Neutron Transport: Theory, Modeling, and Computations (Graduate Texts in Physics) [1st ed. 2023] 3031269314, 9783031269318

Table of contents :
Preface
Contents
1 Neutron Transport—The Neutron Transport Equation
1.1 Fundamental Concepts
1.2 Derivation of the Neutron Transport Equation
1.2.1 Boundary Conditions
1.3 Simplified Forms of the Neutron Transport Equation
1.4 The Integral Method
References
2 Neutron Diffusion—The Finite Difference Solution of the Neutron Diffusion Equation
2.1 Finite Difference Formulation of the One-Dimensional One-Group Neutron Diffusion Equation
2.1.1 Left Boundary
2.1.2 Right Boundary
2.2 Solution Algorithm
2.2.1 Boundary Conditions
2.2.2 Methods of Solution
2.2.3 Power Distribution
2.3 Finite Difference Formulation of the One-Dimensional Multi-group Neutron Diffusion Equation
2.4 Finite Difference Formulation for the Multi-group Two-Dimensional ( x - y ) Neutron Diffusion Equation
2.4.1 Method of Solution
References
3 Spherical Harmonics—The PN Method
3.1 Derivation of the PN Equations in Plane Geometry
3.1.1 The PN Approximation
3.2 Derivation of the PN Equations in Spherical Geometry
3.3 Boundary and Interface Conditions
3.3.1 Free Surface Boundary Condition
3.3.2 Interface Condition in Plane Interface
3.3.3 Interface Condition in Curved Interface
3.3.4 Yvon’s Method
3.4 Difference Scheme for the One-Dimensional One-Speed P1 Equations
3.4.1 Plane Geometry
3.4.2 Difference Equations in Spherical Geometry
3.5 The Elastic Transfer Cross Section
References
4 The Adjoint Transport Equation—The Equation of Neutron Importance
4.1 Mathematical Definitions [1, 2]
4.2 Derivation of the Equation of Neutron Importance (The Adjoint Function)
4.2.1 The Equation of Conservation of Importance
4.2.2 Boundary Condition
4.2.3 Final and Initial Conditions
4.3 The Adjoint Neutron Diffusion Equation
4.3.1 The Energy-Dependent Neutron Diffusion Equation (DE)
4.3.2 The One-Speed Neutron Diffusion Equation
4.3.3 The Multi-Group Neutron Diffusion Equation
4.3.4 The Two-Group Neutron Diffusion Equation
4.4 The Adjoint One-Speed P1 Equations
4.5 Derivation of the Kinetic Parameters
4.5.1 Transport Model
4.5.2 One-Node Model
4.5.3 Diffusion Model
4.5.4 Two-Node Model
References
5 Perturbation Theory
5.1 Perturbation of the Effective Multiplication Factor (k)
5.2 Perturbations in the One Speed Diffusion Theory
5.3 Multigroup Perturbation Theory
5.4 Perturbation of the Multiplication Rate Constant (α)
References
6 The Method of Discrete Ordinates: The SN Method
6.1 Fundamental Concepts
6.2 The Combined Discrete Ordinates and Legendre Polynomials Equation
6.2.1 Gauss Quadrature Set
6.2.2 Anisotropic Scattering
6.2.3 Multi-group DO Equations with Anisotropic Scattering and Fission Source
6.2.4 The Double-PN Method in Discrete Ordinates
6.3 Numerical Solution of the Discrete Ordinate Equation
6.3.1 Plane Geometry
6.3.2 Spherical Geometry
6.3.3 General Geometry
6.4 Discrete Ordinates Computer Codes
References
7 Computer Simulation of Neutron Transport—The Monte Carlo Method
7.1 Basic Statistical Definitions
7.2 Particle History Generation
7.2.1 History Generator
7.3 Modeling of Radiation Transport Parameters [4–6]
7.3.1 Source Parameters
7.3.2 Path Length
7.3.3 Collision Parameters
7.4 Variance Reduction Methods
7.4.1 Particle Weight
7.4.2 Source Biasing Parameters
7.4.3 Variance Reduction Techniques
7.4.4 Neutron/Photon Problem
7.5 Criticality Calculations
7.5.1 keff Cycle
7.5.2 keff Estimators
7.6 Particle Scoring
7.6.1 Neutron Tallies
7.7 Accuracy, Precision, Relative Error, and Figure of Merit
7.7.1 Precision
7.7.2 Relative Error
7.7.3 Figure of Merit
References
8 Neutron Transport—The Variational Methods
8.1 Variational Calculus
8.1.1 Functionals
8.1.2 The Euler–Lagrange Equation
8.2 Variational Principles
8.2.1 Sturm–Liouville Differential Equations
8.2.2 Fredholm Integral Equations with Symmetric Kernels
8.2.3 Eigenvalue Problems
8.2.4 Inhomogeneous Problems
8.3 Application of Variational Methods to Transport Theory
References
9 Selected Problems Solutions
A The concept of Directional Derivative
A.1 Directional Derivative
A.1.1 ∇ in Planar Geometry
A.1.2 in Spherical Geometry
B Solid Angle and Differential Velocity
B.1 Solid Angle
B.2 Differential Velocity
C Radiative Transport
C.1 Applications of the Transport Theory
C.1.1 Neutron Transport Equation
C1.2 Photon Transport Equation: (Low Energy e.g. Light)
C1.3 Photon Transport Equation (High-Energy, e.g. Gamma and X-rays)
C.1.4 Radiative Transport Equation
C.1.5 Electron Transport Equation
C.1.6 Ionized Gases and Plasma Transport Equation
D Input Description of the 1D Program
D.1 Input Description
D.2 Running the 1D Program
D.3 Output Description
E Sample Problems and Exercises for the 1D Program
F Legendre Polynomials

Citation preview

Graduate Texts in Physics

Ramadan M. Kuridan

Neutron Transport Theory, Modeling, and Computations

CONSDIDEE I

Graduate Texts in Physics Series Editors Kurt H. Becker, NYU Polytechnic School of Engineering, Brooklyn, NY, USA Jean-Marc Di Meglio, Matière et Systèmes Complexes, Bâtiment Condorcet, Université Paris Diderot, Paris, France Sadri Hassani, Department of Physics, Illinois State University, Normal, IL, USA Morten Hjorth-Jensen, Department of Physics, Blindern, University of Oslo, Oslo, Norway Bill Munro, NTT Basic Research Laboratories, Atsugi, Japan Richard Needs, Cavendish Laboratory, University of Cambridge, Cambridge, UK William T. Rhodes, Department of Computer and Electrical Engineering and Computer Science, Florida Atlantic University, Boca Raton, FL, USA Susan Scott, Australian National University, Acton, Australia H. Eugene Stanley, Center for Polymer Studies, Physics Department, Boston University, Boston, MA, USA Martin Stutzmann, Walter Schottky Institute, Technical University of Munich, Garching, Germany Andreas Wipf, Institute of Theoretical Physics, Friedrich-Schiller-University Jena, Jena, Germany

Graduate Texts in Physics publishes core learning/teaching material for graduateand advanced-level undergraduate courses on topics of current and emerging fields within physics, both pure and applied. These textbooks serve students at the MS- or PhD-level and their instructors as comprehensive sources of principles, definitions, derivations, experiments and applications (as relevant) for their mastery and teaching, respectively. International in scope and relevance, the textbooks correspond to course syllabi sufficiently to serve as required reading. Their didactic style, comprehensiveness and coverage of fundamental material also make them suitable as introductions or references for scientists entering, or requiring timely knowledge of, a research field.

Ramadan M. Kuridan

Neutron Transport Theory, Modeling, and Computations

Ramadan M. Kuridan Nuclear Engineering University of Tripoli Tripoli, Libya

ISSN 1868-4513 ISSN 1868-4521 (electronic) Graduate Texts in Physics ISBN 978-3-031-26931-8 ISBN 978-3-031-26932-5 (eBook) https://doi.org/10.1007/978-3-031-26932-5 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Cover image: IAEA This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

To Fauzia, Khaled, Abdulrahim, Abdulalim

Preface

The aim of this book is to provide a thorough explanation of the physical concepts and present the general theory of neutron transport processes in nuclear reactors and emphasize the numerical computing methods that lead to the prediction of neutron behavior. Detailed derivation and thorough discussion are the prominent features of this book instead of brevity and conciseness a characteristic of most available textbooks on the subject. The titles covered in this book are suitable for mainly graduate students in the fields of nuclear engineering and physics with sufficient background in reactor theory and calculations during the undergraduate level. Therefore, basic concepts of reactor physics such as the fission process, cross sections, and neutron diffusion are bypassed in this book. An adequate knowledge of mathematical subjects such as partial differential equations, eigenvalue problems, numerical analysis, and vector calculus is required. This book came about after many years of teaching the topics on neutron transport to graduate students in the nuclear engineering department at the University of Tripoli. Certainly, I have benefited from their discussions and I am so thankful to them. In particular M. Ben Ghzail and H. Aburebaiya. Special thanks to Professors Naeem Abdurrahman and Wajdi Ratemi from the nuclear engineering department at the University of Tripoli; Dr. Omran Abuzid, a senior research scientist at the Libyan Atomic Energy Establishment; Prof. Mohamed Elsawi from Argonne National Laboratories for their fruitful advices and comments of the manuscript. A particular acknowledgment and gratitude must be expressed to the professors from the nuclear engineering department at the University of Michigan, who provided me with the knowledge and understanding of nuclear reactor physics and neutron transport during my BSc and MSc study programs. In particular, professors J. Duderstadt, W. Martin, and J. Lee, my gratitude extends to a long list of other scientists who provided the insight and inspiration through their books and research papers. Janzour, Tripoli, Libya

Ramadan M. Kuridan

vii

Contents

1 Neutron Transport—The Neutron Transport Equation . . . . . . . . . . . . . . 1.1 Fundamental Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Derivation of the Neutron Transport Equation . . . . . . . . . . . . . . . . . . . . 1.2.1 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Simplified Forms of the Neutron Transport Equation . . . . . . . . . . . . . 1.4 The Integral Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Neutron Diffusion—The Finite Difference Solution of the Neutron Diffusion Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Finite Difference Formulation of the One-Dimensional One-Group Neutron Diffusion Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Left Boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Right Boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Solution Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Methods of Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 Power Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Finite Difference Formulation of the One-Dimensional Multi-group Neutron Diffusion Equation . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Finite Difference Formulation for the Multi-group Two-Dimensional (x − y) Neutron Diffusion Equation . . . . . . . . . . . 2.4.1 Method of Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Spherical Harmonics—The PN Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Derivation of the PN Equations in Plane Geometry . . . . . . . . . . . . . . 3.1.1 The PN Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Derivation of the PN Equations in Spherical Geometry . . . . . . . . . . . 3.3 Boundary and Interface Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Free Surface Boundary Condition . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Interface Condition in Plane Interface . . . . . . . . . . . . . . . . . . . . . 3.3.3 Interface Condition in Curved Interface . . . . . . . . . . . . . . . . . . . 3.3.4 Yvon’s Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 8 12 14 20 25 27 28 32 33 35 36 39 43 44 45 56 59 61 61 66 70 74 74 76 78 78 ix

x

Contents

3.4 Difference Scheme for the One-Dimensional One-Speed P1 Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Plane Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2 Difference Equations in Spherical Geometry . . . . . . . . . . . . . . 3.5 The Elastic Transfer Cross Section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 The Adjoint Transport Equation—The Equation of Neutron Importance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Mathematical Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Derivation of the Equation of Neutron Importance (The Adjoint Function) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 The Equation of Conservation of Importance . . . . . . . . . . . . . 4.2.2 Boundary Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.3 Final and Initial Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 The Adjoint Neutron Diffusion Equation . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 The Energy-Dependent Neutron Diffusion Equation (DE) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 The One-Speed Neutron Diffusion Equation . . . . . . . . . . . . . . 4.3.3 The Multi-Group Neutron Diffusion Equation . . . . . . . . . . . . . 4.3.4 The Two-Group Neutron Diffusion Equation . . . . . . . . . . . . . . 4.4 The Adjoint One-Speed P1 Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Derivation of the Kinetic Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.1 Transport Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.2 One-Node Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.3 Diffusion Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.4 Two-Node Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

83 83 89 92 97 99 99 101 101 105 105 107 107 108 109 111 111 114 114 116 117 120 124

5 Perturbation Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Perturbation of the Effective Multiplication Factor (k) . . . . . . . . . . . . 5.2 Perturbations in the One Speed Diffusion Theory . . . . . . . . . . . . . . . . 5.3 Multigroup Perturbation Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Perturbation of the Multiplication Rate Constant (α) . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

125 126 128 130 132 134

6 The Method of Discrete Ordinates: The SN Method . . . . . . . . . . . . . . . . . 6.1 Fundamental Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 The Combined Discrete Ordinates and Legendre Polynomials Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 Gauss Quadrature Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.2 Anisotropic Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.3 Multi-group DO Equations with Anisotropic Scattering and Fission Source . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.4 The Double-PN Method in Discrete Ordinates . . . . . . . . . . . .

135 136 139 142 144 145 147

Contents

xi

6.3 Numerical Solution of the Discrete Ordinate Equation . . . . . . . . . . . . 6.3.1 Plane Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.2 Spherical Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.3 General Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Discrete Ordinates Computer Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

147 147 156 167 167 169

7 Computer Simulation of Neutron Transport—The Monte Carlo Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Basic Statistical Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Particle History Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.1 History Generator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Modeling of Radiation Transport Parameters . . . . . . . . . . . . . . . . . . . . . 7.3.1 Source Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.2 Path Length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.3 Collision Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Variance Reduction Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.1 Particle Weight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.2 Source Biasing Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.3 Variance Reduction Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.4 Neutron/Photon Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5 Criticality Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.1 keff Cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.2 keff Estimators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6 Particle Scoring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6.1 Neutron Tallies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.7 Accuracy, Precision, Relative Error, and Figure of Merit . . . . . . . . . 7.7.1 Precision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.7.2 Relative Error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.7.3 Figure of Merit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

171 172 177 178 179 179 182 185 190 190 190 191 193 194 194 196 196 197 198 199 199 199 200

8 Neutron Transport—The Variational Methods . . . . . . . . . . . . . . . . . . . . . . 8.1 Variational Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.1 Functionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.2 The Euler–Lagrange Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Variational Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.1 Sturm–Liouville Differential Equations . . . . . . . . . . . . . . . . . . . 8.2.2 Fredholm Integral Equations with Symmetric Kernels . . . . . 8.2.3 Eigenvalue Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.4 Inhomogeneous Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Application of Variational Methods to Transport Theory . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

203 203 203 205 208 208 212 213 218 222 226

9 Selected Problems Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227

xii

Contents

Appendix A: The concept of Directional Derivative . . . . . . . . . . . . . . . . . . . . . 251 Appendix B: Solid Angle and Differential Velocity . . . . . . . . . . . . . . . . . . . . . . 261 Appendix C: Radiative Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263 Appendix D: Input Description of the 1D Program . . . . . . . . . . . . . . . . . . . . . . 269 Appendix E: Sample Problems and Exercises for the 1D Program . . . . . . 277 Appendix F: Legendre Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281

1

Neutron Transport—The Neutron Transport Equation

The neutron transport equation is the mother of all equations in reactor physics. All approximate analytical and numerical forms are derived from it. Therefore, the beginning chapter of the book has been allocated to the neutron transport equation. In the first section, the fundamental concepts of neutron density, flux, and current are presented. In the second section, a rigorous derivation is formulated along with the boundary and source conditions. Then simplified and approximate forms of the equation are discussed in Sect. 1.3. In Sect. 1.4, the integral method is applied to the non-multiplying and multiplying slab. The concept of directional derivatives, differential velocity, and solid angle important to the mathematics of the neutron transport equation are included in appendices A and B. Finally, in appendix C, other types of radiative transport are summarized [1–5].

1.1

Fundamental Concepts

In this section, we introduce the basic definitions describing the neutron and its motion which will assist in the derivation of the neutron transport equation [6]. 1. Neutron density Neutron density n is defined as the number of neutrons in a unit volume in a manner similar to nuclear density N . We designate the neutrons by their position r and kinetic energy E = 21 mυ 2 , where mandυ are the neutron mass and speed, υ respectively, and direction of a unit vector Ω = υ in spherical coordinates (θ, ψ). the neutron density using the fore mentioned indepenTherefore, we can express ) ) dent variables, e.g. n r, E, Ω, t at time t (Fig. 1.1). In addition to the units of volume extra units for[ the other ]variables are needed, i.e. a unit of energy, and a unit of solid angle cmneutrons 3 −eV−st . Therefore, the neutron density is defined as

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 R. M. Kuridan, Neutron Transport, Graduate Texts in Physics, https://doi.org/10.1007/978-3-031-26932-5_1

1

2

1

Neutron Transport—The Neutron Transport Equation

) ) Fig. 1.1 The angular density n r , E, Ω, t

) ) n r , E, Ω, t d V d EdΩ [neutrons]: The expected number of neutrons at position r in d V at energy E in d E in the direction Ω in the solid angle dΩ = sinθ dθ dψ (Appendix B) at time t. The word expected indicates the statistical nature of the neutron density; hence, we ) always )mean the average density of neutrons and not the actual density. n r , E, Ω, t is referred to as the angular density and, if integrated over direction, it becomes the total density (

) ) n r , E, t =

) ) dΩn r , E, Ω, t

4π

And, it is still called the total density if integrated over energy ) ) n r, t =

(∞

) ) d En r , E, t

0

The angular dependence of the neutron density is more apparent near boundaries or neutron sources. 2. Neutron flux Neutrons are always in a state of motion and hence their speed and direction change continuously as a result of interactions. The neutron density alone does not represent this physical situation precisely. Therefore, we have to incorporate the neutron speed along with the density to obtain a new physical quantity such as the flow rate of neutrons or the commonly used terminology, neutron flux: ϕ = υn or the angular flux

1.1 Fundamental Concepts

3

Fig. 1.2 The concept of angular flux

] [ ) ) ) ) n ϕ r , E, Ω, t = υn r , E, Ω, t and units of cm2 −sec−eV−st Accordingly, we can define the neutron flux [ ϕ as ]the number of neutrons crossing a unit area perpendicular to Ω per sec: cm2n−sec . It is a scalar quantity unlike the concept of flux used in electromagnetic theory which is a vector quantity closer to the concept of neutron current. Figure 1.2 shows the concept of angular flux. If the angular flux is integrated over all directions, we obtain the total flux which can be defined as the total number of neutrons crossing the unit area per second regardless of direction. ( ) ) ) ) φ r , E, t = dΩϕ r , E, Ω, t (1.1) 4π

If the angular neutron flux is weakly dependent on angle (isotropic) which means that the neutrons move in different directions with equal probability, then ) ) ) φ r , E, t ) ϕ r , E, Ω, t = (1.2) 4π 3. Neutron current density If the neutron velocity has replaced the neutron speed in the definition of the angular flux, we obtain the angular current density ) ) ) ) ) ) ) ) j r , E, Ω, t = υn r , E, Ω, t = Ωυn r , E, Ω, t = Ωϕ r , E, Ω, t

4

1

Neutron Transport—The Neutron Transport Equation

Or we call it by the unused terminology (vector flux). The angular current density has the same value and units as the angular flux | | | | | | | | | | j | = Ωϕ | = |Ω ||ϕ| = ϕ The current density is defined as ) ) J r , E, t =

(

) ) dΩ j r , E, Ω, t

4π

And ) ) J r, t =

(∞

( dE

0

4π

) ) dΩ j r , E, Ω, t =

(∞

) ) d E J r , E, t

(1.3)

0

) ) ) ) Notice that J r , t or J r , E, t are vector quantities. 4. Net neutron current density Now, since the current density is a vector quantity, we may be able to utilize it in expressing some physical quantities such as the net leakage rate through surfaces, unlike the scalar flux which may be utilized in the calculation of reaction rates. Consider an arbitrary differential surface area d A which can be expressed in a vector form d A = d An, where n is the out normal vector to the differential area as shown in Fig. 1.3. Here, we want to draw the attention of the reader to the confusion which may happen between the normal vector and the neutron density. The latter rarely appears again in equations to come. Suppose there is a neutron current making an angle θ with the unit normal vector n so what would be the number of neutrons which can pass through the area d A. Actually, only a fraction of the neutrons is capable of passing through because not all of them can see the whole area. If the angle θ = 90◦ , none will pass; on the other hand, if θ = 0◦ , all can pass. Therefore, we can express this argument for the net current density mathematically using the dot product of vectors such as | ) )|| | ) ) J r , t · d A = | J r , t ||n |d A cos θ

(1.4)

which can be expressed as the net number of neutrons passing through the differential surface area d A oriented in any direction per sec. Integrating this quantity assuming a unit area of the surface

1.1 Fundamental Concepts

5

Fig. 1.3 The concept of the net neutron current

) ) J r, t · n =

(∞

( dE

0

4π (2π

(∞ =

) ) dΩ n. j r , E, Ω, t =

dE 0

(∞ 0

(π dψ

0

( dE

) ) ) ) dΩ n.Ω ϕ r , E, Ω, t

4π

) ) sin θ dθ cos θ ϕ r , E, θ, ψ, t

0

Hence, ) ) J r, t · n =

(∞

(2π dE

0

(+1 ) ) dψ dμμϕ r , E, μ, ψ, t

(1.5)

−1

0

where, n.Ω = cos θ and μ = cos θ 5. Partial neutron current density The partial current density is related to the net current density which corresponds to the total rates at which neutrons pass through a unit area from left to right (J+ ) and from right to left (J− ) or from bottom to top or top to bottom, respectively. This means separating the net current into two components ) ) J r, t = ±

(∞

(

) ) dΩ n. j r , E, Ω, t

dE 0

n.Ω >

cosθ )positive or negative which imply the range of < 0 indicate ) /weather ) ) / integration 0, π 2 or π 2, π . Using the definition of the solid angle dΩ = sin θ dθ dψ and taking account of the relations: μ = cos θ ; dμ = −sin θ dθ ) ) ) ) ϕ r , E, Ω, t d EdΩ = ϕ r , E, θ, ψ, t d E sin θ dθ dψ ) ) ) ) ϕ r , E, θ, ψ, t sin θ dθ dψ = −ϕ r , E, μ, ψ, t dμdψ We can express the partial currents as ) ) J+ r, t =

(∞

(2π dE

0

0

(∞

(2π

=

0

dE

(0 dψ

) ) (−dμ)(+μ)ϕ r , E, μ, ψ, t

+1

0

0

(∞

(2π

=

π ( /2 ) ) dψ sin θ dθ (+cos θ )ϕ r , E, θ, ψ, t

dE

(+1 ) ) dψ dμ μϕ r , E, μ, ψ, t

0

0

0

(∞

(2π

(π

(1.7)

and ) ) J r, t = −

dE

dψ π/ 2

0

0

(∞

(2π

=

dE 0

0

(∞

(2π

=

dE 0

) ) sin θ dθ (−cos θ )ϕ r , E, θ, ψ, t

(−1 ) ) dψ (−dμ)(−μ)ϕ r , E, μ, ψ, t 0

(−1 ) ) dψ dμ μϕ r , E, μ, ψ, t

0

(1.8)

0

Notice that the cosine is positive in the upper half of the spherical geometry and negative in the lower half. Figure 1.4 shows the partial currents. Writing the partial currents in one equation ) ) J r, t = ±

(∞

(2π dE

0

0

(±1 ) ) dψ dμ μϕ r , E, μ, ψ, t 0

1.1 Fundamental Concepts

7

Fig. 1.4 The concept of partial currents

Subtracting Eq. (1.8) from (1.7) +

J −J

−

(∞ =

(2π dE

0

(+1 ) ) dψ dμ μϕ r , E, μ, ψ, t

0

(∞ −

0

(2π dE

0

(−1 ) ) dψ dμ μϕ r , E, μ, ψ, t

0

0

Changing the integral limits +

J −J

−

(∞ =

(2π dE

0

(+1 ) ) dψ dμ μϕ r , E, μ, ψ, t

0

(∞ +

0

(2π dE

0

(0 dψ

) ) dμ μϕ r , E, μ, ψ, t

−1

0

Summing up the two integrals in one integration J+ − J− =

(∞

(2π dE

0

dψ 0

(+1 ) ) dμ μϕ r , E, μ, ψ, t −1

Comparing the result to Eq. (1.5), we obtain ) ) ) ) ) ) J r, t · n = J+ r, t − J− r, t which is called the net current density.

