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LEGAL NOTICE
This report was prepared as an account of
Government sponsored work. · Neither the
United States, nor the Commission, nor any
person acting on behalf of the Commission:
A. Makes any warranty or representa-
tion, expressed or implied, with respect to
the accuracy, completeness, or usefulness
of the information contained in this report,
or that the use of any information, appa-
ratus, method, or process disclosed in this
report may not infringe privately owned
rights; or .
B. Assumes any liabilities with respect
to the use of, or for damages resulting from
the use of any information, apparatus,
method, or process disclosed in this report.
As used in the above, "person acting on
behalf of the Commission” includes any em-
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employee of such contractor, to the extent
that such employee or contractor of the
Commission, or employee of such contractor
prepares, disseminates, or provides access
to; any information pursuant to his employ-
ment or contract with the Commission, or
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ORNI-P-148 DTIE
CONF-638-3:
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AUG 1 1 1864
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A THERMAL COMPARATOR APPARATUS FOR THERMAL CONDUCTIVITY
MEASUREMENTS FROM 50 TO 400°C*
T. G. Kollie, D. L. McElroy,
R. S. Graves, and W. Fulkerson
Metals and Ceramics Division
Oak Ridge National Laboratory
Oak Ridge, Tennessee, USA
11
Paper to be presented at the
National Physical Laboratory
Thermal Conductivity Conference
Teddington, England
July 15–17, 1964

Focsimile Price $_Hube
Microfilm Price $ 116,
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Available from the
Office of Technical Services
Department of Commerce
Washington 25, D. C.
*Research sponsored by the U. S. Atomic Energy Commission under
contract with the Union Carbide Corporation.

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A THERMAL COMPARATOR APPARATUS FOR THERMAL CONDUCTIVITY
EASUREMENTS FROM 50 TO 400°C
T. G. Kollie, D. L. McElroy,
R. S. Graves, and W. Felkerson
Metals and Ceramics Division
Oak Ridge National Laboratory
is
..
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ABSTRACT
The experimental details, mathematical models, and typical data for
.
>
>
a rapid comparative method for thermal conductivity measurements are
presented. The method consists of measuring the temperature change of a
small silver sphere after it is brought in contact with a small disk-
shaped specimen which was initially at a higher temperature. This temper-
ature change was calibrated in the range of 50 to 400°C by making measure-
ments on samples of known thermal conductivity. Results at 75 and 300°C
on uranium sulfide, thorium sulfide, uranium nitride, and uranium dioxide
are presented to demonstrate the usefulness of the method. The accuracy
of this technique was shown to be better than 110% with a reproducibility
of at least 12.5%. Using known transport mechanisms for heat conduction
in solids and the temperature dependence of the electrical conductivity,
A means to judiciously extrapolate thermal conductivity data obtained
m.
between 50 and 400°C to higher temperature is presented.
INTRODUCTION
The first object of this work was to develop an apparatus capable of
making rapid, reproducible, and moderately accurate thermal conductivity
measurements over a modest temperature range on small specimens, 1/4 to
1 in. in diameter and 1/4 in. thick.
This is one phase of a program to
study solids of nuclear and scientific interest to determine the effect of
controlled variables on the various heat transport mechanisms.
To perform
such a study, one is faced with choosing a method of measuring the thermal
conductivity. Each of the two alternate approaches, steady state or non-
steady state, has merit. Commonly used steady-state methods, such as
radial or longitudinal heat flow techniques, 1-3 yield accurate and repro-
ducible thermal conductivity data; however, the methods have the dis-
advantages of requiring large specimens and of being slow. Commonly
selected transient methods use smaller specimens and an appropriate time-
varying heat input-7 and, though fast and reproducible, are not as accurate
as steady-state methods. One reason for this is that transient methods most
often yield thermal diffusivity data that require a knowledge of specific
heat and density in order to obtain thermal conductivity values.
The method described here, which was developed by Powell, 8–10 is a
transient comparative method which gives thermal conductivity, not
diffusivity, and can be used to nondestructively test small specimens.
The
principle of the method is the same as that of a qualitative handling test
used for many years in the steel industry to distinguish carbon steels from
high-alloy steels because the former feels cooler to the touch. This difference
in feel is a direct consequence of the higher thermal conductivity of the carbon
steel.
In Powell's own words, 8 his comparator "is a device which is more sensi-
tive than most hands and seeks to translate the sensation of 'warmth' and.
'cold' to readily measurable quantities." His criginal device consisted of
two 1/4-in.-diam silver spheres which were instrumented with thermocouples and
mounted in a block of balsa wood. One sphere protruded through the bottom of
the block to permit the sphere to contact the disk-shaped specimen when the
device was placed on it. In order to reduce the load on the sphere, a three-
point contact with the specimen was made by mounting two pins on the bottom of
.
the block. Measurements were made by first heating the sphere assembly about
40°C hotter than the specimen and measuring the differential temperature
change between the spheres as a function of time after the device was placed
on the specimen. The temperature change of the sphere in contact with the
specimen was due to three effects: (1) heat flow through the contact by con-
duction, (2) the radiant energy loss by the sphere to its surroundings, and
(3) the convection and conduction loss to air; whereas, the temperature change
of the noncontacting sphere was proportional only to the latter two effects.
The differential response was proportional to the heat flow through the contact
area and thus was proportional to the thermal conductivity of the specimen,
rather than to the specimen thermal diffusivity. This method was comparative
and a calibration curve was obtained by plotting k1/2 vs the differential tem-
perature change at 10 sec for materials of known thermal conductivity (k).
Dahl and Jones 1.1 1.nvestigated several modifications of Powell's method and
gave a simple analysis of the operation of the comparator which indicated
that under certain conditions the initial differential cooling rate was a
measure of the thermal conductivity of the specimen. Their analysis, using
an electrical analogy of a sphere with limited contact at the surface of a
.
.
semi-infinite solid, indicated that there was a transient involving thermal
diffusivity but that this transient was of relatively short duration (less
than 0.1 sec in their case).
They pointed out that the thermal resistance
in the semi-infinite solid is mostly in a small region near the contact,
but that the heat capacity of this region is so small that the region heats
to a quasi-steady condition in a very short time. For some time after this
starting transient, the heat flow is mostly by steady state conduction. In &
discussion of this work, Ginnings 12 substantiated this conclusion with heat
transfer calculations.
From a consideration of the overall heat transfer processes in the
Powell comparator, Ginnings 12 suggested certain limitations and possible ways
to increase its utility. His model showed that, when the specimen dimensions
are large compared to the sphere contact radius, the error due to the initial
transient state can be made less than 1%. In addition, he showed that
gaseous heat conduction is predominant for low thermal conäuctivity specimens,
and that the contact area as well as the second sphere spacing might be used
to increase the sensitivity of the method.
The above-mentioned studies have shown that the data obtained using the
thermal comparator method are strongly dependent on (1) the initial tempera-
ture difference between the specimen and the spheres, (2) the area of contact
and any variable that affects this, (3) the measuring atmosphere, (",) the
specimen size, and, of course, (5) the specimen thermal conductivity. The
following sections describe in detail the current version of the modified
Powell thermal comparator developed at the Oak Ridge National Laboratory.
.
DESCRIPTION OF COMPONENTS
The current thermal comparator is the result of experience gained during
the past three years with a number of modifications of the basic components.
The present apparatus has the following features: (1) small specimens with
a wide range of thermal conductivity values can be measured from 50 to
400°C, (2) the measurement is rapid, nondestructive, and the apparatus is
capable of being operated by one person, (3) specimens can be introduced
easily into the chamber without contamination of the atmosphere, and (4) the
method is modestly accurate and the results are reproducible. The system is
different from Powell's original thermal comparator in several ways. First,
measurements can be made to 400°C, in a purified argon atmosphere, whereas
Powell's original comparator was limited to room-temperature operation in air.
Second, the noncontacting sphere has been deleted because its response was
found to be independent of the specimen under test.
The ORNL apparatus is
discussed below and the numbers in parentheses are keyed to the isometric
sketch of Fig. 1.
Specimen and sphere temperatures are achieved by two tandem-mounted
cylindrical Nichrome-wound copper furnaces (13) and (18), separated from each
other by movable radiation shields. The upper furnace containing the sphere
was operated 8 to 20°C cooler than the lower furnace containing the specimen,
depending on the temperature of investication. The temperature of each
furnace was controlled using a Chromel-R-Constantan thermocouple (23) and
(25), connected in series opposition to an adjustable Zener diode voltage
source. The output of the diode was set so that the differential vol.tage
would be zero when the desited temperatures were reached. The differentials
were amplified 200x by direct-current microvolt amplifiers and fed zero-
center, 15 mv span, recorders. The upper furnace recorder fed a controller
UNCLASSIFIE.O
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SPHERE SUPPORT SHAFT
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& WATER COOLED JACKET
7. WATER COOLED JACKET SUPPORT PLATE
SPHERE THERMOCOUPLE
9. PIVOT
10 THERMAL RADIATION SHIROS
il. NCHROME WNOINGS
12 SCHERE FURNACE SUPPORT PLATE
12 SPHERE FURNACE
12 MOVABLE RADIATION SHIELD
15. RADIATION SHELD SUPPORT ARM
12 SPHERE ASSEMBLY
17. SPECIMEN RECEIVER CUP
12 SPECIMEN FURNACE
SPECIMEN FURNACE SUPPORT PLATE
PLUNGER ROD
21. SUPPORT CHANNELS
PNEUMATIC PISTON
22 SPHERE CONTROL THERMOCOUPLE
24 COURO WRE
SECMEN CONTROL THERMOCOUPLE
SPECIMEN THERMOCOUPLE
LAWTE DISK
A STAINLESS STEEL 100
2. NO, HSULATION
1 XT SET
1. SPECSEN
SHERE
LAMTE AP MOLDER
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Fig. 1. Isometric Drawing of the Thermal Comparator.
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which pulsed power from a high-low autotransformer power supply to achieve
temperature control.
The lower furnace recorder fed a controller which
varied the output of a solid-state power supply.
These control systems
allowed the temperature difference between the furnaces to be maintained
constant to £0.04°C.
The specimen was contained in the lower furnace in
a copper receiver cup (17), which could be raised into the upper furnace by
means of an external, double-acting pneumatic piston (22). The system
requires a disk-shaped specimen (31) with a thickness between 0.22 and
0.27 in. and a diameter between 0.25 and 1 in.
The upper furnace contained a 1/4-in.-dian Council-D* silver sphere
(22) mounted on a 0.086-in.-diam stainless steel rod (28) which was bolted
onto a Lavite** disk (27) attached to the titanium support shaft(4). The
sphere temperature change was indicated by a 30-gage Chromel-p-Constantan
thermocouple (8) silver soldered to the sphere center, and its output was
connected in series opposition with an adjustable Zı'ner diode. This
differenti.al signal was amplified 80x by a direct-current amplifier and
fed à multispan recorder which had a 1/4-scc full-scale response, a chart
speed of 1 in./sec, and was readable to 0.1 chart division or +0.6 uv.
-
-
The sphere support shaft (4) was positioned in the upper furnace
-
by a water-cooled ball bushing (5) which was suspended above the upper
furnace by a metal plate (7).
The sphere support shaft was supported by
a wire-suspended rod (1) and (3). During a test, the specimen was raised
into the upper furnace and when the pneumatic piston had completed its
full stroke the suispension wire was heated electrically; the resulting
expansion of the wire lowered the suspending rod to allow contact of the
sphere and the specimen.
*Council-D is a dispersion-hardened silver.
(Handy-Harmon Company)
**lavlte is a commercial trade name for an aluminε-silicate compound.
(American Laval Corporation)

