole system and solved on the computer. IL^J Figure 9. Heat balance for a point in the wall. T Q , Tj, T~ .... : Temperature in equi- distant points. G : Air mass flow. X : Heat conductivity of insulation material. Ax : Network modulus. B : Width of the constant geometry section. c„ : Spec, heat of air. Heat balance: X • B (T1+T2+T3+T4 -4To)= %G • c p (T2-T4). For no air flow: (Tj+T,+T 3 +T,-4T ) = 0. 504 100 *10 6 80 40 20 D h / ^ / \A(B) C^t N. ^^ 40 60 80 100 120 140 k 9/m3 Figure 10. A: Effective hydraulic diameter for two mineral fibre materials at different density. B: Fibre diameter distribution for the materials investigated. 0,11 Figure 11. Apparent mean conductivity for walls insulated with different mineral fibre materials with different density. Curve A: Loose fill insulation, material A. Curve B: Mat insulation, material A. Curve C: Bat insulation, material C. 0.04 0,03 0.02 505 Figure 12. Schematic slab arrangement and slit geometry for test walls with rigid board insulation. C: D: Figure 13. Apparent mean heat conduc- tivity for a polystyrene foam wall insulation with two values of the average slit opening. Results for the same walls in a convec- tion free horizontal posi- tion are also shown. The increase due to convection is about 17 percent at AT = 70 °C for 2.5 mm slits. Apparent mean heat conduc- tivity for a polystyrene foam wall insulation with 2.5 mm average slits. The improvement by sealing the surface seams on one side is shown. Results for an "onazote" wall insulation. Computer calculated appar- ent heat conductivity as a function of slit opening for different values of AT. The influence of radiation is excluded. 0.05 0.04 0.03 0.02 0.01 • 2,5mm silts, horizontal ^— — o 1.5 mm slits, vertical • 1,5mm slits, horizontal 0.05 W /m-C 0.04 0.03 0,02 -o No joint sealing, vertical -• No joint sealing, horizontal -o Taped seams one layer, vertical -• Taped seams one layer, horizontal -a Plastered, vertical -a Plustered,horizontal B 20 40 60 80 At °C -o 3,3 mm slits, vertical • 3.3 mm silts, horizontal o 2,9 mm silts, vertical » 2,0 mm silts, horizontal 20 40 60 80 M 'C 506 Thermal Diffusivity Measurements from a Step Function Change in Flux Into a Double Layer Infinite Slab E. K. Halteman and R. W. Gerrish, Jr. 1 Pittsburgh Corning Corporation Pittsburgh, Pa. 15239 A step function change of flux into an infinite slab with the output face held at constant temperature was used to determine thermal diffusivity by using a graphical and a time integral method. Samples of gum rubber and foamed polyurethane were used to show deviations from the expected result of the single layer infinite slab theory. Use of a low mass, low heat capacity heater reduced the deviations. A double layer infinite slab analysis was used so as to include the heat capacity of the heater. The error in the diffusivity using the single layer model was determined as a function of the ratio of the heat capacity per unit area of the sample and heater. Key Words: Double layer infinite slab, heater correction, thermal diffusivity. 1. Introduction The thermal diffusivity of ceramic and organic insulating materials is most readily obtained from transient linear heat flow through an infinite slab. The relative ease of fabricating the sample in the form of a slab makes this geometry attractive. Plummer, Campbell and Comstock [1] developed a method based on a constant flux into a thick slab of material which was treated as a semi-infinite solid. This method was further refined by Harmathy [2] who also developed a pulse heating scheme. Steere [3] used the constant flux method with samples of plastic assembled from multilayers of thin films. In all cases the finite samples were considered to be infinitely thick during the time when measurements were taken. Also, in each case the heat capacity of the heater was shown to be a small fraction of the heat capacity of the sample and was therefore not included in the analysis. When the constant flux input method is used with a low density, low specific heat capacity, and low conductivity insulator such as foamed polyurethane, difficulties arise. The conductivities of many solid and foamed insulators are approximately proportional to their densities; hence, their dif fusivities are similar. But the heat capacity per unit volume of the sample can vary widely since it depends upon density and specific heat capacity. Thus, for low density organic insulators the heat capacity of the heater may represent an appreciable fraction of the heat capacity of the sample. In such cases it is necessary to treat the heater as a separate layer with its own thermal properties and to determine the diffusivity of the sample from an analysis of a double layer infinite slab model. 2. Theory The temperature distribution, 6(x,t), within an infinite slab of thickness, L, is given by the solution of the one dimensional equation of linear heat flow with specified boundary conditions. a 6 (x,t) = 8 (x,t) for < x < L (1) The distance x is measured from the input face and the thermal diffusivity, a, is assumed to be independent of position, time, and temperature. The subscripts denote differentiation with respect to a particular variable. The boundary condition at the input face, x = 0, will be an input flux, = -A 6 , which experiences a step function change at time zero. The temperature of the output face will be held" constant. The Physicists 2 Figures in brackets indicate the literature references at the end of this paper. 507 temperature of the output face will be held constant. The temperature of the slab will be uniform and equal to the output face temperature at time zero. These boundary conditions can be written as 4>(0,t) = -A6 (0,t) = 0, t < >(0,t) = -A6 (0,t) = $ , t > (2) 6(L,t) = 0, t >, (3) 9(x,t) = 0, t « . (4) The solution for the homogeneous single layer has been given by Carslaw and Jaeger [4] as , , UL - x) 8$ o L S exp(-(2n + l) 2 TT 2 at/4L 2 ) cos(2n + 1)ttx/2L ... ti(,) A V^fc (2n + l) z * (5) At the input face, x = 0, and for large values of time the series can be truncated at one term to give 9(0,t) = 6 oo {l-8Ti" 2 exp(-iT 2 at/4L 2 )} , (6) where 9 = $ L/A . The value of a can be determined from eq (6) or by the use of the time integral which is defined as TI(x) = ?{l-e(x,t)/e(x,»)}dt . For the input face, (7) TI(0) = L 2 /3a at the center of the slab, TI(L/2) = 1.375(L 2 /3a) and at the output face TI(L) = 1.5(L 2 /3a). When the heat source for the infinite slab is in physical contact with the slab and has heat capacity itself, the conditions used in deriving eq (5) are not exactly fulfilled as it had been assumed that the heat flux came from a source with no heat capacity. Whenever an electrically energized heat source is used, the power is dissipated in a conducting element which may require a substratum for support. Thin sheets of chromel [1], constantan [2,3], palladium [2], graphite coated asbestos, and Fe-Ni evaporated on plastic have been used as heaters. If the heat capacity of the heater is an appre- ciable fraction of the heat capacity of the sample, it is necessary to use a double layer infinite slab for a model. For an infinite slab composed of two layers, each of uniform thickness, an additional pair of boundary conditions are required; namely, the flux and temperature must be continuous at the interface. If the numerical subscripts refer to the sequence of layers from the front face, the boundary conditions may now be expressed as e^Lj.t) = e 2 (L 1 ,t) (8) A^xCL^t) = A 2 6 2x (L 1 ,t) (9) e 2 ((L! + L 2 ),t) =0, t 5- (10) 8i(x,t) =0, t ^ 0, < x < Lj (11) e 2 (x,t) =0, t £ 0, h l < x < (L x + L 2 ) • 508 This boundary value problem can be solved by the use of the theory of Laplace transforms which converts the partial differential equation in 6(x,t) to the ordinary differential equation in U(x,p) by the use of the relation U(x,p) = 7 exp(-pt)© (x,t)dt . The transformed equations and boundary conditions become u ixx( x »p) ~ qi 2 Ui(x,p) = 0, < x < L (la) U2xxU,p) - q 2 2 U (x,p) =0, Lj < x < (Lj + L 2 ) (lb) Uix(O.p) = "WAiP (2a) Ui(Li,p) = U 2 (L 1)P ) (8a) \ 1 V 1 (L 1>P ) = A 2 U 2 ( L i-P) ( 9a > U 2 ((L! + L 2 ),p) = 0, (10a) where qj 2 = p/a^ and q 2 2 = p/a 2 . Applying the boundary conditions to the general solution of the transformed equation gives $ a sinhqj (Li - x) coshq 2 L 2 + coshqi (Lj - x) sinhq 2 L 2 Ui(x,p) = 7- [ — r — : r — ; — - — — — ; r~r — : 1 < x < Li 1 r pqi^i coshqiLj coshq 2 L 2 + sinhqiLi smhq 2 L 2 ' L $ a sinhq 2 ((Lj + L 2 ) - x) U 2 (x,p) = ■ — [ — ; — : — — ; — — r— ] , Li < x < (Li + L?) , " r pqiAj a coshqiLj coshq.L. + sinhqjLi sinhq 2 L 2 ' L L *■' * 111 where a = q 2 A 2 /qiAj = [A 2 d 2 C 2 /AidiCi ] . (12) The solution is given by the inverse transform of eq (12) which may be written as (x,t) = £ (residues of exp(pt) U(x,p)) _ y (13) n where the summation is taken over all of the singular points p of U(x,p). These will be at p = and the zeros of the term in brackets of the denominator of eq (12; . This term can be zero only if the arguments are imaginary, therefore a = tancxmLi tana 2n L 2 (14) when q = ia and it is now necessary to find the n roots of this transcendental equation. Since o is always positive, a root can appear for only those values of «jLi and ot 2 L 2 for which the product of the tangents is positive. Rewriting eq (14) as a = A 2 LiA 2 /AiL 2 Ai = tanAi tanA 2 , roots will occur at the intersections of lines of constant a and the line with slope of AiL 2 /A 2 Li. Using these values of a and evaluating the residues of the singularities, the temperature in layer one is obtained as 509 (Lj - x) L 2 „' 5 + I - 1 O A J A 2 )j(x,t) = $„[ + — ] 2$ <» (a 2n A 2 sinai n (Li - x)cosa 2n L 2 + aiAjcosamCLi - x)sina 2I1 L 2 ) expC-ajajnt) tl ai D , < x < L and in layer two as (15) $ [Li + L 2 - x] o° sina 2n (Li + L 2 - x) exp(-a l0ll £t) e 2 (x,t) = -2 - 2 $ o J — , Li < x < (Lj + L 2 ) where D = [(a ln a 2n X 2 L 1 + a in a 2n A 1 L 2 ) sina ln Li cosa 2n L2 + ( a inAiLi + a?£A 2 L 2 ) cosa ln Li sin<* 2n L2] " 3. Experimental The experimental arrangement as shown in Figure 1 consisted of a central heater, a pair of identical samples and a pair of constant temperature heat sinks. A high heat capacity heater was fabricated from an asbestos heating paper consisting of a graphite conducting layer between two identical covering sheets of asbestos. The uniformity of the power generation per unit area was sufficient to produce temperature differences of less than 3% under steady state conditions when measured at a dozen points on a square foot sample. Power was supplied to the heater from a regulated AC supply. 3 A low heat capacity heater was fabricated from a sheet of 0.025 mm polyimide plastic upon which a coating of 0.008 mm iron nickel alloy had been vacuum deposited . A grid of 3.2 mm strips was formed by preferential etching of the Fe-Ni coating. The high heat capacity samples consisting of 0.305 meter squares with a thickness of 0.0102 meter were cut from a continuous strip of gum rubber. The low heat capacity samples consisted of 0.305 meter square of .0506 m thick foamed polyurethane. A differential thermopile of two junctions of number 30 copper constantan wire was used to measure the temperature. To measure temperatures at interior points in built-up layer samples, the number of couples was increased so that the thermopile outputs were all about equal. Temperatures were recorded with a 12 point recorder at 6 sec. intervals. The recorded temperatures at the front face of the sample were used to produce a graph of lnCe^ - 8 t )/(8 a) - 6 ) vs time from which a could be determined from the slope. The time integrals were also determined from the recorder record by the use of numerical integration. The heat sinks were fabricated from surface ground slabs of .0254 meter aluminum plate and were 0.61 meters square. A labyrinth of 0.00954 meter aluminum with 0.0318 meter channels was bolted to the rear of the heat sink. Thermostated, refrigerated water was circulated through the labyrinth in each sink. The sinks and samples were enclosed with 0.153 meter of foamed polyurethane and placed within an angle iron frame and at a pressure of 300 newtons/square meter applied by means of screw and torque wrench. To shield it from air currents the whole assembly was enclosed within a shroud. 4. Results Graphical analysis of 9(0,t) for a sample of gum rubber heated by means of a graphite-asbestos heater confirmed to the expected exponential approach to equilibrium predicted by eq (5). A slight discrepancy was noted between the observed intercept of 0.82 and the expected value of 8/tt 2 = 0.8105. Changing to the smaller heat capacity heater of Fe-Ni on plastic lowered the intercept to 0.815 and decreased the time integral from 0.1320 hrs. to 0.1237 hrs . The diffusivity derived from the slope remained essentially the same in the two cases. The reduction in the time integral indicates that less total energy was needed to reach equilibrium as the heat capacity of the heater was reduced. The graphite-asbestos heater was used with a sample of foamed polyurethane, a light weight insulating material having a thermal conductivity of 0.016 to 0.022 watts meter" 1 deg~ 1 [5] and a density of 25 to 32 kg/m [5]. Graphical analysis of this case as shown in Figure 2 indicated an intercept of 2 Cellotherm, Chemelex, Inc., Mineola, New York 3 Kapton, Type H, E. I. duPont de Nemours and Company, Wilmington, Delaware **Lashclad X, Lash Laboratories, San Diego, California 510 0.88 and a diffusivity of 5.71 x 10~ 7 m 2 /sec. Upon changing to the Fe-Ni plastic heater, the intercept dropped to 0.83 and the diffusivity increased to 6.77 x 10~ 7 m 2 /sec. The time integral was reduced from 0.482 hr. to 0.348 hr. indicating the proportionally higher ratio of the heat capacity of the heater to the sample. To account for the heat capacity of the heater, it is necessary to consider the experiment as a double layer infinite slab. The flux is assumed to be generated in a plane at x = by Joule heating of an electrical conducting coating having no thickness. The substratum supporting this coating was therefore assumed to be the first layer and to have a thickness of 1/2 of the total thickness of the asbestos-graphite-asbestos heater element. The sample is then assumed to be the second layer. The temperature and, hence, the time integral was measured at the interface between the asbestos and the sample at x = Li . In order to use eq (15) to find the value of a 2 , it is necessary to know the value of a it the diffusivity of the heater substratum. This was done by preparing a sample assembled from a stack of heater elements, thereby, making the properties of layers 1 and 2 identical. The temperatures and time integrals were measured between the first four layers. Eq (15) was simplified by setting a\ = a 2 = a and, then, this was used to obtain a with the appropriate lengths used for each case. This was accom- plished by using a computer program to calculate a time integral for three different values for a, parabolic interpolation to find a new value for a and a new time integral, then using the three closest a's, interpolated to find another new a and then repeating until the fractional change in a was less than lO -4 . This usually required 4 or 5 interpolations. A value of 1.28 + .03 x 10~ 7 m 2 /sec was found for the diffusivity of the graphite-asbestos heater. The thermal conductivity of the heater was deter- mined from a steady state experiment using a heat flow meter. After determining the necessary thermal properties of the heater, the foamed polyurethane graphite- asbestos heater case was re-examined as a two layer case. By using the experimentally determined time integral of 0.4823 hours and the computer program for evaluating eq (15), a diffusivity of 6.815 x 10 m 2 /sec. was obtained. This result agrees with and is slightly larger than the value obtained by the use of the equation a = L 2 /3(TI). The small difference indicates that some error could still be present when using the Fe-Ni plastic heater. Through use of this value for the diffusivity of foamed polyurethane, a value of 1075 Joule kg deg is obtained for the specific heat; this is in fair agreement with a reported range of 840 to 1045 Joule kg -1 deg -1 [6], 5. Beater Error Analysis From examination of the results obtained by graphical and time integral analysis of the transient heat flow in a low heat capacity sample, it has been shown that large errors are introduced when the heat capacity of the heater is ignored. When discussing this type of error, it was found more convenient to use the extensive variables of conductance, A/L, and heat capacity per unit area, dCL, rather than the intensive variables of conductivity, A, and diffusivity, a. The conductance ratio, £ , will be defined as A2L1/A1L2 and the ratio of the heat capacities per unit area, p, as d2C2L2/d^CiLi . From parametric studies of eq (15), it can be shown that at constant conductance ratios, £ , the heat capacity ratio p, is given by p = mfTIoAi/djC^Lj 2 ] + b where m and b are functions of E, and TI is the time integral measured at x = 0. As the conductance ratio £ becomes smaller, the slope approaches 3£ and the intercept approaches -3. For £ less than 0.01 the heat capacity ratio approaches 3£ [TI(0) A^djCjLj 2 ] - 3. This can be reduced to E = 3/p where E is the error in the time integral due to the presence of the heater; ie. , [TI (Heater + Sample) - TI (Sample) ]/TI (Sample). TI (Sample) is the time integral of the sample with a massless heater measured at its input race and would be given by TI (Sample) = L 2 2 /3a 2 . Table 1. Thermal dif fusivities determined a step function change in flux units of meter /sec. Graphical Time Double Layer Method Integral Method Method Sample Heater Eq (6) Eq (7) Eq (15) Gum Rubber Graphite, Asbestos 7.97 x 10~ 8 7.31 x 10" 8 Fe-Ni, Polyimide 7.97 x 10" 8 7.81 x 10" 8 Foamed Polyurethane Graphite, Asbestos 5.71 x 10~ 7 4.92 x 10~ 7 6.815 x 10~ 7 Fe-Ni, Polyimide 6.77 x 10" 7 6.805 x 10~ 7 511 References [1] Plunnner, W. A., Campbell, D. E. and Comstock, Method of measurement of thermal diffusivity to 1000°C, J. Am. Ceram. Soc. 45, 310 (1962). [2] Harmathy, T. Z., Variable-state methods of measuring the thermal properties of solids, J. Appl. Phys. 35, 1190 (1964). [3] Steere, R. C. , Thermal properties of thin- film polymers by transient heating, J. Appl. Phys. 3_7_, 3338 (1966). [4] Carslaw, H. S. and Jaeger, J. C, Conduction of Heat in Solids (Oxford University Press, Oxford, England, 1959), p. 113. [5] Norton, F. J., Thermal conductivity and life of polymer foams, J. Cell. Plast. _3,23 (1967). [6] Editors of Modern Plastics, Plastio in Building (McGraw-Hill Inc., New York, 1966) p. 28. SINK,0 = O ////////// /////////////// SAMPLE SUBSTRATUM X = X=L, CONDUCTOR HEATER SUBSTRATUM SAMPLE X=L ' + L * 7777777777777777777777777 SINK, 0=0 Figure 2. Fractional temperature changes at input face of a foamed ureathane sample. Figure 1. Experiment configuration for double layer infinite slab. 1000 2000 TIME IN SECONDS 512 A Guarded Hot Plate Apparatus for Measuring Thermal Conductivity From -80 + 100 °C Jean-Claude Rousselle Laboratoire National d'Essais (L.N.E.) Paris, France The described apparatus is used for measuring the thermal conductivity of insulating materials from -80 to + 100 °C. Contrary to the usual methods under which two different types of apparatus are dealt with, the one with guarded hot plate for testings above ambient temperature, the other calorimetric for testings at low temperature, the proposed apparatus can be operated continuously at all temperatures from -80 to +100 °C. It is characterised by a conditioning temper- ature maintained lower than the lowest temperature prevailing within the sample. This is obtained by means of a controlled gas current obtained, according to the test conditions, either by spraying of liquid nitrogen or hot air circulation. Samples may be placed within plastic air tight bags and maintained in a gaseous medium different from the one used for the thermal conditioning of the surround- ing box. We give a somewhat detailed description of the dimensional and constructive characteristics of the apparatus. For instance, the results of tests performed on a polyurethane sample between -80 and +40 °C are presented. This test has been run with the sample maintained within an air tight envelope enclosing atmo- spheric air, and with the chamber thermally conditioned by means of liquid nitrogen spraying. Keywords: Conductivity, guarded hot plate, polyurethane, thermal conductivity. 1. Introduction For the measurement of the thermal conductivity of insulating materials from -80° to +100° C, the Laboratoire National d'Essais uses, since the beginning of 1967, two identical apparatuses. These ap- paratuses have been constructed to satisfy the increasing requirement of measurements on refrigeration industry insulating materials above and below the ambient temperature, mainly in the range -80 to +100 °C. In their general arrangement, they are similar to guarded hot plate apparatuses designed for work- ing above ambient temperature. Nevertheless, they include several distinctive features which have been studied to reduce, indeed to remove, the operative difficulties of the standard type arrangement below the ambient temperature. Indeed the guarded hot plate method may even be suitable in case of a sur- rounding temperature lower than the temperatures prevailing in the samples, and specially the coldest face temperature of these samples. When this condition is realized, the lateral thermal losses appear- ing on the edges of the sample may be balanced by superheating the guard heater. On the other hand, if the surrounding temperature remains above the sample temperatures (and particularly the hot face temper- ature) a reheating of the edges takes place and a parasitic thermal flow appears in the central measure- ment area which, theoretically, implies an adjustment by underheating of the heater edges. This is certainly the reason for which the ASTM Standards recommend two different types of apparatuses on each side of the ambient temperature; that is to say: the guarded hot plate apparatus above this limit (ASTM Standard C 177-63 [1] ) and the calorimeter making use of the vaporisation latent heat of liquid nitro- gen for low temperatures (ASTM Standard C 420-62 T [2]). Engineer Chief of the Thermal Conductivity Department. 2 Figures in brackets indicate the literature references at the end of this paper. 513 To make use of two different test set-ups is not always convenient: a. The setting up and taking down of the specimens on two different apparatuses does not ensure the same thermal contacts on their cold and hot faces. b. The calorimetric arrangement comprises a guard ring, but as a general rule, there is no balance of the heating element edge heat losses. c. The guarded hot plate method makes use of two symmetrical specimens while the calorimeter uses only one sample. d. The heat flow measurement is entirely different: gas flow measurement with the calorimeter, and Joulean-dissipated energy measurement in case of the guarded hot plate. Laboratories working with the guarded hot plate method recommend particular building arrangements which permit extension of the method to the low temperatures. The easiest way consists in the use of a large temperature ratio between the two faces of the specimens. In this case, the middle temperature may become higher than the surrounding temperature, and it is then possible to control the superheating. This proceeding is usable only for temperatures close to ambient temperature, and allows only the determination of mean values of thermal conductivity. Two variations of the following proceeding were also tested: The stack, including heater, samples and cold plates, is first enclosed within a casing made of high thermal conductivity material such as copper or aluminum. The walls of this casing are in close thermal contact with the cold plates so that their average temperatures are close to the cold plates ones. Obviously, the metal casing and the cold plates are carefully insulated and great care is taken against moisture. For the second version, the general arrangement is identical with the first one, but the four walls of the metal casing are fitted with a cooling coil. This coil is connected with the cold plates outlet circuit. Both arrangements allow the guarded hot plate method to be used for temperatures as low as -40 °C. Nevertheless, they constitute a poor improvement since it is impossible to obtain a surrounding temperature lower than the cold plates one. Again, appears the disadvantage of using large temperature variation between the two faces of the specimen. Because of the disadvantages inherent in these arrangements, the Laboratoire National d'Essais has fitted two of their own apparatuses with the aim of realizing a surrounding temperature always lower than the temperature prevailing within the sample. This result is obtained by means of a thermally in- sulated external casing controlled in temperature either by a sweeping of gaseous nitrogen resulting from liquid nitrogen evaporation, or by a hot air current. 2. Experimental Equipment 2.1. Specimens The apparatus is convenient on insulating materials having thermal conductivity values varying from 0.015 to 2 kcal/h m °C. The specimens have to be presented in square shaped panels (500 x 500 mm) of uniform thickness (preferably from 10 to 50 mm). Thickness and surface finish uniformity requirements are those prescribed in ASTM Standard C 177. The specimens are generally enclosed within thin plastic welded bags. As a general rule, the filling fluid is atmospheric air. £.2. The Apparatus Features a. Plates Stacking The stacking constituting the main part of the apparatus is of the conventional type of guarded hot plate set up. It is formed with the following parts, all square shaped (see figure 1): One electrical heating plate Two samples on each side of the heater Two cooling plates in close contact with the cold faces of each sample Four sheets fitted with thermocouples (these sheets are sandwiched between the metal plates and the samples). 514 The stacking may be disposed either vertically or horizontally. In horizontal position, in pre- ference, a well known weight loading ensures a good thermal contact between the samples and the heating or cooling plates. A tie-rods and scaled-spring arrangement may also ensure it, particularly in the vertical position. The stacking is placed within a parallelepiped-shaped copper casing, the copper plate ensuring a uniform wall temperature. Thermal insulation between stacking and casing is realized by means of a mineral fibers filling (20 mm thickness) enclosed within air tight envelopes of silicone coated glass cloth. b. Conditioned Casing (See figures 2 and 3) The copper casing enclosing the stacking is set up inside a cubic chamber resting on a base-plate. The walls and base-plate are fitted with a steam baked cork insulation 22 cm thick. Sweeping-gas coils (air, nitrogen), liquid coils (oil, methanol, liquid nitrogen) and electrical wiring (heating and mea- surement) are embedded within the base-plate thermal insulation. Thus it is possible to take down the chamber without disconnecting. From ambient temperature to -80 °C, the chamber is conditioned by evaporation of liquid nitrogen sprayed by a distributor fitted in the upper part of the chamber. The liquid nitrogen flow is control- led by a solenoid valve and a regulating cock. A platinum probe regulator governs the solenoid valve. A small drain connects the chamber with the atmosphere. A fan ensures the temperature homogeneity of the chamber. Between ambient temperature and +100 "C the temperature is controlled by an electrical heating resistor coupled with the fan agitating the ambient air. The resistor is controlled by the above mentioned regulator. The chamber is screwed on the base-plate with intercalation of a silicone gasket. The relative pressure inside the chamber does not exceed 100 mm WC. c. The Features of the Guarded Hot Plate (See figures 4 and 5) This plate (500 x 500 mm) is realized by inserting a heater between two anodic oxidized matte black painted aluminum plates (5 mm thick). The central measurement area is 250 x 250 mm square, insulated by an air gap (1.2 mm wide). When fabricating the plates, eight bridges have been kept; they ensure the mechanical bonding of the measure- ment area with the guard ring (each bridge is about 4 or 5 mrn^ section). The heater constitutes of a wire wound in a double square shaped spiral placed between two asbestos paper sheets (1 mm thick). The heating wire is 0.8 mm diameter Driver-Harris Nichrome V. The coil pitch is 10 mm. There are two separate coils as far as electric energy is concerned. The first one heats the central area and a bordering part of the guard ring. This arrangement avoids discontinuity of specific heating energy in the air gap zone. The second one (4 whorls) constitutes the guard ring. The Joulean-dissipated energy measurement is performed by means of voltage taps located on a level with the air gap. Two sheets of silicone coated glass cloth bearing the superheating control thermopiles are tighten- ed between the heater and the two metal plates (0.3 to 0.4 mm thick). These thermopiles are made of 0.2 mm chromel-alumel wire and each comprises 20 pairs of weldings symmetrically disposed 15 mm apart along the margin of the square measurement area of the metal plates (See figure 6), d. Cooling Plates The two cooling plates are made of copper plates (5 mm thick each). Two superpositioned coils are thermally bounded with the copper plate by means of Thermon (conductive cement including graphite). The coils are wound in a double, square shaped spiral (copper pipe 10 x 12 mm in diameter, 20 mm pitch). One of the two coils is connected with a methanol circuit controlled by a diaphragm flowmeter. The methanol circulation is provided by a Lauda Cryostat controlled from -80 to +20 °C with ±0.2 to 0.5 °C accuracy. The second coil is connected with a Haake thermostat controlled from +20 to +100 °C with ±0.1 °C accuracy. So we are able to control the temperature of the cold plates between -80 to +100 °C without removing the specimens. e. Sample Face Temperature Measurement Sheets Glass cloths (0.2 mm thick) are generally used. The thermocouples wires are insulated with sili- cone and the junctions remain free. Each sheet bears five 0.2 mm chromel-alumel thermocouples disposed according to the R.I.L.E.M. arrangement [3]. That is to say on each face there is one thermocouple in the middle and four thermocouples in the middle of the half-diagonals. These last four thermocouples are series connected and measure the temperatures difference between the faces of the sample. These sheets are useful with specimens such as mineral wool. For hard materials, rubber plates of known thermal resistance are placed between measurement sheets and specimens. 515 f. Measurement Equipment (See figure 7) a. Superheating Control. The two thermopiles of the heating plate are connected separately or in series. A Leeds and Northrup amplifier (type CAT 9835B) and a single channel recorder measure their emf. The guard heater control is manually operated to nearly cancel the unbalance between the central area and the guard ring. b. Surface specimen temperature measurements. The temperature of each face and the four-times- amplified temperature drop of each specimen are measured. The values are recorded on a multichannel automatic potentiometer. The same apparatus records the temperatures of the copper box and of the chamber. When steady state is reached the surface temperatures of the samples are measured by means of a precision potentiometer (P-12 type A. 0.1. P. -1 |j,V sensitivity). c. Joulean-dissipated energy measurement in the central area. The heating plate is supplied with direct current. Automatic regulation of the generator is ±0.05% of the fixed value. Standard resistors and the precision potentiometer measure the emf and the current intensity in the measurement area. 3. Tests Run Presently, tests are being run in the apparatus on polyurethane specimens undergoing different aging treatments. To specify the apparatus working conditions, we describe the test measurements technique and results for a polyurethane specimen. 3.1. Test Description The test is run on expanded polyurethane having the following characteristics: Mean density before test: 35.95 kg/nr Expansion agent: freon (exact type not stated) Aging conditions: 15 months in ambient air and temperature Specimen sizes: 500 x 502 x 50 mm, 500 x 502 x 50 mm Total weight of the two samples: 0.905 kg (The two samples are enclosed within air tight bags made of welded plastic film (0.07 mm thick, Manufactured by "La Cellophane" under the name of "Therthene")). Test in horizontal position Loading pressure 0.06 kg/cm Inside the bags, the specimens remain in contact with the atmospheric air initially introduced. During the test no gas sampling has been effected. It may be supposed that during the test an air- freon blend takes place with the temperature changes. Thickness variations of the specimens versus the temperature are measured before starting the test within an insulated glass walled cold chamber. The two specimens remain in their bags and mean temperature and loading conditions are reapplied. Thick- ness variations are metered with a cathethometer. Before test the specimens have been submitted to a cycling ambient temperature down to -80 °C and back to ambient temperature. 3.2. Test Results In the test described, the conditioned chamber is always adjusted relative to the cold plates temperature. A small difference between these temperatures is kept for all test points. The measured values during the test are grouped in table 1. 4. Conclusions These new apparatuses have the following advantages: A small temperature ratio between the stacking and the surrounding, the surrounding temperature being always below that of the cold plates. Only a small amount of thermal insulation is required around the stacking (2 cm of fibrous mineral wool). A very homogenous surrounding temperature, thanks to the copper box. Hence, it follows that: A small disadjusting of the temperature unbalance between the central area and the guard ring, from one temperature test level to the next. The testing run with a constant central energy, this unbalance needs only a small super- heating correction from one point to another. 516 A much smaller thermal inertia than with a large amount of lateral thermal insulation. An eminent time benefit. Each point needs, on the average, less than 48 hours instead of a minimum of 72 hours with the previously used procedures. Adjustments which are as easy at the limits of the temperature range as at ambient temperature. Testing continuity on either side of the ambient temperature without dismounting the stack. The cooling by evaporation of liquid nitrogen shows ease of utilization and reliability. 5. Acknowledgments The author wishes to express his appreciation to the management for its acceptance and encouragement concerning the present paper. The author is further indebted to the team of engineers and technicians having performed the fine adjustments of these apparatuses. 6. References [1] ASTM, C177-6J Thermal Conductivity of Materials by Means of the Grarded Hot Plate. [2] ASTM, C420-62T. Thermal Conductivity of Insulating Materials at Low Temperatures by Means of the Wilkes Calorimeter. [3] R.I.L.E.M. , Method of Test for Thermal Conductivity of Building Materials by Means of the Guarded Hot Plate. September 1962. Table 1. Values Obtained for the Thermal Conductivity of the Sample of Polyurethane Foam. ■1 1 II Mean Temperature of Unbalance 1 1 n 1 «i Mean Tempera- ture in Samples Between Central & Guard Area Temperature Difference ^xl00 A9 R k.10" 8 Point N° Hot Face Cold Face Chamber Ambient Specimen Thickness 9 °C m 9 H °C 9 °C c °C At °C A9 °C cm 8 °C w w cm °C cm 1 -69.3 -60.3 -78.3 -85 +O.OO36 18 0.020 1.740 0.0287 4.99 5 2 -50.2 -41.3b -59.1 -66.5 +0.0084 17.8 0.047 1.715 0.0291 4.99 s 3 -31.7 -24.0 -39.4 -46.5 -0.0024 15.4 0.016 I.636 0.0305 4.99a 4 -18.8 - 9.4 -28.2 -35 -0.0045 18.8 0.024 1.751 0.0285 5.01 5 - 7.0 + 3-8 -17.8 -24 +0.0018 21.6 0.008 1.959 0.0255 5.01b 6 + 0.9 6 +11.6 - 9.7 -17.5 -0.0006 21.36 0.003 1.932 0.0258 5.0Ls 7 + 14. % +25.9 + 4.0 - 3-2 0.0 21.8b 1.997 0.0250 5.02 8 +29.7 +39. 7 B + 19.6 +12.5 +O.OO36 20. Is 0.018 1.826 0.0273 • 5.02s 517 Tie - Rods and Springs- Copper Bo arded Plate 50x50 cm.) Figure 1. Arrangement of the staking in its copper box. Figure 2. General features of the insulated conditioned chamber. 1 - Insulated base-plate containing the different circuits. Insulated chamber. Stacking in his copper box in hori- zontal position. The same in vertical position. Electrical leads and measurements wires. Solenoid-valve on the liquid nitrogen pipe. - Liquid nitrogen inlet. Oil inlet. Oil outlet. Manuel oil flow control cocks. Oil float flowmeters. Methanol circuit. Methanol flow rate measurement diaphragms. 13 - Manual methanol flow control cocks. 14 - Diaphragms U tubes. 15 - Distributors. 2 • 3 ■ 3'- 4 ■ 6 7 8 9 10 11 12 Figure 3. Conditioned chamber with liquid nitrogen tank and oil thermostat in the rear. 518 Figure 4. Exploded view of the guarded hot plate. 1 - Heater. 2 - Metal plate. 3 - Thermopile support siliconed glass cloth. 4 - Thermopile. 5-6 - Heating wires connections. 7 - Thermopiles wire connections. Figure 5. Guarded hot plate. Figure 6. The thermopile and its glass cloth support siliconed both alike. 519 DC Regulator ' Multi -channel Recorder ( Temperatures) Energy Control h 1 Precision Energy Measurement in the Central Heater Manual Com - mutator ' Precision Potentiometer Galvanometer — »■ Ice Box I Guarded Hot Plate DC. Amplificator Recorder Thermostats + 20 to-80"C. + 20 to +100'C *" i ' I Cold Plates A utomatic Control A Liquid Nitrogen Tank Fan and Heater 1 \ J Figure 7. General chart of the guarded hot plate apparatus (range -80 to - 100 C) ♦ 520 A Study of the Effects of Edge Insulation and Ambient Temperatures On Errors in Guarded Hot-Plate Measurements H. W. Orr Division of Building Research, National Research Council of Canada Ottawa The error due to edge effects of the guarded hot-plate apparatus has limited the thickness of specimens that may be tested to approximately one third the dimension of the central section of the heating unit. Samples of known conductivity were tested using various thicknesses of edge insulation and various ambient temperatures. It was found possible to determine the optimum edge conditions so that errors due to edge effects are negligible for very thick specimens. KeyWords: Conductivity, errors, guarded hot-plate, heat conduction, standard test, thermal conductivity, thermal insulation. 1. Introduction The most widely used and precise method of determining thermal conductivity is with the guarded hot-plate and a standard method using this equipment has been adopted by ASTM (1) . This method of test limits the thickness of specimen to one third the linear dimension of the heating unit. In order to test thick specimens using this method, guarded hot-plates with proportionately large dimensions are required. Another approach is to slice the material and produce a number of thinner specimens that may be tested in a smaller guarded hot-plate. A third approach is to test thick specimens in a small guarded hot-plate apparatus and reduce errors due to edge effects by control of the boundary conditions. Each of the three methods has its limitations, A large hot-plate is expensive, difficult to build and operate, and not suitable for the majority of specimens. Because of this many laboratories are un- likely to have one. Slicing samples to make specimens thin for testing in smaller guarded hot-plates is not always convenient or desirable and testing thick specimens in small plates can result in large errors due to edge effects. An experimental program was proposed to evaluate the possibility of re- ducing edge errors so that thick specimens can be tested on small guarded hot-plates. The guarded hot-plate method for measurement of thermal conductivity is based on the assumption that isotherms in the central area of the test specimens are planes parallel to the hot and cold-plates, i. e. , that the heat flow is perpendicular to the faces of the sample. This assumption of one-direction- al heat flow is valid only for thin specimens unless the edge of the sample has a temperature profile that is nearly the same as that at the centerline of the sample. The normal procedure for the guarded hot-plate is to limit the heat gained or lost from the edge of the sample by using edge insulation, by maintaining the ambient temperature at an appropriate level ~{Z), and by limiting the thickness of the specimen (3) so that errors due to edge effects are negligible. With thick specimens the isotherms become distorted if heat is allowed to enter and/or leave the edge of the specimens. This may result in an erroneous measurement of thermal conductivity if the edge conditions are not carefully controlled. If the ambient temperature is much above the sample temperature, heat will flow into the edge of the specimen, distorting the isotherms and making the Figures in brackets indicate the literature references at the end of this paper. 521 apparent thermal conductivity low. An ambient temperature much below the sample temperature will give the opposite effect. It would appear, therefore, that there is some optimum temperature that will give a minimum error. Since the apparent thermal conductivity is affected by heat entering or leaving the edge of the specimen, edge insulation should help to reduce the flow of heat and consequently reduce the error in the thermal conductivity. The object of the experimental work reported in this paper was to determine the effectiveness of using edge insulation and of controlling ambient temperature in reducing errors when measuring the thermal conductivity of thick specimens. 2. Experimental 2. 1 Equipment The guarded hot-plate equipment used for the tests was similar to one designed by the National Bureau of Standards in Washington (4). The test area is 10 cm square with a 5 -cm-wide guard area that is automatically kept at the same temperature as the test area. The hot-plate was operated in a horizontal position in an insulated box. The temperature in the box was controlled by circulating fluid at a controlled temperature through heat exchanger plates that covered three walls of the box. The air in the box was stirred with a large-diameter, low- speed propeller fan. 2.2 Specimens Polyurethane foam and silicone rubber were used to make up the samples for test. Thick samples were made by stacking homogeneous material in slices 13 mm to 19 mm thick. The actual thermal con- ductivity was determined by adding the previously determined resistances of pairs of slices and convert- ing the total resistance to thermal conductivity. The resistance of each pair of slices was determined in two guarded hot-plates, in the plate used for the thick specimens, and in a plate of the same size, similar in design to one developed at the University of Saskatchewan and described by Woodside and Wilson (5). The values obtained with the two guarded hot-plates had a maximum deviation of 0. 06 per cent and a mean deviation 0. 03 per cent for the polyurethane foam samples and 0.46 per cent and 0.21 per cent respectively for the silicone rubber. 2.3 Method The first series of tests was carried out to check the effect of varying the thickness of edge in- sulation. Specimens of polyurethane foam and silicone rubber 85 mm thick were used for these tests. The ambient temperature was controlled close to the mean temperature of the sample 24 C; the temperature difference across the sample was maintained at 22 C. A series of tests was run starting with no edge insulation and applying 13 mm, 25 mm, 51 mm, and 76 mm of edge insulation to the edge of the specimen covering the hot-plate completely and extending 20 mm beyond the faces of the cold- plates. The test was repeated with an ambient temperature of 38 C; all other conditions remained the same. The edge insulation had a thermal conductivity approximately equal to the specimen. The poly- urethane samples had edge insulation of polyurethane with a X = 0. 0230 W m'^deg" while the silicone rubber samples had neoprene rubber for the edge insulation with a \ = 0.304. After the completion of these two series an approximate optimum ambient temperature for minimum error was determined by a linear interpolation of the previous results and a third series was run using this ambient temperature. The results of these three tests are given in Table 1 for polyurethane foam and in Table 2 for silicone rubber. The ambient temperature has been converted to a nondimensional temperature index (ATI) by relating it to the temperatures of the hot- and cold-plate, i. e. ATI = <{T - T ) / (T, - T ). Where T is the ambient temperature and T and T are hot- and cold-plate temperatures respect- ively. 2.4 Results The results shown in Tables 1 and 2 have been plotted in figures 1 and 2. The error in thermal conductivity has been plotted against the thickness of edge insulation for corrected ATI. For each thickness of edge insulation the best straightline fit was used to determine the optimum ATI for mini- mum error and to correct the ATI to 0. 5, 1.10, and a value near the optimum for plotting. The values of optimum ATI have been plotted against the thickness of edge insulation in figures 3 and 4. 522 Table 1 Experimental values of error in thermal conductivity measurements for 85. 05 mm polyurethane foam samples Polyurethane foam samples: \ - 0.230 d = 31.89; NBS guarded hot-plate, in a horizontal configuration: 203. 2 mm x 203. 2 mm, test area 101.6 mm x 101.6 mm. Edge Mean Temperature Ambient Insulation Temperature, Difference, Temperature Error, thickness, •c °C Index per cent mm Series 1 23. 89 22.26 0.479 6.4 12.7 24. 12 22.86 0.479 3.5 25.4 24. 19 22.94 0.476 2.8 50. 8 24.28 23. 16 0.508 1.7 76.2 24.34 23.21 0.502 1.5 Series 2 23.91 22. 11 1. 103 -27. 65 12.7 24.25 22.99 1. 100 -16.26 25.4 24. 13 22.77 1. 121 -13.27 50.8 24.39 23. 16 1.081 -7.84 76.2 24. 18 22.78 1.098 -6.33 Series 3 24. 08 22.56 0. 649 -0.6 12.7 24.08 22.61 0. 646 -0.7 25.4 24. 12 22. 28 0. 668 -2.2 50.8 24.29 23. 14 0.631 0. 1 76.2 24.23 23. 12 0. 626 0.2 The ambient temperature for minimum error is between the mean temperature of the sample and the temperature of the hot-plate. This is the expected result for a guarded hot-plate where the heat is metered at the hot-plate side. If the heat is metered at the cold-plate side the optimum ambient temperature would be between the mean and cold-plate temperatures. When the ATI is not the opti- mum, edge insulation reduces the error as expected. With thick edge insulations the ambient temper- ature is not as critical. The optimum ATI appears to be a linear function of the edge insulation thick- ness. Thicker edge insulation requires a higher optimum ATI. Specimens of high thermal conducti- vity require higher ATI for minimum error for all thickness of edge insulation. Using the results of these series of tests an attempt was made to see how thick the specimens could be made and still maintain a reasonable degree of accuracy. The thickness of polyurethane foam specimens was varied from 75 mm to 189 mm while the edge insulation thickness was kept constant at 57 mm and the ATI was near optimum as determined in the first series. As specimen thickness was increased, the measured error was larger than expected when compared with the tests done with 85- mm samples. This would seem to indicate that the optimum ATI is also a function of specimen thick- ness. The results are shown in Table 3. Conclusion The error due to the edge effects of the guarded hot-plate may be reduced with edge insulation but only becomes zero when the ambient temperature is at one specific value. The optimum ATI appears 523 Table 2 Experimental values of error in thermal conductivity measurements for 84.46 mm silicone rubber samples 1190 kg/m 3 Neoprene rubber edge insulation: \ = 0.304, d = 1400 kg/m NBS guarded hot-plate, in a horizontal configuration: 203.2 mm sq, test area 101. 6 mm sq Silicone rubber samples: \ = 0. 246 , d Edge Mean Temperature Insulation Temperature, Difference thickness, °C °C mm Series 1 23. 87 22. 12 12.7 23.89 22. 10 25.4 24.01 22.24 50.8 23.94 22. 10 7 6.2 23.82 22. 08 Ambient Temperature Index Error, per cent 0.508 0.499 0.502 0.512 0.500 3.62 3.28 3.22 3.31 3.86 Series 2 12.7 25.4 50.8 76.2 23.82 23.75 23.99 23.93 23.84 21.94 22. 17 22. 17 22.07 21.99 1. 150 1. 127 1. 164 1.166 1. 150 ■18.4 •12.32 ■10.49 -6.32 -4.46 Series 3 12.7 25.4 50.8 76.2 24.02 23.89 23.92 23.92 23.83 22.52 22.25 22. 11 22.05 22.07 0.604 0. 603 0.608 0. 600 0. 610 0. 18 0.77 0.97 2. 07 2. 14 Table 3 Experimental values of error in thermal conductivity measurements for polyurethane foam samples tested in an NBS 203. 2 mm x 203. 2 mm guarded hot-plate using 57 mm of polyurethane foam edge insulation Sample Mean Temperature Thickness, mm Temperature, Difference, °C °C 75.29 23.91 22.25 75.25 23.93 22.26 75.25 23.92 22.21 75.25 23.-87 22. 12 75.25 23.95 22.25 75.25 23.89 22. 14 75.25 23.88 22.13 94.06 23.93 22.28 112.89 23.85 22. 14 112.85 23.96 22.26 112.85 23.93 22.24 131.68 23.89 22.24 150.47 23.91 22.28 150.42 23.92 22.26 150.42 23.90 22.21 150.42 23.89 22. 16 150.42 23.92 22.22 150.42 23.89 22. 17 150.42 23.89 22. 17 169.24 23.97 22.40 169.23 23.86 22. 16 169.23 23.92 22.25 169.23 23. 80 22.01 Ambient Error, Temperature per cent Index 0.647 -1.0 0.596 -0.6 0.618 -1.4 0. 666 -1.9 0. 680 -1.8 0.619 -1.4 0.594 -0.9 0.661 -2.4 0. 659 -3.5 0.594 -3.0 0.585 -2.3 0. 660 -4.0 0. 628 -5.0 0. 569 0.4 0.564 0.2 0.568 0.3 0.598 -2. 1 0.597 -1.9 0.602 -3. 1 0.625 -4. 1 0.643 -6.7 0.628 -6.4 0.627 -5.4 524 to be a function of the edge insulation thickness, sample thermal conductivity, and thickness. In the work reported here the edge insulation had a thermal conductivity the same as the sample. Edge insulation with a thermal conductivity lower than the sample will probably reduce the error due to edge effects, but the optimum ATI will doubtless change. This paper deals mainly with a sample thickness that is 0. 84 times the linear dimension of the central area of the hot-plate. From the values of error obtained with much thicker samples it appears that different values of specimen thickness have different values for the optimum ambient temperature index. It would appear that samples that are as thick as the width of the central area of the hot-plate may be tested with little additional error provided edge insulation is used and the ambient temperature is maintained at the optimum value. More experience is needed, however, before thick specimens can be tested with assurance that errors due to edge effects have been eliminated. It should be noted that homogeneous samples were used in this study and the results are not necessarily applicable to non- homogeneous specimens. This is a contribution from the Division of Building Research, National Research Council of Canada, and is published with the approval of the Director of the Division. 4. References (1) Method of Test for Thermal Conductivity of (4) Ibid, Note 1, p. 17. Materials by Means of the Guarded Hot-Plate (C177 - 6?), 19 66 Book of ASTM Standards, (5) Woodside, W. and Wilson, A. G. Unbalance Part 14, p. 17. errors in guarded hot-plate measurements. In Symposium on Thermal Conductivity (2) Ibid, Paragraph 4(k) p. 22. Measurements and Applications of Thermal Insulations, A. S. T.M. Special Technical (3) Ibid, Paragraph 6(a) p. 25. Publication No. 217, p. 32-46 (1957). 525 "" — I — r K 0.50 0.62' 1. 10 AMBIENT TEMPERATURE INDEX -28' I I I I I I— 20 40 60 J L 80 100 EDGE INSULATION (POLYU RETHANE FOAM) MM Figure 1. Error in X vs edge insu- lation thickness shoving effect of ambient temperature indexes for 85 mm samples of polyurethaae foam. 4J= -4 12 16 1- 1 1 1 ' 1 1 1 I =- °- 5 .- — 0.7 m ^^^m k* - ^'^\. 1 - /^—AMBIENT - / TEMPERATURE - / INDEX ■ - 1 . 1 1 20 - -24 h -28 20 40 60 80 100 EDGE INSULATION (NEOPRENE RUBBER) MM Figure 2. Error in X vs edge insu- lation thickness showing effect of ambient temperature indexes for 85 mm samples of silicone rubber. 1.2 20 40 60 80 100 EDGE INSULATION (POLYU RETHANE FOAM) MM 20 40 60 80 100 EDGE INSULATION (NEOPRENE RUBBER) MM Figure 3. Optimum ambient temper- ature index vs edge insulation thickness for 85 mm samples of polyurethane foam. 526 Figure 4. Optimum ambient temper- ature index vs edge insulation thickness for 85 mm samples of silicone rubber. Ring Heat Source Probes for Rapid Determination of Thermal Conductivity of Rocks 1 Wilbur H. Somerton and Mohammad Mossahebi University of California, Berkeley National Iranian Oil Company, Tehran, Iran A ring heat source probe has been developed for the rapid deter- mination of thermal conductivity of rocks. In addition to the short period of time required to run the test (2-3 minutes), sample prepa- ration is minimal, requiring only that parallel flats approximately two cm in diameter be cut or ground on opposite sides of the sample. k. calibration chart can be provided with each probe so that thermal conductivity can be obtained from simple experimental data without elaborate calculation. Key Words: Conductivity, diffusivity, probe, ring heat source, rocks, thermal conductivity, thermal properties of rocks. ■'■Paper published in: The Review of Scientific Instruments, Vol. 38, no. 10 (October, 1967) 527 Thermal Conductivity and Thermal Diffusivity of Green River Oil Shale S. S. Tihen, H. C. Carpenter, and H. W. Sohns 1 Laramie Petroleum Research Center Bureau of Mines, U.S. Department of the Interior Laramie, Wyoming 82070 Information on the thermal properties of oil shale is necessary for process design of both aboveground and in situ processes. This paper presents data on the thermal conductivities and thermal diffusivities of various grades of oil shale, of retorted shales, and of burned shales which are free of organic matter. The thermal properties are correlated with shale grade, temperature, and bedding plane orientation. Determinations were made both perpendicular and parallel to the bedding planes, and at several different temperature levels. Conductivity values ranged from about .69 W m~l deg~l for 58. 6-gallon-per-ton shale to about 1.56 W m _i deg" 1 for a lean, 8. 6-gallon-per-ton shale. Conductivities of the retorted or spent shales from these raw shales ranged from .26 to 1.38 W m'*- deg"l and of the burned shales from less than .17 to 1.21 W m"l deg"l. An equation was developed for calculating the thermal conductivity of raw, retorted, and burned Green River oil shales. Thermal diffusivities were determined experimentally on twelve different grades of raw shale assaying from 8.1 to 59.1 gallons of oil per ton and on the retorted and the burned shales from these raw shales. Thermal diffusivities of raw shales ranged from 2.6 x 10"' to 9.8 x 10"' m^/sec. Diffusivities of the 7 7 9 retorted shales ranged from 1.3 x 10"' to 8.8 x 10"' m^/sec, and of burned shale from 1.0 x 10" 7 to 7.2 x 10" 7 m 2 /sec. Key Words: Burned shale, oil shale, retorted shale, thermal conductivity, thermal diffusivity, transient line source method. 1. Introduction During the last decade, oil-shale research and development in this country have progressed to the point where oil-shale processing appears to be an emerging industry. A number of aboveground retorting processes have been demonstrated on a pilot or prototype scale, and efforts to develop in place, or in situ, processing methods are currently receiving much attention. The operability of any in situ process is largely dependent on the physical structure and thermal properties of oil shale. Information on the thermal properties of oil shale is .necessary for process design of both aboveground and in situ processes. This paper presents data on the thermal conductivities and thermal diffusivities of oil shales of various grades, of retorted or spent shales, and of burned shales which are free of organic matter. These thermal properties are correlated with shale grade, temperature, and bedding plane orientation. 2. Thermal Conductivity 2.1 Apparatus and Procedure The apparatus was designed to utilize the transient line source method of measuring thermal conduc- tivity. In this procedure heat is supplied at a constant rate from a line source in the interior of the material being tested and temperature rise at the heat source is measured as a function of time. These data are then used to calculate thermal conductivity. This method, adaptable to a wide variety of materials and conditions, has been used and discussed by a number of investigators (1,2,3). Oil shale Research Chemist, Project Leader, and Project Coordinator, respectively. 2 Figures in parentheses indicate the literature references at the end of this paper. 529 is a heterogeneous material which decomposes as it is heated and undergoes changes in physical and chem- ical properties. Because of these phenomena the transient line source method is particularly suitable for determining thermal conductivity of oil shale. a. Apparatus A specially designed oven was used to maintain the oil shale sample at the desired temperature. This oven was designed in such a way that the shale samples could be surrounded by an inert atmosphere, and that inert sweep gas could be used to remove the combustible pyrolysis products during heating. A direct-current, constant -current power supply was used to heat the line source. An ammeter sen- sitive to 1 milliampere in the to 1 ampere range was used to measure the current, and a strip recorder with 1-millivolt full-scale sensitivity was used to measure and record temperature. b. Sample Preparation The test pieces, 7.5-cm cubes, were prepared from 7.5-cm by 7.5-cm by 18-cm blocks of shale which were carefully chosen for uniformity and absence of fractures. Two 7.5-cm cubes were cut from each block and the remaining material was assayed. These blocks were cut in such a way that the same bedding planes occurred in all three pieces. A face-centered hole was drilled through each of the two cubes to accommodate a slender, multi-hole porcelain tube containing a heating wire and a butt-welded thermo- couple. In one cube the hole was parallel to the bedding planes and in the other the hole was perpen- dicular to the bedding planes. Iron plates were placed on the faces parallel to the bedding planes and bolted together, to restrict separation of the layers during heating (fig. 1). The restraining plates can be seen in position on the blocks of 58. 6-gallon-per-ton shale. Subsequent removal of these plates allowed the blocks of rich shale to disintegrate completely. After conductivity determinations had been made on the raw shale, the cubes were retorted in an inert atmosphere at 482° C, and used for retorted shale conductivity determinations. These cubes were then heated in a stream of air to remove all organic matter and subsequently were used for burned shale conductivity determinations. Figure 1 pic- tures the cubes of burned shale, and clearly shows the disintegration that took place in the shale structure of the richer shales during heating. The lean, 8 . 6-gallon-per-ton shale developed only tiny cracks during heating. These cannot be seen in the photograph. The 11.4-gallon-per-ton shale developed visible shallow cracks, while the 19.0-gallon-per-ton shale developed larger, deeper cracks. c. Experimental Procedure Each pair of shale samples was placed in the oven and the heating wire and thermocouple connections were made, following which the oven was brought to the desired temperature. After the sample had reached constant temperature, current was supplied to the heating wire at constant rate and the temperature rise, with respect to time, was measured. Five replicates were made on each sample at each of the following ambient temperatures: 38°, 149°, and 260° C. A nitrogen atmosphere was used for the determination at 260° C. Retorted shale determina- tions were made at 38°, 149°, 260°, 371°, and 482° C, again using a nitrogen atmosphere for temperatures above 149° C. Conductivity determinations on burned shale were made in air at the same temperatures as those on retorted shale, and also at 593° C. 2.2 Calculations and Discussion of Results The transient line source technique involves calculation of thermal conductivity from power input to the heater wire, the temperature rise of the specimen, and the time. This relationship is expressed by a complex mathematical eq (1) which may be approximated to within less than 1 percent error by the following expression: 2 9 = - — &— (y + In ^_ ), (1) 47T/1 4at r2 if the value of — — is held equal to or less than 0.0058, 4a t where 8 = the temperature rise in °C, Q = heat supplied per unit length per unit time in J m sec , A = thermal conductivity in W m deg" , Y= Euler's constant: 0.5772, R = distance from the line source in meters, a = thermal diffusivity in m^/sec, and t = time in seconds. 530 Comparing temperatures at two different times, e 2 - 9i = — 2_ (lnJL_ - In J*— ) = SL In -1 . (2) 4'TTX 4ati 4at 2 4 TT A tl ' Then /I = 2 In _?- (3) 4 7T(92 - Ql) ti R 2 Restriction of — — to 0.0058 is easily met in the laboratory within short periods of time because the 4a t , 7 o values of a for most rocks in the temperature range covered lie between 3.84 x 10" D to 2.58 x 10 m z /sec and R can be held to values of 3.05 x 10 m or less. Equation (3) requires only the heat input and time- temperature values for calculating thermal con- ductivity. Temperature rise is plotted as a function of time on semilog paper. After an initial tran- sient period, the values of any two points on straight portion of the curve, along with Q, calculated from the current and resistance per unit length of the heater wire, are used to calculate thermal conduc- tivity. Because oil shale has a tendency to crack and separate into layers upon heating, only the first portion of the heating curve could be used to determine the slope. Perfect contact between shale and the heating source is not attainable; therefore, the first part of the curve must be corrected for the error which imperfect contact introduces. This can be done by several methods. The one used here con- sisted of choosing a time, x sec, such that the temperature rise between (8 + x) sec and (32 + x) sec equaled the rise between (32 + x) sec and (128 + x) sec. These points, when plotted against temperature on semilog paper, then fall in a straight line. Therefore, they can be substituted in eq (3) to calcu- late thermal conductivity. a. Calculation of the General Equation Statistical and computer techniques were used to develop exponential, quadratic, and second-order equations from the A values obtained above for raw shale, retorted shale, and burned shale perpendicular to and parallel to the bedding planes. Statistical examination of these equations showed that the second-order equation gave the lowest standard error of estimate. The following equation, therefore, is the one recommended for calculating the thermal conductivities of Green River oil shales assaying between 8.6 and 58.6 gallons per ton within the temperature ranges of 38° to 260° C for raw shale, 38° to 482° C for retorted shale, and 38° to 593° C for burned shale: ^= c l + C 2F + C3T + C4F 2 + C5T 2 + C6FT, (4) where A is the thermal conductivity of the shale in W m deg , Cl through Cg are constants, F is the Fischer assay of the original shale in gallons per ton, and T is the temperature in °C. Table 1 lists the values of the constants in the above equation for the various conditions under which thermal conductivities were determined, and also the standard error of estimate of A for each of the conditions. The two curves of figure 2 show the relationship of thermal conductivity at 38° C of raw shale to Fischer assay. The three curves of figure 3 show the thermal conductivities of raw, retorted, and burned shales at 38° C. These figures show that thermal conductivity of Green River oil shale varies inversely as shale grade, and that conductivities of retorted and burned shales are generally lower than those of the raw shales from which they were obtained. The burned shales, in turn, have a lower conductivity than the corresponding retorted shales. This can be accounted for by the fact that the mineral matter is a better conductor of heat than the organic matter and that the organic matter is a better conductor than the voids created by its removal. 