8

1

Neutron Transport—The Neutron Transport Equation

6. Neutron reaction rates ) ) ) ) The macroscopic cross section is Σ r , E = N r σ (E), where it is defined as the probability that a neutron makes an interaction as it moves a 1 cm distance in the medium, where N is the nuclear density and σ is the microscopic cross section for an interaction with a single nucleus. However, if neutrons are represented by the neutron flux in this medium then the reaction rate density will be ] [ ) ) ) ) ) ) interactios R r , E, t = Σ r , E φ r , E, t cm 3 − sec − eV or in a more general form, the interaction rate at time t (∞

( R(t)

dV

dE 0

1.2

(

] [ ) ) ) ) interactios dΩΣ r , E ϕ r , E, Ω, t sec

(1.9)

4π

Derivation of the Neutron Transport Equation

There are a few reasonable assumptions which have to be mentioned before we proceed with the derivation [7]: 1. The average neutron density and not the actual neutron density is used in the derivation where the statistical fluctuations are lower compared to the actual density. 2. Only neutron–nucleus (n-N) reactions are considered and neutron–neutron reactions are ignored for a simple reason: the neutron density is so low (109 cmn 3 ) compared to nuclear density (1022 nuclie ). Therefore, the probability that a cm3 neutron encounters a neutron is so negligible. 3. The neutron displacement during an n-N reaction is neglected because the reaction time (10−14 s) as compared to the time taken between two consecutive reactions: (10−3 s to 10−6 s). 4. Neglecting natural radioactive decay being a slow process compared to the fast neutron chain reaction. 5. Neglecting attraction and electromagnetic forces because they are so weak compared to nuclear forces 6. Neutron polarization and coherent scattering have no effect on neutron transport in the reactor. 7. Delayed neutrons become important only during transients where such short transient times are comparable with the half-life of neutron precursors which produces delayed neutrons. Such a small number of delayed neutrons compared to a large number of prompt neutrons become important in determining the reactor response.

1.2 Derivation of the Neutron Transport Equation

9

Fig. 1.5 Differential elements for the neutron conservation

The aim of the derivation of the neutron transport equation is to establish a balance or imbalance of the mechanisms of production or loss that is the balance or imbalance of through differentials of all variables volume in space, energy, solid angle d V , d E, dΩ and at r , E, and Ω (Fig. 1.5). Therefore, the gain or loss of neutrons is not only entering or leaving d V but also entering or leaving d E and dΩ mainly through scattering processes. For example, a neutron scattered from energy E and direction Ω in d E and dΩ, respectively, to other energies and directions E , , and Ω , in d E , and dΩ , is considered loss and inversely true for gain when neutrons are scattered from other energies and directions E , , and Ω , in d E , and dΩ , to energy E and direction Ω in d E and dΩ . of neutrons in all differential elements is just ) ) The number n r , E, Ω, t d V d E dΩ at time t. The total number of neutrons in volume V in d E in dΩ is just: ⎤ ⎡ ( ) ) ⎣ d V n r , E, Ω, t ⎦d EdΩ V

And, the rate of change of this number is ⎤ ⎡ ( ) ) ∂⎣ n r , E, Ω, t d V ⎦d EdΩ = gain rate in V and d EdΩ ∂t V

− loss rate from V and d EdΩ The source term is treated in the same way as the density term; hence, the total number of neutrons emitted by the source in volume V in d E in dΩ is just: ⎤ ⎡ ( ) ) ⎣ S r , E, Ω, t d V ⎦d EdΩ V

10

1

Neutron Transport—The Neutron Transport Equation

The scattering rates in V from all other energies and directions to d EdΩ . ⎡ ⎣

(

(∞ dV

d E,

0

V

⎤ ) ) ) ) dΩ , υn r , E , , Ω , , t Σs r , E , → E, Ω , → Ω ⎦d EdΩ

( 4π

Rate of loss via absorption and scattering of neutrons in V from d E from dΩ is just ⎤ ⎡ ( ) ) ) ) ⎣ υn r , E, Ω, t Σt r , E, Ω d V ⎦d EdΩ V

The net flow of neutrons through the differential surface d S into and out of V is given by ) ) ) ) d S · Ωυn r , E, Ω, t = n · j r , E, Ω, t d S, and the total over the surface of V is the integration over the surface ⎤ ⎡ ( ) ) ⎣ d S · Ωυn r , E, Ω, t ⎦d EdΩ S

We change the integration from surface to volume integration using the Gauss theorem ⎤ ⎡ ( ) ) ⎣ d S · Ωυn r , E, Ω, t ⎦d EdΩ S

⎡

=⎣

(

⎤ ) ) d V ∇ · Ωυn r , E, Ω, t ⎦d EdΩ

V

Rewriting the balance equation, we get ⎡ ⎤ ⎤ ( ) ) ∂n ⎣ d V ⎦d EdΩ = ⎣ S r , E, Ω, t d V ⎦d EdΩ ∂t V V ⎤ ⎡ ( (∞ ( ) ) ) ) + ⎣ d V d E , dΩ , υn r , E , , Ω , , t Σs r , E , → E, Ω , → Ω ⎦d EdΩ ⎡

(

0

V

⎡

−⎣

(

V

4π

⎤ ) ) ) ) υn r , E, Ω, t Σt r , E, Ω d V ⎦d EdΩ

1.2 Derivation of the Neutron Transport Equation

11

⎡

⎤ ( ) ) − ⎣ d V ∇ · Ωυn r , E, Ω, t ⎦d EdΩ V

Canceling out d EdΩ and collecting terms integrated ( )over ) ) use ) d V and making of the relation ϕ = nυ the reduction takes the form f r d V = 0 or f r = 0, we obtain the general form of the transport equation: ) ) 1 ∂ϕ = S r , E, Ω, t + υ ∂t (∞ ( ) ) , dE dΩ , ϕ r , E , , Ω , , t Σs (r , E , → E, Ω , → Ω) 0

4π

) ) ) ) ) ) − ϕ r , E, Ω, t Σt r , E, Ω − Ω · ∇ϕ r , E, Ω, t

(1.10)

This is of course an integral differential equation] of many variables. For exam[ ple, in a cartesian space, r : x, y, z; E; Ω : θ, ψ; t and can be applied for many different situations. The source term can be external or localized with all or less variables, a point, line, surface or distributed, and so forth. In the case of a nuclear reactor core, it will be a fission source, e.g. (∞ ) ) ) ) ) ) ) ) (E) ( dΩ , d E , ν E , Σ f r , E , ϕ r , E , , Ω , , t S r , E, Ω, t = χ4π 4π

0

noticing that the fission spectrum χ (E) is isotropic and independent of the energy of the neutron causing fission (E , ). If short-time transients are considered the prompt and delayed neutrons are expressed separately in the fission source term and another equation is added for the rate of change of precursors. Some of the objectives for solving the transport equation are to obtain the multiplication factor, the neutron flux, the power distribution, reaction rates, etc. Solutions are almost all through numerical methods, except for the Monte Carlo method, for all possible core geometries such as: 1. 2. 3. 4. 5. 6. 7.

Multigroup methods, Escape probabilities, Spherical harmonics expansions, Diffusion approximation, Discrete ordinates method, Adjoint equation, perturbation theory, and variational methods. …….

Limited analytical solutions are possible and can help understand the physics of the problem and perhaps reduce the computational effort through approximations and cancellation of unnecessary mathematical terms.

12

1

1.2.1

Neutron Transport—The Neutron Transport Equation

Boundary Conditions

Generally, as a mathematical condition, the flux must be positive, real, and finite and if the solution comes up negative, imaginary, or infinite the solution should be rejected. The boundary conditions are necessary for the solution to be determined exactly otherwise it will be general. 1. Initial condition: this is valid when the flux is time-dependent ) )] ) ) ϕ r , E, Ω, t t=t = ϕ0 r , E, Ω 0

which implies that the angular flux distribution should be well defined at the beginning of counting time. 2. Free surface condition: the free surface is the one which the neutrons can’t return to after they leave it (a non-reentrant surface). An example is a surface surrounded by vacuum (Fig. 1.6) ) ) ϕ r s , E, Ω, t = 0 if n · Ω < 0 for all r s on S In addition to a condition of zero return current at the surface (Fig. 1.7), the return partial current is zero ) ) J rs, t = −

(∞

( dE

0

n·Ω

φ 2,4

3

4

12

φ 3,4 = φ 12

S 3,4 = S 12

13

φ 4,4

=

φ 13

S 4,4 = S 13

=

φ8

S 5,2 = S 8

4 5

2

8

φ 5,2

5

3

11

φ 5,3 = φ 11

|

| Σa1,3

+

D 1,3

4

| | | | | | φ 7 + −D 1,2 φ 4 + −D 1,4 φ 10 + −D 2,3 φ 8 = S 7 |

| Σa2,4

+

D 2,4

Σa3,4

+

D 3,4

Σa4,4

+

D 4,4

|

|

|

|

| | | | | | φ 11 + −D 2,5 φ 14 + −D 1,4 φ 10 + −D 2,3 φ 8 | | + −D 3,4 φ 12 = S 11

| | | | | | φ 12 + −D 2,4 φ 11 + −D 4,4 φ 13 + −D 3,3 φ 9 | | + −D 3,5 φ 15 = S 12

| | | | | | φ 13 + −D 3,4 φ 12 + −D 5,4 φ 14 + −D 4,3 φ 10 | | + −D 4,5 φ 16 = S 13 |

| Σa5,2

+

S 5,3 = S 11

D 5,2

| | | | | | φ 8 + −D 4,2 φ 7 + −D 6,2 φ 9 + −D 5,1 φ 5 | | + −D 5,3 φ 11 = S 8

| | | | | | | | 5,3 5,3 φ 11 + −D 4,3 φ 10 + −D 6,3 φ 12 + −D 5,2 φ 8 Σa + D

2.4 Finite Difference Formulation for the Multi-group Two-Dimensional (x − y) …

53

| | + −D 5,4 φ 14 = S 11 |

| Σa2,2

|

+

D 2,2

| | | | | | | Σa3,2 + D 3,2 φ 6 + −D 3,1 φ 3 + −D 3,3 φ 9 + −D 2,2 φ 5 | | + −D 4,2 φ 7 = S 6 |

| Σa4,2

|

| | | | | | φ 5 + −D 2,1 φ 2 + −D 2,3 φ 8 + −D 1,2 φ 4 | | + −D 3,2 φ 6 = S 5

+

D 4,2

| | | | | | φ 7 + −D 4,1 φ 4 + −D 4,3 φ 10 + −D 3,2 φ 6 | | + −D 5,2 φ 8 = S 7

| | | | | | | Σa2,3 + D 2,3 φ 8 + −D 2,2 φ 5 + −D 1,3 φ 7 + −D 3,3 φ 9 | | + −D 2,4 φ 11 = S 8 | Σa3,3

|

+

| | | | | φ 9 + −D 3,2 φ 6 + −D 2,3 φ 8 | | | | + −D 4,3 φ 10 + −D 3,4 φ 12 = S 9

D 3,3

| | | | | | | Σa4,3 + D 4,3 φ 10 + −D 4,2 φ 7 + −D 4,4 φ 13 + −D 5,3 φ 11 | | + −D 3,3 φ 9 = S 10

| | i, j i, j The notation Σa + D i, j = Da is used to reduce the matrix size for the above equations. The fluxes in the boundary points should be known using boundary or reflective conditions depending on the location of the I × J mesh structure whether they are in a full core, a quarter core, or else. The solution algorithm is similar to the one-dimensional case.

54

2

Neutron Diffusion—The Finite Difference Solution …

Fig. 2.12 A (3 × 2) mesh spacing

The matrix form is

Example 2 Write Eq. 2.21 in a two-group form and then construct the matrix for the 3 × 2 mesh structure shown in Fig. 2.12 with I = 4 and J = 3. i, j D Rg φgi , j

| +

i , j−1 −Dg

φgi , j

|

|

+

|

i+1, j −Dg

| φgi+1, j

+

| +

i , j+1 −Dg

|

i, j i, j −Σsg−1→g φg−1

φgi , j+1

| =

|

| +

i−1, j −Dg

φgi−1, j

1 i, j S . . . . . . . . . (23) λ g

| | i, j i, j i, j i, j i, j where D Rg = Σ Rg + D i, j and Σ Rg = Σtg − Σsgg by definition and G E

|

h j−1 i , j−1 h i h j−1 + νg' Σ f g' 2 2 2 g ' =1 | i−1, j h i−1 h j i, j h i h j + νg' Σ f g' + νg' Σ f g' 2 2 2 2

Sgi , j = χg

i, j

φg'

i−1, j−1 h i−1

νg' Σ f g'

2

2.4 Finite Difference Formulation for the Multi-group Two-Dimensional (x − y) …

55

Assumptions for the two-group theory: (i) Only directly coupled groups are considered. (ii) Up-scattering is ignored, hence only down-scattering into adjacent groups is allowed. (iii) Fission neutrons are born only in the fast group implying χ1 = 1 and χ2 = 0. (iv) The removal cross section as known from before. i, j

i, j

i, j

i, j

i, j

i, j

i, j

i, j

i, j

i, j

i, j

Σ R1 = Σt1 − Σs11 = Σa1 + Σs1 − Σs11 i, j

= Σa1 + Σs11 + Σs12 − Σs11 = Σa1 + Σs12 i, j

i, j

i, j

i, j

i, j

i, j

i, j

i, j

i, j

Σ R2 = Σt2 − Σs22 = Σa2 + Σs22 + Σs21 − Σs22 = Σa2 and Σs21 = 0

where no up-scattering is assumed. Considering fast and thermal energy groups, then Eq. 2.23 becomes g=1:

i, j i , j D R1 φ1

i, j i , j

g = 2 : Da2 φ2

| | | | | i, j i, j+1 i, j+1 i−1, j i−1, j φ1 + −D1 φ1 φ1 + + −D1 | | 1 i, j i+1, j i+1, j φ1 + −D1 = S1 k | | | | | | i, j−1 i, j i , j+1 i , j+1 i−1, j i−1, j φ2 + −D2 φ2 φ2 + −D2 + −D2 | | | | i+1, j i+1, j i, j i, j φ2 + −D2 = Σs1→2 φ1 |

i , j−1 −D1

where the source of neutrons in the thermal group is the slowing down source from fast group and the only neutron source in the fast group is the fission source. The matrix equation for fast group is

(2.24)

56

2

Neutron Diffusion—The Finite Difference Solution …

And the matrix equation for the thermal group is

(2.25)

2.4.1

Method of Solution

The methodology of solution is quite similar to that of the two-group onedimensional case in Sect. 2.3 [4, 7]. Step 1: Make an initial guess of the fast group source vector S1 , and λ(0) , where (0) indicates the initial guess for commencing the iteration procedure. Using this guess for the source Eq. 2.24 is solved for the first iteration fast flux (1) ϕ1 : i, j(0)

A1 φ 1(1) =

1 (0) S λ(0) 1

Substituting φ 1(1) into Eq. 2.25 and solving for the thermal flux φ 2(1) A2 φ 2(1) = E

1→2

φ 1(1)

2.4 Finite Difference Formulation for the Multi-group Two-Dimensional (x − y) … (1)

Step 2: Calculate a new source value from Eq. 2.21 S 1

| v

El, j f

|

57

φ (1) using φ (1) 1 1

calculated in Step 1. Substituting back into 24 we obtain a new φ (2) and λ(1) . 1 We continue the iterative scheme until the n th iteration where a criterion for accuracy is satisfied such as the percentage change in the eigenvalue between two successive iterations is less than a small preset acceptable convergence limit, e.g.

| (n+1) | |λ − λ(n) || | | | < eλ λ(n+1) A pointwise criterion can be applied as well, | (n+1) (n) || |S | 1i , j − S 1i, j | | | < es (n+1) | | S 1i, j The same procedure applies to obtain the fluxes in higher energy groups. Equations 2.24 and 2.25 can be solved using different methods which can be found in the literature of numerical methods starting from forward and backward substitutions in the Gaussian elimination method, the Jacobi method, Gauss–Seidel of the L-U factorization of matrix A, convergence acceleration techniques, etc. Several codes have been written to solve the neutron diffusion equation for a variety of energy groups and dimensions reading directly the collapsed cross section from spectrum codes and doing burn-up calculation as well [8, 9]. A widely used code by students is 2DB. It can do two-group or four-group two-dimensional core (Cartesian and cylindrical) along with burn-up and adjoint calculations. The elements of the matrix A and others are filled by the nuclear properties and dimensions which are the characteristics of the reactor core. However, it is not a simple task because it has to undergo a huge process of homogenization and elementary cross-sectional data averaging using the spectra of the fundamental units of fuel cells. The spectral codes such as LEOPARD are assigned such a task. Exercises 1. Choose a typical PWR where the raw data and geometry for the fuel cells, assemblies, and core are known. (a) Run LEOPARD code or any other spectrum code to obtain the cross sections for the fuel cell types and reflector. Ignore accounting for poison and non-lattice materials. (b) Try to link the cross sections from LEOPARD [10] to 2DB [11] and hence run 2DB or any other two-dimensional multi-group codes for the flux. You may as well link the available spectrum code to a space-dependent code of your choice.

58

2

Neutron Diffusion—The Finite Difference Solution …

2. Write down the matrices of a three-group 2D mesh structure of Fig. 2.9. 3. Assume a simple case of homogenous non-multiplying slab surrounded by vacuum with a uniformly distributed neutron source. (a) Rederive the difference equations for this case assuming one-group diffusion. (b) Using the Taylor expansion, | | Δ2 d 2 φ || dφ || + + ··· d x |i 2 d x 2 |i | | dφ || Δ2 d 2 φ || = φ(xi−1 ) = φi − Δ + + ··· d x |i 2 d x 2 |i

φi+1 = φ(xi+1 ) = φi + Δ φi−1

where Δ is the mesh spacing. Add the two expansions to obtain | d 2 φ || ∼ φi+1 − 2φi + φi−1 + ··· = d x 2 |i Δ2 Substituting in the diffusion equation to obtain the difference form. Compare with the result in (a). 4. Run the ONED code or any other one-dimensional diffusion codes for the bare slab core with the following nuclear cross sections:

D

Ea

Ef

0.9

0.066

0.07

By making small-sized meshes iterate on the critical core size. All the information you need for ONED (1D) [2, 12] code is in Appendices D and E. 5. Consider a bare slab core and the cross sections in Problem 4. Write a computer program which solves the one-group neutron diffusion equation. Draw the neutron flux and power profiles. (i) Compare the criticality constant and the flux and power profiles with the ONED results in part (a). (ii) Obtain an analytical solution for this simple case and compare the multiplication factor and the flux.

References

59

References 1. R.M. Kuridan, Nuclear reactor theory. National Bureau of Research and Development, Tripoli, Libya, (2004). (in Arabic) 2. J.C. Lee, The ONED code, lecture notes, on reactor theory. University of Michigan, Nuclear Engineering, 1975–1976 3. J.C. Lee, Nuclear reactor physics and engineering (Wiley, NJ, 2020) 4. B. Quintero-Leyva, Numerical solution of the multi-group integro-differential equation of the neutron diffusion kinetics in 2D-Cartesian geometry. Ann. Nucl. Energy. 48, (2012) 5. J. Duderstadt, L. Hamilton, Nuclear reactor analysis (Wiley, New York, 1976) 6. A. Henry, Nuclear reactor analysis (MIT Press, Mass, 1982) 7. W. Cheney., D. Kincaid, Numerical mathematics and computing, 6th edn (Thomson Brooks/Cole, 2008) 8. A. Hébert, DRAGON5: Designing computational schemes dedicated to fission nuclear reactors for space. in Proceedings of nuclear and emerging technologies for space. (Albuquerque, NM, 2013) 9. A. Hébert, TRIVAC: A modular diffusion code for fuel management, (1987) 10. R.F. Barry, Leopard: A spectrum dependent non-spatial depletion code. WCAP-3269–26, (1963) 11. A 2DB-UM, A two-dimensional neutron diffusion burn up code. University of Michigan, Nuclear Engineering version 12. R.M. Kuridan, ONED-LOPARD link for depletion calculation (1DB code). Master thesis, University of Michigan, Nuclear Engineering, unpublished, 1980

3

Spherical Harmonics—The PN Method

The PN method is based on the expansion of the angular distribution of the neutron flux (ϕ), i.e. the dependence on the direction Ω which is a function of the angles θ, ψ in a complete set of orthogonal functions, namely, the Legendre polynomials. The angular flux and the transfer cross section are weakly dependent on the azimuthal angle ψ. Therefore, it is neglected from the analysis throughout the chapter. Actually, the Legendre polynomials are considered the best of all polys. Such that the difference between the value of the function before and after the expansion is small as compared to other polys. The resulting form of the transport equation is made practical for a solution by truncating the expansion to a practical number of terms. We will consider both the plane and curved geometries separately. In Sect. 3.1, the PN equation for plane geometry is derived. From which the famous P1 equations are obtained. Followed by the derivation the PN equation in spherical geometry in Sect. 3.2. Marshak and Mark boundary conditions and interface conditions are presented in Sect. 3.3 for a free surface and plane and curved interfaces. Due to discontinuity the concept of double PN , the so-called Yvon’s method is introduced. In Sect. 3.4, the finite difference scheme is used for the solution of P1 equations in plane and curved geometries. Finally, in Sect. 3.5, the elastic transfer cross section is considered since its dependence on angle and energy requires a special treatment.

3.1

Derivation of the PN Equations in Plane Geometry

The general form of expansion is in the components θ, ψ of the direction Ω in what is called spherical harmonics [1, 2] ( ) Ynm Ω = Ynm (θ, ψ) = Ynm (μ, ψ) =

{

(2n + 1)(n − m)! (n + m)!

