TRANSPORT MODEL BASED ON 3-D CROSS-SECTION GENERATION FOR TRIGA CORE ANALYSIS

Transcription

1 The Pennsylvania State University The Graduate School College of Engineering TRANSPORT MODEL BASED ON 3-D CROSS-SECTION GENERATION FOR TRIGA CORE ANALYSIS A Thesis in Nuclear Engineering by Nateekool Kriangchaiporn 2006 Nateekool Kriangchaiporn Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of philosophy May 2006

2 The thesis of Nateekool Kriangchaiporn was reviewed and approved* by the following: Kostadin Ivanov Professor of Nuclear Engineering Thesis Advisor Chair of Committee Alireza Haghighat Professor of Nuclear Engineering C. Frederick Sears Senior Scientist Affiliate Professor of Nuclear Engineering Ludmil Zikatanov Assistant Professor of Mathematics Yoursry Azmy Professor of Nuclear Engineering Jack Brenizer Professor of Nuclear Engineering Chair of Nuclear Engineering *Signatures are on file in the Graduate School. ii

3 ABSTRACT This dissertation addresses the development of a reactor core physics model based on 3-D transport methodology utilizing 3-D multigroup fuel lattice cross-section generation and core calculation for PSBR. The proposed 3-D transport calculation scheme for reactor core simulations is based on the TORT code. The methodology includes development of algorithms for 2-D and 3-D cross-section generation. The fine- and broad- group structures for the TRIGA cross-section generation problems were developed based on the CPXSD (Contributon and Point-wise Cross-Section Driven) methodology that selects effective group structure. Along with the study of cross section generation, the parametric studies for S N calculations were performed to evaluate the impact of the spatial meshing, angular, and scattering order variables and to obtain the suitable values for cross-section collapsing of the TRIGA cell problem. The TRIGA core loading 2 is used to verify and validate the selected effective group structures. Finally, the 13 group structure was selected to use for core calculations. The results agree with continuous energy for eigenvalues and normalized pin power distribution. The Monte Carlo solutions are used as the references. iii

4 TABLE OF CONTENTS List of Tables vii List of Figures xii Acknowledgement xiv CHAPTER 1 Introduction Background Research Objectives... 3 CHAPTER 2 Literature Review and Methodology TRIGA Review The Forward Neutron Transport Equation The Multigroup Discrete Ordinates Equations Discrete Ordinate Quadrature Sets Level (Fully) Symmetric (LQ N ) Quadrature Square Legendre-Chebyschev (SLC) Quadrature Set Resonance Treatment Flux Calculator The Bondarenko Method CENTRM Group Structure Selection Methodology Code Description DORT TORT Applications of Discrete Ordinates Method to Criticality Calculations A Sub-Critical C28 and a Critical Assembly The C5G7 MOX Benchmark CHAPTER 3 Cross-Section Generation Methodology Cross Section Generation Procedure and Studies The Weight Function Study The Corner-Material Study Resonance Treatment Study Fine Group Structure Selection Extension of the CPXSD Methodology to Criticality Problem Cross-Section Collapsing and Homogenization Fine- to Broad-Group Collapsing Cross-Section Homogenization Summary CHAPTER 4 Two-Dimensional Cross Section Generation Two-Dimensional Model for Cross-Section Generation Fine Group Structure for TRIGA Fast Range Group Refinement Epithermal Range-Group Refinement iv

5 4.2.3 Thermal Range-Group Refinement Parametric Studies Spatial Mesh, Angular Quadrature, and Scattering Order Studies Qudrature Order Determination Cross-Section Collapsing Fast Range-Group Collapsing Epithermal Range-Group Collapsing Thermal Range-Group Collapsing Two-Dimensional Cross Section Generation for Other Materials Graphite Control Rod Cross-Section Homogenization Summary CHAPTER 5 Three-Dimensional Cross Section Generation Three-dimensional model for Cross-Section Generation for Fuel Element Parametric Studies Spatial Mesh, Angular Quadrature, and Scattering Order Studies Qudrature Order Determination Fine- Group Structure for TRIGA Fast Group Refinement Epithermal-Group Refinement Thermal-Group Refinement Cross-Section Collapsing Fast-Group Collapsing and Axial Nodalization Study Epithermal Energy Range: Thermal Energy Range: Three-Dimensional Cross Section Model for Materials with Non-Fissile Element Control Rod Two-Dimensional vs. Three-Dimensional Cross Sections Two-Dimensional vs. Three-Dimensional Flux Distribution Collapsing Two-Dimensional vs. Three-Dimensional Group Structure Summary CHAPTER 6 Core Simulation Mini-Core Simulation Mesh Size Study Mini-Core Results Coarse Group Study Core loading 2 Simulations Mesh Size Study Core Reflector Thickness Study Core Loading 2 ARI Core Loading 2 ARO Summary v

6 CHAPTER 7 Conclusions and Future Research Conclusions Future Research References APPENDIX A. TORT INPUT SAMPLE FOR TRIGA APPENDIX B. TRIGA FISSION SPECTRUM vi

7 LIST OF TABLES Table 3-1: Results of eigenvalue calculation using MCNP...32 Table 3-2: Reaction rates with continuous energy cross-section library in MCNP...36 Table 3-3: Reaction rates with 238-group cross-section library using the Bondarenko method...37 Table 3-4: Percent deviations from MCNP...37 Table 3-5: Reaction rates with 238-group cross-section library using Flux- Calculator in NJOY...38 Table 3-6: Percent deviations from MCNP...38 Table 3-7: Reaction rates with 238-group cross-section library using CENTRM...39 Table 3-8: Percent deviations from MCNP...39 Table 3-9: Reaction rates with 238-group cross-section library using Flux Calculator in NJOY for U Table 3-10: Percent deviations from MCNP...41 Table 3-11: Reaction rates with 238-group cross-section library using Centrm treatment for Zr and U Table 3-12: Percent deviations from MCNP...42 Table 3-13: Reaction rates with 238-group cross-section library using Centrm treatment in Zr, U 238, and Fe Table 3-14: Percent deviations from MCNP...43 Table 3-15: Reaction rates with 238-group cross-section library, cells:...44 Table 3-16: Percent deviations from MCNP...45 Table 3-17: Reaction rates with 253-group cross-section library...46 Table 3-18: Percent deviations from MCNP...46 Table 4-1: Material density of the fuel elements...53 Table 4-2: Cladding composition...54 Table 4-3: Fine groups selected in the fast energy range...55 Table 4-4: Eigenvalue results of fine group energy for 8.5% wt. case...57 Table 4-5: Eigenvalue results of fine group energy for 12% wt. case...57 Table 4-6: Fine groups generated in the epithermal energy range...58 Table 4-7: Eigenvalue results of fine group energy...59 Table 4-8: Eigenvalue results of fine group energy...59 Table 4-9: Reaction rate comparison for 8.5% wt. case...59 Table 4-10: Reaction rate comparison for 12% wt. case...59 Table 4-11: Fine groups generated in the thermal energy range...60 Table 4-12: Eigenvalue results for fine group energy...61 Table 4-13: Eigenvalue results for fine group energy...62 Table 4-14: Reaction rate comparison of 8.5% wt. case...62 Table 4-15: Reaction rate comparison of 12% wt. case...62 Table 4-16: DORT results with 280-energy group XS and 1554 cells, Level- Symmetric...65 Table 4-17: DORT results with 280-energy group XS and 6132 cells, Level- Symmetric...65 vii

8 Table 4-18: DORT results with 280-energy group XS and cells, Level- Symmetric...65 Table 4-19: DORT results with 280-energy group XS and cells, Level- Symmetric...66 Table 4-20 DORT results with 280-energy group XS and 1554 cells, Level- Symmetric...66 Table 4-21: DORT results with 280-energy group XS and 6132 cells, Level- Symmetric...66 Table 4-22: DORT results with 280-energy group XS and cells, Level- Symmetric...67 Table 4-23: DORT results with 280-energy group XS and cells, Level- Symmetric...67 Table 4-24: DORT results with 280-energy group XS and 1554 cells, SLC...69 Table 4-25: DORT results with 280-energy group XS and 6132 cells, SLC...70 Table 4-26: DORT results with 280-energy group XS and cells, SLC...70 Table 4-27: DORT results with 280-energy group XS and cells, SLC...70 Table 4-28: DORT results with 280-energy group XS and 1554 cells, SLC...71 Table 4-29: DORT results with 280-energy group XS and 6132 cells, SLC...71 Table 4-30: DORT results with 280-energy group XS and cells, SLC...71 Table 4-31: DORT results with 280-energy group XS and cells, SLC...72 Table 4-32: DORT results with 280-energy group XS and 6132 cells, SLC...74 Table 4-33: Neutron-production reaction rates from MCNP and DORT...78 Table 4-34: Percentage of relative deviation from MCNP...78 Table 4-35: Comparison between 229G and 280G for 8.5% wt. case...80 Table 4-36: Comparison between 229G and 280G for 12% wt. case...81 Table 4-37: k inf comparison between 229G and 127G for 8.5% wt. case...81 Table 4-38: k inf comparison between 229G and 127G for 12% wt. case...82 Table 4-39: Reaction rate comparison between 229G and 127G for 8.5% wt. case...82 Table 4-40: Reaction rate comparison between 229G and 127G for 12% wt. case...82 Table 4-41: Result comparison in thermal energy range for 8.5% wt. case...83 Table 4-42: Result comparison in thermal energy range for 12% wt. case...84 Table 4-43: DORT results with 280-energy group XS and 6132 cells for 8.5% wt. case...85 Table 4-44: DORT results with 12-energy group XS, 6132 cells for 8.5% wt. case...85 Table 4-45: DORT results with 280-energy group XS and 6132 cells for 12% wt. case..86 Table 4-46: DORT results with 12-energy group XS, 6132 cells for 12% wt. case...86 Table 4-47: DORT calculation with 280-group cross section library for 8.5% wt. case...87 Table 4-48: DORT calculation with 12-group cross section library for 8.5% wt. case...88 Table 4-49: Reaction rates deviation between 280G and 12G for 8.5% wt. case...88 Table 4-50: DORT calculation with 280-group cross section library for 12% wt. case...89 Table 4-51: DORT calculation with 12-group cross section library for 12% wt. case...89 Table 4-52: Reaction rates deviation between 280G and 12G for 12% wt. case...90 Table 4-53: Eigenvalue results for graphite cross section generation model...92 Table 4-54: MCNP reaction rates...92 Table 4-55: DORT, 280GP3 reaction rates...92 Table 4-56: DORT, 280GP1 reaction rates...93 viii

9 Table 4-57: DORT, 12GP1 reaction rates...93 Table 4-58: Percentage deviation between DORT, 280GP3 and MCNP...93 Table 4-59: Percentage deviation between DORT, 280GP1 and MCNP...94 Table 4-60: Percentage deviation between DORT, 12GP1 and MCNP...94 Table 4-61: Percentage deviation between DORT, 12GP1 and 280GP Table 4-62: Eigenvalues calculated by DORT and MCNP...96 Table 4-63: Reaction rates calculated by MCNP...96 Table 4-64: The reaction rates calculated by DORT with 280 groups, S10 quadrature order...96 Table 4-65: Percent deviation of reaction rates between DORT 280G and MCNP...97 Table 4-66: Percent deviation of reaction rates between DORT 280GP1 and 280GP Table 4-67: K inf results predicted by DORT with 280G...99 Table 4-68: Reaction rates calculated by DORT with 280 groups, S10 quadrature order for Model# Table 4-69: Reaction rates calculated by DORT with 280 groups, S10 quadrature order for Model# Table 4-70: Percent deviation of reaction rates between DORT 280G S10 Model #2 and MCNP Table 4-71: Percent deviation of reaction rates between DORT 280G S10 Model #3 and MCNP Table 4-72: Eigenvalues calculated by DORT and MCNP Table 4-73: Reaction rates calculated by DORT with 12 groups, S10 quadrature order and P3 scattering order Table 4-74: Percent deviation of reaction rates between DORT 12GP3 and 280GP Table 4-75: K inf calculated by DORT (3 region combination) Table 4-76: K inf calculated by DORT (4 region combination) Table 5-1: Material density of the fuel elements Table 5-2: TORT results with 238-energy group XS, Level-Symmetric Table 5-3: TORT results with 238-energy group XS, SLC Table 5-4: TORT results with 238-energy group XS, S8 (SLC), P Table 5-5: TORT results with 238-energy group XS, S8 (SLC), P Table 5-6: TORT results with 238-energy group XS and 48x59x55 cells Table 5-7: Neutron production reaction rate and percent deviations Table 5-8: Fine groups generated in the fast energy range Table 5-9: Eigenvalue results of fine group energy for 8.5% wt. case Table 5-10: Fine groups generated in the epithermal energy range Table 5-11: Eigenvalue results of fine group energy Table 5-12: Reaction rate comparison for 8.5% wt. case Table 5-13: Fine groups generated in the thermal energy range Table 5-14: Eigenvalue results for fine group energy Table 5-15: Reaction rate comparison of 8.5% case Table 5-16: Group structure of the 280 fine groups Table 5-17: Number of groups for each energy range Table 5-18: Eigenvalue results for 3D, 8.5% fuel cell Table 5-19:The minimum and maximum of mesh-wise reaction rate deviations for each layer between case 2 and case ix

10 Table 5-20:The minimum and maximum of mesh-wise reaction rate deviations for each layer between case 3 and case Table 5-21: Number of groups for each energy range Table 5-22: Eigenvalue results for 3D, 8.5% fuel cell Table 5-23: Number of groups for each energy range Table 5-24: Eigenvalue results for 3D, 8.5% fuel cell Table 5-25: Reaction rate comparison for broad group in epithermal range Table 5-26: Number of groups for each energy range Table 5-27: Eigenvalue results for 3D, 8.5% fuel cell Table 5-28: Result comparison in thermal energy range Table 5-29: Energy boundaries of 26-group structures Table 5-30: Eigenvalues calculated by TORT and MCNP Table 5-31: MCNP calculation with continuous cross section library Table 5-32: TORT calculation with 280-group cross section library Table 5-33: TORT calculation with 26-group cross section library Table 5-34: Reaction rates deviation between 280G and MCNP Table 5-35: Reaction rates deviation between 26G and MCNP Table 5-36: Reaction rates deviation between 26G and 280G Table 5-37: Eigenvalues calculated by TORT and MCNP Table 5-38: Reaction rates calculated by MCNP Table 5-39: Reaction rates calculated by TORT with 26 groups, S8 quadrature order and P1 scattering order Table 5-40: Reaction rates calculated by TORT with 26 groups, S8 quadrature order and P3 scattering order Table 5-41: Percent deviation of reaction rates between TORT 26GP1 S8 and MCNP.146 Table 5-42: Percent deviation of reaction rates between TORT 26GP3 S8 and MCNP.146 Table 5-43: Eigenvalue results 2-D vs 3-D flux distribution collapsing cases Table 5-44: Percentage deviation of reaction rates between 2-D and 3-D cross-section collapsing cases Table 5-45: Number of groups placed in each energy range Table 5-46: Eigenvalue results 2-D vs 3-D group structure cases Table 5-47: Percentage deviation of reaction rates between 2-D and 3-D group structure cases Table 6-1: Eigenvalues calculated from TORT Table 6-2: Percentage deviation of reaction rates between 2 nd model and 1 st model Table 6-3: Percentage deviation of reaction rates between 3 rd model and 1 st model Table 6-4: Percentage deviation of reaction rates between 4 th model and 1 st model Table 6-5: Percentage deviation of reaction rates between 5 th model and 1 st model Table 6-6: Eigenvalues calculated from TORT Table 6-7: Percentage deviation of reaction rates between 2 nd model and 1 st model Table 6-8: Percentage deviation of reaction rates between 3 rd model and 1 st model Table 6-9: Eigenvalues calculated from TORT and MCNP Table 6-10: MCNP reaction rates Table 6-11: TORT reaction rates for P1 case Table 6-12: TORT reaction rates for P3 case Table 6-13: Percentage deviation of reaction rates between Tort-P1 and MCNP x

11 Table 6-14: Percentage deviation of reaction rates between Tort-P3 and MCNP Table 6-15: Eigenvalues calculated from TORT Table 6-16: Percentage deviation of reaction rates between 12G and 26G cases Table 6-17: Energy boundaries of 12-group structures Table 6-18: Eigenvalues calculated by TORT Table 6-19: Percentage deviation of reaction rates between 13G and 26G cases Table 6-20: Energy boundaries of 13-group structure Table 6-21: Eigenvalues calculated by TORT Table 6-22: Percentage deviation of reaction rates between 2 nd model and 1 st model Table 6-23: Percentage deviation of reaction rates between 3 rd model and 1 st model Table 6-24: Percentage deviation of reaction rates between 4 th model and 1 st model Table 6-25: Eigenvalues calculated from TORT Table 6-26: Percentage deviation of reaction rates between 2 nd model and 1 st model Table 6-27: Percentage deviation of reaction rates between 3 rd model and 1 st model Table 6-28: Percentage deviation of reaction rates between 4 th model and 1 st model Table 6-29: Eigenvalues calculated by TORT Table 6-30: Percentage deviation of reaction rates between 2 nd model and 1 st model Table 6-31: Percentage deviation of reaction rates between 3 rd model and 1 st model Table 6-32: Percentage deviation of reaction rates between 4 th model and 1 st model Table 6-33: Eigenvalues calculated from TORT Table 6-34: Percentage deviation of reaction rates between 2 nd model and 1 st model Table 6-35: Percentage deviation of reaction rates between 3 rd model and 1 st model Table 6-36: Eigenvalues calculated by TORT Table 6-37: Eigenvalues calculated from TORT and MCNP Table 6-38: Eigenvalues calculated from TORT and MCNP xi

12 LIST OF FIGURES Figure 2-1 : Core cycle Figure 3-1: Procedure for generating cross section library...28 Figure 3-2: Unit cell for TRIGA fuel element...30 Figure 3-3: MCNP-predicted TRIGA spectrum...31 Figure 3-4: Cells Models for MCNP...32 Figure 3-5: Flux distribution in fuel Region...33 Figure 3-6: Pointwise absorption cross section of Zr...34 Figure 3-7: Pointwise absorption cross section of U Figure 3-8: Mesh Model from 1554 cells to cells...44 Figure 3-9 Fuel cell homogenization...50 Figure 4-1: Cross section generation model...53 Figure 4-2: Importance of groups of 238G and 246G libraries...56 Figure 4-3: Importance in groups of 246G library...58 Figure 4-4: Importance in groups of 246G, 254G and 280G libraries...61 Figure 4-5: P1 scattering order with level symmetric quadrature order...68 Figure 4-6: P3 scattering order with level symmetric quadrature order...68 Figure 4-7: P1 scattering order with Square Legendre-Chebyshev quadrature order...72 Figure 4-8: P3 scattering order with Square Legendre-Chebyshev quadrature order...73 Figure 4-9: Flux distribution of group 23 rd...75 Figure 4-10: Flux distribution for group 242 nd...76 Figure 4-11: Detector locations...77 Figure 4-12: 2-D model for graphite XS generation...91 Figure 4-13: 2-D model for control rod XS generation...95 Figure 4-14: Absorption reaction rate as a function of B 4 C radius...98 Figure 4-15: Three-Region Homogenization Figure 4-16: Four-Region Homogenization Figure 5-1: 3D cross section generation model Figure 5-2 Eigenvalue behavior under variation of scattering order and level symmetric quadrature Figure 5-3 Eigenvalue behavior under variation of scattering order and Square Legendre-Chebyshev quadrature Figure 5-4: Eigenvalue behavior with different radial-mesh model Figure 5-5: Eigenvalue behavior with different axial-mesh model Figure 5-6: Flux distribution for each quadrature order Figure 5-7: Importance in groups of 238G and 246G libraries Figure 5-8: Importance in groups of 246G libraries Figure 5-9: Importance in groups of 246G, 254G and 280G libraries Figure 5-10 Axial mesh size used in nodal length collapsing study Figure 5-11: 3-D model for control rod XS generation Figure 5-12: A pin cell model in axial direction Figure 6-1: Configuration of Mini-core Figure 6-2: Importance distribution of 26-group structure Figure 6-3: TRIGA core loading xii

13 Figure 6-4: Pin cell in axial direction Figure 6-5: The studied models Figure 6-6: The studied models Figure 6-7: Core loading 2 with 15 cm reflector thickness Figure 6-8: Radial-cross-section view of ARI Figure 6-9: Axial-cross-section view of ARI Figure 6-10: Normalized pin-power distribution for ARI Figure 6-11: Radial-cross-section view of ARO Figure 6-12: Axial-cross-section view of ARO Figure 6-13: Normalized pin-power distribution for ARO Figure B-1: 8.5% wt. fuel fission spectrum of 280 groups Figure B-2: 8.5% wt. fuel fission spectrum of 26 groups 196 Figure B-3: 8.5% wt. fuel fission spectrum of 13 groups 197 Figure B-4: 12% wt. fuel fission spectrum of 280 groups Figure B-5: 12% wt. fuel fission spectrum of 26 groups. 198 Figure B-6: 12% wt. fuel fission spectrum of 13 groups. 198 xiii

14 ACKNOWLEDGEMENTS I express deep thanks to my family, especially my father, Kriang Kriangchaiporn, and my mother, Sangiam Kriangchaiporn, for their love, prayer, support and encouragement throughout long journey of my study. I also would like to express my gratitude to my academic advisor, Dr. Kostadin Ivanov, for his motivation, enthusiasm and guidance, which played a major role in the successful completion of the work. I would like to especially thank Dr. Frederick Sears for all his questions during the meetings, which have been very useful for this research project. I gratefully acknowledge for all the suggestions of the committee, Dr. Alireza Haghighat, Dr. Yoursry Azmy, and Dr. Ludmil Zikatanov. I would like to thank the Radiation Science and Engineering Center (RSEC) for financial support of this research project. I would like to extend my thanks to my best friend, Dr. Sathaporn Opasanon, my wonderful roommates Hathairat Maneetes (Bell) and Tianboon Soh (Chris) for their friendship, help, comfort and always being there whenever I needed. Lastly, I thank God who has provided me with all opportunities and blessings. xiv

15 CHAPTER 1 Introduction The demand for accurate simulations of nuclear reactors is increasing to enable improving the reactor design, safety and economy. The computer simulation of a reactor core is an important aspect of both designing new reactors and analyzing the safety of existing reactors. Innovative three-dimensional (3-D) core models are necessary to achieve the desired accuracy. Since these types of numerical simulations tend to be computationally expensive, further developments are needed to address both accuracy and efficiency. Recent progress in computer technology combined with new methods and code developments makes feasible new calculation schemes capable of providing accurate solutions in an efficient manner. 1.1 Background Generally, the reactor core physics calculation process contains two main steps. The first step is to compute the group cross sections for the various regions of a nuclear reactor. The second is to employ these cross sections by using varying methods to analyze the reactor core. This modeling approach is applied to both steady state and transient calculations. Most of the core analysis methodologies utilize approximate methods to simplify the complex problems associated with reactor core modeling. The cross sections are generated in two-dimensional (2-D) instead of three-dimensional (3-D) geometries. In addition, the diffusion theory methodology, which is derived from a transport equation, is used to analyze the full core using the 2-D cross-section library. These approaches have three main weaknesses. The first weakness is in the cross-section 1

