Nuclear Reactor Physics and Engineering

John C. Lee, PhD, has been on the nuclear engineering faculty at the University of Michigan since 1974 and served as the department chair for six years. He has published two Wiley books, Risk and Safety Analysis of Nuclear Systems (with N.J. McCormick, 2011, 2017 second printing) and Nuclear Reactor Physics and Engineering (2020). He has served on two U.S. National Academy of Sciences committees, including the recent committee on advanced nuclear reactors and used nuclear fuel. Dr. Lee is a Fellow of the American Nuclear Society.

List of Tables xv

List of Figures xvii

Preface xxix

Preface to the Second Edition xxxi

Permissions and Copyrights xxxiii

About the Companion Website xxxv

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 Systems 4

1.4 Boiling Water Reactor Systems 10

1.5 Advanced Reactor Designs 17

References 30

Problems 32

2 Neutron–Nucleus Reaction and Neutron Cross Section 33

2.1 Neutron–Nucleus Reaction Probability and Neutron Cross Section 34

2.2 Mechanisms of Neutron–Nucleus Interaction 35

2.3 Nuclear Fission Process 38

2.4 Two-Body Collision Mechanics and Center-of-Mass System 44

2.5 Single-Level Breit–Wigner Formula for Resonance Reaction 48

2.6 Differential Scattering Cross Section and Scattering Kernel 51

2.7 Further Remarks on Neutron Cross Section 55

References 60

Problems 61

3 Neutron Flux, Reaction Rate, and Effective Cross Section 65

3.1 Neutron Flux and Current 66

3.2 Rate of Neutron–Nucleus Interaction 72

3.3 Neutron Energy Distribution and Effective Thermal Cross Section 75

3.4 Application to a 1∕V-Absorber 79

References 80

Problems 80

4 Derivation of the Neutron Diffusion Equation 83

4.1 Basic Assumptions for Neutron Balance Statement 84

4.2 Neutron Balance Equation 85

4.3 Neutron Source Term 89

4.4 Fick’s Law of Neutron Current 90

4.5 Neutron Transport Equation and P1 Approximation 93

4.6 Remarks on Diffusion Coefficient 98

4.7 Limitations of Neutron Diffusion Theory 100

4.8 One-Group Neutron Diffusion Equation 100

4.9 Summary Discussion of Diffusion Equation 102

References 102

Problems 103

5 Applications of the One-Group Neutron Diffusion Equation 105

5.1 Boundary Conditions for Diffusion Equation 106

5.2 Solution of Steady-State Diffusion Equation 110

5.3 Neutron Flux in Multiplying Medium and Criticality Condition 120

5.4 Four- and Six-Factor Formulas for Multiplication Factor 129

5.5 Concluding Remarks 131

References 131

Problems 132

6 Numerical Solution of the Neutron Diffusion Equation 137

6.1 Finite Difference Form of Diffusion Equation 138

6.2 Flux Solution Algorithm: Inner Iteration 142

6.3 Boundary Conditions for Difference Equation 144

6.4 Source or Outer Iteration 146

6.5 Relative Power Distribution and Overall Flow Chart 148

6.6 Single-Channel Flux Synthesis 151

6.7 Multidimensional Finite Difference Formulation 154

6.8 Coarse-Mesh Diffusion Equation Solver 160

6.9 Krylov Subspace Method as a Diffusion Equation Solver 164

References 168

Problems 169

7 Applications of the Two-Group Neutron Diffusion Equation 171

7.1 Derivation of Multigroup Neutron Diffusion Equation 172

7.2 Steady-State Multigroup Diffusion Equation 176

7.3 Two-Group Form of Effective Multiplication Factor 178

7.4 General Two-Group Diffusion Analysis 181

References 184

Problems 184

8 Nuclear Reactor Kinetics 189

8.1 Derivation of Point Kinetics Equation 190

8.2 Solution of Point Kinetics Equation without Feedback 194

8.3 State Space Representation of Point Kinetics Equation 205

8.4 Point Kinetics Equation with Feedback 208

8.5 Reactivity Measurements 214

8.6 System Stability Analysis 217

8.7 Point Reactor and Space-Dependent Reactor Kinetics 221

References 223

Problems 223

9 Fast Neutron Spectrum Calculation 227

9.1 Neutron Balance Equation and Slowing Down Density 228

9.2 Elastic Scattering and Lethargy Variable 232

9.3 Neutron Slowing Down in Infinite Medium 234

9.4 Resonance Escape Probability 243

9.5 Doppler Broadening of Resonances 250

9.6 Fermi Age Theory 256

9.7 Comments on Lattice Physics Analysis 260

References 261

Problems 262

10 Perturbation Theory and Adjoint Flux 265

10.1 Operator Notation for Neutron Diffusion Equation 265