}1/2 Pnm (μ)ex p i mϕ

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 R. M. Kuridan, Neutron Transport, Graduate Texts in Physics, https://doi.org/10.1007/978-3-031-26932-5_3

61

62

3

Spherical Harmonics—The PN Method

where n is real, positive, or zero and −n ≤ m ≤ n and Pnm (μ) are the associated Legendre polynomials. The flux expansion is as follows: ∞ E n ) E ( ) ( φ(r , E)Ynm Ω ϕ r , E, Ω = n=0 m=−n

We will consider the one-speed transport equation in one dimension in order to make the problem simpler (look up Chap. 1). The energy dependence is not affected by the Legendre expansion. μ

∂ ϕ(z, μ) + Σt ϕ(z, μ) ∂z {2π {+1 ) ( ) ( = dψ dμ' Σs Ω ' → Ω ϕ z, μ' + S(z, μ)

(3.1)

−1

0

{ 2π where φ(z, μ) = 0 dψφ(z, μ, ψ) assuming no dependence on the azimuthal angle. The expansions are defined as ϕ(z, μ) =

∞ E (2n + 1) n=0

S(z, μ) =

4π

∞ E (2n + 1) n=0

4π

Pn (μ)φn (z)

(3.2)

Pn (μ)Sn (z)

(3.3)

( ) ( ) ( ) Es Ω ' → Ω = Es Ω ' .Ω = Es μ' .μ = Es (μ0 ) =

∞ E (2n + 1) n=0

4π

Esn Pn (μ0 )

The spherical harmonics satisfy the addition theorem let two vectors on the unit sphere having coordinate angles as shown in Fig. 3.1 (such vectors could be the neutron direction before and after scattering (Ω ' , Ω) then, ) ( cosθ0 = cosθ cosθ ' + sinθ sinθ ' cos ϕ − ϕ ' The addition theorem states ( ) Pn (cosθ0 ) = Pn (cosθ )Pn cosθ ' or ( ) Pn (μ0 ) = Pn (μ)Pn μ' + · · ·

3.1 Derivation of the PN Equations in Plane Geometry

63

′

′

′

Fig. 3.1 Coordinates for the addition theorem

where μ0 = cosθ0 . Substituting the first term into the scattering x-section above, we get ∞ E ) ( ( ) (2n + 1) Es Ω ' → Ω = Es (μ0 ) = Esn Pn (μ)Pn μ' 4π

(3.4)

n=0

Substituting 2, 3, and 4 into the transport Eq. (3.1) μ

∞ ∞ E ∂ E (2n + 1) (2n + 1) Pn (μ)φn (z) + Σt Pn (μ)φn (z) ∂z 4π 4π n=0

n=0

{+1

∞ E (2n + 1)

−1

n=0

dμ'

= 2π

+

∞ E (2n + 1) n=0

4π

4π

( ) ( ) Esn Pn (μ)Pn μ' ϕ z, μ'

Pn (μ)Sn (z)

First term: Substituting the recursion relation for the Legendre Polynomials (appendix F) into the first term in order to get rid of μ

64

3

μPn (μ) = ∞ E (2n + 1)

4π

n=0

n n+1 Pn+1 (μ) + Pn−1 (μ) 2n + 1 2n + 1

[μPn (μ)] =

Spherical Harmonics—The PN Method

dφn dz

∞ | dφn 1 E| (n + 1)Pn+1 (μ) + n Pn−1 (μ) 4π dz n=0

The first term on the right side will not be affected if the series starts at n = 1. This will require a modification of the derivative ∞ ∞ 1 E dφn dφn−1 1 E = [(n)Pn (μ)] (n + 1)Pn+1 (μ) 4π dz 4π dz n=0

n=1

∞ 1 E dφn−1 = [(n)Pn (μ)] 4π dz n=0

where we have returned the beginning to n = 0 since the first term in the expansion is zero anyway. The second term on the right side with similar steps is ∞ ∞ | dφn 1 E| dφn+1 1 E n Pn−1 (μ) = [(n + 1)Pn (μ)] 4π dz 4π dz n=0

n=−1

∞ 1 E dφn+1 = [(n + 1)Pn (μ)] 4π dz n=0

Adding terms, we get the first term of the transport equation | ∞ | 1 E dφn−1 dφn+1 + (n + 1)Pn (μ) (n)Pn (μ) 4π dz dz n=0

Second term: As it is. Third term: {1 2π

dμ −1

2π

'

|∞ E (2n + 1) n=0

|∞ E (2n + 1) n=0

4π

4π

| ( ') ) ( Σsn Pn μ Pn (μ) ϕ z, μ' = | {1

Σsn Pn (μ) −1

( ) ( ) dμ' Pn μ' ϕ z, μ'

3.1 Derivation of the PN Equations in Plane Geometry

65

Substituting for the angular flux from 2, we get

2π

|∞ E (2n + 1) 4π

n=0

| {1 Σsn Pn (μ) −1

∞ ( )E (2n + 1) ( ' ) dμ' Pn μ' Pn μ ϕn (z) 4π n=0

( ) In the integrand, Pn μ' multiplies all series terms and only one term is with the same order (the nth term) such that {1 −1

|

| ( ') ( ') 1 (2n + 1) ( ' ) ( ' ) dμ Pn μ P0 μ φ0 (z) + · · · + Pn μ Pn μ φn (z) + · · · 4π 4π '

In this case, we employ the orthogonality property as {1

dμ' Pm (μ)Pn (μ) =

−1

2δmn , δmn = 2n + 1

{

0 m /= n 1m=n

δmn is known as the Kronecker delta. Accordingly, all terms in the integrand must vanish except the nth term {1

dμ'

−1

2 (2n + 1) ( ' ) ( ' ) Pn μ Pn μ φn (z) = φn (z) 4π 4π

The third term becomes ∞ E (2n + 1) n=0

4π

Σsn Pn (μ)φn (z)

Fouth term: As it is. Collecting terms, the transport equation becomes 1 4π

| ∞ | E (n)Pn (μ) dφdzn−1 + (n + 1)Pn (μ) dφdzn+1

n=0

+Σt =

∞ E

(2n+1) 4π Pn (μ)φn (z)

n=0 ∞ E (2n+1) (2n+1) Σ P + (μ)φ (z) sn n n 4π 4π Pn (μ)Sn (z) n=0 n=0 ∞ E

66

3

Spherical Harmonics—The PN Method

Multiplying the equation by 4π Pn (μ), integrating over μ, and using the orthogonality, for example, {1 dμPn (μ)

4π −1

∞ dφn−1 1 E 2n dφn−1 n Pn (μ) = 4π dz 2n + 1 dz n=0

The neutron transport equation NTE becomes n dφn−1 n + 1 dφn+1 + + (Σt − Σsn )φn (z) 2n + 1 dz 2n + 1 dz = Sn (z); n = 0, 1, 2, · · · , N

(3.5)

This is the PN equation. Multiplying Eq. (3.2) by 4π Pn (μ), integrating over μ, and using the orthogonality, we will be able to define φn (z) {+1 4π dμPn (μ)ϕ(z, μ) −1

{+1 ∞ E (2n + 1) = 4π dμPn (μ) Pn (μ)φn (z) 4π −1

n=0

= 2φn (z) or φn (z) = 2π

3.1.1

{ +1

−1 dμPn (μ)ϕ(z, μ)

The PN Approximation

There are N + 1 equations making up Eq. (3.5) (n = 0, 1, 2, . . . , N ), however, it contains N + 2 coefficients φn (z) starting from φ0 (z) knowing that φ−1 (z) = 0 and therefore the number of unknowns exceeds the number of equations by one and they can be made equal using dφ N +1 (z) =0 dz In other words, dφn =0 ; n>N dz And, this is called the PN approximation.

(3.6)

3.1 Derivation of the PN Equations in Plane Geometry

67

In the same fashion, we can write {+1 φ N +1 (z) = 2π dμPN +1 (μ)ϕ(z, μ) −1

PN +1 (μ) oscillates fast at large values of N (see the figure in the appendix of Fig. 6.2), where it changes sign N + 1 times in the interval −1 ≤ μ ≤ 1; hence, we can say that φ N +1 (z) is so small and the PN approximation becomes reasonably accurate. We can define Sn (z) and Σsn using orthogonality and Eqs. (3.3) and (3.4) { Sn (z) = 2π

+1 −1

dμPn (μ)S(z, μ)

{+1 Σsn (z) = 2π dμΣs (μ0 )Pn (μ0 ) −1

Example Derive the P1 equations. Substituting n = 0, 1 in Eq. (3.5), we get dφ1 + (Σt − Σs0 )φ0 (z) = S0 (z) dz 1 dφ0 2 dφ2 + + (Σt − Σs1 )φ1 (z) = S1 (z) 3 dz 3 dz According to the PN approximation Eq. (3.6): we require is greater than the P1 order of (1). The P1 equations are

dφ2 (z) dz

= 0 because 2

dφ1 + (Σt − Σs0 )φ0 (z) = S0 (z) dz 1 dφ0 + (Σt − Σs1 )φ1 (z) = S1 (z) 3 dz where {+1 {+1 φ0 (z) = 2π dμP0 (μ)ϕ(z, μ) = 2π dμϕ(z, μ) = φ(z) −1

−1

{+1 {+1 S0 (z) = 2π dμP0 (μ)S(z, μ) = 2π dμS(z, μ) = S(z) −1

−1

(3.7)

68

3

Spherical Harmonics—The PN Method

{+1 {+1 Σs0 (z) = 2π dμΣs (μ0 )P0 (μ0 ) = 2π dμΣs (μ0 ) = Σs −1

−1

are the flux, source strength, and scattering x-section, respectively. The last equation can be explained as follows if we write: ( ) Σs (μ0 ) = Σs R(μ0 ) = Σs R μ' · μ ( ) where R μ' · μ is the probability that the neutron will scatter between the angles { +1 ( ) θ, θ ' and normalized to 1 as 2π −1 dμR μ' · μ = 1. Σs0 (z) = 2π

+1 { −1

dμΣs (μ0 )P0 (μ0 ) = 2π +1 {

+1 { −1

( ) dμΣs R μ' · μ P0 (μ0 )

( ) = 2π Σs dμR μ' · μ [1] −1 | | +1 { ( ' ) = Σs 2π dμR μ · μ = Σs [1] = Σs −1

The other quantities in the P1 equations are {+1 {+1 φ1 (z) = 2π dμP1 (μ)ϕ(z, μ) = 2π dμμϕ(z, μ) = J (z) −1

−1

{+1

{+1 dμP1 (μ)S(z, μ) = 2π dμμS(z, μ) = u(z)

S1 (z) = 2π −1

−1

{+1 {+1 Σs1 (z) = 2π dμ0 Σs (μ0 )P1 (μ0 ) = 2π dμ0 μ0 Σs (μ0 ) −1

⎤ {+1 ( ' ) = Σs ⎣2π dμ0 μ0 R μ · μ ⎦ = μ0 Σs ⎡

−1

−1

The net current, (physically un-defined source), and the scattering x-section multiplied by the average cosine of the scattering angle, respectively, Σs1 (z) can be understood if we define: { +1 2π −1 dμ0 μ0 R(μ0 ) μ0 = { +1 2π −1 dμ0 R(μ0 )

3.1 Derivation of the PN Equations in Plane Geometry

If the source is isotropic such that S(z, μ) = equation for S1 (z), we get

69 S(z) 2 ,

so by substitution into the

{+1 {+1 S(z) S(z) S1 (z) = 2π dμ μ dμ μ = 0 = 2π 2 2 −1

−1

Question: Why not considering the flux to be isotropic? Answer: If so, there will be no need for the PN method. It is introduced to eliminate the direction variable. Rewriting the P1 equations, dJ + (Σt − Σs )φ(z) = S(z) dz 1 dφ + (Σt − μ0 Σs )J (z) = 0 3 dz

(3.8)

using Σt − Σs = Σa and Σt − μ0 Σs = Σtr Eq. (3.8) becomes dJ + Σa φ(z) = S(z) dz 1 dφ + Σtr J (z) = 0 3 dz

(3.9)

From the second equation, J (z) = −

1 dφ 3Σtr dz

Substituting in the first equation, we get | | 1 dφ d − + Σa φ(z) = S(z) dz 3Σtr dz If the diffusion coefficient is defined as D = diffusion equation: −D

1 3Σtr

d 2φ + Σa φ(z) = S(z) dz 2

=

λtr 3

,we arrive at the

(3.10)

Note: The P1 Eq. (3.8) are reduced to the one-speed diffusion equation by the virtue of the assumption that the source is isotropic. Let us take a look at the flux expansion Eq. (3.2) when the P1 approximation is considered: ϕ(z, μ) =

1 E (2n + 1) n=0

4π

Pn (μ)φn (z)

70

3

Spherical Harmonics—The PN Method

1 3 P0 (μ)φ0 (z) + P1 (μ)φ1 (z) 4π 4π 1 = [φ(z) + 3μJ (z)] 4π

=

Recall the definition of the partial currents from Chap. 1 and substitute from the expansion {±1 {±1 1 J (z) = 2π dμμϕ(z, μ) = 2π dμμ [φ(z) + 3μJ (z)] 4π ±

0

1 2

0

{±1

1 dμμ[φ(z) + 3μJ (z)] = 2

0

J ± (z) =

|

{±1 | | dμ μφ(z) + 3μ2 J (z) 0

φ(z) J (z) ± 4 2

|

From which, the net current is J (z) = J + (z) − J − (z), and using the diffusion approximation, J (z) |= −D dφ(z) dz

J ± (z) =

3.2

φ(z) 4

∓

D dφ(z) 2 dz

|

(3.11)

Derivation of the PN Equations in Spherical Geometry

Consider the transport equation in spherical geometry (problem 7 in Chap. 1). Derivation of the PN form follows a similar procedure as in the plane geometry. We start by the substitution of the expansions of the flux, and anisotropic scattering cross section in Legendre polynomials as we have done before. Notice that fission is isotropic by nature and hence the fission cross section is not a function of μ. The fission source in one-speed form is just [1, 3] ( ) νΣ f S f r, t = 4π

{

( ) dΩ ' ϕ r , Ω ' , t

4π

And assuming symmetry in the azimuthal angle and steady state, then ( ) νΣ f Sf r = 2

{+1 ( ) dμ' ϕ r , μ' −1

3.2 Derivation of the PN Equations in Spherical Geometry

71

Upon substitution of the above expansions, we get ∞ E (2n + 1) ∂φn

+ Σt

∞ E n=0

=

∞ E n=0

+

∂r

4π

n=0

∞ E ) d Pn (μ) (2n + 1) φn (r ) ( 1 − μ2 4π r dμ

μPn (μ) +

n=0

(2n + 1) Pn (μ)φn (r ) 4π

(2n + 1) Esn Pn (μ) 4π

{+1 ( ) ( ) dμ' Pn μ' ϕ r , μ'

−1

νΣ f 2

{+1

∞ E (2n + 1)

−1

n=0

dμ'

4π

( ) φn (r )Pn μ'

(3.12)

Consider the recursion relation: μPn (μ) =

n n+1 Pn+1 (μ) + Pn−1 (μ) 2n + 1 2n + 1

and the recursion relation ( ) ∂ Pn (μ) n(n + 1) 1 − μ2 = [Pn−1 (μ) − Pn+1 (μ)] ∂μ (2n + 1) Substituting the last two recursion relations in Eq. (3.12), +

| ∞ E (2n + 1) φn (r ) n(n + 1) | n=0

4π

(2n + 1)

r

Pn−1 (μ) − Pn+1 (μ)

| |

| | ∞ E n (2n + 1) ∂φn n + 1 Pn+1 (μ) + Pn−1 (μ) 4π ∂r 2n + 1 2n + 1 n=0

+ Σt

∞ E (2n + 1)

4π

n=0

=

∞ E (2n + 1) n=0

4π

Pn (μ)φn (r )

Σsn Pn (μ)φn (r )

{ ∞ ( ) νΣ f E (2n + 1) + φn (r ) dμ' Pn μ' 2 4π +1

n=0

−1

72

3

Spherical Harmonics—The PN Method

Collecting similar terms | | ∞ ∂φn 1 E φn (r ) −n (n + 1)Pn+1 (μ) 4π ∂r r n=0 | | ∞ ∂φn 1 E φn (r ) n Pn−1 (μ) + + (n + 1) 4π ∂r r n=0

+ Σt

∞ 1 E (2n + 1)Pn (μ)φn (r ) 4π n=0

=

1 4π

∞ E

(2n + 1)Σsn Pn (μ)φn (r )

n=0

{ ∞ ( ) 1 νΣ f E + (2n + 1)φn (r ) dμ' Pn μ' 4π 2 +1

n=0

−1

We change the indexes of the polynomials Pn±1 to Pn and the rest of the indexes as well | | ∞ ∂φn−1 1 E φn−1 (r ) n Pn (μ) − (n − 1) 4π ∂r r n=0 | | ∞ ∂φn+1 1 E φn+1 (r ) + + (n + 2) (n + 1)Pn (μ) 4π ∂r r n=0

+ Σt

∞ 1 E (2n + 1)Pn (μ)φn (r ) 4π n=0

∞ 1 E = (2n + 1)Σsn Pn (μ)φn (r ) 4π n=0

{ ∞ ( ) 1 νΣ f E + (2n + 1)φn (r ) dμ' Pn μ' 4π 2 +1

n=0

−1

Multiplying by 4π Pn (μ), integrating over μ and making use of orthogonality as before, however, we first treat the fission term separately and we imbed P0 (μ) inside the summation and inside the integral since it is equal to one and will have no effect 4π νΣ f 4π 2

{+1 {+1 ∞ E ( ) ( ) dμPn (μ) (2n + 1)P0 (μ)φn (r ) dμ' P0 μ' Pn μ' −1

n=0

−1

3.2 Derivation of the PN Equations in Spherical Geometry

73

The orthogonality requires that {+1 ( ) ( ) 2δon dμ' P0 μ' Pn μ' = = 2 since δon = 1 for n = 0 2n + 1

−1

Therefore, the fission term becomes 2νΣ f ϕ0 (r ) = 2νΣ f ϕ(r ) | | ∂φn−1 2n φn−1 (r ) − (n − 1) 2n + 1 ∂r r | | 2(n + 1) ∂φn+1 φn+1 (r ) + 2(Σt )φn (r ) + + (n + 2) 2n + 1 ∂r r = 2Σsn φn (r ) + 2νΣ f φ(r ) Multiplying by 2n + 1 and dividing by 2, we obtain | | | | ∂ϕn+1 ∂ϕn−1 ϕn−1 (r ) ϕn+1 (r ) + (n + 1) n − (n − 1) + (n + 2) ∂r r ∂r r + (2n + 1)(Σt − Σsn )φn (r ) = (2n + 1)νΣ f φ(r ) Rearranging, we obtain the PN equation for a spherical reactor | | d (n − 1) φn−1 (r ) n − dr r | | d (n + 2) φn+1 (r ) + (n + 1) + dr r +(2n + 1)(Σt − Σsn )φn (r ) = (2n + 1)νΣ f φ(r ); n = 0, 1, 2, . . . , N

(3.13)

Finally, we obtain the P1 equations as ) ( d 2 J (r ) + Σa φ(r ) = νΣ f φ(r ) + dr r dφ(r ) + 3(Σt − Σs1 )J (r ) = 3νΣ f φ(r ) dr Substituting for the cross sections in the second equation, we get (d dr

)

2 r J (r ) + Σa φ(r ) = νΣ f φ(r ) 1 dφ(r ) 3 dr + Σtr J (r ) = νΣ f φ(r )

+

(3.14)

74

3

Spherical Harmonics—The PN Method

The free surface condition (follows in the next section) is as in the planar geometry in addition to the zero net current condition at the center of the sphere, i.e. J − (R) = 0 J (0) = 0 where R is the radius of the sphere.

3.3

Boundary and Interface Conditions

3.3.1

Free Surface Boundary Condition

Diffusion theory: When the medium is bounded by a void or a black absorber, where the neutrons escape and are not able to come back, we can either use the ˜ = 0) or use a zero extrapolated boundary at which the neutron flux vanishes (φ(c) negative partial current at the boundary z = c (J − (c) = 0) as shown in Fig. 3.2. Applying the condition at the boundary (Eq. 3.11), | φ(c) D dφ(z) || J (c) = =0 + 4 2 dz |z=c −

Fig. 3.2 Free surface boundary condition

Multiplying medium void Diffusion flux

The true flux

3.3 Boundary and Interface Conditions

75

or | dφ(z) || φ(c) =− dz |z=c 2D

(3.15)

In order to estimate the extrapolated distance z 0 , we extend the diffusion flux (with a negative slope near the boundary) until it crosses the z-axis. From the figure, the slope would be | dφ(z) || φ(c) =− dz |z=c z0

(3.16)

Substituting 3.15 into 3.16, we get −

φ(c) φ(c) =− 2D z0

Therefore, z 0 = 2D = 23 λtr = 0.666λtr as compared to z 0 = 0.7104λtr from transport calculation which indicates the invalidity of diffusion theory near the boundary and the true flux is as shown should remain constant simply because nothing should cause the flux to vanish. P1 approximation: For the PN equations in general, the one choice of the boundary conditions would be the Marshak boundary conditions Referring to Fig. 3.3, a semi-infinite slab surrounded by vacuum, +1 { 0 {0 −1

dμ ϕ(0, μ)Pn (μ) = 0 for n = 1, 3, . . . , N odd dμ ϕ(a, μ)Pn (μ) = 0

Fig. 3.3 Free surface boundary condition (P1 approx.)

Multiplying medium void

void

76

3

Spherical Harmonics—The PN Method

In particular, for the P1 approximation, the zero incoming current is applied at the free surface J + (0) = 2π

−

{+1 {+1 1 dμμϕ(0, μ) = 2π dμμ [φ(0) + 3μJ (0)] = 0 4π 0

0

{0

{0 dμμϕ(a, μ) = 2π

J (a) = 2π −1

dμμ −1

1 [φ(a) + 3μJ (a)] = 0 4π

Integrating over μ leads to J (0) = −

φ(0) φ(a) ; J (a) = 2 2

Or more generally, n. J = φ2 , where n is the outward normal vector. For vacuum (free surface) boundary, it is convenient and sufficiently accurate for most P1 calculations simply to set φ(0) = φ(a) = 0 at the extrapolation distance on both sides [3]. Notice: We have used the same condition in the diffusion theory with the definition of the incoming current using the diffusion approximation. An alternative choice known as the Mark boundary conditions sets ϕ(0, μi ) = ϕ(a, −μi ) = 0, i = 1, 2, . . . ,

N +1 2

where μi are the positive zeros of PN +1 (μ) = 0.

3.3.2

Interface Condition in Plane Interface

At the interfaces between different regions with different nuclear properties where the cross sections change, the angle-independent flux derived from diffusion theory is always continuous across interfaces because neutrons diffuse either way due to neutron density difference until equilibrium is attained at the interface. However, the angular flux at the planar interface will be discontinuous at μ = 0 and continuous otherwise. It is explained as follows: Firstly, the flux is said to be continuous only when the fluxes have equal values at the interface. Secondly, considering Fig. 3.4a, both fluxes are at positions on both sides of the interface and in the same direction (μ /= 0); however, they originate from the same medium A and therefore they are equal (continuous).

