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ARTICLE Monte Carlo Shielding Calculations for a Spent Fuel Transport Cask with Automated Monte Carlo Variance Reduction 1,* 22 Mitsufumi ASAMI, Hidenori SAWAMURAand Kazuya NISHIMURA 1 National Maritime Research Institute, 6381, Shinkawa, Mitaka, Tokyo, 1810004, Japan2 MHI Nuclear Engineering Co., Ltd, 31, Minatomirai 3chome, Nishiku, Kanagawa, 2208401, Japan For the purpose of performing reasonable shielding calculation of a spent fuel transport cask, the use of Monte Carlo methods has been proposed for solving the radiation transport problem on a detailed structure of the transport cask considering fixed neutron sources. A SMIRE (Simplified MCNPANISN_W Variance Reduction) system has been developed in the present study, which is possible to generate automatically the lower weight boundary of the weight window for each mesh based on the Consistent Adjoint Driven Importance Sampling (CADIS). Compared with the case of the importance based on the empirical formula, the figure of merit is increased by a factor of 25. In this system, it is possible to calculate the weight suitable for the distantlypositioned detector point from the fuel ef fective region to introduce the relaxation factor which relaxes the increase of the particle numbers at the boundary of weight window meshes that are generally caused by the large attenuation of adjoint flux. This system is used for a va riety of the radiation transport problems as well as the transport cask. KEYWORDS: CADIS methodology, relaxation factor, weight window, figure of merit, Monte Carlo shielding analysis, empirical formula, adjoint flux1 I. Introductionsorption. The tracking on the particle is continued until the particle is eliminated from the considered region or until the The neutron flux distribution of some complicated shiel tracking on the particle reached the number of the particles ding structures can be determined precisely by solving the that have already been set in the calculation. In order to ex integral Boltzmann transport equation in the Monte Carlo ecute simulation with nonanalog Monte Carlo techniques method. Additionally, the Monte Carlo method with conti1) efficiently, importance samplingis indispensable. In im nuousenergy crosssection data was considered to be the portance sampling, importance functions are selected for most accurate method for performing these shielding ana suitable variance reduction. After that, many random walks lyses. Moreover, the use of the Monte Carlo method is are rationally executed in the objective phase space (space exempt from constructing spaceenergyangle grids for and energy). In particular, a weight window importance pa modeling problems, and hence, no discretization errors are24) rameter consistsof upper and lower boundary for a introduced into solving problems. Therefore, the Monte particle’s statistical weight in each phase space region. In the Carlo simulation is relatively straightforward in both physics weight window method, the lower weight boundary of each and geometry modeling. cell which composes the calculation model is set so that the However, it is difficult to solve radiation transport calcu collision density of the particles entering each cell is kept lations in practical problems with thick shielding and large constant. In order to set the variance reduction parameter source volume by the Monte Carlo method. Thick shielding appropriately, particles entering each cell are to be split or results in significant attenuation of radiation intensity from Russianrouletted correctly. the source to the tally. Large source volume causes an addi In the past, the lower weight boundary of the weight tional selfshielding effect for the inner source region. These window has been determined from the experiences of the situations make it more difficult to do sampling effectively Monte Carlo practitioners. This way may lead to inefficient of the source variables. These arguments point to a need for variance reduction parameters of the calculations. For exam the nonanalog Monte Carlo techniques are indispensably. ple, if the statistical weights of the source particles