como . - . I OFL ORNL P | 482 . *SO 11:25 | 1.4 LE MICROCOPY RESOLUTION TEST CHART NATIONAL BUREAU OF STANDARDS - 1963 Ornih up-1482 FREQUENCY RESPCASE SENSITIVITIES IN REACTOR DYNAMICS STUDIES* - LEGAL NOTICE - T. W. Kerlin Oak Ridge National Laboratory Oak Ridge, Tennessee J. L. Lucius Oak Ridge Gaseous Diffusion Plant Oak Ridge, Tennessee TWI report ou ponudu na mocow of Govoru spousond wort. Welchor the Valid Homo, nor the counton, nor any purrou noties omhello the countestens A. Makes uy murruty or more station, arriend or implied, nii rupect to the accu racy, complerenons, or wo fulwows on the walorusion contained in this report, or that the wo of way lalonuevo, sportu, wached, or process declared to do report wy not latringo i primuly oned rou; or D. Antoni vay waluuios no ment to the woo, a lor su really from the woolway latorustion, imuntus, aethed, or proowus dicloud uwis report. As word to the abomo, "porno estag a ball of the Counselo" includes my a. Wisus or cosinuc'ar of the Counselon, or augloyw of such contractor, lo onluat was auch maplush or controlor the coulaskou, or employees of auch coauructor propert, domainsloo, or provides wocows 10, way tularation pure to unployunl or contract viu dhe Counselon, or Mocmployw.al vill such utructor. AUG 2 1965 It is common. practice to include theoretical frequency response analysis in the design and evaluation of reactor systems. This analysis is usually performed for a range of values on important system parameters. This permits the analyst to assess the effect of uncertainties in design parameters on dynamic performance, to establish allowable tolerances on system components, and to interpret any discrepancies between theoretical predictions and ex- perimental results which might be encountered. Several general-purpose computer codesł,2 have been written for calcu- lating frequency response, and one has a provision for automatically varying one parameter at a time to provide results which indicate the effect of parameter changes. However, it is desirable that this numerical procedure for one-at-a-time evaluation be replaced by an analytical procedure which simultaneously evaluates the effects of individual changes in each of the parameters. Such a procedure 18 described in this paper using the notion of sensitivity coefficients; (or parameter influence coefficients*). RHEASED FOR ANNOUNCEMENT , IN NUCLEAR SCIENCE ABSTRACTS A sensitivity coefficient is defined as the partial derivative of a selected measure of performance with respect to a selected system parameter In the present application, the selected measure of performance is the frequency response and the selected system parameters are coefficients in the system dynamic equation. * Research sponsored by the U. 8. Atomic Energy Commission under contract with the Union Carbide Corporation, ON be te AND IFS CONTAMOTORS GREY .-...... Supermotora como : The linear, constant coefficient, ordinary differential equation chosen to represent the system 18 + oo, waere x = the solution vector. · The components of this vector are such things as power level, precursor concentrations, and temperatures, t = time, A = the system matrix. This is a constant, square, real matrix whose non-zero components are the coefficients in the dynamics equations, a = the selected independent (scalar) variable, ñ = a vector of coefficients of a. Equation (1) can be Laplace transformed and re-arranged to give the transfer function equation: (e) ſco) = - (A - 81] [, (2) where G(8) = the transfer function vector. The components of G(s) are X, lã, 8 = the Laplace transform variable, x = Laplace transform of X, ã = Laplace transform of a. The frequency response is obtained by replacing the parameter, 8, by jw, where j = -1 and w = the angular frequency. The operation indicated in Equation (2) involving inversion of a matrix with complex elements is perfectly suitable for digital computer evaluation. However, in more general terms, it 18 convenient to think of G8) as being obtained by applying some unspecified operator on the forcing vector, 5. Thus, Equation (2) would be written GCB) = B(s) , (3) where B(8) = an operator equivalent to -[A - 81]". . The sensitivity equation simply obtained by differentiating Equation (2) with respect to element &, of the system matrix, A. This gives enco) - [A – 81]** 4; (e), (4) where 44 = a matrix with unity in row 1 and columa j and zeros elsewhere. 'In operator notation, Equation (4) may be written on = B(8) 4: ) (5) The important feature of this equation is that it uses exactly the same operator as was used to obtain the frequency response. The only change being that the forcing function, 4, G(8.), replaces in Equation (3). The forcing function, 42, G(s), 18 readily available once the frequency response vector, G(s), 18 calculated since 4 y operating on the Gf 8) vector only serves to give a vector with one non-zero element in row 1 whose value is the component from row j of G(s). Thus, it is clear that once the operator, B, has been evaluated in calculating the frequency response; the additional labor required to calculate the sensitivities (which requires only a simple matrix multiplication) 18 small. Equation (5) gives all the components of OG(8)/84,,. The following formula 18 a special case of Equation (5) which is useful if only one component of 80(8)/8027 18 desired: .. Det(8) 6,(o) , (6) where G (8) = component of G(s) in row l, ber(8) = component of B in row l and columa i, G,(8) = component of G(8) in row j. The above procedure for frequency response sensitivity analysis bas been implemented with a computer code for the IBM 7090 at Oak Ridge National Laboratory. The code is currently in use, and has proved useful in the dynamic analysis of the Molten Salt Reactor Experiments where sensitivity data provided Information which aided in developing an under- standing of the physical causes for the calculated dynamic performance. References 1. MUELLER, G. O. and WARRINGTON, J. A., "GALS-4 - General Analysis of Linear Systems," USAEC Report KAPL-M-6433, Knolls Atomic Power Laboratory, December 15, 1964. KATZ, . M. and ST. JOHN, D. 8., "LASS, An IBM 704 Program for Calculating System Stability," USAEC Report DP-894, E. I. du Pont de Nemours and Co., July 1964. 3. TOMOVIC, R., Sensitivity Analysis of Dynamic Systems, McGraw-H121 Book Co., Inc., New York, 1963. 4. MWSSINGER, 1. F., "The Use of Parameter Influence Coefficients in Computer Analysis of Dynamic Syistems, " Simulation, 3(2): 52 (1964). 5. BALL, S. J. and KERLIN, T. W., "MSRE Stability Analysis," USAEC Report OFNL-IM-1070, Oak Ridge National Laboratory. To be published. . -LEGAL NOTICE - The maport we prepared u aa score of Gomarimeat sponsored work. Molchor the United Males, nor the Commission, nor w porno active on behalf of the Counloddon: A. Makas may notuty or represcalation, proased or leaplied, with respect to the accr- msy, completes, or wefalous of the taformation contaland la duo report, or that the we of any balorukom, appuntos, melhad, or proces declared in the report may not latriage privately owned relo; or B. Asema nay las luas will repect to the one of, or for dengue rening from the w of any buforution, appuntu, method, or procou diecloud la wito report. As wed us the above, "porno actions on detall of the Commission" tncludes may re- ployee of contractor of the Counterton, or employw al much atractor, to the test dat met umployee or contractor of the Coantastoo, or employu al rock contructor prepurua, donenaren, or provides neuen bo, un culormation permetto No aplogant or contract via the Coundation, or Mo saployment with such contractor, Y, 16 . WIN ! T WWW. S. I 1 . END . et no DATE FILMED 11/ 19 /65 A 5 : 1.