3.3 Boundary and Interface Conditions

77

Medium

Medium

Medium Medium

(a)

(b)

Fig. 3.4 a Both fluxes originated from medium A. b Fluxes originated from media A and B

Thirdly, considering Fig. 3.4b, both fluxes are at positions on both sides of the interface and in opposite directions (μ /= 0); however, they originate from different media A and B and therefore they are not equal (discontinuous). Fourthly, considering Fig. 3.5a, both fluxes are at positions on both sides of the interface and in the same direction (μ = 0); however, they originate from different media A and B and therefore they are not equal (discontinuous).

Medium Medium Medium

Medium

(a)

(b)

Fig. 3.5 a Fluxes originating from media A and B. b Both fluxes originating from media B

78

3

3.3.3

Spherical Harmonics—The PN Method

Interface Condition in Curved Interface

A simple and obvious example of discontinuity in the angular flux for μ = 0 at an interface arises in the case of a free (planar) surface. If, in the figures, medium B on the right produces neutrons, while medium A is a vacuum, it follows that φ B (z 0 , μ) is finite for all values of μ > 0, but is zero for all μ < 0. Clearly, there must be a discontinuity in the angular flux for μ = 0 at a free surface. By applying the arguments developed above it can be shown that at a curved interface, the angular flux will not be discontinuous as a function of direction, μ. Considering Fig. 3.5b for a curved boundary, both fluxes are at positions on both sides of the interface and in the same direction (μ = 0) and they originate from the same media B, therefore they are equal, hence continuous. Although the angular flux is not discontinuous at μ = 0, its derivative with respect to μ is discontinuous and the flux may change rapidly with μ near μ = 0.

3.3.4

Yvon’s Method

When separate Legendre expansions are used for the two half-ranges in μ at a plane interface, the treatment is known as the double-PN approximation or J. J. Yvon’s method. In this approximation, it is possible to satisfy the free surface boundary condition exactly and also to allow for discontinuities at interfaces. As a result, the method is remarkably accurate in plane geometry. In order to examine the double-PN equations, a time-independent, one-speed transport problem will be considered in-plane geometries with no source, i.e. Eq. (3.1); thus, [1, 4] μ

∂ ϕ(z, μ) + Σt ϕ(z, μ) ∂z

=

∞ E (2k + 1) k=0

2

{+1 ( ) ( ) Esk Pk (μ) dμ' Pk μ' ϕ z, μ'

(3.17)

−1

In the double PN approximation, it is assumed that φ(z, μ) =

N E

(2n + 1)[φn+ (z)Pn+ (2μ − 1)

n=0

+ φn− (z)Pn− (2μ + 1)]

(3.18)

With the following definitions, Pn+ (2μ − 1) = Pn (2μ − 1) =0

μ≥0 μ0

n.Ω>0

in spherical coordinates n.Ω > 0(cos θ +)and n.Ω < 0 (cos θ −), where n.Ω = cosθ . Therefore, the integral covers up and down or right and left, i.e. inward and

4.3 The Adjoint Neutron Diffusion Equation

107

outward directions. n is the normal to the differential surface on the volume from which the neutron current passes. Applying the divergence theorem, we get ˚

˚ ∇.Ωϕϕ ∗ d V d Edy = − n.Ωϕϕ ∗ d Sd Edy ⎫ ⎧ ⎪ ⎪ { ¨ ⎨ { ( ) ⎬ ) ( ∗ ∗ ϕ n.Ωϕ d S + ϕ n.Ωϕ d S =− d Edy ⎪ ⎪ ⎭ ⎩

/\ = −

n.Ω0

The first term represents the inward current which is zero (b.c) and the second term represents the outward importance equal to zero because it is lost (b.c). Therefore /\ = 0 which satisfies the requirement. The adjoint operator L ∗ will not be self-adjoint even if only one term in the operator is not self-adjoint or even only the sign is opposite to the sign of the corresponding term. Now we are in the position to examine the adjoint equations to some familiar forms of the neutron transport equation.

4.3

The Adjoint Neutron Diffusion Equation

We recall the equations from Chap. 1 without going into details of their derivations.

4.3.1

The Energy-Dependent Neutron Diffusion Equation (DE) ( ( ) ( ) ) ( ) − ∇ · D r , E ∇φ r , E + Σt r , E φ r , E {∞ ( ) ) ( − d E ' Σs r , E ' → E φ r , E ' 0

{∞ − χ (E)

) ( ) ( ) ( d E 'v E ' Σ f r , E ' φ r , E ' = 0

0

(4.13) ( ( ) ( ) ( ) ) ( ) L di f f φ r , E = −∇ · D r , E ∇φ r, E + Σt r , E φ r , E {∞ ( ) ) ( − d E ' Σs r , E ' → E φ r , E ' 0

{∞ − χ (E) 0

) ( ) ( ) ( d E 'v E ' Σ f r , E ' φ r , E '

108

4 The Adjoint Transport Equation—The Equation of Neutron Importance

) ( ) ( where φ r , E = ∫ ϕ r , E, Ω dΩ. By comparison with the transport equation 4π

(TE) we may deduce the adjoint operator for the (DE) such that 1. The transfer terms’ scattering and fission are similar. 2. The directional dependence is absent in the DE because it is based on the assumption of isotropic scattering and flux. 3. The boundary conditions of the outgoing zero importance and the non-reentrant surface made the streaming term not self-adjoint in the TE. 4. In the DE, the conditions in (3) are not applicable. Therefore, the streaming term is self-adjoint, hence we can write ( ( ( ) ( ) ) ) ( ) L ∗di f f φ ∗ r , E = −∇ · D r , E ∇φ ∗ r , E + Σt r , E φ ∗ r , E {∞ −

( ) ( ) d E ' Σs r , E → E ' φ ∗ r , E '

0

) ( − v(E)Σ f r , E

{∞

( ) ( ) d E 'χ E ' φ∗ r , E '

0

4.3.2

The One-Speed Neutron Diffusion Equation

Recall from Chap. 2 the one-speed (one-group) neutron diffusion equation ( ) ( ) ( ) ( ) ( ) ( ) (4.14) −∇ · D r ∇φ r + Σa r φ r = vΣ f r φ r The operator form is ( ) ( ) ( ) ( ) ( ) ( ) ( ) L 1speed φ r = −∇ · D r ∇φ r + Σa r φ r − vΣ f r φ r And the adjoint form can just be written down as ( ) ( ) ( ) ( ) ( ) ( ) ( ) L ∗1speed φ ∗ r = −∇ · D r ∇φ ∗ r + Σa r φ ∗ r − vΣ f r φ ∗ r Notice that to) the transport equation ( ( at the ) left ( the)boundary condition applied ∗ r , y = 0 for y < 0 or ϕ ∗ r , y = 0 y > 0 and ϕ boundary ϕ r 0 , y ( = 0 for 0 0 ) for y < 0 and ϕ ∗ r 0 , y = 0 for y < 0. The unit vector y causes the streaming term in the TE to be of opposite sign. This theory ( is not ( in) the diffusion ( ) ( ) ) the case j r , y r , y and J r r = yϕ = yφ where the flux is isotropic. Notice also ) ( ) ( are both vectors and ϕ r , y and φ r are both scalars. The boundary conditions to the DE are ) ) ( ( J− r s , E = 0 and J+∗ r s , E = 0

4.3 The Adjoint Neutron Diffusion Equation

109

Proof Applying requirement II to DE and its adjoint: ( ∗ ) ) ( φ , −∇ · D∇φ + Σa φ − vΣ f φ − φ, −∇ · D∇φ ∗ + Σa φ ∗ − vΣ f φ ∗ The absorption and fission terms are clearly self-adjoint and no need to show the proof. We prove only the streaming term: {

−φ ∗ ∇ · D∇φ +

{

φ∇ · D∇φ ∗

Integration by parts for the two integrals, first integral is u = −φ ∗ ; dv = ∇ · D∇φ du = −∇φ ∗ ; v = D∇φ Apply the identity ∫ −φ ∗ ∇ · D∇φ = uv − ∫ vdu = −φ ∗ D∇φ + ∫ D∇φ∇φ ∗ d V −φ ∗ D∇φ + φ ∗ D∇φ = 0 and the second integral is likewise. Therefore, the streaming term is self-adjoint and hence the one-speed diffusion equation.

4.3.3

The Multi-Group Neutron Diffusion Equation

The steady-state multi-group DE with no up-scattering ( ) ( ) ( ) 1 EG ( ) ( ) ( ) ν ' E f g ' r φg ' r − ∇ · Dg r ∇φg r + Σ Rg r φg r = χg ' =1 g g k G E ( ) ( ) Esg' g r φg' r , g = 1, 2, 3, · · · , G + g ' /=g;g ' >g

where E Rg = Etg − Esgg = Eag + Esg − Esgg = Eag + In a matrix form: A8 =

1 F8 k

where A and F and 8 are given explicitly as

G E g ' =1

( ) Esgg' r − Esgg

110

4 The Adjoint Transport Equation—The Equation of Neutron Importance

⎡

−∇ · D1 ∇ + E R1 0 0 ⎢ −Es12 −∇ · D2 ∇ + E R2 0 ⎢ ⎢ ⎢ −Es13 −Es23 −∇ · D3 ∇ + E R3 A=⎢ ⎢ . . . ⎢ . . . ⎣ . . . −Es2G −Es3G −Es1G

⎡

χ1 v1 E f 1 χ1 v2 E f 2 ⎢ χ2 v1 E f 1 χ2 v2 E f 2 ⎢ ⎢ F = ⎢ χ3 v1 E f 1 χ3 v2 E f 2 ⎢ .. .. ⎣ . . χ G v1 E f 1 χ G v2 E f 2

χ1 v3 E f 3 χ2 v3 E f 3 χ3 v3 E f 3 .. .

χ G v3 E f 3

... ... ... .. .

0 0 0 . . . . . . −∇ · DG ∇ + E RG

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

⎡ ⎤ ⎤ φ1 · · · χ1 v G E f G ⎢ φ2 ⎥ · · · χ2 v G E f G ⎥ ⎢ ⎥ ⎥ ⎥ ⎥ · · · χ 3 v G E f G ⎥; o = ⎢ ⎢ φ3 ⎥ ⎢ ⎥ ⎥ . . .. .. ⎣ .. ⎦ ⎦ . · · · χG vG E f G φG

Reversing the energy transfer terms, we can readily obtain the adjoint form ( ) ( ) ( ) ( ) 1 ( ) ( ) EG χg' φg∗' r − ∇ · Dg r ∇φg∗ r + Σ Rg r φg∗ r = νg E f g r g ' =1 k G E ( ) ( ) Esgg' r φg∗' r , g = 1, 2, 3, · · · , G + g/=g ' ;g>g '

And the matrix form is (notice the same eigenvalue) A∗ o∗ =

1 ∗ ∗ F o k

where A∗ and F∗ are just the transpose of the matrices A and F, respectively, and o∗ /= 8, thus ⎡ ⎢ ⎢ ⎢ ⎢ ∗ A =⎢ ⎢ ⎢ ⎣

⎡

−∇.D1 ∇ + E R1 −Es12 −Es13 0 −∇.D2 ∇ + E R2 −Es23 0 0 −∇.D3 ∇ + E R3 . . . . . . . . . 0 0 0

χ 1 v1 E f 1 χ2 v1 E f 1 ⎢ χ1 v2 E f 2 χ2 v2 E f 2 ⎢ ⎢ F ∗ = ⎢ χ1 v3 E f 3 χ2 v3 E f 3 ⎢ .. .. ⎣ . . χ1 v G E f G χ2 v G E f G

χ3 v1 E f 1 χ3 v2 E f 2 χ3 v3 E f 3 .. .

χ3 v G E f G

··· ··· ··· .. .

−Es1G −Es2G −Es3G . . . . . . −∇.DG ∇ + E RG

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

⎡ ∗⎤ ⎤ φ1 · · · χ G v1 E f 1 ⎢ φ∗ ⎥ · · · χ G v2 E f 2 ⎥ ⎢ 2⎥ ⎥ ⎢ ∗⎥ · · · χ G v3 E f 3 ⎥ ⎥; o∗ = ⎢ φ3 ⎥ ⎢ . ⎥ ⎥ .. .. ⎣ .. ⎦ ⎦ . . ∗ · · · χG vG E f G φG

4.4 The Adjoint One-Speed P 1 Equations

4.3.4

111

The Two-Group Neutron Diffusion Equation

The matrix form is straightforward from the multi-group DE and its adjoint. The forward equation is |

|| | −∇.D1 ∇ + E R1 φ1 0 −∇.D2 ∇ + E R2 φ2 −Es12 | || | 1 χ1 ν1 E f 1 χ1 ν2 E f 2 φ1 = k χ2 ν1 E f 1 χ2 ν2 E f 2 φ2

And the adjoint equation is |

|| ∗ | φ1 −Es12 −∇.D1 ∇ + E R1 0 −∇.D2 ∇ + E R2 φ2∗ | || | 1 χ1 ν1 E f 1 χ2 ν1 E f 1 φ1∗ = k χ1 ν2 E f 2 χ2 ν2 E f 2 φ2∗

Most of the fission neutrons are born in the fast energy range, therefore χ1 = 1 and χ2 = 0. Also, from the definition of the removal of x-section E R1 = Ea1 + Es12 and E R2 = Ea2 + Es21 , however, Es21 = 0 because we have ignored up-scattering, therefore E R2 = Ea2 and we can rewrite the equations again |

| || | || | 1 ν1 E f 1 ν2 E f 2 φ1 −∇.D1 ∇ + E R1 φ1 0 = −∇.D2 ∇ + Ea2 φ2 0 0 φ2 −Es12 k | | || ∗ | || ∗ | 1 ν1 E f 1 0 φ 1 −∇.D1 ∇ + E R1 φ1 −Es12 = 0 −∇.D2 ∇ + Ea2 φ2∗ k ν2 E f 2 0 φ2∗

Calculation of fluxes and adjoints can be performed using either deterministic methods or Monte Carlo methods through well known codes. One can use a onedimensional or two-dimensional code such as ONED or 2DB to obtain fluxes and adjoints of a reflected core and plot them to see the major differences between them. Referring to Fig. 4.5, we firstly notice the differences between fast and thermal fluxes, especially at the reflector peak and the higher fast flux in the core. Secondly, the differences between the fast and thermal adjoints where the adjoint of thermal neutrons is higher than the adjoint of fast neutrons indicating higher importance since it is more likely to induce fission as we have discussed earlier [5, 6, 7].

4.4

The Adjoint One-Speed P 1 Equations ( ) ( ) ( ) ∇ · J r + (Σt − Σs )φ r = S0 r ( ) ( ) ( ) 1 ∇φ r + (Σt − μ0 Σs )J r = S1 r 3

112

4 The Adjoint Transport Equation—The Equation of Neutron Importance Fast flux

Fast Adjoint

Thermal Adjoint

Thermal flux

0.035

0.010 0.008

0.030

0.006 0.025 0.004 0.020

Thermal flux

fast flux and therm al & fast adjoints

0.012

0.002

0.015

0.000 0

4

8

12

16

hight (cm) Fig. 4.5 Relative two-group flux and adjoint for the Tajoura Nuclear Research Center Research Reactor (TNRCRR)

which reduces to ( ) ( ) ( ) ∇ · J r + Σa φ r = S r

(4.15)

( ) ( ) ( ) 1 ∇φ r + Σtr J r = S1 r 3

(4.16)

They satisfy the boundary condition (from Chap. 3) J (0) = −

φ(0) φ(a) ; J (a) = 2 2

or n.J =

φ 2

We can use the zero reentrant partial current boundary condition (J− = 0) and zero outgoing adjoint (J+∗ = 0) since the partial currents are angle integrated so we arrive at the adjoint form of Eq. 4.15 and Eq. 4.16 as ( ) ( ) ( ) −∇ · J ∗ r + Σa φ ∗ r = S ∗ r

(4.17)

113

( ) ( ) ( ) 1 − ∇φ ∗ r + Σtr J ∗ r = S1∗ r 3

(4.18)

They satisfy the boundary condition J ∗ (0) = +

φ∗ φ ∗ (0) φ ∗ (a) ; J ∗ (a) = − and n.J ∗ = − 2 2 2

Proof Use requirement (2) and the following steps: (i) assume isotropic sources (S1 = 0) and (S = 0); (ii) multiply Eq. 4.15 by φ ∗ and Eq. 4.16 by J ∗ ; (iii) multiply (4.17) by φ and (4.18) by (J ); (iv) subtract iii from ii; and (v) add the resulting equations. Prove that /\ = 0. Steps (i), (ii), (iii) | ( ) | ( ) ( )| ( )| φ ∗ ∇ · J r + Σa φ r ; J ∗ ∇φ r + 3Σtr J r and ( ) ( )| | ( ) ( )| | φ −∇ · J ∗ r + Σa φ ∗ r ; J −∇φ ∗ r + 3Σtr J ∗ r Subtract | ( ) | ( ) ( )| ( )| φ ∗ ∇ · J r + Σa φ r − φ −∇ · J ∗ r + Σa φ ∗ r | ( ) | ( ) ( )| ( )| J ∗ ∇φ r + 3Σtr J r − J −∇φ ∗ r + 3Σtr J ∗ r Add the two equations {

| | | | φ ∗ [∇ · J ] + J ∗ [∇φ] + φ ∇ · J ∗ + J ∇φ ∗ d V { = ∇ · [φ ∗ J ]+∇·[φ J ∗ ]d V { = ∇ · [φ ∗ J ]+∇·[φ J ∗ ]d V

Using the divergence theorem {

∗

∗

∇ · [φ J ]+∇·[φ J ]d V =

{

) ) | |( ( n · J φ∗ + n · J ∗ φ d S

Using the boundary conditions above { |

| φφ ∗ φ∗φ dS = 0 − 2 2

114

4 The Adjoint Transport Equation—The Equation of Neutron Importance

4.5

Derivation of the Kinetic Parameters

4.5.1

Transport Model

In this section, we establish the methodology for a generic definition of the kinetic parameters used in the kinetic equations by weighing such parameters on the adjoint (importance function). The kinetic parameters are going to be derived from the neutron transport and the adjoint equations [8]. The transport equation is written explicitly for the prompt and delayed neutrons as (see Chap. 1) ( ) ( ( ) ) 1 ∂ϕ = −Ω · ∇ϕ r , E, Ω, t − Σt r , E, Ω ϕ r , E, Ω, t υ ∂t {∞ { ) ( ( ) ' dΩ ' Σs r , E ' → E, Ω ' → Ω ϕ r , E ' , Ω ' , t + dE 0

4π

{ + (1 − β)χ (E) ×

dΩ 4π

+

6 E

'

{∞

( ) ( ) ) ( d E 'ν E ' Σ f r , E ' ϕ r , E ', Ω ', t

0

( ) λi Ci r , t χi (E)

(4.19)

i=1

The last term is the number of delayed neutrons produced from precursor decay and the preceding term represents the prompt fission neutrons. The familiar precursor equation from previous knowledge of reactor kinetics is ∂ ( ) Ci r , t = ∂t

{

dΩ '

{

) ( ( ) ( ) ( ) d E ' βi v E ' Σ f r , E ' ϕ r , E ' , Ω ' , t − λi Ci r , t

4π

(4.20) The adjoint equation for a critical unperturbed reference reactor can be written in analogy to the above transport Eq. 4.19 as ( ) ( ) ) ( − Ω · ∇ϕ0∗ r , E, Ω + Σt r , E ϕ0∗ r , E, Ω {∞ { ( ) ( ) − d E ' dΩ ' Σs r , E → E ' , Ω → Ω ' ϕ0∗ r , E ' , Ω ' 0

4π

( ) 1 = ν(E)Σ f r , E k +

6 E i=1

(

βi χi E

'

)

{

{∞

| ( ) d E ' (1 − β)χ E '

0

dΩ ' ϕ0∗

) ( r , E ', Ω '

| (4.21)

4.5 Derivation of the Kinetic Parameters

115

The overall system is subject to the boundary conditions discussed before. The derivation proceeds as follows: ( ) ∗ Equation 4.19 is multiplied ( by ϕ0 )r , E, Ω and integrated over r , E, Ω and Eq. 4.21 is multiplied by ϕ r , E, Ω, t and integrated over r , E, Ω then the two are subtracted giving {

{ ( ) ϕ0∗ ϕ d V d Edy = − Ω · ∇ ϕ0∗ ϕ d V d Edy υ { { ( )| | k−1 d E ' dΩ ' ϕ0∗ r , E ' , Ω ' X ' + k ˚ ( ) ( ) × ν(E)Σ f r , E ϕ r , E, Ω, t d V d Edy

∂ ∂t

{ −

dΩ '

4π

{∞

i=1

0

˚

( ) ( ) ν(E)Σ f r , E ϕ r , E, Ω, t d V d Edy

× +

6 E

| 6 | E ( ) ( ) d E ' ϕ0∗ r , E ' , Ω ' βi χi E '

{ λi

( ) ( ) ϕ0∗ r , E, Ω Ci r , t χi (E)d V d Edy

(4.22)

i=1

where

X'

| = (1 − β)χ

(

E'

)

+

6 E

βi χi

(

E'

)

| .

i=1

The total reaction rate terms will cancel each other by subtraction and the scattering terms are like so since integration over E and E ' will remove the difference between the two terms and the direction of energy transfer wouldn’t matter.) The ( , E, Ω, t = 0 ϕ r boundary conditions on the exterior of the reactor volume, i.e. s ( ) for n · Ω < 0 and ϕ ∗ r s , E, Ω, t = 0 for n · Ω > 0 can be applied if the integration over r extends to include the whole reactor core. As a result, the steaming term will disappear. {

( ) Ω · ∇ ϕ0∗ ϕ d V d Edy =

{

( ) Ω · ϕ∇ϕ0∗ + ϕ0∗ ∇ϕ d V d Edy = 0

(4.23)

( ) ( ) where we have used the identity ∇ ϕ0∗ ϕ = ϕ∇ ϕ0∗ + ϕ0∗ ∇(ϕ). However, if this integration is limited to a subregion of the reactor as in the case of core and reflector regions, there is no reason for this term to disappear. For example, at the core–reflector interface, a fraction of neutrons leaving the core will re-enter the core at a later time, hence their importance is non-zero. The shape function: The angular flux is separated into an amplitude factor and a shape function: ) ( ) ( ϕ r , E, Ω, t = n(t)ψ r , E, Ω, t

(4.24)

116

4 The Adjoint Transport Equation—The Equation of Neutron Importance

For the following cases, where the point reactor model is accurate, hence, the shape function is not strongly varying function with time and it can be calculated using eigenvalue calculations: 1 The transients have died out. 2 The reactor is not anywhere near or above prompt critical. 3 The reactor is near critical and the perturbations are small. Hence, we can write ) ( ) ( ψ r , E, Ω, t ∼ = ψ r , E, Ω

4.5.2

(4.25)

One-Node Model

The core and reflector are considered as one node; therefore, the integration is over the reactor volume. First, we examine the left side of Eq. 4.22 by substituting for the flux in Eq. 4.24 ∂ ∂t