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The entire apparatus was surrounded by a vacuum-tight, water-cooled
stainless steel chamber. This chamber, which was filled with a purifled
.
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argon atmosphere during measurements, was fitted with two glove ports,
a sight port, and a double-door entrance port so that specimens could be
,
IN
introduced into the system without contamination of the argon atmosphere
while the system was at temperature. In order to prevent the gloves from
..
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being heated, a swinging-door type radiation shield made of stainless
steel was inserted between the apparatus and the chamber as shown in
20
Fig. 2.
A gas-purification furnace containing pure zirconium strips was
FA
suspended in the chamber between the swinging-door radiation shield
5
and the chamber wall diametrically opposite the gloves. The zirconium
strips were maintained at 800°C to getter impurities in the argon gas.
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Fig. 2. Photograph of the Apparatus with the Vacuum-Tight Chamber
Removed. The swinging-door radiation shield was open. The sphere and
specimen furnaces were as they would be during temperature equilibration
prior to a run.
-
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PROCEDURE AND DATA INTERPRETATION
Experience dictated that the sphere should be polished and the zirconium
getter strips replaced every two weeks, which required removal of the chamber.
When these tasks were completed, the chamber was replaced, the system atmo-
sphere purged, the furnaces were brought to the desired temperatures, and
the thermal comparator was ready for use. A specimen was polished and placed
in the entry port, which was then evacuated and purged three times with argon
from the chamber. The swinging-door radiation shield was opened, the upper
furnace rotated, and the specimen cup elevated to a position between the
two furnaces. Metal tongs were used to move the specimen from the port to
the proper position in the specimen cup. The upper furnace was realigned
with the lower furnace and the swinging-door radiation shield was closed.
Approximately 45 min were required to obtain the desired temperature stability
after a specimen was installed.
The differential temperature between the two
furnaces was checked with a K-3 potentiometer, and the specimen pneumatic
piston was activated. The specimen contacted the sphere as previously
described, and the sphere temperature change (response, as it shall here-
tofore be referred) was recorded for about 40 sec. The swinging-door radiation
shield was then opened and the specimen cup elevated to a position between the
two furnaces to allow the specimen to be rotated in the cup, permitting a dif-
ferent area to be co tacted on subsequent determinations. About 30 min. were
required for temperature stability between determinations. Between 6 and 10
determinations were usually made on each specimen, depending on the reproducibilit
12
After this, another specinen was introduced into the entrance port and the
above procedure was repeated.
A plot of a typical sphere response is shown in Fig. 3. The method of
determining the starting time and the sphere response at any time is also
shown. At 75°C, the responses 20 and 30 sec after contact were used. The
method of determining the response was identical for specimens of known
and unknown thermal conductivity. Calibration curves were obtained from
the response for a specimen of known thermal conductivity and these curves
will be discussed.
VARIABLES EFFECTING THE SPHERE TEMPERATURE CHANGE
There were five variables that controlled the sphere response: (1) the
initial temperature difference between the sphere and specimen, (2) the area
of contact of the sphere and specimen, (3) the chamber utmosphere, (4) th:
specimen size, and (5) the specimen thermal conductivity.
Since this was a comparative technique, the first four variables had to
be maintained constant in order that the fifth would be controlling. Needless
to say, great time and effort were devoted to this end. However, a detailed
description of these efforts would not be within the scope of this paper.
The following is a description of current methods used to control these
variables along with some appropriate comments.
Specimen-Sphere Temperature Difference
Powell's' work showed the sphere response as a fixed time after contact
was a linear function of the initial specimen-sphere temperature difference.
Only in the early models of the ORNL thermal comparator was this true, but
as long as the initial temperature difference was constant, the nonlinearity
was unimportant. The selection of the absolute value of the temperature .
UNCLASSIFIED
ORNL-DWG 64-1880

