Full Core, Heterogeneous, Time Dependent Neutron Transport Calculations with the 3D Code DeCART by Mathieu Hursin A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy In Engineering - Nuclear Engineering in the GRADUATE DIVISION of the UNIVERSITY OF CALIFORNIA AT BERKELEY Committee in charge: Professor Ehud Greenspan, Chair Professor Per F. Peterson Professor Thomas Downar Professor Jon Wilkening Fall 2010 ABSTRACT Hursin, Mathieu. PhD, University of California at Berkeley, December 2010. Full Core, Heterogeneous, Time Dependent Neutron Transport Calculations with the 3D Code DeCART. Major Professors: Tom Downar and Ehud Greenspan. The current state of the art in reactor physics methods to assess safety, fuel failure, and operability margins for Design Basis Accidents (DBAs) for Light Water Reactors (LWRs) rely upon the coupling of nodal neutronics and one-dimensional thermal hydraulic system codes. The neutronic calculations use a multi-step approach in which the assembly homogenized macroscopic cross sections and kinetic parameters are first calculated using a lattice code for the range of conditions (temperatures, burnup, control rod position, etc...) anticipated during the transient. The core calculation is then performed using the few group cross sections in a core simulator which uses some type of coarse mesh nodal method. The multi-step approach was identified as inadequate for several applications such as the design of MOX cores and other highly hetereogeneous, high leakage core designs. Because of the considerable advances in computing power over the last several years, there has been interest in high-fidelity solutions of the Boltzmann Transport equation. A practical approach developed for high-fidelity solutions of the 3D transport equation is the 2D-1D methodology in which the method of characteristics (MOC) is applied to the heterogeneous 2D planar problem and a lower order solution is applied to the axial problem which is, generally, more uniform. This approach was implemented in the DeCART code. Recently, there has been interest in extending such approach to the simulations of design basis accidents, such as control rod ejection accidents also known as reactivity initiated accidents (RIA). The current 2D-1D algorithm available in DeCART only provide 1D axial solution based on the diffusion theory whose accuracy deteriorates in case of strong flux gradient that can potentially be observed during RIA simulations. The primary ojective of the dissertation is to improve the accuracy and range of applicability of the DeCART code and to investigate its ability to perform a full core transient analysis of a realistic RIA. The specific research accomplishments of this work include: • The addition of more accurate 2D-1D coupling and transverse leakage splitting options to avoid the occurrence of negative source terms in the 2D MOC equations and the subsequent failure of the DeCART calculation and the improvement of the convergence of the 2D-1D method. • The implementation of a higher order transport axial solver based on NEM-Sn derivation of the Boltzmann equation. 1 • Improved handling of thermal hydraulic feedbacks by DeCART during transient calculations. • A consistent comparison of the DeCART transient methodology with the current multistep approach (PARCS) for a realistic full core RIA. An efficient direct whole core transport calculation method involving the NEM- Sn formulation for the axial solution and the MOC for the 2-D radial solution was developed. In this solution method, the Sn neutron transport equations were developed within the framework of the Nodal Expansion Method. A RIA analysis was performed and the DeCART results were compared to the current generation of LWR core analysis methods represented by the PARCS code. In general there is good overall agreement in terms of global information from DeCART and PARCS for the RIA considered. However, the higher fidelity solution in DeCART provides a better spatial resolution that is expected to improve the accuracy of fuel performance calculations and to enable reducing the margin in several important reactor safety analysis events such as the RIA. 2 To my mother for her unwavering support, And to my father for sharing his passion. iii TABLE OF CONTENTS ABSTRACT ___________________________________________________________ 1 TABLE OF CONTENTS ________________________________________________ iv LIST OF FIGURES ___________________________________________________ vii LIST OF TABLES ____________________________________________________ viii AKNOWLEDGEMENTS _______________________________________________ ix Introduction ___________________________________________________________ 1 1.1. Motivation _____________________________________________________ 1 1.2. Issues and Undertakings __________________________________________ 2 1.2.1. Key issues _________________________________________________ 2 1.2.2. Improvements to DeCART undertaken in this work _________________ 3 1.3. Dissertation Outline _____________________________________________ 3 2. The DeCART Integral Transport Code ________________________________ 5 2.1. Overview ______________________________________________________ 5 2.1.1. 