are not Actually, they have been used with great care to prevent within the weight window, the particles are split or Rus getting unreliable results. sianrouletted immediately in an effort to bring their weights To date, a large number of the nonanalog Monte Carlo into the weight window. This event results in unnecessary techniques has been developed and improved. In the degradation in computational efficiency. Therefore, respec nonanalog Monte Carlo techniques, particles have the sta tive techniques must be invariably consistent with one tistical weights. The statistical weight of the particle is another when we conduct the shielding analysis with some reduced through physical events such as collision and ab variance reduction techniques. The purpose of the present study is to perform, compare *Corresponding author, Email:asami@nmri.go.jp
Data Extraction
Superimposed Importance Mesh Geometry
Weight Window Lower Weight Bound
(MCNP4C Input data)
PRESMIRE (Data Extraction)
Group Constant DLC23F/Cask Librar
MCNPANISN_W (Adjoint Calculation)
Adjoint flux
POSTSMIRE (fluxWeight Window)
Continuousenergy Cross Section Data (JENDL3.3)
MCNP4C (Forward Calculation) END
Data Extraction
Material Card
Source Biasing
Fig. 1flow of the lower weight boundary of the Calculation weight window and forward calculation
and analyze the shielding calculations for a transport cask in detail with the variance reduction technique based on the Consistent Adjoint Driven Importance Sampling (CADIS) 59) 10) methodology andthe empirical formula.CADIS me thodology makes use of the adjoint function that is associated with particle importance which is the contribution of a particle with respect to the objective. The Monte Carlo method with variance reduction shown in the present study should be helpful to those in performing calculations for similar some shielding structures. II. SMIRE System
1. Description of the SMIRE System Figure 1shows the schematic flow of the SMIRE System to generate the lower weight boundary of the weight win dow. 11) MCNP4C codeis provided with “superimposed impor tance mesh” which can create the space partition based on mesh that is independent of the geometrical cell. In the present study, the meshbased weight window parameter generation system has been developed. In this system, com plicated cell partitions for the variance reduction are not necessarily required. The lower weight boundary of the weight window in each mesh is determined as follows: a)Each coordinate of the superimposed importance meshes is calculated from the mesh information in the MCNP input data. b)Each of these adjoint fluxes in the superimposed impor tance meshes is calculated by onedimensional 12) deterministic code, MCNPANISN_W.This code makes it possible to solve the onedimensional neutron transport problem specified by MCNP input data.Fig ure 2shows the flow of the MCNPANISN_W:
START
Extraction of the source information from MCNP input ・The location of the source region ・Source strength distibution
Particle production from source region NPS=NPS+1, NPS：Number of particle histories
ITAL=ITAL+1 ITAL：The number of mesh
Extraction of the geometry information from MCNP input ・The coordinate of the intersection between the straight-line and each cell boundary ・Material composition ・Atom density
Preparation of the input data for ANISN-W ・Preparation of the mixing table for each mesh ・Determination of mesh coordinate
One-dimentional transport calculation by ANISN-W
Determination of source
Generation of random number for decision of the generation-position of particle Preparation of the input data for ANISN-W
Neutron transport Calculation
N ITAL=ITAL ? max Y N NPS=NPS ? max Y Calculation of the adjoint flux END Fig. 2 Flowchartof the MCNPANISN_W calculation
