Modelling of Nuclear Reactor Multi-physics: From Local Balance Equations to Macroscopic Models in Neutronics and Thermal-Hydraulics 0128150696, 9780128150696

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Modelling of Nuclear Reactor Multi-physics

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Modelling of Nuclear Reactor Multi-physics From Local Balance Equations to Macroscopic Models in Neutronics and Thermal-Hydraulics Christophe Demazie`re

Academic Press is an imprint of Elsevier 125 London Wall, London EC2Y 5AS, United Kingdom 525 B Street, Suite 1650, San Diego, CA 92101, United States 50 Hampshire Street, 5th Floor, Cambridge, MA 02139, United States The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, United Kingdom Copyright © 2020 Elsevier Inc. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher. Details on how to seek permission, further information about the Publisher’s permissions policies and our arrangements with organizations such as the Copyright Clearance Center and the Copyright Licensing Agency, can be found at our website: www.elsevier.com/permissions. This book and the individual contributions contained in it are protected under copyright by the Publisher (other than as may be noted herein). Notices Knowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary. Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein. In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility. To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein. Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library ISBN: 978-0-12-815069-6 For information on all Academic Press publications visit our website at https://www.elsevier.com/books-and-journals Publisher: Joe Hayton Acquisition Editor: Maria Convey Editorial Project Manager: Aleksandra Packowska Production Project Manager: Sruthi Satheesh Cover Designer: Greg Harris Typeset by TNQ Technologies

To Anna, Clara, Noah and Elsa To Marcelle and Jean

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Contents List of Abbreviations 1. Introduction

xiii 1

1.1 Topics covered in the book

1

1.2 Structure of the book

2

1.2.a Contents

2

1.2.b Pedagogical approach

3

1.3 Notations and conventions used in the book

4

1.4 Reminder about some useful mathematical concepts

5

1.4.a Calculus on scalars, vectors and tensors

5

1.4.b Spherical coordinates and solid angles

8

1.4.c Gauss divergence theorems References

2. Transport phenomena in nuclear reactors 2.1 Nuclear reactors as multi-physics and multi-scale systems

10 11

13 13

2.1.a Multi-physics aspects

13

2.1.b Multi-scale aspects

19

2.2 Neutron transport

25

2.2.a Introduction

25

2.2.b Derivation of the neutron transport equation

33

2.3 Fluid dynamics

43

2.3.a Mathematical formalism

43

2.3.b Generic differential conservation laws

46

2.3.c Mass and momentum differential conservation equations

48 vii

viii Contents

2.4 Heat transfer

50

2.4.a Heat transfer by conduction

51

2.4.b Heat transfer by convection

52

2.5 Overview of the modelling strategies

57

2.6 Deterministic and macroscopic modelling of nuclear systems

60

2.6.a Equations governing the neutron flux

60

2.6.b Equations governing the temperature and flow fields

65

2.6.c Coupling between the neutron kinetic and thermalhydraulic modellings

68

2.7 Conclusions

69

References

70

3. Neutron transport calculations at the cell and assembly levels 3.1 Representation of the energy dependence

73 73

3.1.a Multi-group formalism

73

3.1.b Nuclear data libraries

77

3.2 Treatment of resonances

81

3.2.a Introduction

81

3.2.b Neutron slowing-down without absorption

83

3.2.c Neutron slowing-down with absorption

92

3.3 Resolving the energy dependence

108

3.4 One-dimensional micro-group pin cell calculations

118

3.4.a Introduction

118

3.4.b Transport correction

119

3.4.c Method of collision probabilities

122

3.4.d Properties of the probabilities

127

3.4.e Application of the method of collision probabilities

129

3.4.f Rational approximation

132

Contents

3.5 Two-dimensional macro-group lattice calculations

ix

138

3.5.a Introduction

138

3.5.b Method of characteristics

138

3.5.c Discrete ordinates (SN) method

143

3.5.d Interface current method

152

3.5.e Acceleration methods

160

3.6 Criticality spectrum calculations

164

3.6.a Introduction

164

3.6.b Properties of integral operators in infinite and homogeneous media

164

3.6.c Integral operators in critical systems

165

3.6.d Homogeneous B1 method

167

3.6.e Homogeneous P1 method

173

3.6.f Fundamental mode method

174

3.7 Cross-section homogenization and condensation

175

3.8 Depletion calculations

178

3.9 Cross-section preparation for core calculations

184

3.10 Conclusions

189

References

191

4. Neutron transport calculations at the core level 4.1 Angular discretization of the neutron transport equation

193 193

4.1.a Spherical harmonics (PN) method

193

4.1.b Diffusion theory

202

4.1.c Simplified PN method (SPN)

208

4.1.d Boundary conditions

213

4.2 Spatial discretization of the neutron transport equation

218

4.2.a Introduction

219

4.2.b Finite difference methods

221

x Contents

4.2.c Nodal methods

223

4.2.d Finite elements

227

4.3 Determination of the steady-state core-wise solution

231

4.3.a Introduction

231

4.3.b Direct methods

232

4.3.c Iterative methods

232

4.4 Determination of the non-steady-state core-wise solution

237

4.4.a Introduction

237

4.4.b Analysis of the balance equations with respect to the prompt neutrons

241

4.4.c Analysis of the balance equations with respect to the delayed neutrons

247

4.5 Conclusions

248

References

249

5. One-/two-phase flow transport and heat transfer

251

5.1 Tools required for flow transport modelling

251

5.1.a Introduction

251

5.1.b Two-phase flow regimes

252

5.1.c Mathematical tools

254

5.2 Derivation of the space- and time-averaged conservation equations for flow transport

258

5.2.a Introduction

258

5.2.b Space-averaging of the local conservation equations

258

5.2.c Time-averaging of the space-averaged conservation equations

262

5.2.d Equations to be solved

263

5.3 Flow models

277

5.3.a Two-fluid model

277

5.3.b Mixture models with specified drift velocities

283

5.3.c Homogeneous equilibrium model

288

Contents

xi

5.4 Spatial and temporal discretizations of the flow models

289

5.5 Modelling of heat conduction in solid structures

300

5.6 Conclusions

307

References

308

6. Neutronic/thermal-hydraulic coupling 6.1 Introduction

311 311

6.2 Modelling of the dependencies of the nuclear material data 314 6.2.a Introduction

314

6.2.b Data functionalization on base and partial values

316

6.2.c Tree-leaf representation

317

6.2.d Polynomial fitting

318

6.3 Spatial coupling

320

6.3.a Thermal-hydraulic to neutronic coupling

320

6.3.b Neutronic to thermal-hydraulic coupling

320

6.3.c Coupling coefficients

321

6.4 Temporal coupling

323

6.4.a Introduction

323

6.4.b Operator Splitting approaches

324

6.4.c Integrated approaches

329

6.5 Conclusions

335

References

336

7. Conclusions

337

7.1 Summary

337

7.2 Outlook

339

References

342

Index

345

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List of Abbreviations BOC Beginning of Cycle BWR Boiling Water Reactor CFD Computational Fluid Dynamics CFL Courant-Friedrichs-Lewy (condition) DSA Diffusion Synthetic Acceleration ENDF Evaluated Nuclear Data File EOC End of Cycle JEF Joint Evaluated File JEFF Joint Evaluated Fission and Fusion JFNK Jacobian-Free Newton Krylov FIV Flow-Induced Vibration FSI Fluid-Structure Interaction GMRES Generalized Minimal Residual method HEM Homogeneous Equilibrium Model IR Intermediate Resonance LWR Light Water Reactor NR Narrow Resonance OS Operator Splitting PWR Pressurized Water Reactor RI Resonance Integral RPV Reactor Pressure Vessel SOR Successive Over Relaxation WR Wide Resonance

xiii

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Introduction

1

1.1 Topics covered in the book This book deals with the macroscopic modelling of nuclear reactor cores, with emphasis on neutron transport, fluid dynamics, heat transfer, and the interdependencies between these three areas of physics. Due to the complexity and large size of a nuclear reactor core, a modelling of all physical processes on all scales represents a challenge from a computational viewpoint. Methods were thus specifically derived to simplify the problem at hand by averaging the true local balance equations onto sufficiently large ‘meshes’ for each of the variables used to describe each of the physics. As a result, the corresponding methods are referred to as macroscopic approaches. Such methods, because of their moderate computing costs, are routinely used by the nuclear industry to characterize the behaviour of a core during both normal and abnormal situations not leading to core damage. The averaging procedure used to transform the actual local balance equations into their macroscopic equivalent form is far from trivial. In addition, many approximations are introduced when performing the averaging of the local balance equations. The purpose of the present book is to present, in a comprehensive manner, how to derive from the true local balance equations the macroscopic balance equations the modelling tools used by the nuclear industry solve. The corresponding methods and algorithms are thus thoroughly presented, together with their approximations and limitations. The primary objective with this book is to give the readers a description of such methods and algorithms, so that the readers can later use the codes based on those methods with confidence and for situations falling into the range of validity of the algorithms. The book also tackles the modelling of all physical phenomena of importance and their interdependencies. This guarantees a holistic approach to the field of nuclear reactor modelling. This is of particular importance in the area of nuclear reactor transients, where the analysts need to be equally knowledgeable in neutron transport, fluid dynamics and heat transfer. Since the emphasis of the book is on methods used by the industry for performing routine calculations, the methods have to be fast running. On the neutron transport side, this means that only deterministic methods are presented. Probabilistic methodseMonte Carloeare thus not dealt with. On the fluid dynamics side, only system code approaches are described. ‘True’ Computational Fluid Dynamics (CFD) methods are not tackled in this book due their high computational cost when modelling large systems as nuclear reactors. Modelling of Nuclear Reactor Multi-physics. https://doi.org/10.1016/B978-0-12-815069-6.00001-5 Copyright © 2020 Elsevier Inc. All rights reserved.

1

2 Modelling of Nuclear Reactor Multi-physics

Although the book can also be considered as dealing with the modelling of nuclear reactor multi-physics and their corresponding multi-scale phenomena, only neutron transport, fluid dynamics and heat transfer are presented. Other physics such as fuel behaviour, structural mechanics, coolant and radiation chemistry and radionuclide transport are not described in this book. In some specific situations, those phenomena might need to be considered. The methods presented in the book can thus be used to model the behaviour of nuclear reactor cores during normal operation and during reactor transients, i.e., during abnormal situations not leading to core damage. All methods are presented in a generic manner and are not specific to any particular code. With the objective of being as generic as possible, the book does not either contain any macroscopic correlations used in the modelling of fluid dynamics and heat transfer. Although such correlations are the backbone of the macroscopic modelling tools and are thus essential for applying the methods presented in this book, it was a deliberate choice not to include any correlation in this book. The interested reader is instead referred to the very abundant literature on this subject. Likewise, on the neutronic side, no values of nuclear cross-sections are given. Nuclear cross-section data should be obtained using suitable nuclear data libraries. Finally, it should be emphasized that the methods described in this book mostly target light-water reactor (LWR) systems, although some of the methods could be used for or extended to other reactor types. The book is primarily addressed to the readers wishing to get a comprehensive view on the modelling of nuclear reactors, starting from the true local balance equations to approximate macroscopic balance equations. Although many methods are presented, the reader interested in the implementation details of a specific method might need to find complementary information from some other sources.

1.2 Structure of the book 1.2.a

Contents

The book is organized in seven chapters, whereof this chapter is the first one. In the second chapter, the governing equations for neutron transport, fluid transport, and heat transfer are derived, so that the readers not familiar with any of these fields can comprehend the following chapters without difficulty. The peculiarities of nuclear reactor systems, i.e. their multi-physics and multi-scale aspects, are dealt with. An overview of the modelling strategies is thereafter given. On the neutronic side, deterministic methods are emphasized, whereas on the thermalehydraulic side, system code modelling approaches are given particular attention. In the third chapter, the computational methods for neutron transport at both the pin cell and fuel assembly levels are presented. The chapter is aimed at following the solution procedure in fuel pin/lattice codes as much as possible. This includes resonance calculations of the cross-sections, the determination of the micro-region micro-fluxes,

Chapter 1
Introduction

3

and of the macro-region macro-fluxes, and finally spectrum correction. The chapter ends with the preparation of the macroscopic cross-sections for subsequent core calculations, where the effect of burn-up is also detailed. In the fourth chapter, the computational methods in use for core calculations are presented. In the first part of this chapter, the treatment of the angular dependence of the neutron flux is described. In the second part, the treatment of the spatial dependence of the neutron flux is outlined. Thereafter, the solution procedure for estimating the core-wise position-dependent multi-group neutron flux is described. Finally, the methodology used for determining the core-wise space- and time-dependent neutron flux in case of transient calculations is derived. The fifth chapter of the book focuses on the computational methods used for one-/ two-phase flow transport and heat transfer. From the local governing equations describing fluid dynamics and heat transfer, macroscopic governing equations are derived and the underlying assumptions clearly emphasized. The discretization of those equations with respect to space and time are considered, with emphasis on the convergence, stability and consistency of the corresponding numerical schemes. The different flow models commonly used in nuclear engineering are also introduced, models having various levels of sophistication. The sixth chapter deals with the coupling between neutron transport, fluid dynamics and heat transfer solvers for coarse mesh macroscopic models. Various aspects of multiphysics coupling are addressed: segregated versus monolithic approaches, numerical coupling, code implementation, coupling terms and the corresponding non-linearities thus introduced and data exchange. Special emphasis is then put on the preparation of the macroscopic cross-sections. The issue of geometrical coupling is thereafter presented. Finally, the numerical techniques for solving the multi-physics problem either in a segregated or monolithic manner are detailed. The last chapter of the book gives a quick and concise overview of the main steps of the modelling techniques presented in the book and their inter-relation. Some outlook on current efforts on beyond state-of-the-art modelling techniques relying on more integrated multi-physics and multi-scale approaches is also given.

1.2.b

Pedagogical approach

Special emphasis was put in the preparation of this book on maximizing the reader’s learning outcomes. The book is thus made of several components:
This book itself (available in printed and electronic formats).
A set of short videos accompanying the book (available in electronic format at https://www.elsevier.com/books-and-journals/book-companion/9780128150696).
A set of quizzes accompanying the book (available in electronic format at https:// www.elsevier.com/books-and-journals/book-companion/9780128150696). The details of the methods are presented in the book, whereas the videos aim at extracting the key elements from those methods, their approximations and limitations.

4 Modelling of Nuclear Reactor Multi-physics

The videos should thus allow the readers to build a conceptual understanding of the topics presented in the book, together with their cross-dependencies and possible hierarchical structure. The quizzes, on which the readers can train a multiple number of times, are designed so that the readers must reflect on the learned concepts. The corresponding progression builds upon the observation that learning is process going from low-order thinking skills (such as remembering and understanding new concepts) to higher-order thinking skills (such as applying, analysing and evaluating such concepts and creating new ones) (Krathwohl, 2002). In terms of learning sequence, it is thus recommended to first read a section of a book, watch the corresponding short video(s) and then work on the quizzes. Beyond understanding the methods presented in the book together with their approximations and limitations, the book could also be used as a solid basis for designing computational tasks aimed at implementing those methods. This procedural understanding nevertheless requires the support from experienced modellers and is best carried out under the teacher’s supervision. The book and its associated electronic resources (videos and quizzes), in addition to offering self-paced learning, could be used as a preparation to possible in-class sessions that focus on the implementation of the methods in a given computing environment. These more engaging activities would promote a deeper learning of the topics.

1.3 Notations and conventions used in the book In this book, different quantities are introduced and used: scalars (generically denoted c hereafter), vectors (generically denoted c) and tensors (generically denoted c). Both vectors and tensors are indicated in bold face, whereas scalars are indicated in italics. A tensor c can be considered as a collection of physical quantities depending on a basis. In a three-dimensional Cartesian coordinate system (x1,x2,x3), an orthonormal basis (e1,e2,e3) can be constructed, with one unit vector ei associated to each of the corresponding coordinates xi,i ¼ 1,2,3. Correspondingly, a vector can then be seen as a 3  1 array, i.e. 3 c1 6 7 c ¼ 4 c2 5 ¼ ðci Þi¼1;:::;3hðci Þ c3 2

(1.1)

with i being the row index and ci representing the component of the vector c on the vector ei of the orthonormal basis. For the sake of simplicity, the explicit numbering of the row indexes is dropped and a vector c is simply denoted (ci). A tensor can be seen as a 3  3 array, i.e. c11 6 c ¼ 4 c21

c12 c22

3 c13 7 c23 5 ¼ ðcij Þi;j¼1;:::;3hðcij Þ

c31

c32

c33

2

(1.2)

Chapter 1
Introduction

5

with i being the row index and j being the column index. For the sake of simplicity, the explicit numbering of the row/column indexes is dropped and a tensor c is simply denoted (cij). Surface integrals on a surface S are integrals on a two-dimensional spatial domain and are thus denoted in this book as: Z

:::d 2 r

(1.3)

S

whereas volume integrals on a volume V are integrals on a three-dimensional spatial domain and are consequently denoted as: Z

:::d 3 r

(1.4)

V

In the equations above, d2r represents an elementary surface (of the surface S) and d r represents an elementary volume (of the volume V). 3

1.4 Reminder about some useful mathematical concepts 1.4.a

Calculus on scalars, vectors and tensors

Some reminder on basic calculus with scalars, vectors and tensors is given hereafter. In the following, the Einstein (summation) notation is used, i.e. any index appearing twice in a single term implies the summation over all of its possible values. For instance, 3 P ck xk is simply written as ckxk. k¼1

Calculus on scalars, vectors and tensors can then be performed according to the following rules:
The sum of a vector v¼(vi) and a vector w¼(wi) is a vector given by v þ w ¼ ðvi þ wi Þ

(1.5)


The multiplication of a vector v¼(vi) by a scalar l is a vector given by lv ¼ ðlvi Þ

(1.6)


The scalar product (or dot product) of a vector v¼(vi) and a vector w¼(wi) is a scalar given by v $ w ¼ vk w k

(1.7)

6 Modelling of Nuclear Reactor Multi-physics


The vector product (or cross product) of a vector v¼(vi) and a vector w¼(wi) is a vector given by 3 v2 w3  v3 w2 7 6 v  w ¼ 4 v3 w1  v1 w3 5 ¼ ðεikl vk wl Þ v1 w2  v2 w1 2

(1.8)

where the Levi-Civita symbol is defined as εijk

8 > < þ1 if ði; j; kÞ ¼ ð1; 2; 3Þ or ð2; 3; 1Þ or ð3; 1; 2Þ ¼ 0 if i ¼ j or j ¼ k or i ¼ k > : 1 if ði; j; kÞ ¼ ð3; 2; 1Þ or ð2; 1; 3Þ or ð1; 3; 2Þ

(1.9)


The tensor product (or dyadic product or outer product) of a vector v¼(vi) and a vector w¼(wi) is a tensor given by v 5 w ¼ ðvi wj Þ

(1.10)


The sum of a tensor t¼(tij) and a tensor s¼(sij) is a tensor given by t þ s ¼ ðtij þ sij Þ

(1.11)


The multiplication of a tensor t¼(tij) by a scalar l is a tensor given by lt ¼ ðltij Þ

(1.12)


The scalar product (or double dot product) of a tensor t¼(tij) and a tensor s¼(sij) is a scalar given by t: s ¼ tij sij

(1.13)


The tensor product (or single dot product) of a tensor t¼(tij) and a tensor s¼(sij) is a tensor given by t $ s ¼ ðtik skj Þ

(1.14)

Chapter 1
Introduction

7


The vector product (or dot product) of a tensor t¼(tij) and a vector v¼(vi) is a vector given by alternatively t $ v ¼ ðtik vk Þ

(1.15)

v $ t ¼ ðvk tki Þ

(1.16)

or The differential operator V (‘del’), which is a vector defined in a Cartesian coordinate system as V¼(v/vxi)¼(Vi), can also be applied to scalars, vectors and tensors following the rules given above. This results in the following relationships.
The gradient of a scalar s is a vector given by Vs ¼ ðVi sÞ

(1.17)


The divergence of a vector v¼(vi) is a scalar given by V $ v ¼ Vk vk

(1.18)


The curl (or rotor) of a vector v¼(vi) is a vector given by V  v ¼ ðεikl Vk vl Þ

(1.19)


The gradient of a vector v¼(vi) is a tensor given by V 5 v ¼ ðVi vj Þ

(1.20)


The divergence of a tensor t¼(tij) is a vector given by V $ t ¼ ðVk tki Þ

(1.21)

Finally, the unit tensor (or matrix) is given by I¼(dij) where the Kronecker delta is defined as 

dij ¼

1 if i ¼ j 0 if isj

(1.22)

8 Modelling of Nuclear Reactor Multi-physics

1.4.b

Spherical coordinates and solid angles

Some reminder about the spherical coordinate system and solid angles is given hereafter. Any point r in a three-dimensional system can be either described by a triplet of coordinates (x, y, z) in a Cartesian coordinate system or by the triplet of coordinates (r,q,4) in a spherical coordinate system, with x ¼ r sin q cos 4

(1.23)

y ¼ r sin q sin 4

(1.24)

z ¼ r cos q

(1.25)

r is the distance from the origin of the coordinate systems, q is the polar angle and 4 is the azimuthal angle, as represented in Fig. 1.1. The polar and azimuthal angles also define the direction U, which is a unit vector aligned with the vector r giving the position of the considered point. Infinitesimal variations in each of the spherical coordinates are then applied, i.e. r is replaced by r þ dr, q is replaced by q þ dq and 4 is replaced by 4 þ d4, as illustrated in Fig. 1.2. The variation with respect to r defines a segment of length dr from the point r (represented in blue in Fig. 1.2). The variation with respect to q defines an arc of length rdq from the point r (represented in red in Fig. 1.2). The variation with respect to 4 defines an arc of length r sin q d4 from the point r (represented in green in Fig. 1.2). The triplet of infinitesimal variations (dr,dq,d4) in spherical coordinates defines from the point r an infinitesimal volume d3r (represented in grey in Fig. 1.2) that is thus given as d 3 r ¼ r 2 sin qdrdqd4

(1.26)

One also notices from the above that if only the polar angle q and the azimuthal angle 4 are varied, one would obtain an infinitesimal surface d2r that can be expressed as d 2 r ¼ r 2 sin qdqd4

FIGURE 1.1 Representation of a point r in a spherical coordinate system.

(1.27)

Chapter 1
Introduction

9

FIGURE 1.2 Result of infinitesimal variations in spherical coordinates from a point r. The variation with respect to r are represented in blue. The variation with respect to q are represented in red. The variation with respect to 4 are represented in green.

FIGURE 1.3 Illustration of the infinitesimal solid angle d2U. The unit sphere corresponds to the 4p solid angle.

which can be rewritten as d2 r ¼ r 2 d2 U

(1.28)

d 2 U ¼ sin qdqd4

(1.29)

with 2

d U represents the infinitesimal solid angle, which is the cone of space defined by the simultaneous infinitesimal variations of the polar angle q and of the azimuthal angle 4, as represented in Fig. 1.3. If the entire range of variation for the polar and azimuthal angles

10 Modelling of Nuclear Reactor Multi-physics

was considered, i.e. q ˛[0;p] and 4 ˛[0;2p], the corresponding solid angle would be Zp Z2p sin qdqdf ¼ 4p 0

(1.30)

0

This solid angle is correspondingly represented in Fig. 1.3 by the sphere centred at the origin of the coordinate system and having a unit radius (since U is a unit vector).

1.4.c

Gauss divergence theorems

In the following, the Gauss divergence theorem and its generalized form are recalled. For a volume V bounded by a surface S, the Gauss divergence theorem states that Z V

V $ cðrÞd 3 r ¼

I

cðrÞ $ Nd 2 r

(1.31)

S

if c(r) is a vector or a tensor. In the equation above, N represents the unit outward vector normal to the elementary surface d2r.

FIGURE 1.4 Schematic representation of a control volume encompassing solids (in dark grey) and two phases (with the liquid phase represented in light grey). The interface between the two phases is indicated by the dashed lines.

In addition to Gauss divergence theorem, another useful theorem is the generalized Gauss divergence theorem (Analytis, 2003). This theorem will be useful when dealing with fluid dynamics. If one considers a control volume V as the one represented in Fig. 1.4, this control volume might encompass solids (such as vertical fuel rods in the case of, e.g., a nuclear reactor) and one or two phases. If one denotes by Skw the interface between the phase k and the solid walls and by Ski the interface between the two phases, the generalized Gauss divergence theorem states that

Chapter 1
Introduction Z

VcðrÞd 3 r ¼ V

Vk

if c is a scalar,

Z

V $ cðrÞd 3 r ¼ V $

Z

3

Z

V $ cðrÞd r ¼ V $

Z

cðrÞd 3 r þ

Z

Z

3

cðrÞNd 2 r

cðrÞ $ Nd 2 r þ

Z

cðrÞ $ Nd 2 r

(1.33)

cT ðrÞ $ Nd 2 r

(1.34)

Skw

Z

2

Z

c ðrÞ $ Nd r þ T

Ski

(1.32)

Skw

Ski

cðrÞd r þ Vk

cðrÞNd 2 r þ

Ski

Vk

if c is a vector and

Vk

cðrÞd 3 r þ

Vk

Vk

Z

Z

11

Skw

if c is a tensor. In Eq. (1.34), the superscript T represents the transpose. Eq. (1.34) can be obtained from Eq. (1.33) by applying Eq. (1.33) to each of the column vectors cj¼(cij) of the tensor c. In the above three equations, Vk is the volume occupied by the phase k within the control volume V.

References Analytis, G.T., 2003. Lectures on Nuclear Engineering. Lecture Notes. Chalmers University of Technology, Gothenburg, Sweden. Krathwohl, D.R., 2002. A revision of Bloom’s taxonomy: an overview. Theory Into Practice 41 (4), 212e218.

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2

Transport phenomena in nuclear reactors 2.1 Nuclear reactors as multi-physics and multi-scale systems Nuclear power plants are large and complex systems meant at converting the thermal energy released in the nuclear fuel assemblies into electrical energy. Many reactor types and generations exist and covering all modelling aspects for all of them is beyond the scope of this book. As illustrative examples, we will nevertheless focus on light-water reactors (LWRs), in which ordinary water is used for transporting the heat from the fuel assemblies to steam turbines (Demazie`re, 2013). Two main types of LWRs are in use worldwide: pressurized water reactors (PWRs) and boiling water reactors (BWRs). The reasons why these two reactor types are worth mentioning are twofold. First, these two types represent the largest fraction of the current fleet of operating commercial reactors. Second, the physics of BWRs is particularly challenging: whereas in PWRs the water remains in liquid state when extracting the heat from the nuclear fuel assemblies, the water is partially vaporized in BWRs. Compared to PWRs, the modelling of BWRs thus requires special attention.

2.1.a

Multi-physics aspects

This section aims at giving some orientation to the readers not necessarily familiar with the physics of nuclear reactors and is therefore simplistic. Readers interested in a more complete phenomenological description of nuclear reactor systems are referred to introductory books in nuclear engineering, such as Lamarsh (1975) or other equivalent books. In order to better highlight the multi-physics aspects of nuclear reactors, simplified sketches of a PWR and a BWR with typical operating pressures and temperatures throughout the entire systems are given in Fig. 2.1. As can be seen in this figure, the steam power cycles in the PWR and BWR cases, respectively, are more or less identical, the main differences coming from how the steam is produced. In the PWR case, the steam is produced on the secondary side of the steam generators while the fluid on the primary side is passing through the steam generators. The coolant in the entire primary system is always in liquid state by means of the pressurizer which maintains a very high pressure (about 155 bar). In the BWR case, the steam is directly produced in the reactor Modelling of Nuclear Reactor Multi-physics. https://doi.org/10.1016/B978-0-12-815069-6.00002-7 Copyright © 2020 Elsevier Inc. All rights reserved.

13

14 Modelling of Nuclear Reactor Multi-physics

FIGURE 2.1 Schematic representation of the two most common types of light-water reactors (LWRs) (pressurized water reactor (PWR) case on the top and boiling water reactor (BWR) case on the bottom e the numerical data are only indicative values). Derived from Drevon, G., 1983. Les réacteurs à eau ordinaire. Collection du Commissariat à l’Energie Atomique, Série Synthèses, Eyrolles, Paris, France.

Chapter 2
Transport phenomena in nuclear reactors

15

pressure vessel, since the operating pressure is much lower (about 70 bar). Steam driers/ separators are thus installed in the reactor pressure vessel in order to produce dry steam to be sent to the turbines. Such steam driers/separators are installed on the secondary side of the steam generators in PWRs. The specific character of nuclear power plants compared to other types of power plants (coal, oil, gas) comes from the fact that the energy released by the fuel assemblies is produced by nuclear fission. The part of the reactor pressure vessel containing the nuclear fuel assemblies is referred to as the nuclear core. Depending on the reactor type and reactor design, the core typically contains a few hundred of nuclear fuel assemblies. Water passes through these fuel assemblies to extract the heat and to transport it to the steam power cycles, either directly in the BWR case or indirectly via the steam generators in the PWR case. Water is thus referred to as the coolant, since it cools down the fuel assemblies. In reactor designs other than LWRs, the coolant is a fluid that can be a liquid metal, a gas, a salt or a supercritical fluid. In LWRs, water enters the nuclear core in liquid state (sub-cooled conditions) for both PWRs and BWRs. Due to the nuclear heating, the enthalpy of the fluid increases while passing through the core. Whereas in the PWR case, the water remains in liquid state, in the BWR case, the water start boiling already at the bottom of the fuel assemblies. At the top of the fuel assemblies, the void fraction, defined as the ratio (in percentage) between the volume occupied by vapour in a given control volume (in which both liquid and vapour might be present) and this control volume, reaches typically around 70%. Fig. 2.2 shows typical axial variations of the density of the water while flowing through the core in both PWRs and BWRs, respectively, with the corresponding void fraction increase in the BWR case. The axial distributions are given at beginning of cycle (BOC), i.e. when the reactors are started, and at end of cycle (EOC), i.e. typically before the reactors are shut down for refuelling. The water density strongly influences the nuclear heating in the fuel assemblies. In the case of reactors having a thermal spectrum, neutrons released through nuclear fissions in the fuel assemblies are emitted at fast energies (average energy around 2 MeV). These neutrons will lose energy down to thermal energies (average energy around 0.0379 eV) by mostly collision with the nuclei of the atoms constituting the water molecules. The loss of energy through these collisions is usually referred to as moderation or slowing-down, and as such, the water is referred to as a moderator. In the case of LWR systems, water is thus both a coolant and a moderator. In other reactor designs, the coolant and the moderator can be two different media. The probability for neutrons to induce new fissions at thermal energies is much higher than at fast energies, and thus the use of water as a moderator allows having a lower enrichment of the fuel assemblies. The slowing-down of neutrons from fast energies to thermal energies (at which the probability of inducing new fission reactions is higher), followed by the release of neutrons at fast energies, is often referred to as the neutron cycle. This is conceptually and simplistically represented in Fig. 2.3 where the neutron cycle is depicted together with a measure of the probability to induce fission reactions on 235 U (the so-called microscopic fission cross-section e defined in Section 2.2). It is to be

16 Modelling of Nuclear Reactor Multi-physics

FIGURE 2.2 Typical axial variations of the properties of water while flowing through a nuclear core and of the fuel temperature (pressurized water reactor (PWR) case on the left-hand side and boiling water reactor (BWR) case on the right-hand side e the numerical data are only indicative values). BOC, Beginning of cycle; EOC, End of cycle.

Chapter 2
Transport phenomena in nuclear reactors

17

FIGURE 2.3 Conceptual illustration of the neutron cycle in a thermal reactor, superimposed to the energy dependence of the microscopic fission cross-section for 235 U (obtained from the JEFF 3.1 neutron data library).

noted that in fast-spectrum reactors on the other hand, slowing-down is much less pronounced than in thermal spectrum reactors. As a consequence, the fuel needs to be greatly enriched in those reactors: the amount of 235U needs to be increased in order to compensate for the rather low value of the microscopic fission cross-section at high energies. Turning back to LWRs, a consequence of the moderation properties of water is that the number of water molecules per unit volume and the temperature have a direct influence on the energy loss of the neutrons and, as a result, on the number of fission reactions induced by thermal neutrons and on the energy released in the nuclear fuel assemblies by nuclear fission. There is thus a strong impact of the water density and temperature on the nuclear heating of the fuel assemblies. The axial variation of the radially averaged fuel temperature is also given in Fig. 2.2 for both PWRs and BWRs. The fuel temperature is dependent on the energy produced by nuclear fission in the fuel assemblies, as well as on the energy that the coolant can extract from the fuel. The fuel temperature also influences the absorption of the neutrons in the fuel assemblies, since the probability of neutrons to interact with the fuel assemblies is strongly temperaturedependent. A key characteristic of nuclear reactors is thus their multi-physics aspects because of the following:
The axial variation of the properties of the coolant while heated by contact with the fuel assemblies depends on the nuclear heating in the fuel assemblies, and the axial variation of the fuel temperature depends on the nuclear heating in the fuel assemblies and on the heat transferred from the fuel to the coolant.

18 Modelling of Nuclear Reactor Multi-physics


The nuclear heating depends on the density and temperature of the moderator and on the fuel temperature. In practice, the multi-physics character of nuclear reactors means that the neutron density field (i.e. the spatial and temporal distribution of the neutron densities throughout the core) can only be determined if the density and temperature fields of the moderator/ coolant (i.e. the spatial and temporal distribution of the density and temperature of the coolant) and the temperature field of the fuel (i.e. the spatial and temporal distribution of the temperature of the fuel throughout the core) are known, and that the density and temperature fields of the coolant/temperature field of the fuel can only be determined if the neutron field is known. Consequently, the determination of all fields needs to be carried out simultaneously if one wants to determine the behaviour of nuclear reactors. The determination of the neutron field is usually referred to as reactor physics or reactor kinetics or neutronics calculations, whereas the determination of the density/velocity/enthalpy fields of the coolant and temperature field of the fuel is usually referred to as thermal-hydraulics calculations. The use of the terminology ‘thermal-hydraulics’ to describe solving a fluid dynamics and heat transfer problem is specific to nuclear engineering. The strong coupling between the neutron kinetics and thermal-hydraulics is a unique feature of nuclear reactors, which makes the calculation of their behaviour a very challenging task. The focus of this book is on neutronics and thermal-hydraulics in conditions not leading to core damage. Other fields that might be important to consider depending on the application areas or the type of scenarios are
Fuel behaviour. Because of the irradiation and huge temperature gradients the fuel is exposed to, the physical and thermomechanical properties of the fuel are greatly modified (Graves, 1979). With associated time scales varying from microseconds to years and characteristic length scales varying from atomic scales to metres, the modelling of such phenomena is a far from trivial task. Because the modelling of heat transfer depends on the properties on the fuel and since the heat source is directly related to the energy released by fission and to the capability of the nuclear system to extract heat, a direct coupling exists between neutronics, thermalhydraulics, fuel behaviour/performance and material physics.
Structural mechanics. Since nuclear power plants contain many rotating machineries, vibrations are an inherent part of the system. In addition, vibrations can be induced by the fluid usually moving at a high velocity and the flow might itself be influenced by vibrating structures (flow-induced vibration, FIV, and fluidestructure interaction, FSI). Since nuclear reactors are large and complex systems made of many components, such phenomena require the development of large mechanical and structural models (Fujita, 1990). Although the modelling of FSI and FIV is not a nuclear-specific problem (Blevins, 2001), the mechanical properties of the components are also influenced by irradiation at the atomic scale and by thermal fatigue. The modelling of such phenomena thus includes some coupling with neutronics and thermal-hydraulics, as well as to material physics.

Chapter 2
Transport phenomena in nuclear reactors

19


Coolant and radiation chemistry. Although the coolant chemistry is strictly controlled in a nuclear power plant, the fluid used to extract the heat from the system has some chemical properties that might influence the distributions of neutrons and how heat can be removed from the nuclear fuel assemblies. Likewise, temperature and radiation have a direct influence on the chemical species constituting the coolant (Takagi et al., 2010). An interplay thus exists between neutronics, thermal-hydraulics, coolant and radiation chemistry.
Radionuclide transport. In case of incidental and accidental situations, the release of radionuclides into the primary system, the containment and possibly the atmosphere is possible. The tracking of such radionuclides is of utmost importance from a radiobiological viewpoint. Beyond the complexity of modelling the chemical reactions taking place and the transport of the radionuclides with a moving fluid, many different scales need to be accounted for, especially when a release to the containment or the atmosphere takes place (Neeb, 1997). A strong interdependence thus exists between radiation chemistry and thermal-hydraulics. In addition, since the build-up of radionuclides occurs mostly during the operation of the reactor, the accurate modelling of fuel burn-up (i.e. neutronics) is required. In case of situations where the core is damaged (often referred to as severe accidents), other physical mechanisms might need to be taken into account depending on the severity of the accident (Sehgal, 2012).

2.1.b

Multi-scale aspects

Another characteristic feature of nuclear reactors is their multi-scale aspects, i.e. phenomena occurring at different scales. The multi-scale character of nuclear systems is explained by the fact that nuclear reactors are large-scale systems being strongly heterogeneous on small scales and by the fact that phenomena involving different characteristic lengths play role in the systems. The nuclear core is the part of the system where the heterogeneity is best illustrated, with typically a few hundred fuel assemblies constituting the core. Taking LWRs as illustrative examples, the system characteristic dimensions of both the reactor pressure vessels (RPVs) and cores are represented in Fig. 2.4. It can be noticed that the RPV of BWRs is much larger than the RPV of PWRs since it also embeds the steam driers/separators located just above the nuclear core. Otherwise, the nuclear cores have roughly the same height in both cases (around 4 m) and a diameter ranging from 3.5 m in the PWR case to 4.5 m in the BWR case. PWR cores contain typically approximately 150 fuel assemblies, whereas BWR cores contain typically approximately 700 fuel assemblies. Each of the fuel assemblies is itself constituted by typically less than 300 solid fuel pins containing species that can lead to fission reactions. These species have possibly various enrichments, both radially and axially. Some fuel assemblies might also contain partlength fuel rods. The characteristics dimensions of the nuclear cores, nuclear fuel assemblies, fuel rods and fuel pellets are given in Fig. 2.5, where it can be noticed that the

20 Modelling of Nuclear Reactor Multi-physics

FIGURE 2.4 Schematic representation of the reactor pressure vessels (RPVs) of the two most common types of light-water reactors (LWRs) (pressurized water reactor (PWR) case on the left-hand side and boiling water reactor (BWR) case on the righthand side e the numerical data are only indicative values). The data/ arrows in black correspond to the RPVs, whereas the data/arrows in red correspond to the cores. Used with permission from Westinghouse Electric Sweden AB.

characteristic lengths vary from the metre range at the core level to the centimetre range at the fuel pellet level. In addition to the heterogeneous configuration of the nuclear reactor cores and the different scales existing in such systems by design, physical phenomena also involving different scales are governing the behaviour of the systems. On the thermal-hydraulic side, the distributions of the coolant and fuel properties are strongly heterogeneous within the cores. Table 2.1 provides some typical axial and radial variations of the fuel temperature and moderator density for LWRs as illustrative examples. The axial variations were obtained by radially averaging the fuel temperature and the moderator density on the fuel and moderator regions, respectively, thus leading to a one-dimensional axial reactor model. This model thus gives the overall range of variations of radially averaged fuel temperature and moderator density. The radial variations were estimated at mid-core elevation without averaging and thus represent the heterogeneities on small scales on the fuel and moderator regions, respectively. For both the PWR and BWR cases, the change of the fuel temperature on short distances is very large in the radial direction (change of about 600 K on 4e5 mm, which is the typical radius of a fuel pellet). Such an extreme temperature gradient is the result of the very large power density associated to nuclear fission. The axial gradient, on the other hand,

Chapter 2
Transport phenomena in nuclear reactors

21

FIGURE 2.5 Characteristic dimensions of nuclear cores, nuclear fuel assemblies, fuel rods and fuel pellets in lightwater reactors (LWRs). (A) Nuclear core; (B) nuclear fuel assemblies; (C) fuel rods; (D) fuel pellets. BWR, boiling water reactor; PWR, pressurized water reactor. Figure (B) used with permission from Westinghouse Electric Sweden AB. Figure (C) used with permission from Chalmers Industriteknik. Figure (D) used with permission from Analysgruppen (2009) e Energiföretagen.

is much smaller. The radial characteristic length for the variation of the fuel temperature is thus very short and much shorter than the axial characteristic length. Concerning the spatial variation of the coolant density, PWRs and BWRs have to be discussed separately, since the heterogeneities are much more pronounced in BWRs than in PWRs.

22 Modelling of Nuclear Reactor Multi-physics

Table 2.1 Typical axial and radial variations of the thermal-hydraulic variables in light-water reactors (LWRs) (the numerical values are only indicative). PWRs Axially (i.e., radially averaged quantities) Coolant density 0.65e0.75 (g/cm3) Fuel temperature 700e900 (K)

BWRs Radially (at midcore elevation)

Axially (i.e., radially averaged quantities)

Radially (at midcore elevation)

0.70e0.73

0.25e0.76

0.35e0.74

800e1400

600e850

800e1400

In PWRs, the density of the coolant decreases from the bottom of the nuclear core to its top, due to nuclear heating, as represented in Fig. 2.2. Such a decrease of the density is rather mild and smooth since the coolant always remains in single-phase conditions e even if sub-cooled boiling might exist in the uppermost part of the fuel assemblies (Zhang et al., 2015). Radially, the coolant density varies in proportions comparable to the axial variations, but on much shorter distances. Therefore, the radial characteristic length is much shorter than the axial one. The short radial characteristic length is the result of turbulence, since the coolant is circulating inside the core at a rather high velocity (approximately 4e5 m/s). This can be clearly noticed in Fig. 2.6 giving the radial distribution of the turbulent kinetic energy of the coolant. The turbulent kinetic energy, estimated as the kinetic energy per unit mass of the velocity fluctuations due to turbulence, represents the extent to which the flow is turbulent. Turbulence is responsible for the formation of eddies at many different length scales, where most of the kinetic energy is contained in the large eddies. The energy ‘cascades’ from the large-scale structures to the smaller scale structures, creating a hierarchy of eddies. Although the coolant in PWRs remains in single-phase conditions, the turbulent character of the flow requires taking many different scales into account. FIGURE 2.6 Typical radial distribution of the turbulent kinetic energy of the coolant between pressurized water reactor (PWR) fuel pins represented in grey (figure derived from (Demazière and Mattsson, 2006)). The turbulent kinetic energy was estimated from a k-ε turbulence model using a computational fluid dynamics (CFD) approach.

1.44e-01 1.38e-01 1.32e-01 1.26e-01 1.20e-01 1.14e-01 1.08e-01 1.02e-01 9.60e-02 9.01e-02 8.41e-02 7.82e-02 7.22e-02 6.63e-02 6.03e-02 5.44e-02 4.84e-02 4.25e-02 3.65e-02 3.06e-02 2.46e-02

[m2.s-2]

Y Z

X

Chapter 2
Transport phenomena in nuclear reactors

annular flow regime

23

liquid droplets in vapour core

bubbly flow regime

bubbles in continuous liquid

Void fraction [1] 5.4937E–01 4.5853E–01 3.6769E–01 2.7685E–01

subcooled liquid core

1.8601E–01 9.5168E–02 4.3270E–03

FIGURE 2.7 Typical radial and axial distributions of the thermal-hydraulic variables in boiling water reactors (BWRs). The radial distribution of the void fraction in a BWR fuel assembly (1/4 of a fuel assembly) is given on the left figure, whereas the axial distribution of the main flow patterns in a BWR fuel bundle is represented on the right figure. In the right figure, liquid is represented in light grey and vapour is represented in white. Left figure used with permission from H. Anglart (Windecker, G., Anglart, H., 1999. Phase distribution in BWR fuel assembly and evaluation of multidimensional multi-field model. In: Proceedings of the 9th International Meeting on Nuclear Reactor Thermal-Hydraulics. NURETH-9, 3-8 October, 1999, San Francisco, CA, USA. American Nuclear Society, La Grange Park, USA).

A major difference between BWRs and PWRs lies with the vapour produced inside the core in BWRs. The production of vapour makes the coolant strongly heterogeneous, as can be seen in Fig. 2.7. When the coolant enters the bottom of the core, the coolant is in liquid phase. Since the outer cladding temperature is slightly above the saturation temperature, small vapour bubbles are produced at the outer radius of the fuel pins. The vapour bubbles nevertheless condense inside the core of the coolant that is in sub-cooled conditions. Such a condensation increases the enthalpy of the coolant and thus its temperature. When the coolant temperature becomes equal to the saturation temperature as the coolant flows upward, no condensation of the vapour bubbles in the liquid core is any longer possible (the corresponding flow pattern is called bubbly flow e see Section 5.1.b of Chapter 5). At some point, the vapour bubbles coalesce to form a central vapour core, with the bulk of the liquid flowing in the form of a film on the wall (the corresponding flow pattern is called annular flow e see Section 5.1.b of Chapter 5). Other types of flow patterns are possible and will be more thoroughly described in Section 5.1.b of Chapter 5. The spatial heterogeneity due to vapour production is thus very high. Furthermore, within the same fuel assembly, the radial distribution of the vapour bubbles or vapour core might be different, resulting in a strong spatial heterogeneity, as can be noticed in Fig. 2.7. The characteristic lengths (both axial and radial) are thus very short.

24 Modelling of Nuclear Reactor Multi-physics

When considering neutron transport, several scales or characteristic lengths are also present. Neutrons, having a diameter in the order of 1015 m, interact with the different nuclei constituting the system. Due to the heterogeneous arrangement of the fuel pellets surrounded by the cladding and moderator, neutrons produced by fission will interact with nuclei belonging to these different media, either directly after their emission by fission or after several scattering events. Because of the tight spacing between fuel rods and between fuel assemblies, neutrons emitted from a given fuel rod can also interact with other fuel rods, belonging either to the same fuel assembly or to another one. The fission chain reactions and scattering reactions thus create a strong coupling between each of the fuel rods and the rest of the core. The mean distance between the generation and absorption of a neutron in a nuclear core gives an indication of the characteristic length for the transport of neutrons. In LWRs, such a mean distance is in the range of several centimetres, to be compared to the size of the core being in the range of several metres. The heterogeneous arrangement of fuel rods and moderator and the mean distance mentioned above result in two characteristic lengths for the spatial distribution of the neutron density throughout the core, as represented in Fig. 2.8: a rather smooth variation at the core level, to which a variation at the level of the fuel pins is superimposed. This figure represents in a schematic manner the thermal neutron flux that is related to the spatial distribution of the density of the thermal neutrons throughout the system. At the core level, having a neutron flux as ‘flat’ as possible is desirable, so that the fuel temperature is as homogeneous as possible in the fuel assemblies and the effect of hot spots can be minimized during plant operation (the fuel temperature being related to the energy released by fission, and thus to the neutron flux/density). At the fuel pin level nevertheless, the amplitude of the spatial variations of the neutron flux might be rather large, with maxima of the neutron flux being observed in the moderator and minima observed in the fuel. Most of the fast neutrons emitted by fission slow down in the moderator. When neutrons are thermalized, they have a higher probability of

FIGURE 2.8 Idealized radial distributions of the thermal neutron flux in light-water reactors (LWRs): at the core level on the left and at the pin level on the right (the fuel rods are represented in grey, whereas the coolant is represented in white).

Chapter 2
Transport phenomena in nuclear reactors

25

inducing new fission reactions in the fuel. Because some nuclides have very large probabilities of absorbing thermal neutrons, the mean free path of thermal neutrons will be rather short and most of the thermal neutrons re-entering the fuel pellets will be absorbed at the periphery of the fuel pellets. This explains the observed dip of the thermal neutron flux in the fuel pellets, an effect referred to as spatial self-shielding. It should also be noted that even in a homogeneous medium, dips of the neutron flux are observed at the energies for which the probability of absorption reaction is large, an effect referred to as energetic self-shielding. Both the spatial and energetic self-shielding effects are inter-related (Knott and Yamamoto, 2010). Self-shielding will be given special attention in Section 3.2 of Chapter 3, when dealing with the treatment of the high values for some probabilities of neutron interactions, referred to as resonances.

2.2 Neutron transport In this section, the neutron transport equation, also called the Boltzmann equation, is derived. This equation describes the true behaviour of neutrons without any need to introduce any approximation. After defining some quantities useful for neutron transport, the neutron transport equation is first derived in its integro-differential form, then in its integral form and finally in its characteristic form.

2.2.a

Introduction

Some elementary concepts in neutron and reactor physics are presented hereafter. The goal is to introduce the concepts required to derive and understand the quantities appearing in the neutron transport equation. The readers interested in getting further details about such concepts are referred to, e.g., Lamarsh (2002) and Lewis (2008).