3. Thermal Diffusivity 3.1. Apparatus and Procedure The apparatus used for determining thermal dif fusivities of Green River oil shale was essentially the same as that used for the thermal conductivity determinations. However, sample preparation and laboratory procedure differed somewhat. a. Sample Preparation The samples were prepared by splitting 7.5-cm cubes of the shale into two equal parts by sawing 531 TABLE 1. - Constants and standard errors of estimate for the calculation of thermal conductivity of Green River oil shale using the equation: %= Ci + C2F + C3T + C4F 2 + C5T 2 + C6FT Standard Cl C2 C3 C4 C5 C6 error of estimate Raw Shale: Case A a +1.8081 -3.698xl0" 2 +1.980xl0" 3 +3.056xl0" 4 -5.184xl0" 6 -1.872xl0" 5 0.173 Case B b +2.0670 -5.779xl0" 2 +1.572xlO" 3 +5.686xl0" 4 -4.585xl0 -6 -1.470xl0" 5 .172 Retorted Shale: Case A +1.8246 -4.4844xl0" 2 -2.309xl0~ 4 +3.652xl0" 4 +1.067xlO" 5 .133 Case B +1.5355 -5.923xl0~ 2 +3.030xl0" 5 +6.250xl0" 4 -2.935xl0" 9 +2.698xl0 -6 .162 Burned Shale: Case A +1.5583 -4.753xl0' 2 +8.724xl0" 5 +3.651xl0" 4 -8.483xl0" 7 +1.142xl0" 5 .157 Case B +1.2111 -4.927xl0" 2 +1.181xl0" 4 +5.376xl0" 4 -5. 00 9x10" 7 +1.827xl0" 6 .122 a Probe perpendicular to shale bedding planes. ° Probe parallel to shale bedding planes. them parallel to the bedding planes. The cut faces were polished to provide good contact when they were placed together. Two small parallel grooves were cut in one of the faces with a scribe mounted in a milling machine. One groove was cut in the center of the block and the other 0.64 cm from it. The heater wire was placed in the center groove and the thermocouple was placed in the outer groove after which the two halves of the block were clamped together securely. b. Experimental Procedure The millivolt recorder was adjusted to a 1-millivolt full-scale response and to zero reading. Chart speed was set at 2.54 cm (1 inch) per minute, and as the pens passed one of the time-indicating lines of the chart, current was applied to the heater wire and allowed to flow for 10 minutes. After the block had cooled, the experiment was repeated. Determinations were made at each of five heat input levels on each shale sample. 3.2. Calculations and Discussion of Results The following eq was used to calculate dif fusivities from recorder readings taken at 1-minute intervals: In a 94 7T A + V + In *_ 4t (5) where the symbols have the same meaning as in eq (1). This eq was obtained by solving eq (1) for In a. The A values were from eq (4). Dif fusivities which did not vary from the asymptotic value by more than 5 x 10" 8 m 2 /sec were averaged to obtain the diffusivity for that particular run. The standard deviation of a was about 2.3 x 10 m /sec for ten runs. In order to obtain a further indication of the accuracy of the experimentally determined values, they were compared with calculated values obtained by using the following eq: d c r (6) where and a = thermal diffusivity in m 2 /sec, A = thermal conductivity in W m"-'' deg" , d = density in kg/m3, : p = specific heat in J kg -1 deg -1 . The values of A were again obtained from eq (4). Densities and specific heats were obtained from curves which correlated these properties with Fischer assay. Densities of the retorted and of the 532 burned shales were obtained by assuming that 85 percent of the organic matter was removed by retorting without any change in shale volume, and that burning removed all of the organic matter without any change in volume. The latter densities are in reasonable agreement with those recently reported by Tisot (4). Table 2 compares the experimentally determined thermal dif fusivities with those obtained by use of TABLE 2. - Thermal diffusivity of Green River oil shale, 10" 7 m 2 /sec Oil yield Raw sha le Retorted shale Burned shale gal /ton Experimental Calculated Experimental Calculated Experimental Calculated 8.1 9.0 7.2 4.9 5.7 3.4 4.1 9.7 9.8 7.0 7.2 4.9 6.2 3.9 12.9 8.3 6.2 8.8 4.4 7.2 3.4 13.3 6.7 6.2 3.6 4.4 3.1 3.4 22.6 4.1 4.9 2.3 3.1 1.5 2.1 26.1 3.1 4.4 2.1 2.6 1.5 1.8 28.5 4.1 4.1 - 2.3 2.1 1.5 28.5 4.1 4.1 2.6 2.3 1.8 1.5 48.1 2.6 2.8 1.3 1.-5 1.0 .8 . 50.7 3.4 2.8 2.1 1.5 - .8 55.0 3.9 3.1 - 2.1 - 1.3 59.1 4.6 3.1 - 2.6 - 1.8 eq (6), which does not require a direct measurement. The reliabilities of the parameters used in eq (6) are as follows: A, + 0.10 W m -1 deg -1 , d, + 160 kg/m3, and Cp, + J kg~l deg"l. For a 28-gallon-per-ton shale after retorting, the true value of the above parameters could lie anywhere within the following ranges: A, 0.47 - 0.67 W m" 1 deg" 1 , d, 1730 - 2050 kg/m 3 and c p , 0.38 - 0.92 J kg" 1 deg"l. Substituting the limits of these parameters in eq (6) gives the following limits for a: 2.58 x 10"? to 4.90 x 10"? m^/sec. Considering the heterogeneity of oil shale, this spread is not at all surprising, and the differences between the values calculated by eq (5) and those calculated by eq (6) are well within the limits of accuracy that can be expected. 4. Summary Thermal conductivities were experimentally determined on several different grades of raw shale and on their respective retorted shales and burned shales. Determinations were made with the heat source both perpendicular and parallel to the bedding planes, and at several different temperatures. Conduc- tivity values ranged from about 0.69 W m~l deg~l for 58. 6-gallon-per-ton shale to about 1.6 W m~l deg'l for a lean, 8.6 gallon-per-ton shale. Conductivities of the retorted or spent shales ranged from 0.26 to 1.4 W m"l deg"l, and of the burned shales from less than 0.17 to about 1.2 W m~l deg"l. An eq was developed for calculating the thermal conductivity of raw, retorted, and burned Green River oil shales. Thermal dif fusivities were determined experimentally on twelve different grades of raw shale assaying from 8.1 to 59.1 gallons of oil per ton and on the retorted and the burned shales from these 533 raw shales. Thermal dif fusivltles of raw shales ranged from 2.6 x 10"' to 9.8 x 10"' m 2 /sec. Diffus- _ ivlties of the retorted shales ranged from 1.3 x 10 to 1.0 x 10-7 t o 7.2 x 10-7 m 2 /sec 8.8 x 10"' m^/sec and of the burned shales from 5. Acknowledgment The research on which this report is based was done under a cooperative agreement between the Bureau of Mines, U.S. Department of the Interior, and the University of Wyoming. 6. References (1) Marovelli, R. L. and Veith, K. F. Thermal (3) conductivity of rock: measurement by the transient line source method, BuMines Rept. of Inv. 6604 (1965). (2) Lentz, C. P. A transient heat flow method of determining thermal conductivity: application (4) to insulating materials, Can. J. of Technol., 30, No. 6, 153-166 (1952). Underwood, W. M. and McTaggart, R. B. The thermal conductivity of several plastics, measured by an unsteady state method, Chem. Eng. Prog. Symposium Series, Heat Transfer, 56, No. 30, 261-268 (1960). Tisot, P. R. Alterations in structure and physical properties of Green River oil shales by thermal treatment, J. of Chem. and Eng. Data, 22, No. 3, 405-411 (July 1967). FIGURE 1. - Burned shale test pieces used for thermal conductivity determinations. 534 CO ro in CO in fO CO ro o in to oo CM CD to CD in to to ro r~- CD o ( _6ap j_uj m =x r~- co in or r— o — _J Q LU LU LU = z —I Z LU^ -J •ri o Pi CJ> o LU" c 01 > O - in ° ••-I ai a> c a. a) C1J a u O °" o **" LU T3 _J c - cu ori c\j to n CO 3 CO C\J in r^ ir> ro cr> ro CVJ CD in CNJ Grad [b] , Koga [5], and others re-examined the mathematical implica- tions of the Boltzmann and Fokker-Planck equations and concluded that the Boltzmann equation was valid for ionized gases after all. Grad stated that ". . .The critical point here is that, although the two physical pictures are entirely dif- ferent, their mathematical descriptions are identical! The net effect of many successive inde- pendent small impulses is the same as many simultaneous independent small impulses, provided only that the means and variances of the two impulse distributions are the same (actually, the entire probability distributions were taken to be the same). Thus we conclude, without setting pencil to paper, that the Fokker-Planck equation, which is an immediate consequence of the simultaneous grazing impulse model, must yield results identical with those obtained from the Boltzmann equation, provided that an appropriate grazing collisions approximation is made and the same cut-off is used in the latter computation." The equivalence between the Boltzmann and Fokker-Planck equation for the case of a fully ionized gas was farther demonstrated by the excellent agreement of the two sets of values of transport coefficients based on these equations [2] . A more rigorous theory for fully ionized gases can be derived from Bogolyubov-Born-Green-Kirkwood- Yvon hierarchy of equations [6] which account for simultaneous but dependent interactions (i.e. many particle correlations). An analysis by Sundaresan and Wu [7] has shown that the thermal conductivity from the B-B-G-K-Y approach is almost identical to that for the Fokker-Planck approach. The accuracy of the Boltzmann equation was demonstrated in two recent experiments by Emmons [8] and Morris [91 • The goal of .their experiments was the measurement of the electrical and thermal conductivity of partially ionized gases. Although the thermal conductivity could not be measured accurately at large degrees of ionization because of the masking effects of thermal radiation, the electrical conductivity was measured. In figures 1 and 2 the experimental values of the electrical conductivity are compared with values based on the Boltzmann equation (second -order Chapman-Enskog formulation). It can be seen that the agreement is fairly good over the entire range of temperatures (degree of ionization ranging from 1 percent at 9000°K to 100 percent at 22,000 K) . Since the predominant cross section for the elec- trical conductivity is the well -verified Coulombic cross section, the agreement shown in figures 1 and 2 is a good indication of the validity of the Chapman-Enskog formulation of the Boltzmann equation for all degrees of ionization. Unfortunately, this is no guarantee that the values of the thermal conductivity can be accurately predicted at these high degrees of ionization since the predominant cross sections (e.g. atom-ion elastic and charge -exchange cross sections) are not known accurately, and since there are different types of diffusional effects present in thermal conduction. This means that measurements of thermal conductivity are still necessary. 2. Derivation of Total Thermal Conductivity The analytical and experimental work described in the previous section indicate that the Boltzmann equation can be used as a starting point to calculate the total thermal conductivity. The approach described in this paper is the same as the Chapman-Enskog formulation applied to the case of neutral monatomic gas mixtures by Hirschfelder, Curtiss, and Bird [10] and will be examined to determine its suitability for a partially ionized gas. Besides the difference in the range of intermolecular forces a partially ionized gas differs from a neutral gas in two other respects. First, a charge separation field arises because of local differences in the ion and electron concentrations, and second, the ratio of the mass of the heaviest particle to that of the lightest particle increases by at least three orders of mag- nitude due to the presence of free electrons. A great amount of foresight was used in the derivation of the complete Chapman-Enskog formulation, for there exist terms which account for these two differences. In principle the charge separation field can be calculated since a macroscopic force term is included in the Chapman-Enskog formulation. The much larger mass ratio in partially ionized gases is not really a fundamental difference, but it does mean a re-evaluation of certain computational simplifications which were carried over from the calculation of the thermal conductivity of neutral gas mixtures. For example, it has been customary to discard certain Sonine expansion terms in the translational thermal conductivity and the multicomponent diffusion coefficients, and to omit effects of thermal diffusion on the total thermal conductivity [10]. An examination of thermal diffusive effects does show that the effects are small for neutral gas mixtures, but it also shows that the effects become increasingly important as the The Boltzmann approach is presently the only one available for the calculation of transport coef- ficient of partially ionized gases, although many competing approaches are available for fully ionized gases. 3 Figures in brackets indicate the literature references at the end of this paper. 552 ratio ^eavy/miigirt increases. Consequently, the author [ll and 12] showed that it was necessary to include the discarded Sonine expansion terms and to include thermal diffusive effects. The results of these calculations will he described in the next section. The calculation of the total thermal conductivity is hased on the expression for the heat flux vector q. - ) n.h.V. - A + |£ - nkT > -i- D T d . , (l) i i where Di is the thermal diffusion coefficient, and hi is the total enthalpy of a particle of species i. The quantity At is the thermal conductivity of a spatially homogeneous gas mixture without concentration gradients. The diffusion velocity, V i5 is defined as V- --^ Vm^d, --^-D T ^ (2) — i n ± p /_, J iJ— J n-jm-jT i £ r J where Dji is the multicomponent diffusion coefficient and the "forcing potential" for a partially ionized gas in the absence of pressure gradients is defined as d- = r-i - -^i(-£- e.E. - > n e E 3) - 1 Sr PP \ m i 1_1 L A j-jj J where €j_ is the charge, Z±e, for a particle of species i, and Eg is the electric field generated by a difference in ion and electron concentrations. Equation (3) differs from the expression for dj. in [10] in that ZieEg describes an internal macroscopic force whereas the corresponding quantity in [10] describes an external macroscopic force. However, the conceptual difference is only apparent as Eg always stems from an external energy source (e.g. heating coils). Assume that this energy is imposed on the system such that the average temperature is high enough to ionize the gas, but the temperature gra- dient is small enough to justify linearization procedures. The temperature gradient induces concentration gradients, o^xi/or, and a charge separation field, Eg. If it is assumed that both dxi/dr and Eg are proportional to the temperature gradient, it can be seen from eqs (2) and (3) that Vi and cj^ are also proportional to the temperature gradient. The heat flux vector can then be expressed as q , -(A r + A t + A d ) | = -A total | (10 The reactive component of the thermal conductivity, A r , derives its name because of the addition of the reaction energy to the enthalpy of the individual species [13] • This mode of heat transfer can be described in terms of a diffusion cycle. In the higher temperature region the concentration of ions and electrons are larger than in the lower temperature region, forcing tne charged particles to diffuse toward the lower temperature region. In this region, the ion recombines with an electron, thereby releasing the ionization energy (i.e. transport of energy). The cycle is completed when the atom is forced by the atomic concentration gradient to diffuse toward the higher temperature region where the ionization process occurs. The translation component, A^, is the conductivity of a spatially homogeneous gas mixture without concentration gradients, and is the only component explained by simple kinetic theory [10], For obvious reasons, the third component of the thermal conductivity, A d , is called the thermal diffusive component. Unfortunately, no simple physical picture can describe this mode of heat transport. The calculation of A-fc is straightforward and is described by Hirschfelder, Curtiss, and Bird [10]. Before numerical values for A r and A d can be calculated it is necessary to obtain values for the con- centration gradient of each species and the charge separation field, all in terms of the temperature gradient. The simplest case which can be examined is that for a gas undergoing the reaction A ■+ I + e (i.e. a ternary mixture of atoms, A, ions, I, and electrons, e). The solution of the problem can be expressed in terms of four dependent variables - the concentration gradients for the atom, ion, and electron, and the charge separation field, all expressed in terms of the temperature gradient. However, there are only three independent equations relating these four quantities. Two of these equations are statements of the flux conservation of elemental particles as formulated by Butler and Brokaw [13] • For the ternary mixture the elemental particles are defined as a singly charged ion and an electron. Then flux conservation requires that the diffusion velocities be related as follows 4 : 4 More explicit forms of (5) and (6), where V.± is expressed in terms of d^, was used in the calculation of [lk], 553 x A V A + ^ = 0, (5) X A^A + X ele = 0. (6) The third equation can be derived from the expression for the equilibrium constant, h n (x iP ) £ i=i (7) where the aj_'s are the stoichiometric coefficients for the reaction (a A = -1, aj =1, a e = l). Combina- tion of the gradient of eq (7) with explicit expressions for the charge separation force results in dT ' S r X A X I Xe (8) The author [12] discussed the difficulties in solving eqs (5), (6), and (8) for obc^/dr and IL, but could not offer a solution. Subsequently, Meador and Staton [15] circumvented the problem of too many vari- ables by assuming that the concentration gradient of the ion was equal to that for the electron. They justified this assumption by combining concepts from electrostatics and plasma physics with expressions from non-equilibrium thermodynamics, then using order of approximation arguments. The accuracy of this approach will be discussed later. Meador and Staton used the equality of obcy/dr and obcg/dr in eqs (5), (6), and (8) to solve for obc A /dr and Eg. Their expression for the reactive component of the thermal conductivity is nn e n A kT D AI (n e + n A ) 2 3 In 5,. &T (9) which is identical to the Butler and Brokaw expression for dissociating gases [13] if it is assumed that the multicomponent diffusion coefficient D A j can be approximated by the binary diffusion coefficient %j« This equation has several implications: (l) The reactive component is completely dominated by the binary diffusion between the ion and atom, (2) The reactive component is independent of the motion of the electron, and (3) The reactive component is not affected by thermal diffusive effects. In a subsequent paper [lb] the author also derived an expression for ths total thermal conductivity. In contrast to the Meador and Staton work, it was not necessary in [lk] to rsly on any subsidiary assump- tions, and the derivation was exact within the framework of the Chapman-EnsXOg formulation. The crux of this approach is not to separate the forcing potential, dj_, of eq (3) into dx±/br and Eg components, but to solve for d_i directly by making use of the conservation of the net flux for each elemental par- ticle. Physically, this means that regardless of the value of the charge separation field, the concen- tration gradients will readjust themselves so that the net flux is conserved (eqs (5) and (6)). Therefore, the combined effects of Sx^/Sr and Eg in the form of dj_ is a more logical dependent variable than its components. The solution for the various dj^'s is then obtained from eqs (5), (6), and (8). The determinantal expression for the various d^'s is , _St dl "ar an ai2 a 13 -Cl a 2 i a 22 a 2 3 -c 2 a3i a32 a33 -C3 &ii °i2 s i3 an a 2 i a3i ai 2 a 22 a32 ai3 a 2 3 a33 = 8i ar" (10) 554 where an = (n 2 /?)^^, a 12 (tP/p)™^^, ft "e D Ae a 23 = (n 2 /p)m e D Ae , c 2 = (D^/m A T) + (Dg/m e T), L = (n 2 /p)m e D ao + (n 2 /p)m e D Ie , ci = (D A /m A T) + (D^/iiLj-T), a 2 i = (nVpJm^D^, a3i a32 = x^ , a33 = Xe" > a 22 = (n 2 /p)m I D AI + (n^p^D^, c 3 = d in Kp/i ai. (11) A coniblnation of eqs (l), (2), (10), and (ll) gives the final expression for the total thermal conductivity: A, +. , = K + total t 2 2 2 _h I 7~ (m A D IA S A + m e D Ie 8 e) " h A 7~ (m I D AI 5 I + m eD Ae 8e) "he T (m A D eA 8 A + ^el 5 ^ T T T h A D A h I D I h e D e + ST" + — =~ + m A T mJT , D I 5 A D I 5 I D e 5 e + nkT + + n^ n Q m, e e (12) where A+ is the translational thermal conductivity, the sum of all terms containing the various h^'s is the reactive thermal conductivity, and the sum of the remaining terms is the thermal conductivity due to thermal diffusion. Note that the reactive component has two modes. The first is by multi component diffusion and the second by thermal diffusion. For a fully ionized gas (i.e. no neutral particles) eg (12) reduces to A total A t " nD el m\2 D \ V.) (13) where u is the reduced mass of the electron-ion system. Since the terms in eq (12) are so complex it would be impossible to single out by inspection any one mechanism (e.g. thermal diffusion) as being the chief contributor to A r or Ad- This can be done only by an inspection of the numerical values which are described in the next section. However, it can be seen from eq (12) that there is a definite need for accurate values of the multicomponent and thermal diffusion coefficients since A r and Ad are determined by Dij and D?, and since Si (the combined rn effect of the concentration gradients and the charge separation field) are also determined by D^ \ and Dj (eqs (10) and (ll)). 3. Numerical Values of Higher Order Transport Coefficients T The importance of higher order Sonine expansion terms in the expressions for A-t> Dij> and DJ will be described first. The expressions for At, Dij, and D? [10] are the ratio of two determinants. If the first and second Sonine expansion terms are used (second approximation) then both determinants contain four subdeterminants which are designated as the 00, 01, 10, and 11 subdeterminants, where the cor- responds to the first Sonine polynomial and the 1 corresponds to the second Sonine polynomial. In the calculation of the thermal conductivity of neutral gases the 00, 01, and 10 subdeterminants are normally discarded (first approximation) with very little loss in accuracy [10] . Unfortunately, this simplifica- tion was carried over into the calculation of the thermal conductivity of partially ionized gases. The result of this approximation is shown in figure 3 for partially ionized argon. The second approximation of At [12] is larger than the first approximation by 30 percent at 50-percent ionization and larger by 50 percent at complete ionization. An inspection of the numerical values of the elements in the At sub- determinants [12] show that the increase in At in going from the first to the second approximation can be attributed to the inclusion of additional terms in the 00 subdeterminant arising from interactions between unlike particles (A-I, e-A, and e-l) . Subsequent analyses by DeVoto [16 and 17] showed that third order Sonine expansion terms for the translational thermal conductivity were also appreciable. Figure 3 shows that the difference between the third and second approximations for At is roughly the same as the difference between the second and first approximations. DeVoto' s calculations, however, show very little difference between the third and fourth approximations. Landshoff's [18] calculations for an electron gas show roughly the same trends in going from the second to the fourth approximation. 5 The author [12] erroneously identified At with Spitzer's [2] field free thermal conductivity. The error was resolved by the analysis of Meador and Staton. 555 Mult i component diffusion coefficients are usually approximated by retaining only the first Sonine expansion terms (i.e. using only the 00 subdeterminant) . These first-order coefficients are compared with the second-order coefficients (i.e. both first and second Sonine expansion terms) in figure k. The second-order coefficients D e _A(2) and D e _j(2) are larger than the corresponding first-order coefficients by 25 percent at 50-percent ionization and by ^5 percent at complete ionization. An inspection of the numerical values of the elements in the Dij subdeterminants show that the increase in Dij in going from the first to the second approximation can be attributed to the inclusion of terms in the 11 sub- determinant arising from interactions between like particles (A-A, I -I, and e-e). The electron-electron interaction is especially important since a simple hard sphere model shows that Dij is inversely pro- portional to the product of the collision cross section and the reduced mass. In contrast, the first- and second -order coefficients for diffusion between heavy particles (e.g. Da-i) differ by only a few percent. DeVoto's calculations show that there is little difference between second- and third-order mult i component and thermal diffusion coefficients. It is essential that second-order thermal diffusion coefficients be used since these coefficients are identically zero in the first approximation. Typical values of the second-order thermal diffusion coefficients for the argon atom, ion, and electron from [ll] are shown in figure 5« In the expression for the total thermal conductivity (eg (12)), D? always occur in combination with the particle mass in the denominator. Although the electron thermal diffusion coefficient, E^, is at most two orders of magnitude smaller than those for the atom and ion, the electron mass is four orders smaller. Conse- quently, thermal diffusive effects can be attributed almost entirely to the electron term. Another interesting conclusion can be reached by an inspection of figure 5« It can be seen that the atom and ion thermal diffusion coefficients are within 1 percent of each other in magnitude from a few percent ioniza- tion up to extremely high degrees of ionization (approximately 95_percent) . Beyond this point iff decreases in value, changes sign, then approaches the value of Dg near 100-percent ionization. These variations imply that the diffusive motion of both the atom and ion are essentially independent of that for the electron, up to large degrees of ionization. The diffusive motion of the electrons, in turn, is dictated by the ion rather than the atom because of the greater magnitude of the Coulombic forces 6 . These same conclusions were reached from an examination of the second-order multicomponent diffusion coefficients. T The previous discussion of second-order values of Dij and I>± leads towards the calculation of A r and Ad from eqs (10), (ll), and (12). The author recently made a series of calculations for hydrogen, nitrogen, and argon for a pressure of one atmosphere and temperatures corresponding to 50-percent ioniza- tion. The values of A r were compared with those based on the expressions derived by Meador and Staton [15] • The values of A r from the two sets of calculations agreed within a few percent. However, this is not necessarily a verification of the Meador and Staton approach, as calculations have shown that any non-trivial assumed values of dxj/dr and 3x e /dr will result in the same values of A r and A^. The agreement can be explained as follows. The values of A r and'A^ are determined not by the separate effects of the concentration gradients and the charge separation field, but by their combined effects. Consequently, if erroneous values of the concentration gradients we're initially assumed the constraints of the problem (eqs (5), (6), and (8)) would compensate for this error in the resulting value of the charge separation field. A more critical comparison could come from the determination of the predominant mechanism for the reactive component of the thermal conductivity. The reactive component from eq (12) can be resolved into two components: (l) the sum of the hin 2 mjDijSj/p terms which shall be called the Dij component for , . rp , identification, and (2; the sum of the hjTi^/m^l} terms which shall be called the Di component. The results are summarized in the following table. Table I. Thermal Conductivity Components for Partially Ionized Gases (p = 1 atm, 50$ ionization) Gas A t (2) W m" 1 deg" 1 A r (2) W m' 1 deg -1 A d (2) W m" 1 deg" 1 Di 4 component T D . component Hydrogen Nitrogen Argon 1.30 1.31* 1.31 5-77 O.87 -0.17 1.78 1.30 1.32 -0.20 -0.25 -0.29 As a result, the values of D e calculated for partially ionized argon can be used for other partially ionized gases to a good degree of accuracy. 556 For hydrogen the largest contribution comes from the B±j component, and the predominant term in this component is the h-rn idaDiaSa/p term as predicted by Meador and Staton. However, the thermal diffusive contribution is not negligible, "but comprises 25 percent of the value of A r . As the mass of the heavy- particle increases, thermal diffusive effects "become relatively more important. For example, in nitrogen the D-E. component of A r is larger than the Dj_j component. In argon the D±j component is almost zero due to the cancellation of the ion term "by the atom and electron term. Consequently, the reactive thermal conductivity for argon can be attributed almost entirely to the e£ component, and the predomi- nant term in this component is the hgDe/meT term. This result is in direct contradiction to those predicted by Meador and Staton and indicates that one or more of their initial assumptions may be erroneous . This paper has described the analytical work done in the past few years to clear up some of the uncertainties in the expression for the total thermal conductivity of a partially ionized gas. The author feels that the theoretical basis for this expression is currently on substantial ground. However, there are several deficiencies which must be resolved before accurate values can be calculated. The first of these is the lack of measured or calculated cross sections for collisions between an atom which is chemically unstable at low temperatures (e.g. nitrogen or oxygen) and an electron, ion, or another chemically unstable atom. A second deficiency is the absence of terms in the expressions for the trans- port coefficients which account for inelastic collisions, and the lack of measured or calculated cross sections for these inelastic collisions. A third deficiency is the absence of any experimental verifica- tion of the total thermal conductivity, especially in the region of 50-percent ionization where A-fcbtal peaks because of the reactive component. A method which shows some promise is the ultrasonic absorption technique as used by Carnevale et al. [19] • Consequently, large amounts of analytical and experimental effort must still be expended to remedy these deficiencies before accurate values of the total thermal conductivity of a partially ionized gas can be obtained. 4. References [l] Cohen, Robert S., Spitzer, Lyman, Jr., and Routly, Paul Mc R. : The Electrical Conductiv- ity of an Ionized Gas. Phys. Rev., vol. 80, no. 2, Oct. 15, 1950, pp. 230-238. [2] Spitzer, Lyman, Jr., and Harm, Richard: Trans- port Phenomena in a Completely Ionized Gas. Phys. Rev., vol. 89, no. 5, March 1953, pp. 977-981. [3] Burgers, Johannes, ed.: Statistical Plasma Mechanics. Symposium of Plasma Dynamics, Addis on -Wesley Publishing Co., Reading, Mass., i960, pp. 16^-169. [ 4] Grad, H. : Modern Kinetic Theory of Plasmas . Proc. Fifth Int. Conf. on Ionization Phenomena in Gases, Vol. II, North-Holland Publ. Co., Amsterdam, 1962, pp. 163O-1649. [5] Koga, Toyoki: The Generalized Validity of the Boltzmann Equation for Ionized Gases. Proc. Third Int. Symposium on Rarified Gas Dynamics, Vol. I, Academic Press, N. Y., 1963, pp. 75-93- [6] Wu, Ta-You: Kinetic Equations of Gases and Plasmas. Add is on -Wesley Publishing Co., Reading, Mass., 1966. [7] Sundaresan, M. K., and Wu, Ta-You: Thermal Conductivity of a Fully Ionized Gas. Can. J. Phys., vol. 42, no. 4, April 1964, pp. 79^-813 • [8] Emmons, Howard W.: Arc Measurements of High- Temperature Gas Transport Properties. Phys. Fluids, vol. 10, no. 6, June 1967, pp. 1125- 1136. [9] Morris, James: Private Communication [10] Hirschfelder, Joseph 0., Curtiss, Charles F., and Bird, R. Byron: Molecular Theory of Gases and Liquids. John Wiley and Sons, Inc., N. Y., 1954. 557 [ll] Ahtye, Warren F. : A Critical Evaluation of Methods of Calculating Transport Coefficients of a Partially Ionized Gas. Proceedings of the 1964 Heat Transfer and Fluid Mechanics Institute, Stanford Univ. Press, 1964. [12] Ahtye, Warren F. : A Critical Evaluation of Methods for Calculating Transport Coefficients of Partially and Fully Ionized Gases. NASA TN D-2611, 1965. [13] Butler, James N., and Brokaw, Richard S.: Thermal Conductivity of Gas Mixtures in Chemical Equilibrium. J. Chem. Phys., vol. 26, no. 6, June 1957, pp. 1636-1643. [l4] Ahtye, Warren F. : Total Thermal Conductivity of Partially and Fully Ionized Gases. Phys. Fluids, vol. 8, no. 10, Oct. 1965, pp. 1918- . 1919 (Erratum, vol. 9, no. 1, Jan. 1966, pp. 224). -[15] Meador, W. E., Jr., and Staton, L. D. : Electrical and Thermal Properties of Plasmas. Phys. Fluids, vol. 8, no. 9, Sept. 1965, pp. 1694-1703. [l6] DeVoto, R. S.: Transport Properties of Par- tially Ionized Monatomic Gases. Dept. of Aeronautics and Astronautics, Stanford Uni- versity Rep. SUDAER No. 207, 1964. [17] DeVoto, R. S.: Argon Plasma Transport Prop- erties. Dept. of Aeronautics and Astronau- tics, Stanford University Rep. SUDAER No. 217, 1965. [18] Landshoff, Rolf: Convergence of the Chapman - Enskog Method for a Completely Ionized Gas. Phys. Rev., vol. 82, no. 3, May 1, 1951, p. 442. [19] Carnevale, E. H., Lynn-worth, L. C., and. Larson, G. S.: Ultrasonic Determination of Transport Properties of Monatomic Gases at High Temperatures. J. Chem. Phys., vol. H-6, no. 8, April 1967, pp. 3C40-3C47. T 10"- .01 .05 .1 .3 FRACTION IONIZED I .5 .7 I .9 l- o Q Z o o < o I- o LjJ _l UJ NITROGEN p= I atm THEORY EXPERIMENT (ref. 9) lO 3 ^ 9 I I 10 II 12 13 14 15 TEMPERATURE, T, °K 16 I 17 18x10^ Figure 1.- Comparison of experimental and theoretical electrical conductivity of nitrogen. 558 r E " 1 1 1 1 1 1 1 .01 .05 .1 .3 .5 .7 .9 FRACTION 1 MLUnT u EXPERIMENT (ref. 8) cj EXPERIMENT (ref. 9) >-" \- £ io 4 - h- - ^ «*■■ O .^ *^ - ^-— Z> - ^-<^-^"^ Q Z o — -. ^ o _l < ,4^ ARGON E H O a/ p= 1 atm / Ul _l I0 3L i i i i i i i i i i 8 10 12 14 16 18 I I .95 .975 i l l I 20 22xlO ,: TEMPERATURE, T,°K Figure 2.- Comparison of experimental and theoretical electrical conductivity of argon. i- 3.0- I- o 3 2 5- z o o III III I II .01 .05 .1 .3 .5 .7 .9 .95.975 FRACTION IONIZED / FIRST APPROX. SECOND APPROX. THIRD APPROX. / / / / / 10 15 20 TEMPERATURE, T, °K 25xl0 3 Figure 3>- Comparison of various approximations for calculating the translational thermal conductivity of argon . 559 o (/> CM E u to o y u. Q icr iii i i i i i r .01 .05.1 .5 .9 .975 .99 FRACTION IONIZED 10" j I _|_ I ^SECOND ORDER FIRST ORDER 10 14 18 22 TEMPERATURE, T,°K 26 30x10- Figure k.- Comparison of various approximations for calculating multicomponent diffusion coefficients of argon. 10-5,^ III ill .01.05.1 .3.5.7 I i I 9 .975 .997 .998 .999 FRACTION IONIZED ARGON 14 18 24 TEMPERATURE, T, °K Figure 5«- Thermal diffusion coefficients of argon. 560 30x10^ Thermal Conductivity of the Alkali Metal Vapors and Argon Chai-sung Lee and Charles P. Bonilla Liquid Metals Research Laboratory, Department of Chemical Engineering Columbia University, New York, N.Y. 10027 The frequency response to alternating' current of a fine wire surrounded by alkali metal vapor was studied theoretically and ex- perimentally for determining the thermal conductivity of the vapor. Data were obtained for cesium and rubidium vapors at pressures from 0.0829 to 0.214 atm. A new method was employed of correcting for temperature jump and vapor volumetric heat capacity. In addition, the thermal conductivity of the alkali metal vapors was calculated by gas kinetics theory, including the effects of dimerization and heat of association. Good agreement between experiment and theory was obtained considering the possible errors. The total probable experimental error is estimated at 6.1%, and the uncertainty In the theoretical analysis at about 25$. Reliability of the method is indicated by agreement ;vith literature values for argon up to IO85 C. The standard error of the argon correlation is estimated at 1.6$. Key Words: Conductivity, heat conductivity, thermal con- ductivity, argon, lithium vapor, sodium vapor, potassium vapor, rubidium vapor, cesium vapor, hot wire cell. 1. Introduction Measurement of the thermal conductivity of alkali metal vapors is difficult because of the high temperatures required, involving radiation between the surfaces, and because of the chemical activity of the alkali metals on containers, gases, and other materials which they may contact. Thus, only preliminary or approximate results are to be found in the literature. Gottlieb and Zollweg (l) measured heat dissipation from tungsten ribbon at a single temperature in cesium, rubidium and potassium vapors at low pressures. On extrapola- ting to infinite pressure for each vapor they obtained an approximate value of the monomer conductivity. Stefanov et al (2) employed a concentric tube geometry with sodium and potassium vapors, obtaining the temperature difference by differential thermal expansion of the tubes, as originally proposed by Timrot et al (3). Radiation has a large effect in this geometry, and reached 85$ of the heat transmitted. Achener (4) is employing laminar flow in a long hot isothermal tube and measuring the average temperature rise. Theoretically the Nusselt number approaches 3.658. The above results all seem somewhat low, or else inconsistent. In view of considerable interest in the thermal conductivity of the alkali metal vapors, particularly for kinetic theory studies, correlating heat transfer data, and calculating results of hypothetical fast reactor transients, a previously developed transient hot-wire method for measuring the thermal conductivity of gases at high temperatures was adapted to these vapors. 2. Mathematical Model for the Transient Hot-Wire Method As in previous studies in this laboratory (5*6,7) a concentric wire in a cylindrical shell is employed, the annulus between being filled with the test gas or TT7 Figures in parentheses indicate the literature references at the end of this paper. 561 vapor. The radius of the wire is r, and of the shell r~, and the wire must have a substantial temperature coefficient of resistance. Pure sinusoidal current of a single frequency i =\|2~ I sine is passed through the wire and the IR drop across it is measured. The equation governing conduction in the annulus is: c f\'c k ($* +± st \ -21. forr,l-kj^ -hZ7rr,L£(t -t z ) at r=r, (3) The wire is so fine that its average temperature can be taken as t, , and the tempera- ture variation is small so that the radiant heat transfer coefficient h and the electrical resistance R are assumed constant for this purpose. In the previous studies on permanent gases (7) it was possible to adjust the pressure to high values, maintaining the mean free path and the temperature jump small as well as substantially constant. Accordingly, the thermal conductivity in eq (3) was that of the gas, as In eq (l) . In this work with alkali metals the pressure must be below saturation, and thus is relatively low, and temperature jump is appreciable at the inner wall. An exact statement of this boundary condition can be written that includes the temperature jump as an equivalent series heat transfer coefficient, however it is considerably more complicated and cannot be solved analytically. Therefore a means of correcting the boundary condition for temperature jump in a simpler way was sought. This was accomplished by substituting k for k in the heat balance, where k is sufficiently reduced that the correct heat conduction term is calculated when t, is the wire temperature rather than the vapor temperature. The expression for k is obtained from temperature jump theory (0) as: The solution of these equations can thus be given in the same form as previously (6,7), but indicating with the superscript the dimensionless ratios in which k appears. The S/olution is: Capital initials have been used for readier recognition of the dimensionless ratios. These ratios, and others shortly to be used, are defined as follows: 2 See section 8. for nomenclature. 562 AR° Amplitude Ratio (CR°/^)Vsd° + b D° (where g D and b^ are, like AR° itself, complex functions of CR, CDR, DR, RR (or QR) * and TJP) CDR Capacity Density Ratio o. x p x /cp ~ CR Conductivity Ratio 2wc'P'r 1 /k DR Dimension Ratio r 2 //r l RR° Radiation Ratio hr^/k QR Fraction Radiative Heat q rad/' q tot TJF Temp. Jump Factor a C^TAf^W I TJC Temp. Jump Correction Term (AR° - AR)/AR RR Radiation Ratio for TJF=0 QR/((l-QR) In (DR)) Derivation of the equations for AR and of $ f° r the case of zero temperature jump (AR and ) has already been carried out analytically (5), and values have been calculated for various ranges of the other dimensionless ratios (6,7). In this study it was necessary to carry the calculations to still higher values of CDR and lower values of CR, due to the lower values of thermal conductivity and density encountered with the alkali metals. Figure 1 shows the results for the case of zero radiation correction (RR = 0) and for the dimension ratio DR = 1000, a satisfactory value for all of the work here described. The turn-up of the curves for high CDR below CR = 0.1 had not been expected. Figure 2 gives the temperature jump correction, TJC, with which AR can be cal- culated from AR, and vice versa. In the actual interpretation of experimental runs the computer program (9) calculates the value of TJC for each specific run. The temperature difference t, - t2 in eq (5) Is obtained from an experiment by observing the departure of the IR drop across the fine wire from the pure sine wave that it would be if R were constant. The IR drop across the fine wire can be shown (9) to be: E = J5lsineoe[R* + ^(t t -t;)] (6) = yjz E, sin^OO -\l2E 3 slnCcoG -A- 4> ) + feE 3 sin (3*06 + ] (9) where the wire resistance Re = (R,/R )R,. The decade capacitors are used to balance out the equivalent reactive impedance or the wire and yield a sharper null point. They would also be read if the phase-shift method were being employed. Operation of the system will be described under Experimental Procedure. 563 An "absolute" determination of thermal conductivity by the above procedures is impractical for many reasons, and in particular because of the difficulty in obtain- ing an accurate value of r, , the wire radius. If a quantity designated by GP is defined by: GP = CR x k x 10- (10) and experimental curves of AR vs. GP are drawn for two different gases, the ratio of the two GP's corresponding to any single value of AR is the same as the ratio of the thermal conductivities of the two gases. If one gas is a reference gas of known thermal conductivity the thermal conductivity of the test gas can be readily obtained, This is the method employed in this study. Two types of ce Cell No. 1 (fig. 4) machined from SS-310 consisted of frozen 1/16 inch OD x 1/64 meable, was found to operation, the cell the cell heater both evacuation stopped. 3. Cell Design and Operation 11 were tested, employing different seals for the insulated lead. consists of a boiler, measurement cell, and seal complex, all and connection lines. The seal in contact with the vapor space alkali metal inside and outside of a fused alumina (Lucalox tube) inch ID. Commercial sintered alumina, though designated imper- short-circuit by permeation when used below the freeze-seal. In was thoroughly evacuated cold and then the freeze-seal water and turned on. In due course the freeze-seal was established and the This cell yielded the rubidium data. Cell No. 2 (fig. 5) employed a dry seal consisting of a washer and a disk of Nb-1$ Zr sheet brazed to the opposite ends of a Lucalox sleeve, as developed for sodium vapor lamps. This seal was tight, but was more fragile and was limited to some 800 C. The cesium data were obtained with this cell. To resist corrosion by the alkali metals tungsten wires were employed, rather than platinum, the other metal so far studied. The finest wires available (0.00012 inch in diameter) were used, since they permit higher frequencies to be utilized. With the wire diameter fixed, the temperature jump and radiation corrections can be substantially reduced by employing higher frequencies, and thereby working In a higher range of CR as shown in figure 2. This, however, is in contradiction with the fact that AR vs. CR curves are linear only in the low range of CR (fig. l). The necessary frequency to obtain a convenient compromised value of CR is given in Table 1, from preliminary estimates. Table 1. Frequency for a Conductivity Ratio of 0.1 Frequency of Current at Diameter of Tungsten Wire, In. (0.0003) (0.00012) Gas or k/k Cs at 750° C Vapor He 64.2 N A Z 11.0 4.68 Cs 1.00 Rb 1.46 K 2.48 Na 4.88 Li 13.7 57 10 7.4 1 1.5 2.5 4.6 13.4 348 61 4 1 6.1 9.2 15.3 28.1 81.8 Although the mathematical model assumes concentricity, this is not important because of the relative method employed and the high ratio DR. Furthermore, no amount of tension could be applied to these fine wires. Accordingly, joints to the nickel leads were made by gently squeezing the lead around the loose tungsten wire with ad- justable vice-pliers. In view of the fine size of the tungsten wire, and its oxidizability, it was necessary to eliminate all possible oxygen from the system. This was done by adding several grams of Ti-Zr chips ("Getterloy" ) to the cell before charging it with alkali metal, and initially heating the cell slowly under argon pressure to prevent the 564 vapor from rising. When this is done carefully with fresh metal, each new 0.00012 inch wire will last for a number of runs up to 1000 C or higher. The cells were mounted in two independent vertical furnaces. The lower one set the liquid alkali metal temperature, and therefore the pressure of the vapor, and the upper one set the vapor temperature or superheat. 4. Experimental Procedure In general, the sequence of tests in a complete run included: ia) Insertion of the wire and sealing of the cell. bj Charging the cell with alkali metal. c) Measuring the electrical resistance of the wire at room temperature with a low current in the Wien bridge at a selected frequency in argon to prevent in-leak- age and calculating the length of the wire from the known resistance per unit length. (d) Measuring the resistance of the wire up to high cell temperatures in argon to determine dR/dt as a function of temperature. (e) Cool the cell to room temperature and adjust the argon pressure to atmos- pheric, then carrying out a calibration run with argon over a range of about 5 to 100 cycles per second (CR about 0.05 to 1.0 and CDR about 3,000). (f) Recheck the room temperature resistance of the wire (if drift occurs it may not invalidate previous data) . (g) Set the hot-wire current up to the operating range and read off the resis- tance again from the bridge. (hjl Read the effective voltage across the hot wire with the thermocouple voltmeter. (l) Set the band-pass filter at 3 times the impressed frequency. (j) Pass the bridge output through the decade amplifier and the filter, and measure it, EL R , on the VTVM. (k) Turn the switch from test (l) to calibration (2) position and set the oscillator to 3 times fundamental frequency. (l) Apply about 1 volt, measured by the thermocouple voltmeter, to the calibra- tion circuit. (id) Adjust the voltage divider until the same output E_ R is read on the VTVM. This yields the absolute value of E 3R . JK (n) Prom EL R we calculate EL By eq (9) and then AR by eq (8) . (o) Employing the computer program for TJC, this quantity is computed by itera- tion, as well as thermal conductivity k, and radiation correction {with a=0.9(l3)) , (p) Corrections for a small drift of CDR from 3,000, a value chosen as the reference, and end conduction are also calculated using the formulas developed in the previous work (7). (q) The corrected amplitude ratio, AR, is now obtained by applying the above correction terms to AR . This yields one point of the corrected AR vs. the corres- ponding GP, as obtained with argon but valid for any gas or vapor. (r) Repeat the above steps from (f ) on at a number of other frequencies to ob- tain a calibration curve for AR vs. GP. The alkali metSl part of a run takes the following steps using the same wire: Es) Evacuate the apparatus at least to 3 x 10~^mm Hg. t) Gradually heat the boiler and cell, and attain steady temperatures while keeping the cell temperature at least about 100°C higher than the boiler temperature. The vacuum line is disconnected at the moment when the boiler reaches about 150°C. (u) Repeat steps from (f) through (r) for five different frequencies, 5 to 50 cps., to obtain the AR vs. GP curve for the alkali vapor. The correction for the large difference between volumetric heat capacity ratio CDR and the reference value is made this time by means of an experimentally determined set of data by a com- puter program (9). These are low pressure data where CDR is large. (v) Finally, take the ratio of AR oh the above curve to that of the calibration curve at GP =0.1, and multiply the ratio by the thermal conductivity of argon at the temperature of calibration to obtain the desired thermal conductivity of the alkali vapor . Since the above ratios were found to be significantly dependent on the value of GP in its high range, the ratio was taken in the low range of GP where it is no longer sensitive to the values of GP. An example of the value of GP in such a range is 0.1. 565 5. Experimental Results Experimental results for argon are shown in figure 6. These points were the last and most reliable made, and were all obtained on one wire (No. 12). The pressure was varied from 1 to 10 atmospheres to keep CDR constant, as these pressures have shown no effect on the thermal conductivity of argon. The solid lines represent the first series of consecutive runs., After a recalibration run the dashed lines were obtained. The small change noted between the two calibration runs shows that the properties and dimensions of the wire changed very little during each set. The spacing of the curves at GP = 0.1 yields the following thermal conductivity relative t& that at the calibration temperature of 32.6°C. t, °C V k 32.6 Table 2. Thermal Conductivity of Argon Relative to 32.6 C. Ob 1.132 160 216 3^6 459 1.351 1.470 1.706 1, 569 707 855 987 IO85 2.062 2.350 2,558 2.786 2.859 These values agree very closely (9) with those of Nuttall (10) with Liley's correla- tion (ll) and with Peterson's earlier points by this method (f) and are within 2% of Vargaftik and Zimina (12). Applying a cubic equation in t to the above points, making it pass through unity at 32.6°C, and extrapolating to 0°C yielded k /k^„ g = O.91588. Dividing the equation through by this quantity gives the correlation^ ^ (k fc /k Q ) = 1 + 2.91287 x 10" 3 t 1.37670 x 10" 6 t 2 + 4.8245 x 10~ 9 t 3 (11) for C and cal/sec. cm c, with a standard deviation of 0.04l. The good internal con- sistency of these results, and their agreement with the others above cited seem to validate the experimental and computational procedures here employed. The effect of temperature on the thermal conductivity of an alkali metal vapor requiring perfect evacuation of the cell and deoxidation of the alkali metal would evidently be some- what more difficult to measure successfully. However, such a measurement by itself would not be adequate. Successful experimental results with the alkali metals relative to argon were still more difficult to achieve, as they required consecutive successful tests with argon and with the alkali metal on the same wire. Furthermore, at low pressures the temperature jump correction was undesirably high. However, successful runs were carried out with cesium and rubidium in the 0.1 to 0.2 atmos- phere range. These runs are shown in figure 6 and the calculated results are listed in Table 3. Table 3. Thermal Conductivity of Cesium and Rubidium Vapors. Impurities, parts/million Rb Thermal Estd. Alkali % Source or Tempera- Pressure, Conduct ivity, cal./sec, cm C 0.1587 Error Metal Purity 99 . 999 Dow Li 1 Na K CS 8 8 10 ture, oc. 615 atmos . 0.214 * Cs 4.6 Cs 554 0.0984 0.2060 11 Cs 547 0.0974 0.2120 11 Rb 99.8+ Kawecki 10 70 500 700 772 0.0882 0.2280 5.1 Rb 826 0.0829 0.2158 n These results are plotted in figures 7 and good, under the circumstances. These metal analyzed typically as shown in Table 3. A may be calculated assuming ideal liquid and data, and the relative thermal conductiviti purities have higher thermal conductivities cesium the correction is entirely negligibl total of only 440 parts per million of the subtracted. 8. Internal consistency is seen to be s were of highest commercial purity, and correction on the thermal conductivity vapor mixtures, available vapor pressure es of Table 1. Although the lighter im- they also are less volatile. For the e. For the rubidium it amounts to a thermal conductivity, thus was not 566 Gottlieb and Zollweg's data are at such low pressures (less than 3 torr) that their thermal conductivities, though extrapolated to eliminate temperature jump, can be ascribed to monomer. Their rubidium point is seen in figure 8 to agree fairly well with the present results, but their cesium point is high in figure 7. However, their own conclusion was that their rubidium value was low and their cesium value reasonable. Achener's results on rubidium are for essentially saturated vapor at higher pressures (k) . They are seen to be loiver than these, but are considered only preliminary and are expected to rise somewhat (l4) . 6. Calculation of Thermal Conductivity from Kinetic Theory for Associating Vapor Calculations of the thermal conductivity of alkali metal vapors have been carried out by Davies et al (13) for vapor consisting exclusively of monomer atoms. However, this is not the case, particularly for vapor at or near saturation and at substantial pressures, as of interest in fast breeder reactor coolant studies. Even small con- centrations of dimer considerably increase the thermal conductivity due to the heat of change in equilibrium composition with temperature, which adds diffusion of the heat of reaction to the conduction of sensible heat. The procedure outlined by Vanderslice et al (15) for predicting the transport properties of hydrogen at high temperatures includes the effect of dissociation. It was decided to apply this procedure to the alkali metal vapors. Potential energy functions for the monomer atoms were taken from Davies' table (13). The potential energy functions and collision integrals for monomer-dimer and dimer-dimer inter- actions were calculated by Vanderslice' s perfect pairing procedure (l6,17). Next the translational conductivity of the binary mixture was calculated (l8), then the internal conductivity of the mixture (19) , and then the conductivity due to chemical reaction (19,20). Pull details are available elsewhere for the potential energy functions and equilibrium and frozen thermal conductivity results (9), as well as for viscosity and the thermodynamic properties (21) . 7. Conclusions and Acknowledgements Pinal results are given in Tables 4 through 8 and figures 7 through 10. The trend with temperature of the thermal conductivity of slightly superheated vapor is particularly noteworthy. It is seen that the thermal conductivity shows a minimum value at a temperature several hundred degrees above saturation, increasing dimer contents at lower temperatures substantially increasing the thermal conductivity at all but the highest pressures. With lithium the minima are still higher. Agreement between the calculated and experimental results for cesium and rubidium is seen to be good. The results of Stefanov et al on sodium and potassium are also seen to be in fair agreement with the calculated results, particularly as to showing a similar trend of increasing thermal conductivity with decreasing temperature. It is concluded that the results calculated by this method closely represent the equili- brium thermal conductivities of the alkali metal vapors. The authors are happy to acknowledge support by the U.S. Atomic Emergy Commission, Division of Research, under Contract AT(30-1)-2S60 with Columbia University, and donation of the cesium by the Dow Chemical "Company. The AEC serial number of this publication is CU- 2660-37. 8. Nomenclature a Accommodation coefficient c Specific heat at constant pressure of the test gas cal/gm °C c 1 Specific heat of fine wire " c Specific heat at constant volume of the test gas " f Frequency of the heating current cps k Thermal conductivity of the test gas cal/sec.cm. °C k Thermal conductivity of a fictitious gas " m Pine wire mass per unit length gm/cm p Pressure of the test gas atm 567 Table k Equilibrium Thermal Conductivity of Cesium Vapor (cal/sec.cm.°C) x ic4 P ressure , atm Satu- T(°K) Monomer 0.02 0.05 0.1 0.2 0.5 1 2 5 10 ration 500 0.0810 0.1023 600 0.0920 0.1341 700 0. 1026 0.1349 0.1655 800 0.1129 0.1232 0.1375 0.1585 0.1923 0.1922 900 0.1230 0.1270 0.132 9 0.1422 0.1590 0.1977 0.2129 1000 0.1328 C.1348 0.1376 0.1421 0.1506 0.1729 0.2013 0.2281 1100 0.1^26 0.1436 0.1451 0.1476 0.1523 0.1653 0.1837 0.2112 0.2394 120P 0.1523 0.1529 0.1537 0.1552 0.1580 0.1660 0.1779 0.1976 0.2347 0.2481 .300 0. 1619 0.1623 0.1628 0.1637 0.165 5 0.1707 0.1787 0.1926 0.2224 0.2494 0.2550 1400 0.1714 0.1717 0.1721 0.1727 0.1739 0.1774 0.1829 0.1930 0.2162 0.2405 0.2610 1500 0.1810 0.1811 0.1814 0.1818 0.182 7 0.1852 0.1892 0.1965 0.2146 0.2355 0.2666 1600 0. 1905 0.1906 0.1908 0.1911 0.1917 0.1936 0.1965 0.2021 0.2163 0.2338 0.2719 1700 0.2000 0.2001 0.2002 0.2005 0.2009 0.2023 0.2046 0.2088 0.2201 0.2347 0.2772 1800 0.2095 0.2096 0.2097 0.2099 0.2102 0.2113 0.2131 0.2164 0.2255 0.2377 0.2827 1900 0.2190 0.2191 0.2192 0.2193 0.2196 0.2205 0.2219 0.2246 0.2320 0.2422 0.2882 2000 0.2286 0.2287 0.2287 0.2288 0.2291 0.2298 0.2309 0.2331 0.2392 0.2479 0.2940 2100 0.2382 0.2383 0.2383 0.2384 0.2386 0.2391 0.2401 0.2419 0.2470 0.2544 0.3000 2200 0.2478 0.2478 0.2479 0.2480 0.2481 0.2486 0.2494 0.2509 0.2552 0.2615 0.3061 2300 0.2575 0.2575 0.2575 0.2576 0.2577 0.2581 0.2588 0.2601 0.2638 0.2692 0.3125 2400 0.2672 0.2672 0.2673 0.2673 0.2674 0.2677 0.2683 0.2694 0.2726 0.2773 0.3190 2500 0.2769 0.2770 0.2770 0.2770 0.2771 0.2774 0.2779 0.2788 0.2816 0.2857 0.3258 2600 0.2867 0.2867 0.2868 0.2868 0.2869 0.2871 0.2875 0.2884 0.2908 0.2944 0.3327 2700 0.2965 0.2966 0.2966 0.2966 0.2967 0.2969 0.2973 0.2980 0.3001 0.3033 0.3397 2800 0.3064 0.3064 0.3064 0.3065 0.3065 0.3067 0.3070 0.3077 0.3095 0.3124 0.3469 2900 0.3163 0.3163 0.3163 0.3164 0.3164 0.3166 0.3169 0.3174 0.3191 0.3216 0.3543 3000 0.3263 0.3263 0.3263 0.3263 0.3264 0.3265 0.3268 0.3273 0.3287 0.3310 0.3618 Table $ Equilibrium Thermal Conductivity of Rubidium Vapor (cal/sec.cm.°C) x 10^ Pressure , atm Satu- T(°K) Monomer 0.02 0.0$ 0.1 0.2 0.5 1 2 5 10 ration 500 0.1118 0.1325 600 0.1271 0.1702 700 0.1419 0.1946 0.2105 800 0.1562 0.1725 0.1951 0.2286i 0.2485 900 0.1703 0.1765 0.1856 0.1999 0.2259 0.2812 1000 0.1841 0.1869 0.1911 0.1979' 0.2109 0.2451 0.2897 0.3081 1100 0.1978 0.1992 0.2014 0.2050 0.2120 0.2315 0.2595 0.3028 0.3297 1200 0.2113 0.2122 0.2134 0.2155 0.2196 0.2313 0.2491 0.2791 0. ,3388 0.3469 1300 0.2247 0.2253 0.2260 0.2274 0.2299 0.2374 0.2491 0.2698 0. ,3162 0. ,3617 0.3610 1400 0.2381 0.2335 0.2390 0.2398 0.2416 0.2466 0.2546 0.2692 0. ,3045 0. 3439 0.3729 1500 0.2514 0.2517 0.2520 0.2526 0.2538 0.2574 0.2630 0.2736 0. ,3006 0. ,3334 0.3832 1600 0.2647 0.2649 0.2652 0.2656 0.2665 0.2690 0.2732 0.2311 0. ,3019 0. 3288 0.3926 1700 0.2780 0.2782 0.2784 0.2787 0.2793 0.2813 0.2844 0.2905 0, ,3067 0. ,3288 0.4014 1800 0.2913 0.2914 0.2916 0.2918 0.2923 0.2938 0.2963 0.3010 0. ,3139 0. ,3321 0.4099 1900 0.3046 0.3047 0.3049 0.3050 0.3054 0.3066 0.3086 0.3123 0. ,3228 0. ,3379 0.4183 2000 0.3180 0.3181 0.3181 0.3183 0.3186 0.3196 0.3211 0.3242 0. ,3328 0. ,3454 0.4266 2100 0.3313 0.3314 0.3315 0.3316 0.3319 0.3327 0.3339 0.3365 0. ,3436 0. ,3542 0.4350 2200 0.3448 0.3448 0.3449 0.3450 0.3452 0.3458 0.3469 0.3490 0. ,3550 0, ,3641 0.4436 2300 0.3582 0.3582 0.3583 0.3584 0.3586 0.3591 0.3600 0.3618 0, ,3669 0. ,3747 0.4522 2400 0.3717 0.3717 0.3718 0.3719 0.3720 0.3725 0.3732 0.3748 0. ,3791 0, ,3859 0.4610 2500 0.3852 0.3854 0.3853 0.3854 0.3855 0.3859 0.3866 0.3879 0, ,3917 0. ,3975 0.4700 2600 0.3988 0.3990 0.3989 0.3990 0.3991 0.3994 0.4000 0.4011 0. ,4044 0, ,4096 0.4791 2700 0.4125 0.4126 0.4125 0.4126 0.4127 0.4130 0.4135 0.4145 0, ,4174 0, ,4219 0.4884 2800 0.4262 0.4262 0.4262 0.4263 0.4263 0.4266 0.4271 0.4279 0. ,4305 0, .4345 0.4979 2900 0.4399 0.4401 0.4400 0.4400 0.4401 0.4403 0.4407 0.4415 0. ,4438 0, .4473 0.5075 3000 0.4537 0.4538 0.4538 0.4538 0.4539 0.4541 0.4544 0.4551 0, ,4571 0, ,4603 0.5172 568 Table 6 Equilibrium Thermal Conductivity of Potassium Vapor (cal/sec.cm.°C) x 10^ Pressure , atm Satu- T(°K) Monomer 0.02 0.05 0.1 0.2 0.5 1 2 5 10 ration 500 0.1851 0. 1979 600 0.2100' 0.2450 700 0.2339 0.3015 800 0.2571 0.3422 0.2931 0.4130 0.3618 900 0.2797 0.3121 0.2929 0.3420 0.3954 0.4197 1000 0.3019 0.3163 0.3077 0.3302 0.3563 0.4241 0.4710 1100 0.3238 0.3311 0.3267 0.3382 0.3521 0.3902 0, 4441 0.5142 1200 0.3454 0.3495 0.3471 0.3535 0.3615 0.3840 0. 4179 0, 4741 0.5497 1300 0.3668 0.3693 0.3678 0.3718 0.3767 0.3908 0, 412 7 0. 4513 0. 5362 0.5786 1400 0.3881 0.3898 0.3888 0.3914 0.3945 0.4038 0. 418 6 0. 4455 0. 5099 0. 5803 0.6025 1500 0.4093 0.4104 0.4098 0.4115 0.4137 0.4201 0. 4304 0. ^497 0. 4984 0.5570 0.6226 1600 0.4304 0.4312 0.4307 0.4320 0.4336 0.4382 0. 4456 0. 4598 0. 4970 0.5449 0.6401 1700 0.4515 0.4521 0.4518 0.4527 0.4538 0.4572 0. 4628 0. 473 5 0. 5023 0.5413 0.6558 1800 0.4726 0.4730 0.4728 0.4735 0.4743 0.4769 0. 4812 0. 4895 0. 5121 0.5439 0.6704 1900 0.4936 0.4940 0.4938 0.4943 0.4950 0.4970 0. 5004 0. 5069 0. 5250 0.5512 0.6843 2000 0.5147 0.5150 0.5149 0.5153 0.5158 0.5174 0. 5201 0. 5253 0. 5400 0.5617 0.6978 2100 0.5358 0.5361 0. 5359 0.5363 0.5367 0.5380 0. 5402 0. 5445 0. 5566 0. 5747 0.7112 2200 0.5570 0.5572 0.5570 0.5573 0.5577 0.5588 0. 5606 0. 5641 .5742 0.5895 0.7246 2300 0.5782 0.5784 0.5783 0.5785 0. 5788 0.5797 0. 5812 0. 5842 0. 5927 0.6057 0.7381 2400 0.5995 0.5996 0. 5997 0.5997 0.6000 0.6007 0. 6020 0. 6045 0. 6118 0.6230 0.7517 2500 0.6208 0.6210 0.6210 0.6210 0.6212 0.6219 0. 6230 0. 6251 0. 6314 0.64H 0.7656 2600 0.6422 0.6423 0.6423 0.6424 0.6426 0.6432 0. 6441 0, 6459 0. 6513 0.6598 0.7797 2700 0.6637 0.6638 0.6639 0.6639 0.6640 0.6645 0. 6653 0. 6669 .6716 0.6790 0.7940 2800 0.6853 0.6854 0.6855 0.6854 0.6856 0.6860 0. 6867 0. 6881 .6922 0.6987 0.8085 2900 0.7069 0.7070 0.7072 0.7071 0.7072 0.7075 0. 7082 0. 7094 .7130 0.7188 0.8233 3000 0.7286 0.7287 0.7288 0.7288 0.7289 0.7292 0. 7297 0. 7308 .7341 0.7392 0.8383 Table 7 Equilibrium Thermal Conductivity of Sodium Vapor (cal/sec.cm.°C) x 104- Pressure, atm Satu- T(°K) Monomer 0.02 0.05 0.1 0.2 o.5 1 2 5 10 ration 500 0.3072 0.4909 600 0.3489 0.7525 700 0.3889 1.0384 800 0.4277 1.2891 900 0.4656 0.9611 1.4624 1.4761 1000 0.5027 0.6772 0.9022 1.2008 1.6067 1.5093 1100 0.5393 0.6092 0.7073 0.8547 1.0995 1.5736 1.6719 1200 0.5754 0.6072 0.6535 0.7266 0.8595 1.1729 1.5100 1.7093 1300 0.6111 0.6273 0.6512 0.6898 0.7630 0.9539 1.1985 1 .5136 1.7244 1400 0.6466 0.6556 0.6690 0.6908 0.7332 0.8496 1.0141 1 .2610 1 .6487 1.7264 1500 0.6819 0.6873 0.6954 0.7085 0.7343 0.8076 0.9170 1 .0973 1 .4398 1 6965 1.7214 1600 0.7171 0.7206 0.7256 0.7341 0.7506 0.7985 0.8724 1 .0015 1 .2798 1 ,5369 1.7134 1700 0.7523 0.7547 0.7579 0.7636 0.7747 0.8072 0.8584 ,9513 1 .1693 1, 4016 1.7047 1800 0.7874 0.7890 0.7913 0.7953 0.8030 0.8258 0.8623 .9301 1 .0989 1 ,2980 1.6968 1900 0.8225 0.8238 0.8253 0.8282 0.8338 0.8503 0.8770 .9275 1 .0584 1 .2244 1.6904 2000 0.8577 0.8588 0.8599 0.8620 0.8661 0.8784 0.8984 0, .9367 1 ,0390 1 .1759 1.6859 2100 0.8930 0.8935 0.8947 0.8962 0.8994 0.9088 0.9241 ,9537 1 ,0345 1 .1471 1.6836 2200 0.9284 0.9289 0.9297 0.9310 0.9334 0.9407 0.9526 .9759 1 .0406 1 .1334 1.6833 2300 0.9639 0.9643 0.9650 0.9660 0.9679 0.9736 0.9832 1 0018 1 .0541 1 ,1311 1.6850 2400 0.9996 1. 0000 1.0005 1.0013 1.0028 1.0074 1.0151 1 .0302 1 .0731 1 .1373 1.6887 2500 1.0355 1.0360 1.0362 1.0368 1.0381 1.0419 1.0482 1 ,0606 1 .0961 1 .1501 1.6941 2600 1.0715 1.0717 1.0722 1.0727 1.0737 1.0768 1.0820 1 ,0924 1 ,1221 1 ,1678 1.7011 2700 1.1077 1.1078 1.1084 1.1086 1.1095 1.1122 1.1165 1 ,1252 1 .1503 1 1893 1.7096 2800 1.1441 1.1448 1.1446 1.1450 1.1456 1.1479 1.1516 1 1590 1 .1804 1 .2138 1.7195 2900 1.1808 1.1815 1.1813 1.1815 1.1821 1.1840 1.1871 L, 1935 1 .2118 1 .2407 1.7305 3000 1.2176 1.2187 1.2180 1.2182 1.2187 1.2204 1.2231 1. 2285 1 ,2444 1 .2696 1.7426 569 Table 8 Equilibrium Thermal Conductivity of Lithium Vapor (cal/sec.cm.°C) x 10^ Pressure j i atm Satu- T(°K) Monomer 0.02 0.05 0.1 0.2 0.5 1 2 5 10 ration 500 0.5720 0.5421 600 0.5934 0.6713 700 0.6619 0.8695 800 0.7282 1.1479 900 0.7929 1.4953 1000 0.8562 1.8850 11C0 0.9195 2.2845 12C0 0.9800 2.6653 1300 1.0409 2.0199 3.0075 1400 1. 1012 1.5540 2.1472 2.Q534 3.3004 150C 1. 1613 1.3864 1.7006 2. 1690 2.9377 3.5416 1600 1.2211 1.3414 1.5146 1.7852 2.2667 3.3504 3.7335 170C 1. 