Y READING AFTER 30 sec
= 58.2 div
RESPONSE = 58.2 -
9.2 = 49.0 div
O READING AFTER 20 sec :
46.7 div
RESPONSE = 46.7 - 9.2 =
37.5 div
TIME (sec) (RECORDER CHART 4 in. = 4 sec )
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EXTRAPOLATED TIME + = 0
SPHERE HEATS DUE TO CLOSE PROXIMITY OF
THE SPECIMEN
SPHERE COOLS AS COLD GAS RUSHES IN WHEN
MOVABLE RADIATION SHIELDS ARE OPENED
--EQUILIBRIUM SPHERE READING = 9.2 div
80
100
20
40
60
RESPONSE (DIVISIONS) (4 div = 6uv ON RECORDER)
Fig. 3. A Plot of the Sphere Response for Armco Iron Showing the
Method Used for Extrapolating to Time Zero. The sensitivity of the chart
(not shown) was 10.6 uv (+0.009°C at 75°C).
.
*
**
.
19
difference was somewhat arbitrary. It was desired to make the temperature
difference large enough to miminize the percentage error in the determination
of the response, but it was also necessary to keep the temperature difference
small to minimize the heat transferred to the sphere by radiation. The
choice of from 8 to 10°C, depending on the temperature, was made since it
best fuirilled the above two conditions. The sphere was arbitrarily run
colder than the specimen.
Area of Contact Between the Sphere and Specimen
There were four variables that controlled the area of contact between
the sphere and specimen: (1) load on the sphere, (2) diameter of the sphere,
(3) sphere and specimen physical properties, and (4) sphere and specimen
surface roughness. Before discussing these, it is advantageous to state
that a sphere was chosen as the detector because various mathematical models 20-12
have been presented on this technique and have illustrated that the contact
area must be small in order for the variable temperature state of the con-
tacting surface to be of short duration. If the contact area were large,
the variable state would be long and the sphere response would then be
proportional to the thermal diffusivity of the specimen and not the thermal
conductivity. A mathematical model derived by the authors is described in
Appendix A. This model shows that, with contact areas of the order of that
obtained by a sphere in contact with a slab, the variable state is of short
duration and the sphere response is thus proportional to the thermal con-
ductivity of the specimen.
The radius of contact, r, of the sphere and specimen is given by: 13
2 = 0.72 [ PD (4.2 + 2x)]" ,
(1)
where the radius of contact 18 in inches, P 18 the load on the sphere in
pounds, D is the diameter of the sphere in incl.es, y 18 Poisson's ratio,
and E is Young's modulus in pounds per square inch. The subscripts 1 and
2 refer to the sphere and specimen, respectively. This equation applies
only if the load is less than the elastic limit of both materials.
Equation 1 predicts that the area of contact 18 proportional to
(PD)". The compressive stress at the interface of the sphere and specimen
18 proportional to p2/3/D2/3. Here an optimization in the choice of P and
D had to be made because it was necessary to keep the compressive stress
and area of contact as small as possible. Thus, it was obvious that the
load should be minimized while the sphere size should optimize the two
factors. The smallest practical sphere-assembly weight was found to be
about 28 g. Thus, the load was somewhat less than this since the sphere
assembly was not exactly perpendicular to the specimen, and a small amount
of counterbalancing was created by the thermocouples. Since the response
on a given specimen was reproducible to +2.5%, it was clear that load
variations between runs were not large.
The diameter of the sphere was chosen as 1/4 in. When the physical
properties of the silver sphere and specimens of type 304 stainless steel
and aluminum were substituted into Eq. (1), the radii of contact were calcu-
lated to be 8.2 x 10-4 and 9.5 x 10-4 in., respectively, for a 25-8 209d.
The compressive stresses were calculated as 14,900 and 10,940 psí for the
25-8 load and for a hemispherical contact. The compressive yield strengths
for most metallic alloys and ceramics are above 15,000 psi, and the yield
strengths for most pure metals are above 10,500 psi. Materials such as
-
-
.
graphite have yield strengths below this and were found to be indented by
the sphere with resulting erroneously high sphere responses.
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Although the physical properties of materials vary appreciably,
Eq. (1.) predicts that the radius of contact varies about 27% over a wide
range of materials.
This is because Poisson's ratio is about the same for
most materials and their Young's moduli are higher than that of the sphere.
Thus, the radius of contact 18 proportional to (c +2,7/3, where C and D
are constants and E, 18 the Young 's modulus of a specimen. Since C 1s
greater than D/E, by at least a factor of 2 for most specimens, commonly
encountered variations in E do not significantly vary the sum. The one-
third exponent of the sum further reduces the sensitivity of the radius of
contact to the variation of E.
Equation 1 is highly idealized and does not consider the variation of
the response due to the specimen- or sphere-surface finish. Powell' showed
that the response at a given time increased 33% between surface finishes
having center-line average values of 68 and 6 uin. He also stated that this
2
dependence decreases with increasing load. Dahill stated that above 200 g
his 3-A comparator response was independent of load. Table 1 contains data
on two different surface finishes on Armco iron and steel No. 17. The
rougher finish consisted of polishing the specimen with 4-0 emery paper,
ard the finer finish was a polish through Linde B* on a silk-cloth polishing
wheel.
The latter had a surface similar to a metallographic specimen prior
to etching. The thermal comparator results indicate at most a 2.6% increase
in response for the finer surface. This, however, corresponded to a difference
of 10% in the thermal conductivity. Thus, it was necessary to have similar
*Trade name for a fine particle size Al2O3 polishing material.
44
Viti
UL.
17
Table 1. Effect of Surface Finish on the Response after
30 sec at 75ºC on Armco Iron and Steel No. 17 in Argon
Average Response in Recorder Divisions
Polished with
Polished with
4-0 Paper
Linde B
Percent
Increase
Material
Armco Iron
Steel No. 17
36.80
27.30
37.77
27.57
2.6
1.0
Temperature change of sphere in recorder divisions.
finishes on all materials under test. The finish obtained using the 4-0 emery
paper was by far the most reproducible, and all materials were given this
polish or its equivalent.
Chamber Atmosphere
The higher the thermal conductivity of the gas in the chamber, the
larger was the contribution of the heat transferred to the sphere by gaseous