Elementary concepts in neutron physics The main mechanism responsible for the production of neutrons in a nuclear reactor is nuclear fission, according to which a heavy nucleus, when bombarded with neutrons, can be split into two lighter nuclei, called the fission fragments. Such a splitting is accompanied by the release of new neutrons 107 s after the scission and 2  1014 s later by the release of g-rays. Such neutrons are called the prompt neutrons. The fission fragments are neutron-rich and in excited states and thus decay into less excited species via b decay and/or g emission. The fission fragments and their radioactive decay products are referred to as fission products. Some of the fission fragments lead by b decay to species that can themselves emit neutrons. Such neutrons are called delayed neutrons, and the corresponding fission fragments are called precursors of delayed neutrons. When considering non-steady-state systems, special care has thus to be taken for the modelling of the source of neutrons, since the prompt neutrons appear typically within 107 s of the fission event, whereas the delayed neutrons appear after a b decay of some of the fission fragments, leading to excited nuclei above the so-called virtual energy level,

26 Modelling of Nuclear Reactor Multi-physics

FIGURE 2.9 Energetic diagram showing the different energy levels of the precursor Br-87 and of its daughter nuclides. Derived from Lamarsh, J.R., 2002. Introduction to Nuclear Reactor Theory. American Nuclear Society, Inc., La Grange Park, USA.

defined as the energy level above which neutron emission is possible. As an illustrative example, the production of delayed neutrons from Br-87, which is one of the possible fission fragments of U-235, is given in Fig. 2.9. Once a fission fragment has decayed above the virtual energy level, neutron emission is almost instantaneous. As a result, the delayed neutron emission is equal to the number of fission fragments decaying above the virtual energy level and, as such, can be modelled by tracking the fission fragments or precursors of delayed neutrons. Many precursors of delayed neutrons exist, each having its own yield from fission and decay constant. Usually, similar precursors (similar in terms of their decay constant) are grouped into several groups described by their average properties and six groups are typically used. For each of these groups, the following parameters are defined:
The fraction of delayed neutrons bi ðr; EÞ, representing the number of delayed neutrons emitted by the precursors belonging to group i divided by the total number of neutrons emitted (all prompt neutrons and all delayed neutrons). The fraction of delayed neutrons depends on the energy E of the neutrons leading to the fission reactions.
The decay constant li ðrÞ of the precursors belonging to the group i. The decay constant is used to describe the time-evolution of the concentration of the precursors

Chapter 2
Transport phenomena in nuclear reactors

27

according to an exponential decay law of the form exp½ li ðrÞt. The half-life is given by Ti ðrÞ ¼ lnð2Þ=li ðrÞ and corresponds to the time required for the considered decaying species to reach half of its initial content.
The space- and time-dependent concentration Ci(r, t) of the precursors belonging to group i. The total fraction of delayed neutrons is then given by: bðr; EÞ ¼

Nd X

bi ðr; EÞ

(2.1)

i¼1

where Nd represents the number of groups of precursors of delayed neutrons (typically Nd ¼ 6). Consequently, the fraction of prompt neutrons is given by 1  bðr; EÞ. Six-group delayed data for fission on U-235 are given in Table 2.2. It should be noted that several species, having various concentrations with respect to space, lead to fission reactions. The above quantities (fraction of delayed neutrons, decay constant and half-life) that will be used in the methods presented in this book for a mixture of species have thus a dependence on position in the most general case. Delayed neutrons are emitted with a spectrum very much different from the one of the prompt neutrons. The spectra of the prompt and delayed neutrons of group i are referred to as cp ðr; EÞ and cdi ðr; EÞ, respectively, so that cp ðr; EÞdE and cdi ðr; EÞdE represent the fraction of prompt and delayed neutrons of group i, respectively, emitted with an energy between E and E þ dE at point r. These two spectra are normalized to unity, i.e. Z

N

0

Z 0

N

cp ðr; EÞdE ¼ 1; cr

(2.2)

cdi ðr; EÞdE ¼ 1; cr

(2.3)

It should also be noted that several species, having various concentrations with respect to space, lead to fission reactions. The above quantities (prompt and delayed neutron spectra) that will be used in the methods presented in this book for a mixture of species have thus a dependence on position in the most general case. Table 2.2

Delayed neutron data for fission induced by thermal neutrons on U-235.

Group i

Decay constant li (sL1)

Half-life Ti (s)

Fraction of delayed neutrons bi (1)

1 2 3 4 5 6

0.0124 0.0305 0.111 0.301 1.14 3.01

55.72 22.72 6.22 2.30 0.610 0.230

0.000215 0.001424 0.001274 0.002568 0.000748 0.000273

Total fraction of delayed neutrons b (1)

0.0065

Data derived from Lamarsh, J.R., 2002. Introduction to Nuclear Reactor Theory. American Nuclear Society, Inc., La Grange Park, USA.

28 Modelling of Nuclear Reactor Multi-physics

With respect to neutrons, fission is not the only type of reactions of importance. Other nuclear reactions need to be properly accounted for. In reactor physics, these reactions are put in two categories, as follows:
Scattering reactions, where a neutron collides on a target nucleus and might change its energy and direction after the collision. The scattering reaction might also lead to the formation of a compound nucleus before neutron emission. The scattering reactions are denoted as (n, n) for elastic collisions (i.e. when the energy of the neutron in the centre-of-mass reference system is unchanged) or as (n, n0 ) for inelastic collisions (i.e. when the energy of the neutron in the centre-of-mass reference system is changed). Whereas the laboratory reference system is the system corresponding to the observer being at rest, another reference system, referred to as the centre-of-mass reference system, is also used in neutron physics. In this system, the observer is moving at the velocity of the centre-of-mass and the coordinates and velocities are then defined with respect to this centre-of-mass. The laboratory and centre-of-mass reference systems are more thoroughly introduced in Section 3.2.b of Chapter 3. In reactor physics, scattering reactions are indicated by the subscript s.
Absorption reactions, where a neutron is absorbed by a target nucleus. Two possibilities then exist: either the compound nucleus thus formed does not emit neutrons or the compound nucleus thus formed does emit neutrons. The former category is referred to as capture, and is denoted as ðn; gÞ if the compound nucleus de-excites by g-ray emission, or ðn; aÞ if the compound nucleus de-excites by emitting a particles, etc. The second category is referred to as fission and is denoted as (n, nf 0 ), or (n, 2nf 0 ) if the second chance fission threshold is exceeded. Captures and fissions are thus special types of absorption reactions. In reactor physics, absorption reactions are indicated by the subscript a, capture reactions are indicated by the subscript c and fission reactions are indicated by the subscript f. Another type of reactions possibly having an important effect is scattering reactions leading to the release of more than one neutron, e.g. (n, 2n0 ), (n, 3n0 ), etc. In principle, such reactions could be treated in the same manner as fission reactions. Nevertheless, in reactor applications, the (n, 3n0 ) reactions have a negligible effect compared to (n, 2n0 ) reactions and can be disregarded. Reactor physics calculations usually consider the (n, 2n0 ) reactions by subtracting correspondingly the absorption reactions, since a (n, 2n0 ) reaction is a reaction leading to the emission of one net neutron (Knott and Yamamoto, 2010). In this approximated treatment, the spectrum of the emitted neutrons in such (n, 2n0 ) reactions is thus assumed to be identical to the spectrum of the absorbed neutrons.

Useful quantities in reactor physics Prior to deriving the governing equation for neutron transport, different quantities have to be defined:
The (space-, angle-, energy- and time-dependent) neutron density is denoted as n(r, U, E, t). Correspondingly, n(r, U, E, t)d3rd2UdE represents the number of

Chapter 2
Transport phenomena in nuclear reactors

29

neutrons at time t, contained in an infinitesimal volume d3r around the position r, having a direction U contained in the infinitesimal solid angle d 2 U, and an energy E contained in the infinitesimal energy bin dE. The neutron density has for dimension a number of neutrons per unit volume, solid angle and energy.
The (space-, angle-, energy- and time-dependent) angular neutron flux is denoted as jðr; U; E; tÞ. It is defined as: jðr; U; E; tÞ ¼ vðEÞnðr; U; E; tÞ

(2.4)

where v is the neutron speed, i.e. the magnitude of the velocity vector v, so that v ¼ kvk. The angular neutron flux has for dimension a number of neutrons per unit area, solid angle, energy and time. The angular flux thus gives a measure of the flow of neutrons in a given direction U contained in the infinitesimal solid angle d 2 U and having an energy E contained in the infinitesimal energy bin dE.
The (space-, energy- and time-dependent) scalar neutron flux is denoted as fðr; E; tÞ. It is defined as: Z

fðr; E; tÞ ¼

jðr; U; E; tÞd 2 U ¼

ð4pÞ

Z

vðEÞnðr; U; E; tÞd 2 U

(2.5)

ð4pÞ

The scalar neutron flux fðr; E; tÞ has for dimension a number of neutrons per unit area, energy and time.
The (space-, energy- and time-dependent) neutron current density vector is denoted as J(r, E, t). It is defined as: Z

Jðr; E; tÞ ¼ ð4pÞ

Ujðr; U; E; tÞd 2 U ¼

Z

UvðEÞnðr; U; E; tÞd 2 U

(2.6)

ð4pÞ

It has to be emphasized that J(r, E, t) is a vector, since the integration in Eq. (2.6) is performed with respect to the direction U, which itself is a vector. The neutron current density vector has for dimension a number of neutrons per unit area, energy and time.
The (energy-dependent) microscopic cross-section is denoted as saX ðEÞ for nuclide X and reaction type a. It represents the ‘area of interaction’ per target nucleus. The microscopic cross-section has thus for dimension an area. Since such areas of interactions are very small, a common unit used for microscopic cross-sections is the barn (b), which corresponds to 1024 cm2.
For neutron-induced nuclear reactions producing neutrons, the (angle- and energy-dependent) differential microscopic cross-section is denoted as saX ðU0 /U; E 0 /EÞ for nuclide X and reaction type a. It represents the ‘area of interaction’ per target nucleus for neutrons having a direction U0 and energy E0 before interaction and having a direction U and energy E after interaction.

30 Modelling of Nuclear Reactor Multi-physics


The (space-, energy- and time-dependent) macroscopic cross-section for reaction of type a is defined as: Sa ðr; E; tÞ ¼

X NX ðr; tÞsaX ðEÞ

(2.7)

X

where NX(r, t) represents the (space- and time-dependent) atomic density of the species X. The macroscopic cross-section represents the probability of interaction per unit path length and has thus for dimension the reciprocal of unit length. As a result, the number of reactions of type a per unit volume, solid angle, energy and time is given, according to the definition of the angular flux, as Sa ðr; E; tÞjðr; U; E; tÞ. Furthermore, if one considers an infinitesimal path length dr from point r along the direction U, the probability of interaction on that infinitesimal path length is given by Sa ðr; E; tÞdr. The variation of the angular neutron flux along such an infinitesimal path would thus be djðr; U; E; tÞ ¼ Sa ðr; E; tÞjðr; U; E; tÞdr, from which one concludes that the attenuation of the angular neutron flux along that infinitesimal path is given by exp½ Sa ðr; E; tÞdr. This quantity thus also represents the probability of non-interaction on the infinitesimal path length dr.
For neutron-induced nuclear reactions producing neutrons, the (space-, energyand time-dependent) differential macroscopic cross-section is denoted as Sa ðr; U0 /U; E 0 /E; tÞ for reaction of type a. The differential macroscopic crosssection represents the probability per unit path length that neutrons having a direction U0 and energy E0 before interaction emerge with a direction U and energy E after interaction. It is customary to write the differential macroscopic cross-section as: Sa ðr; U0 /U; E 0 /E; tÞ ¼ Sa ðr; E 0 ; tÞns ðr; E 0 ; tÞfa ðr; U0 /U; E 0 /E; tÞ 0

0

(2.8)

2

where fa ðr; U /U; E /E; tÞd UdE is the probability for neutrons having a direction U0 and energy E0 to emerge, if there is interaction, with a direction U contained in the infinitesimal solid angle d 2 U, and an energy E contained in the infinitesimal energy bin dE. This probability fulfils the following normalization condition: Z Z

ð4pÞ

0

0

N

fa ðr; U0 /U; E 0 /E; tÞd 2 UdE ¼ 1

(2.9)

In Eq. (2.8), ns(r, E , t) is the number of secondaries (i.e. number of neutrons emitted by the nuclear reaction). The number of reactions of type a per unit volume, solid angle, energy and time leading to neutrons with a direction U contained in the infinitesimal solid angle d2 U, and an energy E contained in the infinitesimal energy bin dE, is thus given, according to the definition of the angular flux, as Sa ðr; U0 /U; E 0 /E; tÞjðr; U0 ; E 0 ; tÞd 2 UdE. Furthermore, if one considers an infinitesimal length dr, the probability of interaction on that infinitesimal path length is given by Sa ðr; U0 /U; E 0 /E; tÞdr. Correspondingly, the probability of non-interaction on that infinitesimal path length is exp½ Sa ðr; U0 /U; E 0 /E; tÞdr.

Chapter 2
Transport phenomena in nuclear reactors

31

For scattering reactions where one neutron emerges for each neutron colliding with a nucleus (i.e. ns(r, E 0 , t) ¼ 1), one has, using Eqs. (2.8) and (2.9): Z Z

ð4pÞ

0

N

Ss ðr; U0 /U; E 0 /E; tÞd 2 UdE ¼ Ss ðr; E 0 ; tÞ

(2.10)

For fission reactions, it can be assumed that the prompt neutrons are isotropically emitted in the laboratory reference system, so that one has: Sf ðr; U0 /U; E 0 /E; tÞd 2 UdE ¼

1 nðr; E 0 Þcp ðr; EÞSf ðr; E 0 ; tÞd 2 UdE 4p

(2.11)

where nðr; E 0 Þ represents, at point r, the average number of neutrons emitted by fission (prompt and delayed neutrons) that are induced at the energy E0 and cp ðr; EÞ is the prompt fission spectrum. Since several species, having various concentrations with respect to space, lead to fission reactions, the average number of neutrons emitted by fission and the prompt fission spectrum, as used above in a macroscopic sense, have a dependence on position in the most general case. The average number of neutrons emitted by fissions n should not be mistaken for the neutron speed v. For other nuclear reactions producing ns(r, E 0 , t) neutrons (such as the (n, 2n0 ) reaction for which ns(r, E0 , t) ¼ 2, the (n, 3n0 ) reaction for which ns(r, E 0 , t) ¼ 3, etc.), Eqs. (2.8) and (2.9) would lead to: Z Z

ð4pÞ

N 0

Sa ðr; U0 /U; E 0 /E; tÞd 2 UdE ¼ Sa ðr; E 0 ; tÞns ðr; E 0 ; tÞ

(2.12)

As earlier mentioned, such reactions are seldom modelled separately in reactor physics calculations. A correction of the absorption cross-section is typically used instead. In the following of this book, (n, 2n0 ), (n, 3n0 ), etc. reactions are thus not explicitly taken into account. Based on the above definitions, a total macroscopic cross-section can also be introduced as: St ðr; E; tÞ ¼ Sa ðr; E; tÞ þ Ss ðr; E; tÞ

(2.13)

where the absorption cross-section can be split into capture and fission, i.e. Sa ðr; E; tÞ ¼ Sc ðr; E; tÞ þ Sf ðr; E; tÞ

(2.14)

Physical significance of the neutron current density vector In order to better illustrate the physical meaning of the neutron current density vector, an elementary surface d2r, as the one represented in Fig. 2.10, is considered. A slant cylinder of length v(E) is then constructed along a direction U. This cylinder is tilted by an angle w as compared to the unit vector N normal to the elementary surface d2r. All neutrons contained in the slant cylinder of length v(E) with ends of surface d2r and

32 Modelling of Nuclear Reactor Multi-physics

FIGURE 2.10 Illustration of the physical meaning of the flow of neutrons through a surface.

having the direction U have to pass through the surface d2r per unit time. Since the volume of this cylinder is given by vðEÞd 2 r cos w, and since the neutron density nðr; U; E; tÞ represents the number of neutrons per unit volume, solid angle and energy, vðE Þd 2 r cos w nðr; U; E; tÞ thus represents the number of neutrons flowing through d2r, i.e. the number of neutrons passing through the surface d2r per unit solid angle, time and energy. Since one also has U $ N ¼ cos w (with $ representing the scalar product), one further has vðEÞd2 r cos wnðr; U; E; tÞ ¼ d 2 r N $ Ujðr; U; E; tÞ. Consequently, one can state that: d 2 r N $ Ujðr; U; E; tÞ ¼ number of neutrons at r; having direction U; and energy E flowing through d 2 r ðhaving a normal unit vector NÞ per unit time; solid angle; and energy

(2.15)

It has to be emphasized that the scalar neutron flux defined by Eq. (2.5) does not represent a flow of neutrons through surfaces. The appellation of ‘neutron flux’ should thus not be mistaken for ‘neutron flow’. Nevertheless, the angular flux defined by Eq. (2.4) can be interpreted as a measure of the flow of neutrons in a given direction U contained in the infinitesimal solid angle d 2 U and having an energy E contained in the infinitesimal energy bin dE, since in this case N $ U ¼ 1 and the angular neutron flux gives a measure of the flow of neutrons through elementary surfaces normal to the direction of neutron travel. It can also be concluded that the integration with respect to the direction U of d 2 r N $ Ujðr; U; E; tÞ gives the net flow rate of neutrons passing through d2r per unit energy. Based on Eq. (2.6), it follows that: Jðr; E; tÞ $ Nd 2 r ¼ d 2 r N $

Z

ð4pÞ

Ujðr; U; E; tÞd 2 U

¼ net number of neutrons at r; having energy E flowing through d 2 r ðhaving a normal unit vector NÞ per unit time and energy

(2.16)

Chapter 2
Transport phenomena in nuclear reactors

2.2.b

33

Derivation of the neutron transport equation

Based on the quantities defined above, a balance equation describing the behaviour of neutrons with respect to their position, their direction, their energy and time can be derived. This equation is referred to as the neutron transport equation or Boltzmann equation. There are several ways to derive the neutron transport equation, and each represents a given form of the transport equation: the integro-differential form, the integral form and the characteristic form. All forms are equivalent.

FIGURE 2.11 Balance with respect to space for the number of neutrons contained in a volume V bounded by a surface S.

Integro-differential form If one considers a volume V bounded by a surface S as depicted in Fig. 2.11, a balance equation per unit solid angle and energy for such a volume and within a given time interval dt can be written in terms of number of neutrons contained in this volume, i.e. Z

nðr; U; E; t þ dtÞd 3 r 

V

Z

nðr; U; E; tÞd 3 r ¼

V

v vt

Z

nðr; U; E; tÞdtd 3 r

V

¼ production of neutrons in V during dt per unit solid angle and unit energy

(2.17)

 disappearance of neutrons in V during dt per unit solid angle and unit energy  transfer of neutrons through S during dt per unit solid angle and unit energy

This equation simply states that the time-variation of the number of neutrons per unit solid angle and energy contained in the volume V is due to the imbalance between the number of neutrons appearing in V and disappearing from V per unit solid angle and energy during the time dt. Since neutrons can also enter into/leave from the volume V by the surface S, such a contribution is explicitly written in Eq. (2.17). The transfer of neutrons through S per unit solid angle and energy is given, according to Eq. (2.15) by: transfer of neutrons through ZS during dt per unit solid angle and energy Z ¼ N $ Ujðr; U; E; tÞdtd 2 r ¼ V $ Ujðr; U; E; tÞdtd 3 r S

(2.18)

V

with N representing the outward unit vector normal to the elementary surface d2r. The last equality was obtained making use of the divergence theorem. The operator V being a differential operator with respect to space, one further has

34 Modelling of Nuclear Reactor Multi-physics

V $ ½Ujðr; U; E; tÞ ¼ U $ Vjðr; U; E; tÞ. Eq. (2.18) can thus be rewritten as: transfer of neutrons through S during dt per unit solid angle and energy Z ¼ U $ Vjðr; U; E; tÞdtd 3 r

(2.19)

V

The disappearance of neutrons from V per unit solid angle and energy (other than by transfer through S) can be accounted for by using the total macroscopic cross-section St ðr; E; tÞ as: disappearance of neutrons in V during dt per unit solid angle and energy Z ¼ St ðr; E; tÞjðr; U; E; tÞdtd 3 r

(2.20)

V

since neutrons interacting via any nuclear reaction in position r and energy E at time t are ‘removed’ from the balance of neutrons existing at position r, energy E and direction U. Finally, the appearance of neutrons into V (other than by transfer through S) has two origins. First, neutrons having any energy E0 and having a direction U0 can be scattered into the energy E and direction U. Second, neutrons can be produced by fission reactions, either as prompt neutrons or as delayed neutrons. Therefore, the appearance of neutrons into V per unit solid angle and energy (other than by transfer through S) can be written using the differential scattering macroscopic cross-section, the fission cross-section and the concentration of the precursors of delayed neutrons as: production of neutrons in V during dt per unit solid angle and energy Z Z Z N ¼ Ss ðr; U0 /U; E 0 /E; tÞjðr; U0 ; E 0 ; tÞdtd 3 rd 2 U0 dE 0 V ð4pÞ

0

Z Z Z þ V ð4pÞ

þ

0

N

½1  bðr; E 0 ÞSf ðr; U0 /U; E 0 /E; tÞjðr; U0 ; E 0 ; tÞdtd 3 rd 2 U0 dE 0

(2.21)

Z Nd 1 X cdi ðr; EÞ li ðrÞCi ðr; tÞdtd 3 r 4p i¼1 V

where the factor 1=4p in the last term on the right-hand side of Eq. (2.21) accounts for the fact that delayed neutrons are emitted isotropically in the laboratory reference system. The integral in the second term on the right-hand side of this equation represents all neutrons produced by fission. Only a fraction 1  bðr; EÞ leads to prompt neutrons at the instant of fission, explaining why the integral is multiplied by such a fraction. The delayed neutrons can be properly considered by counting the decay rate of the corresponding precursors Ci. Each time a precursor decays above the virtual energy level leading to neutron emission, this secondary emission is occurring in extremely

Chapter 2
Transport phenomena in nuclear reactors

35

short times compared to the decay. As a consequence, the decay rate of the neutron precursors corresponds to the production rate of the delayed neutrons. Because of the introduction of the concentration of the neutron precursors in Eq. (2.21), a separate equation is also required for each group of precursors. Such additional equations can be simply written as: vCi ðr; tÞdt Ci ðr; t þ dtÞ  Ci ðr; tÞ ¼ vt Z Z N bi ðr; EÞnðr; EÞSf ðr; E; tÞjðr; U; E; tÞdtd 2 UdE  li ðrÞCi ðr; tÞdt; i ¼ 1; . ; Nd ¼ ð4pÞ

(2.22)

0

These balance equations simply state that any time dependence in the concentration of the precursors of delayed neutrons belonging to group i results from an imbalance between the number of precursors decaying at a rate of li ðrÞCi ðr; tÞ and the number of neutron precursors produced. Since the precursors of delayed neutrons are produced at the instant of fission, and since they eventually lead to a fraction bi ðr; EÞnðr; EÞ of delayed neutrons (emitted almost instantaneously after the decay of the corresponding fission fragments), bi ðr; EÞnðr; EÞ times the number of fission reactions during the time dt, i.e. R RN 2 0 bi ðr; EÞnðr; EÞSf ðr; E; tÞjðr; U; E; tÞdtd UdE, gives the number of precursors of ð4pÞ

delayed neutrons from the group i produced during time dt. Recalling the definition of the scalar flux (i.e. Eq. 2.5), Eqs. (2.21) and (2.22) can be rewritten, using Eq. (2.11), as: production of neutrons in V during dt per unit solid angle and energy Z Z Z N ¼ Ss ðr; U0 /U; E 0 /E; tÞjðr; U0 ; E 0 ; tÞdtd 3 rd 2 U0 dE 0 0

V ð4pÞ

þ

þ

cp ðr; EÞ 4p 1 4p

and vCi ðr; tÞdt ¼ vt

Nd X

Z Z V

0

0

½1  bðr; E 0 Þnðr; E 0 ÞSf ðr; E 0 ; tÞfðr; E 0 ; tÞdtd 3 rdE 0 Z

cdi ðr; EÞ

i¼1

Z

N

(2.23)

li ðrÞCi ðr; tÞdtd 3 r

V

N

bi ðr; EÞnðr; EÞSf ðr; E; tÞfðr; E; tÞdtdE  li ðrÞCi ðr; tÞdt; i ¼ 1; . ; Nd

(2.24)

For the sake of ease of the notations, we introduce: e bi ðrÞ ¼

RN 0

bi ðr; E 0 Þnðr; E 0 ÞSf ðr; E 0 Þfðr; E 0 ÞdE 0 RN nðr; E 0 ÞSf ðr; E 0 Þfðr; E 0 ÞdE 0 0

(2.25)

and define accordingly: e bðrÞ ¼

Nd X i¼1

e bi ðrÞ

(2.26)

36 Modelling of Nuclear Reactor Multi-physics

Eqs. (2.23) and (2.24) thus result in: production of neutrons in V during dt per unit solid angle and energy Z Z Z N ¼ Ss ðr; U0 /U; E 0 /E; tÞjðr; U0 ; E 0 ; tÞdtd 3 rd 2 U0 dE 0 0

V ð4pÞ

Z Z N  cp ðr; EÞ  e 1  bðrÞ þ nðr; E 0 ÞSf ðr; E 0 ; tÞfðr; E 0 ; tÞdtd 3 rdE 0 4p 0

(2.27)

V

Z Nd 1 X d þ c ðr; EÞ li ðrÞCi ðr; tÞdtd 3 r 4p i¼1 i V

and vCi ðr; tÞdt ¼ e bi ðrÞ vt

Z

N

nðr; EÞSf ðr; E; tÞfðr; E; tÞdtdE  li ðrÞCi ðr; tÞdt; i ¼ 1; . ; Nd

0

(2.28)

Combining Eqs. (2.17), (2.19) and (2.20), and (2.27), one notices that all terms are written as volume integrals, that can therefore be dropped. The resulting equation becomes, using Eq. (2.4): 1 v jðr; U; E; tÞ þ U $ Vjðr; U; E; tÞ þ St ðr; E; tÞjðr; U; E; tÞ vðEÞ vt Z Z N ¼ Ss ðr; U0 /U; E 0 /E; tÞjðr; U0 ; E 0 ; tÞd 2 U0 dE 0 ð4pÞ

þ

 cp ðr; EÞ  1e bðrÞ 4p

0

Z

N 0

nðr; E 0 ÞSf ðr; E 0 ; tÞfðr; E 0 ; tÞdE 0 þ

(2.29)

Nd 1 X cd ðr; EÞli ðrÞCi ðr; tÞ 4p i¼1 i

where the equation for the precursors of delayed neutrons is given, using Eq. (2.24), as: vCi ðr; tÞdt ¼ e bi ðrÞ vt

Z

N 0

nðr; EÞSf ðr; E; tÞfðr; E; tÞdE  li ðrÞCi ðr; tÞ; i ¼ 1; . ; Nd

(2.30)

Eq. (2.29) represents the time-dependent integro-differential form of the neutron transport equation, which has to be solved with Eq. (2.30). Because of its simpler structure (no spatial derivatives), Eq. (2.30) can be directly integrated (using e.g., the variation of constants method), such that the precursor density can be expressed as: Ci ðr; tÞ ¼ e bi ðrÞ

Z

t

N

Z

0

N

0

nðr; EÞSf ðr; E; t 0 Þeli ðrÞ½tt  fðr; E; t 0 Þdt 0 dE

(2.31)

Eq. (2.29) can then be re-arranged into: 1 v jðr; U; E; tÞ þ U $ Vjðr; U; E; tÞ þ St ðr; E; tÞjðr; U; E; tÞ vðEÞ vt Z Z N ¼ Ss ðr; U0 /U; E 0 /E; tÞjðr; U0 ; E 0 ; tÞd 2 U0 dE 0 ð4pÞ

0

t Z N   1 nðr; E 0 ÞSf ðr; E 0 ; t 0 Þfðr; E 0 ; t 0 Þ 1  e bðrÞ cp ðr; EÞdðt  t 0 Þ þ . 4p N 0 ) Nd X 0 cdi ðr; EÞli ðrÞe bi ðrÞeli ðrÞ½tt  dt 0 dE 0

Z

þ

i¼1

(2.32)

Chapter 2
Transport phenomena in nuclear reactors

37

In case of steady-state behaviour, Eq. (2.30) simplifies into: li ðrÞCi ðrÞ ¼ e bi ðrÞ

Z

0

N

nðr; EÞSf ðr; EÞfðr; EÞdE

(2.33)

which can be used in Eq. (2.29) to give: U $ Vjðr; U; EÞ þ St ðr; EÞjðr; U; EÞ Z Z N Ss ðr; U0 /U; E 0 /EÞjðr; U0 ; E 0 Þd 2 U0 dE 0 ¼ 0

ð4pÞ

( þ

Nd  cp ðr; EÞ  1 X cd ðr; EÞe bi ðrÞ 1e bðrÞ þ 4p 4p i¼1 i

(2.34)

)Z

N 0

nðr; E 0 ÞSf ðr; E 0 Þfðr; E 0 ÞdE 0

Eqs. (2.32) and (2.34) represent the transport equations, for time-dependent and time-independent systems, respectively, written in their integro-differential form. If the system is not critical, a time-independent transport equation can still be obtained by re-normalizing the fission source by a factor k as: U $ Vjðr; U; EÞ þ St ðr; EÞjðr; U; EÞ Z Z N Ss ðr; U0 /U; E 0 /EÞjðr; U0 ; E 0 Þd 2 U0 dE 0 ¼ ð4pÞ

0

(

d   X 1 cp ðr; EÞ 1  e þ bðrÞ þ cdi ðr; EÞe bi ðrÞ 4pk i¼1

N

(2.35)

)Z

N 0

nðr; E 0 ÞSf ðr; E 0 Þfðr; E 0 ÞdE 0

Integrating this equation with respect to r on the reactor volume V, with respect to U on all directions, and with respect to E on all energies allows writing: Z Z

k

¼ Z Z Z V

ð4pÞ

N 0

N

0

V

nðr; E 0 ÞSf ðr; E 0 Þfðr; E 0 Þd 3 rdE 0

½U $ V þ St ðr; EÞjðr; U; EÞd 3 rd 2 UdE 

Z Z Z

N

0

V ð4pÞ

Ss ðr; E 0 Þjðr; U0 ; E 0 Þd 3 rd 2 UdE 0 (2.36)

where Eqs. (2.2) and (2.3), as well as (2.10), were used. Using the fact that the operator V is a differential operator with respect to space, one has U $ Vjðr; U; EÞ ¼ V $ ½Ujðr; U; EÞ in the denominator of the previous equation. Making use of the divergence theorem and recalling the definition of the scalar neutron flux given by Eq. (2.5) and of the neutron current density vector given by Eq. (2.6), one can then write: Z Z

k

¼ Z Z S

0

N

V 2

N 0

Z Z

Jðr; EÞ $ Nd rdE þ V

0

nðr; E 0 ÞSf ðr; E 0 Þfðr; E 0 Þd 3 rdE 0

N

St ðr; EÞfðr; EÞd 3 rdE 

Z Z V

0

N

Ss ðr; E 0 Þfðr; E 0 Þd 3 rdE 0

(2.37)

38 Modelling of Nuclear Reactor Multi-physics

where S represents the outer surface of the system. Using finally Eq. (2.13) leads to: Z Z

k¼ Z Z S

0

N

V

0

N

nðr; E 0 ÞSf ðr; E 0 Þfðr; E 0 Þd 3 rdE 0

Jðr; EÞ $ Nd 2 rdE þ

Z Z V

0

N

Sa ðr; EÞfðr; EÞd 3 rdE

(2.38)

k is the ratio between the number of neutrons released by fissions per unit time and the sum between the number of neutrons leaking out of the system per unit time and the number of neutrons absorbed per unit time. This factor is associated in classical reactor physics to the ratio between the number of neutrons in any neutron generation and the number of neutrons in the immediately preceding generation, when referring to the neutron cycle (as simplistically depicted in Fig. 2.3). Consequently, k is also the effective multiplication factor keff of the system.

Integral form The neutron transport equation can also be written in a pure integral form. The integral form of the neutron transport equation is, in some occurrences, easier to use than the integro-differential form of the neutron transport equation. The integral form of the neutron transport equation is derived by estimating the angular neutron flux from the neutron emission density q(r, U, E, t), giving the number of neutrons emitted at time t, at position r, in direction U, with an energy E either by scattering or fission reactions. Following the same basic principles as for the integrodifferential form, the neutron emission density can be written as: qðr; U; E; tÞ Z Z N ¼ Ss ðr; U0 /U; E 0 /EÞjðr; U0 ; E 0 ; tÞd 2 U0 dE 0 ð4pÞ

0

Z

t Z N   1 nðr; E 0 ÞSf ðr; E 0 Þfðr; E 0 ; t 0 Þ 1  e bðrÞ cp ðr; EÞdðt  t 0 Þ þ . 4p N 0 ) Nd X 0 d l ðrÞ½tt ci ðr; EÞli ðrÞe bi ðrÞe i dt 0 dE 0

þ

(2.39)

i¼1

For the sake of simplicity, it is assumed that the macroscopic cross-sections are timeindependent. Allowing the cross-sections to be time-dependent significantly complicates the derivation presented hereafter. If one considers a given direction U and a given point A(r) in position r on that direction, as represented in Fig. 2.12, the neutrons existing in that position are the ones

FIGURE 2.12 Diagram for deriving the integral form of the neutron transport equation.

Chapter 2
Transport phenomena in nuclear reactors

39

that were emitted upstream from that position and that did not interact on the path between their emission site and point A(r). If P(r, s, U, E) denotes the probability of non-interaction between a point M(r  sU) upstream from point A(r) by the distance s along the direction U, then the angular flux observed in point A(r) can be formally estimated as: Z

jðr; U; E; tÞ ¼

N 0

 Pðr; s; E; UÞq r  sU; U; E; t 

 s ds vðEÞ

(2.40)

since the neutrons that arrive in point A(r) at time t were necessarily emitted in all upstream positions M(r  sU) at time t  s/v(E), where v(E) represents the neutron speed (related to the neutron energy as E ¼ mv2(E)/2 with m being the mass of a neutron). Using the total macroscopic cross-section earlier introduced, the probability of noninteraction between a point M(r  sU) upstream from point A(r) by the distance s along the direction U is given by: 2

Pðr; s; E; UÞ ¼ exp4 

3

Z

s 0

0

St ðr  s U; EÞds

05

(2.41)

Combining Eqs. (2.40) and (2.41) leads to the integral form of the neutron transport equation: Z jðr; U; E; tÞ ¼

0

8

N
0, a loss of F occurs through Sm(t), whereas when F$N < 0, a gain of F occurs.

Chapter 2
Transport phenomena in nuclear reactors

Using Eq. (2.62) written for ðrf Þ, one also has: D Dt

Z

ðrf Þðr; tÞd 3 r ¼

Vm ðtÞ

Z

Vm ðtÞ

vðrf Þ ðr; tÞd 3 r þ vt

I

ðrf vÞðr; tÞ $ Nd 2 r

47

(2.66)

Sm ðtÞ

Combining Eq. (2.64), (2.65) and (2.66), one obtains the following balance equation: Z Vm ðtÞ

vðrf Þ ðr; tÞd 3 r þ vt

I

Z

2

ðrf vÞðr; tÞ $ Nd r ¼ Sm ðtÞ

I

3

Fðr; tÞ $ Nd 2 r

ðrqÞðr; tÞd r  Vm ðtÞ

(2.67)

Sm ðtÞ

if f is a scalar, and Z Vm ðtÞ

vðrfÞ ðr; tÞd 3 r þ vt

I

ðrf5vÞðr; tÞ $ Nd 2 r ¼

Sm ðtÞ

Z

ðrqÞðr; tÞd 3 r 

Vm ðtÞ

I

Fðr; tÞ $ Nd 2 r

(2.68)

Sm ðtÞ

if f is a vector. In Eq. (2.67), q is a scalar and F is a vector, whereas in Eq. (2.68) q is a vector and F is a tensor. Using Gauss’ theorem in Eq. (2.67) allows writing: Z Vm ðtÞ

vðrf Þ ðr; tÞd 3 r þ vt

Z

Z

V $ ðrf vÞðr; tÞd 3 r ¼

Vm ðtÞ

Z

ðrqÞðr; tÞd 3 r 

Vm ðtÞ

V $ Fðr; tÞd 3 r

(2.69)

Vm ðtÞ

if f is a scalar, and in Eq. (2.68): Z Vm ðtÞ

vðrfÞ ðr; tÞd 3 r þ vt

Z Vm ðtÞ

V $ ðrf5vÞðr; tÞd 3 r ¼

Z Vm ðtÞ

ðrqÞðr; tÞd 3 r 

Z

V $ Fðr; tÞd 3 r

(2.70)

Vm ðtÞ

if f is a vector. Since this equation is valid for any arbitrary volume Vm(t), the volume integration can be removed. This gives: vðrf Þ ðr; tÞ þ V $ ðrf vÞðr; tÞ ¼ ðrqÞðr; tÞ  V $ Fðr; tÞ vt

(2.71)

vðrfÞ ðr; tÞ þ V $ ðrf5vÞðr; tÞ ¼ ðrqÞðr; tÞ  V $ Fðr; tÞ vt

(2.72)

if f is a scalar, and

if f is a vector. Eqs. (2.71) and (2.72) are referred to as the local (differential) conservation or balance equation for the specific value f or f of the extensive property F or F, respectively. We will see in the following of this chapter that the balance Eq. (2.71) can be written for expressing the conservation of mass and energy, whereas the balance Eq. (2.72) can be used for expressing the conservation of linear momentum.

48 Modelling of Nuclear Reactor Multi-physics

2.3.c

Mass and momentum differential conservation equations

Mass conservation equation The first equation to be written considers mass conservation. In such a case, the extensive property is F ¼ m and correspondingly, one has f ¼ 1. In Eq. (2.67) and per definition of the material volume, one has F ¼ 0 since there is no transfer of mass through the corresponding surface, and one has q ¼ 0 since there is no production/ disappearance of mass within the volume. The conservation of mass thus gives the following local conservation equation: vr ðr; tÞ þ V $ ðrvÞðr; tÞ ¼ 0 vt

(2.73)

This conservation equation can also be written as: Dr ðr; tÞ þ rðr; tÞV $ vðr; tÞ ¼ 0 Dt

(2.74)

making use of the fact that V $ ðrvÞðr; tÞ ¼ vðr; tÞ $ Vrðr; tÞ þ rðr; tÞV $ vðr; tÞ and of Eq. (2.56).

Momentum conservation equation The conservation of momentum is an expression of Newton’s second law, according to which the rate of change of the momentum is due to the forces applied to the fluid. The forces that need to be considered in the present case are the forces due to pressure, stresses and gravity. The stress forces are due to viscosity and can be defined via the use of a stress tensor s that is defined in a Cartesian coordinate system as: sxx 6s s ¼ 4 yx

sxy syy

3 sxz syz 7 5

szx

szy

szz

2

(2.75)

and that completely specifies the stresses at a point in the fluid by its nine components, as represented in Fig. 2.15. This figure shows an infinitesimal volume that one should consider shrinking to a point, since stresses are defined at points. A normal stress and a shear stress are applied on each of the faces, where the normal stresses are normal to the faces, whereas the shear stresses are tangential to the faces. The shear stress is further resolved into two components in the directions of the axes. The first index of sij indicates the direction of the normal to the surface on which the stress is applied, and the second index indicates the direction along which the stress acts. A stress represents a force per unit surface area. The amplitude of the infinitesimal force dFij corresponding to the stress sij , applied in the direction j, can thus be expressed as: dFij ¼ sij d2 r

(2.76)

where d2r is the elementary surface normal to the direction i. One thus notices that the

Chapter 2
Transport phenomena in nuclear reactors

49

FIGURE 2.15 Representation of the stresses on an infinitesimal volume (for clarity, components on only three of the six faces are shown).

multiplication of the stress tensor with the outward unit vector N normal to the surface d2r, i.e. s$Nd2r is a vector having for components the projection, on each of the axes, of the force dF. For the problems investigated in nuclear applications, it could be further demonstrated that the stress tensor is symmetric (Kundu and Cohen, 2004), i.e. sij ¼ sji

(2.77)

Regarding the pressure, the elementary force dF related to the pressure P applied to an elementary surface d2r is given by dF ¼ PNd2r ¼ P(I$N)d2r with I being the identity matrix. The minus sign comes from the fact that N is the outward normal vector of the elementary surface d2r (delimiting the considered material volume), whereas the force is acting towards the surface, as illustrated in Fig. 2.16.

FIGURE 2.16 Representation of the pressure force on an infinitesimal volume.

50 Modelling of Nuclear Reactor Multi-physics

Consequently, the elementary forces applied to a surface d2r due to stresses and pressure are given by s$Nd2r  P(I$N)d2r ¼ (s  PI)$Nd2r. Finally, for the forces due to gravity applied on an elementary volume d3r, one has dF ¼ rgd 3 r where g is the gravitational acceleration. Comparing the expression of the forces dues to stresses and pressure to the second term on the right-hand side of Eq. (2.68), one notices that for the momentum balance equation, following Newton’s second law, one should have F ¼ s  PI. Likewise, comparing the expression of the forces due to gravity to the first term on the right-hand side of Eq. (2.67), one should have q ¼ g. The conservation of (linear) momentum F ¼ mv, for which correspondingly one has f ¼ v, can thus be expressed by the following local conservation equation: vðrvÞ ðr; tÞ þ V $ ðrv5vÞðr; tÞ ¼ V $ sðr; tÞ  VPðr; tÞ þ rðr; tÞg vt

(2.78)

using the fact that V $ ½Pðr; tÞI ¼ Pðr; tÞV $ I þ VPðr; tÞ $ I ¼ VPðr; tÞ

This equation can also be written as: rðr; tÞ

vv ðr; tÞ þ ðrvÞðr; tÞ $ V5vðr; tÞ ¼ V $ sðr; tÞ  VPðr; tÞ þ rðr; tÞg vt

(2.79)

making use of the fact that V $ ðrv5vÞðr; tÞ ¼ V $ ðv5rvÞðr; tÞ ¼ vðr; tÞV $ ðrvÞðr; tÞ þ ðrvÞðr; tÞ $ V5vðr; tÞ

and of Eq. (2.73). Because of Eq. (2.57), Eq. (2.79) can also be written as: rðr; tÞ

Dv ðr; tÞ ¼ V $ sðr; tÞ  VPðr; tÞ þ rðr; tÞg Dt

(2.80)

The mass and momentum local conservation equations can be expressed using the generic local conservation equations as given in Eqs. (2.71) and (2.72), in which the expressions for the surface and volumetric terms are given in Table 2.3.

2.4 Heat transfer In this book, emphasis is put on the nuclear core, i.e. the part of the nuclear power plants containing the nuclear fuel assemblies. In normal and incidental situations, two main Table 2.3 Definition of the variables appearing on the right-hand side of Eqs. (2.71) and (2.72) for the conservations of mass and momentum. f or f F q or q

Conservation of mass

Conservation of momentum

1 0 0

v -(s  PI) g

Chapter 2
Transport phenomena in nuclear reactors

51

mechanisms are responsible for the transfer of heat: conduction and forced convection. The heat transfer by radiation typically from the fuel pellets to the cladding in such conditions is negligible (Tong and Weisman, 1996). Nevertheless, in accidental situations, the transfer of heat by radiation also becomes important, for instance when insufficient cooling leads to high temperatures of the fuel and no liquid is in contact with the cladding of the fuel (Tong and Weisman, 1996). Since the focus of this book is on normal and incidental situations, the heat transfer by radiation is not touched upon in this book, and the interested reader is instead referred to the existing literature on that subject. The governing equations describing heat transfer by conduction and convection are presented hereafter.

2.4.a

Heat transfer by conduction

Conduction refers to the transfer of energy arising from temperature differences within a body. This transfer occurs by interaction at the microscopic scale between adjacent particles within the body (molecules, atoms, electrons, etc.). Heat transfer by conduction is characterized by Fourier’s law of conduction, according to which one has: q00 ðr; tÞ ¼  kðr; T Þ $ VT ðr; tÞ

(2.81)

00

where q is the surface heat flux, T is the temperature, and k is the thermal conductivity. In the most general case, the thermal conductivity is a tensor. Nevertheless, for isotropic media, the thermal conductivity can be taken as a scalar. If the medium is also homogeneous, Eq. (2.81) then simplifies into: q00 ðr; tÞ ¼  kðT ÞVT ðr; tÞ

(2.82)

In the following, one considers a volume V, bounded by a surface S, as represented in Fig. 2.17, and encompassing a body considered as a homogeneous and isotropic me000 dium, and possibly containing a volumetric heat source with q being the rate of heat production per unit volume. In the absence of work, a balance equation can be written in terms of specific internal energy u (i.e. internal energy per unit mass) during the infinitesimal time dt as:

FIGURE 2.17 Balance with respect to the thermal energy contained in a volume V bounded by a surface S.

