A 2D/1D Neutron Transport Method with Improved Angular Coupling by Michael Gregory Jarrett A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy (Nuclear Engineering and Radiological Sciences) in the University of Michigan 2018 Doctoral Commitee: Dr. Brendan M. Kochunas, Co-Chair Professor Edward W. Larsen, Co-Chair Professor Thomas J. Downar Professor Vikram Gavini Dr. Shane G. Stimpson, Oak Ridge National Laboratory Michael Gregory Jarrett [email protected] ORCID iD: 0000-0003-1834-0719 c Michael Gregory Jarrett 2018 To my late parents, Mark and Mary Ann, who continue to inspire and encourage me every single day. ii Acknowledgements Obviously, I could not have completed my thesis alone. Many people have helped me academically, professionally, and personally along the way. I would like to thank my entire thesis committee for helping me to learn and grow as a scientific researcher. I would like to express my gratitude to my multiple co-advisors, Professor Edward Larsen, Professor Thomas Downar, and Dr. Brendan Kochunas for their invaluable mentorship and guidance. Also, a special thanks is owed to Dr. Shane Stimpson, whose doctoral thesis gave me a strong foundation upon which to base my own work. I must express thanks to everyone involved with MPACT, both at Michigan and at Oak Ridge National Laboratory, for their helpfulness, diligence, and collaboration. I am very fortunate to have had the opportunity to contribute to and use this code for my thesis research. I would also like to thank the Department of Energy (DOE), for funding almost all of my thesis research, first through the DOE Nuclear Energy University Programs Graduate Fellowship, and later through the Consortium for Advanced Simulation of Light Waters Reactors (CASL) under DOE contract number DE-AC05-00OR22725. Most importantly, I would like to thank my family. I am extraordinarily lucky to have been raised by my late parents, who inspired and encouraged me to set ambitious goals. I am immensely grateful to the rest of my family, who have been unbelievably supportive, present, and loving. They have done countless things, both big and small, to help me through difficult personal tribulations and stay on track academically and professionally. I am very appreciative of my girlfriend, Carly, whose companionship, advice, and assistance made graduate study substantially more enjoyable. iii Table of Contents Dedication . ii Acknowledgements . iii List of Tables . vii List of Figures . ix List of Acronyms . xii Abstract . xiii Chapter 1: Introduction 1 1.1 Motivation . 1 1.2 History of the 2D/1D Method . 5 1.3 Dissertation Layout . 8 Chapter 2: Computational Neutron Transport Theory 10 2.1 The Boltzmann Transport Equation . 10 2.2 keff Eigenvalue Problems . 12 2.3 Monte Carlo Methods . 14 2.4 Discretization Methods . 15 2.4.1 The Multigroup Approximation . 16 2.4.2 Angular Discretization . 19 2.4.3 Spherical Harmonics Expansion . 22 2.4.4 Scattering Approximations . 25 2.5 Spatial Discretization and Solution Methods . 28 2.5.1 The Method of Characteristics . 29 2.5.2 Coarse Mesh Finite Difference (CMFD) . 32 2.5.3 Nodal Methods . 33 2.6 Summary . 38 Chapter 3: The 2D/1D Method 39 3.1 2D/1D Equations . 39 iv 3.1.1 2D Radial Equations . 40 3.1.2 1D Axial Equations . 42 3.1.3 Azimuthal Expansion . 43 3.1.4 2D to 1D Homogenization . 45 3.1.5 Within-pin Spatial Shape of Axial TL . 50 3.1.6 Radial Transverse Leakage Interpolation . 52 3.1.7 P3 Expansion of the 1D Transport Equation . 54 3.2 Other Aspects of 2D/1D in MPACT . 57 3.2.1 Transverse Leakage Splitting . 57 3.2.2 2D/1D Iteration Scheme . 59 3.2.3 2D/1D Relaxation . 62 3.2.4 2D/1D Methods . 63 3.3 Homogenization of XS for 1D Solution . 64 3.4 2D/1D Summary . 68 Chapter 4: Numerical Results 69 4.1 1D/1D SN Demonstration . 69 4.1.1 1D/1D SN Equations . 70 4.1.2 1D LWR Results . 71 4.2 C5G7 Pin Cell . 73 4.3 Homogeneous Fuel Test Problem . 75 4.4 3x3 Partially Rodded Lattice . 78 4.4.1 2D X-Z Slice . 