{

{ ϕ0∗ ϕ n(t)ψϕ0∗ ∂ d V d Edy = d V d Edy υ ∂t υ { { ψϕ0∗ ψϕ0∗ ∂n ∂ = d V d Edy + n(t) d V d Edy ∂t υ ∂t υ

Using the approximation in Eq. 4.25, the integration in the last term in the substitution above is constant with time and hence the derivative is zero { ∗ { ϕ0 ϕ ψϕ0∗ ∂ ∂n d V d Edy = d V d Edy ∂t υ ∂t υ Substituting in Eq. 4.22 and dividing the equation by ∫ Eq. 4.23 and defining dξ = d V d Edy, we get

ψϕ0∗ υ d V d Edy

and using

{{

( )| | ˝ ( ) ) ( d E ' dΩ ' ϕ0∗ r , E ' , Ω ' X ' ν(E)Σ f r , E ψ r , E, Ω dξ { ψϕ0∗ υ dξ { {∞ )|E6 ) ( ( ' )| ˝ ( ) ( ∗ ' ' ' ' ν(E)Σ f r , E ψ r , E, Ω dξ i=1 βi χi E 4π dΩ 0 d E ϕ0 r , E , Ω − n(t) { ψϕ0∗ υ dξ { ∗( ) ( ) E6 C , E, Ω , t χ λ ϕ r r (E)dξ i i i 0 + i=1 { ψϕ0∗ υ dξ

k−1 ∂n = n(t) ∂t k

From (4.26), we can designate ρ = 1 = /\

{{

k−1 k

and

) ( ( )| | ˝ ( ) d E ' dΩ ' ϕ0∗ r , E ' , Ω ' X ' ν(E)Σ f r , E ψ r , E, Ω dξ { ψϕ0∗ υ dξ

(4.26)

4.5 Derivation of the Kinetic Parameters βe f f = /\ ci (t) =

{ 4π

{

dΩ '

{∞ 0

( d E ' ϕ0∗ r , E ' , Ω

117

)|E6 '

i=1

{

) ( ) ( ϕ0∗ r , E, Ω Ci r , t χi (E)dξ { ψϕ0∗ υ dξ

( βi χi E '

)| ˝

) ( ( ) ν(E)Σ f r , E ψ r , E, Ω dξ

ψϕ0∗ υ dξ

( ) The precursor Eq. 4.20 is also multiplied by ϕ0∗ r , E, Ω χi (E) and integrated over all variables and divided by ∫

ψϕ0∗ υ d V d Edy,

and we obtain

{

( ) ( ) ϕ0∗ r , E, Ω χi (E)Ci r , t dξ { ψϕ0∗ υ dξ { { ( ) ) ( ) ( ) ( ∗ ∫ ϕ0 r , E, Ω βi χi (E)dξ 4π dΩ ' d E ' ν E ' Σ f r , E ' ψ r , E ' , Ω ' = n(t) { ψϕ0∗ υ dξ { ∗( ) ( ) ϕ0 r , E, Ω χi (E)Ci r , t dξ (4.27) − λi { ψϕ0∗ dξ υ

∂ ∂t

where we define { { ( ) ( )˝ ( ) ) ( ( ) ' d E 'ϕ∗ r , E ', Ω ' β χ E ' v(E)Σ f r , E ψ r , E, Ω dξ βi i i 0 4π dΩ = { ψϕ0∗ /\ e f f dξ υ

Rewriting Eqs. 4.26 and 4.27 using the notations defined above and collecting terms, we obtain the single node or point kinetic equation 6 E ( ) ρ − βe f f ∂n λi ci r , t = n(t) + ∂t /\ i=1 ( ) ( ) ( ) ∂ci r , t βi n(t) − λi ci r , t = ∂t /\ e f f

4.5.3

(4.28)

Diffusion Model

From part A above in the section on the diffusion equation, the non-steady-state form is ( ) ( ) ) ( ) ( 1 ∂φ = ∇ · D r , E ∇φ r , E − Σt r , E φ r , E υ ∂t {∞ ) ( ( ) + d E ' Σs r , E ' → E φ r , E ' 0

118

4 The Adjoint Transport Equation—The Equation of Neutron Importance

χ (E) + k

{∞

) ( ) ( ) ( d E 'ν E ' Σ f r , E ' φ r , E '

0

And the adjoint diffusion equation for a critical reference reactor is ( ) ) ) ( ) ( ( 0 = ∇ · D r , E ∇φ0∗ r , E − Σt r , E φ0∗ r , E {∞ ( ) ( ) + d E ' Σs r , E → E ' φ0∗ r , E ' 0

( ) + ν(E)Σ f r , E

{∞

) ( ) ( d E ' χ E ' φ0∗ r , E '

0

We follow the same derivation steps performed based on the transport equation: ∂ ∂t

{

φ0∗ φ dV d E = υ

{

| ∗( ) ( )| φ0 r , E ∇ · D∇φ r , E d V d E { ( | ( ) )| − φ r , E ∇ · D∇φ0∗ r , E d V d E ˚ { { )| | ( ( ) ( ) k−1 + ν(E)Σ f r , E φ r , E, t d V d E d E ' φ0∗ r , E ' X ' k | 6 |˚ {∞ ) ( ) ( ) E ( ) ( − d E ' φ0∗ r , E ' βi χi E ' v(E)Σ f r , E φ r , E, t d V d E i=1

0

+

6 E

{ λi

) ( ) ( φ0∗ r , E Ci r , t χi (E)d V d E

i=1

Integrating by parts the first two terms on the right-hand side following the same steps in the proof of the one-speed diffusion equation being self-adjoint leads to φ ∗ D∇φ −φ ∗ D∇φ = 0. The second term is also zero likewise. The above equation becomes { ∗ φ0 φ ∂ dV d E ∂t υ { { ˚ )| ' | ( ( ) ( ) k−1 ' ∗ ' d E φ0 r , E X ν(E)Σ f r , E φ r , E, t d V d E = k | 6 |˚ {∞ E ( ) ( ) ( ) ( ) ν(E)Σ f r , E φ r , E, t d V d E − d E ' φ0∗ r , E ' βi χi E ' i=1

0

+

6 E i=1

{ λi

( ) ( ) φ0∗ r , E Ci r , t χi (E)d V d E

4.5 Derivation of the Kinetic Parameters

119

We finally obtain the kinetic parameters based on the diffusion theory: k−1 k { { { {{ ) ( ) ( )| | ( ν(E)Σ f r , E ψ r , E d V d E d E ' φ0∗ r , E ' X ' 1 = { ψφ0∗ /\ υ dV d E |E {∞ ( ) ( )| { { { ( ) ( ) 6 ' ∗ ' ' ν(E)Σ f r , E ψ r , E d V d E i=1 βi χi E 0 d E φ0 r , E βe f f = { ψφ0∗ /\ υ dV d E { ∗( ) ( ) φ0 r , E Ci r , t χi (E)d V d E ci (t) = { ψφ0∗ υ dV d E ρ=

where we define { ) ( ){ { { ( ( ) ( ) ( ) d E ' φ0∗ r , E ' βi χi E ' ν(E)Σ f r , E ψ r , E d V d E βi = { ψφ0∗ /\ e f f dV d E υ

3 Two-dimensional multi-group form for the diffusion model We can further obtain the multi-group form of the kinetic parameters for a two-dimensional reactor EG E N E M EG i, j i , j ' ∗ i, j i=1 j=1 g ' =1 X g ' φ0 g ' /\g ' g=1 νg E f g ψg /\g /\i /\ j 1 = E N E M EG 1 ∗ i, j i, j /\ φ ψg /\g /\i /\ j i=1

|

and X g' ' = (1 − β)χg' + βe f f = /\

g=1 υg

j=1

6 E

g'

βi χi

i=1

.

E N E M EG i=1

j=1

∗ i, j g ' =1 φ0 g ' /\g ' j=1

E N E M EG ci=1,6 (t) = (

βi /\

E6

E N E M EG i=1

g' i=1 βi χi

= ef f

i=1

EG

i, j i , j g=1 νg E f g ψg /\g /\i /\ j

1 ∗ i, j i, j g=1 υg φ0 g ψg /\g /\i /\ j

∗ i , j i, j g g=1 φ0 g Ci=1,6 (t)χi=1,6 /\g /\i /\ j E N E M EG 1 ∗ i, j i , j i=1 j=1 g=1 υg φ0 g ψg /\g /\i /\ j

i=1

j=1

E N E M EG

)

0g

|

j=1

∗ i, j g' g ' =1 φ0 g ' βi=1,6 χi=1,6 /\g '

E N E M EG i=1

j=1

EG

i , j i, j g=1 νg E f g ψg /\g /\i /\ j

1 ∗ i, j i, j g=1 υg φ0 g ψg /\g /\i /\ j

So far, we have derived the kinetic equations and the kinetic parameters associated with it. The kinetic parameters are calculated from forward and adjoint transport or diffusion codes. More nodes such as two nodes for the core and reflector can be considered or two core nodes in what is known as the coupled reactor kinetics model, especially for certain reactor core types.

120

4 The Adjoint Transport Equation—The Equation of Neutron Importance

4.5.4

Two-Node Model

In the one-node model, the spatial effects are ignored though the reactor which is large in size where any disturbances at a given point are not felt further away in another point until later. Therefore, multi-nodes are more representative to the spatial effects; however, we will limit ourselves to a two-node model one for the core and one for the reflector. Example: A two-node reactor The reactor is to be divided into two nodes: core node and a reflector node (Fig. 4.6). The integrals over the core and reflector are expressed, respectively, ∫ d V d Edy. Dealing with the integration as ∫ = ∫ d V d Edy and ∫ = c

cor e

r

r e f lector

over the core first; Eq. 4.26 becomes { ( ) ϕ0∗ ϕ d V d Edy = − Ω · ∇ ϕ0∗ ϕ d V d Edy υ c c { { { ˚ ( )| | ( ) ( ) k−1 d E ' dΩ ' ϕ0∗ r , E ' , Ω ' X ' + ν(E)Σ f r , E ϕ r , E, Ω, t d V d Edy k

∂ ∂t

{

c

{ { −

dΩ '

c 4π

+

6 E i=1

{∞

| 6 |{ { { ( ) E ( ) ( ) ( ) ν(E)Σ f r , E ϕ r , E, Ω, t d V d Edy d E ' ϕ0∗ r , E ' , Ω ' βi χi E ' i=1

0

{ λi

( ) ( ) ϕ0∗ r , E, Ω Ci r , t χi (E)d V d Edy

c

Now we follow the same procedure as before in the one-node model by subψφ ∗ stituting with the shape function and dividing throughout by ∫ υ 0 d V d Edy. c

However, the streaming term will not disappear since the core is bounded by the reflector and in the previous argument 22 for a void boundary is not valid for this case, therefore we get |( | ) |n · y| ϕ ∗ ψ d Sd Edy 0 { ψϕ0∗ c υ dξ |( ∗ ) | |n · y| ϕ ψ d Sd Edy 0 n.y0

Fig. 4.6 Two-node model

C core

reflector

void

4.5 Derivation of the Kinetic Parameters

121

{ {{

) ( ( )| | { { { ( ) d E ' dΩ ' ϕ0∗ r , E ' , Ω ' X ' ν(E)Σ f r , E ψ r , E, Ω dξ k−1 n c (t) c + { ψϕ0∗ k c υ dξ | { { {∞ ) ( ( ) E6 ( ' )| ˝ ( ) ∗ ' ' ' ' ν(E)Σ f r , E ψ r , E, Ω dξ i=1 βi χi E c 4π dΩ 0 d E ϕ0 r , E , Ω − n c (t) ∗ { ψϕ0 c υ dξ { ∗( ) ( ) E6 λi c ϕ0 r , E, Ω Ci r , t χi (E)d V d Edy + i=1 { ψϕ0∗ c υ dξ

(4.29)

where S is the surface of the core region and the coefficients except the two streaming terms to the right and left are defined as before. The continuity of the partial currents at the interface is given by ) ) ( ( J+c r s , E = J+r r s , E ( ( ) ) J−c r s , E = J−r r s , E where r s is at the core-reflector interface. The condition of continuity can be expressed using the definition of the partial currents {

{

|( | ) |n · y| φ ∗ ψ d Sd Edy 0

n c (t) S n.y>0

{

{

= nr (t)

| |( ) |n · y| φ ∗ ψ d Sd Edy 0

S n.y ζ ≥ 550 then the interaction is capture; 50 and if 1 > ζ ≥ 550 then it is fission. The choice of collision parameters is not an easy task. First, a large amount of x-section data has to be formatted and coded properly in libraries with the associated interpolation subroutines.

7.3.3.3 Parameters After Collision After determining the path length, we automatically assume a collision has taken place so the type of nucleus is determined followed by the type of collision. If the type of collision leads to the termination of particle history, then tracking is

7.3 Modeling of Radiation Transport Parameters [4–6]

187

stopped. Secondary particles if created are banked for later tracking. Otherwise, a scattering collision is assumed and the parameters after collision must be determined. Neutron transport in the reactor and hence criticality calculations are our main concern, therefore only MC applications to the interactions of neutrons are discussed in this chapter and MC simulation of gamma and electron interactions are excluded.

1. Elastic scattering: (σe ) In elastic scattering, the incident neutron gives off some of its energy to the recoiling nucleus. It is isotropic in the COM for light elements. For heavy elements, it becomes more complicated and forward. Anisotropic scattering becomes important as the energy of the incident neutrons increases. The elastically scattered neutron energy is given by E' A2 + 2 A cos θc + 1 = E ( A + 1)2 The angles in both systems are related by cos θ L = (

A cos θc + 1

A2

)1/2 + 2 A cos θc + 1

A schematic diagram for elastic scattering in the LAB and COM before and after collision is shown in Fig. 7.14. The final energy is minimum (maximum energy loss) when the neutron is backscattered (θC = π ), and hence EEmin = | |2 A−1 A+1 . In the case of isotropic scattering in the COM the average cosine of the 2 scattering angle in the LAB is given by cosθ L = 3A . The angle in the COM is selected randomly from μ = cosθc = 2ζ −1. For other distributions, the differential angular scattering cross section σ (θc , E) is used for random selection of the angle. The CDF is calculated from

F(θc , E) =

∫θ0c σ (θc , E)dθc =ζ ∫π0 σ (θc , E)dθc

Once the scattering angle in the COM is obtained, the energy loss is calculated from relations which were presented previously. Once the angle and energy are obtained, the azimuthal angle ϕ (0 ≤ ϕ ≤ 2π ) should be randomly selected from an isotropic distribution, hence ϕ = 2π ζ . Figure 7.15 shows the steps of random calculations for elastic scattering angles and emerging energies.

188

7

Computer Simulation of Neutron Transport—The Monte Carlo Method

Fig. 7.14 Elastic scattering in the LAB and COM before and after collisions

Fig. 7.15 Random selection of elastic scattering angles from the differential cross section

2. Inelastic scattering: (σin ) In inelastic scattering, the neutron is absorbed by the target nucleus forming the compound nucleus which gives off a neutron and remains in the excited state which subsequently decays by photon emission. Neutron energies must be high enough to cause an inelastic reaction. It requires higher energies for light nuclei and lower energies for heavier nuclei. It is worth mentioning that the energy loss mechanism is via elastic scattering in the case of collisions with light nuclei and via inelastic scattering in the case of heavy nuclei. The total inelastic cross section is divided into two components firstly, σnn ' (L) which is the probability that the nucleus is excited to any level L from which

189

gamma rays are emitted and secondly, σnn ' (C) which is the probability that the nucleus is excited to the region where levels are closely spaced and the emission is continuous. The kinematic equation for inelastic scattering is given by | | ( ( E' 1 ε) ε )1/2 2 1 + A cosθ = + 2 A 1 − 1 − i c i E (Ai + 1)2 E E ( )1/2 1 + Ai cosθc 1 − Eε cosθ L = | ( ) ( )1/2 |1/2 1 + Ai2 1 − Eε + 2 Ai cosθc 1 − Eε The neutron energy required for an excitation of particular level is E > [( A + 1)/ A]ε where angles are defined as before and ε is the excitation energy of the target nucleus. Sampling from inelastic scattering cumulative distribution function is treated in the same manner as in elastic scattering. 3. Calculation of the emergent direction cosines: If the direction cosines of the incident particle before the reaction are (α, β, γ ) and the direction cosines after scattering are (α ' , β ' , γ ' ), and the scattering angle in the LAB is θ L and the azimuthal angle is ϕ the relations between the two cosines are given by α ' = αcosθ L + γ α

sinθ L cosϕ sinθ L sin ϕ −β 2 1/2 (1 − γ ) (1 − γ 2 )1/2

β ' = βcosθ L + γβ

sinθ L cos ϕ sinθ L sin ϕ −α (1 − γ 2 )1/2 (1 − γ 2 )1/2

γ ' = γ cosθ L + (1 − γ 2 )1/2 sinθ L cos ϕ ( ) if 1 − γ 2 → 0 α ' = sinθ L cos ϕ β ' = sinθ L sin ϕ γ ' = γ cos ϕ

190

7

Computer Simulation of Neutron Transport—The Monte Carlo Method

7.4

Variance Reduction Methods

7.4.1

Particle Weight

If the simulation is exactly a physical transport, then each simulated particle would represent one physical particle and would have a unit weight. However, for computational efficiency, the simulation allows many techniques that do not exactly simulate physical transport. For instance, each simulated particle might represent a number w of particles emitted from a source. This number w is the initial weight of particle simulation. The w physical particles all would have different random walks, but the one simulated particle representing these w physical particles will only have one random walk. Clearly this is not an exact simulation; however, the true number of physical particles is preserved in the simulation in the sense of statistical averages and therefore in the limit of a large number of simulated source particles (of course including particle production or loss if they occur). Each simulated particle result is multiplied by the weight so that the full results of the w physical particles represented by each simulated particle are exhibited in the final results (tallies). This procedure allows users to normalize their calculations to whatever source strength they desire. A second normalization to the number of Monte Carlo histories is made in the results so that the expected means will be independent of the number of source particles actually initiated in the calculation. Therefore, the result is given per one source particle and the more histories simulated the better the statistics. The real number of particles at any point is just [3, 7, 8, 9]: [the actual sour ce str ength × MC r esult] Noting that: Sometimes the number of sampled source particles are artificially increased in some direction or energy bins or position by a specified factor hence increasing the number of histories in order to improve statistics. At the end of the random walk, bias is neutralized by dividing the MC results by the same factor.

7.4.2

Source Biasing Parameters

In certain calculations it may be desirable to prejudice the selection of one or more source parameters to favor those most likely to contribute to the quantity of interest such as shield leakage or detector response. This can be done by selecting a larger number of source particles in the favored direction and assigning to each particle a number called its weight to adjust for a bias that was introduced. The benefit is to reduce drastically the tracking time.

7.4 Variance Reduction Methods

191

Fig. 7.16 Biased tracking toward point B

Table 7.1 Weight balance in a biased case Unbiased case No. of histories

Biased case Weight (w)

No. of histories

Weight (w)

Histories in /\o

2000

1

8000

1/4

Histories outside /\o

9000

1

3000

3

Total

11,000

11,000

Example Suppose there is a total of 10,000 source particle histories to be tracked from which 2000 in /\o (Fig. 7.16). However, the detector recording at B is the main objective of the calculation. Therefore, we will bias the flux in /\o and make it 6000 and leave the 4000 outside /\o. This will require weight of (1/3) in /\o and weight of (2) in the outside in order to neutralize the bias. In order to obtain a near true measurement at the detector, /\o should be carefully chosen. Table 7.1 explains the weight balance in this biasing case. Situations where biasing might be used include: (1) selecting more source points near the periphery of the reactor (2) selecting more particles with an initial direction toward the shield or detector (3) selecting more particles with a high energy or with an energy corresponding to a low total cross section. It is important to remember that an adjustment of one category (e.g. angle bin or energy group) effects not only the bin or group but the entire distribution.

7.4.3

Variance Reduction Techniques

When most of the computation time is spent on more probable particle histories that do not contribute to the desired result, extra effort is required to increase the sample size seeking accuracy in the less probable more important rare events. The wasteful effort may lead to exceeding the Random Number Generator (RNG) capability giving rise to correlated samples. Therefore, not just computer time and

192

7

Computer Simulation of Neutron Transport—The Monte Carlo Method

economics are of concern, rather the RNG finite period may be exceeded. Therefore, is it possible to reduce the relevant numerical and statistical uncertainty errors associated with the MC calculation without increasing sample size? The answer is through the variance reduction technique, the computer time can be reduced and still obtain results of sufficient precession. Common variance reduction techniques in Monte Carlo calculations are listed below: 1. Energy cutoff Particles are terminated when their energy falls below a predetermined energy cut off. It should be noted that: It is a user supplied low limit on energy. • If particle energy falls below the specified limit it is killed with zero probability. • It should be used only when it is known that the low energy particle is not important. Remembering that: 1. Low energy neutrons may produce high energy ones e.g. fission. 2. Low energy neutrons may be in some regions have more importance e.g. reactor core. 3. Neglecting weight of low energy particles by zero kill probability may make the answer biased low. Russian roulette (RR) is played to preserve weight. 4. If particle is not killed with RR, it is continued but weight is increased by the reciprocal of the survival probability to conserve weight. 2. Time cutoff Particles are terminated when their time exceeds the time cutoff. It is for time dependent problems. 3. Importance We define the importance of a cell as the expected score when a unit weight particle will generate after entering a cell. total score generated by particles (and thier progony) entering the cell Importance(expected score)= total weigt entering the cell After the importances have been generated the weights are assigned inversely proportional to the importances. Zero importance regions imply killing particles e.g. a surrounding infinite void where tracking is no longer desired.

7.4 Variance Reduction Methods

193

Fig. 7.17 Particle splitting

4. Geometry splitting/Russian roulette Particles are increased in number when move to important directions (Fig. 7.17). '

1. When m' > m, the particle is split into mm ∼ 2 or 4 . . . splits, however weight is reduced by mm' to become W' = mm' W. 2. When m' < m ( Russian ) roulette is played and the particle is killed with a probability of 1 − m' m . 3. Or followed further with a probability m' /m and weight W' = mm' W. Russian roulette procedure (play) is used to randomly terminate particle history moving toward less important phase space regions. The importance number assigned to a region is dictated by how important it is and the number assigned will determine the action taken weather it is 1 or (2 or 3). 5. Energy splitting/Russian roulette In space energy problems, particles are more important in some energy ranges than in others. For example, in a reactor core where fission occurs by thermal neutrons with a high probability so in the case of small number of fissions are taking place. Once a neutron falls below a certain energy level it can be split into several neutrons with appropriate weight adjustment. 6. Weight cutoff It is a lower bound on weight e.g. if the weight of a particle reaches a preset lower bound Russian roulette is played and its weight is transferred to other particles.