2808 1.3494 1.4498 1.6106 1.9098 2.6569 3.5428 3.8817 1800 1.3404 1.3818 1.4430 1.542 3 1.7318 2.235? 2.9023 3.8067 3.9929 1900 1.4001 1.4264 1.4652 1.5290 1.652* 1.9Q36 2.4782 3.2129 4.0736 2000 1.4599 1.4770 1.5030 1.5454 1.6284 1.8633 2.2127 2.7838 3, ,8531 4. 1299 2100 1.5198 1.5319 1.5493 1.5785 1.6360 1.8012 2.0546 2.4QC7 3, .4032 4.1669 2200 1.5798 1.5881 1 .6007 1.6215 1.6623 1.7812 1.9672 2.2992 3, ,0522 3.8104 4.1890 2300 1.6402 1.6461 1.6554 1.6704 1.700 2 1.7875 1.9262 2.1801 2, ,7913 3.4717 4.1997 2400 1.7008 1.7057 1.7120 1.7232 1.7455 1 .9109 1.9159 2.1118 2, ,6047 3.1970 4.2015 2500 1.7617 1.7658 1 .7702 1.7788 1.7957 1.8455 1.9263 2.0791 2, ,4764 2.9828 4.1966 2600 1.8229 1.8266 1.8295 1.8362 1.8492 1.8879 1.9509 2.0713 2, ,3927 2.8214 4.1867 2700 1.8845 1.8866 1 .889 3 1.8949 1.9051 1.9357 1.9855 2.0815 2, ,3*27 2.7040 4.1728 280C 1.9465 1.9494 1.9507 1.9550 1.9629 1.9873 2.0272 2.1045 ?, ,3182 2.6223 4.1559 2900 2.0089 2.0102 2.0128 2.0157 2.C2 2 3 2.0419 2.0741 2.1370 2, ,3130 2.5692 4.1366 300C 2.0717 2.0742 2.0746 2.0771 2.0826 2.0986 2.1250 2.1766 2.3225 2.53«8 4.1153 57Q r Radial distance from the center of the fine wire r. Pine wire radius r~ Outer shell radius t Instantaneous temperature t. Instantaneous temperature of the wire ty Outer shell temperature E IR drop across the fine wire E, Major rms fundamental voltage E, Rms third harmonic voltage E^ R Bridge output rms third harmonic voltage I Rms current passing through the fine wire L Length of the fine wire R Gas constant, instantaneous electrical resistance R Average electrical resistance of the fine wire R.. Thermocouple voltmeter resistance dR First derivative of electrical resistance with respect to the aT average wire temperature T Absolute temperature Y Ratio of specific heats 6 Time f Density of the test gas f Density of the fine wire ^ Phase shift angle to Angular frequency, 2irf cm volts amps cm ohms ohms/ C °K sec gm/cm" radians radians/sec 9. References (8) (1) Gottlieb, M. and R. J. Zellweg, J. Chem. Phys. 39, 10, 2773-2774(Nov.l963) . (2) Stefanov, V. I., D. L. Timrot, E. E. (9) Totskee and Ch. Venkhao, Thermophyslcs of High Temperature 4 , l4l(l966j, Academy of Science, USSR. (3) Timrot, D. L. and E. E. Totskee, Thermophysics of High Temperature, 3, (10) No. 5(1965). (4) Achener, P. Y., AGN Report No. 8222, • (ll) Contract AT(o4-3)-368, p. 1 , Aerojet- General Corp., San Ramon, Calif. (May 1967). (12) (5) Lee, C. S. and Bonilla, C. P., Techni- cal Report No. 1, Contract Nonr 266 (11), Columbia Univ., N.Y.(l952). (13) (6) Tarmy, B. L., PhD Dissertation, Dept . of Chemical Engineering, Columbia Univ. (1956). (14) (7) Peterson, J. R., Eng.Sc.D. Dissertation, Dept. of Chemical Engineering, Columbia (15) Univ. (1963); Peterson, J. R. and Bonilla, C. F., Third Symposium on Thermophysical Properties. ASME, pp. 264-276(March 22-25, 1965) . Kennard, E. H., Kinetic Theory of Gases, McGraw-Hill, N.Y. 1938 Lee, C. S.j Eng.Sc.D. Dissertation, Dept. of Chemical Engineering, Columbia Univ.(l968). To be avail- able through University Microfilms, Inc., Ann Arbor, Mich. Nuttall, R. L., Nat'; Bur. Standards Circ. 564 , Washington, D.C. (1955). Liley, P. E. , Private communication (Dec, i960) . Vargaftik, N. B. and Zimina, N. Kh., Teplofizika Vysokikh Tempera tur, 2, 5, 716-724(1964). Davies, R. H., Mason, E. A. and Munn, R. J., Phys. Fluids, 8, 3, 444(1965) . Achener, P. Y. , Private communication (Nov.l, 1967). Vanderslice, J. T., Weissman, S., Mason, E. A. and Fallon, R. J., Phys. Fluids, 5, 2, 155(1962) . 571 '(16) Vanderslice, J. T., Mason, E. A. and (19) Hirschf elder, J. 0., J. Chem. Phys . , Lippincott, E. R.,J. Chem. Phys., 30 26, 27^(1957). 129(1959). (17) Vanderslice, J. T. and Mason, E. A. J. Chem. Phys., 13, 492 ( I960) . (18) Muckenfass, C. and Curtiss, C. P., J. Chem. Phys., 29, 1273(1958). (20) Butler, J. N. and Brokaw, R. S., J. Chem. Phys., 32, 1005 (i960). (21) Lee, D. I., Eng.Sc.D. Dissertation, Dept. of Chemical Engineering, Columbia Univ.(l967). To be avail- able through University Microfilms, Inc., Ann Arbor, Mich. 1 1 — 1 1 11 1 1 1 1 1 — 1 1 1 1 1 1 1 1 1 — 1 1 1 1 1 1 1 1 1 — 1 1 1 1 I LI 2 10 100 500 = 1,000 = 5,000 = 10,000 = 30,000 = 70, 000 Conductivity Ratio, CR 1 i i i L i I 1 1 1 1 1 1 1 | | I I I I II 10 100 Figure 1. Theoretical Frequency Response Curves for Radius Ratio (DR) of 1000, and Zero Temperature Jump (TJF), and Thermal Radiation (QR). 572 0.1 0.01 i — I — I I I 1 1 1 1 1 — I — I I I I u -_UF=0. 32^ SR=0, CDR=3QQQ g"" TJF=a 32] "0R= 0. 25, C DR = 3000 :jg TJF= 0.08, CR=0, CDR =3000 ~ or -- j__ -j S TJF=0.08, 0R=0.06, CDR=3000 a. -2 —i I TJF=0,Q2, Ofi=0, CDRr3Q 3Q j TJFzO.02, QR=0.25, COR= TJF = 0.02, 0R=0, COR = TJF=0.02, QR=0, TJF=0.02, (5R 0.001 0.001 0.01 100 Figure 2. Fractional Temperature Jump Correction Term as a Function of Conductivity Ratio (CR) for Various Values of Temperature Jump Factor (TJF), Thermal Radiation (QR), and Volumetric Heat Capacity Ratio (CDR). VACUUM TUB! VOLTMETER BAND-PASS FILTER Z] — 1= DECADE AMPLIFIER SWA2?" "1 AUDIO POWERi— AMPLIFIER X .2 TRANSFORMER ^-DECADE RESISTORS ~*s/W* k*w wh+ UaW »vs^a— - IK 0.1 K 10ft 1A 0.1ft \3WAJ STEPS it ii it ,i \ -r Q ■ ,R a -8K 10 K 5 K 2 K io7» n i swc R d -800A 10 6 fl -*^wvw*/vw*- DECADE CAPACITORS #####1 THERMO- COOPLE VOLT- METER -IT? _L X OSCILLATOR SWAiy" — -O O- 1 K 1 2 VOLTAGE DIVIDER 10 5 10^ lO^ 10 2 10 MF rr~ it ii ii || STEPS WIEN BRIDGE CALIBRATION CIRCDIT Figure 3. Power, Mearurement, and Calibration Circuit of a Hot Wire Transient Thermal Conductivity Cell. 573 TO MEASURE- MENT CELL THERMO- J COUPLE WELL' WELD BOILER BODY NICKEL LEAD COMPRE- SSION NUT FOLLOWER TEFLON- CERAMIC SEAL EXTENSION TUBING BOILER MEASUREMENT CELL FROZEN SEAL Figure k. Hot Wire Thermal Conductivity Cell with Frozen Metal Seal, 574 SWAGELOK SEAL LDCALOX Nb-lZr SEAL RING FOLLOWER NICKEL WIRE TUNGSTEN WIRE EXTENSION TUBING \ CAP CAP ADAPTER MEASUREMENT CELL THERMOCOUPLE WELL VACUUM- ARGON- ALKALI LINE\ BOILER - DETAILED VIEW OVERALL VIEW Figure 5. Hot Wire Thermal Conductivity Cell with Brazed Seal. 575 0.4 0.3 0.2 0.1 0.08 0.06 0.04 CESIUM 547 < 0.0974 DC I O Z> 0.1 L5 ^ 0.08p 0.06- 0.04 0.03 0.02 0.01 RUBIDIUM 0.03 0.05 0.2 0.3 0.5 0.7 1 Figure 6. Working Curves for Thermal Conductivity Tests on Argon, Rubidium and Cesium. 576 .1 " r— x Curve No. 1 2 34 56789 10 Press., gtm 0.01 0.02 0.05 0.1 0.2 0.5 1 2 5 10 O 0.214 atm This work D 0.0984 •• O 0.0974 •• ? ■• Gottlieb &Zollweg(1) C&&8&- p ; >\ Siikooled x R Curve No. (theor) I ;' / yV, (exper ) ' T> / /'/', \\- Press., atm *.' ; / -'; '/> \ \ + Experimental Curve No. (theor) 1 2 3 4 56789 10- (exper) 4 a be Press., atm 0.01 0.02 0.05 0.1 0.2 0.5 1 2 5 10 >; .t»! ;', 3. /Subcoojea; Rsg&ru;^ o ; ; ; ; y./* «/\ A v_.\_-i*__> OH i ; J. '/ ',/. Vl-V-A"^ X by Stefanov et al (2) Saturation Curve --^Moooroer Temperature, °K x 500 1000 1500 2000 2500 3000 Figure 10. Experimental and Calculated Thermal Conductivity Results for Sodium Vapor. 5 78 Recent Developments at Bellevue on Thermal Conductivity Measurements of Compressed Gases B. Le Neindre, P. Bury, R. Tufeu, P. Johannin, and B. Vodar Laboratoire des Hautes Pressions - C.N.R. S. 1, Place A. Briand - Bellevue - (France) In this paper we give a description of an apparatus adapted to the measurement of the thermal conductivity of gases in the temperature range from 25 to 700°C and at pressures up to 120xl0 6 N/m 2 (1200 bar). The concentric cylinder method was used with different gaps (from 0.2 to 0.25 mm) and with a temperature difference from 1 to 4 deg C between cylinders. Tables giving experimental thermal conductivity values are presented for argon, nitrogen, methane, carbon dioxide near 25°C, and helium near 30 C C in the pressure range up to 120xl0 6 N/m 2 (1200 bar). The data are estimated to be accurate within 1.5 per cent; however, in general, the data are consistent within 0.5 per cent. Key words: Gases, pressure, radial heat flow. 1. Introduction Over the last decade there has been interest in thermal conductivity coefficients of the dense gaseous and liquid phases. The relevant experimental data were needed for the development of indus- trial plants working at high pressure and temperature. Therefore, the study of thermal conductivity has been undertaken for water (1), heavy water (2) in the liquid and gaseous phases, helium, and carbon dioxide. On the other hand, in order that the validity of any given theory of the non- equilibrium state might be assessed, there is a need for experimental data of thermal conductivity coefficients of the simplest molecules, i.e. argon, nitrogen and argon-helium mixtures. 2. Description of Method and Details of Apparatus A vertical concentric cylinder method has been used for these measurements. The experimental method and the design and assembly have been described in several papers (1, 2, 3, 4, 5). Some con- structional details and modifications of the apparatus are summarized here. In this method heat is generated in the inner cylinder (or emitter) and passes outward through a narrow gap (between 0.2 and 0.25 mm) to the outer system, consisting of the outer cylinder and of two centering pieces (fig. 1). The fluid under test is contained within the annulus. The use of end guards at the inner cylinder has been avoided by judicious location of thermo- couples, and by proper nonuniform heating of the inner cylinder to compensate for end effects. Figures in parentheses refer to literature references cited at the end of this paper. 579 The inner cylinder has been mounted within the outer cylinder by five insulating plugs. From the emitter heat is transferred by conduction, convection and radiation through the fluid and conduction through the insulating plugs, thermocouples and resistance leads. At ambient temperature, and far from the critical point, heat is transferred almost entirely by conduction through the fluids, the leads, and the ceramic supports. The experiments show that the thermal contact resistance between ceramic supports and silver is not completely negligible. Therefore, it is not possible to calculate the heat losses due to these supports. The thermal resistance of the system has been treated as if it were composed of two thermal conductances in parallel. With insulating plugs of quartz the "parallel" heat transfer by leads and ceramic supports has been found equal to 1.8x10"^ W m"deg"^ for a gap of 0.25 mm using the value 17.35x10 m" deg -1 for the thermal conductivity of argon at 25°C and O.lxlO 6 N/m 2 . Several experiments have been done with different numbers and kinds of plugs, and different types of gases. These experiments have shown that the "parallel" heat transfer is a constant at 25°C. The electrical capacitance of the cell measured in air at room temperature determines the goemet- rical constant of the cell which enters into the calculation of the thermal conductivity coefficient. The thermal conductivity cell is placed in a high pressure vessel heated externally by a vertical solid thermostat in copper (fig. 2). The heating unit consists of six electrical heaters, each heater is associated with a hand operated variable transformer and a Pt - Pt 107„ Rh thermocouple. Power to each element is hand controlled so that the longitudinal gradient is negligible. Input voltage to the variable transformers is controlled by an automatic temperature regulator. The sensing element of the temperature regulator is a Pt - Pt 10% Rh thermocouple located on the outside of the central heater so as to minimize the time-lag in regulation; temperature stability was better than 0. 1 deg C. The large thermal inertia of the high pressure vessel and the small time-lag in temperature regulation insure a complete absence of detectable temperature fluctuations in the thermal conductivity cell. The pressure in the high pressure vessel is measured by means of Bourdon type gauges manufactured by Heise Co. (USA), Temperature differences between emitter and receiver are accurately measured by means of eight Pt - Pt 107„ Rh thermocouples connected in series. The emf of the thermocouples is measured by means of a Wenner potentiometer (Leeds & Northrup Co.) with a Keithley null detector, the sensitivity of which is better than 0.1 hV. 3. Measurements and Results Since stabilization of temperature requires a rather long period of time, all measurements have been made at constant temperature and varying pressure. A variation of pressure also produces a slight change in the temperature equilibrium due to the heat of compression and the modification of the temperature distribution in the high pressure vessel. The time necessary to reach equilibrium under these conditions is highly dependent on the thermal conductivity of the gas used in the experiment; it takes about one hour for argon and less for the other gases. Measurements have been made in general at decreasing pressures with some verification points at increasing pressures. The aim of this work was to provide thermal conductivity coefficients of fluids in large tempera- ture and pressure ranges. Because of difficulties in making measurements under high pressure at constant temperature, an attempt was made to find empirical relations for the variation of thermal conductivity coefficients with pressure. As it will be shown subsequently, the relation AX = X - X = f(p) (1) P,T P,T o,T 2 is true for a gas in a large temperature and pressure range (where X. T is the thermal conductivity coefficient at density p and temperature T, X T is the thermal conductivity coefficient at density zero and temperature T. ) It is possible to use this relation to interpolate and extrapolate the experimental thermal conductivity coefficient of a gas. On the other hand, assuming that this relation can be applied to another gas, a measurement of thermal conductivity as a function of temperature up to T°C and another measurement as a function of pressure up to P N/m along an isotherm provide thermal conductivity coefficients in the pressure- temperature range desired. It may be recalled here that for a larger range (1600 bar and 700°C) some deviations from this sample relation were observed for nitrogen (3) (4). 580 All measurements have been plotted in the (X,p) diagram. The first measurements have been carried out for different gases near 25°C at pressures up to 120x10^ N/m 2 (1200 bar). The thermal conductivity coefficient has been calculated at 25°C using eq (1) at constant density: X P,T _ X o,T = X p,25°C " X o,25°C ( 2 ) ^p,T " *p,25°C = X o,T " X o,25°C (3) It is found that the variation of the thermal conductivity coefficient at atmospheric pressure is a linear function of the temperature in a large interval around 25°C: *o,T = *o,25°C + a < T " 25 > <*> so that eqs (3) and (4) give X p,T " x p,25°C = a control thermocouples; T c , measuring thermocouple; I isolating piece. Figure 1. The thermal conductivity cell. Cx, external cylinder; C 2 , internal cylinder; Gi, and G2, centering pieces; R, heater; T, thermocouple hole; A, isolating plug', P, ceramic piece; F, spring. 589 W m deg I I I I I I I I I 80 A i x10* / /: / 60 / - 40 / - Michels & Others 20 I I I — This investigation > Q i i i i i i 200 400 600 800 g/m 3 Fig. 3 - The thermal conductivity coefficient of argon at 25°C versus density Xx10 3 Wm-'°C-' ^ kg/r Fig. 4 Isotherms of thermal conductivity coefficients of argon versus density from 25°C to 700°C 5 90 -u~= ( Wrrfdeg 80 60 40 20 Ax10 : / y • / ,"'" ./ ■>>**' Michels and Botzen -•-Misic and Thodos — This work + 9 100 2 00 300 A 00 kg/m 3 Fig. 5 - The thermal conductivity coefficient of nitrogen at 25°C versus density Wrrfdeg 130 110 90 70 50 30 *x10 / _ / /// />' .■// ■■*■ Golubev '// ~— Misic & Thodos — — This wor k 100 200 300 kg/m* Fig. 6 - The thermal conductivity coefficient of methane at 25°C versus density 591 Wrrf'deg 1 110 90 70 _] 10 ] i r Ax10 200 25 C Michels e.a 25°C— This work 132°C — This work 400 600 kg/W Fig. 7 - The thermal conductivity coefficient of carbon dioxide versus density at 25°C and 132°C kc/m a Fig. 8 The thermal conductivity coefficient of helium at 30°C versus density 5 92 Wrn'deg 160 120 80 '♦0 Ax10 : ■tc-i — r Mixtures Mol fraction, x A He 1 0.204 0796 2 0.415 0.585 3 0.4955 0.5045 4 0.5605 0.4395 5 0.792 208 40 80 120 ■(A-He), - Here M is the molecular weight, T( °K) the temperature, and o(A) is a parameter representing the distance between two molecules. The term q< 2 ' 2 )* (rp*) j_ s called the collision integral: the entity involving the collision dynamics and the intermolecular potential between two molecules. The collision integral is determined as a function of reduced temperature, T . The reduced temperature is defined by the relation, T = T/(e/k), where e Is the maximum energy of attraction between the molecules, and k is Boltzmann's constant. Equation (l) can, however, only be applied to experiment if we know the intermolecular potential function. Only then can we compute q( 3 » 2 )* an £ obtain values for a and e/k. Quantum mechanics has yet to derive a function from first principles so we are required to use a function based on a model. Hence the ability to use eq (l) to correlate and predict experimental data rests on a proper choice of the model function. 1 This article is essentially that of H. J. M. Hanley and G. E. Childs. Science , 159, 1114 (1968), copyright 196£ by the American Association for the Advancement of Science. Figures in brackets indicate the literature references at the end of this paper. 5 97 In the past, one has usually relied on intuition to make this choice. Recently, however, we have clarified the overall relation of model functions, theoretical expressions (such as eg (l)), and experi- ment [2-4] • Two points from this study are relevant here so we list them; 1. Potential functions have been derived and classified in terms of families. For example we have the m-6 family where m is a family parameter. If U(r) is the interaction potential of two molecules separated by distance r, the functions of the m-6 family are written: 6 m -«■.[(§)■- cf) 6 ]/tcD SI5 - ar 5 ]. where the distance parameter a is equal to the distance separating the molecules when U(r) = 0. The family parameter, m, depicts the repulsive part of the interaction. When m = 12, we have the famous 12-6 or Lennard- Jones potential. The exponential: 6 (Exp: 6) family is frequently used. The characteristic family parameter here represents the steepness of the repulsive part of the interaction potential and is given the symbol a. Other families of interest are the Kihara and the Morse . Now we have shown that all families are essentially part of the same large family with respect to their relation to experiment. This means that if we can fit a member of a given family to experimental data via a theoretical expression, we can get the same fit from a member of another family. Alterna- tively, members of different families can be interchanged without materially altering the fit. There is nothing special about any of the families known at this time. They are equivalent. 2. We can only make a significant selection of a function and its parameters from experiment if we have data outside the reduced range of about 2 < T* 2 _ 6 < 5 • (3) Here T 12 _ 6 is given by T l9 ^ e = T/(e/k) ls _ 6 , where (e/k) ls _ 6 is the well-depth parameter for the gas from the 12-6 function-'. Any function will fit data in the range of eq (3)- We must have data outside this range if we are to find a function suitable for extrapolation purposes. We give an example for the case of argon. Since (e/k) lg _ 6 for argon is about 120°K, the range of eq (3) is approximately 240 < T°K < 600 . We have found that all the functions of all the families tested can, via eq (l) , fit experimental viscosity data in this range. An example has been published as figure 2 of reference [5] (see also [3, 4]). The point we wish to make is that only one member of a family can fit this data and data above 600°K. Furthermore, if we fit data above 600°K with a given function, we will automatically fit data in the range 240 < T°K < 600 . We mention the above two points because our discussion makes use of fits of kinetic theory expres- sions to experimental data. We feel that the fits given below are satisfactory and that any conclu- sions are not influenced by our choice of potential functions. 3- Experimental Discrepancies There is a disagreement between two sets of experimental viscosity coefficient data for several gases. The disagreement is depicted for the case of argon in figure 1. One set, open circles, repre- sents data points which are generally accepted as correct and which are used to compute the tabulations available . Here we have fitted this set to eq (l) with the 40-6 potential function. The references for these points are given in another publication [5], but data above 500°K are mainly from the work of Trautz and Vasilesco [6]. The other set, closed circles, are points from the work of Kestin and co- workers [7-9] (300 - 570°K) and from recent work by a group at the Los Alamos Scientific Laboratory, which we designate by G. M. W. [10] (1100 - 2100°K). These closed circle points were also fitted to the 40-6 function. It is very obvious that the discrepancy is striking. Up to this point the poten- tial function plays no special role with respect to the discrepancy. The potential function is merely being used to represent data as far as figure 1 is concerned. A similar diagram would be obtained if the average of the open circle data set was used as a reference for figure 1. This deviation is made only for convenience. The factor (e/k) ls _ s is available for most substances. There is nothing special about the 12-6 function here. 5 98 The discrepancy between the work of Kestin and the work of others has been known, of course, for some time. Mason also reported several years ago [ll] that the high temperature viscosity data of Trautz and Vasilesco could be in error. For tabulation purposes, however, the discrepancy has usually been overlooked, possibly due to the simple fact that Kestin 's results are different. In fairness to this negative point of view, it is only recently that high temperature data have become available to supplement Kestin's work. Also much of Kestin's results lie in the temperature range of eq (3). This being the case it is difficult to pin down a correlating function from these data alone (point 2 above) and the discrepancy could have been caused by systematic error. The situation is different when we take Kestin and G. M. W. together. One can show by plotting their data on the same graph that their data are compatible and, furthermore, here we have data which are definitely outside the range of eq (3) and so can make positive statements on the correlating potential function. Figure 2 shows the fit of Kestin and the G. M. W. data alone. We found that an m of about 12 or 15 was the best . Eventually we used the Exp: 6 function with a = 15 • This is equivalent to an m of about 13. (Remember, point 1 above, there is absolutely no significance in changing from one family to another. We changed to the Exp: 6 family only because we did not have available collision integrals for m = 13 or 1*0 . k. Analysis We now show that there is no justification for omitting the Kestin (and now G. M. W.) work. Indeed, we give substantial evidence that this work is correct. Part of our analysis is based on the fact that kinetic theory predicts that the thermal conduc- tivity of a dilute monatomic gas, X, is linearly related to the viscosity by the expression: X = f c y Tl (I*) where c v is the specific heat per gram at constant volume and f is a constant. Theory predicts that this constant is very close to 2. 50. Explicitly the kinetic theory expression for X is [l]: o 2 n C2,2) * (t ) If one can properly fit experimental viscosity data to eq (l) , then one must be able to represent experimental thermal conductivity data by eq (5) without altering the correlation parameters. We thus determined theoretical thermal conductivity coefficients for argon from eq (5) using the function and parameters found suitable for the open circle viscosity data. These values were compared to experi- mental values taken from references [12-17]. Figure 3 shows the deviation curve. This figure clearly indicates that the correlation of X is not satisfactory, even allowing for the wide scatter of the thermal conductivity data. There is a systematic trend of the data away from a proper fit. This deviation has been observed before but usually has been attributed to f rather than to errors in T| or X. There is no theoretical evidence that f (i.e., eqs (l) and (5)) can change by an amount necessary to cause the systematic deviation of figure 3- Let us see what happens when we fit the thermal conductivity to the Exp: 6 function with a = 15, i.e., "fit" on the basis of the Kestin and G. M. W. data. The result, given as figure h, shows a substantial improvement over figure 3- (it should be noted that the parameters used could be adjusted so as to balance the deviations of fig. k around zero. In particular, this could be done while keeping the fit of the viscosity data to within the experimental precision. We have not done thisj our only purpose was to check the thermal conductivity on the basis of a fit to viscosity which was not pre- judiced by other considerations . We stress that any adjustment would involve a small change in e/k or a, not in a or m) . We have another simple, but less exact, procedure which does not involve a potential function. Assuming that the gases obey the principle of corresponding states, a reduced viscosity, T| g , should be a universal function of reduced pressure p/p e and reduced temperature t/t c . Here p c and T are the critical pressure and temperature respectively. The reduced viscosity is then given by TL =T, m-^p" 2 / 3 (RT)^ . (6) R expt c v c' 599 We know that the principle of corresponding states is not precisely obeyed for the noble gases and obeyed only approximately for polyatomic gases of simple structure. Nevertheless eq (6), and others like it, provide a simple method to correlate data. Figure 5 shows a plot of the reduced viscosity of argon from Trautz and the workers who agree with him (open circles) , and from Kestin and G. M. W. (closed circles). Also included are reduced viscosities of neon (squares) and hydrogen (triangles). These latter viscosities are reduced from measurements taken below 400°K and are assumed not to have the systematic deviation represented by figure 1. Because of the approximate nature of the principle of corresponding states and uncertainties in the values of p e and T (especially for neon) we can only expect to see a trend from figure 5- This is what we get: No points fall on a single line, but it is apparent that the closed circles are closer to such a line than are the open circles. k.l Evidence from Elastic Scattering From figure 3 and, to a much lesser extent, figure 5, the evidence is in favor of the Kestin and G. M. W. data. However, we have found that both the 40-6 and 13 (approx.)-6 functions fit the two sets of viscosity data. Can we decide between the two functions from an independent experiment? We think we can. The repulsive part of the intermolecular potential can be estimated from precise and powerful experiments on the elastic scattering of high energy beams of neutral atoms. Results from these experiments for argon indicate that the repulsive exponent is between 7 and 1^ [19] ■ Hence, if we assume that the k-0 and 13 exponents of the hO-6 and 13-6 functions respectively are indica- tive of the true repulsion, then the conclusion from scattering experiments is that one must favor the 13-6 potential and rule out the 40-6 potential. 5 • Conclusion We have demonstrated that the work of Kestin and G. M. W. cannot be excluded from future estimates of the viscosity coefficient of argon above room temperature. We have based our arguments on argon because a) eqs (l) and (h) are only valid for monatomic gases, b) thermal conductivity data for the other monatomic gases are not available in the required temperature range to allow us to repeat figures 3 and h, and c) the elastic scattering of the argon-argon system has been particularly well studied. A further argument in favor of the Kestin and G. M. W. results can be based on the fact that not only are the two experiments independent, but the methods are entirely different. Kestin' s measure- ments were made with an oscillating disc viscometer while the Los Alamos measurements were made with a capillary viscometer. Each of the methods is presumably subject to different systematic errors. The presence of large systematic errors of the same sign and magnitude in both experiments would seem highly improbable . Our conclusion is not restricted to argon, however. Trautz and his co-workers have had a tremen- dous influence on high temperature viscosity estimates for many gases. If he (and other authors who agree with him) is incorrect for argon, he is possibly incorrect for the other gases. We mention, in this respect, that Kestin and co-workers report a disagreement between their results and Trautz 's and other workers for several gases [9] (e.g., C0 S , N s , Ne, Hg , He, Kr, air). The Los Alamos group |~10]> as well, report results for hydrogen, helium and nitrogen which differ from the previously accepted values, but which seem to agree with Kestin 's results. Hence we believe that the dilute gas viscosity coefficients above room temperature for all simple gases (Ar, Ne, Kr, He, Hg , S , N s , etc.) should be reexamined both from the experimenter's and cor- relater's point of view. Not only are new experiments clearly required, but an analysis of the old experiments, with an eye to discover possible overlooked corrections, is necessary. 6 . Acknowledgement This work was supported by the National Aeronautics and Space Administration's Office of Advanced Research and Technology, Contract number R-06-006-046. 7- References [1] Hirschfelder, J. 0., Curtiss, C F., and [3] Hanley, H. J. M. , and Childs, G. E. , The Bird, R. B., Molecular Theory of Gases and viscosity and thermal conductivity coeffi- Liquids, 2nd edition (Wiley, 196*0. cients of dilute neon, krypton, and xenon, Nat. Bur. Std. Technical Note 352 (March [2] Klein, M. , Determination of intermolecular 1967) . potential functions from macroscopic measure- ments, J. Res. Nat. Bur. Std. 70A, 259 (1966) . [k~\ Hanley, H. J. M. , and Klein, M. , On the selection of the intermolecular potential function: application of statistical mech- anical theory to experiment, Nat. Bur. Std. /qq Technical Note 36O (November 1967) . [5] Hartley, H. J. M. , Comparison of the Lennard- Jones , exp-6, and Kihara potential functions from viscosity data of dilute argon, J. Chem. Phys. 44, 4219 (1966). [6] Trautz, M. , and Zink, R., Die reibung warme- leitung und diffusion in gasmischungen. XII. Gasreibung bei hoheren temperaturen (The viscosity, heat conduction and diffusion of gas mixtures. XII. The viscosity of gases at higher temperatures), Ann. Physik J_» 427 (l930)j Vasilesco, V., Recherches experi- mentales sur la viscosite des gaz aux tempera- tures elevees (Experimental research on the viscosity of gas at high temperatures), Ann. Phys. (Paris) 20, 137 0-945)- [7] Kestin, J., and Leidenfrost, W. , An absolute determination of the viscosity of eleven gases over a range of pressures, Physica 25_, 1033 (1959)- [8] Kestin, J., and Whitelaw, J. H. , A relative determination of the viscosity of several gases by the oscillating disk method, Physica 29, 335 (1963). [9] DiPippo, R. , An absolute determination of the viscosity of seven gases to high tempera- tures, Ph.D. Thesis, Brown Univ. (June I966) . [10] Guevara, F. A., Mclnteer, B. B. , and Wageman, W. E., private communication. These authors have made their results on hydrogen, helium, argon, and nitrogen available prior to submitting them for publication. The data may still be subject to minor modifica- tions (± 1%). [11] Mason, E. A., and Rice, W. E. , The inter- molecular potentials for some simple non- polar molecules, J. Chem. Phys. 22, 843 (1954) ["12] Michels, A., Sengers , J. V., and Van de Klundert, L. J. M. , The thermal conductivity of argon at elevated densities, Physica 29 , 149 (1963). [13] Eucken, A. , Uber die temperaturabhangigkeit der Warmeleitfahigkeit einiger gase (Con- cerning the temperature dependence of thermal conductivity of several gases) , Physik. Z. 12, 1101 (1911). [14] Keyes, F. G., and Vines, R. G., The thermal conductivity of nitrogen and argon, J. Heat Transfer 87, 177 (1965)- [15] Kannuluik, W. G. , and Carman, E. H. , The thermal conductivity of rare gases, Proc. Phys. Soc. (London) B65, 701 (1952). [16] Rothman, A. J., Thermal conductivity of gases at high temperatures, U. S. Atomic Energy Comm. UCRL 2339, (1953)- [17] Vargaftik, N. B. , and Zimina, N. Kh., Heat conductivity of argon at high temperatures, High Temperature 2, 645 (1964) , translated from Teplofiz. Vysok. Temp. 2, 716 (1964). [18] Johnston, H. L. , and McCloskey, K. E., Viscosities of several common gases between 90 °K and room temperature , J . Phys . Chem . 44, IO38 (1940); Buddenburg, J. W. , and Wilke, C. R., Viscosities of some mixed gases, J. Phys. Colloid Chem. 55, 1491 (1951); Michels, A., Schipper, A. C. J., and Rintoul, W. H., The viscosity of hydrogen and deuter- ium at pressures up to 2000 atmospheres , Physica 19, 1011 (1953)- [19] Amdur, I., and Mason, E. A., Scattering of high-velocity neutral particles. III. Argon-argon, J. Chem. Phys. 22, 670 (1954) J Amdur , I . , and Jordan , J . E . , Molecular Beams, Editor, J. Ross (interscience, 1966) . 601 o o f* 1 1 10 9 8 7 6 5 o n 4 (?■ 1 1 1 1 1 1 1 1 1 1 — • — • . . • • _ • • • — • — • • •;: ' ^° On 000 ( p n r p00Q0" ° ° o o o o c ^ ° 200 400 600 800 1000 1200 1400 TEMPERATURE, °K 1600 1800 2000 2200 Figure 1. Argon viscosity percent deviation curve. T| e »ie represents the calculated viscosity coefficients from eq.(l)using the kO-6 function with a = 3. 15 A and e/k » 224. 1°K. \ xv represents the experimental viscosity coefficients. Open circles represent data of Trautz and Vasilesco [6] and other workers [5]. Closed circles represent data of Kestin and co-workers [7-9] and Wageman and co-workers [10]. O O o o P" P" o p^- 1|W 22- «w- T ▼T -+-*- ♦♦ ♦♦ *♦♦♦♦♦* ♦ ♦ * ♦ ♦ 200 400 600 800 1000 1200 1400 TEMPERATURE, °K 1600 1800 2000 2200 Figure 2. Argon viscosity percent deviation curve. \ne represents the calculated viscosity coefficients from eq(l) using the Exp: 6 function with a = 15 , • [15], D[16],X [IT]- O o x o o s< o u Figure k. 200 400 600 800 1000 TEMPERATURE, °K 1200 1400 Argon thermal conductivity percent deviation curve X »ic an< ^ ^ Jp represent the calculated and experimental thermal conductivity coefficients with respective parameters suitable for the viscosity correlation of figure 2. Symbols as in figure 3. 603 % T/T Figure 5. Plot of the viscosity reduced by critical values, eq (6), versus T/T where T is the critical temperature. The closed circles are reduced argon viscosities from Kestin and G. M. W. , open circles are reduced from Trautz and the workers who agree with him. The squares are reduced viscosities of neon taken from reference [3]. The triangles are viscosities for hydrogen [18]. Both the neon and hydrogen data are measurements taken below ijO0°K. 604 Thermal Conductivity of Binary, Ternary and Quaternary Mixtures of Polyatomic Gases S.C. Saxena Thermophysical Properties Research Center, Purdue University, West Lafayette, Indiana and G.P. Gupta Department of Physics, University of Rajasthan, Jaipur, Rajasthan, India The thermal conductivity of argon, hydrogen, oxygen, nitrogen, argon -hydrogen, argon -nitrogen, argon-oxygen, argon-hydrogen-nitrogen and argon-hydrogen-nitrogen- oxygen systems are reported at temperatures 40°, 65° and 93° C, and for mixtures at several compositions. The measurements are taken on a thick -wire cell of the hot-wire type. The experimental results are compared with the predictions of rigorous theory as well as with the values obtained on approximate, semitheoretical, and Lindsay-Bromley methods. The competence of some simpler procedures to predict thermal conductivity of multicomponent mixtures is also investigated with conclusions of some practical importance. Keywords: Gases, kinetic theory, multicomponent mixtures , thermal conductivity, thick -hot -wire cell. 1. Introduction In the last few years we have been engaged in a program of measurement of thermal conductivities of rare gases and their binary and ternary mixtures [1-3] > and deuterium and its binary and ternary mixtures with rare gases [4,5]. The method adopted was the thick-wire variant of the hot-wire cell and measurements covered the temperature range 30° to 100° C. The details of the apparatus, theory of the method and various corrections which need to be applied are being reported in a recent paper [6]. The general incentive behind the work has been: (a) to check the rigorous kinetic theories in their appro- priateness to explain heat transfer through gases and gas mixtures, (b) to examine and develop new cor- relation procedures so that reliable estimation may become possible wherever direct data are not avail- able, and (c) to resolve in due course some of the discrepancies observed in the existing data [7-10], The present paper reports as a continuing effort ox this program thermal conductivity data on Ar, H 2 , N 2 , 2 , Ar-H 2 , Ar-N 2 , Ar-0 2 , Ar-H 2 -N 2 and Ar-H 2 -N 2 -0 2 systems at temperatures of 40°, 65°, and 93° C. These are compared with the values obtained from rigorous kinetic theory, approximate, semitheoretical, Lindsay-Bromley and simpler methods of computing thermal conductivity of mixtures. A critical comparison of the observed and various calculated values enables some interesting conclusions to be drawn regarding the appropriateness of different procedures in correlating or predicting the thermal conductivity of gas mixtures. 2 . Apparatus The principal component of the experimental arrangement is the hot-wire cell. It consists of a precisely bored stainless steel tube having an effective length of 10.44 cm and an internal diameter of 0.6412 cm. The bore is found to be uniform within one part in 600. Along the axis is run a platinum wire 0.05076 cm in diameter as measured by a profile projector having the least count of 0.002 cm. The effective length was determined by measuring the resistance. The top end is closed by silver soldering a brass cap and the bottom end by a kovar cap through a c-40 glass seal. This arrangement provides the necessary electrical insulation between the main metal body of the cell and the axial platinum wire. Current and potential leads are soldered at the two metal end caps, while a side connection enables the cell to be either evacuated or loaded with any desired gas. An efficient high 1 Figures in brackets indicate the literature references at the end of this paper. 605 vacuum pumping system evacuates the conductivity cell up to 1CT 6 cm of mercury. A suitably designed gas handling system is used to prepare precise mixtures of different known compositions and also to charge the cell and compress it to any desired value up to an atmosphere. The cell is vertically mounted in a thermostat bath which can be main- tained at any temperature from room to 100° C within an accuracy of ± 0.01. This is obtained by efficient stirring, distributed heaters energized by regulated electrical power, a contact thermometer, and a sensitive relay and proportional control system. A Tinsley vernier potentiometer of the type 4363 D, reading directly up to 0. 1 microvolt is employed for electrical measurements. 3. Theory and Corrections The theory of the thick-wire cell is well developed [11-13] and consequently only the necessary working relations are reproduced here. The differential equation for the radial heat flow through the gas is, 77r 2 K ^| - 277 ri he + I 2 P (l+a9) = , ( 1) where h = X[r^n (r/rj)]-!, ( 2) K and Xare the thermal conductivity values of the wire material and test gas respectively, r t the radius of the wire, r 2 the inner radius of the cell tube, p the resistance per unit length of the wire at the bath temperature, the temperature excess of the wire over that of the bath at a distance z from the center and a the temperature coeffi- cient of resistance of the wire. Assuming the ends of the wire to be at the bath temperature, the solution of eq (1) is, f_l.\ l~i _taahlj,-j_ 2irr?K(R fi - R„) . 3 , where 2h I 2 R a * rjK 2irifK-t ' V ; Here R0 is the resistance of the wire when a current I is flowing, R is the resistance at the bath temperature and 21 is the length of the cell wire. In practice the heat flow lines through the gas are slightly curved at the ends . The exact solution then is given by, ff i R p " R?) = C lNol - i C 3 N 03 + l C 5 N 05 - - - , (5) 2 R o a The defining relations for C ^ and N on are given in reference [11]. The correction for the nonradial flow of heat is the difference in the two conductivity values as obtained from eq (2) and eq (5) respectively. This correction is negligible for values of \ below 0.04 yW m" 1 deg~ I , about 0.7%for X ranging from 0.04 to 0.08, 1% for X between 0.08 and 0.12, and 1.2% for X values above 0.12. By careful design and installation of the conductivity cell and selection of operating conditions, the convection effect is found to be negligible in our measurements. The temperature jump effect is also likewise reduced. In a note [14] both these corrections are discussed and it is found that the measurements taken at gas pressures varying between 8 and 16 cm of mercury are not contaminated by these uncertainties. The radiation correction is evaluated experimentally by taking measurements on the highly evacuated cell for different bath temperatures. I, R and Rq are thus experimentally determined and the corresponding Xr determined from eqs (2) to (4V For example, the correction at about 100°C is 0.9 and 6% for He and Xe respectively. The wall effect is negligible for our cell and the correction amounts to only 0.02%. Again the noncentering of the wire has no serious effect on our values. Calculations reveal the correction to be 0.04, 0.98 and 4.06% only for wire displacements from the true center by 0.01, 0.05 and 0. 1 cm respectively. We thus claim our measure- ments to be absolute. 606 For the gases and gas mixtures examined here the thermal energy absorption is negligible and we have ignored this correction. 4. Results Spectroscopically pure argon supplied by M/S British Oxygen Co. has been used in these measurements. The resistance of the cell wire with zero current at different bath temperatures R , is determined by highly evacuating the cell_and measuring resistance as a function of current. R is then obtained by extrapolation of the plots of I 2 against Rg -1 to I = 0. a is obtained by measuring the resistance of a specimen of platinum wire at ice, steam and boiling sulphur temperatures and evaluating the constants A and B of the relation, R = R [1 +A(T -273.16) + B (T - 273. 16) 2 ] 6) oi is then evaluated at any temperature by 1 dR ffi "KdT ( 7) T is the temperature in ° K. The thermal conductivity of the platinum wire at 40° C is determined in the same set up by taking measure- ments on a highly evacuated cell and using eq (3) . h is now h r = 4eaT 3 , e and a are the emissivity of the platinum wire and Stefan's constant respectively. At higher temperatures K is obtained on the basis of this value and its temperature dependence from the literature [ 15] . Now all that remains to be determined for calculating the gas conductivity is the cell wire resistance Rg , when it is heated a couple of degrees above the bath temperature (4 to 6°) , by passing a current I in the presence and absence of the gas. This \ value is assigned the average temperature as the arithmetic mean of the wire and wall temperatures. The X values so obtained, after corrections for the various undesirable but unavoidable side effects are recorded in tables 1 to 4 for pure, binary, ternary and quaternary mixtures, respectively. Table 1 Comparison of presently measured y, values (yW nr'deg -1 ) with literature and theoretical values. .Gas Temp.o, Ar H, N, O, 40 65 93 0.0185 0.182 0.0268 0.0281 Present 0.0185 0.185 0.0268 0.0278 Literature 0.0 +1.6 0.0 -1.1 % dev 0.0183 0.192 0.0277 0.0280 Theoretical -1.1 +5.5 +3.4 -0.4 % dev 0.0193 0.194 0.0290 0.0291 Present 0.0196 0.198 0.0283 0.0296 Literature +1.5 +2.1 -2.4 +1.7 %dev 0.0195 0.202 0.0294 0.0298 Theoretical +1.1 +4.1 +1.4 +2.4 %dev 0.0210 0.204 0.0312 0.0313 Present 0.0209 0.209 0.0301 0.0318 Literature -0.5 +2.5 -3.5 +1.6 % dev 0.0208 0.214 0.0312 0.0317 Theoretical -1.0 +5.0 0.0 +1.2 % dev The precision of our measurements is on the average about 1% depending upon the value of thermal conduc- tivity. A detailed error analysis reveals that a gas thermal conductivity value within 2% of the true value at the worst is obtained if no error is made in determining the thermal conductivity of the wire. It also reveals that if the gas is pumped out so that the electrical heat generated can only flow along the wire (neglecting radiation) an uncer- tainty of nearly 1% at the worst will exist in K. Then the resulting uncertainty in \ could be as much as 4%. How- ever, this represents the most unfavourable case in which all errors are assumed to contribute in the same direc- tion. On the average, a considerable cancellation of such uncertainties will occur so that one can expect \ to be accurate to about 1 to 2%. This can be further reduced by multiple measurements. This estimate does imply that uncertainties arising from convection etc. , are negligible. The three diatomic gases as supplied by M/S British Oxygen Co. were used. These were stated to be 99. 9% pure and we made no effort to further purify them. 607 Table 2 Comparison of experimental and various calculated X values for binary mixtures in j/W rri^deg -1 . Entries in columns 4 to 7 are percentage deviations. Gas pair Ar Rigor. Approx. Semitheor (temp.) mole fraction Exptl. eq(10) eq(ll) Lindsay-Bromley eq(13) H 2 -Ar 0.202 0. 1245 44.1 +4.8 +0.6 +1.2 (40°C) 0.388 0.0891 +1.8 +5.4 -0.6 - 0.614 0.0557 +2.1 +7.5 +0.8 +1.3 (65°C) 0.202 0.1325 +3.5 +4.7 +0.5 +1.0 0.388 0.0955 +0.5 +4.3 -1.6 -1.1 0.614 0.0589 +2.2 +7.4 +0.8 +1.2 (93°C) 0.202 0.1395 +4.4 +5.0 +0.8 +1.4 0.388 0.1026 -0.8 +2.6 -3.3 -2.7 0.614 0.0640 -0.1 +4.9 -1.7 -1.2 N 2 -Ar 0.188 0.0255 +1.3 -2.6 -0.7 -0.8 (40°C) 0.492 0.0227 +0.2 -3.1 +0.1 - 0.767 0.0207 -1.8 -3.4 -1.0 -1.0 (65°Ci 0.188 0.0269 +1.7 -0.7 +1.1 +1.1 0.492 0.0241 +0.4 -2.7 +0.6 +0.5 0.767 0.0220 -1.9 -4.4 -1.9 -2.0 (93°C) 0.188 0.0292 -0.5 -1.2 +0.6 +0.5 0.492 0.0254 +1.1 -0.1 +3.2 +3.1 0.767 0.0234 -1.8 -2.3 +0.2 +0.2 Oj-Ar 0.249 0.0252 +1.2 -0.4 +2.0 +0.2 (40°C) 0.484 0.0228 -0.9 -0.9 +2.5 - 0.753 0.0208 -1.2 -2.6 +0.1 -1.9 (65°C) 0.249 0.0272 -0.8 -4.6 -2.3 -4.0 0.484 0.0245 -0.5 -4.5 -1.2 -3.6 0.753 0.0221 -1.7 -4.9 -2.3 -4.2 (93°C) 0.249 0.0292 -1.4 -4.1 -1.8 -3.5 0.484 0.0262 -0.1 -3.1 +0.2 -2.2 0.753 0.0234 -0.4 -2.1 +0.6 -1.4 Table 3 Comparison of experimental and various calculated X values for ternary mixtures of Ar-N 2 -H 2 in y W m -1 deg -1 . In columns 5, 6 and 7 are recorded the percentage deviations. 'emp. mole fraction Exptl. Approx °C Ar N 2 eq(H) 40 0.298 0.501 0.0417 +0.3 0.103 0.289 0.0918 +4.5 65 0.298 0.501 0.0439 +1.6 0.103 0.289 0.1021 +0.1 93 0.298 0.501 0.0471 +1.0 0.103 0.289 0.1067 +1.1 Lindsay-Bromley -2.9 -1.0 -1.7 -5.1 -2.3 -4.3 Semitheor. eq(13) -0.4 +2.8 +0.9 -1.5 +0.4 -0.5 Table 4 Comparison of experimental and various calculated \ values for quaternary mixtures of Ar-N 2 -0 2 -H 2 in y W m -1 deg -1 . Columns 6, 7 and 8 give the percentage deviations. Temp. mole fraction Exptl. Approx. Lindsay-Bromley Semitheor °C Ar O2 N 2 eq(H) eq(13) 40 0.132 0.364 0.125 0.0615 +4.7 -0.2 +1.0 0.250 0.244 0.374 0.0360 -0.8 -1.7 -1.2 65 0.132 0.364 0.125 0.0671 +1.4 -3.3 -2.2 0.250 0.244 0.374 0.0383 -1.3 -2.2 -1.6 93 0.132 0.364 0.125 0.0717 +0.7 -4.0 -2.8 0.250 0.244 0.374 0.0423 -4.3 -5.2 -4.6 608 5. Comparison with Theory In table 1 our results are compared with the smooth values recommended on the basis of available measure- ments [8,9]. The agreement may be regarded as satisfactory and it seems the estimate of disagreement between different measurements as 2% is reasonable. Considerable careful work will be required to improve this to within 1%. In table 1 we also record the theoretically computed X values. For Ar as well as for the frozen part (X°) we use the expression given by Hirschfelder, Curtiss and Bird [ 16] , 0.083266/TM , f 3 (T * a) . ( 8) CT 2 Q (2,21* (T * ( q,) Here M is the molecular weight of the gas and the reduced omega collision integral is a function of the reduced temperature T* = k T/g . k is the Boltzmann constant, and a , £ and a are the potential parameters, f X 3 is a correction factor and its value is determined by T* and Q . We have used the modified Buckingham exp-six potential for which all the necessary collision integrals are tabulated by Mason [17], The potential parameters are known for the gases of our current interest [ 18-20] and are reproduced in table 5, Table 5 Exponential -six potential parameters and Sutherland constants for pure gases. o Gas a e/k, °K r m , A S H 2 14.0 37.3 3.337 83 N 2 17.0 101.2 4.011 103 O z 17.0 132 3.726 138 Ar 14.0 123.2 3.866 147 For polyatomic gases we use the expression derived by Hirschfelder [21], X = X° [0.115 + M§4_v ^ ^ ( g) y is the ratio of the specific heat of the gas at constant pressure to that at constant volume. Two more formulations are available [22,23] but for our present work eq (9) is found adequate. In general, we find that the theoretically predicted values for pure gases, table 1, agree with our measure- ments within the limits of uncertainty in the data. Further, for polyatomic gases the theoretical estimates are not very precise and the agreement between theory and experiment is worst for hydrogen and the difference is appre- ciable in magnitude. Unforunately, the refined formulations [22,23] which consider the relaxation of internal- translational energy are also not able to improve upon the disagreement and special consideration is due for this gas. In this paper the deviations are designated as positive or negative whenever the values are greater or smaller than the corresponding measured values respectively. In table 2, our results for the three binary systems are presented and are compared with the pre- dictions of theory and other procedures for computing X. The rigorous calculations are based on the expression derived by Hirschfelder [24] as, n ° n + .£ , (* i - X i)[l+ E 1=1 \n D 11 X mix = *mix + 1 .S 1 (M-^i)[l+ 1 L 1 JJJj^] • (10) Here X m ix refers to the thermal conductivity of the mixture of n components, D to the diffusion coefficient, and Xj to the mole fraction of the i - th component in the mixture. The frozen thermal conductivity of the mixture Xjj^ is obtained from the expression derived by Muckenfuss and Curtiss [25] with the modification suggested by Mason and Saxena [26]. Di j are also computed and we use the exp-six potential with parameters reproduced in table 6 and obtained on the basis of combination rules of Mason and Rice [ 18] . X and X are also computed from eqs (9) and (8) respectively. The calculated Xmix values are not reported but instead their percentage deviations from the experimental values for convenience of evaluation. It is to be noted that this simple theory [24] does quite well in reproducing the experimental results and therefore more complicated theories developed by Monchick, Pereira and Mason [27], and Saxena, Saksena, Gambhir and Gandhi [28] are not considered. The average absolute deviation is 609 1.4% for 27 mixtures of table 2. A slightly enhanced discrepancy for H 2 - Ar mixtures may also be partly due to translational relaxation, Saksena, Saxena and Mathur [29]. This correction is significant only when the masses involved differ considerably and consequently it is likely to be appreciable only for this system. Table 6 Exponential -six potential parameters and Sutherland constants for binary gas pairs. Gas pair H 2 -Ar N 2 -Ar Q 2 -Ar a 12 14.0 15.5 15.5 e a A, u k 69.7 111.8 131.2 ( r m)i2. A 3.574 3.951 3.785 a 12 111 123 142 Mason and Saxena [30] simplified the expression obtained by Hirschfelder, eq (10), and instead suggested the following simpler but approximate relation; n n x i *mix - E Ai[l+ E q u -J ] i=l i = l J x i r (11) where 'i] 1 Ofi^i M i 1,ub5 (1 + — L\ 2/2" [ M j ' 7[i+(- r 1 ).i Mi t _ 2 (mT) 4 ^ (12) The ipj j for each system were computed at the lowest temperature and the same were used at the two higher tempera- tures. The (/?ij values are reported in table 7 and A. m ix values in column 5 of table 2. The agreement between theory and experiment is only fair, the average absolute deviation for all the 27 mixtures is 3.5%. Table 7 Values of cp^; at 40° C. Gas pair Approximate ?2 M M. TJ2 (13) and one ^mix value. Here TJj is the coefficient of viscosity of the pure component i. This method is found very sucessful in predicting thermal conductivity of binary and ternary mixtures [33-35] . Calculated values of 0\ j at 40° C are reported in table 7 and those of ^mlx in column 7 of table 2. The experimental values are reproduced within an average absolute deviation of 1.7%. Similar calculations and comparison against the directly measured values for ternary and quaternary mix- tures are reported in tables 3 ahd 4 respectively. The approximate, Lindsay-Bromley and semitheoretical proce- 610 dures predict for the two systems values which agree with the experiments within an average absolute percentage deviations of 1.4, 2.9 and 1. 1; and 2.2, 2. 8 and 2.2 respectively. Thus, on the whole the approximate and Lindsay and Bromley methods are of competing accuracy, and the former is slightly more preferable because one needs less initial information and computation is simpler. The semitheoretical method seems to most reliable and should be preferred when one Xmix value for all involved binary systems is known. These conclusions are in conformity with earlier more detailed investigations. It may be pointed out that the relation (11) is approximate for polyatomic gas mixtures, by considering a more rigorous theoryF28], Saksena and Saxena[~35] have derived its alternative form. Simpler methods, like simple mixing rules, reciprocal mixing rules, a combination of the two, and a quadratic expression, are also used for computing Xmix [37J. The values so obtained are designated as \SM, ARM, XC, and XQ respectively and are reporduced in table 8. The quadratic function does not seem to be adequate for reproducing the composition dependence, and the success of the combination method for a quick and approximate estimate is notable. The method reproduces the data for binary, ter- nary and quaternary mixtures within an average absolute deviation of 4.2, 4.8 and 2.0% respectively. Table 8 Comparison of experimental and various calculated X values according to simpler procedures for binary systems. Computed percentage deviations from experimental values are tablulated in columns 3 to 6. +3.6 -6.2 +4.0 -4.8 +5.2 -4.1 -0.8 -1.1 +0.8 -2.4 -1.1 -2.2 +0.3 -2.0 -1.6 -1.4 -1.9 +0.4 Relations for predicting diffusion and viscosity coefficients on the basis of thermal conductivity and other experimental data are known [38-41]. In table 9 we report such indirectly generated Dj j and Tf m ix values. The former are compared with the directly measured values as well as with the predictions of rigorous theory [16], The measured values of TJmix for H 2 -Ar system agree with the generated values within an average absolute devia- tion of 1.3%. Thus, we find that such approaches are quite dependable and act as valuable procedures for generat- ing data on other transport properties. Gas pair (temp.) Ar mole fraction H 2 -Ar 0.202 (40°C) 0.388 0.614 (65°C) 0.202 0.388 0.614 (9SPC) 0.202 0.388 0.614 N 2 -Ar 0.188 (40°C) 0.492 0.767 (65°C) 0.188 0.492 0.767 (93°C) 0.188 0.492 0.767 Ojj-Ar 0.249 (40°C) 0.484 0.753 (65°C) 0.249 0.484 0.753 (93°C) 0.249 0.484 0.753 X SM X RM x c +19.7 -47.5 -13.8 +33.2 -53.8 -10.3 +46.8 -49.0 - 1.1 +19.7 -48.2 -14.3 +32.0 -55.0 -12.3 +47.0 -49.8 - 1.3 +20.0 -46.8 -13.4 +29.4 -54.6 -12.6 +42.3 -49.7 - 3.4 - 0.8 - 2.8 - 1.8 + 0.1 - 3.3 - 1.6 - 1.0 - 3.3 - 2.1 + 1.1 - 1.4 - 0.1 + 0.6 - 3.5 - 1.4 - 1.9 - 4.8 - 3.3 + 0.6 - 1.8 - 0.6 + 1.5 - 0.7 + 1.2 + 0.2 - 2.5 - 1.1 + 2.2 - 1.0 + 0.6 + 2.9 - 1.4 + 0.7 + 0.4 - 2.8 - 1.2 - 2.1 - 5.1 - 3.6 - 0.9 - 4.9 - 2.9 - 2.0 - 5.0 - 3.5 - 1.6 - 4.5 - 3.0 + 0.5 - 3.3 - 1.4 + 0.9 - 2.0 - 0.6 611 Table 9 Indirectly generated coefficients of diffusion (Dn), {fi m 2 /s) and viscosity Tjmix » (y Ns/m 2 ) for binary gas systems. Gas pair (temp.) H 2 -Ar (40°C) (65°C) (93°C) N 2 -Ar (40°C) (65°C) (93°C) 02-Ar (40°C) (65°C) (93°C1 Ar Generated mole fraction D U 0.202 90.7 0.388 93.1 0.614 90.9 0.202 103.1 0.388 107.6 0.614 103.2 0.202 117.6 0.388 127.6 0.614 123.0 0.188 22.6 0.492 21.7 0.767 22.4 0.188 23.8 0.492 24.4 0.767 25.9 0.188 28.2 0.492 27.0 0.767 28.6 0.249 20.2 0.484 20.4 0.753 21.4 0.249 26.1 0.484 25.0 0.753 25.8 0.249 29.9 0.484 28.3 0.753 28.0 Mean D ij 92 Measured 105 123 22.2 24.7 27.9 20.7 25.6 28.7 i] D 93 105 123 21.2 24.8 21.3 24.4 Theoretical D ij 87.7 100 115 21.0 24.1 27.7 21.0 23.9 27.8 Generated ^mix 1729 2053 2253 1832 2183 2385 1942 2309 2505 1967 2140 2262 2066 2243 2390 2176 2346 2491 2169 2209 2270 2371 2440 2488 2490 2546 2588 [ 1] Gambhir, R.S. and Saxena, S.C., Thermal conductivity of binary and ternary mixtures of krypton, argon and helium, Mol. Phys. 11 , 233(1966). [ 2] Gandhi, J.M. and Saxena, S.C., Thermal con- ductivity of binary and ternary mixtures of helium, neon and xenon, Mol. Phys. 12_, 57 (We? 1 !. [ 3] Mathur, S., Tondon , P.K. and Saxena, S.C., Thermal conductivity of binary, ternary and quaternary mixtures of rare gases, Mol. Phys. 12, 569 (1967). [ 4] Gambhir, R.S. and Saxena, S.C., Thermal conductivity of the gas mixtures; Ar-D 2 , Kr-Dj andAr-Kr-Dj, Physica 32, 2037 (1966). [ 5] Gandhi, J.M. and Saxena, S.C., Thermal con- ductivities of the gas mixtures; D 2 -He, D 2 -Ne and E^-He-Ne, Brit. J. Appl. Phys. 18, 807 (1967) [ 6] Gambhir, R.S. , Gandhi, J.M. and Saxena, S.C. , Thermal conductivity of rare gases, deuterium and air, Indian J. Pure & Appl. Phys. , in press. [ 7] Saxena, S.C, Gandhi, J.M., Thermal con- ductivity of multicomponent mixtures of inert gases, Rev. Mod. Phys. 35, 1022 (1963). References [ 8] Gandhi, J.M. and Saxena, S.C, Thermal con- ductivity of monatomic gases and binary gas mixtures, University of Rajasthan Studies (Physics ^ 1, 7 (1965). [ 9] Gambhir, R.S. and Saxena, S.C, Thermal con- ductivity of common nonpolar polyatomic gases, Suppl. Def. Sci. J. 17, 35(1967). [10] Saxena, S.C, Mathur, S. and Gupta, G.P., The thermal conductivity data of some binary gas mix- tures involving polyatomic gases, Supp. Def. Sci. J. 16, 99 (1966). [11] Kannuluik, W.G. and Martin, L.H., The thermal conductivity of some gases at 0° C, Proc. Roy. Soc. A 144 , 496 (1934). [12] Kannuluik, W.G. and Carman, E.H., The thermal conductivity of rare gases, Proc. Phys. Soc. B65 , 701(1952). [13] Srivastava, B.N. and Saxena, S.C, Thermal con- ductivity of binary and ternary rare gas mixtures , Proc. Phys. Soc. B70 , 369(1957). [14] Gupta, G.P. and Saxena, S.C, Pressure depen- dence of thermal conductivity in a hot-wire type of cell, Can. J. Phys. 45, 1418 (1967). [15] Powell, R.W. and Tye, R.P., The promise of platinum as a high temperature thermal conduc- tivity reference material, Brit. J. Appl. Phys. 14, 662 (1963). 612 [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] Hirschf elder, J.O., Curtiss, CF. and Bird, R.B. , Molecular theory of gases and liquids (John Wiley & Sons, Inc., New York, 1954). Mason, E.A. , Transport properties of gases obeying a modified Buckingham (exp-six) poten- tial, J. Chem. Phys. 2£, 169 (1954). Mason, E.A. and Rice, W.E., The intermolec- ular potentials of helium and hydrogen, J. Chem. Phys. 22, 522 (1954). Mason, E.A. and Rice, W.E., The intermolec- cular potentials for some simple nonpolar mole- cules, J. Chem. Phys. 22, 843 (1954). Vanderslice, J.T. , Mason, E.A. and Maisch, W.G. , Interactions between oxygen and nitrogen; O-N, 0-N 2 , and0 2 -N 2 , J. Chem. Phys. 31, 738 r33 -, (1959). L J [29] Saksena, M.P., Saxena, S.C. and Mathur, S., Thermal conductivity of mixtures of monatomic gases and translational relaxation, Trans. Faraday Soc. 63, 591(1967). [30] Mason, E.A. and Saxena, S.C, Approximate formula for the thermal conductivity of gas mixtures, Phys. Fluids 1., 361(1958). [31] Lindsay, A.L. and Bromley, L.A. , Thermal conductivity of gas mixtures, Ind. Eng. Chem. 42_, 1508(1950). [32] Chapman, S. and Cowling, T.G., The mathe- matical theory of nonuniform gases (The Univer- sity Press, Cambridge, 1953). Hirschfelder, J.O. , Heat conductivity in poly- atomic or electronically excited gases, n, J. Chem. Phys. 26, 282 (1957). Mason, E.A. and Monchick, L., Heat conductiv- ity of polyatomic and polar gases, J. Chem. Phys. 36, 1622 (1962). Saxena, S.C, Saksena, M.P. and Gambhir, R. S . , The thermal conductivity of nonpolar poly- atomic gases, Brit. J. Appl. Phys. 15_, 843 (1964). Hirschfelder, J.O. , Heat conductivity in poly- atomic, electronically excited, or chemically reacting mixtures, in, Sixth International Combustion Symposium, p. 351 (Reinhold Pub- lishing Corporation, New York, 1957). Muckenfuss, C and Curtiss, C.F., Thermal conductivity of multicomponent gas mixtures , J . Chem. Phys. 29, 1273 (1958). Mason, E.A. and Saxena, S.C, Thermal con- ductivity of multicomponent gas mixtures, n, J. Chem. Phys. 31, 511 (1959). Monchick, L., Pereira, A.N.G. and Mason, E. A. , Heat conductivity of polyatomic and polar gases and gas mixtures, J. Chem. Phys. 42 , 3241(1965). Saxena, S.C, Saksena, M.P., Gambhir, R.S. and Gandhi, J. M. , The thermal conductivity of nonpolar polyatomic gas mixtures, Physica 31 , 333(1965). Mathur, S. and Saxena, S.C. , Methods of calcu- lating thermal conductivity of binary mixtures involving polyatomic gases, Appl. Sci. Res. 17, 155 (1967^. [34] Gupta, G.P. and Saxena, S.C, Calculation of thermal conductivity of polyatomic gas mixtures at high temperatures , Def. Sci. J. 16, 165(1966). [35] Mathur, S. and Saxena, S.C, Calculation of thermal conductivity of ternary mixtures of poly- atomic gases, Indian J. Pure & Appl. Phys. 5_, 114 (1967). [36] Saksena, M.P. and Saxena, S.C, Thermal con- ductivity of polyatomic gas mixtures and Wassiljewa form, Appl. Sci. Res. 17, 326(1967). [37] Gandhi, J.M. and Saxena, S.C, Thermal con- ductivity of multicomponent mixtures of inert gases; Part III - some simpler methods of com- putation, Indian J. Pure & Appl. Phys. 4, 461 (1966). [38] Gandhi, J.M. and Saxena, S.C, Correlation between thermal conductivity and diffusion of gases and gas mixtures of monatomic gases, Proc. Phys. Soc. 87, 273 (1966). [39] Mathur, S. and Saxena, S.C, Relations between thermal conductivity and diffusion coefficients of pure and mixed polyatomic gases, Proc. Phys. Soc. 89, 273 (1966). [40] Gupta, G.P. and Saxena, S.C, Prediction of thermal conductivity of pure gases and mixtures Supp. Def. Sci. J. 17, 21(1967). [41] Saxena, S.C andAgrawal, J. P., Interrelation of thermal conductivity and viscosity of binary gas mixtures, Proc. Phys. Soc. 80, 313(1962). 613 Thermal Conductivities of Gaseous Mixtures of Diethyl Ether with Inert Gases P. Gray, S. Holland and A. 0. S. Maczek School of Chemistry The University Leeds 2 U.K. Thermal conductivities of binary mixtures of diethyl ether with four inert gases, helium, argon, neon and nitrogen have been measured at sub-atmospheric pressure at 50 C and 100 C (ca. 0.5$ error). In the composition dependence curves, measured thermal conductivities were always less than molar average values. The difference was most marked for helium and least for nitrogen, and pronounced minima were displayed by (C^H^JD-Ar. At 100 C the (CpHj-KO-Ar system showed a point of inflexion. The experimental values have been applied to test the success of Hirschf elder and Eucken's approach to the thermal conductivity of binary mixtures containing polar polyatomic molecules. When empirical estimates of the thermal conductivity of mixing have to be made from properties of pure constituents the most successful form of equation is that due to Wassiljewa (1904). It is capable of accommodating varied behaviour including maxima, minima and points of inflexion. The conditions necessary for these phenomena are investigated briefly. Physically, its success arises from the simple way in which it takes account of processes in mixing of gases. A., are coefficients giving relative efficiencies in impeding trtnsport of energy. J Empirical formulae depend for their success on reliable estimates for A. .. The formulae of Lindsay and Bromley, normally successful, here fail to give good predictions and errors of about 5$ in thermal conductivity at equimolar concentrations occur. Key Words: Binary mixtures, diethyl ether and inert gases, gas phase, Hirschf elder approximation, thermal conductivity, Wassiljewa equation. 