convection and conduction. Ideally, a vacuum of about 10-6 torr should have
been used to eliminate all of the gaseous components of the response, but
this would increase the conudct film resistance. Vacuum operation was
attempted, but was not practical since precise control of the initial specimen-
sphere temperature difference in a vacuum proved to be very difficult and the
changing of specimens could not be accomplished without cpening the system.
Other factors considered in selecting a chamber atzosphere for testing
to 400°C included inertness and purity.
Initial tests were made using
helium, and its gaseous component was found to be forbiddingly large. Tests
using gas-train purified nitrogen indicated a lower gaseous heat transfer
component than the tests using helium, and successful operation was found to
be possible at 125°C; however, at 400°C an Armco iron specimen was observed
to develop a perceptible film within 15 min. Tests using argon containing
L
.
-
-
.
.
.
about 1 ppm O2 indicated an even lower ga seous heat transfer coefficient than
1
Az
WE
18
nitrogen; however, the film on iron was noticed in 30 min.
To circumvent
this, the previously described zirconium gettering furnace was installed,
and no visible oxidation occurred in an 8-hr period on Armco iron in the
argon atmosphere at 400°C; during this period, the sphere response repro-
ducibility was +1.5% at 30 sec. These encouraging results led to the
acceptance of argon* as the system atmosphere. No quantitative comparison
can be made regarding the sphere response in the above-mentioned gases
because other changes were being made to the detector system during this
period. Even in the purified argon atmosphere, the silver sphere slowly
c.evelops a film that in three or four weeks affects the response on good
conductors but is less effective on poor conductors. To circumvent this
effect, the zirconium strips are replaced every two weeks and at this time
the sphere is polished with a diamond compound and cleaned with ethyl alcohol.
Specimen Size Effect
Powell10 has shown the differential sphere response to be affected by
the specimen size up to some ciitical value of diameter and thickness which
depends on the specimen thermal conductivity. For any specimen thicker than
0.128 in., he found that the response was independent of thickness regardless
of the specimen conductivity. His measurements showed that the minimum
diameter for copper was 0.3 in. and for borosilicate glass was 2.5 in.
Diameter effects were noted in the ORNL thermal comparator on l- and
1/4-in.-diam, 1/4-in.-thick specimens of 1100 AL, INOR-8, ** and Pyrex.
During testing the 1/4-in.-diam specimens were placed in a 1/4-in.-diam hole
in the center of a l-in.-diam holder of type: 347 stainless steel. The results
are shown in Table 2. Since the 1/4-in.-diam aluminum specimen had a thermal
mypical tank analysis of impurities by mass spectrometer in parts per
million: 11 Hz, 2 CH, 2 02, 25 (N2 + CO), and 1 C02.
.
.
.
...
.
.
**N1 -Mo-Cr-Fe alloy, commercially available as Hastelloy N.
.
19
19
Table 2. Effect of Specimen Size on Comparator
Response after 20 sec at 125°C
Thermal
Conductivity
(w/m°c)&
Sphere Response in
Recorder Divisions
174 in. 1 in.
Percent Difference
In Thermal
Response Conductivity
In
Material
nae
17.56
16.8
+4.0
+25.0
26.5
Pyrex
INOR-8
1100 AL
6061 Al
1.22
21.74
211..
1972.
-3.0
26.25
50.66
47.70,
4.5
--23.0
53.0
49.0
--13
2
b
watts per meter °C
ºl-in.-diam holder made of type 347 stainless steel with a central
1/4-in.-diam hole, k approximately 15 w/mºc.
€1-in.-diam holder made of a mild steel with a central 0.8-19.-diam
hole, k approximately 35 w/mºc.
40.8-in.-diam specimen.
.
conductivity higher than its holder, the sphere response was lower than that
in
1
for the l-in.-diam aluminum specimen. Likewise the response of the 1/4-in.-
4.
diam Pyrex was higher than the l-in.-diam Pyrex specimen, because the thermal
L
,
conductivity of the holder was higher than the specimen. In the case of the
i
1-
m
1/4-in.-diam INOR-8 specimen, the specimen and its holder were about the same
wherr,
o
thermal conductivity and the results were only slightly lower than the l-in.-
T
.
1
.
diam specimen, the small difference probably being due to a film resistance
.
.
between the holder and specimen. Similar results on 0.8- and l-in.-diam
.
.
.
6061 aluminum alloy specimens in a mild steel holder are listed in Table 2.
.
The average presented for the smaller 6061 aluminum specimen is based on only
- .
..* : : .
two runs; whereas six were made on the larger specimen. The two averages
could be part of the same data set and reflect no difference in response.
iri,
If the difference is real, then the results indicate a size effect still
exists for a 0.8-in.-diam specimen.
20
The above results agree with Powell's findings in that the magnitude
and sign of the diameter size effect are dependent on the thermal conductivity
of the surrounding material. However, these results do not indicate a depend-
ence of the specimen diameter on the specimen thermal conductivity as
found by Powell.20 These results do show that Pyrex and aluminum are
affected about equally, though understandably in opposite directions. This
is consistent with the electrical analogue of Dahl and Jones 11 provided the
contact radius for Pyrex is the same as for aluminum, which is to be expected
from Eq. (1).
Although there may be a size effect at 0.8 in., our tests indicate
that at i in. the size effect is negligible compared to the overall accuracy.
Furthermore, since the standards are 1 in. in diameter, any size effect will
be included in the calibration curve. When measurements were required on
substandard size specimens, the holder material was matched with the specimen
but a poorer absolute accuracy was expected.
Specimen Thermal Conductivity and Calibration Curve
Meaningful results with the comparator are obtained only when operating
procedures control the above-mentioned variables so that a calibration
C
.
curve relating the sphere response to thermal conductivity at a given
operating temperature can be established. However, for intercomparison of
basically similar specimens, which differ by only 5 to 10% at the same tempera-
ture, only the relative shape of the calibration curve is required. Table 3
contains tabulated calibration data, and the results are plotted in Fi.gs. 4
*-*
and 5 for 75 and 300°C. Eleven materials* were used to establish the
ON
IN
*These specimens and their temperature dependent thermal conductivity
values were obtained through the courtesies of D. R. Flynn, National Bureau
of Standards and R. W. Powell, National Physical Laboratory.
mm
2
KE
M
1
AL
UNCLASSIFIED
ORNL-OWG 64-1879