16 generation. Current lattice physics codes suffer from a combination of some of the following shortcomings: 2-D geometry approximation, shape leakage approximation, and approximated self-shielding trestment for the burnable-poison-containing fuel rods. The reason for the aforementioned shortcomings is in the fact that lattice physics codes are generally based on the collision probability method (CPM). This method is practical only in 2-D geometries, since it becomes cumbersome and impractical in 3-D geometries for arbitrary boundary conditions, combined geometric shapes, or where detailed information of problems is required. The second weakness is in the current cross-section modeling approach, which is based on cross-section parameterization and functionalization techniques. These techniques cause uncertainties in the evaluated cross sections from the cross-section libraries, which are used later in the core simulations. The third weakness is the diffusion approximation in the full core calculations. This approximation is not accurate at the interfaces between different dissimilar assemblies. Transport effects are of particular importance in highly heterogeneous cores where the traditional procedure of applying various transport corrections is unsatisfactory. In order to achieve the desired accuracy, a three-dimensional (3-D) model based on the exact transport theory method is necessary to simulate the real problems. In the past, 3-D numerical simulations were computationally expensive and impractical. However, recent advancements in computer technology, combined with new methods and code developments makes it feasible to develop novel calculation schemes capable of providing accurate solutions in an efficient manner. The Pennsylvania State University Breazeale Reactor (PSBR) is a TRIGA Mark III research reactor designed for 1 MWt power generation. It is a light water cooled, pool 2

17 type reactor, which utilizes U-ZrH 20% enriched fuel elements containing 8.5 wt% and 12 wt% uranium [Ref.14]. The uniform lattice in PSBR has a hexagonal shape and the PSBR core has relatively small dimensions as compared to the commercial light water reactors (LWRs). The other differences from LWRs especially those affecting the neutronics characteristics are as follows: TRIGA is an over-moderated reactor, where the majority of neutron moderation occurs in the fuel meat (UZrH), TRIGA core has more pronounced upscattering effects, due to the Zr-H mixture crystalline structure, which cause hardening of the neutron spectrum as compared to LWRs The above-mentioned facts show the uniqueness of this reactor type and require the development of its own core analysis methodology and cross-section library. Since TRIGA has a relatively small reactor core, it s modeling even using higher order transport methods does not require prohibitly large computer resources. This fact along with the availability of measured data makes TRIGA core an appropriate test environment for testing new methodologies. 1.2 Research Objectives This research addresses the development of a state-of-the-art reactor core physics model based on 3-D transport methodology utilizing 3-D multigroup fuel lattice crosssection generation and core calculation. The focus of the proposed research is a new methodology for enhanced core physics simulation of the PSBR. The proposed 3-D transport calculation scheme for reactor core simulations is based on the TORT code. The complete methodology includes development of 3

18 algorithms for 3-D cross-section generation and modeling. This will solve several major weaknesses of the current reactor core analysis methodology (the diffusion approximation of the whole core calculations), the shortcomings of generation of multigroup cross section, and the approximations introduced with cross-section parameterization and functionalization. In fact, in this research instead of proposing incremental improvements to the TRIGA (research reactor) analysis methodology, the performed research results in a new generation of reactor core analysis methods. The objectives of the proposed research are formulated as development and implementation of (a) An efficient transport method for 3-D reactor core simulation for steady-state calculations. (b) An innovative algorithm for 3-D cross-section generation and modeling. (c) An effective group structure for the TRIGA reactor. (d) A systematic validation of the new calculation scheme against Monte Carlo results for the PSU TRIGA reactor. The expected outcome of the above-described objectives is the development of new methodology for accurate 3-D reactor core analysis in an efficient manner. This methodology will be validated for the TRIGA reactor and can be later expanded for power reactor applications. Increasing the accuracy and efficiency of core analysis methodologies can directly improve both safety and economy of nuclear power reactors. 4

19 CHAPTER 2 Literature Review and Methodology This chapter describes the TRIGA reactor and several research studies that have been performed for analysis of this reactor. The linear Boltzmann equation in multigroup form is presented along with the discrete ordinates method used for its solution. Two types of quadrature order techniques are discussed: i) level (full) symmetry, and ii) Legendre-Chebyshev. Several techniques of resonance treatment are explained for accounting of self-shielding effect in cross sections. Finally, the group-structure selection to generate a multigroup cross section library is discussed. 2.1 TRIGA Review The PSBR is a TRIGA Mark III research reactor manufactured by General Atomic. It has been operated since 1965, when the core was upgraded from MTR type fuel. The PSBR is a light water cooled, pool type reactor designed for 1 MW (t) steadystate power operation (up to 2000 MW when pulsing) with natural circulation cooling. It is used for experimental, training, educational and service purposes. The PSBR core system was first loaded in 1965 with only 8.5 % wt ZrH x -U fuel. Since July 1972, the core has been reloaded with fresh 12 % wt ZrH x -U fuel elements, six at each reload. Currently, the PSBR is operated using core cycle 52 as shown in Figure 2-1. The uniform lattice in PSBR is formed in hexagonal shape. The center of the core is the location of the central thimble (the water rod), which is surrounded by hexagonal rings. The rings running from the center outward are designated B, C, D, E, F and so on, respectively. There are 102 fuel rods; 34 of them are 12 % wt. and 68 of them are 8.5 % wt. both with a 20% uranium-235 enrichment. Three control rods (shim, regulating and safety) are fuel follower control rods driven by motor. They are composed of graphite at 5

20 the top and bottom, fuel and absorber (borated graphite) are in the middle. The fourth control rod is the transient rod (air rod), the only control rod without fuel material driven by an electro-pneumatic during the steady state. The neutron source used in PSBR is a 3- Curie americium-beryllium (Am-Be) neutron source doubly encapsulated in type 304L stainless steel. A B C D E F G H I air air SA SH RR TR 8.5 wt % 12 wt% C.R. Source Figure 2-1 : Core cycle 52 6

21 In the past, the TRIGA core management model (TRICOM) [Ref.18] was developed based on old codes like PSU-LEOPARD, EXTERMINATOR2 and MCRAC. The core fuel management plan of PSBR has been developed and verified based on TRICOM during the years by the researchers and staff of the reactor for fuel management and safety analyses. However, these outdated tools have modeling limitations that introduce large uncertainties in the calculated parameters. Subsequently, the calculated results have to be normalized to the measured data in order to be used for analysis. In 1994, analytical models of the TRIGA core configuration based on the Monte Carlo Method were developed and applied in the framework of Y.S. Kim s Master thesis [Ref.20]. The reactor core power distribution was examined using MCNP code for criticality simulations and ORIGEN2 for the depletion calculations. The results indicated that a maximum of 21% of the U 235 was depleted in 8.5% fuel rods, and a maximum of 15% of the U 235 was depleted in 12% fuel rods. In the analyzed core configuration, the power peaking factor was extremely high, but it can be reduced by using a proper core configuration. Thus, the improvement of the core configuration was investigated with a goal to gain lower peaking factor (lower maximum temperature) and minimal change of reactivity relative to previous configuration. However, the Monte Carlo based calculation method is not practical for routine use because it is very time-consuming, but it can be used to generate reference results for verification of the more efficient deterministic codes. In 2000, a new Advanced Fuel Management System (AFMS) [Ref.14] was developed based on the HELIOS lattice-physics code and the multi-dimensional nodal diffusion code ADMARC-H. The modeling deficiencies of the old TRICOM code system 7

22 are corrected on both levels; the cross-section generation and the core simulation. The HELIOS code was used to generate the cross-section library. HELIOS improved the geometry modeling by explicitly modeling the hexagonal unit cell, and therefore allowing for a better thermalization model. The transport theory and CCCP methods in HELIOS are superior to slowing down theory approximations in LEOPARD, especially in the case of TRIGA, which uses a hydride fuel. The ADMARC-H code uses a 3-D full core hexagonal geometry and a 3-D macroscopic semi-implicit burnup model. It yields more accurate results than those predicted by the 2-D finite-difference MCRAC code in onequarter rectangular core geometry. The cross-section generation and modeling in the aforementioned research was developed in 2-D geometry approximation and using off-line calculations. Furthermore, the diffusion approximation in the full core calculations is the cause of degradation in accuracy at the interfaces between different regions. In summary, the following shortcomings of the current core analysis methodology have to be addressed: the use of diffusion approximation of the whole core calculations, 2-D cross-section generation and depletion, and cross-sections parameterization. Hence, we further develop new algorithms and methods for fuel management based on 3-D transport theory. These methodologies can be applied for accurate determination of flux/power distribution and isotropic depletion of PSBR in an efficient manner, which is the goal for this research. 8

23 2.2 The Forward Neutron Transport Equation The neutron transport equation is given by the linear form of the Boltzmann equation. The linear form is derived by ignoring neutron-neutron interactions. The timeindependent neutron transport equation with no external source is given below [Ref.5]. Ωˆ v Ψ Ωˆ v v ( r, E, ) + σ ( r, E) Ψ( r, E, Ωˆ ) = t de 0 4π Ω ˆ v Ω ˆ Ωˆ v d σ ( r, E E, ) Ψ( r, E, Ω ˆ ) s χ( E) + 4π v de νσ ( r, E ) f 0 4π v dω Ψ( r, E, Ω ˆ ) Equation 2-1 The terms on the left hand side of Equation 2-1 represents the loss, and the right hand side represents the gain of the neutrons in a phase space. Each term is explained [Ref.5] as follows. Streaming Term: Ω ˆ v Ψ( r, E, Ωˆ ) dedωdv This term gives the flow of neutrons. Ωˆ is the unit vector that gives the direction of a particle and Ψ( r v, E, Ωˆ ) is the angular flux. Angular flux is defined as the expected rate of particles crossing a d 2 r at position r v, with energies between E and E+dE, traveling in directions Ωˆ d about Ωˆ. v v Collision Term: σ ( r, E) Ψ ( r, E, Ω ˆ ) dedωdv t This term gives the removal rate of neutrons due to all types of interactions in a volume element d 3 r, about r v, with energies between E and E+dE, traveling in directions d Ωˆ about Ωˆ. Interactions include scattering (elastic and inelastic) and 9

24 v absorption ((n,f),(n,2n),(n,p),(n,γ),etc.). σ ( r, E) is the total interaction macroscopic t cross section at position r v and energy E. It gives the probability per unit length that a neutron will have an interaction of any type. Scattering Term: de dω ˆ v v σ s ( r, E E, μ ) Ψ( r, E, Ω ˆ dedωdv 0 4π 0 ) This term gives the rate of scattering of particles (in a volume element d 3 r, about r v, with energy between E and direction Ω ˆ ) into energies between E and E+dE, traveling in directions d Ωˆ about Ωˆ, in dv about r v. Integration is performed over all v incoming energies and directions. σ r, E E, μ ) is the macroscopic differential s ( 0 scattering cross section and defines the probability per unit length that neutrons, at position r v, energy E, direction Ω ˆ are scattered into de about E, and d Ωˆ about Ωˆ. Note that the scattering cross section does not depend on the initial and final directions separately, but rather on the angle between the incident and emerging particle (i.e., μ = Ω ˆ ˆ ). 0 Ω Fission Term: χ( E) v v de νσ f ( r, E ) Φ( r, E ) dedωdv 4π 0 This term gives the rate of fission neutrons generated in de about E, Ωˆ d about Ωˆ, in dv about r v. χ(e) is the fraction of fission neutrons emitted per unit energy. ν is the average number of neutrons emitted per fission event. σ ( r v, E ) is the macroscopic f v fission cross section. The scalar flux is formulated as Φ = Ω Ψ ˆ v ( r, E ) d ( r, E, Ω ˆ ). 4π 10

25 The Multigroup Discrete Ordinates Equations In order to solve the transport equation with a deterministic computational method, discretization of the energy, angular and spatial variables are applied to the transport equation (Equation 2-1). First, we present the multigroup equations for timeindependent criticality or eigenvalue problems. It is derived by integrating the linear Boltzmann equation over each energy interval g, as given below. ) ˆ,, ( ), ( ) ˆ,, ( ˆ Ω Ψ + Ω Ψ Ω E r E r de E r de t g g v v v σ = Ω Ψ Ω = g s G g g E r E E r d de de π μ σ ) ˆ,, ( ),, ( ˆ v v = Φ + g f G g g E r E r de E de ), ( ), ( ) ( v v νσ χ π Equation 2-2 Equation 2-2 is re-written by preserving reaction rates in each term: = Ω Ψ Ω = Ω Ψ + Ω Ω Ψ G g g g g s g t g r r d r r r 14 0, ) ˆ, ( ), ( ˆ ) ˆ, ( ) ( ) ˆ, ( ˆ π μ σ σ v v v v v = Φ + G g g g f g r r 1, ) ( ) ( 4 v v νσ π χ Equation 2-3 The group flux ) ˆ, ( Ω Ψ r g v is defined as: Ω Ψ = Ω Ψ g g E r de r ) ˆ,, ( ) ˆ, ( v v Equation 2-4 The total, scattering and fission group constants are given by Equations 2-5, 2-6 and 2-7, respectively.

26 v v deσ t ( r, E) Ψ( r, E, Ωˆ ) v g σ t, g ( r ) dω = v Equation 2-5 dω deψ( r, E, Ωˆ ) de s v g g s, g g ( r, μ0 ) dω = v g v v de σ ( r, E E, μ ) Ψ( r, E, Ω ˆ ) σ Equation 2-6 dω de Ψ( r, E, Ω ˆ ) f v g f, g ( r) = v g g v v de νσ ( r, E ) Φ( r, E ) νσ Equation 2-7 de Φ( r, E ) 0 Finally, the group fission spectrum is defined in Equation 2-8. χ = de χ( E) Equation 2-8 g g The Discrete Ordinate Method (Sn) is one of the most widely used techniques to solve the Linear Boltzmann equation in terms of discretization of the angular variable. In this method, the Boltzmann equation is solved for a number of discrete directions Ωˆ m, to each of which is associated a weight w m. Each weight represents a segment or area ΔΩ ˆ m on the unit directional sphere. Normally, these areas are expressed in units of 4π, so that w m ˆ m ΔΩ = and 4π wm = 1 Equation 2-9 m 12

27 The Boltzmann equation for arbitrary direction Ωˆ m is given by Ωˆ m Ψ v ( r, Ωˆ m v v ˆ v ) + σ ( r,) Ψ( r, Ω ) = q( r, Ωˆ ), Equation 2-10, t m m v where the group index g is suppressed and q r, Ωˆ ) ( m includes scattering from other energy groups, scattering at the given energy from other directions, fission and any other particle sources. 2.4 Discrete Ordinate Quadrature Sets The choice of ordinate sets for discrete ordinates approximation is one of the major modeling methods used to apply the S N method for solving different problems. There are several techniques for the generation of discrete ordinates and associated weights. Here, we present two techniques of discrete quadrature orders, level (fully) symmetric quadrature (LQ N ) and Square Legendre-Chebyshev quadrature (SLC) Level (Fully) Symmetric (LQ N ) Quadrature Level symmetric quadratures are used for general applications. Full symmetry requires that Ωˆ be invariant under all 90 rotations about any axis. Hence, each set of coordinates must be symmetric with respect to the origin and the set of points on each axis must be the same. For N levels, the total of N(N+2) directions are on the unit sphere (N(N+2)/8 per octant) with the same set of N/2 positive values of the direction cosines with respect to each of the three axes. There is only one degree of freedom in determining the direction cosines of the ordinates, the choice of μ 1. Then, the other values of μ n are i + j k determined based on Equation 2-11 by considering μ η + ξ = 1 and 13

28 + + = N i j k 2 + 2, where N refers to the number of levels and i,j,k are indices for direction cosines. 2 i 2 1 μ = μ + ( i 1) Δ Equation (1 3μ1 ) where Δ = and 2 i ( N 2) 2 N 2 ;0 < μ 2 1 The weights associated with directions are obtained as follows: 1 3 M w i i= 1 = 1.0 Equation 2-12 M i= 1 n i M w μ = w η = w ξ = 0.0 for n odd Equation 2-13 i i= 1 i n i M i= 1 i n i M M M n n n 1 wi μ i = wiη i = wiξi = for n even Equation i= 1 i= 1 i= 1 n + Equation 2-12 is a normalization condition for the weights, Equation 2-13 and Equation 2-14 represent the odd-moment and even-moment conditions, respectively. The odd-moment condition is automatically satisfied over the entire range of μ because of symmetry. The even-moment condition in Equation 2-14 is required in order to properly integrate the Legendre polynomials. This technique is limited to order 20, because beyond this order some of weights become negative Square Legendre-Chebyschev (SLC) Quadrature Set The SLC methodology has been derived in order to relax the constraints imposed by the LQ N method. The ξ levels are set on the z-axis equal to the roots of Legendre 14

29 polynomials and the azimuthal angles on each level are calculated by the roots of the Chebyschev polynomials. Points lie on the unit sphere on ξ levels but not on μ or η levels, and point weights are the product of Legendre and Chebyschev weights. The use of the same Chebyschev quadrature on each ξ level gives μ and point weights as the following formulation: μ 2 i0 1 ξi = p i = 0 μ ij = 1 ξ 2 i 2n 2 j + 1 cos( π ) 2n wi pi = Equation 2-15 n i = 1,2,,n/2 and j = 1,2,,n The μ points with zero weights are those incoming directions used as starting directions in the current version of the S N discrete ordinates transport code. For the same ξ i and w i the use of a different order Chebyschev quadrature on each ξ level gives μ and point weights in Equation μ 2 i0 1 ξi = p i = 0 μ ij = 2 2n 4i 2 j ξi cos( π ) 2n 4i + 4 p i wi = Equation 2-16 n + 2 2i i = 1,2,,n/2 and j = 1,2,,(n+2-2i) This SLC technique gives a significant improvement over level symmetric quadrature, since the SLC quadraure set completely satisfies the even-moment condition for all axes. It is another option to study the effect of types of quadrature set on our problem. 15

30 2.5 Resonance Treatment In the resonance energy region, from roughly 1 ev to 100 kev, the main absorption of neutrons by heavy nuclei takes place at pronounced peaks or resonances of cross section. The shielding effects are presented in this region because of the flux dip at resonances. The resonance structure can be separated into two regions, resolved and unresolved. In resolved resonance region, the resonances are wide when compared to the scattering ranges for the mixtures in a particular configuration. It is in the range of ev up to a few kev. This region is significant for thermal reactors. In the unresolved resonance region, the resonances are not able to achieve adequate resolution of the individual resonances. The neutron absorption in this region is important for fast reactors. An appropriate treatment of the resonance absorption is needed in order to obtain more accurate solutions. The three selected methods for resonance shielding treatment are explained as follows Flux Calculator The narrow resonance approach is quite useful for practical fast reactor problems. However, for nuclear systems sensitive to energies from 1 to 500 ev, there are many broad- and intermediate-width resonances, which cannot be self-shielded with sufficient accuracy using the Bondarenko approach. The flux calculator option of GROUPR module in NJOY is designed to solve such problems. The infinite-medium neutron spectrum equation is expressed as 0 Σt ( E) Φ( E) = de' Σ s ( E' E) Φ( E') + S( E) Equation

31 where the term on the left hand side of Equation 2-17 represents the collision, the integral on the right hand side is the scattering source, and S(E) the external source. Next, Equation 2-17 is written considering a homogeneous medium consisting of two materials: an absorber and a moderator, represented by A and M, respectively in Equation Elastic scattering cross sections that are isotropic in the center of mass are used. Neutron slowing down in a single resonance of the absorber material is assumed. Σ ( E) Φ( E) = t E / α E M M Σ s ( E') de' Φ( E') + (1 α ) E' M E / α E A A Σ s ( E') de' Φ( E') (1 α ) E' A Equation 2-18 where α M and α A are the moderator and absorber collision parameters, respectively, defined as: A 1 α = A Equation 2-19 where A is the atomic mass in Equation 2-19 The following approximations are introduced to Equation 2-18 : The moderator scattering cross-section is assumed to be constant and equal to the potential scattering cross-section: i.e. M Σ ( ') s E = Σ M p The moderator absorption cross-section is assumed to be negligible; i.e. M Σ ( ') t E = Σ M p The narrow resonance approximation is used for the moderator. This states that the resonance width is very small compared to the energy loss from scattering with the moderator nucleus. Therefore, the flux distribution is the moderator integral is 17

32 assumed to have an asymptotic form. In general, the moderator integral is assumed to be a smooth function of energy represented as C(E). The moderator is assumed to represent all nuclides other than the absorber. This enables the inclusion of the dilution microscopic cross-section of the absorber, σ o, in Equation The dilution (or background) cross section of an isotope i is defined to be all cross sections representing isotopes other than the isotope i. The dilution cross-section is a measure of energy self-shielding. It determines the significance of a resonance compared to other cross sections. If the dilution cross-section (σ o ) is small, it indicates that the resonance has a significant impact on the flux and a large self-shielding effect exists. If σ o is very large (infinite dilution), the cross sections of the absorber do not affect the flux spectrum, and the flux may be represented as a smooth function of energy. Including the above approximations, Equation 2-18 becomes: E / α A A A σ ( E') [ σ σ ( E) ] Φ( E) = C( E) σ + de' s Φ( E' ) o + t o (1 α ) E' E A Equation 2-20 The dilution cross-section for an isotope i is given as: σ o = 1 ρ i j i ρ σ j j t Equation 2-21 Where i and j represent isotope indexes and ρ is atomic density. Equation 2-20 is the simplest form used in NJOY for computing the flux with the flux calculator option. In NJOY, several dilution cross sections are provided as input. Depending on a system of interest, the cross sections corresponding to the appropriate dilution cross-section are used. 18

33 2.5.2 The Bondarenko Method The Bondarenko method is obtained by using the narrow resonance approximation in the absorber scattering integral of Equation 2-22, which is derived from neutron slowing down equation in Equation E / α A A A σ ( E') [ σ σ ( E) ] Φ( E) = C( E) σ + de' s Φ( E' ) o + t o (1 α ) E' E A Equation 2-22 The practical width of a resonance of the absorber is considered to be much smaller than the energy loss due to a collision with the absorber. This enables the absorber integral to be represented as a smooth function of energy. Therefore, the flux is represented by: C( E) Φ( E) = Equation 2-23 A ( σ ( E) + σ ) t o If σ 0 is larger than the tallest peaks inσ, the weighting flux φ is approximately proportional to the smooth weighting function C(E). This is called infinite dilution; the cross section in the material of interest has little or no effect on the flux. On the other hand, if σ 0 is small with respect toσ, the weighting flux will have large dips at the t locations of the peaks inσ, and a large self-shielding effect will be expected. This t treatment is good for the unresolved region (high energy resonances). Since resonance width in this region is very small. t 19