1 . Introduction The present paper is concerned with the dependence of thermal conductivity on the composition of binary mixtures of a polar polyatomic gas, diethyl ether, with the non polar gases, helium, neon, argon and nitrogen. Previous work [1,2,3] on this type of mixture has shown a non-linear variation of conductivity with composition which can neither be represented successfully by empirical equations nor be predicted adequately by present kinetic theory. This is in contrast to mixtures of polar gases only or to mix- tures of non polar gases only. In these cases conductivity-composition dependence deviates less from molar averages and empirical or theoretical estimations. The first approximation to the rigorous kinetic theory of gases [4-9] accounts successfully for the thermal conductivities of mixtures of monatomic gases. For mixtures of diatomic or of polyatomic gases Hirschf elder [10] has assumed that the total thermal conductivity of each consituent is separable into translational and internal (rotational and vib- rational) contributions. K = K. + K, . (1) trans int v ' Figures in brackets indicate the literature references at the end of this paper. 615 The translational contribution to the thermal conductivity of the mixture is calculated as if the indiv- idual gases were all monatomic. Each pure gas is regarded as having a translational thermal conduct- ivity which is related to the viscosity K 1 trans ^Vvl = 3>?5 1^ (2) The first approximation to rigorous theory for a binary mixture then gives K,. = (K, . *2 + a + b K„ . X 1 , „ X 2 + c + b *Jk ,,» trans 1 trans — 2 trans — )/(— — ) (3) X 1 x 2 1 2 where the values of the constants a, b, c are well established [4]. The internal contribution to the thermal conductivity of the mixture is calculated from the equation (4) This theory gives satisfactory agreement in a number of cases [ 10-13] and considerable discrepan- cies in others [1,10-14]« Subsequent attempts have been made to take into account the interchange at collision between translational and internal energy [14]- For example, allowance has been made for inelastic collisions. The corrections are small and may be in the wrong sense. For strongly polar gases, theory and experiment have been compared only in a few cases [14-16]« Of empirical equations the most successful is due to Wassiljewa [17] K = f Ki /( If 'Jj A,, x./x.), ( 5 ) 1 int " 1 + D 11 X 2 D 12 X 1 + K 2 int int 1 + D 22 X 1 D 12 X 2 where K and the mole fra K. are the thermal conductivities of the mixture and pure component i respectively; x. is action of component i; A. . is one of a set of adjustable parameters. The A. . have a simple physical significance [18-20]. They are a measure of the ratio of the eff- iciencies with which molecules j and molecules i impede the transport of heat by molecules i. Equat- ion (5) may be written in an alternative form for a binary mixture as K = 1 + 1 (6) R 1 + a 12 x 2 / ' X 1 *2 + a 21 x / x 2 ' where R- = l/K, at = A _/K 1 and a. = A ?1 /K ? . Thus a. is a single resistance expressing the extent to which molecules 2 impede the transport by molecules 1 . R. expresses the extent to which molecules 1 impede transport by their own kind. Thus if molecules 2 are more efficient than molecules 1 in impeding transport of heat by 1 then <* will be greater than R. . This is, however, an oversimplification of the effect of molecular interaction on transport phenom- ena. It overestimates impedance of transport of one molecule by another and neglects smaller effects such as transfer of transport. For a binary mixture it has been shown [19, 21 ] thtt the first approx- imation to rigorous theory reduces to eq. (5) with A. . slightly dependent on composition. When the molecular weights of the components of the gas mixture are very different, a formula of the Wassiljewa type is theoretically possible [21 ] but errors are greater. The Wassiljewa equation is also flexible algebraically. This is not the case for a linear or quadratic expression where the scope is more limited. Thus eq.(5) can represent not only (i) a minimum or (ii) a maximum but also (iii) a point of inflexion and (iv) the intersection of the molar average line by the conductivity-composition curve. Estimates of A. . for polar gases may be made from the empirical equation of Lindsay and Bromley [22] in terms of viscosity "h , Sutherland's constant S, and molecular weight of pure components: 616 . i j, + n pp (^A)!) 2 2LL_ s ii (7) ij 4 {_' -7. ^M ± J 4 + S. J J T + Sj » w; 1 I where S , = (S S.) 2 if i and j are both non polar, and S. . = 0.733 (S.S.) 2 if one of i or j is polar. Francis [23] suggested an expression A ij y 2 (-A-)te-)-. where S is the collision diameter for unlike molecules and Si , . ' ' is a collision integral tabu- lated [41 as a function of the reduced temperature. The aims of the paper are a) to provide reliable experimental data for binary polar-nonpolar gas mixtures b) to use these data to estimate resistances to heat transfer c) to test Hirschf elder' 3 approx- imation and d) to test empirical equations of the Wassiljewa form. 2. Nomenclature The components of the mixture are identified by the suffixes: 1 = (C 2 H 5 ) 2 0; 2 = Ar; 3 = He; 4 = Ne; 5 = N ? Thus K is the thermal conductivity of pure diethyl ether, x_ is the mole fraction of argon. Bin- ary diffusion and Wassiljewa coefficients are identified by a pair of suffixes; A, . is the coefficient of the quotient x./x. and A., is that of its reciprocal. J 3. Experimental Method 3-1. Apparatus The thermal conductivity cell employed [25] was of the two wire type; a heating wire through which a known current is passed to supply heat to the cell, and a measuring wire which acts as a resistance thermometer indicating the rise in temperature of the gas at a point in the cell as a change in resist- ance. Both wires were thin tungsten spirals (30 ohm at 20 C) stretched longitudinally down the cell some 2-3 mm apart. Mercury cells of high stability were used to supply current both to the heating cir- cuit and the Wheatstone bridge. The thermostat temperature was kept constant to about 0.02 C. 3-2. Procedure The difference, A, between the resistances of the wire with two different standard heating currents inversely proportional to the thermal conductivity of the gas in the cell. & is reproducible^to better than 0.0008 ohm for the pure gases used in this work.- This represents, at best, a precision of -<).2£ for diethyl ether at 50°C, and, at worst, a precision of ±1% for helium at 100°C. 3. 3- Calibration A was determined at yft and 100°C for various gases (He, Ne, Ar, Kr, Xe) whose thermal conductivit- ies are known. The greatest weight was given to the data of Kannuluik and Carman [26] and the 'best* value from reference [27]. The product K& was not constant, but fell smoothly by some 20,S between helium and xenon. At each temperature a plot of KA against A was constructed. The absolute accuracy of our measurements depends on the accuracy of the values used for the calib- rating gases. In the region of interest in this work, the accuracy of the data of Kannuluik and Carmen [26] is ca. U. 617 4. Results A was measured at 50 C and 100 C for the four binary mixtures of diethyl ether with helium, neon, argon and nitrogen and the thermal conductivities deduced. These are shown in figs. 1-4. Also shown in these figures are the predictions of Hirschf elder and of the Lindsay-Brc-.ley [22] equation. Tables 1 and 2 give a summary of the parameters used in deducing these predictions and table 3 shows their average discrepancies. Table 1. Pure component properties. Gas T (°C) ( cal km s degK (g km D ii (C) 2 ( 1 atm c :. s" 1 ) K (d) trans (cal.lcn K. + (e) int v ' f 1 s" 1 deg K _1 ) 8 (f) 6 ii 8 M (gm) (C 2 H 5 ) 2 50 100 4.15 5-31" 8.18 9-31 O.O383 0.0501 -0.8221 -0.9356 3.328 ] 4.375 J 0.08 5-539 74.123 Ar 50 100 5-04 2 23.93 27.07 0.2049 0.2682 -4.491 -5.042 ! } 3.465 39.948 He 50 100 38.0 41.6° 20.56 23.23 1.7959 2.2755 -38.OO -41.65 °o] 2.576 4.003 He 50 100 12.3 6 33-35 36.90 0.5750 0.7331 -12.36 -13.57 \ J 2.858 20.183 N 2 50 100 6.58. 7.385 18.80 21.01 O.2307 O.2982 -4.999 -5.587 1.5857 1.796 ] 3-749 28.014 (a) Experimental values. (b) [4] and [42]. (d) From eq. 2. reference [4] Experimental data [41 (e) From eq. 1 , 27]. (c) Calculated from data in reference (f) From reference [4] and [33]- (g) From Table 2. Mixture properties. system (i) + (j) T (°C) D..=D.. (a) 1J i x -i 1 atm.cm s ) » A ij » (c) ij (c) (c) a -1 (cal. km -1 -1 s deg K (d) L . . ij in' (d) ji int Mass ratio (C H ) + Ar 2 (if (2) (C-Hj.O + He 2 (?) 2 (3) (C_H,.)_0 + Ne 2 (?) 2 (4) (C 2 H^) 2 ■ « (5) 50 100 50 100 50 100 50 100 0.0897 0.1176 O.4077 O.5208 0.1667 0.2144 0.1022 0.1332 1.096 1.094 1.102 1.103 1.096 1.097 1.094 1.094 1.137 1.126 8.751 9.906 1.091 134.62 1.090 1.099 1.093 1.123 1.112 19-311 27.579 29.902 9.208 10.257 3.744 3-734 143.609 18.610 7.902 7.681 3.564 3.518 3-979 3-973 9.066 8.923 5.967 5-902 3.918 3.895 0.395 2.284 0.087 4.405 0.212 3-449 0.346 2.257 0.394 2.281 0.089 4.369 0.216 3-419 0.348 2.239 1.85 18.52 3-67 2.65 (a) Calculated from reference [4]. (b) Date from reference [4] and L33J. (c) Parameters of calculated according to reference [7]. (d) A ij int = D ii int / D i: int' Deviations from the molar average are in all cases negative ranging from 1£ for (CpH,-)~0-Ar at 50 C (C^HpJ^O-N^ shows less deviation from the linear mean than (C 2 H,-) n 0-Ne jrkedlv un symmetrical. to 43X for (C_H K j o 0-He at 100°C. 2T5 but both curves appear fairly symmetrical. "(C 2 H 5 )„0-Ar and (C 2 H ) 0-He are n veen 50* and 60,6 diethyl ether at (C^Hp^pO-Ar shows a broad shallow minimum between 50$ and 60,6 diethyl ether at 50"C and a sharper minimum at 15* diethyl ether at 100 C. This curve also shows the unusual phenomenon of an inflexion point occuring between Ifi% and 60^ diethyl ether at 100°C. 618 Tsble 3- Average discrepancies from observed thermal conductivity System (i) + (j) (CHj-O + Ap n (2) (CX),0 + He h$ 2 (3) (CUO o + Ne Tip (4) (C H ) + N ?ip 2 (!) ij Wassiljewa + experimental A.. (mean *) +0.05 +0.01 -0.73 +0.41 -0.01 -0.08 +0.12 +0.10 :;;rrr: r-r^rr r^r-^r- — =: :=_r-r=-r^r: ==:; Hirschf elder approximation (mean &) Wassiljewa Lindsey-Bromle (mean *) +3.01 +2.31 +4.82 +3.22 +4.32 +2.98 -2,63 -4.81 +3.15 +2.37 +4.16 +2.36 +2.28 +2.09 +3.80 +2.63 . „ — 1 A. . Wassiljewa + Francis A. . (meen %) 1J +3.62 +3.11 +3.69 +2.76 +4.67 +4.34 +3-H +3-93 5. Discussion There are no previous experimental date on the binary mixtures discussed here. For pure components previous experimental data on thermal conductivity of diethyl ether [28, 29] and nitrogen [30-34] agree well. Binary mixtures of diethyl ether ( fK = 1.15 Debye) with other weakly polar molecules, chloroform and diethylamine [2], show an almost linear relationship between theimal conductivity end composition. This is also found to be the case with binary mixtures of diethyl ether and the more strongly polar ammonia (/< = 1.47 Debye) [29]. Mixtures, of nonpolar gases (argon and neon) with ammonia give non-lin- ear curves in which the thermal conductivity falls below the molar average [39 ]• Methanol (/a = 1.7 Debye) and argon mixtures show a maximum in the conductivity-composition curve; in this case the thermal conductivity lies well above the molar average [2]. This is also true in binary mixtures of nitrogen [353 or of air [3] with ammonia. Minima in conductivity-composition curves have also been found for Hp- He, N ? - Ar, Op - Ar [36»37]> end are accounted for by relaxation of rotational energy. In each of these cases the thermal conduct- ivities of the pure components lie close to each other. Consequently, a minimum might be expected in the (CpH )pO-Ar system. In the N ? - Ar and Op- Ar systems [37] the minima occur at a small percentage of the diatomic molecule end their depth increases with increasing temperature. In (CpH^pO-Ar the trend is not so consistent} the position and depth of the minimum is very sensitive to the temperature. There is an overall similarity between the four conductivity-composition ci rves studied here. They show a certain amount of resemblance to the non polar binary mixture benzene-argon [2], Mixtures of n-heptene with inert gases might provide a better comparison since n-heptane end diethyl ether have similar molecular weights, viscosities end molecular structure. 5.1. Hirschfelder Approximation The paremeters for the calculation of thermal conductivity from eqs. (3) and(4) are shown in tables 1 and 2. In determining values of A. . and B. . tabulations of Hirschfelder [4j were used for the Lennard-Jones potential (6=0). In the case of dietliyl ether 6 = 0.08 and A., and B . , tabulations of the Stockmeyer potential were used. J Hirschfelder' s calculations are based on the assumptions that the^e is no interchange between trans- lational and internal energy transport and that values adopted for parameters expressing unlike inter- actions have to be based on combining rules. The combination rules are s = ( e +$*..)/ 2 ij li ij" £,. = (£.. £..) 2 ij 11 \y (10) ■■■ V "***„>*/ V M B„ = (6.. 5„)S [(d\« *„)*/ tf*,] 3 619 From teble 3 it may be seen that Eirschf elder's approximation overestimates thermal conductivity for each binary mixture. This is most for (C„H,-)„0-He and least for (C ? H^)^0-Np with the gr^etest deviations occuring at 50°C and in compositions of between lfij> and 60% diethyl ether. 5.2. Wassiljewa coefficients 5.2.1. Introduction "Best" values of A, . have been estimated from the experimental results for the four binary mixtures. If A., and A., are both considered freely disposable there exists a large range of pairs of values which give 1 ^ equally good fit with experimental thermal conductivity. If an independent general relation between the two were available the prediction of the A., would be easier. Such a relationship [19] is A. . _1J. = A.. (12) There is good reason to believe that v> = 4[19>24] and our "best" A. . are calculated on this assumption. Values of A. . for experiment, Lindsay-Bromley and Francis ere shown in table 4* From tables 3 anc ^ Table 4. Values of A ij Experimental A. . Lindsay-Bromley A. . Francis A. . 50°C 100°C 50°C 100°C 50°c 100°C 4i 0-504 0.491 2.340 2.270 0.439 2.039 0.446 2.064 0.355 2.647 O.36O 2.599 A 3i 0.254 0.242 5.810 5-400 0.265 6.061 0.274 6.091 0.030 6.189 0.032 6.151 \i O.352 O.348 3.8lO 3.660 0.336 2.978 O.326 3.426 0.150 4.250 0.155 4.185 A* A 51 0.474 O.468 2.260 2.190 0.427 2.035 0.437 2.045 0.275 2.733 0.28l 2.611 4 it may be seen that "best" A. . fit observed results well and show only small dependence on temperature. They are also consistent with minima and an inflexion for (CpH^pO-Ar system (Appendix). 5. Empirical estimates The experimental date permits a test of Lindsay-Bromley A. . (table 3) • In all cases except (GJEL-J-O-He.the thermal conductivities are overestimated by an ev verage of 34 which is about \% worse For (C_Hj-)pO-He the thermal conductivity is underestimated by about 3$. The greatest deviations occur for mixtures between 40$ and 60$ diethyl ether. For example, the maximum deviation for (C-RV^O-Ar at 50°C is 46.71 at 60$ diethyl ether and for (C,H )„0-He it is -7.3$ 2 5' 2 than Hirschf elder's approximation. at 60$ diethyl ether. -2-5/2*. '2" 5 '2^ Thus in all cases except (C_H^),,0-He the Lindsay-Bromley equation underestimates A... 0.733 [38] in the equation ° 1J The factor 'ij 0.733 (Vj )? (13) has no rigorous foundation. It was introduced by Lindsay and Bromley in an attempt to reduce the high values of A^, which their formula gave for mixtures of polar and nonpolar gases. In the case of diethyl 620 ether with neon or argon or nitroge n much improved values of A. . are obtained if diethyl ether is consid- ered to be non polar, (S. . =1.0 xys.S.)» Such an assumption, however, provides even less adequate values of A. . for mixtures of diethyl ether with helium. We conclude that the Lindsay-Bromley equation is very unreliable where there is a very great difference in the masses of the constituents of a binary mixture, and that the combining rule for S. . depends too critically on the decision whether a given gas is polar or not. The A. . predicted by the Francis equation uniformly overestimate thermal conductivities of our mixtures by between 2,£ and ly% (tables 3 an ^ 4)- It is useful, however, even when the masses of con- stituents differ greatly. 5.2.3. "Best" Experimental values As already stated values of A. . may be given a simple interpretation. They are a quotient of efficiencies of impeding transport of heat. In all cases studied here, one of the Wassiljewa coeffic- ients is much greater than one, and the other is much smaller than one. Diethyl ether is more effective in impeding the transport of heat by the inert gas than the inert gas is in impeding the transport of heat by diethyl ether. As most of tne heat is transferred by the inert gas this causes the thermal conductivity of diethyl ether-inert gas mixtures to fall below molar averages. When considering impedance of heat transport, mass ratios and relative sizes must be considered (tables 1 and 2). These two factors cannot be distinguished in the case studied here since they effect- ively follow similar trends. Thus helium which is small and light is most effectively impeded by the large, heavy diethyl ether. Consequently the thermal conductivity of the mixture can fall by as much as 40/o below molar average. The effect of replacing helium by neon, nitrogen and argon reflects the increased masses and collision diameters of these molecules; negative deviations decrease in proportion to the mass ratios. However, even for mixtures of diethyl ether with argon in which the mass ratio (1.85) is not very large, the combined effect of relative mass and size results in diethyl ether being about five times more effective in impeding the transport of heat by the other molecule than is argon. 6. Appendix As already stated, the Wassiljewa equation is capable of accommodating turning points and inflexions. The conditions for these are derived below. On introducing z = x,/x. and assuming i refers to the component with lower thermal conductivity eq. (5) becomes K. K. K s— * + •> . (16) 1 + A. .z 1 + 1/A, .z For a turning point dK/dz = or dK/dx = so that dK A., 2 A..K. (1-A.,) 2 - A..K. (A. ,z 4 A. .A . . ) 2 dz = (17) (1 - A.jZ) ( Alj . 4 A ljAji ) therefore, , „ _ Jk, .A..K./K. - 1 V " iJ J 1 1 3 - . (18) 1 V1/A..A...K./K. A. ., A.., K., K. and z are all positive. If the right hand side of eq.(l8) is negative there can be no turning point, since this implies a negative (physically impossible) value of z. There must there- fore be a monotonic variation of K with z and V K i> Vji> K i /K j • For a turning point the right hand side of eq. (18) must be positive so Vji < V K j or A ij A ji > K A 2 In collaboration with Dr. J » h . Sutton, N.E.L., East Kilbride, Glasgow, O.K. 621 „ To, obtain the conditions for a maximum or minimum dK/dz - is substituted into the value of d^K/dz'n where 2A. . A 2 V K, ( A, .A..-1 d K i J iXJ- 1 dz 2 (/..z+1) 2 I (A. .Z+1)(A. .z+A. .A,.) (19) The sign of the equation is determined by the sign of ({..I-.. - 1) . If A. .A.. > 1, =» idnimum is obtained and i i ii If A.jA. . • 1 , J-' ^xiraum is obtained. For each binary mixture experimental A., values, K./K. and K./K. are shown in table 5> Table 5 System (i) 4 (J) A. .A., ij Ji 50°G A. .A.. 1J Ji K./K. 1 J 00°C K./K. 1 J K./K. J 1 0A. .A... ij Ji K./K. J 1 0A. .A.. (C 2 H 5 ) 2 0-/.r 1.180 0.924 1.082 1.020 1.204 1.115 0.947 1-053 0.993 1.107 (C 2 H 5 ) 2 0-He 1.476 0.109 9.157 1.902 2.807 1.408 0.127 7.842 1.832 2.579 (C 2 H 5 ) 2 0-Ne 1.34? O.336 2.978 1.363 1.829 1.274 0.391 2.555 1.306 I.664 (C 2 H 5 ) 2 0-N 2 1.071 O.630 1.586 I.160 1.242 1.024 0.719 1.390 1.115 1.142 it is clear that the Wassiljewe expression for (CpfO^O-Ke, (C„H_) ; - ) 0-Ne and 3 e monotonic variation of thermal conductivity with^x^ The mixtures of (C, both^at 50^"C ^nd 100 C satisfy the conditions for a minimum value in the thermal conductivity-composition From the table (C„H,-)„0-N,, predicts e , 2 R J 2 0-Ar curve. These are calculated at 57 % and 14? diethyl ether at 50 C and 100 C respectively which are in agreement with experiment (fig.l). In order to derive the condition for a turning point, eq.(l8) is expressed in terms of resistances as explained earlier. Thus if R. . = 1/K. and a., - A../K. etc. eq. (18) becomes e 11 1 ij 13 1 a. .z n..a..a (1-R. .//a7,a.. ) ii ij j i li /tf xj 3 1 ' R. .R.. fr57.~. / R. . - 1) 11 JJ ij Ji JJ (20) Thus a minimum occurs when jl. .0. . . > R. R . or when the product of resistances for unlike interactions ii ii ii ii is greater than that for like interactions. 9 P 9 P A point of inflexion occurs when d^K/dz or d K/dx =0. An inflexion in the K - z plot does not necessarily coincide with the inflexion in the K - x plot, thus & = & [2(1+.) dK + (Hz) 2 df| ], dx^ dx dz dz (21) where d 2 K/dz 2 is given by eq.(l9) and dK/dz by eq.(l7). When d 2 K/dx 2 = 0. (0 - 1) (22) 622 where 3 = _J_ . . h m J^_:Ji_ . (23) A.. A.. K. (A., -A. .A..) The conditions become > 1 > 0A i[l + (2/5) S (C/F)] =Xf' H (2) where C is the rotational specific heat and /vis the thermal conductivity excluding rotational energy transport; f \\ is called the "modified" Eucken correction and differs from the "Eucken factor", f, generally encountered in the literature [k] . Hi rschf el der ' s [3} expression for <§ is C> H = (V5b^L (2 ' 2) y_^ (1 ' n * = (2/3) P D/^l (3) (1 1 )* (2 2)* where D is the coefficient of self-diffusion, p is the gas density and_C2- ' and-^2- ' are collision integrals \S] • Eucken's original equation is a special case of eq (2) obtained by letting p D/J"| = 1, in which case o |_| takes the value c? H = 2/3 = S E (k) _ Scientist Figures in brackets indicate the literature references at the end of this paper. 627 Thermal conductivity measurement provides a value of the Eucken factor, f, from which the value of 8 may be estimated. However, the accuracy of this estimate is severely affected by the fact that the term involving S in eq (2) is always a relatively small contribution and is especially small at low temperatures, where the rotational motion is restricted by quantum effects. For hydrogen below 100°K the term involving S contributes less than 10% so that a 1% error in measuring thermal con- ductivity becomes a 10% error in determining S • Low temperature thermal conductivity data for hy- drogen [6] shows a spread of about 5%; and the corresponding values of & differ by a factor of two. Consequently, although the thermal conductivity of hydrogen is used to detect the ortho-to-para con- version [7] it has not been possible for students of that phenomenon to correlate their results with theoretical or experimental results such as those cited above. Such correlation is also hampered by the fact that most of the investigators cited above have not considered orthohydrogen and parahydrogen as separate constituents. In this paper a method is described with which o may be accurately deter- mined at 77°K from the relative change in thermal conductivity associated with the ortho-to-para con- version. Only an outline of the method, and the results are given here. Details of the apparatus and procedure and further discussion of the results will be published elsewhere. 2. Method For a binary gas mixture, Hi rschf el der ' s [8Q expression for thermal conductivity is ^',2 -^12 + ( ^'l "^1>/[ 1 + (V*1>< D 11 /D 12>] + (>-' 2 - ^ 2 )/[> + (x 1 /x 2 )(D 22 /D 12 )] (5) where the subscripts 1, 2 and 12 denote gas #1 , gas #2 and the mixture, respectively, and x refers to mole fraction. )^ includes the effects of rotational energy and X does not. For a mixture of ortho- and parahydrogen, since the difference between these two species is purely rotational, it is approx- imately true (denoting ortho and para by subscripts o and p respectively) that y = y = y. sk (6) op ' o ' p ' v ' If it is true that D Cs'd a* D (7) 00 pp op y where the subscripts o and p refer to ortho- and parahydrogen respectively, then substituting (6), (7) and (2) into (5) and using the fact that x + x p = 1, one obtains: A'o P = >£ + ( 2/ 5R)(c r ) £ + x p b^ (8) where b- (2/5R) [(C r ) p £ p - (C r ) £ ] (9) The quantity b may therefore be determined experimentally by measuring the rel at i ve rate of change (l/>. )(b V op /3*p)- Once b has been determined experimentally, eq (9) provides a relationship between and O . If sufficient additional information regarding either of these is available from other experimental data or from theory, then one may determine their values individually. At 100°K or lower, orthohydrogen be- haves almost like a monatomic gas [i.e., (C r ) o ^'0.01 cal/mole}. Therefore, at such temperatures (9) yields S p = [ R /( c r ) p ] (5/2)b (T< 100°K) (10) Equation (10) is used in determining the results to be reported here. 628 To measure thermal conductivity, a cy 1 i ndri cat 1 y symmetrical hot-wire cell was used, for which it is well-known that, neglecting radiation and end-effects, the rate of heat transfer Q out through the cell is [9] '■' K l ' X 'op (T) dT K = 2*L/in(r /r) (11) where L is the length of the cell and T r and T are the temperatures at the wire surface (radius r) and at the outer wall (radius r Q ), respectively. From eq (11) it can be shown that OV-c^Jv-^ [?"Wo>]j (12) To avoid measurement errors that occur at small values of (T r - T ) it is best to plot Q /(T r - T ) vs. (T r - T ) and extrapolate linearly to (T r - T ) = 0. At 100°K or lower temperatures it is possible to determine Afrom the experimental data. Since at these temperatures the rotational motion of orthohydrogen is "frozen", the thermal conductivity of pure orthohydrogen (A 'qq) ' s equal to ^ . Since ^ „ is a linear function of x p [see eq (8)J , /^'o. 0^/0.25 - A a' - Therefore, ^>-'o.25 " A / V (13) Figure 1 shows a pi ot ofl 3. Results for one set of experimental data. The lower points and correspon- ding straight line are for normal hydrogen; the upper points and straight line are for 50.3% para- hydrogen. Table 1 shows experimental values of X'o.5 ar) d A 0.25' [calculated by extrapolating the straight lines to T r - T = according to eq (12)]; and /> from eq (13). Table 1. Values of thermal conductivity and 6 obtained from ten experiments at 77°K. n A0.5 mw./cm. deg. K. A' 0.25 A 77> ' theor • 5920 .5870 • 5878 .5861 • 5786 • 5803 • 5738 • 5761 .5789 .5787 • 5796 • 5750 • 575** • 5730 .5661 • 5678 • 5617 • 5631 .5664 .5669 • 567 ■ 563 .563 .560 ■ 553 ■ 555 550 • 550 553 555 .506 .506 0863 • 731 0843 ■ 714 0869 • 736 0922 .782 0897 .760 0886 ■ 751 O869 • 737 0931 .789 0895 • 758 0841 • 713 Mean values and 99-9% confidence limits • 747 + 04 The fourth column shows a theoretical value of X calculated from kinetic theory [j o] . The correspon- 629 ding values of b and 6 p are given in columns five and six. The value of C r used in these calculations was obtained by calculating from Farkas's equations [7] the rotational energy of parahydrogen at fifty equally spaced temperatures between 50°K and 100°K. These points were least-square fitted by a fourth degree polynomial which was then differentiated at 77'2°K to obtain C r = 0.5862 cal/mole. The first line in the table is calculated from the data of Fig. 1. Lines two through ten of Table 1 show the results of nine more runs. All data was taken during a three-month interval in which the equipment was shut down twice, once to replace the platinum wire and once to modify the Baratron connections. The last line in Table 1 shows the mean values and 99-9% conf i dence limits [l lj for the value of o p . k . Di scussion The mean value of O differs significantly from the value O ^r^uO.%% calculated by Hirschfelder for T* = 2 (which corresponds to liquid nitrogen temperature for H2, since T* = T/ (6 /k) and £ /k ~>37° for H2, where €: is the well-depth parameter of the i ntermol ecul ar potential and k is Bolzmann's constant). Some values of O for hydrogen can be calculated using absolute measurements of viscosity and thermal conductivity. From the data of Stolyarov and of Hi 1 senrath [j 2] , Liley [6] gives values of 2.310 and 2.175 respectively for the Eucken factor, f, at 77°K. The corresponding jV] values of f ' are .9^2 and . 9 1 from which eq (8) yields negative values of So- At '0°° K the data of Johnston £■ Grilly \\ 3] gives f' = 1.03 from which £ p = O-^ f° r a normal mixture of three parts ortho- and one part parahydrogen. The variation among these values of o p illustrates the previously mentioned difficulty in using absolute thermal conductivity measurements, which must be extremely accurate to give reliable values of O , especially at low temperatures. At higher temperatures, Srivastava and Srivastava [)!+] found 8 = O.76 at 3 1 I ° K and they calculated that 8 = 0.7^ according to data of Johnston £■ Grilly [13] at 300°K. At the same temperature Hi rschfel der ' s [}] calculations give 8^0.89. Interestingly, it is possible to use the thermal conductivity of monatomi c-di atomi c gas mixtures in a manner entirely analogous to that which was described above for mixtures of ortho- and para- hydrogen. Calling the monatomic gas #1 and the diatomic gas #2, using eqs (2) and (6), differentiating with respect to X2 and evaluating the derivative at X2 = (and noting that X| + X2 = 1), one obtains ^2 = [y^r^] (5/2)b ' ( ' 6) where b' ■ (D 22 /0, 2 , ( L ^ I iU , „ (17) Equations (16) and (17) have the considerable advantage that they are not limited to any particular temperature range or any particular diatomic gas. To use them, however, one must know (d ^ . /dx„), which may be found either theoretically (j 5J or from thermal conductivity measurements at temperatures low enough that (C r )2~0- The value of £ obtained from (16) and (17) depends not on the absolute value of thermal conductivity, but only on its relative variation with mixture ratio. Srivastava and Srivastava Ql4] measured the thermal conductivity of mixtures of hydrogen and argon at 311°K. From their data, eqs (16) and (17) yield Ql6] ij = O.76. The agreement between this value of and that determined from the absolute thermal conductivity measurements is excellent. Hirschfelder [3] pointed out that O can be calculated from the Prandtl number, and using Keyes's [17] data for hydrogen, he found that S = 0.64 at 173°K and cS = 0.77 at 273°K. 5. Conclusions A new method has been devised with which Hi rschfelder 's [3] parameter O for rotational heat transfer can be determined from the rel at i ve change of thermal conductivity of gas mixtures. This method was used to evaluate o for parahydrogen at 77°K using mixtures of ortho- and parahydrogen; the value so obtained is O p (77°K) = 0.75- This result is not sensitive to systematic errors in the measurement of absolute thermal conductivity and is probably more reliable than any of the values obtained previously from absolute measurements of thermal conductivity. When applied to the thermal conductivity p4] of a mixture of hydrogen and argon at room temperature, the new method yields O = O.76 for normal hydrogen. This value is in excellent agreement with the values 0.7*+ and O.76 obtained from absolute thermal conductivity measurements ['3,1'*] and the value 0.77 from measurement of Prandtl number |J3,17]. 6- Acknowledgment An experimental study of the heterogeneous catalysis of the ortho-to-parahydrogen conversion was 630 suggested to me by W. Roth of Xerox Corporation. 8. References [)] Hi rschfelder, J.O., C.F. Curtis and R.B. Bird; Molecular Theory of Gases and Liquids (John Wiley & Sons, N.Y., 196*0 p. 53*+. [2] Eucken, A., Phys. I. , \k, 329 (1913). [3] HI rschfelder, J.O., J. Chem. Phys. 26, 282 (1957). See Ref. 1, p. 527 for the second equality in equation (3). [*+] Ref. 1, pp. *+98ff., 53*+, 57*+. Many authors use the well-known "Eucken factor " (designated f), which is defined by the relation A 1 = f C v ^ /M. I shall follow the practice of Hi rschfelder and use the "Eucken cor rect ion " defined by the first and last members of eq (2), denoting it by f' instead of f. The relation between the two is f'/f = (*+/15)C v /R. For values of C v , see ref. 7, pp. 19, 26. [5] Ref. 1, p. *+8*+. [6] The literature on this subject is surveyed by P.E. Li ley in Thermodynamic Transport Prop- erties: Gases, Liquids, Solids papers (Symposium, Lafayette, Indiana, 1959). Other data is compiled by S.C. Saxena and J. P. Agrawal Qj. Chem. Phys. 35, 2107 (1961)]. [7] Farkas, A., Orthohydrogen, Parahydrogen and Heavy Hydrogen (Cambridge University Press, 1935). [8] Hi rschfelder, J.O., in Sixth Symposium (international) on Combustion , (Rei nhold Publ ishing Corp., N.Y., 1957), p351. [9] Zemansky, M.W., Heat and Thermodynamics , 3rd ed., McGraw Hill, 1951, p. 83. Do] DO Ref. 1, p. 53*+. Confidence limits are obtained using the "Student-t" distribution. See A.M. Mood, Introduction to the Theory of Statistics , 1st ed., McGraw-Hill, 1950, p. 225- [12] Hilsenrath, et. al., NBS circular 56*+ (1955); Stolyarov, E.A., Zhur. Fiz. khim., 2*+, 279 (1950), both quoted by Li ley in ref. 10. [13] Johnston, H.L. and E.R. Grilly, J. Chem. Phys. }k, 233 (19^6). [l*+] Srivastava, B.N. and R.C. Srivastava, J. Chem. Phys. 30, 1200 (1959). [15] Ref. 1, Chapter 8. [16] The cited paper does not quote a value of D22- I used the value 1.285 cm2sec - ' at 273°K [P. Hartek and H.W. Schmidt, Z. Physik. Chem., 2JJB_, *t*+7 (1933)]- To correct it to 311°K, I used the formula D = const. x/t3/^.(I,0"(p'-) [see Ref. 1, p. 527] which gives D22 = 1-61 cm^sec - ' at 311°K. C r was taken equal to R, which is very nearly true for T ^ 300°K. [17] Keyes, F.G., SQUID Technical Report 37, MIT (April 1, 1952); Trans. ASME, 76, 809 (195*+) and 77, 1395 (1955). 631 5.7 5 r 5.50- «A 0) 0) k. o> o> E 5.25- 5.00- 4.75 (T r -T ) (degrees K) Figure 1. Thermal conduct i"ity data corresponding to the first line of Table 1. Upper points and straight line are for a 50-50 ortho-para mixture. Lower points and line are for normal hydrogen. 632 Prediction of Minor Heat Losses in a Thermal Conductivity Cell and Other Calorimeter Type Cells D. R. Tree and W. Leidenfrost Mechanical Engineering Purdue University Lafayette, Indiana 47907 In recent years there has been an increasing number of reported thermal conductivity measurements of liquids, gases and gas mixtures. The major emphasis of these works has been to obtain thermal conductivity values with a high degree of accuracy. Several authors have reported values which they claim have an error of less than 1%. In the present work some of the factors affecting the accuracy of thermal conductivity measurements in a calorimeter type thermal conductivity cell are considered. In this work a calori- meter type cell is defined to be a cell where the inner body is completely surrounded by the outer body and the fluid to be tested is placed within a cavity between these two bodies. The inner body is always at a higher temperature than the outer body. The steady state heat flow by conduction for such a cell is q, = k BAT where k is the thermal conductivity of the fluid, B is a geometrical constant of the cell, and AT is the temperature difference between the two bodies. A method is developed for very accurately measuring the geometrical constant. This method has yielded the geometrical constant with an error of less than 0.01 per- cent. A method is also developed that allows the total heat input to the cell to be measured with a high degree of accuracy. Unfortunately, all of the heat input to the inner body is not transferred by conduction through the test layer. There are at least three other ways that heat can be transferred from the inner body to the cold outer body. They are: 1) by free convection in the test fluid; 2) by conduction along the devices used to position the two bodies with respect to each other, by conduction along the wires used to supply power to the hot inner body, and by conduction along the wires used to obtain the temperature of the inner body; and 3) by radiation between the two bodies. A detailed analysis of each of these methods of heat transfer is made, and a method of predicting or measuring each is obtained. The analysis shows that by proper design of the instrument that heat transfer by free convection is so small that if it is neglected the error in the thermal conductivity is negligible. This is not true of the other two forms of heat transfer if accuracies of 1% or better are desired. The heat loss along the connecting devices and lead-in wires is shown to produce the largest error and its value is the most difficult to obtain. The neglecting of this method of heat transfer can cause error of more than 1% in the measured thermal conductivity value. It appears that the only way of obtain- ing accurate predictions of the lead-in loss heat transfer 633 is by direct measurements. In the literature there are reported two different methods of obtaining the radiation heat transfer in this type of cell. These two methods give quite different results for certain fluids. These two methods are reviewed and equations for one of the methods are developed. Experimental data is obtained to support the theoretical equations developed. Key Words: Convection, geometric constant, lead-in losses, minor heat losses, radiation, thermal conductivity, thermal conductivity cells (hot-wire, parallel plates, concentric cylinders) . 1. Introduction For numerous applications in physics and engineering the exact value of the transport and thermodynamics properties must be known. One of the major problems connected with the precise experimental measurements of these properties has been to determine the minor heat losses from the testing cell. In this paper these losses are determined for a thermal conductivity cell, but the methods can be used for any calorimetric instrument. In the past, the three most common types of thermal conductivity cells used have been 1) the hot-wire or Scheirmacher ' s cell, 2) the horizontal parallel plate arrange- ment and 3) the concentric cylinder cell. The hot-wire apparatus consists of a fine wire, generally platinum, axially stretched between the ends of a closed cylinder. The area between the wire and the cylinder is filled with the fluid to be tested. The wire serves as the heater and also as a thermometer and the cylinder as a constant temperature heat sink. The main disadvantages of this method are the difficulties or placing and keeping the wire exactly on the axis of the cylinder, heat losses from the ends of the wire and an inhomogeneous temperature field. The inhomogeneous temperature field can produce free convection currents. The convection currents as well as the heat losses from the end of the wire are greatly influenced by the radial and axial dimensions of the annulus. The second method uses two horizontal parallel plates with the material to be tested between them. To eliminate free convection the upper plate serves as the heat- er and the lower plate as the constant temperature heat sink. One of the main advantages of this type of cell is the fact that the plate separation and the temperature difference can be made large without introducing free convection. This is not true of any other system. Its main disadvantage is the heat loss from the edges of the plate. This effect is increased as both the plate separa- tion and the temperature difference are increased. To reduce this effect most parallel plate cells are built with some type of a guard heater. While these guard heaters do reduce heat loss from the edges of the plate, it is very difficult to control their temperature, and it has been impossible to keep them at the same temperature as the cell. This results in two major problems. One, the time to reach steady state condition is greatly increased and two, a temperature difference exists along the wires used to supply power to the cell and the wires used to measure temperatures in the cell. These temperature gradients introduce another path for heat flow. To try to reduce the edge effects of the parallel plate cell, the third system, that of two concentic cylinders, has been used. This system consists of two, generally metallic, cylinders. The larger cylinder has a hole drilled through it that is slightly larger than the outside diameter of the smaller cylinder. The smaller cylinder is placed in this hole and the material to be tested is placed between the two cylinders. The inner cylinder serves as the heater and the outer cylinder as the constant temperature heat sink. The main disadvantage of this system is to assure perfect alignment of the axes 634 of the cylinders. Although edge effects are much smaller then in the parallel plate cell, they still exist and guard heaters are still required. Another disadvantage is that the distance between the two cylinders and the temperature difference must be small in order to prevent convective currents. In order to overcome some of the disadvantages of the above named cells, especially the edge effects and the need for guard heaters, many different types of cells were considered. A few of these cells are listed below. In 1948, Riedel [1] made modifications to the concentric cylinder cell by placing a right circular cone on each end of the cylinder and making a matching cavity in the outer cylinder. This elimated any edge effects but greatly increased the problem of making and assembling the instrument. It also introduced inhomogenities in the temperature field. Riedel [2], because of the large disagreement between his earlier work and other reported values of the thermal conductivity of several hydrocarbons, also constructed a concentric sphere apparatus. The inner sphere was the heater and the outer sphere served as the constant temperature heat sink. This cell, like his modified circular cylinder arrangement, eliminated any edge effect. Its great disadvantages were in construction, especially making sure that the spheres were truly concentric, and in obtaining isothermal surfaces. In addition the method of supporting the inner sphere greatly increased heat transfer along the supporting members. In 1953, Schmidt and Leidenfrost [3] reported thermal conductivity values obtain- ed with an arrangement that employed concentric cylinders with hemispherical ends. Like the conventional concentric cylinder cell, this cell used the inner cylinder as the heater and the outer cylinder as a heat sink. It also eliminated any edge effects, and Leidenfrost [4] showed that the main parts of this cell could be very accurately positioned with respect to each other. Although the last three cells have eliminated any edge effect, they all have introduced a new problem. Each has the inner body as a heater element and each requires that the inner body be held in a fixed position with respect to the outer body. This requires some type of a support and wires running from the outer cold body to the inner hot body. Thus introducing a path for heat flow and a new unknown in determining thermal conductivity. The purpose of this work is to examine this loss along with the other heat losses due to convection and radiation, and to examine some other factors which affect the accuracy of thermal conductivity measurements. 2. Description of Apparatus 2. 1 Basic Unit Figure 1 shows a schematic drawing of the thermal conductivity cell. The equations given in this paper are for this instrument, but the methods used for calculating the minor heat losses can be applied for any calorimeter type device where the cold body completely surrounds the hot body and where the cold body has a connection to the hot body through a holding rod (or rods) and lead-in wires. The cell is made from two concentric cylinders with hemispherical ends. The inside or heater element, also called hot body, 1 , and the outside or cold body, 2, are separated by a gap, nominally 0.05 cm. They are made from pure copper and are gold plated on all parts in contact with the fluid to be tested. The low strength of copper, especially at high temperatures and pressures, makes it necessary to follow special precautions in order to avoid damage to the cell. Thus the unit, as shown in Figure 1, is placed inside a high pressure vessel which makes it possible to maintain the pressure on the outside of the cold body the same as exists in the gap of the cell. The high pressure vessel is surrounded by a constant tempera- ture bath. 1 Figures in brackets indicate the literature references at the end of this paper. 2 Number refers to number on Figure. 635 3. Basic Equations and Correction Factors The steady state heat flow by conduction for any configuration with two isothermal surfaces can be written as q k = k L BAT (1) where q. heat flow by conduction k = thermal conductivity of the fluid between the isothermal surfaces B = geometric constant AT = temperature difference between isothermal surfaces The geometric constant in equation 1 is a function of the shape of the isothermal surfaces and the distance between these surfaces. Since for most cases the shape and/ or the distance between the surfaces changes with temperature, the geometric constant changes with temperature. Equation 1 is solved for the thermal conductivity k to give Xj k L = BAT (2) To determine thermal conductivity, the heat transfer by conduction, the geometric constant and the temperature difference must be measured or determined. 3.1 Geometric Constant If all heat flow is radial the steady state heat transfer equation by conduction through a fluid layer in the form of a cylindrical annulus with hemispherical ends can be written as = k T AT 2TT r h 1 D ln d D d - D-d- (3) where h = length of cylinder d = inside diameter D = outside diameter Comparing Eq. 3 with Eq. 4, the geometric constant B is B = 2tt[- ln f D + D-d J (4) Several disadvantages exist in determining the geometric constant in this manner, First, the equation assumes that all heat flow is radial, but in the combined arrange- ment used here there exists a thermal inhomogeneity in the transition from the cylindrical to the spherical portion of the annulus due to the fact that at this point the temperature profile across the gap changes from a logrithmic to a hyperbolic one. Leidenfrost [4] stated that this effect can be made small by a proper choice of a diameter and gap width, but even with this error made small, direct measurements of the length and diameters prove to be extremely difficult and they are functions at the temperature and pressure. No matter how careful one is and how sensitive are the instruments used, it is still impossible to determine the geometric constants exactly because it is impossible to account for surface roughness and other inhomogeneties [4], A better way of determining B is to use an analog between the geometric constant of the heat transfer equation and the geometric constant in the equation of electrical capacitance of the cell assembly. Due to the similarity between the electrical and the thermal fields all non-homogeneties will be included. (See detailed information in reference 4. ) Leidenfrost [4] has shown that the geometric constant determined in this manner can be obtained with an error of less than 0. 01 percent. The value of B as determined by the electrical capacitance analog will be the same as the value of B in the heat transfer equation only if the relative position of the hot and cold bodies are exactly the same in both measurements. Thus, there is a need for some device to accurately position these two units (with respect to each other). This is accomplished by the two tubes (numbers 4 and 5) shown in Figure 1. 636 3 . 2 Heat Input Heat is supplied to the unit by an electrical heater inside the heater element. The total heat input to this heater is found by using the electrical circuit shown in Figure 2. R, , R„ and R, are standard resistors, R is the resistance of the heater and Rj. is the resistance of all lead-in wires. The current I, was determined by measuring the voltage drop across R, and current I- by measuring the voltage drop across R„. All voltages are measured with a six-dial Rubicon potentiometer. The current through the heater is I = I x - I 2 (5) and the voltage drop across the heater is V = I 2 (R 2 + R 3 ) - IBj. (6) The total heat input becomes or q T = VI (7) q T = i 2 R (8) If equation 8 is used, the resistance of the heater as a function of temperature is required but if Eq. 7 is used only the change in the lead-in resistance as a function of temperature is needed. Since all parts of the lead-in wire except for a small amount passing through the centering rod are outside the instrument, it can be held at a constant temperature. The lead-in wire inside the instrument is made from constantan wire and the changes in its resistance over the temperature range used are negligibly small. Therefore, by using Eq. 7 the need to know the temperature dependence of any resistance is eliminated. The constantan wire also serves another purpose. Because of its low thermal conductivity and its high electrical resistance as compared to copper, it serves to reduce heat transfer by conduction along the centering rod and lead-in wires. This effect is discussed later in section 3.4. Unfortunately, all of the heat input is not transferred by conduction through the test layer. There are at least three other major ways that heat can be transferred from the heater element to the cold body. They are: 1) by free convection in the fluid, 2) by conduction along the centering rod and lead-in wires, and 3) by radiation across the gap. The total heat input q can be written as q T = q k t q r + q c ± q x + . . . (9) q = heat transfer by free convection in the fluid layer q. = heat transfer by conduction through centering rod and lead-in wires q = Heat transfer by radiation across the gap 3.3 Heat Transfer by Free Convection In any arrangement, except a horizontal layer of test fluid uniform heat from above, free convection motion will be present. However, it has been shown that for very slow motion, heat transfer by free convection adds nothing to the total heat transfer. For example, Kraussold [5] showed that free convection heat transfer in a cylindrical liquid layer had no contribution to total heat flow if the product of the Grashof number (Gr) and the Prandl number (Pr) was below 600. Schmidt and Milverton [6] reported a theoretical product of Gr. Pr of 1709 below which the heat transfer by convection of a fluid enclosed between two horizontal plates heated from below 637 contributed nothing to the total heat flow. They also reported an experimental value of 1770 + 140. Schmidt and Leidenfrost [3] established a critical Gr. Pr product for the configuration of the combined arrangement of the present test cell of 1200. The product of the Grashof number and the Prandtl number is Gr Pr = E a e at p 2 s 3 ] [LfE] (10) M k L where g = acceleration due to gravity (3 = coefficient of volumetric expansion p = density of fluid S = average thickness of fluid layer [i = dynamic viscosity c = constant-pressure specific heat P By careful design of the cell and by controlling the temperature difference between the two isothermal surfaces the Grashof-Prandtl product can always be kept below its critical value. Thus, the effect of free convection can be reduced to a point where it is of no importance. 3.4 Lead-in Losses Figure 3 shows a schematic drawing of a typical device used to hold the hot body in the desired position with respect to the cold body (see Figure 1). In many instances two or more devices are required to assure perfect positioning. Generally, by careful design, the heat transfer by conduction along all but one of the devices can be reduced to an insignificant amount, by making this device (or these devices) from a poor conducting material such as nickel-steel alloys, making the tube (or tubes) thin walled, and drilling many small holes in the tube wall (or walls) to further reduce the heat transfer area. Quite different is the situation in the case of the top tube (4) in Figure 1. This tube not only serves as a positioning device but also as an enclosure for all lead-in wires. In this case the centering rod consisted of: 1) a tapered hollow stainless steel tube Stl, 2) two insulated constantan wires carrying current to the heater element. The insulation of these wires is aluminum-oxide and serves as an electrical insulator but has a rather high thermal conductivity (approximately 0.22 watts cm C~ at 25°C). The ceramic and heater wires are enclosed and electrically shielded by a stainless steel tube (St2). This unit (constantan wires, insulator and stainless steel tube) is referred to as the "heater lead-in unit. " 3. threethermocouples enclosed in insulators. This unit is referred to as "thermocouple unit." To calculate the heat loss through the centering rod the following assumptions are made: 1) That each of the units considered above (stainless steel tube Stl. , heater lead-in unit, and thermocouple unit) is a homogeneous material with constant properties. 2) That the stainless steel tube Stl is not tapered. 3) That each end of the centering rod is held at a constant temperature; the lower end at 'the temperature of the hot body t. and the upper end at the temperature of the cold body t . Even with these assumptions it is impossible to calculate the lead-in losses. If two additional assumptions are made it becomes mathematically possible to evaluate these losses. These assumptions are: 1) That the temperature of any cross section of any one of the three units described above is a constant. This assumption does not imply that the 638 temperature of all three cross sections are the same. 2) That the heat transfer between each section of each unit and its surround- ing can be written as q = C . . (T. - T) where C. . is a constant which depends only on the geometry and the- 1 average- 1 temperature of the centering rod. T. and T. are the temperatures of the cross section and its surrounding respectively. Figure 4 shows a schematic diagram of a section of the centering rod. Referring to this figure the following equations result; q r = C lc (T.- T c ) dZ (ID %1 = C 12 (T 1 - T 2> d? ' + °13 (T i - T 3 ) dZ U2) *h2 = C 21 (T 2 " T l> dZ + °23 (T 2 " T 3 ) dZ (1 ^ %3 = °31 (T 2 " T l } dZ + °32 (T 3 " T 2 } (14) q. = -K. A. dTi d 5 ) ^ 1 x dZ - dT dT. q. + dq.= - K,A. £ + d_ (-K.A _Jl dZ) (16) 1 x 1 X dZ dZ 1 dZ ( a) where i-l, 2, or 3. therefore d? *± C lc (T L - TJ dz 2 K i A i + C 12 (T K 1 A 1 - T_) + C 13 K 1 A 1 d T 2 = C 21 (T„ - T x ) dZ 2 K 2 A 2 + °23 (T 2 - K 2 A 2 T 3 J (b ) 2 d T 3 + g_= °31 (T 3 - T x ) + C 32 (T 3 - T 2 ) ( c ) dZ 2 K 3 K 3 A 3 K 3 A 3 Theoretically equation 16a, b, and c can be solved simultaneously for T , T and T,, Once these temperature distributions are known the heat transfer along the center- ing rod can be found. The big problem with this approach is the fact that the C. . are not known and they are very difficult if not impossible to determine. - 1 Therefore a simpler approach is needed. This is accomplished by looking at three special cases listed below. It was hoped that by looking at these three cases the lead-in loss could be bracketed. These three cases are: 1) there is no internal radial heat flow i.e. q, , = q, „ = q, , = 0. This gives the minimum heat loss. 2) there is no internal resistance to radial flow from the heater lead-in unit, but there is not other internal radial heat flow i.e. q, . = q, „ = while q, 3 = C. (T - T ). This case should give the maximum loss. 3) tne centering rod is regarded as a solid homogeneous, slender rod with uniform heat generation. This is an ideal case. Case 1 For case 1 the following assumptions are made (a) The heater wires, ceramic insulators and stainless steel tube St2 (heater lead-in unit) are considered as one material with thermal conductivity k^ and resistance R. . It is assumed that this unit generates heat uniformly but has 639 no radial heat flow, k. is defined as VSi = *h — 4^" = 2kconAcon + k Cer A Cer + k Stl A Stl where the subscripts: Con = constantan Cer = ceramic St = stainless steel and R = total resistance of heater lead-in unit (b) The thermocouples and their insulator are one material with conductivity k (with no radial heat flow, and no heat generation). k m is defined as ~ "tc^c = k Tc — f— = 3 < k P l A Pl + k Rd A Rd + k Ins A Ins ) 4 where Pi = platinum Rd = platinum - platinum 10 percent rhodium Ins = insulation (c) The stainless steel tube Stl has radial heat flow only to the fluid being tested and none toward the inside of the tube. The differential equation and boundary conditions for case 1 and assumption a are ^+ J*- = (17) dZ 2 *h where q = heat generated in the wires and t = t, when Z = t = t when Z = L (18) where Z is the distance along the axes of the centering rod. Equation 17 with boundary condition 18 is solved to give ■ „2 , ' - fc h ~ t c, _ (19) gZ + ( qL ) Z + t "e. 2k, h h t ~ 2k, 2k, " - ~ h where L = length of centering rod. Letting 9 = t - t and At = t, - t and noting that . I R. c n c where A, = area of heater lead-in unit Equation 19 becomes .2 I R. „ ,2 *h 2Lk hAh Z 2 + 2k h\ L The amount of heat transferred from the heater element is t 1 "h - At) Z + At (20) q %£ Z = = - k hAh r 1 *h - At > Vh At - . J \ (21) L 2k *A. L J L 2 i.--Vh4ft z-o--ic.il r 1 VS1 The differential equation for case 1 and assumption b is ^-|=0 (22) dZ^ with boundary conditions 18. Equation 22 with boundary conditions 18 is solved to give 640 e ib = 3~ Z + At (23) and the heat loss from the heater element is Tc A Tc *lb " JE p £ At (24) The differential equation for case 1 and assumption c is d 2 t _ k L (t ~ t c ) 2n (25) dz 2 k stl A stl lx ^a i where D , D. = diameter of annulus as shown in Figure 3 with D. = D. and the 3 1 13 3 \7G boundary condition are given by Eq. 18. 2 2 *" k T 1 Letting M = ± — and = t - t the k A D a c *Stl A Stl In -2- differential Eq. 25 is £-1 - M 2 = (26) dZ with boundary conditions 9 = At at Z = (a) 6 = at Z = L (b) (27) — MZ MZ The general solution to 26 is q. = c e + c„ e Using the boundary conditions of Eq. 27, the above equation is solved and yields r MZ 2ML -MZ ] At [_e - e e J **lc , 2ML 1-e This equation is reduced to give q lc = sinhrM(L-ZV] At (28) 1C sinh (ML) ' The heat transfer from the heater element is q lc = M k Stl A Stl At coth (ML) (29) The total heat transfer from the heater element for case 1 is the sum of Eqs. 21, 24, and 29 or q x - [Vh + ^TcN-c + M k Stl A Stl COth (ML) ] AT * X ** (30) If the heat transfer from the heater element by convection and radiation is assumed to be negligibly small, then the total heat input can be written as 641 q T = I 2 R = k L BAT + q L 2 3 (31) where q „ - is the heat transfer by conduction along the centering rod and lead-in wires. ' ' The subscripts 1, 2, and 3 refer to the three cases considered and only one at a time can be used. 2 Solving for I from Eq. 31 gives c 2 m k L BAT q l,2,3 I = ^- + -^ R -^ (32) Substituting Eq. 32 into Eq. 30 and using q gives q, = f VSi + k Tc\c + M k Q . tA-. . coth (ML) - V ^lW f"l + M (33) ■ 1 L l 2 R L 2R For the special case where K.— KD (This is the case where all energy generated in the hot body is transfered to the cold body by radiation through the gap and by conduct- ion along the centering rod. This condition is experimental , obtainable by drawing a vacuum in the space between the hot and cold bodies. ) q-, is 2 q (0) = pA + JW^TC + k Stl A Stl ] At - T \ (34) (See reference 10 for proof of equation 34. ) Case 2 The assumptions made for case 2 are: a) The heater lead-in unit is defined as in case la but is loosing heat from its outer surface at the rate D 2tt1c (t ? - t )/(ln =— ) where t_ is the temperature, as defined in assumption i 2b below. b) The remainder of the centering rod is one solid material having thermal but loosing as much heat at its outer conductivity k and temperature t„ surface as it gains from the heate r lead-in unit. k„ is defined as 2 k 2 A 2 = k 2 nD i ave = 3 k pl A pl + 3 k Rd A Rd + 3 k^A^ + k sti A sti In order to obtain temperature t„> case 2 and assumption b must be solved first. The differential equation for this case is ^4=0 (35) dZ with the boundary conditions (18), viz t = t, when Z = 2 h (18) t„ = t when Z = L 2 c The solution to Eqs. 35 and 18 is .. „ t 2 =^p + t h (36) 642 and the heat transfer from the heater element is k_A~ *2b=- J TT At < 37 > The differential equation for case 2 and assumption a is Vn^t = ^ ' l2R h since q = — - — equation 40 becomes ri <*2a = Vh At - !!i (41) The total heat transfer in case 2 is ^9 = ^5=. + 3 ok = (k^A, + k„A ) T 2 R 2 2a 2b JO! L_2 + NL k h A h ; At - * \ (42) As in case 1, if the heat transfer by convection and radiation are neglected , k BAT q„ 1 2 --i— + t- ^ and equation 42 is q 2 -phVV2 + 2TTLk L - V fhl At/fl + \1 L L D 2 R J L 2rJ 3 ln_^ D. l Looking again at the special ease where k -*0, equation 42 becomes 643 q 2 (0) =f Vh + k 2 A 2 ] At - l2R l L T. J o (44) Case 3 Leidenfrost [4] assumed that the entire centering rod is a slender rod with uniform heat generation and arrived at the following equation At sinh fP (L-Z) 1 * CQSh & ( l * Z > J 6 < 45 > sinh (PL) - 2 CQsh ( |L } + k p 2 S 4. S where 2Trk_ P 2 = ^ D k A lnr 2 - s s D. irD. A = average cross-section area of the rod = — -r k = apparent thermal conductivity of the rod and is defined as follows: k s A s = k Cer A Cer + 3k Pl A Pl + 3k Rd A Rd + 2k Con A Con + k Stl A Stl + k St2 A St2 + 3k Ins A Ins D . D. are the diameters of the annulus as shown in Figure 3 with D. = D. ave. a' l ^11 Using Equation 45, the heat transfer from the heating element for Case 3 is: q3 . Bj /^ t„h ,a, [ ^lp at- -^ ] ,46, where R 2 _ 2nk A B i s s In a D. l B 2nL 2 , D I n a D. l Neglecting heat transfer by convection and radiation equation 46 may be written as BR , x ^ t-h (^(MtJg -df-> ««» -tanh (— ) 2 . 1 .Hi. k T tanh -PL. 1 + Bj I—) L (— ) Leidenfrost [4] points out that by correctly designing the thermal conductivity cell Eq. 47 will always be zero. This is accomplished by making PL large enough so that + J ° Xn X (T X )E bX< T X )E 2 (51) where R. . _>.. = radiosity or radiant energy leaving (emitted and reflected) a surface, n, = index of refraction E, , = emissive power of a black body radiating into a fluid having an idea r x = optical depth; r^ = $Y ^ dy TV = optical thickness; r, = i s -, Xo r Xo J /c, dy k = absorption coefficient E (X) = exponential integral function defined by e"(X) = j\x n - 2 exp(-^)dn Equation 51 stated in words is q . = energy leaving surface times its attenuation + energy emitted by the fluid layer times its attenuation - energy leaving surface i, which is simply the net heat transfer by radiation from the it* 1 surface. The radiosity at surface i is T oX r _2 R iX = e iX n X ( ° )E bX (0) + 2(1 - £ iX> [ R oX E 3 (T oX ) + J n X (T X )E bX (T)E 2 (5 2) < T x> dT xJ where e. = emissivity of the surface i. The first term on the right had side of equation 52 represents the emission of energy by surface i and the second term gives the fraction of the energy incident on i that is reflected from it. The equation for the energy leaving is similar to equation 52, viz. R oX= e oX n X< T oX )E bX [ R iX E 3< T oX> + r°" n X (T X )E bX (r)E 2 (T oX- T X )dT X } (53) where e . = emissivity of the surface 0. OA Equation 52 and 53 could be solved simultaneously and the resulting radios ities introduced into equation 51. However, the evaluation of this equation would be very complex due to the double integration that results. The assumption is made that e, and n. can be replaced by an appropriate average value and that an average absorption coefficient is given by Planck's mean value 647 TX X bX I X bX K = J °-7- = J ° X— (54) J E bx d; IX aT I ' .A o where X = wave length E, , = monochromatic emissive power of a black body. Then equation 53 becomes R o = *o n \ {T J + 2(1_£ o ) [ R i E 3 (T o> + n ' r° E b U)E 2 (T o " T)dT ] = e o n2oT o + 2(1 " e o ) [V3 (T o> + n2a J^° T4(T)E 2 (T o " T)dT J or R Q = c o n 2 oT^ + 2(l-, ) [^V + n 2 P o ] (55) P ' = f-°T (t)E-(t - T)d T o J /. o Equation 52 becomes , T o £i n^E b (0) + 2(l- e .) [R E 3 (t o ) + n 2 | E b ( t) E 2 ( t) dT ] i R i - e i ^oT 4 + 2(l- e .) [r q E 3 (t) + n 2 a/ T °T 4 (T)E (t) dr ] o J "X + 2(1 - e i> [ R E 3 = 2 [R E 3 (t o ) + n 2 a P : -±R. ] Introducing the following dimensionless variables, T <*ri . r = D 1 n oT i n Z oT equation 57 becomes 648 cp = q ri = 2 Po E^(t o ) + "'< - i J*i I = 24 L 2 4 24 2 ~~ 2 4~~ J n^oT^ r/aT* noT* 2 n 2 oT 4 ii l i where 2 0o E 3< T o> + P i 'I h] T T P i - Ii = J ° T 4 (T) E (T)dT = f °9 4 (T)E 9 (T)dT (58) T 4 " ° T 4 i l Equation 55 becomes R o_ - e o n gj + 2 ,. . r R. ^[Jl-.,!.^^.] 2_„4 n 2 ^ n 2 oT 4 i i n or . n oT . 