75°C
20 sec
30 sec
1100 All
H MAGNOX -B
VK (watts/meter och
NOWN
ARMCO IRON
F STEEL NO. 2
-
H STEEL NO. 10 –
H STEEL NO. 17
1
SED
F347 STAINLESS STEEL
HINOR -8
HYLITE 55
truth PYROCERAMT
MM PYREX
2
I PUNKNOWN
1R20
TR30
TRUNKNOWN
10
20
30
40 50
RESPONSE (divisions)
60
70
:
,
-
Fig. 4. Calibration Curves at 75°C for 20 and 30 sec after Contact. The
procedure used to determine the thermal conductivity of an unknown is shown. The
brackets show the standard deviation of the response.
2
ng
Kal
.
.
M
T
.
LLLL
UNCLASSIFIED
ORNL-DWG 64-1959A

300°C
ARMCO
IRON
STEEL NO. 2.)
o 20 sec
30 sec
STEEL NO.
VK (watts /meterocybe
STEEL NO. 17
22
347 STAINLESS STEEL
SINOR-8
HYLITE
*
*
• PYROCE RAM
..
.
-
--
- t
-
.
-
J PYREX
24 28
-
16
20
-
32
52
36 40 44 48
RESPONSE ( divisions )
56
60
64
.
.
.
-
.
-
:
-
Fig. 5. Calibration Curves at 300°C for 20 and 30 sec after Contact.
..
-
F.
.:
»
:
.
Table 3. Sphere Response Calibration Data at 20 and 30 sec
after Contact at 75 and 300 °C in Argon
20-sec 30-sec
Response Response
(w/m°c)+/- (division) (division)
k7/2
30-sec
Response
Interpolated
(division)
Material
(whose
Percent
Deviation
(w/m°C) &
1.152
3.646
75°C Data, Initial Temperature Difference - 10.65°C
1.07 13.0 17.7
17.80
1.91 16.48 22.66 22.35
2.82 19.74 26.78 27.13
3.33 22.33 30.14 29.73
3.80 23.55 31.67 31.98
Pyrex
Pyroceram
Hylite 55
INOR-8
Type 347
stainless
steel
7.95
-0.56
+1.39
-1.29
+1.38
-0.97
11.08
14.44
26.28
5.13
Steel
No. 17
28.54
37.58
37.50
+0.21
35.45
5.96
Steel
No. 10
30.69
40.53
40.71
-0.44
58.20
7.63
35.78
46.75
46.75
O
Steel
No. 2
Armco Iron
Magnox B
1100 AL
70.30
123.1
201.6
o
+0.14
-0.11
1.46
3.38
11.4.
14.1
18.23
8.38 37.77
49.17
49.18
11.09 45.08 57.28
57.20
14.20 53.07 65.92 65.99
300 °C Data, Initial Temperature Difference : 9.06°C
1.21 17.3 24.3
1.84 22.0 30.0
3.38
3.38 29.2 39.5
3.75 32.6 43.8
4.27 3404 45.9
Pyrex
Pyroceram
Hylite 55
INOR-8
Tyne 347
stainless
steel
Steel
No. 10
Steel
No. 17
Steel
No. 2
Armco Iron
37.0
6.08
42.6
55.2
27.6
5.25
39.4
51.5
49.5
7.04
48.5
61.7
55.5
7.45
51.6
65.1
L
.
.
watts per meter °C
A
to
calibration curve at 75 °C and 9 materials at 300°C. The average
responses at 75°C are based on six to ten runs on each specimen of known
thermal conductivity, and Fig. 4 shows the standard deviation observed. The
average responses at 300 °C are based on only four runs on each specimen, and
this is not sufficient to yield maaningful standard deviations. One smooth
curve was drawn through the data and Table 3 indicates the maximum deviation
at 75 °C between the interpolated curve and the measured response is 11.39%,
which in the case of Pyroceram corresponds to a +6% deviation in thermal
conductivity.
If the noncontacting spheres of the previous comparators behaved similarly
to the noncontacting sphere of the original ORNL comparator, that is, its
response was independent of the specimen being tested, then intercomparisons
of the shapes of the calibration curves of previous studies may be meaningful.
The calibration curves at 75 and 300°C exhibit a nonlinear relation between
k' and the sphere response, and the slope increases sharply near 16 w/mºc.
Powell's results showed two linear regions when the differential response
and kt/2 were compared, with a break near 11 w/mºc. This behavior is to be
compared with the results of Dahl and Jones11 that showed that within +10%
the differential response was linear with the logarithm of k. The ORNL data
for the response of a single sphere show the form predicted by Ginnings 12
for thermal conductivities up through Armco iron but do not show the predicted
tendency to reach a limiting value at higher thermal conductivities. This
deviation is in the correct direction to be due to the relative softness
at 75°C of the Magnox B and 1100 alumimum known thermal conductivity specimens,
although specimen indentation was never observed for these materials. Similar
.
.

15
T41
on
re
Mo
C
22
4
.
ant?
.
Wir
**
.
42
.
.
..
UNCLASSIFIED
ORNL
}
2
.
_
LAN
148
20F2
7
SET
URUS
ON
1
.
.
.
!!
.
7
1.
49
V
+
9
,-
"T
..
]
1
1915
.
I
.
..
"W"..
s
,
.
.
ly
-
-
'n
.
*
#

.
WM
.
.
N'
s
?
.
'
-..
DTIE
MICROCARD
ISSUANCE
DATE
.
.
.
9/00
.
.
...
1964
.
:
U2
w
Y
1
...
.

A
ILLU
Y
.
discrepancies were noted at room temperature on soft materials by the previous
studies. Calibration tests on hard materials of high thermal conductivity
are needed to resolve this effect.
REPRODUCIBILITY AND ACCURACY
The reproducibility of the thermal comparator was assessed by deter-
mining the percent standard deviation of the sphere response on homogeneous
specimens and this was found to be better than 12.5% from 50 to 400°C.
Table 4 shows the 30-sec sphere response data, the average response, and the
standard deviation for Armco iron at 75, 300, and 400°C. These data show that
the percent standard deviation generally decreases with increasing temperature.
Furthermore, since the sphere response record can be read to only 0.1 division
and since a given determination requires two readings, then a maximum read-
ability variation could be 10.2 division. The fact that the observed
standard deviation is larger than this is not surprising. The data of 21 runs
on type 347 stainless steel at 75°C at 30 sec are shown in Table 5 and are
grouped to demonstrate that the deviation from the average tends to follow
the normal distribution law and could be considered random scatter. The 95%
confidence limit of the average of six to ten runs made on most specimens
was usually below +2.0%. It is believed that most of the scatter, in excess
of that due to readability, was due to load and specimen surface variations
encounterea by the various determinations.
The absolute accuracy was evaluated experimentally, rather than by an
error analysis, by establishing the calibration curve using some of the
standards and then using this curve to determine the thermal conductivity
of the remaining standards. At 75°C, the absolute accuracy proved to be
better than $10%, agreeing in magnitude with the best smooth curve approach
26
Table 4. Sphere Response Data at 30 sec after contact on Armco Iron
at 75, 300, and 400°C
Sphere Response
in divisions at:
75°C
300°C
62.6
50.2
49.8
62.3
400°C
66.3
66.3
66.1
66.7
65.0
49.4
62.4
49.1
62.4
62.0
48.1
48.6
49.0
63.2
66.1
65.5
66.0
Average response
Standard Deviation (division)
Percent Deviation
49.2
10.7
62.5
10.4
10.6
10.6
+0.9
11.4
Table 5. Groupings of the Response in 0.2-Div. Intervals for Type 347
Stainless Steel at 30 sec at 75°Cå
30.7–30.
9
31.0-31.
2
Division Intervals
31.3-31.5 31.631.
8
31.932.
1
32.2-32.4
31.0
31.6
30.7
30.8
30.8
31.3
31.4
31.6
31.9
31.9
32.0
32.0
32.1
32.3
32.4
32.4
31.6
31.6
31.7
31.8
32.1
31.8
Average: 31.7; standard deviation: 10.5 div. or 11.6%; 95% confidence:
10.2 div. or +0.69%.
which was previously shown to be within 6% of k for all of the standards.
Further support for the absolute accuracy of the thermal comparator was
obtained by measurements on several specimens of unknown thermal con-
ductivity, and the results agreed within the above limits with those of
absolute measurements, as discussed below.
RESULTS AND DISCUSSION
The thermal conductivity of many solids may be conveniently divided
into two parts:
k = ke + ko,
where k, is the electronic portion due to the transport of heat by
(2)
electrons and k
is the lattice portion due to the transport of heat by
phonons. The Wiedemann-Franz-Lorenz (W-F-L) relation can be used to
determine k from the electrical resistivity by:
ke - LT/ ,
(3)
where p is the electrical resistivity at T, the absolute temperature, and
L is 2.445 x 10-6 w2/m• cm °C, the theoretical W-F-L constant. The lattice
portion of the thermal conductivity is not so simply related to other
properties of the solid. One common approach is to divide the phonon thermal
resistance, l/k, into portions which are due to & particular phonon
scattering mechanism and then attempt to calculate each separately. Thus
one writes the phonon thermal resistance, R, as:
= R + R + Ppei
(4)
where Ry, Ry, and Roe are the thermal resistances due to Umklapp, impurity,
and electron scattering of phonons, respectively. For a solid well above
pe
NY
28
the Debye temperature, Ry should be a constant and R, should be propor-
tional to T. The temperature dependence or R
is not clear, but if one
pe
assumes it to be constant, the thermal conductivity results on many
electrically conducting solids can be explained. With this assumption,
the phonon thermal resistance is:
R = A + BT
(5)
where A and B are constants and the total thermal conductivity is:
16)
Thermal conductivity measurements at moderate temperatures, in the
range of the thermal comparator, can be used with high-temperature electrical
resistivity values to provide a reasonable extrapolation of k to
higher temperatures on the basis of Eq. (6).
This procedure is useful
because high-temperature thermal conductivity measurements are much more
difficult and less precise than are high-temperature electrical resistivity
measurements.
Thermal conductivity measurements were made with the ORNL thermal
comparator at 75 and 300°C on UN, US, Ths, and UO2 with the results shown
in Teble 6. The measured values were corrected to theoretical density,
and the corrected values are plotted in Figs. 6, 7, 8, and 9 as
a function of temperature. Previous thermal conductivity measurements
have been made on UN, 14 us, 15 and 102, 2 and comparison data are included in
these figures.
The VO2 and 10:2-X specimens were cut from one of the UO z disks used
for measurements in the ORNL radial heat flow apparatus.?
The VO2
M
...
Windo
.....
.
.
..
..
UNCLASSIFIED
ORNL-DWG 64-1963R

