The Pennsylvania State University. The Graduate School THE DEVELOPMENT AND APPLICATION OF AN IMPROVED REACTOR ANALYSIS MODEL FOR FAST REACTORS

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1 The Pennsylvania State University The Graduate School THE DEVELOPMENT AND APPLICATION OF AN IMPROVED REACTOR ANALYSIS MODEL FOR FAST REACTORS A Dissertation in Department of Mechanical and Nuclear Engineering by Jia Hou c 2013 Jia Hou Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy December 2013

2 The dissertation of Jia Hou was reviewed and approved by the following: Kostadin Ivanov Distinguished Professor of Nuclear Engineering Dissertation Advisor, Chair of Committee Maria Avramova Assistant Professor of Nuclear Engineering Massimiliano Fratoni Assistant Professor of Nuclear Engineering Fuqing Zhang Professor of Meterology Hangbok Choi Special Member Senior Scientist, General Atomics Arthur Motta Professor of Nuclear Engineering and Materials Science and Engineering Chair of Nuclear Engineering Signatures are on file in the Graduate School.

3 Abstract Accuracy in neutron cross sections calculation and consistency in reactor physics are fundamental requirements in advanced nuclear reactor design and analysis. The work presented in this dissertation focuses on the development and advanced application of a reactor analysis model with updated cross section libraries that is suitable for online cross section generation for fast reactors. Research has been performed in two areas of interest in reactor physics. The first target of the research is to develop efficient modeling capacity of the 1- D lattice code MICROX-2 for its neutron spectrum calculation based on Collision Probability Method (CPM). Expanded master cross section libraries have been generated based on updated nuclear data and optimized fine-group energy structure to accommodate both thermal and fast reactor spectra as well as to comply with the need for advanced fuel cycle analysis. After verifying the new libraries, the solution methods have been reviewed and updated, including the update of interpolation scheme for resonance self-shielding factors and improvement of spatial self-shielding models for various fuel assembly geometries. The assessment of the iii

4 updated lattice calculation models has shown that the prediction accuracy of lattice properties represented by the eigenvalue and reaction rate ratios is improved, especially for fast neutron spectrum lattices of which the importance of neutrons in the unresolved energy range is high. The second target of the research is to improve the accuracy of few-group nuclear cross section generation for the reactor core calculation. A 2-D pin-by-pin lattice model has been developed based on embedded CPM within the framework of the Nodal Expansion Method (NEM), which is capable of modeling the heterogeneity of the fuel assembly. Then, an online cross section generation methodology along with discontinuity factors has been developed based on Iterative Diffusion- Diffusion Methodology (IDDM), which can minimize the inconsistency in physics parameters by feeding the actual core condition into the cross section generation by the 2-D lattice code. In order to facilitate the iterative scheme between the 2-D lattice and core calculation, appropriate interface routines are used for accurate and consistent data transfer. Finally the overall physics method, starting from the 1-D lattice calculation to the core calculation, has been validated against benchmark problems, and promising results have been observed in core eigenvalue and power distribution comparisons. iv

5 Table of Contents List of Figures List of Tables List of Abbreviations List of Symbols Acknowledgments ix xi xiii xv xvii Chapter 1 Introduction Fast reactors Reactor physics calculation methodology Neutron transport theory Nodal diffusion methods Online cross section generation process Objective of this research Synopsis Chapter 2 Literature Review Cross section library Governing equations in reactor physics Transport equation k-eigenvalue problems Diffusion equation Basic approximations v

6 2.3.1 The multigroup approximation Spherical harmonic (P N ) approximation B 1 method for the asymptotic spectrum The B n leakage calculation The homogeneous fundamental mode Homogenization Equivalent homogenized cross sections Collision probability (CP) method ABH method Discontiuity factor MICROX-2 code Nodal expansion method Nodal balance equation Transverse integration procedure Polynomial expansion method Embedded lattice approach Chapter 3 Generation of Master Cross Section Library Fine-group energy mesh generation Methodology Selection of optimized energy group structure Master library generation Code system upgrade Library generation procedure Benchmark models Calculation models Calculation results Homogeneous lattice cases Heterogeneous lattice cases Finite lattice cases Analysis of comparison results Summary and conclusions Chapter 4 1-D Lattice Physics Model Resonance self-shielding Model Bondarenko method Power fitting self-shielding method Numerical benchmark models vi

7 4.1.4 Model verification Spatial self-shielding model Statement of the problem Wigner rational approximation and Dancoff factor Embedded Monte Carlo model Model verification Critical flux model Stamm ler s method Model verification Summary and conclusions Chapter 5 2-D Lattice Calculation Model Nodal expansion method Coupled calculation scheme Coupling method Assembly homogenization Interface routines Calculations and results The FA-BAPL benchmark The C5G7 benchmark Online cross section generation The fixed k eff algorithm A MICXN-MICXN calculation scheme Iterative Diffusion-Diffusion Methodology (IDDM) Chapter 6 Reactor Core Calculations Reactor modeling using off-line approach LWR benchmark Fast reactor benchmark Discussion Application of IDDM Data exchange in IDDM D C5G7 modeling results Conclusion and discussion Chapter 7 Conclusion and Future Work Conclusions vii

8 7.2 Recommended future work Bibliography 185 Appendix A MICROX-2 Master Cross Section Library 191 A.1 PSUmesh A.2 NJOY input file A.3 MICROR input files A.4 Library summary Appendix B A PyMicrox Manual 208 B.1 PyMicrox instance variables B.2 PyMicrox methods B.3 Execution of PyMicrox B.4 Sample PyMicrox scripts Appendix C More about calculation model 218 C.1 Evaluation of multi-collision probabilities C.2 Cross section file in ISOTXS format C.3 Sample MICXN input file C.4 Managing partial current from DIF3D viii

9 List of Figures 1.1 Calculational flow scheme of reactor physics design Division of the energy range into G energy groups Example two-region lattice pin cell Thermal flux distribution in two assemblies without burnable absorbers Thermal flux distribution in assemblies with and without burnable absorbers Discontinuous flux distributions in assemblies with and without burnable absorbers Discontinuous flux distributions in assemblies Discontinuous flux distributions and reconstructed fluxes Utilization of cross section libraries in MICROX Nodal modal nomenclature Comparison of the normalized fluxes for GFR cases Comparison of the normalized fluxes for PWR-UO 2 cases Iterative procedure in energy structure optimization Cross section library generation for MICROX Comparison of homogeneous and heterogeneous lattice calculations U radiative capture cross section Linear interpolation of the dilution parameter Monotonic scheme to fit σ 0 variation of Bondarenko factors Iterative scheme of the new self-shielding factor calculation routine Comparison of MICROX-2 resonance self-shielding methods Example of rectangular fuel lattices Dancoff factor calculation using Monte Carlo method Iteration history of total cross section of moderator region Comparison of k with different Dancoff factor calculation scheme. 108 ix

10 4.10 Convergence history of critical buckling B 2 with original and improved search models for TRX-1 case Schematic data flow of MICXN Reflector modeling in MICXN Configuration of the FA-BAPL benchmark problem C5G7 2-D LWR benchmark C5G7 UO 2 assembly configuration C5G7 MOX assembly configuration The albedo problem Calculation strategy for fixed k eff calculation with fixed boundary condition The MICXN-MICXN test scheme for fixed k eff calculation approach Pin fluxes differences of single UO 2 and MOX assembly Pin fluxes differences of UO 2 and MOX assembly in C3 benchmark C5G7 benchmark in MICXN-MICXN calculation scheme Pin fluxes differences of C5G7 benchmark (combined view) Formation of the partial currents as boundary conditions C3 2-D benchmark Power distribution of C3 benchmark C5G7 2-D benchmark Power distribution of C5G7 benchmark /12 core layout of modified JOYO benchmark Heterogeneous model of core fuel or blanket assembly Radial power distribution of simplified JOYO core Diagram of IDDM iterative procedure In-current shape functions of A2 lattice surfaces in C5G7 benchmark Convergence history of IDDM in C5G7 benchmark Power distribution of C5G7 case using IDDM Schematic data flow of the online cross section generation method. 184 C.1 Augment H 1 (τ) for cylinders, spheres, and slabs C.2 DIF3D node order in 2-D Cartesian geometry C.3 Partial currents on east side of assembly A x

11 List of Tables 3.1 Summary of the GCR and GFR calculations Comparison of objective cross sections Characteristics of benchmark problems Benchmark calculation results for TRX/BAPL homogeneous lattices Benchmark calculation results for ICSBEP homogeneous lattices Benchmark calculation results for power reactor homogeneous lattices Benchmark calculation results for TRX/BAPL heterogeneous lattices Benchmark calculation results for ICSBEP heterogeneous lattices Benchmark calculation results for power reactor heterogeneous lattices Comparison of finite lattice calculations for TRX and BAPL Comparison of k and reaction ratios for TRX/BAPL homogeneous lattices Comparison of k and reaction ratios for ICSBEP homogeneous lattices Comparison of k and reaction ratios for power reactor homogeneous lattices Summary of benchmark calculations using Wigner approximation and Monte Carlo methods for the Dancoff factor calculations Numbers of iterations of critical buckling search for selected cases Card 0 of MICXN input deck Material composition of assembly case FA-BAPL Comparison of k results of FA-BAPL case Average isotopic atom densities used in FA-BAPL calculation Homogenization of principle cross sections in NEM with data from MICROX-2 for FA-BAPL case Homogenization of scattering matrices in NEM with data from MICROX-2 for FA-BAPL case Homogenized isotopic microscopic cross sections from MICXN and MCNPX for FA-BAPL case xi

12 5.8 Homogenized isotopic microscopic cross sections from DRAGON and MCNPX for FA-BAPL case Assembly homogenized macroscopic cross sections for FA-BAPL case Cell dimension of C5G7 benchmark Isotopic distribution for each medium C5G7 benchmark calculation of fully reflected assembly problems C5G7 benchmark calculation of partially reflected assembly problems Cross section verification for fixed k eff calculation scheme Calculation results of C3 Benchmark problem Calculation results of C5G7 Benchmark problem Dimension data for heterogeneous model of core and assembly Isotopic distribution per medium in JOYO benchmark Comparison of k for JOYO fuel assembly and blanket assembly Calculation results of JOYO Benchmark problem Comparison of diffusion coefficient from lattices calculations Relative assembly power results of JOYO benchmark IDDM convergence criteria k eff convergence in C5G7 case using IDDM Relative error of converged in-currents in C5G7 case using IDDM. 174 A.1 Energy boundaries in the optimized PSUmesh structure A.2 Dilution cross sections at which Bondarenko factors are tabulated in the MICROX-2 library B.1 Default values for instance variables used in NJOY process xii

13 List of Abbreviations 1-, 2-, or 3-D One-, Two, or Three-Dimensional ADF ANL CPM CSEWG ENDF FD FEM GENDF GFR HTR HWR ICSBEP IDDM ITDM LFR LWR Assembly Discontinuity Factor Argonne National Laboratory Collision Probability Method Cross Section Working Group Benchmark Specifications Evaluated Nuclear Data File Finite Difference Finite Element Method Group-wise Evaluated Nuclear Data File Gas-cooled Fast Reactor High Temperature Reactor Heavy Water Reactor International Criticality Safety Benchmark Evaluation Project Iterative Diffusion-Diffusion Method Iterative Transport-Diffusion Method Lead-cooled Fast Reactor Light Water Reactor xiii

14 MCNP MOC MOX NEM NR PENDF PSI PSU PWR RBC RDFMG SFR SHEM S N SP 3 UOX WBC Monte Carlo N-Particle computer code Method of Characteristics Mixed-Oxide Nodal Expansion Method Narrow Resonance Point-wise Evaluated Nuclear Data File Paul Scherrer Institute Pennsylvania State University Pressurized Water Reactor Reflective Boundary Condition Reactor Dynamics and Fuel Management Group Sodium-cooled Fast Reactor Santamarina-Hfaiedh Energy Mesh Discrete ordinates Simplified P 3 approximation Uranium-Oxide White Boundary Condition xiv

15 List of Symbols B 2 C E F H J +, J J k k eff l N p esc Q V w(e) Y m n (ˆΩ) β Bucklings (leakage) parameter Ratio of 238 U capture to 235 U fission Neutron energy Bondarenko factor Augment function Outgoing, incoming partial currents Angular neutron current density Infinite multiplication factor Effective multiplication factor Mean chord length Neutron density Escape probability External neutron source Spatial domain Condensation weighting function Spherical harmonics Blackness xv

16 δ 25 δ 28 ζ ρ 28 Σ x σ x τ φ χ(e) ψ Epithermal-to-thermal 235 U fission ratio Ratio of 238 U to 235 U fission Flux disadvantage factor Epithermal-to-thermal 238 U capture ratio Macroscopic cross section of reaction type x Microscopic cross section of reaction type x Optical thickness Scalar neutron flux Fission spectrum Angular neutron flux xvi

17 Acknowledgments Completing my Ph.D. degree is probably the most challenging and joyful task of the first 31 years of my life. It has been a great privilege to share years of hard work with people in the Department of Mechanical and Nuclear Engineering at the Pennsylvania State University (PSU). First and foremost I would like to thank my advisor, Prof. Kostadin Ivanov, for the patient guidance, encouragement and advice he has provided throughout my time as his student. I have been extremely lucky to have an advisor who has been totally supportive yet given enough freedom to me to conduct research independently. He has been an incredible role model to me as a successful member of academia because of his passion and commitment to research and education. One simply could not wish for a better advisor and mentor. My gratitude goes out as well to my Ph.D. committee, Dr. Maria Avramova, Dr. Massimiliano Fratoni, Dr. Fuqing Zhang and Dr. Hangbok Choi for their insights and valuable feedback on this dissertation. Specifically, my sincere thanks go to Dr. Choi, for his guidance on this research and hard work that helps me finish my recent publications. I would like to acknowledge the General Atomics for providing the funding which allows me to undertake this project in the doctoral xvii

18 program and offering me the internship opportunities. Completing this work would have been much more difficult were it not for the support and friendship provided by the current students and alumni of the Reactor Dynamics and Fuel Management Group at PSU. In particular I would like to thank Dr. Zainuddin Karriem, Dr. Shadi Ghrayeb and Adam Rubin for their technical and personal help. Last but not least, to my family. I would like to express my gratitude to my mother, Lai Zhang and my grandmother, Meifang Liu. Their love and unconditional support in me inspire my life and create who I am. I must also thank my wife, Yilun Li, for her confidence in me and the sacrifice she has made during all the ups and downs of my graduate study. I m looking forward to our future. I would also like to dedicate this work to my lost relatives including my father Tongrui Hou and grandfather Linchao Zhang. I hope that this little achievement makes you proud. xviii

19 Lovers, if they knew how, might utter strange things in night air. Since it seems everything hides us. Look, trees exist; houses, we live in, still stand. Only we pass everything by, like an exchange of air. And all is at one, in keeping us secret, half out of shame perhaps, half out of inexpressible hope. Duino Elegies by Rainer Maria Rilke ( ) xix

20 Chapter 1 Introduction This work is concerned with the development and application of a coupled 1- D lattice transport and 2-D lattice nodal diffusion model for advanced reactor physics calculations. In this chapter a short overview of nuclear reactor types relevant to this thesis is given first, followed by a brief overview of the reactor physics calculation methodology and highlight the importance of the cross section generation. Thereafter an overview of the transport method and nodal diffusion method, as well as the neutron cross section generation process is presented. The last section is denoted to the introduction of the goal of this research. 1.1 Fast reactors Although the basic sources of nuclear energy, uranium, is plentiful, it is difficult to use the natural uranium directly in the fission process. In fact, most reactors nowadays are fueled with enriched uranium, in which the concentration of 235 U is increased above its natural value, because fission reactions can be triggered for 235 U by neutrons with energies in the low and intermediate to fast ranges, i.e., at

21 2 thermal energies (< 1 ev) and at energies at around 1 MeV. Same feature is found in 233 U and 239 Pu. These isotopes are called fissile isotopes. Some other isotopes, including 238 U and 232 Th, would fission only when struck by higher energy neutrons, of the order of 1 MeV or more. It is known that 238 U and 232 Th could capture neutrons at energies below the 1 MeV range and thereby convert 238 U into 239 Pu and 232 Th into 233 U in the neutron transmutation reactions: 238 U 239 U 239 Np 239 Pu 232 Th 233 Th 233 Pa 233 U Therefore, they are called fertile isotopes. If more fissile isotopes can be produced from fertile isotopes than are destroyed in the chain reaction, it would be possible to utilize the abundant fertile isotopes and transmit them into fissile isotopes. This process is referred as the breeding. A key parameter in a breeding process is the average number of neutrons released per fission per neutron absorbed, or η [1], which is dependent on the neutron energy E. For example, the eta value for 239 Pu is higher in a fast neutron spectrum than in a thermal neutron spectrum, which means more neutrons would be available for conversion of 238 U to 239 Pu in fast spectrum. Therefore, a breeder reactor operating on fast spectrum neutrons would utilize 238 U efficiently than one operating on thermal spectrum neutrons. Fast spectrum reactors, also known as fast reactors, are the most efficient system for the effective utilization of uranium resource, due to its capability of using the uranium left in the used fuel to be recycled while producing energy. It is also possible to burn plutonium along with uranium stored as tailings from enrichment plants. With these unique features, the energy potential of uranium increase signif-

22 3 icantly (by a factor of approximately 60) compared to light water reactors (LWRs). In addition, the radioactive wastes containing long lived minor actinides becomes practically insignificant. Hence, fast reactors can minimize the mining effort and also reduce storage space and the time required in a used fuel repository to reduce the waste radiotoxicity to the level of the natural uranium. 1.2 Reactor physics calculation methodology Although the discipline of reactor physics that deals with the design and analysis of such reactors encompasses several areas in science and engineering, the reactor physics has matured on its own and established a unique field; and thus in particular, reactor analysis and methods development may be characterized as a discipline concerning determination and prediction of the states of a reactor that sustains chain reaction by balancing neutron production by fission and loss by absorption and leakage. An important part of reactor design is steady state reactor analysis. This entails the solution to the neutron transport equation, which describes the neutron population in the reactor core. From the neutron population or flux, one can derive such design quantities as power distribution, burnup rates in the fuel pins and reactivity coefficients for use in the dynamics studies. The input data are, on the other hand, the cross section library and dimensions of the reactor core or of its fuel assemblies, material compositions, power level, etc [2]. A direct solution of the entire core is not feasible, the limits to calculating the neutron flux in a reactor is not only the accuracy of our methods, but practicality. In general one cannot solve complex phenomena exactly and has to turn to numerical methods not just to obtain a fast solution, but it is often the only approach

23 4 to a solution. For this reason early and present approaches in computational methods for reactor physics [1, 3] was to solve the problem in a stepwise manner. This gave raise to the development of many numerical methods that essentially arose from applying different assumptions to the transport equation in order to obtain a solution for a specific application or physical phenomena. In reactor analysis the approach is to start with a detailed transport problem and then proceed through a series of computational steps during which the problem resolution is decreased, but with the intention of preserving the accuracy of the solution. The primary drive behind this approach is to ensure relatively short computation times so that many core design iterations may be possible. Figure 1.1 [2] shows the different stages of the typical calculation flow involved in reactor core design and analysis. Part A represents the generation of the master cross section library, which is a collection of nuclear data that forms a suitable stating pointing for reactor design calculations. The energy range of interest, usually 0-20 MeV, is discretized into thousands of groups and the group-averaged cross sections are given. Transport methods applied in this regime may be 0- or 1-D and the transport methods may be methods like the Collision Probability Method (CPM) or some other simplified transport treatment. The general philosophy of the calculation flow of the second and third stages is to start with small systems, such as pin-cell (Part B1), and to proceed via intermediate systems, like a fuel assembly (Part B2), to the entire reactor core (Part C). In the second stage (Part B), 1- and 2-D transport methods such as CP and Discrete Ordinates (S N ) methods are often used, but the Diffusion Nodal method has also been added to this regime. Detailed information obtained from this calculation is used to synthesize an equivalent homogeneous representation

24 5 ENDF/B LIBRARY CONSTR. LIBRARY CONSTR. A XS LIBR. XS LIBR. ENGINEERING INPUT CELL CALCULNS B1 BURNUP CONDENSE HOMOGENIZE ASSEMBLY CALCULNS B2 FEW-GR XS CONDENSE HOMOGENIZE FEW-GR XS ENGINEERING INPUT CORE CALCULN DYNAMICS C Figure 1.1: Calculational flow scheme of reactor physics design. of the detailed problem, with a reduced number of energy groups and geometric detail. The flux obtained in each phase is used to produce, by energy condensation or collapsing and geometric homogenization, the cross sections for the calculation

25 6 for the next phase. The third stage corresponds to the use of the nuclear data in the whole core calculation (Part C) to find multiplication factor and other physics parameters such as power distribution of the reactor core. These calculations maybe in 1-D, 2- D or 3-D. At this level, neutron diffusion theory methods such as Finite Difference (FD) method and nodal method are usually applied due to the size of the problem. Although each step in this scheme is important, the cross section generation process, therefore also the accuracy of neutron spectrum solution used in this process, is the key to accurate core design. The accuracy of the neutron transport solution during the cross section generation process and the accuracy of the generated cross sections is one of the focuses of this study. 1.3 Neutron transport theory Analytic (exact) solutions to the transport equation can only be obtained for simplified versions of the transport equation. Although these simplified problems often have limited physical application, analytic transport methods do play a vital role in the development of numerical methods, providing important insights into the transport problem. Computational methods in reactor physics are typically based on numerical methods. These methods entail the development of a numerical representation of the transport equation that can be implemented on a computer. This process involves the discretization of phase space from a continuous (infinite) to a discrete (finite) representation. This discretization can be done by the direct discretization of each independent variable and a numerical representation of the differential and integral operators in the transport equation to obtain a set of linear equations which

26 7 can be solved on a computer [1, 3]. Desirable properties of a numerical method is that it should converge to the true solution with increasing discretization and that convergence ultimately ensures a positive solution which obeys a neutron balance [4]. In practice one would also like to maintain these two properties for a coarse discretization, but this is often not possible. There are several discretizations for all independent variables, but we will describe the techniques that are commonly used in most transport codes today. The energy variable is treated with the multigroup approximation, which will be introduced in detail in the next chapter. Assuming we know how to treat the spatial and angular variable, the approach in the multigroup approximation is to solve the transport equation for each group, which amounts to a one-energy group calculation. The groups are coupled through neutron scattering (and fission) between groups and hence the solution of one group feeds into the other groups. Each group is solved successively and on completion of all groups, the source (scattering source iteration) is updated, and the process repeated until the solution of the flux converges to some user specified criteria. Important issues with the multigroup approach are the choice of the number of groups and the energy group boundaries (structure). Some of the most commonly used neutron transport methods are the Collision Probability Method (CPM), Discrete Ordinates (S N ) method and the Method of Characteristics (MOC). 1.4 Nodal diffusion methods Even after the local fuel pin, clad, coolant, and so on, heterogeneity is replaced by a homogenized representation, a reactor core remains a highly heterogeneous

27 8 medium because of the intra-assembly and assembly-to-assembly variation in fuel composition, burnable poisons, control rods, water channels, structure and so on. The mesh spacing in a conventional few-group finite-difference model of such a core must be sufficiently fine to represent to remaining spatial heterogeneity adequately. It also must not be larger than the shortest group diffusion length in order to avoid numerical inaccuracy. A few-group finite-difference model that could adequately describe such a core might well have 10 5 to 10 6 unknowns (the fluxes in each group at each mesh point). The direct solution of such a problem, even in diffusion theory, remains a formidable computation. For calculations such as fuel burnup or transient analysis, where many such full-core solutions are needed, direct few-group finite-difference solutions remain impractical. A large number of approximation methods have been developed to enable a more computationally tractable solution for the effective multiplication constant and neutron flux distribution in reactor cores. Nodal methods is one of them and became highly competitive in light water reactor analysis. Number of production codes based on nodal diffusion methods has been developed in last decades. Nodal methods characterize the global neutron flux distribution in terms of a small number of parameters in each of several large region, or nodes, into which the reactor core is subdivided for this purpose. Such methods generally require detailed heterogeneous intranodal flux distributions to construct homogenized parameters for each of the many nodes into which a reactor core may be divided and to calculate coupling parameters that link the average flux solutions in adjacent nodes. The heterogeneous flux distribution can be reconstruct by combining the global average nodal fluxes and the nodal surface flux from heterogeneous flux. The recent nodal methods are characterized by the systematic derivation of the relationship between the flux inside the node and the currents on its sides. Modern

28 9 nodal methods have three common features: 1. The unknowns are defined in terms of volume-averaged fluxes and/or surface averaged partial or net currents. 2. The node fluxes and surface currents are related through auxiliary one- dimensional equations obtained by integrating the multidimensional diffusion equation over transverse coordinate directions. 3. The transverse leakage term that appears in those auxiliary equations is approximated by a polynomial (typically quadratic) fit over consecutive nodes. Early generation nodal models assumed that the currents between nodes could be related to the difference between fluxes in neighboring nodes, that is, simple finite difference or modified coarse-mesh finite-difference approximations. The recent nodal models, however, make no such approximation, and the coupling relationships can be derived directly from the neutron diffusion equation. 1.5 Online cross section generation process The cross section generation process has already been mentioned in Section 1.2 and involves the needed few-group nuclear data sets for the diffusion calculation at the core level. At presently two approaches for the preparation of nuclear cross section data for reactor design. The first is pre-tabulation of data and the second online cross section generation. The former has been the traditional method, but interest in online cross section generation is gaining momentum. The process of pre-tabulation involves the transport simulation of representative geometries (fuel assembly) with often ad hoc approximations and boundary conditions to simulate the neutronic conditions of

29 10 the fuel in the reactor core [2]. The main reason for the move to online cross section generation strategies is to eliminate errors that result from approximations that are made in the cross section generation process. The intention is thus to generate cross section data that include more realistic core conditions by iterating between the transport (lattice) calculation and the core calculation. This essentially gives rise to an a whole core multigroup transport core calculation. 1.6 Objective of this research The objective of this research includes development of a embedded lattice calculation methodology for accurate and efficient cross section generation and its application to reactor physics calculations, primarily for fast reactors. The first aspect of the research is concerned with the modeling capacity and efficiency of the 1-D neutron spectrum calculation model MICROX-2, which is based on the collision probability method (CPM). New master cross section libraries using recent ENDF/B-VII.0 nuclear data and optimized fine-group energy structure are generated for this model to accommodate both thermal and fast reactor spectra as well as to comply with the need for advanced fuel cycle analysis. The second aspect of this research is to develop a 2-D lattice calculation model based on the embedded CPM within the Nodal Expansion Method (NEM) framework with interface routines for the utilization of cross section data in the reactor core solver. An online cross section and discontinuity factor generation scheme is also designed to eliminate potential errors introduced by the conventional two-step reactor analysis scheme due to the non-leakage approximation.