1 i = e o 9 4 + 2(l-e o ) Wt) + 2 (l-« o ,P o (5g) where P =!p_ -/V^E^,- T)dT= J T o e 4 (T) T)dT T . T o ° i i and T. l and Eq. 56 becomes Pi = R i = e i n2oT i + 2(l-e.) TVliV + " 2 ° p 'j 1 2^4 2 4 x L~2 7 -5 — 2 J n oT. n oT n 2 oT 4 n 2 aT 4 11 11 (60) e. + 2(l-e.)P. 1 11 Solving for £ and p. from Eqs. 59 and 60 gives e o = £ o e o +2 h {1 -^ E 3 {l o ) + 4 ( 1 -«q) (^j) VyPj + 2(l-c n ) P ^ l-4(l-e ) (l-e^E 2 ^) " (61) Since € = € . = € equation 61 reduces to O = ee 4 + 2(1-Ue E 3 < T o )f 4 (1_e) ^3 ( T o ) P i + 2 ^-^ p l-4(l- e ) 2 E 3 (T o ) (62) h = £i + 2(1 - £ i )E 3 (T o )e o 9 o *_ 4 ( 1 - £ i ) < 1 - e o^ E 3< T o )P o + 2 ( 1 ^j) p i M(l-c o )(l- e .)E2(T o ) {63j and since e- = e = e equation 63 reduces to 649 0. = e t 2e(l-e)E 3 (T o )e^ I 4 (l- e ) 2 E 3 ( t q ) P q + 2(l-e)P i 1_4(1_ g ) 2 E 3 2 (t o ) (64) Before the values of P and P. can be obtained, the temperature distribution between the plates is needea. In general the temperature can be found by the solution of the conservation of energy equation. This has been done by Viskanta and Grosh [8] and by Viskanta [9]. Viskanta [9] pointed out that if (K//4n 2 aT 3 ) were large the temperature distribution was very little affected by the presence of radiation. He showed that as this term approached infinity the heat transfer is only by conduction and for the case where the term is zero heat transfer is only by radiation. Leidenfrost [4] stated that when this term was larger than or equal to 10 the conduction temperature profile can be used. Reference [4] also states that regard- less of what (K//4n aT ) is, if the temperature difference between the plates is small, the linear temperature profile of conduction can be used. In this work both the above conditions are met. Thus the temperature profile is given by T = T. + (T - T.) 7? (65) l o l S or h = ^ > & ~ h I - L + (G o - x) I = x + (Q o - x) w i ii t (66) or 6(t) = 1 + ( - 1) O T Introducing equation 66 into the equations for P. and P gives 4(e c -n P, ==i-E 3 (T ) + -^-[i- T o E 3 (T o ) -B 4 (T )] 6(9 ^- ir [I- ,h 3 i, Q ) -2x o E 4 ( To ) -2B 5 (T )] 2 T O + + 3 -J. J 1 --^IL [f - ^ E3 ( To ) - 3^ 4 ( To) - 6To E 5 ( To ) - 6E 6 (T o )] [^ - T*B 3 (T ) - 4^E 4 (t o ) - 12r 2 o E 5 i, o ) - 24^^) + T O (e o -D 4 4 T o 24E 7 (T o?] (67) and V=f - E 3 ( v + 6 (e o -D 2 [l-f^j.- 2e 5 ( To )] T o 650 , 4 Figure 5. Lead in loss versus Rh for a thermal conductivity of 0.0007 ™ a A£s cm-c K L = 0.0035 -*^- L cm-c Figure 6. Lead in loss versus Rh for a thermal conductivity of 0.0035 ™££5 . cm-c 655 o_ Rh= 1.36 ohms 1.2 8 .4 l.l.l. 20 40 60 Thermal Conductivity x I0 4 ( WQJ Figure 7. Precent uncertainty versus thermal conductivity. cm-c Ih "7! --« /I Tc C o '■*— o O or >» -O i_ 0) c o a 0) X Poltz Leidenfrost Log Optical Thickness Figure 8. Heat transfer by radiation versus optical thickness. |— >-Y, Figure 9. Physical model considered for radiation heat transfer. 656 CM O ■«- c o o Si o <0 X (A co ^) C o "vt c a> E Figure 10. Dimensionless heat transfer by radiation. 2.2 2.1 2.0 1.9 1.8 1.7 1.6 1.5 1.4 AT = 5°C «r = 0.5 1 Total Equation 2 Leidenfrost's Approximation ( AT) 3 ,(AT) 4 Terms Dropped 3 Poltz's Approximation (AT), (AT) 3 , (AT) 4 Terms Dropped 01 T = 25°C Optical Thickness t Figure 11. _Comparison of radiation heat transfer T = 25 °C. Optical Thickness r 65 7 A.2 A3 A4 Wave Length (X) Figure 12. Monochromatic absorptivity versus wavelength. O o a> O a. t- a> «♦- w c o i- H o a> X "5 o 0.001 0.01- 0.01 0.1 I 10 Pressure (yx Hg) 100 1000 Figure 13. Vacuum heat transfer data. 0.12 T = 60 / y^ L 0.10 - CO 0.08 — T=20°C a. 0.06 c l-H - g 0.04 X 3 0.02 r- - / 0.00 1 1 1 1 1 2 4 6 8 10 Temperature Difference (°C) Figure 14. Heat transfer at a pressure of 0.2/i Ug. 658 The Thermal Conductivity of Pure Organic Liquids 1 J. E. S. Venart and C. Krishnamurthy 2 Faculty of Engineering The University of Calgary Calgary, Alberta, Canada Unsteady state absolute measurements are presented for the thermal conductivity, of pure organic liquids in the temperature range 15 to 130°C. The measured results are thought to be accurate to ±0.5%. Key Words: Conductivity, heat conductivity, liquids, organic liquids, thermal conductivity. 1. Introduction In recent years there has been a rapid development in the sophisticated analytical and numerical methods used in heat transfer and flow studies. The efficient and economic use of these tools by research and industry implies the availability of accurate values of the thermophysical properties for the fluids involved — density, specific heat, viscosity, thermal conductivity and their variation with temperature and pressure — unfortunately this is not the case except for a few fluids. In addition to the above the physical and theoretical chemist wishes today to extend his knowledge of the structure of liquids; accurate equilibrium and noneauilibrium thermoDhvsical property data can not help but assist in this area. Accurate measurements of density, specific heat, and viscosity can now be attained to a pre- cision of one part in a thousand; with thermal conductivity this is no longer true — for liquids deviations in reported data can exceed 10% even with reliable apparatus and experienced investigators [1,2]. 3 It is the purpose of the present investigation to provide some data for water and several pure organic liquids. 2. Source and Purity of Materials With the exception of water, all liquids studied in the present investigation were pure organic liquids of analytic (A.R) quality, meeting the American Chemical Society (A.C.S) specifications. Some of these liquids are of A.R quality supplied under the designation of British Pharmacopoeia (B.P) standards. Details of purity of the various liquids are listed in Table 1. The water employed in the present investigation was triple distilled water containing no dissolved gases. Work performed with the support of the National Research Council of Canada under Grant Number NRC.A.2788. 2 Department of Mechanical Engineering. 3 Figures in brackets indicate the literature references at the end of this paper. 659 Table 1. Source and purity of liquids. Compound Purity Reagent Quality Manufacturing Pharmaceutical Company Standard Specifications Methanol Ethanol 100% N-Butanol Ethylacetate Glycol Glycerol N-propanol Aniline Benzaldehyde Acetic acid 100% A.R. ! A.R. A.R. A.R. Fischer B.D.H. Matheson Coleman & Bell Company Allied Chemical Company of Canada A.C.S. B.P.+ Standards A.C.S. A.C.S. ''A.C.S. means American Chemical Society Standard Specification. tB.P. means British Pharmacopoaia Standards. *A.R. means analytical reagents with maximum purity. 3. Method and Apparatus An absolute transient line source method of measuring the thermal conductivity of liquids was utilized in the present investigation. Unlike steady-state methods, the transient method requires only simple apparatus and yields accurate results in a relatively short time. 3.1 Basic Theory In transient measurements of thermal conductivity, temperature is a time dependent variable. The development of this temperature is obtained by passing a continuous electric current through a thin straight axial wire immersed in the homogeneous medium maintained initially in thermal equilibrium. After a short interval of time the temperature rise at the surface of the wire will be proportional to the logarithm of time. The entire arrangement, to a first approximation, can be treated as an infinite line source with constant heat generation in an extended or infinite medium. The solution of the one dimensional radial heat conduction equation, / 3 2 T 1 8T\ _ 1 /3T\ ^3r 2 r 3r' a \3t' (1) with the boundary conditions t = 0; r/O, T = t > 0; r=», T=0 t > 0; r ■* 0, -2TrrAf— J = f- = Constant (la) (lb) (lc) yields on solution V'.t) =4^U Ei (-4^) (2) 660 where T (r,t) is referred to the temperature rise in any radial position r. Here -Eil-r — ] is the exponential integral defined by ' ' -Ei (-x) = | v dz = _0 - 5772 + ln (x) + rrr - rsr + (3) For x << 1, the infinite series may be truncated after the first two terms to a reasonable degree of accuracy. The resulting equation thus acquires the form T -fr*« = 4^(-°- 5772 + ln ^l) (4) The difference between two temperatures Tj and T 2 at two times t, and t ? is therefore given by AT = Tjr,t 2 ) - Tjr,tO = ^_ ln (li) AT = «fe Mnt (5) From eq (5) it can be readily seen that a plot of AT vs. Alnt will give a straight line with slope , J * thus, knowing the energy liberated per unit length and unit time at the wire, the thermal conductivity of the substance can be obtained from the slope. With a line source of finite diameter, the inner boundary condition, eq (lc) must be modified to: t > 0, r = r Q , f = -2irr x(g) + Trr 2 C p (ff) = Constant since in the theoretical case the medium situated in the region r < r Q will take up a certain amount of energy depending on its heat capacity, which is different than that for the fluid at r i r Q , the solution of eq (4) will embody a term dependent on the heat capacities of the wire and medium. Assuming a constant amount of heat q generated in a perfectly conducting wire with no contact resistance between the wire and the medium, the solution as given by Carslaw and Jaeger [3] is, t t .-\ 1 Hi u , ■, 4 , r , (w-2)r 2 L 4at T (r,t) = j^- lnt + ln y + -5— + -^r — '- — In 4tt a£ I 2 2at 2ioat ' (-iH (6) where y = 0.5772 is Euler's constant and 'io' is twice the ratio of the heat capacity of an equivalent volume of the medium to that of the wire C p P C P P w (6a) For -y— — << 1 the last three terms in eq (6) tend to zero and hence eqs (6) and (4) become identical. However, since convection usually limits the times of measurement to short intervals it is necessary to make this correction. 661 3.2 Apparatus The apparatus used was similar to that employed by Horrocks and McLaughlin [4]. Only those items of significant difference will be discussed. The cell design (fig. 1) used in this work was similar to that of [4], with the exception of the provision for evacuation and filling. In addition the provisions made for a double thermostated bath and cell resistance calibrations were very much as has been described by [4]. The major problem involved in the transient method is the measurement of the unsteady temperature. This time dependent temperature is measured by the variation of the electrical resistance of the heating wire. For eq (5), A = ~^- — — — , and the small temperature rise (0.5°C) experienced in the present work, dR the temperature coefficient of the resistance, — of the wire can be treated effectively as a constant during each measurement. Therefore, JL ~ _L ( &L\ AT AR \dTJ ' Substitution yields X = -y-*- — — — ■ as q = V I , the heat liberated, differentiation of Ohm's law gives 4tt8. AR dT AR m I(AV) - V(AI) resulting in VI Alnt dR . . 4tt£/AV V , T \ dT ' ( ' '(¥-? ") where R is the resistance of the source of length I between the potential leads through which a current I flows under a potential V. The current is measured in terms of the voltage V developed across a standard 25 ohm resistance in series with the cell. Both potentials V and Vj were measured by three dial, 0.03% accurate, potentiom- eters (Honeywell 2746). The transient voltages, AVj = 25 I and AVy, across the standard resistance and wire were determined simultaneously by a two channel strip chart recorder (HP Mosley Model 7100B) with a full scale deflection of lmV using a chart speed of 1 or 2 inches/sec. Current was supplied to the dummy resistance, R-g, and wire by a six volt storage battery and adequate current control was maintained by the adjustable resistance R.. To prevent voltage fluctua- tions in the electrical circuit on 'switching, a ballast resistor was used to match the cell resistance. Switching was accomplished by a manually operated fast response relay. The complete circuit arrangement is shown in figure 2. 4. Corrections and Sources of Error As discussed previously, the major source of error is due to the finite wire diameter which neces- sitates a correction for the thermal capacity of the heater. Considering the last two terms of the eq (6) C(t) ri + Si 2Lai ln 2at 2toat M (8) eq (5) can be written to include this correction term as i (9) 6 62 tj*^ - T „( r » t i) = aTu [ Alnt + c ( t 2 ) - C(t l ) ] a^ VI Alnt AT - mJ x -T- 1 + c(t 2 )-c(t 1 ) Alnt (10) Subsequent substitutions and reduction allows eq (7) to be written as X = VI 4tt£ Alnt /AV I 1 V T I AI dR dT 1 + C(t 2 ) - C(tj) Alnt (11) — 3 —\ The specific heat correction for the organic liquids is approximately 3 x 10 W m deg decreasing with an increase in temperature, test result. The calculated corrections were applied to each liquid Other sources of possible error can be cited: (a) distortion of the one dimensional time-dependent temperature field by the finite length of the wire , (b) effect of an outer boundary, (c) convection, (d) radiation. With careful regard to the physical design of the apparatus all these corrections, with the exception of the radiation error, may be reduced to less than 0.2% [4]. The calculation of the heat transferred by radiation unfortunately poses a more difficult problem; Poltz [5] has dealt with this problem for the steady state situation of parallel flat plates; an extension of his work to the transient radial case has not as yet been made. Therefore for all results considered here the fluids were assumed either transparent or richly absorbent and the resultant correction negligible, therefore for the fluids reported all measurements do not require correction for any effect due to the absorptivity of the fluid within the range of accuracy stated. In addition to the above the only other major correction necessary is due to the finite time lag of the recorder, that is, the recorded voltage lags of the applied voltage. The lag of the recorder was determined by the utilizing phase relationship between two wave forms of the same frequency and ampli- tude. Wave forms derived from a function generator and from a tapping on the recorder slide wire were analyzed by means of the lissajous pattern on an oscilloscope. The sine of the phase angle existing between the two being obtained from the maximum horizontal deflection and the projected length on the x axis of the lissajous pattern. For the frequencies of interest the lag was obtained as a linear function of the velocity of the recorder pen. Taking into account all effects the results presented are believed accurate to ±0.5%. 5. Results A compilation of measured values of thermal conductivity at the different test temperatures are presented in Table 2. Each determination reported here is a mean of two to three measurements. With the exception of water, the temperature dependence of thermal conductivity data for all liquids was found to be, within the experimental error, a linear relationship of the form A Q + (#)' (12) where T is in °C. Here the temperature coefficient of thermal conductivity, — , was negative for all liquids with the exception of ethylene glycol and glycerol. The constants in the above equation, X n and A A — were found by means of a linear least squares analysis applied to the experimental values. Values of conductivity, temperature coefficient so found and the applicable range of temperature are listed in Table 3. Water showed an increasing thermal conductivity over a temperature range - 80°C with non- linear temperature dependence. The temperature conductivity data was best fitted by a parabolic curve, A„ + A,T + A T^ W m deg (13) 663 where the computed coefficients and the corresponding uncertainties are A Q = 564.8 (±3.4)10" A x = 1.99 7 (±0.17 2 )10 -3 Aj = -0.0085 7 (±0.0017 5 ) 10 -3 Table 2. Conductivity Data for Pure Organic Liquids and Water Temperature °C Thermal Conductivity Calculated from Equation (7) (Xcal 10 3 ) W m _1 deg -1 Water 13.5 590.0 13.5 592.2 20.8 596.7 19.0 597.8 31.0 619.8 30.5 619.5 32.0 623.0 42.0 629.7 41.5 629.7 62.0 657.5 62.0 656.8 82.5 670.3 82.5 670.3 Methanol 20.0 197.8 20.2 198.0 25.0 195.3 25.0 195.2 34.9 193.4 34.6 193.4 43.5 191.8 43.5 191.7 52.0 189.6 53.2 189.4 61.5 186.5 61.0 186.4 Ethanol 17.0 163.2 17.2 164.8 23.5 160.8 25.0 160.2 36.0 152.9 38.0 152.5 46.5 146.6 46.0 147.0 57.0 141.7 58.0 141.6 68.0 135.9 67.5 135.6 76.5 131.2 N-Propanol 16.0 151.4 15.7 151.3 25.0 149.8 25.0 150.8 38.5 145.1 40.0 144.9 55.0 141.2 Specific Heat Correction Calculated from Last Term of Equation (11) (Xcp 10 3 ) W m _1 deg -1 Total Thermal Conductivity (Xcal + Xcp) = X T 10 3 W m -1 deg -1 0.17 0.17 0.18 0.18 0.20 0.20 0.20 0.22 0.22 0.27 0.27 0.33 0.33 2.49 2.50 2.47 2.47 2.47 2.47 2.48 2.48 2.45 2.43 2.43 2.44 2.62 2.62 2.46 2.45 2.31 2.31 14 14 99 99 88 89 80 2.47 2.47 2.36 2.36 2.19 2.19 2.00 590.2 592.4 596.9 600.1 620.0 619.7 623.2 629.9 630.0 657.7 657.1 670.6 670.6 200.3 200.5 197.8 197.7 195.9 195.9 194.3 194.2 192.0 191.9 188.9 188.8 165.9 167.4 163.3 162.7 155.2 154.8 148.7 149.1 143.7 143.6 137.8 137.5 133.0 153.9 153.7 152.2 153.2 147.3 147.1 143.2 664 Table 2 (continued) Temperature °C. Thermal Conductivity Calculated from Equation (7) (Acal 10 3 ) W m -1 deg-1 N-Propanol (cont •d) 55.0 141.0 65.5 137.9 66.0 137.2 N-Butanol 15.2 151.5 15.2 151.5 20.5 151.0 20.5 151.0 34.5 146.5 34.0 146.4 47.5 144.1 64.5 140.9 64.5 140.8 82.0 138.4 101.5 134.6 Glycol 16.0 248.5 16.0 248.7 31.0 251.0 31.0 251.0 61.5 256.5 61.5 256.0 75.0 258.0 76.0 258.2 91.5 262.1 92.0 260.6 103.8 262.8 103.7 262.8 103.0 266.9 132.85 266.7 Glycerol 17.4 284.9 35.9 286.6 58.0 288.7 58.0 288.5 76.4 289.7 76.5 290.0 95.5 291.5 114.0 293.4 113.6 293.2 Ethylacetate 16.0 146.4 16.4 146.2 26.3 143.0 25.2 143.4 39.0 138.7 39.0 139.1 48.2 135.4 47.8 135.3 56.0 132.6 57.0 132.6 Benzaldehyde 16.0 149.5 16.0 150.1 34.5 142.4 34.5 142.3 51.6 136.0 Specific Heat Correction Calculated from Last Term of Equation (11) (Acp 10 3 ) W m -1 deg -1 Total Thermal Conductivity (Xcal + Xcp) = A T 10 3 W m" deg -1 2.00 1.99 1.99 2.52 2.52 2.42 2.42 2.34 34 22 20 20 87 69 1.71 1.72 1.66 1.66 1.54 1.54 1.46 1.46 1.38 1.38 1.35 1.35 1.24 1.24 1.40 1.26 1.15 1.15 1.03 1.03 0.95 0.86 0.86 2.37 2.35 2.34 2.34 2.34 2.32 2.34 2.35 2.32 2.32 2.51 2.52 2.52 2.52 2.52 143.0 139.9 139.2 154.0 154.0 153.4 153.4 148.9 148.8 146.3 143.1 143.0 140.3 136.3 250.3 250.4 252.7 252.7 258.1 257.5 259.4 259.7 262.5 262.0 264.2 264.1 268.1 267.9 286.3 287.9 289.9 289.6 290.7 291.0 292.4 294.3 294.1 148.8 148.6 145.4 145.7 141.0 141.4 137.7 137.6 134.9 135.0 152.0 152.6 145.0 144.9 138.5 665 Table 2 (continued) Temperature °C. Thermal Conductivity Cal :ulated from Equation (7) (A cal 10 3 ) W m~l deg - l Benzaldehyde (cont'd) 52.0 135.8 67.2 132.1 67.5 132.0 Aniline 21.5 160.3 21.0 160 . 1 41.2 159.8 41.2 159.3 57.0 158.7 57.0 158.3 77.5 157.5 77.5 157.4 93.0 155.9 93.0 156.7 100% Acetic Acid 17.5 156.4 17.5 156.4 30.5 154.9 30.5 154.9 47.5 151.2 47.5 152.2 63.0 148.8 63.0 148.7 79.0 146.5 Specific Heat Correction Calculated from Last Term of Equation (11) (Acp 10 3 ) W m - l dee~l Total Thermal Conductivity (Acal + Acp) = X T 10 3 W m~l deg~l 2.54 2.56 2.56 2.21 2.21 2.17 2.17 2.14 2.14 2.09 2.09 2.09 2.09 18 18 15 15 11 2.12 2.08 2.07 2.02 138.4 134.7 134.6 162.5 162.3 161.9 161.5 160.8 160. 159. 159. 157. .5 .6 .4 .9 158.8 158.6 158.6 157.1 157.1 153.3 154.3 150.9 150.8 148.5 Table 3. Fitted Values of Conductivity and Temperature Coefficients Name of the Liquid Thermal Conductivity of Liquid at 0°C. with un- certainty in Aq x 10 W m~l deg -1 Temperature Coefficient of Range of Thermal Conductivity with Measurements dA , ,. 3 C • uncertainty in -==■ 10 3 W m" 1 deg~2 Methanol Ethanol N-Propanol N-Butanol Ethylene Glycol Glycerol Ethylacetate Benz aldehyde Aniline Acetic Acid 205.0 176.4 159.2 156.9 (±0.5) (±0.5) (±0.5) (±0.4) 248.1 (±0.2) 285..0 (±0.2) 154.4 157.4 163.9 (±0.2) (±0.7) (±0.3) 161.9 (±0.4) -0.256 (±0.011) -0.573 (±0.009) (±0.011) (±0.007) (±0.002) (±0.002) (±0.005) (±0.014) (±0.004) (±0.007) -0.295 -0.208 +0.153 +0.080 -0.344 -0.349 -0.568 -0.171 20 - 61 17 - 76 16 - 66 15 - 101 16 - 133 17 - 113 16 - 57 16 - 67 21 - 93 17 - 79 6. Discussion In order to check, the reliability and accuracy of the apparatus, conductivity measurements were made first on water and methanol for which thermal conductivities are well established to at least ±1% over the temperature range - 100°C. 6 . 1 Water The usage of water as a thermal conductivity standard was first proposed by Riedel [6], To assure the validity of the apparatus, conductivity measurements were carried out on water over a temperature 666 range - 80°C. A comparison of this work with earlier research was limited to seven papers published since 1950. Figure 3, a deviation plot for eq (13), indicates that at 0°C. the values reported by Riedel [6,7, 8] and the present work are virtually the same. Slightly higher values are reported by LeNeindre and Vodar [9] in 40 - 70°C. range. Excepting the temperature range - 20°C, consistantly higher values are reported by Challoner and Powell [l] . The agreement with Poltz [5] is excellent, and in addition, in very good agreement with the equation recommended by the Sixth International Conference [11] on the properties of steam in the temperature range, which is also shown in Figure 3. 6.2 Methanol Twelve thermal conductivity determinations were made on Methanol. A comparison of this work with four major investigations is shown in figure 4. The agreement with Poltz [5] and Fritz and Poltz [16] is 0.1 and 0.4 per cent respectively. 6.3 Other Liquids The above agreement is evidence that the apparatus will yield reproducible and accurate results for most liquids and strongly supports the work of Poltz [10] since the measurements are made with two entirely different apparatus. 7. References Challoner, A. R. , and Powell, R. W. , Pro. Roy. Soc. (London) A238, 90 (1956). Riedel, L. , Chem. Ingr. Tech., 23 (19), 465- 484 (1951). C.R. Acad. So. Paris, Vol. 258, 3277 (1964). [10] Poltz, H., and Jugel, R. , "The Thermal Con- ductivity of Liquids IV," Int. J. Heat Mass Transfer, 10, 1075 (1967). Carslaw, H. S., and Jaeger, J. C. , "Conduction [11] The Thermal Conductivity Panel of "The 6th of heat in solids," Oxford University Press (1959). Horrocks, J. K. , and McLaughlin, E., Pro. Roy. Soc. (London), A273, 259 (1963). Poltz, H., "Die Warmeleitfahigkeit von Fliissigkeiten III," Int. J. Heat Mass Trans- fer, Vol. 8, 609-620, Pergamon Press (1965). International Conference" on the properties of steam, Paris, June 22-24, 1964, Technical Report No. 15. [12] Schmidt, E. , and Leidenfrost, W. , Chem. Ingr. Tech., 26, 35 (1964). [13] Vargaftik, N. B., and Oleshchuk, 0. N., Teploenerg. , 6, 70 (1959). Riedel, L. , Chem. Ingr. Tech., 23, 321 (1951). [14] Challoner, A. R. , Gundry, H. A., and Powell, R. W. , Proc. Roy. Soc. A, 245 (1241) 259-261 Riedel, L. , Forsch Gebiete, Ingenieurw, 11, (1958). 340 (1940). [15] Mason, H. L. , Trans. Amer. Soc. Mech. Engrs. Riedel, L. , Arch. Tech. Messen. , 1, 273 76(5), 817-821 (1954). (1954). [16] Fritz, W. , and Poltz, H. , Int. J. Heat Mass LeNeindre, B., Johanin, P., and Vodar, B. , Transfer, 5(2), 307 (1962). 667 WW — i cc < vV\^/V cc 4 /V > or cc hi Q cc 8* LlI oc CD > > tu U C IJ tu •H 4-1 4J 5 4J tn o •H to p. 4-> 3 .-h o o rH r-4 42 M to Ph •H XI rH O ^) to (O a .-h « 1 — 1 •h n) p-i p. o »! >>tH tu 4-1 U O (11 4-1 C o X. M o tO c a a) 4-1 to 4-1 m H tn 4-1 ■H o a) •H tn n tn ■H in 4-4 T3 tu tn 3 > O 4-1 M (d >^ tu (U rH 10 -H rH T1 0) i-i a XI >-i u o tO to o tn 4-4 TJ W ■•H tn tn a 4-1 to 3 tO A to 4-4 !-i S T3 tn 0) tu to 4-1 x: C/3 (1) o < « Fi en d o •H (1) 4J P-i u c CNJ M c 0) 01 to 4-1 UJ Q 2.0 20 LEGEND a RIEDEL [8] a CHALLONER 8 POWELL [I] + LENEINDRE.JOHANIN 8 VODAR [9] ■ SCHMIDT 8 LEIDENFROST 02] • VARGAFTIK a OLESHCHUK [13] A POLTZ [5] X CHALLONER.GUNDRY a POWELL f4] ® THIS WORK 30 40 50 60 70 80 TEMPERATURE C Figure 3. Deviation plot for the thermal conductivity of water against temperature + % < > UJ Q , 1 o 1 o 1 O 1 o 8 1 O 9 9 1 • *K 2 9 * • 2.0 • • 4.0 o • LEGEND RIEDEL [6] MASON [15] — a FRITZ 8 POLTZ [16] 6.0 p n X 9 POLTZ [5] THIS WORK 1 1 1 1 1 1 20 30 40 TEMPERATURE °C 50 60 70 Figure 4. Deviation plot for the thermal conductivity of methanol against temperature 669 Pressure and Temperature Dependence of the Thermal Conductivity of Liquids McLaughlin Chemical Engineering Department Louisiana State University Baton Rouge, Louisiana Differences in behaviour between the coefficients of shear viscosity and thermal conductivity of liquids with respect to changes in pressure, temperature and molecu- lar structure are reviewed. A lattice model is used to interpret the behaviour of the coefficient of thermal conductivity and comparisons are made with results based on the Enskog theory. Keywords: Enskog theory, effect of pressure on thermal conductivity, effect of temperature on thermal conductivity, lattice model, liquids, thermal con- ductivity. In the study of the pressure and temperature dependence of viscosity T| and thermal conductivity X of liquids, most attention has been devoted to the effect of temperature, as distinct from pressure, and in addition to viscosity rather than thermal conductivity. This has possibly arisen because of the exponential type dependence of viscosity on temperature at atmospheric pressure. Tl = 71 o exp (E^/RT) (1) which invited discussion in terras of the activation energy Ej, for viscous flow [_lj- The coefficient of thermal conductivity on the other hand being approximately linear in temperature at atmospheric pressure \ = \ Q - AT (2) and more difficult to measure precisely, tended to be neglected. It is the purpose of the present paper to summarize briefly some of the main differences in behaviour of the coefficients of viscosity and thermal conductivity and attempt to explain these in terms of a simple model which can also be used to determine the factors which control the pressure and temperature dependence of thermal conductivity. In comparing the experimentally determined behaviour of the two coefficients the main differences are as follows: 1. Firstly, as mentioned above, at atmospheric pressure the temperature dependence of viscosity is much steeper than the temperature dependence of thermal conductivity. For example, comparative figures for benzene show that the viscosity drops by 50% between 20 and 70°C while the thermal conductivity falls by only about 11%. 2. Secondly, the thermal conductivity of a liquid is insensitive in magnitude to changes in molecular structure whereas the viscosity of a liquid is very sensitive to such changes. This can best be illustrated by looking at the magnitude of both coeffi- cents for a series of closely related molecules. Table 1 lists values [2] for such 1 2 N.S.F. Senior Visiting Foreign Scientist Permanent address: Department of Chemical Engineering and Chemical Technology, Imperial College, London S.W.7., England Figures in brackets indicate the literature references at the end of this paper. 671 a series of aromatic hydrocarbons at the melting point and shows that although all the thermal conductivities lie within a few percent of each other substantial dif- ferences occur in the viscosities particularily on going from para - to meta - to ortho - terphenyl where there is a fortyfold increase. Table 1. Viscosities and Thermal Conductivities of a Series of Liquid Aromatic Hydrocarbons at the Melting Point benzene diphenyl p-terphenyl m-terphenyl o-terphenyl Tl x 10 4 Kg m" 1 sec" 1 8.2 14.3 8.0 53.2 322.0 X. x 10 2 W m" 1 deg" 1 14.8 13.8 12.8 13.7 13.2 A 0.032 0.014 0.011 0.0036 0.0069 3. Thirdly, although the thermal conductivities of most organic liquids are roughly the same lying between 8 x 10" 2 to 30 x 10 W m"'- deg~l there are significant differences in the temperature dependence of the coefficient. Again this can be illustrated with the series of aromatic polyphenyl hydrocarbons discussed above for which the coeffi- cent A of eq (2) has been listed in Table 1 on taking Aq as the conductivity at 0°C. The results show that A is greatest for the simplest molecule of the series, benzene , and decreases with increasing molecular complexity to give the smallest values for ortho - and meta - terphenyl. This again contrasts with the temperature depen- dence of the coefficient of viscosity which is steepest for the most complex mole- cules and lowest for the simplest. 4. Finally, although the coefficient of thermal conductivity at atmospheric pressure decreases with increasing temperature for normal liquids, Bridgman [3] showed that for a number of liquids he studied, the temperature dependence reversed sign above about 5000 bars beyond which the coefficient increased with increasing tempera- ture at constant high pressure. Again in contrast, no reversal in the sign of the temperature dependence of viscosity was obtained for the same liquids at high pres- sures. To set up a model to use in an attempt to explain these features of the behaviour of the thermal conductivity of a liquid it is necessary to make assumptions about (i) the structure of the liquid (ii) the molecular mechanism of transport (iii) the molecular shape (iv) the intermolecular forces In order to keep the model simple, the assumptions will be necessarily crude. Firstly, the liquid is assumed to be a quasi-crystalline assembly. This is not an unreasonable assumption for liquids well away from the critical region in view of the small volume change on fusion which usually occurs, of the order of 10%, and the fact that study of the X-ray patterns of liquids reveal only a small decrease in co-ordination number compared with the solid. For example, for argon [4] decreasing from 12 to 10.4 in the liquid close to the freezing point. In addition, the specific heats of solids and liquids close to the freezing point are not substantially different. This evidence suggests that energy- and volume-wise a solid-like model for the liquid would not be grossly in error. In theories of thermal conduction derived from statistical mechanics the structure of the fluid turns up in terms depending on the radial distribution function g * •'(r). In the absence of experimental values of g ' '(r) such terms can only be readily determined from experimental data if they are thermodynamic in form such as: £V_ NKT Jr 0'(r) g Q (2) (r) d 3 r (3) or for an s, q potential 0(r) = Ar" s - Br" q (s>q) Na 672 where U 7 is the conf igurational energy of the liquid [5], a the molecular diameter, and the other symbols have their usual significance. In the cell model of the liquid adopted, it is natural to assume that the molecules oscillate about fixed sites with a mean frequency v determined by the properties of the system. In a temperature gradi- ent the excess energy 1 du/dx, between molecules in layers a distance 1 apart, is handed on by 'collisions' with neighbors at the extremity of the vibration. The heat flux J can therefore be written [6] dT dx -n. 2 vl 7" A dx sr c - — v a dx (5) hence = /r C /a v (6) where n^ is the number of molecules per unit area of the liquid, C v the specific heat at constant volume per molecule, and a the nearest neighbor distance. In eqs (5) and (6) the lattice geometry has been assumed to be f.c.c. so that a-* = /2 v where v = V/N is the volume of the system per molecule and 1 = a/2/2. From eq (6) it is now possible [7] [8] to determine the factors which control the pressure and temperature dependence of thermal conductivity. In the first approximation we will assume C v is constant, hence on differentiation we obtain the set of equations: 1 / ax \ r 1 / a in v \ n x \ a? ) ~ -« L I ' \ a in v J J (7) I / SL \ = R Tl ( S In y \ 1 \ \ 3p L p t L 3 v a in v ; J (8) ) ay \ aT / (9) where a is the coefficient of thermal expansion and |3_ the isothermal compressibility. The quantity - ( a In v/ a lnv) is known in solid state theory as Gruneisen's constant and if this is assumed constant (1/X) (a\/aT)p should be linear in cf. As the coefficient of thermal expansion is largest for simple molecules, these have the steepest temperature dependence of thermal conductivity. Conversely as the more complex molecules have lower coefficients of thermal expansion they have a lower temperature dependence of thermal conductivity. Examination of a plot of (l/x)(aX/aT) against a for a series of liquids, however, shows [7] that although the results all lie on a single curve, this curve does not pass through the origin but cuts it at a ~ 0.5 x 10"^ deg"-'-. It is to be expected therefore that for liquids with a less than this value the coefficient of thermal conductivity would increase with increasing tem- perature. This has been shown to be the case with liquids at high pressure [8], where beyond about 5000 bars a values are generally less than 0.5 x 10"^ deg "1. Likewise it has been shown that the pressure dependence of thermal conductivity is substantially controlled by the isothermal compressibility as plots of available data [3] in the form (1/X) ( aX/3p) T are linear in £L. These conclusions therefore provide partial answers to points 3 and 4 raised above. With regard to point 2, calculated vibrational frequencies in liquids are not sensitive to molecular structure. Hence, provided the specific heat term in eq (6) does not include rotational or internal vibrational modes as is suggested by studies of the experimental behaviour of the ratio mX/kT] [9], and the transport properties of isotopic molecules [10] but is purely a lattice term, then insensitivity to molecular structure in X is to be expected. To summarise therefore, as eq (1) indicates, the temperature dependence of viscosity is controlled by energy parameters and hence one would expect sensitivity of this coefficient to changes in pressure, tem- perature, and molecular structure. Such changes would, in addition, be greatest for the most complex molecules. In contrast, as eq (7) and (8) indicate the pressure and temperature dependences of thermal conductivity are substantially controlled by volume parameters and hence, compared with viscosity, rela- tively insensitive to changes in pressure and temperature. 673 In view of the comparative success of the model it is pertinent to examine the relationship of the present results to those obtained from statistical theories. This is possible for hard sphere mole- cules of diameter a where the model corresponds to that of a particle in a rectangular cell potential with a frequency v given by [7] /8kT _L v = V 77 m 4(a - a) so that (d In v/S In v) can be evaluated to yield [7] Hi ) ■ -[i-i + H* + i >] <»> where y = (pV/RT) - 1 = o~/(a-cj) has been obtained from the free volume theory of a dense fluid of hard spheres [11]. Existing statistical theories of thermal conductivity of hard spheres, in the approximation that only binary collisions are important, all reduce essentially to the form first proposed by Enskog [11] X = ¥ [{MM!(i+tO + °-W] « 2 > where \ is the thermal conductivity of the corresponding dilute gas and b the hard sphere second virial coefficient. The first term of eq (13) arises from the diffusive type displacement of molecules which is small for dense fluids (y » 1). On discarding this term to bring the models on to the same com- parative basis it is found on differentiation that i(S) - ■« C 1 - ife + \ (y + 1) f(y) ] (13) where f(y) -» 1 for y >» 1. Eqs (12) and (13) are therefore seen to be substantially of the same form. A similar correspondence is found for the case of the pressure dependence of thermal conductivity which on the present model is given by: Hf) T - •,[>**♦«] (14) Broad agreement with experiment and close correspondence in form with the Enskog results therefore suggest the essential correctness of the conclusion that the pressure and temperature dependences of thermal conductivity of liquids are determined by the coefficients of isothermal compressibility and thermal expansion respectively. Finally it is well known that for a number of simple fluids when X is plotted against density for various temperatures the isotherms are either co-incident or very close together. This also is in agreement with results obtainable from the present model as from eq (10) (S).-fe (15) and if evaluated on the simple harmonic oscillator model for v [6] f & \ dT ( ■& ) = (16) The rectangular cell potential can be looked on as the limiting case of anharmonicity and hence (SX/dT) for liquids would be expected to lie within the limits given by eqs (15) and (16) and hence closely spaced together or coincident, which is in agreement with experiment [8], 674 References [1] Glasstone, S. , Laidler, K. J., and Eyring , [7] "The Theory of Rate Processes", McGraw-Hill, [2] Horrocks, J. K. and McLaughlin, E. Proc. [8] Roy. Soc. (London), A 275 . 259 (1963). [3] Bridgman, P. W. "The Physics of High Pressure," [9 J Bell and Sons, London (1949). [4] Eisenstein, A. and Gingrich, N. S. Phys. Rev. [10] 62, 261 (1942). [5] Rowlinson, J. S. "Liquids and Liquid Mixtures," Butterworths, London (1959). [11] [6] Horrocks, J. K. and McLaughlin, E. Trans. Faraday Soc, 56, 206 (1960). Horrocks, J. K. and McLaughlin, E. Trans. Faraday Soc. 59, 1709 (1963). Kamal, I. and McLaughlin, E. Trans. Faraday Soc. , 60, 809 (1964). McLaughlin, E. Chem. Rev., 64, 389 (1964). Horrocks, J. K. , McLaughlin, E., and Ubbelohde, A. R. Trans. Faraday Soc, _59, 1110 (1963). Hirschfelder , J. 0., Curtis, C. F. and Bird, R. B. "Molecular Theory of Gases and Liquids' Wiley, New York (1954). 675 Thermal Conductivity of Several Dielectric Liquids of Low Boiling Point R. C. Chu, O. Gupta, J. H. Seely International Business Machines Corporation Systems Development Division, Poughkeepsie, New York This paper presents recent thermal conductivity measurements for three, low- saturation temperature, dielectric liquids: FC-78, FreonC51-12, and Freon 113. The measurements were made through the mechanism of steady-state, radial heat- flow in a thin annulus. The system was calibrated using air and distilled water. Emphasis is placed on comparing the results obtained against data of other investi- gators and the measurement techniques used. An assessment of the accuracy of the measurements and the effect of natural convection and air solubility are also discussed. The experimental apparatus used and the details of instrumentation are illustrated by schematics and photographs. Key Words: Thermal conductivity, heat transfer, coolant property, measurement technique. 1. Introduction Developments in electronic components and packaging techniques have created a dual problem in cooling. On one hand, heat fluxes have been increased by orders of magnitude, so that high values of heat transfer coefficients are needed. In addition, when the response times of circuits are measured in nanoseconds or picoseconds, the temperature difference between the components becomes an im- portant consideration. The above criteria are best satisfied by liquid cooling techniques, particularly subcooled-flow- boiling systems and subcooled-pool or saturated-pool-boiling systems. Intimate contact between liquid and components is often necessary in such systems, so that dielectric properties, chemical inertness, and material compatibility must also be considered. A further restriction exists in elec- tronic cooling; namely, silicon and germanium semiconductor devices must be maintained at relatively low temperatures (100°C or less). In recent years, several manufacturers have developed fluorinated heat-transfer liquids that satisfy most requirements. Because these products have not had a wide- spread use, some of their physical properties are not well-established. In particular, thermal con- ductivity, because it is of paramount importance in a thermal system and because it is somewhat diffi- cult to measure, has been of concern to designers of heat transfer systems. For comparing thermal characteristics of these liquids, we decided to perform our own conductivity measurements, pri- marily intending to obtain accurate results for correlating heat transfer data. After an extensive searching of literature in the field, the "thin annulus" technique which is most compatible with our instrumentation capability was selected. 677 2. Experimental Techniques 2. 1 Theoretical Considerations The "thin annulus" technique for measurement of thermal conductivity is based on the following radial heat-flow equations: 2,XMAT) =CXAT Ln r ? /r. where: length of annulus, m r , r = inner and outer radius of annulus, m AT = temperature difference across the liquid annulus, °C k - thermal conductivity w m~ deg C = constant of test configuration, m 2 IT L Ln v 2 /r l Rearranging equation (1), gives X = CA* (2) Equation (2) is derived with the assumptions that heat flow takes place purely in the conduction mode. It is obvious that this assumption is somewhat unrealistic, as natural convection always exists in such a configuration. However, if AT and annulus thickness are properly selected, it is possible to minimize the natural convection effect to a negligible degree. Indeed, this is the only way one can make the "thin annulus" method work. For our measurements, an annulus thickness of 0. 025 and 0. 080 inch and a 5. inches length was selected. The reason for choosing two different annulus di- mensions is to determine the natural convection effect due to annulus thickness. 2. 2 Description of Apparatus The apparatus, which is made up of a series of concentric cylinders, is shown in figure 1. The innermost cylinder is made of a ceramic material and has four inner passageways, each of which houses a 0. 005 inch diameter nichrome wire. This arrangement ensures a uniform heat flux along the length of the heat source. The ceramic heater is clad with a copper cylinder which serves as the inner surface of the annular test section. The outer surface is formed with another copper cylinder which is positioned by means of two nylon rings. A high degree of concentricity is maintained between the rings and the cylinder to guarantee a uniform annulus. The outer portion of the apparatus forms a water jacket which serves as a heat sink. Provisions are made for mounting eleven iron-constantan thermocouple junctions. Two sets of diametrically opposed junctions are mounted to record the internal temperature of the copper cylinder cladding around the heater -- one set near the top and one set near the bottom of the apparatus. Another pair of junctions, 180° apart, are located at the center to record the temperature of the inner surface of the test annulus. The five remaining thermocouples are used for monitoring the outer section of the annulus and water temperatures. Arrangements are provided for filling the test section with liquid and for purging the system of air. A schematic of the liquid conductivity experimental testing system is shown in figure 2. The experimental equipment is shown in figure 3. 678 2. 3 Test Procedure Liquid is poured into the test section through one of the venting tubes to form a liquid annulus. When the filling operation is completed, the system is turned on, power and temperature readings are taken when thermal equilibrium is established. Various heat-flux levels are used to evaluate the natural convection effect. Two levels of sink temperatures are applied for determining the tempera- ture coefficient of conductivity. Similar tests are repeated for all three liquids. 2. 4 Results Thermal conductivity measurements obtained on three dielectric liquids, FC-78, (a product of the 3M Company), Freon C-51-12, Freon 113 (products of the duPont Company) are shown in figure 4 as a function of heat flux. Conductivity versus temperature data of the three liquids are compared with data of other investigators (see figure 5). It is evident that the data obtained from the 0. 080 inch thick annulus test section shows a sub- stantial natural convection effect. However, it is equally evident that the thinner annulus (62 = 0. 025 inch) has virtually eliminated the natural convection heat transfer in the system. These comparative results easily suggest the postulation that the discrepancies of other reported data may very well be caused by this important factor. 3. Discussions 1. One of the reasons that we chose two different annulus thicknesses was to see how this critical dimension affects measurements on various kinds of fluids. The data shown in figure 4 can be used to estimate, quantitatively, the natural convection effect due to wider annulus thickness. Further- more, if we extend the 6? line, shown in figure 4-A, we will see that it intersects with the 6j line at about 100 W/M^ heat flux, which can be interpreted as the maximum heat flux allowable to avoid natural convection for the annulus of 62 thickness. 2. One of the most striking characteristics of these dielectric liquids is their relatively high air solubilities (see example in figure 6). Our experience indicates that it is extremely difficult, if not impossible, to maintain a consistency of these liquids in air solubilities, as they are extremely tem- perature dependent at a given pressure which for our case is atmospheric. Because these liquids all have negative temperature coefficients in air solubilities, this can cause an increase of observed thermal conductivity when liquid temperature is raised to a higher level. This effect seems to be particularly pronounced with wider annulus thickness (see results in figure 4). 3. Since natural convection is affected not only by annulus thickness (6), but also by AT and liquid properties (see Table 1) such as thermal expansion coefficient ()3) and kinematic viscosity (y) , it is generally true that with a given 6 and AT, better accuracy can be expected for liquids with higher kinematic viscosity and lower coefficient of thermal expansion, like water, and will show considerable error for the converse case; that is, lower y and higher j3 like air. We did observe these aforemen- tioned results during our system calibration, using water and air. 4. Conclusions and Recommendations The thermal conductivity values summarized in this paper represent our recent measurements in this area. Based on our instrumentation capability and extremely good repeatibility of data, it is our judgment that these results do have good absolute accuracy. We have found the information to be quite meaningful and useful in making heat transfer correlations of these liquids. We would like to suggest that these liquids be further evaluated with broader temperature range and various solubilities, to determine the effects of these variables on thermal conductivity. 5. Acknowledgment The authors wish to thank all members of IBM's Poughkeepsie Heat Transfer Development De- partment for their encouragements and discussions during the pursuing of this work. 679 Table 1. Properties of the three dielectric liquids. Manufacturer Nominal Boiling Point F Density at 77°F - lb /ft Specific Heat at 77°F - Btu/lb-°F Thermal Conductivity at 77 F - Btu/hr-ft- F Kinematic Viscosity at 77 F-c. s. Heat of Vaporization at the Boiling Point - Btu/lb Coefficient of Expansion ft /ft - F Surface Tension at 77 F - dynes/cm Dielectric Constant at 77 F and lkc Dielectric Strength at 77 F rr— e mil Solubility of air in liquid at 1 atm. & 77 F mol. % FC-78 Freon-113 Freon C51-12 3M Co. DuPont DuPont 122 117. 5 113. 5 106. 97. 7 104. 4 . 248 . 218 . 266 See ; figure 5 . 44 . 42 . 61 41. 63. 1 40. . 009 . 00078 . 00073 13. 19. 11. 6 1. 81 2. 41 1. 80 430 310 420 . 50 . 12 . 15 (estimated) 6. Bibliography [1] Grimm, T. C. , and Kreder, K. R. , "Appara- tus for Measuring Thermal Conductivity of Liquids From -70°C to +200°C," Proceed- ings of the Conference on Thermal Con- ductivity, 1965. [2] Ingersoll, L. R. , Zobel, O. J. , and Ingersoll, A. C. , "Heat Conduction, " McGraw-Hill Book Company, New York, 1954. [3] Jakob, M. , "Heat Transfer, " Vol. 1, John Wiley &c Sons, Inc. , New York, 5th Printing Sept. 1956. [4] Kern, D. Q. , "Process Heat Transfer, " McGraw-Hill Book Company, New York, 1950. [5] Mason, H. L. , "Thermal Conductivity of Some Industrial Liquids from °C to 100 C, " ASME Transactions, lb_, 1954. [6] Technical Information by the Chemical Divi- sion of the 3M Company, "3M Brand Inert Fluorochemical Liquids," 1966. [7] Technical Bulletin by the Freon Products Division of duPont, "Freon C-51-12 Fluoro- carbon, " 1967. [8] Van der Held and Van Drumen, "A Method of Measuring the Thermal Conductivity of Liq- uids, " Physica, 15, 1945. 680 H 2 OUT VENT 8 LIQUID LEVEL HEATER LEADS NYLON RING -NO I CYLINDER NO 2 CYLINDER NO 3 CYLINDER NO 4 CYLINDER \ T x THERMOCOUPLE LOCATIONS A WATER CIRCULATION TEST LIQUID H 2 IN NYLON RING Figure 1. Liquid thermal conductivity apparatus. CROSS- SECTION A-A POWER SUPPLY \ no V.A.C. IIOVA.C. MULTISWITCH LIQUID HERNIAL CONDUCTIVITY APPARATUS 1 10 VAC. K3-UNIVERSAL POTENTIOMETER Figure 2. Schematic of the liquid conductivity testing system. 681 Figure 3. Photograph of the experimental equipment. A) Test section. B) Experimental system. 682 r 4 i r (A) 8 1 = 0.080 J L BEFORE DEGASSING (22° TO 26°C) AFTER DEGASSING (22° TO 26°C) AFTER DEGASSING (35° TO 40°C) BEFORE DEGASSING (29° TO 34°C) AFTER DEGASSING (39° TO 45°C) AFTER DEGASSING (29° TO 33°C) J I I I I I I I 4 5 6 7 8 9 1000 HEAT FLUX ( W/m 2 ) 3 4 5 6 7 8 9 10 4 BEFORE DEGASSING (23° TO 26°C) AFTER DEGASSING (23° TO 26°C) AFTER DEGASSING (35° TO 40°C) BEFORE DEGASSING (27° TO 32°C) AFTER DEGASSING (38° TO 44°C) AFTER DEGASSING (28° TO 32°C) J I I I I I I 3 4 5 6 7 89I0 4 HEAT FLUX (W/m*) BEFORE DEGASSING (23° TO 26°C) AFTER DEGASSING (23° TO 26°C) AFTER DEGASSING (35° TO 40°C) BEFORE DEGASSING (27° TO 30°C) AFTER DEGASSING (40° TO 45°C) AFTER DEGASSING (29° TO 32°C) J I I I I I 4 5 6 7 8 9 1000 HEAT FLUX ( W/m 2 ) 3 4 5 6 7 8 9 10 4 Figure 4. A) Thermal conductivity of FC-78 vs. heat flux. B) Thermal conductivity of C51-12 vs. heat flux. C) Thermal conductivity of Freon 113 vs. heat flux. 683 FC-78 (EARLY 3M DATA) F- 113 (EARLY OUPONT OATA ) F 113 . (RECENT DUPONT OATA) F-II3 ( AUTHORS' DATA) >- (AUTHORS' DATA) G- (RECENT 3M DATA) - (AUTHORS' DATA ) Figure 5. Conductivity vs. temperature of the three liquids compared with data of other investigators. C5I- 12 (EARLY DUPONT DATA) TEMPERATURE (°C) Figure 6. Air solubility of FC-78. FC-78 (ESTIMATED) 90 100 TEMPERATURE (°F) 120 130 684 Thermal Conductivity of Two-Phase Systems 1 2 Alfred L. Baxley , Nicholas C. Manas , and James R. Couper Department of Chemical Engineering University of Arkansas Fayetteville, Arkansas An experimental-theoretical study was conducted to establish a theoretically sound and numerically accurate technique for predicting two-phase system thermal conductivity. Experimental data were obtained with the aid of newly designed concentric-sphere type apparatus. Thermal conductivities were measured for repre- sentative volume fractions of several liquid-solid and liquid-liquid systems. A digital simulation technique was proposed for the prediction of two-phase thermal conductivities. The model used as a basis a constant density function random placement of the discontinuous phase within a large cubical system. In the design of the model, consideration was given to three-dimensional heat conduction, potential field interaction, the thermal conductivities of both phases, and the volume fraction of the discontinuous phase. The model was tested along with the models of eight other investigators on data of five investigations, including the authors' data. The authors' model predicted thermal conductivities with a mean error of twenty percent, an error variance of 1.35, and it was biased below the experimental values by an average of eleven percent. Key Words: Conductivity, thermal conductivity, two-phase, liquid-solid systems, liquid-liquid systems, suspensions, emulsions. 1. Introduction In the design of heat transfer equipment the evaluation of heat transfer coefficients for hetero- geneous two-phase liquid-liquid or liquid-solid systems has been troublesome. In such a design it is necessary to know the thermal conductivity, specific heat, viscosity, density and coefficient of ther- mal expansion of the heterogeneous system. During the past eighty years, numerous methods have been proposed for the calculation of thermal conductivity of two-phase systems. These methods [3, 4,5, 6, 8, 10, 11, 13, 14,15,16,17, 19,20, 21, 23, 24, 25, 26,28] have most commonly assumed that the effective thermal conductivity of a real two-phase system is the same as for a system of simple geometry having the same volume fraction of the two phases. The effective conductivity of the simplified model is then found from the analytic solution of Fourier's equation with appropriate boundary conditions. In general, most of these techniques have been found to be approximate and unrelated to each other. The assumptions of thermal conductivity not varying with temperature, orderly arrays of well-defined geometric configurations in established distributions, and particle non-interaction are the most important of those which have been used in the development of theoretical models. The experimental data are meager and the experimental techniques require considerable patience and care. Even under the most desirable conditions, the accuracy of existing correlations cannot be Presently employed at Celanese Chemical Company, Corpus Christ! , Texas Presently employed at Hunble Oil and Refining Company, Bay town, Texas 685 expected to be better than about + 20%. As a result, the design engineer occasionally uses the ther- mal conductivity of a liquid phase which is usually the continuous medium and over designs the heat transfer, a costly compromise brought about by the lack of understanding of the existing phenomena and the lack of experimental data. 2. Theoretical Analysis Thennal conductivity is defined for a homogeneous, isotropic medium by the Fourier equation: X = . -Q/ A (1) where Q is the heat flow rate, A is the area of heat transfer, aT/aX is the applied temperature gra- dient, and X is the thermal conductivity. According to the Fourier equation, the concept of thermal conductivity dictates that it represent a ratio of heat transfer through a constant area to an applied constant temperature potential. The definition further implies that the material is homogeneous and isotropic. One of the results of the above definition is that the thermal conductivity is a basic intensive property of the material. As the basic concept of thermal conductivity is applied to a two-phase system, some of the above criteria are immediately violated. Two-phase systems are neither isotropic nor homogeneous , and the resultant thermal conductivity cannot be defined as an intensive property of the material. Although thermal conductivity for two-phase systems does not adhere to some of the original criteria, it is nevertheless a useful concept. A photomicrograph of a liquid-solid suspension revealed that the suspension was a randomly oriented unsymmetric spatial array of irregularly shaped particles in a continuous medium. Further, some of the particles seemed to be in contact, and there appeared to be no basic repeated structure. Observations such as these have led historically to consideration of the following system variables: liquid and solid thermal conductivity, volume fraction of solids, particle orientation, particle dis- tribution, particle shape, and particle "bridging". The classical approach has been to combine all or part of the above variables into a predictive type "equation". Such equations have usually been developed considering the basic repeated unit of spatial configuration, for example, a cubic array of cubes. However, the photomicrograph revealed that such a basic unit is not easily defined and the authors believe that an entirely different approach would prove more satisfactory. A detailed survey and critical review of the literature [7] clearly indicated the need for a numerical-statistical approach for predicting the conductivity of suspensions and emulsions. Deissler and Eian [4] applied a numerical technique to a two-dimensional array of cylinders. The temperature distribution at regular intervals in the model were determined using finite difference techniques and a relaxation procedure. They suggested that their model could be improved by considering a random distribution of solid cubes in a three-dimensional cubical system. Such an approach would be based on the premise that any liquid-solid system can be represented to any degree of approximation by a three-dimensional cubical system composed of small liquid and solid cubes. Effectively, any irregu- larly shaped particle can be "built" to any degree of approximation by arranging a sufficient number of small solid cubes according to a predetermined format. Further, particles could be placed into the large cubical system according to any specified statistical distribution. Following such a system synthesis would be the application of isothermal and insulation requirements to the proper planes, the application of a temperature potential, the determination of system temperature profiles, and finally, the calculation of heat flows and thermal conductivity. 2.2. Problem Documentation The first step in the procedure is to establish system dimensions, boundary conditions, and initial system thermal conductivity. The physical system was "built" to the limit of the computing machine capacity and was represented as a three-dimensional matrix. (The computer available was an IEM 7040 with a 16K memory.) The dimensions are shown in figure 1 along with the directional convention. The front and rear temperatures (TFRONT and TREAR) are isothermal, TFRONT being larger numeri- cally than TREAR. In this particular study, TFRONT = 1.10°C (2°F) and TREAR = 0.55°C (1°F). The planes IK and JK are insulated so that all net heat flows in the K direction. Initially, all positions 686 in the system are occupied by the conductivity of the continuous phase, making the initial system thermal conductivity equal to that of the pure liquid. Following this the number of solid cubes was determined and each was randomly placed. The system dimensions were selected to provide the largest conventional three-dimensional matrix which would fit into the computer memory. The dimensions of the matrix were 8 x 8 x 11 resulting in 704 system cubes. Therefore, the number of solid cubes required to build a given volume fraction (f) was 704 times f- Programming features were provided to neglect fractions of solid cubes. Random coordinates (I, J, K) were produced by a 0-1.0 constant density random number generator. For each solid cube to be placed in the system, three random numbers between 0-1.0 are generated in the subroutine. By manipulation in the simulation program, these three random numbers are converted into I, J, K coordinates. If a random set of coordinates is duplicated, it is discarded and another set selected. When a solid cube is placed into the system, the solid thermal conductivity replaces the liquid thermal conductivity stored previously in that location in the computer program. Step three is to determine the average working thermal conductivities and assign the initial temp- erature distribution. The equation for the series model, as given by Jakob [9], is used to compute the average working conductivities between adjacent cubes. _ x c x d (2) x e o.5X c + 075X~[ where X is the average thermal conductivity, X is the thermal conductivity of the continuous phase, and X^ is the thermal conductivity of the discontinuous phase. Next, an initial temperature distri- bution is assigned to the system cubes, assuming them as the pure liquids. In this particular problem solution, a 0.55°C (1.0°F) AT with a 0.055°C (0.1°F) AT per k-plane was used. However, any convenient temperature drop could be assumed. Step four is to compute the actual temperature distribution. As the program is documented, there exists a unique solution for system temperatures. The condition which must be met by every cube temperature for a solution to exist is the Laplace equation 3?T . + 3 2 T + 3 2 T . (3) 3l2 3J2 3k2 Also, each temperature in the system must be bounded by TFRONT and TREAR. In application, Laplace's equation merely requires that the temperature at every cube be constant. This condition is satisfied only when there is no net heat accumulation in each cube. Therefore, a heat conduction balance can be written around every cube in the system, excluding the front and rear K planes. The basic system section around which each heat balance is written is shown in figure 2. Each cube represented by the numbered circles is assumed to be isothermal. Heat flow to the center cube, T7, is regarded as posi- tive. The resulting heat balance is i_^6 CO R 7 T. *. (T, - T 7 ) where L is the distance between the cube being considered and cube 7- Ry is the residual heat flow in the relaxation scheme. Application of this balance to all system cubes, 'except the 128 front and rear cubes, results in a system of 576 simultaneous, linear equations. Since all conductivities are known, each equation contains 7 unknowns, and no two equations contain the same 7 unknowns. A solution to such a large number of linear equations can be accomplished only by a numerical relaxation technique. Further, to program the technique of relaxation requires superimposing on it an iterative scheme. This scheme is described by Varga [27] ■ In applying Varga 's scheme to the solution of the 576 heat balances, the following basic conver- gence equation results ~i = 6 Zm 7 (t i - V (5) 687 Equation 5 states that a better estimate of any cube temperature results from adding a group of terms to an initially specified cube temperature. This group of terms is seen to be a constant divided by the sum of the surrounding working thermal conductivities and multiplied by the thermal residue, R. This is quite logical when it is realized that a negative R results frcm T 7 being too high. The appro- priate action is thus to reduce T7 by the aforementioned amount. If a positive R occurs, T7 is too low and should be increased in the same manner. This basic convergence equation is applied to each cube temperature repeatedly until the sum of the absolute values of the residuals is less than or eaual to 0.55°C (1°F). The final step in the program is to determine heat flows and effective system thermal conductivity. The solution for temperature distribution results in the resolution of heat conduction from each cube into three orthogonal vectors. Since for the entire system, all net heat is being conducted in the K direction, this is the quantity of heat on which the system thermal conductivity should be based. Further, with the correct temperature distribution established, it follows that heat conducted through all K-planes is identical. The net K-plane heat flow is easily calculated by sunming K-directed heat flows between every cube in adjacent K-planes. With this heat flow determined, effective system ther- mal conductivity is readily obtainable from a modified Fourier equation: , = 11_Q e (8)(8)(TFRONT - TREAR) (6) Initially, no model was proposed specifically for liquid-liquid systems, for it was believed that the model developed might be satisfactory or could be extended to include liquid-liquid systems with minor modifications. 3 . Experimental 3-1 Thermal Conductivity Measurements A concentric-sphere type of thermal conductivity cell was designed and built by Baxley and Couper [1]. A schematic drawing of the cell is shown in figure 3- The annular space, which was 0.444 x 10~2 m. (0.175 inches) between the inner and outer spheres, contained the systems to be tested. Heat was generated electrically at the cell center by passing direct current through a specially con- structed carbon resistance heater. The heat generated passed radially through the cell and the test systems. Thermal conductivity of the experimental systems was calculated frcm the equation for three- dimensional radial heat conduction in a sphere. The rate of neat generation was obtained by measuring the current flow through and the voltage drop across the carbon resistance heater in the inner sphere. The meters used were certified as accurate to + 0.25% of full-scale reading. The temperature difference across the test systems, of the order of 0.55°C (1°F), was determined by a Leeds and Northrup K-3 Potentiometer, using copper-constantan thermocouples. The entire thermal conductivity apparatus was immersed in a constant temperature oil bath during test runs so that the temperature distribution through the cell was uniform. Large excursions in bath temperatures were accomplished using an immersion heater or a refrigeration unit. The bath temperature was controlled in a classical fashion by a laboratory mercurial thermoregulator-relay- light bulb circuit. The resulting bath temperature variation was maintained at less than + 0.0055 °C (0.01°F). A recording potentiometer was used to continuously monitor temperatures throughout the systems making possible the observation of transient temperatures and indicating when steady-state had been attained. The average error expected in the measured thermal conductivity values was H%. A detailed error analysis and a more thorough description of the test apparatus appears in a previous publication [1]. 