Applied Reactor Physics (3rd Ed.)
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Langue : Anglais

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Ouvrage 410 p. · 16.2x23.8 cm · Broché · 
ISBN : 9782553017353 EAN : 9782553017353
Presses internationales Polytechnique

Reactor physics is the discipline devoted to the study of interactions between neutrons and matter in a nuclear reactor. This third edition of Applied Reactor Physics, addresses the fundamentals of reactor physics. Legacy numerical techniques are introduced with sufficient details to help readers implement them in Matlab. The fundamental approaches presented here provide a solid foundation for the more advanced and proprietary techniques that readers may encounter in production environments.

Applied Reactor Physics emphasizes the algorithmic nature of the numerical solution techniques used in reactor physics. Many numerical solution approaches described in the book are accompanied by Matlab scripts, and readers are encouraged to write short Matlab scripts of their own in order to solve the end-of-chapter exercises.

The third edition includes key improvements in the text, a completely rewritten appendix on numerical methods, new end-of-chapter exercises, and an extended bibliography.

Contents

1 Introduction 1

2 Cross sections and nuclear data 5

2.1 Solid angles and spherical harmonics . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2 Dealing with distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.3 Dynamics of a scattering reaction . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.3.1 Collision of a neutron with a nucleus initially at rest . . . . . . . . . . . . 16

2.4 Definition of a cross section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.5 Formation of a compound nucleus . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.5.1 The single level Breit-Wigner formulas . . . . . . . . . . . . . . . . . . . . 28