(1)A detector locating point is arranged at the center of each mesh which constitutes the geometric form described in MCNP input. (2)The location of the particle production is stochas tically decided from the source region and the source strength distribution described in MCNP input. (3)Particle energy is stochastically decided from the distribution of source energy spectrum described in MCNP input. (4)The distance in a straight line between the loca tion of the particle production and the detector location is decided from the geometry data de scribed in MCNP input. (5)The coordinate of the intersection between the straight line and each cell boundary is decided from the geometry data described in MCNP input. The material composition and the atom density each cell is also obtained from the MCNP input. (6)The onedimensional model for the deterministic Sn transport code is made from the calculation conditions acquired from the process (2) to (5). The transport calculation is performed by 13) ANISNW code. (7)Calculation conditions for ANISNW are as fol lows: Basic geometry form: sphere Source: Shell source problem
500.0 Left boundary condition: reflection Right boundary condition: vacuum (no reflection)283.0 Outer iteration: 1252.0 214.5 199.3 (8)The adjoint flux in a certain mesh is obtained by182.4 onedimensional transport calculation. (9)The adjoint flux of each mesh is obtained by the process from (2) to (8). 0.0 (10)The process of the adjoint flux calculation in a certain mesh from (2) to (9) is iterated by the number of particle histories. (11)The end result of the adjoint flux in a certain mesh -182.4 -193.3 is obtained averaged over the number of particle-203.3 histories.-238.8 (12)The adjoint fluxes in all of the mesh are obtained. -500.0 c)The codegenerated adjoint flux obtained from b) is normalized for each phase space by the adequate method. d)The reciprocal of the adjoint flux is the lower weight Fig. 3geometry of the transport cask and adjoint Calculation boundary of the weight window of each superimposed1718) source region importance mesh. 2. Calculation on Weight Window by the SMIRE System ௦ and energy group݃, respectively.߶ ൫ܧ ൯ isthe adjoint The weight window is a variance reduction scheme in flux at energy group݃ and spatial mesh݅. which each region of phase space݅assigned an upper is 14)Here, the lower weight boundary of the weight window is and lower weight boundary.Particles entering a phase ത ௦ set by the use offollows: Let the energy group as ߶ ൫ܧ ൯ space region with a weight outside the boundaries are either corresponding to the maximum of the source energy spec split or Russianrouletted which is performed to bring their ୢୣ trum distribution before biasing beܧ୫ୟ୶. The lower weight weights into conformity. The weight window can be used in boundary of the weight window corresponding to the energy any dimension of phase space. The lower weight boundary ୢୣ ܧ୫ୟ୶ wasassumed to be the standard of the lower weight of the weight windowܹspatial mesh for݅ andenergy 15)boundary of the weight windows of other energy groups; the group݃ are calculated from the adjoint flux as follows: lower weight boundary of the weight window on the source ܥ region was set as the reciprocal of the adjoint flux at the ܹ ݅,݃ሻ ,ሺ ൌற (1) ሾ߶ሺݎ, ܧሻሿ݀ݎ݀ܧ ౨ ୢୣ ܧ ாenergy୫ୟ୶ averagedover the real source region, , ௦ ୢୣ ത 1⁄߶ ൫୫ୟ୶ൌ0.5. In the case of the transport cask such as ܧ ൯ ற where߶ ሺݎ,ܧሻ is the adjoint flux at pointݎ and energyܧ,ୢୣ ܧ containing the spent fuels, the energy group୫ୟ୶ is ܸ aܧ ndthe volume of spatial mesh represent݅ and 1.111.83 MeVwith the energy group structure of energy group݃, andܥ isthe constant for normalization,16) DLC23/CASK. ܹ making in the source regions to be a half of the biased weight of source particle. With finite weight window mesh III. Validation of SMIRESystem sizes, significant increase of the particle numbers at the boundary of the weight window meshes is generally causedThe SMIRE system is tested by applying to an analytical by the large attenuation of adjoint flux. In this case, loweredmodel of a cask, as shown inFig. 3. The model was pre efficiency of calculations occurs with an increase of CPUpared in considerable detail, especially in the fuel basket. 15) ݊The main specifications of the cask are as follows; total time per history. Therefore, the relaxation factor,୰ is introduced to improve the calculation efficiency for theseweight is 115.0 tons, outer diameter is 2.6m and height is cases. 