3.2 Experimental Systems Each suspension used in this study was observed to experience no appreciable settling at 10°C (50°F) for a one hour period. One hour was sufficient time to make all thermal conductivity measure- ments for any suspension. The conductivities were experimentally determined for suspensions of polypropylene glycol (PPG), aluminum oxide-PPG and selenium-PPG. The suspensions were prepared by 688 slowly mixing predetermined weights of the finely divided powders into the liquid. After each suspen- sion was charged to the cell, it was evacuated to 100 microns absolute pressure to remove any trapped air. The volume fraction of solids was varied from 0.1 to 0.5- Data were taken on all three systems between 1.7°C (35°F) and 10°C (50°F). In the emulsion phase of the work, solutions of hydroxyetbyl cellulose were dispersed in poly- propylene glycol and in a commercial plasticizer known as Flexol. The volume fraction varied from 0.1 to 0.3 for the dispersed phase. The experimental temperature range was from 4.5°C (40°F) to 60.0°C (140°F). 4.0 Results 4.1 Liquid-Solid Systems The thermal conductivities obtained from the model were compared with those predicted by eight other models. The eight other models were those developed by Russell [25], Lichtenecker [15] , Jefferson [11], Maxwell [17], Rayleigh [23], Bruggeman [3], Meredith and Tobias [19,20,21], and the series model as shown by Jakob [9]. All nine models were tested on liquid-solid data of Lees [14], Jefferson [11], McAlister and Orr [18], Johnson [12], and the authors' data. Frcm Table 1 it can be seen that experimental values of suspension thermal conductivity are more closely related to liquid thermal conductivity and volume fraction than to thermal conductivity of the solid. For example, compare the results of the aluminum-water system (X„ = 0.657, X d = 204, f = 0.175) with those for the aluminum-grease system (X c = 0.284, X d = 204, f = 0.180). If the same solid is used in almost the same volume fraction, an increase in liquid thermal conductivity by a factor of 2.3 increases the effective conductivity by a factor of 4.0. Further, for the aluminum-water system, a change in f from 0.055 to 0.210 changes the effective conductivity by a factor of 2.3. Now consider the zinc-grease system (X c = 0.282, X^ = 112, f = 0.22) and the aluminum-grease system (X„ = 0.284, X,j = 204, f = 0.24) and note that an increase in the solid conductivity by a factor of 1.82 causes the effective conductivity to increase from 0.420 Wm -1 deg -1 to 0.429 Wm-ldeg-1. This Increase is not significant . TABLE 1 Comparison of Other Investigators' Data With Authors' Model Liquid-Solid Systems at 10°C SYSTEM Wm~ldeg~l / d 1 Wm~- L deg -i - Wm' -fiP -1 ^ofS 1 -1 ^deg ± Wm -^deg x Aluminum-Water 0.657 204 0.055 0.115 0.175 0.210 0.761 0.969 1.401 1.730 0.724 0.802 0.976 1.042 Aluminum-Grease 0.284 204 0.070 0.120 0.180 0.240 0.320 0.320 0.330 0.349 0.429 0.586 0.328 0.354 0.429 0.500 0.972 Zinc-Grease 0.282 112 0.020 0.080 0.160 0.220 0.330 296 322 380 420 557 0.290 0.323 0.392 0.498 0.795 Copper-Water 0.657 382 0.055 0.115 0.175 0.210 0.245 0.295 0.744 0.934 1.141 1.280 1.436 1.826 0.721 0.820 0.932 1.159 1.273 1.730 For data on other systems and other models see reference [2] 689 If one compares the thermal conductivities of the aluminum-water (X c = O.657, X, = 204) and copper-water (X c = 0.657, X = 382) systems, it is seen that the effective conductivity is higher for the system with the lower solid conductivity. This result dramatically emphasizes the influence of system variables other than X c , x^ and f. These variables may include particle description, spatial distribution, and particle-particle contact. Table 2 is presented to demonstrate the comparison between Baxley's model and experimental data obtained by use of the spherical cell. TABLE 2 Comparison of Authors' Data with Authors' Model Liquid-Solid Data SYS™' 1 *d x c f *exp * M 0d el Wm~ldeg - 1 V4n -1 deg~l Wn-ldeg~l Wm~ldeg-1 Aluminum Oxide-PPG 38.9 0.140 10°C 0.100 0.198 0.166 0.200 0.429 0.219 0.300 0.741 0.338 0.400 1.192 0.635 0.500 1.840 1.398 0.100 0.230 0.160 0.200 0.465 0.228 0.300 0.879 0.610 0.400 1.580 1.717 0.100 0.180 0.167 0.200 0.219 0.208 0.300 0.315 0.274 0.400 0.422 0.374 0.500 0.486 0.512 Aluminum-PPG 216 0.137 1.7°C Selenium-PPG 5-19 0.140 10° C For data on other systems and other models see reference [2] On the 51 systems tested, the authors' model gave predicted conductivities with a mean error of 20.5$, an error variance of 1.350 and is biased below the experimental value by an average of 10.9555. Since the predictability of a model improves as the mean error and error variances are reduced, the authors' model is slightly better than the best model of any previous investigation [10,11]. The mean errors of all other models except the series model and Russell's model compare reasonably well with the authors' model. These two models each had a mean error in excess of 32%. It was observed that both of these models have high errors even for systems of low severity , i.e., low values of X(j/X c or low values of f . It was quite unexpected that Jefferson's model, with limiting assumptions of linear heat flow and a cubical array of spherical particles would predict conductivity as well as the authors' model. It was also unexpected that Jefferson's model was better than Maxwell's model. Prior to this study, it was generally expressed in the literature that Maxwell's model was the best model for all types of systems . In general, the authors' model was found to experience larger errors as severity parameters X d /X c and f were increased. In thirty of the 57 cases tested, the authors' model predicted thermal conductivity with a cumulative average error of less than 10%. Extensive information on the liquid- solid suspensions may be found in reference [2]. 4.2 Liquid-Liquid Systems None of the theoretical models predicted the thermal conductivity with consistent accuracy. Pre- dicted results based on all models with the exception of the authors' gave approximately linear func- tions of temperature. As a typical example of such behavior, see figure 4. For more complete results on the liquid-liquid systems, the reader is directed to reference [22]. 690 The authors' model was tested using the liquid-liquid data obtained. It yielded data with the mean error approximately the same as for liquid-solid systems. However, Jefferson's model [11] gave effective thermal conductivity values at low volume fractions less than 0.3, with an error less than 11.4$. This result might have been expected since Jefferson's model involved the distribution of spherical particles in a cubical section. Research is presently underway to improve the authors' model. 5. Conclusions A stable, explicit, numerical solution has been developed for the thermal conductivity of three- dimensional, nonhcmogeneous media. The model represents a fresh approach to the problem in that it simulates heterogeneous system thermal conductivity. The model is a general numerical solution to potential field problems, and is particularly attractive for heterogeneous systems. The model is therefore not restricted to heat conduction, but can be readily extended without hesitation to any heterogeneous system property for which the Laplace equation is applicable. The authors ' model was found to predict thermal conductivity for all cases with a mean error, error variance, and error bias of 20.5$j 1.35%, and 10.95$ s respectively. The model was found to predict thermal conductivity slightly better than the best model of previous development [10,11]. The variance of the predicted thermal conductivities for a suspension of a given volume fraction solids, due to a variety of solid cube placements, was found to be generally quite small. It was less than 0.010 for 69$ of the cases tested. Variances larger than 0.010 were found for systems with large values of X d /A c and f . 6 . Acknowledgement The authors wish to acknowledge the financial support of this project by the National Science Foundation through grants NSF GP-1152 and GK-1151. 7. References [1] Baxley, A. L. and J. R. Couper, Thermal Conductivity of Two-Phase Systems: Design and Construction of a Thermoconductimetric Apparatus, University of Arkansas Engineering Experiment Station Research Report No. 6, Fayetteville, Arkansas January (1955). [2] Baxley, A. L. and J. R. Couper, Thermal Conductivity of Two-Phase Systems: Thermal Conductivity of Suspensions, University of Arkansas Engineering Experiment Station Research Report No. 8, Fayetteville, Arkansas, November (I966). [3] Bruggeman, D. A. G., Dielectric Constant and Conductivity of Mixtures of Isotropic Materials, Ann. Physlk, 24, 636-79 (1935). [4] Deissler, R. G. and C. S. Elan, Investiga- tion of Effective Thermal Conductivity of Powders, NACA RM E52C05 (1952). [5] Fricke, H., The Electrical Conductivity of Suspensions of Homogeneous Spheroids, Phys. Rev., 24, 575-87 (1924) [6] Goldsmith, A., et al, Handbook of Thermo- physical Properties of Solid Materials, The MacMlllan Co., New York (1961). [7] Griffis, C. L. , N. C. Nahas, and J. R. Couper, Thermal Conductivity of Two-Phase Systems: A Survey and Critical Evaluation of the Literature, University of Arkansas Engineering Experiment Station Research Report No. 5> Fayetteville, Arkansas, March (1964). [8] Hamilton, R. L. and 0. K. Crosser, Thermal Conductivity of Heterogeneous Two Component Systems, Ind. Eng. Chem. Fund., 1, 187-90 (1962). [9] Jakob, M., Heat Transfer, Vol. I, p. 88, John Wiley & Sons, Inc., New York (1950). [10] Jefferson, T. B. et al, Thermal Conductivity of Graphite-Silicone Oil and Graphite-Water Suspensions, Ind. Eng. Chem., 50, 1589-1592 (1958). [11] Jefferson, T. B. , Thermal Conductivity Deter- minations for Suspensions of Graphite in Water and Oil, Ph.D. Dissertation, Purdue University (1955). [12] Johnson, F. A., U. K. Atomic Energy Estab. Lab., Reprint ASEE No. R/R 2578. [13] Kannuluik, W. G. and L. H. Martin, Conduction of Heat in Powders, Proc. Roy. Soc. (London), A141, 58 (1933). 691 [14] Lees, C. H., On the Thermal Conductivity of Single and Mixed Solids and Liquids and Their Variation with Temperature, Proc. Roy. Soc. (London), A191, 339-440 (1898). [15] Lichtenecker, K. , The Electrical Conducti- vity of Periodic and Random Aggregates , Phys. Z., 10 1005-1008 (1909). [16] Loeb, A. L.j The Theory of Thermal Conduc- tivity of Porous Materials, J. Am. Ceram. Soc. 37, 96-9 (1954). [17] Maxwell, J. C, A Treatise on Electricity and Magnetism, Vol. I, 435-49, Oxford University Press, London (1946). [18] ?4cAlister, A. and C. Orr, Thermal Conduc- tivity of Aluminum and Zinc Powder Suspen- sions , J. Chem. and Eng. Data, 9 (1964). [19] Meredith, R. E. Studies on the Conductivi- ties of Dispersions, Ph.D. Dissertation, University of California, Berkeley (I960). [20] Meredith, R. E. and C. W. Tobias, Conducti- vities in Emulsions, J. Electrochem. Soc, 108 , 286-90 (1961). [21] Meredith, R. E. and C. W. Tobias, Resistance to Potential Plow Through a Cubic Array of Spheres, J. Appl. Phys., 31, 1270-1273 (I960). [22] Nahas, N. C. and J. R. Couper, Thermal Conductivity of Two-Phase Systems: Thermal Conductivity of Emulsions, University of Arkansas Engineering Experiment Station Research Report No. 7, Payetteville, Arkansas, March (1966). [23] Rayleigh, R. J. S. , On the Influence of Obstacles Arranged in Rectangular Order Upon the Properties of a Medium, Phil. Mag., 34, 481-507 (1892). [24] Runge, I., On the Electrical Conductivity of Metallic Aggregates, Tech. Physik, 6, 61-8 (1925). [25] Russell, H. W. , Principles of Heat Plow in Porous Insulators, J. Am. Ceram. Soc, 18, 1-5 (1938). [26] Tsao, G. T. , Thermal Conductivity of Two- Phase Materials, Ind. Eng. Chem. 53, 395-7 (1961). [27] Varga, R. S., Matrix Iterative Analysis, Prentice Hall, Inc., Englewood Cliffs, N. J. (1962). [28] Von Frey, G. S. , The Electrical Conducti- vity of Binary Aggregates , Zeitschrift fur Electrochemie , 38, 260-74 (1932). ^ TREAR / T FRONT - S\\ 1 1 1 1 1 L^il Figure 1 Digital Simulation System Dimensions 692 Figure 2 Heat Balance Basic System Section SPACING PINS THERMOCOUPLE 2 .6350 cm. SCREWED RING 1905 cm. 1.270 cm. 1.280 cm. El NTRAL HEATI NG EMENT THERMOCOUPLE I Figure 3 Concentric Sphere Thermal Conductivity Cell II 693 I o» •o T E u 3 c o o o E 0.300 0.280 0.260 0.240 0.220 0.200 0.180 0.160 Predicted: 1 Parallel 2 Maxwell 3 Jefferson 4 Series Experimental © Author _| I 10 20 30 40 Temperature, °C 50 60 Figure 4 Experimental and Predicted Results for Cellosize in EPO, f = 0.3 694 Measurements of the Thermal Conductivity of Granular Carbon and The Thermal Contact Resistance at the Container Walls C. A. Fritsch and P. E. Prettyman Bell Telephone Laboratories, Incorporated Whippany, New Jersey Experimental measurements of the thermal resistance of gran- ular carbon in a brass container under room conditions are presented for both loose and compacted states. The thermal resistance is described as consisting of two parts: the resist- ance due to the effective thermal conductivity of the "continuum" and a contact resistance due to the interface between a layer of granules and the container walls. Results indicate that the thermal resistance depends on the packing and, to some extent, on the wall geometry relative to the grain size. The contact resistance can be a factor of 2 greater than the continuum resistance across a path corresponding to one grain diameter. Key Words: Carbon, conductivity, contact conductance coefficient, granular carbon, heat conductivity, particulate medium, thermal conductivity, thermal contact resistance . 1. Introduction Heat transfer through particulate media, of intese^t in innumerable applications, is generally characterized by an effective thermal conductivity. The multiphase system is treated as a continuum. However, in those situations where the particles occupy a significant percentage of the transfer path, one might suppose that an additional thermal resistance is experienced at the container walls. A significant reduction in contact area may occur there since the particle arrangement is influ- enced by the wall. Hence, the region close to the wall may more properly be described in terms of a contact resistance coefficient. A study of the thermal resistances in particulate systems having large grain sizes should be of interest in such fields as packed-bed heat exchangers, regenerative heat exchangers, and ordnance applications. Another is the standard Bell System telephone transmitter. Studies at Bell Telephone Laboratories have indicated that the thermal response of the granular carbon microphone to the imposition of a bias current has a direct bearing on the efficiency of the microphone as a transmitter. For an understanding of this problem, detailed knowledge of the thermal resistance of the granular carbon and its container is necessary. To the authors' knowledge, the measurement of a contact type resistance between a wall and a particulate medium had heretofore not been performed. Thus the experimental study presented here was undertaken to measure both the continuum resistance and the wall contact resistance of granular carbon. Both the loose and compacted states for the system were considered at atmospheric pressure and temperatures near room conditions. Member of Technical Staff and Technical Aide, respectively. 695 2 . Apparatus Radial heat flow through a series of concentric cylinders was first taken as the logical choice for the experimental configuration. (Heat losses were more readily- controlled in this arrangement than in an apparatus using axial conduction.) The total thermal resistance for each cylindrical container in terms of the thermal con- ductivity A, and the contact conductance coefficient h , can be expressed as follows: _ ln(D Q A) 1 ) + _^_ + ^_ _ AT (1) 27AL h c A o h c A.. " Q Where D denotes the cylinder diameter, A the area and L the length. The subscripts o and i denote the outer and inner walls, respectively, of each cavity. To determine the resistance, R, both the radial heat flow, Q, and the temperature difference, AT, between adjacent cylinders had to be measured. Hence, a monitored wirewound resist- ance heater was inserted in the innermost cylinder. Thermocouples in a radial line, positioned on the cylinders, were used to determine the temperature drop. Measurements for two different cavities were used to obtain two independent equa- tions for the simultaneous solution of A and h . Unfortunately, the process of solving these two equations produced a serious loss in accuracy so that modification of the apparatus was required. A vertical cross section of the final apparatus is shown in figure 1. The carbon granules were contained in the annular space between the two brass cylinders, and the styrofoam insulator blocks at each end. Two rows of thermocouples were immersed in the granular carbon. The thermal resistance corresponding to the temperature differ- ences of the immersed thermocouples consisted of the continuum resistance only, i.e., from eq (1) , ln(D /D. ) Once A was determined from this measurement the thermal resistance between an immersed thermocouple and one on a wall was used to evaluate h . In this case eq (1) became ln(D /D. x o 1 2ttAL h A c o (3) Placed within the inner cylinder was a heater, nichrome wire wrapped on a phenol- fiber tube. Centrally located voltage taps were attached at a spacing of 5 .08 cm. A separate guard heater was provided at each end to control the heat losses. The 2.54 cm outside-dia'meter brass inner cylinder consisted of a 5- 08 cm centrally located test section. A 2 . 5^ cm length of tubing of the same diameter was glued with an epoxy cement to each end. The inner cylinder was made in three electrically iso- lated pieces to allow differential thermocouple readings. The wall thickness was 0.159 cm (1/16 inch) . The 7.62 cm outside-diameter brass outer cylinder had a wall thickness of 0.159 cm and was 10. 80 cm long. The outside surface was painted flat black to maximize the radiative heat transfer from the cylinder. Chromel-alumel thermocouples were positioned in two vertical planes 90 degrees apart. Most of the thermocouples were in the plane illustrated in figure 1. The principal thermocouple, designated REF , was attached to the center of the test section. A similar row of two thermocouples was placed in the annulus at 90 degrees to the vertical plane containing REF. The actual positions of the resistance measuring thermocouples are listed in Table 1. These positions were measured to within 0.025 cm accuracy. 696 Table 1. Thermocouple locations, Thermocouple Designation Location Radial Position fern) Azimuthal Position w.r.t. REF REF Test section 1.27 -- 4 Carbon medium 1.70 90° 5 Carbon medium 2.77 90° 6 Carbon medium 1.75 0° 7 Carbon medium 2.81 0° 9 Outer cylinder 3.66 0° Thermocouples at the extremities of the inner cylinder were used to adjust the guard heaters. An additional couple was placed in the 90 degree plane for tests for varia- tion around the test section. Couples were added at the ends of the outer cylinder to indicate the extent of axial variations . The REF thermocouple was connected through a Leeds and Northrup ten-position thermocouple switch to an ice junction and differentially to each thermocouple in the system. A Leeds and Northrup K-3 potentiometer and a type-E galvanometer were used to read the EMFs. The input to the portion of the heater under the test section was measured by two Simpson voltmeters. The current was calculated by measuring the voltage drop across a 10 ohm resistor in series with the heater. The voltage was measured by using the voltage taps on the test section heater. 3. Characterization of Medium Western Electric Company standard telephone transmitter carbon, sieve size -55 to +70, was used in these studies. Figure 2 is a photomicrograph of the granules with a 25 |Xm (one-mil) wire as a reference scale. The photograph shows the irregular shapes of the granules. The size and eccentricity of a statistical sampling of the carbon is given in figure 3^- First the density of the granular carbon in both loose and totally compacted states was determined in a 50 cm3 graduated cylinder and with a calibrated balance. The loose state was accomplished by putting carbon in the graduated cylinder and gently tumbling the graduate end-over-end a number of times. The volume was observed and the carbon weighed; a density of 900 kg/nP was calculated. The totally compacted state was obtained by tapping the graduate until no further change in volume was 3 Q observed. The carbon was then weighed and a density of 1000 kg/m J was computed. Taken with permission from unpublished work of M. C. Huffstutler, Jr. and Miss B. T. Kerns at Bell Telephone Laboratories. The eccentricity is the ratio of the longest axis to the dimension perpendicular to it, on the same surface. 3 After the rather mild tapping, no further effect was produced by striking the graduate vigorously. 697 Then, measurements were made with the carbon granules in various states of com- paction from loose to totally compacted. The loose state was attained by carefully filling the annular space between the test section and the outer cylinder, leaving a small space at the top. The apparatus was then gently tumbled to loosen the parti- cles. The partially compacted state was accomplished by pouring the carbon granules freely into the annular space until it was filled - pouring tends to compact the granules. The totally compacted state was achieved by filling the annular space while tapping the outer cylinder , until no more carbon could be added. The density of the partially and totally compacted states in the test apparatus was determined by computing the volume and weighing the carbon granules. For the totally compacted state the computation gave a density of 1000 kg/m^ in agreement with the previous determination. The density of the loose state was not determined from the in situ measurements since the size of the small space left to insure a loose state could not be accurately measured. 4. Experimental Results The experimental results for the states tested are presented in Table 2. The ac- tual temperatures at each thermocouple location can be computed by subtracting the appropriate temperature differences from the temperature of test section, T„ Ep . The different heat inputs listed were chosen so that sufficiently large temperature differ- ences could be measured, yet the temperature of the continuum never exceeded 4l deg C. The temperatures read differentially with T-o-p-p were accurate to within 0.05 deg. Con- sequently, the other temperature differences could be as much as 0.1 deg in error. The power input to the heater was accurate to within 2 percent. All thermocouples aid voltmeters were calibrated against laboratory standards. Table 2. Exper: Lmental measurement s and results. Density (kg/m ) Run 1 2 3 4 5 6 7 900 900 900 960 960 1000 1000 Heat input, Q (watts) 0.797 0.797 0.440 O.838 0.445 0.445 0.442 T REp (deg C) 38.39 38.39 33.17 40.28 34.72 32.78 33.00 Tref - T 4 (deg) 2.81 2.88 1.64 2.62 1.68 1.58 1.51 T 4 - T 5 (deg) 5.21 5.22 2.80 4.79 3.06 2.73 2.67 T REF " T 6 < deg ) 2.73 2.74 1.62 2.71 I.69 1.50 1.56 Tg - T ? (deg) 4.83 4.94 2.81 4.88 2.94 2.67 2.53 T ? - T g (deg) 2.89 2.90 I.67 2.88 1.86 1.55 1.54 A (4-5) (W m" deg " ) O.232 0.232 0.241 0.266 0.222 0.247 0.251 A (6-7) ( W m " deg " ) 0.246 0.239 0.234 0.256 0.225 0.247 0.260 h c (7-9) ( W m " 2 deg _1 ) 370 430 230 260 160 390 270 698 Equations (2) and (3) were used to compute the values of A and h . The X values were determined using two adjacent thermocouple readings in the continuum, e.g., TV and T,-, or TV and T„ . (In this determination, the contact resistance terms were omitted since the 30 gauge thermocouple wires - 250 |im in diameter - were about the same size as the carbon granules and hence did not act like a wall.) The average value of X from each run was used to determine h with the temperature difference of T ? minus T„ in eq (3) 5. Experimental Error Analysis The possible sources of experimental errors in radial-flow thermal conductivity 4 apparatus have been carefully described by Flynn [1] . Of these sources, some can be shown to be negligible while some contribute only to the absolute uncertainty. Considering first the question of whether or not steady-state had been reached when the measurements were taken , most of the runs were made at least 5 hours after the heaters were turned on. However, in some cases the apparatus was left operating overnight resulting in essentially no change in the measured parameters. The length of the test section as determined by the location of the voltage taps was measured to within 2%. This factor affects the absolute uncertainty only since it was constant for all runs. The relative radial location of the inner and outer cylinders was measured to the same accuracy. Hence, an eccentricity of 0.03 was possible. According to Flynn [l] this eccentricity contributes error of less than 0.1$. In a given run the total power generated was measured to within 2%. The location of the thermocouples was determined so that the logarithm of the ratios of the appro- priate diameters varied by less than 3%- The fact that the thermocouples were em- bedded in the brass cylinders and were not exactly on the surface of the walls could account for less than 0.2$ error [1] for the heat flows used here. As mentioned ear- lier, the temperature differences were accurate to within 0.1 deg which corresponds at most to a 5$ error. Lastly, since the outer cylinder was not guard heated but rather it stood ver- tically in air one might rightfully ask what were the effects of axial heat flow. The inverse of our situation was solved by Flynn [1] but his results are also applicable here. Using the temperature differences measured by the thermocouples placed on the ends of the outer cylinder the maximum correction for the resistance across the cavity due to axial heat flow was always less than 1$. Since all the X values were determined from the thermocouples immersed in the granular carbon, the actual error is probably somewhat less than 1%. Combining the absolute values of all these errors the maximum amount that the X values could be in absolute error is 13$- The relative error within a given run is 8$. Since the contact resistance was on the order of one tenth of the total thermal resistance measured, the error in measuring h would be amplified by a factor of 10. 6. Discussion of Results and Conclusions The averages of the measured thermal conductivity of granular carbon for the loose state (9°0 kg/iiP), the partially compacted state (9^0 kg/m3) and the totally compacted state (1000 kg/m^) were 0.237, 0.244 and 0.251 W m deg , respectively. These values indicate an increase in X almost in proportion to the increase in density (compaction). Due to the scatter in h , no trend in that coefficient could be ascer- tained. However, the average value is about 300 W m~ 2 deg -1 . This value of h corresponds to a thermal resistance at the wall, a factor of 2 greater than the con- tinuum resistance across a path corresponding to one grain diameter. 4 Figures in brackets indicate the literature references at the end of this paper. 699 It is of interest to compare the measurements reported here to some of the ana- lytical models for thermal conductivity of particulate media. Apparently, all of the analytical description had assumed at most point contact between the solid particles immersed in a fluid. The exact solution, such as that by Deissler and Boegli [2], have considered idealized grain shapes and have computed the attendant temperature profiles. The approximate solutions, such as that of Russell [3], have considered the heat to flow in parallel through the discontinuous (solid) and continuous (fluid) phases and then in series through the fluid phase. The heat flow through a continuous aggregate of solid is neglected. Godbee and Ziegler [4] have modified Russell's equa- tion to account for a statistical distribution of particle sizes. They have also taken into account free molecular effects in gases for sufficiently small grain sizes and low pressure. For the conditions of interest here the free molecular effects can be neglected. A calculation of the Godbee and Ziegler ' s statistical parameter, a, for the distribu- tion of figure 3 yields, a = 0.99* an d for values of a near unity the Godbee and Ziegler model reduces to Russell's equation. In addition, for large ratios of solid conductivity to fluid conductivity (here, it is nearly l600) Russell's equation becomes independent of the solid conductivity yielding, A = A . air i-vH :*) where V, is the volume fraction of solid. Taking the solid density of the carbon grains to be 2240 kg/m one obtains V, = 0.4 for the loose state. Hence, Russell's equation predicts A = 0.10 W m deg . As in other studies, Russell's equation underpredicts A by a factor of 2.4 for the loose state. It is quite possible that this difference can be attributed to the existence of solid to solid conduction that was neglected in the analytical models. An interesting aspect of this study comes out of the thermal resistance values between the test section and thermocouples 4 and 6. If one applies the values of 11 C7-q x i tabulated in Table 2 to eq (3) with A replaced by A. and calculates a value of A for this region, a value of about 0.34 W m deg is obtained. (Conversely, if one takes the values of A/h(-\ or A,£„* and tries to compute h /u\ or h r^\ he will find that h will be negative.) Apparently the thermal resistance in this apparatus is a function of radial position. Note that the A value of 0.34 represents a h0% in- crease over the average value for the loose state. Hence, such a marked change cannot be attributed to experimental error. In addition, this "inconsistency" was present in all of measurements made in this work. Deissler and Boegli [2] made experimental measurements of the temperature as a function of radius and their reported values indicate that A is independent of radius. However, their ratio of grain size to wall radius was a factor of 20 smaller than the condition considered here. Since the design of this experiment was not sufficient to assess this effect the answer must come from future studies. A study could center around two precisely instrumented cylindrical apparatus arrangements, one consisting of a row of several thermocouples immersed in the continuum at small and large radii, the other a series of concentric cylinders at the same radial positions as the thermocouples in the first apparatus. The measurements from the first apparatus for the effective thermal con- ductivity at each radial section could then be used in the second apparatus to deter- mine h , as a function of radial position. Because Western Electric standard carbon c ' r is precision manufactured and of a narrow distribution, this medium is a good candi- date for such future studies. In summary, the thermal conductivity of granular carbon has been measured and the presence of a thermal contact resistance at a wall has been demonstrated. The wall contact resistance is important only when a single grain occupies a significant portion of the heat transfer path between container walls. A dependence of these coefficients on radius of curvature of the multiphase system seems to exist. 700 8. References [ l] Flynn, D. R. , A radial-flow apparatus for determining the thermal conductivity of loose-fill insulations to high temperatures, J. Research NBS 67C (Eng. and Instr.) 129 (1963). [2] Deissler, R. G. and Boegli, J. S., An investigation of effective thermal conduc- tivities of powders in various gases, Trans. Am. Soc. Mech. Eng. 80 1417 (1958). [3] Russell, H. W. , Principles of heat flow in porous insulators, J. Am. Ceram. Soc. JJ5 1 (1935). [4] Godbee, H. W. and Ziegler, W. T. , Thermal conductivities of MgO, A1~0 , and ZrC>2 powders to 850° C. II, Theoretical , J. Appl. Phys. 37 56 (1966). REF Figure 1. Vertical Cross Section of Experimental Apparatus. 1. Guard heaters. 2. Test section heater. 3. Voltage taps. 4. Annular cavity for granular carbon. 5- Outer brass cylinder. 6. Thermocouples. J. Inner brass cylinder. 8. Styro- foam insulator. 9. Phenolfiber alignment tubes. 10. Support blocks. 701 Figure 2 . Photomicrograph of Typical Carbon Granules en ui o < 20 0. O | 10 o UI cr 1.0 2.0 ECCENTRICITY 3.0 en ID o cr 20 < Q. >- O UJ z> o UJ cr 250 375 500 AVERAGE DIMENSION, ^m Figure 3. Size and Eccentricity Distribution of Granular Carbon. 702 The Thermal Conductivity of Thoria Powder From U00 to 1200 C 1 In Various Gases At Atmospheric Pressure A. D. Feith 2 Nuclear Materials and Propulsion Operation General Electric Company- Cincinnati, Ohio 1(5215 The thermal conductivity of powdered thoria at 75? of theoretical density was measured in a radial heat flow apparatus. Several different types of gas (He, N , He + 52? N_, Ar) were introduced into the powder matrix to study the effect of the conductivity of the gas phase on the conductivity of the compact. The results of this work indicate that the thermal conductivity of the system varies only slightly, if at all, from one gas to another, but considerably lower conductivity values are obtained when the pore phase is evacuated. With a gas present, values of about 1.25 Win -1 deg _1 at 1*00 °C, 1.05 Wm _1 deg at 1000 °C , and 1.5 Wm _1 _deg _1 at 1200 C were obtained whereas in vacuum the values were about 0.5 Wm~ deg~ in the U00 to 750 C temperature range. The estimated precision of the data including the effect of the different gases was about ±10?. Key Words: Conductivity, gaseous atmospheres, radial heat flow, powder, thoria. 1. Introduction The purpose of this work was to determine the thermal conductivity of thoria powder which was vibra-packed to 75? of theoretical density and to determine the effect of the type of gas present in the voids on the thermal conductivity of the compact. Although the results of this study are to be applied to production engineering problems, they may also contribute to the understanding of heat flow in powder- gas systems at high volume fraction solid. A detailed study by Godbee (l) 3 on MgO, Zr0„, and AI2O3 powders of considerably lower volume fraction solid than used in this study demonstrates that the con- ductivity of a powder-gas system can be predicted from the conductivity of the solid and the conduc- tivity of the gas. The success of such a prediction is based on the knowledge of several other param- eters including particle size, shape, and distribution; contact area and inter-particle spacing. It may be concluded from a discussion by Kennard (2) that in powder-gas systems where the inter-particle spacing is very small the conductivity of the system is not dependent upon the thermal conductivity of the bulk gas but rather on the free molecular conduction. The results of this current study which are described below seem to confirm, qualitatively at least, this latter concept. 2 . Equipment The measurements described in this report were performed in a radial heat flow apparatus which has been previously described in detail (3). The equipment was developed specifically for measuring the thermal conductivity of solid materials in inert or reducing atmospheres. Previous measurements on relatively high density compacts of ZrOg, UO2 , BeO, and Mo C+,5), have been shown to be reproducible and in good agreement with the work of other laboratories. The estimated precision of the apparatus is about ±6? above 1000 C. However, the precision is more nearly ±10? below 1000 C. Due to the nature of the specimen used in this study, some modifications in the usual experimental setup and procedures were necessary; these are described below. 1 This work was performed under U.S.A.E.C. Contract No. AT(07-2)-l. Principal Engineer Figures in parentheses indicate the literature references at the end of this paper. 703 3. Specimen and Container Property measurements on powdered materials pose many problems which are not associated with solid materials. The major problem is to maintain the geometric requirements of the measuring system. For the radial heat flow system, it is necessary to contain the powder in a cylindrical can with a center hole of a specific size to fit the apparatus. The specimen container used in this study is shown in detail in figure 1. Thermocouple wells were placed at three; different radial locations in a plane parallel to the centerline of the can to avoid having them in the same radial plane thereby minimizing the effect of the presence of the tubes on the geometry of the isotherms. One additional tube extended from the top of the can to facilitate the required changes in the atmosphere. Careful measurements of the radial positions of the ends of the thermocouple wells were made prior to filling the can with the powder. These measurements were necessary for the determination of the value ln(r /r ) which is used in calculating the thermal conductivity (3). The free volume of the can was determined by weighing it empty and then filled with distilled water. correcting the density of the water for its temperature. The amount of Th0 2 powder required to give an average density of 75? to 80? of theoretical density was determined from the free volume. A vibrational packing system was then used to load the powder in the can. The resulting specimen had a gross density of 75.02? theoretical. Radiographs were made to determine that the thermocouple wells were still located in their original positions. A Type 30 f i stainless steel extension tube, sufficiently long to protrude through the top of the furnace, was welded to the can. This tube was then connected to a manifold which had gas and vacuum facilities connected to it. The manifold also had a vent to the atmosphere so that the pressure of the gas in the can would remain close to one atmosphere (see figure 2). The gases were supplied from individual bottles which could be exchanged while the system was evacuated. Valving in the manifold isolated the gas system from the vacuum, system. Flow rates of the gases were adjusted by use of a metering valve in the manifold and a valve in the vent line so that a pressure slightly above one atmosphere (^7^5 Torr) was maintained in the manifold. A portable mechan- ical vacuum pump, having a pumping speed of U25 liters/min., was used to exhaust the system to 3.5 x 10 - ^ Torr when changing from one gas to another. Two or three purges were made each time the gas was changed. 3.1. Thoria Powder The specimen material used in this work had the particle size distribution listed in table 1. While this is not a fine powder, about 75? of the particles are less than 150 microns in diameter. Table 1. Particle size distribution of thoria powder (typical) Sieve Size Particle Size (mm) m -325 O.OUh 30.5 -200 +325 0.07 1 * 17 -100 +200 0.1U9 27.5 -50 +100 0.297 17.5 -20 +50 0.8)t0 6.5 +20 0.81*0 ]. h. Thermocouples Calibrated chromel-alumel thermocouples, constructed from 0.025*+ cm diameter wire and insulated with alumina, were used to measure the temperature in the thermocouple wells. These thermocouples were constructed in a manner such that the beads entered the small diameter tube at the bottom of the wells. Calibration indicated that the couples were in agreement with each other and within 1 C of a standard couple up to 800 °C, 2 °C at 800° and 900 °C , and 3 °C at 1000 °C . Above 1000 °C the departure from the standard increased and some variations in the couples occurred. The indicated temperatures during test were corrected by the calibration errors in determining the true temperatures and temperature differences in the specimen. 704 5 . Test Procedure The specimen was loaded into the furnace and the thermocouple and gas connections were made. Helium was introduced into the furnace volume surrounding the specimen container while the specimen container was being evacuated overnight. The pressure in the can reached 3.5 x 10~2 Torr as a result of this pump down. Helium was then introduced into the specimen container until an equilibrium pressure of about T65 Torr was reached. The system was evacuated again and refilled with helium before bringing the specimen to temperature. After maintaining the furnace at TOO C overnight, measurements were made in the 700° to 1050 °C temperature range on heating and in the 600 to *t00 C range on cooling. These values are shown in table 2 and in figure 3. _2 Measurements were also performed with the specimen evacuated to about 3.5 x 10 Torr in the tem- perature range of ^30 to 750 C in order to establish a value for the conductivity of the powder phase alone and to provide a basis for determining the effect of the different atmospheres. The values obtained in vacuum are also shown in table 2 and figure 3. Following the vacuum measurements, tests were performed using a gas mixture of He + 52$ Ng (v/o). The mixed gas was changed twice by cycling from vacuum to gas a total of three times before measurements were made. During this time the center heater was energized and qualitative observations were made of the voltage drop and current in the heater, and measurements of the specimen AT were made. When the gas was evacuated, the AT in the specimen increased and the operating temperature of the center heater rose causing a larger voltage drop due to its higher resistivity at the higher temperature. Re-introducing the gas caused the operating conditions to restore the same values obtained in the previous observation. This exercise provided qualitative evidence that the gas did permeate the powder matrix. Quantitative evidence was given by the conductivity data. Measurements with the mixed gas present were made during heating and cooling between H00 and 1200 C. These data are shown in table. 2 and figure 3. A second set of measurements was obtained in helium from U00 to 1200 C. The data bel for this run indicate that virtually no change occurred in the conductivity as a result of the prior thermal cycling; however, the data in the 1000 to 1150 C range were somewhat higher ("\<20$) than when He + 52$ N 2 was present. 5.1. Additional Measurements After completing the above measurements and observing that the type of gas made little difference, measurements were made using 100$ nitrogen and then argon. These gases have much lower thermal con- ductivities than those previously used, as shown in table 3. The results of the measurements in these two gases are compiled in table 2 and figure 3. 5.2. Discussion of Results The data in table 2 have been summarized by drawing a smooth curve through each set of data and are tabulated in table h at even increments of temperature to show the quantitative effect of the gas. It will be observed that the conductivity of the compact (with a gas present) varies a maximum of ±10$ about a mean value for any temperature regardless of the type of gas present. If this system were sensitive to the conductivity of the gas and obeyed the model proposed by Godbee (l), the conductivity at 500 C would vary from 1.91 Wm -1 deg~l for helium down to 0.37 Wm~-^ deg for argon or a ratio of approximately 6 to 1; this was not experienced in this study. Close examination of the data in table 2 may cause speculation that the conductivity of the compact was changing with time and temperature cycling since the final measurements in helium (2nd cycle on cooling) are higher by about 25$ in the 1000 to 1050 C range than the initial values in the mixed gas. This is probably due to experimental error rather than densification by sintering. Separate studies on similar systems indicated that sintering does not occur at temperatures below 1500 C. A review of the literature describing thermal conductivity studies performed on two-phase (powder- gas) compacts (l) suggests that the effect of the conductivity of the gas may be limited by the distances between the solid particles. It could, in fact, cause the gas to behave in three distinctly different modes as a heat conductor. 1. Where the spacing between the particles is large in relation to the mean free path of the gas, the contribution of the conductivity of the gas would be that of the bulk gas. 2. Where the spacing is very small in relation to the mean free path, i.e., approximately equal to it, especially surrounding the points of contact between the particles, the thermal trans- port is more closely related to the free-molecular heat conduction which has been shown by Kennard (2) as being related to the specific heat, pressure, and temperature. 705 Table 3. Thermal conductivity of gases at one atmosphere of pressure (Win -1 deg -1 ) Temp . , °C (a) Argon (a) Nitrogen He + 52$ N g Helium' a ' Uoo 3.27 x 10 -2 1*.80 x 10~ 2 10.9 x 10~ 2 27.0 x 10~ 2 1+50 3.1*1* 5.05 11.6 28.5 500 3.60 5.28 12.2 29.9 550 3.76 5-52 12.8 31.3 600 3.91 5.75 13.1* 32.7 650 1+.05 5.97 lU.O 3l+. 1 TOO k.19 6.19 1U.6 35.5 750 I+.30 6.l»o 15.2 36.9 800 k.h-J 6.61 15.8 38.2 850 l*.6o 6.82 16.1* 39.6 900 k.-Jk 7.02 17.0 1*0.9 950 it. 88 7.22 17.6 1*2.2 1000 5.01 7.**3 18.2 1*3.6 1050 5.15 7.61* 18.8 1*1*. 9 1100 5.28 7.85 19.1* 1*6.2 1150 5.fcl 8.08 20.0 1*7.5 1200 5.5k 8.20 20.6 1*8.8 1250 5.68 5.81 x 10 8.5U 8.79 x 10 21.2 21.8 x 10 50.0 1300 51.1* x 10 (a) Powell, R. W.\ Ho, C. Y. and Liley, P. E. , "Thermal Conductivity of Selected Materials," NSRDS-NBS-8, U.S. Dept . of Commerce, National Bureau of Standards, Nov. 1966. (b) After Wright, J. M. , "Calculated Thermal Conductivities of Pure Gases and Gaseous Mixtures at Elevated Temperatures," AECD-1*197 , Hanford Works, Richland, Wash., Dec. 1955. Computerized by R. Pugh, GE-NMPO. Table 1*. Thermal conductivity of powdered thoria at 75? theoretical density with various gas atmospheres (Wm -1 deg" 1 ) (a) Temp . , °C Helium He + 52? N 2 Nitrogen Argon b Vacuum 1*00 1.28 1.30 1.22 1.07 0.1* 500 1.22 1.22 1.18 1.01 0.1*3 600 1.19 1.16 1.07 1.00 0.1*7 700 1.18 1.13 1.01+ 1.05 0.5 800 1.18 1.13 1.03 1.25 900 1.22 1.17 1.06 1000 1.28 1.22 1100 1.37 1.32 1200 1.50 1.1*6 (a) Values from smooth curve through each set of data. (b) At approximately 3.5 x 10~2 Torr. 3. At intermediate particle spacings there would be conduction by both mechanisms. When relating the considerations above to the system under study, it is reasonable to consider the structure as having a random distribution of particle spacings ranging from contact to several hundred microns. Extensive measurements and mathematical analyses will be required to determine the statis- tical distribution of particle spacings and the contact area between the particles to estimate the contribution of the gas conductivity and the solid conduction in this system. However, the results of these measurements would seem to indicate that the mode of heat conduction in the gas is primarily by free molecular heat transport since differences in the conductivities of the system when filled with the various gases are too small to be explained by the bulk conductivities of the gases. Furthermore, the values measured in vacuum are higher than would be expected in a system which would have large inter-particle distances and small contact area. The recommended values for high density (< figure 3 for comparison. of theoretical) sintered thoria (6) are shown in 706 Table 2. Thermal conductivity data for thoria powder in various .cases (Wra -1 deg -1 ) Temp . , °C 1st Cycle 2nd Cy d.<»> Heating Cooling Heating Coolinn Max. Min. 1 Avg. Max. | Min. | Avg. Max. j Min. | Avg. Max . | Min. Avg. 377 Helium 1.39 1.10 1.20 386 1.31+ 1.25 1.29 1*20 1.1+2 1.35 1.39 1*29 1.10 1.36 1.21 U92 1.31+ 1.07 1.20 506 1.50 1.1*5 1.1+8 529 1.23 1.06 l.ll* 571 1.26 1.25 1.21+ 708 1.31 1.07 1.18 711 1.32 1.09 1.22 750 1.26 1.22 1.2l* 772 1.16 1.12 1.11+ 797 1.12 1.01 1.06 828 1.29 1.25 1.28 989 l.ll* 1.11 1.13 1002 1.58 1.33 1.1+7 1130 I.69 1.39 1.56 1138 1.65 1.1+0 1.51) 1181 1.62 1.25 1.1+5 Vacuum UlU 0.663 0.571 0.616 1*36 0.1*50 0.399 0.1*23 1*1+5 0.1*1+5 0.1*01 0.1*1*2 517 0.1*1*1* 0.1*37 0.1+1*0 650 0.1*23 0.1*09 0.1+17 752 0.610 0.525 O.565 391+ 1.33 He + 52$ N g 1.12 1.12 1.39 1.30 1.35 1*1*6 1.35 1.10 1.21 1*1*9 1.38 1.27 1.32 556 1.23 1.08 1.15 6ll* 1.25 1.12 1.18 637 1.21+ 1.20 1.22 65I* 1.21+ 1.20 1.22 731 1.2l* 1.08 1.15 788 1.18 1.09 1.11+ 823 1.13 1.07 1.10 87!* 1.37 1.33 1.35 930 1.12 1.11 1.11 971 1.27 1.19 1.25 1076 1.22 1.18 1.20 1092 1.25 1.17 1.22 lll*7 1.1*0 1.30 1.35 1157 i.ut 1.1+1+ 1.1+6 1197 1.63 1.31+ 1.50 1201 1.61 1.52 1.57 1*06 1.25 I litrogen 1.17 1.22 1*1*2 1.20 1.12 1.17 513 1.17 1.05 1.12 529 l.ll* 1.01+ 1.10 692 1.13 1.01 1.08 735 1.06 0.971* 1.02 911 1.07 1.02 1.05 Argon 1*60 1.01+ 1.03 1.03 582 0.973 0.91*6 0.962 698 0.976 0.971 O.965 731* 1.12 1.09 1.11 787 1.31 1.28 1.30 . co< r 707 6. Conclusions and Recommendations The results of these measurements indicate that the conductivity of the powder-gas compact used in this study will not be greatly affected by a change in composition of the gas as long as the gas pressure does not vary. Stated in another manner, helium with its high conductivity does not signi- ficantly increase the conductivity of the powder-gas compact over the values obtained with argon which has a much lower conductivity. Air, which has a conductivity approximating that of nitrogen, would contribute equally as well as helium or any other gas to the conductivity of the compact. This, of course, assumes that the system is dry and that sorbed water vapor and other fluids are kept to a minimum; Godbee (l) points out that such materials would enhance the thermal conductivity of the powders. Further studies in which gas pressure, particle size, and bulk density are variables may shed new light on the contribution of the gas phase on the conductivity of the two-phase comnact . 7. References (1) Godbee, H. W. , "Thermal Conductivity of MgO, Al o and ZrO Powders in Air at Atmospheric Pressure from 200°F to 1500°F," Oak Ridge National Laboratory, ORNL-353 0, April 1966 , and private communication April 12, 1967. (2) Kennard, E. H., K inetic Theory of Gases , McGraw-Hill Bool- Company, Inc., New York, 1938, p. 315. (3) Feith, A. D., "A Radial Heat Flow Apparatus for High Temperature Thermal Conductivity Measurements," GE-NMPO, GEMP-296, August 1963 . CO Feith, A. D., "The Thermal Conductivity of Several Ceramic Materials to 2500 C," Advances in Thermophysical Properties at Extreme Temperatures and Pressures, ASME, March 1965. (5) Feith, A. D., "Measurements of the Thermal Conductivity and Electrical Resistivity of Molybdenum," GE-NMPO, GE-TM 65-10-1, October 1965 • (6) Powell, R. W., Ho, C. Y., and Liley, P. E. , "Thermal Conductivities of Selected Materials," NSRDS-NBS-8, U.S. Department of Commerce, National Bureau of Standards, November 1966. Figure 1. Specimen container used to determine the thermal conductivity of thoria powder. 708 vacuum gage Figure 2. Schematic diagram of gas supply and vacuum system. 1200 Figure 3. Thermal conductivity versus temperature for powdered thoria at 75$ theoretical density in various atmospheres. 709 Thermal Conductivity of a 587„ Dense MgO Powder in Nitrogen J. P. Moore, D. L. McElroy, and R. S. Graves Oak Ridge National Laboratory Oak Ridge, Tennessee A technique for measuring the thermal conductivity of small quantities (as 220 crrH) of powders from 300 to 1300 C K is described in detail. A thorough analysis is made of the determinate and indeterminate errors of the technique. Results of measurements on a 58% dense MgO powder in N2 gas as a function of temperature at 1 atm absolute gas pressure are presented. These results are compared to data obtained by previous investigators for a similar powder. The experimental results and this comparison reflect serious error sources for this radial technique. Key Words: Magnesium oxide, nitrogen, powders, radial heat flow, two-phase media Research sponsored by the U. S. Atomic Energy Commission under contract with the Union Carbide Corporation. 711 Measurements of the Thermal Conductivity of Glass Beads in a Vacuum at Temperatures from 100° to 500°K R. B. Merrill Space Sciences Laboratory The George C. Marshall Space Flight Center Huntsville, Alabama, 35812 A method of measuring thermal conductivity has been developed that is suitable for measuring evacuated powders. This method gives consistent results, it is fast, and the initial and boundary conditions are easily met. The temperature dispersion during any one measurement is small enough to make measurements as a function of temperature meaningful. The conductivity was measured for glass beads in a vacuum of at least 2 Xl0" 6 torr (3X lO^N/m 2 ) or better over a temperature' range of 100°- 500°K. The particle sizes used were 10jj - 20)j, 38/i - 53|J, and 125jLt - 243fX. The conductivity can be represented by an expression of X = AT + B where A and B are constants and T is the absolute temperature. The value of A does not change significantly over the range in powder sizes from 10/i to 243jj. Key Words: Differentiated Line Heat Source method, evacuated powder, glass beads, Line Heat Source method, radiative transfer, temperature dependence, thermal conductivity. 1. Introduction The theory of heat transfer through powders in a vacuum considers two modes. One mode in- volves radiative transfer. Wesselink [l] , Laubitz [2 ] , Schotte [3 J, and many others have derived theoretical expressions for this mode. One thing these expressions have in common is that the radiant conductivity is proportional to the temperature raised to the third power. The other mode concerns solid conduction. Halajian et al [4] derived an expression in which the value is nearly independent of temperature. Watson [5 ] was one of the first to suggest that the conductivity could be represented by the equation A = AT 3 + B, (1) where X is the thermal conductivity, T is the absolute temperature, and A and B are constants. The theoretical derived equations state that the conductivity should be temperature dependent. Watson [5 ] assumed this relationship and measured the effect of changing the porosity and particle size for spherical glass beads, olivine powders, and several other materials. But Bernett et al [6] measured olivine basalt powders and silica powder and found little temperature dependence. Recently Wechsler [7 ] has shown that there is indeed a temperature dependence. The above conflicting evi- dence suggests that the thermal conductivity should be measured as a function of temperature, but this is a very difficult thing to do. One reason for the difficulty is that the temperature excursion for one measurement usually is so large as to obscure the temperature that properly should be assigned to the measured conductivity. ^Physicist Figures in the brackets indicate the literature references at the end of this paper. 713 The goals of this research were: (1) to find a method suitable for measuring the conductivity as a function of temperature, (2) to measure the conductivity, and (3) to compare these measurements with theoretic values and other measurements reported in the literature. 2. Method of Measurement 2. 1. Configuration The method used to make measurements, developed especially for the work, is called the "Differentiated Line Heat Source" method (abbreviated as DLHS). This method has the same geometry as the Line Heat Source (LHS) method, but deviates in that the temperature change is differentiated with respect to time. The geometry of this method is cylindrical, A long heater wire runs along the axis, and parallel to it is a thermocouple. The construction used to hold these wires and leads is called a heater and thermocouple assembly. The container which holds the powder and which supports the heater and thermocouple assembly is called the sample holder. The heater and thermocouple assembly is constructed as shown in figure 1. The heater is a constantan wire one-thousandth of an inch in diameter, attached to copper posts on a glass base. The other parallel wire is an iron-constantan thermocouple, also one -thousandth of an inch in diameter. The thermocouple junction is located midway between the two posts. The iron lead is attached to an iron post and the constantan lead to a constantan post. Similar wires run to the exit electrical con- nector in the vacuum system. The two wires are about 0. 06 centimeter apart and about twenty centi- meters long. The supports are made of glass and are bonded together at both ends with a ceramic -type cement. The wire heater and thermocouple assembly are placed in the glass tubes as depicted in figure 2; they open into the vacuum chamber through the base plate. The powder is put into a container inside the vacuum chamber, and the chamber is evacuated. The tube is brought to a temperature of about 500 K by a nichrome heater surrounding the tube. To ensure thorough outgassing, the powder (while still in the vacuum) is poured into the tube from the container over an eight-hour period. The temperature of the tube is controlled by immersing it in liquid nitrogen or by heat supplied from the nichrome heater. The rate of temperature change is controlled by the radiation shielding of a dewar, evaporating liquid nitrogen, and/or the heater. A block diagram of the entire measuring system is depicted in figure 3. The voltage applied across the line heater is supplied by a precision d. c. power supply. Constantan has a low thermal coefficient of resistivity so that the power input remains constant during a measurement. The tem- perature of the powder is measured at the junction through balancing a Leeds and Northrup K-3 uni- versal potentiometer until a null condition is reached. The reference junction is maintained at the triple point by a thermocouple reference system. Amplifiers (1) and (2) are Leeds and Northrup d. c. microvolt amplifiers model 9835. The differentiating circuit differentiates the zero to one-volt out- put of amplifier (1). Figure 4 is a schematic of the differentiating circuit with the appropriate values of the components marked. The constant rate of change term is initially bucked to a null condition with potentiometer (2), which is constructed from a battery and a variable resistor. Figure 5 is a typical trace of a run by the strip chart recorder, showing all the pertinent data taken. A series of runs is begun as follows. After the sample has been outgassed, the temperature is changed to some initial temperature. The electronics system is allowed to warm up and stabilize during this time. The scale factors on amplifiers (1) and (2) are set so that they will not overrun during a test. The rate of> change is next subtracted from the output of the differentiating circuit by setting potentiometer (2). The temperature is measured by potentiometer (1), and then the run is initiated almost immediately by applying a voltage across the line heater. 2. 2. Theory of Measurement The solution of the heat transport equation is written as v^T - T = g(t) + / 4T ^_ t>) exp[-b 2 /4a(t-t')]dt', (2) 714 where T is the initial temperature, F is the heat flux per unit length of the heater wire, X is the thermal conductivity, t is the time, b is the separation distance between the heater wire and the ther- mocouple, and a is the thermal diffusivity. The second term in eq (2) is readily recognized as the solution for the LHS method where uniform initial conditions prevail. The first term is apparent from Green's method of solving the heat transport equations. The dependence of the temperature change due to the initial conditions is completely described by the function g(t). This function can be expanded in a Taylor's series, and if the time interval is small enough, the higher order terms can be neglected. This means that the solution to the heat transfer equation may be written as v = A + A,t + / 4?r x ^ t _ t>) exp [ -b E /4a(t-t') ] dt' , (3) where the time is sufficiently small and where A and A, are constants. The DLHS method considers the derivative of the temperature change; hence, f - A > + TVXI «P(-^/4at) (4) and X = Fexp{-b 2 /4at)/47r (dv/dt - Aj)t . (5) The conductivity can be measured with this arrangement while the temperature is changing uni- formly. A typical measurement takes only a few minutes. Three methods are used to calculate the conductivity. The first two are based upon the following considerations. For the case in which the maximum change of temperature with respect to time occurs, X = F/4tt e(dv/dt - A,^^, (6) where (dv/dt - A ) m denotes the maximum change of temperature with respect to time and t m refers to the time at which this occurs. The maximum happens to be fairly broad. A more precise way to calculate the conductivity is to use the time ti where (dv/dt - A x ) - % (dv/dt - Aj) . This condition is satisfied when t, = 0. 37337t m and when ti' = 4. 3l6t m . Methods 1 and 2 now may be defined as follows. Method 1 uses the first halfway point, tt , so that X = 0. 37337F/4 7T e(dv/dt - A[) m ti . Method 2 uses the second halfway point, t/ , so that X = 4. 316F/4 7re(dv/dt - A ) tj . Method 3 involves fitting the curve expressed by eq (4) to the strip chart record. Two unknown parameters, x, and x , (defined as: x = X and x = b 2 /4a) are found by this method. The min- imized function is of the least-squares form, and the minimization is performed by a numerical method called "A Variable Metric Minimization Scheme" [8,9]. 3. Data The conductivity versus temperature is plotted in figures 6 through 9. The abscissa is the abso- lute temperature in degrees Kelvin, and the conductivity is expressed along the ordinate in units of 10" 4 watt m -1 °K _1 . The selection of a semi-log graph presentation allows all of the conductivity data for one particle size to be conveniently located on one graph. Figure 6 is a plot of the conductivity data for the 10jj - 20jj size powder. Method 2 gives the smoothest data. The maximum chart reading for these measurements was reached in a time "t m " on the order of ten seconds. This means that the uncertainty in this time is large for the usual method 1 data-reduction procedure. The uncertainty explains the scatter in the data. The initial points near t m were used by method 3. 715 Figure 7 is a plot of the data for the 38ji - 5 3p. size powder. The data represented by the half- moons were found in the literature [10] . Table 1 gives the pertinent properties of these measurements. Table 1. Glass beads, 50/^ average particle size T gmperature G as P ressure Method °K torr N/m 2 LHS 298 8 X 10" 5 1 X 10" 2 Probe 298 5X 10" 4 7 X 10" 2 Guarded cold plate 190 2 X 10" 4 3 X 10" 2 X (X 10" 3 ) watt m- 10 K-' 2. 5 3. 2 1. 7 These values are greater than the other values reported. The relatively high gas pressure may have resulted in an appreciable gas conduction . The data represented by the squares were also found in the literature [11] . These data are for 44/J - 74ji glass beads at a pressure of less than 2 X 10" 6 torr (3 X 10~ 4 N/m 2 ) and were gathered by an LHS method. The system was stabilized for at least twenty-four hours. Figure 8 is a plot of the 1 2 5 /J- - 243/J data (sample number 5) and of data found in the literature for powders in this size range. The data represented by the half-moons have the properties as given in table 2 [lO] . Table 2. 1 50 /J. average particle size Temperature G as P ressure X(X 10" 3 ) Method °K torr N/m 2 watt m" 1 °K _1 LHS 298 3X 10" 4 4X 10" 2 5.0 Probe 298 3 X 10" 5 4X lO" 3 5.0 Guarded cold plate 190 3 X 10" 4 4 X 10" 2 4.6 The data represented by the squares was also obtained from the literature [l 1 ] . Figure 9 is a plot of the remaining 125ji - 243/i data of sample numbers 3 and 4. The solid line in this diagram describes the data of sample number 5. The conductivity versus temperature data, summarized in figures 6 through 9, are used to de- termine the constants A and B of eq (1) by the least-squares method. Table 3 is a summary of the values of A and B for the various particle sizes. Some of the conductivity data appears disjointed between 200 K and 300°K. Separate calculations were made for all the data below and above 300°K. The quantity represented by "f" is the sum of the squares of the deviation. 716 Table 3. Values of the constants A and B found by the least-squares method Particle Sample Size, fi No. 10-20* 7 10-20* 7 10-20* 7 38-53* 6&8 38-53** 6&8 38-53*** 6&8 125-243* 5 125-243** 5 125-243*** 5 125-243* 3 125-243* 2 Method of No. of B( XlO" 3 ) A( XlO" 1 °) Data Reduction Experiments watt m- 1 °K"' watt m" 1 " K- 4 f 2 11 . 484 . 230 . 0155 3 11 . 346 . 252 . 0556 11 . 472 . 276 . 213 24 . 454 . 327 . 0241 17 . 153 . 358 . 0143 7 . 437 . 452 . 0083 41 . 071 . 542 . 107 28 -. 132 . 565 . 151 13 . 0870 . 639 . 00034 27 . 151 . 310 . 0037 17 -.093 . 417 . 00522 * All the data points used in computing the values ** All the data points above 300°K used in computing the values *** All the data points below 300°K used in computing the values 4. Conclusions Method 3 gives nearly the same results as the numerically simpler methods 1 and 2. Applying the first halfway point as in method 1 is useful because the length of the run is shorter, and hence the uniform rate of change of the temperature, which is initially determined, need hold for a shorter time when compared to method 2. Method 2 becomes more useful, however, when the uncertainty in the first halfway point becomes unacceptable as demonstrated by the conductivity data of the 10 fi - 20fJ. powder. The percent difference between the measured and the calculated conductivity from eq (1), using the values of A and B as given in table 3, is always within the estimated random error. Therefore, the conductivity can be represented within the experimental error by the expression A = AT 3 + B. The value of A does not change significantly over the range of powder sizes from 10)j to 243ji. This means that the value of A is not adequately described for these particle sizes by the theories of Wesselink, Laubitz, and Russell. Laubitz's expression is more descriptive when Watson's data are included. 