7.4.4

Neutron/Photon Problem

When photons are produced in a neutron problem where photons are also tracked, such as in shielding problems, the weight of photons is WP =

W n σγ σt

194

7

Computer Simulation of Neutron Transport—The Monte Carlo Method

where, Wn σγ σt

neutron weight photon production x-section total neutron x-section

σγ , σt are calculated at the energy of the incident neutron. 1. Capture (1) In analog capture, the particle is killed with a probability σσat and σa = σn,γ , σn,α , σn,d , · · · . Neutron energy and weight are deposited in the collision cell when particle is killed. ' (2) In Implicit capture, weight is reduced to Wn as follows: Wn'

| | | σa σs ' Wn or Wn = Wn = 1− σt σt |

'

| It| represents the probability of survival (no' capture) and Wn − Wn = σa σt Wn is deposited in addition to energy. If Wn is equal to a problem weight cut off (on a cut card), Russian Roulette is played and of course its weight is transferred to other particles resulting in fewer particles with larger weight. (3) Implicit capture along a flight distance to scatter l = − E1s ln(1 − ζ ). The particle weight is reduced at the scattering point by the capture loss W ' = W e−Ea l since capture did not take place. A final remark on the variance reduction methods is that all the schemes employed aim at varying the importance sampling. One should be cautious in implementing schemes because they may increase variance. Fortunately, in criticality problem calculations we only need some simple schemes of cell importance.

7.5

Criticality Calculations

Nuclear criticality is the ability to sustain a chain reaction by fission neutrons, and characterized by keff is thought of as the ratio between the number of neutrons in successive generations, with the fission process regarded as the birth event that separates generations of neutrons. In criticality applications, the effective multiplication factor of an assembly is of primary interest [3, 10, 11].

7.5.1

keff Cycle

A group of neutron histories is often referred to as a keff cycle or neutron generation with the multiplication factor of the assembly given by the ratio of the

7.5 Criticality Calculations

195

number of neutrons generated at the end of the keff cycle (i.e., those created in fission events in this cycle) to the number of neutrons whose histories are evaluated in this cycle (i.e., the number at the start of a generation). In short, keff =

number of neutrons in a generation number of neutrons in a previous generation

A generation is defined as the period between birth of a neutron to its death by escape, parasitic capture, or absorption leading to fission. It is the computational equivalent by MC to a fission generation. (n, 2n), (n, 3n) are not considered as termination and are internal to the cycle. It is optional to include delayed neutrons. The expected value of the multiplication factor is then estimated by averaging over the events in the keff cycle. In the same way, the expected value of the leakage probability or the fraction of events leading to capture can also be obtained. The relative error in the estimate of the effective multiplication factor will usually decreases as the number of keff cycles increase. Thus, numerous cycles are necessary to arrive at a good estimate of keff . In addition, the first few cycles are inaccurate because the spatial neutron source has not converged. Because the distribution of source (fission) neutrons in a system is dependent on the eigenvalue of the system and on its geometry, it takes a number of inactive cycles for the Monte Carlo spatial neutron distribution to approach the converged distribution. For this reason, the first few cycles are ignored in the final estimate of keff . The estimates of keff from the remaining cycles are averaged to obtain a mean value for the effective multiplication factor. Criticality calculation in a Monte Carlo code such as MCNP (A General Monte Carlo N-Particle Transport Code) is performed through a special card known as KCODE card which describes the initial source distribution, skipped cycles, total cycles, etc. For example, let’s say we evaluated G generations and discarded the first D of them. (It is recommended that G − D > 100 to observe any trends in the calculations.) Then the estimated effective multiplication factor of the system is given by keff =

G E 1 i keff (G − D) i=D+1

i is the multiplicawhere keff is the estimated system multiplication factor and keff tion factor determined from the ith cycle. The repeatability of the estimate (i.e., if the same calculation is performed with different random numbers, how much different will the estimate of keff be?) is determined from the estimated standard deviation of the mean. The standard deviation of the mean is calculated using the standard deviation, σ S , of the distribution of keff -values.

| | | σS = |

G E ( i )2 1 keff − keff (G − D − 1) i=D+1

196

7

Computer Simulation of Neutron Transport—The Monte Carlo Method

For a valid Monte Carlo calculation, the range keff − σ to keff + σ should include the precise keff result about 68% of the time. The final result of the Monte Carlo calculation would be reported as keff ± σ for a nominal 68% confidence interval,keff ± 2σ for 95% and keff ± 2.6σ for a 99% confidence interval for large N. These percentages refer to the fraction of the time the precise value of keff included in a confidence interval. MCNP has three different estimators for keff : collision, absorption, and track length between collisions. A statistically combined average is used as the final keff .

7.5.2

keff Estimators

There are several estimators for the multiplication factors, such as collision, absorption, track length, and prompt removal lifetime. The difference between A and collision k C estimators is that in the absorption estiimplicit absorption keff eff mator only the nuclides involved in collision are used for the estimation of the keff rather than an average of all nuclides in the material for the collision estimator. TL is accumulated every time the neutron traverses The track length estimator keff a distance in a fissionable material cell. Monte Carlo codes Sampling the probability distributions during particle history simulation are taken from the detailed physics of the problem. They are deducted from the deterministic formulae of the relevant physics of interaction. This direct approach is called the physical analogue approach which is quite lengthy and time-consuming. The modern MC codes rather develop sampling procedure from the integral form of the transport equation. The reader may consult specialized references for the mathematical details. There are many codes based on the Monte Carlo method. They have developed so rapidly as the capabilities of computers evolved. Some of these codes are MORSE, MCN, NCNG (LASL), Tripoli, TART (LLL), and more recently the well-known MCNP (LASL) series. Figure 7.18 shows the general organization of an MC code.

7.6

Particle Scoring

The discussion so far has been concerned with the generation of source particles, tracking to the first collision and the determination of the new particle direction and energy following a collision. This is continued until particle history is terminated. Large number of histories and variance reduction methods must be employed in order to obtain acceptable statistics for the physical quantities desired in the energy, angle bins and spatial cells defined. Such physical quantities are [3]: (1) (2) (3) (4)

Flux density as a function of space, energy, and direction The penetration dose or flux Time dependence of arriving particles The energy and angular distributions of the penetrating particles

7.6 Particle Scoring

197

Fig. 7.18 The general organization of an MC code

7.6.1

Neutron Tallies

A Monte Carlo code like MCNP provides: 1. A • • • • •

standard summary information that includes: Creation and loss of tracks and their energies Tracks entering and reentering a cell and track population in the cell Number of collisions in the cell Average weight, mean free path, energy tracks in the cell Activity of each nuclide in cell (interactions of with each nuclide in the cell, not radioactivity) • A complete weight balance for each cell 2. Seventeen standard tally types: 7 (neutron tallies); 6 (photon tallies); 4 (electron tallies). They can be modified by the user in many ways. They are normalized per starting particle except in KCODE criticality problem, which are normalized to be per fission neutron generation. There is also a plotter for displaying results. Table 7.2 lists the basic standard tally types. N, E, P refer to neutrons, electrons, and photons respectively. Table 7.3 shows the mathematical definitions of the physical quantities defined by the tally. The starred tallies are the same physical quantities in units of energy. Note that cell flux becomes a surface flux when the cell becomes so thin. Also, the units of the flux follow the units of the source, e.g. S (n/s) give a flux in units of (n/cm2 -s). In criticality calculations since the sources are fission sources the

198

7

Computer Simulation of Neutron Transport—The Monte Carlo Method

Table 7.2 MCNP standard tallies Unit

Fn F1:N,E,P

Name

*Fn

Unit

Surface current

E

MeV

F2:N,E,P

1/cm2

Surface flux

E

MeV/cm2

F4:N,E,P

=

Track length estimate of cell flux

E

=

F5a:N,P

=

Flux at a point or ring detector

E

=

F6:N,E,(N,P)

MeV/g

Track length estimate of energy deposition

1.6022E-22

Jerks/g

F7:N

=

Track length estimate of fission energy deposition

1.6022E-22

=

F8:N,E,P,(P,E)

Pulses

Pulse height tally

E

MeV

Table 7.3 MCNP mathematical definitions of the tally physical quantities

Fn

Physical quantity { { { { J = A d A μ dμ t d t E d E|μ| AΦ(r, E, μ, t) { { { 8s = A dAA t d t E d EΦ(r, E, t) { { { 8 V = V dVV t d t E d EΦ(r, E, t) { { 8 P D = t d t E d EΦ(r, E, t)

F1 F2 F4 F5

Physical quantity × MeV { { { { A d A μ dμ t d t E d E E|μ| AΦ(r, E, μ, t) { { dA { A A t d t E d E EΦ(r, E, t) { dV { { V V t d t E d E EΦ(r, E, t) { { t d t E d E EΦ(r, E, t)

*Fn *F1 *F2 *F4 *F5

units are deduced from the reactor power. The following conversion is used: (

J/s w−s

)(

1 MeV 1.602 × 10−13 J

)(

fissions 180 MeV

) = 3.467 × 1010 fissions/w − s

Therefore, in order to produce a power( of P watts we ) need (3.467 × 1010 P) fissions/s. this power level produces 3.467 × 1010 P ν neutrons/s which is the neutron source strength per second for a power level P in watts.

7.7

Accuracy, Precision, Relative Error, and Figure of Merit

A systematic error (true value of x − x) is a measure of how close is the expected value of (x) to the true value of x, which is seldom known [12].

7.7 Accuracy, Precision, Relative Error, and Figure of Merit

MC estimate

True value

Fig. 7.19 The meaning of precision and accuracy

199

Systematic error

Table 7.4 Interpretation of the relative error

7.7.1

Range of R = Sx /x Quality of tally 0.5–1

Completely unacceptable

0.2–0.5

Still unacceptable

0.1–0.2

Questionable (undependable)

0. The unit vector Ω causes the _

streaming term in TE to be of opposite signs. This ( in) the diffusion ( ) ( is not ) the case , Ω r , Ω and J r = j r = Ωϕ theory where the flux is isotropic. Notice also ) ( ) ( ( ) Ωφ r are both vectors and ϕ r , Ω and φ r are both scalars. Applying requirement II to DE and its adjoint, we get (

) ) ( φ ∗ , −∇ · D∇φ + Σa φ − νΣ f φ − φ, −∇ · D∇φ ∗ + Σa φ ∗ − νΣ f φ ∗

The absorption and fission terms are clearly self-adjoint and no need to prove that. Only the streaming term needs a proof. (

−φ ∗ ∇ · D∇φ +

(

φ∇ · D∇φ ∗

Integration by parts for the two integrals; first integral: u = −φ ∗ ; dv = ∇ · D∇φ

238

9

Selected Problems Solutions

du = −∇φ ∗ ; v = D∇φ Apply the identity

(

( ( −φ ∗ ∇ · D∇φ = uv − vdu = −φ ∗ D∇φ + D∇φ∇φ ∗ d V −φ ∗ D∇φ + φ ∗ D∇φ = 0

And, the second integral is likewise. This proves that the streaming term is self-adjoint. 4.2 Examine the adjoint of the time-dependent DE by proving Δ = 0. Solution The time rate of change in the TE: Applying requirement II, (φ ∗ , Lφ) = (φ, L ∗ φ ∗ ) or (φ ∗ , Lφ) − (φ, L ∗ φ ∗ ) = 0 and, for the time-dependent ] term, we should prove [ 1 ∂ϕ ∗ 1 ∂ϕ ∗ (ϕ υ ∂t − −ϕ υ ∂t ) = 0 and in the explicit form: ˝ ∗ 1 ∂ϕ ∗ ϕ υ ∂t + ϕ υ1 ∂ϕ ∂t d V d EdΩ is equal to zero. Notice the integrand does not include the time derivative; therefore, ˚ ˚ 1 ∂ 2 ∂ ϕ ∗ ϕ + ϕϕ ∗ d V d EdΩ = ϕ ∗ ϕd V d EdΩ υ ∂t υ ∂t The initial ( apply to the ) flux and adjoint, respectively. For ( and final) conditions example, ϕ r , E, Ω, ti = ϕi ; ϕ ∗ r , E, Ω, t f = ϕ ∗f have constant distributions at the beginning and at the end, hence 2 υ

( ) ˚ ∂ ϕ ∗ ϕi f ∂t

d V d EdΩ = 0

4.4 Write down the energy-dependent P1 equations and from which try to deduce the adjoint form.

Solution The Legendre equation with fission term from the chapter on spherical harmonics Example: sphere (rewrite it for a plane) [ ] ] ∂ϕn+1 ∂ϕn−1 ϕn−1 (r ) ϕn+1 (r ) + (n + 1) − (n − 1) + (n + 2) ∂r r ∂r r + (2n + 1)(Σt − Σsn )φn (r ) = (2n + 1)νΣ f ϕ0 (r )

[ n

9

Selected Problems Solutions

The P1 equations ] [ ∂φ1 φ1 (r ) + (Σt − Σs0 )φ0 (r ) = νΣ f φ0 (r ) +2 ∂r r ∂φ + 3(Σt − Σs1 )φ1 (r ) = 3νΣ f φ0 (r ) ∂r The energy form ] [ ∂ϕ1 ϕ1 (r , E) + Σt φ0 (r , E) +2 ∂r r ( ( ) ) ( − d E , Σs0 E , → E φ0 r , E , ( ) ( ) ( ) ( = χ (E) d E , ν E , Σ f E , φ0 r , E , ( ( ) ) ( ∂ϕ0 + 3Σt φ1 (r , E) − d E , Σs1 E , → E φ1 r , E , ∂r ( ) ( ) ( ) ( = 3χ (E) d E , ν E , Σ f E , φ0 r , E , or

] ∂ J (r , E) J (r , E) + Σt φ(r , E) +2 ∂r r ( ( ( ) ) ) ( ( ) ( ) ( − d E , Σs0 E , → E φ r , E , = χ (E) d E , ν E , Σ f E , φ0 r , E ,

[

( ( ) ) ( ∂φ + 3Σt J (r , E) − d E , Σs1 E , → E J r , E , ∂r ( ) ( ) ( ) ( = 3χ (E) d E , ν E , Σ f E , φ r , E , The energy-dependent adjoint P1 equations ] [ ∗ ∂ J (r , E) J ∗ (r , E) + Σt φ ∗ (r , E) − +2 ∂r r ( ( ) ( ) − d E , Σs0 E → E , φ ∗ r , E , ( ) ( ) ( = ν(E)Σ f (E) d E , χ E , φ ∗ r , E , ( ( ) ( ) ∂φ ∗ ∗ − + 3Σt J (r , E) − d E , Σs1 E → E , J ∗ r , E , ∂r ( ( ) ( ) = 3ν(E)Σ f (E) d E , χ E , φ ∗ r , E ,

239

240

9

Selected Problems Solutions

4.5 Find out the adjoint angular and space-dependent discrete ordinate equation. Solution The DO equation is

μ

∂ Σs ϕ(z, μ) + Σt ϕ(z, μ) = ∂z 2

(+1 ) ( dμ, ϕ z, μ, + S(z, μ) −1

The adjoint form is ∂ Σs −μ ϕ ∗ (z, μ) + Σt ϕ ∗ (z, μ) = ∂z 2

(+1 ( ) dμ, ϕ ∗ z, μ, + S ∗ (z, μ) −1

It is originally derived from the TE where the major difference is in the first term (streaming term) in the negative direction. Try to obtain the energy-dependent adjoint form. 4.6 Derive the last term in the kinetic equation: ( ) ν(E)Σ f r , E

(∞ 0

+

6 Σ

(

βi χi E

) ,

[ ( ) d E , (1 − β)χ E ,

](

( ) dΩ , ϕ0∗ r , E , , Ω ,

i=1

Solution The TE with delayed neutrons ( ) ) ( ) ( 1 ∂φ = −Ω · ∇φ r , E, Ω, t − Σt r , E, Ω φ r , E, Ω, t υ ∂t (∞ ( ( ) ( ) , dΩ , Σs r , E , → E, Ω , → Ω φ r , E , , Ω , , t + dE 0

4π

( + (1 − β)χ (E)

dΩ

4π

+

6 Σ i=1

( ) λi Ci r , t χi (E)

,

(∞ 0

) ( ( ) ( ) d E ,ν E , Σ f r , E , φ r , E ,, Ω ,, t

9

Selected Problems Solutions

241

All terms are familiar except the last two terms ( + (1 − β)χ (E)

dΩ

+

(∞

( ) ( ) ) ( d E ,ν E , Σ f r , E , φ r , E ,, Ω ,, t

0

4π 6 Σ

,

( ) λi Ci r , t χi (E)

i=1

which are meant for the accounting for delayed neutrons. First term represents the rate of prompt neutron production rate from fission, where β is the delayed neutron fraction; and (1 − β) is the remaining prompt neutrons. Secomd term is the delayed neutron production rate. How? Fission produces fragments. Some of them are converted to other products via decay. Those who decay by neutron emission are classified into six groups. They are delayed relative to prompt neutrons with known decay times and energy spectra χi (E). Therefore, ( ) λi Ci r , t is the decay rate of group (i) precursors ⇔ (number of delayed neutrons( in )group (i)). λi Ci r , t χi (E) the delayed neutrons emitted with energy E. ( ) Σ6 i=1 λi Ci r , t χi (E) is the total number of delayed neutrons emitted by all the precursors at energy E. The last term of the adjoint form: ( ) ν(E)Σ f r , E

(∞

[ ( ) d E , (1 − β)χ E ,

0

+

6 Σ

(

βi χi E

,

)

](

( ) dΩ , φ0∗ r , E , , Ω ,

i=1

Proof: Normally, in the process of fission neutrons causing fission are chosen on the energy scale E , and the outcome is selected at energy E. The concept of fission in the adjoint system is the reversal of the actual process. We start from the end of the fission process (the scale is E , where we have the importance fractions of prompt and delayed neutrons (∞

( ) ( )( d E , (1 − β)χ E , dΩ , φ0∗ r , E , , Ω , and 6 (∞ ) ( )( ( Σ delayed: 0 d E , βi χi E , dΩ , φ0∗ r , E , , Ω , . Prompt:

0

i=1

242

9

Selected Problems Solutions

) ( both of which resulted from the fission of one neutron ν(E)Σ f r , E at energy E, then ⎡∞ ( ) ( ( ) ν(E)Σ f r , E ⎣ d E , (1 − β)χ E , 0

(∞ +

dE 0

,

6 Σ

(

) ,

⎤

βi χi E ⎦

(

( ) dΩ , φ0∗ r , E , , Ω ,

i=1

( ) ν(E)Σ f r , E

(∞

[ ( ) d E , (1 − β)χ E ,

0

+

6 Σ

]( ( ,) ( ) dΩ , φ0∗ r , E , , Ω , βi χi E

i=1

4.8 Let J (E)d E be the source of neutrons slowing down to the energy interval between E and E + d E from collisions with a moderator of mass A which scatters neutrons isotropically in the center of mass system. J A (E) can be viewed as an integral operator acting on the flux φ(E) E ( ,) ( /α H ( ) , Σs E J A (E) = Lφ(E) = dE φ E, , (1 − α)E E

where symbols are familiar from neutron slowing down theory. Determine the adjoint L ∗ . Solution The slowing down source where neutrons slow down to energy E. Fig. 9.3 shows the limits of lower energies upon slowing down or the upper limit of initial neutron energy. So, what would be the adjoint operator) L ∗ E ( ,) ( /α A ( ) , Σs E φ E, J A (E) = Lφ(E) = dE (1 − α)E , E

/ where neutrons scattered from E , to E with E α as the upper limit (top part of the figure). On the other hand, if a neutron at E is scattered downward, α E would be the lower limit.

9

Selected Problems Solutions

243

Fig. 9.3 Elastic scattering of neutrons

The transfer scattering cross section can be written as ΣsA

(

)

,

E →E =

ΣsA

(

( ) ) ( , ) ΣsA E , E P E →E = (1 − α)E , ,

The scattering probability is always dependent on the neutron energy before the scattering event. Therefore, J A (E) can be rewritten as E ( /α ) ( ) ( J A (E) = Lφ(E) = d E , ΣsA E , → E φ E , E

The adjoint form is just reversing the energy transfer J A∗ (E)

∗ ∗

(E

= L φ (E) =

( ) ( ) d E , ΣsA E → E , φ ∗ E ,

αE

J A∗ (E)

∗ ∗

(E

= L φ (E) = αE

J A∗ (E) (E

αE

d E,

ΣsA (E) ∗ ( , ) φ E (1 − α)E

ΣsA (E) = L φ (E) = (1 − α)E ∗ ∗

(E

( ) d E ,φ∗ E ,

αE

( ) d E , φ ∗ E , is the total importance of the emergent neutrons from scattering.

244

9

Selected Problems Solutions

Chapter 5: Perturbation theory 5.3 The accurate flux and the adjoint flux in a subcritical system can readily be calculated. The configuration of another subcritical system is only slightly different. Find an expression for the desired reaction rate integrated over the second system, such that the expression has only second-order errors and does not depend on variations from the flux and adjoint flux of the first system. Solution The expression has only second-order errors: To explain the order of errors, assume the quantities φ+0 φ + δφ + 0 φ + δφ + δφδφ + 0 The first quantity all orders of error higher than the zero order are neglected (assumed exact). The second quantity all orders of error higher than the first order are neglected. The third quantity all orders of error higher than the second order are neglected. , However, the first-order perturbation theory the perturbed flux and adjoint φ = φ + δφ ∼ = φ ∗ (negating the small = φ such that δφ = 0 and φ ∗ = φ ∗ + δφ ∗ ∼ differences as if first-order errors and higher are neglected). When the expression has only second-order errors imply only first-order error is accepted. A subcritical system is kept critical by a source S(t) [reactor kinetics] The configuration of another subcritical system is only slightly different: (1) Could mean only the geometry is slightly different like in the case of approximations to suit different codes or (2) Could mean everything is slightly different. Variations from the flux and adjoint flux of the first system: It may depend on φ and φ ∗ but does not depend on the variations δφ and δφ ∗ . Find an expression for the desired reaction rate integrated over the second system? Assuming one-speed theory where the flux is self-adjoint, the total importance of the reaction rate integrated on the second system is ( V2

( )( ( ) ( )) ( ) d V φ ∗ r Σa r + δΣa r φ r

9

Selected Problems Solutions

245

If the cross section is uniform and the system is a slab (a

[ ] d x(Σa + δΣa ) φ 2 (x)

−a

φ(x) is the solution of the diffusion equation (as we have mentioned the criticality is sustained by an external supposedly isotropic source) −D

d 2φ + Σa φ(x) = νΣ f φ(x) + Sext dx2

Actually, it is a space–time-dependent problem in reactor kinetics, 1 ∂φ ∂ 2φ = νΣ f φ(x, t) + Sext (t) + D 2 − Σa φ(x, t) υ ∂t ∂x However, we ignore the time dependence and consider a moment in time ] [ νΣ f − Σa d 2φ d 2φ Sext φ(x) = + + Bm2 φ(x) = − 2 2 dx D dx D φ(x) = −

Sext + A cos Bm x D Bm2

Substituting in the integral, (a −a

[ ]2 Sext + A cos B x d x(Σa + δΣa ) − m D Bm2

5.4 Demonstrate the validity of problem (2) by considering the following example: An infinite medium extends from x = 0 to ∞ and contains a plane source at x = 0. The flux is to be described by the one group diffusion theory. The first system has an absorption cross section Σa and a diffusion constant D. The second system has the same source and diffusion constant, but an absorption cross section Σa + δΣa . Calculate the absorption rate throughout exactly and according to the derived perturbation procedure. Then show that results differ by second-order terms in δΣa . Solution The exact solution The unperturbed system flux by solving the one-speed neutron diffusion equation is

246

9

Selected Problems Solutions

The perturbed system φ , (x) =

SL, exp −x/L , ; 2D

D Σa,

L ,2 =

The absorption rate before and after (∞

SL d xΣa exp −x/L = 2D

0

(∞

d xΣa,

SL, S exp −x/L , = 2D 2

0

Meaning that neutrons produced by source are eventually absorbed in both cases. Perturbation theory: (∞

d xΣa, φ , (x)

(∞ d x(Σa + δΣa )(φ(x) + δφ(x))

=

0

0

(∞ =

d x[Σa φ(x)] + [δΣa φ(x)] + [Σa δφ(x)] + [δΣa δφ(x)] 0

Second-order errors δΣa δφ(x) = 0 (∞ =

d x[Σa φ(x)] + [δΣa φ(x)] + [Σa δφ(x)] 0

First-order perturbation δφ(x) = 0 (∞ =

d x[Σa φ(x)] + [δΣa φ(x)] =

S + 2

0

( ) δΣa S δΣa S S 1+ = + = 2 Σa 2 2 Σa ( ( ) δΣa S a is equivalent to 2S 1 + δΣ 2 1 + Σa Σa +

d x[δΣa φ(x)] 0

(δΣa )2 Σa

a) zero as compared to the exact solution term (δΣ Σa other as δΣa → 0 2

(∞

) + . . . with the second order S 2.