34 2.5.3 CENTRM CENTRM (Continuous Energy Transport Module) is the new method existing in SCALE 5.0 (Ref.17). It computes continuous-energy neutron spectra in zero- or onedimensional systems, by solving the Boltzmann Transport Equation using a combination of pointwise and multigroup nuclear data. Several calculational options are available, including discrete ordinates in slab, spherical, or cylindrical geometry; collision probabilities in slab or cylindrical coordinates; and zone-wise or homogenized infinite media. In SCALE, CENTRM is used mainly to calculate problem-specific fluxes on a fine energy mesh (>10000 points), which may be used to generate self-shielded multigroup cross section for subsequent criticality or shielding analysis. CENTRM avoids many of the inherent assumptions by calculating a problemdependent flux profile, thus making it a far more rigorous cross-section treatment. Effects from overlapping resonances, fissile material in the fuel and surrounding moderator, and inelastic level scattering are explicitly handled in CENTRM. Another advantage of CENTRM is that it can explicitly model rings in a fuel pin to more precise model the spatial effect on the flux and cross sections. CENTRM enables problem-dependent multigroup cross sections to have the flexibility and accuracy of pointwise-continuousenergy cross sections for criticality analyses. 2.6 Group Structure Selection Methodology In 2003, Alplan and Haghighat developed the Contributon and Point-wise Cross Section Driven (CPXSD) methodology and its application focused on the shielding problem [Ref.2]. The CPXSD methodology constructs fine- and broad- 20

35 group structures considering two criteria: i) importance of groups and ii) pointwise cross sections of an isotope/material mixture of interest. The importance of the groups is determined using the group-dependent response flux formulation (or contributon) given by Equation C g L l 2 l + 1 m +, m Vs Ψl, g, sψl, g, s s D l= 0m= 0 4 = π Equation 2-24 In Equation 2-24, V s is the volume of the sub-domain, l and m are azimuthal and polar indices for the spherical harmonic polynomial, g refers to energy group, Ψ is the angular flux, and Ψ + is the adjoint ( importance ) function. The CPXSD methodology constructs group structures by refining an initial arbitrary group structure, considering the two aforementioned criteria. First, the objectives are calculated using the cross section library having the initial group structure. The importance values of all groups are calculated and the most important group is identified. Depending on the point-wise cross sections of the important isotope/mixture and/or group of isotopes/mixture, sub-groups are placed in the most important group, considering the resonance and non-resonance behavior of cross sections. After the number of sub-divisions in the most important group is obtained, the number of subdivisions in other groups are determined based on the ratio of their Cg, to the maximum Cg. Sub-divisions in other groups are performed and a new group structure is generated. The refinement process continues until a convergence criterion on the objectives is achieved. The CPXSD methodology was applied to a reactor pressure vessel problem using TMI-1 to generate new group structures for the fast neutron dosimetry applications. It 21

36 was demonstrated that the broad-group libraries containing the CPXSD generated group structures are in close agreement with their fine-group libraries (within 1-2%). Also, comparing with continuous energy Monte Carlo predictions, Alplan and Haghighat used the CPXSD methodology to generate new broad-group libraries, which have significantly fewer groups, but yield more accurate results than the standard BUGLE libraries. Their analyses demonstrated that the group structures constructed by the CPXSD methodology can significantly improve the efficiency and accuracy of shielding calculations. 2.7 Code Description DORT DORT [Ref.4] is a 2-D discrete ordinates code (it also has a 1-D slab option) that is suitable for XZ, RZ, or R-Θ geometry. It can be used to solve either the forward or the adjoint form of the Boltzmann transport equation. The Boltzmann transport equation is solved, using either the method of discrete ordinates or diffusion theory approximation. In the discrete ordinates method, the primary mode of operation, balance equations are solved for the flow of particles moving in a set of discrete directions in each cell of a space mesh and in each group of a multigroup energy structure. Iterations are performed until all implicitness in the coupling of cells, directions, groups, and source regeneration has been resolved. Several methods are available to accelerate convergence i.e., single group-wise rebalance factor, diffusion acceleration, and partial current rebalance. Anisotropic cross sections can be expressed in a Legendre expansion of arbitrary order. Output data sets can be used to provide an accurate restart of a previous problem or to deliver information to other codes. Several techniques are available to remove the effects 22

37 of negative fluxes caused by the finite difference approximation and of negative scattering sources due to truncation of the cross-section expansion. The space mesh can be described such that the number of first-dimensional (i) intervals varies with the second dimension (j). The number of discrete directions can vary across the space mesh and with energy. Direction sets can be biased, with discrete directions concentrated such as to give fine detail to streaming phenomena TORT TORT [Ref.4] is a 3-D discrete ordinates code that is suitable for cylindrical (RΘZ) or Cartesian (XYZ) geometry, as well as several two-dimensional subsets. It calculates the neutron flux and/or photons throughout three-dimensional systems due to particles incident upon the system's external boundaries, due to fixed internal sources, or due to sources generated by interaction with the system materials. The Boltzmann transport equation is solved using the method of discrete ordinates to treat the directional variable. The weighted difference, nodal, or characteristic methods are available to treat spatial variables. Energy dependence is treated using a multigroup formulation. Anisotropic scattering is treated using a Legendre expansion. Iterations are used to resolve implicitness caused by scattering between directions within a single energy group, by scattering from one-energy group to another group previously calculated, by fission, and by certain boundary conditions. Methods are available to accelerate convergence. Fixed sources can be specified at either external or internal mesh boundaries, or distributed within mesh cells. 23

38 2.8 Applications of Discrete Ordinates Method to Criticality Calculations A Sub-Critical C28 and a Critical Assembly Sjoden and Haghighat selected two criticality safety problems from KENO multigroup Monte Carlo code standard set [Ref.13]. The first problem is a sub-critical 2C8 enriched uranium cylinder, and the second problem is a critical assembly composed of an enriched uranium annular ring with an offset cylindrical inside the ring. These two problems were solved using the PENTRAN 3-D Cartesian parallel discrete ordinates code with 16-group Hansen-Roach multigroup cross section library, assuming a zero potential dilute absorber treatment. Besides considering the eigenvalue of the problems, the calculations were also performed to examine the effect of quadrature set, spatial differencing scheme, and grid refinement on the eigenvalue solutions. PENTRAN results were compared to KENO Monte Carlo Code, MCNP code in Multigroup mode (using an independent 30-Group Library), and MCNP using the standard Continuous Energy mode. The results are quite consistent with all Monte Carlo code results for both problems. It is also found that PENTRAN computed k eff values depend on the order of the angular quadrature, the spatial grid interval, and the spatial differencing scheme used The C5G7 MOX Benchmark The seven-group form of the C5 MOX fuel assembly (C5G7MOX) is a transport benchmark problem [Ref.11]. The model includes two MOX-fuel assemblies and two UO 2 -fuel assemblies, which are partitioned in a square lattice and surrounded by water. 24

39 Each fuel assembly is 17x17 lattice of fuel cells. This benchmark was designed to test the performance of deterministic transport methods and codes in solving reactor physics problems. Three papers about application of S N methods to this benchmark problem have been reviewed as follows. In the first paper [Ref.1], Haghighat, Ce Yi, and Sjoden developed models for three study cases used in PENTRAN; (i) a fuel cell model (ii) a fuel cell assembly and (iii) a full C5G7MOX model. The problems were solved in 2-D geometry by using reflective boundary conditions on the top and bottom boundaries. They examined different angular qudrature orders between S8 and S20, and various mesh sizes between cm. and cm. for a fuel cell case. The PENTRAN results were compared with the reference MCNP solutions for k eff. The fuel cell results indicate that S16 with 0.09 cm mesh size model is adequate for this simulation. For a fuel cell assembly and a full C5G7MOX model cases, PENTRAN yields accurate solutions with an error less than 0.1% on k eff. The relative differences of power distribution in C5G7MOX model vary in a range of ~-3% to ~+2%, the larger differences can be attributed to higher uncertainties in the Monte Carlo predictions. In the second paper [Ref.8], Klingensmith, Azmy, Gegin, and Orsi used TORT- MPI to solve the 3-D C5G7MOX problem and compared with the KENO (Monte Carlo reference solution). The problem was solved on a sequence of refined spatial grids using increasing orders of angular quadratures (S6, S12, and S16), and it was observed that the eigenvalue converges. The results show that the obtained eigenvalues on various meshes and angular quadrature orders is accurate if compared to the expectations in reactor applications with an error less than 0.2%. 25

40 In the third paper [Ref.10], Dahl and Alcouffe used PARTISN (PARallel Time Dependent S N ) to perform this problem in 2-D and 3-D Cartesian grid with various mesh refinement. They used the angular quadrature orders of the square TChebyshev-Legendre to solve each problem with diamond spatial differencing. The results were compared for different mesh refinement and quadrature orders but not compared with any reference code. It was found that the angular dependence was stronger than the spatial. At present, there is no such a complete calculation scheme for core analysis that performs both cross-section generation and core simulation based on 3-D S N transport method in a consistent manner. 26

41 CHAPTER 3 Cross-Section Generation Methodology The multigroup cross sections are very important data for the nuclear reactor analyses. Standard cross-section generation techniques involve three major steps. The first one is to generate a fine-group cross-section library from the ENDF/B-VI data using a piecewise linear energy weighting function generated from theoretical spectrum approximations. The cross sections are processed with the appropriate resonance treatment method. Second, infinite array unit cell calculations using the fine-group library are performed to get the spatial flux distribution. These weighting flux functions are used to collapse the fine-group library to a broad-group library. The third step involves spatial homogenization of the unit cell in the framework of the broad group structure. In this chapter the developed cross-section generation methodology for the TRIGA core analysis is described, including the selection of fine and broad energy group cross-section structures for the TRIGA core analysis. 3.1 Cross Section Generation Procedure and Studies The NJOY code version [Ref.12] is used for cross section processing followed with the AMPX module from the SCALE code package [Ref.17] for postprocessing of cross sections. The standard cross section generation procedure contains several steps as follows: Step 1: NJOY generates multigroup cross section in Group-wise Evaluated Nuclear Data Format (GENDF) format. Step 2: SMILER converts the NJOY (GENDF) files to the AMPX master library format Step 3: AJAX combines each AMPX master library file of isotopes to a single file. 27

42 Step 4: BONAMI performs resonance self-shielding effect with Bondarenko factors. Step 5: NITAWL converts AMPX master library to AMPX working library format. Step 6: ALPO converts AMPX working library format to standard ANISN format. Step 7: GIP generates mixture cross-section library. Step 8: Utilize the multigroup cross section library with transport code e.g., DORT and TORT The cross section generation flow chart is presented in Figure 3-1. NJOY Fine group XS SMILER AMPX MASTER LIBRARY AJAX AMPX MASTER LIBRARY BONAMI AMPX MASTER LIBRARY (self-shielding XS for selected region) NITAWL AMPX WORKING LIBRARY ALPO ANISN FORMAT GIP Mixture XS TRANSPORT CODE Figure 3-1: Procedure for generating cross section library 28

43 The isotopes that are used to generate a multigroup cross-section library for TRIGA core analysis are listed below categorized by the elements. 1) Zirconium Zr 2) Boron Carbide B10, B11, C12 3) UZrH U234, U235, U236, U238, Zr, H1 4) Graphite C12 5) SS304 Fe54, Fe56, Fe57, Fe58, Cr50, Cr52, Cr53, Cr54, Ni58, Ni60, Ni61, Ni62, Ni64, Si, Mn 6) H 2 O H1, O16 7) Al Al27 In the NJOY process, all nuclides, except for hydrogen, zirconium in UZrH, and graphite, are processed at 300, 600, 1000, and 2100 K. Hydrogen, zirconium in UZrH and graphite are processed for the temperatures available in the ENDF tape that contains thermal neutron scattering data. These temperatures are 296, 400, 500, 600, 700, 800, 1000 and 1200 K. It should be noted that in this phase of study, calculations are performed based on an 8.5% wt. single unit fuel element cell model (shown in Figure 3-2) with fresh fuel and at cold conditions (300 K). This model is used to establish the cross-section generation methodology for deterministic transport S N -based TRIGA core analysis, which later will be applied to other material compositions in the present TRIGA core. 29

44 Clad Fuel Zr Coolant Figure 3-2: Unit cell for TRIGA fuel element The Weight Function Study The accuracy of a set of multigroup constants is determined by the selected energy group structure and the utilized weight function. It is necessary to have a weight function that represent as accurate as possible the flux distribution as a function of energy in the nuclear reactor core of interest. GROUPR in NJOY provides the in-code built weight functions that represent a few typical nuclear systems including the thermal reactor spectrum. The later weight function combines a thermal Maxwellian at low energies, a 1/E function at intermediate energies, and a fission spectrum at high energies. In GROUPR, user has freedom to choose the temperatures of the Maxwellian and fission parts and the energies where the spectra join. A quarter of 8.5%wt. fuel TRIGA cell was modeled in MCNP to study for the weighting function spectrum that will be applied in NJOY. The energy tally card was used to tally neutron flux for each of 238-group energy bins. This 238-group structure is the group structure of the library in the SCALE package. Figure 3-3 shows the predicted 30

45 with MCNP neutron flux distribution per unit lethargy as function of energy. It has the shape of the thermal reactor spectrum that is available in GROUPR. The cutoff energies between spectra were determined by using the Maxwell- Boltzmann distribution function for low energies and a 1/E function for intermediate energies. As a result, the function consists of 1. A Maxwellian spectrum (peak at 0.07eV) from 10-5 to 0.3 ev 2. An 1/E spectrum from 0.3 ev to 20.0 kev 3. A fission spectrum from 20.0 kev to 20 MeV. 1.E+01 1.E+00 Flux per unit lethargy 1.E-01 1.E-02 1.E-03 1.E-04 Maxwellian Spectrum 1/E Spectrum Fission 1.E-05 1.E-03 1.E-02 1.E-01 1.E+00 1.E+01 1.E+02 1.E+03 1.E+04 1.E+05 1.E+06 1.E+07 1.E+08 Energy (ev) Figure 3-3: MCNP-predicted TRIGA spectrum The Corner-Material Study TRIGA core has a hexagonal unit cell lattice type, which cannot be modeled explicitly in the S N transport codes such as DORT, TORT or PENTRAN. Therefore, the model has to be generated in a rectangular geometry. This study is performed to 31

46 determine the suitable material that should be used in the corner of the TRIGA cell for the cross-section generation process. Four study cases have been considered as shown in Figure ) None (Real Model) 2) Void 3) Water 4) Graphite Figure 3-4: Cells Models for MCNP All cases were modeled in a quarter sector of symmetry. Case 1 represents the real model in the TRIGA cell; the reflective boundary was applied on the hexagonal surface. In cases 2, 3 and 4, the void, water, and graphite were filled in the corner, respectively. The reflective boundaries were applied on the rectangular surface. The calculations for all cases were performed by using MCNP4C2 with cycles with 100 inactive cycles and 5000 histories/cycles. Table 3-1 shows the k inf and the percentage of deviations from the real model. Figure 3-5 depicts the neutron spectrum in fuel region for each case. Table 3-1: Results of eigenvalue calculation using MCNP Case k inf Dev. in pcm Real ± (3σ) - Void ± (3σ) 18 Water ± (3σ) Graphite ± (3σ)

47 Flux in fuel region 2.E+00 1.E+00 1.E+00 Flux/lethargy 1.E+00 8.E-01 6.E-01 Void Water Real Graphite 4.E-01 2.E-01 0.E+00 1.E-10 1.E-09 1.E-08 1.E-07 1.E-06 1.E-05 1.E-04 1.E-03 1.E-02 1.E-01 1.E+00 1.E+01 1.E+02 Energy (MeV) Figure 3-5: Flux distribution in fuel Region The relative differences of eigenvalue results from the real cell model are 18 pcm for void case, pcm for water case, and -62 pcm for graphite case. Besides the k inf, the neutron spectrum of the void case is closer to the real case than other cases. Hence, the model with void is selected for cross-section generation Resonance Treatment Study One of the most important issues to be considerated in criticality calculations is the energy self-shielding in the resonance region for multigroup cross sections. The method utilized for treatment of energy self-shielding is one of the factors in a multigroup cross-section generation that may have a significant impact on the multiplication factor and also on the absorption reaction rate predictions, mostly in the epithermal region. 33

48 Here, we study the effect of different self-shielding methods for Zr and U 238 that are present in the TRIGA fuel cell. These two isotopes have significant resonances in the energy range of ev to KeV as illustrated in Figure 3-6 and Figure 3-7. The calculations involve the use of GROUPR module in NJOY code (version 99.81) to calculate Zr and U 238 self-shielded cross sections in an infinite homogeneous medium. The Bondarenko, flux calculator and CENTRM methods were used for self-shielding calculations. The 238-group structure of the library in the SCALE package was utilized in NJOY. Figure 3-6: Pointwise absorption cross section of Zr 34

49 Figure 3-7: Pointwise absorption cross section of U 238 The study is performed with DORT using S10 quadrature order and P1 scattering order. The k inf and reaction rates from DORT were compared with the reaction rates obtained using the continuous energy MCNP calculation. The reaction rates calculated by MCNP have less than 0.1% of statistical uncertainty. Table 3-2 shows the reaction rates for each energy range from the MCNP calculation. 35

50 Table 3-2: Reaction rates with continuous energy cross-section library in MCNP Reaction Energy range Zr rod Fuel meat Cladding Water Absorption Fast 1.37E E E E-04 Epithermal 4.10E E E E-04 Thermal 1.08E E E E-02 Total 1.62E E E E-02 nu-fission Fast 0.00E E E E+00 Epithermal 0.00E E E E+00 Thermal 0.00E E E E+00 Total 0.00E E E E+00 Total Fast 8.28E E E E+00 Epithermal 6.83E E E E+00 Thermal 6.67E E E E+00 Total 2.18E E E E+01 The resonance treatments for Zr and U 238 were studied separately. First, we concentrated on the Zr in the Zr rod region. The Bondarenko, flux calculator, and CENTRM methods were utilized to calculate self-shielded cross sections of Zr from ENDF/B-VI. For other nuclides, the Bondarenko Method was used. Table 3-3 shows the reaction rates for each energy range from 238-group cross section library with the Bondarenko method for Zr. Table 3-4 presents the percent deviations from the MCNP calculation. 36

51 Table 3-3: Reaction rates with 238-group cross-section library using the Bondarenko method Reaction Energy range Zr rod Fuel meat Cladding Water Absorption Fast 1.37E E E E-04 Epithermal 4.95E E E E-04 Thermal 1.10E E E E-02 Total 1.73E E E E-02 nu-fission Fast 0.00E E E E+00 Epithermal 0.00E E E E+00 Thermal 0.00E E E E+00 Total 0.00E E E E+00 Total Fast 8.30E E E E+00 Epithermal 7.08E E E E+00 Thermal 6.61E E E E+00 Total 2.20E E E E+01 Table 3-4: Percent deviations from MCNP Reaction Energy range Zr rod Fuel meat Cladding Water Absorption Fast Epithermal Thermal Total nu-fission Fast Epithermal Thermal Total Total Fast Epithermal Thermal Total Table 3-5 shows the reaction rates for each energy range from 238-group cross section library with Flux Calculator in NJOY for Zr. Table 3-6 shows the percent deviations from the MCNP calculation. 37

52 Table 3-5: Reaction rates with 238-group cross-section library using Flux- Calculator in NJOY Reaction Energy range Zr rod Fuel meat Cladding Water Absorption Fast 1.37E E E E-04 Epithermal 4.91E E E E-04 Thermal 1.10E E E E-02 Total 1.72E E E E-02 nu-fission Fast 0.00E E E E+00 Epithermal 0.00E E E E+00 Thermal 0.00E E E E+00 Total 0.00E E E E+00 Total Fast 8.30E E E E+00 Epithermal 7.08E E E E+00 Thermal 6.61E E E E+00 Total 2.20E E E E+01 Table 3-6: Percent deviations from MCNP Reaction Energy range Zr rod Fuel meat Cladding Water Absorption Fast Epithermal Thermal Total nu-fission Fast Epithermal Thermal Total Total Fast Epithermal Thermal Total In order to implement the CENTRM method in NJOY, the CENTRM code in SCALE5 package (Ref.17) was used for the TRIGA fuel cell. ENDF/B-V data were used for this calculation, since there is no ENDF/B-VI pointwise data available in SCALE package. The average scalar flux spectrum in the Zr rod region was introduced in the GROUPR module of NJOY as a weighting function to calculate multigroup cross 38

53 sections. Table 3-7 shows the reaction rates for each energy range from 238-group crosssection library with CENTRM method for Zr. Table 3-8 shows the percent deviations from the MCNP calculation. Table 3-7: Reaction rates with 238-group cross-section library using CENTRM Reaction Energy range Zr rod Fuel meat Cladding Water Absorption Fast 1.37E E E E-04 Epithermal 4.09E E E E-04 Thermal 1.09E E E E-02 Total 1.64E E E E-02 nu-fission Fast 0.00E E E E+00 Epithermal 0.00E E E E+00 Thermal 0.00E E E E+00 Total 0.00E E E E+00 Total Fast 8.30E E E E+00 Epithermal 6.99E E E E+00 Thermal 6.61E E E E+00 Total 2.19E E E E+01 Table 3-8: Percent deviations from MCNP Reaction Energy range Zr rod Fuel meat Cladding Water Absorption Fast Epithermal Thermal Total nu-fission Fast Epithermal Thermal Total Total Fast Epithermal Thermal Total

54 From the analysis of the above-presented results, it is found that the deviation of Zr absorption rate in epithermal range decreases from 21% to -0.3% when using the CENTRM method for resonance treatment. Now we focus on the resonance treatment of U 238 in the fuel meat in the epithermal range. Zr in the Zr rod was treated with CENTRM method. Since the Bondarenko method was applied previously in Table 3-7 and Table 3-8, the Flux Calculator, and CENTRM methods were utilized to calculate self-shielded cross sections of U 238. For other nuclides, the Bondarenko method was used. Table 3-9 shows the reaction rates for each energy range from 238-group cross-section library with the Flux Calculator method for U 238. Table 3-10 shows the percent deviations from the MCNP calculation. Table 3-9: Reaction rates with 238-group cross-section library using Flux Calculator in NJOY for U 238 Reaction Energy range Zr rod Fuel meat Cladding Water Absorption Fast 1.37E E E E-04 Epithermal 4.09E E E E-04 Thermal 1.09E E E E-02 Total 1.64E E E E-02 nu-fission Fast 0.00E E E E+00 Epithermal 0.00E E E E+00 Thermal 0.00E E E E+00 Total 0.00E E E E+00 Total Fast 8.30E E E E+00 Epithermal 6.99E E E E+00 Thermal 6.60E E E E+00 Total 2.19E E E E+01 40

55 Table 3-10: Percent deviations from MCNP Reaction Energy range Zr rod Fuel meat Cladding Water Absorption Fast Epithermal Thermal Total nu-fission Fast Epithermal Thermal Total Total Fast Epithermal Thermal Total We applied the CENTRM method for U 238 in the fuel meat region. Table 3-11 shows the reaction rates for each energy range from 238-group cross-section library with CENTRM method for Zr Rod and fuel meat. Table 3-12 shows the percent deviations from the MCNP calculation. The deviation decreases from 3.31% to 1.76% in the fuel meat region. Table 3-11: Reaction rates with 238-group cross-section library using Centrm treatment for Zr and U 238 Reaction Energy range Zr rod Fuel meat Cladding Water Absorption Fast 1.37E E E E-04 Epithermal 4.09E E E E-04 Thermal 1.09E E E E-02 Total 1.64E E E E-02 nu-fission Fast 0.00E E E E+00 Epithermal 0.00E E E E+00 Thermal 0.00E E E E+00 Total 0.00E E E E+00 Total Fast 8.30E E E E+00 Epithermal 6.99E E E E+00 Thermal 6.61E E E E+00 Total 2.19E E E E+01 41