30 Synopsis Chapter 2 gives a brief description of the neutron transport and diffusion problem and common numerical methods that are used for solving it. The generation of the master cross section library is introduced in Chapter 3. The improvement of 1-D neutron spectrum calculation model is shown in Chapter 4. Chapter 5 discusses the development of the 2-D embedded pin-by-pin lattice model and the online cross section and discontinuity factor generation scheme. The validation and verification of the developed model are introduced in Chapter 6, where benchmark calculations for the selected reactor core problems are performed. Chapter 7 concludes with a summary of the goals, status and the work that lie ahead.

31 Chapter 2 Literature Review This chapter introduces the neutron transport problem and methods used to solve it. A literature review is presented on cross section library, important topics in lattice calculation, reactor core solver, and recent developments on online cross section generation methodologies are highlighted. 2.1 Cross section library Before finally reaching the reactor core calculation, which is depicted as Part C in Figure 1.1, the reactor physics design is companied with the cross section generation with the following two steps. The first step is to use the fundamental cross section data that have been measured and/or calculated using basic theoretical principles. Such data as exists in the literature have been studied, interpreted for their validity, and compiled for use in nuclear codes in the Evaluated Nuclear Data Files/B (ENDF/B data library files). ENDF/B data files are developed and updated in the USA. There are similar data sets in Europe - JEFF (Joint European Fission Files) and in Japan - JENDL.

32 13 In general, for most applications, the ENDF/B data are much too detailed for direct use with nuclear reactor design codes. Consequently intermediate processing codes such as NJOY [5] are used to generate multigroup cross section libraries for the lattice physics codes. The multigroup (or fine-group) cross section library included in the lattice cell programs is where the cross sections and other quantities have been averaged for. Typically the library would have about energy groups covering the energy range from zero to MeV with noticeably less resolution than that in the original data files. This calls for an averaging process. Letting σ(e) denote any of the microscopic data species the averaging procedure for the energy group n is of the type: σ n = E n w(e)σ(e)de E n w(e)de (2.1) where w(e) denotes a weighting function. For fission related quantities the fission spectrum is an applicable weighting, whereas the slowing down region employs usually the asymptotic 1/E weighting. Similarly for thermal neutrons the Maxwellian spectrum is used. The next step is to compute the few group parameters for the actual fuel assemblies, represented as unit volumes in homogeneous regions, using the lattice physics codes and the multigroup libraries generated in the first step. There are large variations in macroscopic cross section Σ both as a function of position within the fuel assembly, and as a function of neutron energy. Two processes are involved in generating few-group assembly-homogenized cross sections: energy collapsing and spatial homogenization. The spatial homogenization is done on two levels pin cell (unit cell) and assembly homogenization.

33 Governing equations in reactor physics One (perhaps the ultimate) objective of the reactor physics is to determine the neutron distribution in a reactor. For this, we need to model the behavior of nuclear reactors under various conditions, or more specifically, to predict the streaming, scattering, capture, and fission interactions of neutrons with the reactor material. Thus, analytical and numerical techniques are necessary for solving relevant neutron transport, slowing-down, and thermalization problems, and providing physical insights into the process of neutron (and photon) transport in nuclear reactors Transport equation An exact equation describing the neutron distribution in a medium such as reactor has been derived by balancing the various mechanisms by which neutron can be gained or lost from the system: 1 ψ v t (r, ˆΩ, E, t) + ˆΩ ψ(r, ˆΩ, E, t) + Σ t (r, E)ψ(r, ˆΩ, E, t) = Σ s (r, ˆΩ ˆΩ, E E)ψ(r, ˆΩ, E, t)dω de 0 + χ(e) 4π 4π 0 4π νσ f (r, E )ψ(r, ˆΩ, E, t)dω de + Q(r, ˆΩ, E, t) (2.2) where Σ x is the macroscopic cross section of reaction type x, which is obtained from either experimental measurements or simulation results. This equation is called the time-dependent linear Boltzmann transport equation. The angular neutron flux ψ is defined as ψ(r, ˆΩ, E, t) = vn(r, ˆΩ, E, t) (2.3)

34 15 where v = 2E/m is the neutron speed and the angular neutron density N has the following meaning: N(r, ˆΩ, E, t)dv dωde = the incremental number of neutrons in dv about r, traveling in directions in dω about Ω, with energies in de about E, at time t The angular flux ψ is a function of seven independent variables: 3 spatial variables for position, 2 angular variables for direction of flight, and the rest two are for energy time. Each term of Eq. (2.2) represents a specific physical process that describes a (rate of) gain or loss of neutrons from each increment of phase space dv dωde about (r, ˆΩ, E) in some volume V enclosed by the boundary R. The second term on the left side of the equation is the net leakage term. The third term on the left side is the collision term. The first term on the right side is the scattering source. The second term on the right side is the fission source; and the last term is the external source. Since the equation contains both derivatives in space and time as well as integral over angel and energy, it is known as an integro-differential equation. The angular flux ψ must satisfy the boundary condition: ψ(r, ˆΩ, E, t) = ψ b (r, ˆΩ, E, t), r R, ˆΩ ˆn < 0, 0 < E <, 0 < t (2.4) where ψ b is specified. There are several types of boundary conditions used in the neutron transport problem: 1. Vacuum boundary condition refers cases in which ψ b in Eq. (2.4) is equal to

35 16 zero for convex bodies. 2. Albedo boundary condition refers to the circumstances where the incoming flux on a boundary is set equal to a known isotropic albedo, α(e), times the outgoing flux on the same boundary in the direction corresponding to spectral reflection, ψ(r, ˆΩ, E, t) = ψ(r, ˆΩ, E, t), r R, ˆΩ ˆn < 0 (2.5) Here, ˆΩ is the reflection angle corresponding to an incident angle ˆΩ, and ˆΩ = ˆΩ 2(ˆΩ ˆn)ˆn (2.6) is referred to as a specular reflective boundary condition. 3. White boundary conditions is a reflective boundary condition where all particles passing out of V turn back to V with an isotropic angular distribution. 4. Periodical boundary conditions corresponds to the case where the flux on one boundary is equal to the flux on another parallel boundary in a periodical lattice grid, i.e., ψ(r, ˆΩ, E, t) = ψ(r + r, ˆΩ, E, t) Also, ψ must satisfy the initial condition: ψ(r, ˆΩ, E, 0) = ψ i (r, ˆΩ, E), r V, Ω 4π, 0 < E < (2.7) where ψ i is specified. In general, the angular flux ψ is obtained by solving Eq. (2.2) in the physical system V, subject to the boundary condition (2.4) and initial

36 17 condition (2.7). In steady-state problems (the main topic of this research), the time derivative in Eq. (2.2) is set to zero and we obtain the steady-state neutron transport equation: Ω ψ(r, ˆΩ, E) + Σ t (r, E)ψ(r, ˆΩ, E) = Σ s (r, Ω Ω, E E)ψ(r, ˆΩ, E )dω de 0 4π + χ(e) νσ f (r, E )ψ(r, 4π ˆΩ, E )dω de + Q(r, ˆΩ, E) 0 4π (2.8) The boundary and initial conditions can be obtained from Eqs. (2.4) and (2.7), respectively. Angularly, the differential scattering cross section Σ s (r, E E, µ 0 ) is only dependent on the cosine of the scattering angel µ 0 = Ω Ω, which satisfies 1 µ 0 1. This makes it possible to be expressed by Legendre polynomials, i.e., the differential scattering cross section can be expanded as: Σ s (r, E E, µ 0 ) = n=0 2n + 1 4π Σ sn(r, E E)P n (µ 0 ), 0 n (2.9) where P n is the Legendre polynomial of the n-th order. Two special cases are of interest: isotropic scattering, in which only the first term in the expansion is present: Σ s (r, E E, µ 0 ) = 1 4π Σ s0(r, E E) (2.10) and linear anisotropic scattering, in which only the first two terms are present: Σ s (r, E E, µ 0 ) = 1 4π [Σ s0(r, E E) + 3µ 0 Σ s1 (r, E E)] (2.11) where Σ s0 and Σ s1 are the Legendre coefficients of 0-th and 1-th order, respectively.

37 k-eigenvalue problems If the neutron flux distribution within a system does not change with time, or a time-independent nonnegative solution to the transport equation (2.2) can be found, the system is considered to be critical. This can be understood by the fact that the neutron production rate is in equilibrium with the absorption and leakage rates. In fact, one can assume that the balance is achieved by adjusting the value of ν, the average number of neutrons per fission. Therefore, we replace ν with ν/k and rewrite Eq. (2.2) as Ω ψ(r, ˆΩ, E) + Σ t (r, E)ψ(r, ˆΩ, E) = Σ s (r, ˆΩ ˆΩ, E E)ψ(r, ˆΩ, E )dω de 0 4π + χ(e) νσ f (r, E )ψ(r, 4πk ˆΩ, E )dω de 0 4π (2.12) This equation always have the zero solution. Our goal here is to find the largest value of k such that the nonzero solution ψ exists. In this case, k is called the criticality eigenvalue and the resulting neutron flux ψ is called the eigenfunction. Clearly the system is critical if the largest value of k = 1. The value of k < 1 implies that the number of neutron per fission needs to be increased to maintain the criticality, thus the system is subcritical. In the contrast, for the case of k > 1, the hypothetical number of neutrons per fission, ν/k, required to make the system critical, is smaller than ν. Hence the system is supercritical.

38 Diffusion equation Due to the fact that the solution of Eq. (2.8) is difficult to obtain in general, and the neutron scalar flux φ(r, E) = ψ(r, ˆΩ, E)dΩ (2.13) 4π is directly related to the physical parameters of interest in most applications, it is the scalar flux that is desired unknown in the practical calculations. Also define the neutron current density J(r, E) = 4π ˆΩψ(r, ˆΩ, E)dΩ (2.14) First, if we assume that angular flux is only weakly dependent on the angel, or it is a linear function of ˆΩ, then ψ can equivalently be expressed in terms of the scalar flux and current by: ψ(r, ˆΩ, E) = 1 [ ] φ(r, E) + 3ˆΩ J(r, E) 4π (2.15) Second, if the neutron scattering is linear anisotropic as shown in Eq. (2.11), the scattering source in Eq. (2.8) can be expressed in terms of the scalar flux and current by: 0 4π = 1 4π = 1 4π Σ s (r, ˆΩ ˆΩ, E E)ψ(r, ˆΩ, E )dω de [Σ s0 (r, E E) + 3Σ s1 (r, E E)ˆΩ ˆΩ ]ψ(r, ˆΩ, E )dω de 0 0 4π [Σ s0 (r, E E)φ(r, E ) + 3Σ s1 (r, E E)ˆΩ J(r, E)]dE (2.16)

39 20 We then substitute Eq. (2.16) into Eq. (2.8) and operate on it by ( )dω and 4π 4π Ω( )dω. The resulting two equations are called P 1 equations. From the second P 1 equation, we can derive the important relation between the scalar flux and neutron current, also called the Fick s Law: J(r, E) = D(r, E) φ(r, E) (2.17) where the operator D(r, E) is called the diffusion coefficient: D(r, E) = 1 3 [Σ t(r, E) Σ s1 (r, E)] 1 = 1 3 [Σ t(r, E) µ 0 Σ s (r, E)] 1 = 1 3Σ tr (2.18) where Σ tr is the transport cross section and µ 0 is the mean scattering angle. Finally, combining the two P 1 equations gives the continuous-energy diffusion equation: D(r, E) φ(r, E) + Σ t (r, E)φ(r, E) = 0 + χ(e) 4π Σ s0 (r, E E)φ(r, E )de 0 νσ f (r, E )φ(r, E )de + Q(r, E) (2.19) Again, appropriate boundary condition needs to be applied to solve this problem. It should be noted that Eq. (2.19) is derived based on the assumption that the neutron angular flux and the differential scattering cross section can both be expressed by Legendre polynomial expansion. A generalization of Legendre polynomial expansions defined on the unit sphere are spherical harmonic expansions;

40 21 therefore, this method is called the spherical harmonics, or P n method. The simplest P n approximation is the P 1 approximation or the diffusion approximation. The P 1 approximation is adequate for many important practical problems, in particular, for the simulation of many types of nuclear reactor cores. Physical regions with significant neutron absorption, or with significant neutron streaming, are problematic for the P 1 approximation. Thus P 1 solutions are not accurate in the regions near strong neutron absorber or source, or in problems containing void. 2.3 Basic approximations In this section, various common approximations and simplifications to the steadystate neutron transport equation developed in the last section are introduced. The most important of these approximations are the multigroup approximation to the energy variable, and the spherical harmonic (P n ) approximation to the angular variable The multigroup approximation In the multigroup approximation [3], the continuous energy range is discretized into a specified number (G) of energy groups, as shown in Figure 2.1. E G = E min E G 1 E g E g 1 E 1 E 0 = E max Energy group g Figure 2.1: Division of the energy range into G energy groups.

41 22 For each energy group, we define: ψ g (r, ˆΩ) = χ g (r) = Q g (r) = Eg 1 E g ψ(r, ˆΩ, E)dE (2.20) Eg 1 E g χ(r, E)dE (2.21) Eg 1 E g Q(r, E)dE (2.22) where ψ g (r, ˆΩ), χ g (r) and Q g are the multigroup angular flux, fission spectrum and source, respectively. Suppose that within each energy group the angular flux can be approximated as the product of a known function of energy and the group flux. We also define the multigroup cross sections: Σ g (r) = Eg 1 E g Σ g (r, E)ψ(r, E)dE Eg 1 E g ψ(r, E)dE (2.23) By integrate Eq. (2.8) over energy range E g to E g 1 and introducing these approximations, we obtain the steady-state multigroup transport equations: Ω ψ g (r, ˆΩ) + Σ t,g (r)ψ g (r, ˆΩ) G = Σ s,g g(r, Ω Ω)ψ g (r, ˆΩ )dω g =1 + χ g(r) 4π 4π G g =1 νσ f,g (r) ψ g (r, ˆΩ )dω + Q(r), 4π 1 g G (2.24) Assuming we know how to treat the spatial and angular variable, the approach in the multigroup approximation is to solve the transport equation for each group, which amounts to a one-energy group calculation. The groups are coupled though

42 23 neutron scattering and fission between groups and hence the solution of one group depends on that of the other groups. Each group is solved successively and on completion of all groups, the source (scattering source iteration) is updated, and the process repeated until the solution of the flux converges to some user specified criteria. Important issues with the multigroup approach are the choice of the number of groups and the energy group boundaries (structure) Spherical harmonic (P N ) approximation The approximation to the angular variable ˆΩ is achieved by the spherical harmonic (P N ) approximation. Earlier, we have shown that the Legendre Polynomials as being useful in the expansion of the differential scattering cross section, Σ s (µ 0 ), where 0 µ 0 1. In fact, a generalization of Legendre polynomial expansions to functions f(µ, γ) = f(ˆω) defined on the unit sphere are spherical harmonic expansions, which are defined in terms of spherical harmonic functions. These functions, written as Y m n (Ω), are defined in terms of the associated Legendre functions. Any function f(ˆω) defined on the unit sphere has the expansion: f(ˆω) = where (overbar = complex conjugate) f n,m = n n=0 m= n 4π f n,m Y m n (ˆΩ) (2.25) f(ˆω)ȳ m n (ˆΩ)dΩ (2.26) Therefore, the group angular flux ψ g (r, ˆΩ) in Eq. (2.24) has the expression ψ g (r, ˆΩ) = n n=0 m= n ψ n,m (r)y m n (ˆΩ)

43 24 To approximately determine the (unknown) expansion coefficients ψ n,m (r), we specify a positive integer N and set ψ n,m (r) = 0 when n > N; this indicates that ψ g (r, ˆΩ) = N n n=0 m= n ψ n,m (r)y m n (Ω) (2.27) The main advantage of the P N equations is the good approximation of the scattering term. Also, when n = 0, the balance equation is derived, and when n = 1, the diffusion equations combined with Fick s Law is derived. If N is taken to infinity, the true transport solution is recovered. However, there are two disadvantages: the treatment of the vacuum boundary conditions, which are typically used, and the difficulty of solving the multi-dimensional P N equations. 2.4 B 1 method for the asymptotic spectrum The B n leakage calculation In lattice calculations, we need to determine the neutron fluxes, leakage and reaction rates of a unit cell or assembly, without the knowledge of the exact environment in the reactor such as the operation condition or surrounding materials. However, we can always assume that the real flux of the cell is under steady-state critical conditions (k eff = 1). We achieve this by assuming all the surrounding unit cells or assemblies are identical to the one being considered and to adjust the neutron leakage in each group in such a way that k eff = 1. In most lattice codes, a fundamental mode approximation is introduced to present the neutron flux as the product of a macroscopic distribution in space ϕ(r) with a homogeneous or periodic fundamental flux Ψ(E, Ω), i.e., the solution to the steady-state Boltzmann function can be expanded

44 25 in modes that are separable in space on one hand, and in energy and angel on the other hand: ψ(r, E, Ω) = ϕ(r)ψ(e, Ω) (2.28) In the context of the fundamental mode approximation applied over a global reactor featuring a periodic lattice of unit cells or assemblies, the macroscopic distributions is assumed to be a property of the complete reactor and to be the solution of a Laplace equation: 2 ϕ(r) + B 2 ϕ(r) = 0 (2.29) where buckling B 2 is a real number that is used to adjust the curvature of ϕ(r) in such a way that k eff = 1. The buckling is positive or negative if the lattice is supercritical or subcritical. Without any knowledge of the complete reactor geometry, we use the following generic solution of Eq. (2.29): ϕ(r) = ϕ 0 e ibr (2.30) The neutron flux will therefore be factorized as ψ(r, E, Ω) = Ψ(E, Ω)e ibr (2.31) The homogeneous fundamental mode This model assumes that the leakage rates can be computed in a unit cell (or assembly) completely homogenized in space. The derivation of the neutron transport equation for the finite and homogeneous medium is demonstrated in Section 2.2.1,

45 26 the corresponding homogeneous B 1 equations are obtained by substituting the flux factorization (2.31) into the steady-state neutron transport equation, [Σ t (E) + ib Ω]Ψ(ˆΩ, E) = Σ s (ˆΩ ˆΩ, E E)Ψ(ˆΩ, E )dω de 0 + χ(e) 4πk eff 4π 0 νσ f (E )φ(e )dω de (2.32) where the integrated fundamental flux is given in terms of the angular fundamental flux using φ(e) = Ψ(ˆΩ, E)dΩ 4π Now, we introduce the Legendre expansion of the differential scattering cross section using zero-th and first order Legendre polynomials as shown in Eq. (2.11), and the fundamental current defined in terms of the fundamental flux J(E) = 4π ˆΩΨ(ˆΩ, E)dΩ After a simple integration over Ω, the first B 1 equation is obtained: Σ t (E)φ(E) + ibj(e) = 0 + χ(e) k eff Σ s0 (E E)φ(ˆΩ, E )de 0 νσ f (E )φ(e )dω de (2.33) Next, divide Eq.(2.32) by [Σ t (E) + ib Ω] and then integrate it over Ω. After simplifications relating to parity properties, we obtain the second B 1 equation: { ij(e) B = 1 1 Σ t (E)γ[B, Σ t (E)] } Σ s1 (E E) ij(e ) B de (2.34)

46 27 where γ is a function of B and Σ t (E). The coupled B 1 equations (2.33) and (2.34) can be used to solve for critical buckling, which is familiar from the elementary reactor theory. It leads to the system being critical or k eff = 1. Remark The pricipal approximation involved in derivation of the P 1 equations is the assumption of linear anisotropy in the angular dependence of the neutron flux made in Eq. (2.15). B 1 equations, instead, doesn t expand the fundamental angular flux in terms of angular variable. 2.5 Homogenization Nuclear reactors are composed of a large number of fuel assemblies, each containing a large number of discrete fuel elements. Those fuel elements consist of fuel (with various composition), cladding, gap, moderator, coolant, structural elements, control rods, burnable poisons, and so on. Thus, a high degree of heterogeneity exists in the reactor. As introduced above, it is impractical to obtain the detailed neutron flux distribution of each of the fuel elements across the whole reactor system even with the help of the advanced solution methods and increased computational power. The methods employed to replace a heterogeneous lattice of materials of differing properties with an equivalent homogeneous mixture of these materials is referred to as homogenization theory. A two-step procedure is usually involved in the homogenization of a heterogeneous assembly: 1) the transport calculation of the neutron flux on the local effects for pin cells, followed by the use of the flux distribution to collapse and homogenize the cross sections; 2) the detailed calculation of the neutron flux for fuel assemblies to obtain the average homogeneous

47 28 cross sections for the assembly. It should be noted that several calculations in each of the two steps may be required as the properties of the pin cells in assembly may vary and the properties of the assemblies in the core can differ from each other. We have discussed previously the energy collapsation of the cross sections; here we are going to focus on the spatial-averaging techniques used to obtain homogenized cross sections appropriately averaged over spatial details Equivalent homogenized cross sections The general procedure of cross section homogenization can be demonstrated in a simple pin cell model, which is composed of the fuel and moderator regions with volumes V F and V M, respectively. The average absorption cross section for the pin cell is Σ a cell = ΣF a φ F V F + Σ M a φ M V M = ΣF a + Σ M a (V M /V F )ζ φ F V F + φ M V M 1 + (V M /V F )ζ (2.35) where ζ = φm φ F (2.36) is referred to as the flux disadvantage factor, and φ M and φ F are the average flux in the moderator and fuel region, respectively. The homogenized cell average cross section of Eq. (2.35) is obtained by preserving the absorption reaction rates in the cell. The same type of definition is also suitable for the equivalent cell average fission and scattering cross sections, while it is not appropriate for the use to calculate the cell average diffusion coefficient, which must represent the net leakage from the cell. Therefore, the cell homogenization problem reduces to the problem of determining the flux disadvantage factor ζ and the equivalent diffusion coefficient. Note

