Bartine, D. E.

Childs, R. L.

Engle, W. W., Jr.

UNT Digital Library

^'? 6 66 £•¥-PERTURBATION THEORY AND SENSITIVITY ANALYSIS FORTWO-DIMENSIONAL SHIELDING CALCULATIONS'1'R. L. Childs*D. E. Bartine**W. W. Engle, Jr.**Union Carbide Corporation - Nuclear DivisionOak Ridge, Tennessee 37830ABSTRACTThe purpose of this work is to describe the development and imple-mentation of perturbation theory methods for two-dimensional shieldingproblems. Cross section sensitivity results are obtained by calculatingthe effect of varying each individual cross section parameter.- NOTICE-This report was prepared as an account of worksponsored by the United States Government. Neitherthe United States nor the United States EnergyResearch and Development Administration, not any oftheir employees, nor any of their - contractors,subcontractors, or their employees, makes anywarranty, express or implied, or assumes any1 legalliability or responsibility for the accuracy,completenessor usefulness of any information, apparatus, product orprocess disclosed, or represents that its use would notinfringe privately owned rights.Research sponsored under the Union Carbide Corporation's contract withthe Energy Research and Development Administration.•Computer Sciences Division, Oak Ridge National Laboratory.**Neutron Physics Division, Oak Ridge National Laboratory.RECEIVED BY TIC FEB 1 2 19752Changes in a response» R, can be calculated using the perturbationtheory relationship1AR s / / /$*(r ,E,n) [ff&Zs(T,E'+Etfc*Q)n ' - SIt(r,E)<j>(r,E,fi)]dEdndr, (1)where the perturbation is represented by the changes in the total andscattering cross sections and <J> and 4>* are the forward and adjoint fluxes.This paper will deal with the evaluation of the above equation with for-ward and adjoint fluxes calculated using the discrete ordinates computerprogram DOT.2 The spatial and energy integrations are easily evaluated byreplacing the integrals with summations over energy groups and a spacemesh. The angular integrations are more complicated and will be consideredin more detail.The scattering term in Equation (1) requires the evaluation of anintegral of the form/4ir 4irThe cross section may be expanded in a Legendre series asA £ ? ' ) , (2)£=0/a(n»nV)Po(fl-fif>dJL4where cr^  = (2A+1)  Similarly, the angular fluxes maybe expanded in a spherical harmonic series asandHZ Z (-l)H2l+l)i!LnAl(Si), (4)where*"* 4irf f dn dif,•'0 •'-I'Ztr r.ldft = |4irandThe form, of the above equations was chosen to conform with the equationsprogrammed in DOT. The (-1)^ and -Q factors arise because the adjoint prob-lem i s solved far <J>*(-Q).The addition theorem for legendre polynomials i s (ignoring terms thatvanish under integration):£= E A"(fl)A"(B'). (5)m=0 * %Substitution of Equations (2), (3), (4) , and (5) into the desired integraland use of the orthogonality relationship,*£•£•$»,••• ( 6 )x.,x, m,myields the final result:j j f i = 2 z C - U ^ j ^ * ^ . (7)The total cross section term in Equation (1) requires the evaluationof an integral of the form/4TT4This integral can be evaluated using the angular fluxes from the discreteordinates calculation as has been done in one-dimensional calculations.^»*However, because of the tremendous quantity of numbers to be stored in twodimensional problems (70 million for the example which follows), it ishighly desirable to avoid storing the angular flux if possible. A way toavoid the angular flux may be found by substituting Equations (3) and (4)into the desired integral and using Equation (6) to obtain%J S I (-l)JL(2£+l)jJljmj£,m. (8)4iT fc=0 m=0Thus* Equation (1) can be evaluated using only the flux moments j#, m andprovided the series in Equations (7) and (8) can be truncated after afew terns without serious error. Present experience indicates that athird order expansion (through it=3) is sufficient for many problems.The approach outlined above has been implemented using the computerprograms Vip5 and SWAHLAKE**. The method was tested for a deep penetrationsodium problem and for an infinite air calculation (including a first col-lision source term) by comparing with 1-D SWANLAKE calculations. Agreementwas satisfactory in both cases.The first realistic two-dimensional sensitivity analysis performed inthis study was for the calculation of neutron and gamma doses at the main-tenance deck of the FTR.7 The DOT calculations were performed with a 21-18coupled neutron-gamma ray cross section set, a 166 angle quadrature setsuitable for streaming in annular gaps, and P3 scattering.Results obtained from the sensitivity analysis include the sensitivityof neutron and gamma dose to changes in cross sections as a function ofposition* energy, and element. For this problem, the important gamma ray5sources are iron (n>Y) reactions (primarily in the stainless steel head)which are responsible for 55 percent of the gamma dose and boron (r|,y)reactions (in a borated polyethylene layer under the maintenance deck)which contributes 28 percent of the gamma dose. A more detailed study in-dicates that 31 percent of the gamma dose is due to gamma ray productionin the 7 to 8 MeV group, which contains strong iron (n»Y) production linesat 7.63 and 7.65 MeV. The gamma dose due to boron production is the resultof the (iWY) 0.48 MeV line.This work demonstrates the feasibility of performing sensitivitystudies and perturbation calculations for two-dimensional shielding prob-lems. The FTR example shows the usefulness of calculations of this typefor realistic shielding applications.* f « f-Si*i= ri i"-.~.<6REFERENCES1. T. J. Hoffman, J. C. Robinson, P. N. Stevens, "The Adjoint DifferenceMethod and Its Application to Deep-Penetration Radiation Transport,"Nucl. Sex. Eng., 48, 179 (1972).2. W. A. Rhoades, F. R. Mynatt, "The DOT III Two-Dimensional Discretelprd±aates':Transp6rt'Xode'*"^ pRNL-TM-4280,''' Oak Ridge National Laboratory(1973)T3. D. E. Bar tine, et al., "ORNL Cross Section Sensitivity Analysis Appli-cations for Radiation Shielding," American Nuclear Society TopicalMeeting for Advanced Reactors: Physics, Design, and Economics, Atlanta,Georgia (1974).4. R. L. Childs, W. W. Engle, Jr., J. C. Robinson, F. R. Mynatt, "TheDevelopment and Application of a Discrete Ordinates Adjoint DifferenceMethod for One-Dimensional Shield Weight Optimization," ORNL-TM-4196,Oak Ridge National Laboratory (1973).5. R. L. Childs, "VIP, A Computer Program Using Two-Dimensional DiscreteOrdinates Transport Calculations for Cross-Section Sensitivity Analysis,"UCCND-CSD-1 (to be published).6. D. E. Bartine, F. R. Mynatt, and E. M. Oblow, "SWANLAKE, A Computer CodeUtilizing ANISN Transport Calculations for Sensitivity Analysis," ORNL-TM-3809, Oak Ridge National Laboratory (1973).7. F. R. Mynatt ejL jil^ ., "Fast Reactor Analytical Shielding Progress Reportfor May and June 1973," ORNL-TM-4401, Oak Ridge National Laboratory(1974).

Perturbation theory and sensitivity analysis for two-dimensional shielding calculations

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Perturbation theory and sensitivity analysis for two-dimensional shielding calculations

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