The results approach each

5.5 A critical system consisting of a slab surrounded on both sides by infinite reflectors (Fig. 9.4) is to be described by two group diffusion theory. The center third of the core suddenly has its thermal group absorption cross section increased slightly. Find the change in criticality factor by perturbation theory. Compare to the immense amount of work would be done if the perturbation theory is not used.

9

Selected Problems Solutions

247

Core

Core

Core

Reflector

Reflector

Fig. 9.4 Reflected slab

Solution The original system: Two-group theory for the core and reflector Derivation in the notes gives ( 2a/3

∗ −2a/3 d xφ2 (x)δΣa2 φ2 (x) ∗ ∗ −2a/3 d xφ1 (x)ν1 Σ f 1 φ1 (x) + φ2 (x)ν2 Σ f 2 φ1 (x)

Δρ = − ( 2a/3

Derivation in the notes gives ( 2a/3

∗ −2a/3 d xφ2 (x)δΣa2 φ2 (x) ∗ ∗ −2a/3 d xφ1 (x)ν1 Σ f 1 φ1 (x) + φ2 (x)ν2 Σ f 2 φ1 (x)

Δρ = − ( 2a/3

Needed are the thermal flux φ2 and adjoint φ2∗ in the central portion of the core and fast flux φ1 ; and adjoint φ1∗ in the same portion. perturbation theory saves plenty of time and effort in contrast to detailed calculation of the fluxes and adjoints fast and thermal in all regions with boundary and interface conditions. Chapter 6: Discrete ordinates 6.4 Derive the discrete ordinates equations in 1D slab geometry for the case of N = 2 (e.g. the S2 equations). Assume isotropic scattering and choose the appropriate angles and weights. Compare the structure of these equations with the P1 equations. Assume a steady-state and a homogenous medium for convenience. 6.5 Using the S2 equations developed in problem 1, derive equations satisfied by the sum and difference of the two angular flux components ϕ(z, μ1 ) = ϕ(z, μ2 ). Then show that with appropriate choice of μ1 the flux will have the exact asymptotic diffusion length.

248

9

Selected Problems Solutions

Solution The DO equation with isotropic scattering d φ˜ n+1 d φ˜ n−1 +n + (2n + 1)Σt φ˜ n (z) = (2n + 1)Σsn δn0 φ˜ n (z) dz dz +(2n + 1) S˜n (z); n = 0, 1, 2, · · · , M − 1

(n + 1)

The S2 equations For M = 2 implies n = 0,1 n=0 ... d φ˜ 1 + Σa φ˜ 0 (z) = S˜0 (z) dz

(9.13)

d φ˜ 0 + 3Σt φ˜ 1 (z) = 3 S˜1 (z) dz

(9.14)

n=1

where φ˜ 0 (z) = 2π [w1 ϕ(z, μ1 )P0 (μ1 ) + w2 ϕ(z, μ2 )P0 (μ2 )] φ˜ 1 (z) = 2π [w1 ϕ(z, μ1 )P1 (μ1 ) + w2 ϕ(z, μ2 )P1 (μ2 )] Again, φ˜ 0 (z) = 2π [w1 ϕ(z, μ1 ) + w2 ϕ(z, μ2 )] φ˜ 1 (z) = 2π [μ1 w1 ϕ(z, μ1 ) + μ2 w2 ϕ(z, μ2 )] We know w1 = w2 and μ1 = −μ2 , substituting φ˜ 0 (z) = 2π w1 [ϕ(z, μ1 ) + ϕ(z, −μ1 )]

(9.15)

φ˜ 1 (z) = 2π μ1 w1 [ϕ(z, μ1 ) − ϕ(z, −μ1 )]

(9.16)

By solving Eqs. (9.13) and (9.14), we can solve Eqs. (9.15–9.17) for ϕ(z, μ1 ) and ϕ(z, −μ1 )

9

Selected Problems Solutions

249

The P1 equations .... dφ1 + Σa φ0 (z) = S0 (z) dz

(9.17)

... dφ0 + 3Σt φ1 (z) = 3S1 (z) dz

(9.18)

which are identical to Eqs. (9.13) and (9.14), except for the definition of ( +1 φ˜ 0,1 (z) and φ0,1 (z), where φ0 (z) = 2π −1 dμϕ(z, μ) and φ1 (z) = ( +1 2π −1 dμμϕ(z, μ) Assuming no isotropic sources Eqs. 9.13 and 9.14 can be solved as , / / ∼ S˜0 +Cex p − z , and φ˜ 1 (z) = − 1 d φ˜0 ≈ −D , d φ˜0 = C D, ex p − z , φ˜ 0 (z) = L L Σa 3Σt dz dz L 1 where D , = 3Σ is the isotropic diffusion coefficient and (Σs1 = 0). t Adding Eqs. 9.15 and 9.16

φ˜ 0 (z) +

φ˜ 1 (z) = 4π w1 ϕ(z, μ1 ) μ1

or and solving for ϕ(z, μ1 ) ( ) φ˜ 1 (z) 1 φ˜ 0 (z) + ϕ(z, μ1 ) = 4π w1 μ1 ( ) / 1 D, z/ , Cex p − z L , + ≈ Cex p − L 4π w1 μ1 L , ( ) / D, C 1+ ex p − z L , ϕ(z, μ1 ) = , 4π w1 μ1 L ( ) / Σa C 1+ ex p − z L , ϕ(z, μ1 ) = 4π w1 μ1 with appropriate choice of μ1 the flux will have the exact asymptotic diffusion length.

A

The concept of Directional Derivative

A.1

Directional Derivative

In short, the concept of directional derivative can be defined as taking the derivative in a particular direction. Partial derivatives Partial derivatives with respect to the axis x, y, z can be considered as the directional derivatives in the directions i, j, k, respectively. Therefore, ∂ f (x,y,z) implies ∂x assuming the variables y, z as constants and taking the derivative of the function f in the x-direction (i) which is defined as the slope in the x-direction. Derivatives with respect to the other directions are like so will produce slopes in the directions of y and z. In the following, we restrict ourselves to the Cartesian coordinates. The gradient (∇) The grad (∇) is defined as an operator that consists of the partial derivatives in the direction of the three-unit vectors i , j, k ∇=

∂ ∂ ∂ i+ j+ k ∂x ∂y ∂z

and operates on a scalar quantity, e.g. f (x, y, z) and the grad of f is ∇f =

∂f ∂f ∂f i+ j+ k ∂x ∂y ∂z

The slope in an arbitrary direction (the directional derivative): Now, we seek derivatives of the function f (x, y, z) in the direction of the unit vector Ω = Ωx i + Ω y j + Ωz k. Referring to Fig. A.1, the coordinates of the point Q is [x(s), y(s), z(s)] dependent on the value of s and the vector r like so,

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 R. M. Kuridan, Neutron Transport, Graduate Texts in Physics, https://doi.org/10.1007/978-3-031-26932-5

251

252

Appendix A: The concept of Directional Derivative

Fig. A.1 Directional derivative in a 3D cartesian geometry

r = x(s)i + y(s) j + z(s)k. The directional derivative can be written using the chain rule as ∂f ∂ f ∂ x(s) ∂ f ∂ y(s) ∂ f ∂z(s) = + + ∂s ∂ x ∂s ∂ y ∂s ∂ z ∂s which is just the scalar product between the two vectors: [ ][ ] ∂f ∂f ∂f ∂ y(s) ∂ z(s) ∂f ∂ x(s) = i+ j+ k . i+ j+ k ∂s ∂x ∂y ∂z ∂s ∂s ∂s

(A.1)

The quantity in the first bracket is ∇ f ; however, the second one is the derivative of r with respect to s as we can see: r ' = x ' (s)i + y ' (s) j + z ' (s)k. The vector a is just a = x0, i + y0 j + z0 k and hence, from the triangle of vectors r = a + sΩ and taking the derivative w.r.t. s, then r ' = Ω because a' = 0 independent of s. Rewriting Eq. A.1 ∂f = ∇ f .r ' = ∇ f .Ω = Ω.∇ f ∂s

(A.2)

This is just the streaming term in the neutron transport equation. Since Ω is a three-dimensional unit vector, we obtain an infinite number of directional derivatives unless we specify a particular direction. Example Find the directional derivative ∂∂sf at point P(1, 1, 1) in the direction of the vector   2i − j + 5k given that f (x, y, z) = f 0 x 2 y + z . Hence, ∇ f = 2 f 0 yxi + f 0 x 2 j + f 0 k

Appendix A: The concept of Directional Derivative

253

∇ f = 2 f0i + f0 j + f0 k 2i− j+5k √2 The unit vector is Ω = |2i− j+5k| = 30 i − At point P, the directional derivative is

√1 30

j+

√5 k 30

check?

  4 ∂f 8 1 5 = √ f0 = Ω.∇ f = f 0 √ − √ + √ ∂s 30 30 30 30 Positive means an increasing f in the direction of Ω.

A.1.1 ∇ in Planar Geometry In rectangular coordinates as in Fig. A.2, the unit vector Ω is in a phase space independent of the coordinate system of the position vector r , e.g. r : x, y, z and Ω : χ , ϑ. Consider a one-dimensional semi-infinite slab in which the neutron flux is varying with the z-axis. We know that the slab extends to infinity in the other directions x and y. Therefore, neutron motion is restricted to right and left on the z-axis and in the plane y–z (x = 0) or the plane x–z (y = 0) such that the dependence is only on ϑ and not on the azimuthal angle χ . Figure A.2 in 3D reduces to 1D Fig. A.3. Applying the above analysis for the directional derivative to this onedimensional case, ∂φ ∂φ ∂ z(s) = ∂s ∂ z ∂s

Fig. A.2 Rectangular coordinate system

254

Appendix A: The concept of Directional Derivative

Fig. A.3 Directional derivative in a 1D slab

where φ is the scalar flux. From the triangle in the figure,

∂z(s) ∂s

= cosϑ and hence,

∂φ ∂φ ∂φ = Ω.∇φ = cosϑ = μ ∂s ∂z ∂z

A.1.2 ∇ in Spherical Geometry The spherical dimensions are r : r , θ, ψ and Ω : ω, μ, where μ is the cosine of the angle between Ω and r as shown in Fig. A.4. The vector r is defined in cartesian coordinates as r = xi + y j + zk From Fig. A.4, x = r sin θ cos ψ, y = r sin θ sin ψ, z = r cos θ therefore, r = r sin θ cos ψi + r sin θ sin ψ j + r cos θ k

(A.3)

| | / |r | = x 2 + y 2 + z 2 = r ˆ So, what are the unit vectors in spherical coordinates rˆ , θˆ , and ψ? Let us first consider this simple example. Taking the derivative of r w.r.t x in the formula of the vector r above, we get ∂∂rx = i (in the direction of increasing

Appendix A: The concept of Directional Derivative

255

x). Therefore, the unit vector rˆ is just the derivative of r in the direction of r in Eq. A.3 or rˆ =

∂r = sin θ cos ψi + sin θ sin ψ j + cos θ k ∂r

which is equal to rr . The other unit vectors are like so, ∂r = r cos θ cos ψi + r cos θ sin ψ j − r sin θ k ∂θ | | / |θ | = [r cos θ cos ψ]2 + [r cos θ sin ψ]2 + [r sin θ]2 = r θ=

θ And, the unit vector θˆ = |θ | = In a similar fashion,

ψ= and

θ r

= cos θ cos ψ i + cos θ sin ψ j − sin θ k.

∂r = −r sin θ sin ψi + r sin θ cos ψ j ∂ψ

| | / | | |ψ | = [−r sin θ sin ψ]2 + [r sin θ cos ψ]2 = r sin θ

Fig. A.4 Spherical coordinate system

256

Appendix A: The concept of Directional Derivative

And, we get −r sin θ sin ψi + r sin θ cos ψ j ψ ψ ψˆ = | | = = | | r sinθ r sinθ |ψ | = −sin ψi + cos ψ j Collect the results for the spherical geometry unit vectors ⎫ rˆ =sin θ cos ψ −i + sin θ sin ψ j + cos θ k− ⎪ ⎪ − ⎪ ⎪ ⎬ θˆ = cos θ cos ψ −i + cos θ sin ψ j − sin θ k− − ⎪ ⎪ ⎪ ⎪ ˆ ⎭ ψ = − sin ψ i + cos ψ j −

(A.4)

−

The following derivatives are going to be employed in the upcoming derivations leading to the gradient. ⎫ ∂x ⎪ ⎪ =sin θ cos ψ ⎪ ⎪ ∂r ⎪ ⎪ ⎬ ∂x =r cos θ cos ψ ⎪ ∂θ ⎪ ⎪ ⎪ ∂x ⎪ ⎭ = − r sin θ sin ψ ⎪ ∂ψ

⎫ ∂y ⎪ =sin θ sin ψ ⎪ ⎪ ⎪ ∂r ⎪ ⎪ ⎬ ∂y =r cos θ sin ψ ⎪ ∂θ ⎪ ⎪ ⎪ ∂y ⎪ ⎭ =r sin θ cos ψ ⎪ ∂ψ

⎫ ∂z ⎪ ⎪ =cos θ ⎪ ⎪ ∂r ⎪ ⎪ ⎬ ∂z = − r sin θ ⎪ ∂θ ⎪ ⎪ ⎪ ∂z ⎪ ⎪ ⎭ =0 ∂ψ

(A.5)

The chain rule: ∂ ∂r ∂ ∂θ ∂ ∂ψ

= ∂∂rx = ∂∂θx ∂x = ∂ψ

∂ ∂x ∂ ∂x ∂ ∂x

+ + +

∂y ∂ ∂z ∂ ∂r ∂ y + ∂r ∂z ∂y ∂ ∂z ∂ ∂θ ∂ y + ∂θ ∂z ∂y ∂ ∂z ∂ ∂ψ ∂ y + ∂ψ ∂z

⎫ ⎪ ⎬ (A.6)

⎪ ⎭

Substituting A.5 into A.6, we get

∂ ∂θ

⎫ = sin θ cos ψ ∂∂x + sin θ sin ψ ∂∂y + cos θ ∂∂z ⎪ ⎬ ∂ = r cos θ cos ψ ∂∂x + r cos θ sin ψ ∂∂y − r sin θ ∂z ⎪ ⎭ ∂ ∂ ∂ ∂ψ = −r sin θ sin ψ ∂ x + r sin θ cos ψ ∂ y

∂ ∂r

(A.7)

Now, obtain ∂∂x from the first equation of A.7 in terms of the rest and substitute in the second and the third equations of A.7. Do the same for ∂∂y and ∂∂z . The reader can do this as an exercise. We finally obtain cos θ cos ψ ∂ sin ψ ∂ ∂ ∂ ∂ x = sin θ cos ψ ∂r + r ∂θ + r sin θ ∂ψ cos θ sin ψ cos ψ ∂ ∂ ∂ ∂ ∂ y = sin θ sin ψ ∂r + r ∂θ + r sin θ ∂ψ ∂ ∂ sin θ ∂ ∂ z = cos θ ∂r − r ∂θ

⎫ ⎪ ⎬ ⎪ ⎭

(A.8)

Appendix A: The concept of Directional Derivative

257

Back to the expressions for the unit vectors in Eq. A.8. We use the orthogonality to get: ⎫ ⎫ ⎫ i .ˆr = sin θ cos ψ ⎬ j.ˆr = sin θ sin ψ ⎬ k.ˆr = cos θ ⎬ i .θˆ = cos θ cos ψ j.θˆ = cos θ sin ψ k.θˆ = −sin θ ⎭ ⎭ ⎭ i.ψˆ = −sin ψ j .ψˆ = cos ψ k.ψˆ = 0

(A.9)

  For example, i.ˆr means the component of i onto rˆ and so on for the rest of dot product. This set produces the following set: ⎫ i = sin θ cos ψ rˆ + cos θ cos ψ θˆ − sin ψ ψˆ ⎬ j = sin θ sin ψ rˆ + cos θ sin ψ θˆ + cos φ ψˆ ⎭ k = cos θ rˆ − sin θ θˆ

(A.10)

Substituting A.10 and A.8 into Cartesian gradient, ∇=

∂ ∂ ∂ i+ j+ k ∂x ∂y ∂z

We get for each component ] [ ∂ ∂ cos θ cos ψ ∂ sin ψ ∂ i = sin θ cos ψ + + ∂x ∂r r ∂θ r sin θ ∂ψ   sin θ cos ψ rˆ + cos θ cos ψ θˆ − sin ψˆ ] [ ∂ ∂ cos θ sin ψ ∂ cos ψ ∂ j = sin θ sin ψ + + ∂y ∂r r ∂θ r sin θ ∂ψ   sin θ sin ψ rˆ + cos θ sin ψ θˆ + cos ψ ψˆ ] [  ∂ ∂ sin θ ∂  cos θ rˆ − sin θ θˆ k = cos θ − ∂z ∂r r ∂θ Summing up (exercise) and rearranging, we get a surprisingly short expression for the gradient in the spherical coordinates system: ∇=

1 ∂ ∂ 1 ∂ θˆ + ψˆ rˆ + ∂r r ∂θ r sin θ ∂ψ

(A.11)

Obtaining the directional derivative in 3D is a bit complicated ( ∂φ ∂s = Ω.∇φ); therefore, we will limit ourselves to spherical symmetry (r , μ) only. Just to get the feeling of complexity the streaming term in all dimensions and directions (general) is √ 1 − μ2 sinω ∂ϕ ∂ϕ 1 − μ2 ∂ϕ Ω.∇ϕ = μ + + ∂r r ∂μ r sinθ ∂ψ

258

Appendix A: The concept of Directional Derivative

√ +

1 − μ2 ∂ϕ cosω − r ∂θ

√ 1 − μ2 ∂ϕ sinωcotθ r ∂ω

Based on the assumption, the flux varies with r and therefore Ω,as a result, varies with μ as defined at the beginning and not equal to cosθ . We consider the simplifying equation A.4 where small changes in the angle θ affects the geometry as shown. Using the definition of the directional derivative and equation A.10, it can be written in spherical coordinates as ] [ ∂ψ(s) ∂f ∂r (s) ∂θ (s) θˆ + ψˆ = Ω.∇ f = rˆ + ∂s ∂s ∂s ∂s ] [ 1 ∂f ∂f 1∂f θˆ + ψˆ . rˆ + ∂r r ∂θ r sin θ ∂ψ ] [ ∂ f ∂r (s) 1 ∂ f ∂θ (s) 1 ∂ f ∂ψ(s) Ω.∇ f = + + ∂r ∂s r ∂θ ∂s r sin θ ∂ψ ∂s We drop the last term for the spherical symmetry ∂f ∂ f ∂r (s) 1 ∂ f ∂θ (s) = Ω.∇ f (r , θ ) = + ∂s ∂r ∂s r ∂θ ∂s and ∂f ∂ f ∂r (s) ∂ f ∂μ(s) = Ω.∇ f (r , μ) = + ∂s ∂r ∂s ∂μ ∂s The Ω component on rˆ is Ωr = Ω.ˆr = cosϑ = μ. Substituting for the differential triangle in Fig. 1.26, ∂r (s) dr ∂θ (s) dθ (s) sinϑ ≈ = cosϑ = μ and ≈ = ∂s ds ∂s ds r ∂ f ∂r (s) ∂f =μ ∂r ∂s ∂r and dμ(s) dμ dθ (s) sinθ 1 − μ2 = =− (−sinθ ) = ds dθ ds r r ∂ f ∂μ(s) 1 − μ2 ∂ f = ∂μ ∂s r ∂μ ∂f ∂f 1 − μ2 ∂ f = Ω.∇ f (r , μ) = μ + ∂s ∂r r ∂μ

∂r (s) ∂s

from

Appendix A: The concept of Directional Derivative

Fig. A.5 Directional derivative in a 3D spherical geometry

When f = φ, then (Fig. A.5) Ω.∇φ = μ

∂ 1 − μ2 ∂ φ(r , μ) + φ(r , μ) ∂r r ∂μ

259

B

Solid Angle and Differential Velocity

B.1

Solid Angle

It is the differential area d S on a spherically shaped surface encapsulated by the differential movement of the unit vector Ω in 3D space and referred to as dΩ. The differential surface area is estimated by approximating its dimensions in the phase space. Referring to Fig. B.1, the approximate dimensions are 1dθ ; 1(sin θ )dψ; hence, dΩ = sin θ dθ dψ The area drawn by the solid angle if θ and ψ is extended to their limits π and 2π , respectively, is the area of the unit sphere 4π or 

2π dΩ =

4π

Likewise, if θ = hemispherical area

π/2

0

and ψ = 2π, then the motion creates the upper 2π dΩ =

B.2

sin θ dθ = 4π

dψ 0

 2π

π

0

π/2 dψ sin θ dθ = 2π 0

Differential Velocity

The derivation of the differential velocity dυ is similar to that of the solid angle if we replace the unit radius by υ. Figure B.2 illustrates this quantity represented by the volume resulting from differential changes in the velocity and the angles θ and ψ. Therefore, the volume created is (υ sin θ dψ)(υdθ )(dυ) or dυ = υ 2 dυsin θ dθ dϕ = υ 2 dυdΩ

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 R. M. Kuridan, Neutron Transport, Graduate Texts in Physics, https://doi.org/10.1007/978-3-031-26932-5

261

262

Fig. B.1 The solid angle dΩ

Fig. B.2 Differential velocity dυ

Appendix B: Solid Angle and Differential Velocity

C

Radiative Transport

Transport theory refers to the mathematical description of the transport of particles through a host medium. In this section, we survey the various transport processes through the governing equations to emphasize three things besides informing the reader about the differences between the physical and mathematical characteristics: firstly, the applications of transport theory. Secondly, the size of difficulty when using the deterministic numerical methods as a method of solution. Thirdly, the type of probabilities involved in the balance equations are taken into account in random sampling. Figure C.1 shows different particles which may be transported through the medium.