52 Modelling of Nuclear Reactor Multi-physics Z

3

Z

uðr; t þ dtÞrðT Þd r  V

3

Z

vu ðr; tÞd 3 rdt vt V I Z ¼ q000 ðr; tÞd 3 rdt  q00 ðr; tÞ $ Nd 2 rdt

uðr; tÞrðT Þd r ¼ V

rðT Þ

V

(2.83)

S

with q00 being the surface heat flux, N representing the outward unit vector normal to the elementary surface d2r and r being the density of the medium. This equation simply states that any variation in the internal energy over the volume is the result of an imbalance between the heat produced in the volume during the infinitesimal time dt and the heat loss at the boundary of the volume during the infinitesimal time dt. If one considers an incompressible medium, one further has: duðr; tÞ ¼ cP ðT ÞdT ðr; tÞ

(2.84)

where cP(T) is the specific heat at constant pressure. Eq. (2.83) can thus be re-arranged as: Z

cP ðT ÞrðT Þ V

vT ðr; tÞd 3 rdt ¼ vt

Z

q000 ðr; tÞd 3 rdt 

V

I

q00 ðr; tÞ $ Nd 2 rdt

(2.85)

S

This equation can be rewritten using Gauss’ divergence theorem as: Z

cP ðT ÞrðT Þ V

vT ðr; tÞd 3 rdt ¼ vt

Z

q000 ðr; tÞd 3 rdt 

V

Z

V $ q00 ðr; tÞd 3 rdt

(2.86)

V

Using Fourier’s law of conduction, i.e. Eq. (2.82), one finally finds: Z

cP ðT ÞrðT Þ V

vT ðr; tÞd 3 rdt ¼ vt

Z

q000 ðr; tÞd 3 rdt þ

V

Z

V $ ½kðT ÞVT ðr; tÞd 3 rdt

(2.87)

V

Since this equation is valid for any volume V, the volume integration can be dropped, and one obtains the heat conduction equation that can be written as: cP ðT ÞrðT Þ

2.4.b

vT ðr; tÞ ¼ q000 ðr; tÞ þ V $ ½kðT ÞVT ðr; tÞ vt

(2.88)

Heat transfer by convection

Convection refers to the process by which heat applied to a fluid is transferred by movement of the fluid. In natural convection, the movement of the fluid is due to density gradients. In forced convection, the movement of the fluid is not primarily due to buoyancy effects, but due to external devices such as pumps. Using the local differential equation (i.e. Eq. 2.71), the conservation of energy gives the following local conservation equation:

Chapter 2
Transport phenomena in nuclear reactors

vðreÞ ðr; tÞ þ V $ ðrevÞðr; tÞ vt ¼ V $ q00 ðr; tÞ þ q000 ðr; tÞ þ V $ ðs $ vÞðr; tÞ  V $ ðPvÞðr; tÞ þ ðrg $ vÞðr; tÞ

53

(2.89)

where e is the specific energy (energy per unit mass), sum of the specific internal energy u (internal energy per unit mass) and the specific kinetic energy (kinetic energy per unit mass), i.e. e¼u þ

v2 2

(2.90)

This equation states that the rate of change of the energy, i.e. the power, is due to the work rate of the elementary forces dF applied to the fluid (the forces being related to pressure, stresses and gravity) expressed as dF$v, as well as any volumetric heat gen000 eration rate q and surface heat flux q00 . The corresponding expressions for f, q and F to be used in Eq. (2.71) are given in Table 2.4. Since the gravity vector g is related to the (gravitational) potential energy j ¼ gz (with z being the elevation) as g ¼ Vj , Eq. (2.89) can be recast, using the mass conservation equation (i.e. Eq. 2.73) and the fact that the potential energy j is time-independent, into: vðreT Þ ðr; tÞ þ V $ ðreT vÞðr; tÞ vt ¼ V $ q00 ðr; tÞ þ q000 ðr; tÞ þ V $ ðs $ vÞðr; tÞ  V $ ðPvÞðr; tÞ

(2.91)

2

where eT ¼ e þ j ¼ u þ v2 þ j is the total specific energy. Eq. (2.91) thus expresses a conservation of the total energy eT, whereas Eq. (2.89) expresses a conservation of the energy e as defined in Eq. (2.90) (i.e. the total energy minus the potential energy). Eqs. (2.89) and (2.91) can alternatively be written as: ve ðr; tÞ þ ðrvÞðr; tÞVeðr; tÞ vt ¼ V $ q00 ðr; tÞ þ q000 ðr; tÞ þ V $ ðs $ vÞðr; tÞ  V $ ðPvÞðr; tÞ þ ðrg $ vÞðr; tÞ

rðr; tÞ

(2.92)

making use of the fact that V $ ðrevÞðr; tÞ ¼ ðrvÞðr; tÞ $ Veðr; tÞ þ eðr; tÞV $ ðrvÞðr; tÞ and of Eq. (2.73). Because of Eq. (2.56), this equation can also be written as: De ðr; tÞ Dt ¼ V $ q00 ðr; tÞ þ q000 ðr; tÞ þ V $ ðs $ vÞðr; tÞ  V $ ðPvÞðr; tÞ þ ðrg $ vÞðr; tÞ rðr; tÞ

Table 2.4 Definition of the variables appearing on the right-hand side of Eq. (2.71) for the conservations of energy. Conservation of energy f F q

e q00  (s  PI),v q000 þ g:v r

(2.93)

54 Modelling of Nuclear Reactor Multi-physics

Usually, the use of the enthalpy is preferred to the use of the internal energy and the total energy for describing open systems. A conservation equation for the enthalpy can be easily obtained from Eq. (2.93) in the following manner. One can first write that: V $ ðPvÞðr; tÞ     P P P rv ðr; tÞ ¼ ðr; tÞV $ ðrvÞðr; tÞ þ ðrvÞðr; tÞ $ V ðr; tÞ ¼ V$ r r r   P vr P ðr; tÞ ¼  ðr; tÞ ðr; tÞ þ ðrvÞðr; tÞ $ V r vt r

(2.94)

where the last equality was obtained using Eq. (2.73). One further has:

      v v P P vr v P Pðr; tÞ ¼ r ðr; tÞ ¼ ðr; tÞ ðr; tÞ þ rðr; tÞ ðr; tÞ vt vt r r vt vt r

(2.95)

Combining Eqs. (2.94) and (2.95) thus gives: V $ ðPvÞðr; tÞ

    vP v P P ðr; tÞ þ rðr; tÞ ðr; tÞ þ ðrvÞðr; tÞ $ V ðr; tÞ vt vt r r   vP D P ðr; tÞ ¼  ðr; tÞ þ rðr; tÞ vt Dt r

¼

(2.96)

where the last equality was obtained using Eq. (2.56). Taking now the scalar product between Eq. (2.80) and v(r, t) gives: ðrvÞðr; tÞ $

Dv ðr; tÞ ¼ vðr; tÞ $ ½V $ sðr; tÞ  vðr; tÞ $ VPðr; tÞ þ vðr; tÞ $ rðr; tÞg Dt

(2.97)

This equation can be re-arranged into: rðr; tÞ

  D 1 2 v ðr; tÞ Dt 2

(2.98)

¼ V $ ðs $ vÞðr; tÞ  sðr; tÞ: ½V5vðr; tÞ  vðr; tÞ $ VPðr; tÞ þ rðr; tÞg $ vðr; tÞ

making use of the fact that V $ ðs $ vÞðr; tÞ ¼ vðr; tÞ $ ½V $ sðr; tÞ þ sðr; tÞ: ½V 5vðr; tÞ. Eq. (2.98) represents a local conservation equation written in a Lagrangian sense for the kinetic energy. Since the specific enthalpy is defined as h ¼ u þ P=r and since e ¼ u þ v2/2, Eq. (2.98) can be subtracted from Eq. (2.93) to obtain: rðr; tÞ

Dh ðr; tÞ Dt

¼ V $ q00 ðr; tÞ þ q000 ðr; tÞ þ sðr; tÞ: ½V5vðr; tÞ þ

vP ðr; tÞ þ vðr; tÞ $ VPðr; tÞ vt

(2.99)

where Eq. (2.96) was used. Finally, recalling Eq. (2.56), one finally finds that: rðr; tÞ

Dh DP ðr; tÞ ¼  V $ q00 ðr; tÞ þ q000 ðr; tÞ þ sðr; tÞ: ½V5vðr; tÞ þ ðr; tÞ Dt Dt

(2.100)

Chapter 2
Transport phenomena in nuclear reactors

55

which is a local conservation equation written in a Lagrangian sense for the enthalpy. In some occurrences, it is more practical to write the energy/enthalpy conservation equation in terms of temperature directly. A relationship between the enthalpy and temperature of the fluid can be derived in the following manner. One first recalls that according to the definition of the enthalpy, the specific enthalpy (i.e., enthalpy per unit mass) can be written when no shaft work is present as: dhðr; tÞ ¼ dqðr; tÞ þ

1 dPðr; tÞ rðr; tÞ

(2.101)

where dq is the infinitesimal heat per unit mass. This infinitesimal heat per unit mass can be expressed in the (P, T) phase space as: dqðr; tÞ ¼ cP ðr; tÞdT ðr; tÞ þ h0 ðr; tÞdPðr; tÞ

(2.102)

where cP and h0 are specific quantities, i.e. quantities per unit mass. cP is the specific heat at constant pressure already introduced in Eq. (2.84) and h0 is a calorimetric coefficient. Furthermore, one has:  dq  h0 ðr; tÞ ¼  ðr; tÞ vP T

(2.103)

Assuming thermodynamic reversibility, the infinitesimal heat per unit mass dq can be expressed via the specific entropy s using the second law of thermodynamics as: dqðr; tÞ ¼ T ðr; tÞdsðr; tÞ

(2.104)

This equation allows transforming Eq. (2.103) into:  vs  h0 ðr; tÞ ¼ T  ðr; tÞ vP T

(2.105)

Finally, using the so-called Maxwell relation applied to the Gibbs free energy, one has:   vs  vð1=rÞ ðr; tÞ ¼  ðr; tÞ vP T vT P

(2.106)

which, put in Eq. (2.105), leads to:

 vð1=rÞ Tb h0 ðr; tÞ ¼  T ðr; tÞ ¼  ðr; tÞ vT P r

(2.107)

with b being the volumetric thermal expansion coefficient defined as: bðr; tÞ ¼ 

 1 vr  ðr; tÞ r vT P

(2.108)

Combining Eq. (2.101), (2.102) and (2.107), one finally obtains: dhðr; tÞ ¼ cP ðr; tÞdT ðr; tÞ þ

1 ½1  T bðr; tÞdPðr; tÞ rðr; tÞ

Dividing by an infinitesimal time interval dt, one gets:

(2.109)

56 Modelling of Nuclear Reactor Multi-physics

vh vT 1  T bðr; tÞ vP ðr; tÞ ¼ cP ðr; tÞ ðr; tÞ þ ðr; tÞ vt vt rðr; tÞ vt

(2.110)

Since Eq. (2.110) represents a relationship between intrinsic properties of the fluid, this equation is also valid in terms of Lagrangian time-derivatives, i.e. Dh DT 1  T bðr; tÞ DP ðr; tÞ ¼ cP ðr; tÞ ðr; tÞ þ ðr; tÞ Dt Dt rðr; tÞ Dt

(2.111)

Using this expression into Eq. (2.100), one obtains: ðrcP Þðr; tÞ

DT DP ðr; tÞ ¼  V $ q00 ðr; tÞ þ q000 ðr; tÞ þ sðr; tÞ: ½V5vðr; tÞ þ ðT bÞðr; tÞ ðr; tÞ Dt Dt

(2.112)

Using Fourier’s law of conduction given by Eq. (2.82) within the fluid, one finally gets: ðrcP Þðr; tÞ

DT ðr; tÞ Dt

¼ V $ ½kðr; tÞVT ðr; tÞ þ q000 ðr; tÞ þ sðr; tÞ: ½V5vðr; tÞ þ ðT bÞðr; tÞ

DP ðr; tÞ Dt

(2.113)

This is the equation to be solved if one wants to determine the temperature distribution within the fluid. For fluids flowing in nuclear cores, three equations thus need to be solved for estimating the characteristics of the fluid. These equations correspond to the conservation of mass (expressed for instance as Eq. 2.74), the conservation of momentum (expressed for instance as Eq. 2.80) and the conservation of energy or enthalpy (expressed for instance as Eq. (2.93) or Eq. (2.100)), or alternatively Eq. (2.113) for the fluid temperature. The energy/enthalpy/temperature equation is only required if the fluid is heated. In case the energy or enthalpy equations are used, the sets of equations contain six unknowns: r, v, e or h, P, s and q00 (the quantity q000 is assumed to be known). Consequently, three additional equations are required to close the system of equation: the equation(s) of state for the fluid (being a thermodynamic equation describing the state of the fluid under a given set of physical conditions), e.g. rðr; tÞhr½Pðr; tÞ; hðr; tÞ

(2.114)

and two constitutive equations that relate the stress tensor s and the heat flux q00 to unknown or given quantities, i.e. sðr; tÞhs½Pðr; tÞ; hðr; tÞ; vðr; tÞ

(2.115)

q00 ðr; tÞhq00 ½Pðr; tÞ; hðr; tÞ; T ðr; tÞ

(2.116)

Constitutive equations are fluid-specific relationships linking some of the fluid properties between each other based on either first principles or phenomenological principles. The equation of state is also a constitutive equation. In case the temperature equation is used, the sets of equations contain seven unknowns: r, v, T, P, s, cP and k (the quantity q000 is assumed to be known). Consequently,

Chapter 2
Transport phenomena in nuclear reactors

57

four additional equations are required to close the system of equations: the equation(s) of state for the fluid, e.g. Eq. (2.114), together with cP ðr; tÞhcP ½T ðr; tÞ

(2.117)

kðr; tÞhk½T ðr; tÞ

(2.118)

and one constitutive equation that relates the stress tensor s to unknown or given quantities, i.e. sðr; tÞhs½Pðr; tÞ; hðr; tÞ; vðr; tÞ

(2.119)

2.5 Overview of the modelling strategies Due to their multi-scale and multi-physics nature, modelling the behaviour of nuclear reactors requires special modelling techniques. Using general purpose multi-physics tools usually does not allow properly capturing the multi-scale aspects of nuclear reactors. The common modelling strategies all rely on separate modelling tools for resolving the different fields (the neutron density, the temperature of the fuel and the density, velocity and enthalpy of the coolant and of the moderator) and possibly the different scales. The interdependence between the different fields/scales is usually accounted for by software coupling, although more integrated approaches are being developed (see e.g. (Fiorina et al., 2015) and (Jareteg et al., 2015)). For the neutron transport calculations, i.e. the calculations of the neutron density field, assuming that the fuel temperature field as well as the coolant/moderator density, velocity and enthalpy fields are known, two classes of techniques are usually used: probabilistic and deterministic methods. In the probabilistic method (or Monte Carlo method), neutron transport problems are solved by compiling the life histories of individual neutrons in the system (Lux and Koblinger, 1991). A set of neutron histories is generated by following individual neutrons through successive collisions, which may result in scattering, radiative capture or fission. The probabilities of the various interactions depend on the neutron energy and the composition of the system and can be determined from the energy-dependent microscopic cross-sections of the different species. Such microscopic cross-sections, available from various nuclear databases, are used to sample neutron life histories throughout the system. The locations of actual collisions and the results of such collisions (direction and energy of the emerging neutrons, if any) are thus estimated from the range of probabilities by sets of random numbers. By following the behaviour of the neutrons until they are either absorbed or escape, the characteristics of the system can be evaluated by performing a statistical average of many neutron histories. Such a probabilistic approach is extremely computer-intensive, since many neutron histories are required in order to

58 Modelling of Nuclear Reactor Multi-physics

obtain results having a statistical significance and since nuclear cores are large systems. Usually, such calculations are used as reference calculations for benchmarking less computer-intensive techniques. The clear advantage of the Monte Carlo method is that all scales of the neutron density field can be resolved. Nevertheless, due to the computational cost of such a method, only steady-state conditions or slowly varying conditions (e.g. due to fuel burn-up) are usually considered. Updating the fuel temperature and coolant enthalpy fields from the results of the calculations giving the neutron density field is far too intensive in terms of computer resources. Research on trying to couple Monte Carlo with thermal-hydraulics modelling tools for both steadystate and transient calculations is on-going (see e.g. Gill et al., 2017; Leppa¨nen et al., 2015; Sjenitzer et al., 2015). Monte Carlo methods are also used for radiation shielding applications, i.e. for determining the spatial and energy distributions of neutrons and gammas due to radioactive sources (e.g. nuclear reactors). The knowledge of such distributions is essential for biological protection. In the deterministic method, the equation describing the behaviour of neutrons, i.e. the transport or Boltzmann equation, is solved explicitly. Only approximated forms of the transport equation are solved, since solving the Boltzmann equation is very difficult. The modelling complexity is explained by the integro-differential nature of the Boltzmann equation, the large size of the system being modelled, the highly heterogeneous nature of nuclear cores and the number of independent variables appearing in the Boltzmann equation. In addition, the use of the Boltzmann equation is complicated by the fact that the nuclear data vary as a function of neutron energies in an irregular manner. Due to the multi-scale character of the system to be modelled, the neutron transport equation is solved from the meso-scales (pin cell and then fuel assembly) and finally to the macro-scales (core). Such calculations are performed in a successive manner starting from the meso-scales, for different sets of values of macro-scales variables, such as the fuel temperature, the moderator/coolant density and the fuel burn-up, for a restricted part of the system. Some boundary conditions are applied to represent in a very approximated manner the effect of the surroundings in the meso-scale calculations. The results of such calculations are typically tabulated as functions of these sets of values or polynomials representing such dependencies are built. The tabulated data or functions are thereafter used as input in the macro-scale calculations where the interdependencies resulting from the modelling of the entire system are properly accounted for. In the macro-scale calculations, both the neutron density field and the fuel temperature and coolant/moderator enthalpy fields are simultaneously determined. Due to this successive approach for covering the different scales, the use of these methods is far from being straightforward. Such calculations are meant to be used for predicting the behaviour of nuclear cores under all foreseeable circumstances and thus provide results relying on ‘reasonable’ computer resources. These calculations include both production and safety analyses. The limitations of such methods come from the fact that they are tailored to a specific geometry and/or reactor design and that the chaining of the calculations from the meso-scales to the macro-scales does not allow

Chapter 2
Transport phenomena in nuclear reactors

59

taking the interactions between neighbouring fuel assemblies at the meso-scale level into account. Although deterministic methods rely on many approximations, they nevertheless provide a sufficiently accurate modelling of the behaviour of nuclear reactor systems. The modelling of all commercial nuclear reactors for routine calculations is based on such methods. It should also be mentioned that if a reactor transient requires the determination of the temperature and flow fields throughout the entire plant, the calculation of the fuel temperature and coolant enthalpy field can be provided in this case by a thermal-hydraulic code, as the ones corresponding to the methods described below. For the thermal-hydraulic calculations, i.e. for calculations of the fuel temperature and of the coolant/moderator density, velocity and enthalpy fields throughout the nuclear core, two classes of techniques are used: system code modelling and computational fluid dynamics modelling. In both cases, the equations solved are basically the NaviereStokes equations with an energy equation for describing the fluid and Fourier’s law of conduction for describing heat conduction. For the case of system code modelling, the entire nuclear power plant is usually modelled, whereas the nuclear heating is modelled in a simplified manner. If such a need arises, the detailed information about the nuclear heating throughout the core can be provided by a deterministic neutron transport code. Only the macroscopic scales are actually resolved in this type of approach, relying on the use of empirical correlations for the modelling of the micro-/meso-scales. Only modelling the macro-scales allows performing calculations in a ‘reasonable’ computing time. The disadvantage of the system code approach is that the models (i.e. input decks) are usually very large and complex, requiring expert users for using these codes. In addition, the development of the empirical correlations requires having access to a very large number of experiments that should cover all situations encountered while operating the plant for all types of fuel assemblies loaded in a core. The use of system codes is particularly adequate when complex reactor transients involving a strong interdependence between the different components of the plant need to be investigated. Such codes are also used for the design of nuclear power plants, as well as for production purposes when running nuclear units. In case of computational fluid dynamics (CFD) modelling, both meso- and macroscales are resolved. In essence, the techniques used in the system codes and CFD approaches are the same, except that the size of the mesh used in the CFD approach is much smaller, thus allowing an explicit modelling of the meso-scales. The reliance on experimentally derived correlations is thus kept at a strict minimum. The CFD technique is a multi-purpose technique, i.e. not specific to nuclear systems, and can be used for any fluid flow problem. As such, CFD techniques can handle any system and thus represent a very flexible modelling tool. Since the meso-scales are resolved, the computing cost of such calculations increases dramatically. The calculations are usually restricted to computational domains of small size, i.e. parts of the nuclear systems. They can also be used for modelling large systems if both the necessary computing resources are available and the long waiting time before getting the results of the simulations is

60 Modelling of Nuclear Reactor Multi-physics

affordable. CFD techniques are therefore meant at providing a reference solution and allow benchmarking the solutions provided by system codes. The areas of specific applications are turbulence and mixing phenomena in one-phase flow conditions. It has to be emphasized that CFD tools are under development for modelling two-phase flow conditions, and thus the use of CFD methods for two-phase flows should not yet be considered as readily available. Large efforts are currently spent on developing CFD twophase flow capabilities (see e.g. Bestion, 2012). The different techniques presented above are summarized in Table 2.5 together with their respective advantages and disadvantages.

2.6 Deterministic and macroscopic modelling of nuclear systems Due to the multi-scale and multi-physics characteristic features of nuclear reactors, only deterministic and macroscopic methods can be used for covering all operational situations in a reasonable computing time. This is the reason why this book only focuses on deterministic methods on the neutron transport side and on system code modelling on the thermal-hydraulics side. The purpose of this section is to briefly present the problem at hand. The derivation of all equations given in this section will be thoroughly presented in the following chapters. The deterministic modelling of nuclear reactors is, as earlier mentioned, usually carried out via separate tools aiming at determining the neutron flux on the one hand (neutron transport tools), and the temperature and fluid flow fields on the other hand (thermal-hydraulic tools) (Demazie`re, 2013). With respect to neutron transport, the multi-scale aspects are in addition handled by separate tools, with one tool dealing with one specific scale. The interdependence between these different fields/scales is usually (but not necessarily) accounted for by code coupling.

2.6.a

Equations governing the neutron flux

The neutron transport equation is far too complicated to be solved on a large computational domain, as the one corresponding to a nuclear core. Instead, the time- and space-dependent neutron flux is in most cases determined using an approximated form of the neutron transport equation, namely the diffusion equation. The derivation from the transport equation of the balance equations in the diffusion approximation will be presented in detail in the following of this book. At this stage, the diffusion approximation is introduced with the sole purpose to better highlight the overall procedure used to model nuclear reactors. It should be mentioned that various methods with various levels of sophistication and refinement exist for modelling nuclear reactor systems. Nevertheless, the diffusion approximation is one of the most used approximations for routine calculations of commercial nuclear reactor cores. In most cases, the neutron energy range is divided into G bins, or energy groups, and all quantities are averaged on

Chapter 2
Transport phenomena in nuclear reactors

Table 2.5 systems. Methods

61

Summary of the different methods used for modelling nuclear reactor Field of physics

Scales that are covered Advantages

Monte Carlo or Neutron All transport probabilistic methods

Any complex geometry can be modelled

Areas of Disadvantages applications Computerintensive

Reference calculations Radiation shielding

Deterministic methods

Neutron All (but transport different calculations performed for each relevant scale)

Modelling steps Production Safety not straightforward analyses

System codes

Thermalhydraulics

Very large and complex models Specific to nuclear reactor systems

Computational Thermalfluid dynamics hydraulics

Provide results in a ‘reasonable’ computing time Can be externally coupled to thermalhydraulic tools for specific reactor transients Provide results in a Explicit modelling of ‘reasonable’ the macro- computing time Can be externally scales coupled to deterministic neutron transport codes for specific reactor transients Can model any Explicit modelling of system/geometry the mesoand macroscales

Computerintensive

Limitations No update of the thermal-hydraulic variables Steady-state or slowly varying conditions Specific to a given geometry/nuclear reactor type

Nuclear heating modelled in a simplified manner (if no coupling to a deterministic neutron transport code) No explicit modelling of the micro-/mesoscales Reference No explicit modelling calculations of the micro-scales (for turbulence Usually ‘small’ and mixing in computational domains one-phase Two-phase flow flows) models under development Design and production Safety analyses

the energy intervals, using proper weighting functions. If g represents the group index, the multi-group diffusion equation written for each energy group g ˛ [1, ., G] reads as: G G X   X   V $ Dg ðr; tÞVfg ðr; tÞ þ Ss0;g 0 /g ðr; tÞfg 0 ðr; tÞ þ 1  e bðrÞ cpg ðrÞ ng 0 ðrÞSf ;g 0 ðr; tÞfg 0 ðr; tÞ g 0 ¼1

þ

Ng X i¼1

1 vfg ðr; tÞ li ðrÞcdi;g ðrÞCi ðr; tÞ  St;g ðr; tÞfg ðr; tÞ ¼ vg vt

g 0 ¼1

(2.120)

62 Modelling of Nuclear Reactor Multi-physics

with the concentration of the neutron precursors given by: G X vCi ðr; tÞ ¼ e bi ðrÞ ng 0 ðrÞSf ;g 0 ðr; tÞfg 0 ðr; tÞ  li ðrÞCi ðr; tÞ; i ¼ 1; . ; Nd vt g 0 ¼1

(2.121)

In these two sets of equations, all quantities having the subscript g represent groupaveraged quantities. The first term on the left-hand side of Eq. (2.120) represents the neutron streaming, formally described by V $ Jg ðr; tÞ. In the diffusion approximation, the current density vector in group g is approximated by: Jg ðr; tÞ ¼  Dg ðr; tÞVfg ðr; tÞ

(2.122)

where Dg is the so-called diffusion coefficient. Such an equation constitutes an approximation of the actual neutron streaming. An analogy with Fourier’s law of conduction (i.e. Eq. 2.82) can be noticed. Due to the large size of the core, Eqs. (2.120) and (2.121) are solved on a relatively coarse mesh, typically 20 to 30 meshes axially and 1 to 4 meshes radially per fuel assembly (assumed in the present case to have a rectangular shape). Integrating these equations on each of such elementary volumes, often referred to as nodes, leads, in a Cartesian coordinate system, to: ‫א‬ ‫א‬ G X X Jg;n ðtÞ  Jg;n1 ðtÞ

1 vfg;n bn Ss0;g 0 /g;n ðtÞfg 0 ;n ðtÞ þ cpg;n 1  e ðtÞ ¼   St;g;n ðtÞfg;n ðtÞ þ D‫א‬ vg vt ‫¼א‬x;y;z g 0 ¼1



G X g 0 ¼1

ng 0 ;n Sf ;g 0 ;n ðtÞfg 0 ;n ðtÞ þ

Nd X

cdi;g;n li;n Ci;n ðtÞ

i¼1

(2.123)

and G X vCi;n ng 0 ;n Sf ;g 0 ;n ðtÞfg 0 ;n ðtÞ  li;n Ci;n ðtÞ; i ¼ 1; . ; Nd ðtÞ ¼ e bi;n vt g 0 ¼1

(2.124)

where the subscript n represents node-averaged quantities relative to a given node n. ‫ ¼ א‬x; y; z represents each of the three possible directions of the Cartesian coordinate ‫א‬ system. Jg;n thus refers to the net neutron current along the direction ‫ א‬with respect to node n averaged on the face normal to the direction ℵ and taken at the boundary between node n and node n þ 1 along that direction. For LWRs, Eqs. (2.123) and (2.124) are typically solved for two energy groups, i.e. G ¼ 2. Much more energy groups are nevertheless required for fast reactor systems, typically G ¼ 33. One difficulty with solving this system of equations is the fact that the macroscopic cross-sections and diffusion coefficients are dependent on the local conditions throughout the core. Such local conditions include history effects (burn-up inclusive), as well as instantaneous effects. The

Chapter 2
Transport phenomena in nuclear reactors

63

macroscopic cross-sections for the reaction type a and in the energy group g can be written in a generic way as: Sa;g ðr; tÞhSa;g ½bðr; tÞ; Hðr; tÞ; aCR ðr; tÞ; Dm ðr; tÞ; Cb ðr; tÞ; Tf ðr; tÞ

(2.125)

where b represents the burn-up, H represents history effects other than burn-up (typically history of the moderator density, history of the control rod insertion and boron concentration in the specific case of BWRs and PWRs, respectively, history of the fuel temperature), aCR is the instantaneous control rod insertion, Dm is the instantaneous moderator density, Cb is the instantaneous boron concentration in the specific case of PWRs and Tf is the instantaneous fuel temperature. The same dependence for the diffusion coefficients is also used, i.e. Dg ðr; tÞhDg ½bðr; tÞ; Hðr; tÞ; aCR ðr; tÞ; Dm ðr; tÞ; Cb ðr; tÞ; Tf ðr; tÞ

(2.126)

The determination of such local conditions requires the estimation of history effects (burn-up inclusive), as well as instantaneous effects. For the latter, thermal-hydraulic calculations are necessary, whereas for the former, both thermal-hydraulic and neutronic calculations are necessary. In other words, the neutron balance equations are coupled to the thermal-hydraulic equations via the dependence of the macroscopic cross-sections and diffusion coefficients on both history and instantaneous effects. One classical technique for solving this type of coupled problem from the neutron transport viewpoint is to parameterize the macroscopic cross-sections and diffusion coefficients in advance, i.e. to determine the dependence of the macroscopic crosssections and diffusion coefficient as functions of the history and instantaneous variables. Such a task is not trivial since only the microscopic cross-section saX ðE; T Þ corresponding to the reaction type a for a specific nucleus X is known from the nuclear data libraries, as functions of the energy E and of the temperature T. It will be later demonstrated that the diffusion coefficients also depend on the microscopic crosssections. This means that the microscopic cross-sections need to be condensed (i.e. averaged in energy) into a multi-group representation with very few groups, as well as homogenized (i.e. averaged in space) since the macroscopic cross-sections and diffusion coefficients used in Eqs. (2.120) and (2.121) are equations spatially averaged on a coarse mesh that does not account for the full geometrical complexity of the fuel assemblies. This condensation and homogenization must be performed while preserving the reaction rates and the neutron currents, as will be later explained. Consequently, a good estimate of the neutron flux in a detailed representation of its energy and spatial dependence is required. Such an estimate of the flux is based on solving the transport equation, which is recalled in its integro-differential form below: U $ Vjðr; U; EÞ þ St ðr; EÞjðr; U; EÞ Z Z N Ss ðr; U0 /U; E 0 /EÞjðr; U0 ; E 0 Þd 2 U0 dE 0 ¼ ð4pÞ

þ

0

"

d   X 1 bi ðrÞ cdi ðr; EÞe cp ðr; EÞ 1  e bðrÞ þ 4pk i¼1

N

(2.127)

#Z

N 0

nðr; E 0 ÞSf ðr; E 0 Þfðr; E 0 ÞdE 0

64 Modelling of Nuclear Reactor Multi-physics

Due to the complexity of Eq. (2.127), different approximations are made for solving it, and this will be the subject of Chapter 3. Furthermore, the system which is modelled only includes a given axial cross-section of one fuel assembly and correspondingly appropriate radial boundary conditions are applied (infinite medium approximation). Calculations are then performed at the pin cell/fuel assembly levels in order to estimate the energy and spatial distribution of the angular/scalar flux using Eq. (2.127). The microscopic cross-sections are then condensed from the micro-groups g to the macrogroup G according to: P

Sa;i;G ¼

Sa;i;g fi;g P fi;g

(2.128)

X NX ;i saX ;g

(2.129)

g˛G

g˛G

with Sa;i;g ¼

X

and NX,i being the atom density of the nuclei X present in a given region i. Since the spatial mesh is fine, such regions are referred to as micro-regions. The condensation is thereafter followed by a homogenization of the macroscopic cross-sections from the micro-region i into the macro-region I according to: Sa;I;G

P Sa;i;G Vi fi;G ¼ i˛I P Vi fi;G

(2.130)

i˛I

with Vi being the volume of the micro-region i. Such transport calculations are performed for all possible (but reasonable) combinations of the history and instantaneous variables, so that the macroscopic cross-sections and diffusion coefficients can be functionalized according to Eqs. (2.125) and (2.126). The strategy for determining the spatial, temporal, and energy-distribution of the neutron density throughout the reactor can be summarized as being multi-scale. This is explained by the fact that the neutron transport equation is first solved at the pin cell and the fuel assembly levels, and thereafter the diffusion equation (which represents an approximation of the transport equation) is solved at the core level. An interesting aspect of this multi-scale strategy is that the number of unknowns to be determined at each step of the calculations is kept at a reasonable level, thus allowing computing the solution at each step in a minimum time. In this respect, the transport calculations are performed in a very detailed representation of the energy dependence of the neutron flux and of the geometry of the fuel assemblies, but for a very small part of the system. The diffusion calculations, on the other hand, are carried out in a coarse representation of the energy dependence of the neutron flux and of the geometry of the fuel assemblies, but for the whole core. The modelling procedure highlighted above is thus based on a multi-step computational approach. A very small part of the computational domain is first considered, and

Chapter 2
Transport phenomena in nuclear reactors

65

assumptions on the surrounding of this domain and on the entry of the neutrons at the boundary of this domain are made. The neutron transport equation is then used to estimate the neutron flux, so that the macroscopic cross-sections and diffusion coefficients can be properly homogenized and condensed. The main approximations and limitations used in the multi-level computing scheme are manifold. The most important one lies with the fact that the calculations performed over a restricted part of the domain rely on many intertwined steps where some assumptions about the number, energy, and direction of the re-entering neutrons at the boundaries of the domain and about the surrounding medium are necessary. Such approximations are considered to be the main limitations in reaching highly accurate results for deterministic neutron transport (Roberts et al., 2010).

2.6.b

Equations governing the temperature and flow fields

The determination of the thermal-hydraulic conditions throughout the core is performed by solving two sets of equations: one set describing the transfer of heat produced by nuclear reactions through the fuel pins to the coolant, and one set of equations describing the transport of the cooling flow through the whole plant. The heat transfer through the fuel pins to the coolant is modelled by the heat conduction equation, which is recalled below: cP ðT ÞrðT Þ

vT 000 ðr; tÞ ¼ q ðr; tÞ þ V $ ½kðT ÞVT ðr; tÞ vt

(2.131)

and where the power density is given by q000 ðr; tÞ ¼

X kðrÞSf ;G ðr; tÞfG ðr; tÞ

(2.132)

G

In Eq. (2.132), k represents the average energy released per fission event. Eq. (2.132) clearly illustrates the coupling between the thermal-hydraulics and the neutronics, since the heat source in Eq. (2.131) is directly related to the neutron flux. The modelling of heat transfer through the fuel pins to the coolant is solved by dividing the considered medium into sub-volumes Vi and by integrating Eq. (2.131) on each of the sub-volumes. As will be seen later in this book, this leads to (see Chapter 5): 1 Vi

Z

rðT Þcp ðT Þ V

vT 1 ðr; tÞd 3 r ¼ vt Vi

Z

Si

kðT ÞVT ðr; tÞ $ Nd 2 r þ

1 Vi

Z

q000 ðr; tÞd 3 r

(2.133)

Vi

where Si represents the outer surface associated to the volume Vi. Although the modelling of heat conduction might appear, at least conceptually, rather simple, the parameters appearing in the heat conduction equation, such as the density, specific heat and thermal conductivity of the materials, depend on the temperature being resolved and on many other phenomena related to fuel irradiation: thermal distortion, pellet cracking, fuel densification, fuel swelling, release of gaseous fission products, etc. (Graves, 1979). Other phenomena related to the cladding and being of importance are thermal stresses, pressure stresses, fuel/pellet cladding interaction stresses, etc. The

66 Modelling of Nuclear Reactor Multi-physics

determination of such variables is often difficult and requires the coupling to other physics solvers. In order to solve Eq. (2.133), a boundary condition between the outer cladding of the fuel pins and the coolant is necessary in order to resolve the temperature distribution within the fuel pellets and the surrounding cladding. This boundary condition is given as the heat flux q00 taken at the outer wall of the cladding and expresses the heat transferred from the fuel pins to the coolant. This boundary condition thus defines a physical coupling with the coolant, for which a second set of equations is required. This second set of equations expresses the conservation of mass, momentum and energy/enthalpy for the fluid. These equations are not written in their local differential form (as given generically by Eq. (2.71) or (2.72)), since such equations cannot be solved in a reasonable computing time on a large computational domain as the one representing a nuclear reactor. Instead, the local differential conservation equations are spatially averaged on volumes Vi and time-averaged on proper time intervals Dt, as will be explained in Chapter 5. This double-averaging process is equivalent to filtering out small scale and high-frequency phenomena, in order to reduce the computational burden incurred by the modelling of such systems. This leads to the following generic conservation equations: 2 * 0 2 * 2 * 0 1+3 1+3 1+ * 0 +3 v4 b ak @rk fk A 5 þ V $ 4b ak @rk fk v k A 5 ¼ b ak @rk qk A  V $ 4b ak Fk 5 vt



1 Vi

Z

½rk fk ðv k  v S Þ þ Fk  $ Nd 2 r 

Sk;int

1 Vi

(2.134)

Z

Fk $ Nd 2 r

Sk;wall

if fk is a scalar, and

2 * 0 2 * 0 1+3 1 +3 1+ 2 * * 0 +3 v4 b ak @rk f k A 5 þ V $ 4b ak @rk f k 5v k A 5 ¼ b ak @rk qk A  V $ 4b ak Fk 5 vt

1  Vi

Z Sk;int

1 ½rk f k 5ðv k  v S Þ þ Fk  $ Nd r  Vi

(2.135)

Z

2

2

Fk $ Nd r Sk;wall

if fk is a vector, and where hxk i ¼

xk ¼

xk ¼

1 Vi 1 Dt

1 Dtk

Z

xðr; tÞd 3 r

(2.136)

xk ðr; tÞdt

(2.137)

xk ðr; tÞdt

(2.138)

Vi;k

Z Dt

Z Dtk

Chapter 2
Transport phenomena in nuclear reactors

67

In the equations above, the subscript k refers to a given phase (liquid or vapour) of the coolant; the subscript i refers to a given node or volume; the subscript k, int denotes the interface between the phase k and the other phase; and the subscript k, wall refers to the interface between the phase k and the solid walls. Vi,k represents the volume occupied by the phase k within the volume Vi and Dtk denotes the cumulative time within Dt during which the phase k is present. b is the porosity, giving the ratio between the volume occupied by the fluid and the chosen control volume. ak is the phase density function, determining whether the phase k is present in a given phase space point. The different quantities fk or fk, qk or qk, and Fk are given in Table 2.6, where it was further assumed that both phases share the same pressure P. The derivation of these equations from the local differential equations will be presented in detail in Chapter 5. Concerning the size of the volumes chosen in Eqs. (2.134)e(2.136), the core is in practice axially modelled by 10e30 volume elements. Radially, it is modelled by one volume per fuel assembly or a couple of volumes per fuel assembly (sub-channel modelling). Sometimes, one volume can radially represent an average for a given number of fuel assemblies. In this respect, the thermal-hydraulic modelling is referred to as macroscopic modelling, since the detailed information about the fluid flow within each of these volumes is lost. Although the volume- and time-averaging of the local balance equations lead to equations to be solved on volumes Vi and time intervals Dt sufficiently large so that an entire reactor core can be modelled, this double-averaging procedure also comes with some limitations. Namely, transfer terms between the phases and between each phase and the walls appear in the balance equations. Such terms need to be determined if one wants to use the system of equations represented by Eqs. (2.134) and (2.135). Despite the filtering of small scales and high-frequency phenomena the modelling technique presented above represents, the transfer terms contain, among other things, some information about the effect of such small scales and high-frequency phenomena onto the macroscopic balance equations given by Eqs. (2.134) and (2.135). Due to the complexity of the flow encountered in nuclear reactor cores, such transfer terms can seldom be determined from first principles. One instead relies on empirical models. Such models are based on many experiments performed in dedicated research facilities and specific

Table 2.6 Definition of the variables appearing on the right-hand side of Eqs. (2.134) and (2.135) for the conservations of mass, momentum and energy. fk or fk F q or q

Conservation of mass

Conservation of momentum

Conservation of energy

1 0 0

vk (sk  PI) g

ek q00k ðsk  PIÞ , vk q000 k þg , vk rk

68 Modelling of Nuclear Reactor Multi-physics

to a reactor type and to a fuel assembly type. In addition, and in the case of two-phase flows, the validity of such correlations strongly depends on the morphology of the flow (which will be referred to in Chapter 5 as flow regimes) for which the correlations were derived. This means that determining the flow regime should be part of the solution procedure when solving Eqs. (2.134) and (2.135).

2.6.c

Coupling between the neutron kinetic and thermal-hydraulic modellings

As earlier stated, the neutron transport modelling is coupled to the thermal-hydraulic modelling via Eqs. (2.125) and (2.126), whereas the thermal-hydraulic modelling is coupled to the neutron kinetic modelling via Eq. (2.132). An essential step in the process of preparing models for coupled calculations is thus the mapping between the neutron transport and the thermal-hydraulic codes. More precisely, two mappings need to be defined: one mapping from the hydrodynamic structures to the neutron transport code and one mapping from the neutron transport solution to the heat structures in the thermal-hydraulic code. The first mapping allows specifying in the neutron transport code which fuel temperature and moderator temperature/density from the thermalhydraulic code need to be associated with a specific node in the coarse mesh neutron transport solver, so that the corresponding macroscopic cross-sections and diffusion coefficients can be retrieved from the pre-generated sets of data using Eqs. (2.125) and (2.126). The second mapping allows specifying in the thermal-hydraulic code which fission power needs to be associated with a specific thermal-hydraulic volume, using Eq. (2.132). For that purpose, both the solution to the neutron transport problem (i.e. the neutron flux fG ) and the values of the macroscopic material data (i.e. the macroscopic fission cross-section Sf ;G and the average recoverable energy per fission event k) need to be retrieved. It has to be highlighted that most of the neutron transport core simulators have an embedded and simplified modelling of the conservation equations associated to fluid dynamics and heat transfer. Likewise, most of the thermal-hydraulic codes have a simplified modelling of neutron transport phenomena (e.g. using point-kinetics). Although using such embedded and simplified solvers might provide some indication of the behaviour of the system where the interplay between neutron transport, fluid dynamics and heat transfer phenomena is important, state-of-the-art modelling techniques and best estimate methods rely on a dedicated neutron transport solver coupled to a dedicated thermal-hydraulic solver (see e.g. (Akdeniz et al., 2010; Todorova et al., 2002)). One further difficulty when using different codes for modelling nuclear reactor transients lies with the exchange of information between the various mono-physics solvers. Because of the non-linear nature of the coupling terms appearing in the equations solved by each code, such coupling terms are usually not fully resolved in case of time-dependent simulations. In practice, the different codes solve their respective field of physics using non-linear terms estimated by the other physics solver at the previous

Chapter 2
Transport phenomena in nuclear reactors

69

time step (Pope and Mousseau, 2009). Fully resolving all non-linear terms at each time step has a too high computational cost and is thus not routinely used when modelling complex reactor transients.

2.7 Conclusions In this chapter, the governing equations describing neutron transport, fluid dynamics and heat transfer were derived from first principles. Such equations were derived without any approximation but are only valid in a local sense. They would thus need to be solved in every single point in the phase space domain defined by the variables appearing in these equations. Due to the size of nuclear systems and their complexity, an averaging process with respect to each of the variables is necessary. Although the thusobtained governing equations are much simpler and can be more easily solved, they contain some quantities that need to be provided to core-mesh solvers. On the neutron transport side, such quantities are typically space- and energy-averaged macroscopic cross-sections and diffusion coefficients. The averaging of those data with respect to space (homogenization) and energy (condensation) requires the prior estimation of the neutron flux on much smaller parts of the system (typically a fuel assembly) with boundary conditions mimicking an infinite medium. On the thermal-hydraulic side, the quantities to be provided are experimentally derived correlations representing the exchange of mass, momentum and energy between each of the two phases possibly present and between a given phase and the solid walls, as well as the thermal properties of the fuel. The modelling of the multi-physics dependencies is typically carried out by coupling at the macroscopic level mono-physics solvers, between which information is exchanged in order to resolve in an approximated manner the non-linear dependencies. Phenomena at small scales are either computed in advance in the case of neutron transport or provided by experimental correlations in the case of fluid dynamics and heat transfer. The influence of large-scale phenomena onto small scales is thus not taken explicitly into account. This methodology, which is summarized in Fig. 2.18, represents

FIGURE 2.18 Overall principle of deterministic and macroscopic reactor modelling.

70 Modelling of Nuclear Reactor Multi-physics

the most used methodology for modelling the multi-physics and multi-scale features of nuclear reactor systems in practical applications. The remaining of this book thus focuses on the details of the modelling along such a modelling philosophy.

References Akdeniz, B., Ivanov, K.N., Olson, A.M., 2010. Boiling water reactor turbine trip (TT) benchmark e volume IV: summary results of exercise 3. OECD Nuclear Energy Agency. NEA/NSC/DOC(2010)11. Analysgruppen, 2009. Fra˚n brytning av uranmalm till fa¨rdigt ka¨rnbra¨nsle. Bakgrund 22 (1), 4. Bestion, D., 2012. Applicability of two-phase CFD to nuclear reactor thermal hydraulics and elaboration of Best Practice Guidelines. Nuclear Engineering and Design 253, 311e321. Blevins, R.D., 2001. Flow-Induced Vibration, second ed. Krieger Publishing Company, Malabar, USA. Demazie`re, C., 2013. Multi-physics modelling of nuclear reactors: current practices in a nutshell. International Journal of Nuclear Energy Science and Technology 7 (4), 288e318. Demazie`re, C., Mattsson, H., 2006. Determination of the fuel heat transfer dynamics via CFD for the purpose of noise analysis. In: Proceedings of the 5th International Topical Meeting on Nuclear Plant Instrumentation, Controls, and Human Machine Interface Technology. NPIC&HMIT 2006, 12e16 November 2006, Albuquerque, NM, USA. American Nuclear Society, La Grange Park, USA. Drevon, G., 1983. Les re´acteurs a` eau ordinaire. Collection du Commissariat a` l’Energie Atomique. Se´rie Synthe`ses, Eyrolles, Paris, France. Fiorina, C., Clifford, I., Aufiero, M., Mikityuk, K., 2015. GeN-Foam: a novel OpenFOAM based multiphysics solver for 2D/3D transient analysis of nuclear reactors. Nuclear Engineering and Design 294, 24e37. Fujita, K., 1990. Flow-induced vibration and fluid-structure interaction in nuclear power plant components. Journal of Wind Engineering and Industrial Aerodynamics 33, 405e418. Gill, D.F., Griesheimer, D.P., Aumiller, D.L., 2017. Numerical methods in coupled Monte Carlo and thermal-hydraulic calculations. Nuclear Science & Engineering 185, 194e205. Graves, H.W., 1979. Nuclear Fuel Management. John Wiley & Sons, Inc., New York, USA. Jareteg, K., Vinai, P., Sasic, S., Demazie`re, C., 2015. Coupled fine mesh neutronics and thermalhydraulics e modelling and implementation for PWR fuel assemblies. Annals of Nuclear Energy 84, 244e257. Knott, D., Yamamoto, A., 2010. Lattice physics computations. In: Cacuci, D.G. (Ed.), 2010. Handbook of Nuclear Engineering e Volume 2: Reactor Design. Springer, New York, USA. Kundu, P.K., Cohen, I.M., 2004. Fluid Mechanics e Third Edition. Elsevier Academic Press, Amsterdam, the Netherlands. Lamarsh, J.R., 1975. Introduction to Nuclear Engineering. Addison-Wesley Publishing Company, Reading, USA. Lamarsh, J.R., 2002. Introduction to Nuclear Reactor Theory. American Nuclear Society, Inc., La Grange Park, USA. Leppa¨nen, J., Hovi, V., Ikonen, T., Kurki, J., Pusa, M., Valtavirta, V., Viitanen, T., 2015. The numerical multi-physics project (NUMPS) at VTT Technical Research Centre of Finland. Annals of Nuclear Energy 84, 55e62. Lewis, E.E., 2008. Fundamental of Nuclear Reactor Physics. Academic Press, Burlington, USA. Lux, I., Koblinger, L., 1991. Monte Carlo particle transport methods: neutron and photons calculations. CRC Press, Boca Raton, USA.

Chapter 2
Transport phenomena in nuclear reactors

71

Neeb, K.H., 1997. The Radiochemistry of Nuclear Power Plants with Light Water Reactors. Walter de Gruyter & Co, Berlin, Germany. Pope, M.A., Mousseau, V.A., 2009. Accuracy and efficiency of a coupled neutronics and thermal hydraulics model. Nuclear Engineering and Technology 41 (7), 885e892. Reuss, P., 1998. The´orie du transport des neutrons e Tome I: Simulations, mises en e´quations et traitements nume´riques. Institut National des Sciences et Techniques Nucle´aires, Saclay, France. Reuss, P., 2008. Neutron Physics. EDP Sciences, Les Ulis, France. Roberts, D.R., Ouisloumen, M., Kucukboyaci, V.N., Ivanov, K.N., 2010. Development of iterative transport e diffusion methodology for LWR analysis. In: Proceedings of the International Conference on the Advances in Reactor Physics to Power the Nuclear Renaissance. PHYSOR 2010, 9e14 May 2010, Pittsburgh, PA, USA. American Nuclear Society, La Grange Park, USA. Sehgal, B.R. (Ed.), 2012. Nuclear Safety in Light Water Reactors e Severe Accident Methodology. Elsevier Inc., Amsterdam, The Netherlands. Sjenitzer, B.L., Hoogenboom, J.E., Escalante, J.J., Sanchez Espinoza, V., 2015. Coupling of dynamic Monte Carlo with thermal-hydraulic feedback. Annals of Nuclear Energy 76, 27e39. Slattery, J.C., Gaggioli, R.A., 1962. The macroscopic angular momentum balance. Chemical Engineering Science 17, 893e895. Takagi, J., Mincher, B.J., Yamaguchi, M., Katsumura, Y., 2010. Radiation chemistry in nuclear engineering. In: Hatano, Y., Katsumura, Y., Mozumder, A. (Eds.), Charged Particle and Photon Interactions with Matter, Recent Advances, Applications, and Interfaces. CRC Press, Boca Raton, USA. Todorova, N., Taylor, B., Ivanov, K., 2002. Pressurised water reactor main steam line break (MSLB) benchmark e volume III: results of phase 2 on 3-D core boundary conditions model. US Nuclear Regulatory Commission and OECD Nuclear Energy Agency. NEA/NSC/DOC(2002)12. Tong, L.S., Weisman, J., 1996. Thermal Analysis of Pressurized Water Reactors. American Nuclear Society, La Grange Park, USA. Windecker, G., Anglart, H., 1999. Phase distribution in BWR fuel assembly and evaluation of multidimensional multi-field model. In: Proceedings of the 9th International Meeting on Nuclear Reactor Thermal-Hydraulics. NURETH-9, 3-8 October, 1999, San Francisco, CA. USA. American Nuclear Society, La Grange Park, USA. Zhang, R., Cong, T., Tian, W., Qiu, S., Su, G., 2015. CFD analysis on subcooled boiling phenomena in PWR coolant channel. Progress in Nuclear Energy 85, 254e263.

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3

Neutron transport calculations at the cell and assembly levels 3.1 Representation of the energy dependence As seen in Chapter 2, the neutron transport equation depends on many variables, that represent the position, direction and energy of the neutrons, and time. The energydependence of the neutrons can be relatively easily handled by a so-called multigroup formalism, which is presented thereafter. The preparation of the nuclear data according to this formalism is then touched upon.

3.1.a

Multi-group formalism

The multi-group formalism consists in dividing the possible range of variation of the neutron energy [Emin; Emax] into several G energy bins, i.e., 1

½Emin ; Emax  ¼

W½E ; E g¼G

g

g1 

(3.1)

so that neutron balance equations can be written on each of these energy groups. It should be noted that the numbering of the energy groups is made in the order of decreasing neutron energy, i.e., the group corresponding to the highest energies [E1; E0] is the group number 1 whereas the group corresponding to the lowest energies [EG; EG1] is the group number G. Before transforming the transport equation in a multi-group formalism, it should be emphasized that the quantities that need to be preserved when transforming the neutron transport equation from a continuous energy representation to a discrete energy representation (i.e., multi-group formalism) are the reaction rates, which are quantities of the form Sa ðr; E; tÞjðr; U; E; tÞ or Sa ðr; U0 / U; E 0 / E; tÞjðr; U0 ; E 0 ; tÞ;

and the neutron currents. Such quantities appear in the neutron transport equation. As a result, any transformation of the neutron transport equation, when treating the energy dependence of the neutrons, will not modify the neutron balance. Moreover, preserving the reaction rates and neutrons currents will lead to a preservation of the effective multiplication factor of the system, as can be seen in Eq. (2.38) in Chapter 2. In addition, the only measurable quantities in a nuclear core are the number of reactions per unit time in a detector and the number of neutrons crossing a surface per unit time (the Modelling of Nuclear Reactor Multi-physics. https://doi.org/10.1016/B978-0-12-815069-6.00003-9 Copyright © 2020 Elsevier Inc. All rights reserved.