48 29 that only the relative values of neutron flux is necessary in the evaluation of cross sections using Eq. (2.35) Collision probability (CP) method Before introducing the methods for determining the disadvantage factor ζ, it is necessary to first provide some basic identities such as the blackness identity, the mean chord length of a material system, and an approximation: the collision probability (CP) method, which relates some of the quantities in these identities. Consider a two-region model consisting of the fuel (R F ) and moderator (R M ) regions. We first focus on the fuel region and define the following two important quantities that one usually encounters in integral transport theory: β F = blackness = probability that a random neutron entering R F through S does not leak out of this region (2.37) and p F esc = escape probability = probability that a random neutron born in R F will eventually leak out through S (2.38) Then the following Blackness identity relates β F and p F esc ( 4V β F F = S ) Σ a p F esc = τ F p F esc (2.39) An analogous blackness identity holds for the moderator region R M : ( 4V β M M = S ) Σ a p M esc = τ M p M esc (2.40)

49 30 It is obvious that if one of the blackness or escape probability is known, we can use the blackness identity to yield the other. Next, we show that p esc can be estimated by using the CP method. In the fuel region R F, we define the first-flight escape probability p F esc: p F esc = probability that a neutron born uniformly and isotropically in R F will have its first collision in R M (2.41) then 1 p F esc = probability that a neutron born uniformly and isotropically in R F will have its first collision in R F If the scattering ratio c F = Σ F s /Σ F t is known, the total probability that the neutron escapes from the fuel into the moderator, or p F esc, may be written p F esc = p F esc = [ ] 1 + c F (1 p F esc) + c F 2 (1 p F esc) 2 + p F esc 1 c F (1 p F esc) (2.42) ABH method The ABH collision probability method (named after its originators [6]) is widely used to determine the disadvantage factor ζ for thermal neutrons without performing detailed transport calculation for the fuel assembly and provides physical insight into the pin cell transport problem. A pin cell model of fuel (R F ) and moderator (R M ) with pure reflective boundary condition is considered, as depicted in Figure 2.2. Note that the shape of the outer boundary doesn t necessarily have to be a square, i.e., the fuel pins can be arranged in square or hexagonal lattices. The

50 31 model is usually cylindricized into an equivalent model with reflective boundary condition. Moderator Fuel V F, Q F J + S V M, Q M J Figure 2.2: Example two-region lattice pin cell. The following assumptions are necessary for ABH method: 1. There is no net flow of neutrons between fuel cells. (This is valid for infinite lattice.) 2. The neutron slowing-down source is uniform in the moderator and zero in the fuel. The following symbols are used to describe the cell dimensions and the physical characteristics: V F (or V M ) = volume of fuel (or moderator) S = surface area of the fuel region lf = 4V F S lm = 4V M S = mean chord length in fuel region = mean chord length in moderator region

51 32 Q F (or Q M ) = spatially averaged source density in fuel (or moderator) J (E) = neutron current through S from R M into R F J + (E) = neutron current in the opposite direction of J (E) If the cross section of a certain type for one region is Σ x (x stands for t, a, f, and s etc.), the optical thickness is defined as τ x = lσ x. Besides the blackness identity given in Eqs. (2.39) and (2.40), there are two others to describe the model, both based on simple physical principles. The first identity is: Σ F a V F φ F = Q F V F (1 p F esc) + J Sβ F = Q F V F + (J J + )S (2.43) This implies that the absorption rate in the fuel is the combination of the absorption rate in fuel due to neutrons born in fuel that never escape and the rate at which neutrons that enter the fuel region from the moderator are absorbed. Note we drop the energy sign (E) from the equation for the sake of simplicity. An analogous relation is given for the moderator region as follows: Σ M a V M φ M = Q M V M (1 p M esc) + J + Sβ M = Q M V M + (J + J )S (2.44) Use Eqs. (2.43) and (2.44) to solve for J + and J, we get [ ] Q J + F V F p F esc + (1 β F )Q M V M p M esc 1 = β F + β M β F β M S [ ] Q J M V M p M esc + (1 β M )Q F V F p F esc 1 = β F + β M β F β M S (2.45) (2.46)

52 33 Let R denote the volume source ratio and R = V F Q F V M Q M (2.47) Inserting Eqs. (2.45) and (2.46) to the right-hand side of Eqs. (2.43) and (2.44) leads to ( ) Σ F ζ(e) = a V F Rβ M p F esc + (β F + β M β F β M β F p M esc) Σ M a V M R(β F + β M β F β M β M p F esc) + β F p M esc (2.48) Recall that Q F = 0 is one of the assumptions made for ABH method thus R = 0 and the expression above becomes ( ) Σ F ζ = a V F β F + β M β F β M β F p M esc Σ M a V M β F p M esc (2.49) Next, apply the relation V F /V M = l F /l M and the blackness identities (2.39) and (2.40) to eliminate β F and β M in the last equation, we obtain, ζ = 1 + τ a F p F esc τa M [ 1 p M esc 1 τ M a ] (2.50) To make practical use of this formula, we must determine p F esc and p M esc. ABH method assumes that Diffusion theory is adequate to approximate p M esc, and the Collision Probability (CP) method is adequate to approximate p F esc. To obtain p M esc, we recall that the angularly flux is spatially nearly flat and angularly nearly isotropic in the cylindricized moderator region R M. This allows us to approximate p M esc by solving the 1-D diffusion equation. Take the slab geometry for example, where the moderator boundary satisfies a < x < b, we have

53 34 the following equation and boundary condition: D M 2 φ(x) + Σ M a φ(x) = q M, x R M φ(a) d dφ dx (a) = 0 (RF -F M interface) dφ dx (b) = 0 (Cell boundary) (2.51) where d is the extrapolated distance. p M esc can be evaluated by the solution of Eq. (2.51) using the following relation p M esc = 1 Σ M a V M φ(x)dv and we get 1 p M esc = d 4D M τ M a + E ( ) a L, b M L M (2.52) where E is a lattice function whose expressions for geometries can be found in [7] and L M = D M /Σ M a is the diffusion length of the moderator region. Now we must obtain from Eq. (2.50) a accurate expression for p F esc. The scalar flux is not spatially flat in R F, so the use of diffusion theory is not justified in R F. Using instead the result from the Collision Probabilities method, or Eq. (2.42), we have 1 p F esc = 1 cf (1 p F esc) p F esc (2.53) then Eqs. (2.50), (2.52), and (2.52) yield the thermal disadvantage factor ζ ζ = 1 cf (1 p F esc) p F esc + τ F a ( d 4D 1 + E 1 ) M τa M (2.54)

54 35 Homogeneous Flux Heterogeneous Flux Homogeneous Flux Flux Form Function Thermal flux distribution in two assemblies without burnable ab- Figure 2.3: sorbers Discontiuity factor Consider the ideal case of two identical assemblies, neither containing burnable poisons (BP), in which the few-group diffusion pin-by-pin solution is known. Let us examine the flux distribution in these assemblies. The heterogeneous flux is represented schematically in Figure 2.3. In the nodal analog to this problem, the nodal flux is smoothly varying because cross sections and diffusion coefficients have been homogenized and are spatially constant (or smoothly varying) within the assembly. Figure 2.3 compares the nodal (homogenized) flux to the pin-by-pin (heterogeneous) flux, and the two flux distributions are different.

55 36 However, it is probable that the node-averaged fluxes and net currents (and flux derivatives) at the assembly interface will be nearly the same. In fact, the heterogeneous flux can be accurately approximated as the product of a homogeneous flux and a form function, as shown in the bottom of Figure 2.3. The form function is assumed to be the distribution of fluxes from the assembly spectrum calculation, which used zero-current boundary conditions. Figure 2.4 depicts a case in which one assembly contains burnable absorber pins and the other does not. Here, the derivative of the homogeneous flux and the heterogeneous flux are quite different at the assembly interface, and the net current between assemblies is miss-predicted in the nodal model. Unless the neutron currents are predicted correctly on all assembly interfaces, the goal of preserving reaction rates in the nodal model cannot be achieved. The use of conventional homogenized cross sections and diffusion coefficients is a fundamental problem leading to significant miss-predictions of neutron currents between dissimilar assemblies. In addition, approximation of the heterogeneous flux and the assembly form functions produces two distinctly different values for the flux at the assembly interface. This is clearly unphysical, and if left unresolved, would adversely affect the reconstruction of pin-by-pin fluxes or powers. The breakthrough in resolving the homogenization problem was found by K. Koebke, who recognized that the homogenization difficulties were not due to the definition of homogenized cross sections or diffusion coefficients, but rather, were due to fact that nodal fluxes are modeled as continuous at assembly interfaces. The conventional continuity condition is motivated by the fact that actual (heterogeneous) flux is continuous at assembly interfaces. The homogeneous fluxes, even in the simple cases shown in Figure 2.3, do not match the heterogeneous fluxes at the assembly interfaces, and there is no physical reason that the homogenized

56 37 Homogeneous Flux Heterogeneous Flux Flux Form Function w/ B.P. Flux Form Function Assembly Average Flux Figure 2.4: Thermal flux distribution in assemblies with and without burnable absorbers. fluxes be continuous at the interfaces. In fact, if the heterogeneous flux, depicted in Figure 2.4, is to be continuous at the interface when approximated by a product of the homogenized flux and the assembly form function, the homogenized fluxes must necessarily be discontinuous at the assembly interface. Figure 2.5 depicts the flux shapes near the assembly interface, which would be required in order for the reconstructed fluxes to be continuous. Note that the derivatives of the discontinuous fluxes at the assembly interface are different from the derivatives obtained by imposing conventional continuity conditions on the scalar flux. This implies that the net current at the interface is

57 38 Discontinuous Flux Heterogeneous Flux Homogeneous Flux Discontinuous Flux Figure 2.5: Discontinuous flux distributions in assemblies with and without burnable absorbers. different when discontinuity is permitted, and in fact, the current is closer to that of the heterogeneous model. The concept of discontinuous homogenized fluxes arises from the assumption that the heterogeneous flux is composed of smooth global (homogeneous) flux and an assembly heterogeneous form functions. The use of discontinuous fluxes in the nodal model requires only that the magnitude of the discontinuity in homogenized fluxes be known and that the scalar flux interface condition be specified. Knobke s work was generalized by Smith who introduced quantities called discontinuity factors defined by: DF s g = Heterogeneous Surface Flux Homogeneous Surface Flux (2.55) There is a unique discontinuity factor associated with each surface of each assembly, as shown in Figure 2.5, and the discontinuity factors may be quite different for the two sides of the interface. The required interface condition (relating the surface homogeneous fluxes in nodes m 1 and m), results from the fact that the heterogeneous flux must be

58 39 continuous at the interface: DF s+,m 1 g φ m 1 g (u + ) = DFg s,m 1 φ m 1 g (u ) (2.56) One significant difficulty with the definition of the discontinuity factors in Eq. (2.56) is that the true heterogeneous flux must be known in order to determine the value of the discontinuity factors. The desire is to be able to establish a nodal model without knowledge of the heterogeneous reactor solution. Smith demonstrated that discontinuity factors could be accurately approximated by the ratios of the heterogeneous and homogeneous surface fluxes from isolated assembly spectrum calculations. The plausibility of this can be seen by comparing the flux distributions in the top and bottom portions of Figure 2.6. In the case in which discontinuity factors are approximated by single assembly calculations, the discontinuity factors are referred to as Assembly Discontinuity Factors (ADFs) and they are computed simply as the ratio of the assembly surface flux to the assembly averaged flux. Thus, ADFs can be computed directly from assembly calculations, and knowledge of the heterogeneous reactor solution is not required. In fact, the ADFs are treated as additional homogenization parameters (like cross sections); and they are edited from information available in lattice physics. Given homogenized cross section and ADFs, the nodal diffusion model can be completely specified, and heterogeneous fluxes can be approximated by use of the assembly form functions, as depicted in Figure 2.7.

59 40 Homogeneous Flux Form Function Homogeneous Flux Homogeneous Flux Flux Form Function w/ B.P. Flux Form Function Assembly Average Flux Figure 2.6: Discontinuous flux distributions in assemblies. Homogeneous Flux Form Function Homogeneous Flux Homogeneous Flux Figure 2.7: Discontinuous flux distributions and reconstructed fluxes.

60 MICROX-2 code The MICROX-2 code [8] is an integral transport code. It prepares broad group neutron cross sections for use in diffusion- and/or transport-theory codes from an input library of fine group and point-wise cross sections. The MICORX-2 code is an improved version of MICROX code [9], which has been successfully used for the design of High Temperature Reactor (HTR), such as the Peach Bottom 1 ( ) and the Fort St. Vrain ( ) nuclear generating stations. It can explicitly account for the overlap and interference effects between resonances in both the resonance and thermal neutron energy ranges and allows for simultaneous treatment of leakage and resonance self-shielding in doubly heterogeneous lattice cells. The neutron weighting spectrum is obtained by solving the B 1 neutron balance equations in about 10,000 energies for a one-dimensional (planar, spherical or cylindrical), two-region unit cell. The regions are coupled by collision probabilities based upon spatially flat neutron emission. Energy-dependent Dancoff factors and buckling data can correct the one-dimensional cell calculations for multi-dimensional lattice effects. A critical buckling search option is also included. The inner region may include two different types of fuel particles (grains), i.e., a second level of heterogeneity may be treated. Efficient temperature interpolation algorithms are included to facilitate temperature coefficient calculations. Interpolation between user specified sets of dilution-dependent fine group data is also allowed at energies above the resonance range (the point-wise resonance- and thermal-neutron energy region input data are not dilution-dependent). The user may specify arbitrary combinations of fission spectra for several (up to 13 in the present code) different fissionable isotopes in each spatial region. To reduce execution time and to simplify user input preparation, spectrum and broad-group

61 42 cross section calculations for several (maximum of 11 in the present code) different two region lattice cells (mixtures) may be performed in parallel during a single computer run. Microscopic cross section data used by MICROX-2 consists of a FDTAPE file (temperature and dilution dependent data sets covering energy between 14.9 MeV and 2.38 ev) in the fast energy range and a GGTAPE file (temperature-dependent data sets below 2.38 ev) in the thermal energy range. In the epithermal resolved energy range, the spectrum calculation is performed on an ultra-fine energy grid using the GARTAPE file (temperature-dependent and Doppler-broadened resonance cross sections at 15,000 mesh points below 3-8 kev) as a complement to the FDTAPE for the important resonance nuclides, called primary nuclides. All the nuclear data libraries are the output file of the conversion code MICROR. [10] Figure 2.8 schematically shows how the cross section libraries are utilized in MICROX-2. [9] The experience of working with MICROX-2 in the Reactor Dynamics and Fuel Management Group (RDFMG) has been accumulated recently through research activities. It has been utilized for the generation of multigroup cross section libraries for various types of reactors under either steady state or transient conditions. The MICROX-2 solves B 1 equations (2.33) and (2.34) for each region and the whole cell, with the neutron source coupled by collision probabilities as introduced in Section In fast energy range, those equations for fuel region (with the subscript 1 ), moderator region (with the subscript 2 ) and the whole cell (with the subscript c ) are: (Σ 1 Σ ss,0,1 )φ 1 V 1 + B J c V 1 = P 1,1 Q 0,1 V 1 + P 2,1 Q 0,2 V 2 (2.57)

62 43 15 MeV For all nuclides 100 kev 3-8 kev FDTAPE For secondary nuclides For primary nuclides GARTAPE 2.38 ev GGTAPE For all nuclides Figure 2.8: Utilization of cross section libraries in MICROX-2. (Σ 2 Σ ss,0,2 )φ 2 V 2 + B J c V 2 = P 1,2 Q 0,1 V 1 + P 2,2 Q 0,2 V 2 (2.58) (B 2 / B )φ c V c + 3(γΣ c Σ ss,1 J c V c ) = 3Q 1,c V c (2.59) where the fine group index g has been omitted and Σ (g g) ss,l,j V j =volume of region j (j = 1, 2) V c = V 1 + V 2 = unit cell volume Σ j = total cross section in region j (j = 1, 2) Σ c = volume-averaged total cross section in unit cell P i,j = P 0 source transfer probability from the region i to region j = l th Legendre moment of the scattering cross section for

63 44 Σ ss,l,j = Σ (g g) ss,l,j transfer from group g to group g in region j φ l,j = l th Legendre moment of the neutron flux in region j φ 0,c = cell-averaged scalar flux φ 1,c = cell-averaged current B 2 = buckling (leakage) parameter Q l,j = l th Legendre moment of the in scatter plus fission source in region j γ = parameter as a function of B /Σ c The governing equations in resonance and thermal energy range are similar to those equation above. 2.7 Nodal expansion method Modern nodal methods are commonly used in the reactor core calculations as they can solve the static and transient neutron diffusion equations with low computational cost and produce results with fairly good accuracy. In general, these methods subdivide the reactor core into homogenized nodes where each node usually corresponds radially to a single fuel assembly so that the number of spatial meshes in the problem is greatly reduced. Among the nodal code developed in recent years, Nodal Expansion Method (NEM) [11] is a few group static and transient nodal core model developed, tested, and maintained at the Pennsylvania State University (PSU). The theory of NEM code for the Cartesian geometry, as depicted in Figure 2.9 will be briefly discussed here. With the energy and space discretization, the steady-state continuous-energy

64 45 (J l gx+) in (n x + 1, n y, n z ) (J l gx+) out z 2 (n x, n y + 1, n z ) x 2 z 2 (n x, n y 1, n z ) x 2 y 2 (n x, n y, n z 1) y 2 (n x 1, n y, n z ) Figure 2.9: Nodal modal nomenclature. diffusion equation Eq. (2.19) becomes D g 2 φ g + Σ rg φ g + G g =1 Σ sg gφ g + χ g k G ν gσ sg φ g = 0 (2.60) g =1 where all the notations have their usual meanings but for group g. Eq. (2.60) can be written in two energy groups for an arbitrary node with constant neutronic properties and dimensions x, y and z, as 2 2 Dg l x 2 φl g(x, y, z) Dg l y 2 φl g(x, y, z) Dg l z 2 φl g(x, y, z) + A l g(x, y, z)φ l g(x, y, z) = Q l g(x, y, z), (x, y, z) V l, g = 1, 2 2 (2.61) where A l 1 = Σ l a1 + Σ l 12 1 k νσl f1

65 46 A l 2 = Σ l a2 Q l 1(x, y, z) = 1 k νσl f2φ l 2(x, y, z) Q l 2(x, y, z) = Σ l 12φ l 1(x, y, z) V = x y z = volume of node l Σ ag = group g absorption cross section Σ 12 = group 1 to 2 scattering cross section Using Fick s Law, which is written in the x-direction as jgx(x, l y, z) = Dg l x φl g(x, y, z) (2.62) where j l gx(x, y, z) = x-component of the net neutron current Eq. (2.61) may be written as x jl gx(x, y, z) + x jl gx(x, y, z) + x jl gx(x, y, z) + A l gφ l g(x, y, z) (2.63) =Q l g(x, y, z), (x, y, z) V l Nodal balance equation Assuming that the coordinate origin is at the center of cell l, Eq (2.63) can be integrated over the volume of the cell to obtain a local neutron balance equation. This balance equation is expressed as 1 x (J l gx+ J l gx ) + 1 y (J l gy+ J l gy ) + 1 z (J l gz+ J l gz ) + A l g φ l g = Q l g (2.64)

66 47 where φ l g = 1 V l x/2 x/2 y/2 z/2 y/2 z/2 = node volume-average flux Q l g = 1 V l x/2 x/2 y/2 z/2 y/2 z/2 = node volume-average flux 1 x (J gx+ l Jgx ) l = 1 y/2 z/2 V l y/2 z/2 x/2 x/2 φ l g(x, y, z)dxdydz Q l g(x, y, z)dxdydz x jl gx(x, y, z)dxdydz J l gx± = average x-directed net current on node faces ± x/2 The average net current of y and z direction follows the same expression as the counterpart of x direction Transverse integration procedure In order to solve for the spatial neutron flux distribution in a medium consisting of neutronically homogeneous nodes, one must derive a relationship between the node-averaged flux and the face-averaged net currents. In NEM code this coupling relationship is provided by a series of three consistently derived onedimensional polynomial flux expansions. In order to implement these polynomial flux expansions, the transverse integration approximation must be employed. This approximation requires that Eq (2.63) be spatially integrated over the two dimensions transverse to the particular direction of interest. Such an approximation is motivated by the simple observation that it is generally easier to solve three one-dimensional equations than to solve one three-dimensional equation. For the x-direction, the transverse integrated diffusion equation within node l

67 48 takes the form d dx jl gx(x) + A l gxφ l gx(x) = Q l gx(x) 1 y Ll gy(x) 1 z Ll gz(x) (2.65) where φ l gx(x) = 1 y/2 y z d dx jl gx(x) = 1 y z Q l gx(x) = 1 y z L l gy(x) = 1 y z z/2 y/2 z/2 y/2 z/2 y/2 z/2 y/2 z/2 y/2 z/2 y/2 z/2 y/2 z/2 φ l g(x, y, z)dydz x jl g(x, y, z)dydz x Ql g(x, y, z)dydz x jl gy(x, y, z)dydz = y-direction transverse leakage L l gz(x) = 1 y z y/2 z/2 y/2 z/2 x jl gz(x, y, z)dydz = z-direction transverse leakage Transverse integration for the y- and z-directions yields similar results Polynomial expansion method In the polynomial flux representation of NEM, the one-dimensional flux that appears in the x-directed transverse integrated diffusion equation is expanded in series as φ l gx(x) = φ N l gx(x) + a l gxnf n (x) (2.66) In the fourth-order approximation, the series is truncated after the first four n=1

68 49 basis function, which are given by f 1 (x) = x x = ξ (2.67) f 2 (x) = 3ξ 2 1 ( 4 f 3 (x) = ξ ξ 1 ) ( ξ + 1 ) 2 2 ( f 4 (x) = ξ 2 1 ) ( ξ 1 ) ( ξ + 1 ) (2.68) (2.69) (2.70) These basis functions can be shown to satisfy the following criteria: 1 x/2 f n (x)dx = 0; n = 1, 4 (2.71) x x/2 ( ) ± x f n =; n > 2 (2.72) 2 Polynomial expansion in the y- and z-directions follows a similar procedure. The expansion coefficients a l gxn need to be determined. The first two expansion coefficients can be found by evaluating Eq. (2.66) at the endpoints of the node (± x/2). The remaining two expansion coefficients are determined by a weighted residual procedure, which is explained in details in [11]. Solution of Eq. (2.65) requires further approximation for the x-denpendence of the x-direction transverse leakage (and similarly for the y- and z-direction transverse leakage terms). A number of approximations have been used, but the most successful has been the quadratic approximation L l gy = L l gy + p l gy1f 1 (x) + p l gy2f 2 (x) (2.73)

69 50 with L l gy = average y-directed leakage in node l and the expansion functions f 1 (x) and f 2 (x) defined as in Eqs. (2.67) and (2.68). This assumes that the evaluating of moments of the transverse leakage extends over node l and the two node adjacent to node l in the x-direction (nodes l 1 and l + 1). If the node-averaged transverse leakages are assumed to be known, the transverse integrated flux distribution within a node can be determined by substituting Eqs. (2.66) and (2.73) into Eq. (2.65) and solving the resulting polynomial form of the transverse integrated diffusion equation. The iterative solution to the NEM nodal equations is obtained by use of the non-linear iteration calculation scheme. 2.8 Embedded lattice approach Although the conventional two-step off-line reactor calculation approach as described in Section 1.2 can produce accurate results in an efficient way and it has been widely used in the past decades, the infinite environment approximation applied to the lattice calculation introduces uncertainties in the cross section homogenization and condensation, that is, the parameterized few-group cross sections generated in the lattice physics calculation are not able to capture the actual condition in the reactor core. An even rigorous assumption made here is that the assembly of interest is surrounded by the same type of assembly, which is accepted in large thermal reactor core, but not valid any more for fast reactors that usually have smaller sizes. With the new advanced core design proposed that leads to more heterogeneous core configuration compared to the current core loadings,

70 51 the current two-step or pre-generation methodology will be challenged. Solutions have been proposed to overcome this problem, such as replacing the infinite lattice approximation with leakage correction in the lattice calculation [12] as a feedback from the core calculation. This method also differs from the two-step approach by using a non-linear iteration between lattice code and core solver. In recent years, research has been conducted in the Reactor Dynamics and Fuel Management Group (RDFMG) at the Pennsylvania State University (PSU) to develop the embedded lattice calculation approaches as an alternative to the pre-generation approach. Part of the study presented in this thesis is the continuation of the research to be discussed below. Ivanov [13] investigated the embedded transport calculations as the advanced core calculation methodology. Two different approaches were investigated. The first approach is based on embedded finite element method (FEM), simplified P 3 approximation (SP 3 ), fuel assembly homogenization calculation within the framework of the diffusion core calculation with NEM code (Nodal Expansion Method). The second approach involves embedded FA lattice physics eigenvalue calculation based on collision probability method (CPM) again within the framework of the NEM diffusion core calculation. The second approach is superior to the first because most of the uncertainties introduced by the off-line cross section generation are eliminated. In this method, the albedo obtained in the core simulation is used as the boundary condition in the lattice calculation; however, difficulties have been observed in the spatial reconstruction of the boundary conditions. The other problem observed in the application of the embedded lattice methodology was revealed when reflector is introduced in the model and would cause convergence problems. These problems were expected to be overcome if the boundary conditions for the embedded single assembly calculations are changed from albedos to incoming partial currents.