2D-1D Solution Approach ____________________________________ 5 2.1.2. Modular Ray Tracing ________________________________________ 6 2.2. Resonance Treatment ____________________________________________ 7 2.2.1. General Treatment __________________________________________ 7 2.2.2. The Subgroup Method ________________________________________ 8 2.2.3. Determination of the Equivalence Cross Section ___________________ 9 2.2.4. Determination of the effective macroscopic resonance cross section. __ 10 2.2.5. Calculation sequence for resonance treatment ____________________ 11 2.3. Steady State Methodology _______________________________________ 12 2.3.1. The 2-D Method of Characteristics. ____________________________ 12 2.3.2. 1-D axial diffusion solvers ___________________________________ 15 2.3.2.1. Nodal Expansion Method (NEM) _______________________________________ 15 2.3.2.2. Semi Analytical Nodal Method (SANM) __________________________________ 16 2.3.3. 3-D Global Solution Strategy _________________________________ 16 2.3.4. Known Issues of original 2D/1D formulation _____________________ 18 2.4. Transient Methodology __________________________________________ 18 2.4.1. Governing equations ________________________________________ 19 2.4.1.1. Time discretization ___________________________________________________ 19 2.4.1.2. Delayed Neutron Source Approximation __________________________________ 20 2.4.1.3. Multi-group CMFD transient fixed source formulation _______________________ 21 2.4.1.4. MOC transient fixed source formulation __________________________________ 21 2.4.2. Calculation flow ___________________________________________ 23 2.4.3. Kinetics parameters ________________________________________ 23 2.4.4. Limitations _______________________________________________ 24 3. Improvement of the 2D/1D coupling __________________________________ 26 3.1. Investigation of the potential issues with the current 2D-1D coupling _____ 26 3.1.1. Test Case Description _______________________________________ 27 3.1.2. Results and Discussion ______________________________________ 28 3.1.2.1. Reduced Mesh Size Case ______________________________________________ 28 3.1.2.2. Low Density Case ___________________________________________________ 29 3.2. Formulation of the new 2D/1D coupling approaches ___________________ 30 3.2.1. Isotropic Leakage Coupling __________________________________ 30 iv 3.2.2. P1 coupling _______________________________________________ 31 3.3. Transverse Leakage Splitting Formulation ___________________________ 32 3.3.1. Partial Current Splitting _____________________________________ 33 3.3.2. Isotropic Leakage Splitting ___________________________________ 33 3.3.3. P1 Splitting _______________________________________________ 34 3.3.1. Approximation involved in transverse leakage splitting _____________ 35 3.4. Results _______________________________________________________ 36 3.4.1. Foreword about CMFD and convergence _______________________ 36 3.4.2. Mesh refinement problems ___________________________________ 36 3.4.2.1. Water Hole case _____________________________________________________ 36 3.4.2.2. Control Rod case ____________________________________________________ 40 3.4.3. Low density case. __________________________________________ 43 3.4.4. Impact of improvements on calculation cost _____________________ 46 3.5. Summary _____________________________________________________ 47 4. Improvement of the axial solver _____________________________________ 49 4.1. Literature Review ______________________________________________ 49 4.2. NEM-Sn derivation _____________________________________________ 50 4.2.1. Legendre Polynomials ______________________________________ 50 4.2.2. Discrete Ordinate Method ___________________________________ 51 4.2.3. Nodal Expansion Method ____________________________________ 51 4.3. Numerical NEM-Sn implementation _______________________________ 52 4.3.1. Choice of angular quadrature ________________________________ 52 4.3.2. Transverse leakage _________________________________________ 53 4.3.2.1. Formulation of the modified 1D equation with transverse leakage ______________ 53 4.3.2.2. Transverse Leakage Approximation _____________________________________ 54 4.3.2.3. 2D-1D modified