ALL DATA CORRECTED TO 100% DENSITY
A SMI, FIRST RUN
• BMI, SECOND RUN
O ORNL THERMAL COMPARATOR
* (wotts/meter ·°C)
EXTRAPOLATION OF
ORNL DATA
|
200
200
800
400 600
TEMPERATURE (°C)
9000
Fig. 6.
The Thermal Conductivity of UN Corrected to 100% Density Between 0 and 1000°C.
-
::
:
.
:
.
......
.
.
.
---
.
e
.
condo
....
NY
-
--
-
-
-
-
-
-
UNCLASSIFIED
ORNL-DWG 64 - 1960A

US
EXTRAPOLATION
OF ORNL DATA
emerom
....
K, THERMAL CODUCTIVITY ( watts/meter:°C)
-
30
ALL DATA CORRECTED TO 100% DENSITY
• R. DUNWORTH, ANL
O ORNL THERMAL COMPARATOR
200
600
800
400
TEMPERATURE (°C)
1000
Fig. 7. The Thermal Conductivity of US Corrected to 100% Density Between 0 and 1000°C.
........... -
.........
.......
.................
2
.
W
Les
UNCLASSIFIED
ORNL-DWG 64-1962A

ThS
DATA CORRECTED TO 100% DENSITY
• ORNL COMPARATOR DATA
K, THERMAL CONDUCTIVITY ( watts/meter •°C)
31
EXTRAPOLATION
OF ORNL DATA
200
200
400 600
TEMPERATURE (°C)
800
9000
Fig. 8.
The Thermal Conductivity of ThS Corrected to 100% Density Between 0 and 1000°C.
.
..
.
M
.
UNCLASSIFIED
ORNL-DWG 64-1961R

UO2
i
K, THERMAL CONDUCTIVITY ( wotts/meter.°C)
DATA CORRECTED TO 100% DENSITY
• ORNL RADIAL HEAT FLOW
O ORNL THERMAL COMPARATOR VO,
ORNL THERMAL COMPARATOR VO2-X
- 100
100
200
TEMPERATURE (°C)
300
400
Fig. 9. The Thermal Conductivity of VO2 and 102-X Corrected to 100% Density
Between -100 and 400°C.
S
W
1
.
.
12

Table 6. Thermal Conductivity Results obtained by the
ORNL Thermal Comparator at 75 and 300 °C
Theoretical
Density
(%)
Measured
Thermal
Conductivity
(w/m°C)
75°C 300°C
Corrected
Thermal
Conductivity
(w/m°C)
75°C 300°C
Material
Ins
UN
US
VO2
U”2-X
95.59
94.56
90.2
93.4
93.4
4.8
12.85
9.55
7.16
7.18
34.3
14.2
10.8
9.95
5.78
46.8
13.6
20.6
7.67
7.70
35.9
15.1
11.9
5.30
6.19
corrected to theoretical density: knees, where D is
the fraction of theoretical density.
specimen yielded k values which agree with the previous determinations to
within 5% at 75 and 300°C. Thermal conduction in this specimen is solely
due to phonon transport. The VO2-x specimen was heated to 2050 °C for
30 min in a 2 x 10-5 torr vacuum and cooled rapidly in the furnace. This
treatment produced a substoichiometric defect structure with an oxygen-to-
uranium ratio slightly less than 2. At room temperature, the excess uranium
is a metallic phase dispersed in stoichiometric UO 2.26 Upon heating, the
free uraniw dopes the lattice and produces n-type centers which can increase
the electronic contribution to thermal conduction. This explains why the
300°C k value for this specimen is 17% high. The agreement at 75°C for the
two specimens substantiates the above observations in that this temperature
is too low to cause appreciable solution of the uranium. These tentative cor-
clusions require additional testing for final verification.
+
7
--
.
*
!
The corrected thermal conductivities of UN and US at 75 and 300 °C
are slightly lower than the previous measurements on these materials, lugas
and this may be due to use of an incorrect correction for the density or
to st.gnificant specimen differences. No previous measurements exist on
Tns, and the high measured value for this material at 75 and 300°C certainly
warrants further interest. In fact, since previous measurements have shown
increases in k in the compounds US, UN, and UC as the anion is changed from
Group VI to IV, then one might expect very interesting k behavior for the
series: Ths, ThN, and ThC.
It is interesting to note the US, Ths, and UN all have large electronic
and phonon heat transport contributions. Thus these materials are in a very
interewting transition region between good metals and semiconductors. This
is a further incentive, besides the nuclear one, for more work to be done
on careful preparation and property measurements of these materials.
Table 7 contains the results of calculations based on Eq. (6) needed
to extrapolate the thermal comparator data on US, THS, and UN to higher
temperatures. Included in the table are the electrical resistivity data
on UN?" and the electrical resistivity data of Argonne National Laboratory
on US and Ths. 17 The constants A and B of Eq. (6) are tabulated in Table 8
for these compounds.
These were obtained using the assumption that the
lattice portion of the thermal resistance follows Eq. (5). The electronic
portion of the thermal conductivity was subtracted from the measured k
values for 75 and 300 °C to obtain k. The lattice thermal resistance,
1/k, at these two temperatures was plotted and a straight line drawn
through them to give A and B. The values of k, reported in Table 7 for
temperatures other than 75 and 300 °C were calculated using Eq. (5).
Table 7. Extrapolation of Comparator Thermal Conductivity Values
to 300°C for US, Ths, and UN