71 52 Colameco [14] studied embedded approachs based on the Iterative Transport- Diffusion Method (ITDM) by loosely coupling the transport code PARAGON [15] with nodal code NEM. PARAGON was updated to accept incoming partial current as boundary condition and perform fixed source calculation for this study. This method shows that the use of incoming partial current as the boundary condition was a major improvement over albedos. The methodology was proven to work for different 2-D mini-cores without reflectors. However, the addition of the reflector and effectively leakage, showed unfavorable results as compared to 2-D reflective cases, as well as the 3-D cases. Karriem [16] developed the MOC in HELIOS [17] for the purpose of online cross section generation and investigated theory and implementation of fixed k eff calculation scheme, which is the fundamental step of developing the online cross section generation. To ensure the correct implementation of the fix k eff model, the HELIOS-HELIOS calculation scheme was developed, which included the transfer of the angular current conditions across the assembly boundaries. The implementation was then tested in HELIOS on a series of problems, spanning several group structures and geometric arrangements for 2-D benchmark problems.

72 Chapter 3 Generation of Master Cross Section Library This chapter gives the results of the generation of the master cross section library. The MICROX-2 [8] code is an updated version of the MICROX [9] code, which has been successfully used for the design of High Temperature Reactor (HTR), such as the Peach Bottom 1 ( ) and the Fort St. Vrain ( ) nuclear generating stations. The MICROX-2 physics method is basically the same as that of the MICROX code: an integral transport theory to solve the neutron slowingdown and thermalization equations in a detailed energy interval for a two-region lattice cell. One distinct feature of these codes is their capability of handling the second level of heterogeneity: the fuel can contain dispersed spherical particle (grains) of two different types, which is a unique feature of the HTR fuels. The MICROX-2 code has been used for cross section generation of various reactor types. However, it was noticed that the use of MICROX-2 code is restricted by the number of nuclides especially when advanced fuel cycles are analyzed, including an ultra-long fuel cycle and recycling used nuclear fuels. For example, a

73 54 30-year fuel cycle simulation resulted in a few percent loss of fuel mass at the end of cycle due to lack of detailed burnup chain and lumped fission products. In order to comply with the need for advanced fuel and fuel cycle analysis and updating the nuclear data itself, a new master cross section library of MICROX-2 code was generated. When generating the new master library, the number of energy groups was also expanded to accommodate both the thermal and fast reactor spectra. The energy group structure was chosen by examining various lattice models, which are described in Section 3.1. The code system and procedure of master library generation are explained in Section 3.2. The adequacy of the master library generation and the performance of the library were assessed for variety of benchmark problems by comparing the results of MICROX-2 calculation to those of continuous-energy Monte Carlo calculations and measurement data. Finally, the conclusions and future work are discussed in Section Fine-group energy mesh generation The current energy group structure used in MICROX-2 library was originally developed by General Atomics with 193 energy groups below MeV, including 92 fast groups mostly equally spaced in lethargy. With the improvement in computing power, it is now feasible to use fine energy group structure especially for the epithermal range, which in principle improves the performance of neutron slowing down calculations. In this study, the Contributon and Point-wise Cross Section Driven (CPXSD) method was used for the selection of energy group boundaries [18].

74 Methodology The CPXSD method was originally developed for applications to shielding problems. It 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 method was derived based on the contributon response theory (the product of the forward and adjoint angular fluxes) [19] to calculate the importance of groups and point-wise cross sections when obtaining the sub-group boundaries. In this theory, the energy-dependent response flux, i.e., the contributon is given by C(E) = V dr dω Ψ(r, E, ˆΩ)Ψ + (r, E, ˆΩ) (3.1) 4π where Ψ(r, E, ˆΩ) is the angular flux and Ψ + (r, E, ˆΩ) is the adjoint function dependent on position r, energy E and direction ˆΩ. Considering spherical harmonics expansion of flux and its adjoint, and using orthogonality, the group-dependent contributon is written as C g = s D V s L l l=0 m=0 2l + 1 4π Ψm l,g,sψ m,+ l,g,s (3.2) where V s is the volume of the sub-domain, l and m are polar and azimuthal indices for the spherical harmonic polynomial, and g refers to energy group. Ψ m l,g,s and Ψ m,+ l,g,s are angular flux and adjoint function, respectively, at position r and direction ˆΩ in group g. The CPXSD method was validated for criticality problems. For instance, it was used to develop the fine- and broad-group energy structures for TRIGA reactor calculation and analysis [20]. The numerical results show good consistency with Monte Carlo method, limiting the relative deviation of k less than 10 pcm and

75 56 those of objective reaction rates less than 1%. In this study, the CPXSD method is used to obtain the optimized fine group structure by examining variations in eigenvalue and reaction rates. It is an iterative scheme that expands the existing energy mesh as follows: Step 1 : Selection of reference group structure. Step 2 : Cross section library generation with the reference group structure for a given lattice model of interest. Step 3 : Calculation of importance for each energy group by performing forward and adjoint transport calculations. Step 4 : Identifying higher importance energy groups and splitting importance by subdividing them into 2 or 3 groups with equal lethargy. Step 5 : Cross section generation for the new group structure and importance calculation for the next test problem. Repeating Steps 4 and 5 to obtain a fine group structure for all the test problems of interest Selection of optimized energy group structure The Santamarina-Hfaiedh Energy Mesh (SHEM) [21] was selected as the initial energy group structure. It contains 361 energy groups with the lethargy width always thinner than 0.2, which allows accurate slowing down for the fast breeder reactor calculations. In order to obtain a flexible fine group structure that can be applied to various reactor types, both the thermal and fast reactor lattice models with various isotopic contents have been selected, including Light Water Reactor

76 57 (LWR), Heavy Water Reactor (HWR), Gas Cooled Reactor (GCR), Sodium-cooled Fast Reactor (SFR), Lead-cooled Fast Reactor (LFR), and Gas-cooled Fast Reactor (GFR) with uranium dioxide, mixed oxide, mixed zirconium, transuranic nitride, mixed carbide, etc. An iterative procedure is employed in the energy structure refinement as depicted in Figure 3.3. A transport code DRAGON Version 4 [22] was used to obtain forward and adjoint fluxes, as the MICROX-2 code is not capable of computing the adjoint flux. It should be noted that the DRAGON code is eligible for this task only when DRAGON calculated fluxes is consistent with those from MICROX-2 calculations. Therefore, the rationale of such an alternative calculation was first assessed by the comparison of forward fluxes from DRAGON and MICROX-2 for all lattice models considered in this study except for the HWR case as a simplified two-region model of a cluster-type HWR fuel bundle by the MICROX-2 will introduce large discrepancies. The selective flux comparisons of PWR-UO 2 and GFR are given in Figure 3.1 and 3.2, respectively, to represent thermal and fast lattices. Good agreements were observed in the comparison, especially for PWR-UO 2 case with well-thermalized spectrum. For GFR model with relatively hard spectrum, slight differences are observed in the lower energy ranges (ev ranges), but they won t lead to bias in the energy group importance calculation because the fraction of lower energy neutron population is extremely small. Therefore, the use of DRAGON code as a substitute of MICROX-2 is valid in this application. The final group structure was searched iteratively to include all the selected lattice models. During the course of energy group refinement, the criteria of energy group subdivision in terms of group importance changed in order not to increase the number of energy groups indefinitely. The final energy group structure was optimized to have 1010 fast

77 DRAGON MICROX Normalized flux Energy (ev) Figure 3.1: Comparison of the normalized fluxes for GFR cases DRAGON MICROX Normalized flux Energy (ev) Figure 3.2: Comparison of the normalized fluxes for PWR-UO 2 cases. groups and 163 thermal groups with a thermal energy cutoff of 2.38 ev. The upper energy boundary of each group is listed in Table A.1 of Appendix A.1. In order to confirm the adequacy of the final energy group structure for the application to both thermal and fast reactor analyses, computational benchmark

78 59 Initial energy mesh Transport calculation Forward flux Ψ Adjoint flux Ψ + Group importance Mesh expansion Generation of new library N Last case? N Y Meet ɛ? Y Final energy mesh Figure 3.3: Iterative procedure in energy structure optimization. calculations were performed by the DRAGON code using the fine-group library generated for room temperature. The benchmark problems selected are GCR and GFR pin cell models. The results of DRAGON calculation were compared with those of continuous-energy Monte Carlo calculation done by MCNPX code [23]. The comparisons were made for infinite multiplication factor (k ) and reaction

79 60 Parameter Table 3.1: Summary of the GCR and GFR calculations GCR GFR MCNPX DRAGON MCNPX DRAGON k ± a (0.001%) b ± (0.002%) Fast 238 U production ± (-0.93%) ± (-0.81%) Epithermal 238 U capture ± (0.35%) ± (-0.03%) Thermal 235 U production ± (0.27%) ± (-0.84%) a One standard deviation due to statistical uncertainty. b ( ) difference calculated as 100 (x DRAGON x MCNPX )/x MCNPX. rates of fast, epithermal, and thermal groups. Two reasonably small acceptance limits were chosen to determine whether or not the optimized fine-group structure is acceptable: 0.01% for k and 1% for reaction rates, respectively. The reaction rates used for comparison are neutron production rate of 238 U, radioactive capture rate of 238 U, and neutron production rate of 235 U for the fast, epithermal, and thermal energy groups, which are summarized in Table 3.1. It can be seen that the results of DRAGON calculation with the optimized group structure are consistent with those of MCNPX [23] calculation: the maximum differences of k and reaction rate are 0.002% and 0.93%, respectively. 3.2 Master library generation The Evaluated Nuclear Data File (ENDF) is generally considered as the standard source for nuclear data in a computer-retrievable form. The evaluations have been continuously updated and the most recent version ENDF/B-VII was released for open distribution in It is a long process to convert raw nuclear data into the cross sections that can be directly used by the reactor physics codes. In this study, the NJOY [24] system and MICROR code [10] are used to process the ENDF and generate cross section sets in GAM-II format for the neutron spectrum code

80 61 MICROX Code system upgrade NJOY was first released in 1977 at the Oak Ridge National Laboratory (ORNL) Radiation Shielding Information Center (RSICC) and at the National Energy Software Center at Argonne National Laboratory (ANL). NJOY94.61, the starting version that was used in this study, was released in December 1996 at RSICC. MICROR was developed under the joint General Atomic Technologies and Swiss Federal Institute for reactor researches. The current version of MICROR was released in September MICROX-2 was developed and tested at the Paul Scherrer Institute (PSI) in Switzerland. All three codes were originally designed to run on a CRAY machine. Transferring the system of codes to run on the Personal Computer (PC) platform is challenging and has been accomplished. The adoption of the newly developed 1173-group PSUmesh energy structure makes it urgent to update the code system because the data generated and transferred has already exceeded the dimension limit of the codes. For instance, the maximum allowable number of energy groups is 641 in NJOY versions prior to version , which obviously needs to be increased. The upgrade of the NJOY code is simply accomplished by preparing the official update directives along with other suggested directives regarding to the compilation environment, and executing the UPD program included in the code package. A NJOY executable with addition PSUmesh energy structure is then obtained by compiling the source using a FORTRAN compiler. The primary modifications to the MICROR and MICROX-2 include the extension of the major dimension limits of important variables (such as total allowable

81 62 nuclides and energy groups), and dynamic memory allocation capability. No details will be discussed in this section Library generation procedure The library generation process starts with the NJOY code system that reads in ENDF and produces Point-wise Evaluated Nuclear Data File (PENDF) and Groupwise Evaluated Nuclear Data File (GENDF). The NJOY consists of a number of modules, each operating as a separate computer program, and conducting a welldefined task. Specifically, PENDF is generated by RECONR, BROADR, UNRES- R/PURR, and THEMR modules, while GENDF is produced from the PENDF and ENDF by GROUPR module. A sample NJOY input file for 235 U is given in Appendix A.2. Once the PENDF and GENDF are generated, they are then converted into FDTAPE, GGTAPE, and GARTAPE data form for MICROX-2 code by a PSI version of the original coupling and reformatting module MICROR. The MICROR input files for 3 He, 16 O and 235 U are provided in Appendix A.3 for examples. Recall that the FDTAPE contains the fine group dilution- and temperature-dependent cross sections in the fast energy range; the GARTAPE contains point-wise Doppler broadened resonance cross sections in the resolved resonance range; and GGTAPE contains of infinite dilution cross sections in the thermal energy range. The data processing procedure has been verified and validated in the previous research [25], which is given in Fig In the present study, an up-to-date NJOY and an upgraded MICROR were used. The library generation was performed on the HP Z400 workstation under the Windows R operating system. The temperature-dependent cross sections were processed at various points

82 63 ENDF/B-VII MODER PENDF BROADR PENDF UNRESR PENDF THERMR PENDF GROUPR GENDF MICROR FDTAPE GARTAPE GGTAPE MICROX-2 Figure 3.4: Cross section library generation for MICROX-2. between K and > 2000 K. For the most important resonance isotope 238 U, PURR module was used for the treatment of unresolved resonances. The PENDF was collapsed to GENDF using a combination of the Bondarenko approach and the flux calculator option to approximate the self-shielding effect. A 1/E + fission spectrum + thermal Maxwellian weight was employed with the flux calculator option. Thermal scattering laws were taken from separate data files for major moderator nuclides such as hydrogen in water, and processed at 4-5 temperature points. Using the aforementioned scheme and these options, the master 1173-group cross section library was generated based on ENDF/B-VII release 0 data. The dilution-dependent cross sections were processed corresponding to the infinite dilution for all nuclides, while for those with relatively significant resonance

83 64 behavior, the Bondarenko factors were tabulated at various dilution (background) cross sections, as shown in Appendix A.4. The generation of such a large cross section library can be tedious and human errors are almost inevitable during this procedure. To reduce the processing time and ensure the accuracy, a Python script PyMicrox was developed to automatically process the nuclear data and converting the format by executing NJOY and MICROR for the nuclides in any specific order defined by the users. A concise manual of PyMicrox is given in Appendix B. In order to confirm the library generation process, the cross sections of the new MICROX-2 library were directly compared to the GENDF and PENDF data for selected nuclides. For simplicity, the neutron energy was divided into three ranges with the energy cutoffs kev and 2.38 ev. The group-wise fast and thermal cross sections were compared with the GENDF data, while the equallethargy point-wise resonance cross sections of the library were compared with the PENDF data. The comparisons of fission and capture cross sections for 235 U, 238 U and 239 Pu are given in Table 3.2, which shows that the difference of cross sections is negligible in fast range, and is also reasonably small in the thermal range. In resonance range, however, the maximum variance is 9.93% for 238 U capture cross section. In MICROR, the resonance cross sections were obtained by reconstructing the PENDF data using a screening method, which results in fewer energy points and increases discrepancy of cross sections especially in the unresolved resonance region.

84 65 Table 3.2: Comparison of objective cross sections Energy 235 U 238 U 239 Pu range δσ f (%) a δσ γ (%) δσ f (%) δσ γ (%) δσ f (%) δσ γ (%) Fast Epithermal Thermal a ( ) difference calculated as 100 (xs MICROX-2 xs ENDF )/xs ENDF Benchmark models The accuracy of the newly generated MICROX-2 cross section library was verified for a series of benchmark problems against the Monte Carlo code MCNPX and the experimental data. MCNPX code was used because of its capability of modeling exact geometry, robust solution method, and handling continuous-energy cross sections. The ACE-formatted continuous-energy neutron data library ENDF70 and thermal S(α, β) data for the cold state (293.6 K) are available in the distributed code package. The pin cell lattices are modeled in a cylindrical geometry with a white boundary condition. The spectral indices are obtained from the reaction rate tallies of MCNPX. Benchmark problems were selected to represent a relatively wide variety of neutron spectra, that can be categorized into three groups as described below. They are all pin cell lattices, characterized by the fuel, clad, coolant, and moderator. The first group includes two thermal reactor benchmark cases, TRX and BAPL, from the Cross Section Working Group Benchmark Specifications (CSEWG) [26]. They are both light water moderated, fully reflected, single lattices reduced from a full reactor models operating at room temperature. The fuel materials of the TRX and BAPL lattices are 1.3 wt% enriched uranium metal and uranium dioxide, respectively. The moderator-to-fuel volume ratios of TRX-1 and TRX-2 lattices

85 66 are 2.35 and 4.02, respectively. The same ratios of BAPL-1, BAPL-2 and BAPL-3 lattices are 1.43, 1.78 and 2.40, respectively. In the second group, a series of critical assembly problems were chosen from the Benchmark Handbook issued by the International Criticality Safety Benchmark Evaluation Project (ICSBEP) [27]. The critical assemblies such as ZEUS, HUG, ZEBRA, ROVER, and RBMK were simplified to pin cell models. The third group includes pin cell lattices of commercial power reactors and Generation-IV type reactors such as Pressurized Water Reactor (PWR) [28], Sodium-cooled Fast Reactor (SFR) [29], Lead-cooled Fast Reactor (LFR) [30], and Gas-cooled Fast Reactor (GFR) [31]. A summary of numerical benchmark problems is given in Table Calculation models In this study, MCNPX code was used to produce the reference solution of the numerical benchmark because of its capability of modeling exact geometry, robust solution method, and handling continuous-energy cross sections. The cross section and thermal S(α, β) data at room temperature are from the standard MCNP package. All pin cell benchmark problems were modeled explicitly in two-dimensional geometry with appropriate boundary condition: the models with cylindrical boundary were applied with the white boundary condition (WBC), which corresponds to a reflection with a cosine direction distribution on the surface, while the specular reflective boundary condition (RBC) were applied to the models with other boundaries. In the MICROX-2, a pin cell lattice is simplified as a 2-region cell: region 1 is the fuel and region 2 is the moderator either in slab or cylindrical geometries.

86 67 Table 3.3: Characteristics of benchmark problems Category Model Fuel type CSEWG benchmark ICSBEP benchmark Power reactor Enrichment (wt% 235 U) Clad Coolant/ Moderator V F /V M a TRX-1 U metal 1.3 Al Water TRX-2 U metal 1.3 Al Water BAPL-1 UO Al Water BAPL-2 UO Al Water BAPL-3 UO Al Water ZEUS U metal 1.33 Graphite Graphite HUG UO 2 92 Al Graphite ZEBRA MOX b 7.6 Stainless steel Graphite/Sodium ROVER U metal 2.06 Graphite Graphite/Water RBMK UO Zirc-110 Graphite/Water PWR-UOX UO Zirc-4 Water PWR-MOX MOX 4.0 Zirc-4 Water a Fuel-to-moderator volume ratio b Mixed oxide c MA = Minor actinides SFR U-Pu-10%Zr 30 HT9 Sodium LFR (U,Pu)N 9.07 HT9 Pb-Bi GFR (U,Pu,MA)C c 3.6 Stainless steel He/SiC However the code explicitly accounts for the overlap and interference effects between resonances in both the resonance and thermal neutron energy ranges and allows for simultaneous treatment of leakage and resonance self-shielding in doubly heterogeneous lattice cells. For a finite lattice, energy-dependent Dancoff factor and geometric buckling are used to consider multi-dimensional lattice effects. The deterministic code DRAGON is used in this study for inter-comparison of calculation results. For the DRAGON code, the cross section library is also generated in 1173 energy groups based on ENDF/B-VII release 0 to be consistent with the MICROX-2 master library. In this process, a new module DRAGR is used to produce the DRAGLIB interface file from the NJOY intermediate cross section

87 68 library. The DRAGLIB format is designed to store multigroup isotopic nuclear data to be used in a lattice code such as DRAGON. This module is not included in the standard NJOY distribution but available in the DRAGON code package [22]. The DRAGON solutions were obtained by the collision probability method along with the self-shielding module based on the subgroup method [32, 33]. WBC and RBC were applied to models with cylindrical boundaries and those with other boundaries, respectively. It should be noted that the purpose of these calculations is not to assess the DRAGON code but to produce additional data for code-to-code comparisons. The multiplication factors, spectral indices (given below) as well as the fastto-thermal flux ratio (R) were used to test the new MICROX-2 master library. Except for the problem set of the first group, the thermal energy cutoff is 2.38 ev, which is a default value in the GGTAPE. ρ 28 : epithermal-to-thermal 238 U capture ratio, δ 25 : epithermal-to-thermal 235 U fission ratio, δ 28 : ratio of 238 U to 235 U fission, and C : ratio of 238 U capture to 235 U fission Calculation results Homogeneous lattice cases The results of TRX and BAPL infinite lattice benchmark calculations are given in Table 3.4. The eigenvalues predicted by the MICROX-2 and DRAGON codes generally agree well with the reference values obtained from MCNPX calculations. Compared with the MCNPX results, MICROX-2 always overestimates the k

88 69 value. For the MICROX-2, the difference of k increases with increasing lattice pitch for both cases, with an average difference of 0.15 For the simplified lattices of the second group, as shown in Table 3.5, the k of MICROX-2 is consistent with that of MCNPX with the average difference of 0.29% and the maximum error of -1.05% for the ZEBRA, which is a fast reactor fuel pin with MOX fuel and sodium coolant. The largest error of the reaction rate ratio is found for epithermal-to-thermal 238 U capture ratio of RBMK lattice, which consists of UO 2 fuel and water coolant. Table 3.6 presents the calculation results for typical power reactor fuel lattices. The average difference of k is 0.6% δk, while the maximum difference is 0.8% δk for the PWR lattice with MOX fuel. The maximum error of spectral index is 9.32% for the SFR lattice with U-Pu-10%Zr fuel and sodium coolant Heterogeneous lattice cases The results of TRX and BAPL lattice calculations are summarized in Table 3.7, in which the eigenvalue and spectral indices predicted by the MICROX-2 and DRAGON codes agree well with the MCNPX. Comparing to the MCNPX calculation, the average errors of k is 0.32% and 0.33% δk for MICROX-2 and DRAGON, respectively. Definite trends have been found for the MICROX-2 calculations: k and ρ 28 are always over-predicted and the differences increase with the lattice pitch, while C is always under-predicted and the relative error increases with lattice pitch. The relation with lattice pitch is similar for ρ 28 and δ 25 : over-estimation for TRX cases and under-estimation for BAPL cases and the difference increases in magnitude as the lattice pitch increases. The maximum error of the spectral indices is 5.31% for ρ 28 in TRX-2 case. It is also found that the fast-to-thermal flux ratio is always overestimated by a few percent by MICROX-2.