56 Table 3-12: Percent deviations from MCNP Reaction Energy range Zr rod Fuel meat Cladding Water Absorption Fast Epithermal Thermal Total nu-fission Fast Epithermal Thermal Total Total Fast Epithermal Thermal Total This study illustrates that the CENTRM method treats the energy self-shielding resonance cross section better than the Bondarenko method and Flux Calculator method in NJOY for Zr in the Zr rod and U 238 in the fuel meat. The reaction rates agree well with MCNP results except the absorption reaction rate of cladding in the epithermal energy range. Different approaches are applied to solve the large deviation of absorption reaction rate of cladding in the epithermal energy range. First, the CENTRM method is used to treat Fe 56, which is the main resonance isotope in cladding. Table 3-13 shows the reaction rates for each energy range from 238-group cross-section library with the CENTRM method in the Zr, U 238, and Fe 56. Table 3-14 shows the percent deviations from the MCNP calculation. The deviation decreases from -7.35% to -5.86%. 42

57 Table 3-13: Reaction rates with 238-group cross-section library using Centrm treatment in Zr, U 238, and Fe 56 Reaction Energy range Zr rod Fuel meat Cladding Water Absorption Fast 1.37E E E E-04 Epithermal 4.09E E E E-04 Thermal 1.09E E E E-02 Total 1.64E E E E-02 nu-fission Fast 0.00E E E E+00 Epithermal 0.00E E E E+00 Thermal 0.00E E E E+00 Total 0.00E E E E+00 Total Fast 8.30E E E E+00 Epithermal 6.99E E E E+00 Thermal 6.60E E E E+00 Total 2.19E E E E+01 Table 3-14: Percent deviations from MCNP Reaction Energy range Zr rod Fuel meat Cladding Water Absorption Fast Epithermal Thermal Total nu-fission Fast Epithermal Thermal Total Total Fast Epithermal Thermal Total After that we increase the mesh model from 1554 cells to cells as illustrated in Figure 3-8 to observe the physical effect. Table 3-15 shows the reaction rates for each energy range from 238-group cross-section library with the CENTRM method for Zr rod and Fuel meat, for the cell model. The Bondarenko method was 43

58 used for other nuclides. Table 3-16 shows the percent deviations from the MCNP calculation. The deviation does not change at all. Figure 3-8: Mesh Model from 1554 cells to cells Table 3-15: Reaction rates with 238-group cross-section library, cells: Reaction Energy range Zr rod Fuel meat Cladding Water Absorption Fast 1.37E E E E-04 Epithermal 4.09E E E E-04 Thermal 1.09E E E E-02 Total 1.64E E E E-02 nu-fission Fast 0.00E E E E+00 Epithermal 0.00E E E E+00 Thermal 0.00E E E E+00 Total 0.00E E E E+00 Total Fast 8.30E E E E+00 Epithermal 6.99E E E E+00 Thermal 6.60E E E E+00 Total 2.19E E E E+01 44

59 Table 3-16: Percent deviations from MCNP Reaction Energy range Zr rod Fuel meat Cladding Water Absorption Fast Epithermal Thermal Total nu-fission Fast Epithermal Thermal Total Total Fast Epithermal Thermal Total Then a number of energy groups were increased from 238 to 253 by refining only in the epithermal range. Table 3-17 shows the reaction rates for each energy range from 253-group cross-section library with the CENTRM method used for Zr and U 238 treatment. The Bondarenko method was used for other nuclides. Table 3-18 shows the percent deviations from the MCNP calculation. The deviation slightly decreases from % to -5.72%; however, this approach affects the absorption reaction rate of cladding and water in fast energy range. 45

60 Table 3-17: Reaction rates with 253-group cross-section library Reaction Energy range Zr rod Fuel meat Cladding Water Absorption Fast 1.37E E E E-04 Epithermal 4.10E E E E-04 Thermal 1.09E E E E-02 Total 1.64E E E E-02 nu-fission Fast 0.00E E E E+00 Epithermal 0.00E E E E+00 Thermal 0.00E E E E+00 Total 0.00E E E E+00 Total Fast 8.28E E E E+00 Epithermal 7.00E E E E+00 Thermal 6.60E E E E+00 Total 2.19E E E E+01 Table 3-18: Percent deviations from MCNP Reaction Energy range Zr rod Fuel meat Cladding Water Absorption Fast Epithermal Thermal Total nu-fission Fast Epithermal Thermal Total Total Fast Epithermal Thermal Total With the CENTRM resonance treatment, the deviation of absorption reaction rate of cladding in epithermal energy range improves a few percent. With the mesh refinement and energy group refinement, the deviation of absorption reaction rate of cladding in epithermal energy range does not improve or otherwise slightly improve but affect the reaction rate in other ranges of energy and materials. Therefore, the problem in cladding region will be resolved using the CENTRM for resonance treatment in Fe

61 3.2 Fine Group Structure Selection The CPXSD methodology developed by Alplan and Haghighat [Ref.2] is an iterative method that selects effective fine- and broad-group structures for a problem of interest, depending on the objectives of the problem. This methodology was derived based on the contribution theory (the product of the forward and adjoint angular fluxes) [Ref.19] to calculate the importance of groups and point-wise cross sections to obtain the sub-group boundaries. The energy dependent response flux, i.e., the contributon is given by: v C E = + dr dωψ r E Ωˆ v ( ) (,, ) Ψ ( r, E, Ωˆ ) v 4π Equation 3-1 In Equation 3-1, Ψ( r v, E, Ωˆ ) is the angular flux and Ψ + ( r v, E, Ωˆ ) is the adjoint function dependent on position r v, energy E and direction Ωˆ. Considering spherical harmonics expansion of flux and its adjoint, and using orthogonality, the groupdependent contributon is given by: C g s D L l 2l + 1 m m, + s Ψl, g, sψl, g, s l= 0m= 0 4π = V Equation 3-2 In Equation 3-2, (, Ω ) ψ r ) is the angular flux and ψ + ( r, Ω ) ) is the adjoint function g g dependent on position r r, and direction Ω r in group g. This CPXSD methodology was applied and validated for the shielding problem but not yet for the criticality problem. Here, we extend this methodology to the criticality problem based on the TRIGA cell/core. The objective is to generate a group structure to determine an accurate eigenvalue and multigroup flux and power distributions. 47

62 3.2.1 Extension of the CPXSD Methodology to Criticality Problem The procedure of the CPXSD methodology for generating fine-group structures, which was adapted for criticality problem is as follows: 1. An initial group structure is selected. The initial group structure can be the existing group structure or arbitrary one. 2. Cross sections are processed for the initial group structure with the established procedure of cross section generation. 3. The importance of groups in the initial group structure is calculated by performing forward and adjoint transport calculations to calculate the group-dependent response flux. The adjoint function for criticality problem is obtained by setting the adjoint source equal to production cross section ( ν Σ ). f 4. The group that has the maximum importance is identified. 5. The group that has the maximum importance is refined by the resonance structure of an objective isotope with an arbitrary number 6. The number of sub-divisions in other groups is set relative to their importance to the maximum importance. 7. After the refinement process is completed for all groups, the new group structure is used for cross-section generation process. The new cross-section library is used to calculate the objectives of a problem of interest. 48

63 8. In order to test the new library, a finer group structure is derived by repeating step 5 through 7 with higher arbitrary number to generate a finer group structure. 9. Calculated objectives are compared with the previous library. If results are within a specified tolerance, the procedure ends; otherwise, steps 5 through 8 are repeated. 3.3 Cross-Section Collapsing and Homogenization It is considered impractical to model a reactor core for routine repetitive design and depletion calculations with its full geometrical detail employing multigroup neutron transport theory. Therefore, the standard approach in core analysis is to combine geometrical details as well as to collapse the energy group structure of cross section library for whole core calculations. The purpose of the cross section homogenization and collapsing is to preserve the sub-region average reaction rates and fluxes, while improving the computation efficiency Fine- to Broad-Group Collapsing The Procedure of the CPXSD Methodology The procedure of the adapted CPXSD methodology for generated broad-group libraries for criticality problem is as follows: 1. An initial broad-group structure is selected. The initial broad-group structure can be selected in evenly partitioned. 2. Fine-group cross sections are collapsed to broad-group and a transport calculation in performed with the broad-group library to calculate the objectives. 49

64 3. The group that has the maximum importance is refined by even partition using the constructed fine-group structure with an arbitrary number. 4. The fine-group library is collapsed to the new broad-group library, and step 2 and 3 are repeated until a user-specified convergence criterion is achieved Cross-Section Homogenization A typical fuel cell comprises of three explicit regions- fuel, clad, and coolant. It can be reduced to an equivalent cell of simpler geometry to expedite calculations as shown in Figure 3-9. The concept of the homogenization is to preserve all of the reaction rates in the problem from the detailed "heterogeneous'' transport calculation. We utilize the scalar flux weighting method to combine the material regions as shown in Equation 3-3. With this method, the multigroup cross sections characterizing materials in the cell are spatially averaged over the cell. Figure 3-9 Fuel cell homogenization Σ g = E nzone g 1 i= 1 Eg Vi E nzone g 1 i= 1 3 i r r de d rσ (, E) φ(, E) Eg 3 r de d rφ (, E) Vi Equation

65 3.4 Summary In this chapter we determined the flux-weighting spectrum that is applied in NJOY to generate the fine group cross-section library. The void was selected to be the material in the corner of the cell for cross section generation process. The resonance treatment in epithermal energy range was studied using different methods i.e. Bondarenko, Flux Calculator and CENTRM. It was found that the CENTRM method is the most effective technique to treat the energy resonance without further refining the energy group structure. The CPXSD methodology was adapted to generate fine- and broad- group structures for criticality problem. 51

66 CHAPTER 4 Two-Dimensional Cross Section Generation In this chapter, the CPXSD methodology, adapted for criticality problem, is applied to study the 2-D cross section generation in order to verify and validate the methodology prior to application to the actual 3-D cross section generation, which will be too costly in terms of computational time and resources. The other objective of generating 2-D cross sections is to compare them with 3-D cross-sections in 3-D core calculations. The 8.5% and 12% wt. TRIGA fuel cells are modeled for this study. The 2- D fine group structure is constructed, and then the optimization study on the parameters of S N method is performed. The 2-D broad group structure is established based on the 2-D fine group structure. Other non-fuel material cross sections are generated with the same fuel-studied structure. Finally, the cross sections are homogenized and compared with the heterogeneous cases. 4.1 Two-Dimensional Model for Cross-Section Generation One quarter of a hexagonal unit cell has been modeled for the 2-D cross section generation study by taking advantage of the model symmetry as illustrated in Figure 4-1. The void is filled in the corner of the rectangular geometry model based on the results from the study presented in Chapter 3. The reflective boundary condition is applied to all of the surfaces. Table 4-1 and Table 4-2 show the material compositions that have been used in the cross-section generation calculations. 52

67 Figure 4-1: Cross section generation model Table 4-1: Material density of the fuel elements Nuclide Density (atoms/barn-cm) Fuel 12 wt.% 8.5 wt.% H Zr U U U U Reflector H O Zr Rod Zr SS304 (Cladding) SS

68 Table 4-2: Cladding composition ISOTOPE %Wt. Fe Fe Fe Fe Cr Cr Cr Cr Ni Ni Ni Ni Ni Mn Si Fine Group Structure for TRIGA By using the procedure of the CPXSD methodology, adapted for criticality problem, the fine group structure for TRIGA cross-section generation is obtained. The 238-group SCALE library is used as a starting group structure. The 238-group cross sections are generated. Initially, the 238-group structure was divided into 3 major ranges of energy: fast (0.1 MeV to 20 MeV.), epithermal (3 ev to 0.1 MeV), and thermal (1E-05 ev to 3 ev). We established two criteria for obtaining a fine group structure. The first criterion is 10 pcm relative deviation of Δk/k and the second criterion is 1% relative deviation of objective reaction rates. The objective reaction rates are different for each range of energy. Using the flux and adjoint function moments computed from the transport calculations with DORT [Ref.4], the contribution function - Cg s are calculated. Depending on the magnitude of the Cg s per group, the group structure is refined for each energy range. The groups corresponding to large C g s were partitioned into more groups. 54

69 The group with the highest Cg was subdivided by the resonance structure of an objective isotope into a number of groups and the remaining groups were divided into fewer groups based on the ratio of their Cg to the maximum Cg Fast Range Group Refinement In this section a group structure in the fast energy range between 0.1 and 20 MeV is derived. The 238-group SCALE library is used as a starting group structure with 44 groups in the fast energy range, 104 groups in the epithermal range, and 90 groups in the thermal range. The 238-group Cg s are calculated using the normalized production cross sections (νσ f ) as the adjoint source to perform the adjoint transport calculation. The pointwise cross section of U 238 (n,f) is used to select the group boundaries. The objectives are eigenvalue and neutron production reaction rate of U 238. The new group structures are generated. Table 4-3 shows the number of groups in the fast energy range that are obtained from the group refinement process. Table 4-3: Fine groups selected in the fast energy range Group Structure Number of Groups in Different Energy Ranges Number Fast Epithermal Thermal Total

70 The importance of different energy groups in the fast energy range, between 0.1 and 20 MeV, of 238-group and 246-group structures are plotted in Figure 4-2. The plot shows that when the groups that have higher importance are refined, the importance of those groups is decreased. 4.00E E E E-02 Importance (E) 2.00E E groups 246 groups 1.00E E E+00 1.E-01 1.E+00 1.E+01 1.E+02 Energy (MeV) Figure 4-2: Importance of groups of 238G and 246G libraries The eigenvalues are calculated and compared between the group structures. For 246-group and 274-group comparison, Table 4-4 and Table 4-5 demonstrate that the relative difference of Δk/k is less than 10 pcm and the percentage relative deviation of U 238 (νσ f ) reaction rate are 0.167% for 8.5% wt. and 0.165% for 12% wt. cases. Consequently, we selected the 246-group structure, which contains 52 groups in the fast energy range, for further group refinement in the epithermal energy range. 56

71 Table 4-4: Eigenvalue results of fine group energy for 8.5% wt. case Group k inf (S10P1) Rel. Dev. in pcm of Δk/k With previous νσ f rate of U238 > 0.1 MeV %Rel. Dev. With previous group group Table 4-5: Eigenvalue results of fine group energy for 12% wt. case Group k inf (S10P1) Rel. Dev. in pcm of Δk/k With previous νσ f rate of U238 > 0.1 MeV %Rel. Dev. With previous group group Epithermal Range-Group Refinement In this section a group structure in the epithermal energy range between 3eV and 0.1 MeV is derived. The 246-group structure from the fast group refinement is used as a starting group structure with 52 groups in fast energy range, 104 groups in epithermal range, and 90 groups in thermal range. The 246-group Cg s are calculated using the summation of the normalized νσ f and down-scattering cross section of H in ZrH from epithermal group to thermal group as the adjoint source to perform the adjoint transport calculation. The absorption point-wise cross section of U 238 is used to select the group boundaries. The objectives are eigenvalue, down-scattering reaction rate of H in ZrH from epithermal energy range to thermal energy range and absorption reaction rate of U

72 Table 4-6 shows the number of groups in epithermal energy range that are obtained from the group refinement process. The importance of groups in epithermal energy range, between 3 ev and 0.1 MeV, of the 246-group structure are plotted in Figure 4-3. Table 4-6: Fine groups generated in the epithermal energy range Group Structure Number of Groups in Different Energy Ranges Number Fast Epithermal Thermal Total E E E-02 Importance(E) 8.00E E E E E E E E E E E E+00 Energy(MeV) Figure 4-3: Importance in groups of 246G library 58

73 Table 4-7: Eigenvalue results of fine group energy for 8.5% wt. case Group k inf (S10P1) Rel. Dev. in pcm of Δk/k With previous group Table 4-8: Eigenvalue results of fine group energy for 12% wt. case Group k inf (S10P1) Rel. Dev. in pcm of Δk/k With previous group Table 4-9: Reaction rate comparison for 8.5% wt. case Down-scat. %Rel. Dev. U238(n,abs) %Rel. Dev. Group Of H in ZrH With previous group With previous group Table 4-10: Reaction rate comparison for 12% wt. case Down-scat. %Rel. Dev. U238(n,abs) %Rel. Dev. Group Of H in ZrH With previous group With previous group The eigenvalues were calculated and compared between the 246-group and 294- group structures in Table 4-7 for 8.5% wt. case and Table 4-8 for 12% wt. case. The relative difference of eigenvalues are less than 10 pcm and the percentage relative deviation of U 238 absorption reaction rate and down-scattering reaction rate of H in ZrH from epithermal range to thermal range are less than 1.0% as given in Table 4-9 and 59

74 Table Consequently, we selected the 246-group structure, which contains 104 groups in epithermal energy range, for further group refinement in the thermal energy range Thermal Range-Group Refinement In this section a group structure in the thermal energy range between 1E-5 ev to 3 ev is derived. The 246-group structure from the fast and epithermal range group refinements is used as a starting group structure with 52 groups in the fast energy range, 104 groups in the epithermal range, and 90 groups in the thermal range. The 246-group Cg s are calculated using the summation of the normalized νσ f and up-scattering cross section of H in ZrH as the adjoint source to perform the adjoint transport calculation. The inelastic scattering point-wise cross section of H in ZrH is used to select the group boundaries. The objectives are eigenvalue, neutron production reaction rate of U 235, and up-scattering reaction rate of H in ZrH in the thermal energy range. Table 4-11 shows the number of groups in the thermal energy range that are obtained from the group refinement process. The importance of groups in the thermal energy range, between 1E-5 and 3 ev, of 246-group, 254-group and 280-group structures are plotted in Figure 4-4. Table 4-11: Fine groups generated in the thermal energy range Group Structure Number of Groups in Different Energy Ranges Number Fast Epithermal Thermal Total

75 4.00E E E-02 Importance(E) 2.50E E E G 254G 280G 1.00E E E E E E E E E-05 Energy(MeV) Figure 4-4: Importance in groups of 246G, 254G and 280G libraries Table 4-12: Eigenvalue results for fine group energy in thermal range 8.5% wt. case Group k inf (S10P1) Rel. Dev. in pcm of Δk/k With previous group

76 Table 4-13: Eigenvalue results for fine group energy in thermal range 12% wt. case Group k inf (S10P1) Rel. Dev. in pcm of Δk/k With previous group Table 4-14: Reaction rate comparison of 8.5% wt. case Group Up-scat. Of H in ZrH %Rel. diff. With previous group U235(n, νσ f ) %Rel. diff. With previous group : : : Note: 1 The reaction rate was calculated in a group-collapsing method to be compared with the previous group structure Table 4-15: Reaction rate comparison of 12% wt. case Group Up-scat. Of H in ZrH %Rel. diff. With previous group U235(n, νσ f ) %Rel. diff. With previous group : : : Note: 1 The reaction rate was calculated in a group-collapsing method to be compared with the previous group structure 62

77 Table 4-12 and Table 4-13 compare the eigenvalues for different energy group structures. Table 4-14 and Table 4-15 compare the rate of up-scattering of H in ZrH and neutron-production reaction rates of U 235 for each energy group structure. The 280-group structure is selected because its relative difference compared to the 336-group case satisfies the set criterion. The 280-group cross-section library was selected to be a fine group structure for the TRIGA reactor based on the CPXSD methodology in 2-D geometry. In conclusion, a methodology is established to generate the fine-group cross-section library and applied to 8.5% and 12% wt. TRIGA fuel cells. 4.3 Parametric Studies Increasingly, the discrete ordinates method has become the dominant means for obtaining numerical solutions to the integrodifferential form of the transport equation. The discrete ordinates (S N ) methods require a suitable multigroup cross section library, and a reasonably accurate combination of spatial discretization, angular quadrature set, and scattering order of cross sections. Assuming the cross sections are reliable, the accuracy of an S N calculation is impacted by the aforementioned modeling factors. The investigation of the parameters in transport calculations is performed to obtain the effective value for each parameter in the view of minimizing computer memory and time requirements for the problems, while maintaining the desired level of accuracy. These effective parameters are obtained for the TRIGA cell cross-section generation process. 63

78 4.3.1 Spatial Mesh, Angular Quadrature, and Scattering Order Studies In this section, we perform sensitivity studies for spatial meshes, angular quadrature set, and scattering order. Four 2-D fine-mesh models have been developed with different uniform grid intervals. 1) 1554 cells: 37 x-axis, 42 y-axis 2) 6132 cells: 73 x-axis, 84 y-axis 3) cells: 109 x-axis, 125 y-axis 4) cells: 144 x-axis, 168 y-axis Two different quadrature techniques, level (fully) symmetric and Square Legendre-Chebyshev (SLC), are used to study this problem with various orders. The S4, S6, S8, S10, and S16 orders are examined for fully symmetric and square Legendre- Chebyshev, respectively. The scattering orders that are used to perform sensitivities studies are P1 and P3. The 8.5% wt. fuel element cell model was used to perform this set of study. The calculations were performed with DORT using different combinations of spatial meshes, angular quadrature and scattering order. The obtained results were compared with continuous energy MCNP results, which are used as a reference solution in this study. The MCNP eigenvalue is ± (3σ). The MCNP calculation was performed for 9000 cycles with 100 skipped cycles and 5000 histories/cycle. The DORT results and relative deviations in pcm are provided in Table 4-16 through Table 4-19 for P1 scattering order and Table 4-20 through Table 4-23 for P3 scattering order with fully symmetric quadrature order. 64

79 Table 4-16: DORT results with 280-energy group XS and 1554 cells, Level- Symmetric S N Order Scattering k inf Conv. Out. Iter. Rel. Dev. in pcm of Δk/k S4 P E S6 P E S8 P E S10 P E S16 P E Table 4-17: DORT results with 280-energy group XS and 6132 cells, Level- Symmetric S N Order Scattering k inf Conv. Out. Iter. Rel. Dev. in pcm of Δk/k S4 P E S6 P E S8 P E S10 P E S16 P E Table 4-18: DORT results with 280-energy group XS and cells, Level- Symmetric S N Order Scattering k inf Conv. Out. Iter. Rel. Dev. in pcm of Δk/k S4 P E S6 P E S8 P E S10 P E S16 P E

80 Table 4-19: DORT results with 280-energy group XS and cells, Level- Symmetric S N Order Scattering k inf Conv. Out. Iter. Rel. Dev. in pcm of Δk/k S4 P E S6 P E S8 P E S10 P E S16 P E Table 4-20 DORT results with 280-energy group XS and 1554 cells, Level- Symmetric S N Order Scattering k inf Conv. Out. Iter. Rel. Dev. in pcm of Δk/k S4 P E S6 P E S8 P E S10 P E S16 P E Table 4-21: DORT results with 280-energy group XS and 6132 cells, Level- Symmetric S N Order Scattering k inf Conv. Out. Iter. Rel. Dev. in pcm of Δk/k S4 P E S6 P E S8 P E S10 P E S16 P E

81 Table 4-22: DORT results with 280-energy group XS and cells, Level- Symmetric S N Order Scattering k inf Conv. Out. Iter. Rel. Dev. in pcm of Δk/k S4 P E S6 P E S8 P E S10 P E S16 P E Table 4-23: DORT results with 280-energy group XS and cells, Level- Symmetric S N Order Scattering k inf Conv. Out. Iter. Rel. Dev. in pcm of Δk/k S4 P E S6 P E S8 P E S10 P E S16 P E