C.1

Applications of the Transport Theory

They include the following areas: Nuclear reactors • Neutron flux distribution in reactor cores. • Shielding against neutron and gamma radiation. Astrophysics • Diffusion of light through stellar atmosphere.

Transport Processes of Particles

Neutrons Photons Electrons Ions Waves (less than a mfp) Gas molecules

Fig. C.1 Particles transported through the medium © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 R. M. Kuridan, Neutron Transport, Graduate Texts in Physics, https://doi.org/10.1007/978-3-031-26932-5

263

264

Appendix C: Radiative Transport

• Penetration of light through planetary atmospheres. Rarified gas dynamics • Upper atmosphere physics. • Sound propagation. • Diffusion of molecules in gases. Charged particle transport • • • • • • •

Multiple scattering of electrons. Gas discharge physics. Diffusion of holes and electrons in semiconductors. Development of cosmic ray showers. Transport of electromagnetic radiation. Multiple scattering of radar waves in a turbulent atmosphere. Penetration of X-rays through matter.

Plasma physics • Microscopic plasma dynamics micro-instabilities. • Plasma kinetic theory. Other • Traffic flow (transport of vehicles along highways). • Molecular orientation of macromolecules. • Particle transport equations.

C.1.1

Neutron Transport Equation

The general form of the neutron transport equation is   1 ∂φ + Ω · ∇φ + Σt φ r , E, Ω, t υ ∂t ∞        ' = dE dΩ ' Σs E ' , Ω ' → E, Ω φ r , E ' , Ω ' , t + S r , E, Ω, t 0

Appendix C: Radiative Transport

265

The neutron source can be localized or due to fission. The general source term is expressed as ∞      χ (E)     S r , E, Ω, t = d E ' dΩ ' ν E ' Σ f r , E ' φ r , E ' , Ω ' , t 4π 0   + Sl r , E, Ω, t Sample problems: • • • • • • •

Sources in an infinite media (plane or point). Behavior of flux near a free surface (Milne problem). Reflection of neutrons from interfaces or surface (albedo problem). Finite geometry problems (slabs or spheres). Criticality problems. Time-dependent problems. Initial value problem (pulsed neutrons).

C1.2 Photon Transport Equation: (Low Energy e.g. Light) We define the photon energy intensity as     Iγ r , E, Ω, t = (hν)cn r , E, Ω, t = photon energy times the photon flux. h is Planck’s constant ν is the photon frequency c is the speed of light n is the number of photons/cm2 s.

C1.3 Photon Transport Equation (High-Energy, e.g. Gamma and X-rays) Steady-state Boltzmann equation for gamma rays:     Ω · ∇ Iν r , λ, Ω + ∑(λ)Iν r , λ, Ω   ' '    ' ' δ 1+λ −λ−Ω ·Ω dΩ Ne k λ , λ Iν r , λ , Ω = dλ 2π 0 4π   + S r , λ, Ω λ

'



'



      and k λ' , λ = 2π λλ' σ λ' , θ

'

266

Appendix C: Radiative Transport

where:   Iν r , λ, Ω d EdΩ: energy flux density   Iν r , λ, Ω = E γ φγ : MeV/cm2 s λ: gamma ray energy after scattering expressed in terms of its Compton wavelength. E = hν, ν = c/λ λ' : gamma ray wavelength prior to scattering Ω ' · Ω = cos θ : cosine of the scattering angle between initial and final unit direction vectors Σ(λ): total macroscopic x-section at energy corresponding to the gamma ray wavelength Ne : electron density [no. of electrons/cm3 ] = ZN N: atom density   σ λ' , θ : microscopic x-section per electron for Compton scattering given by Klein–Nishina formula.   δ 1 + λ' − λ − Ω ' · Ω : dirac delta function that prescribes the angular change Ω ' ·Ω be consistent with the change in wave length λ' −λ as given by the Compton scattering equation λ' − λ = 1 − Ω ' · Ω).   S r , λ, Ω d EdΩ: energy emission by gamma ray sources [MeV/cm3 s].

C.1.4

Radiative Transport Equation         1 ∂ Iν + Ω · ∇ Iν = ρ r , t −κν r , Ω, t Iν r , Ω, t + εν r , Ω, t c ∂t

  ρ r , t : material density   κν r , Ω, t : absorption coefficient   εν r , Ω, t : emission coefficient.

Appendix C: Radiative Transport

C.1.5

267

Electron Transport Equation

The electron transport equation is similar to that of neutrons except for two physical differences. 1. strong infrequent collisions with heavy ions (nuclei) in which little energy transfer occurs (large-angle deflection)   1 ∂φ + Ω · ∇φ + ∑t φ r , E, Ω, t υ ∂t        = dΩ ' ∑s Ω ' → Ω φ r , E, Ω ' , t + S r , E, Ω, t 2. frequent weak collisions with atomic electrons which give rise to very irregular particle trajectories (small-angle deflections) It gives rise to energy loss. The energy loss over a given path length ξ . The differential energy loss is given by the Beth formula for ddξE and using dξ = υdt, hence,

1 ∂ υ ∂t

=

∂ ∂ξ

to get

  ∂φ + Ω · ∇φ + ∑t φ Ω = ∂ξ



    dΩ ' Σs Ω ' → Ω φ Ω ' + S

Monte Carlo is used because of multi-scatter events involved which are difficult or impossible to solve adequately by other mathematical techniques.

C.1.6

Ionized Gases and Plasma Transport Equation

The dynamics of ionized gases and plasmas is influenced by the long nature of Coulomb interaction.     ∂n e ∂n e ∂n e ∂n e e   = − +υ · E r, t + υ × B r, t . ∂t coll ∂t ∂r m ∂υ where n e : electron density E: electric field vector B: magnetic field vector υ: electron velocity.

D

Input Description of the 1D Program

D.1

Input Description

The 1D program solves the steady neutron diffusion equation in one-dimensional geometry using the finite-difference method. It can be run either with one energy group or two energy groups and either in a slab or cylindrical geometry. Temperature feedback can be simulated in terms of changes in absorption cross sections as a function of local power density. The input file has five data arrays, NDAT, ADAT, XSEC, AHX, and ILX in a free format. (1) DATA1 1. NMAX, maximum number of mesh intervals, ≤100 2. NG, number of energy groups, ≤2 3. NREG, number of different material regions, ≤ 4 4. NPG, geometry, 0 for a slab, and 1 for cylinder 5. BC(1), Left boundary condition at x = x0 , 0 = zero flux, 1 = zero current 6. BC(2), right boundary condition at x = x NMAX+1 , 0 = zero flux, 1 = zero current (2) DATA2 1. APOUT, total core output, (MW). Need only for feedback calculation. Feedback calculation will be bypassed if APOUT = 0 2. AREA, transverse cross-sectional area of core, (cm2 ); input only when APOUT /= 0 3. BSQ, transverse buckling 4. ελ , eigenvalue convergence criterion; default = 1.0 × 10–4 5. ε p , point-wise source convergence criterion; default = 1.0 × 10–4 6. λ, initial eigenvalue guess, default = 1.0 ∑ ∂ a(cm−1)  , input only when APOUT / = 0  7. APC1, power coefficient, w ∂P

cm3

(3) AHX1 Mesh spacing for N = 1,2, ….. NMAX. (4) ILX1 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 R. M. Kuridan, Neutron Transport, Graduate Texts in Physics, https://doi.org/10.1007/978-3-031-26932-5

269

270

Appendix D: Input Description of the 1D Program

Material region number for each mesh interval, N =1, 2, 3, …. NMAX. (5) XSEC One group cross sections: ∑a

D

Identifier #

ν∑ f

Two group cross sections, Identifier #

D1

∑a1

∑s12

ν∑ f 1

D2

∑a2

ν∑ f 2

The identifier corresponds to the material region number given in the array ILX designated by a set of cross sections. Sample input text file: A bare core without a reflector (one region slab core), one group cross sections with a fixed mesh size, and zero flux boundary conditions. The slab corresponds to the axial direction of the reactor core. File name1.inp NMAX, NG, NREG, PG, BC 26 1 1 0 0 0 POUT, AREA, BSQ, EPSE, EPSP, LAMDA 0.0 0.0 0.0 1.0e-4 1.0e-4 1.0 (AHX(I), I=1, NMAX) 26*6.0 (ILX(I), I=1, NMAX) 26*1 (XSEC(I), I=1, NMG) 1,9.21,0.1532,0.157,

D.2

Running the 1D Program

There are two ways of running the 1D program: (1) Once the input file is prepared in a notepad file you may run (execute) the 1D program by double clicking the.exe file or (application file: the compiled source Fortran file). Both files can be downloaded from the publisher assigned site. You will be prompted for the input file name (name1.inp) and then you will be prompted for the output file name (name2.out). name2.out contains a detailed output. Finally, you will be prompted for another output file name (name3.out). The output file name3.out contains the power and relative flux profiles prepared for plotting using excel or other software.

Appendix D: Input Description of the 1D Program

271

The output file name2.out consists of an edit of the DAT and other arrays, flux, and power profiles. Flux is normalized to one neutron produced per second in the core. Power distribution is also normalized so that the power averaged over the core is unity. The eigenvalue which is equivalent to the effective multiplication factor is also printed out. (2) Using the file 1D.mlapp (interface window) working in a MATLAB environment. You will be prompted for the input items and then RUN the program Note: You can execute all the jobs without reference to the. For file unless you want to modify it, then you have to compile it using a Fortran compiler and produce a new .exe file such as the tasks described in some problems in the exercises. Sample output text file:

D.3

Output Description

The output file name2.out consists of an edit of the DAT and other arrays, flux, and power profiles. Flux is normalized to one neutron produced per second in the core. Power distribution is also normalized so that the power averaged over the core is unity. The eigenvalue which is equivalent to the effective multiplication factor is also printed out.

272

Appendix D: Input Description of the 1D Program

File name2.out NMAX 26 1 POUT 0.000000E+00 1.000000E-04 AHX(I 6.000000 6.000000 6.000000 6.000000 6.000000 6.000000 6.000000 ILX(I 1 1 1 1 1 1 1 1 1 1 XSEC(I 1.000000

1

0

0

0

0.000000E+00 0.000000E+00 1.000000 6.000000 6.000000 6.000000 6.000000 6.000000 6.000000 6.000000 1 1 1 1

6.000000 6.000000 6.000000 6.000000 6.000000 6.000000 1 1 1 1

9.210000

1 1 1 1

1.532000E-01

1.000000E-04

6.000000 6.000000 6.000000 6.000000 6.000000 6.000000 1 1 1 1 1.570000E-01

**** ONED DIFFUSION CODE **** MAXIMUM NO. OF MESH INTERVAL NMAX= 26 NO. OF ENERGY GROUPS NG= 1 NO. OF DIFFERENT MATERIAL REGIONS NREG= 1 GEOMATERY 0/1=SLAB/CYLINDER PG= 0 LEFT BOUNDARY CONDITION 0/1 FLUX/CURRENT BC(1)= 0 RIGHT BOUNDARY CONDITION 0/1 FLUX/CURRENT BC(2)= 0 TOTAL CORE POWER OUTPUT (MW) POUT= .00000E+00 TRANSVER X-SECTIONAL AREA OF CORE AREA= .00000E+00 TRANSVER BUCKLING (CM-2) BSQ= .00000E+00 EIGEN VALUE CONVERGENCE CRITEION = .10000E-03 POINT WISE SOURCE CONVERGENCE = .10000E-03 INITIAL EIGEN VALUE GUESS = 1.00000 I

ILX

1 2 3

1 1 1

AHX(CM) 6.00000 6.00000 6.00000

Appendix D: Input Description of the 1D Program 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

6.00000 6.00000 6.00000 6.00000 6.00000 6.00000 6.00000 6.00000 6.00000 6.00000 6.00000 6.00000 6.00000 6.00000 6.00000 6.00000 6.00000 6.00000 6.00000 6.00000 6.00000 6.00000 6.00000 **** ONED DIFFUSION CODE ****

CROSS SECTIONS ----------------------------------------------|REGION| D | SA | NUSF | ----------------------------------------------| 1 | .92100E+01| .15320E+00| .15700E+00| ----------------------------------------------| 0 | .00000E+00| .00000E+00| .00000E+00| ----------------------------------------------| 0 | .00000E+00| .00000E+00| .00000E+00| ----------------------------------------------| 0 | .00000E+00| .00000E+00| .00000E+00| ----------------------------------------------*** FLUX HAS CONVERGED AFTER 24 ITERATION *** WITH EIGEN VALUE= 1.0004330 POWER AND ONE GROUP FLUX VS POSITION

273

274

Appendix D: Input Description of the 1D Program -------------------------------------|MESH |POSITION| POWER | FLUX | -------------------------------------| 1 | .00 | .09482 |.00000E+00| -------------------------------------| 2 | 6.00 | .28308 |.77464E-02| -------------------------------------| 3 | 12.00 | .46719 |.15379E-01| -------------------------------------| 4 | 18.00 | .64447 |.22788E-01| -------------------------------------| 5 | 24.00 | .81233 |.29862E-01| -------------------------------------| 6 | 30.00 | .96831 |.36501E-01| -------------------------------------| 7 | 36.00 | 1.11015 |.42605E-01| -------------------------------------| 8 | 42.00 | 1.23579 |.48088E-01| -------------------------------------| 9 | 48.00 | 1.34339 |.52869E-01| -------------------------------------| 10 | 54.00 | 1.43140 |.56879E-01| -------------------------------------| 11 | 60.00 | 1.49854 |.60059E-01| -------------------------------------| 12 | 66.00 | 1.54385 |.62364E-01| -------------------------------------| 13 | 72.00 | 1.56667 |.63761E-01| -------------------------------------| 14 | 78.00 | 1.56667 |.64228E-01| -------------------------------------| 15 | 84.00 | 1.54385 |.63761E-01| -------------------------------------| 16 | 90.00 | 1.49854 |.62364E-01| -------------------------------------| 17 | 96.00 | 1.43140 |.60059E-01| -------------------------------------| 18 | 102.00 | 1.34339 |.56879E-01| -------------------------------------| 19 | 108.00 | 1.23579 |.52869E-01| -------------------------------------| 20 | 114.00 | 1.11015 |.48088E-01| -------------------------------------| 21 | 120.00 | .96831 |.42605E-01| -------------------------------------| 22 | 126.00 | .81233 |.36501E-01| -------------------------------------| 20 | 114.00 | 1.11015 |.48088E-01| -------------------------------------| 21 | 120.00 | .96831 |.42605E-01| -------------------------------------| 22 | 126.00 | .81233 |.36501E-01| --------------------------------------

Appendix D: Input Description of the 1D Program

275

| 23 | 132.00 | .64447 |.29862E-01| -------------------------------------| 24 | 138.00 | .46719 |.22788E-01| -------------------------------------| 25 | 144.00 | .28308 |.15379E-01| -------------------------------------| 26 | 150.00 | .09482 |.77464E-02| -------------------------------------| 27 | 156.00 | .00000 |.00000E+00| -------------------------------------**** ONED DIFFUSION CODE ****

File name3.out POSITION POWER FLUX .00 .09482 .00000E+00 6.00 .28308 .77464E-02 12.00 .46719 .15379E-01 18.00 .64447 .22788E-01 24.00 .81233 .29862E-01 30.00 .96831 .36501E-01 36.00 1.11015 .42605E-01 42.00 1.23579 .48088E-01 48.00 1.34339 .52869E-01 54.00 1.43140 .56879E-01 60.00 1.49854 .60059E-01 66.00 1.54385 .62364E-01 72.00 1.56667 .63761E-01 78.00 1.56667 .64228E-01 84.00 1.54385 .63761E-01 90.00 1.49854 .62364E-01 96.00 1.43140 .60059E-01 102.00 1.34339 .56879E-01 108.00 1.23579 .52869E-01 114.00 1.11015 .48088E-01 120.00 .96831 .42605E-01 126.00 .81233 .36501E-01 132.00 .64447 .29862E-01 138.00 .46719 .22788E-01 144.00 .28308 .15379E-01 150.00 .09482 .77464E-02 156.00 .00000 .00000E+00

This sample problem is a criticality search problem where the core dimensions and the nuclear properties such as nuclear densities of the isotopes making up the core and the reflector are known and the task is to find the eigenvalue of the system (the criticality constant). The problem could be reversed if the reactor is known to be critical (the eigenvalue ∼ = 1) and the nuclear densities are unchanged however the dimensions need to be found. In this case, it is a critical size search problem where the mesh number and spacing are varied in a systematic way until the eigenvalue is nearly equal to one with an acceptable deviation. The fissile density could also be varied through enrichment and the densities of absorber material such as control absorber, soluble Boron, or burnable absorbers can also

276

Appendix D: Input Description of the 1D Program

be varied and therefore the nuclear cross sections in the input file are altered and the effect on criticality can be observed and hence studied. Notice that ∑ = N σ , where N is the isotopic density and σ is the microscopic cross section.

E

Sample Problems and Exercises for the 1D Program

1. Bare (un-reflected) slab research reactor core: The slab represents the core axially in harmony with the expected cosine shape solution (Fig. E.1). The input file with one group cross sections is shown in Fig. E.2. The mesh spacing is varied with thicknesses: 2; 4; 6; 12 cm and the resulting fluxes are plotted along with the analytic solution [Ccos(π/H )]. In all the cases, the mesh spacing need not to be equal (Fig. E.1). The relative fluxes all coincide with the analytic solution which implies the accuracy of the numerical scheme (Fig. E.3). The variation of the mesh spacing does not alter the shape and the eigenvalues are almost unaffected as expected. However, if on the other hand the cross sections are modified the eigenvalues will change (Figs. E.2 and E.3).

top reflector

core

Slab height

Radial reflector

bottom reflector

Fig. E.1 Core and reflectors in cylindrical geometry

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 R. M. Kuridan, Neutron Transport, Graduate Texts in Physics, https://doi.org/10.1007/978-3-031-26932-5

277

278

Appendix E: Sample Problems and Exercises for the 1D Program

NMAX, NG, NREG, PG,BC 26 1 1 0 0 0 POUT, AREA, BSQ, EPSE, EPSP, LAMDA 0.0 0.0 0.0 1.0e-4 1.0e-4 1.0 (AHX(I), I=1, NMAX) 26*6.0 (ILX(I), I=1, NMAX) 26*1 (XSEC(I), I=1, NMG) 1 9 21 0 1532 0 157 Fig. E.2 Bare slab core input file

(∆x=2 cm)

(∆x=4 cm)

(∆x=6 cm)

0.070

Normalized flux

0.060 0.050 0.040 0.030 0.020 0.010 0.000 78

84

90

96 102 108 114 120 126 132 138 144 150 156

Core hight (cm) Fig. E.3 Relative fluxes in the upper part of the bare slab core

2. Reflected slab core: It is reflected from above and below. A three-region core needs to be defined (reflector–core–reflector) and one group cross sections are chosen. The input file is shown in Fig. E.4. 3. Reflected cylindrical core with two energy group cross sections: Because, of the radial symmetry of the cylinder two regions can be described, the core and its reflector. Therefore, the boundary conditions are zero current at the center and zero flux on the outside and the printout follow accordingly. The input file is presented in Fig. A.2.e. The eigenvalue is about 1.26 because it is a core-free poison (Fig. E.5). Exercises 1. You may test the effect of changing the nuclear cross sections and mesh spacing on criticality. Also, you can calculate the reflector savings on the core size by considering a bare critical core and then repeating the calculation after introducing a reflector. The eigenvalue will increase because some neutrons are reflected

Appendix E: Sample Problems and Exercises for the 1D Program

279

NMAX, NG, NREG, PG, BC 38 1 3 0 0 0 POUT, AREA, BSQ, EPSE, EPSP, LAMDA 0.0 0.0 0.0 1.0e-4 1.0e-4 1.0 (AHX(I), I=1, NMAX) 6*2.0 26*6.0 6*2.0 (ILX(I), I=1, NMAX) 6*1 26*2 6*3 (XSEC(I), I=1, NMG) 1,1.328,.0181,0, 2,9.21,0.1532,0.157, 3,1.328,.0181,0, Fig. E.4 Reflected slab core input file

NMAX, NG, NREG, PG, BC

39 2 2 1 1 0 POUT, AREA, BSQ, EPSE, EPSP, LAMDA 0.0 0.0 0.0 1.0e-4 1.0e-4 1.0 (AHX(I), I=1, NMAX) 34*5.0 5*5.0 (ILX(I), I=1, NMAX) 34*1 5*2 (XSEC(I),I=1,NMG) 1,1.2627 0.01207 .02619 .008476 .3543 .121 .18514 2,1.13 0.0004 0.0494 0.000 0.16 0.0197 0.000 Fig. E.5 Reflected cylindrical core input file

2. 3. 4. 5. 6.

7.

back to the core enhancing it to become supercritical. In order to regain criticality the core size is reduced and the amount of volume removed is the savings as a result of reflector presence. Improve the printout of the text output file to look more elegant. Develop an interface window to use Matlab for input/output data. Modify the 1D program so that the normalized flux and normalized power output become in MW and n/cm2 s, respectively. Modify the 1D program to perform a depletion calculation (the numerical scheme is in Chap. 2). Modify the 1D program to perform an adjoint calculation (theory is in Chap. 4). (Notice that the eigenvalues should be equal in both the forward and the adjoint versions). Convert the FORTRAN file to PYTHON programing language.

F

Legendre Polynomials

Legendre differential equation   1 − x 2 y '' − 2x y ' + n(n + 1)y = 0 where n is a real number in the range (−1, 1) l

Pn (x) =

≤2 ∑

(−1)l

l=0

Pn (x) =

(2n − 2l)! x n−2l 2n l!(n − l)!(n − 2l)! 1

2n n!

dn 2 (x − 1)n dxn

Properties: Conjugate Legendre polynomials series: m/2

Pnm (x) = (1 − x 2 )

dm Pn (x), −n ≤ m ≤ n dxm

P0 (x) = 1, (x) = x, P4 (x) = P5 (x) =

P2 (x) =

 1 2 3x − 1 2

1 (35x 4 − 30x 2 + 3), 8

1 (63x 5 − 70x 3 + 15x) . . . . 8

Orthogonality: 

1 d x Pn (x)Pm (x) = −1

0 2 2n+1

m /= n m=n

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 R. M. Kuridan, Neutron Transport, Graduate Texts in Physics, https://doi.org/10.1007/978-3-031-26932-5

281

282

Appendix F: Legendre Polynomials

Recursion relations: (n + 1)Pn+1 (x) − (2n + 1)x Pn (x) + n Pn−1 (x) = 0 ' Pn+1 (x) − x Pn' (x) = (n + 1)Pn (x)



1 − x2

 ∂ Pn (x) ∂x

=

 n(n + 1)  Pn−1 (x) − Pn+1 (x) (2n + 1)