73

74 Modelling of Nuclear Reactor Multi-physics

angular neutron flux jðr; U; E; tÞ used in the neutron transport equation is not directly measurable). Preserving the reaction rates and neutron currents is thus practically advantageous. The multi-group neutron transport equation can thus be obtained by integrating the neutron transport equation in, e.g., its integro-differential form (i.e., Eq. (2.32) in Chapter 2) on a given energy bin, or so-called energy group, while preserving the reaction rates. With this constraint, the macroscopic cross-sections should, in principle, be integrated with respect to the angular flux. One would then obtain: 1 v j ðr; U; tÞ þ U $ Vjg ðr; U; tÞ þ St;g ðr; U; tÞjg ðr; U; tÞ vg ðr; U; tÞ vt g Z G X Ss;g 0 /g ðr; U0 /U; tÞjg 0 ðr; U0 ; tÞd 2 U0 ¼ g 0 ¼1

ð4pÞ

G Z t n  1 X þ ðnSf Þg 0 ðr; t 0 Þfg 0 ðr; t 0 Þ 1  e bðrÞ cpg ðrÞdðt  t 0 Þ þ . 4p g 0 ¼1 N ) Nd X d li ðrÞ½tt 0  e dt 0 c ðrÞli ðrÞbi ðrÞe

(3.2)

i;g

i¼1

with the different energy-integrated quantities defined as: Z

jg ðr; U; tÞ ¼

Eg1

Z fg ðr; tÞ ¼

1 ¼ vg ðr; U; tÞ

R Eg 0 1 S

s;g 0 /g

0

ðr; U / U; tÞ ¼

Eg 0

fðr; E; tÞdE

(3.4)

Eg1

R Eg1 1 jðr; U; E; tÞdE Eg vðEÞ jg ðr; U; tÞ St ðr; E; tÞjðr; U; E; tÞdE

Eg

jg ðr; U; tÞ

jðr; U0 ; E 0 ; tÞ

R Eg1 Eg

Ss ðr; U0 /U; E 0 /E; tÞdE 0 dE

jg 0 ðr; U0 ; tÞ R Eg1

ðnSf Þg ðr; tÞ ¼

(3.3)

Eg

R Eg1 St;g ðr; U; tÞ ¼

jðr; U; E; tÞdE

Eg

Eg

nðr; EÞSf ðr; E; tÞfðr; E; tÞdE fg ðr; tÞ Z

cpg ðrÞ ¼

Eg1

Eg

cp ðr; EÞdE

(3.5)

(3.6)

(3.7)

(3.8)

(3.9)

Chapter 3  Neutron transport calculations at the cell and assembly levels Z cdi;g ðrÞ ¼

Eg1 Eg

cdi ðr; EÞdE

75

(3.10)

It can be noticed that this way of averaging the neutron transport equation automatically preserves the neutron current, since one has, according to the definition of the neutron current density vector given by Eq. (2.6) in Chapter 2: Z

Jg ðr; tÞ ¼ Z ¼

Z U

ð4pÞ

Eg1

Z Z

Jðr; E; tÞdE ¼

Eg Eg1

Eg

ð4pÞ 2

Eg1

Eg

Z

jðr; U; E; tÞd U dE ¼

Ujðr; U; E; tÞd 2 U dE Ujg ðr; U; tÞd 2 U

(3.11)

ð4pÞ

There are several drawbacks with using the angular flux as a weighting function for integrating the macroscopic cross-sections:  The group-averaged neutron velocity vg depends on position, angle, and time.  The group-averaged total macroscopic cross-section St;g now depends on the angle, whereas its energy-continuous version does not.  In case of isotropic media, the energy-continuous differential scattering macroscopic cross-section does not depend explicitly on U0 but only on the scalar product between U0 and U. In the multi-group case, the group-averaged differential scattering macroscopic cross-section Ss;g 0 /g is no longer rotation-invariant and depends explicitly on the direction of incidence U0 of the neutrons. The consequence of using the angular flux as a weighting function is that one loses the properties of isotropy of the medium being considered, i.e., the medium has become anisotropic. Nevertheless, when examining the group-averaged fission cross-section, one notices that the isotropy of the medium is preserved when the scalar flux is used for averaging the macroscopic cross-section. This is why the scalar neutron flux is used as a weighting function instead of the angular neutron flux when considering the averaging of macroscopic cross-sections. Moreover, since both the angular neutron flux and the scalar neutron flux are unknown quantities, the formulation of the multi-group neutron balance equations poses problem. The multi-group macroscopic data cannot be estimated before the neutron scalar or angular neutron flux is determined, whereas those fluxes represent the ultimate quantities to be determined when performing reactor calculations. In practice, this means that a ‘good enough’ guess of the energy-dependence of the scalar neutron flux has to be used for averaging the macroscopic cross-sections with respect to energy in the multi-group neutron transport equation. Such a flux spectrum will be referred in the following to as fw ðEÞ. Experience shows that if the energy group structure is properly chosen, and if the flux spectrum fw ðEÞ is not too different from the actual flux spectrum that exists in the reactor, then the macroscopic cross-sections averaged with the flux spectrum fw ðEÞ are relatively insensitive to the chosen flux spectrum. Taking light-water reactors (LWRs) as an illustrative example, the energy-

76 Modelling of Nuclear Reactor Multi-physics

dependence of the flux in this type of systems is dominated by three main phenomena, each having its own energy range: fission at high energies, slowing-down at intermediate energies, and thermalization at low (thermal) energies. To each of these phenomena corresponds a typical energy-dependence of the neutron flux, which is summarized hereafter: 8 pffiffiffiffiffiffiffiffiffiffiffiffiffi > fexp ð1.036EÞsinh 2.29E ; fission spectrum ðhigh energiesÞ with E in MeV > > > < 1 fw ðEÞ ¼ f ; slowing  down spectrum ðintermediate energiesÞ > E > > > : fE expðE=kB T Þ; thermalization ðMaxwellÞ spectrum ðlow energiesÞ

(3.12)

In the above equation, kB is the Boltzmann constant. Using the above flux spectrum fw ðEÞ for averaging the neutron transport equation leads to the following multi-group neutron transport equation: 1 v j ðr; U; tÞ þ U $ Vjg ðr; U; tÞ þ St;g ðr; tÞjg ðr; U; tÞ vg vt g Z G X Ss;g 0 /g ðr; U0 /U; tÞjg 0 ðr; U0 ; tÞd 2 U0 ¼ g 0 ¼1

ð4pÞ

G Z t n  1 X þ ðnSf Þg 0 ðr; t 0 Þfg 0 ðr; t 0 Þ 1  e bðrÞ cpg ðrÞdðt  t 0 Þ þ . 4p g 0 ¼1 N ) Nd X d li ðrÞ½tt 0  e dt 0 c ðrÞli ðrÞbi ðrÞe

(3.13)

i;g

i¼1

with the different energy-integrated quantities defined as: Z

jg ðr; U; tÞ ¼

jðr; U; E; tÞdE

(3.14)

fðr; E; tÞdE

(3.15)

Eg

Z fg ðr; tÞ ¼

Eg1

Eg1

Eg

R Eg1 1 f ðEÞdE Eg 1 vðEÞ w ¼ R Eg1 vg fw ðEÞdE Eg R Eg1 St;g ðr; tÞ ¼ R Eg 0 1 0

Ss;g 0 /g ðr; U / U; tÞ ¼

Eg 0

Eg

St ðr; E; tÞfw ðEÞdE R Eg1 fw ðEÞdE Eg

fw ðE 0 Þ

(3.16)

(3.17)

R Eg1 Eg

Ss ðr; U0 /U; E 0 /E; tÞdE 0 dE R Eg 0 1 fw ðE 0 ÞdE 0 Eg 0

(3.18)

Chapter 3  Neutron transport calculations at the cell and assembly levels

R Eg1 ðnSf Þg ðr; tÞ ¼

Eg

nðr; EÞSf ðr; E; tÞfw ðEÞdE R Eg1 fw ðEÞdE Eg

Z cpg ðrÞ ¼

(3.19)

cp ðr; EÞdE

(3.20)

cdi ðr; EÞdE

(3.21)

Eg

Z cdi;g ðrÞ ¼

Eg1

77

Eg1 Eg

It should be emphasized that Eq. (3.13) is an approximation of the true multi-group neutron transport equation, i.e., Eq. (3.2). Nevertheless, the approximation used for deriving Eq. (3.13), i.e., replacing the unknown angular flux jðr; U; E; tÞ by a typical (scalar) flux spectrum fw ðEÞ when estimating the energy-averaged quantities as explained above, leads to negligible errors and to group-averaged data preserving the isotropic properties of the medium. In the following of this chapter, Eq. (3.13) is the equation that one will try to solve.

3.1.b

Nuclear data libraries

Eq. (3.13) can be solved as soon as the energy-averaged macroscopic cross-sections are given. With the exception of the multi-group differential macroscopic scattering crosssection Ss;g 0 /g (for which further explanations are given later in this section), all the energy-averaged macroscopic cross-sections can be deduced from the point-wise nuclear data libraries (i.e., the continuous energy-dependent cross-sections). When the concentrations NX of the different species X present in the system are known, Eqs. (3.17) and (3.19) can be generically written: R Eg1

Sa;g ðr; tÞ ¼

Eg

Sa ðr; E; tÞfw ðEÞdE X NX ðr; tÞ ¼ R Eg1 fw ðEÞdE X Eg

R Eg1 Eg

saX ðEÞfw ðEÞdE R Eg1 fw ðEÞdE Eg

(3.22)

where the last equality was obtained using Eq. (2.7) in Chapter 2. All the microscopic cross-sections are available from nuclear databases as a function of the energy of the incoming neutrons, as well as a function of the temperature of the target. These cross-sections are determined from measurements and from calculations using nuclear models. They are then assembled in libraries, together with comprehensive information on the measurements and on the calculations. Because of the huge amount of sometimes contradictory nuclear data, the data need to be evaluated before they can be used for any reactor physics calculations. The evaluations can be carried out in different ways, such as the intercomparison of data, the use of data to calculate benchmark experiments, the critical assessment of statistical and systematic errors, the check for internal consistency and consistency with standard neutron cross-sections, and the derivation of consistent preferred values by appropriate averaging procedures (Stacey, 2001). Once evaluated, the nuclear data are added to the evaluated nuclear data

78 Modelling of Nuclear Reactor Multi-physics

files. Several of those exist. The ones mostly used for reactor physics calculations are as follows:  The United States Evaluated Nuclear Data File (ENDF/B) (Chadwick et al., 2011);  The Joint Evaluated File of the Nuclear Energy Agency Countries (JEF) (OECD/NEA Data Bank, 2000) and the Joint Evaluated Fission and Fusion library of Nuclear Energy Agency Countries (JEFF) (OECD/NEA Data Bank, 2009). The evaluated nuclear data files are available in point-wise format (i.e., where the cross-sections are given as functions of the energy) or in group-wise format (i.e., where the cross-sections are averaged on energy groups). A processing of the point-wise data is necessary to prepare the data in a group-wise format according to Eq. (3.22). The multi-group differential macroscopic scattering cross-section Ss;g 0 /g ðr; U0 /U; tÞ needs special attention, since, in addition to the dependence of the cross-section on energy, the cross-section also depends, in the most general case, on the direction of the incoming neutron as well as on the direction of the outgoing neutron. For isotropic media, the differential cross-section nevertheless only depends on the scalar product between the direction U0 of the incoming neutron and the direction U of the outgoing neutron. This scalar product is referred to as m, thus defined as: m ¼ U0 $ U

(3.23)

and also represents the cosine of the deviation angle. In the following, the representation of the differential macroscopic scattering cross-section with respect to m is explained. For the sake of simplicity, the possible dependence of the macroscopic cross-sections on time is dropped (such a dependence only arises from the variation of the concentration of the considered species). Starting with the energy-dependent macroscopic crosssection written as Ss ðr; U0 /U; E 0 /EÞ, an expansion of this cross-section on Legendre polynomials with respect to m is usually carried out, and reads as: Ss ðr; U0 / U; E 0 / EÞ h Ss ðr; m; E 0 / EÞ ¼

N X 2l þ 1 l¼0

4p

Ssl ðr; E 0 / EÞPl ðmÞ

(3.24)

where the expansion coefficients are expressed as: Ssl ðr; E 0 / EÞ ¼ 2p

Z

1

1

Ss ðr; m; E 0 / EÞPl ðmÞdm

(3.25)

Pl is the Legendre polynomial of order l, l being a positive integer (that can be equal to zero), fulfilling the following properties (Reuss, 2008):  Pl ðmÞ is a polynomial in m having real values.  Pl ðmÞ is of parity (1)l, i.e., Pl ð mÞ ¼ ð1Þl Pl ðmÞ.  The Legendre polynomials are orthogonal and normalized according to: Z

1

1

Pl ðmÞPk ðmÞdm ¼

2dlk 2l þ 1

(3.26)

Chapter 3  Neutron transport calculations at the cell and assembly levels

79

where dlk is the Kronecker delta defined as: 

dlk ¼

1 if l ¼ k 0 if lsk

(3.27)

 The Legendre polynomials are given as: P0 ðmÞ ¼ 1 Pl ðmÞ ¼

(3.28)

l 1 dl  2 m  1 for l  1 l l 2 l! dm

(3.29)

and one also has: Pl ð1Þ ¼ 1; cl

(3.30)

 The Legendre polynomials fulfil the following recurrence relations: ðl þ 1ÞPlþ1 ðmÞ  ð2l þ 1ÞmPl ðmÞ þ lPl1 ðmÞ ¼ 0 

m2  1

h i  dPl ðmÞ ¼ l mPl ðmÞ  Pl1 ðmÞ dm

(3.31) (3.32)

 The Legendre polynomials constitute a complete set of orthogonal functions for functions in m defined on the interval [1;1]. Using Eqs. (3.28) and (3.29), one also notices that: P1 ðmÞ ¼ m

(3.33)

The two first Legendre moments of the differential scattering cross-sections are thus given, using Eqs. (3.25), (3.28) and (3.33), as: Ss0 ðr; E 0 / EÞ ¼ 2p Ss1 ðr; E 0 / EÞ ¼ 2p

Z

1

Ss ðr; m; E 0 / EÞdm

(3.34)

Ss ðr; m; E 0 / EÞmdm

(3.35)

1

Z

1

1

One also notices from Eqs. (3.34) and (3.35) that the ratio of Ss1 and Ss0 defines the average of the cosine of the deviation angle m using the differential macroscopic crosssection Ss ðr; m; E 0 /EÞ as a weighting function, i.e., mðr; E 0 / EÞ ¼

2p

R1

2p

R11

Ss ðr; m; E 0 /EÞmdm

1

Ss

ðr; m; E 0 /EÞdm

¼

Ss1 ðr; E 0 /EÞ Ss0 ðr; E 0 /EÞ

(3.36)

80 Modelling of Nuclear Reactor Multi-physics

In reactor calculations, sufficient accuracy is usually obtained by limiting the development of the differential scattering cross-section on the two first Legendre polynomials. Some specific applications might nevertheless require higher moments of the expansion. Restricting the development to the two first moments in the following and using Eq. (3.24), one thus obtains: Ss ðr; m; E 0 / EÞz

1 ½Ss0 ðr; E 0 / EÞ þ 3mSs1 ðr; E 0 / EÞ 4p

(3.37)

which can equivalently be rewritten, using Eq. (3.36), as: Ss0 ðr; E 0 /EÞ ½1 þ 3mmðr; E 0 / EÞ Ss ðr; m; E 0 / EÞz 4p

(3.38)

Using Eq. (3.37) into Eq. (3.18), the multi-group differential scattering cross-section can be estimated as: Ss;g 0 /g ðr; U0 /U; tÞhSs;g 0 /g ðr; m; tÞ Z Eg 0 1 Z Eg1 1 fw ðE 0 Þ ½Ss0 ðr; E 0 /E; tÞ þ 3mSs1 ðr; E 0 /E; tÞdE 0 dE 4p Eg 0 Eg z Z Eg 0 1 fw ðE 0 ÞdE 0

(3.39)

Eg 0

¼

1 ½Ss0;g 0 /g ðr; tÞ þ 3mSs1;g 0 /g ðr; tÞ 4p

with R Eg 0 1 Ss0;g 0 /g ðr; tÞ ¼

Eg 0

RE fw ðE 0 Þ Egg1 Ss0 ðr; E 0 /E; tÞdE 0 dE R Eg 0 1 fw ðE 0 ÞdE 0 Eg 0

(3.40)

RE fw ðE 0 Þ Egg1 Ss1 ðr; E 0 /E; tÞdE 0 dE R Eg 0 1 fw ðE 0 ÞdE 0 Eg 0

(3.41)

and R Eg 0 1 Ss1;g 0 /g ðr; tÞ ¼

Eg 0

The multi-group macroscopic (isotropic) cross-section Ss0;g 0 /g and (anisotropic) cross-section Ss1;g 0 /g can be constructed from the corresponding microscopic (isotropic) cross-section ss0X ;g 0 /g and (anisotropic) cross-section ss1X ;g 0 /g of the various species X present in the system having a concentration NX(r,t) as: Ss0;g 0 /g ðr; tÞ ¼

X NX ðr; tÞss0X ;g 0 /g

(3.42)

X

Ss1;g 0 /g ðr; tÞ ¼

X NX ðr; tÞss1X ;g 0 /g X

(3.43)

Chapter 3  Neutron transport calculations at the cell and assembly levels

81

with R Eg 0 1 ss0X;g 0 /g ¼

Eg 0

R Eg 0 1 ss1X;g 0 /g ¼

Eg 0

RE fw ðE 0 Þ Egg1 ss0X ðE 0 /EÞdE 0 dE R Eg 0 1 fw ðE 0 ÞdE 0 Eg 0

(3.44)

RE fw ðE 0 Þ Egg1 ss1X ðE 0 /EÞdE 0 dE R Eg 0 1 fw ðE 0 ÞdE 0 Eg 0

(3.45)

ss0X ;g 0 /g and ss1X ;g 0 /g , often referred to as scattering isotropic and anisotropic, respectively, matrices, are also available from the evaluated nuclear data files. Some higher moments might also be included in the libraries.

3.2 Treatment of resonances 3.2.a

Introduction

One case in which the above data are of limited use is for resonant species, i.e., species for which resonances in some microscopic cross-sections exist. A typical example of such a species is 238U, for which the energy-dependence of the microscopic absorption cross-section is given in Fig. 3.1. Four regions can be clearly distinguished on this figure:  A typical 1/v-behaviour at low energies (below approximately 1 eV), which in the double-logarithmic plot of Fig. 3.1 appears as a straight line.

FIGURE 3.1 Energy-dependence of the microscopic absorption cross-section for neutron data library.

238

U. Obtained from the JEFF 3.1

82 Modelling of Nuclear Reactor Multi-physics

 A region referred to as the resonance region where a series of spectacular resonances can be observed (between approximately 1 eV and 20 keV). In the lower end of this energy range, the resonances can be resolved (region of resolved resonances) whereas in the upper end of this energy range, the resonances cannot be individually resolved by the actual measurement techniques (region of unresolved resonances).  A region referred to as the continuum region where the resonances are so close to each other that they overlap and result in a smooth variation of the cross-section as a function of energy (approximately above 20 keV). The effect of resonances on reactor calculations arises mostly from neutron absorption, which includes both radiative capture and fission reactions (Bell and Glasstone, 1970). In the resonance region, the slowing-down of neutrons represent nevertheless another important phenomenon. Resonance absorption and neutron slowing-down are thus two intricate and inter-related phenomena that need to be treated simultaneously in order to properly capture the physics at hand. While an appropriate energy group structure would allow a proper representation of some of the resonances at low energy, most of the resonances cannot be accounted for by a multi-group formalism without the need of using several thousands of energy groups. Clearly, another approach is required for providing adequate multi-group crosssections for resonant species. It relies on the preservation of the reaction rates in the resonance region without the need to exactly compute the detailed energy structure of the neutron flux in the resonance region. In order to better comprehend the methodology, the physics of neutron slowingdown without any absorption is first briefly recalled, before looking at the combined effect of resonance absorption and slowing down. The slowing-down of neutrons occurs via scattering, mostly in an elastic and isotropic manner (with isotropy assumed in the centre-of-mass reference system) (Lamarsh, 2002). Emphasis will thus be put on elastic and isotropic scattering in the remaining of this section. Accurately computing resonance absorption and correspondingly prepare the groupwise cross-sections is of utmost importance, since the temperature dependence of resonance absorption, also known as the Doppler effect, has major implications on reactor safety. In the following, a simplistic and brief description of this effect is given, in order to better highlight the importance of properly modelling resonance absorption. Further details about the Doppler effect can be found in any reactor physics books, such as, e.g., the book by Lamarsh (2002). The Doppler effect comes from the fact that the target nucleus is not at rest when an absorption reaction occurs. Since the velocity distribution of the target nucleus is temperature-dependent, the interaction rate is temperature-dependent as well. When the temperature of the target nucleus increases, it could be shown that the resonances become wider, while the peaks of the resonances get smaller. Nevertheless, the absorption rate increases when the temperature increases: as the resonance becomes wider, neutrons losing energy in discrete steps have a higher

Chapter 3  Neutron transport calculations at the cell and assembly levels

83

probability to interact at energies where the resonance plays some role. As a result, more neutrons are absorbed in the resonances. Whether the broadening of the resonances contribute to an increase or decrease of the number of neutrons depends on the relative contributions to either capture (leading to a decrease of the number of neutrons) or fission (leading to an increase of the number of neutrons). The goal of reactor design is to guarantee that an increase in temperature leads to an overall decrease in the number of neutrons, thus creating an inherent negative feedback highly desirable for having a safe reactor.

3.2.b

Neutron slowing-down without absorption

In the following, the physics of slowing-down without absorption is first described according to a classical reactor physics approach, such as the one of Lamarsh (2002). Because of the simpler kinematics of the scattering reactions in the centre-of-mass reference systems than in the laboratory system, the centre-of-mass reference system is used. The laboratory reference system is the system corresponding the observer being at rest. A neutron of mass m and a target nucleus of mass M are represented in Fig. 3.2. If the subscripts l and c refer to the laboratory and centre-of-mass reference systems, respectively, the centre-of-mass of the system is defined as: rl ¼

mrl þ MRl mþM

(3.46)

The coordinates of the neutron and nucleus are given in the centre-of-mass reference system by rc ¼ rlrl and Rc ¼ Rlrl, respectively. Correspondingly, the velocities of the neutron and nucleus in the centre-of-mass reference system are given by vc ¼ vlv0 and Vc ¼ Vlv0, respectively, where v0 is the velocity of the centre of mass defined as: v0 ¼

mv l þ MV l mþM

(3.47)

Eq. (3.46) is equivalent to mrc þ MRc ¼ 0, which is a more common definition of the barycentre in mathematics. The observer in the centre-of-mass reference system is then moving with the velocity v0, and the velocities seen by this observer in the centre-ofmass reference system are thus given by vc and Vc. If the target nucleus is at rest in the laboratory reference system, then Vl ¼ 0, from which it results, according to Eq. (3.47), that:

FIGURE 3.2 Illustration of the laboratory and centre-of-mass reference systems. Derived from Lamarsh, J.R., 2002. Introduction to Nuclear Reactor Theory. American Nuclear Society, La Grange Park, USA.

84 Modelling of Nuclear Reactor Multi-physics

v0 ¼

m vl mþM

(3.48)

Accordingly, one finds that: vc ¼ vl  v0 ¼

M vl mþM

(3.49)

m vl mþM

(3.50)

Vc ¼  v0 ¼ 

The total momentum in the centre-of-mass reference system, defined as pc ¼ mvc þ MVc, gives, according to Eqs. (3.49) and (3.50), pc ¼ 0. This means that in the centre-of-mass reference system, the vectors vc and Vc are colinear. In the case of elastic scattering reactions, the kinetic energy and the linear momentum are kept constant during the collision. It could be demonstrated that the conservation of these two quantities results in the conservation of the moduli of the velocities in the centre-of-mass reference system (Lamarsh, 2002), i.e., in case of elastic scattering one has: vc ¼ vc 0

(3.51)

Vc ¼ Vc 0

(3.52)

where the prime refers to quantities taken after the collision. The kinematics of collisions is thus much simpler in the centre-of-mass reference system since, in case of elastic scattering, the moduli of the velocities of the neutron and nucleus are unchanged by the collision and the velocities are colinear before and after interaction. These conclusions are schematically summarized in Fig. 3.3. Before interaction, the neutron and nucleus are travelling towards each other, whereas after interaction, the neutron and nucleus are travelling back to back. The only quantity that remains unknown in the centre-of-mass reference system is the angle of deviation qc between the direction of incidence of the

FIGURE 3.3 Kinematics of elastic scattering in the laboratory and centre-of-mass reference systems, respectively. Derived from Lamarsh, J.R., 2002. Introduction to Nuclear Reactor Theory. American Nuclear Society, La Grange Park, USA.

Chapter 3  Neutron transport calculations at the cell and assembly levels

85

FIGURE 3.4 Vector diagram relating the scattering angles in the laboratory and centre-of-mass reference systems. Derived from Lamarsh, J.R., 2002. Introduction to Nuclear Reactor Theory. American Nuclear Society, La Grange Park, USA.

incoming neutron and the direction of emergence of the neutron in this reference system. The relationship between the deviation angle qc in the centre-of-mass reference system and the deviation angle ql in the laboratory reference system can be easily found based on the following vector-based equation: v c0 ¼ v l0  v 0

(3.53)

and the corresponding vector diagram given in Fig. 3.4. Projecting on two orthogonal directions leads to: tan ql ¼

sin qc g þ cos qc

(3.54)

and g þ cos qc cos ql ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi g2 þ 2g cos qc þ 1

(3.55)

with g¼

kv 0 k kv c0 k

(3.56)

In case of elastic scattering in the centre-of-mass reference system, kv c0 k ¼ kv c k, and g simplifies into g ¼ m=M because of Eqs. (3.48) and (3.49). Furthermore, the rela tionship between the energy El ¼ mv 2l 2 of a neutron in the laboratory reference system  before interaction and its energy El0 ¼ mv 02 l 2 in the laboratory reference system after interaction can be derived from Fig. 3.4 as follows. Since v 0l ¼ v 0c þ v 0 , taking the scalar product between v 0l and itself leads to: 2 2 v 02 l ¼ v c þ v 0 þ 2kv 0 kkv c kcos qc

(3.57)

where the fact that the scattering is elastic in the centre-of-mass reference system, i.e., kv c0 k ¼ kv c k, was used. Using Eqs. (3.48) and (3.49), one finds that:

86 Modelling of Nuclear Reactor Multi-physics

1 El 0 ¼ El ½ð1 þ aÞ þ ð1  aÞcos qc  2

where a¼

(3.58)

2 A1 Aþ1

(3.59)

M m

(3.60)

and A¼

A is thus approximately the mass number of the target nucleus. From Eq. (3.58), one    notices that El 0 is bounded between El 0 min ¼ aEl (obtained for qc ¼ p) and El0 max ¼ El (obtained for qc ¼ 0). For a given target nucleus characterized by its mass M, there is a unique relationship between the distribution of the deviation angle qc in the centre-ofmass system and the energy El 0 of the neutron after collision in the laboratory reference system. In case of isotropic scattering in the centre-of-mass reference system, there is equiprobable distribution of any direction of emission U (to which correspond a polar angle qc and an azimuthal angle 4c ) in the centre-of-mass reference system. The probability of emission along any direction U is thus: PU ðUÞ ¼

1 4p

(3.61)

Due to the rotational invariance with respect to 4c , one also has for the probability of emission along any azimuthal angle 4c : P4c ð4c Þ ¼

1 2p

(3.62)

One can further write that: PU ðUÞd 2 U ¼ Pqc ðqc ÞP4c ð4c Þdqc d4c

(3.63)

with d 2 U ¼ sin qc dqc d4c , and where Pqc ðqc Þ represents the probability of emission along the polar angle qc . One further deduces that: Pqc ðqc Þ ¼

sin qc 2

(3.64)

From Eq. (3.58), one notices that, for a given energy El of the neutron before interaction in the laboratory reference system, the energy El 0 of the neutron after interaction in the laboratory reference system is entirely defined by the value of the deviation angle qc in the centre-of-mass reference system. If P(El / El 0 )dEl 0 represents the probability for a neutron with the energy El to emerge after collision with an energy between El 0 and El0 þ dEl0 , one has: PðEl / El 0ÞdEl 0 ¼  Pqc ðqc Þdqc

(3.65)

Chapter 3  Neutron transport calculations at the cell and assembly levels

87

where the minus sign accounts for the fact that the energy after interaction El0 and the angle of deviation qc vary in opposite directions. Noticing from Eq. (3.58) that: dEl0 ¼

a1 El sin qc dqc 2

(3.66)

one finds, combining Eqs. (3.64) and (3.65), that:

PðEl / El 0Þ ¼

8 > > > >
El

1 for aEl < El 0 < El El ð1  aÞ > > > > : 0 for 0 < El 0 < aEl

(3.67)

There is thus equiprobable distribution of the energy El0 of the neutron after interaction in the laboratory reference system in the interval ½aEl ; El . Such a probability distribution is schematically represented in Fig. 3.5.

FIGURE 3.5 Probability distribution of the neutron energy in the laboratory reference system after an elastic scattering reaction assumed to be isotropic in the centre-of-mass reference system. Derived from Lamarsh, J.R., 2002. Introduction to Nuclear Reactor Theory. American Nuclear Society, La Grange Park, USA.

It has to be emphasized that isotropy in the centre-of-mass reference system (i.e., qc ¼ 0, with the bar denoting the average value) does not necessarily mean isotropy in the laboratory reference system, i.e., ql s0. In case of isotropy in the centre-of-mass reference system, one could actually demonstrate that (Lamarsh, 2002): ql ¼

2 3A

(3.68)

This ratio decreases for increasing values of A, and thus the scattering reaction in the laboratory reference system becomes only isotropic for an infinitely heavy target nucleus. Likewise, equality of the neutron speeds before and after interaction in the centre-ofmass reference system (i.e., kvc0 k ¼ kvck) does not necessarily imply that kvl0 k ¼ kvlk in the laboratory reference system, as Eq. (3.58) demonstrates. kvl0 k / kvlk only when a/1, i.e., when the target nucleus becomes infinitely heavy, according to Eq. (3.59). As can also be seen from Eq. (3.58), the energy loss per collision in the laboratory reference system, i.e., ElEl0 depends on the energy El before interaction. On the other hand, the ratio of the energy El before interaction and the energy El0 after interaction is

88 Modelling of Nuclear Reactor Multi-physics

energy-independent. Consequently, the neutron slowing-down is most often characterized by a quantity called the lethargy u(El), which is related to the neutron energy El in the laboratory reference system as: uðEl Þ ¼ ln



El;0 El

(3.69)

where El,0 is a reference energy, usually taken as the energy of the most energetic neutrons in the system being considered. While slowing-down, the neutrons lose energy and, therefore, gain lethargy. A practical quantity, based on the change of representation from the energy variable to the lethargy variable, is the average increase of lethargy per collision, defined as: Z

x ¼ Du ¼

uðaEl Þ  uðEl Þ

      u0 El0  uðEl Þ P uðEl Þ / u0 El0 du0

(3.70)

Making the change of variable P[u(El) / u0 (El0 )]du0 ¼ P(El / El0 )dEl0 , where the minus sign accounts for the fact that lethargies and energies vary in opposite directions, one finds, based on Eq. (3.69), that: Z

x¼

El aEl



 El  ln 0 P El / El0 dEl0 El

(3.71)

In case of elastic and isotropic scattering in the centre-of-mass reference system, Eq. (3.67) can be used to demonstrate that (Lamarsh, 2002): x¼1 

ðA  1Þ2 Aþ1 ln A1 2A

(3.72)

2 A þ 2=3

(3.73)

for which one further has: lim x ¼

A/N

In the remaining of this section, an expression giving the energy-dependence of the neutron flux during slowing-down is derived in the absence of absorption. The system being considered is infinite and contains a uniformly distributed source of neutrons emitted at energies much higher than the energy considered (the energies that are considered hereafter belong to the so-called asymptotic region that will also be defined). As a result, all quantities are space-independent. Furthermore, the scattering is assumed to be elastic and isotropic in the centre-of-mass reference system. For the sake of simplicity, the subscript l referring to the laboratory reference system is dropped, and thus all energies are assumed to be taken in this reference system. Defining the collision density F(E) as the number of collisions per unit volume, unit time, and unit energy in the infinitesimal energy bin [E, E þ dE], one has: FðEÞ ¼ Ss ðEÞfðEÞ

where fðEÞ is the scalar neutron flux per unit energy.

(3.74)

Chapter 3  Neutron transport calculations at the cell and assembly levels

89

Another useful quantity to be defined for treating neutron slowing-down is the slowing-down density q(E) representing the number of neutrons per unit volume, whose energy falls below the energy E per unit time. An expression for fðEÞ can be derived by writing two balance equations in case of steady-state conditions: one balance equation for F(E), and one balance equation for q(E). The difficulty in treating the slowing-down of neutrons lies with the fact that the expression for the collision density F(E) is very complicated for energies E close to the energies of the neutron source, since the number of collisions required to scatter a neutron from the energy E0 to the energy E below E0 cannot be determined: when E is sufficiently close to the energy E0, a neutron can reach the energy E either directly after one collision from the energy E0, or after multiple collisions. Nevertheless, for energies E sufficiently well below the lower energy bound of the source denoted as the energy Es (formally when E < a3 Es ), it can be assumed that neutrons cannot scatter directly from any energy above Es to the energy E. This defines the so-called asymptotic region. In this region, the slowing-down and collision densities, denoted as Fas(E) and qas(E), respectively, can be estimated relatively simply. The interested reader is referred to Lamarsh (2002) for further details on the asymptotic region. The information provided above is nevertheless sufficient to comprehend the remaining of the derivation of the expression for the neutron flux in the asymptotic region. According to the definition of the asymptotic region given as the energy region below a3 E0 , such a region does not exist for hydrogen, since a ¼ 0 in such a case. Nevertheless, it could be demonstrated that the relationship giving the neutron flux as a function of energy in the asymptotic region and derived hereafter in this section, i.e., Eq. (3.81), holds for all energies in the case of hydrogen. The interested reader is referred to Lamarsh (2002) for the corresponding demonstration. In the asymptotic region, neutrons scattered to the energy bin [E, E þ dE] had, before interaction, at the most an energy E=a, as is represented schematically in Fig. 3.6. In steady-state conditions, the number of neutrons scattered in the energy bin [E, E þ dE] should be equal to the number of neutrons scattered to that energy bin, i.e., Z

Fas ðEÞ ¼

E=a

Fas ðE 0 ÞPðE 0 / EÞdE 0

(3.75)

E

FIGURE 3.6 Diagram illustrating the calculation of the collision density in the asymptotic region. Derived from Lamarsh, J.R., 2002. Introduction to Nuclear Reactor Theory. American Nuclear Society, La Grange Park, USA.

90 Modelling of Nuclear Reactor Multi-physics

FIGURE 3.7 Diagram illustrating the calculation of the slowing-down density in the asymptotic region. Derived from Lamarsh, J.R., 2002. Introduction to Nuclear Reactor Theory. American Nuclear Society, La Grange Park, USA.

with P(E0 / E) given by Eq. (3.67) and accounting for the fact that not all neutrons colliding at the energy E0 actually lead to neutrons in the energy interval [E, E þ dE]. One could verify that: Fas ðEÞ ¼

C E

(3.76)

is solution to Eq. (3.75), where C is a constant. Likewise, in steady-state conditions, neutrons scattered below the energy E are the results of collisions in the energy interval ½E; E =a. Nevertheless, not all collisions in the energy interval ½E; E =a lead to neutrons of energy below E. Therefore, the balance equation for the slowing-down density results in: Z

qas ðEÞ ¼

E=a

Fas ðE 0 ÞPðE 0 / E 00 < EÞdE 0

(3.77)

E

where P(E0 / E00 < E)dE0 represents the probability for a neutron within the energy interval [E0 , E0 þ dE0 ] to be scattered at energies E00 below the energy E. Due to the assumption of isotropic scattering in the centre-of-mass reference system, there is equiprobability for neutrons to be scattered from the energy E0 to an energy in the interval ½aE 0 ; E 0 . Based on the schematic representation given in Fig. 3.7, one thus finds that: PðE 0 / E 00 < EÞdE 0 ¼

E  aE 0 dE 0 E 0  aE 0

(3.78)

Furthermore, since there is no absorption, one also has: Z

qðEÞ ¼

N

SðE 0 ÞdE 0 for E < Es

(3.79)

Es

where S(E0 ) represents the energy spectrum of the source of neutrons, i.e., S(E0 )dE0 neutrons are emitted per unit time and volume in the energy bin [E0 , E0 þ dE0 ]. Combining Eqs. (3.75e3.79), one could demonstrate that: RN

C¼

Es

SðE 0 ÞdE 0 x

(3.80)

Chapter 3  Neutron transport calculations at the cell and assembly levels

91

Using this result in Eqs. (3.76) and (3.74), one finally finds that: RN Es

fas ðEÞ ¼

SðE 0 ÞdE 0

xESs ðEÞ

(3.81)

in the asymptotic region. The previous result for the asymptotic flux can be extended without problem to the case of neutron slowing-down in a mixture of species X, as follows. The collision density of neutrons elastically scattered by the species X can be written as: FX ðEÞ ¼ SsX ðEÞfðEÞ

(3.82)

Because of Eq. (3.74), this collision density can also be rewritten as: FX ðEÞ ¼

SsX ðEÞ Fas ðEÞ Ss ðEÞ

(3.83)

The scattering reactions might occur on any of the species X. In steady-state conditions, the number of neutrons scattered in the energy bin [E, E þ dE ] should be equal to the number of neutrons reacting in that energy bin, which can be expressed in the asymptotic region as: XZ

Fas ðEÞ ¼

E=aX

E

X

FX ðE 0 ÞdE 0 ð1  aX ÞE 0

(3.84)

It could be verified that: FX ðEÞ ¼

CX E

(3.85)

is solution to Eq. (3.84), where CX is a constant. Likewise, in steady-state conditions, neutrons scattered below the energy E are the results of collisions in the energy interval ½E; E =a. Nevertheless, not all collisions in the energy interval ½E; E =a lead to neutrons of energy below E. Therefore, the balance equation for the slowing-down density results in: qas ðEÞ ¼

XZ

E=aX

FX ðE 0 Þ

E

X

E  aX E 0 dE 0 E 0  aX E 0

(3.86)

Because of Eq. (3.85), one could demonstrate that: qas ðEÞ ¼

X CX xX

(3.87)

X

Furthermore, since there is no absorption, one also has: Z

qas ðEÞ ¼

N

SðE 0 ÞdE 0 for E < Es

(3.88)

Es

where S(E0 ) represents the energy spectrum of the source of neutrons, i.e., S(E0 )dE0 neutrons are emitted per unit time and volume in the energy bin [E0 , E0 þ dE0 ]. Using Eqs. (3.85), (3.87) and (3.88), one obtains: Z

qas ðEÞ ¼

N Es

SðE 0 ÞdE 0 ¼ E

X xX FX ðEÞ X

(3.89)

92 Modelling of Nuclear Reactor Multi-physics

Finally, because of Eq. (3.83), this equation can be expressed as: Z

qas ðEÞ ¼

N

SðE 0 ÞdE 0 ¼

Es

EFas ðEÞ X x SsX ðEÞ Ss ðEÞ X X

(3.90)

and one thus obtains, since Fas ðEÞ ¼ Ss ðEÞfas ðEÞ: RN

fas ðEÞ ¼

Es

SðE 0 ÞdE 0

xðEÞSs ðEÞE

(3.91)

with xðEÞ ¼

3.2.c

1 X x SsX ðEÞ Ss ðEÞ X X

(3.92)

Neutron slowing-down with absorption

The physics of neutron slowing-down in the resonance region is one of the most challenging topics in reactor physics. Due to the complexity of the different phenomena at hand, the presentation in this section is kept at a rather elementary level in order to highlight the main features of resonance absorption. The interested reader is referred to the vast literature on this subject for further details (see, e.g., Knott and Yamamoto, 2010 for a complete overview of recent advances in this area). First, neutron slowing-down in the resonance region will be investigated in homogeneous media and then in heterogeneous media. In both cases, one will explain how the resonance data available in the evaluated nuclear data files can be used for treating resonance absorption and determining the multi-group macroscopic cross-sections of resonant absorbers. Finally, the limitations of the methodology presented in this section will be touched upon.

Resonance modelling in homogeneous media In the following, a homogeneous infinite medium made of different species X is considered. The system contains a uniformly distributed source of neutrons emitted at energies much higher than the energy considered hereafter (one thus assumes to be in the asymptotic region). As a result, all quantities are space-independent. Furthermore, the scattering is assumed to be elastic and isotropic in the centre-of-mass reference system. For the sake of simplicity, the subscript l referring to the laboratory reference system is dropped, and thus all energies are assumed to be taken in this reference system. Following the same procedure as for neutron slowing-down without absorption, a balance equation can also be established in the asymptotic region. In this region, neutrons scattered to the energy bin [E, E þ dE] had, before interaction, at the most an energy E=a, as is represented schematically in Fig. 3.6. In addition, the scattering reactions might occur on any of the species X present in the system. In steady-state

Chapter 3  Neutron transport calculations at the cell and assembly levels

93

conditions, the number of neutrons scattered in the energy bin [E, E þ dE] should be equal to the number of neutrons reacting in that energy bin, i.e., St ðEÞfðEÞ ¼

XZ X

E

E=aX

SsX ðE 0 ÞfðE 0 ÞdE 0 ð1  aX ÞE 0

(3.93)

Formally, this equation allows solving for the neutron flux fðEÞ. If one assumes that the neutron flux is given for energies E above the upper bound Eres of the resonance region by the asymptotic flux in the absence of resonance (i.e., Eq. 3.81) and is thus known, Eq. (3.93) could be used to numerically evaluate the neutron flux at the energy EresdE, where dE is an infinitely small energy interval. This procedure could be repeated in an iterative manner to resolve the neutron flux in the entire resonance region. Nevertheless, such a procedure is too computationally intensive for being used routinely in practical reactor calculations. As will be seen in the following, resonance absorption can be more easily determined based on some approximations leading to acceptable results. These approximations are based on the comparisons between the maximum energy loss by collision of a neutron of energy E, given by Eð1 aX Þ, and the practical width Gp of a resonance, with the practical width of a resonance being defined as the energy range over which the resonance cross-section is larger than the non-resonance part of the cross-section of the considered nuclide. The practical width of a resonance is illustrated in Fig. 3.8. Three approximations are usually encountered: the narrow resonance approximation, the wide resonance or infinite mass approximation and the intermediate resonance approximation (Stamm’ler and Abbate, 1983).

FIGURE 3.8 Diagram illustrating the practical width of a resonance.

In the narrow resonance approximation (NR) with respect to a specific species X, one considers that the resonance of the mixture of species fulfils the following condition: Gp  E  ð1  aX Þ

(3.94)

In virtue of Eq. (3.59), this approximation usually holds for light nuclei. In practice, this means that the neutron flux is not affected by the presence of resonances, since the resonance is very small compared to the maximum energy loss by collision. As a result and as illustrated in Fig. 3.9, the neutron flux should be the neutron flux in the absence of the

94 Modelling of Nuclear Reactor Multi-physics

FIGURE 3.9 Diagram illustrating the maximum energy loss per collision Eð1 aX Þ in relation to the practical width Gp of a resonance in case of the narrow resonance (NR) approximation.

considered resonance, i.e., the neutron flux given by Eq. (3.91), and the scattering crosssection SsX ðE 0 Þ in Eq. (3.93) can be taken equal to the potential scattering cross-section SpX . The potential scattering cross-section corresponds to the cross-section in case of scattering reactions not involving any creation of a compound nucleus. It is thus the scattering cross-section in the absence of resonance and is energy-independent. Consequently, the term related to the slowing-down on the species X in Eq. (3.93) can be replaced by: Z

E

E=aX

SsX ðE 0 ÞfðE 0 ÞdE 0 zSpX ð1  aX ÞE 0

Z

E

E=aX

fas ðE 0 ÞdE 0 ð1  aX ÞE 0

(3.95)

In virtue of Eq. (3.91), the asymptotic flux can be expressed as: fas ðEÞ ¼

C E

(3.96)

where C is a normalization constant. For the sake of simplicity, C is re-normalized to C ¼ 1. Using Eq. (3.96) into Eq. (3.95) leads to: Z

E

E=aX

SsX ðE 0 ÞfðE 0 ÞdE 0 SpX z ð1  aX ÞE 0 E

(3.97)

In the wide resonance approximation (WR) with respect to a specific species X, one considers that the resonance of the mixture of species fulfils the following condition: Gp [Eð1  aX Þ

(3.98)

In virtue of Eq. (3.59), this approximation usually holds for heavy nuclei. This approximation is thus sometimes referred to as the infinite mass approximation, since Eq. (3.98) is equivalent to taking aX /1, which, according to Eq. (3.59), is equivalent to an infinite mass for the species X. In practice, this means that the interval of integration in the term related to the species X in Eq. (3.93) is very small compared to the resonance being considered. As a result and as is illustrated in Fig. 3.10, one can assume that, for energies E 0 ˛½E; E =aX , one can replace SsX ðE 0 ÞfðE 0 Þ=ðE 0 Þ by SsX ðEÞfðEÞ=E. Consequently, the term related to the slowing-down on the species X in Eq. (3.93) can be replaced by:

Chapter 3  Neutron transport calculations at the cell and assembly levels

95

FIGURE 3.10 Diagram illustrating the maximum energy loss per collision Eð1 aX Þ in relation to the practical width Gp of a resonance in case of the wide resonance (WR) approximation.

Z

Z E=aX SsX ðE 0 ÞfðE 0 ÞdE 0 SsX ðE 0 ÞfðE 0 ÞdE 0 z lim aX /1 E ð1  aX ÞE 0 ð1  aX ÞE 0 E Z E=aX SsX ðEÞfðEÞ dE 0 SsX ðEÞfðEÞ E lim lim ¼ z ¼ SsX ðEÞfðEÞ aX /1 E aX /1 aX E E ð1  aX Þ E=aX

(3.99)

In the intermediate resonance approximation (IR) with respect to a specific species X, the term related to the slowing-down on the species X in Eq. (3.93) is expressed as: Z

E=aX E

SpX SsX ðE 0 ÞfðE 0 ÞdE 0 þ ð1  lX ÞSsX ðEÞfðEÞ ¼ lX ð1  aX ÞE 0 E

(3.100)

where lX ˛½0;1 and is a parameter to be determined. The IR approximation was originally proposed by Goldstein and Cohen (1962). When lX /1, the NR approximation is obtained (i.e., Eq. 3.97), whereas when lX /0, the WR approximation is obtained (i.e., Eq. 3.99). Using Eq. (3.100) into Eq. (3.93), one obtains: St ðEÞfðEÞ ¼

X SpX lX þ ð1  lX ÞSsX ðEÞfðEÞ E X

(3.101)

from which the neutron flux can be determined as: P

P lX SpX 1 1 X X P P ¼ fðEÞ ¼ St ðEÞ  ð1  lX ÞSsX ðEÞ E Sa ðEÞ þ lX SsX ðEÞ E lX SpX

X

(3.102)

X

where the last equality was obtained by using the fact that [see Eq. (2.13) in Chapter 2]: St ðEÞ ¼ Sa ðEÞ þ Ss ðEÞ ¼ Sa ðEÞ þ

X SsX ðEÞ

(3.103)

X

Based on Eq. (3.22), the group-averaged macroscopic cross-section for reaction type a of a single resonant species X mixed with different moderating species Y can be estimated as:

96 Modelling of Nuclear Reactor Multi-physics P lY SpY dE Y P saX ðEÞ Sa ðEÞ þ lX SsX ðEÞ þ lY SsY ðEÞ E Y P lX SpX þ lY SpY R Eg1 dE Y P Eg Sa ðEÞ þ lX SsX ðEÞ þ lY SsY ðEÞ E lX SpX þ

R Eg1 Eg

SaX ;g ¼ NX

(3.104)

Y

where the expression given by Eq. (3.102) was used for the weighting flux fw ðEÞ. Introducing the so-called background cross-section as: sb ¼

X NY Y

NX

(3.105)

lY spY

assuming that only the resonant absorber is contributing to the absorption cross-section, i.e., Sa ðEÞ ¼ SaX ðEÞ ¼ NX saX ðEÞ, and that the scattering on the moderating species Y in the energy range of interest is only via potential scattering, i.e., SsY ðEÞ ¼ SpY ¼ NY spY , Eq. (3.104) can be rearranged into: R Eg1 Eg

SaX ;g ¼ NX

saX ðEÞ

R Eg1 Eg

lX spX þ sb dE P NY saX ðEÞ þ lX ssX ðEÞ þ NX lY spY E Y

lX spX þ sb dE P saX ðEÞ þ lX ssX ðEÞ þ NNYX lY spY E

(3.106)

Y

s0sX ðEÞ

Defining the anomalous scattering cross-section as the difference between the actual scattering cross-section ssX ðEÞ and the potential cross-section spX in the vicinity of the resonance, i.e., ssX 0 ðEÞ ¼ ssX ðEÞ  spX

(3.107)

Eq. (3.106) can be finally written as: R Eg1 Eg

SaX ;g ¼ NX

lX spX þ sb dE saX ðEÞ þ lX s0sX ðEÞ þ lX spX þ sb E R Eg1 lX spX þ sb dE Eg saX ðEÞ þ lX s0sX ðEÞ þ lX spX þ sb E saX ðEÞ

(3.108)

In order to obtain a somewhat simpler expression, a further approximation is usually made by assuming that the anomalous scattering cross-section s0sX ðEÞ is negligible in comparison with the background cross-section sb , so that the denominator of Eq. (3.108) can be rewritten as: Z

Eg1

Eg

Z

z

lX spX þ sb dE saX ðEÞ þ lX s0sX ðEÞ þ lX spX þ sb E

Eg1

Eg

lX spX þ sb þ lX s0sX ðEÞ dE saX ðEÞ þ lX s0sX ðEÞ þ lX spX þ sb E

(3.109)

Chapter 3  Neutron transport calculations at the cell and assembly levels

97

which can be rearranged into: Z

lX spX þ sb dE saX ðEÞ þ lX s0sX ðEÞ þ lX spX þ sb E Z Eg1  saX ðEÞ dE z 1 0 s E ðEÞ þ l s ðEÞ þ l s þ s aX X sX X pX b Eg

Z Eg1 Eg1 lX spX þ sb 1 dE ¼ ln  saX ðEÞ Eg saX ðEÞ þ lX s0sX ðEÞ þ lX spX þ sb E lX spX þ sb Eg Eg1

Eg

(3.110)

Defining the resonance integral (RI) for the species X and reaction type a as: Z

RIaX ;g ðsb Þ ¼

Eg1

saX ðEÞ

Eg

lX spX þ sb dE saX ðEÞ þ lX s0sX ðEÞ þ lX spX þ sb E

(3.111)

The group-averaged macroscopic cross-section for the resonant species X can be finally written, combining Eqs. (3.108), (3.110) and (3.111), as: SaX;g ¼

N RI ðs Þ X aX ;g b Eg1 RIaX;g ðsb Þ  ln lX spX þ sb Eg

(3.112)

As can be seen from Eq. (3.111), the RI for the species X only depends on the species itself and the background cross-section sb . Such RIs are pre-determined once for all as a function of the resonant species X and of the background cross-section sb in the following manner (Stamm’ler and Abbate, 1983). Eq. (3.93) is solved without any approximation for the energy-dependent flux fðEÞ for a homogeneous mixture between the resonant species X and the moderating species (hydrogen in the case of LWRs). Thereafter, the RI is evaluated as: Z

RIaX ;g ðsb Þ ¼

Eg1

saX ðEÞfðEÞdE

(3.113)

Eg

since following the derivation above, the RI is nothing else than the reaction rate on the resonant species divided by its concentration. The RIs are then tabulated in the evaluated nuclear data files as functions of the background cross-section and for each resonant species. When the group-averaged macroscopic cross-section is needed, Eq. (3.112) can simply be used using the tabulated RIs from the cross-section library. It should be mentioned that instead of containing the RIs, some nuclear data files provide relationships to directly retrieve the group averaged microscopic cross-section saX ;g (Knott and Yamamoto, 2010). In nuclear reactors, the medium is typically not a homogeneous mixture of fuel and moderator. Rather, the medium is made of a heterogeneous arrangement of fuel pins surrounded by moderator. In the following, the procedure for producing the multi-group data for resonant species in such a case is highlighted. Two methods are considered: the equivalence method and the subgroup method.