10.2 Adjoint Operator and Adjoint Flux 267

10.3 First-Order Perturbation Theory 269

10.4 Adjoint Flux for Control Rod Worth Calculation 271

10.5 Adjoint Flux for Variational Formulation 273

10.6 Adjoint Flux for Detector Response Calculation 274

10.7 Adjoint Formulation for Flux Perturbation Calculation 276

10.8 Concluding Remarks on Adjoint Flux 280

References 280

Problems 281

11 Lattice Physics Analysis of Heterogeneous Cores 283

11.1 Material Heterogeneity and Flux Distribution in Unit Cell 285

11.2 Neutronic Advantages of Fuel Lumping 287

11.3 Diffusion Theory Model for Thermal Utilization 291

11.4 Improved Method for Thermal Disadvantage Factor 296

11.5 Resonance Escape Probability for Heterogeneous Cell 300

11.6 Thermal Spectrum Calculation 310

11.7 Integral Transport Methods 313

11.8 B1 Formulation for Spectrum Calculation 316

11.9 Lattice Physics Methodology for Fast Reactor 322

11.10 Monte Carlo Lattice Physics Analysis 325

11.11 Overall Reactor Physics Analysis 326

References 327

Problems 330

12 Nuclear Fuel Cycle Analysis and Management 333

12.1 Nuclear Fuel Cycle Analysis 334

12.2 Nuclear Transmutation Formulation 337

12.3 Equilibrium Cycle and Mass Balance 347

12.4 Simplified Cycling Model 353

12.5 Fission Product Xenon Buildup 359

12.6 General Incore Management Considerations 365

12.7 Fission Products and Radioactive Waste 371

12.8 Management of Used Nuclear Fuel 376

12.9 Key Considerations for Geological Disposal 396

References 404

Problems 409

13 Thermal-Hydraulic Analysis of Reactor Systems 413

13.1 Empirical Laws for Energy and Momentum Transport 415

13.2 Derivation of Fluid Conservation Equations 418

13.3 Simple Solutions of Fluid Conservation Equations 426

13.4 Conservation Equations for Channel Flow 443

13.5 Axial Temperature Distribution in Reactor Core 446

13.6 Boiling Heat Transfer and Two-Phase Flow 456

13.7 Thermal Hydraulic Limitations and Power Capability 471

13.8 Thermal-Hydraulic Models for Nuclear Plant Analysis 479

13.9 Advances in Nuclear Plant Modeling System 488

References 489

Problems 493

14 Power Coefficients of Reactivity 499

14.1 Physical Phenomena Affecting Core Reactivity 500

14.2 Relationship Between Reactivity Coefficients 502

14.3 Two-Group Representation of Reactivity Feedback 503

14.4 Parametric Dependence of LWR Reactivity Coefficients 505

14.5 Reactivity Coefficients in Sodium-Cooled Fast Reactor 508

14.6 Reactivity Feedback Model for Sodium-Cooled Fast Reactor 510

References 512

Problems 513

15 Nuclear Energy Economics 515

15.1 Electrical Energy Cost 516

15.2 Overview of Engineering Economics 519

15.3 Calculation of Nuclear Electricity Generation Cost 521

15.4 Impact of Increased Capital and O&M Costs 530

15.5 Energy Generation Efficiency of Various Technologies 533

References 536

Problems 539

16 Space-Time Kinetics and Reactor Control 541

16.1 Space-Time Reactor Kinetics 542

16.2 Space-Time Power Oscillations due to Xenon Poisoning 551

16.3 Time-Optimal Reactor Control 564

16.4 Model-Based Reactor Control 572

16.5 Alternate Reactor Control Techniques 581

16.6 Kalman Filtering for Optimal System Estimation 585

References 588

Problems 592

17 Elements of Neutron Transport Theory 595

17.1 Collision Probability Method 595

17.2 First-Flight Escape Probability and Dirac Chord Method 600

17.3 Flux Depression Calculation and Blackness 605

17.4 Numerical Solution of Neutron Transport Equation 610

References 621

Problems 623

Appendix A Key Physical Constants 627

Appendix B Comparison of Major Reactor Types 629

References 633

Appendix C Special Mathematical Functions 635

C.1 Gamma Function 635

C.2 Legendre Polynomial and Spherical Harmonics 637

C.3 Bessel Function 639

C.4 Dirac Delta Function 642

References 642

Appendix D Integral Transforms 643

D.1 Laplace Transform 643

D.2 Fourier Transform 645

D.3 Jordan’s Lemma 645

References 647

Appendix E Calculus of Variation for Optimal Control Formulation 649

E.1 Euler–Lagrange and Hamilton Equations 649

E.2 Pontryagin’s Maximum Principle 650

References 656

Appendix F Kalman Filter Algorithm 657

F.1 Linear Kalman Filter 657

F.2 Unscented Kalman Filter 660

References 662

Answers to Selected Problems 663

Index 679