89 70 Table 3.4: Benchmark calculation results for TRX/BAPL homogeneous lattices Case Source k ρ 28 δ 25 δ 28 C R TRX-1 TRX-2 BAPL-1 BAPL-2 BAPL-3 MCNP DRAGON MICROX-2 MCNP DRAGON MICROX-2 MCNP DRAGON MICROX-2 MCNP DRAGON MICROX-2 MCNP DRAGON MICROX a b a One standard deviation due to statistical uncertainty. b ( ) difference calculated as 100 (x Code x MCNP )/x MCNP.

90 71 Table 3.5: Benchmark calculation results for ICSBEP homogeneous lattices Case Source k ρ 28 δ 25 δ 28 C R ZEUS HUG ZEBRA ROVER RBMK MCNP DRAGON MICROX-2 MCNP DRAGON MICROX-2 MCNP DRAGON MICROX-2 MCNP DRAGON MICROX-2 MCNP DRAGON MICROX a b a One standard deviation due to statistical uncertainty. b ( ) difference calculated as 100 (x Code x MCNP )/x MCNP.

91 72 Table 3.6: Benchmark calculation results for power reactor homogeneous lattices Case Source k ρ 28 δ 25 δ 28 C R PWR-UOX PWR-MOX SFR LFR GFR MCNP DRAGON MICROX-2 MCNP DRAGON MICROX-2 MCNP DRAGON MICROX-2 MCNP DRAGON MICROX-2 MCNP DRAGON MICROX-2 a One standard deviation due to statistical uncertainty. b ( ) difference calculated as 100 (x Code x MCNP )/x MCNP a b

92 73 Table 3.7: Benchmark calculation results for TRX/BAPL heterogeneous lattices Case Source k ρ 28 δ 25 δ 28 C R TRX-1 TRX-2 BAPL-1 BAPL-2 BAPL-3 MCNP DRAGON MICROX-2 MCNP DRAGON MICROX-2 MCNP DRAGON MICROX-2 MCNP DRAGON MICROX-2 MCNP DRAGON MICROX a b a One standard deviation due to statistical uncertainty. b ( ) difference calculated as 100 (x Code x MCNP )/x MCNP.

93 74 Table 3.8 shows the calculation results of simplified pin-cell models derived from ICSBEP full core benchmark models, except for HUG case which is originally a homogeneous model. Compared to the results for the first category, relatively larger errors are found for the k from the MICROX-2 calculations against MCNPX with an average error of 0.58% δk. The largest error of the reaction rate ratio was found for the ratio of 238 U-to- 235 U fission of the ZEUS lattice. It should be noted that the MCNPX tally for ZEBRA lattice didn t completely converge due to dominant fast neutron population, especially for ρ 28 and δ 25 that include reaction rates of thermal neutrons. The dominance of fast neutron flux can be clearly seen from the flux ratio R. The calculation results of typical power reactor pin-cell models are summarized in Table 3.9. Some of the parameters for MCNPX calculations of the LFR and SFR cases are not provided here due to large statistical errors. As for the infinite multiplication factor from MICROX-2 calculations, the results show underestimation by 0.46%, 0.26% and 0.14% δk for SFR, LFR and GFR, respectively. On other hand, it was overestimated by 0.16% and 0.13% k for PWR-UOX and PWR- MOX models. The maximum error of spectral indices is % for ρ 25 in the PWR-MOX case. Again, the MCNPX results with large statistical errors are not used in the comparison Finite lattice cases For the TRX and BAPL, experimental data are available for the critical lattices. For MICROX-2, the effective multiplication factor keff is obtained using homogenized buckling for both fast and thermal groups. As shown in Table 3.10, MICROX-2 models under-predict k eff for all, indicating the code overestimates the leakage overall. It is probable that the uniform buckling values from the exper-

94 75 Table 3.8: Benchmark calculation results for ICSBEP heterogeneous lattices Case Source k ρ 28 δ 25 δ 28 C R ZEUS ZEBRA ROVER RBMK MCNP DRAGON MICROX-2 MCNP DRAGON MICROX-2 MCNP DRAGON MICROX-2 MCNP DRAGON MICROX a b a One standard deviation due to statistical uncertainty. b ( ) difference calculated as 100 (x Code x MCNP )/x MCNP. iment can t exactly reproduce the spectrum shift in fast and thermal ranges. In general, the prediction error of keff is slightly lower (0.32% δk on average) for the finite lattices when compared to the infinite cases.

95 76 Table 3.9: Benchmark calculation results for power reactor heterogeneous lattices Case Source k ρ 28 δ 25 δ 28 C R PWR-UOX PWR-MOX SFR LFR GFR MCNP DRAGON MICROX-2 MCNP DRAGON MICROX-2 MCNP DRAGON MICROX-2 MCNP DRAGON MICROX-2 MCNP DRAGON MICROX-2 a One standard deviation due to statistical uncertainty. b ( ) difference calculated as 100 (x Code x MCNP )/x MCNP a b

96 77 Table 3.10: Comparison of finite lattice calculations for TRX and BAPL Case Source k ρ 28 δ 25 δ 28 C Experiment DRAGON TRX a MICROX Experiment DRAGON TRX MICROX Experiment DRAGON BAPL MICROX Experiment DRAGON BAPL MICROX Experiment DRAGON BAPL MICROX a ( ) difference calculated as 100 (x Code x Experiment )/x Experiment.

97 Analysis of comparison results Due to the diversity of the model in material composition and geometry configuration, it is difficult to draw a single conclusion from the calculation results. However, a general trend can be observed from the comparative analysis of the calculation results. For the homogeneous lattice cases, the infinite multiplication factor is over-predicted when the lattice is fully thermalized (e.g., TRX, BAPL and PWR cases), while it is under-predicted when the fast-to-thermal flux ratio is large (e.g., ZEBRA, SFR and LFR). For the heterogeneous lattice cases, the trend of the difference between the reference value and MICROX-2 result is similar to that of the homogeneous lattice case but with relatively larger errors of k and other physics parameters. The comparison of the homogeneous and heterogeneous lattice calculation is shown in Figure 3.5 for the prediction of k by MICROX-2. Same relation of discrepancy of k with neutron spectrum can also be found for the heterogeneous lattice cases, with an average error of 1.1% for 14 cases and the maximum error of 3.6% for RBMK. The fast-to-thermal flux ratio is always overestimated in all cases for both homogeneous and heterogeneous lattices, except for the heterogeneous PWR lattice with MOX fuel. In general, the MICROX-2 results are more close to the DRAGON results than to the MCNPX results. For the heterogeneous lattices, it is found that the prediction of k is more closely related to C (relative conversion ratio), which directly depend on 238 U capture, than to other spectral indices. The prediction error of C is opposite to that of k, namely the underestimation of C leads to the overestimation of k, and vice versa. This parameter is a rough indication of the balance of neutron loss due to capture and production from their primary sources, 238 U and 235 U; thus it reflects the global effects of the solution method of the code. It is expected that the

98 Homogeneous cases Heterogeneous cases 1.0 Relative difference of k (%) TRX-1 TRX-2 BAPL-1 BAPL-2 BAPL-3 ZEUS HUG ZEBRA ROVER RBMK Benchmark models PWR-UOX PWR-MOX SFR LFR GFR Figure 3.5: Comparison of homogeneous and heterogeneous lattice calculations. resonance self-shielding model has a dominant effect on the spectrum calculations, which will be briefly discussed below. In the MICROX-2 calculations, the cross sections for various dilutions and temperatures for each nuclide are available in the fine group calculations. Semilogarithmic interpolation of the fine group data that is based on the Bondarenko approach is used for both the dilution and the temperature interpolations. On the other hand, the point-wise data is not dilution-dependent and no dilution interpolation is available in the point-wise resonance (GARTAPE). The MICROX- 2 code performs detailed point-wise resonance calculations for all specified nuclides over a shared energy range from an upper limit of a few kev down to a few ev. The original code has no capacity to compute the dilution parameters in the point-wise energy range or to allow the use of both dilution-independent pointwise and dilution-dependent fine group data for the same nuclide in the point-wise

99 80 energy range. In addition, the two-point logarithmic interpolation scheme used in the fine group calculations is considered to be inaccurate in many cases. It is understood that the discrepancy of the MICROX-2 calculations for the fast reactor lattices is larger, since the unresolved resonance treatment is significant for the fast reactors. Moreover, this also explains why the calculation of physics parameter is poor for the heterogeneous lattices, especially when the fuel fraction increases. As such, it is expected that the increase of resonance absorption of 238 U will decrease the resonance escape probability and depress the thermal flux in the MICROX-2 calculations, especially for the models with relative hard spectra such as LFR, SFR and GFR. Although this is consistent with the overestimation of the flux ratio R, the MICROX-2 calculations do not consistently underestimate the k s. The overestimation of the multiplication factor for the lattices with wellthermalized neutron spectrum such as TRX and BAPL implies that the MICROX- 2 solution method has certain influence on the results, but it compensates for the effect of the resonance absorption treatment; i.e., the code slightly overestimates the thermal neutron flux. The thermal fission of 235 U will become stronger, as seen from the large underestimation of ρ 25 in ROVER and RBMK cases, and increase the neutron production. This competing effect depends on the degree of neutron spectrum thermalization, because the dominant reaction type changes accordingly. The physics calculation becomes more complex when the fuel includes plutonium. The thermal fission of the Pu nuclides results in a larger increase of the thermal flux and an underestimation of the fast-to-thermal flux ratio. This explains the over-prediction of k. The large relative errors of the spectral indices of uranium in this case are the reflection of the spectrum shift instead of the global effects because the majority of the fission and absorption comes from plutonium. In previous results, a small increase of the resonance absorption has been ob-

100 81 served when comparing the cross sections in the new library and the original nuclear data, as shown in Table 3.2. This also slightly decrease the resonance escape probability and depress the thermal flux; however, considering the magnitude of errors and the trend associated with the fuel-to-moderator volume ratio, it is believed that the solution method, including self-shielding model, dominates the error. On the other hand, the continuous-energy ENDF70 library used in the MCNPX calculations was processed by NJOY Version 248 and the consistency was checked to test the processed data [34]. It was reported that modifications were made to the original evaluated nuclear data and resulting ACE files for a couple of isotopes due to various issues found during the checking process. [35] There are also uncertainties in the NJOY input parameters, such as convergence tolerance and weighting function, used for nuclear data processing. Therefore the discrepancy of the MICROX-2 results with the reference values is partially due to the fact that some of key information is missing regarding the generation of MCNPX library. 3.3 Summary and conclusions New master libraries of MICROX-2 code were generated from the ENDF/B-VII release 0 with a fine-group energy structure of 1173 groups based on CPXSD method. The new library offer a relatively wide range of variety in the number of nuclides, available temperature and dilutions (or background cross sections). The performance of the MICROX-2 code with new libraries was assessed by comparing results of the MICROX-2 calculation to those of MCNPX code, DRAGON codes, and measurement data for a series of pin cell lattices developed from CSEWG, ICSBEP and typical power reactor models. The results of MICROX-2 calculation with the new library have shown good

101 82 agreement with the reference results for the homogeneous lattice cases with an arithmetic average error of 0.31% for k. For the water reactor finite lattice cases, the arithmetic average error of k eff is also reasonably small, 0.32%. The results of heterogeneous lattice cases have shown a relatively larger k error of 0.44%, which is mostly due to the solution method and possibly due to the cross section generation options of the master library itself to a certain extent. Further investigations are required on following issues to identify the source of error and improve the performance of MICROX-2 with the new library: The cross section library generation procedure needs to be confirmed for the neutron capture cross section of 238 U and thermal scattering data of moderating nuclides, respectively, in conjunction with the solution method of MICROX-2 code. The resonance and spatial self-shielding model of the MICROX-2, including the effect of upper and lower boundaries, needs to be examined in conjunction with the cross section generation procedure.

102 Chapter 4 1-D Lattice Physics Model The results of the benchmark calculations shown in the last Chapter indicated that the calculation error stems primarily from the solution method of MICROX-2, of which the self-shielding model plays an important role. More specifically, the resonance self-shielding model is responsible for computing the effective dilution used to interpolate the cross sections, while the spatial self-shielding model computes the correction factors to take into account of the nonlinear condensation effects for heterogeneous cases. [36] In addition, the critical flux model also needs updates to improve the efficiency. In this Chapter, the resonance self-shielding model of MICROX-2 is introduced in Section 4.1 along with the model updates and the verification results of homogeneous benchmark problems. Section 4.2 describes the implementation of the new model for spatial self-shielding factor, as well as the numerical benchmark of the heterogeneous lattices. The improvements of the critical flux model will be discussed in Section 4.3. The conclusions and summaries are given in Section 4.4.

103 Resonance self-shielding Model When the relative energy of an incident neutron and a nucleus plus the neutron binding energy match an energy level of the compound nucleus that would be formed upon neutron capture, the probability of neutron absorption is quite large. This is phenomenon is called the resonance, as shown between 1 ev and a few thousand ev in Figure for 238 U radiative capture 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 the resolved resonance range (ev up to a few kev), resonances are wide when compared to the scattering ranges for the mixtures in a particular configuration. This region is significant for thermal reactors. In the unresolved resonance region, on the other hand, it is difficult to achieve an adequate resolution of individual resonances. However, the neutron absorption in this region is important especially for fast reactors. In the vicinity of a resonant energy region of an isotope that has a high concentration in the material mixture, the total cross section Σ t varies significantly, resulting in a large impact on the neutron flux and the corresponding reaction rates as well. This effect complicates the evaluation of the effective multigroup cross sections given in Eq. (4.1). σ x,g = g deσ x(e)φ(e) g deφ(e) (4.1) where σ x is the microscopic cross section of reaction type x and φ(e) is the neutron spectrum. The objective of resonance treatment is to evaluate effective cross sections for the resonance nuclides, in all cell regions and resonance groups, while 1 Source: National Nuclear Data Center.

104 Source: National Nuclear Data Center Cross Section (barns) Incident Energy (MeV) Figure 4.1: 238 U radiative capture cross section. the problem-independent fine-group cross section data σ x is prepared prior to the calculation of Eq. (4.1). Thus the resonance self-shielding calculation implies determination of the effective flux φ(e). Since the flux will remain unknown until the whole calculation is completed, an additional step is required to estimate the effective cross sections. Two methods are generally used in the reactor physics codes: the Bondarenko method [37] and the subgroup method [32, 33]. The current version of MICROX-2 code is based on the first method Bondarenko method The Bondarenko method, which is based on the narrow resonance (NR) approximation, assumes that collision density is unaffected by a very narrow resonance, leading to the modification of Eq. (4.1) for nuclide i in the unresolved resonance

105 86 region: σ i g = = g deσi (E)C(E)/Σ t (E) dec(e)/σ g t(e) C(E) g deσi (E) σt(e) i + σ0(e) i g de C(E) σt(e) i + σ0(e) i (4.2) where Σ t (E) is is the total macroscopic cross section for the material and C(E) is a smooth weight function such as 1/E. [5] The σ0 i term, normally called the dilution parameter (or dilution cross section), is the cross section per atom of resonant nuclide i for all nuclides in the mixture other than nuclide i itself. The dilution cross section of a homogeneous mixture is given as follows: σ i 0 = j i N j σ j t (E) N i (4.3) where N i is the corresponding number density. In this way, the self-shielding effect of the flux is obtained for nuclide i by representing all other nuclides with a single dilution cross section. Therefore the self-shielded cross section becomes a function of three variables (other than the nuclide s own cross sections): energy group g, temperature T, and dilution parameter σ 0. For resonant nuclide i, if σ 0 is larger than the peak value of σ t, it is called infinite dilution (σ 0 = ), and the cross section of the material of interest has little or no effect on the flux. On the other hand, if σ 0 is small with respect to σ t, it is called finite dilution, and a large self-shielding effect is expected. For a finite value of σ 0 at temperature T, the Bondarenko factor or f-factors, F g (σ 0, T ), is defined as σ g (σ 0, T ) = F g (σ 0, T )σ g (, T ) (4.4)

106 87 This method allows one to pre-calculate cross sections for several values of σ 0 and temperature without having to know the detailed composition of the material of interest, interpolate cross sections based on σ 0 and T, and get desired self-shielded group constants. In practice, f-factors are generated for important reaction types (generally fission, capture, elastic scattering, transport, and total) at five to eight σ 0 s, spanning the typical range of cross section values encountered for the nuclide, and at three or four temperatures. The major advantages of the Bondarenko approach are its simplicity and speed. It is especially attractive for many fast reactor applications, which operate in an energy regime where the narrow resonance approximation is apt to be appropriate. In the resonance calculation of MICROX-2, as shown in Fig. 2.8, the cross section library prepared by the MICROR module normally contains smooth dilutiondependent data for the FD fine group data (fast group) and smooth infinite dilution for the GAR point-wise file (resonance group) [10]. For the unresolved resonance treatment, the user may specify fine group data for different dilutions and/or temperatures for each nuclide in the FD fine group calculations. The code estimates the dilution cross section in the mixture for each nuclide and a semi-logarithmic interpolation of the fine group data is used for the corresponding Bondarenko factors based on following formula: F (σ 0 ) = F (σ 1 )s(σ 0 ) + F (σ 2 )[1 s(σ 0 )] (4.5) s(σ 0 ) = ln(σ 2/σ 0 ) ln(σ 2 /σ 1 ) (4.6) A preliminary investigation showed that the accuracy of the current interpolation scheme is very sensitive to the magnitude of user-provided dilution cross sections

107 88 Bondarenko factor σ 0 Figure 4.2: Linear interpolation of the dilution parameter. and has a large effect on the calculation results. The existing interpolation method of the MICROX-2 is a two-point semi-logarithmic scheme (see Figure 4.2) and the discrepancy of the calculation result is primarily due to the way that the cross section table is interpolated for a non-linear variation of the self-shielding factor. That is, the linear interpolation based on two data points doesn t always fit the data, especially when the data points are scattered over a large range Power fitting self-shielding method In order to improve the interpolation scheme such that the fitting function is continuous, converged, and monotonic, a new monotonic scheme based on power functions was proposed, in which a curve is always found to fit three monotonic points [38]. The analytical expression of such a scheme for interpolating the Bondarenko factor is written as F (σ 0 ) = F (σ 1 ) + σp 0 σ p i σ p i+1 σp i [F (σ i+1 ) F (σ i )] (4.7)

108 p=0.830 Bondarenko factor x =log(σ 0 /0.1) +1.0 Figure 4.3: Monotonic scheme to fit σ 0 variation of Bondarenko factors. The data point σ 0 is in the interval (σ i, σ i+1 ), while the Bondarenko factor is in [F (σ i ), F (σ i+1 )], and p is a power fitting constant for a point σ 0 to fit three points in a curve, of which the value can be determined as follows: p(σ 0 ) = p(σ 1 ) + σ 0 σ i σ i+1 σ i [p(σ i+1 ) p(σ i )] (4.8) For each fine-group calculation of MICROX-2, after the cross section data is read and transformed into logarithmic coordinates, nuclide-dependent monotonic power fits (p) are prepared based on the dilution-bondarenko factor pairs. Then iteration is performed to adjust p until it satisfies Eq. (4.7) for a given intermediate σ 0. Note that the temperature interpolation is applied before the dilution interpolation if necessary. Fig. 4.3 shows a typical fit of three dilution cross section points developed by the power function fitting method. Next, the code estimates the dilution parameter σ 0 for each nuclide using

109 90 Eq. (4.3) and search for the smallest input dilution interval (σ i, σ i+1 ) that covers σ 0. Eq. (4.8) is then used to calculate the corresponding power fit p(σ 0 ). When all the information is obtained, the nuclide- and energy-dependent Bondarenko factor is computed to update the objective cross sections and to calculate the dilution cross section σ 0 again; and this iteration process continues until σ 0 converges. As depicted in Fig. 4.4, this self-shielding calculation scheme covers all the nuclides in each fine energy group and cell regions. This model follows and extends the current model by fitting the existing data structure to the fine group calculation. It also enables the code to make use of all the self-shielded cross sections tabulated as a function of the dilution parameter and avoids the possibility of extrapolation Numerical benchmark models In order to assess the new self-shielding model of MICROX-2, numerical calculations were performed for selected benchmark problems as introduced in Section Similarly, the multiplication factors, spectral indices (given below), and fast-to-thermal flux ratio (R) are used for the verification. Among all the spectral indices, the epithermal-to-thermal 238 U capture ratio ρ 28 will be examined carefully as it is directly related to the resonance absorption. Note that the thermal energy cutoff is 2.38 ev, which is the default value of the MICROX Model verification In order to exclude the spatial self-shielding effect from the lattice calculation, the pin cell models were further simplified to homogeneous lattices. The results of TRX and BAPL infinite lattice benchmark calculations are given in Table 4.1. The

110 91 Cross section data read-in Itarative scheme Power fitting constant p calculation Group-dependent σ 0 calculation Interpolation of Bondarenko factor Objective cross section update σ 0 converged? N Y Last nuclide? Y Last group? Y N Next nuclide N Next group Figure 4.4: Iterative scheme of the new self-shielding factor calculation routine eigenvalues predicted by the MICROX-2 using the existing and new self-shielding models generally agree well with the reference values obtained from the MCNPX calculations, while the errors from the existing model is slightly smaller than the new model in general. Compared with the MCNPX results, these two models always overestimate the k value, with a root-mean-square (rms) error of 0.15% and 0.19% δk, respectively. Most of the spectral indices show good agreements with the reference value within 1.0%. The new model has a larger discrepancy in

111 92 Table 4.1: Comparison of k and reaction ratios for TRX/BAPL homogeneous lattices Case Codes k ρ 28 δ 25 δ 28 C R MCNPX TRX-1 Existing model a New model MCNPX TRX-2 Existing model New model MCNPX BAPL-1 Existing model New model MCNPX BAPL-2 Existing model New model MCNPX BAPL-3 Existing model New model a Difference calculated as 100 (x MICROX-2 /x MCNPX 1). ρ 28 for all lattices except for TRX-1, but the average error is smaller than that of the existing model. For the second group shown in Table 4.2, the k of the MICROX-2 with the new model produces a better agreement with that of the MCNPX with an average error of 0.12%, while the results with the existing model have an average error of 0.37%. The error of ρ 28 is also reduced for ZUES, HUG, and RBMK models, although there is a slight increase in case of ROVER. The largest error of the reaction rate ratio still exists in the epithermal-to-thermal 238 U capture ratio of the RBMK lattice, but it is reduced with the new resonance self-shielding model. Table 4.3 presents the calculation results of typical power reactor fuel lattices. It can be seen that the estimation of k is improved when the new resonance self-shielding model is applied. The average error has been reduced from 0.42%

112 93 Table 4.2: Comparison of k and reaction ratios for ICSBEP homogeneous lattices Case Codes k ρ 28 δ 25 δ 28 C R MCNPX ZEUS Existing model a New model MCNPX HUG Existing model New model MCNPX ZEBRA Existing model New model MCNPX ROVER Existing model New model MCNPX RBMK Existing model New model a Difference calculated as 100 (x MICROX-2 /x MCNPX 1). to 0.30% δk, when compared with the existing model. The comparison of ρ 28 is available only for the PWR lattice because of large statistical errors in the MCNPX calculations of other lattices; and it is consistently over-predicted in both the existing and new self-shielding models. A deterministic code DRAGON [22] was also used in this study for intercomparison of calculation results. The cross section library of the DRAGON was generated in 1173 energy groups based on ENDF/B-VII.0, which is consistent with the MICROX-2 master library. The code also provides various options for the resonance self-shielding calculations such as the universal self-shielding (USS) module, which is used in this study. The USS is based on the subgroup method with corrections on the non-correlation approximation of the resonances in the scattering sources and main collision terms of the transport equation for the thermal lattice