Temp
(°c)
me asured 100% Dense
(u ohm came)
emo
USC
75
290
247
3.5
10.6°
100
298
253
10.7
200
310
264
4.4
11.3
300
318
5.1
7.1
7.1
6.9
6.8
6.6
6.5
6.4
6.3
323
275
6.0
11.96
12.6
13.2
14.1
500
325
276
6.7
600
327
278
7.7
700
328
279
8.5
14.8
800
328
9.6
6.1
279
279
15.5
16.3
200
10.3
6.0
328
328
1000
11.
5.9
17.1
Thea
65
.12.2
14.0
46.5
32.8
30.0
22.8
46.86
44.5
100
14.5
200
16.4
39
300
18.0
35.96
400
101
500
110
33.4
600
119
32.8
17.9
19.2
20.3
21.2
21.8
22.7
23.3
23.6
700
128
109
15.3
13.1
11.6
10.3
9.6
8.4
7.7
32.1
800
137
116
32.1
900
146
124
31.7
1000
155
132
31.3
*
...
17
.
AL
12
21
Table 7 continued
measured
enee
Temp
(°C)
_
_
Tu obar cm)
UNC
193
166
13.1
199
17
13.66
9.1
8.6
8.5
7.9
100
201
173
13.8
24.4
200
208
179
300
212
183
7.4
15.76
4.03
4.98
5.28
6.47
7.67
8.86
10.0
11.1
12.2
13.3
400
216
186
15.9
500
220
6
.6
7.0
6.6
6.3
6.0
600
224
192
16.6
17.4
18.2
19.0
700
227
195
6.0
231
231
198
800
900
5.7
235
202
14.2
5.4
19.6
1000
240
206
15.1
5.2
20.3
*85% dense, M. Tetenbaum: "Thermoelectr! c Parameter Studies on US,
Ths, and UC-ThS Solid Solutions," Seminar on Groups V and VI Anions,
Argonne National Laboratory, February 26, 27, 1964 (in press).
Experimental. values.
€86% dense, measured by a 6-probe technique in a vacuum of
10-5 torr to 1000°C at ORNL.
Table 8. Derived Constants for
Thermal Conductivity Extrapolation
B
(mºK/w)
US
UN
0.130
0.086
-0.008
0.0000314
0.0000835
0.0001080
Ths
The extrapolation to 1000 °C of k for these materials 18 shown in
F188. 6, 7, and 8. This extrapolation 18 25% below the measured values
for UN reported by BMI at 1000 °C and within 3% of the values for US reported
by ANL at 1000°C. In this extrapolation all electrical resistivity data and
thermal conductivity data were corrected to theoretical density using the
-
relations:
Pcorr * P meas *D
where D 1s the fraction of theoretical density. The validity of these
corrections is uncertain but the change 1.s in the proper direction which
is better than no correction. It should be mentioned that the quality
of the specimens of these materials on which measurements were made was not
ideal. Since the specimens were polycrystalline and porous (90 to 96% of
theoretical density) of unknown purity, then the differences between the
ORNL results and those reported on us25 and UN24 could be accounted for by
slight specimen differences. It is noteworthy that the extrapolation did
give the proper temperature dependence of k. Ideally, thermal conductivity
and electrical resistivity measurements should be made on the same sample.
Specimen non-ideality is probably more detrimental in the case of electricas
resistivity measurements than for thermal conductivity measurement. This
18 because thermal conductivity 18 less susceptible to change by grain
boundary resistance than 18 electrical resistivity. In our measurements of
o of UN, we found a jolarization effect which 18 thought to be due to save
of the grains being electrically insulated by an oxdde in the grain boundary.
Microstructural examination has revealed oxide inclusions. Nevertheless,
our electrical resistivity values for UN agree reasonably well with those
of BMI when both are corrected to theoretical density by Eq. (8).
The theory on which our thermal conductivity extrapolation is based
18 for bulk material. The data reported on non-ideal specimens must be
considered preliminary. The measurements need to be repeated on fully
dense specimens of known purity before careful comparison with theory can
be made.
The electronic portion, k, of UN, US, and Ths increases with increasing
temperature according to the W-F-L relation. It should be pointed out that,
although the electrical conductivity of most metals decreases with increasing
temperature, the W-F-L relation does not necessarily predict that this will
be true for the electronic portion of the thermal conductivity. If the
electrical resistivity can be expressed as a linear function of temperature
in some range, such as:
Depo(1 + a[T - T.))
(9)
where T. 18 the temperature corresponding to the beginning of the linear
region and a is the temperature coefficient of resistivity, one can
differentiate the W-F-L relation with respect to temperature using the above
expression for p and determine by the sign of the derivative whether k increases
or decreases with increasing temperature. The result of this differentiation
is:
1(1 - )
pofl + a[T - T])2
(10)
This derivative is positive if or <1, regative if of > 1, and zero
12 * . For Us, ThS, and UN, the electrical resistivity is linear
between 673 and 1173°K, 273 and 1373°K, and 473 and 1273°K, respectively,
with a values of 2.9 x 10-5, 1.4 x 10-3, and 1.9 x 10", respectively, giving
values of one of 1.96 x 10-2, 3.8 x 10-7, and 9.0 x 10-2, respectively.
Since all of these or values are less tiian 1, then Eq. (10) predict: that
k
should increase with increasing temperature in these temperature ranges
for the three materials, as indeed it does.
CONCLUSIONS
A single-sphere thermal comparator apparatus was developed to allow
rapid and nondestructive determinations on small specimens which were
1/4 in. thick and 1 in. or less in diameter with a wide range of thermal
conductivity values in the range 50 to 400°C in an inert atmosphere. Pro-
cedures were developed to maintain constant those variables effecting the
sphere response, such as the initial temperature difference between the
sphere and the specimen, the area of contact of the sphere and the specimen,
ani the specimen preparation. These variables must remain constant in order for
the specimen thermal conductivity to control the sphere response. The resulting
reproducibility of the sphere response proved to be better than +2.5% with an
attendant absolute accuracy of thermal conductivity of better then 10%.
Calculations based on several models of the thermal comparator proved this
transient method does indeed measure thermal conductivity and that the sphere-
specimen contact radius is of the order of 20 H.
40
Calibration curves were established at 75 and 300°C using specimens of
known thermal conductivity and determinations on US, UN, and UO2 agreed
with previous absolute measurements to better than +20% at 300°c. Original
data obtained on Ths indicated a thermal conductivity of 46.8 w/mºc at 75°C
and 35.9 w/mºc at 300°C. Measurements on UO2 heat treated to produce 102-X
indicated as much as a 17% improvement in the thermal conductivity may be
expected at 300°C.
Using these low-temperature thermal conductivity measurements, high-
temperature electrical resistivity measurements and expected heat transport
mechanisms, reasonable extrapolations were made of the thermal conductivity
of these materials to temperatures as high as 1000°C. A differentiation
of the Weidmann-Franz-Lorenz relation was performed which shows that the
electronic portion of the thermal conductivity may increase, decrease, or
not change with increasing temperature, depending on the magnitude of the
electrical resistivity and its temperature coeflicient.
41
APPENDIX A
Many studies 10–12 have predicted that the response of the Powell
comparator, a few milliseconds after contact, is proportional to the
specimen thermal conductivity and not the thermal diffusivity. This result
was reaffirmed by the authors using a mathematical model which was an
12
extension and modification of one used by Ginnings. +2
The solution of the model gives the temperature of the contacting
sphere as a function of time after contact.
This temperature response can
be shown to be a function of the thermal conductivity of the specimen, not
the thermal diffusivity, if the contact area is small enough. The response
is the same under this condition as would be obtained by assuming that
heat flow in the specimen occurs under steady-state conditions. It was