2.5.2 Low-energy variation of cross sections . . . . . . . . . . . . . . . . . . . . 31

2.6 Thermal agitation of nuclides and binding effects . . . . . . . . . . . . . . . . . . 33

2.6.1 Numerical convolution of cross sections . . . . . . . . . . . . . . . . . . . 34

2.6.2 Convolution of Breit-Wigner cross sections . . . . . . . . . . . . . . . . . 36

2.6.3 Convolution of a constant cross section . . . . . . . . . . . . . . . . . . . . 39

2.6.4 Convolution of the differential scattering cross section . . . . . . . . . . . 41

2.6.5 Effects of molecular or metallic binding . . . . . . . . . . . . . . . . . . . 46

2.7 Expansion of the differential cross sections . . . . . . . . . . . . . . . . . . . . . . 50

2.8 Calculation of the probability tables . . . . . . . . . . . . . . . . . . . . . . . . . 53

2.9 Production of an isotopic cross-section library . . . . . . . . . . . . . . . . . . . . 54

2.9.1 Photo-atomic interaction data . . . . . . . . . . . . . . . . . . . . . . . . . 59

2.9.2 Delayed neutron data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

2.9.3 An overview of DRAGR . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

3 The transport equation 69

3.1 The particle flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

3.2 Derivation of the transport equation . . . . . . . . . . . . . . . . . . . . . . . . . 72

3.2.1 The characteristic form of the transport equation . . . . . . . . . . . . . . 74

3.2.2 The integral form of the transport equation . . . . . . . . . . . . . . . . . 75

3.2.3 Boundary and continuity conditions . . . . . . . . . . . . . . . . . . . . . 76

3.3 Source density in reactor physics . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

3.3.1 The steady-state source density . . . . . . . . . . . . . . . . . . . . . . . . 77

3.3.2 The transient source density . . . . . . . . . . . . . . . . . . . . . . . . . . 80

3.4 The transport correction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

3.5 Multigroup discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

3.5.1 Multigroup steady-state transport equation . . . . . . . . . . . . . . . . . 85

3.5.2 Multigroup transient transport equation . . . . . . . . . . . . . . . . . . . 87

3.6 The first-order streaming operator . . . . . . . . . . . . . . . . . . . . . . . . . . 89

3.6.1 Cartesian coordinate system . . . . . . . . . . . . . . . . . . . . . . . . . . 89

3.6.2 Cylindrical coordinate system . . . . . . . . . . . . . . . . . . . . . . . . . 91

3.6.3 Spherical coordinate system . . . . . . . . . . . . . . . . . . . . . . . . . . 93

3.7 The spherical harmonics method . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

3.7.1 The Pn method in 1D slab geometry . . . . . . . . . . . . . . . . . . . . . 95

3.7.2 The Pn method in 1D cylindrical geometry . . . . . . . . . . . . . . . . . 99

3.7.3 The Pn method in 1D spherical geometry . . . . . . . . . . . . . . . . . . 104

3.7.4 The simplified Pn method in 2D Cartesian geometry . . . . . . . . . . . . 106

3.8 The collision probability method . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

3.8.1 The interface current method . . . . . . . . . . . . . . . . . . . . . . . . . 110

3.8.2 Scattering-reduced matrices and power iteration . . . . . . . . . . . . . . 113

3.8.3 Slab geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

3.8.4 Cylindrical 1D geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

3.8.5 Spherical 1D geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

3.8.6 Unstructured 2D finite geometry . . . . . . . . . . . . . . . . . . . . . . . 126

3.9 The discrete ordinates method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

3.9.1 Quadrature sets in the method of discrete ordinates . . . . . . . . . . . . 133

3.9.2 The difference relations in 1D slab geometry . . . . . . . . . . . . . . . . 138

3.9.3 The difference relations in 1D cylindrical geometry . . . . . . . . . . . . . 140

3.9.4 The difference relations in 1D spherical geometry . . . . . . . . . . . . . . 144

3.9.5 The difference relations in 2D Cartesian geometry . . . . . . . . . . . . . 146

3.9.6 Synthetic acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

3.10 The method of characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

3.10.1 The MOC integration strategy . . . . . . . . . . . . . . . . . . . . . . . . 151

3.10.2 Unstructured 2D finite geometry . . . . . . . . . . . . . . . . . . . . . . . 157

3.10.3 The algebraic collapsing acceleration . . . . . . . . . . . . . . . . . . . . . 162

3.11 The multigroup Monte Carlo method . . . . . . . . . . . . . . . . . . . . . . . . . 167

3.11.1 Mathematical background . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

3.11.2 Rejection techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

3.11.3 The random walk of a neutron . . . . . . . . . . . . . . . . . . . . . . . . 176

3.11.4 Criticality calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

3.11.5 Monte Carlo reaction estimators . . . . . . . . . . . . . . . . . . . . . . . 189

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

4 Elements of lattice calculation 195

4.1 A historical overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196

4.2 Neutron slowing-down and resonance self-shielding . . . . . . . . . . . . . . . . . 199

4.2.1 Elastic slowing down . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201

4.2.2 A review of resonance self-shielding approaches . . . . . . . . . . . . . . . 204

4.2.3 The Livolant-Jeanpierre approximations . . . . . . . . . . . . . . . . . . . 205

4.2.4 The physical probability tables . . . . . . . . . . . . . . . . . . . . . . . . 209

4.2.5 The statistical subgroup equations . . . . . . . . . . . . . . . . . . . . . . 212

4.2.6 The multigroup equivalence procedure . . . . . . . . . . . . . . . . . . . . 215

4.3 The neutron leakage model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217

4.3.1 The Bn leakage calculation . . . . . . . . . . . . . . . . . . . . . . . . . . 218

4.3.2 The homogeneous fundamental mode . . . . . . . . . . . . . . . . . . . . . 219

4.3.3 The heterogeneous fundamental mode . . . . . . . . . . . . . . . . . . . . 224

4.3.4 Introduction of leakage rates in a lattice calculation . . . . . . . . . . . . 227

4.3.5 Introduction of leakage rates with collision probabilities . . . . . . . . . . 229

4.3.6 Full-core calculations in diffusion theory . . . . . . . . . . . . . . . . . . . 231

4.3.7 Full-core calculations in transport theory . . . . . . . . . . . . . . . . . . 232

4.4 The SPH equivalence technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234

4.4.1 Definition of the macro balance relations . . . . . . . . . . . . . . . . . . . 235

4.4.2 Definition of the SPH factors . . . . . . . . . . . . . . . . . . . . . . . . . 236

4.4.3 Iterative calculation of the SPH factors . . . . . . . . . . . . . . . . . . . 240

4.5 Isotopic depletion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241

4.5.1 The power normalization . . . . . . . . . . . . . . . . . . . . . . . . . . . 244

4.5.2 The saturation model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245

4.5.3 The integration factor method . . . . . . . . . . . . . . . . . . . . . . . . 247

4.5.4 Depletion of heavy isotopes . . . . . . . . . . . . . . . . . . . . . . . . . . 249

4.6 Creation of the reactor database . . . . . . . . . . . . . . . . . . . . . . . . . . . 252

4.6.1 Selected information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252

4.6.2 Database information structure . . . . . . . . . . . . . . . . . . . . . . . . 254

4.7 A presentation of DRAGON . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255

4.7.1 A DRAGON tutorial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259

5 Full-core calculations 271

5.1 The steady-state diffusion equation . . . . . . . . . . . . . . . . . . . . . . . . . . 274

5.1.1 The Fick law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274

5.1.2 Continuity and boundary conditions . . . . . . . . . . . . . . . . . . . . . 276

5.1.3 The finite homogeneous reactor . . . . . . . . . . . . . . . . . . . . . . . . 278

5.1.4 The heterogeneous 1D slab reactor . . . . . . . . . . . . . . . . . . . . . . 280

5.2 Discretization of the neutron diffusion equation . . . . . . . . . . . . . . . . . . . 284

5.2.1 Mesh-corner finite differences . . . . . . . . . . . . . . . . . . . . . . . . . 285

5.2.2 Mesh-centered finite differences . . . . . . . . . . . . . . . . . . . . . . . . 288

5.2.3 A primal variational formulation . . . . . . . . . . . . . . . . . . . . . . . 290

5.2.4 The Lagrangian finite-element method . . . . . . . . . . . . . . . . . . . . 292

5.2.5 The analytic nodal method in 2D Cartesian geometry . . . . . . . . . . . 296

5.3 Generalized perturbation theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 303

5.3.1 Mathematical background . . . . . . . . . . . . . . . . . . . . . . . . . . . 303

5.3.2 State variables and reactor characteristics . . . . . . . . . . . . . . . . . . 306

5.3.3 Computing the Jacobian using the implicit approach . . . . . . . . . . . . 308

5.3.4 Computing the Jacobian using the explicit approach . . . . . . . . . . . . 309

5.4 Space-time kinetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310

5.4.1 Point-kinetics equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311

5.4.2 The implicit temporal scheme . . . . . . . . . . . . . . . . . . . . . . . . . 316

5.4.3 The space-time implicit scheme . . . . . . . . . . . . . . . . . . . . . . . . 318

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321

Answers to Problems 327

A Tracking of 1D and 2D geometries 335

A.1 Tracking of 1D slab geometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335

A.2 Tracking of 1D cylindrical and spherical geometries . . . . . . . . . . . . . . . . . 337

A.3 The theory behind sybt1d . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 340

A.4 Tracking of 2D square pincell geometries . . . . . . . . . . . . . . . . . . . . . . . 341

A.5 The theory behind sybt2d . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346

B Special functions with Matlab 349

B.1 Function taben . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349

B.2 Function akin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350

C Numerical methods 353

C.1 Solution of a linear system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353

C.1.1 Gauss elimination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353

C.1.2 Cholesky factorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355

C.1.3 QR factorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 358

C.1.4 Iterative approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 360

C.2 Solution of an eigenvalue problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 367

C.2.1 The inverse power method . . . . . . . . . . . . . . . . . . . . . . . . . . . 368

C.2.2 The inverse power method without inversion . . . . . . . . . . . . . . . . 374

C.2.3 The preconditioned power method . . . . . . . . . . . . . . . . . . . . . . 380

C.2.4 The Hotelling deflation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381

C.2.5 The multigroup partitioning . . . . . . . . . . . . . . . . . . . . . . . . . . 383

C.2.6 Convergence acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . . 385

C.3 Solution of a fixed-source eigenvalue problem . . . . . . . . . . . . . . . . . . . . 388

C.3.1 The inverse power method . . . . . . . . . . . . . . . . . . . . . . . . . . . 389

C.3.2 The preconditioned power method with variational acceleration . . . . . . 391

Bibliography 394

Index 405

This book is intended for graduate-level readers who have no existing knowledge of reactor physics. It was initially written as support for graduate-level courses offered in the regular program of the Institut de génie nucléaire (nuclear engineering institute) at Polytechnique Montréal. The book contains sufficient material for an instructor to build three or four graduate courses based on its content.

Alain Hébert has been a professor in the Institut de génie nucléaire at Polytechnique Montréal since 1981. From 1995 to 2001, he worked at the Commissariat à l'Énergie Atomique, located in Saclay, France. During this period, he led the team that developed the APOLLO2 lattice code, an important component of the Science and Arcadia packages at Areva. Back in Montréal, he participated in the development of the DRAGON lattice and TRIVAC reactor codes, both available as open-source software.