5. Acknowledgements The author gratefully acknowledges the advice and encouragement given by Dr. Daniel Decker of Brigham Young University. The author also wishes to acknowledge the support given to him by Mr. G. Heller and Mr. J. Fountain of Space Sciences Laboratory, MSFC. 6. References [l] Wesselinck, A. J. , Astron. Inst, of Nether- lands 10, 351 (1948). [2] Laubitz, M. J., Can. J. Phys. 37, 798 (1959). [3] Schotte, W. , A. I. Ch. E. J. 6, 798 (1959). [4] Halajian, J. D. , Reichman, J. , and Karafiath, L. L. , Correlation of Mechanical and Ther- mal Properties of Extraterrestrial Materials, Grumman Research Department Report RE- 280, 17 (Jan. 1967). [5] Watson, K. , The Thermal Conductivity of Selected Silicate Powders in Vacuum From 150° - 350°K, Thesis, The California Institute of Technology, Pasadena (1964). • [6] Bernett, E. C. , Wood, H. L. , Jaffe, L. D. , and Marlens, H. E. , A.I. A.A.J. 1_, 1402(1963). [7] Wechsler, A. E. , Glass Beads — A Standard for the Low Thermal Conductivity Range?, paper No. 7 in this volume. [8] Davidson, W. C. , Variable Metric Method for Minimization, Argonne National Laboratory, Lemont, 111. (1959). [9] Decker, D. , Private communication, Brigham Young University, Provo, Utah. 717 [10] Wechsler, A. E. and Glaser, P. E. ICARUS 4, 335 (1965). [ll] Wechsler, A. E. and Simon, I,, Thermal Conductivity and Dielectric Constant of Silicate Materials, Arthur D. Little Co. Report, 91 (1966). glass beams Figure 1. Line Heater and Thermocouple Assembly- Figure 2. Sample Holder THERMOCOUPLE bell jar base plate Figure 3. Block Diagram of the Measuring System 718 Input ]\ 7.31/it R 9329Q = C 2 200/if Output Figure 4. Schematic of the Differentiating Circuit Figure 5. Typical Trace of Run by the Strip Chart Recorder 70^ r r\ RUN NO. 352 bU SAMPLE NO. 5 TFMPFRATIIRF aco 01/ -50— TIME 12.0 sec \ ti/ ? = I2 se ECM (scale reading) 0.49 volt S, 1000 \ S 2 20C 1 tu V 1.70 volts I 12 1 mi Iliac" V ^ n L 13.57cm \ 3U \ i n \ i r\ l « ■ TIME 5 sec 1 U k START 100 o * 10 8 • o * «| o jf 100 200 300 400 TEMPERATURE IN °K 500 Figure 6. A Plot of the Conductivity for the 1 fi. - 20^ Size Powder 719 100 50 20 £ 10 o _ =50 DA E — -.44 #ECHSLER WECHSLER -74^ et ol [10] ,et ol [II] AVERAGE PARTICLE PARTICLE SIZE size: , /' - a I s ° . i Figure 7. A Plot of the Conductivity for the 38/i - 53fi Size Powder 100 200 300 400 TEMPERATURE IN °K 500 Figure 8. A Plot of the Conductivity for the 125LI - 243LI Size Powder of Sample Number 5 100 50 20 o 10 s -«AE WECHSLER, _ = I50p _ DAE WECHSLER, PARTICLE SIZE el ol [10] 'average 'particle size- et ol [10] BASALT POWDER;^ = 104-150^ . jj* ■ °°F D w> t ( 1 % « s 8 W 100 200 300 400 TEMPERATURE IN °K 500 i — ■ SAMPLE no. A SAMPLE NO ? 5 o / i 1 ■ o * 10 1 r u> ,U ^ I J B I 5 '." " ™ X 2 1 Figure 9. A Plot of the Conductivity for the 125fi - 243/J Size Powder of Samples Number 2 and 3 100 200 300 400 TEMPERATURE IN °K 500 720 The Measurement of the Effective Thermal Conductivities of Well-Mixed Porous Beds of Dissimilar Solid Particles by use of The Thermal Conductivity Probe C. S. Beroes and H. D. Hatters Chemical and Petroleum Engineering Department University of Pittsburgh Pittsburgh, Pennsylvania 15213 The effective thermal conductivities of well- mixed porous beds at various compositions of glass and steel beads were measured using two thermal con- ductivity probes. The interstitual gases were dry air and carbon dioxide. The gas pressure was varied from 1.25 mm Hg to atmospheric pressure at 26°C. The effects of volume fraction of steel in the steel-glass mixture of beads is determined and explained. A better understanding of the energy transfer mechanism is provided by analyzing the experimental data with respect to the effects of the pure solid thermal conductivities and the differences in particle surface. A technique has been developed to measure the degree of mixedness of porous beds of dissimilar solids. Key Words: Dissimilar solid particles, glass shot-steel shot, mixed porous beds, thermal conductivity, thermal conductivity probe. 1. Introduction The measurement of the effective thermal conductivities of porous mixtures of dissimilar solid particles is important because thermal con- ductivity data is very necessary in many processes where porous solids must be heated or cooled. Furthermore, such data is important in insula- tion applications. This study was undertaken to establish a reliable technique to provide quick, reproducible measurements of effective thermal conductivities of said porous mixtures. The purpose of this study described in this paper was three-fold: To determine the suitability of the thermal conductivity probe method as a means of measuring the thermal conductivities of porous mixtures of dis- similar solid particles; to compare a rugged thermal conductivity probe, constructed at the University of Pittsburgh Chemical Engineering Laboratory with a commercially available laboratory- type thermal conductivity probe; and to determine the effective thermal conductivities of beds of well-mixed dissimilar solid particles, and relate these conductivity measurements to the thermal conductivities of the pure materials, and the concentration of each material in the mixture. Associate Professor and MS Candidate, respectively. 721 2 . Experimental Measurements 2.1 Measurement Method The thermal conductivity probe method was chosen for thermal conductivity measurements because of its quickness, simplicity, the small volume of sample required, and because it is a direct method (i.e. it allows measurement of thermal conductivities directly, thereby, eliminating the necessity for specific heat and density knowledge) . The thermal con- ductivity probe method has been used by many investigators (1,2,3)^, and the theory (1,4) concerned with its use is well-known. 2.2 Apparatus A Model CS-48 thermal conductivity probe consisting of an 0.12" o.d. bifilar heater coil of 0.001" constantan wire in an 0.020" o.d. x 0.002" wall x 8.75 long stainless steel sheath was first tested in this service. An 0.0005" x 0.005" Chromel "P" thermocouple located at the midpoint inside the probe allows measurement of the temperature of the probe. This probe was built by Custom Scientific Instruments of Kearney, New Jersey, and is a laboratory type instrument of high accuracy and reproducibility. Current measurements of about 7.65 milliamperes were made at a heater resistance of 1,520 ohms with a flux of approximately 0.09 watt in this probe. Instrumentation for this probe system consisted of a twelve volt battery as a power supply, a Sanborn Model 296 Recorder with a low level preamplifier Model 350-1500 was used to record time and temperature. A Leeds and Northrup Model 8690 potentiometer was used to set recorder at full scale deflection for an emf. of 0.5 millivolt and a Simpson Model 260 milliammeter was used for the current measurement. A Shallcross resistance box Model 817-B was used to control the current at the proper level. A more rugged probe (see fig. 1) was constructed at the University of Pittsburgh Chemical Engineering Laboratory by drawing a double length of thirty gauge constantan wire into an 0.072" o.d. x 0.052" i.d. stainless steel sheath 8.75" long. An iron constantan thermocouple was located at the midpoint inside the probe. Current measurements of about 0.25 amperes were made at a heater resistance of 4.10 ohms with a resultant energy flux of approximately 0.2 5 watt. The instrumentation for this probe was the same as that previously described with the following exceptions: A variable DC power supply was used to replace the twelve volt battery and a Westinghouse ammeter replaced the Simpson milliammeter. In both systems, the heater circuits were arranged as shown in figure 2. A vacuum system used with the above system allowed measurement of the effective thermal conductivities of the porous mixture in dry air and car- bon dioxide at various levels of pressure from atmospheric to 1.25 mm Hg absolute. * Figures in parentheses indicate the literature references at the end of this paper. 722 3. Samples The particulate materials selected for use in this study were steel shot and glass shot. These materials were selected because of (a) the difference in the effective thermal conductance of the pure particulate systems, (b) their suitability as typical particulate materials. Table _1 shows the suppliers and particle sizes. Mixtures of 0, 25, 50, 75, and 100 per cent steel shot (by volume) were made using a laboratory constructed spouted bed mixer. A ten hour mixing time was used for all samples. 4. Experimental Procedure Initially, the sample material whose thermal conductivity was to be measured was inserted into the sample hold (reaction kettle) and vibrated by hand to insure good contact between the probe and sample material. The probe and sample material were then allowed to equilibrate (thermally) for a minimum of eight hours. (When vacuum runs where made the system pressure was then reduced and controlled at the desired pressure level for one to one and one-half hours prior to the measurement of thermal conductivity.) When the sample temperature changed less than 0.1°F, steady state conditions were assumed to have been reached, and a thermal conductivity measurement was made. The initial temperature of the sample was determined, heater power applied and measured, and the time- temperature response was recorded for ten minutes. Gas pressure and heater current were maintained at constant levels throughout the duration of each run. 5 . Experimental Design As previously stated, one purpose of this study was to experimentally determine the general effect of mixing on the effective thermal conduc- tivities of porous beds of dissimilar solids. Mixtures of glass shot and steel shot were prepared at 0, 25, 50, 75, and 100 per cent concentrations by volume. Each of the above mixtures was blended for ten hours using a spouted bed mixer in an effort to get uniform mixtures. The average effective thermal conductivities of these mixtures were evaluated at ten levels of pressure (from atmospheric to 1.25 mm Hg) in dry air and in carbon dioxide at room temperature. The average effective thermal conductivities were obtained by averaging together the effective thermal conductivities taken at three points in the mixture. 6. Results Comparison of the thermal conductivity probe made at the Univ. of Pittsburgh Chemical Engineering Laboratory and laboratory-type commercially available probe was carried out prior to conducting the runs on the glass shot-steel shot mixtures . Thermal conductivity values obtained with the two probes were identical, and extremely reproducible. The reproducibility of the more rugged probe is estimated to be approximately 0.2 per cent. Tables 2 and 3 show the experimental average effective thermal conductivities obtained in this study, and compares the experimental values with values calculated from: k eff, mixture = M. k eff,sl + '2 k eff,s2 723 where, k _„ . = effective thermal conductivity of the mixture, eff, mixture k __ = effective thermal conductivity of solid eff, si , , . n material 1. k ^,. = effective thermal conductivity of solid eff,s2 . . material 2. it/ , ^ = volume fractions of solid materials 1 and 2 respectively. It should be noted that this equation allows calculation of the volume fractions of each component in a binary system provided experimental values for k -_ . , k __ n , and k ., _ are known, eff, mixture eff, si eff, s2 From the experimental data, it is observed that the pressure of the interstitial gas affects the effective thermal conductivity of each of the pure particulate components. There is a range of pressures for which the effective thermal conductivity is very nearly constant. Below this range, the thermal conductivity decreases rapidly as the pressure is reduced toward zero. This breakaway pressure for constant conductivity is different for different gases. For glass and steel in dry air at 26°C, this pressure as shown on figure 3 is approximately 2 25 mm Hg . On figure 4, for the same conditions except that the interstitial gas is carbon dioxide instead of dry air, the breakaway pressure is about 125 mm Hg . The observed effects of pressure can be explained by the kinetic theory of gases. For the range where the pressure has no effect on thermal con- ductivity, the mean free path of the gas molecules is small compared with the distances between the particles which are effective in transferring heat. For the long mean free paths which occur at low pressures, the thermal con- ductivity varies with pressure. Deissler and Eian (5) have developed equations which indicate that the pressure at which effective thermal con- ductivity begins to vary with pressure (breakaway pressure) is a function of the Knudsen number, and further, the effective thermal conductivities of powders become nearly independent of pressure at same ratio of mean free path of gas molecules to a characteristic dimension of the powder particles. 7 . Conclusion The thermal conductivity probe method is a highly reproducible and accurate technique for the measurement of the effective thermal conductivity of porous beds of solid particles. The average effective thermal con- ductivity of a mixture of unconsolidated dissimilar solids is closely approximated by the weighted average effective thermal conductivity calculated using the volume fractions and the effective thermal conductivities of the pure components under the same conditions. 8 . Acknowledgment The work described herein was carried out for the Applied Research Laboratory of the United States Steel Corporation under Research Contract No. 5639. 9. References (1) Wechsler, A. E. and Kritz, M. A., for the U.S. Army Cold Regions Development and use of thermal Research and Engineering Lab conductivity probes for soils and under Contract No. D.A-27-021- insulations. Summary report pre- AMC-25(X), (1965). pared by Arthur D. Little, Inc., 724 (2) D'Eustachio, D and Schreiner, R. E., A study of a transient heat method for measuring thermal con- ductivity, Trans, of the Amer. Soc. of Heat, and Vent. Engrs . , 58 , 333-339, (1952). (3) Woodslde, W. and Messmer, J. H., Thermal conductivity of porous, media. I. unconsolidated sands, Jour, of Appl Phys., 32, 1688- 1699, (1961). ~ (4) Carslaw, H. S. and Jaeger, J. 0., Conduction of heat in solids. Oxford: Oxford University Press, (1959). (5) Deissler, R. G. and Elan, C. S., Investigation of effective thermal conductivities of powders, NACA Research Memorandum E52C05, (1952), Table 1. List of Particulate Materials. Material Average Particle Diameter Manufacturer Type 405, Glass IV, Microbeads 306 microns Microbeads, Inc., Jackson, Miss. Size S-70 Steel Shot 334 microns Pangborn "Rotoblast", The Pangborn Corp. , Hagerstown, Md. Pressure (mm Hq) 730 350 225 Table 2 Thermal Conductivities of Mixtures System I: Steel Shot-Glass Shot Mixtures in Dry Air at 26°(-3°)C Eff. Thermal Cond. (k sl ) of pure steel system shot (BTU/hr (°F/in)ft2) % steel in mixture (v ol.) Eff. Thermal Cond. (k S2 ) of pure glass shot system (BTU/hr (°F/in)ft 2 ) Eff. Thermal Cond. (*eff) based on vol. fractions in mixture (BTU/hr (°F/in)ft2) 1.92 1.92 1.92 7 5% 50% 25% 75% 50% 2 5% 75% 50% 2 5% 1.3£ 1.38 1.38 1.78 1.65 1.51 1 1 JL 1.78 1.65 1.51 Average effective thermal conductivity (observed ) 1.81 1.67 1.53 .78 1.80 .65 1.68 .51 1.51 1.74 1.70 1.51 150 1.74 75% 50% 2 5% 1.34 1.64 1.54 1.44 1.64 1.55 1.46 75 1.60 75% 50% 25% 1.28 1.52 1.44 1.36 1.52 1.46 1.32 50 1.45 7 5% 50% 25% 1.22 1.39 1.33 1.28 39 35 29_ 17 10 08 25 1.25 1.18 0.285 75% 50% 25% 75% 50% 2 5% 1.04 0.572 1.15 1.11 1.08 0.357 0.428 0.499 75% 0.865 0.860 10 0.853 50% 0.902 0.877 0.880 2 5% 0.889 0.901 75% 0.478 0.490 3 0.400 50% 0.711 0.555 0.564 2 5% 0.633 0.653 0.370 0.450 0.556 725 Table 3 Thermal Conductivities of Mixtures System II: Steel Shot-Glass Shot Mixtures in Carbon Dioxide at 26"(-3°)C, Eff. Thermal Cond. of pure steel system (k sl ) Eff. Thermal Cond. of pure glass shot (k S2 ) Eff. Thermal Cond. ( k eff) based on vol. fractions in 1.25 0.341 75% 50% 25% 0.560 0.396 0.451 0.505 Average effective thermal Pressure shot (BTU/hr % steel system (BTU/hr mixture (BTUAr conductivity (mm Hq) (°F/in)ft 2 ) in mixture ■ (vol.) (°F/in)ft 2 ) (°F/in)ft 2 ) (observed) 7 5% 1.18 1.18 730 1.24 50% 2 5% 1.03 1.13 1.08 1.16 1.07 7 5% 1.18 1.32 350 1.24 50% 2 5% 1.01 1.12 1.07 1.14 1.06 75% 1.18 1.20 225 1.24 50% 25% 1.01 1.12 1.07 1.13 1.04 75% 1.18 1.19 150 1.24 50% 25% 1.00 1.11 1.06 1.23 1.10 75% 1.16 1.15 75 1.23 50% 25% 0.950 1.09 1.02 1.10 1.02 7 5% 1.04 1.06 50 1.09 50% 25% 0.890 0.988 0.958 0.990 0.961 75% 0.813 0.810 25 0.837 50% 25% 0.745 0.791 0.767 0.816 0.774 75% 0.646 0.665 10 0.653 50% 25% 0.625 0.639 0.632 0.633 0.627 7 5% 0.411 0.403 3 0.355 50% 2 5% 0.580 0.468 0.524 0.475 0.517 0.385 0.435 .0.526 726 24 B+S Copper Wire Heater Lead Sauereisen Cement Asbestos covered 30 gauge iron-constantan TC wire (Hot Junction located at Midpoint of Probe) Asbestos covered 30 gauge constantan heater wire 0.072" o.d. x 0.01" wall S.S. tubing 24 B+S Copper Wire Heater Lead Welds 8.75 Figure 1. Schematic diagram of thermal conductivity probe constructed at the University of Pittsburgh. + 6 D.C. Power Supply "" O Resistor Stabilise Switch Run ytfv Helipot (Resistance Box) To Probe Heater Ammeter Figure 2. Schematic diagram - probe heater circuit. 727 LEGEND u, - 25% Steel u A - 50% Steel :tivity (BTU b □ V - 75% Steel - Pure Glass - Pure Steel 1.0 — 8= A" Figure 4. Effective thermal conductivity versus pressure. System II: Steel Shot-Glass Shot Mixtures with Carbon Dioxide at 26°(±3°)C. St 3X 0% 10 100 P, Pressure (mm Hg) 1000 1.0 1 LEGEND: 1 O - 25% Steel A - 50% Steel □ - 75% Steel ?n — V - Pure Glass - Pure Steel 100% — tJ ^^ r-, 75% ""V >Of """^ ■*.. ? 0% ■r^S^^Ht^ -o 25% si&^^Z. — ° r "" o% 1.0 Figure 3. System I: Effective thermal conductivity versus pressure. Steel Shot-Glass Shot Mixtures with Dry Air at 26° (±3°)C. — 1 1 10 100 1000 P, Pressure (mm Hg) 728 Thermal Conductivity Measurements on a Fibrous Insulation Material R. R. Pettyjohn General Dynamics Corporation Convair Division San Diego, California Apparent thermal conductivities of a high temperature fibrous insulation were determined as a function of temperature, pressure, and density. These measurements were conducted in an air atmosphere at various pressures from 10 "2 to 760 ram Hg, and mean temperatures from 400° to 800°K. Specimen densities were varied from 55 to ll+9 kg/m3. The data obtained from these measurements were analyzed to determine the various modes of heat transfer. The modes investigated were radiation, gaseous and fiber conduction, convection, and improved conduction between the individual fibers due to the presence of the gaseous medium. The apparent thermal conductivity- density products (Kp) were determined for each density and subsequently compared. The scattering cross-sections per unit volume were also determined as a function of density. Key Words: Conductivity, fibrous insulation, heat transfer, insulating materials, insulation, scattering cross-sections, thermal conductivity, thermal conductivity-density products, thermal insulation. 1. Introduction Many of the advanced re-entry and hypersonic cruise vehicles will be exposed to a more severe thermal environment than normally encountered by those of today. Most of these future vehicles will also be required to make multiple missions. Therefore, their thermal protection systems will be required not only to provide higher temperature capabilities, but also to withstand these environ- ments repeatedly. In order for such a system to be developed, a survey was conducted to determine what type of insulation would best meet these requirements. The fibrous insulations were found to offer the greatest potential. Unfortunately, no single insulation provided its best thermal protection capabilities over the entire anticipated temperature range. In general, the thermal physical properties of insulating materials are sacrificed as the tempera- ture capabilities are increased. For example, if a higher maximum continuous use temperature is desired, an insulating material with a higher conductivity and density is normally required. In order to overcome this penalty, the designer can use a composite insulation system. That is, various types of insulations are selected to meet the thermal environment within the composite system. Consequently, not all insulating materials must withstand the maximum environmental temperature, but each must be capable of being an effective insulator over the temperature range at which it will be exposed. Therefore, a thermal insulating system can be optimized by using the more effective lower temperature insulations in the cooler portions of the composite. The evaluation of these lower temperature or back-up insulations was the main objective of this program. From an earlier phase of this program, the weight and dimensional stabilities of each insulation was determined over the anticipated temperature range. The most promising back-up insulation, a fibrous silica blanket, was then selected for this phase of the program. In general, the manufacturers provided adequate conductivity data on their insulations in one atmosphere of air. However, they had very little data on their insulations when exposed to various partial vacuums. Normally, it is desirable for the designers of aerospace vehicles to take advantage of the improved thermal conductivi- ties of these insulations in a space environment. Therefore, this phase of the program was to determine the apparent conductivities of the selected insulation at various reduced pressures, mean temperatures, and bulk densities. 729 The following equations were used to determine the gaseous conduction contribution at all the temperatures and pressures evaluated: - i (1) 1 + P L £2 / T P d T 1 o n + 0.5 1 + P' d Where: n_D (2) Ff = the ratio of the gaseous conduction at the environmental pressure and temperature to that of the gaseous medium at standard conditions, P = standard pressure, 76O mm Hg., P = environmental pressure, L = standard mean free path length, T = standard temperature, 273°K. (492°R.), T = environmental temperature, n = property constant of gas (air = 0.75*0 > d = calculated pore size, defined by eq. (2), D = fiber diameter, 1.3 microns, f = ratio of bulk density to theoretical density. The convection and/or improved fiber conduction, as represented by the increment C in figure 5, was also calculated with the aid of these gaseous conduction parameters since both were considered to be proportional to the number of gas molecules present at the various reduced pressures. Next, the conductivities contributed by the residual gaseous medium, the individual fibers, and radiation were considered. From the above calculations, the residual gas contribution at 0.01 mm Hg. was found to be virtually zero. The fiber conduction was also found to be negligible at these densities Typical conduction values were found to be 0.0002, 0.0003, and 0.0005 W m'^eg" 1 for densities of 55, 98, and 1^9 kg/m3, respectively. The solid conduction of silica was found not to change appreciably over the temperature range investigated (400° to 800 C K ). Therefore, the radiation contributions can be obtained directly from figures 2, 3 and k. Total calculated conduction contributions were then com- pared with the experimental data as shown in figure 6. The primary deviations between the experimental and calculated conductivities occurred at the intermediate pressures (3 to 29 mm Hg. range). This was expected since the greatest rate of change in the apparent conductivity occurred when the pressure at which the mean free path length of the gaseous medium contained within a fibrous insulation equaled the mean interfiber spacing. From eq (l) and the discussion above, the pore size, the conductivity of the gaseous medium, and the mean free path length of the medium appear to be the primary controlling factors of the gaseous con- duction contributions at these intermediate pressures. Therefore, the accuracy at which any one of these factors is determined has a definite influence upon the calculated conductivity. It is fortunate that such a questionable ratio between the mean free path length and the pore size can exist and still produce a reasonably accurate calculated gaseous conduction contribution. The primary reason for this accuracy is that the ratio is contained in the P' portion of eq (1) and, consequently, does not influ- ence the calculation proportionately. The thermal conductivity of the gaseous medium, however, is directly proportional, but fortunately has been verified by several investigators and is assumed to be reasonably accurate. Another series of curves can be generated by combining the data represented in figures 5 and 6. This series is represented by figure 7 • These curves show each conduction contribution as a function of temperature. Figure 7 illustrates how each conduction contribution is related to the mean temperature of the specimen while being exposed to a particular environmental pressure. In this case, the pressure is at one atmosphere. As the pressure is reduced, the gaseous conduction contributions are diminished 730 2. Experimental Procedure and Equipment The thermal conductivity measurements were conducted with the aid of a radial heat flow apparatus . Figure 1 shows the basic construction of the unit and one of its four automatic temperature control systems for the guard heaters. The apparatus consists of essentially two concentric ceramic cores that were heated by six wire wound, resistance heating elements. The inner elements on each core were used to provide the temperature gradient across the test specimen, while the remaining four elements were used to minimize any axial heat flow from the central test section. The basic operation of each guard heater system was as follows : whenever a temperature difference existed between the central and guard heaters, a small D.C. voltage was generated by the series connected differential thermocouples. This voltage was detected by a highly sensitive moving-coil, reflecting- type galvanometer. The voltage applied to the galvanometer caused its suspended mirror to rotate. With the light beam source focused onto this mirror, its rotation was greatly amplified by the deflection of the light beam. When the light beam was deflected onto the light sensitive resistor, the current flow through the resistor was greatly increased. The resulting current flow was fed directly into the D.C. transistor amplifier. The power output of the amplifier was proportional to the resistance change of the light sensitive resistor. This power was fed directly to its corresponding guard heater. Therefore, each system essentially maintained its guard heater at the same temperature as their central heater. Each central heater received its electrical power from a variable direct current power supply that was usually set at a fixed level at the beginning of the test. Temperature gradients of 15°K/cm were nor- mally established through the specimen at each mean temperature. The thickness of the specimen was slightly less than 6 cm. Its length was 28 cm. The conductivity measurements were conducted in a large test chamber that was sealed and evacuated to the desired pressure level. After each of the desired pressure and temperature equilibriums were reached, the respective thermal conductivities were computed and recorded. 3. Discussion and Results The results of the apparent thermal conductivity measurements are graphically illustrated in figures 2, 3> ax ^ L **■• The measurements were conducted in an air atmosphere at various pressures from 10-2 to 760 mm Hg., and mean temperatures from 400° to 800°K. The specimen densities were varied from 55 to lit-9 kg/m3 . Figure 5 represents a reasonable approximation of the various conduction contributions to the apparent conductivities of the fibrous silica blanket at a mean temperature of 778 °K (ll+00°R). Similar curves can also be generated to represent these contributions at each of the mean temperatures and densities evaluated. From this figure, increment D is the major contributor to the total heat transfer. This indicated that the gaseous medium within the highly porous structure of the insulation had a considerable influence upon its apparent conductivity at atmospheric pressure. This was expected because the void volume of all the insulations evaluated exceeded 90 percent. Normally, pressure has very little influence upon the thermal conductivity of an unconfined gaseous medium, because with any change in pressure, the density of the molecules will change proportionally, while their path length changes inversely, yielding essentially a constant heat transfer. Therefore, a gas contained within a fibrous insulating material reacts very similarly to an unconfined gas near atmospheric pressures, because the fiber spacing (or apparent pore size) is normally much larger than the mean free path of the gas molecules. However, as the pressure of the gaseous medium is reduced, the mean free path length of the molecules will increase until it reaches the average diameter of the pore size. As the pressure is reduced further, intermolecular collisions will occur less, while the molecular to fiber collision will occur more often. At the very low pressures where the mean free path is much larger than the pore size, the number of molecules available to transport heat is once again proportional to any further decrease in pressure. Therefore, as the number of molecules approaches zero, the conductivity due to gaseous medium also approaches zero. Thus, the typical "s" shaped curve is obtained when the apparent thermal conductivity of a fibrous insulating material is measured as a function of pressure. From figure 5> it should also be noted that the thermal conductivity values appear to be asymptotically approaching a minimal value at the lower pressures. This remaining conduction is pri- marily due to radiation and, to a minor extent, residual gas conduction and solid conduction of the individual fibers. This amount is represented by A. Increment B represents the conductivity contribu- tion of the air medium. This quantity must be corrected to allow for the volume occupied by the glass fibers. It should be noted that the sum of A and B does not equal the apparent conductivity of the insulation at atmospheric pressure. Other investigators have noted this difference, C, and attribute it to convection and/or improved conduction between the individual fibers due to the presence of the environmental gas. At this time, it should be noted that a theoretical approach to analytically develop an "apparent" thermal conductivity requires a simultaneous solution of all the above-mentioned modes, because each mode is dependent upon the temperature gradient which is controlled by the total heat transfer. However, for this study, it was assumed that an analytical expression could be developed for each mode independent of the others, and that the apparent thermal conductivity would be the sum of the conductivities of each mode. 731 until only the radiation contributions remain. A plot of the radiation contributions shows them to be decreasing as the density of the specimen is increasing (see figure 8). This reduction in radiation was primarily attributed to the increasing scattering characteristics of the denser insulations. The scattering cross sections were calculated from the radiation contribution of the above thermal conduc- tivity measurements. Since the absorption cross section has been shown to be two orders of magnitude less than the scattering cross section, it was omitted from the general equation, leaving the following modified equation for making these calculations: _ ka J 2 Tm 3 (3) K r Where : N = scattering cross section per unit volume, m K = radiation contribution, W m deg , J = index of refraction of silica, o" = Stefan-Boltzmann constant, Tm = mean temperature, °K. The results obtained from these calculations are shown in figure 9- The scattering cross sections were found to increase with increasing temperature and density. Since weight is of primary importance in the design of a flight vehicle, the product of the conduc- tivity and density (Kp) is often used in the aerospace industry to determine the thermal efficiency of an insulation system. These products were determined at the various temperatures, pressures, and densities. With the aid of figures 10 and 11, the density at which the insulation is the most efficient can be selected for the anticipated environmental condition. For example, if the temperature of the environ- ment is approximately 780°K and exists at an altitude where the pressure is below 10 mm Hg., the insula- tion with a density of lU9 kg/m3 has a lower Kp product than either of the lower density insulations. Therefore, the thermal efficiency of the 1^9 kg/m3 insulation at this particular environmental condition exceeds either of the other densities investigated. At pressures down to at least 10 mm Hg., the rate of decline in Kp of the 1^9 kg/nP insulation exceeded the others. This appears to be due to the rate of reduction in the gaseous conduction contribution, because the rate will increase to a maximum as the mean free path length approaches the pore diameter. However, the lower Kp values at pressures below 10 mm Hg. appear to be related to the radiation attentuation characteristics and its relationship with the size of the fiber (or the apparent pore size resulting from the small fiber diameter) . It is recommended that this characteristic be investigated more thoroughly. This is a very interesting characteristic, because if the thermal efficiencies of two insulations at different densities are almost equal, the one with the greater density should be selected. Normally, it will be easier to install and should provide a more reliable composite insulation system. k. Conclusions Gaseous conduction and radiation were found to be the major contributors to the total heat transfer through the fibrous silica insulation in one atmosphere of air at mean temperatures from I4OO to 800°K, and bulk densities from 55 to 149 kg/nP. In a partial vacuum of 10"-'- mm Hg. or less, radiation became the only major contributor. At the intermediate pressures, 3 to 29 mm Hg., the gaseous conduction contributions were found to be reduced to approximately half of those obtained at one atmosphere. An increase in the specimen density (from 55 to lh-9 kg/m3) was found to yield a substantial decrease in the total heat transfer through insulating materials of this type. The primary reduction was found to result from the large decrease in the radiation contribution. The thermal conductivity-density products (Kp) were found to be lower than expected. At the inter- mediate pressures (llj- mm at 780°K), the insulation with a density of lit-9 kg/nP was found to have a higher thermal efficiency (lower Kp) than any of the insulations evaluated. From a practical viewpoint, the higher density insulation would reduce the handling difficulties normally encountered during the assem- bly of an insulation system. Also, for a vehicle requiring a sizeable insulation system, the bulk volume of the vehicle would be reduced considerably. 732 GUARD HEATERS* POWER LINES TO GUARD HEATER DC AMPLIFIER SERIES DIFFERENTIAL THERMOCOUPLE CIRCUIT AUXILIARY HEATER CENTRAL HEATER HEUPOT POTENTIOMETER (GALVANOMETER SENSITIVITY CONTROL) LIGHT SENSITIVITY ~1 LIGHT BEAM SOURCE IDENTICAL GUARD CONTROL CIRCUITS WERE USED FOR REMAINING THREE GUARD HEATERS, AS INDICATED. LIGHT-TIGHT ENCLOSURE^ GALVANOMETER (DC REFLECTING TYPE) Figure 1. Guard hea: control system. J ,0.12 Figure 2. Apparent thermal conductivity of a fibrous silica insulation at various environmental pressures and mean temperatures (density ■ 55 kg/rrr). o o o -I < oc Ul 1.0 10 100 PRESSURE, mmHg 1,000 10,000 733 0.10 0.01 611°K(1,100°R) <£556°K(1,000°R) 500°K(900°R) 1.0 10 100 PRESSURE, mm Hg 1,000 10,000 Figure 3. Apparent thermal conductivity of a fibrous silica insulation at various environmental pressures and mean temperatures (density = 98 kg/m 3 ) ,, Figure 4. Apparent thermal conductivity of a fibrous silica insulation at various environmental pressures and mean temperatures (density = 149 kg/m 3 ). 0.08 0.06 > i- o => a z o 0.02 ex. i±j 0.1 1.0 10 100 PRESSURE, mmHg 1,000 10,000 > l- > l- o Q LlI U.1U 0.08 0.06 0.04 - " _( _ 3 [ ) 1 A. Rfi F iDIATION 1BERS0L AND JD Lll ( INDUCTION 0.02 n C. CO 1 — B" MVECTI0M jASEOUS ouimuu^ i iuim AND IMPROVED FIBER A if i i 1 mi c D. (B) 1 1 I lull DNDUCTIC + (C) N , ,,!..„ i,,i 0.01 0.1 1.0 10 100 PRESSURE, mmHg 1,000 10,000 Figure 5. Accumulative conduction contributions to the apparent thermal conductivity of a fibrous silica insulation in an air atmosphere at various environmental pressures. (Mean temperature = 780 °K and density = 98 kg/m 3 ) „ 734 < 5 ex. LU 0.16 0.12 0.08 0.04 1 1 1 1 1 GASEOUS CONDUCTION CONTRIBUTION 2 CONVECTION AND FIBER CONDUCTION CONTRIBUTION 3 RADIATION CONTRIBUTION 4- 500 600 700 MEAN TEMPERATURE, °K 800 Figure 6. Thermal conductivity of a fibrous silica insulation at various reduced pressures in an air atmosphere, (Density =55 kg/nr) „ Figure 7. Accumulative conduction contribu- tions to the apparent thermal conductivity of a fibrous silica insulation in one atmosphere of air (density = 55 k.g/m 3 ) , 0.14 0.12 0.08 0.04 — 500 600 700 MEAN TEMPERATURE, °K 800 0.10 0.08 0.06 0.04 0.02 778°KU,400°R)< 667°K(l,200°R)i I I _556°K(1,000°R)< Figure 8. Radiation contributions for various densities of the fibrous silica insulation. 20 40 60 80 100 120 140 160 BULK DENSITY, kg/m 3 735 u U 20,000 16,000 12,000 8,000 4,000 Figure 9. Scattering cross section of the fibrous silica insu- lation at various densities. 80 100 BULK DENSITY, kg/m 120 3 160 Figure 10 o Thermal conductivity-density products of the fibrous silica at various environmental pressures and bulk densities in an air atmosphere (mean temperature = 500 °K) . > g> I- -o O ,H Q E % 5 149 kg/rrf 98 kg/rtT 0.01 0.1 55 kg/m3 1.0 10 100 1,000 10,000 PRESSURE, itimHg > ■ 149 kg/m" 98 kg/m- 55 kg/m' I ' I ' l""l i ' i IhiiI i I i I mil i i i iinl i i i hull i i i 1 1 1 1 j 0.01 0.1 1.0 10 100 1,000 10,000 PRESSURE, mmHg Figure 11. Thermal conductivity-density products of the fibrous silica at various environmental pressures and bulk densities in an air atmosphere (mean temperature = 780 °K) . 736 The Effect of Thickness and Temperature on Heat Transfer through Foamed Polymers T.T. Jones Rideal Research Laboratories Monsanto Chemicals Limited Corporation Road Newport, Monmouthshire Great Britain. The mode and extent of heat transfer through polymer foams is difficult to define. While it is relatively easy to eliminate convection effects by choosing sufficiently small cell sizes there is still left the apportioning of the heat transferred between the modes of conduction and radiation. This proportion is a function of the temperature of the determination, the thickness of the sample, its density and its cell size. The apparent thermal conductivity of plane horizontal layers of still air confined between parallel surfaces of varying emissivity is shown to increase with thickness of the air layer, its temperature and the emissivity of the surfaces. This is due to the varying relative effects which the radiated and conducted heat have on the total heat transfer. Similarly a study of foamed polystyrenes of varying density and cell size at different temperatures and thickness shows similar trends to those observed with air layers. The apparent conductivity increases with thickness and temperature but tends to level off with thickness due to the absorption of the radiated heat by the polymer. The influence of the emissivity of the confining surfaces is also not as important as for air layers due to the polymer cell walls adjacent to the surface acting as emitters and absorbers of radiation. Using, as a first approximation, a theoretical model for the foam based on a series of parallel alternate polymer and air layers at right angles to the direction of heat flow, it is possible to arrive at an estimate of the distance apart of such hypothetical layers and relate it to density and cell size. Low density and large cell size combine to produce high radiation components and increase in distance apart of the hypothetical layers. Key Words: Apparent thermal conductivity, cell size, convection, emissivity, foam density, heat transfer, polymer foams, polystyrene foams, thermal conduction, thermal conductivity, thermal radiation. 1. Introduction It is relatively easy to define and understand what is meant by the thermal conductivity of a simple gas, liquid or solid, and experiments can be devised which give an unambiguous estimate of the quantity desired. In the case of gases in particular considerable precautions have to be taken in order to eliminate or correct for the influence of radiation and convection on the determination. If, however, we are dealing with a composite material such as discrete gas globules embedded in a solid continuous phase then the question arises as to what exactly determines the mode of transference of heat through such a medium. If the gas is distributed haphazardly in cells of varying shapes and sizes then in certain cases it is quite certain that part of the heat transmitted through such a medium is carried by convection currents. The simplest way to eliminate this effect is to reduce the size of the cells so as to limit the gas circulation. Research Scientist. 737 It is not so easy however tc eliminate the effect of radiated heat. Moreover, the proportion of heat transmitted by radiation to that by conduction varies with the mean temperature of the determination, the thickness of the sample, its density, its cell size and the nature of the surfaces which are in contact with those of the sample. It follows therefore that if convection is eliminated by reducing the cell size sufficiently then the total heat conducted, namely the apparent conductivity of the material, is also dependent on the above parameters. It is the purpose of this paper to demonstrate the above effects for a number of foamed polystyrene examples, whose cell size is small enough to eliminate the effect of convection. 2. Apparatus and Materials 2.1. Apparatus The apparatus used for the measurements consists of a heat flow meter constructed by Technisch Physische Dienst T.N.O. en T.H. Delft and described by J. de Jong and L. Marquenie (l) . This method was originally described by D.L. Lang (2) and has been shown to give results which correlate well with those of the guarded hot plate, e.g. see J.M. Buist, D.J. Doherty and R. Hurd (3). The heat flow meter of dimensions about 5 cm diameter and 0.32 cm in thickness is placed in the cut-out centre of a 25 cm x 25 cm P.V.C. guard of the same thickness. The test specimen is divided into two 25 cm x 25 cm square specimens of equal thickness which are placed above and below the heat flow meter assembly. The hot source is placed on top of this arrangement and the cold sink below. Both hot source and cold sink consist of plates 22.5 cm in diameter forming part of circulating thermostated water systems respectively controlled to any desired temperature between 5 C and 75 C +_ .05 deg. The thickness of the hot source and cold sink are about 2.5 cms respectively and are encased, except for the working surfaces, in 7.5 cm deep polymer foam lagging. The whole apparatus is enclosed in an air thermostat controlled at the mean temperature of the determination. The heat flow meter output in milli volts measures the heat flux through the system, the calibration factor being 14.47 W m 2 per mV at 20 C. This factor decreases by ifo for each 10 deg. increase in temperature above 20 C. 2.2. Materials Three polystyrene foams were studied in this investigation, namely Samples A, B and C. Sample A was of average density 20.1 Kg/m* and contained cells of widely varying cell size from very fine to very large as is shown in figures 1 and 2. Some cells were over 1 mm in maximum dimension. Sample B was of average density 15.2 Kg/m and contained a narrower distribution of smaller cell sizes as shown in figure 3. Sample C was of average density 28.7 Kg/61 „ . The distribution of cells was rather like sample B, but with slightly smaller cells and thicker cell walls, as shown in figure 4. 3. Heat Transfer through Plane Layers of Air In order to understand why the apparent conduction of heat through foamed polymers is so dependent on temperature, thickness, density etc. the question of the apparent conductivity of plane layers of still air is first considered. If a layer of still air is enclosed between two infinite parallel planes 1 and 2 composed of materials of emissivity £ ( and £ a and reflectivities (l - 6 ,) and (l -€ x ) at absolute temperatures T-i and T2 respectively then it can be shown (e.g. see Poltz (4)) that the flux of radiant heat per square metre, Qg passing from plane 1 to plane 2, providing the intervening air does not absorb any of this heat, is given by q r _ a-e,e 2 (t x 4 - t 2 4 ), (i) _p __p _/ wheretTis Stefan's constant = 5.67 x 10 W m deg. If the distance apart of the two planes is x and the mean thermal conductivity of the air in the layer is A i.e. the conductivity at the mean temperature of the layer, then, provided that the mean difference in temperature (T_ - T„) is not more than say 10 to 15 deg. the flux of conducted heat per square metre, Q^, is given to a very close approximation by 2 Figures in brackets indicate the literature references at the end of this paper. 738 Xmean (T 1 " T 2 } (2) lence the total he; t flux per square metre Q_ is given "by 1_ = Q R + q_= 10 C. In another experiment polystyrene foil was used instead of aluminium foil, the thickness "increased to 2.4 cms and temperature varied from 20 to 70 C. The results of these experiments are shown on figure 6 where the apparent conductivity is plotted against temperature. The marked difference between the effect of polystyrene foils and aluminium foils placed on the flat surfaces is again observed both in the magnitude of the apparent conductivity and the way it varies with temperature. The dotted line on figure 6 is obtained by calculating the value of the constant 4 Sample B>Sample C. The reason for the greater radiation component for Sample A over Sample B of lower density must be associated with its greater cell size cuasing less scattering of radiation. The cell size of sample C is a little smaller than that of sample B also the cell walls are thicker. Both these effects combine to reduce the radiation component in sample C over that in Sample B. The difference between the total heat transmitted and the heat conducted i.e. the radiated heat must bear a relationship to both the cell size which influences the scattering of radiated heat and the amount of solid polymer which absorbs it. The theory given by Poltz (4) combines these two effects. Essentially the foam is considered to be composed of a solid scaffolding composed of very thin parallel layers running at right angles to the direction of heat flow. The layers are supported by parallel layers running at right angles. Each thin layer of emissivity C^is assumed to absorb a small portion of the heat radiation falling normally on it. The radiation is also scattered by reflection, the reflectivity being 1 - €j_ The system is now replaced by a system of very thin radiation impermeable lamellae parallel to the thin layers and replacing a number t\ F of them. These lamellae are at a distance S apart and so S = K-cSt-, where S_, is the distance apart of the original very thin parallel layers. This theory leads to a constant apparent conductivity at very large thicknesses in terms of the actual thermal conductivity of the composite medium and a radiation component. The theory leads to the following expression for the constant apparent conductivity ^n-n-n //£_><») at very large thicknesses Aapp (d-»«») = A + 4CTT" 5 £ l -St- (8) 2 ~K where A is the actual thermal conductivity of the composite medium. This is of exactly the same form as for air layers as given by eq. (7) except that S, is a constant namely the distance apart of the hypothetical radiation impermeable lamellae. It is this quantity S, that is a function of both the density and the cell size. The greater the density and so the greater the absorbed heat the smaller is S L . Again, the smaller the cell size, the greater the scattering of radiated heat and so the smaller is S u . The effect of emissivity of the cell walls is given by the £,./ (2 - € L ) term. The composite term £■- S i_ is the quantity that determines the extent of heat transferred by radiation at any temperature. In order to estimate the above quantity Poltz (4) shows that by plotting the reciprocal of the apparent thermal conductivity ( Xappi against the reciprocal of foam thickness, d , and extrapolating to d = it is possible to arrive at an estimate of ^ a -n-n(d -»■<*• ) • ^he relevant plots are given on figure 8, and values of the reciprocal of ^ avv ld->ao) are obtained by extrapolating to d~ = 0. If estimates of \ , the actual thermal conductivity of the composite medium, can be made then the value of the quantity £1 Si_ may be obtained. The value of A for a polymer foam ; i.e. the conduction contribution, may be approximated by a well established relationship due to Kerner (6) for the properties of composite systems. The relation for the thermal conductivity of a composite system consisting of largely spherical particles of a component of conductivity A 2 , (namely air), immersed in a continuous phase of a component of conductivity A, t (namely polystyrene), is given by ^ = Ai A 2 (2v 2 + 1) + 2 Aid - v 2 ) (9 ) A 2 (1 - v 2 ) +\ 1 (2 + v 2 ) 741 where Vp is the fractional volume of the discrete component. It can be shown in the case of light density foams where V — 1 and (l - V ) =£r that a rough measure of \ is given by A = A,+ 2A i V i (10) 3 where V-, = 1 - V is the fractional volume of the polystyrene. This implies that for low density foams e.g. Vn *k 0.05 the thermal conductivity of the composite is essentially that due to the air + 2/3 that due to the polymer present, the remaining l/3 of the polymer being effectively absent i.e. the portion of the polymer lying at right angles to the direction of flow of heat. If the values of A ]_ and A 2 are taken at the temperature T then a value of A may be calculated for each foam of known density. Table 1 lists the values obtained in this way for €%. Su using the values obtained from figure 8 for A a pp(j^o«5 and for A using Kerner's equation and values of A . and A P taken from the literature, (7) and (8), (9) respectively. Table I also gives the value of S, assuming a' value for ^_ = 0.66. Table 1. Extrapolated estimates of the apparent thermal conductivity of the foam samples at d -*■ 00 and calculated values of -S L . T ^app(d-»«>) W m deg 1 / app(d-»oo) W m deg W A - 1 ^ - ] m deg \pp(a->«j) —A = 4crT3 e c S L e L s L ( Sample B > Sample C which is as expected from the relative values of S u respectively. 5. Summary and Conclusions The combined influence of cell size and density on the apparent thermal conductivity of foamed polymers is shown for three examples by its variation with thickness and temperature. This behaviour is attributed to the effect cell size and density have on the heat radiated through the structure. 6. Acknowledgements I wish to thank the Directors of Monsanto Chemicals Limited for permission to prepare this paper for presentation at the Seventh Conference on Thermal Conductivity. I also wish to thank Mr. S. Baxter for his continued interest and encouragement during the progress of this work. Finally, thanks are due to Mr. T.G. Wigmore who carried out the experimental part of the work and some of the calculations and to Mr. F. J. Parker for preparing the micrographs of foam sections. 7. References (1) de Jong, J. and Marquenie, L. , Heat-flowmeters and Their Applications, Instrument Practice, I6(l) , 45 (1962). (2) Lang, D.L., A Quick Thermal Conductivity Test on Insulating Materials. A.S.T.M. Bulletin, Sept. 1956, p. 58. (3) Buist, J.M. Doherty, D.J. and Hurd, R. , Urethane Rigid foams; Factors Affecting their Behaviour as Thermal Insulants. Progress in Refrigeration Science and Technology, 1965, p. 271. (4) Poltz, H., Einfluss der Warmestrahlung auf die Isolierwirkung por'oser Dammschichten. Allg. Warmetechnik, 11 64 (l962). (6) Kerner, E.H., The Electrical Conductivity of Composite Media. Proc. Phys. Soc. B69 802, 808 (1956). (7) Ueberreiter, K. and Nens, S. , Spezifische Warme , spezifisches Volumen, Tempera tur-und Warmeleitfahigkeit von Hochpolymeren Kolloid Zeit, 123., 92 (l95l). (8) Weast, R.C. and Selby, S.M., Handbook of Chemiotry and Physics, 1966-1967. The Chemical Rubber Co. Kaye G.W.C. and Laby T.H., Tables of Physical and Chemical Constants, 1957, Longmans, Green and Co. (5) Fischer, F. , Die Problematik von Warmeleit- fahigkeitsmessungen an Schaumstof fen. Kunststoffe, $6 321 (1966). (9) McAdams W.H. Heat Transmission, Third Edition McGraw - Hill Book Co. Inc. 743 Figure 1. Micrograph of section of Sample A. Figure 2. Micrograph of section of Sample A. 744 - * %i * } '••V 14*. *!'*&*!. ..." M? •.,*? sv% $m&m jS^j? tc> i V - WW '"IF •- *G* ^ B *^L^^ m Tjr$-^BLT JHM^^L&^j'J • » . ■ * 7 ■•'' 8*"^^!^ m SdiWrX ■f\I, r Tl^^l^r^r. .4flv*I -JKe^^H Figure 3 Micrograph of section of Sample B j^Q^ ^ jL *■«• ^.v ^T^ TV * K/mi 1 1 w, . * ... 4* %" ■ 5\ ^ M^v JM •^ * ''•*' 1 mm . ■•. '1 Figure 4 Micrograph of section of Sample C. 745 n d«g — O— POI .YSTYRENE FOIL >. Ti»ton= 40°C T,-T» & IO dcg t 03 (< >■ > /® THERMAL CONDUCT 9 ro y — A— ALUMINI JM FOILS. z UI < 0. o- 1 ,© Tm«an = 27-5°C. T,-T,C= 15 dag. Q. W ' < / THERMAL C 3NDUCTIVITY ^' ^OF AIR AT t BOUT 40°C A A — /2l_ A i & i i 1 i i i 3 4 5 6 7 E Figure 5. Plot of apparent thermal conductivity of still air layers against thickness showing effect of emissivity of confining surfaces. THICKNESS OF AIR LAYER '• cm. Figure 6. Plot of apparent thermal conductivity of still air layers against temperature showing effect of emissivity of confining surfaces. 30 40 SO 60 MEAN TEMPERATURE: °C 746 WflTd.9 Figure 7 8 10 2 4 6 8 10 2 TOTAL THICKNESS OF FOAM LAYER ; d : «n. The effect of thickness on the apparent thermal conductivity of three polystyrene f oams at three temperatures. Figure 8 0-4 00 0-1 0-2 0-3 0-4 00 Ol RECIPROCAL OF FOAM THICKNESS : d J; em" 0-2 0-3 0-4 Plot of the reciprocal of apparent thermal conductivity of three polystyrene foams against the reciprocal of the thickness at three temperatures. 747 WaT'dcg" TEMPERATURE: C Figure 9. The effect of temperature on the apparent thermal conductivity of three polystyrene foams, at a constant total thickness of 6 cms. 748 Measurements of In-Vivo Thermal Diffusivity of Cat Brain G. J. Trezek, D. L. Jewett, and T. E. Cooper 1 University of California The thermal diffusivity of living brain tissue was studied immediately after cessation of cerebral blood flow by means of the transient thermal response to a cold line source. In cat brain, by numerical fitting of the observed temperature field to the governing heat conduction equation, diffusivity values in the range of 0.125 to 0.110 mm 2 /sec were obtained, these values being close to that of water evaluated at the same temperature. During the succeeding 2 1/2 hours the thermal diffusivity decreased approximately 50%. Key Words: Brain tissue, central nervous system, living tissue, local cooling, thermal diffusivity. 1. Introduction Thermal conductivity and diffusivity are important properties which must be known in order that heat transfer equations can be fully applied to a given system. Accurate knowledge of the heat transfer pro- perties of brain tissue has become particularly important with the advent of both neurosurgical techni- ques utilizing cryoprobes and radio-frequency probes, and other research using localized brain cooling. As an approach to brain heat transfer properties we have chosen to study the transient temperature field generated by a cold line source; this approach differs in principle from the more usual "hot plate" methods used to determine thermal conductivity. Analytical solutions of the governing equations can be generated most easily if the magnitude contribution of each of the terms is known approximately. The problem is simplified if the convective terms due to blood flow can be neglected. Although these terms are probably an order of magnitude less than the conductive terms [l] 2 , they can be completely eliminated by cessation of blood flow. The thermal properties immediately upon cessation of the blood flow closely approximate the in- vivo condition since the brain is still viable for a few minutes. A recent survey by Chato[2] has shown that only a limited amount of thermal conductivity and diffusivity data is available for biological materials and the measurements made on internal organs have been performed only on in-vitro specimens. Comparative information on in- vivo and in-vitro measurements is lacking. Measurements of thermal conductivity in excised dog brain by Ponder [ 3] showed values close to that for water. 2. Analytical Considerations A measurement of thermal diffusivity can be made by observing the transient temperature field in a particular media. As a first approximation, we have considered the brain as a homogeneous media even though there is an inter-mixture of white and gray matter. So as not to induce edge effects and other non-symmetrical effects which would ultimately perturb the temperature field and have to be accounted for analytically, a line source cooling probe was selected for the creation of a temperature field in the brain. The heat conduction equation expressed in cylindrical coordinates takes the form: , 3T , ,3 2 T 1 9T 1 3 2 T d z T, , n > 3t 3r^ r 3r r^ 36^ 3z z Assistant Professor of Mechanical Engineering, Thermal Systems, Berkeley; Assistant Professor of Physiology and Neurosurgery, School of Medicine, San Francisco; NIH Trainee, College of Engineering, Berkeley; respectively. 2 Figures in brackets indicate the literature references at the end of this paper. 749 where T = T(r, 9, z) is the temperature at r, 6, and z; t is time, d is density, c is specific heat, X is thermal conductivity, r is the radial distance from the probe, z is the distance from the surface of the brain, and 9 is the angular location. The thermal diffusivity (a) is equal to A/dc and has the units of mm 2 /sec. Note the absence of a convective term for blood flow. 3. Experimental Apparatus A schematic diagram of the experimental apparatus is shown in figure 1. An anesthetized cat was mounted in a sterotaxic frame. The sterotaxic atlas for cat brain gave a reasonably good indication of where the line source cooling probe should be placed in attempting to avoid such areas as the fluid filled ventricles. A location 6 mm anterior to the ear bars and 8 mm from the mid-line was selected for the experiments so that most measurements would be in white matter. A portion of the skull was removed in this area and the 0.5 mm diameter stainless steel cooling probe was inserted vertically 22 mm into the brain. A linear array of five 0.2 mm diameter stainless steel shielded chromel-constantan thermocouples were mounted in a rotatable three-dimensional traverse which permitted measurement of the field with respect to r, 9, and z. The first thermocouple in the array was positioned 1 mm from the cooling probe, the next two each 1 mm further away and the last two each 2 mm away, allowing temperature measurements up to 7 ■ from the probe. Enough of the skull was removed to permit the thermocouple array to be swung through an angle (9) of 90 degrees. Thus, for a given position of the thermocouples, five values of the temperature field were obtained along the radial distance (r) from the probe at a particular set of 9 and z coordinates. The portion of the exposed brain was covered with absorbent cotton soaked with mineral oil to diminish any vertical gradients due to a convective transfer to the air. Extreme care was exercised in positioning the probe and thermocouples to insure that they were mutually parallel. The thermocouples were calibrated against a mercury thermometer of an accuracy of 0.1°C. Heptane was circulated through the probe from a constant temperature bath. A flow rate of approxi- mately 80 ml/min at 200 psi through the stainless steel hypodermic tubing resulted in a uniform tempera- ture in the z direction on the surface of the cooling probe to within 0.5°C. A twelve channel multipoint recorder was used to moniter the locations shown in figure 1 with an accuracy of ± 0.25°C. During the transient cooling run the recorder was adjusted to record only the five free thermocouples and a single probe temperature repeatedly thereby allowing a more precise specification of the initial transient field. The recording cycle was 1.2 sec/point in either mode. h. Data Analysis and Results Values of the thermal diffusivity (a) were obtained by calculating the value of each of the terms in the transient heat conduction equation and then taking the ratio of the time dependent term to the spacial term. This approach was taken for several reasons, namely, l) An exact solution of the governing equation is feasible for a one dimensional field. Thus some knowledge of the 9 and z variation would be required before simplification to a radial dependence could be made. Even assuming a one dimensional spacial dependence the problem is further complicated by the introduction of a time dependent boundary condition, i.e., the cooling probe requires about ^0 seconds to reach its steady state value. 2) In order to acquire an appreciation for the manner in which the temperature field develops, that is, a knowledge of what time in the transient field do the various spacial dependencies dominate would prove extremely helpful in guiding future analytical endeavors. A typical data analysis proceeded as follows: a point in space was selected and all rates of change of temperature with time and space were evaluated at this point. The evaluation of dT/dt can be obtained directly from the chart record while the spacial dependence of temperature is obtained by cross plotting the data at a given time. Typical sets of radial positions versus temperature curves are shown in figures 2-5. The initial state values shown in figures 3, h, and 5 are decreased due to absence of blood flow as body temperature dropped. A complete set of data was first taken for the live case followed by sets of data for the case immediately after the animal expired, i.e. within 15 minutes of the cessation of blood flow, and at 1 1/2 and 2 1/2 hours after expiration. Comparison of the radial dependence (at the same 9 and z) within 60 seconds and after 15 minutes from the cessation of blood flow showed no significant differences. Using the Newton method, a polynomial was fit to the data around the point of evaluation. Analysis of the data showed the angular dependence to be virtually zero which could partially be expected from the geometrical symmetry. The value of the terms for the data shown in figures 2-5, evaluated at a radial distance (r) of 3 mm from the cooling probe and a vertical distance (z) of 9.1 mm (which was the approxi- mate mid-line of the probe from the brain surface), are summarized in Table 1. Also shown in Table 1 are the values of (a) obtained from the second experimental animal. 750 Table 1. Summary of transient heat conduction terms Immediately after animal expires 2nd animal Time 3T ^C_ 1 3T °C 3 2 T°C 3 2 T°C mmf_ mm 2 t sec 3t sec r 3r mm" 2 " 3r z mm 2 " 3z z mm z sec sec 5 -0.201 0.267 -2.07 0.112 0.118 12.5 -0.2U3 0.789 -3.2U 0.50 0.12U 0.126 20 -0.208 1.12 -U. 20 1.25 1st animal a 2 mm sec .112 12U .113 1 1/2 hours after expiration 5 -0.10l( 0.25 -1.90 0.063 0.06U 2 1/2 hours after expiration 5 -0.0667 o.iu -1.65 o.oUit 0.050 5. Discussion The results of observations made on two animals gave practically identical results for the nature of the temperature field and values of thermal diffusivity even though r,z location was chosen as the evaluation point in the second animal. Analysis of the values shown in Table 1 indicates that the vertical temperature gradient does not contribute in the early part of the transient. In the latter part of the transient (after about 10 sec) heat transfer from the bulk of tissue below the probe destroys the initial one dimensional cooling pattern. Thus, any one dimensional analytical solution would have to be limited to the early transient. Even though the contribution of the various terms varies with time during the transient, it should be noted that the value for (a) remains nearly constant during the transient. The change in thermal diffusivity during the hours of the experiment is shown in figure 6 and can be readily observed from figures 3, k, and 5 as a retardation of the development of a given temperature field. Diffusivity values fairly close to that of water evaluated at body temperature of 35°C were obtained for the tissue immediately after expiration. A drop in diffusivity occurs as the tissue begins to deterior- ate. Even though a decrease in body temperature occurred in the dead animal (from 35°C to 2*I C) the corresponding thermal diffusivity of water is only slightly decreased; thus implying that the physical properties of the tissue must be changing. Since we lack data on d and c it is difficult to ascertain whether it is the thermal conductivity or the density-specific heat product (or both) that result in a decrease in diffusivity with time. In-vivo measurements of each quantity (d and c) would actually be required in order to resolve this point. It is interesting to note that the value of thermal conductivity obtained by Ponder [3] in in- vitro experiments yields thermal diffusivity values (calculated by using the density and specific heat of water) slightly above the value of water. As a first approximation, the brain thermal diffusivity can be considered to be that of water; however, our data indicates that the value is probably different from that of water. Possibly such a difference is due to the microscopic structure of the brain which would invalidate our assumption that it can be treated as a homogeneous medium. 6. Acknowledgements The experimental portion of this study was supported in part by NINDB Grant #NB 05702. 751 7. References [l] Jewett , D. L. and Trezek, G. J., Temperature fields in brain with parallel plane cooling probes , Proceedings 20th Annual Conference on Engineering in Medicine and Biology. [2] Chato, J. C, A survey of thermal conductivity and diffusivity data on biological materials, ASME, Paper #66-WA/HT-37, 1966. [3] Ponder, E. , The coefficient of thermal conduc- tivity of blood and of various tissues, J. of Gen. Physiolgoy, Vol. U5, 1962. TOP VIEW OF BRAIN EAR BARS — v^ /- PROBE •THERMO- COUPLE BANK 12 CHANNEL RECORDER CONSTANT TEMPERATURE BATH '^e\ (r.fl.z) COORDINATE SYSTEM THERMOCOUPLE LOCATIONS T.C. LOCATION * I I mm FROM COOLANT PROBE # 2 2mm FROM COOLANT PROBE # 3 3mm FROM COOLANT PROBE *4 5 mm FROM COOLANT PROBE *5 7mm FROM COOLANT PROBE # 6 BOTTOM OF COOLANT PROBE * 7 10 mm FROM BOTTOM OFPROBE # 8 15 mm FROM BOTTOM OFPROBE * 9 SURFACE OF BRAIN *I0 HEMISPHERE OPPOSITE PROBE ^11 COOLANT BATH ^12 RECTUM Figure 1. Schematic representation of experimental facility. 752 Figure 2. Development of radial temperature field in live tissue. 12 3 4 5 DISTANCE FROM PROBE (mm) Figure 3. Development of radial temperature field in tissue immediately after cessation of cerebral blood flow. VERTICAL POSITION 9. 1 mm FROM SURFACE o INITIAL STATE 2.5 SEC. A 5.0 SEC. V 10.0 SEC. X 15.0 SEC. • 20.0 SEC. ■ 30.0 SEC. r 40.0 SEC. TISSUE IMMEDIATELY AFTER ANIMAL EXPIRES- i.e NO CIRCULATION I 2 3 4 5 6 DISTANCE FROM PROBE (mm) 753 VERTICAL POSITION FROM SURFACE o INITIAL STATE □ 2.5 SEC. SEC. SEC. SEC. SEC. SEC. SEC. 5 10.0 15.0 20.0 30.0 40.0 1/2 HR. DEAD TISSUE I 2 Dl STANCE 3 FROM 4 PROBE 5 (mm ) Figure 5. Development of radial temperature field in tissue 2\ hours after cessation of cerebral blood flow,, O.I9p. 18 0.17 L 0.16- 0.15^. 0.14- 0.13 - 0.12 ; 0.1 I 0.10- 0.09- 0.08- 0.07- 0.06- 0.05- 0.04- 0.03- 0.02- 0.01 PONDERS RESULTS (REF2) FOR X OF DEAD TISSUE - CONVERTED TO a USING d AND c OF WATER (DOG BRAIN) WATER DIFFUSIVITY EVALUATED AT BODY TEMPERATURE CAT BRAIN TISSUE Figure 4. Development of radial temperature field in tissue \\ hours after cessation of cerebral blood flow. DISTANCE FROM PROBE (mm) Figure 6„ Comparison of thermal diffusivity measurements. o I 2 3 TIME (HOURS AFTER ANIMAL EXPIRED) 754 A Correlation for Thermal Contact Conductance of Nominally-Flat Surfaces in a Vacuum C. L. Tien 1 University of California Berkeley, California 94720 A semi-empirical correlation for the thermal conductance of nominally-flat surfaces in a vacuum has been proposed in terms of three dimensionless groups, which characterize, respectively, the thermal contact conductance, the contact pressure, and the surface irregularities. The proposed correlation is shown to be supported quantitatively by previous analytical and experimental investigations. Key Words: Thermal contact conductance, thermal contact resistance, thermal conductivity, heat conduction, heat transfer. 1. Introduction The problem of thermal contact conductance has received considerable attention in recent years. Comprehensive surveys of literature on the subject can be found in references [1,2,3,^1 . In particu- lar, significant progress has been made toward a quantitative analysis of thermal contact conductance in a vacuum environment. Not only is the study of thermal contact conductance in a vacuum of great importance in the thermal design of spacecrafts, but also it serves as a logical starting point for the analysis of the more complex problem involving interstitial fluids. Indeed, impressive analytical groundwork has been laid down by Clausing and Chao [1,5 J for macroscopic constriction resistance due to surface waviness or flatness deviations, and by Yovanovich and Fenech [6] and Mikic and Rohsenow [7] for microscopic constriction resistance due to surface roughness of nominally-flat surfaces. Recent analytical attempts also considered the combined effect of surface roughness and waviness upon the overall thermal contact resistance [7,8]. Cm the other hand, a vast amount of experimental information has become available in recent years. While further analytical and experimental works are needed, the present state of knowledge seems td have reached such a stage that a workable engineering correlation could be constructed for the thermal contact conductance in a vacuum. The present paper is to establish a correlation for the thermal contact conductance of nominally- flat metallic surfaces in a vacuum environment. Accordingly the effect of surface waviness of flatness deviations is neglected. The correlation, which is based on simple dimensional consideration, consists of three dimensionless groups characterizing respectively the thermal contact conductance, the contact pressure, and the surface irregularities. It is shown that the proposed correlation is in quantitative agreement with previous analytical and experimental results. 2. Dimensional Consideration Consider two similar metals of nominally-flat, rough surfaces in contact in a vacuum. For dissimi- lar metals, it is customary to proceed as in the case of similar metals except for the replacement of the metallic physical property by the harmonic mean of those of the dissimilar metals. It is possible, however, to have other complications such as directional effects [9] in the case of dissimilar metals. For this reason, discussions in the present paper will be restricted to the case of similar metals. The rough surfaces under consideration are nominally-flat so that there exists no large-scale waviness Associate Professor of Mechanical Engineering 2 Figures in brackets indicate the literature references at the end of this paper. 755 or flatness deviations. Furthermore, the surface irregularities are assumed to be statistically random and of Gaussian type [10,11]. This is a common assumption for most analyses involving rough surfaces. To describe such a rough surface requires only two statistical parameters, i.e., the rms roughness a and the autoconvariance length a. These two characteristic lengths are related to the rms slope m by the following relation [ll]: 2o 2 = a 2 m 2 (l) When two statistically-independent rough surfaces are put in contact, the two characteristic lengths are defined by [7] : a 2 = 0l 2 + o 2 2 (2) and 2(a } 2 + a 2 2 ) = a 2 ( mi 2 + m 2 2 ) (3) It should be noted that, when a = a, and m, = m 9 , a = /2~ a. = /2~~ a and m = /2~ m, = /2~~ m„ , but still 2a ^ = a^m . To perform a dimensional analysis by use of the Pi theorem [12], it requires first the identifica- tion of primary physical parameters in the physical problem. It is natural to have thermal contact conductance h and thermal conductivity A as two of the primary parameters. The two characteristic lengths a and a for contact surface irregularities must also be included. In addition, the surface deformation as caused by contact pressure P must be taken into account. For the pressure range of practical interest, it has been shown [6] that the deformation is in the plastic range and the charac- teristic material property is the microhardness H, which may be conveniently represented by three times the tensile yield stress, i.e., H = 3F, • From the above consideration, it can be stated that the present problem is characterized by the six parameters, h, A, a, a, P and H. Indeed they represent the three major phases of the problem, thermal (h, a), surface (a , a), and deformation (P,H). A straightforward application of the Pi theorem leads to the conclusion that there exist three dimensionless groups and they can be logically expressed and related as (f) - b (|) C ( |) d 00 where the constants b, c, and d are to be determined from theoretical or experimental results. Based on (l), the above equation may be rearranged as Of-) = fm S (|) d (5) Indeed, an equation of this type has been obtained by Mikic and Rohsenow [7] through their elaborate analysis of the physical problem. Their relation gives f = 0.9, g = 1, and d = l6/l7- It should be realized, however, that their analysis is based on idealized physical models. For actual engineering applications, the validity of the relation (h) or (5); snd. the values of its constants must be deter- mined by the Vast amount of experimental data available in the literature. 3. Correlation of Experimental Data Most existing experimental data do not contain sufficient information to conclusively determine the suggested correlation. In particular, except for the work of MIT group, information concerning a or m is totally missing. Furthermore, all existing measurements of a and m are of doubtful nature, since they are based on profilometer readings. Bennett and.Porteus [ll] have not only demonstrated the defi- ciency in such readings, but also developed an ingenious optical method for the measurements of and m. The lack of information on a and m, however, does not prevent a check on the suggested functional relationship between (h oA) and (p/h). Summarized in Table 1 are experimental investigations with thermal contact conductance data for nominally-flat surfaces in a vacuum. Surfaces are classified here as nominally flat if the rms roughness is greater than one-tenth of the total flatness deviation. Only data for similar metals and for clean surfaces (without plating or oxide films) are included. Actual data points from various investigations are shown in figures 1 and 2 in terms of (ha/A ) and (P/H) . In view of the wide range of conditions (pressure, temperature, surface and material) under which data were obtained by various investigators, figures 1 and 2 indicate quite convincingly, if not con- clusively, that a power relation does exist as (f) « (|) d (6) where d = O.85 approximately. The deviations of experimental data from (6) at low values of (p/h) pro- bably result from the waviness effect, which becomes dominant at low contact pressures. The slight 756 3 0) c •S •p (h CD & U P. a> f-< o c CO f-i •P H CD UN SB Q N O •H K C CO Tl o a NO E-< i UN t- o t— I o I H CO CO 3 3 CO -=t UN o oo ao UN UN UN UN co o CM CO o UN UN UN UN UN H C— NO rH CM 757 variation of data trend at large values of (p/h) in some series of data (notably, CC. and FA.) could be due to a change of deformation characteristic from plastic to elastic range [6]. From their analysis [7], Mikic and Rohsenowhave obtained g=l in (5). This power dependence on rms slope m is indeed in good agreement with their experimental data. This functional form may also be checked qualitatively from the data presented in figures 1 and 2. Assuming g = 1 and d = O.85, it follows that all investigations except for CC. have surfaces with a rms slope m in the range from 0.01 to 0.13, or in terms of angle, from l/2 to 10 degrees. This seems to agree qualitatively with other surface characterization studies [19]. The value of m for CC A data is extremely small (m = 0.0001), and this could be caused by an error in their estimate of a • With given values of g and d, the correlation can now be established from the experimental infor- mation. It is thus proposed the following correlation for the thermal contact conductance of nominally- flat metallic surface in a vacuum: A 0.55 m 0.85 (7) The above correlation differs slightly from the analytical result of Mikic and Rohsenow [7], but appears to be in better agreement with existing experimental information. km Acknowledgment The author wishes to acknowledge the support of the Miller Institute for Basic Research in Science, University of California (Berkeley) , through the appointment of Research Associate Professor during the year of I967-I968. 5. References [l] Clausing, A. M. and Chao, B. T., Thermal contact resistance in a vacuum environment, University of Illinois, Eng. Exp. Sta., Report ME-TN -242-1 (August I963). [2] Fried, E., Study of interface thermal contact conductance, NASA Document Wo. 645D652 (May 1964). [3] Yovanovich, M. M. , Thermal contact conductance in a vacuum, Mech. Eng. Thesis , Mass Inst. Tech. (February 1966). [4] Minges, L. M. , Thermal contact resistance, Vol. I-A review of literature, Tech. Report AFML-TR-65-375 > Air Force Materials Laboratory, Dayton, Ohio (April 1966). [9] Clausing, A. M. , Heat transfer at the inter- face of dissimilar metals — the influence of thermal strain, Int. J. Heat Mass Transfer 9, 691 (1966). [10] Laning, J. H. , Jr. and Battin, R. J., Random Processes in Automatic Control , McGraw-Hill? New York (1956). [5] Clausing, A. M. and Chao, B. T., Thermal con- tact resistance in a vacuum environment, J. Heat Transfer 87c , 243 (1965)- [6] Yovanovich, M. M. and Fenech, H. , Thermal contact conductance of nominally-flat, rough surfaces in a vacuum environment, Thermophysics and Temperature Control of Spacecraft and Entry Vehicles (Ed. G. B. Heller), Academic Press, New York (1966). [7] Mikic, M. M. and Rohsenow, W. M. , Thermal con- tact resistance, Tech. Rept. No. 4542-41, Dept. of Mech. Eng., Mass. Inst. Tech. (Septem- ber 1966). [8] Mikic, M. M. , Yovanovich, M. M. and Rohsenow, W. M. , The effect of surface roughness and waviness upon the overall thermal contact resistance, Tech. Rept. No. 76 361-43, Dept. of Mech. Eng., Mass. Inst. Tech. (October 1966). [11] [12] [13] [14] [15] [16] [17] Bennett, H. E. and Porteus, J. 0., Relation between surface roughness and specular re- flectance at normal incidence, J. Opt. Soc. Am. 51, 123 (1961). Kline , S . J • , Similitude and Approximation Theory , McGraw-Hill, New York (1965). Weiss, V. and Gessler, J. G. (Eds.), Aero- space Structural Metals Handbook , Syracuse Univ. Press, Syracuse, New York (1963). Bloom, M. F. , Thermal contact conductance in a vacuum environment, Mis sle and Space Systems Division, Douglas Aircraft Company Report SM-47700 (December 1964). Cunnington, G. R. , Jr., Thermal conductance of filled aluminum and magnesium joints in a vacuum environment, ASME Paper No. WA/HT-40 (1964). Fried, E. and Atkins, H. conductance in a vacuum, Rockets 2, 591 (1965). , Interface thermal J. Spacecraft and Fried, E. and Kelley, M. J., Thermal conduc- tance of metallic contacts in a vacuum, Thermophysics and Temperature Control of Spacecraft and Entry Vehicles (Ed. G. B. Heller), Academic Press, New York (1966). 758 [18] Henry, J. J., Thermal contact resistance, Sc.D. Thesis, Mass. Inst. Tech. (August, 1964). [19] Funai, A. and Rolling, R. E. , Inspection techniques for characterization of smooth, rough and oxidized surfaces, AIAA Paper No. 67-318, to appear in Thermophysics of Spacecraft and Planetary Bodies (Ed. G. B. Heller), Academic Press, New York (1967). Figure 1. Thermal contact conductance for nominally-flat surfaces in a vacuum (data group 1) . Figure 2. Thermal contact conductance for nominally-flat surfaces in a vacuum (data group 2). 759 Thermal Conductance of Imperfect Contacts Evan Charles Brown, Jr. and Vernon Emerson Holt North Carolina State University- Raleigh, North Carolina A quantum model of the thermal conductance of imperfect contacts with continuum boundary conditions is presented. The case of a junction with finite roughness and with adsorbed gases at the interface is considered. It is shown that a number of parameters such as surface roughness, flatness, oxide layer thickness, adsorbed gas layer thickness, and contact pressure can be represented by a single variable namely the thickness of a compressible layer between adjoining contact materials. Specific solutions obtained with the aid of a computer for representative materials and layer thicknesses are in good agreement with experiment near and above room temperatures. Key Words: Conductance, thermal conductance, thermal resistivity, contact resistance, interface resistance. 1. Introduction The resistance to the flow of heat at the interface of two materials placed together is a well- known and observable phenomena. It is a problem encountered in many applications. Much of the publish- ed work in this field has been of an experimental nature. General correlation of data by empirical or analytical models has been a difficult challenge due to a considerable number and variation of variables. A few of the important variables include the relative flatness of the surfaces of the two materials at the interface, kind of material, contact pressure, type and degree of roughness, chemical condition of the surfaces, interface temperature and gradient, and interstitial material. The available experimental observations are highly specialized and information on important test variables is incomplete because of the complex nature of the thermal contact problem. Rather than fight these variables in the accepted manner one at a time, perhaps it might prove fruitful in this case to join the variables into one parameter and attempt to determine the functional dependence of this parameter in a self-consistent general model. The purpose here is to present a general model for thermal contacts, and to evaluate the model and a few characteristics of the thermal contact problem. Evalua- tion is made on the basis of some of the recent literature, although a survey or critical evaluation of the literature is not attempted here by any means. Before developing a model for an imperfect contact with interstitial gas or oxide, the classical model for a single perfect interface is reviewed. 2. Perfect Interface Consider the thermal resistance imposed by two semi-infinite isotropic similar or dissimilar mate- rials placed together as in figure 1 when a heat flux is maintained across a perfect interface. It is assumed here that the energy transport across the interface is due to phonons having discrete energy of hio rather than electrons. This assumption has some justification in the subsequent considerations of imperfect contacts. The transmitted energy from media 1 to media 2 per element of area and time is el^S (1) where e is the directional transmission coefficient, and I is the energy intensity of the phonons in media 1 striking the interface per elemental solid angle. The integration is over the solid angle S. The energy intensity striking the interface per elemental solid angle in media 1 is 761 where E is the energy density in media 1 and the factor c cos8/4tt gives the fraction of phonons strik- ing the interface per elemental solid angle, time and unit area of interface. Only the longitudinal phonon model with velocity c is indicated here. The elemental solid angle can be expressed as dS = sin6ded<|>. (3) Substituting eqs. (2) and (3), eq. (1) becomes 2tt r-n/2 E c e — — cos6sineded<|>. (4) 0' 0- 1 Equation (4) may be rewritten as where E 1 C 1 q x _ 2 = -±f- r(c) (5) r6max r(e) = ' e cos9 sin6 d6 (6) is the integrated transmission coefficient for the case of the perfect interface as considered by Little [1]1 with 6 determined by sin 6., = c, /c„ for c c„. As indicated in max 112 1 .2 max 1 2 figure 1, the directional transmission coefficient is £ = ^1 Z 2 (z 1+ z 2 ) P 1 C 1 z i = ^rv x etc - where p is the density of media 1. The energy density E of phonons in media 1 as derived in texts such as Kittel [2] for Bose-Einstein energy distribution is 4 4 k TV I *"* 3 E n = ^rH o- x (7) 1 R 3 3 2 I t -7- dx n c 2tt 'o Cexp x -1) where Kui hto_ x = kT , x max max kT and the maximum frequency u is assumed to be determined by the periodic spacing of the structure of the material if known, or by the Debye temperature of the material. The symbol h" denotes Planck's constant divided by 2rr and k is the Boltzmann constant. Figures in brackets indica the literature references at the end of this paper. 762 The derivation of the corresponding equations for each of the two transverse modes is similar to the preceding except that the transverse velocities c and c are substituted for the longitudinal veloc- ities c and c in all of the above expressions. As indicated by Little [1], the above description of quantized energy density seems to be quite accurate within the material but does not describe contact conductances very well. Observed contact conductances are much less than suggested by the single perfect interface model. In order to better represent actual imperfect contacts, a plane layer with two interfaces will be incorporated into the model next. This plane layer will represent interstitial gases or oxides and their effect will be sub- sequently evaluated. 3. Imperfect Contact Represented by a Layer with Two Interfaces Next, consider the resistance caused by two planes with another plane layer of material between them when a heat flux is maintained across both interfaces as indicated in Figure 2. The transmitted energy from media 1 per element of area and time is ! . . i a I^S (8) where a is the overall transmission coefficient for a layer with two interfaces and I and S are as de- fined for the single interface case. Hence, using eqs . (2), (3), and (7) in eq. (8) we have (•2t7 (-6 max E c q x _ 3 = : a -ijp cos9 1 sine i de i d<|>. (9) For plane waves and a plane isotropic non-dissipative layer, the transmission coefficient a according to Holt [3] is 8Z 1 Z 2 Z 3 (10) (z?+zi?)(z:?+z*) + iz7„ 2 z, + iz^-zhizl-zl) cos ( ^ — cose.) 1223 123 1223 c 2 where to is the frequency, c is the characteristic or acoustic velocity, d is the thickness of the layer, and Z is as defined before. The limitation on 8 is taken as sin 8 <_ 1 for c > c sin 8 <_ c /c for c < c . Equation (9) may be written as E 1 C 1 r(-) (id n-3 2 r(o) = i maX L maX a(w,e )E 1 (a ) )cos8 1 sin8 1 d8 1 d U (12) 1 where T(a) is the integrated transmission coefficient for this case. The functional dependence has been indicated in parentheses. Since E appears in both the numerator and denominator of eq. (11), the E coefficient in eqs. (11) and (12) may be evaluated using an infinite upper limit in eq. (7) giving 4 2 (kT V tt E.(-) = i-m . (13) 1 3011 cj 763 Substituting eq. (13) into eqs . (11) and (12) q l-3 r(o) U 9 (kT^V 60R 3 cJ 3 3 30R c 1 ~~ 2 4 - (k V r(a) "max {"l max Rw r 1 -, __ . „■ a j Q A , a ^~2 c Twkr ] ^e^ine^e^ C l 2 ^ e X -l (It) (15) where a is given by (10). The transmission coefficient for the two transverse moder, is found by substi- tuting the transverse velocity of propagation in place of the longitudinal velocity. Equation (15) was integrated with the aid of a computer program for several materials with layer thickness as a parameter. In evaluating the impedances Z , Z and Z , the relationships between the angles of incidence and re- fraction were expressed as sin (— sine, ) C l 1 sin ( — sin sin ( — sin6,)). c c 2 1 Once the transmission coefficient r( = ) is known, the net heat flow q" can be determined as done by Little [1] and Kittel [2] using eq. (14) to obtain (q l-3 " q 3-l } ° r kVr(q) r 4 _ 4 1" ' 3T— L 1 2 J 60R c n (16) c i W/m (17) Expressions for the transverse modes are the same except for the velocity term. In the development of this plane layer model of an imperfect contact, local quantum theory has been applied in evaluating the thermal energy density within the contacting materials, and non-local or "global" continuum boundary conditions have been applied in evaluating the thermal energy transport across the contact interface. It is assumed that phonons are the dominant mode of transport across interstitial gas or oxide. Subsequent comparison with experiment supports this assumption. 4. Comparison with Experiment The value of the transmission coefficient as defined by eq. (17) is tabulated in table 1 for repre- sentative experimental data reported by Barzelay et. al. [4] [5] and Ozisik and Hughes [6]. The contact pressure is indicated. Data presented by Eenech and Rohsenow [7] [8],Rogers[9] and Fried[10] was consid- ered in a study of the relation of layer thickness in the model to the experimental variables. Table 1. Experimental transmission coefficient. The thickness d of an air layer required for agreement of computed values with experimental values of the transmission coefficient is indicated. Case Finish Microns CLA P psi exp Steel-Air-Steel 441 Al-Air-Al 369 SS-Air-SS 369 .75 .9 20 7 7 7.93xl0-8[6] 2.32xl0"^[4] 6.91xl0 _8 r4] 9.8x10-* 8. 30x10"; 9.9xl0" E 764 T n Finish P T d 1 exp Case o Microns psi m CLA SS-Air-Al 375 ~ 1 90 8.41xl0 _8 [5] 5.0xl0" 10 Al-Air-SS 368 ~ 1 90 1.85xl0- 7 [5] 6.0xl0" 10 The thickness d of an air layer required to make computed values of the transmission coefficient agree with experimental values is also tabulated in table 1. This required layer thickness was obtained by plotting computed values of the total transmission coefficient T(<*) as a function of d for each set of materials as illustrated in figure 3, and then the required value of d can be found corresponding to the experimental T(") that satisfied eq. (17). A computed value of the total transmission coefficient r(=) including both longitudinal and transverse modes that could be compared with experimental determi- nations was obtained from r(«) =[r 1 + 2r t e*/c*]. (is) This is the transmission coefficient plotted in figure 3. Note that r(°=) is a function of d for a particular material arrangement. This layer thickness d may in turn be a function of surface flatness, roughness, interstitial gas or oxide, pressure, temperature, temperature gradient, and other variables. Thus, most of the variables affecting thermal conductance as studied by others have been incorporated into one parameter, d. Dr. P. H, McDonald first suggested this approach to us. The physical properties of the materials were obtained from the Handbook of Chemistry and Physics 1966 ed. and from the American institute of Physics Handbook 1963 ed. Values of longitudinal and trans- verse phonon velocities of propagation were obtained from tabulations of acoustic velocities. The results of comparisons such as those in table 1 and figure 3 are surprising in several respects. First, the required air layer thickness appears reasonable and consistent with some mean void that would be physically inherent in contacts pressed together normally. Second, the reduction in conductance observed by Barzelay [5] and Rogers [9] from stainless steel to aluminum compared to the conductance from aluminum to stainless steel (with other variables held essentially the same) is predicted remark- ably well by the model! As one evidence, note the nearly constant value of d obtained for the last two reciprocal cases in table 1. This effect would not occur if there was no air or oxide layer. Also, the effect of contact pressure is predicted with some success by treating the layer as compressible with thickness d inversely proportional to pressure. This has some basis, particularly in the elastic region of behavior of the materials. Relative elastic strengths of dissimilar contacting materials are undoubtedly a significant factor in some instances. In cases such as the specific example in figure 3, the effect of pressure usually appears to be less than it is for smoother surfaces with a smaller "d" where a "compressible layer model" suggests r(°0 or conductance roughly proportional to applied pressure over a somewhat steeper portion of the curve in figure 3 in qualitative agreement with a number of observations. The observation of conductance increasing exponentially with pressure by Weills and Ryder [11] for smooth aluminum and bronze appears consistent with the compressible layer model since d and a steep slope on a curve such as Figure 3 is consistent with their results. Even for thermal contacts in vacuum, some oxide or adsorbed gas will undoubtedly be present at the interface. The "mismatch" of an oxide layer is much less than for the air layer considered above, and calculated conductances are much greater. No such marked increase has been reported for smooth flat sur- faces in vacuum by Fried [10]. Adsorbed gases are inherent even in vacuum, and to a phonon with charac- teristic wavelength less than 100 Angstroms even optically flat surfaces do not look flat. More detailed comparisons and evaluations are being made, particularly for the role of surface roughness on the equivalent layer thickness for the model described here. However, experimental data is not abundant and often is by necessity incomplete or insufficient to establish a definite pattern that either qualifies or disqualifies a theory. The effect of relative surface flatness is a particu- larly distressing variable. If the layer thicknesses on the order of 100 Angstroms indicated by the model developed here are indeed representative of actual contacts , relative flatness may be an important variable as pointed out by Ozisik and Hughes [6] even for "optically flat" surfaces considered by Fried [10] who noted that optically polished contacts are the least reproducible, particularly in a vacuum. This is to be expected according to the case considered in figure 3 where the thinner the layer, the more sensitive the transmission coefficient is to layer thickness. Conversely, for a thicker effective layer such as sawtooth milled surfaces might provide, observations should be more reproducible because the transmission coefficient would be less sensitive to changes in effective layer thickness brought about by changes in contact pressure. This would correspond to the region for d > 10~ 7 m in figure 3. 765 This does not necessarily mean that investigation of this reproducible region will provide greater under- standing of contact heat transfer phonomena. The non-reproducible region may reveal more. The model presented here can be more easily applied to specific cases with some approximation by evaluating T(«) in eq. (17) for normal incidence (6 = 0) and taking an infinite upper limit for oj so that E (to) becomes E (°°) as defined by eq. (13). 5. Summary and Conclusions A general model for thermal contact conductance is presented in terms of one parameter, an equiv- alent effective gas or oxide layer thickness that is a function of the several variables. Local quantum concepts have been applied in determining the thermal energy or phonon density within the material and non-local continuum boundary conditions have been superimposed to evaluate the energy transport across the contact interface. It is shown that some of the wide variation in observations is to be expected due to a marked increase in energy transmission with decreasing effective layer thickness, with trans- mission very sensitive to small changes in the effective layer thickness for the thin layers associated with smooth flat surfaces. The data considered that have a strong dependence on contact pressure fall in this sensitive region and the observed behavior is consistent with treating the layer as being compressible. The effective layer thicknesses required to make the model agree with experiment are roughly on the order of 10-8 m (100 Angstroms) so optically flat surfaces may be far from being ther- mally flat since variations of 10 - 8 m may affect conductance appreciably. The difference reported for the net conductance from aluminum to stainless steel compared to the reciprocal arrangement of stainless steel to aluminum is predicted by the model. A determination of the functional dependence of the effec- tive layer on the several variables has not been attempted, even so the model presented here should be useful in evaluating experiments and in determining how to best change some of the important variables in order to increase or decrease thermal conductances reproducibly . 6. References [1] Little, W. A., The transport of heat between dissimilar solids at low temperatures. Canadian Journal of Physics, Vol. 37, pp. 334-349. (1959). [2] Kittel, C. , Introduction to Solid State Physics, 2nd Ed. John Wiley and Sons, Inc., New York (1956). Also 3rd ed. [3] Holt, V. E. , Energy transport across inter- faces at low temperature , Proceedings 4th annual S.E.S. (October 31, 1966). [4] Barzelay, M. '£., Tong, K. N. and Holloway , G. F. , Thermal conductance of contacts in aircraft joints. National Advisory Committee for Aeronautics, Technical Note 3167, Wash- ington, D. C. (1954). [5] Barzelay, M. E. , Tong, K. N. and Holloway, G. F. , Effect of pressure on thermal con- ductance of contact joints. National Advi- sory Committee for Aeronautics , Technical Note 3295, Washington, D. C. (1955). [7] Fenech, H. and Rohsenow , W. M. 1959. The ther- mal conductance of metallic surfaces in con- tact. Atomic Energy Commission, New York Office, Report 2136. Office of Technical Service, Department of Commerce, Washington, D. C. [8] Fenech, H. and Rohsenow, W. M. 1963. Predic- tion of thermal conductance of metallic sur- faces in contact. Journal of Heat Transfer, Vol. 85, February, pp. 15-24. [9] Rogers, G. F. C, Heat transfer at the inter- face of dissimilar metals. Int. Journal Heat and Mass Transfer, Vol. 2, pp. 150-154 (1961). [10] Fried, E., Study of Interface Thermal Contact Conductance, Contract NAS 8-11247, No. 66SD44 71 (July, 1966). Also No. 65SD4395 (June 1965). [11] Weills, N. D. and Ryder, E. A., Thermal resist- ance measurements of joints formed between stationary metal surfaces. Transactions ASME, Vol. 71, No. 3, pp. 259-267 (1949). [6] Ozisik, M. N. and Hughes, Daniel, Thermal contact conductance of smooth-to-rough con- tact joints. Paper presented at 1966 Winter Annual Meeting of ASME, N. Y., N. Y., 66WA/ HT-54 (1966). Also unpublished data. 766 € * € l-2"«2-l 4Z.Z I *-2 (Z, + Z 2 ) COS0. i- 1,2 Figure 1. Model for the transmission of phonons across a perfect interface between two isotropic materials . />i C i cos 9. MEDIA 3 1-1,2,3 SIN 9, S,N ^, ^ *_ UJ u_ O o 2 »- ri u. 20 - o => a: Q X Z N. O 3 15 O K -1 £ / / INTERFACE NO. 1 ^ " 2 I a: o 10 UJ "~ i * ■ ■"■» •- j= 5 1 1 1 1 FIG. 3 0.2 0.4 0.6 0.8 1.0 ULTRASONIC TRANSMISSION THERMAL CONDUCTANCE VERSUS ULTRASONIC TRANSMISSION FIG. 4 0.2 0.4 0.6 0.8 ULTRASONIC TRANSMISSION RATIO OF EQUIVALENT INTERFACIAL LENGTHS VERSUS ULTRASONIC TRANSMISSION 775 COPPER HEATER NICHROME WIRE HEATING COIL MATED MEMBER A TEST INTERFACE MATED MEMBER B COPPER SINK WATER LINE BASE PLATE PVC INSULATOR TRANSDUCER LEADS FIG. 5 SKETCH OF STACK ARRANGEMENT 776 Thermal Resistance of Sapphire-Sapphire Contact Y. Baer Laboratorium fur Festkorperphysik ETH, Zurich, Switzerland The transport of heat between smooth sapphire surfaces has been measured in ultra-high vacuum in the temperature range from 15 to 300°K. The contact consists of a sphere pressed against a plane with a variable load up to 5 kp. The thermal resistance can be separated into two terms: the well known constriction resistance which is mainly determined by the bulk resistivity and the boundary resistance. A theoretical model has been developed to describe the last term. Two parameters must be introduced to take into account the state of real surfaces. The experimental results as a function of load and tempera- ture are in very good agreement with such a model and the influence of different surface treatments is well explained. Further experi- ments are devoted to contacts between rough surfaces. Key Words: Thermal contact resistance, sapphire (D' (uJ , y , f ) d^J d , i , f ) is the trans- mission coefficient. For a temperature gap AT = 1 deg, the net flow of heat across the film will be dS / (fro)) ho)/kT F ( If , T) dS = ™ S Z/w 2 D(a; ' ^ } duJ (4) 2 (2TT)2 C 2 ^2 kT 2 I (e hu)/kT_ 1) 2 o uJ^ : Debye frequency D(vJ , jf ) is now a directional average of D'(w , £ , f ). The expression (4) is very simi- lar to that of the specific heat of a phonon gas whose spectrum is multiplied with D(oJ , ft ) . It is well known that the specific heat is very insensitive to the detailed shape of the phonon spectrum so that we can describe the transmission coefficient by the simple relation: D(u; , V ) = H ^ 3 ^ - w (5) D(uo ■ f> = H ^ 3 c D(w .*> - uj >a> c uj appears as the maximum vibrational frequency transmitted through the film and H as a mean transmission coefficient for uj - uj . With these definitions, the conductance A s may be written in the convenient form >s = — = A H 9? c(— W a 2 (6) R s k 4 h*j c with A = , = 6(2ir) 2 n 5 c 2 c (-)^i^) 3 / z4e : d Z W) lei r e z_i^2 4 e z ( e z-l)2 In the low temperature limit we obtain with the usual approximation for the specific heat: X s = 2 t, 2 H-^T^TTa 2 (7) s (2Tr) 2 tiV 15 This formula, except for the definition of H, is identical to those obtained by Little [3] and Jones and Pennebaker [4]. It is interesting to notice that the expression (7) no longer depends on the parameter 9 . For practical purposes we define a temperature dependent transmission coefficient: 3 f 9 o H9 c c rr", D(T) = Q . 39 D° 779 6 3 which is H/3 at low temperatures and decreases to reach the value of(g — ) H/3 at very- high temperatures. Using this definition, the following expression for^the conductance may be obtained: X = — c C D(T)TTa 2 (9) ' s 4 v v ' v J ' where C is the specific heat per volume unit. If one writes the bulk thermal conducti- vity in the form o- = 1/3 c C , the final result is v _1_ = \ = 3_£L D ( T ) rr a 2 ( 10) R s 4 A 3.3. The Total Resistance The two resistances R s and R have been described separately and the problem is now to determine how to express the total contact resistance. This question requires a de- tailed discussion which will appear elsewhere [5]. The result is that the total resistance is given simply by the following sum: R = R p + R s (11) This formula is a good approximation except when Rp£^ R s . In that short range the total error will be still less than 15$. The combination of equations (1) and (10) gives a useful relation for studying the behaviour of the total contact resistance: R s R p 3TT D(T) (12) It can be used to define different ranges as a function of the two parameters K and D(T), as shown in figure 2. An ideal contact is characterized by D(T)s 1 so that to any variation of T or a, corresponds a motion along a vertical line. This case has been cal- culated by Wexler [6] which confirms the approximate validity of the relation (11). At high temperatures (ohmic regime) the constriction resistance is dominant while at low temperatures (Knudsen regime) a new term appears which is absolutely independent of A . This resistance is a size effect related to the geometrical shape of a contact and the measurement of the transition range could be used as a direct determination of the mean free path. In the case of a real contact 'D(T) < 1) , the variation of T for a constant radius of contact a, determines now a curve which is no longer vertical and the transi- tion form Rp to R s occurs in the ohmic range. 4. Models of the Transmission Coefficient 4.1. Disturbed Film Model It is impossible to give a detailed description of the phonon scattering in the dis- turbed film. One way to get around this difficulty is to relate the actual disordered structure of the film to statistical parameters. Such a theory has been developed by Ziman [7] who calculated the mean free path A p in irregular materials. He de scribe s the disturbed structure with two parameters: the variance of the sound velocity ( 6 c) x and the correlation length L which determines the scale of the irregularities. In the long wave limit the result of Ziman can be used to calculate the transmission coefficient: ■! . 4Tr 7/2 JML ^^ with A» 2d, L = ex 2d, c< <=, 1. 780 This expression has only a qualitative validity but it shows clearly that the pho- nons are scattered only slightly. As the frequency increases the transmission coefficient will become appreciably smaller than one. That means that the phonons have a non- vanish- ing probability to be scattered more than once, and the definition of D(u-> , jf ) no longer has meaning. In this case the film resistance can be very simply described by the following expression: -±- - A f = l/3cci TTa 2 = o~ JUl ( 14 ) R f f v 2d f 2d A- is the mean free path of -the dominant phonon mode. It must be noticed that this formula can be formally identified with (9) if one puts D(T) =-4 £ (15) 3 2d In ignorance of the numerical factors in (13) it is very difficult to give a cri- m for at about 9, terion for the transition from R to R_ but it seems correct to assume that this occurs s f c 4.2. Interface Resistance The model of a disturbed film does not take explicitly into account the presence of an interface nor does it suppose that its influence is predominant. The potential between two flat surfaces has probably the form of a Lennard-Jones potential and depends only on the mean separation of these surfaces so that the motion of the atoms parallel to the interface will practically not be coupled. Accordingly, one third of the incident energy only has some probability to be transmitted through the interface and the para- meter H will not be greater than one. Little [3] has pointed out a geometrical argument based on the roughness of the surfaces, which shows that the effective transmitting area for a contact between hard crystals is a decreasing function of the frequency. This means in the formalism of this work that, for a constant apparent area of contact, the transmission coefficient is a decreasing function of the frequency. It is a very complex problem to know exactly the scattering at the interface and it would require a detailed description of the force constants that is practically impossible. Ludwig and Pick [8] have calculated the scattering of phonons at defect planes in the simple cubic lattice and an examination of this work shows clearly that the general tendency in all cases is that the high frequencies are more effectively scattered than the low frequencies. It is very difficult to decide at low temperatures what mechanism is the most important because they all are consistent with the definition (5). An observation of pressure dependence of ©c an( i H could lead to a conclusion because the disturbed film resistance alone is certainly independent of the mechanical pressure. 5. Analysis Method of the Experimental Results All the expressions established previously depend on the contact radius a, which is related to the load, for an elastic deformation, by the classic theory of Hertz (see Holm [2]). A direct measurement of the contact radius as a function of load has been carried out under a microscope. The results, as shown in figure 3, are in good agreement with the formula of Hertz and confirm a strictly elastic deformation. The thermal properties of sapphire are well known [9], [10] but it is of interest in connection with this problem to remember the very great variation of the mean free path as a function of temperature. The total contact resistance can be expressed as the following sum R(a,T) = + _ L_^ __ ( 16 ) 2acr(T) fta 2 A H ©3 C(6 /T) L> C 781 Some remarks are required concerning the validity of this formula: a) The simple addition of the two terms may cause a maximal error of 15$ in the range where R o£ R • b) The thermal properties have been considered as independent of the mechanical deformation. c) The parameters 6 C and H are possibly dependent on the load. d) If the film resistance model is valid, R-f (14) must be substituted for R s (6) in the high temperature range. 6. Experimental Results and Discussion 6.1. Polycrystalline Surfaces The surfaces of these samples (S-,) have been submitted to an intensive mechanical polishing which produces a nearly perfect geometrical shape but disturbs strongly the structure to a depth of about 0.5/ /rr) [ll]. Fig. 4 shows the temperature dependence of the contact conductance for different contact radii. It is quite clear in this case that Rj must be used in formula (11) to analyze these results, but cr^ is an unknown function. Therefore the constriction term has been substracted from the total resistance and the resistance per area unit has been calculated, as shown in figure 5. This dependence is characteristic of a highly polycrystalline or nearly amorphous structure. If 2d *** 1 m*k the conductivity of this material is 6 ■ 10 - - 1 - W m~-'-deg~ at 300°K which is a typical value for glasses. It appears that mechanically polished surfaces are not convenient to study the surface resistance and that other treatments must be investigated in order to pro- duce better surfaces. 6.2. Treated Surfaces Two different methods have been used to obtain flat surfaces without perturbing the superficial structure too much. In the first case (samples G) the mechanically polished crystals have been annealed for two hours at 1300OC . Dinichert [ll] has shown that this thermal treatment is sufficient to recrystallize the disturbed region. The other samples (B]_]_) have been etched for 2 minutes in ortho phosphoric acid at 450°C. In such a way a 3//m thick film is removed from the surface, which remains perfectly flat and smooth. These two kinds of samples yield the same results which proves that the surface state is no longer influenced by the mechanical polishing. As indicated in figure 6 the tempera- ture dependence of A is more pronounced and the maxima occur at lower temperatures. An analysis of these results is now possible with the surface resistance model. The low temperature range has been carefully measured and the experimental values fitted to the theoretical expression (6) as shown in figure 7. The characteristic temperature 6 C is small, 6 C = ©d/5, this means that an interface acts really as a high frequency phonon filter. The parameter H is independent of the contact radius and its value is consistent with the theoretical prediction. The whole temperature range can be described now by the sum of the two resistances as shown in figure 8. The resistance R s has been used up to 300°K which is not necessary correct but the possible induced error is small and the agreement is fairly good. The dependence on the contact radius has been measured at high and low temperatures where only one resistance in dominant (see fig. 9 and 10) and the behaviour predicted by equation (16) is fully satisfied. As indicated in figure 10, the points labelled A correspond to a contact closed with a maximal load at room temperature while the points B correspond to a contact opened for 2 minutes at 35°K. The most satis- factory explanation of this effect is that the adsorbed molecules are driven back out of the contact at high temperatures whereas they are strongly bound to the surface at low temperatures. The run A is reproducible so that such adsorption effects have been syste- matically eliminated. Contacts between rough surfaces have also been investigated. The results, which will appear elsewhere [5], also give a very nice confirmation of the surface resistance model. 782 7. Conclusion The thermal contact resistance under well defined conditions has been interpreted with a two-resistance model. A satisfactory description of the phonon diffraction is gi- ven by the parameters C and H which have a simple physical meaning. It is very difficult to decide exactly what mechanism is the most effective, therefore, it seems desirable to vary as much as possible the surface quality and to observe the influence on the trans- mission coefficient. Unfortunately a comparison with the low temperature measurements of Berman and Mate [12] on sapphire-sapphire contact is very difficult because the effective contact area is not known. A deviation from the T^-law is not surprising if one realizes that 6 C may be as small as D /5. The advantage of measuring the thermal resistance with this geometrical shape is that it allows one to study the diffraction on an isolated obstacle independently of any other bulk diffraction. This separation may be used now in varied situations so that it seems permissible to think that this technique will have a scope in the study of many surface properties. I am very grateful to Prof. G. Busch for helping with the choice of problem and for his continuing encouragement of the work. I would like to thank C. Frohlich for many helpful discussions and contribution in instrumentation. The author is also indebted to the Schweizerischer Arbeitsbeschaffungskredit for financial support. 8. References [l] Frohlich, C, Private Communication. [7] Ziman, J.M. , Electrons and Phonons, Oxford (1962). [2] Holm, R. , Electric Contacts Handbook, Springer (1958). [8] Ludwig, W. and Pick, H. , Phys. Status Solidi 19, 313 (1967). [3] Little, W.A., Canad.J.Phys. 22, 334 (1959). [9] Slack, G.A. , Phys. Rev. 126, 427 (1962). [4] Jones, R.E. and Pennebaker, W.B., [10] Holland, M.G., J.Appl.Phys. 22.2910, Cryogenics 3, 215, (1963). (1962). [5] Baer, Y. , to appear in Phys.Kondens. [ll] Dinichert, M.P., Bull, de la Society Materie. Suisse de Chronometrie (1964). [6] Wexler, G. , Proc.Phys.Soc. 89, 927, [12] Berman, R. and Mate C.P., Nature 182 , (1966). ' 1662 (1958). 783 regime of Knudsen Figure 1. Schematic diagram of the experimental arrangement. - — - — '° a I log D (T) real contact | ideal contact Figure 2. Diagram illustrating the different conduction regimes. 0,15 a (mm) 0,10 0,05 «-o- y ^ --• 8 '" (, ^' o o = U ft 066f l/s r 4 f (kp) Figure 3. Contact radius versus load. X(jty 300 Figure 4. Temperature dependence of thermal conductance for different contact radii (polycrystalline surfaces) . 784 R ( iro 2 , mm z °K . W ' 1 Si 1 * a = 0,105 mm 1 I o n = 077 mm 061 mm 1 \ o a = 1 \ o I \ o — c a V . /^° \ P •\b/ c • 100 200 300 T (°K) - Figure 5. Thermal resistivity of the polycrystalline film as a function of temperature. Figure 6. Temperature dependence of thermal conductance for different contact radii (treated surfaces). x_ T(J 2 H 'mm 8 °K' ; oa= 0,045 mm"l Bj e (5) A(^) = A(¥k) *~*,e»o (6) £(5%y~fi c ( r 2-^) *=^,e>o (7) Where the normalized temperature T (x, 9) has been used, ' to - tc -t a - tc (8) The solutions obtained from eqs (1) by the standard technique of separation of variables in product form are well known. With the proper choice of sign on the separation constants the solutions for the two regions are: Region ( I) Region (II) T. tx,f) = K, IX) + 2 %i c -n &*« K* +> M^ *»*3 T a (A,e) = 14 c*) +2 B >i [F 11 H^^+^^^x_]e - v**, e - sfae (9) These solutions have been written as the sum of steady and time-dependent parts. For the latter, summations have been indicated in anticipation of having to expand the initial condition in a series. The steady state solutions are given by - sf^+K wherein S- v Application of eq (4) gives / (10) (11) % «, ^c ' ^z *U a, Next, eq (5) is applied and results in c ..r/Xjtja^/ in which AJ = "4 £» It is assumed that no heat storage occurs in the inf initesimally thin interface. This is equivalent to saying that the time derivative of both sides of eq (6) are equal, or Application of eq (6), together with eqs (13; and (14) yields (15) The eigenvalue equation results from the application of eq (7). After elimination of the derived con- stants the region 1 eigenvalue equation becomes From eq (14) the region II eigenvalues are then determined. Only the series coefficients B are needed to complete the solution. As usual these coefficients are obtained by matching the initial condition in a series expansion. Ordinary techniques fail in this case due to the nonorthogonality of the solution eigenf unctions over the full x-interval (0,L) with respect to the usual weighing factor of unity. However, employing the work of Tittle (1) the B can be found by evaluating the following integral relation. R ~J £, (K)£C>, CtxY^x + -dUA y^xJcJx -hCy^-Zd (■*)[£, -d^ S^X -f-G-r, OM S„ xjd* £} ■ - ,- £. ~~ ~i In this equation the constant, C, called "orthogonality factor" by Tittle is given by Proof of the validity of eq (17) is given by Tittle. However, from eq (17) it can be seen that, alternatively, C could be thought of as a discontinuous weighting function, having a value of unity in (o/ (2 2 ) the temperature drop across the interface contact (AT ) exceeds or overshoots its steady state value for a while during the transient period. The quantity used for comparison in this phenomenon is the time required for the transient AT to reach its maximum value. In the theoretical work it was found that this time depended on h . The experimental values of this parameter are also shown in Appendix B. Small adjustments were also made to this parameter, in the manner discussed above, to provide a set of experi- mental data for a single value of the end conductance, h and h , for each series. The results are plotted in figures 8 through 10, in which the theoretical curves are drawn as solid lines and the exper- imental points are plotted as symbols. It can be seen that the agreement follows about the same pattern as in the time to approach steady state comparison. Note that the error is such (as in the previous case) that the experimental observations appear to be somewhat slower than the theoretical. That is, the experimental time of maximum overshoot occurs later than the theoretical time. This also seems to support the abuve argument in that a temporary dip in driving potential should have a delaying effect on the time to reach maximum overshoot. As in figure 5, figure 9 represents tests with two surface roughness combinations and tests in air apd vacuum. Again no effects due to these quantities were noted. As further evidence in the comparison it is pointed out that only four of the six series produced an overshoot in AT . For the two series (2 and 4) in which ■"£ < 1 no overshoot was found to occur, which agrees with the theoretical work. In view of the above comparisons the agreement between theory and experiment appears to be good. 793 5. Summary and Conclusions Solutions have been derived for the time-dependent temperature distribution in a two layer slab with contact resistance at the interface and contact or convective resistance on the outer boundaries. Experiments have been conducted in which a series of test samples were subjected to thermal transients which closely approximated the boundary conditions used in the theoretical solution. Comparisons made on the basis of time to approach steady state and time of occurrence of maximum overshoot in interface temperature drop show good agreement between theoretical and experimental values. For those cases in which a noted discrepancy in the experimental boundary condition was believed to have only a small effect the agreement was within at least 10%. On the basis of the above comparisons it is concluded that the theoretical solutions presented could be used to predict the transient response of systems to which they are applicable to an accuracy sufficient for most engineering purposes. That is, with these solutions properly programmed on a digital computer, a knowledge of the required thermal properties and an estimate of the contact con- ductance from the existing literature, a design engineer could predict the transient response of a given system. Thus used an accuracy at least as good as his knowledge of the contact conductance could be expected. The work reported here is part of a project carried out under grant NsG-711 for the National Aeronautics and Space Administration whose sponsorship is gratefully acknowledged. 6. References [1] [2] Tittle, C. W. , Boundary value problems in composite media: Quasi-orthogonal functions, J. Appl. Physics, 38, pp. 486-488, April 1965. Tittle, C. 17, 1965. W. Personal communication, April [3] Blum, H. A., and Moore, C. J., Jr., Transient phenomena in heat transfer across surfaces in contact, ASME Paper No. 65-HT-59, August 1965. f4] Moore, C. J., Jr., Heat transfer across surfaces in contact: Studies of transients in one-dimensional composite systems, Ph.D. thesis, Southern Methodist University, Dallas, Texas, March 1967. Also NASA Contractor Report CR-84868. [5] Williams, D. R. , The thermal conductivity of several metals: An evaluation of a method employed by the National Bureau of Standards, Southern Methodist University, Master's Thesis, November 1966. 7. Appendix A Test Specimen Data Specimen No. Material Length Avg. Surface Roughness of Interface End 3 4 5 7 7A 9 2024-T351 2024-T351 303-MA Armco Iron Armco Iron 2024-T351 2.525 5.067 5.055 5.123 5.123 5.080 1.78 x 10 , m (rms) 1.91 x 10" ,m (rms) 0.076 x lO", m (rms) 0.076 x 10", m (rms) 0.762 x 10" m (rms) 0.508 x 10" m (rms) Material Properties Material Armco Iron 2024-T351 Alum. 303-MA St. Stl. Therm. Diffusivity (m /sec) 200 670 37 Thermal Conductivity (w/m-deg) , t in C 78.35-0.0757 t 111.43 + 0.189 t 14.19 + 0.0199 t 794 8. Appendix B Tabulated Data Series Specimen No's. Run No. hot side/cold side No. ad. 3/4 5 241 3620 20000 19300 26.1 23.3 26.6 24.7 6 76 1130 6810 10400 49.4 49.4 44.5 39.7 7 138 2200 11200 18300 36.4 33.9 35.3 31.0 8 345 6300 26000 27800 20.8 13.6 21.7 16.0 17000 4/5 1 221 516 * 19800 167.5 70.7 164.5 65.8 2 90 165 4810 4840 245.0 155.0 224.0 95.8 3 138 278 * 35700 181.0 101.7 182.0 105.5 4 331 499 9680 54500 148.1 72.6 136.5 67.8 28400 4/7 221 324 442 538 152 200 35 726 9490 857 10000 999 12500 1630 14800 692 6200 3830 12000 2460 4960 12800 19100 28600 24200 35200 17700 28100 82.8 79.9 75.0 64.9 99.7 59.1 74.7 114.3 106.5 77.5 73.6 135.5 53.2 67.8 77.4 77.7 77.1 66.3 101.0 58.1 69.5 103.5 101.5 79.3 75.5 129.0 51.1 52.3 17000 4/7A 1 97 908 8180 7860 83.2 111.3 73.6 93.8 2 207 1250 10400 13500 74.5 92.9 71.4 85.1 3 365 1950 * 32300 66.8 67.8 66.8 72.5 4 524 2870 13400 36200 59.1 60.0 62.1 62.9 17000 7A/4 3 152 579 5400 6030 126.7 None 105.5 None 4 248 664 6720 6400 122.0 None 104.5 None 5 379 1110 8270 17000 98.7 None 95.3 None 6 1330 5670 17600 29600 70.2 None 74.5 None 7 490 1660 9710 19400 89.0 None 89.1 None 14200 9/4 1 131 817 5750 7980 72.1 None 62.8 None 2 214 1430 8050 13600 58.1 None 54.8 None 3 317 1940 9540 20500 54.2 None 53.4 None 4 435 1130 12000 15700 52.3 None 53.2 None 5 765 2620 14500 24500 43.6 None 46.5 None 14200 h, = contact pressure (kN/m ) interface conductance (kw/m hot end conductance " cold end conductance -deg) Q = time to reach steady state (sec) 8 = time of maximum overshoot " mo Primes indicate adiusted times for h =h =h , 1 2 ad . No value. Figure 1. Boundary value problem geometry. Interface, h c 795 Nuts Adapter Source- Upper Specimen Lower Specimen Insulation- Load Cell Nuts Spring (3) .Upper Plate .Spherical Joint Insulation Load Rod Hydraulic Cylinder Rods (3) Base Plate Spherical Bearing Figure 2. Test apparatus sectional drawing. Figure 3. Time to approach steady state versus interface contact conductance for series 1, with hj h 2 17,000 kw/m2-deg. CO CO CD 60 - O Series 1 40 - o o 20 1R - Theoretical^"- — I I 1 O , 1,1,1 1 2 3 4 5 6 7 h c (kw/m 2 -deg) 796 (A V) 60- 60- 40- 20 — - Series 2 - o - o O o - Theoretical^ o - 1 . 1 . 1 . 1 1 , 1 0.6 1 2 3 h c (kw/m 2 -deg) Figure 4. Time to approach steady state versus interface contace conductance for series 2, with h^ h 2 14,200 kw/m-deg. Figure 5. Time to approach steady state versus interface contact conductance for series 3, with h. h„ 17,000 kw/m -deg. 150r 100 - O W 50 30 Series 3 O o„ O O O Theoretical- I ■ I ■ I . I 0.6 1 (kw/m 2 -deg) 150r 100 O o Theoretical Series 4 Figure 6. Time to approach steady state versus interface contact conductance for series 4, with h, h- 14,200 kw/m 2 -deg. in V) 50- 30L 1,1,1,1 0.6 i c ( kw/m 2 -deg ) I , I CM O (0 CD Figure 7. Time to approach steady state versus interface contact conductance for series 5, with hj h 2 28,400 kw/m 2 -deg. 4 r 3- 0.1 Series 5 Theoretical 0.2 0.4 0.6 (kw/m 2 - deg ) 797 o E ® - o 1U_U_ Theoretical- Series 1 J i I ■ I.I.I 1 2 3 4 5 6 7 h c (kw/m 2 -deg) Figure 8. Time of r.iaximuui overshoot in interface contact temperature drop versus interface contact conductance for series 3, with hi h 9 17,000 kw/m 2 -deg. Figure 9. Time of maximum overshoot in interface contact temperature drop versus interface contact conductance for series 3, with h^ h 2 17,000 kw/m -deg. © 15r 10 c E 5 Series 3 Theoretical 3 1 I ■ I . I . I , I l L 0.6 1 2 3 4 h c (kvv/m 2 -deg) o en 15r « 10- " Series 5 o - o — Theoretical -^ - i 1 ■ 1 1 . 1 5- 0.1 0.2 0.4 0.6 h c (kw/m 2 -deg) Figure 10. Time of maximum overshoot in interface contact temperature drop versus interface contact conductance for series 5, with hj^ h 2 28,400 kw/m 2 -deg. 798 AUTHOR INDEX Author Ahtye, W. F. Ainscough, J. B. Androulakis , J. Baer, Y. Barisoni, M. Bartram, R. H. Bates, J. L. Baxley, A. L. Beroes, C. S. Bhagat, S. M. Bird, B. L. Blair, M. G. Blum, H. A. Bonilla, C. F. Brazel, J. P. Brendeng, E. Brown, E. C. , Jr. Burckbuchler, F. V. Burns, M. M. Bur stein, E. Bury, P. Carpenter, H. C. Carwile, L. C. K, Catton, I. Chang, H. Chari, M. S. R. Charsley, P. Childs, G. E. Chu, R. C. Clark, N. N. Clough, D. J. Page Author 551 Codegone, C. 467 Cooper, T. E. 337 Couper, J. R. 777 Damon, D. H. 279 Danielson, G. C. 257 Easton, C. R. 477 Erdmann, J. C. 685 Feith, A. D. 721 Ferro, V. 253 Fritsch, C. A. 103 Frivik, P. 355 Gammon , R . W . 349, 787 Gerrish, R. W. , Jr 5 61 Gersi, J. 415 Goff, J. F. 495 Goldsmith, L. A. 761 Graves, R. S. 257 Gray, P. 331 Gruner, P. P. 407 Gueths, J. E. 579 Gupta, G. P. 529 Hager, N. E. , Jr. 59 Hall, C. W. 207 Halteman, E. K. 355 Hanley, H. J. M. 365 Harris, L. B. 131 Hatters, H. D. 597 Ho, C. Y. 677 Hoge, H. J. 257 Holland, S. 491 Holt, V. E. Page 151, 163, 177 749 685 111 331, 381 209 259 371, 703 151, 163, 177 695 495 405 507 111 311 469 711 615 139 257 605, 677 241 455 507 597 627 721 1, 33 59 615 761 799 Author Hust, J. G. Jahoda, J. A. Jewett, D. L. Johannin, P. Jones, T. T. Klein, N. Klein, P. H. Kosson, R. L. Kostenko, C. Krishnamurthy, C. Kruger, 0. L. Kudman , I . Larsen, D. C. Laubitz, M. J. Leaver, A. D. W. Lee, C. S. Leidenfrost, W. LeNeindre, B. Liley, P. E. Lilley, J. R. Little, A. J. Lorentzen, G. McElroy, D. L. McLaughlin, E. Maczek, A. 0. S. Markowitz, D. Martin, J. J. Matthews, F. V. , Jr. Merrill, R. B. Minges, M. L. Mirkovich, V. V. Moore, C. J., Jr. Moore, J. P. Moser, J. B. Page 271 259 749 579 737 407 399 337 769 659 461 97 323 325 131 561 633 579 57, 549 207 469 495 279, 297, 711 671 615 257 381 455 713 77, 197 537 787 297, 711 461 Author Mossahebi, M. Nahas, N. C. Nakata, M. M. Orr, H. W. Pearlman, N. Pearson, G. J. Pettyjohn, R. R. Poltz, H. Powell, R. L. Powell, R. W. Prettyman, P. E. Pyron, C. M. , Jr. Reynolds, C. A. Rousselle, J. C. Roy, D. Sacchi, A. Salter, J. A. M. Saxena , S. C. Schriempf, J. T. Seely, J. H. Sengers, J. V. Shanks, H. R. Shaw, D. Shirfin, G. A. Shirtliffe, C. J. Smith, C. A. Srayly, E. D. Sohns, H. W. Somerton, W. H. Spitzer, D. P. Stephenson, D. G. Styhr, K. H. , Jr. Thodos, G. Tien, C. L. Page 527 685 387, 493 229, 521 103 257 729 47 271 1, 33, 293, 323 695 425 257 513 547 151, 163, 177 131 539, 605 249 677 595 331, 381 469 405 219, 229 387, 493 425 529 527 123 219 415 547 755 800 Author Tihen, S. S. Tree, D. JR. Tr.ezek, G. J. Tufeu, R. Ulbrich, C. W. Ure, R. W., Jr. Van der Meer, M. P. Venart, J. E. S. Vodar, B. Von Gutfeld, R. J. Page 529 633 749 579 257 111 325 659 579 247 Author Wechsler, A. E. Weeks, C. C. Weitzel, D. H. Whreeler, M„ J. Wilkes, K. E. Williams, D. R. Williams, R. K. Winer, B. Wolf, L. , Jr. Page 77, 89 387, 493 271 467 293 349 279, 297 253 769 801 NATIONAL BUREAU OF STANDARDS The National Bureau of Standards 1 was established by an act of Congress March 3, 1901. Today, in addition to serving as the Nation's central measurement laboratory, the Bureau is a principal focal point in the Federal Government for assuring maxi- mum application of the physical and engineering sciences to the advancement of tech- nology in industry and commerce. To this end the Bureau conducts research and provides central national services in three broad program areas and provides cen- tral national services in a fourth. These are: (1) basic measurements and standards, (2) materials measurements and standards, (3) technological measurements and standards, and (4) transfer of technology. 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