82 P1 scattering order Eigenvalues S4 S6 S8 S10 S Meshes Figure 4-5: P1 scattering order with level symmetric quadrature order P3 scattering order Eigenvalues S4 S6 S8 S10 S Meshes Figure 4-6: P3 scattering order with level symmetric quadrature order 68

83 The results presented in Table 4-16 through Table 4-23 are summarized graphically in Figure 4-5 and Figure 4-6. Several tendencies are observed: (i) All the cases (different combinations of spatial meshes, angular quadrature and scattering order) give a deviation of Δk/k less than 150 pcm compared with the MCNP solution. (ii) The fluctuation of the eigenvalues is produced because of using different quadrature orders. (iii) The finer meshes do not yield better results. (iv) The order of scattering anisotropy from P1 to P3 affects the k inf by a maximum amount of 3 pcm. In conclusion, this study shows that the quadrature order of level symmetric techniques does not converge the results in the asymptotic region. Table 4-24 through Table 4-27 show the results for P1 scattering order and Table 4-28 through Table 4-31 display the results for P3 scattering order with square Legendre- Chebyshev quadrature order. Table 4-24: DORT results with 280-energy group XS and 1554 cells, SLC S N Order Scattering k inf Conv. Out. Rel. Iter. Dev. in pcm of Δk/k S4 P E S6 P E S8 P E S10 P E S16 P E

84 Table 4-25: DORT results with 280-energy group XS and 6132 cells, SLC S N Order Scattering k inf Conv. Out. Rel. Iter. Dev. in pcm of Δk/k S4 P E S6 P E S8 P E S10 P E S16 P E Table 4-26: DORT results with 280-energy group XS and cells, SLC S N Order Scattering k inf Conv. Out. Iter. Rel. Dev. in pcm of Δk/k S4 P E S6 P E S8 P E S10 P E S16 P E Table 4-27: DORT results with 280-energy group XS and cells, SLC S N Order Scattering k inf Conv. Out. Iter. Rel. Dev. in pcm of Δk/k S4 P E S6 P E S8 P E S10 P E S16 P E

85 Table 4-28: DORT results with 280-energy group XS and 1554 cells, SLC S N Order Scattering k inf Conv. Out. Iter. Rel. Dev. in pcm of Δk/k S4 P E S6 P E S8 P E S10 P E S16 P E Table 4-29: DORT results with 280-energy group XS and 6132 cells, SLC S N Order Scattering k inf Conv. Out. Rel. Iter. Dev. in pcm of Δk/k S4 P E S6 P E S8 P E S10 P E S16 P E Table 4-30: DORT results with 280-energy group XS and cells, SLC S N Order Scattering k inf Conv. Out. Rel. Iter. Dev. in pcm of Δk/k S4 P E S6 P E S8 P E S10 P E S16 P E

86 Table 4-31: DORT results with 280-energy group XS and cells, SLC S N Order Scattering k inf Conv. Out. Iter. Rel. Dev. in pcm of Δk/k S4 P E S6 P E S8 P E S10 P E S16 P E P1 scattering order Eigenvalues S4 S6 S8 S10 S Meshes Figure 4-7: P1 scattering order with Square Legendre-Chebyshev quadrature order 72

87 P3 scattering order Eigenvalues S4 S6 S8 S10 S Meshes Figure 4-8: P3 scattering order with Square Legendre-Chebyshev quadrature order As it can be seen in Figure 4-7 and Figure 4-8, all the different combinations of fine meshes, angular quadrature and scattering order show a positive bias as compared to the reference MCNP solution by 80 to 97 pcm. The increasing of scattering order from P1 to P3 affects the k inf by maximum 3 pcm of Δk/k. The S4 k inf is higher than any other quadrature order k inf. The results are converged with finer meshes. It was found that k inf converges with the mesh refinement higher than 6132 cells. The scattering order higher than P1 does not have a significant effect on k inf. Even though, the square Legendre-Chebyshev Quadrature technique yields physically well behaved results, the k inf results are not sensitive to the quadrature order beyond S6. As a result, it would be difficult to select the appropriate order of quadrature for our problem by considering only k inf value. 73

88 4.3.2 Qudrature Order Determination From the above study, we concluded that the optimum cells for TRIGA cell are 6132 cells. This study is performed to determine the appropriate order of quadrature set. We utilize the Square Legendre-Chebyshev quadrature type with S4, S6, S8, S10, S12, S14, S16, S20, and S24. Table 4-32 gives the k inf results from DORT calculations and the deviations between DORT and MCNP solutions. Table 4-32: DORT results with 280-energy group XS and 6132 cells, SLC S N Order Scattering k inf Rel. Dev. in pcm S4 P S6 P S8 P S10 P S12 P S14 P S16 P S20 P S24 P The scalar flux distributions are examined visually in Figure 4-9 and Figure 4-10 for the 23 rd energy group as a representative of the fast energy range and the 242 nd energy group as a representative of the thermal energy range. We observe a nonphysical behavior of flux distribution in the cell. This behavior is referred to as ray effects. It results from the inability of low-order S N quadrature to integrate accurately over the 74

89 angular flux. As shown in Figure 4-9, these effects are very strong in S4 and S6 quadrature set orders. S4 S6 S8 S10 S12 S14 S16 S20 S24 Figure 4-9: Flux distribution of group 23 rd 75

90 S4 S6 S8 S10 S12 S14 S16 S20 S24 Figure 4-10: Flux distribution for group 242 nd Six-cell detectors were defined within the fuel region of 6132-cell model to compare the neutron production reaction rate (local parameter) of the selected cells in fuel region as shown in Figure Table 4-33 and Table 4-34 give the neutron 76

91 production reaction rates predicted by MCNP and DORT for each cell detector and the percentage of relative deviation as compared to the MCNP reference case, respectively Figure 4-11: Detector locations 77

92 Table 4-33: Neutron-production reaction rates from MCNP and DORT Cell1 Cell2 Cell3 Cell4 Cell5 Cell6 S S S S S S S S S MCNP ± σ ± σ ± σ ± σ ± σ ± σ Table 4-34: Percentage of relative deviation from MCNP Cell1 Cell2 Cell3 Cell4 Cell5 Cell6 S S S S S S S S S From Table 4-34, it can be observed that the percentage deviation of reaction rate for each cell changes relatively within 0.01% for the quadrature orders higher than S10. As a result, S10 has been selected to be used for further study. 78

93 In this section we have performed sensitivity studies on the spatial meshing of the unit cell, angular quadrature set, and scattering order in order to obtain the effective values for the TRIGA problem. The calculations show that the 6132-cell model, S10 quadrature order of Square Legendre-Chebyshev technique, and P1 scattering order constitute the appropriate model in terms of accuracy and efficiency for further crosssection collapsing and homogenization. 4.4 Cross-Section Collapsing It is considered impractical to model a reactor core for routine repetitive design and depletion calculations with a fine group structure employing multigroup neutron transport theory. Therefore, the standard approach in core analysis is to collapse the energy group structure of cross section library for whole core calculations. The purpose of the cross section collapsing is to preserve the sub-region average reaction rates and fluxes, while improving the computational efficiency. In this section, the 280-group structure was collapsed into a broad group structure. Using the same approach as the one utilized to select the fine group structure, we established two criteria to obtain a broad group structure. The first criterion is 10 pcm relative deviation of Δk/k and the second criterion is 1% relative deviation of objective reaction rates. The objective reaction rates are different for each range of energy. The U 238 (n,f) fission reaction rate is considered in the fast energy range, the down-scattering reaction rates of H in ZrH and U 238 (n,a) absorption reaction rate are considered in the 79

94 epithermal energy range, and the U 235 (νσ f ) reaction rate and the thermal up-scattering reaction rate of H in ZrH are considered in the thermal range. The group collapsing started with fast energies by initiating a very-broad-group structure and using the same fine-group structure in the epithermal and thermal energies. Then, the aforementioned contributon approach was used to refine the broad-group structure. This process is repeated until the two criteria were met, and consequently a new broad-group structure for the fast energies was obtained. With this new fast broad group structure, we continue the same process for the epithermal and thermal energy ranges Fast Range-Group Collapsing In fast energy range, we combined all the energy groups into one group. The new group library contains 229 groups. The eigenvalue is calculated and compared with the 280-group library. Table 4-35 shows that relative difference of eigenvalue is less than 10 pcm and the percentage relative deviation of U238(n,f) is 0.11% for 8.5% wt. case. Table 4-36 shows that relative difference of eigenvalue is less than 10 pcm and the percentage relative deviation of U238(n,f) is 0.17% for 12% wt. case. Consequently, we selected the 229-group structure to be collapsed in the epithermal energy range. Table 4-35: Comparison between 229G and 280G for 8.5% wt. case Group k inf (S10P1) Rel. Dev. In pcm of Δk/k U238 (νσ f ) Above 0.1 MeV %Rel. Dev. Reaction rate of U238(νΣ f )

95 Table 4-36: Comparison between 229G and 280G for 12% wt. case Group k inf (S10P1) Rel. Dev. In pcm of Δk/k U238 (νσ f ) Above 0.1 MeV %Rel. Dev. Reaction rate of U238(νΣ f ) Epithermal Range-Group Collapsing In this step we develop the broad group structure in the epithermal energy range (3.0 ev. to 0.1 MeV). The objective reaction rate for the epithermal energy range is the down-scattering reaction rates of H in ZrH. We have placed two energy groups in this range to separate resolved and unresolved regions and ended up with a 127-group structure. Table 4-37 and Table 4-38 demonstrate that the relative differences of eigenvalues are less than 10 pcm. The percentage relative deviations of down-scattering reaction rate of H in ZrH and absorption reaction rate of U 238 in the epithermal range are less than 1% as given in Table 4-39 and Table As a result, we used 127-group structure for further collapsing in the thermal energy range. Table 4-37: k inf comparison between 229G and 127G for 8.5% wt. case Group Rel. Dev. k inf (S10P1) In pcm of Δk/k

96 Table 4-38: k inf comparison between 229G and 127G for 12% wt. case Group Rel. Dev. k inf (S10P1) In pcm of Δk/k Table 4-39: Reaction rate comparison between 229G and 127G for 8.5% wt. case Down-scat. %Rel. Dev. U238(n,abs) %Rel. Dev. Group Of H in ZrH With previous With previous group group Table 4-40: Reaction rate comparison between 229G and 127G for 12% wt. case Down-scat. %Rel. Dev. U238(n,abs) %Rel. Dev. Group Of H in ZrH With previous With previous group group Thermal Range-Group Collapsing In the last step, the broad group structure in thermal energy range (1.0E-05 ev. to 3.0 ev) is developed. The objective reaction rate of the thermal range is the neutronproduction reaction rate of U 235 and the thermal up-scattering reaction rates of H in ZrH. We initially introduced one energy group in this range and obtained a 4-group structure. Then, we subdivided the most important group into three groups. Each time, we generated a new broad-group structure until the result met the criteria. Table 4-41 indicates that percent relative difference of the eigenvalue of 12-group structure and 14-82

97 group structure is 12 pcm and the percentage relative deviations of U 235 (νσ f ) is 0.01% and upscattering of H in ZrH is 0.00% for the 8.5% wt. case. Table 4-42 shows that percent relative difference of the eigenvalue of 12-group structure and 14-group structure is 9 pcm, and the percentage relative deviation of U235(νΣ f ) is 0.01% and upscattering of H in ZrH is 0.00% for 12% wt. case. Therefore, we select the 12-group structure as the broad group structure for this study. Group Table 4-41: Result comparison in thermal energy range for 8.5% wt. case k inf (S10P1) Rel. Dev. In pcm of Δk/k With Previous group U235 (νσ f ) reaction rate 1E-05 to 3 ev %Rel. Dev. U235(νΣ f ) reaction rate Up-scattering of H in ZrH %MaxRel. Dev. In upscattering rate : : : : :

98 Group Table 4-42: Result comparison in thermal energy range for 12% wt. case k inf (S10P1) Rel. Dev. In pcm of Δk/k With Previous group - U235 (νσ f ) reaction rate 1E-05 to 3 ev %Rel. Dev. U235(νΣ f ) reaction rate Up-scattering of H in ZrH %MaxRel. Dev. In upscattering rate : : : : : In order to verify that we have selected the effective broad-group structure for our problem, MCNP with continuous cross-section library, DORT with 12-broad-group and 280-fine-group cross-section libraries for both 8.5% wt. and 12% wt. cases were performed. The MCNP eigenvalues are ± (3σ) for 8.5% wt. case and ± (3σ) for 12% wt. case. The calculation was performed for 5800 cycles with 900 skipped cycles and 5000 histories/cycle for 12%wt. case. Table 4-43 gives the k inf results calculated by DORT for the 280-fine-group cross-section library and Table 4-44 gives the k inf results calculated by DORT for the 12-broad-group cross-section library with different scattering and angular quadrature orders for the 8.5% wt. case. The Δk maximum of absolute percentage relative deviations in k compared to the 280-group and continuous Monte Carlo calculations are 38 pcm and 136 pcm, respectively. Table 4-45 gives the k inf results calculated by DORT for the 280-fine-group cross-section library, and Table 4-46 gives the k inf results calculated by DORT for the 12-broad-group 84

99 cross-section library with different scattering and angular quadrature orders for the 12% Δk wt. case. The maximum of absolute percentage relative deviations in k compared to the 280-group and continuous Monte Carlo calculations are 30 pcm and 111 pcm, respectively. The deviations between the 12-group and the 280-group structures are less than the deviation between the 12-group structure and the continuous-energy MCNP solution. These differences are identified as the method difference between deterministic (DORT) and statistic (MCNP) including the cross-section library between multigroup and continuous energy. The 12-group structure was selected to be our final broad group structure. Table 4-43: DORT results with 280-energy group XS and 6132 cells for 8.5% wt. case Sn order Scattering k inf Rel.Dev. in pcm of Δk/k(MCNP) S4 P S6 P S8 P S10 P S16 P Table 4-44: DORT results with 12-energy group XS, 6132 cells for 8.5% wt. case Sn order Scattering k inf (12 G) Rel. Dev. In pcm of Δk/k (MCNP) Rel. Dev. In pcm of Δk/k (280 G) S4 P S6 P S8 P S10 P S16 P

100 Table 4-45: DORT results with 280-energy group XS and 6132 cells for 12% wt. case Sn order Scattering k inf Rel.Dev. in pcm of Δk/k (MCNP) S4 P S6 P S8 P S10 P S16 P Table 4-46: DORT results with 12-energy group XS, 6132 cells for 12% wt. case Sn Scattering Rel. Dev. Rel. Dev. k inf order (12 G) In pcm In pcm (MCNP) (280 G) S4 P S6 P S8 P S10 P S16 P The absorption rate, neutron production, and total reaction rates are compared between the two cross-section libraries: 280 groups and 12 groups, in each energy range (fast, epithermal, and thermal) and region (Zr rod, fuel meat, cladding, and water). The DORT-calculations are performed with S10P1. For the 8.5% wt. case, Table 4-47 and Table 4-48 give the reaction rates from DORT for 280-group library and 12-group library, respectively. The percentage of relative deviation between these two libraries is presented in Table For the 12% wt. case, Table 4-50 and Table 4-51 give the reaction rates from DORT for 280-group library and 12-group library, respectively. The 86

101 percentage of relative deviation between these two libraries is presented in Table Overall, it is demonstrated very good agreement for these selected reaction rates in each energy range with less than 0.4% difference. Table 4-47: DORT calculation with 280-group cross section library for 8.5% wt. case Reaction Rate Energy Range Zr Rod Fuel Meat Cladding Water Fast E E E E-04 Epithermal E E E E-04 Absorption Thermal E E E E-02 Total E E E E-02 Fast E Neutron production Epithermal E Thermal E Total E Fast E E E E+00 Epithermal E E E E+00 Total Thermal E E E E+00 Total E E E E+01 87

102 Table 4-48: DORT calculation with 12-group cross section library for 8.5% wt. case Reaction Rate Energy Range Zr Rod Fuel Meat Cladding Water Fast E E E E-04 Absorption Neutron production Epithermal E E E E-04 Thermal E E E E-02 Total E E E E-02 Fast E Epithermal E Thermal E Total E Fast E E E E+00 Total Epithermal E E E E+00 Thermal E E E E+00 Total E E E E+01 Table 4-49: Reaction rates deviation between 280G and 12G for 8.5% wt. case Reaction Rate Energy Range Zr Rod Fuel Meat Cladding Water Fast Epithermal Absorption Thermal Total Fast Neutron Epithermal production Thermal Total Fast Epithermal Total Thermal Total

103 Table 4-50: DORT calculation with 280-group cross section library for 12% wt. case Reaction Rate Energy Range Zr Rod Fuel Meat Cladding Water Fast E E E E-04 Epithermal E E E E-04 Absorption Thermal E E E E-02 Total E E E E-02 Fast E Neutron Epithermal E production Thermal E Total E Fast E E E E+00 Epithermal E E E E+00 Total Thermal E E E E+00 Total E E E E+01 Table 4-51: DORT calculation with 12-group cross section library for 12% wt. case Reaction Rate Energy Range Zr Rod Fuel Meat Cladding Water Fast E E E E-04 Epithermal E E E E-04 Absorption Thermal E E E E-02 Total E E E E-02 Fast E Neutron Epithermal E production Thermal E Total E Fast E E E E+00 Epithermal E E E E+00 Total Thermal E E E E+00 Total E E E E+01 89

104 Table 4-52: Reaction rates deviation between 280G and 12G for 12% wt. case Reaction Rate Energy Range Zr Rod Fuel Meat Cladding Water Fast Epithermal Absorption Thermal Total Fast Neutron Epithermal production Thermal Total Fast Epithermal Total Thermal Total Two-Dimensional Cross Section Generation for Other Materials For other materials, which are not fissile material, we use the color-set approach model for cross section generation. The model contains the fissile material portion in order to produce sources to the problem. Here, 280G (fine group) and 12G (broad group) structures with S10 Square-Legendre-Chebyshev quadrature set are considered. The MCNP eigenvalues and reaction rates are used as the reference Graphite The 2-D color-set graphite model is illustrated in Figure The total number of cells is 148x118 cells. It is modeled with a uniform mesh distribution with 0.03 cm mesh size. The 12G broad-group library is obtained by 280G flux distributions obtained from 280G DORT calculations. 90

105 Figure 4-12: 2-D model for graphite XS generation Table 4-53 compares the eigenvalues from MCNP and DORT calculations. Table 4-54 through Table 4-61 give the reaction rates. These results demonstrate that 280G and 12G structures are in good agreement in eigenvalues and reaction-rate comparisons. The scattering order does not have effect on the 2-D graphite cross-section model. Comparing DORT with MCNP, DORT agrees well with MCNP in eigenvalue. Large deviations take place in fast and epithermal energy ranges for most of regions; however, they are not the main contribution to the problem, which are about 1 or 2 order of magnitude smaller than the reaction rates in the thermal energy range. 91

106 Table 4-53: Eigenvalue results for graphite cross section generation model CODE K INF REL. DEVIATION IN PCM OF ΔK/K MCNP ± (3σ) - DORT-280G,S10P DORT-280G,S10P DORT- 12G,S10P Table 4-54: MCNP reaction rates Gra_left Clad_left Water_left Gra_top Clad_top Water_top Abs_Fast 6.05E E E E E E-04 Abs_Epi 6.13E E E E E E-04 Abs_Thermal 9.54E E E E E E-02 Abs_Total 1.02E E E E E E-02 Tot_Fast 3.26E E E E E E-01 Tot_Epi 5.10E E E E E E+00 Tot_Thermal 1.75E E E E E E+01 Tot_Total 2.59E E E E E E+01 Table 4-55: DORT, 280GP3 reaction rates Gra_left Clad_left Water_left Gra_top Clad_top Water_top Abs_Fast 5.60E E E E E E-04 Abs_Epi 6.07E E E E E E-04 Abs_Thermal 9.58E E E E E E-02 Abs_Total 1.02E E E E E E-02 Tot_Fast 3.27E E E E E E-01 Tot_Epi 5.11E E E E E E+00 Tot_Thermal 1.74E E E E E E+01 Tot_Total 2.58E E E E E E+01 92

107 Table 4-56: DORT, 280GP1 reaction rates Gra_left Clad_left Water_left Gra_top Clad_top Water_top Abs_Fast 5.62E E E E E E-04 Abs_Epi 6.07E E E E E E-04 Abs_Thermal 9.58E E E E E E-02 Abs_Total 1.02E E E E E E-02 Tot_Fast 3.27E E E E E E-01 Tot_Epi 5.11E E E E E E+00 Tot_Thermal 1.74E E E E E E+01 Tot_Total 2.58E E E E E E+01 Table 4-57: DORT, 12GP1 reaction rates Gra_left Clad_left Water_left Gra_top Clad_top Water_top Abs_Fast 5.63E E E E E E-04 Abs_Epi 6.07E E E E E E-04 Abs_Thermal 9.58E E E E E E-02 Abs_Total 1.02E E E E E E-02 Tot_Fast 3.27E E E E E E-01 Tot_Epi 5.11E E E E E E+00 Tot_Thermal 1.74E E E E E E+01 Tot_Total 2.58E E E E E E+01 Table 4-58: Percentage deviation between DORT, 280GP3 and MCNP Gra_left Clad_left Water_left Gra_top Clad_top Water_top Abs_Fast Abs_Epi Abs_Thermal Abs_Total Tot_Fast Tot_Epi Tot_Thermal Tot_Total

108 Table 4-59: Percentage deviation between DORT, 280GP1 and MCNP Gra_left Clad_left Water_left Gra_top Clad_top Water_top Abs_Fast Abs_Epi Abs_Thermal Abs_Total Tot_Fast Tot_Epi Tot_Thermal Tot_Total Table 4-60: Percentage deviation between DORT, 12GP1 and MCNP Gra_left Clad_left Water_left Gra_top Clad_top Water_top Abs_Fast Abs_Epi Abs_Thermal Abs_Total Tot_Fast Tot_Epi Tot_Thermal Tot_Total Table 4-61: Percentage deviation between DORT, 12GP1 and 280GP1 Gra_left Clad_left Water_left Gra_top Clad_top Water_top Abs_Fast Abs_Epi Abs_Thermal Abs_Total Tot_Fast Tot_Epi Tot_Thermal Tot_Total

109 4.5.2 Control Rod The 2-D color-set control rod model is illustrated in Figure The total number of cells is 148x118 cells. It is modeled with a uniform mesh distribution with 0.03 cm mesh size. Using 280G cross section library, DORT calculated the 280G flux spectrum. The 280G cross sections were collapsed to 12G broad group cross sections. Figure 4-13: 2-D model for control rod XS generation Table 4-62 shows the eigenvalues calculated from DORT and MCNP. The MCNP calculation was performed with 3000 cycles with 100 inactive cycles and 5000 histories/cycles. Table 4-63 to Table 4-66 show reaction rates for each energy range and comparisons for each case. The scattering order has the effect on the eigenvalue. The differences of results from MCNP are 524 pcm with P1 scattering order comparing to 66 pcm for P3 scattering order. Thus, we use flux distribution P3 scattering case to collapse to broad group structure. 95