found that the actual experimental response of the contacting sphere was
consistent with this steady-state assumption.
The model applies only to the contacting sphere and the specimen,
not to the noncontacting sphere. The specimen is assumed to be in the
shape of a hemisphere of radius, L, and the contacting sphere touches the
specimen on a small hemispherical surface of radius, a. A film coefficient,
n, is assumed to exist between the contacting surfaces. The contacting
sphere has an infinite thermal conductivity compared to that of the measured
specimen so that there is no temperature gradient in the sphere. The sphere
has a heat content, MC. All the surfaces of the sphere and specimen are
assumed to be insulated with the exception of the common contact area.
Also, the assumption is made that all of the properties of the specimen are
42
constants within the temperature range equal to the initial temperature
difference between the sphere and the specimen. With these conditions,
flow of heat in the sample 18 radial from the contact surface and the
usual Fourier heat flow equations c n be solved exactly for the temperature
at any radius, r, in the specimen and for the temperature of the ball as a
function of time after contact.
For reasonable values for the specimen thermal diffusivity, thermal
conductivity and radius, L, calculations showed that the contact radius, a,
must be less than about 0.015 cm for the contact ball response to be only
a function of the thermal conductivity. Under these conditions, the
reduced sphere temperature may be written approximately as:
T - no
*s
*z 1-6-212x = 0,27 - (028)
-
+...
,
(11)
O
where
2rakt
ajte
MC (1 + a)
T = the sphere temperature at time, t,
TO - the initial sphere temperature,
the initial specimen temperature, and
k = the specimen thermal conductivity.
It is interesting that this result is exactly the same as that which
would be obtained using the same geometrical configuration and boundary
conditions and the assumptions that steady-state conditions exdst
within the sample and that the surface at r = L 18 held at To.
On this basis, a further analysis of the contacting sphere has been
performed using the assumptions that heat flow in the solid is under
steady-state conditions and that the sphere picks up additional heat
from its surroundings at a rate proportional to the temperature of the
ball. This analysis predicts that
L2 = constant
TV-TA
where
2. = the slope of the temperature response curve of the
contacting ball for one material and
= the slope for another material.
If both slopes are taken from the experimental curves at times corresponding
to the same value of To I for the two materials, then the ratio in Eq. (12)
should be a constant.
Experimental data were substituted into the left hand side of Eq. (12)
for Armco iron and Steel No. 2 vs type 347 stainless steel, INOR-8, and
Pyrex; type 347 stainless steel and INOR-8 vs Pyrex.
These calculations
are tabulated in Table 9 and are constant in each case to within the
experimental error of determining the slopes. The constant C is given by
iro,
Inom K2
27
(13)
til
Poz-fiz
02 12
where MC, is the heat capacity of the sphere in joules per °C, re is the
radius of contact of the sphere and specimen in meters, r is the outside
44
Table 9. Experimental Determinations of C and I
of Equation 12 at 75°C in Argon
Time
After
Contact
for
(sec)
Armco Iron
V8
INOR-8
C (1/sec)
Araco Iron
Steel No. 2
vs Type 347 Steel No. 2 vs Type 347
Stainless
VS
Stainless
Steel INOR-8
Steel
C (1/sec) C (1/sec) C (1/sec)
25
0.0106
0.0106
0.0105
0.0104
0.0102
0.0104
9.4 x 200
0.0100
0.0100
0.00994
0.00979
0.00969
0.00989
9.5 x 104
0.00910
0.00920
0.00927
0.00922
0.00900
0.00916
10.4 10-4
0.00853
0.00874
0.00865
0.00858
0.00849
0.00860
10.5 X 1004
10
Average "C"
(cm)
Time
After
Contact
for
(sec)
Armco Iron Steel No. 2
V8
V8
Pyrex
Pyrex
C11/sec) c (1/sec)
Type 347
Stainless
Steel
V6
Pyrex
C (1/sec)
INOR-8
V8
Pyrex
C (1/sec)
0.00671
0.00540
0.0164
25 0.0162
0.0159
0.0155
10 0.0153
Average "C" 0.0158
F (cm) 12.3 x 104
0.0152
0.0149
0.0147
0.0143
0.0141
0.0147
13.8 x 100
0.00671
24.3 x 10-4
0.00540
29.3 % 100%
X
13
45
radius of the specimen in meters, k is the thermal conductivity in w/ °C,
and the subscripts 1 and 2 correspond to different specimens. If the
sphere is assumed to be pure silver, the value of MC is 0.337 Joules/°C.
When r. » F4, Eq. 13 becomes
cara , 62 - ruke ) - 18.7 [ -2,- ].
(24)
An average radius, i, was defined as
ra
18.77 Tkı - k27
(15)
and these calculated values are also contained in Table 9
Equation (1) for a 25-g load and a silver sphere predicted that the
radius should be between 20.8 x 10-4 and 23.9 x 104 cm. Only the values
of 5 of Eq. (15) for type 347 Svainless steel and INOR-8 vs Pyrex agree
closely with these; the others are more than a factor of 2 lower. These
differences may be explained by one of four factors. First, the value of
MC
s at most an estimate since the detector was not a solid silver
sphere as was assumed but rather had two radial holes drilled to its
center - one hole having a 0.085-in.-diam stainless steel rod threaded
into it and the other containing a thermocouple sheathed in 1/16-in.-diam
aluminum oxide insulation. Secondly, Dahl and Jones 11 have pointed out in
their electrical analogue that the equipotential surfaces are elliptical
in shape rather than hemispherical as was assumed in deriving Eq. (13).
Thirdly, Ginnings 12 has pointed out that the effective radius of contact
may be larger for lower conductivity materials due to the heat conducted
to the sphere from the specimen by the gas in the vicinity of the contact.
Lastly, a film coefficient could influence the "effective" radius of
contact as shown here.
46
In spite of this discrepancy, the model approximates the experimental
sphere response since C 18 indeed a constant and thus substantiates the
supposition that the response is proportional to the specimen thermal
conductivity.
RHFERENCES
YTI
.
1. T. G. Godfrey, W. Fulkerson, T. G. Kollie, J. F. Moore, and
D. L. McElroy: USAEC Report, ORNL-3556, April, 1964.
2. 1. W. Watson and H. E. Robinson: Trans. ASME J. Heat Transfer, 1961,
vol. 83, no. 2, pp. 403–08.
3. R. W. Powell: Proc. Phys. Soc. (London), 1934, vol. 46, pp. 659-678.
4. W. J. Parker, R. J. Jenkins, C. P. Butler, and G. L. Abbott: J. Appl.
Phys., 1961, vol. 32, pp. 1679–1684.
5. G. D. Cody, B, Abeles, and D. S. Beers: Trans. Met. Soc., AIME, 1961,
vol. 221, no. 2, pp. 25–27.
6. P. H. Sidles and G, C, Danielson: J. Appl. Phys., 1952, vol. 25, p. 710.
7. J. A. Cape, G. W. Lehman, and M. M. Nakata: J. Appl. Phys., 1963, vol. 34,
no. 12, pp. 3550–3555.
8. R. W. Powell: J. Sci. Instr., 1957, vol. 34, pp. 485492.
9. R. W. Powell and R. P. Tye: Techniques of Non-Destructive Testing,
Butterworths, London, 1960.
10. R. W. Powell: Proceedings of the Black Hills Summer Conference on
Transport Phenomena, South Dakota School of Mines and Technology,
Rapid City, Final Report issued under Office of Naval Research
Contract No. Nonr(G)-00064-62 (15 October 1962) pp. 95–154.
11. A. I. Dahl and D. W. Jones:ASME-AIChe Heat Transfer Conference,
Buffalo, New York, August, 1960, Paper 60-HT-30.
12. D. C. Ginnings: Progress in International Research on Thermodynamic
and Transport Properties, Academic Press, New York, 1962.
13. R. J. Roark: Formulas for Stress and Strain, McGraw-Hill, New York, 1943.
14. E. 0, Speidel and D. L. Keller: USAEC Report, BMI-1633, May, 1963.
15. R. J. Dunworth: Argonne National Laboratory, Private Communication,
October, 1963.
16. E. Rothwell: AERE-R3897, 1961.
17. M. Tetenbaum: "Thermoelectric Parameter Studies on US, Ths, and UC-THS
Solid Solutions, " Seminar on Groups V and VI Anions, Argonne National
Laboratory, February 26, 27, 1964 (in press).

we
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