6.3m, main structure is carbon steel, the fuel basket is com The lower weight boundary of the weight window mustposed of stainless steel both with and without boron, lead is be set close to 1.0 near the cell or mesh of the source regionused for a gammaray shield and NS4FR resin is used for a because the particle’s statistical weight just after being emitneutron shield, and it has cooling fins made of stainless steel. ted from the source is equal to 1.0. The constantܥ in Eq. (1)It is possible to install 14 bundles of pressurized water reac 1718) is the adjusting parameter of the reciprocal of the importance.tor spentfuel assemblies.Figure 3 also shows the source In the present study, the constantܥregion for the calculation of the adjoint flux. Since thedecided as follows. is The adjoint flux at energy group݃source intensity of neutrons in a spent fuel assembly depends averagedover the real source region is given by Eq. (2).strongly on the specific burnup distribution in the axial di rection, the burnup distribution was taken into account in ௦ ߶ ൫ܧ ൯ the present calculation. A peaking factor of 1.15 was as ௦ ത ߶ ൫ܧ ൯ൌ(2) ܸ sumed for the middle part of 10/12 of all fuel assemblies. cation fr,ୣ, of the cask con The effective multipliacto݇ ܸ ܧ݅ Where, and representthe volume of spatial mesh taining 14 PWR assemblies was calculated by the KENO
1.0E05
1.0E06
1.0E07
1.0E08 0
1.0E05
1.0E06
1.0E07
1.0E08
1.0E09
50 100150 200 Distance from center of the cask (cm) History=3000 History=5000 History=10000 (Adjoint flux at 14.9 MeV)
250
1.0E10 0 50100 150 200 250 Distance from Center of the cask (cm) History=3000 History=5000 History=10000 (adjoint flux at energy 1.11MeV)Fig. 4MeV andof adjoint flux at energy 14.9 Comparison 1.11 MeV
19) V.a code,and a݇ୣused to obtain the neu of 0.63 was tron source intensity of the cask. DLC23/CASK library was used for all adjoint flux cal culations. The weight window parameters, source energy biasing and the doserate conversion factor were given in the energy group structure of DLC23/CASK library. The 20) doserate conversion factorwas used for the energy spec trum for the calculation of the adjoint flux. In addition, the final MonteCarlo shielding calculation was performed with 21) the Japanese Evaluated Nuclear Data Library, JENDL3.3. Figure 4the calculation result of the adjoint flux shows for each mesh of the calculation model with a different number of particle histories (3,000, 5,000 and 10,000, re spectively). The fuel effective region was equally divided into three in accordance with axial direction. The detector location was set atሺݔ, ݕ, ݖሻൌሺ0,135,0ሻ. As shown Fig.4, number of particle histories has an insignificant effect on the calculation result of the adjoint flux above 14 MeV. On the other hand, it is shown that the distribution of the adjoint flux below 1.11MeV depends strongly on number of par ticle histories. Analysis shows that number of particle histories requires at least 10,000 to calculate the adjoint flux of each mesh in the shielding configuration such as the transport cask. Therefore, the adjoint flux was conservatively calculated with a history number of 10,000 for the transport cask. IV. Results and Discussion ݊ In the present study, the relaxation factor୰ wasintro
0.20
0.15
0.10
0.05
0.00 Emp irical formulaR.F.=1.0 R.F.=0.5 Case FSD FoM
18 16 14 12 10 8 6 4 2 0 R.F.=0.3
Fig. 5 FoMfor the neutron dose rate at the side surface of the cask calculated with the lower weight boundary of the weight window that is generated by using a different relaxation factor. R.F. denotes the relaxation factor.
15) duced, whichrelaxes the attenuation of the adjoint flux among adjacent meshes: ౨ 11 ౨ ୪୭୵ ୪୭୵ ܹ ן ՜൫ܹ ൯ן ൨ ,݊ 1.0(3) ୰ ற ற ߶ ߶ ୪୭୵ ற Where,ܹthe lower weight boundary and is߶the is adjoint flux. The calculation of the lower weight boundary was per formed in cases of݊୰ൌ1.0, 0.5, re , 0.3spectively, and then the shielding calculation was performed by using these lower weight boundaries of the weight window. The detector loca tion was set atሺݔ, ݕ, ݖሻൌሺ0,135,0ሻ. Figure 5shows the result of the computational efficiency (Figure of Merit: FoM) and the fractional standard deviation. For a detector point near the axial center of the fuel effective length in the transport cask, the computational efficiency ݊ with the relaxation factor୰ൌ1.0better than with is ݊ ୰൏ 1.0. It is shown that the calculation is correctly per formed for transport processes based on the deterministic adjoint flux obtained from these results. The computational efficiency based on the lower weight boundary of the weight window that was generated by the SMIRE system was compared to that based on the impor tance that is based on the empirical formula. The calculation result is shown inFig. 