98 Modelling of Nuclear Reactor Multi-physics

Resonance modelling in heterogeneous media using the equivalence method In the equivalence method, the effect of resonances is modelled by the method of collision probabilities, which is fully explained in Section 3.4c. Here, only the application of this method to resonance species is presented. As will be demonstrated hereafter, an approximation in the representation of the probability for neutrons emitted in the fuel region to have their first collision in the moderator region leads to an expression of the group-averaged macroscopic cross-section equivalent to the one obtained for homogeneous media given by Eq. (3.112). This is the reason why the method presented below is called the equivalence method. If one denotes by F the fuel region having a volume VF and by M the moderator region having a volume VM, a balance equation similar to Eq. (3.93) can be established. Since the effect of resonance occurs in the fuel region, such a balance equation will be written for the fuel region only. Nevertheless, neutrons interacting in the fuel region may originate either from the fuel region or from the moderator region. It will be assumed that the flux in each region is spatially independent (flat flux approximation). In case of steady-state behaviour, the collision rate CM/F in region F due to neutrons emitted from region M can be formally written as: CM/F ðEÞ ¼ PM/F ðEÞVM

XZ XM

E=aX E

SsXM ðE 0 ÞfM ðE 0 ÞdE 0 ð1  aXM ÞE 0

(3.114)

where the subscript XM represents quantities taken in the moderator and PM/F represents the probability for neutrons emitted in the moderator region to have their first collision in the fuel region. Likewise, in case of steady-state behaviour, the collision rate CF/F in region F due to neutrons emitted from region F can be formally written as: CF/F ðEÞ ¼ ½1  PF/M ðEÞVF

XZ XF

E=aX E

SsXF ðE 0 ÞfF ðE 0 ÞdE 0 ð1  aXF ÞE 0

(3.115)

where the subscript XF represents quantities taken in the fuel and PF/M represents the probability for neutrons emitted in the fuel region to have their first collision in the moderator region. 1PF/M thus represents the probability for neutrons emitted in the fuel region to have their first collision in the fuel region. The sum of CM/F and CF/F represents the total collision rate in the fuel region, thus allowing writing: CM/F ðEÞ þ CF/F ðEÞ ¼ VF StF ðEÞfðEÞ X Z E=aX SsXM ðE 0 ÞfM ðE 0 ÞdE 0 ¼ PM/F ðEÞVM ð1  aXM ÞE 0 XM E X Z E=aX SsXF ðE 0 ÞfF ðE 0 ÞdE 0 þ½1  PF/M ðEÞVF ð1  aXF ÞE 0 E XF

(3.116)

As will be seen in Section 3.4d, the collision probabilities PM/F and PF/M furthermore fulfil the following reciprocity relationship: VF StF ðEÞPF/M ðEÞ ¼ VM StM ðEÞPM/F ðEÞ

(3.117)

Chapter 3  Neutron transport calculations at the cell and assembly levels

99

Using Eq. (3.117) into Eq. (3.116) leads to: StF ðEÞfðEÞ

X Z E=aX SsXM ðE 0 ÞfM ðE 0 ÞdE 0 StF ðEÞ PF/M ðEÞ StM ðEÞ ð1  aXM ÞE 0 XM E X Z E=aX SsXF ðE 0 ÞfF ðE 0 ÞdE 0 þ½1  PF/M ðEÞ ð1  aXF ÞE 0 E XF

¼

(3.118)

In order to simplify this expression, the IR approximation is made for the fuel region whereas the NR approximation is made for the moderator region (the NR approximation is valid since the moderator is made of light nuclei). Using Eqs. (3.97) and (3.100) into Eq. (3.118) gives: StF ðEÞfðEÞ

P SpXM StF ðEÞ z PF/M ðEÞ XM StM ðEÞ E  X SpXF þ ð1  lXF ÞSsXF ðEÞfðEÞ lXF þ½1  PF/M ðEÞ E XF

(3.119)

from which the energy-dependent neutron flux can be evaluated as: P P StF ðEÞ PF/M ðEÞ SpXM þ ½1  PF/M ðEÞ lXF SpXF 1 StM ðEÞ XM XF P P fðEÞz StF ðEÞ  ½1  PF/M ðEÞ SsXF ðEÞ þ ½1  PF/M ðEÞ lXF SsXF ðEÞ E XF

(3.120)

XF

This expression will now be used to determine the group-averaged macroscopic cross-section of type a for a single resonant species X in the fuel region mixed with other moderating species XF (with XFsX), whereas the moderator region is made of a mixture of different moderating species XM. It is further assumed that in the moderator region, absorption is negligible, and that scattering is mostly driven by potential scattering, leading to: StM ðEÞ z SsM ðEÞzSpM ¼

X SpXM

(3.121)

XM

Eq. (3.120) can thus be rearranged into:

 P StF ðEÞPF/M ðEÞ þ ½1  PF/M ðEÞ lX SpX þ lXF SpXF 1 XFsX fðEÞz N ðEÞ E

(3.122)

with NðEÞ

"

X

SsXF ðEÞ ¼ StF ðEÞ  ½1  PF/M ðEÞ SsX ðEÞ þ XFsX " # X lXF SsXF þ½1  PF/M ðEÞ lX SsX þ XFsX

# (3.123)

100

Modelling of Nuclear Reactor Multi-physics

In order to transform Eq. (3.122) into a simple enough expression, a so-called rational approximation for estimating PF/M is performed. Namely, PF/M is assumed to be of the form: Se PF/M ðEÞz StF ðEÞ þ Se

(3.124)

Rational approximations will be explained in detail in Section 3.4f, and Se will be then defined. Using Eq. (3.124) into Eq. (3.122) leads, after some algebra, to: lX SpX þ

P

lXF SpXF þ Se 1 XFsX  fðEÞz P P E StF ðEÞ þ Se  SsX ðEÞ þ SsXF ðEÞ þ lX SsX þ lXF SsXF XFsX

(3.125)

XFsX

Assuming that the moderating species XF present in the fuel region and other than the resonant species X do not have any scattering resonance, then SsXF ¼ SpXF for XFsX . Furthermore, since X is the only resonant species, one can reasonably assume that SaF ðEÞzSaX ðEÞ in the resonance region for X. Eq. (3.125) thus gives: lX SpX þ

P

lXF SpXF þ Se 1 XFsX P fðEÞz SaX ðEÞ þ lX SsX þ lXF SpXF þ Se E

(3.126)

XFsX

This equation was obtained using the fact that: StF ðEÞ ¼ SaF ðEÞ þ SsF ðEÞ

where SsF ðEÞ ¼ SsX ðEÞ þ

X

SsXF ðEÞ

(3.127)

(3.128)

XFsX

Based on Eq. (3.22), the group-averaged macroscopic cross-section for reaction type a of a single resonant species X can be estimated as: R Eg1 Eg

SaX ;g zNX

lX SpX þ saX ðEÞ

P

lXF SpXF þ Se dE P þ lXF SpXF þ Se E

XFsX

SaX ðEÞ þ lX SsX XFsX P lX SpX þ lXF SpXF þ Se R Eg1 dE XFsX P Eg SaX ðEÞ þ lX SsX þ lXF SpXF þ Se E XFsX

(3.129)

Chapter 3  Neutron transport calculations at the cell and assembly levels

101

where the expression given by Eq. (3.126) was used for the weighting flux fw ðEÞ. Defining the background cross-section for the heterogeneous case as: s+ b ¼

X NXF XFsX

NX

lXF spXF þ

Se NX

(3.130)

Eq. (3.129) can be rearranged into: R Eg1 SaX ;g zNX

Eg

lX spX þ s+ dE b saX ðEÞ þ lX s0sX ðEÞ þ lX spX þ s+ b E R Eg1 lX spX þ s+ dE b Eg saX ðEÞ þ lX s0sX ðEÞ þ lX spX þ s+ b E saX ðEÞ

(3.131)

where the anomalous scattering cross-section s0sX ðEÞ defined in Eq. (3.107) was used. One notices that Eq. (3.131) is identical in its form to Eq. (3.108). The only difference between the homogeneous and heterogeneous cases is in the definition of the background cross-section, which is given by Eq. (3.105) in the homogeneous case and by Eq. (3.130) in the heterogeneous case. It can be observed that the effect introduced by the heterogeneity is equivalent to adding the quantity Se =NX to the background crosssection of the homogeneous case. Following the same procedure as for the homogeneous case, one finally obtains: SaX ;g ¼

  NX RIaX;g s+ b  

RIaX;g s+ Eg1 b  ln Eg lX spX þ s+ b

(3.132)

The methodology presented above for the heterogeneous case relies on several approximations. First, the method of collision probabilities that is used assumes that the flux is spatially homogeneous in each region, i.e., in the fuel and moderator regions, respectively. In fact, there is a steep flux gradient in the fuel pins surrounded by the moderator, as a result of resonance absorption. Such a steep gradient cannot be properly modelled if the fuel regions are treated as one homogeneous region. In addition, only one resonant species was assumed to be present in the fuel region, whereas several resonant species are usually present at the same time in the fuel. The simultaneous presence of several resonant species might result in some overlapping of the resonances. Moreover, the non-resonant species were assumed to have a constant (with respect to energy) scattering cross-section dominated by potential scattering. Such an approximation is acceptable, especially for LWRs, where slowing-down is mostly dominated by scattering on hydrogen and oxygen, both having fairly constant scattering cross-section. The resonant species were also assumed to have constant scattering cross-sections dominated by potential scattering. This approximation does not always hold, since the considered species might have non-constant scattering cross-sections, especially when resonances in scattering are present. Finally, the assumption of elasticity was used when treating slowing-down. This approximation is acceptable, even if the effect of thermal motion of the target nucleus would need to be taken into account. Corrections to the above model have been proposed to account for such effects (see, e.g., Knott and Yamamoto, 2010 for an overview of those methods). Such corrections are

102

Modelling of Nuclear Reactor Multi-physics

beyond the scope of this book, and the interested reader is thus referred to the existing literature on these topics.

Resonance modelling in heterogeneous media using the subgroup method The difficulty in the treatment of resonances lies with the fact that the microscopic cross-sections in the resonance region exhibit large and complicated variations as a function of the energy. A faithful representation of the cross-section dependence on energy requires many energy groups. The subgroup method relies on the observation that the neutron flux in the resonance region depends mostly on the magnitude of the resonant cross-section, as Eq. (3.126) indicates (Knott and Yamamoto, 2010). It is expected that energy groups having the same magnitude of the cross-section could be grouped together, thus creating subgroups, as conceptually illustrated in Fig. 3.11 (Ribon, 1989). In this illustrative example, one notices that the two resonances in the energy interval [Eg, Eg1], although requiring nine energy groups, could be correctly described by only three group-averaged microscopic cross-sections. These so-called subgroup cross-sections represent cross-sections averaged on discontiguous energy intervals. Formally, the subgroup cross-section on the subgroup k is defined as: R

ssubgroup aX ;k

¼

subgroup k

saX ðEÞfw ðEÞdE

R

subgroup k

with

fw ðEÞdE

o n subgroup subgroup k ¼ Ejssubgroup aX;k;min saX ðEÞ saX ;k;max

(3.133)

(3.134)

FIGURE 3.11 Diagram illustrating the subgroup method using three subgroups. The variation of the cross-sections saX as a function of energy E on the energy interval [Eg, Eg1] is given on the left-hand side, whereas the variation of the probability pðsaX Þ of having a cross-section saX is given on the right-hand side as a function of the cross-section saX .

Chapter 3  Neutron transport calculations at the cell and assembly levels

103

The definition of the subgroup cross-sections depends on the number of crossh i subgr subgr section bands saX ;k;min ; saX ;k;max one chooses to represent the resonances with. Assuming that the subgroup microscopic cross-sections are provided, the steadystate neutron transport equation given in Chapter 2 by Eq. (2.35) can be rewritten as: U $ Vjðr; U; EÞ þ St ðr; EÞjðr; U; EÞ ¼ qðr; U; EÞ

(3.135)

where q(r, U, E) represents the neutron emission density, defined in Chapter 2 by Eq. (2.47). Eq. (3.135) can be integrated with respect to energy on the thus-defined subgroups. This formally results in: U $ Vjsubgroup ðr; UÞ þ Ssubgroup ðrÞjsubgroup ðr; UÞ ¼ qsubgroup ðr; UÞ k t;k k k

with

(3.136)

Z

jsubgroup ðr; UÞ ¼ k

jðr; U; EÞdE

(3.137)

subgroup k

ðrÞ ¼ Ssubgroup t;k

X NX ðrÞssubgroup tX ;k

(3.138)

X

Z

ðr; UÞ ¼ qsubgroup k

qðr; U; EÞdE

(3.139)

subgroup k

The main difference compared to the conventional multi-group neutron transport equation lies with the fact that Eq. (3.136) was obtained after integration on discontiguous energy intervals. The subgroup method has the advantage of requiring much less subgroups than the number of energy groups one would need to correctly represent resonances using a conventional multi-group approach (i.e., using contiguous energy intervals). Eq. (3.136) can then be solved using any transport method (as the ones subgroup to be presented in Sections 3.4 and 3.5) and the subgroup fluxes jk can be obtained. Thereafter, the subgroup fluxes can be used to estimate the multi-group microscopic cross-section of type a in group g for the resonant species X as: P

saX ;g ðrÞ ¼

subgroups k ˛ group g

ssubgroup fsubgroup ðrÞ aX ;k k

P

subgroups k ˛ group g

with fsubgroup ðrÞ ¼ k

Z ð4pÞ

fsubgroup ðrÞ k

jsubgroup ðr; UÞd 2 U k

(3.140)

(3.141)

104

Modelling of Nuclear Reactor Multi-physics

Although the derivation of the subgroup method presented above is straightforward, the subgroup averaged microscopic cross-sections would still have a dependence on the background cross-sections, unless the number of subgroups is drastically increased (Knott and Yamamoto, 2010). Since the evaluation of the background cross-section becomes difficult for general geometries, another approach, referred to as the probability table approach, is preferred. Using a Lebesgue integration technique, integrals with respect to energies on the energy group [Eg, Eg1] are replaced by integrals with respect to cross-sections as follows (Ribon, 1989; He´bert and Coste, 2002; Knott and Yamamoto, 2010): Z

Eg1

Z

FðEÞdE ¼

Eg

saX;max;g

0

f ðsaX ÞpðsaX ÞdsaX

(3.142)

In this equation, saX ;max;g corresponds to the maximum of the microscopic crosssection saX on the energy group [Eg, Eg1] and pðsaX ÞdsaX represents the probability for the microscopic cross-section to have a value between saX and saX þ dsaX on the energy group [Eg, Eg1]. F(E) represents any energy-dependent function, and f ðsaX Þ is its counterpart when the dependence is expressed as a function of the microscopic crosssection saX . The probability density function pðsaX Þ is normalized to unity, i.e., Z

saX ;max;g

0

pðsaX ÞdsaX ¼ 1

(3.143)

Assuming that the energy interval [Eg, Eg1] is split into a number of subgroups, the probability density function pðsaX Þ is replaced by a series of Dirac delta functions and associated weights as: pðsaX Þ ¼

X

subgroups k ˛ group g



psubgroup d saX  ssubgroup aX ;k aX ;k

(3.144)

subgroup

with one Dirac delta function and one associated weight paX ;k on each of the subgroups. These weights are illustrated on the right-hand side of Fig. 3.11. Using Eq. (3.144) into Eq. (3.142) thus leads to: Z

Eg1

Eg

with

X

FðEÞdE ¼

subgroups k ˛ group g

fksubgroup psubgroup aX;k



fksubgroup ¼ f ssubgroup aX ;k

(3.145)

(3.146)

Likewise, using Eq. (3.144) into Eq. (3.143) results in: X

subgroups k ˛ group g

psubgroup ¼1 aX ;k

(3.147)

Chapter 3  Neutron transport calculations at the cell and assembly levels

105

Turning back to the generation of microscopic cross-sections onto an energy group [Eg, Eg1] formally defined as: R Eg1

saX ;g ¼

Eg

saX ðEÞfw ðEÞdE

R Eg1 Eg

using Eq. (3.145) results in: saX ;g ¼

fw ðEÞdE

¼

P subgroups k ˛ group g

R saX;max;g saX fw ðsaX ÞpðsaX ÞdsaX 0R saX;max;g fw ðsaX ÞpðsaX ÞdsaX 0

ssubgroup fsubgroup psubgroup aX ;k w;k aX ;k

P

subgroups k ˛ group g

with

(3.148)

fsubgroup psubgroup w;k aX ;k

 fsubgroup ¼ fw ssubgroup w;k aX ;k

(3.149)

(3.150)

For the sake of simplicity in Eqs. (3.148) and (3.150), the dependence of the weighting flux was denoted with the same function fw , irrespective of whether the dependence was expressed with respect to the energy E or to the microscopic cross-section saX . This weighting flux is obtained from Eq. (3.141) by solving the neutron transport equation given by Eq. (3.136) obtained after energy-integration on the discontiguous subgroups belonging to a given energy group [Eg, Eg1]. In the probability table approach, the group-averaged cross-sections are thus given by Eq. (3.149). This equation can be compared with Eq. (3.140) representing a more intuitive use of the subgroup crosssections, sometimes referred to as the direct approach (Knott and Yamamoto, 2010). The probability table approach requires the determination of both the subgroup subgroup subgroup cross-sections saX ;k and their associated weights paX ;k . Two main techniques are typically used for generating the subgroup cross-sections and their associated weights: the fitting method and the moment method (Ribon, 1989; Knott and Yamamoto, 2010). In the fitting method, the subgroup-averaged microscopic cross-sections are made independent of the background cross-section. Recalling the expression of the neutron flux obtained in the equivalence method given by Eq. (3.102) in the homogeneous case and by Eq. (3.126) in the heterogeneous case, and neglecting the dependence on 1/E, one obtains taking the NR approximation (Ribon, 1989; Knott and Yamamoto, 2010): fw ðEÞ z

spX þ s+ spX þ s+ b b ¼ + 0 saX ðEÞ þ ssX ðEÞ þ spX þ sb stX ðEÞ þ s+ b

(3.151)

s+ b is the background cross-section, given by Eq. (3.105) in the homogeneous case and by Eq. (3.130) in the heterogeneous case. stX ðEÞ represents the total microscopic crosssection, being the sum between the absorption cross-section saX ðEÞ, the potential scattering cross-section spX and the anomalous scattering cross-section s0sX ðEÞ. Neglecting the dependence on 1/E is justified by the fact that the variations of the neutron flux on 1/E e which correspond to the neutron flux in the absence of resonance e are negligible. Using Eq. (3.133) while noticing that (Knott and Yamamoto, 2010):

106

Modelling of Nuclear Reactor Multi-physics

R  +  subgroup k ssubgroup sb ¼ tX ;k R

stX ðEÞ

spX þ s+ b dE stX ðEÞ þ s+ b

spX þ s+ b dE + subgroup k stX ðEÞ þ sb R

+ stX ðEÞ þ s+ b  sb dE + stX ðEÞ þ sb subgroup k ¼ R 1 dE ðEÞ þ s+ s subgroup k tX b Z R 1 + dE  sb dE ðEÞ þ s+ s tX subgroup k b subgroup k ¼ R 1 dE + subgroup k stX ðEÞ þ sb Z psubgroup dE tX ;k

¼

group g

R

1 dE s ðEÞ þ s+ subgroup k tX b

where

R subgroup k psubgroup h R tX ;k group g

 s+ b

dE (3.153)

dE

one deduces that: Z

and

psubgroup tX ;k

Z stX ðEÞ subgroup k

Z

¼ spX þ sb

psubgroup tX ;k

Z

subgroup k

  dE  spX þ s+ b

group g



 +

¼ spX þ sb

Z

dE  s+ b

group g

 +

(3.154)

spX þ s+ b dE stX ðEÞ þ s+ b

 subgroup  ¼ spX þ s+ b ptX ;k



R

dE   spX þ group g + dE ¼ spX þ sb subgroup   + stX ðEÞ þ s+ s+ stX ;k b b þ sb s+ b

subgroup k

(3.152)

 +  subgroup sb ptX ;k ssubgroup tX ;k

Z dE group g

 + sb þ s+ ssubgroup b tX ;k

spX þ s+ b dE stX ðEÞ þ s+ b subgroup s+ b paX;k

Z dE group g

 + sb þ s+ ssubgroup b tX ;k

(3.155)

Chapter 3  Neutron transport calculations at the cell and assembly levels

107

The group-averaged microscopic cross-section thus becomes, using Eqs. (3.154) and (3.155): Z

  ¼ stX ;g s+ b

Eg1

Eg

stX ðEÞfw ðEÞdE Eg1

Eg

P ¼

Z

fw ðEÞdE R

subgroups k ˛ group g subgroup k

stX ðEÞfw ðEÞdE

R

P

subgroups k ˛ group g subgroup k

fw ðEÞdE

(3.156)

 +  subgroup ssubgroup sb ptX ;k tX ;k subgroup  +  sb þ s+ subgroups k ˛ group g stX ;k b P

¼

P subgroups k ˛ group g

psubgroup tX ;k  + sb þ s+ ssubgroup b tX ;k

Forcing the subgroup cross-section to be independent of the background crosssection, Eq. (3.156) becomes: P

ssubgroup psubgroup tX ;k tX ;k

  subgroups k ˛ group g ssubgroup þ s+ b tX ;k saX;g s+ b ¼ subgroup ptX ;k P subgroups k ˛ group g

(3.157)

ssubgroup þ s+ b tX ;k

For a number Nsubgroup,g of subgroups in the energy group g, 2Nsubgroup,g unknowns subgroup exist: Nsubgroup,g unknown subgroup cross-sections stX ;k and Nsubgroup,g unknown subgroup probabilities ptX ;k . 2Nsubgroup,g relationships are thus required:  One relationship is given by Eq. (3.147).  2Nsubgroup,g 1 additional relationships given by Eq. (3.157) are necessary, for 2Nsubgroup,g 1 given values of s+ b , assuming that the left-hand side of this equation is known/given for those values. In the moment method, the following moments of the microscopic cross-section are preserved (Ribon, 1989; He´bert and Coste, 2002; Knott and Yamamoto, 2010): R Eg1

Mn ¼

Eg

½stX ðEÞn dE R Eg1 dE Eg

(3.158)

where n is the order of a moment. Using Eq. (3.145), Eq. (3.158) can be rewritten as:

108

Modelling of Nuclear Reactor Multi-physics

P Mn ¼

subgroups k ˛ group g

ssubgroup tX ;k

P

subgroups k ˛ group g

X

¼

subgroups k ˛ group g

n

psubgroup tX ;k

psubgroup tX ;k

ssubgroup tX ;k

n

(3.159)

psubgroup tX ;k

where the last equality was obtained making use of Eq. (3.147). For a number Nsubgroup,g of subgroups in the energy group g, 2Nsubgroup,g unknowns subgroup exist: Nsubgroup,g unknown subgroup cross-sections stX ;k and Nsubgroup,g unknown subgroup probabilities ptX ;k . 2Nsubgroup,g relationships are thus required:  One relationship is given by Eq. (3.147).  2Nsubgroup,g 1 additional relationships given by Eq. (3.159) are necessary, for 2Nsubgroup,g 1 moments, assuming that the left-hand side of this equation is known/given for those moments.

3.3 Resolving the energy dependence Once all the group-averaged macroscopic cross-sections are determined, either directly from the evaluated nuclear data files (see Section 3.1), or after some processing of the data from the nuclear data files in order to account for the effect of resonances (see Section 3.2), the multi-group neutron transport equation can be used to determine the neutron flux. Such an equation can be expressed in steady-state conditions in its integrodifferential form as: U $ Vjg ðr; UÞ þ St;g ðr; tÞjg ðr; UÞ Z Z G G X cg ðrÞ X Ss;g 0 /g ðr; U0 /UÞjg 0 ðr; U0 Þd 2 U0 þ ðnSf Þg 0 ðrÞ jg0 ðr; U0 Þd 2 U0 ¼ 4pk g 0 ¼1 g 0 ¼1 ð4pÞ

(3.160)

ð4pÞ

or in its integral form as: Z jg ðr; UÞ ¼

N 0

2 exp4 

Z 0

3 s

St;g ðr  s0 UÞds0 5qg ðr  sU; UÞds

(3.161)

or in its characteristic form as: vjg ðr0 þ sU; UÞ þ St;g ðr0 þ sUÞjg ðr0 þ sU; UÞ ¼ qg ðr0 þ sU; UÞ vs

(3.162)

with qg ðr; UÞ Z Z G G X cg ðrÞ X Ss;g 0 /g ðr; U0 /UÞjg 0 ðr; U0 Þd 2 U0 þ ðnSf Þg 0 ðrÞ jg 0 ðr; U0 Þd 2 U0 ¼ 4pk g 0 ¼1 g 0 ¼1 ð4pÞ

ð4pÞ

(3.163)

Chapter 3  Neutron transport calculations at the cell and assembly levels

109

and Nd   X cg ðrÞ ¼ cpg ðrÞ 1  e bðrÞ þ cdi;g ðrÞe bi ðrÞ

(3.164)

i¼1

In its integro-differential, integral or characteristic forms, the multi-group transport equation represents a set of G coupled equations to be solved, with the coupling between the different groups originating from scattering and fission. After energy discretization, the multi-group transport equation can be recast in the following generic form:

1 Lj ¼ H þ F j k

(3.165)

Solving for the energy, angular, spatial and possible time dependence of the angular neutron flux is the primary objective of reactor physics calculations. As will be seen in the following, this is typically achieved by embedding various iterative schemes within each other. This section primarily focused on resolving the energy dependence of the angular flux. To start with and to simplify the discussion, the discretization with respect to space and angle of the angular flux will be disregarded. As a consequence, j represents in Eq. (3.165) a column vector of length G having for components the angular flux averaged on each energy group. Correspondingly, H and F represent the scattering and fission operators, respectively, which can thus be considered as matrices of size G  G. L is the transport operator relating the angular flux in a given group to the source terms from scattering and fission (i.e., the right-hand side of Eq. 3.165) in all groups. Because of the streaming term existing in the transport operator, L cannot be directly expressed as a matrix. It will thus be simply considered in the following that the application of the transport operator L onto the angular neutron flux j fulfils Eq. (3.165). If a large number of energy groups is used, the number of unknowns corresponding to the energy discretization of the angular neutron flux is usually too large to allow solving Eq. (3.165) directly. Iterative techniques are thus used, as explained hereafter. The scattering matrix H is first split into a down-scattering matrix Hd, a self-scattering matrix Hs and an up-scattering matrix Hu, i.e., H ¼ Hd þ Hs þ Hu

(3.166)

The down-scattering matrix is a strictly lower triangular matrix that represents scattering reactions leading to a loss of energy for the neutrons, whereas the upscattering matrix is a strictly upper triangular matrix that represents scattering reactions leading to a gain of energy for the neutrons. Up-scattering reactions are only non-negligible at thermal energies. The self-scattering matrix is a diagonal matrix that represents scattering reactions not changing the energy of the neutrons. Eq. (3.165) can be rearranged into (Sanchez, 1996):

1 ðL  Hd  Hs Þj ¼ Hu þ F j k

(3.167)

110

Modelling of Nuclear Reactor Multi-physics

The transport operator L is a diagonal matrix with respect to energy. The matrix on the left-hand side of Eq. (3.167) is thus a lower triangular matrix. Because of the structure of the matrices, Eq. (3.167) can be easily solved using a Gauss-Seidel iterative method as detailed hereafter. If one assumes that the angular flux at the iteration number p  1, denoted as j(p1), is known, the angular flux at the iteration number p, denoted as j(p), can be determined as: ðL  Hd  Hs ÞjðpÞ ¼ Hu þ

F jðp1Þ ðp1Þ 1

k

(3.168)

The estimation of j(p) is straightforward, since the matrix L  Hd  Hs is a lower ðpÞ

triangular matrix, and as a consequence, the components jg of j(p) corresponding to the energy group g can be easily determined by forward substitution, i.e., by a sweep from the first energy group g ¼ 1 (i.e., the group of neutrons having the highest energy) to the last energy group g ¼ G (i.e., the group of neutrons having the lowest energy), as explained below. When solving Eq. (3.168) for the first energy group g ¼ 1, Ml ¼ L  Hd  Hs has ðpÞ

l only one non-zero component M1;1 corresponding to j1 . Denoting the corresponding ðp1Þ

, which is known from the previous iteration p  1, right-hand side of Eq. (3.168) as S1 the balance equation for the first energy group simply reduces to: ðpÞ

ðp1Þ

l M1;1 j 1 ¼ S1

(3.169)

ðpÞ

from which j1 can be readily obtained. When solving Eq. (3.168) for the second energy l l and M2;2 group g ¼ 2, Ml ¼ L  Hd  Hs has only two non-zero components M2;1 ðpÞ

ðpÞ

corresponding to j1 and j2 , respectively. Denoting the corresponding right-hand side ðp1Þ

, which is known from the previous iteration p  1, the balance of Eq. (3.168) as S2 equation for the second energy group simply reduces to: ðpÞ

ðpÞ

ðp1Þ

l l M2;1 j1 þ M2;2 j 2 ¼ S2

ðpÞ

(3.170) ðpÞ

Since j1 has just been determined from Eq. (3.169), j2 can be readily obtained from Eq. (3.170). Repeating the same procedure for every energy group g until g ¼ G allows updating the entire energy dependence of j, which results in the knowledge of this vector at the iteration p, i.e., j(p). Such a sweep is referred to as an outer iteration. An outer iteration can be complemented by thermal iterations. The matrix Hu is a strictly upper triangular matrix having most of its non-zero elements for the columns and rows corresponding to the thermal groups, since up-scattering occurs mostly for thermal neutrons, and the up-scattered neutrons are still thermal neutrons. In a given ðpÞ

outer iteration p, the angular flux jg in a thermal group g is determined using an upscattering source Huj(p1) not yet updated with the new iterate of the angular flux for groups g 0 > g. The update of the up-scattering source Huj(p1) can nevertheless be accelerated within an outer iteration if iterations on the up-scattering sources or thermal iterations are performed within the outer iterations.

Chapter 3  Neutron transport calculations at the cell and assembly levels

111

Due to the complexity of the transport operator L, the direct inversion of this operator is impossible. Iterative methods are thus used in order to solve for the angular flux assuming that both the fission and up-scattering/down-scattering sources are known. Such iterations of the transport operator are referred to as inner iterations and can be seen as determining the spatial and angular dependence of the angular flux in a given energy group, once the fission source from all energy groups and the up-scattering/ down-scattering sources from all other energy groups are known. Inner iteration methods used for determining the angular flux will be detailed in the remaining sections of this chapter. In summary, inner iterations can be seen as resolving the spatial and angular dependence of the angular neutron flux within each group, assuming that the sources from the other groups are given. Within one outer iteration, the coupling due to downscattering is resolved, complemented by the thermal iterations aimed at updating the up-scattering source Huj (Stamm’ler and Abbate, 1983). Within an outer iteration, it should be noted that the fission source k1 Fj is nevertheless frozen. Between two successive outer iterations, the fission source thus needs to be updated. This includes updating both the k value and the fission operator Fj. In order to illustrate how the updating of those factors is carried out, Eq. (3.165) is first rewritten as: 1 Mj ¼ Fj k

(3.171)

M¼L  H

(3.172)

with Since the fission operator acts on the angular flux integrated on all directions, i.e., on the scalar flux, as can be seen in the expression of the fission operator below: Fj h

Z G G cg ðrÞ X cg ðrÞ X ðnSf Þg 0 ðrÞ jg 0 ðr; U0 Þd 2 U0 ¼ ðnSf Þg 0 ðrÞfg 0 ðrÞhFf f 4p g 0 ¼1 4p g 0 ¼1

(3.173)

ð4pÞ

The right-hand side of Eq. (3.171) can be rewritten as to: 1 1 Mj ¼ Fj ¼ Ff f k k

1 k

Ff f (Sanchez, 1996). This leads (3.174)

or j¼

1 1 M Ff f k

(3.175)

Multiplying the above expression by F and making use of Eq. (3.173), one obtains: Ff f ¼ Fj ¼

1 FM1 Ff f k

(3.176)

Defining the matrix A as: A ¼ FM1

(3.177)

112

Modelling of Nuclear Reactor Multi-physics

and the vector x as: x ¼ Ff f

(3.178)

1 x ¼ Ax k

(3.179)

Ax ¼ kx

(3.180)

Eq. (3.176) can be rewritten as:

or In Eq. (3.180), both x and k represent the solution to the problem at hand and have to be estimated. From a mathematical viewpoint, this equation has only a non-zero solution if and only if the determinant of the matrix A  kI is zero, i.e., jA  kIj ¼ 0

(3.181)

To understand the solution procedure to the problem, the spatial dependence of the neutron flux needs to be accounted for. As explained hereafter, the solutions to Eq. (3.181) are related to the spatial distribution of various modes of the neutron flux. Using G energy groups for the energy discretization and N regions for the spatial discretization, the matrix A is of size n  n, with n ¼ GN. Although expedited in this derivation, the spatial discretization of the neutron transport equation will be dealt with in detail in the following of this chapter. Eq. (3.181) has thus n, in general non-necessarily distinct, roots k1, k2, ., kn. Correspondingly, Eq. (3.180) has n, in general non-necessarily distinct, pairs of solutions [k1, x1], [k2, x2], ., [kn, xn]. xi and ki, with i ¼ [1, 2, ., n], are referred to as the eigenvectors and the eigenvalues, respectively. Not all pairs of eigenvalues and eigenvectors have a physical relevance for steady-state neutron transport equation. If the system was spatially homogeneous and if the diffusion approximation was used, it could be demonstrated that there is only one eigenmode having the same sign throughout the entire spatial domain and that this mode corresponds to the largest ki value (Lamarsh, 2002). Since the neutron flux should be a positive quantity, only this mode represents a solution to the steady-state neutron transport equation. This mode is referred to as the fundamental mode and the corresponding ki value is the effective multiplication factor of the system, i.e., keff. It has to be mentioned that the transport equation is a homogeneous equation. Consequently, any solution multiplied by an arbitrary scalar is still a solution. This explains why the sign and amplitude of the mathematical solution to the transport equation is undetermined. On the other hand, the physical solution to the steady-state transport equation cannot change sign through the spatial domain. Assuming that both the inner and thermal iterations have converged, the new estimate p of the scalar flux resulting from an outer iteration can thus be seen, using Eq. (3.176), as given by: Ff fðpÞ ¼

1 kðp1Þ

FM1 Ff fðp1Þ

(3.182)

Chapter 3  Neutron transport calculations at the cell and assembly levels

113

One notices that the result of an outer iteration can be represented by the action of the operator A, defined in Eq. (3.177), onto the vector x, defined in Eq. (3.178), as: x ðpÞ ¼

1 Ax ðp1Þ kðp1Þ

(3.183)

When the iterative scheme converges, Eq. (3.183) becomes: x ðNÞ ¼

1 Ax ðNÞ kðNÞ

(3.184)

Multiplying in a scalar manner both sides of Eq. (3.184) by x ðNÞ leads to: kðNÞ ¼

x ðNÞ $ Ax ðNÞ x ðNÞ $ x ðNÞ

(3.185)

Based on the expression given by Eq. (3.185) for the converged eigenvalue, it is possible to build-up an iterative scheme for updating the eigenvalue once the new iterate of the vector x has been determined, as follows: k ðpÞ ¼

x ðp1Þ $ Ax ðp1Þ x ðp1Þ $ x ðp1Þ

(3.186)

Because of Eq. (3.183), this expression can also be written as: k ðpÞ ¼ k ðp1Þ

x ðp1Þ $ x ðpÞ x ðp1Þ $ x ðp1Þ

(3.187)

The iterative scheme defined by Eqs. (3.183) and (3.187) is known as the power iteration method. In the following, it will be demonstrated that the power iteration method converges to the eigenvector of the iterative matrix A having the largest eigenvalue. If the system to be modelled was homogeneous and if diffusion theory was used, it could be demonstrated that the eigenvalues are all real, positive and distinct (Lamarsh, 2002). We will assume in the following that these properties of the eigenvalues are also fulfilled in the problem represented by Eq. (3.180). The pairs of eigenvalues ki and eigenvectors xi, with i ¼ [1, 2, ., n], can then be ordered as: k1 > k2 > . > kn

(3.188)

Following the derivation of Nakamura (1977), if the power iteration method is applied to an initial start vector x(0) and a given value for k(0), the p iterate can be written, using Eq. (3.183), as: x ðpÞ ¼

Ap x ð0Þ k ðp1Þ kðp2Þ .k ð0Þ

(3.189)

The initial vector can be expanded on the eigenvectors of the matrix A as: x ð0Þ ¼

X ai x i

(3.190)

i

with the expansion coefficients given by taking the scalar product between the initial vector and the corresponding eigenmode, i.e., ai ¼ x i $ x ð0Þ

(3.191)

114

Modelling of Nuclear Reactor Multi-physics

Using Eq. (3.190) and the fact that the eigenvectors of the matrix A fulfill Eq. (3.180), i.e., Ax i ¼ ki x i

Eq. (3.189) thus leads to: x

ðpÞ

¼

(3.192)

P p ai ki x i i

(3.193)

k ðp1Þ kðp2Þ .kð0Þ

which can be rearranged into: x

ðpÞ

" # X ai ki p a1 k1p x1 þ ¼ ðp1Þ ðp2Þ xi k k .k ð0Þ a k1 i>1 1

(3.194)

According to Eq. (3.188), one has: ki < 1; for i > 1 k1

(3.195)

One thus finds that: lim x ðpÞ ¼

p/N

a1 k1p ðp1Þ k kðp2Þ .k ð0Þ

x1

(3.196)

Using Eq. (3.194) into Eq. (3.187) also leads to:

 P ai ki p a1 k1p x0 þ x $ ðp1Þ ðp2Þ xi k k .k ð0Þ i>1 a1 k1 " # k ðpÞ ¼ k ðp1Þ

P ai ki p1 a1 kp1 x ðp1Þ $ ðp2Þ 1 ð0Þ x 0 þ xi k .k i>1 a1 k1

(3.197)

 P ai ki p x ðp1Þ $ x 0 þ xi i>1 a1 k1 " k ðpÞ ¼ k1 p1 # P ai ki x ðp1Þ $ x 0 þ xi i>1 a1 k1

(3.198)

ðp1Þ

or

Because of Eq. (3.195), one thus finds that: lim kðpÞ ¼ k1

p/N

(3.199)

Eqs. (3.196) and (3.199), therefore, mean that the power iteration method, implemented as given by Eqs. (3.183) and (3.187), leads to a vector aligned with the eigenvector of the iterative matrix having the largest eigenvalue and to a k value equal to the largest eigenvalue. As earlier mentioned, this eigenvector is the only mode having the same sign throughout the reactor, referred to as the fundamental mode. The corresponding eigenvalue is the effective multiplication factor of the system keff. The convergence of the power iteration method is directly related to the ratio between the higher eigenvalues and the first eigenvalue, i.e., li =l1 for i > 1. In the case of nuclear reactor calculations, the eigenvalues are usually clustered eigenvalues, i.e., quite close to each other. This decreases the convergence rate of the power iteration method. This

Chapter 3  Neutron transport calculations at the cell and assembly levels

115

method can nevertheless be accelerated using a Wielandt shift technique, as explained below. The basic idea in Wielandt  shift technique is to subtract from the original problem, as given by Eq. (3.174), Ff f kest , thus resulting in the following modified problem (Nakamura, 1977): Mj 

1 1 1 1 1 Ff f ¼ Fj  Ff f ¼ Ff f  Ff f kest k kest k kest

(3.200)

In this equation, kest is a known (i.e., given) input parameter. Making use of Eq. (3.173) results in: j ¼ F1 Ff f

(3.201)

Eq. (3.200) can thus be rewritten as:





1 1 1  MF1  x I x¼ kest k kest

(3.202)

Aw x ¼ lx

(3.203)

or more compactly as: with Aw ¼ MF1 

1 1 I ¼ A1  I kest kest

(3.204)

and 1 1 l¼  k kest

(3.205)

The power iteration method earlier presented can be applied to the modified problem defined in Eq. (3.203), thus leading to the following iterative scheme: x ðpÞ ¼

1 l

Aw x ðp1Þ

(3.206)

x ðp1Þ $ x ðpÞ x ðp1Þ $ x ðp1Þ

(3.207)

ðp1Þ

and lðpÞ ¼ lðp1Þ

Using Eq. (3.205), the p iterate of the eigenvalue of the original problem given by Eq. (3.180) is thus obtained as:  kðpÞ ¼

1 x ðp1Þ $ x ðpÞ þ lðp1Þ ðp1Þ ðp1Þ x $x kest

1

ðp1Þ ðpÞ 1 1 1 1 x $x þ  kest kðp1Þ kest x ðp1Þ $ x ðp1Þ

 ¼

(3.208)

It can be noticed that the eigenvectors xi, with i ¼ [1, 2, ., n], of the original problem given by Eq. (3.180), are also eigenvectors in the modified problem given by Eq. (3.203).

116

Modelling of Nuclear Reactor Multi-physics

The eigenvalues ki in the original problem are nevertheless replaced, in the modified problem, by eigenvalues defined as: li ¼

1 1  ki kest

(3.209)

The convergence properties of the modified iterative scheme can be examined in the same manner as for the original problem. Namely, applying the modified power iteration method onto an initial start vector x(0) and a given value for lð0Þ , the p iterate can be written, using Eq. (3.183), as: x ðpÞ ¼

l

Apw x ð0Þ ðp1Þ ðp2Þ l

.lð0Þ

(3.210)

The initial vector can be expanded on the eigenvectors of the matrix Aw, which are identical to the eigenvectors of the matrix A, as: x ð0Þ ¼

X ai x i

(3.211)

i

with the expansion coefficients given by taking the scalar product between the initial vector and the corresponding eigenmode, i.e., ai ¼ x i $ x ð0Þ

(3.212)

Using Eq. (3.211) and the fact that the eigenvectors of the matrix Aw fulfil Eq. (3.203), i.e., A w x i ¼ li x i

Eq. (3.210) thus leads to: x ðpÞ ¼

which can be rearranged into: x ðpÞ ¼

P p ai li x i i

lðp1Þ lðp2Þ .lð0Þ

" # Xaj lj p ai lpi x þ x i j a li lðp1Þ lðp2Þ .lð0Þ jsi i

(3.213)

(3.214)

(3.215)

where a given eigenvector xi was purposely singled out. According to Eq. (3.209), one also has: 1 1  lj kj kest kest  ki ¼ ¼ 1 li 1 kest  kj  ki kest

(3.216)

It can then be noticed from Eqs. (3.215) and (3.216) that the iterative algorithm will converge to the eigenvector xi fulfilling the following condition:

Chapter 3  Neutron transport calculations at the cell and assembly levels

117

  lj    0. The probability PS/i for a neutron entering the ‘fuel cell’ by the surface S according to the angular flux distribution jin ðr; U; EÞ to first interact in a given volume Vi can be estimated as: PS/i ðEÞ ¼ S0t;i ðEÞ

Z

Z

jin ðr; U; EÞjU $ Nj Vi

S

exp½  sðr0 ; r; EÞ kr  r0 k2

d 2 rd 3 r0

(3.276)

since all neutrons entering the elementary surface d2r, at a rate of jin ðr; U; EÞjU $ Njd 2 r, have a probability exp½  sðr0 ; r; EÞ of non-interaction between r and r0 . Since only neutrons entering the system are being considered, the surface integral is actually evaluated for U$N < 0. If the entry of the neutrons is spatially homogeneous on the surface S and isotropic, and if the angular neutron flux entering the surface S is given by jin , then jin depends neither on the position nor on the angle. In such a case, the number of neutrons entering the surface S is given by: Z

Z

jUjin ðr; U; EÞ $ Njd 2 rd 2 U

Jin ðEÞS ¼ S ð2pÞ;U $ N0

R

RðUÞjU $ Njd 2 rd 2 U

R

S ð2pÞ;U $ N>0

jU $ Njd 2 rd 2 U

¼

1 pS

Z

Z

RðUÞjU $ Njd 2 r d 2 U

(3.302)

S ð2pÞ;U $ N>0

where a chord is the distance between two points located on the surface S. Since one has d2r0 jU$N0 j ¼ krr0 k2d2U (see Fig. 3.14), Eq. (3.302) can be rewritten as: 1 Pesc ðEÞ ¼ 4pV

Z Z

exp½  sðr0 ; r; EÞd 3 r d 2 U

(3.303)

V ð4pÞ

The integrals in Eq. (3.303) can be evaluated in the following manner (Bell and Glasstone, 1970). For an infinitesimal surface area d2r0 and a direction U, the infinitesimal volume d3r can be replaced by: d 3 r ¼ d 2 r0 jU $ N0 jdl

(3.304)

where dl represents the infinitesimal length on the chord along the direction U. This way of representing the infinitesimal volume d3r is equivalent to partitioning the volume V into infinitesimal volumes constructed as slices of ‘tubes’ along the direction U, for which the cross-sectional surface area is d2r0 jU$N0 j. This is schematically represented in Fig. 3.18. Eq. (3.303) thus becomes: Pesc ðEÞ ¼

1 4pV

Z

Z

Z

0

S ð2pÞ;U $ N >0

0

RðUÞ

exp½  sðr0 ; r; EÞd 2 r0 jU $ N0 jd 2 Udl

(3.305)

Since the volume is assumed to be homogeneous, the optical path length defined in Eq. (3.261) can be written as: sðr0 ; r; EÞ ¼ S0t ðEÞkr  r0 k

(3.306)

134

Modelling of Nuclear Reactor Multi-physics

FIGURE 3.18 Partitioning of the volume V into infinitesimal volumes constructed as slices of ‘tubes’ along the direction U, for which the cross-sectional surface area is d2r0 jU$N0 j.