110 Table 4-62: Eigenvalues calculated by DORT and MCNP K inf Rel. Dev. from MCNP in pcm of Δk/k MCNP ± (3σ) - DORT (280G, S10,P1) DORT (280G, S10,P3) Table 4-63: Reaction rates calculated by MCNP Reaction Type B 4 C Clad (B 4 C) Water (B 4 C) Abs_Fast 1.01E E E-04 Abs_Epi 9.99E E E-04 Abs_Thermal 9.68E E E-03 Abs_Total 2.07E E E-03 Tot_Fast 2.58E E E-01 Tot_Epi 2.55E E E-01 Tot_Thermal 9.87E E E-01 Tot_Total 6.12E E E+00 Table 4-64: The reaction rates calculated by DORT with 280 groups, S10 quadrature order P1 P3 Reaction Type B 4 C Clad (B 4 C) Water (B 4 C) B 4 C Clad (B 4 C) Water (B 4 C) Abs_Fast 1.02E E E E E E-04 Abs_Epi 1.02E E E E E E-04 Abs_Thermal 9.65E E E E E E-03 Abs_Total 2.08E E E E E E-03 Tot_Fast 2.62E E E E E E-01 Tot_Epi 2.60E E E E E E-01 Tot_Thermal 9.84E E E E E E-01 Tot_Total 6.20E E E E E E+00 96

111 Table 4-65: Percent deviation of reaction rates between DORT 280G and MCNP P1 P3 Reaction Type B 4 C Clad (B 4 C) Water (B 4 C) B 4 C Clad (B 4 C) Water (B 4 C) Abs_Fast Abs_Epi Abs_Thermal Abs_Total Tot_Fast Tot_Epi Tot_Thermal Tot_Total Table 4-66: Percent deviation of reaction rates between DORT 280GP1 and 280GP3 Reaction Type B 4 C Clad (B 4 C) Water (B 4 C) Abs_Fast Abs_Epi Abs_Thermal Abs_Total Tot_Fast Tot_Epi Tot_Thermal Tot_Total The reaction rate comparisons between DORT 280GP1 and 280GP3 are in good agreement except the absorption rate of B 4 C in epithermal energy range, which of course affects the value of eigenvalue. The reaction rate comparisons between DORT and MCNP calculations show considerable deviation in cladding region for the whole range of energy and in thermal energy range of water region. This could be expected since the mean-free-path (mfp) of absorber (B 4 C) is very small ~5.0E-4 cm. in 280-group structure but the mesh size in B 4 C region is 0.03 cm, which is about 60 times larger than the mfp. 97

112 The neutrons most likely may not be able to escape from the absorber region once entered. The absorption rates in each energy range are plotted as a function of radius as illustrated in Figure The thermal absorption reaction rate drop dramatically. This phenomenon explains the spatial self-shielding effect in absorber. 1.00E E E-02 Absorption Reaction Rate 1.00E E E-05 Thermal Epithermal Fast 1.00E E E Radius (cm) Figure 4-14: Absorption reaction rate as a function of B 4 C radius In order to demonstrate what we have presumed for deviation in cladding region. The outer bound of the B 4 C rod is refined. It should be note that we are not refining the whole model in order to save memory and computational time and since the thermal flux gets absorbed mostly in the outer bound as shown above. There are three models in this 98

113 study. Model#1 is a base model from previous calculations. It has a uniform mesh distribution throughout the model. The mesh size for this model is cm. Model#2 is the model that cells in the outer bound of B 4 C rod is refined to cm. Model#3 is the model that cells in the outer bound of B 4 C rod is refined to cm. Table 4-67 shows k inf results predicted by DORT with 280 groups and relative deviation from MCNP. The reaction rates for each energy range and comparisons for Model #1 were shown previously in Table 4-64 and Table Table 4-68 to Table 4-71 show reaction rates for each energy range and comparisons for Models #2 and #3. We observed the improvement of results in cladding region when the cells in B 4 C has refined. Table 4-67: K inf results predicted by DORT with 280G K inf (Rel. Dev. from MCNP in pcm of Δk/k) P1 P3 Model #1 (148x118 cells) (-524) (66) Model # (170x138 cells) Model #3 (347x304 cells) (-458) (-447) (134) (144) Table 4-68: Reaction rates calculated by DORT with 280 groups, S10 quadrature order for Model#2 P1 P3 Reaction Type B 4 C Clad (B 4 C) Water (B 4 C) B 4 C Clad (B 4 C) Water (B 4 C) Abs_Fast 1.02E E E E E E-04 Abs_Epi 1.02E E E E E E-04 Abs_Thermal 9.64E E E E E E-03 Abs_Total 2.08E E E E E E-03 Tot_Fast 2.62E E E E E E-01 Tot_Epi 2.60E E E E E E-01 Tot_Thermal 9.83E E E E E E-01 Tot_Total 6.20E E E E E E+00 99

114 Table 4-69: Reaction rates calculated by DORT with 280 groups, S10 quadrature order for Model#3 P1 P3 Reaction Type B 4 C Clad (B 4 C) Water (B 4 C) B 4 C Clad (B 4 C) Water (B 4 C) Abs_Fast 1.03E E E E E E-04 Abs_Epi 1.02E E E E E E-04 Abs_Thermal 9.66E E E E E E-03 Abs_Total 2.09E E E E E E-03 Tot_Fast 2.62E E E E E E-01 Tot_Epi 2.61E E E E E E-01 Tot_Thermal 9.85E E E E E E-01 Tot_Total 6.21E E E E E E+00 Table 4-70: Percent deviation of reaction rates between DORT 280G S10 Model #2 and MCNP P1 P3 Reaction Type B 4 C Clad (B 4 C) Water (B 4 C) B 4 C Clad (B 4 C) Water (B 4 C) Abs_Fast Abs_Ephi Abs_Thermal Abs_Total Tot_Fast Tot_Ephi Tot_Thermal Tot_Total

115 Table 4-71: Percent deviation of reaction rates between DORT 280G S10 Model #3 and MCNP P1 P3 Reaction Type B 4 C Clad (B 4 C) Water (B 4 C) B 4 C Clad (B 4 C) Water (B 4 C) Abs_Fast Abs_Ephi Abs_Thermal Abs_Total Tot_Fast Tot_Ephi Tot_Thermal Tot_Total The 280G cross sections were collapsed to 12G broad group cross sections using Model#3. DORT calculated the 280G flux spectrum. The eigenvalues from 280G-fine groups and 12G-broad groups agree well under 100 pcm as shown in Table Table 4-73 and Table 4-74 demonstrate reaction rates for each energy range and comparisons of 12G. They agree well with the 280GP3 case. Table 4-72: Eigenvalues calculated by DORT and MCNP K inf Rel. Dev. from MCNP in pcm of Δk/k MCNP ± (3σ) - DORT (280G, S10,P3) DORT (12G, S10,P3)

116 Table 4-73: Reaction rates calculated by DORT with 12 groups, S10 quadrature order and P3 scattering order Reaction Type B 4 C Clad (B 4 C) Water (B 4 C) Abs_Fast 1.01E E E-04 Abs_Epi 1.01E E E-04 Abs_Thermal 9.60E E E-03 Abs_Total 2.07E E E-03 Tot_Fast 2.58E E E-01 Tot_Epi 2.61E E E-01 Tot_Thermal 9.79E E E-01 Tot_Total 6.17E E E+00 Table 4-74: Percent deviation of reaction rates between DORT 12GP3 and 280GP3 Reaction Type B 4 C Clad (B 4 C) Water (B 4 C) Abs_Fast Abs_Epi Abs_Thermal Abs_Total Tot_Fast Tot_Epi Tot_Thermal Tot_Total Cross-Section Homogenization After completing the study of broad-group cross-section library for TRIGA fuel pin cell, studies on spatial homogenization were performed. The ideal concept of the homogenization is to preserve all the reaction rates in the problem. We utilize the scalar flux weighting method to combine the material regions as shown in Equation 4-1. Two sets of homogenized cross sections are calculated: i) one which combines 3 regions (Zr Rod, UZrH fuel, and SS304 Cladding) as shown in Figure 4-15 and ii) the other, which 102

117 combines 4 regions (Zr Rod, UZrH fuel, SS304 Cladding, and H 2 O) as shown in Figure 4-16 for 12-energy group structure. Σ g = nzone Eg 1 i= 1 Eg nzone Eg 1 i= 1 3 i r r de d rσ (, E) φ(, E) Vi Eg 3 r de d rφ (, E) Vi Equation 4-1 Zr+UZrH+SS304 Zr+UZrH+SS304 Figure 4-15: Three-Region Homogenization Zr+UZrH+SS304+ H 2 O Figure 4-16: Four-Region Homogenization 103

118 Table 4-75 and Table 4-76 give the k inf of three-region and four-region homogenization approaches using scalar flux weighting, respectively. The differences of results are ~60 pcm comparing the three-region homogenization with heterogeneous geometry, and ~200 pcm comparing with MCNP solution. For the four-region homogenization, the differences are ~200 pcm comparing the four-region homogenization with heterogeneous geometry, and ~350 pcm comparing with MCNP solution. The deviations of homogenized cross sections with MCNP are less than heterogeneous solutions, since the negative deviation between heterogeneous and MCNP compensate with the positive deviation between heterogeneous and homogenized calculations. These differences are higher than three-region homogenization. Another observation is that all quadrature orders including S4 give almost the same results for k inf. As we combine water into the homogenized region, the moderation and scattering properties of water are smeared with the fuel meat and cladding as one material. Table 4-75: K inf calculated by DORT (3 region combination) Heterogeneous XS Homogenized XS k inf Rel. Dev. in pcm of Δk/k (Homo. VS. Het.) Rel. Dev. in pcm of Δk/k (Homo.VS. MCNP) S4P S6P S8P S10P S12P S14P S16P

119 Table 4-76: K inf calculated by DORT (4 region combination) Heterogeneous XS Homogenized XS k inf Rel. Dev. in pcm of Δk/k (Homo. VS. Het.) Rel. Dev. in pcm of Δk/k (Homo.VS. MCNP) S4P S6P S8P S10P S12P S14P S16P Summary In this chapter the fine energy-group and broad-energy-group structures for 2-D cross-section generation have been selected in fast, epithermal, and thermal energy ranges by the CPXSD methodology using different objectives corresponding of each energy range. The scalar flux weighting technique is utilized in collapsing fine- to broadgroup libraries. Results indicate very good agreement between 280 fine- and 12 broadgroup structures. The studied fine- and broad- group structures were also applied in non-fissile material, i.e., graphite and control rod. Results indicate good agreement in eigenvalues compared with MCNP. The scattering order has a strong effect on a control rod model but not in the graphite model. 105

120 For the homogenization, the results are in good agreement for three-region homogenization approach. For four-region homogenization approach, the quadrature order does not affect the solution. 106

121 CHAPTER 5 Three-Dimensional Cross Section Generation In the previous chapter, the 2-D cross section group structure was established with 280 fine groups and 12 broad groups. In this chapter, the same CPXSD methodology, adapted for criticality problem, will be used to study the actual 3-D cross section generation. We start with the parametric optimization study for the S N methods and follow with the fine- and broad-group structure selection process. The 8.5% TRIGA fuel cell is selected for this study. First, the 3-D fine group structure will be constructed. Finally, the 3-D broad-group structure will be established and compared with 2-D broadgroup structure. 5.1 Three-dimensional model for Cross-Section Generation for Fuel Element One eight of a hexagonal unit cell has been modeled for the 3-D cross-section generation study by taking the advantage of the model symmetry as illustrated in Figure 5-1. The reflective boundary is applied for all the surfaces except the top surface. Table 5-1 shows the material data that has been used in the cross-section generation calculations. As the reference, the Monte Carlo MCNP5 calculation is performed with 5000 histories, 3000 cycles and 100 inactive cycles with continuous energy cross section library. The predicted reference eigenvalue is ± (3σ). 107

122 Fuel Graphite Figure 5-1: 3D cross section generation model Unit: cm. Table 5-1: Material density of the fuel elements Nuclide Density (atoms/barn-cm) Fuel 12 wt.% 8.5 wt.% H Zr U U U U Reflector H O Zr Rod Zr SS304 SS Graphite C

123 5.2 Parametric Studies Spatial Mesh, Angular Quadrature, and Scattering Order Studies In this section, the spatial discretization, angular quadrature, and scattering order of cross sections are studied for the 3-D case. The 238-group structure is used throughout this set of studies. Generally, performing this type of studies simultaneously is cumbersome for 3-D problems. Thus, the sensitivity studies for spatial meshes, angular quadrature set, and scattering order were done, separately. A. Scattering Order Study The scattering orders that are used to perform sensitivity studies are P1, P3, and P5. The S4, S6, and S8 orders are examined for fully symmetric and Square Legendre- Chebyshev, respectively. The fine mesh model used in this study is 20x23x37 (x,y,z). The TORT results and relative deviation in pcm from the reference MCNP5 results are provided in Table 5-2 and Table 5-3 for fully symmetric and Square Legendre- Chebyshev quadratures, respectively. They are summarized graphically in Figure 5-2 and Figure

124 Table 5-2: TORT results with 238-energy group XS, Level-Symmetric S N Order Scattering K eff Conv. Out. Iter. Rel. Dev. in pcm of Δk/k S4 P E S6 P E S8 P E S4 P E S6 P E S8 P E S6 P E S8 P E Table 5-3: TORT results with 238-energy group XS, SLC S N Order Scattering K eff Conv. Out. Iter. Rel. Dev. in pcm of Δk/k S4 P E S6 P E S8 P E S4 P E S6 P E S8 P E S6 P E S8 P E

125 Level Symmetric P1 P3 P Eigenvalue S4 S6 S8 Quadrature Order Figure 5-2 Eigenvalue behavior under variation of scattering order and level symmetric quadrature Square Legendre Chebychev P1 P3 P Eigenvalue S4 S6 S8 Quadrature Order Figure 5-3 Eigenvalue behavior under variation of scattering order and Square Legendre-Chebyshev quadrature 111

126 For both level-symmetric and Square Legrendre-Chebyshev types of quadrature, the results show that scattering order has the effect on eigenvalue predictions for TRIGA cell in 3-D geometry with a systematical bias about 160 pcm between P1 and P3. However, using scattering order higher than P3 does not provide a significant improvement on k eff predictions. The P3 and P5 solution yield almost the same value of k eff. As a result, the P3 scattering order should be used in order to get a good solution. B. Spatial Mesh Study In 3-D problems, not only the study in radial directions but also the study in axial direction has to be performed. The Square Legendre-Chebyshev was used to study this problem with S8 order. In order to save computing time, the P1 scattering order was used to perform sensitivity studies. All the results were compared with the continuous energy MCNP result, which is used as a reference solution in this study. First, we performed the study on the radial directions by fixing the number of axial meshes. After we established the radial meshes, then the axial mesh was studied. Three 3-D fine-mesh models have been developed with different grid intervals to study for radial direction refinement. 1) cells: 20 x-axis, 23 y-axis, 37 z-axis 2) cells: 48 x-axis, 59 y-axis, 37 z-axis 3) cells: 54 x-axis, 69 y-axis, 37 z-axis The TORT results and relative deviations as compared to the MCNP results in pcm are provided in Table

127 Table 5-4: TORT results with 238-energy group XS, S8 (SLC), P1 Meshing k eff Conv. Rel. Dev. in pcm of Δk/k 20x23x E x59x E x69x E It was found that k eff converges with the mesh refinement higher than 48x59 cells in radial directions as shown in Figure 5-4. Thus, this radial meshing will be used further for axial mesh study keff x23x37 48x59x37 54x69x37 Mesh Model Figure 5-4: Eigenvalue behavior with different radial-mesh model 113

128 Four 3-D fine-mesh models have been developed with different grid intervals in axial direction. 1) cells: 48 x-axis, 59 y-axis, 19 z-axis 2) cells: 48 x-axis, 59 y-axis, 37 z-axis 3) cells: 48 x-axis, 59 y-axis, 55 z-axis 4) cells: 48 x-axis, 59 y-axis, 70 z-axis The results are presented in Table 5-5. It was found that k eff converges with the mesh refinement higher than 55 cells in axial directions as shown in Figure 5-5. Thus, the optimum mesh-model for 3-D TRIGA cell was established with 48x59x55 cells. Table 5-5: TORT results with 238-energy group XS, S8 (SLC), P1 Meshing k eff Conv. Rel. Dev. in pcm of Δk/k 48x59x E x59x E x59x E x59x E

129 keff x59x19 48x59x37 48x59x55 48x59x70 Mesh Model Figure 5-5: Eigenvalue behavior with different axial-mesh model Qudrature Order Determination The optimum mesh model for 3-D TRIGA cell was established with 48x59x55 cells. Here, we study the optimum quadrature set for 3-D TRIGA cell. The S4, S6, S8, and S10 with SLC were used. Table 5-6 gives TORT calculations and deviations from the MCNP results. The results show that k eff predictions are insensitive to the quadrature order higher than S6. Table 5-6: TORT results with 238-energy group XS and 48x59x55 cells S N order k eff Rel. Dev. in pcm of Δk/k with MCNP S S S S

130 The reaction rate comparison Six-radial detectors were defined in each selected axial mesh within the fuel and graphite region to compare the neutron production reaction rate for fuel region and the absorption reaction rate for graphite region in Table 5-7. Overall, the reaction rates change relatively less for higher quadrature order. As a result, S8 has been selected to be used for further study. Table 5-7: Neutron production reaction rate and percent deviations Position Nu-fission raction rate % Deviation X Y Z S4 S6 S8 S10 S6 VS S4 S8 VS S6 S10 VS S f 2.551E E E E f 3.210E E E E f 2.552E E E E f 2.686E E E E f 3.165E E E E f 3.234E E E E f 2.237E E E E f 2.816E E E E f 2.238E E E E f 2.357E E E E f 2.779E E E E f 2.839E E E E f 1.747E E E E f 2.202E E E E f 1.749E E E E f 1.843E E E E f 2.183E E E E f 2.232E E E E g 1.991E E E E g 2.055E E E E g 1.940E E E E g 2.044E E E E g 2.144E E E E g 2.100E E E E Note: f is the fuel level, g is the graphite level. The following set of figures display the flux distribution for each quadrature to observe the ray effect. We selected group 23 rd to represent the fast range of energy and 116

131 group 212 th to represent the thermal range of energy. Level 15 th represents the fuel part and level 50 th represents the graphite part. S4 S6 1.a) zlev15g23 1.b) zlev15g212 1.c) zlev50g23 1.d) zlev50g212 S8 2.a) zlev15g23 2.b) zlev15g212 2.c) zlev50g23 2.d) zlev50g212 S10 3.a) zlev15g23 3.b) zlev15g212 3.c) zlev50g23 3.d) zlev50g212 4.a) zlev15g23 4.b) zlev15g212 4.c) zlev50g23 4.d) zlev50g212 Figure 5-6: Flux distribution for each quadrature order 117

132 5.3 Fine- Group Structure for TRIGA Using the procedure of the CPXSD methodology developed further for criticality problem, the fine-group structure for TRIGA cross-section generation is obtained. The 238-group SCALE library is used as a starting group structure. The 238-group cross sections are generated. Initially, the 238-group structure was divided into 3 major ranges of energy: fast (0.1 MeV to 20 MeV), epithermal (3 ev to 0.1 MeV), and thermal (1E-05 ev to 3 ev). We established two criteria for obtaining a fine group structure. The first criterion is 10 pcm relative deviation of Δk/k and the second criterion is 1% relative deviation of objective reaction rates. The objective reaction rates are different for each range of energy. Using the flux and adjoint function moments computed from the transport calculations with TORT, the Cg s are calculated. Depending on the magnitude of the Cg s per group, the group structure is refined for each energy range. The groups corresponding to large C g s were partitioned into more groups. The group with the highest Cg was subdivided into a number of groups, and the remaining groups were divided into fewer groups based on the ratio of their Cg to the maximum Cg. This study is performed using 1/8 of 8.5% fuel cell with 48x59x55 fine cells in x, y, and z direction with S8 (SLC) quadrature order and P1 scattering order Fast Group Refinement In this section a group structure in the fast energy range between 0.1 and 20 MeV is derived. The 238-group SCALE library is used as a starting group structure with 44 groups in fast energy range, 104 groups in epithermal range, and 90 groups in thermal 118

133 range. The 238-group Cg s are calculated using the normalized νσ f as the adjoint source to perform the adjoint transport calculation. The point-wise cross section of U 238 (n,f) is used to consider the group boundaries. The objectives are eigenvalue and neutron production reaction rate of U 238. The new group structures are generated. Table 5-8 gives the number of groups in fast energy range that we obtained from the group refinement process. Table 5-8: Fine groups generated in the fast energy range Group Structure Number of Groups in Different Energy Ranges Number Fast Epithermal Thermal Total The importance of groups in fast energy range, between 0.1 and 20 MeV, of 238- group and 246-group structures are plotted in Figure 5-7. The plot shows that when the groups that have more importance are refined, the importance of those groups is decreased. 119

134 4.00E E E-02 Importance (E) 2.50E E E groups 246 groups 1.00E E E+00 1.E-01 1.E+00 1.E+01 1.E+02 Energy (MeV) Figure 5-7: Importance in groups of 238G and 246G libraries The eigenvalues are calculated and compared between the group structures. For the 246G and 274G comparison, Table 5-9 shows that percent relative difference of eigenvalue is less than 10 pcm and the percentage relative deviation of U 238 (n,νσ f ) is 0.199%. Consequently, we selected the 246-group structure, which contains 52 groups in fast energy range, for further group refinement in the epithermal energy range. Table 5-9: Eigenvalue results of fine group energy for 8.5% wt. case Group k inf (S8P1) Rel. Dev. in pcm of Δk/k With previous νσ f rate of U 238 > 0.1 MeV %Rel. Dev. With previous group group E E E

135 5.3.2 Epithermal-Group Refinement In this section a group structure in the epithermal energy range between 3eV and 0.1 MeV is derived. The 246-group structure from the fast group refinement is used as a starting group structure with 52 groups in the fast energy range, 104 groups in the epithermal range, and 90 groups in the thermal range. The 246-group Cg s are calculated using the summation of the normalized νσ f and down-scattering cross section of H in ZrH from the epithermal group to the thermal group as adjoint source to perform the adjoint transport calculation. The absorption point-wise cross-section of U 238 is used to consider the group boundaries. The objectives are eigenvalue, down-scattering reaction rate of H in ZrH from the epithermal energy range to the thermal energy range and absorption reaction rate of U 238. Table 5-10 shows the number of groups in the epithermal energy range that we obtained from the group refinement process. The importance of groups in epithermal energy range, between 3 ev and 0.1 MeV, of 246-group structure are plotted in Figure 5-8. Table 5-10: Fine groups generated in the epithermal energy range Group Structure Number of Groups in Different Energy Ranges Number Fast Epithermal Thermal Total

136 1.40E E E-02 Importance(E) 8.00E E E E E E E E E E E E+00 Energy(MeV) Figure 5-8: Importance in groups of 246G libraries The eigenvalues were calculated and compared between the group structures in Table For the 246G and 294G comparison, the relative difference of eigenvalues are less than 10 pcm and the percentage relative deviation of U 238 (n,abs) and downscattering of H in ZrH from the epithermal range to the thermal range are less than 1.0% as demonstrated in Table Consequently, we selected the 246-group structure, which contains 104 groups in epithermal energy range, for further group refinement in the thermal energy range. 122

137 Table 5-11: Eigenvalue results of fine group energy for 8.5% wt. case Group k inf (S8P1) Rel. Dev. in pcm of Δk/k With previous group Table 5-12: Reaction rate comparison for 8.5% wt. case Down-scat. %Rel. Dev. U 238 (n,abs) %Rel. Dev. Group Of H in ZrH With previous group With previous group Thermal-Group Refinement In this section, a group structure in the thermal energy range between 1E-5 to 3 ev is derived. The 246-group structure from the fast and epithermal group refinements is used as a starting group structure with 52 groups in the fast energy range, 104 groups in the epithermal range, and 90 groups in the thermal range. The 246-group Cg s are calculated using the summation of the normalized νσ f and up-scattering cross section of H in ZrH as the adjoint source to perform the adjoint transport calculation. The inelastic scattering point-wise cross-section of H in ZrH is used to consider the group boundaries. The objectives are eigenvalue, neutron production reaction rate of U 235, and up-scattering reaction rate of H in ZrH in the thermal energy range. 123

138 Table 5-13 shows the number of groups in thermal energy range that we obtained from the group refinement process. The importance of groups in the thermal energy range, between 1E-5 and 3 ev, of 246-group, 254-group and 280-group structures are plotted in Figure 5-9. Table 5-13: Fine groups generated in the thermal energy range Group Structure Number of Groups in Different Energy Ranges Number Fast Epithermal Thermal Total E E E E-02 Importance(E) 2.00E E G 254G 280G 1.00E E E E E E E E E-05 Energy(MeV) Figure 5-9: Importance in groups of 246G, 254G and 280G libraries 124

139 The eigenvalue results and comparisons for fine group energy in thermal range of 8.5% case are given in Table Table 5-15 shows the result of up-scattering of H in ZrH and neutron-production reaction rates from each group structure library and the comparisons. Comparing between 280G and 336G, the percent relative difference of eigenvalues, the U 235 neutron production rate and the up-scattering of H in ZrH are within the criteria. The 280-group structure is selected to be our final fine group structure. Table 5-16 lists energy group boundaries of the 280-group structure. Table 5-14: Eigenvalue results for fine group energy in thermal range 8.5% case Group k eff (S8P1) Rel. Dev. in pcm of Δk/k With previous group Group Table 5-15: Reaction rate comparison of 8.5% case %Rel. Dev. U 235 (n, νσ f ) With previous group Up-scat. Of H in ZrH %Rel. Dev. With previous group E : E E : E E : E Note: 1 The reaction rate was calculated in a group-collapsing method to be compared with the previous group 125

140 Table 5-16: Group structure of the 280 fine groups Energy Group Number Upper Energy(MeV) Energy Group Number Upper Energy(MeV) Energy Group Number Upper Energy(MeV) E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E

141 Energy Group Number Upper Energy(MeV) Energy Group Number Upper Energy(MeV) Energy Group Number Upper Energy(MeV) E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E

142 Energy Group Number Upper Energy(MeV) Energy Group Number Upper Energy(MeV) Energy Group Number Upper Energy(MeV) E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E-08 The 280-fine-group cross-section library was selected to be a fine-group structure for the TRIGA reactor based on the CPXSD methodology. This 3-D fine-group structure is the same as in the 2-D study. In conclusion, a methodology is established to generate the fine-group cross-section library and applied to an example of 8.5% wt. TRIGA 3-D fuel cell. 5.4 Cross-Section Collapsing In this section, the 280-group structure was collapsed into a broad-group structure. With the same approach as developing the fine-group structure, we established two criteria to obtain a broad group structure. The first criterion is 10 pcm relative deviation of Δk/k and the second criterion is 1% relative deviation of objective reaction rates. The objective reaction rates are different for each range of energy. The U 238 (n,νσ f ) is considered in the fast energy range, the down-scattering reaction rates of H in ZrH and U 238 (n,a) are considered in the epithermal energy range, and the U 235 (n,νσ f ) and the thermal up-scattering reaction rates of H in ZrH are considered in the thermal range. The 128

143 group collapsing started with fast energies by initiating a very-broad-group structure and using the same fine-group structure in the epithermal and thermal energies. Then, the aforementioned contributon approach was used to refine the broad-group structure. This process is repeated until the two criteria were met, and consequently a new broadgroup structure for the fast energies was obtained. With this new fast broad group structure, we continue the same process for the epithermal and thermal energy ranges Fast-Group Collapsing and Axial Nodalization Study In fast energy range, we first combined all the energy groups into one group. A new group library contains 229 groups. This group structure was also used to study the axial nodalization for cross section collapsing. Three cases were performed as shown in Figure In Case 1, we used the fluxes of the material-wise full axial length to collapse the cross sections. In Case 2, a 4-cm node was used to collapse the cross sections. In Case 3, a 1-cm node was used to collapse the cross sections. Table 5-17: Number of groups for each energy range Group Structure Number of Groups in Different Energy Ranges Number Fast Epithermal Thermal Total

144 Case1 Case 2 Case 3 Figure 5-10 Axial mesh size used in nodal length collapsing study Table 5-18 shows the eigenvalues for each case. Table 5-19 shows the minimum and maximum of mesh-wise reaction rate deviations for each layer between case 2 and case1. Table 5-20 shows the minimum and maximum of mesh-wise reaction rate deviations for each layer between cases 3 and 2. We observed that the eigenvalue and reaction rate deviations between these three cases are fairly small and less than 10 pcm. From this study, a full axial length can be used to collapse the cross sections. Comparing between 280G and 229G, the relative deviation is large. As a result, the one group structure is not enough in the fast energy range. Further group refinement has to be studied. 130

145 Table 5-18: Eigenvalue results for 3D, 8.5% fuel cell Group k eff (S8P1) Rel Dev. in pcm of Δk/k with 280G (1) (2) (3) Note: (1) using the full length of axial direction to collapse the cross sections (2) using 4-cm node in axial direction to collapse the cross sections (3) using 1-cm node in axial direction to collapse the cross sections Table 5-19:The minimum and maximum of mesh-wise reaction rate deviations for each layer between case 2 and case1 neutronproduction absorbtion rate rate total rate layer min max min max min max

146 Table 5-20:The minimum and maximum of mesh-wise reaction rate deviations for each layer between case 3 and case2 neutronproduction absorbtion rate rate total rate layer min max min max min max

147

148 A refining process was repeated until the criteria were met. Table 5-21 shows the number of groups that we obtained in the fast energy range. Table 5-22 shows the eigenvalue results of each group structure and U 238 (n, νσ f ) above 0.1MeV reaction rate and comparisons. We ended up placing 7 groups in fast energy range and obtaining a 235 group structure. Table 5-21: Number of groups for each energy range Group Structure Number of Groups in Different Energy Ranges Number Fast Epithermal Thermal Total Group Table 5-22: Eigenvalue results for 3D, 8.5% fuel cell k eff (S8P1) Rel Dev. in pcm of Δk/k with previous group U 238 (n,νσ f ) above 0.1 MeV %Rel. Dev.Reaction rate of U 238 (n,νσ f ) E (1) E E E E E Note: (1) using the full length of axial direction to collapse the cross sections 134

149 5.4.2 Epithermal Energy Range: In the next step, we developed a broad group structure in the epithermal energy range (3.0 ev to 0.1 MeV). The objective reaction rate for the epithermal energy range is the down-scattering reaction rates of H in ZrH. We initially have placed two energy groups in this range and ended up with 133-group structure. Table 5-23 shows the number of groups that were studied in the ephithermal energy range. Table 5-24 shows that relative difference of Δk/k is less than 10 pcm comparing between 135G and 137G. The percentage relative deviations of down-scattering reaction rate of H in ZrH and U 238 (n,a) are 0.0% as shown in Table As a result, we used 135-group structure to further collapse in the thermal energy range. Table 5-23: Number of groups for each energy range Group Structure Number of Groups in Different Energy Ranges Number Fast Epithermal Thermal Total Table 5-24: Eigenvalue results for 3D, 8.5% fuel cell Group k eff (S8P1) Rel Dev. in pcm of Δk/k with previous group

150 Table 5-25: Reaction rate comparison for broad group in epithermal range Down-scat. %Rel. Dev. U 238 (n,abs) %Rel. Dev. Group Of H in ZrH With previous group With previous group Thermal Energy Range: In the last step, we developed a broad group structure in the thermal energy range (1.0E-05 ev to 3.0 ev). The objective reaction rates of the thermal range are fission rate of U 235 and the thermal up-scattering reaction rate of H in ZrH. We initially introduced one energy group in this range and obtained 12-group structure. Then, we subdivided in the most important group into three groups and each time we generated a new broadgroup structure until the result met the criteria. Table 5-26 shows the number of groups that were refined in the thermal energy range. Table 5-27 shows that relative difference of the eigenvalue of 26-group structure and 28-group structure is 9 pcm. Table 5-28 demonstrates that the percentage relative deviation of U 235 (n,νσ f ) is 0.01% and the percentage relative deviation of upscattering rate is 0.0%. The 26-group structure was selected to be our final broad group structure for 3-D study, which has 14 groups more than the 12G structure for 2-D study. Table 5-29 lists energy boundaries of the 26-group structure. 136

151 Table 5-26: Number of groups for each energy range Group Structure Number of Groups in Different Energy Ranges Number Fast Epithermal Thermal Total Table 5-27: Eigenvalue results for 3D, 8.5% fuel cell Group k eff (S8P1) Rel Dev. in pcm of Δk/k With previous group

152 Table 5-28: Result comparison in thermal energy range %MaxRel. Dev. U 235 (n,νσ f ) In upscattering reaction rate rate 1E-05 to 3 ev Group Up-scattering of H in ZrH %Rel. Dev. U 235 (n,νσ f ) reaction rate E : E E : E E : E E : E E : E E : E E : E E : E Energy Group Number Table 5-29: Energy boundaries of 26-group structures Energy Group Number Energy Group Number Upper Energy(MeV) Upper Energy(MeV) Upper Energy(MeV) E E E E E E E E E E E E E E E E E E E E E E E E E E E-08 The TORT calculations were performed with S8P1 for both 280-group and 26- group structures. The results are given in Table The absolute relative deviations in Δk/k of 26-group as compared to the 280-group and continuous Monte Carlo calculations 138

153 are 56 pcm and 125 pcm, respectively. The errors from the comparison between the 26- group and the 280-group structure are less than the comparison between the 26-group structure and the continuous-energy MCNP solution. These differences are identified as the method difference between the deterministic (TORT) and statistical (MCNP) and the multigroup and continuous energy cross-section libraries. The absorption rate, neutron production, and total reaction rates are compared between the two cross-section libraries: 280G and 26G, in each energy range (fast, epithermal, and thermal) and region (Zr Rod, fuel meat, clad (fuel), water(fuel), graphite, clad(gra), and water(gra)). Table 5-31 through Table 5-33 give the reaction rates from MCNP for continuous energy, TORT for 280-group library and 26-group library, and their comparisons. The percentage of relative deviation between codes and libraries are presented in Table 5-34 through Table Compared to the MCNP continuous energy results, we observed large differences of absorption reaction rate in fast energy range for graphite region ~10% and in epithermal energy range for cladding region ~11%. We suspect that the cause of the differences may be due to group refinement process that focused on only the neutron production of U 238. Thus, we may obtain good agreement with our objectives while finding the large errors in other regions. However, those reaction rates are insignificant parts of total absorption reaction rates. They are smaller by 1 or 2 orders of magnitude. For the reaction rate comparisons between 280G- fine and 26G-broad groups, they show very good agreement for these selected reaction rates in each energy range with less than 1% difference. 139

154 Table 5-30: Eigenvalues calculated by TORT and MCNP k eff Rel. Dev. from MCNP in pcm of Δk/k Rel. Dev. from 280G in pcm of Δk/k MCNP ± (3σ) TORT (280G, S8,P1) TORT (26G, S8,P1) Table 5-31: MCNP calculation with continuous cross section library Reaction Rate Absorption Neutron production Total Energy Range Fast Epithermal Thermal Total Fast Epithermal Thermal Total Fast Epithermal Thermal Total Zr Rod Fuel Meat Clad Water Graphite Clad Water 6.54E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E

155 Table 5-32: TORT calculation with 280-group cross section library Reaction Rate Absorption Neutron production Total Energy Range Fast Epithermal Thermal Total Fast Epithermal Thermal Total Fast Epithermal Thermal Total Zr Rod Fuel Meat Clad Water Graphite Clad Water 6.52E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E-01 Table 5-33: TORT calculation with 26-group cross section library Reaction Rate Absorption Neutron production Total Energy Range Fast Epithermal Thermal Total Fast Epithermal Thermal Total Fast Epithermal Thermal Total Zr Rod Fuel Meat Clad Water Graphite Clad Water 6.50E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E

156 Table 5-34: Reaction rates deviation between 280G and MCNP Reaction Rate Absorption Neutron production Total Energy Range Fast Epithermal Thermal Total Fast Epithermal Thermal Total Fast Epithermal Thermal Total Zr Rod Fuel Meat Clad Water Graphite Clad Water Table 5-35: Reaction rates deviation between 26G and MCNP Reaction Rate Absorption Neutron production Total Energy Range Fast Epithermal Thermal Total Fast Epithermal Thermal Total Fast Epithermal Thermal Total Zr Rod Fuel Meat Clad Water Graphite Clad Water

157 Table 5-36: Reaction rates deviation between 26G and 280G Reaction Rate Absorption Neutron production Total Energy Range Fast Epithermal Thermal Total Fast Epithermal Thermal Total Fast Epithermal Thermal Total Zr Rod Fuel Meat Clad Water Graphite Clad Water Three-Dimensional Cross Section Model for Materials with Non-Fissile Element Non-fissile materials have to be modeled with the color set approach and with the 280 fine-group structure, which requires too large computational effort; hence, we decided to apply the developed 26 broad-group structure, collapsed from 280 groups, with 2-D flux spectrum for non-fissile material and compare with MCNP Control Rod The 3-D color-set control rod model is illustrated in Figure The total number of cells is 148x118x46. It is modeled with a uniform mesh distribution with 0.03 cm for radial mesh size and 0.5 cm for axial mesh size. 143

158 Fuel / B 4 C Graphite Figure 5-11: 3-D model for control rod XS generation Table 5-37 shows the eigenvalues calculated by TORT and MCNP. Table 5-38 to Table 5-42 show reaction rates for each energy range and comparisons for each case. We observed the same large deviation of reaction rate comparisons between TORT and MCNP calculations in cladding region for the whole range of energy and in thermal energy range of water region as observed in the 2-D case. The scattering order has the effect on the eigenvalue predictions. The differences of results as compared to MCNP are 568 pcm with P1 scattering order and 7 pcm for P3 scattering order. Table 5-37: Eigenvalues calculated by TORT and MCNP K inf Rel. Dev. from MCNP in pcm of Δk/k MCNP ± (3σ) - TORT (26G, S8,P1) TORT (26G, S8,P3)

159 Table 5-38: Reaction rates calculated by MCNP Reaction Type B 4 C Clad (B 4 C) Water (B 4 C) Abs_Fast 4.93E E E-06 Abs_Epi 4.98E E E-06 Abs_Thermal 5.21E E E-04 Abs_Total 1.07E E E-04 Tot_Fast 1.26E E E-02 Tot_Epi 1.26E E E-02 Tot_Thermal 5.31E E E-02 Tot_Total 3.05E E E-02 Table 5-39: Reaction rates calculated by TORT with 26 groups, S8 quadrature order and P1 scattering order Reaction Type B 4 C Clad (B 4 C) Water (B 4 C) Abs_Fast 5.01E E E-06 Abs_Epi 5.09E E E-06 Abs_Thermal 5.21E E E-04 Abs_Total 1.08E E E-04 Tot_Fast 1.28E E E-02 Tot_Epi 1.29E E E-02 Tot_Thermal 5.31E E E-02 Tot_Total 3.11E E E-02 Table 5-40: Reaction rates calculated by TORT with 26 groups, S8 quadrature order and P3 scattering order Reaction Type B 4 C Clad (B 4 C) Water (B 4 C) Abs_Fast 4.98E E E-06 Abs_Epi 5.01E E E-06 Abs_Thermal 5.21E E E-04 Abs_Total 1.07E E E-04 Tot_Fast 1.27E E E-02 Tot_Epi 1.28E E E-02 Tot_Thermal 5.31E E E-02 Tot_Total 3.08E E E

160 Table 5-41: Percent deviation of reaction rates between TORT 26GP1 S8 and MCNP Reaction Type B 4 C Clad (B 4 C) Water (B 4 C) Abs_Fast Abs_Epi Abs_Thermal Abs_Total Tot_Fast Tot_Epi Tot_Thermal Tot_Total Table 5-42: Percent deviation of reaction rates between TORT 26GP3 S8 and MCNP Reaction Type B 4 C Clad (B 4 C) Water (B 4 C) Abs_Fast Abs_Epi Abs_Thermal Abs_Total Tot_Fast Tot_Epi Tot_Thermal Tot_Total Two-Dimensional vs. Three-Dimensional Cross Sections Two-Dimensional vs. Three-Dimensional Flux Distribution Collapsing The 26-group structure 2-D and 3-D cross sections were used to study the effect of 2-D and 3-D flux distribution collapsing. TORT is used to perform the study using S8 Square Legendre-Chevbychev quadrature order and P1 scattering order for a 3-D pin cell as shown in Figure Table 5-43 shows the eigenvalue results of 2-D vs. 3-D flux cross-section collapsing cases. Table 5-44 shows the percentage deviation of reaction rates between 2-D and 3-D flux distribution collapsing cases. The 2-D and 3-D cross- 146

161 section collapsing cases agree well with each other in eigenvalue. The difference between the two cases is observed in the absorption rate in fast energy range of graphite. The 2-D case has less absorption rate than 3-D case by 14.86%. This difference can be attributed to the axial flux distribution effect, which is not present in the 2-D case. Fuel Graphite Figure 5-12: A pin cell model in axial direction Table 5-43: Eigenvalue results 2-D vs 3-D flux distribution collapsing cases CASE k eff Deviation in pcm of Δk/k MCNP ± (3σ) - TORT- 26GS8P1,3-DXS TORT- 26GS8P1,2-DXS Table 5-44: Percentage deviation of reaction rates between 2-D and 3-D crosssection collapsing cases Reaction Type Zr Fuel Clad_fuel Water_fuel Graphite Clad_gra Water_gra Abs_Fast Abs_Epi Abs_Thermal Abs_Total Nu-Fis_Fast Nu-Fis_Epi Nu-Fis_Thermal Nu-Fis_Total Tot_Fast Tot_Epi Tot_Thermal Tot_Total

162 5.6.2 Two-Dimensional vs. Three-Dimensional Group Structure The 2-D, 12-group and 3-D, 26-group structures were used to study the effect of 2-D and 3-D group structure studies. The 3-D flux distribution was used to collapse 280- group cross-section library to 12-group and 26-group structures. The number of groups placed in each energy range is given in Table TORT is used to perform the study using S8 Square Legendre-Chevbychev quadrature order and P1 scattering order for a 3- D pin cell as shown in Figure Table 5-46 shows the eigenvalue results of 2-D vs. 3- D group structure cases. Table 5-47 shows the percentage deviation of reaction rates. The 3-D, 26-group structure agrees better with MCNP than the 2-D, 12-group structure in both eigenvalue and reaction rates. Table 5-45: Number of groups placed in each energy range Number of groups Enery Range 26G 12G Fast Range 7 1 Epithermal Range 4 2 Thermal Range 13 9 Table 5-46: Eigenvalue results 2-D vs 3-D group structure cases CASE k eff Deviation in pcm of Δk/k MCNP ± (3ρ) - TORT- 26GS8P1,3DXS TORT- 12GS8P1,2DXS

163 Table 5-47: Percentage deviation of reaction rates between 2-D and 3-D group structure cases Reaction Type Zr Fuel Clad_fuel Water_fuel Graphite Clad_gra Water_gra Abs_Fast Abs_Epi Abs_Thermal Abs_Total Nu-Fis_Fast Nu-Fis_Epi Nu-Fis_Thermal Nu-Fis_Total Tot_Fast Tot_Epi Tot_Thermal Tot_Total Summary In this chapter, the parametric study has been performed for a 3-D pin cell model. We selected the S8 (SLC) quadrature and P3 scattering order with 48x59x55 (x,y,z) to be a model for fine group study. Then, the fine energy-group and broad-energy-group structures for 3-D cross-section generation have been selected in the fast, epithermal, and thermal energy ranges by the CPXSD methodology using different objectives corresponding to each energy range. The scalar flux weighting technique is utilized in collapsing fine- to broad-group libraries. Results indicate very good agreement between 280 fine- and 26 broad-group structures. Also, we demonstrated that the group structure for 3-D problem should be developed in 3-D geometry not in 2-D geometry. Comparing previous 2-D, 12 groups and 3-D, 26 groups, the obtained results show the significant impact of geometry on the group structure selection. 149

164 CHAPTER 6 Core Simulation In this chapter, we intended to implement the developed 26-group cross-section library to core simulations. The problem is that the use of 26 groups is still computationally expensive for a whole 3-D core calculation. For this reason, the 26- group structure is verified by a mini-core test problem. Coarse group structure has been selected from the 26 broad-group structure in order to make our TRIGA core problem feasible for 3-D transport calculations. The TRIGA core loading 2 is used to verify and validate the selected effective coarse group structure. In both validation efforts, continuous energy Monte Carlo solutions are used as the references. 6.1 Mini-Core Simulation A mini-core test problem was set up to validate the 26-group cross-section library. It consists of 7 fuel elements as shown in Figure 6-1. A 1/8 mini-core was modeled in TORT to take advantage of the core symmetry. The overall size of the 3-D model is x x cm 3. The studies were performed with S8-SLC quadrature set. The flux convergence was set to 1x10-4 and the eigenvalue convergence was set to 1x10-6 for TORT calculations. Fuel Graphit Water Figure 6-1: Configuration of Mini-core 150

165 6.1.1 Mesh Size Study In this part, we performed mesh size study for the mini-core model in both axial and radial direction. In axial-mesh size study, five models with different mesh sizes were examined. The first model has 0.5 cm cell thickness. The second model has 1 cm cell thickness. The third model has 1.5 cm cell thickness. The forth model has 2 cm-cell thickness and the fifth model has 2.5 cm cell thickness. The calculations were performed using P1 scattering order with S8-SLC quadrature order. Table 6-1 shows the eigenvalues calculated from TORT and pcm deviation for each model. Table 6-2 through Table 6-5 show the percentage deviation of reaction rates between each model and the 1 st model. From the results, the 3 rd model is chosen to be the axial mesh size model and will be used further for radial-mesh size study. Table 6-1: Eigenvalues calculated from TORT Model-mesh size (no. of cells) k eff Deviation From the 1 st model in pcm Time (hr) cm (117x114x71) cm (117x114x36) cm (117x114x24) cm (117x114x18) cm (117x114x14)

166 Table 6-2: Percentage deviation of reaction rates between 2 nd model and 1 st model Reaction Type Zr Fuel Clad_fuel Water_fuel Graphite Clad_gra Water_gra Reflector Abs_Fast Abs_Epi Abs_Thermal Abs_Total Nu-Fis_Fast Nu-Fis_Epi Nu-Fis_Thermal Nu-Fis_Total Tot_Fast Tot_Epi Tot_Thermal Tot_Total Table 6-3: Percentage deviation of reaction rates between 3 rd model and 1 st model Reaction Type Zr Fuel Clad_fuel Water_fuel Graphite Clad_gra Water_gra Reflector Abs_Fast Abs_Epi Abs_Thermal Abs_Total Nu-Fis_Fast Nu-Fis_Epi Nu-Fis_Thermal Nu-Fis_Total Tot_Fast Tot_Epi Tot_Thermal Tot_Total Table 6-4: Percentage deviation of reaction rates between 4 th model and 1 st model Reaction Type Zr Fuel Clad_fuel Water_fuel Graphite Clad_gra Water_gra Reflector Abs_Fast Abs_Epi Abs_Thermal Abs_Total Nu-Fis_Fast Nu-Fis_Epi Nu-Fis_Thermal Nu-Fis_Total Tot_Fast Tot_Epi Tot_Thermal Tot_Total

167 Table 6-5: Percentage deviation of reaction rates between 5 th model and 1 st model Reaction Type Zr Fuel Clad_fuel Water_fuel Graphite Clad_gra Water_gra Reflector Abs_Fast Abs_Epi Abs_Thermal Abs_Total Nu-Fis_Fast Nu-Fis_Epi Nu-Fis_Thermal Nu-Fis_Total Tot_Fast Tot_Epi Tot_Thermal Tot_Total In radial-mesh size study, there are three models. The first model has 0.10 cm cell size. The second model has 0.15 cm cell size. The third model has 0.20 cm cell size. The meshes are distributed uniformly and the uniform mesh thickness in axial direction of all four models is 1.5 cm. The calculations were performed using P1 scattering order and S8- SLC quadrature order. Table 6-6 shows the eigenvalues calculated by TORT and pcm deviation for each model. Table 6-7 and Table 6-8 show the percentage deviation of reaction rates between each model and the 1 st model. From the results, the 2 nd radial mesh-size model, 0.15 cm, is chosen for use in mini-core calculations. Table 6-6: Eigenvalues calculated from TORT Model-mesh size (no. of cells) k eff Deviation From the 1 st model in Time (hr) pcm cm (117x114x24) cm (96x91x24) cm (83x75x24)

168 Table 6-7: Percentage deviation of reaction rates between 2 nd model and 1 st model Reaction Type Zr Fuel Clad_fuel Water_fuel Graphite Clad_gra Water_gra Reflector Abs_Fast Abs_Epi Abs_Thermal Abs_Total Nu-Fis_Fast Nu-Fis_Epi Nu-Fis_Thermal Nu-Fis_Total Tot_Fast Tot_Epi Tot_Thermal Tot_Total Table 6-8: Percentage deviation of reaction rates between 3 rd model and 1 st model Reaction Type Zr Fuel Clad_fuel Water_fuel Graphite Clad_gra Water_gra Reflector Abs_Fast Abs_Epi Abs_Thermal Abs_Total Nu-Fis_Fast Nu-Fis_Epi Nu-Fis_Thermal Nu-Fis_Total Tot_Fast Tot_Epi Tot_Thermal Tot_Total Mini-Core Results The selected mesh sizes in previous section (0.15 cm for radial direction and 1.5 cm in axial direction) were used to perform mini-core calculations. The reference solution was obtained by MCNP with 3000 number of histories per cycle, 1000 number of skipped cycles and 4000 number of active cycles. The standard deviation is within 1% for reaction rates. Table 6-9 gives the eigenvalues calculated by MCNP and TORT for P1 and P3 cases. Results indicate that scattering order has a pronouced effect on eigenvalue. 154

169 The P3 case agrees with MCNP better than the P1 case. Table 6-10, Table 6-11 and Table 6-12 display the reaction rates calculated by MCNP and TORT. Table 6-13 and Table 6-14 show the percentage deviations of reaction rates between TORT and MCNP for P1 and P3 cases, respectively. Using P3 scattering order improves also the agreement with MCNP results on reaction rate prediction but much lesser extend than for the eigenvalue prediction. Table 6-9: Eigenvalues calculated from TORT and MCNP k eff Deviation From MCNP In pcm of Δk Time (hr) MCNP (± σ) TORT,S8P TORT,S8P Table 6-10: MCNP reaction rates Reaction Type Zr Fuel Clad_fuel Water_fuel Graphite Clad_gra Water_gra Reflector Abs_Fast 5.28E E E E E E E E-07 Abs_Epi 1.15E E E E E E E E-07 Abs_Thermal 2.79E E E E E E E E-05 Abs_Total 4.47E E E E E E E E-05 Nu-Fis_Fast E Nu-Fis_Epi E Nu-Fis_Thermal E Nu-Fis_Total E Tot_Fast 3.17E E E E E E E E-04 Tot_Epi 1.95E E E E E E E E-03 Tot_Thermal 1.69E E E E E E E E-02 Tot_Total 6.82E E E E E E E E

170 Table 6-11: TORT reaction rates for P1 case Reaction Type Zr Fuel Clad_fuel Water_fuel Graphite Clad_gra Water_gra Reflector Abs_Fast 5.20E E E E E E E E-07 Abs_Epi 1.13E E E E E E E E-07 Abs_Thermal 2.80E E E E E E E E-05 Abs_Total 4.45E E E E E E E E-05 Nu-Fis_Fast E Nu-Fis_Epi E Nu-Fis_Thermal E Nu-Fis_Total E Tot_Fast 3.13E E E E E E E E-04 Tot_Epi 1.95E E E E E E E E-03 Tot_Thermal 1.66E E E E E E E E-02 Tot_Total 6.73E E E E E E E E-02 Table 6-12: TORT reaction rates for P3 case Reaction Type Zr Fuel Clad_fuel Water_fuel Graphite Clad_gra Water_gra Reflector Abs_Fast 5.33E E E E E E E E-07 Abs_Epi 1.15E E E E E E E E-07 Abs_Thermal 2.84E E E E E E E E-05 Abs_Total 4.52E E E E E E E E-05 Nu-Fis_Fast E Nu-Fis_Epi E Nu-Fis_Thermal E Nu-Fis_Total E Tot_Fast 3.20E E E E E E E E-04 Tot_Epi 1.99E E E E E E E E-03 Tot_Thermal 1.68E E E E E E E E-02 Tot_Total 6.87E E E E E E E E-02 Table 6-13: Percentage deviation of reaction rates between Tort-P1 and MCNP Reaction Type Zr Fuel Clad_fuel Water_fuel Graphite Clad_gra Water_gra Reflector Abs_Fast Abs_Epi Abs_Thermal Abs_Total Nu-Fis_Fast Nu-Fis_Epi Nu-Fis_Thermal Nu-Fis_Total Tot_Fast Tot_Epi Tot_Thermal Tot_Total

171 Table 6-14: Percentage deviation of reaction rates between Tort-P3 and MCNP Reaction Type Zr Fuel Clad_fuel Water_fuel Graphite Clad_gra Water_gra Reflector Abs_Fast Abs_Epi Abs_Thermal Abs_Total Nu-Fis_Fast Nu-Fis_Epi Nu-Fis_Thermal Nu-Fis_Total Tot_Fast Tot_Epi Tot_Thermal Tot_Total Coarse Group Study As described in the previous section, a mini-core was used to test the obtained 26- broad group cross section library and the solution agrees well with the MCNP results. However, the calculation time is quite significant even for this mini-core simulation, which is considered to be a very small model. With a problem of running time, it is not practical to perform a whole core calculation with 26 groups. Consequently, we attempt to develop a fewer group structure in order to solve the problem in a reasonable amount of time within the accepted range of results in terms of accuracy requirements. The collapsing process was done in each energy range starting with fast, epithermal, and thermal. We started with the important distribution of 26-group structure as shown in Figure 6-2. The groups that have the most importance were kept with their original same energy interval and the groups that have lower importance were combined together. 157

172 3.50E-03 Thermal Range Ephithermal Range Fast Range 3.00E E E-03 C(E) 1.50E E E E+00 3.E-09 1.E-08 3.E-08 5.E-08 5.E-08 6.E-08 7.E-08 8.E-08 9.E-08 1.E-07 1.E-07 2.E-07 3.E-07 1.E-06 3.E-06 1.E-04 1.E-03 1.E-02 1.E-01 2.E-01 3.E-01 6.E-01 1.E+00 2.E+00 3.E+00 2.E+01 Energy (MeV.) Figure 6-2: Importance distribution of 26-group structure 1) Fuel pin model The 26-broad-group structure has been finally collapsed to 12-coarse-group structure. Fuel pin model calculations have been performed with both 26-group ans 12- group cross sections and the obtained results have benn compared. The eigenvalues are shown in Table 6-15 and the reacton rate comparisons are shown in Table The 12G structure shows a good agreement with 26G structure in both eigenvalue and reaction rate predictions. Table 6-17 lists energy boundaries of 12-group structure. Table 6-15: Eigenvalues calculated from TORT k eff Deviation From 26G in pcm 26G G

173 Table 6-16: Percentage deviation of reaction rates between 12G and 26G cases Reaction Type Zr Fuel Clad_fuel Water_fuel Graphite Clad_gra Water_gra Abs_Fast Abs_Epi Abs_Thermal Abs_Total Nu-Fis_Fast Nu-Fis_Epi Nu-Fis_Thermal Nu-Fis_Total Tot_Fast Tot_Epi Tot_Thermal Tot_Total Table 6-17: Energy boundaries of 12-group structures Energy Group Number Upper Energy(MeV) E E E E E E E E E E E E E-11 2) Control Rod model For control rod modeling, the 26-broad-group structure has been collapsed to 13- coarse-group structure. Calculations were performed with both 26-group ans 13-group cross sections The 13G structure shows a good agreement with 26G structure in both eigenvalue and reaction rates as demonstrated in Table 6-18 and Table As a result, 159

174 the final coarse group structure for the core calculations is the 13-group structure. Table 6-20 lists energy boundaries of the 13-group structure. Table 6-18: Eigenvalues calculated by TORT k eff Deviation From 26G in pcm 26G G Table 6-19: Percentage deviation of reaction rates between 13G and 26G cases Reaction Type B 4 C Clad Water Abs_Fast Abs_Epi Abs_Thermal Abs_Total Tot_Fast Tot_Epi Tot_Thermal Tot_Total Table 6-20: Energy boundaries of 13- group structure Energy Group Number Upper Energy(MeV) E E E E E E E E E E E E E E

175 6.3 Core loading 2 Simulations We selected the TRIGA core loading 2 because it is a fresh core. It consists of % wt. fuel elements with 4 control rods. Figure 6-3 shows the cross sectional view of the core arrangement. The fuel elements were modeled explicitly specifying the detailed structure of the rod to eliminate any homogenization effects. Figure 6-3: TRIGA core loading 2 161

176 6.3.1 Mesh Size Study We attempt to use TORT to simulate core model with the coarse-group cross section library. The mesh size study was performed in order to get optimum mesh size for coarse group structure. The first step is to study on the axial mesh size and later on the radial mesh size Fuel Pin Model A pin cell model with reflector layer on top is used for 12 coarse-group structure. Figure 6-4 shows the configuration in axial direction of the studied model. Fuel Graphite Water Figure 6-4: Pin cell in axial direction Four models with different mesh-sizes were used in this study as shown in Figure 6-5. The first model has 0.5 cm mesh thickness for all layers. The second model has 1 cm 162

177 mesh thickness for all layers. The third model has 2 cm mesh thickness for all layers. The forth model has 2 cm mesh thickness for the fuel layer, 1 cm mesh thickness for the graphite layer, and 0.5 cm mesh thickness for the reflector layer. 0.5cm-mesh thickness 0.5cm.-mesh thickness 0.5cm.-mesh thickness Fuel Graphite Water cm.-mesh thickness 1cm.-mesh thickness 1cm.-mesh thickness Fuel Graphite Water cm.-mesh thickness 2cm.-mesh thickness 2cm.-mesh thickness Fuel Graphite Water cm.-mesh thickness 1cm.-mesh thickness 2cm.-mesh thickness Fuel Graphite Water st model 2 nd model 3 rd model 4 th model Unit:cm Figure 6-5: The studied models Table 6-21 shows the eigenvalues calculated by TORT and pcm deviation for each model. Table 6-22 through Table 6-24 show the percentage deviation of reaction rates between each model and the 1 st model. Even though, the eigenvalue deviation of 2 nd model is smaller than the 4 th model, the deviations of reaction rates in the 4 th model are even out through all regions. Thus, the 4 th axial mesh-size model is chosen for use in core calculations. 163

178 Table 6-21: Eigenvalues calculated by TORT Model (no. of cells) k eff Deviation Time (min) From 1 st model in pcm 1(48x59x71) (48x59x36) (48x59x18) (48x59x35) Table 6-22: Percentage deviation of reaction rates between 2 nd model and 1 st model Reaction Type Zr Fuel Clad_fuel Water_fuel Graphite Clad_gra Water_gra Reflector Abs_Fast Abs_Epi Abs_Thermal Abs_Total Nu-Fis_Fast Nu-Fis_Epi Nu-Fis_Thermal Nu-Fis_Total Tot_Fast Tot_Epi Tot_Thermal Tot_Total Table 6-23: Percentage deviation of reaction rates between 3 rd model and 1 st model Reaction Type Zr Fuel Clad_fuel Water_fuel Graphite Clad_gra Water_gra Reflector Abs_Fast Abs_Epi Abs_Thermal Abs_Total Nu-Fis_Fast Nu-Fis_Epi Nu-Fis_Thermal Nu-Fis_Total Tot_Fast Tot_Epi Tot_Thermal Tot_Total

179 Table 6-24: Percentage deviation of reaction rates between 4 th model and 1 st model Reaction Type Zr Fuel Clad_fuel Water_fuel Graphite Clad_gra Water_gra Reflector Abs_Fast Abs_Epi Abs_Thermal Abs_Total Nu-Fis_Fast Nu-Fis_Epi Nu-Fis_Thermal Nu-Fis_Total Tot_Fast Tot_Epi Tot_Thermal Tot_Total After the axial mesh size for 12 groups was studied, the second step is to study the radial mesh size. Four models with different mesh-size are used in this study as shown in Figure 6-6. The first model has a uniform 0.03 cm mesh. The second model has 0.1 cm mesh. The third model has 0.15 cm mesh. The forth model has 0.2 cm mesh. 165

180 1 st model 2 nd model 3 rd model 4 th model Figure 6-6: The studied models Table 6-25 shows the eigenvalues calculated by TORT and pcm deviation for each model. Table 6-26 through Table 6-28 show the percentage deviation of reaction rates between each model and the 1 st model. From the results, the 3 rd radial mesh-size model of 0.15 cm mesh size is chosen for use in core calculations. 166

181 Table 6-25: Eigenvalues calculated from TORT Model (no. of cells) k eff Deviation From 1 st model in pcm Time (min) 1 (48x59x35) (22x25x35) (16x18x35) (13x14x35) Table 6-26: Percentage deviation of reaction rates between 2 nd model and 1 st model Reaction Type Zr Fuel Clad_fuel Water_fuel Graphite Clad_gra Water_gra Reflector Abs_Fast Abs_Epi Abs_Thermal Abs_Total Nu-Fis_Fast Nu-Fis_Epi Nu-Fis_Thermal Nu-Fis_Total Tot_Fast Tot_Epi Tot_Thermal Tot_Total Table 6-27: Percentage deviation of reaction rates between 3 rd model and 1 st model Reaction Type Zr Fuel Clad_fuel Water_fuel Graphite Clad_gra Water_gra Reflector Abs_Fast Abs_Epi Abs_Thermal Abs_Total Nu-Fis_Fast Nu-Fis_Epi Nu-Fis_Thermal Nu-Fis_Total Tot_Fast Tot_Epi Tot_Thermal Tot_Total

182 Table 6-28: Percentage deviation of reaction rates between 4 th model and 1 st model Reaction Type Zr Fuel Clad_fuel Water_fuel Graphite Clad_gra Water_gra Reflector Abs_Fast Abs_Epi Abs_Thermal Abs_Total Nu-Fis_Fast Nu-Fis_Epi Nu-Fis_Thermal Nu-Fis_Total Tot_Fast Tot_Epi Tot_Thermal Tot_Total Control Rod Model A 3-D color-set control rod model using the 13 coarse-group structure cross sections is used for mesh size study of control rod. In axial direction, the same mesh models as studied in the fuel pin model were used. The first model has 0.5 cm mesh thickness for all layers. The second model has 1cm-mesh thick for all layers. The third model has 2 cm mesh thickness for all layers. The forth model has 2 cm mesh thickness for the fuel layer, 1 cm mesh thickness for the graphite layer. Table 6-29 shows the eigenvalues calculated by TORT and pcm deviation for each model. Table 6-30 through Table 6-32 show the percentage deviation of reaction rates between each model and the 1 st model. The 4 th axial mesh-size model is chosen to use in core calculations. 168

183 Table 6-29: Eigenvalues calculated by TORT Model (no. of cells) k eff Deviation From 1 st model in pcm Time (min) 1 (148x118x46) (148x118x23) (148x118x12) (148x118x14) Table 6-30: Percentage deviation of reaction rates between 2 nd model and 1 st model Reaction Type B 4 C Clad Water Abs_Fast Abs_Epi Abs_Thermal Abs_Total Tot_Fast Tot_Epi Tot_Thermal Tot_Total Table 6-31: Percentage deviation of reaction rates between 3 rd model and 1 st model Reaction Type B 4 C Clad Water Abs_Fast Abs_Epi Abs_Thermal Abs_Total Tot_Fast Tot_Epi Tot_Thermal Tot_Total

184 Table 6-32: Percentage deviation of reaction rates between 4 th model and 1 st model Reaction Type B 4 C Clad Water Abs_Fast Abs_Epi Abs_Thermal Abs_Total Tot_Fast Tot_Epi Tot_Thermal Tot_Total The second step is to study the radial mesh size. Three models with different mesh-size are used in this study. The first model has 0.03cm-mesh. The second model has 0.1cm-mesh. The third model has 0.15 cm-mesh. Table 6-33 shows the eigenvalues calculated from TORT and pcm deviation for each model. Table 6-34 and Table 6-35 show the percentage deviation of reaction rates between each model and the 1 st model. From the results, the 1 rd radial mesh-size model, 0.03 cm. is chosen to use for control rod in core calculations. Table 6-33: Eigenvalues calculated from TORT Model (no. of cells) k eff Deviation From 1 st model in pcm Time (min) 1 (148x114x14) (45x38x14) (35x27x14)

185 Table 6-34: Percentage deviation of reaction rates between 2 nd model and 1 st model Reaction Type B 4 C Clad Water Abs_Fast Abs_Epi Abs_Thermal Abs_Total Tot_Fast Tot_Epi Tot_Thermal Tot_Total Table 6-35: Percentage deviation of reaction rates between 3 rd model and 1 st model Reaction Type B 4 C Clad Water Abs_Fast Abs_Epi Abs_Thermal Abs_Total Tot_Fast Tot_Epi Tot_Thermal Tot_Total Core Reflector Thickness Study Pseudo-core loading 2 was modeled for TORT calculation as shown in Figure 6-7. Four control rods were replaced with 8.5% fuel cell. The fuel part with cm. in axial direction was modeled. This core configuration is used to study the radial thickness of reflector. The mean-free-path (mfp) of water for is about 5 cm for fast group. Thus, we have 3 models of radial thickness based on the mfp, 1 mfp-5 cm, 2 mfp-10 cm, and 3 mfp-15 cm. The reflective boundary condition is applied at the top and bottom of the model and the vacuum boundary condition is utilized at front, back, left, and right of the 171

186 model. The total number of cells for the model with 5 cm reflector thickness is 1,372,410 cells. The total number of cells for the model with 10 cm reflector thickness is 1,918,620 cells.the total number of cells for the model with 15 cm reflector thickness is 2,521,640 cells. The 12G structure library is used for this study. Figure 6-7: Core loading 2 with 15 cm reflector thickness 172

187 The results are presented in Table The eigenvalues are not sensitive comparing the 10-cm reflector thickness model with the 15-cm reflector thickness model. The model with 5-cm reflector thickness differs from the model with 15 cm reflector thickness by 238 pcm in the eigenvalue prediction. Reflector Thickness (cm) Table 6-36: Eigenvalues calculated by TORT k eff Convergence Rel.Deviation (1E-4,1E-6) in pcm with 15 cm Time (hr) reflector thickness model E E E Core Loading 2 ARI A model with 5 cm reflector thickness was used to perform core calculations because of the computational time. It is used in both TORT and MCNP. In this section, we modeled the core with all control rods in (ARI) as shown in Figure 6-8 for radialcross-section view and Figure 6-9 for axial-cross-section view. The 13-coarse group cross section library was used with S8-SLC quadrature set and P1 scattering order. The flux convergence was set to 5x10-4 and the eigenvalue convergence was set to 1x10-5. The selected mesh sizes in previous section were used. For fuel rods, the mesh sizes are 0.15 cm in radial direction and 2,1,0.5 cm mixed model in axial direction. For control rod, the mesh sizes are 0.03 cm in radial direction and 2,1,0.5 cm mixed model in axial direction. 173

188 Figure 6-8: Radial-cross-section view of ARI 174

189 Figure 6-9: Axial-cross-section view of ARI The comparison of the eigenvalue predictions indicates that TORT over-estimates by 9 pcm k eff as compared to the reference MCNP result as shown in Table This deviation is within 3σ. Figure 6-10 shows the normalized power map of MCNP and TORT also the percentage relative difference of TORT results as compared to MCNP results. The normalized power map calculated by MCNP has less than 1% of statistical uncertainty. The relative differences vary in a range of ~-3% to +4%. The maximum differences occur at the core periphery at which the power is low. The agreement in this region can be improved by using 10 or 15 cm reflector thickness in both TORT and MCNP models. 175

190 Table 6-37: Eigenvalues calculated from TORT and MCNP k eff Deviation from MCNP Time (hr) in pcm of Δk MCNP ± (3σ) TORT,S8P SA RR CT SH TR Figure 6-10: Normalized pin-power distribution for ARI x.xxx x.xxx x.xxx MCNP NP TORT NP TORT MCNP MCNP x100% 176

191 6.3.4 Core Loading 2 ARO In this section, we modeled the core all control rods out (ARO) as presented in Figure 6-11 for radial-cross-section view and Figure 6-12 for axial-cross-section view. We performed the calculation with the same parameter values as in the ARI case. Figure 6-11: Radial-cross-section view of ARO 177

192 Figure 6-12: Axial-cross-section view of ARO Comparison of the eigenvalue prediction indicates that TORT over-estimates by 91 pcm as shown in Table Figure 6-13 shows the normalized power maps of MCNP and TORT also the percentage relative difference of TORT results as compared to MCNP results. The relative differences vary in a range of ~-3% to +4%. The maximum difference occurs at the same location as in the ARI case and can be improved by using a thicker reflector. Table 6-38: Eigenvalues calculated from TORT and MCNP k eff Deviation From MCNP Time (hr) In pcm of Δk MCNP ± (3σ) TORT,S8P