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Nuclear Reactor Physics and Engineering

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Nuclear Reactor Physics and Engineering

John C. Lee

ISBN: 978-1-119-58232-8 January 2020 624 Pages

Hardcover
Pre-order
£103.00

Preface ix

Permissions and Copyrights xi

1 Nuclear Power Plants 1

1.1 History and Current Status of Nuclear Power Plants 1

1.2 Basic Features of Nuclear Power Plants 3

1.3 Pressurized Water Reactor System 4

1.4 Boiling Water Reactor System 9

1.5 Advanced Reactor Designs 14

References 20

Problems 22

2 Neutron-nucleus Reactions and Neutron Cross Section 23

2.1 Neutron-nucleus Reaction Probability and Neutron Cross Section 24

2.2 Mechanisms of Neutron-nucleus Interactions 25

2.3 Nuclear Fission Process 27

2.4 Two-body Collision Mechanics and Center-of-mass System 31

2.5 Single-Level Breit-Wigner Formula for Resonance Reaction 36

2.6 Differential Scattering Cross Section and Scattering Kernel 39

2.6.1 Differential Microscopic Scattering Cross Section 39

2.6.2 Scattering Kernel for Isotropic Scattering in CM Frame 40

2.7 Further Remarks on Neutron Cross Section 42

References 45

Problems 48

3 Neutron Flux, Reaction Rate, and Effective Cross Section 51

3.1 Neutron Flux and Current 52

3.2 Rate of Neutron-nucleus Interactions 57

3.3 Neutron Energy Distribution and Effective Thermal Cross Section 59

3.4 Application to a 1/v-Absorber 63

References 64

Problems 64

4 Derivation of the Neutron Diffusion Equation 67

4.1 Basic Assumptions for Neutron Balance Statement 68

4.2 Neutron Balance Equation 69

4.3 Neutron Source Term 72

4.4 Fick’s Law of Neutron Current 73

4.5 Neutron Transport Equation and P1 Approximation 76

4.6 Remarks on the Diffusion Coefficient 80

4.7 Limitations of Neutron Diffusion Theory 81

4.8 One-Group Neutron Diffusion Equation 82

4.9 Summary Discussion on the Diffusion Equation 83

References 83

Problems 84

5 Applications of One-Group Neutron Diffusion Equation 85

5.1 Boundary Conditions for Diffusion Equation 86

5.2 Solution of the Steady-state Diffusion Equation 89

5.2.1 Flux in Non-multiplying Media with Localized Sources 89

5.2.2 Flux in Non-multiplying Media with Distributed Sources 96

5.3 Neutron Flux in Multiplying Medium and Criticality Condition 99

5.3.1 Criticality and Buckling 99

5.3.2 Effective Multiplication Factor 100

5.3.3 Eigenfunctions of Diffusion Equation and Buckling 102

5.4 Four-and Six-factor Formulas for Multiplication Factor 106

5.5 Concluding Remarks 108

References 108

Problems 108

6 Numerical Solution of Neutron Diffusion Equation 113

6.1 Finite Difference Form of Diffusion Equation 114

6.2 Flux Solution Algorithm: Inner Iteration 117

6.3 Boundary Conditions for Difference Equation 119

6.4 Source or Outer Iteration 121

6.5 Relative Power Distribution and Overall Flow Chart 123

6.6 Single Channel Flux Synthesis 126

6.7 Multi-dimensional Finite Difference Formulation 128

6.7.1 Two-dimensional Matrix Formulation 128

6.7.2 Three-dimensional Formulation 132

6.7.3 Convergence Properties of Matrix Iteration Schemes 133

6.8 Coarse-mesh Diffusion Equation Solver 134

6.8.1 Nodal Expansion Method 134

6.8.2 Pin Power Reconstruction Algorithm 136

6.9 Krylov Subspace Method as Diffusion Equation Solver 137

References 140

Problems 141

7 Applications of Two-Group Neutron Diffusion Equation 143

7.1 Derivation of Multi-group Neutron Diffusion Equation 144

7.2 Steady-state Multi-group Diffusion Equation 147