Using this expression in Eq. (3.305) and using the distances defined in Fig. 3.18 to express krr0 k along the chord gives: Pesc ðEÞ ¼

1 4pV

Z

Z

Z

0

S ð2pÞ;U $ N0 >0

1 ¼ 4pS0t ðEÞV

RðUÞ

Z

Z

  exp  S0t ðEÞ½RðUÞ  l d 2 r0 jU $ N0 jd 2 U dl





1  exp 

S0t ðEÞRðUÞ



(3.307) 2 0

0

2

d r jU $ N jd U

S ð2pÞ;U $ N0 >0

Again using the partitioning of the volume as depicted in Fig. 3.18, the average chord length given by Eq. (3.302) can be estimated, permuting the integrals, as: R¼

1 pS

Z

Z

RðUÞjU $ Njd 2 U d 2 r

(3.308)

ð4pÞ S;U $ N>0

The last integral, i.e., the integral of R(U)jU$Nj with respect to d2r, actually represents the volume of the body. Consequently, one finds that: R¼

4V S

(3.309)

This result is sometimes referred to as Cauchy theorem. Because of Eq. (3.309), the escape probability can thus be finally written as: R

Pesc ðEÞ ¼

R

S ð2pÞ;U $ N0 >0

   1  exp  S0t ðEÞRðUÞ d 2 r0 jU $ N0 jd 2 U S0t ðEÞRpS

(3.310)

The product between the macroscopic cross-section S0t ðEÞ and the average chord length R is referred to as the opacity, denoted in the following as: u ¼ S0t ðEÞR

For large opacities, one has

S0t ðEÞRðUÞ[1; cU

(3.311)

and Eq. (3.310) simplifies into:

Chapter 3  Neutron transport calculations at the cell and assembly levels R Pesc ðEÞ z

R

S ð2pÞ;U $ N0 >0

d 2 r0 jU $ N0 jd 2 U ¼

S0t ðEÞRpS

1 S0t ðEÞR

135

(3.312)

For small opacities, one has S0T ðEÞRðUÞ  1; cU and Eq. (3.310) simplifies into: R Pesc ðEÞz

R

S ð2pÞ;U $ N0 >0

(

S0t ðEÞRðUÞ 

 0 2 ) St ðEÞRðUÞ d 2 r0 jU $ N0 jd 2 U 2

S0t ðEÞRpS

(3.313)

where the following second-order development in S0t ðEÞRðUÞ was used:   exp  S0T ðEÞRðUÞ

z1 

S0T ðEÞRðUÞ



2 S0T ðEÞRðUÞ þ ; for S0T ðEÞRðUÞ  1 2

(3.314)

Eq. (3.313) further simplifies into: 1 Pesc ðEÞ z 1  Qu 2

with Q defined as: R

R

S ð2pÞ;U $ N>0

Q¼"

R

½RðUÞ2 d 2 rjU $ Njd 2 U

R

S ð2pÞ;U $ N>0

#2 ¼

R

R

S ð2pÞ;U $ N>0

RðUÞd 2 rjU $ Njd 2 U

(3.315)

½RðUÞ2 d 2 rjU $ Njd 2 U  2 pSR

(3.316)

where the last equality was obtained using Eq. (3.302). It is customary to use an approximated form of the escape probability Pesc instead of its exact expression given by Eq. (3.310). Such an approximated form is referred to as a rational approximation. Several forms exist, the most common one being the Wigner rational approximation, according to which the escape probability is given as: 1 Pesc ðEÞz 1 þ S0t ðEÞR

(3.317)

The Wigner rational approximation allows retrieving the expression given by Eq.   (3.312) for large opacities, i.e., Pesc ðEÞz1 S0t ðEÞR , and the expression given by Eq. (3.315) for small opacities, i.e., Pesc(E) z 1. It should be noted that other approximations of the escape probabilities have been proposed and are aimed at better reproducing the behaviour of the escape probabilities as a function of the opacity. The interested reader is referred to, e.g., Knott and Yamamoto, 2010 for such other approximations. In the case of an infinite lattice of fuel rods surrounded by a moderator, the probability PF/M for neutrons emitted in the fuel region to have their next collision in the moderator region can now be estimated using the escape probability given in the Wigner rational approximation by Eq. (3.317) in the following manner. In order for a neutron born in the fuel region to have its next collision in the moderator region, the neutron should first escape the fuel region and should then collide in the moderator region,

136

Modelling of Nuclear Reactor Multi-physics

either directly or after one or several traversals of fuel cells. As a result, the probability PF/M for neutrons emitted in the fuel region to have their next collision in the moderator region can be written as: PF/M ðEÞ ¼ Pesc;F ðEÞ½PS/M ðEÞ þ PM ðEÞPF ðEÞPS/M ðEÞ

(3.318)

þ PM ðEÞPF ðEÞPM ðEÞPF ðEÞPS/M ðEÞ þ .

where PS/M(E) represents the probability for neutrons entering the moderator in an homogeneous and isotropic manner to first interact in the moderator, and PF(E) represents the probability for neutrons entering the fuel region in a homogeneous and isotropic manner to escape the fuel region without interaction. PM(E) represents the probability for neutrons entering the moderator region in a homogeneous and isotropic manner to escape the moderator region without interaction and is also referred to as the Dancoff correction. The Dancoff correction can also be understood as the probability for neutrons to escape the fuel region, traverse the moderator region without interaction, and enter the fuel region again. In case the system is made of an isolated ‘fuel cell’ for which the moderator is thus infinitely thick, the Dancoff correction is equal to zero. In the case of an infinite lattice of closely packed fuel rods surrounded by a moderator, the Dancoff correction is non-zero and is smaller than unity, expressing the attenuation of neutrons while traversing the moderator. Compared to the isolated system, more neutrons interact in the fuel region. Correspondingly, this leads to an apparent decrease of the neutron escape probability from the fuel region. It should be noted that alternative definitions of the Dancoff correction exist and are typically expressed in terms of the relative attenuation of the neutrons entering a given fuel pin in the multi-cell configuration as compared to the isolated system (shadowing effect). The interested reader is referred to, e.g., Knott and Yamamoto, 2010 on this subject. In Eq. (3.318), the right-hand side represents the sum of a geometric series of ratio PM(E )PF(E ). Eq. (3.318) can thus be rewritten as: "

PF/M ðEÞ ¼ Pesc;F ðEÞ PS/M ðEÞ 

1  lim ½PM ðEÞPF ðEÞn n/N

1  PM ðEÞPF ðEÞ

#

(3.319)

Since in practical cases the product of PM(E) and PF(E) is a quantity strictly smaller than unity, PF/M is simply given by: PF/M ðEÞ ¼ Pesc;F ðEÞ

PS/M ðEÞ 1  PM ðEÞPF ðEÞ

(3.320)

Combining the reciprocity relationship given by Eq. (3.288) and expressed in the present case for the fuel region as:

Chapter 3  Neutron transport calculations at the cell and assembly levels

PS/F ðEÞ ¼

4VF S0t;F ðEÞ Pesc;F ðEÞ S

137

(3.321)

with the complementarity relationship given by Eq. (3.292) and expressed in the present case for the fuel region as: PS/F ðEÞ þ PF ðEÞ ¼ 1

(3.322)

leads to: PF ðEÞ ¼ 1  PS/F ðEÞ ¼ 1 

4VF S0t;F ðEÞ Pesc;F ðEÞ ¼ 1  S0T ;F RF Pesc;F ðEÞ S

(3.323)

where the last equality was obtained using Eq. (3.309). Likewise, the reciprocity relationship (i.e., Eq. 3.288) can be written for the moderator region as: PS/M ðEÞ ¼

4VM S0t;M ðEÞ Pesc;M ðEÞ S

(3.324)

and the complementarity relationship (i.e., Eq. 3.292) can be written for the moderator region as: PS/M ðEÞ þ PM ðEÞ ¼ 1

(3.325)

The two last equations lead to: PM ðEÞ ¼ 1  PS/M ðEÞ ¼ 1 

4VM S0t;M ðEÞ Pesc;M ðEÞ ¼ 1  S0t;M RM Pesc;M ðEÞ S

(3.326)

where the last equality was obtained using Eq. (3.309). Using Eqs. (3.323), (3.325) and (3.326) in Eq. (3.320) finally leads to: PF/M ðEÞ ¼ Pesc;F ðEÞ

S0t;M RM Pesc;M ðEÞ ih i 1  1  S0t;M RM Pesc;M ðEÞ 1  S0t;F RF Pesc;F ðEÞ h

(3.327)

Using the Wigner rational approximation (i.e., Eq. 3.317) for both Pesc,F and Pesc,M, i.e., 1 Pesc;F ðEÞz 1 þ S0t;F ðEÞRF

(3.328)

and Pesc;M ðEÞz

1 1þ

S0t;M ðEÞRM

(3.329)

respectively, leads to the following final expression for the probability PF/M for neutrons emitted in the fuel region to have their next collision in the moderator region: PF/M ðEÞ ¼

S0t;M RM

þ

S0t;M RM 0 St;F RF þ S0t;M RM S0t;F RF

(3.330)

 1 ReF  S0t;F þ 1 ReF

(3.331)

which can be recast into: PF/M ðEÞ ¼

138

Modelling of Nuclear Reactor Multi-physics

with ReF defined as: ReF ¼ RF

1 þ S0t;M RM S0t;M RM

(3.332)

Eq. (3.331) was used in the determination of the macroscopic multi-group resonance  absorption data in Section 3.2c (see Eq. 3.124 where one has Se ¼ 1 ReF ).

3.5 Two-dimensional macro-group lattice calculations 3.5.a

Introduction

One of the major drawbacks of the method of collision probabilities is that the interaction between neighbouring ‘fuel cells’ is treated in an approximative manner, since neutrons entering the ‘fuel cells’ are assumed to be emitted isotropically on the outer surface of the region and the emission is also assumed to be homogeneously distributed on the surface. Furthermore, the interaction between fuel pins having different characteristics cannot be properly taken into account. Finally, the pin cell calculations are performed on an equivalent system for which the square outer boundary was replaced by a circle. This transformation of the outer boundary does not always allow accurately representing the boundary effects in the actual geometry. A remedy is then to complement the micro-group pin cell calculations by a true twodimensional transport calculation performed at the fuel assembly level. Since the complexity of the system being considered increases, the number of energy groups used for such a calculation is usually much less than the number of energy groups used in the pin cell calculations. As a result, the corresponding calculation is referred to as a macrogroup calculation. Several techniques are available for such two-dimensional macro-group calculations, the most common ones being the method of characteristics, the discrete ordinates method and the interface current method. Such methods are explained in detail below.

3.5.b

Method of characteristics

The method of characteristics is based on the characteristic form of the steady-state Boltzmann transport equation (i.e., Eqs. (2.52) and (2.54) of Chapter 2) (Reuss, 1998). Along the characteristic [O;U), the dependence on position is entirely described by the dependence on the abscissa s, as: vj ðr0 þ sU; U; EÞ þ St ðr0 þ sU; EÞjðr0 þ sU; U; EÞ ¼ qðr0 þ sU; U; EÞ vs

with

(3.333)

Chapter 3  Neutron transport calculations at the cell and assembly levels

qðr; U; EÞ Z Z N Ss ðr; U0 /U; E 0 /EÞjðr; U0 ; E 0 Þd 2 U0 dE 0 ¼ ð4pÞ

þ

(3.334)

0

cðr; EÞ 4pk

Z 0

N

139

nðr; E 0 ÞSf ðr; E 0 Þfðr; E 0 ÞdE 0

and cðEÞ given by Eq. (3.251). Eq. (3.333) represents a first-order differential equation with respect to the abscissa s with a non-zero right-hand side. This inhomogeneous equation can be solved by the method of the variation of the constant. Noticing that the homogeneous equation: vj ðr0 þ sU; U; EÞ þ St ðr0 þ sU; EÞjðr0 þ sU; U; EÞ ¼ 0 vs

(3.335)

vj ðr0 þ sU; U; EÞ ¼  St ðr0 þ sU; EÞvs j

(3.336)

can be rewritten as:

This equation can be directly integrated along the characteristic [O;U) between the abscissa sin (representing the position of the incoming neutrons along the characteristic [O;U), i.e., the intersection of the outer boundary of the region being considered and the characteristic [O; U), as can be seen in Fig. 3.19) and s, i.e., 2

6 jðr0 þ sU; U; EÞ ¼ C exp4 

3

Z

s

sin

7 St ðr0 þ s0 U; EÞds0 5

(3.337)

The general solution to the inhomogeneous Eq. (3.333) can thus be written in the most general case as: 2

6 jðr0 þ sU; U; EÞ ¼ CðsÞexp4 

Z

3

s

sin

7 St ðr0 þ s0 U; EÞds0 5

(3.338)

where the function C(s) has to be determined. Using Eq. (3.338) into Eq. (3.333), one finds that the function C(s) fulfils the following equation:

FIGURE 3.19 Representation of the position of neutrons along the characteristic [O; U).

140

Modelling of Nuclear Reactor Multi-physics 2 vC 6 ðsÞ ¼ qðr0 þ sU; U; EÞexp4 vs

3

Z

s

sin

7 St ðr0 þ s0 U; EÞds0 5

Integrating this equation between sin and s leads to: Z CðsÞ ¼

2

s sin

6 qðr0 þ s0 U; U; EÞexp4

Z

s0

(3.339)

3

7 St ðr0 þ s00 U; EÞds00 5ds0 þ Cðsin Þ

sin

(3.340)

Combining Eqs. (3.338) and (3.340) finally gives: 2

6 jðr0 þ sU; U; EÞ ¼ Cðsin Þexp4 

þ

8

< m ¼ Um $ x h ¼ Um $ y > : x ¼ Um $ z

(3.360)

where x, y and z represent the unitary vectors on the three axes. In three-dimensional geometries, it is desirable to have invariance with respect to 90 deg rotations about the axes, as depicted in Fig. 3.22. This way of selecting the directions is referred to as the level-symmetric configuration. Any direction cosine in Eq. (3.360) can thus be expressed in such a case as a function of the direction cosine with respect to the x direction only for instance as: 8 > < m ¼ Um $ x ¼ mi h ¼ Um $ y ¼ mj > : x ¼ Um $ z ¼ mk

(3.361)

In the following, the direction cosines are numbered in increasing order, i.e., mi < miþ1 ; ci

(3.362)

Chapter 3  Neutron transport calculations at the cell and assembly levels

145

FIGURE 3.22 Example of a symmetrical S4 angular discretization.

Because of the symmetry of the problem, i.e., mi ¼  mN þ1i with 1 i N =2

(3.363)

it is sufficient to only study one octant of the unit sphere. The octant for which 1 i N/2, 1 j N/2, 1 k N/2 is chosen and one has: 0 < mi < 1; ci

(3.364)

The strict inequalities are required in the above expression to guarantee that no discretized direction is contained in the [O, x), [O, y) plane, [O, y), [O, z) plane, and [O, x), [O, z) plane or along the x-, y- and z-directions. Since a planar source gives rise to a diverging angular flux in any direction of the emitting plane, not having any direction aligned with this plane allows having an angular flux remaining finite. On the unit sphere, one further has: m2i þ m2j þ m2k ¼ 1

(3.365)

For two neighbouring directions, a change from mi to miþ1 means that simultaneously one should have a change from mj to mj1 or a change from mk to mk1 . Using for instance the first condition and Eq. (3.365), one finds that: m2i þ m2j þ m2k ¼ m2iþ1 þ m2j1 þ m2k

(3.366)

146

Modelling of Nuclear Reactor Multi-physics

for 1 i < N/2, 1 < j N/2 and 1 k N/2. Eq. (3.366) can be rearranged into: m2iþ1  m2i ¼ m2j  m2j1

(3.367)

Since Eq. (3.367) is valid for any combination of i and j such that 1 i < N/2 and 1 < j N/2, one concludes that: m2iþ1  m2i ¼ C

(3.368)

where C is a constant. Using Eq. (3.368) recursively, one finds that: m2i ¼ m21 þ ði  1ÞC

(3.369)

Using Eq. (3.369) into Eq. (3.365) at, for instance, i ¼ 1, j ¼ 1 and k ¼ N/2, which represents the closest direction to the upward direction [O,z) on the unit sphere, results in: C¼

 2  1  3m21 N 2

(3.370)

In the previous derivation, m1 is thus a free parameter. Nevertheless, since mi represents a direction cosine, a closer examination of the possible range of variation of m1 is necessary. Eqs. (3.369) and (3.370) reveal that the sign of 1  3m21 determines whether the m2i are in increasing or decreasing order of magnitude. The convention adopted in Eq. (3.362) implies that one should have: m21 < 1=3

(3.371)

Combining this result with Eq. (3.364) thus gives the following range of variation for the free parameter m1 : 0 < m1 < 1

.pffiffiffi 3

(3.372)

Once the directions Um have been determined, weights wm associated to each of these directions need to be calculated, so that integrals with respect to solid angles can be replaced by a quadrature formula as: Z

f ðUÞd 2 Uz4p

X wm f ðUm Þ

(3.373)

m

ð4pÞ

with the following normalization condition

X wm ¼ 1

(3.374)

m

Eq. (3.373) is usually written:

Z ð4pÞ

X f ðUÞd 2 Uz wm fm m

(3.375)

Chapter 3  Neutron transport calculations at the cell and assembly levels

147

with fm ¼ 4pf ðUm Þ

(3.376)

The weights and directions are usually required to fulfil moment conditions, according to which the quadrature formula should be exact if the function f is a polynomial of low order in the direction cosine, i.e., Z

2p

1 1

Z X mn dm ¼ 4p wm mnm ; 2p

1

1

m

Z X hn dm ¼ 4p wm hnm ; 2p

1

1

m

X xn dm ¼ 4p wm xnm

(3.377)

m

Eq. (3.377) results in:

X X X wm mnm ¼ 0 ; wm hnm ¼ 0 ; wm xnm ¼ 0 for n odd m

m

(3.378)

m

and in: X wm mnm ¼ m

X X 1 1 1 ; ; for n even wm hnm ¼ wm xnm ¼ nþ1 n þ 1 n þ 1 m m

(3.379)

Eq. (3.378) is referred to as the odd-moment conditions, whereas Eq. (3.379) is referred to as the even-moment conditions. As for the choice of the directions Um, the weight should fulfil the following condition in case of symmetry of the problem: wi ¼ wN þ1i with 1 i N=2

(3.380)

Combining Eq. (3.363) with Eq. (3.380), one notices that the odd-moment conditions are automatically satisfied in case of symmetry in the choice of the directions Um and weights wm. Such directions and weights will also be chosen so that as many evenmoment conditions can be satisfied. Several level-symmetric quadrature sets exist (see, e.g., He´bert, 2010). Once the directions Um and weights wm have been chosen, the transport equation in the SN approximation can then be derived. One starts with the time-independent neutron transport equation written in its integro-differential form (i.e., Eq. (2.35) of Chapter 2), which is recalled hereafter: U $ Vjðr; U; EÞ þ St ðr; EÞjðr; U; EÞ Z Z N Ss ðr; U0 /U; E 0 /EÞjðr; U0 ; E 0 Þd 2 U0 dE 0 ¼ ð4pÞ

þ

0

cðr; EÞ 4pk

Z 0

N

(3.381)

nðr; E 0 ÞSf ðr; E 0 Þfðr; E 0 ÞdE 0

where cðr; EÞ is given by Eq. (3.251). Using the expansion of the scattering cross-section on the Legendre polynomials (i.e., Eq. 3.24), one obtains:

148

Modelling of Nuclear Reactor Multi-physics

FIGURE 3.23 Principles and conventions used in the spatial discretization.

U $ Vjðr; U; EÞ þ St ðr; EÞjðr; U; EÞ Z Z N N X 2l þ 1 Ssl ðr; E 0 /EÞPl ðU $ U0 Þjðr; U0 ; E 0 Þd 2 U0 dE 0 ¼ 4p 0 l¼0 ð4pÞ

þ

cðr; EÞ 4pk

Z

N

0

(3.382)

nðr; E 0 ÞSf ðr; E 0 Þfðr; E 0 ÞdE 0

On a given direction Um, this equation becomes: Um $ Vjðr; Um ; EÞ þ St ðr; EÞjðr; Um ; EÞ Z N N X X ð2l þ 1Þ wm0 Pl ðUm $ Um0 Þ Ssl ðr; E 0 /EÞjðr; Um0; E 0 ÞdE 0 ¼ l¼0

þ

cðr; EÞ 4pk

Z 0

0

m0 N

(3.383)

nðr; E 0 ÞSf ðr; E 0 Þfðr; E 0 ÞdE 0

where the quadrature formula (i.e., Eq. 3.373) was used for expressing the integral with respect to the directions U in the scattering term. Based on the definition of the scalar neutron flux (see Eq. (2.5) of Chapter 2), the quadrature formula can also be used to express the scalar neutron flux as: X fðr; EÞ ¼ 4p wm0 jðr; Um0; EÞ

(3.384)

m0

Using this expression in Eq. (3.383) finally leads to the following set of equations in the SN method:

Chapter 3  Neutron transport calculations at the cell and assembly levels

Um $ Vjðr; Um ; EÞ þ St ðr; EÞjðr; Um ; EÞ Z N N X X ð2l þ 1Þ wm0 Pl ðUm $ Um0 Þ Ssl ðr; E 0 /EÞjðr; Um0 ; E 0 ÞdE 0 ¼ 0

m0

l¼0

Z N cðr; EÞ X þ wm0 nðr; E 0 ÞSf ðr; E 0 Þjðr; Um0 ; E 0 ÞdE 0 k 0 0 m

149

(3.385)

Using the convention given by Eq. (3.376), Eq. (3.385) can be rewritten as: Um $ Vjm ðr; EÞ þ St ðr; EÞjm ðr; EÞ ¼

N X

ð2l þ 1Þ

m0

l¼0

þ

Z X wm0 Pl ðUm $ Um0 Þ

cðr; EÞ X wm0 k m0

0

Z

N 0

N

Ssl ðr; E 0 /EÞjm0 ðr; E 0 ÞdE 0

(3.386)

nðr; E 0 ÞSf ðr; E 0 Þjm0 ðr; E 0 ÞdE 0

or in a multi-group formalism: Um $ Vjm;g ðrÞ þ St;g ðrÞjm;g ðrÞ ¼

N X

ð2l þ 1Þ

m0

l¼0

þ

G X X wm0 Pl ðUm $ Um0 Þ Ssl;g 0 /g ðrÞjm0 ;g 0 ðrÞ g 0 ¼1

(3.387)

G X

cg ðrÞ X wm0 ðnSf Þg 0 ðrÞjm0 ;g 0 ðrÞ k m0 g 0 ¼1

Concerning the treatment of the spatial dependence of the quantities appearing in Eq. (3.387), a Cartesian geometry is considered in the following for the sake of simplicity, as represented in Fig. 3.23. The system is assumed to be infinite in the axial direction. Integrating Eq. (3.387) on each of the homogeneous regions, denoted by the set of integers (i, j) in the x- and y-directions, respectively, leads to, using Gauss divergence theorem: X ‫¼א‬x;y

¼

mm;‫א‬

N X l¼0

þ

with

J‫א‬m;g;ði;jÞ  J‫א‬m;g;ði;jÞ1

ð2l þ 1Þ

D‫א‬

þ St;g;ði;jÞ jm;g;ði;jÞ

G X X wm0 Pl ðUm $ Um0 Þ Ssl;g 0 /g;ði;jÞ jm0 ;g 0 ;ði;jÞ m0

g 0 ¼1

(3.388)

G X cg;ði;jÞ X wm0 ðnSf Þg 0 ;ði;jÞjm0 ;g 0 ;ði;jÞ k m0 g 0 ¼1

(

mm;x ¼ Um $ x mm;y ¼ Um $ y

(3.389)

and where ‫ א‬represents the direction (x or y). D‫ א‬is the width of the region in the ‫א‬-direction, and the subscript “-1” is a generic representation of the region adjacent to the

150

Modelling of Nuclear Reactor Multi-physics

region (i, j) along the ‫א‬-direction for decreasing ‫ א‬values, as illustrated in Fig. 3.23. In this equation, the different averaged quantities are calculated as: jm;g;i ¼

1 Dx $ Dy

Z

Z

Dx=2 Dx=2

dx

Dy=2 Dy=2

jm;g ðx; yÞdxdy

8 > > > Z Dy=2 > > 1 > x > > ¼ jm;g ðx; yÞdy J > m;g;i Dy < Dy=2 Z Dx=2 > > 1 y > > J ¼ j ðx; yÞdx > m;g;i > Dx Dx=2 m;g > > > :

(3.390)

(3.391)

Eq. (3.390) defines the region-averaged angular neutron flux in region (i, j), whereas Eq. (3.391) defines the surface-averaged angular neutron flux relative to the region (i, j) across the surface perpendicular to the ‫א‬-direction. Since the system is infinite in the direction transverse to the one represented in Fig. 3.23, the balance equation given by Eq. (3.388) actually corresponds to a so-called finite volume discretization scheme. It should be noted that the balance equation contains both volume-averaged quantities (as given by Eq. 3.390) and surface-averaged quantities (as given by Eq. 3.391). To obtain a set of algebraic equations containing the same type of unknown quantities, the volume-averaged angular fluxes are replaced by linear approximations of the surface-averaged angular fluxes, as: jm;g;ði;jÞ ¼ aJ‫א‬m;g;ði;jÞ1 þ ð1  aÞJ‫א‬m;g;ði;jÞ for ‫˛א‬fx; yg

(3.392)

0 a 1

(3.393)

with If a ¼ 1, one has the step difference scheme, whereas for a ¼ 1/2, one has the diamond difference scheme. Usually, the diamond difference scheme is used in SN computer codes. Eq. (3.388) together with Eq. (3.392) allows determining the volume-integrated angular fluxes and surface-integrated angular fluxes once the incoming angular flux at the boundary of the computational domain is known. As a consequence, the determination of the angular neutron flux has to be carried out in the direction of neutron travel. This means that when m‫ > א‬0, jm;g;ði;jÞ and J‫א‬m;g;ði;jÞ are determined assuming that ‫א‬ Jm;g;ði;jÞ1 is known. Using Eq. (3.392), this reads as: J‫א‬m;g;ði;jÞ ¼

jm;g;ði;jÞ  aJ‫א‬m;g;ði;jÞ1 ð1  aÞ

(3.394)

which can then be used in Eq. (3.388) to determine jm;g;ði;jÞ . Once jm;g;ði;jÞ has been determined, Eq. (3.394) is again utilized to calculate J‫א‬m;g;ði;jÞ , which is then used as boundary condition in the next adjacent region, i.e., the region generically denoted as (i, j) þ 1.

Chapter 3  Neutron transport calculations at the cell and assembly levels

151

When m‫ < א‬0, jm;g;ði;jÞ and J‫א‬m;g;ði;jÞ1 are determined assuming that J‫א‬m;g;ði;jÞ is known. Using Eq. (3.392), this reads as: J‫א‬m;g;ði;jÞ1 ¼

jm;g;ði;jÞ  ð1  aÞJ‫א‬m;g;ði;jÞ a

(3.395)

which can then be used in Eq. (3.388) to determine jm;g;ði;jÞ . Once jm;g;ði;jÞ has been determined, Eq. (3.395) is again utilized to calculate J‫א‬m;g;ði;jÞ1 , which is then used as boundary condition in the next adjacent region, i.e., the region generically denoted as (i, j)1. Concerning the inner iteration strategy for SN methods, the following procedure is adopted. Starting from an assumed distribution of the scattering and fission sources from all groups, the angular neutron fluxes in group g is estimated in the direction of neutron travel from some chosen outer boundaries of the system being modelled to the remaining boundaries of the system. Using boundary conditions on the latter representative of an infinite system allows converting the outward angular fluxes into inward angular fluxes and thereafter estimating the angular neutron fluxes in the direction of neutron travel back to the original boundaries. Such a procedure defines an SN sweep. From this sweep, the self-scattering source in group g is updated and the above process is repeated using boundary conditions representative of an infinite system on the chosen original boundaries. One of the problems of the discrete ordinates methods is the so-called ray effect, according to which only a few discrete directions of the neutrons are permitted in the method. In the case of a purely absorbing medium containing a point-like isotropic source, a spatial discretization of the medium will lead to non-zero angular flux values only in the regions traversed by the discrete directions, as illustrated in Fig. 3.24 in a two-

FIGURE 3.24 Schematic representation of the ray effect in two dimensions for a purely absorbing medium containing a point-like source, with the regions traversed by the discretized directions highlighted in grey. Derived from Reuss, P., 2008. Neutron Physics. EDP Sciences, Les Ulis, France.

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Modelling of Nuclear Reactor Multi-physics

dimensional system, whereas non-zero angular fluxes should be found in all regions surrounding the source. Such a non-physical behaviour can only be alleviated by increasing the number of discrete directions, i.e., by increasing the order of the SN method. Finally, it should be mentioned that an inadequate choice of the mesh spacing D‫א‬ (i.e., a too large mesh spacing) can lead to negative values of the surface-averaged angular fluxes. Such an unphysical behaviour has to be corrected (‘negative flux fixup’) by setting the offending surface-averaged angular flux values to zero and correcting accordingly the freshly calculated region-averaged angular fluxes to preserve neutron balance.

3.5.d

Interface current method

One of the major drawbacks of the method of collision probabilities is that the size of the computational domain cannot be too large, explaining why this method is usually applied to an infinite lattice of a single ‘fuel cell’. Such calculations must be carried out for all possible types of ‘fuel cells’. Nevertheless, the possible interdependence of different types of neighbouring ‘fuel cells’ cannot be taken into account, due to the computing resources such a method would require when applied to an array of different ‘fuel cells’. In the interface current method, method also called the transmission probabilities method, arrays of ‘fuel cells’, which might be different, are modelled on two different meshes: a coarse mesh used to compute the neutron currents between neighbouring cells (assuming that one knows the neutron fluxes within each ‘fuel cell’), and a fine mesh to compute the neutron fluxes (assuming that one knows the neutron currents at the boundary of each ‘fuel cell’) (Knott and Yamamoto, 2010; Stamm’ler and Abbate, 1983; Stacey, 2001). The main difference between the interface current method and the method of collision probabilities is that only within ‘fuel cells’ collision probabilities need to be determined in the interface current method, whereas collision probabilities between all possible ‘fuel cells’ would be required in the method of collision probabilities. The interface current method can be derived from Eq. (3.342), in which only the isotropic macroscopic scattering cross-section is retained. Some degree of anisotropy is introduced by virtue of the transport correction carried out at the level 0, i.e., the multigroup isotropic macroscopic cross-section Ss0;g 0 /g is replaced by S0s0;g 0 /g using Eq. (3.248). In other words, the total and isotropic macroscopic cross-sections are modified using the transport correction, as Eqs. (3.252) and (3.255) indicate. Using the notations introduced in Fig. 3.25, Eq. (3.342) can be rewritten as: Z

jðr; U; EÞ ¼ jðrin ; U; EÞexp½  sðr; rin ; EÞ þ

0

sin

qðr  s0 U; U; EÞexp½  sðr; r  s0 U; EÞds0

(3.396)

Chapter 3  Neutron transport calculations at the cell and assembly levels

153

FIGURE 3.25 Notations used for deriving the interface current method.

with the optical path length defined in Eq. (3.261) and the emission density given by Eq. (3.256). Integrating this equation on all solid angles leads to: Z

jðr; U; EÞd 2 U

ð4pÞ

Z

jðrin ; U; EÞexp½  sðr; rin ; EÞd 2 U

¼ ð4pÞ

Z Z

0

þ ð4pÞ

(3.397)

qðr  s0 U; U; EÞexp½  sðr; r  s0 U; EÞd 2 Uds0

sin

At the interface along the direction [A,U) upstream from point A defined by the position: rin ¼ r  sin U

(3.398)

The infinitesimal surface element spanned by a change of the direction U is given by: 2 2 d 2 r0 jU $ N0 j ¼ sin d U ¼ kr  rin k2 d 2 U

(3.399)

r0 ¼ r  s0 U

(3.400)

At the point defined by: the infinitesimal volume element spanned by a change of the direction U and of the abscissa s0 is given by: 2

d 3 r0 ¼ s02 ds0 d 2 U ¼ kr  r0 k ds0 d 2 U

(3.401)

Using Eq. (3.399), the first integral on the right-hand side of Eq. (3.397) can be replaced by a surface integral, whereas using Eq. (3.401), the second double integral on the right-hand side of Eq. (3.397) can be replaced by a volume integral. Recognizing on the left-hand side of Eq. (3.397) the scalar neutron flux, Eq. (3.397) then reads as: fðr; EÞ Z exp½  sðr; r0 ; EÞ 2 0 ¼ jin ðr0 ; U; EÞ d r jU $ N0 j 2 kr  r0 k S

Z þ V

qðr0 ; U; EÞ

0

exp½  sðr; r ; EÞ 2

kr  r0 k

(3.402)

d 3 r0

Partitioning the volume V into sub-volumes Vi, such that V ¼ WVi , and partitioning i the surface S into sub-surfaces Sa, such that S ¼ WSa , Eq. (3.402) can be rewritten as: a

154

Modelling of Nuclear Reactor Multi-physics

fðr; EÞ XZ expð  sðr; r0 ; EÞÞ ¼ jin ðr0 ; U; EÞ jU $ N0 jd 2 r0 2 kr  r0 k a Sa

þ

XZ i

qðr0 ; U; EÞ

0

exp½  sðr; r ; EÞ 2

kr  r0 k

Vi

(3.403)

d 3 r0

Multiplying this equation by the total (transport-corrected) macroscopic crosssection and integrating on a volume Vj lead to: Z

Vj

¼

S0t ðr; EÞfðr; EÞd 3 r XZ a

þ

jin ðr0 ; U; EÞ

i

S0t ðr; EÞ

exp½  sðr; r0 ; EÞ 2

kr  r0 k

Vj

Sa

XZ

Z

qðr0 ; U; EÞ

Vi

Z

exp½  sðr; r0 ; EÞ

S0t ðr; EÞ

2

kr  r0 k

Vj

d 2 r0 jU $ N0 jd 3 r

(3.404)

d 3 r0 d 3 r

Since the emission density was assumed to be isotropic, Eq. (3.404) can be rewritten as:

Z Vj

¼

S0t ðr; EÞfðr; EÞd 3 r XZ a

þ

jin ðr0 ; U; EÞ

Sa

XZ i

Vi

Qðr0 ; EÞ

Z

Z

S0t ðr; EÞ

exp½  sðr; r0 ; EÞ 2

kr  r0 k

Vj

S0t ðr; EÞ

exp½  sðr; r0 ; EÞ 2

4pkr  r0 k

Vj

d 2 r0 jU $ N0 jd 3 r

(3.405)

d 3 r0 d 3 r

with the solid angle-integrated emission density Q defined in Eq. (3.264). Eq. (3.405) can be written in a more compact form as: S0t;j ðEÞfj ðEÞVj ¼

X X Sa Jin;a ðEÞPa/j ðEÞ þ Vi Qi ðEÞPi/j ðEÞ a

(3.406)

i

where the volume-averaged quantities are calculated, in order to preserve the reaction rates, as: fj ðEÞ ¼

Qi ðEÞ ¼

1 Vj 1 Vi

Z

fðr; EÞd 3 r

(3.407)

Qðr; EÞd 3 r

(3.408)

Vj

Z Vi

Chapter 3  Neutron transport calculations at the cell and assembly levels

S0t;j ðEÞ ¼

1 Vj fj ðEÞ

Z

S0t ðr; EÞfðr; EÞd 3 r

155

(3.409)

Vj

and the area-averaged quantities are defined as: Jin;a ðEÞ ¼

1 Sa

Z

Z

jðr; U; EÞjU $ Njd 2 r d 2 U

(3.410)

Sa ð2pÞ;U $ N 0 lead to: Z

Z

jout ðr; U; EÞjU $ N0 jd 2 rd 2 U

S ð2pÞ;U $ N>0

Z

Z

jðrin ; U; EÞexp½  sðr; rin ; EÞjU $ N0 jd 2 r d 2 U

¼ S ð2pÞ;U $ N>0

Z

Z

þ S ð2pÞ;U $ N>0

jU $ N0 j

Z

0 sin

qðr  s0 U; U; EÞexp½  sðr; r  s0 U; EÞd 2 r d 2 U ds0

(3.415)

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Modelling of Nuclear Reactor Multi-physics

At the interface along the direction [A, U) upstream from point A defined by the position: rin ¼ r  sin U

(3.416)

the infinitesimal surface element spanned by a change of the direction U is given by: 2 2 d 2 r0 jU $ N0 j ¼ sin d U ¼ kr  rin k2 d 2 U

(3.417)

r0 ¼ r  s0 U

(3.418)

At the point defined by: the infinitesimal volume element spanned by a change of the direction U and of the abscissa s0 is given by: 2

d 3 r0 ¼ s02 ds0 d 2 U ¼ kr  r0 k ds0 d 2 U

(3.419)

Using Eq. (3.417), the integral with respect to solid angles in the first term on the right-hand side of Eq. (3.415) can be replaced by a surface integral, whereas using Eq. (3.419), the integrals with respect to solid angles and abscissa in the second term on the right-hand side of Eq. (3.415) can be replaced by a volume integral. Eq. (3.415) then reads as: Z

Z

jout ðr; U; EÞjU $ Njd 2 r d 2 U

S ð2pÞ;U $ N>0

Z

Z exp½  sðr; r0 ; EÞ jU $ Njd 2 r jin ðr0 ; U; EÞ jU $ N0 jd 2 r0 2 kr  r0 k

¼ S

(3.420)

S

Z

Z exp½  sðr; r0 ; EÞ 3 0 d r jU $ Njd 2 r qðr0 ; U; EÞ 2 kr  r0 k

þ

V

S

Partitioning the volume V into sub-volumes Vi, such that V ¼ WVi , and partitioning i the surface S into sub-surfaces Sa, such that S ¼ WSa , Eq. (3.420) can be rewritten as: Z

a

Z

2

2

jout ðr; U; EÞjU $ Njd r d U Sa ð2pÞ;U $ N>0

¼

XZ b

þ

Sa

XZ i

as:

jU $ Njd 2 r

Sa

Z

jin ðr0 ; U; EÞ

Sb

jU $ Njd 2 r

Z

Vi

qðr0 ; U; EÞ

exp½  sðr; r0 ; EÞ 2

kr  r0 k

exp½  sðr; r0 ; EÞ 2

kr  r0 k

jU $ N0 jd 2 r0

(3.421)

d 3 r0

Since the emission density was assumed to be isotropic, Eq. (3.421) can be rewritten

Chapter 3  Neutron transport calculations at the cell and assembly levels Z

Z

jout ðr; U; EÞjU $ Njd 2 r d 2 U

Sa ð2pÞ;U $ N>0

¼

XZ b

157

jU $ Njd 2 r

Sa

Z

jin ðr0 ; U; EÞ

exp½  sðr; r0 ; EÞ 2

kr  r0 k

Sb

jU $ N0 jd 2 r0

(3.422)

Z XZ exp½  sðr; r0 ; EÞ 3 0 þ d r jU $ Njd 2 r Qðr0 ; EÞ 2 4pkr  r0 k i Vi

Sa

with the solid angle-integrated emission density Q defined in Eq. (3.264). Eq. (3.422) can be written in a more compact form as: Sa Jout;a ðEÞ ¼

X X Sb Jin;b ðEÞPb/a ðEÞ þ Vi Qi ðEÞPi/a ðEÞ

(3.423)

i

b

with Qi defined in Eq. (3.408) and Jin,a defined in Eq. (3.410). The averaged outgoing neutron current on the surface area Sa is given by: Jout;a ðEÞ ¼

1 Sa

Pi/a is given by: Pi/a ðEÞ ¼

and Pb/a is given by: Pb/a ðEÞ ¼

1 Vi Qi ðEÞ

Z

1 Sb Jin;b ðEÞ

Z

Z

jout ðr; U; EÞjU $ Njd 2 r d 2 U

(3.424)

Sa ð2pÞ;U $ N>0

Z

jU $ Njd 2 r

Qðr0 ; EÞ

Vi

Sa

jU $ Njd 2 r

Sa

Z

Z

jin ðr0 ; U; EÞ

exp½  sðr; r0 ; EÞ 4pkr  r0 k2

exp½  sðr; r0 ; EÞ kr  r0 k2

Sb

d 3 r0

jU $ N0 jd 2 r0

(3.425)

(3.426)

Since exp½  sðr; r0 ; EÞ represents the probability of non-interaction between r and r0 , Pi/a represents the probability for a neutron emitted in the volume Vi in an isotropic manner and having an emission density given by Q(r0 , E) to escape without interaction through the surface Sa. Likewise, Pb/a represents the probability for a neutron entering through the surface Sb according to jin to escape without interaction through the surface Sa. If one assumes that the emission density Q(r0 , E) is homogeneous within each region, itself assumed to be homogeneous, Eq. (3.425) reduces to: Pi/a ðEÞ ¼

1 Vi

Z

Sa

jU $ Njd 2 r

Z

exp½  sðr; r0 ; EÞ

Vi

4pkr  r0 k2

d 3 r0

(3.427)

If one considers that the system to be studied contains I ¼ 1, ., N sub-systems having volumes VI and surfaces SI, Eqs. (3.406) and (3.423) can be respectively written on each of the sub-systems as: S0t;j ðEÞfj ðEÞVj ¼

X a˛SI

Sa Jin;a ðEÞPa/j ðEÞ þ

X Vi Qi ðEÞPi/j ðEÞ for j ˛VI i˛VI

(3.428)

158

Modelling of Nuclear Reactor Multi-physics

Sa Jout;a ðEÞ ¼

X

Sb Jin;b ðEÞPb/a ðEÞ þ

X Vi Qi ðEÞPi/a ðEÞ for a ˛ SI

(3.429)

i˛VI

b˛SI

or in a multi-group formalism as: S0t;j;g fj;g Vj ¼

X X Sa Jin;a;g Pa/j;g þ Vi Qi;g Pi/j;g for j ˛VI

a˛SI

Sa Jout;a;g ¼

X b˛SI

(3.430)

i˛VI

Sb Jin;b;g Pb/a;g þ

X Vi Qi;g Pi/a;g for a ˛ SI

(3.431)

i˛VI

In the equations above, i ˛ VI and j ˛ VI denote a region i and a region j, respectively, both belonging to the sub-system I. a˛SI and b ˛ SI denote a sub-surface a and a subsurface b of the outer surface, respectively, of the sub-system I. If JIout represents a vector having for components each of the currents Jout,a on all sub-surfaces belonging the boundary of the subsystem I, if JIin represents a vector having for components each of the currents Jin,a on all sub-surfaces belonging the boundary of the subsystem I, and if JIsource represents a vector having for components the contribution to the current Jout,a due to volumetric sources inside the subsystem I, the set of equations (3.431) can be recast in the following vector equation: JIout ¼ RI JIin þ JIsource

(3.432)

where RI is referred to as the response matrix for the subsystem I. Such a vector equation can be written for any subsystem I of the computational domain, and thus one can write: Jout ¼ RJin þ Jsource

(3.433)

for the entire system being modelled. In this equation, R is referred to as the response matrix for the entire system. Since the outgoing current from a given sub-system is equal to the incoming current to its neighbouring subsystem, a topographical relationship exists between Jin and Jout which can be generically written as: Jin ¼ PJout

(3.434)

where P is a matrix defined for the entire system being considered. Usually, the interface current method is applied to a single fuel assembly or to a set of fuel assemblies in an infinite lattice. The symmetry of the lattice allows relating the incoming and outgoing currents on the same or on different faces belonging to the outer boundary of the lattice, and the matrix P also contains such information. Combining Eqs. (3.433) and (3.434) leads to: Jin ¼ PRJin þ PJsource

(3.435)

Eq. (3.435) represents the global problem, i.e., the problem defined for the entire computational domain. Likewise, Eq. (3.430) can also be recast into the following vector equation: fI ¼ SI JIin þ fIsource

(3.436)

Chapter 3  Neutron transport calculations at the cell and assembly levels

159

This equation represents the local problem, i.e., the problem defined for the subsystem I. This means that two different computational grids are used: a coarse grid to resolve the neutron currents and the associated coupling between the coarse regions, and a fine grid to resolve the neutron flux within each region, as illustrated in Fig. 3.26. Solving on the coarse grid requires the knowledge of the emission densities and thus the neutron fluxes within each coarse region, whereas solving on the fine grid requires the knowledge of the neutron currents between the coarse regions. Iterating between the two grids is thus necessary. It should be mentioned that solving the fine mesh problem is carried out assuming that the neutron currents between the coarse mesh regions are given. Each of the coarse mesh regions is thus considered independently from each other while solving the fine mesh problems. This also explains why the interface current method does not require the computation of collision probabilities between regions belonging to different coarse mesh problems.

FIGURE 3.26 Illustration of coarse and fine mesh computational grids that could be used in the interface current method.

The probabilities Pa/j, Pi/j, Pb/a and Pi/a need to be calculated prior to the application of the interface current method in the true geometry of the system, i.e., in two dimensions. The evaluation of such probabilities is thus more involved than for the method of collision probabilities, where the system is modelled as a one-dimensional system. Nevertheless, only probabilities within each subsystem need to be evaluated in the interface current method. Usually, the global problem is solved after a spatial homogenization of each of the coarse mesh regions, and the local problems are solved by replacing the outer square-shaped boundary of the elementary ‘fuel cells’ by a circle (as

160

Modelling of Nuclear Reactor Multi-physics

was done in the method of collision probabilities e see Section 3.e). The spatial homogenization in the global problem and the equivalently defined sets of onedimensional local problems simplify the computation of the various probabilities. In addition, approximations concerning the incoming angular neutron flux are also required to evaluate the surface probabilities. The simplest approximation is to assume that the entry of neutrons is isotropic and homogeneous on each surface. More sophisticated representations of the entry of neutrons are possible, although they complicate the application of the method. How the probabilities are calculated is beyond the scope of these lecture notes. The interested reader is referred to the existing literature on this subject. Once the different probabilities have been evaluated, the interface current method proceeds as follows. From a known distribution of the emission densities for the entire computational domain, the global problem is solved, i.e., the neutron currents between each subsystem are computed using Eq. (3.435). Once the currents have been determined, the local problem is solved for each of the subsystems, i.e., the scalar neutron fluxes within each sub-system are determined using Eq. (3.436). The calculated neutron fluxes within each subsystem are then used to calculate an updated distribution of the emission densities for the next inner iteration.

3.5.e

Acceleration methods

The SN method and the method of characteristics usually converge very slowly, especially for diffusive problems typically encountered in LWRs. The determination of the angular fluxes in those methods can only be achieved if the sources from all groups are known. These also include the self-scattering source in the group being resolved within the given inner iteration. Because of the necessity to iterate on the self-scattering source, the angular fluxes are evaluated assuming a known self-scattering source, which is then updated using the newly calculated angular fluxes, and the process is then repeated until convergence is achieved. This can be formally written as: s ðp1Þ Lg jðpÞ þ Sg g ¼ Hg j g

(3.437)

where Lg represents the transport operator in group g, Hsg represents the self-scattering operator in group g and Sg contains the up-scattering/down-scattering sources from all other groups and fission sources from all groups. The superscripts represent the iteration numbers. The iterative procedure as highlighted above usually converges very slowly for diffusing media and needs to be accelerated. Several acceleration techniques exist. The most common ones are the coarse-mesh re-balancing acceleration technique and the diffusion synthetic acceleration. These two techniques are briefly explained in the following.

Chapter 3  Neutron transport calculations at the cell and assembly levels

Acceleration by coarse-mesh re-balancing

161

ðpÞ

In the coarse-mesh re-balancing acceleration technique, the flux iterate jg at the iteration number p is forced to fulfil the neutron balance equations with the scattering ð pÞ source from the same iteration, i.e., Hsg jg . This ‘re-balancing’ is usually carried out on a coarser spatial mesh than the one originally used in the inner iteration procedure explained above (Sanchez, 1996; Knott and Yamamoto, 2010). Denoting by Vi and Sik one of these coarse mesh volumes and its corresponding surfaces, respectively, the neutron balance equation can be stated on this volume in the following manner. Starting with the neutron transport equation written in its integro-differential form (i.e., Eq. (2.35) of Chapter 2), one has in a multi-group formalism:   e g ðr; UÞ þ S0 ðrÞj e g ðr; UÞ V $ Uj t;g

¼

G G c ðrÞ X 1 X eg 0 ðrÞ þ g eg 0 ðrÞ S0s0;g 0 /g ðrÞf ðnSf Þg 0 ðrÞf 4p g 0 ¼1 4pk g 0 ¼1

(3.438)

where scattering was assumed to be isotropic and the cross-sections were corresponde g ðr; UÞ ¼ ingly transport-corrected. Furthermore, the following identity U $ Vj i h e g ðr; UÞ was used. Integrating this equation with respect to position and angle V $ Uj gives:

Z Z

  e g ðr; UÞ d 3 r d 2 U þ V $ Uj

Vi ð4pÞ

¼

eg ðrÞd 3 r S0t;g ðrÞf

Vi

G Z X g 0 ¼1

Z

eg 0 ðrÞd 3 r S0s0;g 0 /g ðrÞf

Vi

G Z 1X eg 0 ðrÞd 3 r þ cg ðrÞðnSf Þg 0 ðrÞf k g 0 ¼1

(3.439)

Vi

The tilde thus represents the solution to the problem integrated on each of the coarse mesh regions (as opposed to the solution of the SN or characteristic balance equations that need to be fulfilled on each of the fine mesh regions). Using Gauss divergence theorem and defining the partial neutron currents as: out 1 Jeik ¼ Sik

and in 1 Jeik ¼ Sik

Z

Z

e g ðr; UÞjU $ Njd 2 r d 2 U j

(3.440)

e g ðr; UÞjU $ Njd 2 r d 2 U j

(3.441)

Sik ð2pÞ;U $ N >0

Z

Z

Sik ð2pÞ;U $ N > > > > < > > > > > :

þS Lg jg ¼ Hsg jðp1Þ g 1 s h i

s e jg  jðp1Þ e L eg H εðpÞ z H g g g

(3.452)

ðpÞ jðpÞ g ¼ jg  ε

One of the challenges in the application of the DSA method is to have compatible spatial discretization schemes between the transport and diffusion solvers, so that the

164

Modelling of Nuclear Reactor Multi-physics

application of the diffusion-based correction term ε(p) to the transport solution is of value and the DSA algorithm is stable (He´bert, 2010).

3.6 Criticality spectrum calculations 3.6.a

Introduction

Both the one-dimensional micro-group pin cell calculations and the two-dimensional macro-group lattice calculations are performed assuming an infinite lattice of fuel rods, fuel assemblies, respectively. When loaded in a nuclear core at nominal operating conditions, the system is of finite size and is furthermore critical. Although the previous calculations are performed by adjusting the effective multiplication factor, so that the time-independent neutron transport equation can be studied, such calculations do not account for the fact that the spectrum of the system is greatly dependent on neutron streaming. The previous calculations have thus to be complemented with a calculation allowing reproducing the neutron streaming and spectrum representative of the actual critical and finite system. The most common methods to perform such criticality spectrum calculations are the homogeneous B1 method, the homogeneous P1 method and the fundamental mode method (Choi et al., 2016). The adjective ‘homogeneous’ refers to the fact that the spatial dependence of the cross-sections is not accounted for, i.e., the system is first spatially homogenized before the methods are applied.

3.6.b

Properties of integral operators in infinite and homogeneous media

As will be seen in Section 3.6.c, the criticality spectrum methods are based on the observation that the overall spatial dependence of the angular flux in a critical infinite homogeneous system is given by functions of the type exp(ib$r). The angular neutron flux can then be factorized as: jðr; U; EÞ ¼ 4ðb; U; EÞexpðib $ rÞ

(3.453)

Such an observation relies on the properties of integral operators, which are also linear operators (and vice-versa) (Reuss, 2008). Integral operators, generically denoted A in the following, are operators transforming any function f into a function g as: Z

gðrÞ ¼ Af ðrÞ ¼

kA ðr; r0 Þf ðr0 Þd 3 r0

(3.454)

where kA is the so-called kernel of the operator A. In the case of an infinite and homogeneous medium, there is invariance by translation, and the kernel of the operator only depends on rr0 , i.e., kA ðr; r0 ÞhkA ðr  r0 Þ

(3.455)

Chapter 3  Neutron transport calculations at the cell and assembly levels

165

When applied to functions of the type exp(ib$r), the operator A gives: Z A expðib $ rÞ ¼ kA ðr  r0 Þexpðib $ r0 Þd 3 r0 Z ¼ expðib $ rÞ kA ðRÞexpðib $ RÞd 3 R ¼ kbA ðbÞexpðib $ rÞ

(3.456)

It thus results that functions of the type exp(ib$r) are eigenfunctions of the integral operator A. As can be seen in Eq. (3.456), any function exp(ib$r) is transformed into kbA ðbÞexpð ib $ rÞ, where kbA is the Fourier-transform of the kernel kA of the integral operator A.