6. The horizontal axis is the spatial coordinatesሺݔ, ݕ, ݖሻ and the longitudinal axis is the FoM or the fractional standard deviation. This figure shows that the computational efficiency based on the adjoint flux by using the SMIRE system is better than that based on the empirical formula at the center of the fuel effective length. Compared with the case of the importance based on the empirical for mula, the figure of merit is increased by a factor of 25. The computational efficiency is very low at the point away from the fuel effective length. Here, the calculation was performed by the lower weight boundary of the weight window based on the adjoint flux obtained from the adjoint source arranged around the topend of the transport cask. The detector location was set atൌሺ0,135,0ሻሺݔ, ݕ, ݖሻ.Fig ure 7shows the calculation result. As shown in Fig. 7, in the
0.20 4.5 4.0 3.5 0.15 3.0 7 0.10 2.5 6 0.10 2.0 5 1.5 4 0.05 0.05 31.0 2 0.5 1 0.00 0.0 Empirical formulaR.F.=1.0 R.F.=0.5 R.F.=0.3 Case 0.00 0 （0，105，220） （0，105，185） （0，135，0） （0，105，200） （0，105，225） Coordinate (x,y,z) FSD FoM EmpiricalFSD SMIREFSD EmpiricalFoM SMIREFoM Fig. 7 Standarddeviation and Figure of merit at a point away Fig. 6fractional standard deviation (FSD) and the figure of The from the fuel effective length. R.F. denotes the relaxation factor. merit (FoM) in an calculation point on the surface of the trans port cask. The coordinates on the horizontal axis represent the coordinates of the calculation point. Each calculation point cor on the empirical formula, the figure of merit is increased by responds with the position of the transported cask as illustrated a factor of 25. If the detector is located in the side part of the in this figure. transport cask, and at some distance from the both ends of the fuel effective length, the limitation of the onedimensional model is relaxed effectively by the relaxa case of optimum relaxation factor is 0.5, the figure of merit tion factor such as݊୰ൌ0.5. The adjoint flux is corrected is increased by a factor of 50 compared to the case based on appropriately by the relaxation factor, which is useful for the the empirical formula. This is because the SMIRE system variance reduction. As a result, the figure of merit is in makes the onedimensional model, the straight line distant creased by a factor of 50 compared to the case based on the from the neutron generation location in the source region to empirical formula. The SMIRE system is useful for the cal the point detector. In the SMIRE system, it is possible to culation of the dose rate for any point around the cask calculate a distantly positioned detector point from the fuel regardless of experience of the practitioner. effective region to introduce the relaxation factor. If the geometric condition can be modeled by onedimensional Acknowledgment form such as the side surface of the cask, the adjoint flux This study was financially supported by the Budget for which is useful for the variance reduction is obtained with ݊ൌ1.0Research of the Ministry of Education, Culture,). Nuclear out respect to the relaxation factor (୰If the detector Sports, Science and Technology, based on the screening and is located in the side part of the transport cask, and at some counseling by the Atomic Energy Commission. distance from the both ends of the fuel effective length shown in Fig.6, the use of the onedimensional model has References limited flexibility. In the case of the spent fuel transport cask discussed in this paper, the limitation of the onedimensional 1)M. H. Kalos, “Importance Sampling in Monte Carlo Shielding model is relaxed effectively by the relaxation factor such asCalculations,”Nucl. Sci. Eng.,16, 227 (1963). ݊୰ൌ0.5. The adjoint flux is corrected appropriately by the2)T. E. Boothet al., “Importance Estimation for Monte Carlo Calculations,”Nucl. Technol./Fusion,5, 91 (1984). relaxation factor, which is useful for the variance reduction. 3)T. E. Booth, “Automatic Importance Estimation in Forward The optimum value of the relaxation factor for the spent fuel Monte Carlo Calculations,”Trans. Am. Nucl. Soc.,41, 308 transport cask is obtained on a casebycase basis. Each (1982). spent fuel transport cask has similarities with the geometric 4)J. S. Hendricks, “A CodeGenerated Monte Carlo Importance condition and source information. Therefore, the relaxation Function,”Trans. Am. Nucl. Soc.,41, 307 (1982). ൌ0 factor݊୰.5applicability to various casks for the has 5)J. C. Wagner, A. Haghighat, “Automated Variance Reduction of spent fuel. Monte Carlo shielding Calculations Using the Discrete Ordi nates Adjoint Function,”Nucl. Sci. Eng.,128, 186208 (1998). V. Conclusion6)K. Uekiet al., “Continuous EnergyMonte Carlo Analysis of Neutron Shielding Benchmark Experiments with Cross Sec This study demonstrated the variance reduction with the tions in JENDL3,”J. Nucl. Sci. Technol.,30, 339 (1993). adjoint flux calculated with onedimensional discrete ordi 7)J. F. Briesmeister (Ed.),MCNP—A General Monte Carlo nate method for the Monte Carlo shielding calculations of NParticle Transport Code Version 4C, LA13709M, Los the spent fuel transport cask. The SMIRE system has been Alamos National Laboratory (LANL) (2000). developed for automatically calculating the parameters of 8)M. Asami, S. Ohnishi, K. Kawakami, T. Matsumoto, N. Odano, weight window and source biasing for precise shielding cal “Rational Shielding Ability Evaluation for a Modular Type culations. Compared with the case of the importance based Interim Storage Facility,”Trans. At. Energy Soc. Japan,6[4],
460475 (2007), [in Japanese]. 9)R. K. Disney, J. Jedruch, R. G. Soltesz, R. S. Kaiser, S. L. Zeigler,ANISNW, WANLTME2778, Westinghouse Astro nuclear Laboratory (1971). 10)J. S. Hendricks, “A CodeGenerated Monte Carlo Importance Function,”Trans. Am. Nucl. Soc.,41, 307 (1982). 11)S. Mitake, O. Sato, H. Tsunoda, “Application of Biasing Op timization Techniques to Monte Carlo Shielding Analysis of a Transport Cask,”Proc. MC2000Lisbo, October 2326, 2000, SpringerVerlag, 859 (2001). 12)CASK81: 22Neutron, 18GammaRay Group, P3 Cross Sec tions for Shipping Cask Analysis, DLC23, Radiation Safety Information Computational Center, Oak Ridge National La boratory, (1974). 13)K. Ueki, K. Sasaki, “Radiation Shielding Analysis of a Spent Fuel Transport Cask with an Actual Configuration Model Us ing the Monte Carlo Method Comparison with the Discrete Ordinates Sn Method,”Proc. The 14th Int. Conf. on the Packaging and Transportation of Radioactive Materials(PATRAM’04) (2004). 14)Y. Nakahara, K. Suyama, T Suzaki,Technical Development on Burnup Credit for Spent LWR Fuels, JAERITech 2000071 Japan Atomic Energy Research Institute (JAERI) (2000), [in Japanese].
15)L. M. Petrie, N. F. Landers,KENO V.a: An Improved Monte Carlo Criticality Program with Supergrouping, NUREG/CR02000, Rev.6, Vol.2, Sec. F11, ORNL/NUREG /CSD2/R6, Oak Ridge National Laboratory (ORNL) (1998). 16)Atomic Energy Society of Japan,Radiation Dose conversion Coefficients for Radiation Shielding Calculations: 2004, AESJSCR002 (2004), [in Japanese]. 17)K. Shibataet al., “Japanese Evaluated Nuclear Data Library Version 3 Revision3: JENDL3.3,”J. Nucl. Sci. Technol.,39, 1125 (2002). 18)A. Haghighat, J. C Wagner, “Monte Carlo Variance Reduction with Deterministic Importance Functions,”Prog. Nucl. Energy, 42[1), 25–53 (2003). 3 19)A. Haghighat, J. C. Wagner, “Application of AMCNP to Radiation Shielding Problems,”Proc. the Int. Conf. on Monte Carlo for Radiation Physics, Particle Transport Simulation and Applications, Lisbon, Portugal, October 23–26, (2000). 20)M. W. Mickael, “A Fast, Automated, Semideterministic Weight Windows Generator for MCNP,”Nucl. Sci. Eng.,119, 34 (1995). 21)J. S. Tang, T. J. Hoffman, “Monte Carlo Shielding Analyses Using an Automated Biasing Procedure,”Nucl. Sci. Eng.,99, 329 (1988).