113 94 Table 4.3: Comparison of k and reaction ratios for power reactor homogeneous lattices Case Codes k ρ 28 δ 25 δ 28 C R MCNPX PWR-UOX Existing model a New model MCNPX PWR-MOX Existing model New model MCNPX SFR Existing model New model MCNPX LFR Existing model New model MCNPX GFR Existing model New model a Difference calculated as 100 (x MICROX-2 /x MCNPX 1). modeling. [39] In order to understand the relation between the neutron flux and the resonance self-shielding treatment, the benchmark problems are regrouped into two categories: the thermal and fast lattices. The thermal lattices include TRX, BAPL, HUG, ROVER, RBMK and PWR models; and others belong to the fast lattices. The MICROX-2 calculation errors of k and ρ 28 with the existing and new methods are compared from each other in Fig. 4.5, in which the results of two methods have a good agreement for the thermal lattice problems, while the new method clearly shows a reduction in the prediction error for the fast lattice problems. The prediction errors of k and ρ 28 have reduced from 0.47% to 0.18% and from 2.29% to 1.46%, respectively. It is also worth noting that the results of the DRAGON calculations are excellent especially for the light water reactor lattices but are no better than those of the MICROX-2 model for the fast reactor problems investi-

114 k MICROX-2 (existing model) MICROX-2 (new model) DRAGON 0.4 Relative difference (%) Thermal Fast Overall Benchmark models ρ 28 MICROX-2 (existing model) MICROX-2 (new model) Relative difference (%) Thermal Fast Overall Benchmark models Figure 4.5: Comparison of MICROX-2 resonance self-shielding methods gated in this study so far. Earlier studies have shown that the error in the MICROX-2 calculation is due to a combined effect of the solution method and the cross section library. However, considering the magnitude of each error, the first effect dominates the calculation result. In addition, the overestimation of the multiplication factor for the lattices of well-thermalized neutron spectrum (such as the models in Group A) implies that

115 96 the thermal flux calculation has a certain influence on the results, i.e., the code slightly overestimates the thermal neutron flux. This effect compensates for the effect of the existing resonance absorption treatment. By modifying the resonance self-shielding model of the MICROX-2, as shown above, the over-prediction of ρ 28 is reduced and the resonance escape probability is increased, leading to higher k values. For the fast reactor lattices, where the unresolved resonance treatment is significant, the MICROX-2 calculation has been improved, showing reduced errors of the estimated k. In case of the thermal lattices, the thermal flux calculation method is more important, and it is reasonable to see the over-estimation of k. 4.2 Spatial self-shielding model The fuel assemblies in the reactor core consist of fuel, moderator/coolant, clad, gap regions and so on. At lower energies in the 10 ev range, the neutron mean free path becomes comparable to the fuel and moderator dimensions, and it is important to take into account the spatial heterogeneity of the fuel-moderator cell since they will cause a local spatial variation in the neutron flux which may strongly influence core multiplication. In addition, an additional level of heterogeneity can also be introduced if the fuel subregion contains coated fuel particles (grain), such as in the the HTR fuel design. Based on the second equivalence theorem of resonance escape probability, the spatial effect in the heterogeneous model can be included by adding an effective escape cross section to the dilution parameter σ 0. [40] It states that a heterogeneous lattice characterized by an effective escape cross section σ e has nearly the same resonance integral as the homogeneous mixture with an effective dilution cross section, i.e. (σ 0 + σ e ), if the narrow resonance (NR) approximation is valid. [41]

116 97 The escape cross section is closely related to the probability that a random neutron born in the fuel region will experience its first collision in the moderator region. Therefore, Eq. (4.3) is extended as follows: σ i 0 = 1 N i [ Σ i e + j i N j σ j t (E) ] (4.9) In the MICROX-2 code, the escape cross section is not directly used, but in the form of the escape probability p r esc. This will be explained in next section in detail Statement of the problem Consider a two-region cell that consists of the fuel and moderator regions, as shown in Figure 2.2, and use the same notations as introduced in Section By defining the two important identities, blackness and escape probability, and two other simple relations (2.43) and (2.44), we reached the disadvantage factor ζ derived by the ABH method, ( τ F ζ(e) = a τ M a V M ) Rβ M p F esc + (β F + β M β F β M β F p M esc) R(β F + β M β F β M β M p F esc) + β F p M esc (4.10) The assumption Q F = 0 we made before is no longer valid since the fast neutrons will be taken into account here. Next, we define a augment function H r of region r (r = F, M) with the following expression: H r = 1 pr esc τ r ap r esc 1, r = F, M (4.11)

117 98 Then the blackness identity (2.39) or (2.40) yields τ r a β r = 1 + τa(1 r + H r ) p r 1 esc = 1 + τa(1 r + H r ) (4.12) (4.13) Note that if the within-group scattering cross section Σ r ss 0, escape probability p r esc is the multi-collision probability. Recall that p r esc is the corresponding firstflight escape probability (the neutron is considered as absorbed after each collision), and let the function H k be the corresponding first-flight augment, i.e., p r esc = τ r a(1 + H r ) p r esc or H r, which depends on τa r only, can be computed more easily than the multiple-collision probabilities. We attempt to express the H r s (or p r esc s) in terms of the H r s (or p r esc s). Assume the neutrons are emitted isotropically and uniformly (spatially flat) after each with-group scattering collision, p e,k and p e,k are related to each other by p r esc = p r esc 1 c r (1 p r esc) (4.14) and c r is the scattering ratio in region r. Eq. (4.14) is proved in Eq. (2.42) and not repeated here. More derivations show that if c r is small enough p r 1 esc = c r + (1 c r )/ p r esc 1 = c r + (1 c r )[1 + τ a (1 + H r )]

118 τ r a(1 + H r ) We conclude that H r H r, i.e., H k (τ r a) = H k (τ r a) + H r (τ r a) (4.15) where H r accounts for the fact that the neutrons are not emitted uniformly after a within-group scattering collision (i.e., not anywhere in region r). Sauer [42] argues that H k is approximately independent of the geometry, and that it is given fairly accurately by H r (τ r a) c r τ r a ( c r), τ r a < 6 (4.16) When the first-flight augment H r is given (their evaluation is given in the following section), then the escape probability and blackness are known, too. Use Eqs. (4.12) and (4.13) to eliminate β F and β M from Eq. (4.10), we get ζ(e) = 1 + R + τ F a W 1 + R(1 + τ M a W ) (4.17) where W = 1 + H F (τ F a ) + H M (τ M a ) + H F + H M (4.18) Remarks: (i) Note that the geometry of the unit cell is not restricted to what is shown in Figure 2.2 to be valid for the derivation above. The same results will be obtained as long as the same boundary condition is applied to a two-region model.

119 100 (ii) The dependence of the augment H on the scattering ratio c k for τ > 1 (i.e., for large bodies) is not negligible, as shown in Eq. (4.16). This correction is not used in the present version of the MICROX-2 code, i.e., the following equation is used in the code instead of Eq. (4.18). W = 1 + H 1 (τ t,1 ) + H 2 (τ t,2 ) Wigner rational approximation and Dancoff factor It can be seen from the derivation above that the application of the ABH method depends on the determination of p r esc. The Wigner rational approximation [43] of the first-flight escape probability of the fuel region is: p F esc = = l F Σ F t (E) ( l F ) 1 ( l F ) 1 + Σ F t (E) (4.19) = Σ F e Σ F e + Σ F t (E) Since Eq. (4.19) is derived by assuming the moderator size is infinitely large, thus ( l F ) 1 has the following interpretation: Σ F e = = escape cross section 1 lf = the probability per unit distance of flight that a random neutron born in R F will experience its first collision in R M. Recall that l is the mean chord length in the body. It is practical to calculate l for simple convex geometries such as sphere, cylinder and slab.

120 101 Comparing Eq. (4.19) with Eq. (4.13) indicates that the escape probability p esc will reduce to the Wigner ration approximation if the augment H is zero. The Wigner ration approximation is quite accurate if the moderator region is optically thick. In other cases, this approximation will over-estimate the escape probability of the fuel region, because it assumes that all neutrons exiting the fuel region will experience their next collision in the surrounding infinite moderator. However, in the practical situation, the different fuel system are not infinitely far apart, as depicted in Figure 4.6, neutrons that leaks out of one fuel region can have their next collision in a neighboring region. Many physics codes that use the dilution cross section method consider this effect by modifying the escape cross section (Bell factor or Levine factor) [44, 45], or by correcting the interaction between different lumps in the moderating region (Dancoff corrections) [46]. In the MICROX-2, these lattice effects can be taken into account by means of the Dancoff factor, which effectively modifies the evaluation of the augment function H. Note that H measures the deviation from the rational approximation; that s also why it is called the augment. The detailed explanation of usage of H in the code is given in Appendix C.1. The Dancoff factor is defined as C = the probability that a random neutron exiting one fuel region will experiences its next collision in another fuel region. (1 C) can be interpreted as the probability of a random neutron leaking out R F will has its next collision in R M, which is than used to modify the escape cross section: Σ F e = 1 C l F (4.20)

121 102 B A Figure 4.6: Example of rectangular fuel lattices. With this modification, the Wigner Rational approximation becomes: p F esc = 1 C 1 C + τ F t (4.21) The advantage of this equation is the equivalence between the given heterogeneous system and a corresponding homogenized system for which the moderator cross section equals the moderator cross section in the fuel rod of the heterogeneous system plus the effective cross section Σ F e. The physical reasoning of the Dancoff factor is given as the following [47]: when the moderator region becomes increasingly thick (and consequently, the fuel regions become increasingly isolated), C 0; in the opposite limit, in which the moderator region becomes thin, C 1 and p F esc 0. The Dancoff factor must

122 103 satisfy that 0 C 1 and: 0, τt M = l M Σ M t 1, C = 1, τt M = l M Σ M t 1. The above equation indicates the dependency of C on the material and geometry of the moderator region, thus to the augment function H M as well. In MICROX-2, the Dancoff factor can be externally calculated by either analytical expressions or numerical methods. The analytical expression of Dancoff factor is based on geometric simplifications and is applicable only to regular lattices of fuel rods. On the other hand, the second option requires an additional modeling effort, which requires a longer processing time and could potentially lead to a significant error. Therefore it is recommended to implement a new Dancoff factor method without losing the efficiency, accuracy, and consistency of the physics calculation Embedded Monte Carlo model A wide variety of options are provided for different lattices and cell geometries through the use of Dancoff approximations. The DANCOFF-MC is a Monte Carlo (MC) program developed for Dancoff factor calculation of a fuel lattice with either cylindrical or spherical geometry [48]. Calculation of Dancoff factor is based on the collision probability definition, i.e., the program calculates the probability that a neutron emitted isotropically from the surface of the fuel region of the fuel element will have its next collision in the fuel region of any other surrounding fuel element. In this study, an iterative method has been taken to implement the DANCOFF- MC in the MICROX-2 code, as shown in Figure 4.7. The existing Wigner rational approximation of the MICROX-2 is used to compute the initial value of the escape

123 104 MICROX-2 Calculation Σ t,2 N MC method? Y First run? Y C 0 DANCOFF-MC N Σ t,2 converged? N Σ t,2 End Figure 4.7: Dancoff factor calculation using Monte Carlo method. cross section and obtain neutron spectrum. After the initial calculation, the homogenized cross section of the moderator region and geometrical parameters are then sent to DANCOFF-MC module to estimate the Dancoff factor, which will then be used in the next iteration. This process continues until the total cross section is converged. The embedded DANCOF-MC module accepts cylindrical fuel in square, triangular or hexagonal lattice pitch, thus the MICROX-2 models in slab and cylindrical geometries need to be converted first. In addition to the conservation of volume ratio of the fuel region, such a conversion also requires consistency in the mean chord length of the fuel region l F, which represents the average distance that an

124 105 uncollided neutron travels after entering the fuel region until returning to the moderator. By preserving l F, the conversion actually preserves the effective size of the fuel Model verification The convergence rate of this scheme is first tested for the benchmark problems and the results have shown that the total cross section of the moderator region converges within two or three iterations for all 15 cases. For example, the total cross sections of region 2 of the first 6 iterations of the TRX-1 heterogeneous model are plotted in Figure 4.8 as an example. Recall that the TRX-1 is a light water moderated, fully reflected, single lattice reduced from a full reactor model operating at room temperature. The fuel material of the model is 1.3 wt% enriched uranium metal and the moderator-to-fuel volume ratios is It can be seen from the figure that the cross section, which is directly related to the Dancoff factor, drops and converges quickly to cm 1 after second iteration. The reference value, obtained from the MCNPX calculation, is cm 1. The good agreement between the MICROX-2 and MCNPX indicates that the implementation of MC method with an iteration scheme can provide a higher accuracy of the Dancoff factor calculation without sacrificing computing time. Next, the calculation results of the heterogeneous benchmark models with the embedded MC method are compared with those obtained by the original option as summarized in Table 4.4. The results show that the errors of the infinite multiplication factors are consistently reduced and the improvement is significant for fast reactor lattices. The relative errors of k for the Wigner approximation and embedded Monte

125 Total cross section of region 2 Σt,2 (cm 1 ) Iteration Figure 4.8: Iteration history of total cross section of moderator region. Carlo methods are compared in Fig. 4.9 for individual benchmark problems. Again, it can be seen that the updated self-shielding models significantly improve the accuracy of the MICROX-2 calculation especially for the fast reactor lattices because the self-shielding treatment for the unresolved resonance is more important to the relatively hard neutron spectrum lattices. The average percentage error of k has been reduced from 0.44% to 0.22%, making it slightly smaller than that of the DRAGON calculation. 4.3 Critical flux model Normally, the neutron spectrum is evaluated in systems with a reflective outer boundary and zero net in-current. Thus, in calculation on a fuel cell and the control cells, the system is assumed to be infinite. The multiplication factor k (eigenvalue) of such a system is their infinite multiplication factor k, which generally differs

126 107 Table 4.4: Summary of benchmark calculations using Wigner approximation and Monte Carlo methods for the Dancoff factor calculations Case Codes k ρ 28 δ 25 δ 28 C R TRX/ BAPL ICSBEP Power reactors MCNPX DRAGON a Wigner approx MC method MCNPX DRAGON Wigner approx MC method MCNPX DRAGON Wigner approx MC method a Average difference calculated as 100 (x Codes /x MCNPX 1) /5. from unity. So the corresponding infinite-medium neutron spectrum differs from the critical spectrum at k = 1. In an operating reactor each of its subsystems will be critical. Deviation of their k from unity is compensated by out-leakage (k > 1) or in-leakage (k < 1). It is this leakage effect which causes criticality flux and infinite-medium flux to be different. Although this difference is usually ignored when the broad-group cross sections are generated, the critical spectrum is undoubtedly necessary to calculate the burnup reaction rates. In the MICROX-2 code, fine-group dependent bucklings are used to specify the leakage into or out of the region of interest. The bucklings are defined by the relation L = DB 2 φv (4.22) where L is the neutron leakage out of or into the region of interest (in n sec 1 ), D is the diffusion coefficient in the region (in cm), B 2 is the buckling (in cm 2 ), φ is

127 Wigner approx. MC method Relative difference of k (%) TRX-1 TRX-2 BAPL-1 BAPL-2 BAPL-3 ZEUS ZEBRA ROVER Benchmark models RBMK PWR-UOX PWR-MOX SFR LFR GFR Figure 4.9: Comparison of k with different Dancoff factor calculation scheme the average neutron flux in the region (in n cm 2 sec 1 ) and V is the volume of the region (in cm 3 ). Bucklings must be supplied for each fine group above the thermal energy range in MICROX-2. An option is provided to perform a search for the critical buckling. When this option is activated, all input bucklings are multiplied by a constant factor such that the computed k eff is unity. The buckling search option iterative procedure requires about five times more computer time than a standard MICROX- 2 calculation with fixed user supplied bucklings. In general, the code performs a two-level iterative scheme, where the convergences of the effective multiplication factor k eff and true critical buckling are the objectives of the two steps, respectively. More specifically, in the first level (inner iteration), the iteration starts with

128 109 the buckling B 2(1) provided by the user or from last cycle. The effective multiplication factor k eff is calculated using the condensed two-group cross sections and the bucking. The evaluation of k eff is conducted again when the buckling is doubled at the second iteration, i.e., B 2(2) = 2B 2(1). The critical buckling for next iteration is estimated based on the values of k eff as shown in Eq. (4.23). B 2(3) = B 2(1) + P (B 2(2) B 2(1) ) k(1) 1.0 k (2) k (1) (4.23) More general form of the estimation of the buckling for (i + 1) th iteration is given below, B 2(i+1) = B 2(i 1) + P (B 2(i) B 2(i 1) ) k(i 1) 1.0 k (i) k (i 1) (4.24) where i represents the iteration number, P is a dumping factor and k is the effective multiplication factor estimated from the two-group cross sections. The iterative procedure continues before the convergence criterion of k eff is reached and the critical buckling B 2 at the current cycle (outer iteration) is obtained. In the second level (cycle or outer iteration), the critical bucklings from the last two successive cycles are compared. The true critical buckling is obtained if they meet the convergence criterion (ɛ = If not, the bucking from the last cycle is applied to B 1 equations to calculate the fine-group flux, which is later used to condense the cross sections into two groups. The procedure returns to the first step with the updated cross sections and the new bucklings will be evaluated. The MICROX-2 code will be used interactively with the core calculation code for reactor analysis; therefore, the speed of the code is of great importance. It is found in previous research that the iterative method of critical buckling search in MICROX-2 converges slowly primarily due to the following reasons:

129 110 The estimation of critical buckling using Eq. (4.24) assumes that relation of bucklings and k eff can be approximately described as the linear-like relation, which is not precise. The dumping factor P is used to under-relax the iteration and to avoid the divergence, but it also slows down the convergence process. In second step of the inner iteration, the buckling from the first step is doubled to generate another data point for the interpolation in the next step. The point created by this process, however, is usually too far from the true critical buckling (see Figure 4.10) Stamm ler s method In order to circumvent such problems, critical buckling search schemes were studied and compared in previous research. The method derived by Stamm ler in his book [2] has been proved to be stable and fast, and widely used in computational codes, such as HELIOS [17]. The model has been modified to replace the existing model in the inner iteration in MICROX-2. A brief introduction of the implementation is given below. The solution process consists of iterating on B 2 until its corresponding multiplication factor k eff = 1. Assume the transport equations are first solved and k eff is obtained for user provided B 2(1) or the buckling from previous cycle. It is recommend to use zero-buckling (usually B 2(1) = 0) for the input buckling, thus k (1) = k. Next, a not-too-small value is taken for B 2(2). Presently, B 2 = ± cm 2 if this is the initial cycle; a 10% perturbation is applied to B 2(1) if this not the first cycle, i.e., B 2(2) = ±1.1 B 2(1). Note that B 2(2) is set negative if k < 1.

130 111 With this new value of B 2, the effective multiplication factor k eff is calculated again using the condensed two-group cross section. To obtain a good estimate for the sought B 2 (k = 1) the two values B 2(1) and B 2(2) are used in an interpolation or extrapolation based on the well-known relation for the material buckling, k = k /(1 + M 2 B 2 ). This gives B 2 = 1 ( k M 2 db 2 = k M 2 d ) k 1 ) ( 1 k The coefficient k /M 2 is found after the first two steps from k M = B 2(2) 2 1 k 1 (4.25) (2) k (1) If this relation were rigorous, the sought critical buckling in i + 1 th iteration B 2 would be B 2(i+1) = B 2(i) + k ( 1 1 ) M 2 k (i) (4.26) With this B 2, a new effective multiplication factor k (i+1) is obtained. If k (i+1) 1 > ɛ, Eq. (4.26) is used again but this time with k (i+1), to estimate B 2 (k = 1), etc. Normally, very few iterations (< 8) are enough to reach an accuracy of ɛ = Note that the coefficient k /M 2 is evaluated only once. Reevaluating it after each iteration step could lead to divergence when the iterations have almost converged due to a loss of significant digits when k (i 1) k (i) and B 2(i 1) B 2(i). When k eff reaches convergence, the corresponding critical buckling of the current cycle is compared with that from last cycle (if any). If they are not sufficiently close together, the iteration continues and the updated buckling is used to solve

131 112 the fine-group transport equations for new flux spectrum and cross sections. Otherwise, the buckling is considered as the the true critical buckling and MICROX-2 will perform a final calculation with it for critical spectrum. This outer iteration scheme remains untouched as described in the last section Model verification The modified Stamm ler s method has been implemented in MICROX-2 and verified for a number of benchmark problems that have been used in cross section library verification. The critical buckling search calculations were conducted using the original and improved critical buckling models, respectively, and the convergence histories were recorded and compared. Table 4.5 compares the numbers of iterations necessary to achieve the same accuracy for three pin-cell cases: TRX-1, ZEUS, and SFR. N original and N improved refer to the numbers of iterations required using original and improved models, respectively. Comparing to the original model, the improved one effectively reduced solution time by reducing the number of iterations for all cases. Define R as the percentage number of iterations saved by using the improved model against the original model, i.e., R = (N original N improved )/N original. It is observed that the solution time has been decreased by 71.4%, 82.8% and 80.5% for TRX-1, ZEUS, and SFR cases, respectively, and the average save is 78.2%. The convergence history of critical buckling search of TRX-1 pin-cell model is plotted in Figure 4.10 for a detailed interpretation. It takes three cycles for both models to have critical buckling B 2 converged, because the updated B 2 needs to be used in the fine-group transport calculation for the next cycle to find new twogroup cross sections. The solution time required in this level is determined by the

132 113 Table 4.5: Numbers of iterations of critical buckling search for selected cases. Existing model Stamm ler s moethod R (%) TRX ZEUS SFR Average Critical buckling B 2 (cm 2 ) 8 x Original model Cycle 1 Cycle 2 Cycle Iteration 8 x 10 3 Improved model Critical buckling B 2 (cm 2 ) Cycle 1 Cycle 2 Cycle Iteration Figure 4.10: Convergence history of critical buckling B 2 with original and improved search models for TRX-1 case. solution method of transport equations in the code. In the inner iteration, it can be seen that the number of iterations required has been significantly reduced. To achieve the convergence of both k eff and B 2 in TRX-1 case, it takes 19, 15 and 8 iterations in the three cycles (outer iteration), or totally 42 iterations if using the original model; however, it costs only 5, 4 and 3 iterations in the three cycles, or 12 iterations in total if using the improved model. The convergence time has been

133 114 reduced by approximately 71.4%. 4.4 Summary and conclusions In this study, the self-shielding methods have been reviewed and updated for the MICROX-2 code to accurately estimate the effective multigroup cross sections. For the resonance self-shielding model, a new interpolation scheme has been implemented to compute the Bondarenko factor by replacing the existing linear fitting function with a continuous, converged and monotonic curve. An iterative procedure has been developed to ensure that the curve has the best fit to the precalculated dilution-bondarenko factor pairs. The benchmark results against the Monte Carlo calculation for a series of homogeneous lattices show that the prediction error of k has been reduced by 0.1% on average. It is also found that this approach is especially favored for fast lattice calculations where the importance of the unresolved energy range is high. A numerical module DANCOFF-MC, based on the Monte Carlo method, has been embedded in the MICROX-2 code for Dancoff factor calculation. The fast convergence rate of this method enables MICROX-2 to obtain accurate Dancoff factors without significantly increasing the computational cost. It is shown from the benchmark calculations that the estimation of the infinite multiplication factor has been improved by 0.2%. In addition, the critical flux model has also been updated by adopting the interpolation method derived by Stamm ler. The benchmark calculations indicate a reduction of convergence time by approximately 71.4% for selected problems.