3.6.c

Integral operators in critical systems

In an infinite and homogeneous system, the neutron cycle, i.e., the neutron distribution S due to fissions induced by the neutron distribution S0 at the immediately preceding neutron generation can be seen as the application of an integral operator F to the distribution S0 as: Sðr; U; EÞ ¼ FS0 ðr; U; EÞ

(3.457)

F is often referred to as the fission matrix. In case of criticality, one should further have: Sðr; U; EÞ ¼ S0 ðr; U; EÞ

(3.458)

A critical distribution of the fission sources should thus correspond to an eigenfunction of the operator F with an eigenvalue being equal to unity. Based on the properties of the integral operators in homogeneous and infinite media, this means that in case of a critical system, the spatial distribution of the fission sources should be given by functions of the type exp(ib$r). Furthermore, the neutron transport equation in its integro-differential form (see Eq. (2.34) of Chapter 2) can be rewritten as: Ljðr; U; EÞ ¼ Sðr; U; EÞ

(3.459)

with Ljðr; U; EÞ ¼U U; EÞ þ St ðr; EÞjðr; U; EÞ Z $ZVjðr; N  Ss ðr; U0 /U; E 0 /EÞjðr; U0 ; E 0 Þd 2 U0 dE 0 ð4pÞ

(3.460)

0

and Sðr; U; EÞ ¼

cðr; EÞ 4pk

Z 0

N

nðr; E 0 ÞSf ðr; E 0 Þfðr; E 0 ÞdE 0

(3.461)

with cðr; EÞ defined in Eq. (3.251). In addition, the angular neutron flux can be seen as the application of an integral operator G or Green function on the fission source density

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Modelling of Nuclear Reactor Multi-physics

S as:

Z

Gðr; U; Ejr0 ; U0 ; E0 ÞSðr0 ; U0 ; E0 Þd 3 r0 d 2 U0 dE0

jðr; U; EÞ ¼

(3.462)

where the Green function is itself solution of the following equation: LGðr; U; Ejr0 ; U0 ; E0 Þ ¼ dðr  r0 ÞdðU  U0 ÞdðE  E0 Þ

(3.463)

This can be demonstrated by multiplying Eq. (3.463) by S(r0, U0, E0) and integrating the resulting equation with respect to r0, U0 and E0. One would then notice that !G(r, U, Ejr0, U0, E0)S(r0, U0, E0)d3r0d2U0dE0 is solution of Eq. (3.459) and should correspond to the angular neutron flux jðr; U; EÞ. The Green function gives the angular neutron flux at the spatial point r, in the direction U and at the energy E induced by one neutron emitted by fission at the spatial point r0, in the direction U0 and at the energy E0. Based on the previous discussion concerning critical, homogeneous and infinite systems, one knows that the critical distribution of the fission sources should be given by functions of the type exp(ib$r). Consequently, the spatial dependence of the angular neutron flux is also given by exp(ib$r), since such functions are eigenfunctions of the integral operator G. In other words, one can write that in case of a critical, homogeneous and infinite system, one has: Sðr; U; EÞ ¼ sðb; U; EÞexpðib $ rÞ

(3.464)

jðr; U; EÞ ¼ 4ðb; U; EÞexpðib $ rÞ

(3.465)

and thus: Using Eq. (3.465) in the integro-differential form of the steady-state Boltzmann transport equation written hereafter for a homogeneous system: U $ Vjðr; U; EÞ þ St ðEÞjðr; U; EÞ ¼ Z Z N Ss ðU0 /U; E 0 /EÞjðr; U0 ; E 0 Þd 2 U0 dE 0 ð4pÞ

0

cðEÞ þ 4pk

(3.466)

Z Z ð4pÞ

N 0

nðE 0 ÞSf ðE 0 Þjðr; U0 ; E 0 Þd 2 U0 dE 0

one finds that: U $ ib4ðb; U; EÞexpðib $ rÞ þ St ðEÞ4ðb; U; EÞexpðib $ rÞ Z Z N Ss ðU0 /U; E 0 /EÞ4ðb; U0 ; E 0 Þexpðib $ rÞd 2 U0 dE 0 ¼ ð4pÞ

þ

0

cðEÞ 4pk

(3.467)

Z Z ð4pÞ

N 0

nðE 0 ÞSf ðE 0 Þ4ðb; U0 ; E 0 Þexpðib $ rÞd 2 U0 dE 0

Chapter 3  Neutron transport calculations at the cell and assembly levels

167

This equation can be simplified into: U $ ib4ðb; U; EÞ þ St ðEÞ4ðb; U; EÞ Z Z N Ss ðU0 /U; E 0 /EÞ4ðb; U0 ; E 0 Þd 2 U0 dE 0 ¼ ð4pÞ

þ

3.6.d

0

cðEÞ 4pk

Z Z ð4pÞ

0

(3.468) N

nðE 0 ÞSf ðE 0 Þ4ðb; U0 ; E 0 Þd 2 U0 dE 0

Homogeneous B1 method

Since Eq. (3.468) does not depend on space, the geometry for solving the transport equation does not play any role. A one-dimensional infinite slab can thus be chosen for simplicity. In such a case, the angular dependence of the neutron flux can be described by the cosine m between the neutron direction U and the direction characterizing the one-dimensional dependence of the system. It is then possible to write: j ðr; U; EÞ h j ðx; m; EÞ ¼ 4 ðb; m; EÞexpðibxÞ

(3.469)

where the dependence on b now only occurs via its modulus b. The  sign in the previous equation comes from the fact that the angular neutron flux should be a real quantity, and thus the angular neutron flux should be a linear combination of jþ ðx; m; EÞ and j ðx; m; EÞ, i.e., jðr; U; EÞhjðx; m; EÞ ¼ aþ jþ ðx; m; EÞ þ a j ðx; m; EÞ ¼ aþ 4þ ðb; m; EÞexpðibxÞ þ a 4 ðb; m; EÞexpðibxÞ

(3.470)

where the aþ and a are chosen so that a real solution is obtained for j. It is interesting to notice that the spatial dependence of the angular neutron flux, as given in Eq. (3.470), corresponds to sine and cosine functions, which are the spatial eigenmodes of the diffusion equation for finite homogeneous systems (Lamarsh, 2002). The B1 method can thus be considered as forcing the spatial dependence of the neutron flux at the core level to be representative of a finite homogeneous system. Superimposing a spatial dependence at the core level is equivalent to creating a ‘bending’ (or buckling) of the flux at the core level. This allows adjusting neutron leakage in order to achieve criticality (Reuss, 2008). A positive buckling is equivalent to creating a local outflow of neutrons (i.e., leakage of neutrons) and would correspond to the case when the infinite multiplication factor is above unity. On the other hand, a negative buckling is equivalent to creating a local inflow of neutrons and would correspond to the case when the infinite multiplication factor is below unity. Injecting the expression given by Eq. (3.469) into Eq. (3.468) gives:

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Modelling of Nuclear Reactor Multi-physics

ibm4 ðb; m; EÞ þ St ðEÞ4 ðb; m; EÞ Z 2p Z 1 Z N Ss ðm0 ; E 0 /EÞ4 ðb; m0 ; E 0 Þdw0 dm0 dE 0 ¼ 0

þ

1

cðEÞ 2k

Z

0

1 1

Z 0

N

(3.471)

nðE 0 ÞSf ðE 0 Þ4 ðb; m0 ; E 0 Þdm0 dE 0

It should be noted that the differential scattering cross-section depends on the cosine of the deviation angle between the incoming direction U0 and outgoing direction U as: m0 ¼ U0 $ U ¼ m0 m þ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1  m02 1  m2 cosðw0  wÞ

(3.472)

In the above equations, w and w0 represent the azimuthal angles associated to the directions U and U0 , respectively. In the homogeneous BN method, the scattering cross-section is expanded on Legendre polynomials according to Eq. (3.24) and the expansion is truncated at the order N, i.e., the scattering cross-section is approximated by: Ss ðU0 / U; E 0 / EÞz

N X 2l þ 1

4p

l¼0

Ssl ðE 0 / EÞPl ðm0 Þ

(3.473)

Injecting Eq. (3.473) into Eq. (3.471) gives: ½St ðEÞ  ibm4 ðb; m; EÞ Z Z Z N X 2l þ 1 2p 1 N Ssl ðE 0 /EÞPl ðm0 Þ4 ðb; m0 ; E 0 Þdw0 dm0 dE 0 ¼ 4p 0 1 0 l¼0 Z Z cðEÞ 1 N þ nðE 0 ÞSf ðE 0 Þ4 ðb; m0 ; E 0 Þdm0 dE 0 2k 1 0

(3.474)

In the B1 method, only the two first moments of the scattering cross-section are retained, and Eq. (3.474) reduces to: ½St ðEÞ  ibm4 ðb; m; EÞ Z 2p Z 1 Z N 1 ½Ss0 ðE 0 /EÞ þ 3Ss1 ðE 0 /EÞm0 4 ðb; m0 ; E 0 Þdw0 dm0 dE 0 ¼ 4p 0 1 0 Z Z cðEÞ 1 N þ nðE 0 ÞSf ðE 0 Þ4 ðb; m0 ; E 0 Þdm0 dE 0 2k 1 0

(3.475)

where Eqs. (3.28) and (3.33) were used. The spectrum of the neutron scalar flux is then defined as Z

J ðb; EÞ ¼ 2p

1 1

4 ðb; m; EÞdm

(3.476)

and the spectrum of the neutron net current is defined as Z P ðb; EÞ ¼ 2p

1 1

m4 ðb; m; EÞdm

(3.477)

Chapter 3  Neutron transport calculations at the cell and assembly levels

169

Using Eq. (3.472), Eq. (3.475) can be rewritten as: ½St ðEÞ  ibm4 ðb; m; EÞ Z N 1 ½Ss0 ðE 0 /EÞJ ðb; E 0 Þ þ 3Ss1 ðE 0 /EÞmP ðb; E 0 ÞdE 0 ¼ 4p 0 Z cðEÞ N þ nðE 0 ÞSf ðE 0 ÞJ ðb; E 0 ÞdE 0 4pk 0

(3.478)

where the first integral on the right-hand side of Eq. (3.478) was obtained noticing that the integration with respect to w0 of the second term on the right-hand side of Eq. (3.472) was zero. Dividing Eq. (3.478) by St ðEÞ  ibm finally gives: 4 ðb; m; EÞ ¼

1 4p½St ðEÞ  ibm þ

N

Z 0

3m 4p½ST ðEÞ  ibm

Ss0 ðE 0 /EÞ þ

Z 0

N

cðEÞ nðE 0 ÞSf ðE 0 Þ J ðb; E 0 ÞdE 0 k

(3.479)

Ss1 ðE 0 /EÞP ðb; E 0 ÞdE 0

This equation can be solved by expanding the function 4 on Legendre polynomials in a similar manner as for the scattering cross-section, i.e., 4 ðb; m; EÞ ¼

N X 2l þ 1

4p

l¼0

4l; ðb; EÞPl ðmÞ

(3.480)

where the expansion coefficients are expressed as: Z

4l; ðb; EÞ ¼ 2p

1 1

4 ðb; m; EÞPl ðmÞdm

(3.481)

One notices that 40; ðb; EÞ ¼ J ðb; EÞ and 41; ðb; EÞ ¼ P ðb; EÞ. Multiplying Eq. (3.479) by the Legendre polynomial Pl ðmÞ and integrating with respect to m gives, using Eq. (3.481): 4l; ðb; EÞ ¼ Al0; ðEÞ

N

Z 0

þ 3Al1; ðEÞ

Ss0 ðE 0 /EÞ þ

Z 0

N

cðEÞ nðE 0 ÞSf ðE 0 Þ J ðb; E 0 ÞdE 0 k

(3.482)

Ss1 ðE 0 /EÞP ðb; E 0 ÞdE 0

with Alj; ðEÞ ¼

1 2

Z

1

Pl ðmÞPj ðmÞ dm S 1 t ðEÞ  ibm

(3.483)

It can be noticed from Eq. (3.482) that only the knowledge of J and P are required to find any moment l of the angular neutron flux. It will be seen in Chapter 4 that the PN method, which is also using a Legendre expansion of the angular neutron flux, requires the truncation of the Legendre expansion of the angular neutron flux to the order N. In the BN method on the other hand, only the expansion of the scattering cross-section on

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Modelling of Nuclear Reactor Multi-physics

the Legendre polynomials is truncated at the order N. In the B1 approximation, the two equations that need to be solved are thus: J ðb; EÞ

N

Z

¼ A00; ðEÞ

Ss0 ðE 0 /EÞ þ

0

Z þ 3A01; ðEÞ

N

cðEÞ nðE 0 ÞSf ðE 0 Þ J ðb; E 0 ÞdE 0 k

(3.484)

Ss1 ðE 0 /EÞP ðb; E 0 ÞdE 0

0

and P ðb; EÞ ¼ A10; ðEÞ

N

Z

Ss0 ðE 0 /EÞ þ

0

Z

þ 3A11; ðEÞ

N

0

cðEÞ nðE 0 ÞSf ðE 0 Þ J ðb; E 0 ÞdE 0 k

(3.485)

Ss1 ðE 0 /EÞP ðb; E 0 ÞdE 0

Evaluating the Alj; coefficients from Eq. (3.483) and using Eqs. (3.28) and (3.33) gives (Stamm’ler and Abbate, 1983): 8 > > > > > > > > >
0 b St ðEÞ 2 3 k A00; ðEÞ ¼ 1 þ > > 1 6 St ðEÞ7 > 2 2 7 > ln6 > k 5 for k ¼ b > 0 4 > 2k > 1 > > : St ðEÞ

(3.486)

i A10; ðEÞ ¼ A01; ðEÞ ¼ H  ½1  A00 ðEÞSt ðEÞ b

(3.487)

A11; ðEÞ ¼

St ðEÞ ½1  A00 ðEÞSt ðEÞ b2

(3.488)

Eqs. (3.484) and (3.485) represent the two equations to be solved in the B1 homogeneous method. A somewhat simpler set of equations can be obtained in the following manner (Stamm’ler and Abbate, 1983). Multiplying Eq. (3.484) by St (E) and Eq. (3.485) by ib and then adding lead to, using Eqs. (3.487) and (3.488): St ðEÞJ ðb; EÞ  ibP ðb; EÞ Z N cðEÞ Ss0 ðE 0 /EÞ þ ¼ nðE 0 ÞSf ðE 0 Þ J ðb; E 0 ÞdE 0 k 0

(3.489)

Eq. (3.484) by ib and Eq. (3.485) by A00; ðEÞb2   Likewise, multiplying 1 A00; ðEÞSt ðEÞ , and then adding lead to, using Eqs. (3.487) and (3.488):



Chapter 3  Neutron transport calculations at the cell and assembly levels

 ibJ ðb; EÞ þ Z ¼3

N 0

A00; ðEÞb2 P ðb; EÞ 1  A00; ðEÞSt ðEÞ

0

0

Ss1 ðE /EÞP ðb; E ÞdE

171

(3.490)

0

or ibJ ðb; EÞ þ 3aðb; EÞSt ðEÞP ðb; EÞ Z N Ss1 ðE 0 /EÞP ðb; E 0 ÞdE 0 ¼3

(3.491)



2 1 b A00; ðEÞSt ðEÞ 3 St ðEÞ 1  A00; ðEÞSt ðEÞ

(3.492)

0

with aðb; EÞ ¼

Eqs. (3.489) and (3.491) are equivalent to Eqs. (3.484) and (3.485) and represent the system of equations to be solved in the B1 homogeneous method. Another advantage of such a method is that the diffusion coefficient that is mostly needed in full core diffusion calculations can be easily estimated, as explained hereafter. Since the scalar neutron flux and the net neutron current are defined as: Z

fðx; EÞ ¼ 2p

and

Z Jðx; EÞ ¼ 2p

1

j ðx; m; EÞdm

(3.493)

mj ðx; m; EÞdm

(3.494)

1

1 1

respectively, one notices, using Eqs. (3.469), (3.476) and (3.477), that: fðx; EÞ ¼ J ðb; EÞexpðibxÞ

(3.495)

Jðx; EÞ ¼ P ðb; EÞexpðibxÞ

(3.496)

and Fick’s law, which is used in diffusion theory and which will be further explained in Chapter 4, assumes a proportionality between the neutron current density vector and the scalar neutron flux. In a homogeneous medium, this reads as: Jðr; EÞ ¼  DðEÞVfðr; EÞ

(3.497)

where D is the so-called diffusion coefficient. In the present one-dimensional problem, Eq. (3.497) reduces to: Jðx; EÞ ¼  DðEÞ

vf ðx; EÞ vx

(3.498)

Using Eqs. (3.495) and (3.496), the diffusion coefficient can be simply expressed from Eq. (3.498) as:

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Modelling of Nuclear Reactor Multi-physics

D ðb; EÞ ¼

R1 iP ðb; EÞ i 1 m4 ðb; m; EÞdm ¼ R1 bJ ðb; EÞ b 4 ðb; m; EÞdm

(3.499)

1

where the last equality was derived using Eqs. (3.476) and (3.477). Eq. (3.499) thus allows estimating the diffusion coefficient. In multi-group theory, Eqs. (3.489), (3.491) and (3.499) read, respectively, as: St;g J;g ðbÞ  ibP;g ðbÞ ¼

G h i X cg Ss0;g 0 /g þ ðnSf Þg 0 J;g 0 ðbÞ k g 0 ¼1

ibJ;g ðbÞ þ 3ag ðbÞSt;g P;g ðbÞ ¼ 3

G X

Ss1;g 0 /g P;g 0 ðbÞ

(3.500)

(3.501)

g 0 ¼1

and D;g ðbÞ ¼

iP;g ðbÞ bJ;g ðbÞ

(3.502)

Injecting Eq. (3.502) into Eq. (3.501) also allows writing the diffusion coefficient as: 1þ3 D;g ðbÞ ¼

G P g 0 ¼1

Ss1;g 0 /g D;g 0 ðbÞ

J;g 0 ðbÞ J;g ðbÞ

3ag ðbÞSt;g

(3.503)

Eqs. (3.500) and (3.501) represent the system of coupled equations that need to be solved. The main point of the B1 method is that the variable b in the above equations has to be adjusted so that the leakage rate is representative of a critical system. In other words, b has to be adjusted in order to obtain k ¼ 1. Denoting J and P column vectors containing on their g-th row J;g and P,g, respectively, and following the derivation of Stamm’ler and Abbate (1983), Eq. (3.501) can be recast in the following matrix equations: AðbÞ½iP ðbÞ ¼  bJ ðbÞ

(3.504)

where the element of the matrix A on the g-th row and g 0 -th column is equal to: Agg 0 ðbÞ ¼ 3ag ðbÞSt;g dgg 0  3Ss1;g 0 /g

(3.505)

Eq. (3.504) can be formally inverted to give: ibP ðbÞ ¼ b2 DðbÞJ ðbÞ

(3.506)

DðbÞ ¼ A1 ðbÞ

(3.507)

with Eq. (3.506) can also be written as: ibP;g ðbÞ ¼ b2

G X g 0 ¼1

Dgg 0 ðbÞJ;g 0 ðbÞ

(3.508)

Chapter 3  Neutron transport calculations at the cell and assembly levels

173

Injecting this expression into Eq. (3.500) leads to the following matrix equation: 1 BðbÞJ ðbÞ ¼ FJ ðbÞ k

(3.509)

where the element of the matrix B on the g-th row and g 0 -th column is equal to: Bgg 0 ðbÞ ¼ St;g dgg 0  Ss0;g 0 /g þ b2 Dgg 0 ðbÞ

(3.510)

and where the element of the matrix F on the g-th row and g 0 -th column is equal to: Fgg 0 ¼ cg ðnSf Þg 0

(3.511) 2

It can be noticed from the above that Eq. (3.509) only depends on b , and not  ib. This means that the spectrum of the scalar neutron flux is simply given by J(b) ¼ Jþ(b) ¼ J(b). Nevertheless, the spectrum of the neutron net current density vector still depends on  ib, as can be seen from Eq. (3.506), and thus two distinct solutions Pþ(b) and P(b) exist. Regarding the practical implementation of the B1 method, one notices that Eq. (3.509) represents a k-eigenvalue equation, which can be solved iteratively using, e.g., the power iteration method explained in Section 3.3. Starting from a given value for b, the spectrum of the angular neutron flux is iteratively determined together with its corresponding eigenvalue. If the eigenvalue differs from unity, b is adjusted and the corresponding spectrum and eigenvalue are determined. This process is repeated until the spectrum/ eigenvalue corresponds to a critical system. Once the angular neutron spectrum is determined, the spectrum of the net neutron current can be calculated, followed by the determination of the corresponding diffusion coefficient.

3.6.e

Homogeneous P1 method

The spherical harmonics method or PN method will be thoroughly derived in Section 4.1a of Chapter 4. In the following, the resulting balance equations are directly used without any derivation. In the P1 homogeneous method, a spatially homogeneous system is considered, and a one-dimensional infinite slab can be chosen for simplicity. In such conditions, the balance equations in the P1 approximation and in multi-group theory read as: G G X cg X v Ss0;g 0 /g fg 0 ðxÞ þ ðnSf Þg 0 fg 0 ðxÞ Jg ðxÞ þ St;g fg ðxÞ ¼ k g 0 ¼1 vx g 0 ¼1

(3.512)

G X 1 v Ss1;g 0 /g Jg 0 ðxÞ fg ðxÞ þ St;g Jg ðxÞ ¼ 3 vx g 0 ¼1

(3.513)

and

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Modelling of Nuclear Reactor Multi-physics

Using Eqs. (3.495) and (3.496) written in multi-group theory, i.e., fg ðxÞ ¼ J;g ðbÞexpðibxÞ

(3.514)

Jg ðxÞ ¼ P;g ðbÞexpðibxÞ

(3.515)

and Eqs. (3.512) and (3.513) reduce to: St;g J;g ðbÞ  ibP;g ðbÞ ¼

G h i X cg Ss0;g 0 /g þ ðnSf Þg 0 J;g 0 ðbÞ k g 0 ¼1

(3.516)

and ibJ;g ðbÞ þ 3St;g P;g ðbÞ ¼ 3

G X

Ss1;g 0 /g P;g 0 ðbÞ

(3.517)

g 0 ¼1

Comparing the two above equations to Eqs. (3.500) and (3.501) corresponding to the homogenous B1 method, one notices that the homogeneous P1 method can be applied by simply setting ag ðbÞ ¼ 1 in the homogenous B1 method. The solution procedure and the estimation of the diffusion coefficient in the homogeneous P1 method are thus identical to the ones in the homogenous B1 method, with the exception that ag ðbÞ ¼ 1 in all equations (i.e., in Eqs. 3.500e3.511).

3.6.f

Fundamental mode method

In the fundamental mode method, the diffusion approximation, which is explained in detail in Section 4.1.b of Chapter 4, is used. In the following, the resulting balance equations are directly used without any derivation. If the medium is homogeneous and considering a one-dimensional infinite slab for simplicity, the balance equations in multi-group theory are given as: Dg V2 fg ðxÞ þ

G G X cg X ðnSf Þg 0 fg 0 ðxÞ  St;g fg ðxÞ þ Ss0;g 0 /g fg 0 ðxÞ ¼ 0 k g 0 ¼1 g 0 ¼1

(3.518)

with Dg ¼

1 3½St;g  Ss1;g 

(3.519)

Using Eq. (3.495) written in multi-group theory, i.e., fg ðxÞ ¼ J;g ðbÞexpðibxÞ

(3.520)

Eq. (3.518) simply becomes: Dg b2 Jg ðbÞ þ

G G X cg X ðnSf Þg 0 Jg ðbÞ  St;g Jg ðbÞ þ Ss0;g 0 /g Jg 0 ðbÞ ¼ 0 k g 0 ¼1 g 0 ¼1

(3.521)

In the above equation, the  subscripts in J;g were dropped, since both the þ

Chapter 3  Neutron transport calculations at the cell and assembly levels

175

solution and - solution lead to the same balance equations. Both solutions are thus identical. The set of balance equations represented by Eq. (3.521) can be recast into the following matrix equation: 1 BðbÞJðbÞ ¼ FJðbÞ k

(3.522)

where the matrix B is a diagonal matrix, for which the element on the g-th row and g 0 -th column is equal to:   Bgg 0 ðbÞ ¼ St;g þ Dg b2 dgg 0  Ss0;g 0 /g

(3.523)

and where the element of the matrix F on the g-th row and g 0 -th column is equal to: Fgg 0 ¼ cg ðnSf Þg 0

(3.524)

Since Eq. (3.522) represents a k-eigenvalue equation, it can be solved iteratively using, e.g., the power iteration method explained in Section 3.3. Starting from a given value for b, the spectrum of the scalar neutron flux is iteratively determined together with its corresponding eigenvalue. If the eigenvalue differs from unity, b is adjusted and the corresponding spectrum and eigenvalue are determined. This process is repeated until the spectrum/eigenvalue corresponds to a critical system. It should be mentioned that contrary to the homogeneous B1 and P1 methods, the diffusion coefficient is directly given by Eq. (3.519) and is not estimated using the calculated critical spectrum. Although the three methods are routinely used for spectrum corrections in lattice calculations, it should be emphasized that they rely on two major approximations: the medium is assumed to be homogeneous, and the parameter b is assumed to be energy-independent. The validity of these approximations might be questioned (see, e.g., Smith, 2016).

3.7 Cross-section homogenization and condensation Three computational tasks were presented in the foregoing sections: one-dimensional micro-group pin cell calculations (Section 3.4), two-dimensional macro-group lattice calculations (Section 3.5) and criticality spectrum calculations (Section 3.6). Following the methodology used in most lattice physics codes, the so-called micro-flux is determined in the first computational stage; the macro-flux is determined in the second computational stage; and the criticality spectrum is determined in the last computational stage. In this section, the interdependence between the different stages is presented (Demazie`re, 2013; Stamm’ler and Abbate, 1983). The one-dimensional micro-group pin cell calculations are meant at determining the fine structure of the distribution of the neutron flux with respect to energy and space in an infinite lattice of fuel pins. The calculations are performed for each type of fuel pins, treated individually with white boundary conditions, and the energy group structure corresponds to the one of the cross-section library. The flux thus obtained is the micro-

176

Modelling of Nuclear Reactor Multi-physics

flux fi,g in the micro-region i and the micro-energy group g. It should be noted that the flux corresponds to the flux of an infinite medium. The energy group structure of the cross-section library is nevertheless constructed so that the calculated micro-flux is relatively independent of the flux spectrum. As a consequence, the errors introduced by not using the critical flux spectrum are minimized. The two-dimensional macro-group lattice calculations are meant at determining a coarser structure of the distribution of the neutron flux with respect to energy and space in an infinite lattice of fuel assemblies. The flux thus obtained is the macro-flux fI,G in the macro-region I and the macro-energy group G. This flux still corresponds to the flux of an infinite medium. Prior to performing the lattice calculations, the macroscopic cross-sections must be transformed from a fine representation in space and energy (i.e., a micro-region and micro-group representation) to a coarser representation in space and energy (i.e., a macro-region and a macro-group representation). The transformation with respect to space is referred to as cross-section homogenization, whereas the transformation with respect to energy is referred to as cross-section condensation. The homogenization and condensation are performed while preserving the reaction rates. This means that the condensation is carried out according to: P

Sa;i;G ¼

g˛G

Sa;i;g fi;g P fi;g

(3.525)

g˛G

and that the homogenization is carried out according to: P

Sa;i;G fi;G Vi Sa;I;G ¼ i˛I P fi;G Vi

(3.526)

i˛I

with fi;G ¼

X fi;g

(3.527)

g˛G

and where Vi is the volume of the micro-region i. It can be verified from the above equations that the condensation and homogenization preserve the reaction rates, i.e., Sa;I;G fI;G VI ¼

XX Sa;i;G fi;g Vi

(3.528)

g˛G i˛I

where

P fi;G Vi

fI;G ¼ i˛I

and VI ¼

VI

X Vi i˛I

(3.529)

(3.530)

Chapter 3  Neutron transport calculations at the cell and assembly levels

177

The energy group structure for the macro-groups must be chosen in such a way that the condensation does not depend on the flux spectrum and thus on the fact that the flux spectrum used in the condensation is not the critical flux spectrum. In terms of computational costs of the micro-group/micro-region calculations and of the macro-group/macro-region calculations, the number of unknowns at each computational step is of the same order of magnitude. In the micro-group/micro-region calculations, the energy resolution is very high, while the description of the geometrical system, although refined in space, is limited to a small part of the computational domain (i.e., pin cells). In the macro-group/macro-region calculations, the energy resolution is simpler, while the description of the geometrical system, although coarser, includes the entire computational domain (i.e., fuel assembly). Since modelling a fuel assembly in a fine resolution in space and in energy represents a too large task in terms of computing power for production purposes, the micro/macro approach aims at keeping the number of unknowns at each computational step at a moderate level. This guarantees fast running algorithms. This strategy thus represents a compromise between reasonable accuracy and acceptable computational times. The goal of the cell/assembly calculations is to provide to the core simulator (for which the algorithms are described in Chapter 4) macroscopic cross-sections homogenized on axial cross-sections of fuel assemblies and condensed into very few groups. This condensation in very few energy groups strongly depends on the flux spectrum, which itself depends on the fact that the system is finite. Since the micro/macro approach relies on the assumption of the system being infinite, the flux spectrum needs to be corrected in order to account for the effect of leakage. The macro-group calculations are thus complemented with a criticality spectrum calculation, from which the critical neutron spectrum fGB is determined, with GB representing the macro-group in the energy group structure used in the criticality calculation. Consequently, one needs to ‘re-balance’ the macro-flux fI;G with the leakage spectrum fGB and thereafter to superimpose the rebalanced macro-flux on the micro-flux fi;g . It should be noted that the macroscopic cross-sections required in the criticality calculations should be condensed from the micro-/macro-group energy structure to the macro-group energy structure used in the criticality calculations, in a similar manner as the condensation defined by Eq. (3.525). The same remark applies to the homogenization of the cross-sections, with the homogenization performed similarly to Eq. (3.526), with the only exception that the macroscopic cross-section data are fully homogenized on the entire lattice, i.e., no spatial dependence is retained if a homogeneous criticality spectrum calculation method is used. The re-balancing of the macro-flux fI;G with the leakage spectrum fGB is carried out according to the following expression: P

fLI;G

¼

GB ˛G

f GB

fG

fI;G

(3.531)

178

Modelling of Nuclear Reactor Multi-physics

where

P

fI;G VI P fG ¼ VI I

(3.532)

I

It can be noticed from Eq. (3.531) that the macro-groups can be a subset of the group structure used in the criticality calculations. Thereafter, the superimposition of the rebalanced macro-flux fLI;G on the micro-flux fi;g is performed according to: fLi;g ¼ P i0 ˛I

VI fLI;G P f Vi0 fi0 ;g 0 i;g

(3.533)

g 0 ˛G

Finally, the re-balanced micro-flux needs to be scaled to the requested power level of the system, so that further depletion calculations can be performed. If P represents the requested power level of the system, the re-balanced micro-flux fLi;g is scaled to the adequate micro-flux level fL.P i,g using the following relationship: fLi;g

fL;P i;g ¼ P i0

Vi0

P P L P fL fi0 ;g 0 kX NX ;i0 sfX ;i0 ;g 0 i;g g0

(3.534)

X

0

The sum on the regions i should be taken on the fuel regions only, whereas the sum on the nuclides X should be taken on only the nuclides leading to fissions and present in the fuel regions. In the equation above, kX represents the average recoverable energy per fission event on the species X.

3.8 Depletion calculations Due to the exposure to the neutron flux, the isotopic composition of the nuclear fuel assemblies is changing, phenomenon usually referred to as fuel depletion or fuel burnup. Species existing when the reactor was started disappear, and other species are produced. Two types of species need to be tracked: the fission fragments/products (some of them significantly influencing the reactivity of the system), and the heavy metals (or transuranic elements for uranium- or plutonium-fuelled reactors). Regarding the fission fragments/products, they are too many to be all tracked individually. Fortunately, only a few ones play a significant role and needs to be tracked. All the other fission products/ fragments are typically grouped into two pseudo fission products/fragments: a nonsaturating one and a slowly saturating one. In order to properly account for the effect of the heavy metals and fission products/ fragments, the change of the concentration of those species has to be determined. The equations giving the time-dependence of those species are the so-called Bateman

Chapter 3  Neutron transport calculations at the cell and assembly levels

179

equations. Those equations express the fact any time-dependence in the concentration of a given species results from an imbalance between reactions leading to this species and the reactions consuming this species. For the species X having a time-dependent concentration NX ;i ðtÞ in the micro-region i, the Bateman equation reads as: Z N dNX ;i ðtÞ ¼ lX NX ;i ðtÞ  saX ðEÞNX ;i ðtÞfL;P i ðE; tÞdE dt 0 2 Z N X scY ðEÞfL;P þ NY ;i ðtÞ4lY þ i ðE; tÞ dE Z þgY /X

(3.535)

0

Y

0

N

Z sfY ðEÞfL;P i ðE; tÞ

dE þ bY /X

0

N

3 nY ðEÞsfY ðEÞfL;P i ðE; tÞ

dE 5

In this equation, lX NX ;i ðtÞ represents the possible decay rate of the species X by radioactive decay, whereas saX (E ) NX,i (t) fL,P i (E,t) represents its possible neutron absorption rate. The sum on the right-hand side of Eq. (3.535) is a generic sum on all other species Y leading to the species X. The species X can be produced from the species Y having a time-dependent concentration Ny,i(t):  By possible radioactive decay at a rate of lY NY ;i ðtÞ leading to the species X.  By possible neutron capture at a rate of scY (E) Ny,i (t) fL,P i (E,t) leading to the species X .  As a fission product (from the fission of Y ) at a rate of gY /X sfY ðEÞNY ;i ðtÞfL;P i ðE; tÞ.  As a possible precursor of delayed neutrons (from the fission of Y ) at a rate of bY /X nY ðEÞsfY ðEÞNY ;i ðtÞfL;P i ðE; tÞ. In Eq. (3.535), fL;P i ðE; tÞ represents the re-balanced micro-flux scaled to the proper power level. It has to be emphasized that the neutron flux depends on the concentration of the different species and thus is a function of time. In a multi-group formalism, Eq. (3.535) reads as: G X dNX;i ðtÞ ¼ lX NX ;i ðtÞ  saX;g NX ;i ðtÞfL;P i;g ðtÞ dt g¼1 " G X X scY ;g fL;P þ NY ;i ðtÞ lY þ i;g ðtÞ

(3.536)

g¼1

Y

þgY /X

G X

sfY ;g fL;P i;g ðtÞ

þ bY /X

G X

g¼1

# ðnsf ÞY ;g fL;P i;g ðtÞ

g¼1

If Ni is a column vector having for components the concentration of the different species present in the micro-region i, Eq. (3.536) can be formally written as: dNi ðtÞ ¼ A½Ni ðtÞ; t Ni ðtÞ dt

(3.537)

where A is a matrix, which depends on both the time and the neutron flux fL;P i;g . Since the neutron flux is itself dependent on the concentration of the different species, the matrix

180

Modelling of Nuclear Reactor Multi-physics

A thus depends on both the time and on the concentration of the various species Ni , as indicated in Eq. (3.537). Eq. (3.537) is usually solved by using the so-called predictor-corrector method. This method is based on a combination of a forward (explicit) Euler scheme and a backward (implicit) Euler scheme for the numerical integration of Eq. (3.537). Such schemes are detailed below. Using a first-order Taylor expansion with respect to the time tn , the explicit scheme or Euler forward scheme approximates Eq. (3.537) as: Ni ðtnþ1 Þ  Ni ðtn Þ dNi ¼ ðtn Þ ¼ A½Ni ðtn Þ; tn  Ni ðtn Þ Dt dt

(3.538)

resulting in: Ni ðtnþ1 Þ ¼ Ni ðtn Þ þ Dt

dNi ðtn Þ ¼ Ni ðtn Þ þ Dt A½Ni ðtn Þ; tn  Ni ðtn Þ dt

(3.539)

Using a first-order Taylor expansion with respect to the time tnþ1 , the implicit scheme or Euler backward scheme approximates Eq. (3.537) as: Ni ðtnþ1 Þ  Ni ðtn Þ dNi ðtnþ1 Þ ¼ A½Ni ðtnþ1 Þ; tnþ1  Ni ðtnþ1 Þ ¼ dt Dt

(3.540)

resulting in: Ni ðtnþ1 Þ ¼ Ni ðtn Þ þ Dt

dNi ðtnþ1 Þ ¼ Ni ðtn Þ þ Dt A½Ni ðtnþ1 Þ; tnþ1  Ni ðtnþ1 Þ dt

(3.541)

In the previous equations, Dt represents the time-step used in the numerical integration of Eq. (3.537) and tk represents the discrete time at which the concentrations of the different species are evaluated. The explicit scheme is first-order accurate, since in virtue of a first-order Taylor expansion with respect to time tn one has: Ni ðtnþ1 Þ  Ni ðtn Þ dNi ¼ ðtn Þ þ O ðDtÞ Dt dt

(3.542)

Likewise, the implicit scheme is first-order accurate, since in virtue of a first-order Taylor expansion with respect to time tnþ1 one has: Ni ðtnþ1 Þ  Ni ðtn Þ dNi ¼ ðtnþ1 Þ þ O ðDtÞ Dt dt

(3.543)

A better accuracy can be obtained by adding Eqs. (3.539) and (3.541). One then obtains: Ni ðtnþ1 Þ ¼ Ni ðtn Þ þ

 Dt dNi dNi ðtn Þ þ ðtnþ1 Þ 2 dt dt

¼ Ni ðtn Þ þ

Dt fA½Ni ðtn Þ; tn  Ni ðtn Þ þ A½Ni ðtnþ1 Þ; tnþ1  Ni ðtnþ1 Þg 2

(3.544)

Chapter 3  Neutron transport calculations at the cell and assembly levels

181

Comparing this expression to the direct integration of Eq. (3.537) in the continuous case, which can be written as: Z

Ni ðtnþ1 Þ ¼ Ni ðtn Þ þ

tnþ1

A½Ni ðtÞ; t Ni ðtÞdt

(3.545)

tn

one notices that Eq. (3.544) uses a trapezoidal method for approximating the integral in Eq. (3.545). Eq. (3.544) is known as the Crank-Nicholson scheme. Rewriting Eq. (3.537) as: dNi ðtÞ ¼ f i ðtÞ dt

(3.546)

a second-order Taylor expansion in Ni with respect to the time tn allows writing: Ni ðtnþ1 Þ ¼ Ni ðtn Þ þ Dt

  dNi Dt 2 d 2 Ni ðtn Þ þ ðtn Þ þ O Dt 3 2 dt 2 dt

(3.547)

whereas a first-order Taylor expansion in f i with respect to the time tn allows writing: f i ðtnþ1 Þ ¼ f i ðtn Þ þ Dt

  df i ðtn Þ þ O Dt 2 dt

(3.548)

The Crank-Nicholson scheme is equivalent to writing Eq. (3.544) as: Ni ðtnþ1 Þ  Ni ðtn Þ f i ðtnþ1 Þ þ f i ðtn Þ ¼ Dt 2

(3.549)

The left-hand side of Eq. (3.549) can be evaluated from Eq. (3.547) as:   Ni ðtnþ1 Þ  Ni ðtn Þ dNi Dt d 2 Ni ¼ ðtn Þ þ ðtn Þ þ O Dt 2 Dt 2 dt 2 dt   Dt df i ¼ f i ðtn Þ þ ðtn Þ þ O Dt 2 2 dt

(3.550)

where the second equality was obtained using Eq. (3.546). Likewise, the right-hand side of Eq. (3.549) can be evaluated from Eq. (3.548) as:   f i ðtnþ1 Þ þ f i ðtn Þ Dt df i ¼ f i ðtn Þ þ ðtn Þ þ O Dt 2 2 2 dt

(3.551)

Comparing Eqs. (3.550) and (3.551), one then notices that the Crank-Nicholson scheme defined by Eq. (3.549) (or alternatively by Eq. 3.544) is second-order accurate. In order to study the stability of the explicit, implicit and Crank-Nicholson methods, the following first-order differential equation is considered: dn ðtÞ ¼ anðtÞ dt

(3.552)

where a is a constant. A closer examination of the Bateman equations given by Eq. (3.536) or alternatively by Eq. (3.537) would reveal that the general solution is given by a combination of analytical solutions corresponding to a first-order differential equation of the type given by Eq. (3.552), where a is in addition a negative constant (Knott and Yamamoto, 2010).

182

Modelling of Nuclear Reactor Multi-physics

The solution n+ ðtn Þ to the discretized problem can be assumed to contain round-off errors and thus can be assumed to contain some deviation εðtn Þ from the exact solution nðtn Þ to the discretized problem, i.e., n+ ðtn Þ ¼ nðtn Þ þ εðtn Þ

(3.553)

Stability analysis aims at determining whether such round-off errors are damped by the numerical algorithm used or amplified. In the former case, the scheme is said to be stable, whereas in the latter case, it is said to be unstable. Substituting Eq. (3.553) into the explicit, implicit and Crank-Nicholson schemes, respectively, written for Eq. (3.552) and using the fact that both the solutions n+ ðtn Þ and nðtn Þ fulfil those equations, one concludes that the error εðtn Þ should also fulfil these equations. This gives the following:  For the explicit scheme: εðtnþ1 Þ  εðtn Þ ¼ aDtεðtn Þ

(3.554)

εðtnþ1 Þ ¼ ð1 þ aDtÞεðtn Þ ¼ ð1 þ aDtÞ2 εðtn1 Þ ¼ . ¼ ð1 þ aDtÞnþ1 εðt0 Þ

(3.555)

leading to:

 For the implicit scheme: εðtnþ1 Þ  εðtn Þ ¼ aDtεðtnþ1 Þ

(3.556)

leading to εðtnþ1 Þ ¼ ð1  aDtÞ1 εðtn Þ ¼ ð1  aDtÞ2 εðtn1 Þ ¼ . ¼ ð1  aDtÞðnþ1Þ εðt0 Þ

(3.557)

 For the Crank-Nicholson scheme: εðtnþ1 Þ  εðtn Þ ¼

leading to εðtnþ1 Þ ¼

aDt ½εðtnþ1 Þ þ εðtn Þ 2



2

nþ1 2 þ aDt 2 þ aDt 2 þ aDt εðtn1 Þ ¼ . ¼ εðt0 Þ εðtn Þ ¼ 2  aDt 2  aDt 2  aDt

(3.558)

(3.559)

One notices that if a is a negative number, one should have: Dt
N by setting:

Chapter 4  Neutron transport calculations at the core level

v v v ¼ 0; ¼ 0; and 4N þ1;m ¼ 0 4 4 vx N þ1;m1 vy N þ1;m1 vz

199

(4.25)

Nevertheless, the equations to be solved are usually too complicated in the PN approximation for N > 1 in case of general geometries. Only in the case of systems presenting an azimuthal invariance (i.e. no dependence on the azimuthal angle 4) can the PN approximation for N > 1 be used, since one thus has m ¼ 0 and the spherical harmonics reduce to the Legendre polynomials. Noticing from Eq. (4.20) that the current density vector can be expressed as: 2 Z

Ux jðx; y; z; U; EÞd 2 U

3

6 7 6 ð4pÞ 7 6 Z 7 6 7 Z 6 7 2 6 7 U jðx; y; z; U; EÞd U 2 y Jðx; y; z; EÞ ¼ Ujðx; y; z; U; EÞd U ¼ 6 7 6 ð4pÞ 7 6 7 ð4pÞ 6 Z 7 6 7 4 Uz jðx; y; z; U; EÞd 2 U 5 ð4pÞ

2 Z

3 expði4Þ þ expði4Þ sin q jðx; y; z; U; EÞd 2 U 7 6 2 7 6 ð4pÞ 7 6 7 6 7 6 Z 7 6 expði4Þ  expði4Þ 6 2 jðx; y; z; U; EÞd U 7 sin q 7 ¼6 2i 7 6 7 6 ð4pÞ 7 6 7 6 Z 7 6 7 6 2 5 4 cos qjðx; y; z; U; EÞd U

(4.26)

ð4pÞ

one has:

3 2 pffiffiffi  2 7 6 6 2 41;1 ðx; y; z; EÞ  411 ðx; y; z; EÞ 7 7 6 pffiffiffi 7 6 Jðx; y; z; EÞ ¼ 6 i 2  7 7 6 ðx; y; z; EÞ þ 4 ðx; y; z; EÞ 4 1;1 11 7 6 2 5 4 410 ðx; y; z; EÞ

due to the fact that: Y10 ðq; 4Þ ¼ cos q Y11 ðq; 4Þ ¼ 

and

pffiffiffi 2 sin qðcos 4 þ i sin 4Þ 2

pffiffiffi 2 Y1;1 ðq; 4Þ ¼ sin qðcos 4  i sin 4Þ 2

(4.27)

200

as:

Modelling of Nuclear Reactor Multi-physics

From Eq. (4.23), the first equation in the P1 approximation, obtained for l ¼ 0, is given pffiffiffi  pffiffiffi    v v v v 2 2 v 4 ðx; y; z; EÞ þ i 4 ðx; y; z; EÞ þ 41;0 ðx; y; z; EÞ  i vx vy 11 vy 1;1 vz 2 2 vx þSt ðx; y; z; EÞ400 ðx; y; z; EÞ Z N Ss0 ðx; y; z; E 0 /EÞ400 ðx; y; z; E 0 ÞdE 0 ¼ 0

þ

cðx; y; z; EÞ k

Z

N 0

(4.28)

nðx; y; z; E 0 ÞSf ðx; y; z; E 0 Þ400 ðx; y; z; E 0 ÞdE 0

Using Eqs. (4.18) and (4.27), one finds that Eq. (4.28) can also be written, with r h x, y, z, as: V $ Jðr; EÞ þ St ðr; EÞfðr; EÞ Z Z N cðr; EÞ N ¼ Ss0 ðr; E 0 /EÞfðr; E 0 ÞdE 0 þ nðr; E 0 ÞSf ðr; E 0 Þfðr; E 0 ÞdE 0 k 0 0

(4.29)

It should be emphasized that this equation is exact, i.e. no approximation was used to derive it. For l ¼ 1, three additional equations are obtained (i.e. for m ¼ 1, m ¼ 0 and m ¼ 1). From Eq. (4.23), they are given in the P1 approximation as: pffiffiffi   1 2 v v þi 4 ðx; y; z; EÞ þ St ðx; y; z; EÞ41;1 ðx; y; z; EÞ 3 2 vx vy 00 Z N Ss1 ðx; y; z; E 0 /EÞ41;1 ðx; y; z; E 0 ÞdE 0 ¼

(4.30)

0

1 v 4 ðx; y; z; EÞ þ St ðx; y; z; EÞ410 ðx; y; z; EÞ 3 vz 00 Z N Ss1 ðx; y; z; E 0 /EÞ410 ðx; y; z; E 0 ÞdE 0 ¼

(4.31)

0

pffiffiffi   1 2 v v 4 ðx; y; z; EÞ þ St ðx; y; z; EÞ411 ðx; y; z; EÞ  þi 3 2 vx vy 00 Z N Ss1 ðx; y; z; E 0 /EÞ411 ðx; y; z; E 0 ÞdE 0 ¼

(4.32)

0

where the approximations expressed by Eq. (4.25) were used. It has to be mentioned that contrary to Eq. (4.28), the threepEqs. ffiffiffi  (4.30)e(4.32) are not exact. Subtracting Eq. (4.32) from (4.30) and multiplying by 2 2 gives, because of Eqs. (4.18) and (4.27): 1 vf ðx; y; z; EÞ þ St ðx; y; z; EÞJx ðx; y; z; EÞ 3 vx Z N Ss1 ðx; y; z; E 0 /EÞJx ðx; y; z; E 0 ÞdE 0 ¼ 0

(4.33)

Chapter 4  Neutron transport calculations at the core level

201

pffiffiffi Likewise, adding Eqs. (4.30) and (4.32) together and multiplying by i 2 2 gives, because of Eqs. (4.18) and (4.27): 1 vf ðx; y; z; EÞ þ St ðx; y; z; EÞJy ðx; y; z; EÞ 3 vy Z N ¼ Ss1 ðx; y; z; E 0 /EÞJy ðx; y; z; E 0 ÞdE 0

(4.34)

0

Finally, Eq. (4.31) can be rewritten, because of Eqs. (4.18) and (4.27), as: 1 vf ðx; y; z; EÞ þ St ðx; y; z; EÞJz ðx; y; z; EÞ 3 vz Z N Ss1 ðx; y; z; E 0 /EÞJz ðx; y; z; E 0 ÞdE 0 ¼

(4.35)

0

Combining Eqs. (4.33)e(4.35), one obtains the second P1 equation, with r h x, y, z, as: 1 Vfðr; EÞ þ St ðr; EÞJðr; EÞ ¼ 3

Z

N 0

Ss1 ðr; E 0 /EÞJðr; E 0 ÞdE 0

(4.36)

In multi-group theory, the two P1 equations thus read as: V $ Jg ðrÞ þ St;g ðrÞfg ðrÞ ¼

G X g 0 ¼1

Ss0;g 0 /g ðrÞfg 0 ðrÞ þ

G cg ðrÞ X ðnSf Þg 0 ðrÞfg 0 ðrÞ k g 0 ¼1

(4.37)

and G X 1 Vfg ðrÞ þ St;g ðrÞJg ðrÞ ¼ Ss1;g 0 /g ðrÞJg 0 ðrÞ 3 g 0 ¼1

(4.38)

It can be noticed from Eq. (4.38) that the multi-group anisotropic macroscopic crosssections should in principle be prepared from the point-wise nuclear data libraries by using the neutron current as a weighting function. Very often, the anisotropic macroscopic cross-sections are prepared from the libraries using flux-weighting, as for any other cross-section (see Section 3.1 of Chapter 3). From Eq. (4.27), one also notices that: pffiffiffi 8 > < Jx ðr; EÞ þ iJy ðr; EÞ ¼ p2ffiffiffi41;1 ðr; EÞ Jx ðr; EÞ  iJy ðr; EÞ ¼ 2411 ðr; EÞ > : Jz ðr; EÞ ¼ 410 ðr; EÞ