7.3 Two-group Form of Effective Multiplication Factor 149

7.4 General Two-group Diffusion Analysis 152

References 154

Problems 154

8 Nuclear Reactor Kinetics 157

8.1 Derivation of Point Kinetics Equation 158

8.1.1 Representation of Delayed Neutron Production 158

8.1.2 Point Kinetics Approximation 159

8.1.3 One-group Delayed Neutron Model 161

8.2 Solution of Point Kinetics Equation without Feedback 162

8.2.1 Step Insertion of Reactivity 162

8.2.2 Prompt Jump or Zero Lifetime Approximation 165

8.2.3 Inhour Equation 166

8.2.4 Linearized Kinetics Equation and Transfer Function 168

8.2.5 Infinite Delayed Approximation 171

8.3 State Space Representation of Point Kinetics Equation 171

8.4 Point Kinetics Equation with Feedback 174

8.4.1 The Ergen-Weinberg Model 174

8.4.2 The Nordheim-Fuchs Model 177

8.5 Reactivity Measurements 178

8.6 System Stability Analysis 181

8.7 Reactor Noise Analysis and Correlation Functions 184

8.7.1 Reactor Noise Analysis 185

8.7.2 Correlation Function Techniques 188

8.8 Point Reactor and Space-dependent Reactor Kinetics 190

References 191

Problems 191

9 Fast Neutron Spectrum Calculation 195

9.1 Neutron Balance Equation and Slowing Down Density 196

9.2 Elastic Scattering and Lethargy Variable 200

9.3 Neutron Slowing Down in Infinite Medium 201

9.3.1 Slowing Down in First Collision Interval 202

9.3.2 Slowing Down below First Collision Interval 206

9.4 Resonance Escape Probability 209

9.4.1 Effective Resonance Integral 209

9.4.2 Energy Self-shielding Factor 211

9.4.3 Wide Resonance Approximation 212

9.4.4 Probability Table or Subgroup Method 213

9.5 Doppler Broadening of Resonances 215

9.5.1 Qualitative Description of Doppler Broadening 215

9.5.2 Analytical Treatment of Doppler Broadening 217

9.6 Fermi Age Theory 220

9.7 Comments on Lattice Physics Analysis 223

References 224

Problems 224

10 Perturbation Theory and Adjoint Flux 227

10.1 Operator Notation for Neutron Diffusion Equation 228

10.2 Adjoint Operator and Adjoint Flux 228

10.3 First-order Perturbation Theory 230

10.4 Adjoint Flux for Control Rod Worth Calculation 232

10.5 Adjoint Flux for Variational Formulation 234

10.6 Adjoint Flux for Detector Response Calculation 235

10.7 Adjoint Formulation for Flux Perturbation Calculation 236

10.8 Concluding Remarks on Adjoint Flux 240

References 240

Problems 240

11 Lattice Physics Analysis of Heterogeneous Cores 243

11.1 Material Heterogeneity and Flux Distribution in Unit Cell 245

11.2 Neutronic Advantages of Fuel Lumping 246

11.3 Diffusion Theory Model for Thermal Utilization 250

11.4 Improved Method for Thermal Disadvantage Factor 254

11.4.1 Blackness or Simplified Collision Probability Method 254

11.4.2 Amouyal-Benoist-Horowitz Method 255

11.5 Resonance Escape Probability for Heterogeneous Cell 257

11.5.1 Spatial Self-shielding Factor for Heterogeneous Unit Cell 258

11.5.2 Engineering Approaches for Resonance Integral Calculation 262

11.5.3 Implementation in the CPM-3 Code 264

11.6 Thermal Spectrum Calculation 265

11.6.1 Wigner-Wilkins Model 266

11.6.2 Qualitative Behavior of Thermal Neutron Spectrum 267

11.7 Integral Transport Methods 268

11.8 B1 Formulation for Spectrum Calculation 271

11.8.1 Basic Structure of B1 Formulation 271

11.8.2 Numerical Solution of B1 Equations 274

11.9 Lattice Physics Methodology for Fast Reactor 276

11.9.1 Bondarenko Formulation for Self-Shielding Factor 276

11.9.2 MC2-3 Code 278

11.9.3 ERANOS System 278

11.10 Monte Carlo Lattice Physics Analysis 278

11.11 Overall Reactor Physics Analysis 279

References 279

Problems 282

12 Nuclear Fuel Cycle Analysis and Management 285

12.1 Nuclear Fuel Management 286

12.2 Key Nuclide Chains for Nuclear Fuel Cycle 289

12.3 Fuel Depletion Model 290

12.3.1 Fuel Depletion Equation 291

12.3.2 Solution of Pointwise Depletion Equation 292

12.3.3 Fuel Depletion Equation in Global MGD Calculation 293

12.3.4 Simple Model for Fuel Burnup Estimation 296

12.4 Equilibrium Cycle and Mass Balance 297

12.4.1 Nuclide Balance Statement 297

12.4.2 Material Flow Sheet 298

12.4.3 REBUS Equilibrium Inventory Calculation 300

12.5 Simplified Cycling Model 301

12.5.1 Reactivity-based Instant Cycling Method 302

12.5.2 Application of Instant Cycling Method 303

12.6 Xenon Fission Product Buildup 307

12.6.1 Mechanism for 135Xe Production and Balance Equation 307

12.6.2 Time-domain Solution of Xe-I Balance Equation 308

12.6.3 Effect of Samarium Buildup 311

12.7 General Incore Management Considerations 312

12.7.1 Reactivity Variation over Fuel Cycle 312

12.7.2 Thermal-hydraulic Feedback and Power Distribution 313

12.7.3 Control Requirements for Light Water Reactor 313

12.7.4 Power Distribution Control 315

12.8 Radioactive Waste and Used Nuclear Fuel Management 317

12.8.1 Classification of Radioactive Waste 317

12.8.2 Characteristics of Radioactive Waste 317

12.8.3 Status of Used Nuclear Fuel Inventory 319

12.8.4 Partition and Transmutation of Waste 320

References 323

Problems 325

13 Thermal-Hydraulic Analysis of Reactor Systems 327

13.1 Empirical Laws for Energy and Momentum Transport 328

13.1.1 Fourier’s Law of Heat Conduction 329

13.1.2 Newton’s Law of Viscosity 329

13.1.3 Newton’s Law of Cooling 330

13.2 Derivation of Fluid Conservation Equations 331

13.2.1 Equation of Continuity 331

13.2.2 Equation of Motion and Navier-Stokes Equation 332

13.2.3 Equations of Energy Conservation 334

13.2.4 Comments on the Fluid Conservation Equations 337

13.3 Simple Solutions of Fluid Conservation Equations 337

13.3.1 Heat Conduction in Cylindrical Fuel Rod 344

13.3.2 Heat Conduction through Composite Wall 346

13.3.3 Forced Convection in Laminar Flow 348

13.3.4 Velocity Distribution in Turbulent Flow 351

13.3.5 Friction Factor and Hydraulic Diameter 352

13.4 Conservation Equations for Channel Flow 353

13.4.1 Equation of Continuity 353

13.4.2 Equation of Motion and Pressure Drop 354

13.4.3 Equation of Energy Conservation 355

13.5 Axial Temperature Distribution in Reactor Core 356

13.5.1 Power Distribution and Heat Flux in Reactor Core 356

13.5.2 Axial Temperature Profile in PWR Core 357

13.5.3 Axial Temperature Profile in BWR Core 360

13.5.4 Hot Channel Factors 361

13.6 Boiling Heat Transfer and Two-Phase Flow 364

13.6.1 Pool Boiling Regimes 364

13.6.2 Flow Boiling Regimes and Two-Phase Flow Patterns 365

13.6.3 Homogeneous Equilibrium Flow Model 367

13.6.4 Slip Flow Model 368

13.6.5 Drift Flux Model 374

13.7 Thermal Hydraulic Limitations and Power Capability 376

13.7.1 DNB Ratio and Number of Fuel Rods Reaching DNB 376

13.7.2 Non-uniform Heat Flux Correction 378

13.7.3 Iterative Determination of DNB Ratio 380

13.7.4 Power Capability Determination 381

13.8 Thermal-Hydraulic Models for Nuclear Plant Analysis 382

13.8.1 Light Water Reactor System Modeling Codes 384

13.8.2 Subchannel Analysis Codes 387

13.8.3 Sodium-cooled Fast Reactor Codes 387

13.8.4 Containment Analysis Codes 389

13.8.5 Computational Fluid Dynamics Codes 390

13.9 Comments on Thermal-Hydraulic Models 391

References 391

Problems 393

14 Power Coefficients of Reactivity 397

14.1 Physical Phenomena Affecting Core Reactivity 398

14.2 Relationship between Reactivity Coefficients 399

14.3 Two-group Representation of Reactivity Feedback 401

14.4 Parametric Dependence of LWR Reactivity Coefficients 402

14.5 Reactivity Coefficients in Sodium-Cooled Fast Reactor 404

14.6 Quasi-static Reactivity Feedback Model for Sodium-Cooled Fast Reactor 405

References 408

Problems 408

15 Nuclear Energy Economics 411

15.1 Electrical Energy Cost 412

15.2 Overview on Engineering Economics 414

15.3 Calculation of Nuclear Electricity Generation Cost 415

15.3.1 Capital Cost 415

15.3.2 Fuel Cost 416

15.3.3 Operation and Maintenance Cost 420

15.3.4 Decommissioning Cost 421

15.4 Impact of Increased Capital and O&M Costs 422

References 423

Problems 424

16 Space-Time Kinetics and Reactor Control 425

16.1 Space-time Reactor Kinetics 426

16.1.1 Numerical Solution of Space-time Kinetics Equation 426

16.1.2 Direct Solution of Space-time Kinetics Equation 427

16.1.3 Quasi-static Formulation of Kinetics Equation 428

16.1.4 Reactivity Determination from Multiple Detectors 430

16.2 Space-time Power Oscillations due to Xenon Poisoning 433

16.2.1 Modal Analysis of Space-time Xenon-power Oscillations 434

16.2.2 Stability of Space-time Xenon-power Oscillations 438

16.2.3 Space-time Xenon-power Oscillations in X-Y plane 442

16.3 Time-optimal Reactor Control 445

16.3.1 Optimal Control of Xenon-induced Transients 445

16.3.2 Control of Spatial Xenon Oscillations 448

16.4 Model Based Reactor Control 452

16.4.1 Linear Quadratic Regulator 452

16.4.2 H2 Controller 454

16.4.3 H Controller 456

16.4.4 Augmented Plant Representation 457

16.5 Alternate Reactor Control Techniques 459

16.6 Kalman Filtering for Optimal System Estimation 463

References 466

Problems 468

17 Elements of Neutron Transport Theory 471

17.1 Collision Probability Method 471

17.1.1 Integral Transport Equation 472

17.1.2 Reciprocity Relationship 474

17.1.3 Transport Kernel and Collision Probability 474

17.2 First-Flight Escape Probability and Dirac Chord Method 476

17.3 Flux Depression Calculation and Blackness 480

17.3.1 Escape Probability and Flux Depression Factor 481

17.3.2 Net Escape Probability and Collision Probability 482

17.3.3 Dancoff Factor for Fuel Lattice 483

17.4 Numerical Solution of Neutron Transport Equation 485

17.4.1 Collision Probability Calculation for Annular Geometry 485

17.4.2 Discrete Ordinates Method 489

17.4.3 Method of Characteristics 491

17.4.4 Monte Carlo Algorithm 492

References 494

Problems 495

Appendix A: Key Physical Constants 497

Appendix B: Comparison of Major Reactor Types 499

References 500

Appendix C: Special Mathematical Functions 503

C.1 Gamma Function 503

C.2 Legendre Polynomial and Spherical Harmonics 505

C.3 Bessel Function 507

C.4 Dirac Delta Function 510

References 510

Appendix D: Integral Transforms 511

D.1 Laplace Transform 511

D.2 Fourier Transform 512

D.3 Jordan’s Lemma 512

References 514

Appendix E: Calculus of Variation for Optimal Control Formulation 515

E.1 Euler-Lagrange and Hamilton Equations 515

E.2 Pontryagin’s Maximum Principle 517

References 521

Appendix F: Kalman Filter Algorithm 523

F.1 Linear Kalman Filter 523

F.2 Unscented Kalman Filter 527

References 528

Answers to Selected Problems 529

Index 541