134 Chapter 5 2-D Lattice Calculation Model After the few group homogenized cross sections are obtained from the 1-D lattice (pin cell) calculation, as described in the last Chapter, they are then carried to the 2-D lattice (fuel assembly) calculation, which produces the homogenized broad group constants for the core calculation. This chapter introduces the development of the 2-D lattice calculation model by coupling the transport calculation of MICROX-2 with the Nodal Expansion Method code NEM to generate the broad group homogenized cross sections and assembly discontinuity factors (ADFs) for the core calculation analysis. The integrated code MICXN (MICROX-2 and NEM) has the capability to set up pin cells of various geometry and material specifications; read cross section data from the transport calculations and homogenize the group constants over selective regions of the model. A new data interface has been implemented in the MICXN to produce group constants for core analysis code in ISOTXS format. Its capability of producing assembly homogenized cross section has been primarily tested by performing example calculation and comparing group constants and other parameters against the reference MCNPX calculations. Good agreement has been observed

135 116 during the comparison in general and the maximum difference of cross sections in any specific energy group is limited within a few percent. 5.1 Nodal expansion method The Nodal Expansion Method (NEM) diffusion code has been developed, maintained, and continuously enhanced at The Pennsylvania State University (PSU). NEM [49] is a few group (up to 70 energy groups can be simulated) three dimensional (3-D) transient nodal core model with three geometry-modeling options: Cartesian, Hexagonal-Z and Cylindrical (R-θ-Z). NEM is based on the transverse integration procedure and it was recently updated to utilize semi-analytical transverse integrated flux representation and improved transverse leakage approximation [50]. The nodal coupling relationships are expressed in a partial current formulation. The time dependence of the neutron flux is approximated by a first order fully implicit finite-difference scheme (upgraded later with exponential transformation technique), whereas the time dependence of the neutron precursor distributions is modeled by a linear time-integrated approximation. NEM is using the Response Matrix (RM) technique for inner iterations to calculate (update) ongoing partial currents for each spatial node in the framework of each energy group solution. The coarse-mesh rebalance (extended later to multi-grid technique) and asymptotic extrapolation methods are used to accelerate convergence of the outer solution process. Recently, SP 3 transport option was implemented within the framework of NEM [51]. A new and more accurate approach for the utilization of discontinuity factors in the response-matrix formulation of the nodal expansion method was implemented based on physical partial-current formulation. A brief derivation of equations used for the Cartesian geometry has been given in

136 117 Section 2.7. The NEM code was extensively verified for standalone neutronics steady state and time-dependent calculations on representative multi-dimensional test problems (benchmarks) including VVER two- and three-dimensional benchmarks [52]. The code has also been successfully used in the 2-D pin-by-pin diffusion calculation [53] to generate the few-group assembly homogenized cross sections in use of the global core analysis. The NEM code is subjected to PSU Quality Assurance Program and has User s, Theory, and Programming Manuals as well as test matrix and documented verification and validation results. There are aspects particular to a fast reactor that make the NEM (diffusion theory) approach directly applicable to calculating reactor physics design products, particularly the effective eigenvalue of the core as well as the radial and axial power distributions and reflector worth. Diffusion theory works best in applications where the flux is predominantly isotropic and the materials are effectively homogeneous. This is exactly the case in the fast reactors where the large migration length of the neutrons effectively masks the details of the heterogeneity of the fuel geometry, because the average fast neutron track length typically far exceeds the dimensions of the repeating lattice size. In addition, since the fast reactor will eschew moderating materials, the predominance of the materials in the core will be comprised of high Z scatters, lending to an almost completely isotropic neutron scattering. Lastly, with the SP 3 transport option implemented in the most recent version of NEM, the code is now capable of retaining more terms from the transport equation, which can effectively increase the accuracy of the calculation without significantly increase the computational time.

137 118 Based on the applicability of NEM code to these particular types of reactors, it is worthwhile to apply this methodology to the 2-D fuel assembly modeling during the iterative procedure for the consistent cross section generation that will be discussed in the following chapter. 5.2 Coupled calculation scheme The pin-by-pin assembly calculation follows a two-step approach, the transport calculation to collapse fine group cross section into a subset of the original energy structure, and the nodal diffusion calculation to homogenize the cross section over the fuel assembly or selected regions, as well as the internal data transfer and the output of group constants for reactor analysis. As shown in Figure 5.1, the procedure starts with successive standard MICROX-2 calculations for various pin cells, continues with internal data transfer from MICROX-2 to NEM and ends with assembly calculation, and cross section output. 1-D Cell CALCULATIONS 2-D FA CALCULATIONS Fuel Cell T f, T m Few-GR XS Fuel Assembly FA w/ Control Rod ISOBCD Non-fuel Cell T f, T m Few-GR XS Reflector ADF MICROX-2 NEM Figure 5.1: Schematic data flow of MICXN

138 Coupling method Four types of input files are required to perform the MICXN assembly calculation in total: the master cross section libraries (FD-, GAR- and GG-TAPE), 1-D pin cell model specification (MICROX-2 input files), 2-D assembly specification (NEM input deck) and the supplemental data file for generating the ISOTXS output file (4DREC). The 4DREC file will be discussed in the later section. There are no differences between the MICROX-2 input file between the standalone and the coupled versions. The users are allowed to directly use the standalone input files or rename their extensions for identification purpose so long as the length is limited to 20 characters. In NEM input deck, two additional input cards, CARD 0 and 43, are introduced. In Card 0, which is placed at the beginning of the file, the users are asked to specify the identification of file (problem description), the total number of pin cell types to be calculated by MICROX-2 and the associated input names. The code will search for the files in the working directory and perform the standalone 1-D lattice calculations consequently. There are three variables in the aforementioned new input card 0 as shown in Table 5.1. The user should repeat word pair 3 and 4 until all (NXSMRX) pin cell types/names have been specified for assembly calculation. Card 43 specifies the regions (nodes) over which the neutron flux homogenization will take place. The users may either provide the node index as given in Card 36a to determine specific regions or simply place 0 to perform homogenization over entire model. This function enables the code to perform homogenization process for only part of fuel assembly; as such, the cross section generation for non-fuel lattices, including reflectors and control elements is achievable. In this

139 120 Table 5.1: Card 0 of MICXN input deck Word Number Format Variable Name Description 1 12A6 HSETID Identification of file (problem description) 2 I NXSMRX Number of pin cell types 3 I PINTYPE Index of pin cell types 4 A20 PINNAME Name of the MICROX-2 input file corresponding to the pin cell type mode, the code actually solves the 2-D color set problem by accurately capturing the potential spectrum changes at the core-reflector or fuel-control rod interface. For example, by defining the geometry with a reflector assembly attached to a unit fuel assembly, as depicted in Figure 5.2, the MICXN code can perform the color set type calculation with the exact fission source in the fuel and obtain the homogenized reflector group constants by using the actual flux in the reflector area. Unit Fuel Assembly Reflector Figure 5.2: Reflector modeling in MICXN The coupled code system is designed by embedding 1-D spectrum calculation

140 121 into 2-D assembly calculation so that the process is driven by NEM, and the data transfer between MICROX-2 and NEM can be performed internally with high efficiency through an incorporated module. The share data includes microscopic cross sections, isotopic densities, atomic mass, and isotope ID etc. After the required data is transferred and stored in the shared memory, MICXN echoes the 1-D calculation results at the beginning of output edits, including mixture (cell) averaged macroscopic cross sections, multiplication factors, and the nuclide compositions. It will then check the consistency of the broad group structure used in all 1-D calculations. The up-scatter is enabled and the maximum number of up-scatter iteration per outer iteration is set to be 10 by default. If conflicts are found, the assembly calculation will be aborted and warning messages will be provided. Otherwise, the code collects geometry information from the rest input cards, locates and assigns macroscopic cross sections to pin cells or nodes with the data from corresponding cross section set, and performs the standard nodal calculation. Note that many of the intricate details of the code coupling scheme has been omitted in the above discussion, because the specific techniques that are implemented in the code are not essential for understanding the objective in this work Assembly homogenization With the completion of the pin-by-pin assembly calculation, MICXN performs the isotopic cross section homogenization by using the node averaged flux. For nuclide k in broad group g, the average macroscopic cross section of reaction type x is given by Σ k x,g = i V iφ i,g Σ k x,i,g i V iφ i,g = i V iφ i,g N k i σ k x,i,g i V iφ i,g (5.1)

141 122 where V i and φ i,g are the volume and average flux in node i; N k i and σ k x,i,g denote the atom density in node i and the microscopic cross section for reaction type x. Meanwhile, the code evaluates the average density of nuclide k using the following expression, N k = i N k i V i i V i (5.2) and the effective microscopic cross section can be evaluated using Eq. (5.3). σ k x,g = Σ k x,g N k (5.3) The assembly averaged fission spectrum χ is calculated in the following way: χ g = i (V iφ i,g ν i,g Σ i,f,g ) g [ i (V iφ i,g ν i,g Σ i,f,g )] (5.4) where ν i,g is the neutron yield in group g in node i. The calculation of the assembly discontinuity factors (ADFs) is implemented by using the surface-averaged fluxes and the volume-averaged fluxes, which are both available in the heterogeneous flux solutions given by pin-by-pin assembly calculations. As shown in Eq. (5.5), the ADFs are computed as f g = φ s g φ g = i S V iφ i,g i S V / φ g (5.5) i where S is the specific surface of the assembly (north, south, etc.), φ s g is the average scalar fluxes in nodes lying along surface S (heterogeneous flux) of energy group g, φ g is the assembly homogeneous flux of group g. φ i,g is the neutron flux of group g for boundary node (cell) i with volume V i. φg can be computed using the following

142 123 equation: φ g = i V iφ i,g i V i (5.6) Interface routines A major problem associated with the development of complex computer codes is their exportability or exchangeability. One way to minimize the problem of code exchange is to develop codes at designated facilities using standardized techniques and procedures that are generally compatible with large-scale computing environment. Work in this effort has been performed under procedures and guidelines established by the Committee on Computer Code Coordination (CCCC) and the resulting files are in the fullest form defined by the CCCC-III and CCCC-IV standard. [54] In order to produce group constants well suited for the reactor calculation codes, the utility module needs to be established between the pin-by-pin assembly calculation and the reactor calculation code DIF3D [55]. The input data for REBUS3 are of two types: 1) a binary cross section dataset on disk or tape, and 2) BCD input data normally via punched cards or card images. The cross section file contains the group constants to in the ISOTXS format as specified by the CCCC-IV. ISOTXS refers to isotope-ordered multigroup neutron cross sections including cross section versus energy functions for the principal cross sections, group-to-group scattering matrices, and fission neutron production and spectra tables. Efforts have been made to implement an interface module in the MICXN code to let the code generate broad group cross sections in ISOTXS format in CCCC-IV form. This module, called by MICXN, writes the ISOTXS format cross sections to output files ISO.XS in the following structure:

143 124 The ISOTXS cross section file begins with four records: File Identification, File Control (1D Record), File Data (2D Record) and File-wide Chi Data (3D Record). These records are defined for each mixture of nuclides in the assembly (or single MICXN run). Note that the interface module uses the file-wide fission spectrum calculated by Eq. (5.4). The records Isotope Control and Group Independent Data (4D Record), Principle Cross Sections (5D Record) and Isotope Chi Data (6D Record) are repeated once for each of the isotopes. The variables in 4D Record define the reaction types provided in 5D Record. The ISOTXS format attempts to pack scattering matrices efficiently in 7D Record. The scattering data is divided into blocks and sub-blocks. A block is either one of the designated scattering reactions (that is, total, elastic, inelastic, or (n,2n)) and contains all the group-to-group elements and Legendre orders for that reaction, or it is one particular Legendre order for one of the designated reactions and contains all the group-to-group elements for that order and reaction. The 7D Record will be presented for each isotope. Some variables required in the 4D Record have not been carried from the nuclear data when generating the master cross section library. These variables, including the total thermal energy yield per fission, total thermal energy yield per capture, and average effective potential scattering in resonance range, need to be readin from a problem-independent data file 4DREC. The interface routine starts by initializing certain variables and the storage system. Warning or error messages will be generated when errors occur during the preparation of ISOTXS format files. A sample output file corresponding to the sample LWR assembly problem described below is shown in APPENDIX C.2.

144 Calculations and results This section presents the results from the fuel assembly calculations to determine the performance of the coupled transport-diffusion lattice model and the cross section homogenization technique used in the MICXN code. Besides the FA-BAPL numerical model, the UO 2 and MOX assemblies from the C5G7 benchmark problems are also included in the verification The FA-BAPL benchmark The first model, FA-BAPL is a LWR type fuel assembly numerical model, created with the geometry found in [56] and the material composition taken from BAPL-1 benchmark problem used in previous benchmark calculations. Figure 5.3a sketches the geometry of the assembly, where the gray regions represent the fuel pins, as seen in Figure 5.3b, and the blue cylindrical regions are the guide and instrument tubes filled with moderator. The blue regions around the cylindrical cells represent the moderator. The dimension of the square region is cm 2 and the radius of the fuel region is cm, which gives the moderator-to-fuel volume ratios 1.43 for the fuel pin. The material composition of the pin cells are given in Table 5.2 in the unit of (b cm) 1. The continuous-energy Monte Carlo code MCNPX is used to generate the reference solution, and the transport code DRAGON is also used to perform the inter-comparison. The mesh size (identical to the height of the numerical model) is equal to the square lattice pitch in the MICXN calculation and the input decks for this problem are given in the Appendix C.3. A four-group energy structure is used in this calculation with up-scattering permitted. The upper energy limits considered are: 14.9 MeV, 0.11 MeV, 7.1 kev, and 2.38 ev, which are based on the

145 126 Table 5.2: Material composition of assembly case FA-BAPL Fuel pin (pin cell 1) Guide tube (pin cell 2) Nuclide Region 1 Region 2 Homogenized 235 U U H O Al one used in VSOP [57], with additional values at 7.1 kev and 2.38 ev. These additional energy boundaries are introduced because of limitations in the MICROX-2 code. All numerical calculations are performed in the room temperature (293.6K). First, the infinite multiplication factor from MICXN calculation is compared with the reference solution, as shown in Table 5.3, where good agreement is observed. k is overestimated by 350 pcm and underestimated by 222 pcm when compared to MCNPX and DRAGON, respectively. Table 5.3: Comparison of k results of FA-BAPL case Codes k diff. (%) MCNPX ± a DRAGON MICXN a One standard deviation. Next, the homogenization process in the nodal diffusion calculation is examined. Table 5.4 presents the isotopic atom densities used in the transport and diffusion calculations, as listed in column 2/3 and 4. It is seen that the assembly averaged value lies in the range of the two pin cell models, and is almost identical to that

146 127 UO 2 Fuel Guide Tube (a) assembly configuration Fuel-Clad Mix Moderator (b) Fuel cell model Figure 5.3: Configuration of the FA-BAPL benchmark problem from DRAGON calculation, which is found in the last column. Same comparisons between the transport and diffusion calculations are also made for principle cross sections and the P 0 scattering matrix, as shown in Table 5.5 and 5.6, respectively. Note that the unit in these tables is barn(b). The latter values are always bounded by those from former models, indicating that the

147 128 Table 5.4: Average isotopic atom densities used in FA-BAPL calculation MICROX-2 NEM DRAGON Nuclide Pin cell 1 Pin cell 2 Assembly Assembly 235 U U H O Al homogenization numerically provides reasonable results based on the data obtained from the transport calculations. Table 5.7 presents a comparison of the isotopic cross sections in four groups generated by MICXN and the reference code MCNPX. As the standard MCNPX does not provide information on the scattering kernel, the principle few-group cross sections are of particular interest here. In general, a good agreement has been found for most nuclides in the sample problem, especially in the second and third group, which covers the resonance ranges. Relatively large overestimations of the capture cross section in the thermal energy range are found for non-fissile nuclides: the relative error is 13.5%, 7.5% and 13.7% for 1 H, 16 O and 27 Al, respectively. This phenomenon is consistent with the finding in the pin cell benchmark calculation that the MICROX-2 code overestimates the thermal flux in a certain degree. The fission and capture cross sections of 238 U are underestimated in the resonance energy range by 6.1% and 5.6%, respectively. A large difference is observed in the prediction of resonance capture cross section for 27 Al. This is not considered as a discrepancy due to the solution method in MICROX-2 or NEM as the error of the same magnitude (27.9%) is found in the

148 129 Table 5.5: Homogenization of principle cross sections in NEM with data from MICROX-2 for FA-BAPL case MICROX-2 NEM pin cell 1 (fuel) pin cell 2 (guide tube) nuclide group ν σf σc σtr ν σf σc σtr ν σf σc σtr U U H O Al

149 130 Table 5.6: Homogenization of scattering matrices in NEM with data from MICROX-2 for FA-BAPL case MICROX-2 NEM pin cell 1 (fuel) pin cell 2 (guide tube) nuclide group U U H O Al

150 131 comparison of DRAGON and MCNPX as it can be found in Table 5.8. It is suspected that a fundamental difference exists in the 27 Al nuclear data file, which is processed to generate the master cross section library. Because key information about the MCNPX library generation is missing, it is recommended to investigate the MCNPX library beforehand to identify the source of error. The MICXN-calculated assembly average cross sections are then verified against those from MCNPX, which is obtained using the standard flux tally and the tally multiplication card FMn, and the results are shown in Table 5.9. The relative difference of ν is within 1% everywhere except in the fast energy range, where it is 5.9% overestimated by MICXN. The trends of capture and fission cross sections are similar: they are over-predicted in the first and last group. The difference in the energy range just above the thermal energy cutoff is negligible. A noticeable underestimation of the capture cross section is found in the resonance range, to which the 238 U resonance cross section makes the main contribution. By comparing the data in Table 5.7, it is understood that over-prediction of the capture cross section in the last group is due to the overestimation of the thermal flux, which is caused by the discrepancies in the thermal calculation scheme in the pin cell calculations The C5G7 benchmark The second benchmark problem selected is the sixteen assembly (quarter core symmetry) C5G7 LWR problem which was first specified in [58] and later updated in Ref. [59]. It was originally developed to test the ability of modern deterministic transport methods and codes to calculate reactor core problems without using homogenization techniques. The 2-D configuration with minor modifications is

151 132 Table 5.7: Homogenized isotopic microscopic cross sections from MICXN and MCNPX for FA-BAPL case MCNPX MICXN Diff. (%) nuclide group ν σf σc σtr ν σf σc σtr ν σf σc σtr U U H O Al

152 133 Table 5.8: Homogenized isotopic microscopic cross sections from DRAGON and MCNPX for FA-BAPL case MCNPX DRAGON Diff. (%) nuclide group ν σf σc σtr ν σf σc σtr ν σf σc σtr U U H O Al

153 134 Table 5.9: Assembly homogenized macroscopic cross sections for FA-BAPL case Parameter Group MCNPX MICXN Diff. (%) a ν Σ f (cm 1 ) Σ c (cm 1 ) Σ t (cm 1 ) a Difference calculated as 100 (x MICXN /x MCNPX 1). used in this study as shown in Figure 5.4. As indicated, this small core consists of a 2 2 array of UO 2 and Mixed-Oxide fuels (MOX) surrounded by a reflector, with reflective boundary condition (RBC) applied to the north and west sides of the core, vacuum boundary conditions (VBC) beyond the moderator. The overall dimensions of the 2-D configuration as shown are cm 2, while each assembly is cm 2. The fuel assembly is made up of a lattice of square pin cells, with each assembly also including guide tubes and fission chamber.

154 135 Reflective BC UO 2 MOX Reflective BC MOX UO 2 Vacuum BC Moderator Vacuum BC Figure 5.4: C5G7 2-D LWR benchmark Table 5.10: Cell dimension of C5G7 benchmark Medium Fuel cell (cm) Guide tube cell (cm) Clad inner radius Clad outer radius Square lattice pitch The dimensions of the fuel cell (MOX 4.3%, MOX 7.0%, MOX 8.7% and UO 2 ) and guide tube cell are given in Table. Table 5.11 provides the isotopic composition of each mixture. Note that Central guide tube or fission chamber contains moderator (as defined in Table 5.3.2) and atom/(b cm) of 235 U. Figure 5.5 shows two configuration of the UO 2 assembly: (a) infinite array with fully reflective boundary conditions; and (b) partially reflected assembly with introduction of the neutron leakage. Same boundary configurations are also applied to the MOX assemblies as shown in Figure 5.6. The mesh size and height are

155 136 Table 5.11: Isotopic distribution for each medium Concentrations (10 24 at/cm 3 ) Nuclide MOX MOX MOX UO 2 Moderator Zr Clad Al Clad (4.3%) (7.0%) (8.7%) 235 U U Pu Pu Pu Pu Pu Am O H 2 O Nat. B Nat. Zr Al Reflective BC Reflective BC Reflective BC Reflective BC Reflective BC Vacuum BC Reflective BC (a) Vacuum BC (b) Figure 5.5: C5G7 UO 2 assembly configuration

156 137 Reflective BC Reflective BC Reflective BC Vacuum BC Reflective BC Reflective BC Reflective BC (a) Vacuum BC (b) Figure 5.6: C5G7 MOX assembly configuration Table 5.12: C5G7 benchmark calculation of fully reflected assembly problems Assembly case Codes k eff diff. (%) UO 2 DRAGON MCNPX ± a MOX MICXN MCNPX ± DRAGON MICXN a One standard deviation due to statistical uncertainty. identical to the square lattice pitch (1.26 cm) in the MICXN calculation using nodal expansion method for both assembly models. Table 5.12 shows the comparison results of the effective multiplication factors for these configurations. It can be seen that MICXN underestimates k eff of both cases with the relative errors -0.63% and -0.84%, respectively, which are slightly larger than those of DRAGON calculations. Same boundary configurations are also applied to the MOX assemblies as shown

157 138 Table 5.13: C5G7 benchmark calculation of partially reflected assembly problems Assembly case Codes k eff diff. (%) UO 2 MCNPX ± a MICXN MOX MCNPX ± MICXN a One standard deviation due to statistical uncertainty. in Figure 5.6, and Table 5.13 presents the calculation results. Same trends are found in these comparisons, that is, k eff is under-predicted by MICXN for both cases and the relative error of the MOX assembly is larger than that of the UO 2 assembly. Note that the DRAGON results are not available for assembly models with leakage. Overall, the MICXN lattice calculation results presented in Table 5.12 and 5.13 show good agreement with MCNPX, although the difference is not negligible. It is worth noting that the prediction error partially stems from fundamental differences in the master cross section libraries. The same method and energy structure were applied to the library generation for MICXN and DRAGON; however, MCNPX uses a continuous energy library, of which some of the key information is missing in the generation process. 5.4 Online cross section generation One of the key procedures of developing the online cross section generation is to replace the infinite medium or no-current assumption with boundary conditions extracted from the core calculation. To achieve this goal, a fixed k eff calculation scheme is first developed and tested by the MICXN-MICXN approach. The non-

158 139 linear iterative method is then designed to couple the lattice and core calculations with appropriate data exchange scheme. Most of these topics will be discussed in this section The fixed k eff algorithm The development of fixed k eff calculation in MICXN code includes two tasks: the first is the fixing of the incoming partial currents and the second forces the k eff to be constant in the fixed source calculation. A detailed description of this method is given in [16]. Consider a simple homogeneous lattice model in the MICXN calculation, depicted as the shaded region in Figure 5.7; there are three major boundary conditions available in MICXN: vacuum, reflective and zero flux boundary conditions. The vacuum boundary means no neutrons exiting the region will return to the medium. The reflective boundary condition indicates that all neutrons leaving the region will return to the system in the same way as light reflects on a mirror. Thus it is also called the specular reflective boundary condition. In the case of the zero flux condition, the flux vanishes beyond the extrapolated length outside the boundary. In general the boundary conditions can be interpreted using the partial currents through the surface of the geometry, using the concept called reflection coefficient or albedo, which is defined as the ratio the partial current into the region (J ) to the current out of the region (J + ) [1]: α = J J + (5.7) In another word, the parameter α denotes the fraction of the neutrons returning to the system and satisfies 0 α 1. For vacuum boundary conditions alpha = 0;

159 140 y J (0) J + (0) 0 x Figure 5.7: The albedo problem for fully reflective boundary conditions α = 1; while for other boundary conditions 0 < α < 1. Therefore, the application of particular incoming partial currents (J = αj + ) to the boundary of the lattice model defines a specific boundary condition in the lattice calculation and is capable of simulating the effect of leakage. To achieve the goal of fixing the incoming neutron currents as the boundary condition, the NEM code has been recently modified to read the incoming current on for the nodes on the domain boundaries. This capability will also be implemented in the coupled MICXN code. The second part in the implementation is to enforce the k eff computed at the core level to the fixed calculation. Strictly speaking, the calculation is not a fixed source calculation, because it involves update of the fission source. Therefore, the basic calculation approach is the same as that of the eigenvalue iteration, except that one does not update the eigenvalue. The fission source calculation routine that is responsible of computing k eff and updating the fission source will be modified to achieve this goal. The general strategy of the fixed k eff calculation approach takes the form

160 141 Obtain model geometry and composition Read boundary condition J and k eff Inner iterations Update group fission source Q g,f = νχ g k eff G g=1 φ gσ g,f Outer iterations Convergence test N Y Finished Figure 5.8: Calculation strategy for fixed k eff calculation with fixed boundary condition sketched in Figure 5.8. The following session summarizes the approach and describes how the changes are effected in the MICXN calculation scheme. With obtaining the geometry and composition of the model, the code takes the following steps. Step 1: The incoming partial currents J and the eigenvalue k eff are read and remain fixed throughout the calculation. Step 2: The normal lattice calculation using the nodal expansion method with the fixed boundary condition is performed, producing the multigroup neutron flux for each pin cell φ g with the scattering source iteration or the

161 142 inner iteration. During this calculation, the fission source does not change as it would in usual inner iterations. Step 3: The fission source Q g,f for each pin cell is calculated explicitly upon the completion of the flux convergence and scaled by the fixed k eff, using the equation shown in the flowchart. Step 4: Steps 1 3 are repeated until the pin cell fission source Q g,f is converged to some pre-defined convergence criterion. Two criterions are used in current version of MICXN, the point or L 4 norm and the average of L 2 norm of the pin cell fission source, which can be defined in the input deck A MICXN-MICXN calculation scheme The fixed k eff to be implemented in the MICXN will enable the code to couple with core solver such as DIF3D and generate consistent cross sections using the online cross section generation method, which will be discussed in the later section. In order to understand the behavior of the fixed k eff calculation on a basic level and verify the implementation, a calculation scheme is devised in which MICXN plays the role of the core solver as well as the lattice code. This approach does not involve any approximations between the core and lattice code and would give us a clear measure of the performance of this algorithm. Figure 5.9 illustrates the simple calculation scheme created to test the fixed k eff algorithm in MICXN. For the sake of simplicity, the system under consideration consists of only two MICXN structures (assemblies), and the two structures are coupled by the partial current through common interface. The calculation scheme consists of the following steps:

162 143 Step 1: A reference eigenvalue calculation is performed for the system. During this calculation, the reference eigenvalue, incoming partial currents and regions fluxes are stored. Step 2: The global incoming current array is then used to construct the boundary conditions per assembly for the fixed k eff calculation. Step 3: An individual fixed k eff calculation is performed for each assembly. MICXN reads the assembly incoming current J and the k eff from the store data. This step requires separate MICXN model for each assembly as they exist in the full model. On completion of the fixed k eff calculation, the groupwise region fluxes and multiplication factor are stored on a separate file, which can then be compared with the reference k eff calculation data. The calculation results of selected models are shown and discussed in the following sections, including single UO 2 and MOX fuel assemblies, the C3 benchmark and the C5G7 benchmark. Single UO 2 and MOX assembly cases In this section, the single UO 2 and MOX fuel assemblies from the C5G7 benchmark are used to test the fixed k eff implementation in MICXN. The geometry and material specifications of the assemblies are identical to those had been used in the standalone core calculation in the previous study. The reflective boundary condition is applied to the assembly model for the initial MICXN calculation, from which the assembly k eff (k due to non-leakage configuration) and pin incoming partial currents for surface pins on each side in the radial directions are obtained. In the second calculation, k eff and boundary in-currents are forced to be identical

163 144 J J 2 J 1 J 4 k eff In-current J 3 3 J 2 J 1 J 4 Figure 5.9: The MICXN-MICXN test scheme for fixed k eff calculation approach to that in the stored data, while the reflective boundary conditions are still applied to the top and bottom assembly surfaces. The spatial discretization of the model is the same as the physical pin cell boundaries. Since the eigenvalue and the surface in-currents are fixed and will definitely be recovered in the second MICXN calculation, it is reasonable to test the calculation scheme by examining the region fluxes. The total pin fluxes across the assembly from the fixed k eff calculation are compared with those from the initial results, and the percentage differences of the fluxes are plotted in Figure It can be seen that the difference in pin fluxes have the same bowl-shape symmetric distribution for both cases, indicating the fluxes are recovered better on the edge of the assembly than in the center. The average absolute differences are 0.007% and 0.013% for UO 2 and MOX assembly, respectively, while the maximum

164 145 (a) UO 2 assembly (b) MOX assembly Figure 5.10: Pin fluxes differences of single UO 2 and MOX assembly differences are 0.012% and 0.023%, showing that the difference for MOX assembly is slightly higher than UO 2 model by a negligible amount. Therefore, the region fluxes are recovered for both cases with high accuracy. The next comparison focuses on the energy dependency of the calculation scheme. The assembly homogenized cross sections from the fixed k eff calculations are compared with the reference values from the calculations using reflective boundary conditions for both cases, and the relative difference are given in Table Note that the 4-group energy structure is used in the condensation and only the differences large than % are shown. The small value of the differences in the comparison is a clear indication that the performance of the fixed k eff calculation is reliable for the fully reflected fuel assembly models and the homogenized cross sections are well recovered. The C3 benchmark Next, the MICXN-MICXN calculation scheme is tested for C3 benchmark, which is essentially the C5G7 core with reflective boundary conditions on outer

165 146 Table 5.14: Cross section verification for fixed k eff calculation scheme Group Σ c (%) Σ f (%) νσ f (%) Σ tr (%) Σ t (%) χ (%) UO 2 MOX surfaces. It introduces neutron leakage on the assembly interfaces and helps understand the performance of the fixed k eff calculation method in such a condition. With the completion of the initial MICXN calculation of the C3 core, the surface in-currents and region fluxes for each assembly are stored as the reference data along with the global eigenvalue. The assembly-dependent in-currents are then applied to UO 2 and MOX assembly separately with the same k eff. Again, the spatial discretization for the current on the assembly boundaries was according to the pin cell edges, i.e., the current is spatially constant over a pin cell edge. The reflective boundary condition is always applied to axial boundaries. The differences of the pin fluxes for the two types of assembly are shown in Figure The perfect symmetry is observed for both cases, which is due to the symmetric assembly design and boundary condition. The smallest difference is found for pins at the corner of the C3 core, i.e., the (1, 1) location for both cases in the figures, which is similar to the results for single assembly from the last section. The flux differences for pins on the core edge are smaller than the rest in general, but the difference is larger for the edge cells with leakage than the ones without leakage.

166 147 (a) UO 2 assembly (b) MOX assembly Figure 5.11: Pin fluxes differences of UO 2 and MOX assembly in C3 benchmark A larger gradient of the flux difference can be found for the MOX case, because of the higher heterogeneity in assembly design when comparing to the UO 2 case. The average absolute error of the pin fluxes for the UO 2 case is 0.006% with the maximum error 0.019%; these values are 0.017% and 0.035% for the MOX case. The differences are higher than those in the single assemblies, but of the same order of magnitude, showing that the fixed k eff calculation is performing consistently. The C5G7 benchmark The final case in the test series is the C5G7 benchmark. In addition to all the incremental geometry and material complexity that were introduced in the previous models, the C5G7 now adds reflection by a moderating reflector and also leakage outside the reflector. Figure 5.12 shows a simplified schematic diagram of the model with new assembly numbering: A1 and A5 are UO 2 assemblies while A2 and A4 are MOX assemblies. Same calculation scheme is used for C5G7 model, except for the vacuum boundary condition is applied to the moderator surface cells. Note that no post-process of

167 148 Vacuum BC Moderator Refl. BC A4 A1 A5 A2 Vacuum BC MOX Fuel UO 2 Fuel Refl. BC Figure 5.12: C5G7 benchmark in MICXN-MICXN calculation scheme the incoming partial currents is necessary before they are assigned to the fixed k eff calculation in the second step because the core analysis and the lattice calculation use the same solver. The pin fluxes comparison is obtained for each assembly and a combined view is show in Figure The symmetric shape of the difference is seen as expected with a smaller average difference found for the A1 (UO 2 ) assembly and larger errors appearing in the A2 or A4 (MOX) assembly, which is in accordance with previous results. The UO 2 assembly that is exposed to the reflector (A5) show slightly larger flux differences than the one in the center of the core (A1). The shape of the errors within the assembly is similar to that found in the C3 results but with increases at the corner pins, especially for those adjacent to the reflector. This is probably because of mathematical difficulties introduced by the small values of the incoming partial currents from the reflector. However, given the average and maximum difference are 0.012% and 0.104%, respectively, the

168 149 Figure 5.13: Pin fluxes differences of C5G7 benchmark (combined view) prediction error is small enough to be neglected. The test results shown above using the MICXN-MICXN calculation scheme show that the fixed k eff approach is able to maintain the eigenvalue and the incurrents on the lattice boundaries obtained from the core calculation and reproduce the region fluxes. As such, it can take the environment inside the core when performing the lattice calculation and generating few-group cross sections. Therefore, it is valid to be used in the online cross section generation scheme, which will be introduced in the following sections Iterative Diffusion-Diffusion Methodology (IDDM) The iterative Diffusion-Diffusion Methodology (IDDM) has been developed based on the previous studies regarding the embedded lattice calculation conducted in RDFMG. IDDM is named in such a way because diffusion theory is applied in

169 150 both lattice and core calculation using a MICXN-DIF3D approach. The method starts with a conventional once through calculation in which MICXN generates the first set of cross sections in an infinite environment for use in the core nodal solver. The nodal solver determines the 2-D nodal solution, which includes the effective multiplication factor k eff and partial currents on the node boundaries. The iterative nature of the methodology then utilizes the partial currents from the core solver as boundary conditions for the next set of cross section calculations. These partial currents have the added characteristic of physical quantity while the albedos are unit less ratios, which introduce additional degree of freedom thus making complete equivalence difficult to achieve. In addition to partial current boundary conditions, the eigenvalue will be utilized by MICXN as boundary conditions in fixed k eff calculations. New cross sections are developed by MICXN and the core nodal solution is updated. Convergence is measured based on the difference of both eigenvalue and fuel node radial currents in the 2-D nodal diffusion core solution. The incoming partial currents in DIF3D output are calculated for homogeneous medium. They are outgoing partial currents from neighboring nodes, incoming partial currents on the outer XY-plane boundary, or outgoing partial currents across surfaces along a periodic boundary. On the other hand, the partial currents along assembly boundaries obtained in the MICXN are generated for heterogeneous medium under the non-leakage condition. To form the boundary condition that can be applied to the lattice calculation, the partial current distribution from lattice calculation must be multiplied by the average partial current from core solution, which represents the core environment. This procedure is the essential difference from the previously developed MICXN- MICXN calculation scheme and also couples the MICXN calculations throughout

170 151 the core. For example, if MICXN runs three times per DIF3D calculation, the shape (distribution) of the in-current used in the MICXN input will change three times but multiplied by the same partial current magnitude of the previous DIF3D calculation. The shape of the in-current is calculated using the incoming partial currents along a lattice radial surface obtained from the lattice solution with the following equation: p g (r) = j g (r) drj g (r) (5.8) where j g (r) is the incoming partial current in energy group g of pin cell at location r (x or y in the Cartesian geometry), and the shape function p g (r) satisfies r p g(r) = 1 as a distribution function. The limits of summation over space are over all the pin cells along the lattice surface in a specific direction. The partial current magnitude is the surface averaged partial currents along the side of the fuel assembly, as shown in the Eq N A g = Jg,n/N (5.9) n=1 where Jg,n is the incoming partial currents of group g of node n from the core calculation results, and N is the total number of nodes on one side of the assembly. The resulting constant A g is the average in-currents over all node meshes of group g. In general, the MICXN and DIF3D performs calculations using the same energy structure, therefore the integration of the energy is not necessary in the current calculation scheme. Utility tools have been developed using Python script to handle the output files of both lattice and core calculations to extract the shape function and magnitude

171 152 shape function p(r) from lattice calculation magnitude A from core solver Incoming partial current J (r) as boundary condition Pin cell location along lattice surface Figure 5.14: Formation of the partial currents as boundary conditions of partial currents. Figure 5.14 gives an illustration of formation of the partial currents as the boundary condition for a MOX fuel assembly.

172 Chapter 6 Reactor Core Calculations With the completion of the development of the coupled 1-D lattice transport and 2-D lattice nodal diffusion model as well as the iterative online cross section generation scheme, a set of 2-D and 3-D numerical benchmark problems need to be set up to assess the performance of the proposed model in the whole core simulation of both steady state and transient situations. The selection of the benchmark problems should include a certain diversity of the reactor design and also keep its focus on the fast reactors. The benchmark problems should provide detailed information on the fuel composition, geometry specification, and the operation condition. As introduced in previous chapters, the enhancement of the MICXN cross section interface tool enables the generation of the assembly averaged isotopic microscopic cross sections in ISOTXS format. The DIF3D code will be used in the neutronics analysis of the reactor core. DIF3D is the flux and eigenvalue solver in the REBUS-3 code suite. [55] It contains solution options for multigroup steady-state neutron diffusion and transport theory calculations. Cross section data provided in standard format (arbitrary group structure) are used in these calculations. Both nodal and finite-difference

173 154 spatial discretization approaches are available in the code. Collectively, the nodal options solve the diffusion and transport equations in two- and three-dimensional hexagonal and Cartesian geometries. One-, two- and three-dimensional orthogonal (rectangular and cylindrical) and triangular geometry diffusion theory problems are solved by the DIF3D finite difference option. Eigenvalue, adjoint, fixed source and criticality search problems are permitted. Upscattering and internal black boundary conditions are also treated by the code. The broad-group assembly homogenized cross sections generated by MICXN code were employed in the full-core calculations presented in this study. The conventional once-through reactor calculation method is first used to model both LWR and fast reactor benchmark problems, and the results are compared with the Monte Carlo reference solutions. Later on, the online cross section generation technique based on IDDM is applied to the LWR case to verify the improvement. 6.1 Reactor modeling using off-line approach In this section, the C5G7 LWR and JOYO fast reactor benchmark problems are studied using the conventional off-line reactor modeling method based on MICXN- DIF3D approach. The primary objective is to test the MICXN capability of generating homogenized cross sections and understand the core calculation method using DIF3D LWR benchmark The LWR benchmark problem selected in the reactor core calculations is C5G7 benchmark case introduced in the previous chapter. Two types of cross section are generated by the MICXN: the homogenized macroscopic cross section by mixture

174 155 Reflective BC Reflective BC Reflective BC Reflective BC UO 2 Fuel 4.3% MOX Fuel 7.0% MOX Fuel 8.7% MOX Fuel Fission Chamber Guide Tube Figure 6.1: C3 2-D benchmark and the microscopic cross sections by isotope, which are then used by the global DIF3D calculations performed in four groups (energy cutoffs given in Section 5.3) and on a one-node-per-pin mesh. The two-dimensional core calculations were performed using the nodal transport option (DIF3D-VARIANT) of version 8.0 of the DIF3D code. All the cases were solved using the P 0 scattering approximation and fourth-order spatial approximation for the source and flux within a node and the linear approximation for the leakages on the nodal surfaces, along with a simplified spherical harmonic P 3 -angular approximation for the flux. The Monte Carlo code

175 156 Table 6.1: Calculation results of C3 Benchmark problem Codes k eff diff. (%) MCNPX ± DIF3D (DRAGON) DIF3D (MICXN xs by mixture) DIF3D (MICXN xs by isotope) MCNPX is used to produce the reference core solutions. First, the C3 benchmark is modeled, which is essentially the C5G7 core without moderator but with zero-current boundary conditions on the outer surfaces as shown in Figure 6.1. The calculation results are summarized and compared with the reference MCNPX calculation in Table 6.1. Note that cross section is denoted by xs in the table. To better understand the performance of MICXN cross section generation, DRAGON code is also employed to produce microscopic cross sections in ISOTXS format for DIF3D. Good agreements with MCNPX reference results have been found for both approaches, with DRAGON-DIF3D (D-DIF3D) and MICXN-DIF3D (M-DIF3D) underestimate k eff with 0.28% and 0.52% difference, respectively. The difference between the two approaches is 300 pcm. The identical outputs of MICXN-DIF3D using both mixture and isotopic cross sections prove that the cross section homogenization technique used in the MICXN is correct so that DIF3D is able to reproduce the mixture cross sections in the core calculations. The individual assembly power can also be calculated from MCNPX results by multiplying the fission rate in the fuel regions by the fission energy released from the fuel. The fission rates are obtained in the tallies; and averaging over all fissionable nuclides [60] in the fuel by their concentrations will yield the effective

176 (1.88) (-2.15) A (-1.40) (1.60) A (-1.40) (1.60) A (1.88) (-2.15) A2 MCNPX (D-DIF3D) (M-DIF3D) ID Figure 6.2: Power distribution of C3 benchmark fission energy of the fuel kev and kev are calculated for UO 2 and MOX fuel in C5G7 benchmark problems, respectively. Finally, assembly power is then normalized so that the average power becomes unity. The DIF3D relative power distribution is directly obtained from the output edit and compared with the MCNPX reference solution, as shown in Figure 6.2. Each box represents one fuel assembly with the ID in the lower left corner and the percentage prediction error the DRAGON-DIF3D (shown as D-DIF3D) and MICXN-DIF3D (shown as M-DIF3D) approach is given below the calculation results. A symmetric distribution of relative errors are found for both approaches. Comparing to the DRAGON-DIF3D case, the MICXN-DIF3D approach yields relatively greater prediction error with the difference of the UO 2 assembly is 1.60% while that of the MOX assembly is -2.15%. The prediction error of the relative power distribution is related to the lattice calculation through the cross section generation. The multiplication factor is underestimated by DRAGON in UO 2 case and overestimated in MOX case, which yields depression of power for A1/A5 assemblies and increase for A2/A4 assemblies. For M-DIF3D approach, k eff is under-predicted in higher degree for MOX assembly, therefore leads to an opposite distribution of

177 158 Reflective BC Reflective BC Vaccum BC Vacuum BC UO 2 Fuel 4.3% MOX Fuel 7.0% MOX Fuel 8.7% MOX Fuel Fission Chamber Guide Tube Figure 6.3: C5G7 2-D benchmark errors in power map than the D-DIF3D approach, that is, higher power is found in A1/A5 assemblies, while lower power is found in A2/A4 assemblies. Next, the C5G7 problem itself is used as the test case, of which the configuration is shown in Figure 6.3. In addition to the difference in the fuel assembly design, C5G7 now introduces reflection by a moderating reflector and also leakage. The model specifications in DIF3D calculation is the same as that for C3 case, except for the zero-flux boundary conditions are applied to the two outer surfaces with solely moderator. Table 8 presents the results of M-DIF3D and D-DFI3D calcula-

178 159 Table 6.2: Calculation results of C5G7 Benchmark problem Codes k eff diff. (%) MCNPX ± DIF3D (DRAGON) DIF3D (MICXN xs by mixture) DIF3D (MICXN xs by isotope) (3.11) (-2.86) A (-3.02) (3.03) A (1.29) (-2.09) A (3.11) (-2.86) A2 MCNPX (D-DIF3D) (M-DIF3D) ID Figure 6.4: Power distribution of C5G7 benchmark tions and the comparisons to MCNPX. The M-DIF3D approach under-predicts the effective multiplication factor with relatively smaller error of 0.063%, comparing to 0.280% from D-DIF3D approach. Good agreement of the calculation results of the two approaches is observed with the difference of k eff is less than 300 pcm. The relative power distributions obtained by the two approaches are plotted and compared with MCNPX solution in Figure 6.4, from which the symmetry of the error distribution is found due to the symmetric core configuration. For the M-DIF3D approach, the prediction error of assembly A1, A2/A4, A5 is 3.03%, %, and -2.09%, respectively. In another word, there is overestimation of power at the center of the core, and underestimation towards the boundary. The D- DIF3D case has relatively larger error in this comparison, which is consistent with

179 160 the relatively large miss-prediction of k eff Fast reactor benchmark The second reactor core model that will be studied in the current research is based on the Japan s sodium-cooled experimental fast reactor JOYO core. [61] The reactor during its first operational years was called MK-I and it achieved its first criticality in The power was produced in the core fuel region that was surrounded by blanket fuel region. The core was fueled by mixed-oxide (MOX) fuel that consisted of 23% enriched uranium and plutonium that accounted for 17.7% of the weight of all metallic material. The plutonium content for its part included 80.4% fissile plutonium. The material in the blanket region consisted of depleted uranium including 0.2% fissile 235 U. The blanket region was also known as the breeding zone as the breeding of fissionable 238 U to fissile 239 Pu occurred mostly there. The leakage of the neutrons that were not captured in the blanket fuel was tried to stop by reflectors, both removable and fixed. In the beginning of operation the maximum thermal power of the reactor was 50 MWt, and it was soon upgraded to 75 MWt. Since 2003 the plant has been operating with the maximum thermal power of 140 MWt subsequent to the upgrade to MK-III core. The experiments of the study were mostly performed for 64- and 70-fuel-assembly cores. In the current study, a 2-D hexagonal numerical benchmark model has been developed based on the 64-fuel-assembly JOYO core by simplifying its radial configuration to satisfy the exact symmetry. There are totally 271 fuel assemblies, including 61 core fuel assemblies located at the center, surrounded by 156 blanket fuel assemblies and 54 reflector regions on the periphery. A 1/12 core layout is

180 161 Figure 6.5: 1/12 core layout of modified JOYO benchmark Table 6.3: Dimension data for heterogeneous model of core and assembly Dimension (cm) Fuel pin pitch 0.76 Fuel pellet outer diameter 0.56 Cladding outer diameter 0.63 Fuel assembly pitch 7.47 Height in z-direction 2.0 depicted in Figure 6.5. The dimension data listed in Table 6.3 is used to prepare for the numerical models. Nuclear number densities of the heterogeneous assembly model is consistent with that of the 64-fuel-assembly core, which are summarized in Table 6.4, for regions including the core fuel, the radial blanket, and the reflector. The core fuel region (see Figure 6.6) consists of three material zones, including the core fuel, the cladding material and the coolant. The whole layout and all the dimensions of the radial blanket fuel region are exactly the same as those of the

181 162 Table 6.4: Isotopic distribution per medium in JOYO benchmark Concentrations (10 24 at/cm 3 ) Nuclide Core fuel Blanket fuel Cladding Coolant Reflector 235 U U Pu Pu Pu Pu Pu Am O Cr Fe Ni Mo Na core fuel region. The only difference from the core fuel region is the replacement of the composition in the core fuel zone with that of the radial blanket fuel zone. Similar to the C5G7 benchmark, Monte Carlo code MCNPX is used to provide the reference solution for assembly and core calculations, by explicitly modeling the geometry and using the continuous energy ENDF/B-VII.0 cross section library. On the assembly level, both core fuel and blanket fuel assemblies are modeled by MICXN using the hexagonal geometry option and the calculations are performed in the aforementioned 4-group energy structure. The inter-comparison solution is provided by deterministic code DRAGON. Table 6.5 shows the comparison results of the assembly calculations, from which it can be seen that results of MICXN

182 163 Figure 6.6: Heterogeneous model of core fuel or blanket assembly and DRAGON have good agreement with each other, and both produce nonnegligible prediction error from MCNPX results. For the MICXN calculation, the multiplication factor miss-prediction is found to be % and 0.811% for fuel assembly and blanket assembly, respectively. It is worth pointing out that the large percentage error predicted by MICXN for blanket assembly case is primarily due to the small value of k itself. Better agreements are found when compared MICXN results with that of DRAGON, while DRAGON yields relatively smaller prediction error. As the reference core solution, the JOYO reactor has been modeled explicitly using MCNPX at room temperature, with 11 rings, 271 assemblies in total. Deterministic codes DRAGON and MICXN are employed to generate few-group cross sections in ISOTXS format in four energy groups, which are then applied to the