78 4.4.2 3D Partially Rodded 3x3 Lattice . 79 4.5 Azimuthal Cross Section Moments . 83 4.6 3D C5G7 Benchmark . 90 4.7 VERA Progression Problem 4 . 96 4.8 Summary of Numerical Results . 100 Chapter 5: The 2D/1D Polar Parity Method 103 5.1 Polar Parity 2D/1D Equations . 103 5.1.1 MPACT Approximation . 105 5.1.2 2D Coarse-Mesh SN for Odd-Parity Flux . 108 5.1.3 Intermediate-Mesh MOC Solution of the Odd-Parity Equation 114 5.1.4 Full MOC Solution of the Odd Parity Equation . 115 5.2 2D/1D Polar-Parity Results . 116 5.2.1 Homogeneous Fuel Test Problem . 117 v 5.2.2 3D C5G7 Benchmark . 119 5.3 Computational and Memory Cost . 122 5.4 Local Refinement of TL and XS Approximation . 126 5.5 2D/1D Polar Parity Summary . 129 Chapter 6: SP3 limit of the 2D/1D Equations 131 6.1 Asymptotic Limit of the SP3 Equations . 132 6.2 2D/1D SN with Isotropic TL . 135 6.3 PN Transverse Leakage dependence . 141 6.3.1 Analysis . 142 6.3.2 Results . 146 6.3.3 Importance of Azimuthal Moments . 147 6.3.4 Modular Azimuthal Quadrature Sets . 149 6.3.5 Tabuchi-Yamamoto Polar Quadrature . 151 6.3.6 S4 Quadrature . 152 6.4 Numerical Results for SP3 Limit . 153 6.4.1 2D Test Problem . 154 6.4.2 Takeda-Ikeda Benchmark Problem . 156 6.5 Summary . 160 Chapter 7: Conclusion 162 7.1 Summary . 162 7.2 Future Work . 165 7.2.1 More Applications to Real LWR Problems . 166 7.2.2 Extension to PN Scattering . 167 7.2.3 2D/1D Convergence and Transverse Leakage Splitting . 167 7.2.4 Intermediate-Mesh Odd-Parity 2D/1D . 169 Appendix A: 1D/1D SN Demonstration Results 170 Bibliography 176 vi List of Tables 3.1 Eigenvalue results for axial TL shape test . 65 4.1 Transverse 1D angular flux error, UO2 pin, scalar flux homogenization 72 4.2 Transverse 1D angular flux error, UO2 pin, angular flux homogenization 72 4.3 Eigenvalue error and pin power errors for 1D Z, 2D X-Z problem . 77 4.4 Eigenvalue results for 3x3 problem . 87 4.5 Angular flux moments for 3x3 problem . 87 4.6 Angular XS moments for 3x3 problem . 88 4.7 Angular flux moments for plate fuel . 89 4.8 3D C5G7 benchmark errors, SHIFT reference . 92 4.9 3D C5G7 benchmark errors, 2D/1D P3 polar XS, hyper-fine mesh . 95 4.10 Eigenvalue error, VERA Problem 4 . 98 4.11 Pin power errors, VERA Problem 4 . 99 5.1 Eigenvalue error and pin power errors for 1D Z, 2D X-Z problem . 118 5.2 3D C5G7 unrodded benchmark results, SHIFT reference . 120 5.3 3D C5G7 rodded A benchmark results, SHIFT reference . 120 5.4 3D C5G7 rodded B benchmark results, SHIFT reference . 121 5.5 Polar XS homogenization weighting results, rodded B . 122 5.6 Memory requirements for 2D/1D . 124 5.7 2D/1D odd-parity coarse-mesh SN run time . 124 5.8 2D/1D full anisotropic TL run time . 125 5.9 2D/1D odd-parity coarse-mesh SN run time, polar XS . 125 5.10 2D/1D full anisotropic TL run time, polar XS . 125 5.11 2D/1D local refinement results . 128 5.12 MOC time per process, in minutes . 128 6.1 2D/1D SP3 limit coefficients (relative) . 146 6.2 2D/1D SP3 limit coefficients (absolute) . 146 vii 6.3 2D/1D relative SP3 limit with P3 polar and varying azimuthal TL . 148 6.4 Modularized azimuthal quadrature set . 150 6.5 SP3 Limit of 2D/1D, Chebyshev quadrature (64 azimuthal angles) . 150 6.6 SP3 Limit of 2D/1D, modularized Chebyshev quadrature (64 azimuthal angles) . 151 6.7 Optimized Tabuchi-Yamamoto polar quadrature . 151 6.8 2D/1D with Tabuchi-Yamamoto quadrature, coefficients relative to SP3 limit . 152 6.9 2D/1D anisotropic TL limit, S4 quadrature . 152 6.10 2D homogeneous fuel results . 154 6.11 Takeda-Ikeda cross section data . 159 6.12 Takeda-Ikeda rodded keff results . 159 6.13 Takeda-Ikeda rodded keff results, rotated . 160 A.1 Transverse 1D angular flux error, MOX pin . 170 A.2 Transverse 1D angular flux error, control pin . 172 A.3 Transverse 1D angular flux error, assembly . 175 viii List of Figures 1.1 Reactor core geometry and pin mesh [6] . 2 1.2 2D/1D computational flow diagram . 4 2.1 U235 total cross section . 17 2.2 Level-Symmetric quadrature points . ..