(4.39)

In the P1 approximation, only the expansion on the Legendre polynomials of order l ¼ 0 and l ¼ 1 are retained. According to Eq. (4.1), the angular neutron flux is then given as: 1 jðr; U; EÞz 400 ðr; EÞY00 ðq; 4Þ 4p  3  þ 4 ðr; EÞY1;1 ðq; 4Þ þ 410 ðr; EÞY10 ðq; 4Þ þ 411 ðr; EÞY11 ðq; 4Þ 4p 1;1

Using Eqs. (4.18) and (4.39), and the fact that Y00 ðq; 4Þ ¼ 1

(4.40)

202

Modelling of Nuclear Reactor Multi-physics

Y10 ðq; 4Þ ¼ cos q Y11 ðq; 4Þ ¼ 

and Y1;1 ðq; 4Þ ¼

pffiffiffi 2 sin qðcos 4 þ i sin 4Þ 2

pffiffiffi 2 sin qðcos 4  i sin 4Þ 2

Eq. (4.40) can be rearranged into: jðr; U; EÞ 1 z ½fðr; EÞ þ 3Jx ðr; EÞsin q cos 4 þ 3Jy ðr; EÞsin q sin 4 þ 3Jz ðr; EÞcos q 4p

(4.41)

or in a more condensed form: 1 jðr; U; EÞz ½fðr; EÞ þ 3U $ Jðr; EÞ 4p

4.1.b

(4.42)

Diffusion theory

The so-called diffusion approximation is obtained from the P1 equations by further assuming that: Z

0

N

Ss1 ðr; E 0 /EÞJðr; E 0 ÞdE 0 ¼

Z

N

0

Ss1 ðr; E/E 0 ÞJðr; EÞdE 0

(4.43)

or in a multi-group formalism: G X

Ss1;g 0 /g ðrÞJg 0 ðrÞ ¼

g 0 ¼1

G X

Ss1;g/g 0 ðrÞJg ðrÞ

(4.44)

g 0 ¼1

In the two above equations, it is thus assumed that the anisotropic in-scatter of neutrons from any energy to a given energy is exactly equal to the anisotropic out-scatter of neutrons from that specific energy to all other energies. In the most general case, two additional approximations are necessary for obtaining the diffusion approximation from the P1 equations: there should not be any anisotropic neutron source and the timedependence of the neutron current density vector should be negligible (Sanchez, 1996). These two approximations were implicitly assumed when deriving the timeindependent P1 equations in Section 4.1.a. Using Eq. (4.43) in Eq. (4.36) gives: 1 Vfðr; EÞ þ St ðr; EÞJðr; EÞ  Ss1 ðr; EÞJðr; EÞ ¼ 0 3

(4.45)

or in a multi-group formalism: 1 Vfg ðrÞ þ St;g ðrÞJg ðrÞ  Ss1;g ðrÞJg ðrÞ ¼ 0 3

(4.46)

Chapter 4  Neutron transport calculations at the core level

203

where the anisotropic cross-section is defined in a similar manner as in Eq. (2.10) of Chapter 2, i.e. Z

Ss1 ðr; EÞ ¼

N

0

Ss1 ðr; E/E 0 ÞdE 0

(4.47)

or in a multi-group formalism: Ss1;g ðrÞ ¼

G X

Ss1;g/g 0 ðrÞ

(4.48)

g 0 ¼1

From Eqs. (4.45) and (4.46), one notices that the current density vector is related to the scalar neutron flux according to the following relationship, called Fick’s law: Jðr; EÞ ¼  Dðr; EÞVfðr; EÞ

(4.49)

with Dðr; EÞ ¼

1 1 ¼ 3½St ðr; EÞ  Ss1 ðr; EÞ 3½St ðr; EÞ  mðr; EÞSs0 ðr; EÞ

(4.50)

and where the average of the cosine of the deviation angle m ¼ U0 $ U is defined in a similar manner as in Eq. (3.36) of Chapter 3, i.e. mðr; EÞ ¼

2p

RN R1

2p

R0 N R11 0

Ss ðr; m; E/E 0 ÞmdmdE 0

S ðr; m; E/E 0 ÞdmdE 0 1 s

¼

Ss1 ðr; EÞ Ss0 ðr; EÞ

(4.51)

with the isotropic cross-section defined in a similar manner as in Eq. (2.10) of Chapter 2, i.e. Z

Ss0 ðr; EÞ ¼

0

N

Ss0 ðr; E/E 0 ÞdE 0

(4.52)

In a multi-group formalism, Eq. (4.49) reads as: Jg ðrÞ ¼  Dg ðrÞVfg ðrÞ

(4.53)

with Dg ðrÞ ¼

1 1  ¼  3½St;g ðrÞ  Ss1;g ðrÞ 3 St;g ðrÞ  mg ðrÞSs0;g ðrÞ

(4.54)

Ss1;g ðrÞ Ss0;g ðrÞ

(4.55)

and mg ðrÞ ¼

Eqs. (4.50) and (4.54) define the so-called diffusion coefficients. The neutron current density vector can then be eliminated from the first P1 equation (i.e. Eq. 4.29) using Eq. (4.49), leading to: V $ ½Dðr; EÞVfðr; EÞ þ St ðr; EÞfðr; EÞ Z N Z cðr; EÞ N ¼ Ss0 ðr; E 0 /EÞfðr; E 0 ÞdE 0 þ nðr; E 0 ÞSf ðr; E 0 Þfðr; E 0 ÞdE 0 k 0 0

(4.56)

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Modelling of Nuclear Reactor Multi-physics

or in a multi-group formalism:

  V $ Dg ðrÞVfg ðrÞ þ St;g ðrÞfg ðrÞ ¼

G X g 0 ¼1

Ss0;g 0 /g ðrÞfg 0 ðrÞ þ

G cg ðrÞ X ðnSf Þg 0 ðrÞfg 0 ðrÞ k g 0 ¼1

(4.57)

Eqs. (4.56) and (4.57) are usually referred to as the diffusion equations. It has to be mentioned that although these equations are balance equations written for the scalar neutron flux, the anisotropic character of the angular neutron flux is retained, because combining Eqs. (4.42) and (4.49), one has: 1 jðr; U; EÞz ½fðr; EÞ  3Dðr; EÞU $ Vfðr; EÞ 4p

(4.58)

or in a multi-group formalism:  1  jg ðr; UÞz f ðrÞ  3Dg ðrÞU $ Vfg ðrÞ 4p g

(4.59)

Using Eqs. (2.10) and (2.13) of Chapter 2 and Eq. (3.34) of Chapter 3, one has: Z

St ðr; EÞ ¼ Sa ðr; EÞ þ Ss ðr; EÞ ¼ Sa ðr; EÞ þ

and Eq. (4.56) can thus be rewritten as:

Z

V $ ½Dðr; EÞVfðr; EÞ þ Sa ðr; EÞfðr; EÞ þ Z ¼

0

N

Ss0 ðr; E 0 /EÞfðr; E 0 ÞdE 0 þ

cðr; EÞ k

0

Z

N 0

N

0

N

Ss0 ðr; E/E 0 ÞdE 0

(4.60)

Ss0 ðr; E/E 0 Þfðr; EÞdE 0 (4.61)

nðr; E 0 ÞSf ðr; E 0 Þfðr; E 0 ÞdE 0

and in a multi-group formalism as: G G   X cg ðrÞ X V $ Dg ðrÞVfg ðrÞ þ Ss0;g 0 /g ðrÞfg 0 ðrÞ þ ðnSf Þg 0 ðrÞfg 0 ðrÞ k g 0 ¼1 g 0 ¼1

Sa;g ðrÞfg ðrÞ 

G X

S

s0;g/g 0

g 0 ¼1

(4.62)

ðrÞfg ðrÞ ¼ 0

It can be noticed from Eq. (4.43) that the diffusion equations are exact (compared to the P1 equations) in the mono-energetic case, i.e. when there is no energy dependence. In such a case, Eq. (4.61) reduces to: V $ ½DðrÞVfðrÞ þ

nðrÞSf ðrÞ fðrÞ  Sa ðrÞfðrÞ ¼ 0 k

(4.63)

with the diffusion coefficient given, according to Eq. (4.50), by: DðrÞ ¼

1 1 ¼ 3½St ðrÞ  Ss1 ðrÞ 3fSa ðrÞ þ ½1  mðrÞ Ss0 ðrÞg

(4.64)

where one used Eqs. (4.52) and (4.60) in the mono-energetic case, i.e. St ðrÞ ¼ Sa ðrÞ þ Ss ðrÞ ¼ Sa ðrÞ þ Ss0 ðrÞ

(4.65)

Chapter 4  Neutron transport calculations at the core level

205

Using the transport correction defined in Eq. (3.237) of Chapter 3 in the monoenergetic case, the diffusion coefficient is simply expressed as: 1  DðrÞ ¼  3 Sa ðrÞ þ S0s0 ðrÞ

(4.66)

Eq. (4.63) and Eq. (4.66) are similar to the diffusion equations that can be obtained by directly assuming isotropic scattering in the laboratory reference system and by replacing the isotropic cross-section appearing in the diffusion coefficient by the transport-corrected (at the level 0) isotropic cross-section, as explained below. Such a correction is thus referred to as the transport correction. Taking again the time-independent neutron transport equation written in its integrodifferential form (i.e. Eq. (2.35) of Chapter 2) and assuming isotropic scattering in the laboratory reference system, the neutron transport equation reads as: U $ Vjðr; U; EÞ þ St ðr; EÞjðr; U; EÞ Z Z N



0 0 0 0 0 0 Ss r; U /U; E /E j r; U ; E d 2 U dE ¼ ð4pÞ

þ

0

cðr; EÞ 4pk

Z 0

N

(4.67)

  0 0  0 0 n r; E Sf r; E f r; E dE

where cðr; EÞ is given by Eq. (3.251) of Chapter 3 and where the differential scattering cross-section is given, using Eq. (3.24) of Chapter 3 and in the case of isotropy, by: Ss ðr; U0 /U; E 0 /EÞ ¼

Ss0 ðr; E 0 /EÞ 4p

(4.68)

Using the fact that U $ Vjðr; U; EÞ ¼ V $ ½Ujðr; U; EÞ , integrating Eq. (4.67) with respect to the direction U, one obtains: V $ Jðr; EÞ þ St ðr; EÞfðr; EÞ Z Z N     cðr; EÞ N  0 0 0 0 0  0 0 ¼ Ss0 r; E /E f r; E dE þ n r; E Sf r; E f r; E dE k 0 0

(4.69)

where the definitions of the neutron current density vector and of the neutron scalar flux were used. In the mono-energetic case, this equation simplifies into: V $ JðrÞ þ Sa ðrÞfðrÞ 

nðrÞSf ðrÞ fðrÞ ¼ 0 k

(4.70)

where Eq. (4.60) was used. A relationship between the neutron current density vector J and the scalar flux f can be established by further assuming that the medium is infinite and uniform, containing no source of neutrons, and that the neutron flux is a slowly varying function of position, following hereafter the derivation of Lamarsh (2002). In a Cartesian coordinate system as depicted in Fig. 4.2, a surface dAz perpendicular to the z-axis is considered. The flow of neutrons through this surface will be determined, first considering the neutrons flowing downward through this surface, then the neutrons flowing upward through this surface. The difference between these two

206

Modelling of Nuclear Reactor Multi-physics

FIGURE 4.2 Diagram for deriving the relationship between the neutron current density vector and the scalar flux. Derived from Lamarsh, J.R., 2002. Introduction to Nuclear Reactor Theory. American Nuclear Society, La Grange Park, USA.

contributions will result in the net flow of neutrons through this surface, from which a relationship between the neutron current density vector and the scalar neutron flux will be derived.  The neutron flow downward through the surface dAz per unit area, referred to as Jz;0 , can be determined in the following manner. The subscript 0 refers to the origin of the system of axes, chosen to coincide with the location of the surface dAz. Since neutrons are isotropically emitted by scattering, the number of neutrons emitted isotropically per unit time from the infinitesimal volume d3r around the position r is Ss0 fðrÞd 3 r. Only a fraction of these neutronsis emitted in the solid angle d2U corresponding to the surface dAz. This fraction is d 2 U 4p due to the assumption of isotropic scattering in the laboratory reference system, where one further has dAz cos q ¼ r2 d 2 U. In addition, only a fraction of the neutrons emitted in the solid angle d2U survives to the surface dAz, i.e. does not interact before reaching dAz. The corresponding probability of non-interaction is, according to the definition and use of the macroscopic cross-sections (see Section 2.2.a of Chapter 2), expð Sr rÞ. It thus results that: number of neutron scattered per unit time from dV through dAz ¼ Ss0 fðrÞd 3 r expð  St rÞ

dAz cos q 4pr 2

(4.71)

 Integrating Eq. (4.71) on the upper half space allows determining Jz;0 as follows:  Jz;0 ¼

Ss0 4p

Z

2p 0

Z

p=2 0

Z

N 0

expðSt rÞfðrÞcos q sin qdrdqd4

(4.72)

Chapter 4  Neutron transport calculations at the core level

207

Using the assumption of slow variation of the neutron flux as a function of position, a first-order Taylor expansion of the flux with respect to position from the origin can be performed as: fðrÞ z f0 þ x

      vf vf vf þy þz vx 0 vy 0 vz 0

(4.73)

One has x ¼ r sin q cos 4, y ¼ r sin q sin f and z ¼ r cos q. After inserting Eq. (4.73) into Eq. (4.72), it can be seen that the integrals containing the terms containing cos 4 and sin 4 are equal to zero. One thus obtains: Ss0  Jz;0 z 4p

Z

0

2p

Z

0

p=2

Z

0

N

 

vf cos q sin qdrdqd4 expðSt rÞ f0 þ r cos q vz 0

finally resulting in:  Jz;0 z

  Ss0 f0 Ss0 vf þ 2 4St 6St vz 0

(4.74)

(4.75)

þ A similar demonstration for the upward flow Jz;0 through the surface dAz per unit area would result in: þ Jz;0 z

  Ss0 f0 Ss0 vf  2 4St 6St vz 0

(4.76)

The demonstration made for the flow of neutrons through the elementary surface dAz perpendicular to the z-axis can be generalized to any elementary surface dAn perpendicular to a given n-axis, so that one can write: H Jn;0 ¼

  Ss0 f0 Ss0 vf  2 4St 6St vn 0

(4.77)

The net flow of neutrons through the elementary surface dAn is thus given by: þ  Jn;0 ¼ Jn;0  Jn;0 ¼

  Ss0 vf 3S2t vn 0

(4.78)

The neutron current density vector, as defined in Eq. (2.6) of Chapter 2, can consequently be written as: J0 ¼ i Jx;0 þ j Jy;0 þ k Jz;0 ¼ DVf0

(4.79)

with D¼

Ss0 3S2t

(4.80)

and where i, j and k are the unitary vectors on the x-, y- and z-axes, respectively. Eq. (4.79) is referred to as Fick’s law. The assumptions used for deriving this law are usually valid when the medium is not a strongly absorbing medium, i.e. Sa Ss0 . In such a case, Eq. (4.80) reduces to: 1 Dz 3Ss0

(4.81)

208

Modelling of Nuclear Reactor Multi-physics

Comparing Eqs. (4.66) and (4.81) while assuming that Sa Ss0 , one notices that replacing the scattering cross-section Ss0 by S0s0 would allow accounting for the first moment of the angular dependence of the angular flux and of the scattering crosssection, although the derivation above assumes isotropic scattering in the laboratory reference system. As earlier mentioned, replacing Ss0 by S0s0 is called the transport correction. This correction equivalently means that the scattering reactions can be assumed to be isotropic in the laboratory reference system (as in P0 theory), and thus only one equation needs to be solved (i.e. Eq. 4.63), while in fact P1 theory is used. This is achieved by using Eq. (4.66) for defining the diffusion coefficient, where, compared with the diffusion approximation with isotropic scattering, the scattering cross-section Ss0 was replaced by the transport-corrected scattering cross-section S0s0 . Finally, it can also be noticed that the diffusion equations in the non-mono-energetic case can only be obtained from the P1 equations by relying on the approximation given by Eq. (4.43), which states that the anisotropic in-scatter of neutrons from all energies to any energy E is equal to the anisotropic out-scatter of neutrons from the energy E to all energies. Such an approximation is only valid when the absorption is weak. This means that Eq. (4.56) is an approximation of the true P1 equations given by Eqs. (4.29) and (4.36). As explained above, it must be highlighted that the diffusion equation in the mono-energetic case (i.e. Eq. 4.63) is exact in the framework of P1 theory. Although the diffusion equations in the poly-energetic case are not exact in this framework, most of the commercial codes for core calculations rely on the approximation given by Eq. (4.43). This is why the scattering cross-section Ss0 is replaced by the transport-corrected scattering cross-section S0s0 . Since the scattering within groups cancels out in the first P1 balance equation, as can be seen in Eq. (4.62), adding the transport correction to the energy of the incoming neutrons, as was done in Section 3.4.b of Chapter 3, does not modify the first P1 balance equation (i.e. Eq. 4.62), thus explaining why the transport correction is only added to the energy of the incoming neutrons.

4.1.c

Simplified PN method (SPN)

The PN equations in Cartesian three-dimensional geometry represent a complex set of equations. The complexity lies with the coupling existing between the various expansion coefficients of the angular neutron flux on the spherical harmonics and between their spatial derivatives. The simplified PN or SPN method is based on the observation that the PN equations for a one-dimensional slab represent a much simpler set of equations. This simplicity will be utilized for building the SPN approximation in a heuristic manner, as originally proposed by Gelbard (1960). For a one-dimensional slab, one first notices that the angular dependence of the angular neutron flux can be recast into the sole dependence on the cosine m of the angle formed between the neutron direction U and the direction characterizing the

Chapter 4  Neutron transport calculations at the core level

209

one-dimensional dependence of the system. As a result, the angular neutron flux can be expanded on Legendre polynomials as: jðx; m; EÞ ¼

N X 2l þ 1

with

Z jl ðx; EÞ ¼ 2p

jl ðx; EÞPl ðmÞ

(4.82)

jðx; m; EÞPl ðmÞdm

(4.83)

4p

l¼0

1

1

In terms of spherical harmonics and using Eqs. (4.1) and (4.2), the above expansion on Legendre polynomials corresponds to the following expansion on the spherical harmonics: jðx; m; EÞ ¼

N X 2l þ 1 l¼0

with

Z 4l0 ðx; EÞ ¼

Z

2p 0

p 0

4p

4l0 ðx; EÞYl0 ðq; 4Þ

jðx; m; EÞYl0 ðq; 4Þd4 sin qdq ¼ 2p

Z

1

1

(4.84)

jðx; m; EÞPl ðmÞdm

(4.85)

The above two equations were obtained utilizing the fact that Eqs. (4.3) and (4.4) lead, for m ¼ 0, to: Yl0 ðq; 4Þ ¼ Pl0 ðcos qÞ ¼ Pl ðcos qÞ ¼ Pl ðmÞ

(4.86)

In case of a one-dimensional slab, one thus concludes that the expansion coefficients on the Legendre polynomials are simply obtained from the expansion coefficients on the spherical harmonics as: jl ðx; EÞ ¼ 4l0 ðx; EÞ

(4.87)

Using Eq. (4.19) thus written for m0 ¼ 0 gives a balance equation for the expansion coefficients on the Legendre polynomials, which reads for a one-dimensional slab as: v vx ¼ þ

Z

Z

2p

Z

0 N 0

p

0

mjðx; m; EÞYl0 0 ðq; 4Þd4 sin qdq þ St ðx; EÞ4l0 0 ðx; EÞ

Ssl0 ðx; E 0 /EÞ4l0 0 ðx; E 0 ÞdE 0

cðx; EÞ d0l0 k

Z

N 0

(4.88)

nðx; E 0 ÞSf ðx; E 0 Þ400 ðx; E 0 ÞdE 0

Using Eqs. (4.86) and (4.87), Eq. (4.88) simply becomes:

Z 1 v mjðx; m; EÞPl0 ðmÞdm þ St ðx; EÞjl0 ðx; EÞ vx 1 Z N Ssl0 ðx; E 0 /EÞjl0 ðx; E 0 ÞdE 0 ¼ 2p

0

þ

cðx; EÞ d0l0 k

Z 0

N

nðx; E 0 ÞSf ðx; E 0 Þj0 ðx; E 0 ÞdE 0

(4.89)

210

Modelling of Nuclear Reactor Multi-physics

For l0 ¼ 0, using Eq. (4.83),one obtains: v j ðx; EÞ þ St ðx; EÞj0 ðx; EÞ vx 1 Z N Ss0 ðx; E 0 /EÞj0 ðx; E 0 ÞdE 0 ¼ 0

þ

cðx; EÞ k

Z 0

N

(4.90)

nðx; E 0 ÞSf ðx; E 0 Þj0 ðx; E 0 ÞdE 0

since, according to Eqs. (3.28) and (3.33) of Chapter 3, one has P0 ðmÞ ¼ 1 and P1 ðmÞ ¼ m. For l0 > 0, using the recurrence relationship given by Eq. (3.31) of Chapter 3, one obtains: 2p v 2l0 þ 1 vx

Z

1

1

jðx; m; EÞ½ðl 0 þ 1ÞPl0 þ1 ðmÞ þ l 0 Pl0 1 ðmÞ dm

þSt ðx; EÞjl0 ðx; EÞ ¼

Z

N 0

(4.91) 0

0

Ssl0 ðx; E /EÞjl0 ðx; E ÞdE

0

which, in virtue of Eq. (4.83), can also be written as: l0 þ 1 v l0 v jl0 þ1 ðx; EÞ þ 0 j 0 ðx; EÞ þ St ðx; EÞjl0 ðx; EÞ 0 2l þ 1 vx 2l þ 1 vx l 1 Z N ¼ Ssl0 ðx; E 0 /EÞjl0 ðx; E 0 ÞdE 0

(4.92)

0

v In the PN approximation, one assumes, according to Eq. (4.25), that vx jN þ1 ðx; EÞ ¼ 0. Eqs. (4.90) and (4.92) thus represent a system of N þ 1 equations relating the N þ 1 expansion coefficients of the angular neutron flux onto the Legendre polynomials. As Eqs. (4.90) and (4.92) reveal, solving the PN equations for a one-dimensional slab is considerably simpler than solving the PN equations in three dimensions. The former only requires expansions on Legendre polynomials, whereas the latter requires expansions on the spherical harmonics. In the particular case of the P1 approximation, the balance equations in the multidimensional case are given by Eqs. (4.29) and (4.36), which are recalled hereafter:

V $ Jðr; EÞ þ St ðr; EÞfðr; EÞ Z Z N     cðr; EÞ N  0 0 0 0 0  0 0 ¼ Ss0 r; E /E f r; E dE þ n r; E Sf r; E f r; E dE k 0 0

and 1 Vfðr; EÞ þ St ðr; EÞJðr; EÞ ¼ 3

Z 0

N

  0 0 0 Ss1 r; E /E J r; E dE

(4.93)

(4.94)

Chapter 4  Neutron transport calculations at the core level

211

whereas the balance equations in the one-dimensional slab system are given, using Eqs. (4.90) and (4.92), as: v j ðx; EÞ þ St ðx; EÞj0 ðx; EÞ vx 1 Z N Ss0 ðx; E 0 /EÞj0 ðx; E 0 ÞdE 0 ¼ 0

þ

cðx; EÞ k

Z

N

0

(4.95)

nðx; E 0 ÞSf ðx; E 0 Þj0 ðx; E 0 ÞdE 0

and 1 v j ðx; EÞ þ St ðx; EÞj1 ðx; EÞ ¼ 3 vx 0

Z 0

N

Ss1 ðx; E 0 /EÞj1 ðx; E 0 ÞdE 0

(4.96)

As can be seen above, the balance equations in the multi-dimensional case and in the one-dimensional slab system are very similar. This led Gelbard (1960) to propose an ad hoc substitution of some of the terms appearing in the multi-dimensional PN equations as follows (Gelbard, 1960; McClarren, 2010; Larsen et al., 1996; He´bert, 2010):  For odd values of l, jl is replaced by a vector, i.e. by jl.  For odd values of l, vjl =vx is replaced by V $ jl (i.e. the divergence operator).  For even values of l, vjl =vx is replaced by Vjl (i.e. the gradient operator). The balance equations in the simplified PN method or SPN method thus read as: V $ j1 ðr; EÞ þ St ðr; EÞj0 ðr; EÞ Z N Ss0 ðr; E 0 /EÞj0 ðr; E 0 ÞdE 0 ¼ 0

þ

cðr; EÞ k

Z 0

N

(4.97)

nðr; E 0 ÞSf ðr; E 0 Þj0 ðr; E 0 ÞdE 0

and lþ1 l V $ jlþ1 ðr; EÞ þ V $ jl1 ðr; EÞ 2l þ 1 2l þ 1 þSt ðr; EÞjl ðr; EÞ Z N Ssl ðr; E 0 /EÞjl ðr; E 0 ÞdE 0 if l > 0 and l is even ¼

(4.98)

0

or lþ1 l Vjlþ1 ðr; EÞ þ Vjl1 ðr; EÞ 2l þ 1 2l þ 1 þSt ðr; EÞjl ðr; EÞ Z N Ssl ðr; E 0 /EÞjl ðr; E 0 ÞdE 0 ¼ 0

(4.99) if l > 0 and l is odd

212

Modelling of Nuclear Reactor Multi-physics

In multi-group theory, these balance equations read as: V $ j1;g ðrÞ þ St;g ðrÞj0;g ðrÞ ¼

G X g 0 ¼1

Ss0;g 0 /g ðrÞj0;g 0 ðrÞ þ

G cg ðrÞ X ðnSf Þg 0 ðrÞj0;g 0 ðrÞ k g 0 ¼1

(4.100)

and lþ1 l V $ jlþ1;g ðrÞ þ V $ jl1;g ðrÞ 2l þ 1 2l þ 1 þSt;g ðrÞjl;g ðrÞ ¼

G X g 0 ¼1

Ssl;g 0 /g ðrÞjl;g 0 ðrÞ

(4.101) if l > 0 and l is even

or lþ1 l Vjlþ1;g ðrÞ þ Vjl1;g ðrÞ 2l þ 1 2l þ 1 þSt;g ðrÞjl;g ðrÞ ¼

G X g 0 ¼1

Ssl;g 0 /g ðrÞjl;g 0 ðrÞ

(4.102) if l > 0 and l is odd

It should be emphasized that the SPN method is a heuristically introduced method. The SPN and PN equations are not generally equivalent. A true equivalence only holds when very specific conditions are fulfilled. Furthermore, the SPN equations do not converge to the transport equation when N /N. Despite these limitations, the solution to the SPN equations is significantly closer to the true transport solution for low values of N than the P1 solution. The interested reader is referred to the literature on this subject (e.g. (McClarren, 2010; Larsen et al., 1996; Pomraning, 1993)). When anisotropic scattering is only taken into account by a transport correction at the level 0, i.e. Ssl;g 0 /g ¼ 0 for l > 0 and Ss0;g 0 /g is replaced by S0s0;g 0 /g defined by Eq. (3.248) in Chapter 3, the SPN balance equations can be further simplified by first rewriting Eq. (4.102) as: jl;g ðrÞ ¼ 



1 lþ1 l Vjlþ1;g ðrÞ þ Vjl1;g ðrÞ St;g ðrÞ 2l þ 1 2l þ 1

(4.103)

This equation can then be used in Eqs. (4.100) and (4.101) leading to, respectively: 

V $ ¼

G X g 0 ¼1

and if l > 0 and l is even

 1 2 1 Vj2;g ðrÞ þ Vj0;g ðrÞ þ St;g ðrÞj0;g ðrÞ St;g ðrÞ 3 3

Ss0;g 0 /g ðrÞj0;g 0 ðrÞ þ

G cg ðrÞ X ðnSf Þg 0 ðrÞj0;g 0 ðrÞ k g 0 ¼1

(4.104)

Chapter 4  Neutron transport calculations at the core level

  lþ1 1 lþ2 lþ1 V$ Vjlþ2;g ðrÞ þ Vjl;g ðrÞ 2l þ 1 St;g ðrÞ 2l þ 3 2l þ 3 

 l 1 l l1 V$ Vjl;g ðrÞ þ Vjl2;g ðrÞ  2l þ 1 St;g ðrÞ 2l  1 2l  1

213



(4.105)

þSt;g ðrÞjl;g ðrÞ ¼0

Eqs. (4.104) and (4.105) are second-order differential equations in space and are very similar in nature to the type of balance equations obtained in the diffusion approximation. After some re-arrangement of the above equations, a diffusion-based solver can be used to solve the SPN balance equations. This explains why the SPN method is particularly interesting, despite its heuristically derived nature.

4.1.d

Boundary conditions

Different types of boundary conditions with respect to the angular dependence of the neutron flux are usually encountered. Since the modelling of boundary conditions in the SPN method is derived from the modelling of boundary conditions in the PN method, only the PN and diffusion theory methods are discussed hereafter.

Free surface boundary condition In the free surface boundary condition, the reactor core to be modelled is assumed to be surrounded by vacuum and to have a convex shape. As a result, any neutron leaving the reactor core cannot re-enter the system and is lost. One thus has: jðrB ; U; EÞ ¼ 0 for U $ NB < 0

(4.106)

where NB is the outward normal unit vector at the boundary point rB. Whereas the modelling of such a boundary condition is straightforward in methods like the discrete ordinates method or the method of characteristics, the free surface boundary condition leads to some difficulty for the spherical harmonics method (PN) and its simplified forms (SPN and diffusion methods). In such methods, the vacuum boundary condition cannot be fulfilled with an expansion of the angular flux as the one given by Eq. (4.1). This shortcoming lies with the fact that the expansion coefficients 4lm in Eq. (4.1) are defined on 4p, as can be seen in Eq. (4.2), whereas the free surface boundary condition defines a condition for the angular flux on only half the space, i.e. on 2p. The treatment of a free surface boundary condition in the PN method and its simplified forms can thus only be approximative. Two approaches are usually encountered: the Marshak and the Mark boundary conditions. In the Marshak boundary condition, the odd moments of the angular flux are set to zero for the incoming directions of the neutrons (Stamm’ler and Abbate, 1983), i.e. one sets: Z

ð2pÞ;U $ NB < l ¼ 1; 2; . ; N þ 1 with N odd 2 jðrB ; ml ; EÞ ¼ 0 with PN þ1 ðml Þ ¼ 0 for > : ml > 0

(4.114)

Chapter 4  Neutron transport calculations at the core level

215

It could be demonstrated that, in case of planar geometry, the equation fulfilled by the angular flux in PN theory using the Mark boundary condition is identical with the equation fulfilled by the angular flux in SNþ1 theory using discrete angles defined as solution to PN þ1 ðmi Þ ¼ 0 (the discrete directions are then referred to as Gaussian points). This is also the reason why N is taken as an odd number in PN theory, since N þ 1 should be an even number in SNþ1 theory. The interested reader is referred to the existing literature for further details on that subject.   In P1 theory, for which using Eq. (3.29) in Chapter 3 gives P2 ðmÞ ¼ 12 3m2  1 , the Mark boundary condition gives: jðrB ; m1 ; EÞ ¼ 0 with P2 ðm1 Þ ¼ 0 for m1 > 0

(4.115)

pffiffiffi 3 JðrB ; EÞ $ NB ¼ fðrB ; EÞ 3

(4.116)

pffiffiffi The only positive root of P2 ðm1 Þ ¼ 0 is m1 ¼ 1 3. Recalling Eq. (4.110), one obtains for the Mark boundary condition:

In diffusion theory, because of Eq. (4.49), the Mark boundary condition is equivalent, using Eq. (4.116), to: pffiffiffi fðrB ; EÞ þ 3DðrB ; EÞVfðrB ; EÞ $ NB ¼ 0

(4.117)

In diffusion theory, it can be noticed that both the Marshak and Mark boundary conditions can be cast into the generic expression: fðrB ; EÞ þ bVfðrB ; EÞ $ NB ¼ 0

(4.118)

pffiffiffi with b ¼ 2D(rB, E) for the Marshak boundary condition and b ¼ 3DðrB ; EÞ for the Mark boundary condition. The generic boundary condition given by Eq. (4.118) is referred to as a mixed boundary condition.

Reflective boundary condition In the reflective boundary condition, two possible situations are usually encountered: specular boundary conditions and white boundary conditions. In the case of specular boundary condition (i.e. mirror boundary condition), any neutron reaching an outer surface with a given angle q is re-emitted inside the system with the same angle, as illustrated in Fig. 4.3. Using the notations of Fig. 4.3, one has: jðrB ; Uout ; EÞ ¼ jðrB ; Uin ; EÞ

(4.119)

where, if the outgoing direction Uout is represented as a sum between its component normal to the surface at the boundary point rB and a tangential component Uout,k, i.e. Uout ¼ jUout $ NB jNB þ Uout;k for Uout $ NB > 0

(4.120)

the ingoing direction Uin is then given as: Uin ¼  jUout $ NB jNB þ Uout;k

(4.121)

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Modelling of Nuclear Reactor Multi-physics

FIGURE 4.3 Illustration of the specular or mirror boundary condition. Uout represents the outward direction of the neutron reaching the outer surface, whereas Uin gives the inward direction after reflection.

In the case of white boundary condition, neutrons leaving the system through the boundary are isotropically emitted back into the system. In addition, the net current at the boundary should be equal to zero, so that one has: J þ ðrB ; EÞ ¼ J  ðrB ; EÞ

or

Z

jU $ NB jjðrB ; U; EÞd 2 U ¼

ð2pÞ;U $ NB >0

Z

(4.122)

jU $ NB jjðrB ; U; EÞd 2 U

(4.123)

ð2pÞ;U $ NB 0

jU $ NB jjðrB ; U; EÞd 2 U J þ ðrB ; EÞ R for U $ NB < 0 ¼ 2 p jU $ NB jd U

(4.125)

ð2pÞ;U $ NB 0 and U$NB < 0, respectively, leads to: J þ ðrB ; EÞ ¼

fðrB ; EÞ 3 þ 4 4p

Z

fðrB ; EÞ 3 þ 4 4p

(4.130)

jU $ NB jU $ JðrB ; EÞd 2 U

(4.131)

ð2pÞ;U $ NB >0

and J þ ðrB ; EÞ ¼

jU $ NB jU $ JðrB ; EÞd 2 U

Z ð2pÞ;U $ NB 0

(4.132)

U ¼  jU $ NB jNB þ Uk for U $ NB < 0

(4.133)

and one then has:

Z

jU $ NB jU $ JðrB ; EÞd 2 U

ð2pÞ;U $ NB >0

Z

¼

and

2p

Z

0

0

Z

1

 2p m2 NB $ JðrB ; EÞ þ mUk $ JðrB ; EÞ d4dm ¼ NB $ JðrB ; EÞ 3

jU $ NB jU $ JðrB ; EÞd 2 U

ð2pÞ;U $ NB > > > > >
u41 ;m > 7 6 6 > > 5 4N 4N > ; t 42 m 42 > : u42 ;m

! 39 > > > > ; tmþ1 7 > 7> u42 ;mþ1 7= " # !7 7> u41 ;mþ1 7> > > ; tmþ1 5> > ; u42 ;mþ1 u41 ;mþ1

#

(6.16)

In the equation above, it was assumed for the sake of simplicity that there was no cross-dependence between the mono-physics solvers in the linear operator L(t), that could then be rewritten as:


LðtÞ ¼

L41 ðtÞ 0 0 L42 ðtÞ



(6.17)

The possible cross-dependencies are assumed to be all embedded in the non-linear operator N(u,t).

6.4.b

Operator Splitting approaches

Operator splitting or OS approaches rely on the use of each of the mono-physics solvers in their non-altered forms and on some exchange of information/data between the solvers. In the following, three OS approaches are presented: the block Jacobi approach

Chapter 6  Neutronic/thermal-hydraulic coupling

325

or explicit coupling; the block Gauss-Seidel approach or explicit staggered coupling; and fixed-point iterations or semi-implicit coupling (Ragusa and Mahadevan, 2009; Pe´rin, 2016; Watson, 2010). Other approaches have been proposed/used, and the interested reader is referred to the literature on this subject for further details. It should also be emphasized that the terminology used varies greatly between authors. Care has thus to be taken when a coupling scheme is referred to. Based on Eq. (6.16), the true balance equation to be solved for each of the monophysics solver can be written as:

 I  qDtm L4k ðtmþ1 Þ u4k ;mþ1

 ¼ I þ ð1  qÞDtm L4k ðtm Þ u4k ;m " " # ! " # !# u4k ;m u4k ;mþ1 þDtm ð1  qÞN4k ; tm þ qN4k ; tmþ1 u4lsk ;m u4lsk ;mþ1

(6.18)

In the block Jacobi coupling or explicit coupling, each mono-physics solver is solved at tmþ1 using the non-linearities that arise from the other mono-physics solver evaluated at the previous time step tm. This means that, in Eq. (6.18), the term 


N4k

is replaced by

  u4k ;mþ1 ; tmþ1 u4lsk ;mþ1


N4k

  u4k ;mþ1 ; tmþ1 u4lsk ;m

Each of the mono-physics problems is solved according to:

 I  qDtm L4k ðtmþ1 Þ u4k ;mþ1

 z I þ ð1  qÞDtm L4k ðtm Þ u4k ;m " " # ! " # !# u4k ;m u4k ;mþ1 þDtm ð1  qÞN4k ; tm þ qN4k ; tmþ1 u4lsk ;m u4lsk ;m

(6.19)

The exchange of information between the mono-physics solver thus only occurs when the solution to each of the mono-physics problems is available at tmþ1. Since the update of the non-linear coupling terms is never performed for the current time step tmþ1 being solved, the update of such terms lags behind, resulting in the true balance equation given by Eq. (6.18) not being fulfilled. The resulting approximation is often referred to as a non-linear inconsistency being introduced in such a scheme. It should finally be emphasized that despite the name of block Jacobi coupling, there is no iteration between the mono-physics solvers to resolve the inconsistencies. In addition, although the coupling scheme is sometimes labelled as explicit, each mono-physics solver can use some implicitness in the resolution of the time-dependence. In the block Gauss-Seidel coupling or explicit staggered coupling, a mono-physics solver, for instance 41, is first solved at tmþ1 using the non-linearities that arise from

326

Modelling of Nuclear Reactor Multi-physics

the other mono-physics solver 42 evaluated at the previous time step tm. This means that, in Eq. (6.18), the term 


  u41 ;mþ1 ; tmþ1 u42 ;mþ1




  u41 ;mþ1 ; tmþ1 u42 ;m

N41

is replaced by N41

The mono-physics problem 41 is thus solved at tmþ1 according to:

 I  qDtm L41 ðtmþ1 Þ u41 ;mþ1

 z I þ ð1  qÞDtm L41 ðtm Þ u41 ;m " " # ! "  # !# u41 ;m u41 ;mþ1 ; tm þ qN41 ; tmþ1 þDtm ð1  qÞN41 u42 ;m u42 ;m

(6.20)

Thereafter, the mono-physics problem 42 is solved at tmþ1, using the solution u41 ;mþ1 evaluated at tmþ1. This means that, in Eq. (6.18), the term 


  u41 ;mþ1 ; tmþ1 : u42 ;mþ1




  u41 ;mþ1 ; tmþ1 : u42 ;mþ1

N4 2

is replaced by N4 2

The mono-physics problem 42 is thus solved at tmþ1 according to:  I  qDtm L42 ðtmþ1 Þ u42 ;mþ1

 z I þ ð1  qÞDtm L42 ðtm Þ u42 ;m " " # ! "  # !# u41 ;m u41 ;mþ1 ; tm þ qN42 ; tmþ1 þDtm ð1  qÞN42 u42 ;m u42 ;mþ1

(6.21)

The intermediate solution u41 ;mþ1 for the mono-physics solver 41 at time step tmþ1 is thus computed using the non-linear coupling terms with the mono-physics solver 42 at time step tm. Since the update of the non-linear coupling terms in the mono-physics solver 41 is never performed for the current time step tmþ1 being solved, the update of such terms lags behind, resulting in the true balance equation given by Eq. (6.18) not being fulfilled for the mono-physics solver 41. For the mono-physics solver 42, despite the apparent resolution of the non-linear coupling terms at time step tmþ1, the nonlinear coupling term related to the mono-physics solver 41 uses the solution u41 ;mþ1 that is inexact. The resulting approximations in such a scheme are also referred to as non-linear inconsistencies. It should finally be emphasized that despite the name of block Gauss-Seidel coupling, there is no iteration between the mono-physics solvers to resolve the inconsistencies. In addition, although the coupling scheme is sometimes

Chapter 6  Neutronic/thermal-hydraulic coupling

327

labelled as explicit staggered, each mono-physics solver can use some implicitness in the resolution of the time-dependence. The main drawbacks of the block Jacobi and block Gauss-Seidel schemes lie with the fact that the non-linear coupling terms are never fully resolved at the time step being computed. Even if each mono-physics solved might use some implicit timediscretization schemes, small time steps need to be used to damp the effect of the non-linear coupling terms lagging behind, either fully (block Jacobi) or partially (block Gauss-Seidel). In the semi-implicit coupling, iterations on the block Gauss-Seidel coupling scheme explained above are performed in order to resolve the non-linear coupling terms at the time step tmþ1. An outer iteration loop is thus added to the block Gauss-Seidel coupling scheme. For the first iteration, a mono-physics solver, for instance 41, is first solved at tmþ1 using the non-linearities that arise from the other mono-physics solver 42 evaluated at the previous time step tm. This means that, in Eq. (6.18), the term 


  u41 ;mþ1 ; tmþ1 u42 ;mþ1

"

# ! u141 ;mþ1 ; tmþ1 u42 ;m

N41

is replaced by N41

where the superscript 1 in u141 ;mþ1 denotes the first update of the solution for the monophysics problem 41. The mono-physics problem 41 is thus solved at tmþ1 according to:

 I  qDtm L41 ðtmþ1 Þ u141 ;mþ1

 z I þ ð1  qÞDtm L41 ðtm Þ u41 ;m " " # ! " 1 # !# u41 ;m u41 ;mþ1 ; tm þ qN41 ; tmþ1 þDtm ð1  qÞN41 u42 ;m u42 ;m

(6.22)

Thereafter, the mono-physics problem 42 is solved at tmþ1, using the solution u141 ;mþ1 evaluated at tmþ1. This means that, in Eq. (6.18), the term 


N42

is replaced by.

u42 ;mþ1 "

N42

u41 ;mþ1

u141 ;mþ1 u142 ;mþ1

  ; tmþ1 #

!

; tmþ1

where the superscript 1 in u142 ;mþ1 denotes the first update of the solution for the monophysics problem 42.

328

Modelling of Nuclear Reactor Multi-physics

The mono-physics problem 42 is thus solved at tmþ1 according to:

 I  qDtm L42 ðtmþ1 Þ u142 ;mþ1

 z I þ ð1  qÞDtm L42 ðtm Þ u42 ;m " " # ! " 1 # !# u41 ;mþ1 u41 ;m þDtm ð1  qÞN42 ; tm þ qN42 ; tmþ1 u42 ;m u142 ;mþ1

(6.23)

The process is repeated by replacing at tmþ1 in the mono-physics solver 41 the nonlinear coupling term arising from the other mono-physics solver 42 with the intermediate solution u142 ;mþ1 evaluated at the current time step tmþ1. If n ¼ 0, 1, . denotes the outer iteration number, the scheme can be seen as replacing in Eq. (6.18), the term 


N41

by

" N4 1

  u41 ;mþ1 ; tmþ1 u42 ;mþ1

unþ1 41 ;mþ1

#

! ; tmþ1

un42 ;mþ1

when solving the first mono-physics problem 41, and as replacing the term 


N42

by

u42 ;mþ1 "

N4 2

u41 ;mþ1

unþ1 41 ;mþ1 unþ1 42 ;mþ1

  ; tmþ1 #

! ; tmþ1

when solving the second mono-physics problem 42. Each of the mono-physics problems are solved sequentially as:

and

 I  qDtm L41 ðtmþ1 Þ unþ1 41 ;mþ1

 z I þ ð1  qÞDtm L41 ðtm Þ u41 ;m " " # ! " nþ1 # !# u41 ;mþ1 u41 ;m þDtm ð1  qÞN41 ; tm þ qN41 ; tmþ1 un42 ;mþ1 u42 ;m

(6.24)

 I  qDtm L42 ðtmþ1 Þ unþ1 42 ;mþ1

 z I þ ð1  qÞDtm L42 ðtm Þ u42 ;m " " # ! " nþ1 # !# u41 ;mþ1 u41 ;m ; tm þ qN42 ; tmþ1 þDtm ð1  qÞN42 u42 ;m unþ1 42 ;mþ1

(6.25)



Chapter 6  Neutronic/thermal-hydraulic coupling

329

for increasing values of n until some convergence criteria is fulfilled. Since this scheme
 u41 ;mþ1 leads to the successive updates of the solution vector as: u42 ;mþ1 "

u141 ;mþ1 u142 ;mþ1

#

"

/

u241 ;mþ1

#

u242 ;mþ1

"

# un41 ;mþ1 /./ n : u42 ;mþ1

the coupling scheme is also referred to as fixed-point iterations or Picard iterations. Despite its simplicity in nature, many iterations are usually required to converge the non-linear coupling terms. In addition, this scheme is prone to numerical oscillations. The effect of such oscillations can be damped by introducing some relaxation factor. Before a new outer iteration is performed, the solution at the iteration number nþ1 is replaced by: "

unþ1 41 ;mþ1 unþ1 42 ;mþ1

#

"

)u

unþ1 41 ;mþ1 unþ1 42 ;mþ1

#

"

un4 ;mþ1 þ ð1  uÞ n1 u42 ;mþ1

#

(6.26)

n where unþ1 4k ;mþ1 represents the solution to Eqs. (6.24) and (6.25) and u4k ;mþ1 the solution at the outer iteration number n. In Eq. (6.26), u is a user-defined relaxation factor. The fixed-point iterations scheme is sometimes referred to in the literature as an implicit coupling as well.

6.4.c

Integrated approaches

As the OS approaches demonstrate, the convergence of the non-linear coupling terms between the mono-physics solvers is either never achieved (block Jacobi and block Gauss-Seidel coupling schemes) or achieved at the cost of many outer iterations (fixedpoint iterations). One of the approaches solving the multi-physics problem in a more integrated manner is the Jacobian-free Newton Krylov method.

Principle In the Jacobian-Free Newton Krylov method or JFNK method, the original multi-physics problem given by Eq. (6.15) and to be solved at time step tmþ1 is rewritten as (Knoll and Keyes, 2004): Hðumþ1 Þ ¼ 0

(6.27)

with Hðumþ1 Þ ¼ ½I  qDtm Lðtmþ1 Þumþ1

(6.28)

½I þ ð1  qÞDtm Lðtm Þum  Dtm ½ð1  qÞNðum ; tm Þ þ qNðumþ1 ; tmþ1 Þ

H(u) is called the residual. Searching the solution to Eq. (6.27) can thus be seen as minimizing the residual (ideally having it equal to 0). Making a first-order Taylor expansion of H near a given value umþ1,0 leads to: Hðumþ1 Þ z Hðumþ1;0 Þ þ Jðumþ1;0 Þ½umþ1  umþ1;0 

(6.29)

330

Modelling of Nuclear Reactor Multi-physics

where J(umþ1,0) represents the Jacobian of the function H in umþ1,0. J(umþ1,0) is a square matrix having an order equal to the size n of umþ1,0 and is formally given as: 

Jðumþ1;0 Þ ¼

vHi vuj;mþ1;0



(6.30) 1in;1jn

where vHi/vuj,mþ1,0 represents the partial derivative of the i-th component of the vector H(umþ1,0) with respect to the j-th component of the vector umþ1,0. Using Eq. (6.27) and rearranging gives: Jðumþ1;0 Þ½umþ1  umþ1;0  z  Hðumþ1;0 Þ

(6.31)

umþ1 z umþ1;0  ½Jðumþ1;0 Þ1 Hðumþ1;0 Þ

(6.32)

and Assuming that umþ1,0 is a given start vector, Eq. (6.32) defines an iterative process that can be more explicitly written as: umþ1;kþ1 z umþ1;k  ½Jðumþ1;k Þ1 Hðumþ1;k Þ

(6.33)

where k is the iteration number. This iterative scheme, referred to as a Newton or Newton-Raphson method (Ypma, 1995), leads to a new vector umþ1,kþ1 that, under given conditions, should be closer to the actual root of Eq. (6.27) than umþ1,k is. Although this method is often used for solving non-linear problems, it requires at each new iteration both the actual evaluation of the Jacobian and its inversion. Depending of the number of variables, these tasks can be computationally very expensive. The essence of the JFNK method is to reformulate the above problem as: Jðumþ1;k Þdumþ1;k ¼  Hðumþ1;k Þ

(6.34)

dumþ1;k ¼ umþ1;kþ1  umþ1;k

(6.35)

with and not performing the actual evaluation of the Jacobian and its inversion. This is achieved by noticing that Eq. (6.34) is equivalent to a linear algebra problem of the form Av ¼ b where A and b represent a given matrix and a given vector, respectively, and v is thus the vector solution to the equation Av ¼ b. This type of equation can be very efficiently solved using a Krylov subspace method, and more specifically the Generalized Minimal Residual (GMRES) method (Saad and Schultz, 1986), already presented in Section 4.3.c of Chapter 4. This solution procedure is highlighted below. Within each Newton iteration k, a second iterative scheme is thus applied. It consists of the construction of a Krylov subspace of dimension t: