The differential operator perturbation technique has been incorporated into the Monte Carlo N-Particle transport code MCNP and will become a standard feature of future releases. This feature includes first and/or second order terms of the Taylor Series expansion for response perturbations related to cross-section data (i.e., density, composition, etc.). Criticality analyses can benefit from this technique in that predicted changes in the track-length tally estimator of K{sub eff} may be obtained for multiple perturbations in a single run. A key advantage of this method is that a precise estimate of a small change in response (i.e., < 1%) is easily obtained. This technique can also offer acceptable accuracy, to within a few percent, for up to 20-30% changes in a response

Iverson, J. L.

McKinney, G. W.

UNT Digital Library

LOS lamos Natio B I  Lab ratory is TITLE: AUTH 0 R(S): - SUBMITTED TO: perated by the University of California for the United States Department of Energy under contract W-7405-ENG-36. MCNP PERTURBATION TECHNIQUE FOR CRITICALITY ANALYSIS G.W. McKinney J.L. Ive r son  I C N C  Meeting, Albuquerque, NM, Sept .  17-21, 1995 DISCLAIMER This report was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor any agency thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsi- bility for the accuracy. completeness, or usefulness of any information, apparatus, product, Or process disclosed, or represents that its use would not infringe privately owned rights. Refer- ence herein to any specific commercial product, proccss, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recom- mendation, or favoring by the United States Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those. of the United States Government or any agency thereof. ~ - -~ ---____- By acceptance of this article. the publisher recognizes that the U.S. Government retains a nonexclusive. royalty-free license to publish or reproduce the published form of this contribution. or to allow others to do so. for U.S. Government purposes. The Los Alamos National Laboratory requesls that the publisher identify this article as work performed under the auspices of the U.S. Department of Energy. Los Alamos National Laboratory Los Alamos,New Mexico 87545 FORM NO. 836 R4 ST NO 26295181 DISCLAIMER Portions of this document may be illegible ' in electronic image products. Images are produced from the best available original document. C MCNP Perturbation Technique For Criticality Analysis G.W. McKinney and J.L. Iverson Los Alamos National Laboratory Los Alamos, NM 87545 (505)665-8367 (505)665-5538 Fax ' gwm@lnal.gov ABSTRACT The differential operator perturbation tech? nique has been incorporated into the Monte Carlo N-Particle transport code MCNPm [l] and will become a standard feature of future releases. This feature includes first and/or second order terms of the Taylor Series expansion for response perturbations related to cross-section data (i.e., density, composition, etc.). Criticality analyses can benefit from this technique in that predicted changes in the track-length tally e sba to r  of may be obtained for multiple perturba- tions in a single run. A key advantage of this method is that a precise estimate of a small change in response (i.e., < 1%) is easily obtained. This technique can also offer acceptable accuracy, to within a few percent, for up to 20-30% changes in a response. INTRODUCTION As given by the Taylor Series expansion, the evaluation of response sensitivities to cross- section data involves finding the ratio of the change in response to the infinitesimal change in the data. In deterministic methods, this ratio is approximated by performing two calculations, one with the original data and . MCNP is a trademark of the Regents of the Uni- versity of California, Los Alamos National Labo- ratory. . one with the perturbed data. This approach is useful even when the magnitudeQf the per- turbation becomes. very small; In Monte Carlo methods, however, this approach fails as the magnitude of the perturbation becomes small due to the uncertainty associ- ated with the response. For this reason, the differential operator technique was devel- oped. The differential operator perturbation technique as applied to the Monte Carlo method was introduced in the early 1970's [2]. Nearly a decade after its introduction, this technique was applied to perturbations in cross-section data by Hall [3,4] and later Rief [5].  A rudimentary implementation into the Monte Carlo transport code MCNP fol- lowed shortly thereafter [6]. With an enhancement of the user interface and the addition of second order effects, this imple- mentation has evolved into a standard fea- ture of future MCNP releases. In the following sections, the theory of this technique is reviewed, an overview of the MCNP user interface is given, and results for two criticality applications are presented. DIFFERENTIAL OPERATOR TECHNIQUE Derivation Of The Operator In the differential operator approach, a change in the Monte Carlo response c, due to changes in the related data set (represented by the parameter v) is given by a Taylor series expansion dnc A v n +  ... dc 1 d2c 2 d v  2! d$ n! dvn AC = - - A v + - .  -. AV + ... + -. -. where the n* order coefficient, u,, is given by which can be written as where &(h) is some constant, B repre- sents a set of macroscopic cross-sections, and H represents a set of energies or an energyinterval. For a track based re- sponse estimator c = z$qi i where 9 is the response estimator and qj is the probability of path segment j (path segment j is comprised of segment j-1 plus the current track). Equation (1) be- comes where With some manipulations presented in Reference [6], the path segment estima- tor of equation (2) can be converted to a particle history estimator un= C V , i P ;  I where V~ is the nth order coefficient esti- mator for the ith history and is given by Note this sum involves only those path segments j' in particle history i. Equation (3) shows how the history estimator for the n* order coefficient can be comput- ed from the track (or path segment) based operator ynf. The Monte Carlo ex- pected value of un becomes for a sample of N particle histories. Computing the nlh order differential OD- erator ynj' follows. First Order For a first order perturbation, ential operator becomes assuming the response estimator 1. the differ- tj is not a function of xb(h). The path segment proba- bility can be written as the product of track probabilities m qj = n 'k k = O  where rk is the probability of track k and segment j contains m+l tracks. In terms of tracks, the operator becomes or m k = O  where combining equations (4) and (5) gives where p k is calculated from one or more terms o L equation (8) for track k. Defining track probabilities, rk, in terms of Monte Carlo transport parameters is the final step of this derivation. If the kfh track starts with a neutron undergoing reaction type "a" at energy E' and is scat- tered from angle 8' to angle 8 and E, con- tinues for a length hk, and collides, then Second Order For a second order perturbation, the dif- ferential operator becomes again assuming the response estimator tj is not a function of xb(h). Omitting steps pre- sented in Reference [7], the' second order operator becomes where xa(E') is the macroscopic reaction cross-section at energy E', xT(E') is the total cross-section at energy E', and Pa(E'->E;0'->0)dEd8 is the probability distribution function in phase space of the emerging neutron. Equation (5) be- comes where and ha are unity if h=E and b=a, otherwise they vanish. Similarly for other types of tracks (i.e./ collision to boundary,, boundary to collision, and boundary to boundary), leading to one or more of these four terms. Finally, It is evident that y2j9 requires little additional effort to ylj$, namely the computation of aj*k If rk is given by equation (7), then aj*k becomes Once again, for other types of tracks one or more of these four terms is required. The second order coefficient estimator, given by equation (4), becomes MCNP USER INTERFACE General Description The PERT card allows the user to make per- turbations in cell material density, composi- tion, or cross-section data. Using the differential operator technique, the perturba- tion estimates are made without actually changing the input material specifications. Multiple perturbations can be applied in the same run, each specified by a separate PERT card. There is no limit to the number of per- turbations, since dynamic memory is used for perturbation storage. The entire tally out- put is repeated for each perturbation, giving the estimated differential change in the tally or alternately this change added to the unper- turbed tally (see the METHOD keyword). Perturbations to the &g estimator can be made by use of a track-length tally estimate of &p The CELL keyword and either the MAT or RHO keyword are required. PERT Card Keywords CELL - The entry(s) following this keyword indicate which cells are perturbed. MAT - The entry following this keyword specifies the perturbation material number, which must have a corresponding M card. Composition changes can only be made through the use of this keyword. RHO - Specifies the perturbed density of the cells listed after the CELL keyword. METHOD - This keyword specifies the number of terms to include in the perturba- tion estimate. 1 - include 1' and 2nd order (default) 2 - include only 1' order 3 - include only 2nd order A positive entry produces perturbation tallies i which give the estimated differential change in the unperturbed tally (default). A negative entry generates perturbation tallies such that this change is added to the unperturbed tally. The ability to produce first and second order terms separately enables the user to deter- mine the significance of including the sec- ond-order estimator for subsequent runs. ERG - The two entries following this key- word specify an energy range in which the perturbation is applied. The default range in- cludes all energies. RXN - The entries following this keyword must be ENDFB reaction types that identify one or more cross-sections to perturb. This keyword allows the user to perturb a specific cross-section of a single nuclide in a materi- al, as well as to perturb a set of cross-sec- tions for all nuclides in a material. The default is the total cross-section. APPLICATION RESULTS Ten test problems were developed to validate the accuracy of the differential operator tech- nique in MCNP. These problems were taken from the MCNP 25 problem test set and included eight fixed-source problems (neu- tron, photon, and coupled), as well as two criticality problems [7]. Perturbation results for the criticality problems follow. First Problem This problem is INPOB from the MCNP test set and is comprised of a cubic shaped piece of 235U (see Figure 1). The cube is approximately 10 cm on each side and is centered at the origin. It has two rectangular pieces of copper within it and a cone shape hole extending through it. A sphere of air extending 20 cm in radius from the origin surrounds the cube. The density of this air was increased from .Olg/cc to .6, .8, 1.5, 2, and 3 g/cc, which effectively increased the track-length tally of 9%, 16%, 21%, and 30%, respectively. (Tally 14) by 7%, %change Rei m o r ,  Fig. 1. A view of criticality problem W09, showing the 235U cube, copper plates, and conical hole. 8.29 11.11 20.95 27.97 42.02 .0167 -0221 .0411 -0547 .0820 rl Actual and predicted values of oations in air density of INPO9. , I , A:Dens:,(g/;). . I 1 .Ol; 0.6' 0.8 1.5 2.0 '3.0 I I I I I 1.009 11.078 11.100 11.175 11.223 11.308 I I I I I .0001 1.0015 1.0017 1.0021 1.0022 1.0025 . .  f - -b ,/' r. Table I lists the actual and predicted per- I ,.' . * cent changes to Tally 14, along with their rel- calculated by performing separate MCNP runs and correctly combining their relative errors. The predicted changes were calcu- lated all within a single run using five appro- priate PERT cards. Figure 2 plots these results and indicates the accuracy of this method (first and second order) is acceptable up to 10-15% changes in I(ke The deviation from the actual changes is most likely due to ative errors. The actual changes were . 1. L L I  I I1 r. - U r m i t y  (ole) Fig. 2. Actual and predicted percent change in changes in the shape of the eigenfunction which is currently not accounted for. versus air density for INPO9. Second Problem ~, o/a change Rel; Error This problem is INP18 from the MCNP test set and is comprised of a hexagonal prism lattice in the shape of a cylindrical reactor (see figure 3). It is cut in half and uses a reflective plane for simplification. The fuel is 70% enriched uranium in the shape of cylindrical rods. The clad on the fuel is a mixture of zirconium and niobium with a inner liner of tungsten. Inside the clad, water 9.20 16.10 20.68 22.92 .0186 .0349 .0581 .1410 blankets the fuel for cooling. Water is also used as the moderator and heat transfer agent for the reactor outside the fuel rods. There are three half control rods and five whole ones, made of '$, "B, and carbon. The control rods are encased in the same zirco- nium and niobium clad as the fuel, but with- out a liner. The sheath for the control rod is also made of zirconium and niobium and has water that traverses between the sheath and clad. The water is a mixture of heavy and light water. The water density was increased from 1 g/cc to 2, 3, 4, and 6 gkc, which increased the track-length tally of &H (Tally . 14) by 9%, 15%, 21%, and 30%, respec- tively. Table II lists the actual and predicted per- cent changes to Tally 14, along with their rel- ative errors. Once again, the predicted changes were calculated all within a single run using four appropriate PERT cards. Fig- ure 4 plots this data and indicates the accu- racy is quite good up to 20-25% changes in Keff Table II. Actual and predicted values of for perturbations in water density of INP18. ReL'Error Fig. 3. A view of criticality problem INP18, showing the triangular pitched fuel rods, moderator, and control rods. Fig. 4. Actual and predicted percent change in versus air density for INP18. CONCLUSIONS The differential operator perturbation tech- nique implemented into future versions of MCNP will’ provide the radiation transport analyst with a powerful tool for estimating the effect of multiple perturbations within a single run. This technique has shown to pro- vide adequate accuracy for up ’ to 20-30% changes in a response for fixed-source prob- lems and up to 10-20% changes in I(kg for criticality problems. Technique for MCNP,” To be Published, Los Alamos National Laboratory (1995). REFERENCES 1. Briesmeister, J., Editor, “MCNP - A General Monte Carlo N-Particle Transport Code,” LA-12625-M, Los Alamos National Laboratory (1993). 2. Takahashi, H., ”Monte Carlo Method for Geometrical Perturbation and its Application to the Pulsed Fast Reac- tor,’’ Nucl. Sci. Eng. 41, p. 259 (1970). 3. Hall, M. C., ”Monte Carlo Perturba- tion Theory in Neutron Transport Calculations,” Ph. D. Thesis, Univer- sity of London (1980). Hall, M. C., “Cross Section Adjust- ment with Monte Carlo Sensitivities: Application to the Winfrith Iron Benchmark,” Nucl. Sci. Eng. 81, p. 423 (1982). Rief, H., “Generalized Monte Carlo Perturbation Algorithms for Corre- lated Sampling and a Second-Order Taylor Series Approach,” Ann. Nucl. Energy 11, p. 455 (1984). 6. McKinney, G., ”A Monte Carlo (MC- IW) Sensitivity Code Development and Application,” M.S. Thesis, Uni- versity of Washington (1984). 4. 5. 7. McKinney et. al., G., ”Verification of the Monte Carlo Differential Operator 

MCNP perturbation technique for criticality analysis

https://digital.library.unt.edu/ark:/67531/metadc623048/m2/1/high_res_d/100040.pdf

MCNP perturbation technique for criticality analysis

Abstract

Similar works

Full text

Available Versions

UNT Digital Library