Fundamentals of Nuclear Reactor Physics Job Name: 209838T Job Name: 209838T

Job Name: 209838t

Fundamentals of Nuclear Reactor Physics Job Name: 209838t Job Name: 209838t

E. E. Lewis

Professor of Mechanical Engineering McCormick School of Engineering and Applied Science Northwestern University

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To Ann Job Name: 209838t Job Name: 209838t

Contents

Preface ...... xiii

1 Nuclear Reactions ...... 1 1.1 Introduction ...... 1 1.2 Nuclear Reaction Fundamentals ...... 2 Reaction Equations ...... 3 Notation ...... 5 Energetics ...... 5 1.3 The Curve of Binding Energy ...... 7 1.4 Fusion Reactions ...... 8 1.5 Fission Reactions ...... 9 Energy Release and Dissipation ...... 10 Neutron Multiplication ...... 12 Fission Products ...... 13 1.6 Fissile and Fertile Materials ...... 16 1.7 Radioactive Decay ...... 18 Saturation Activity ...... 20 Decay Chains ...... 21

2 Neutron Interactions ...... 29 2.1 Introduction ...... 29 2.2 Neutron Cross Sections ...... 29 Microscopic and Macroscopic Cross Sections ...... 30 Uncollided Flux ...... 32 Nuclide Densities ...... 33 Enriched Uranium ...... 35 Cross Section Calculation Example ...... 36 Reaction Types ...... 36 2.3 Neutron Energy Range ...... 38 2.4 Cross Section Energy Dependence ...... 40 Compound Nucleus Formation ...... 41 Resonance Cross Sections ...... 42 Threshold Cross Sections ...... 46 Fissionable Materials ...... 47

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2.5 Neutron Scattering ...... 48 Elastic Scattering ...... 49 Slowing Down Decrement ...... 50 Inelastic Scattering ...... 52

3 Neutron Distributions in Energy ...... 57 3.1 Introduction ...... 57 3.2 Nuclear Fuel Properties ...... 58 3.3 Neutron Moderators ...... 61 3.4 Neutron Energy Spectra ...... 63 Fast Neutrons ...... 65 Neutron Slowing Down ...... 66 Thermal Neutrons ...... 70 Fast and Thermal Reactor Spectra ...... 72 3.5 Energy-Averaged Reaction Rates ...... 73 Fast Cross Section Averages ...... 75 Resonance Cross Section Averages ...... 78 Thermal Cross Section Averages ...... 79 3.6 Infinite Medium Multiplication ...... 81

4 The Power Reactor Core ...... 85 4.1 Introduction ...... 85 4.2 Core Composition ...... 85 Light Water Reactors ...... 88 Heavy Water Reactors ...... 91 Graphite-Moderated Reactors ...... 92 RBMK Reactors ...... 93 Fast Reactors ...... 94 4.3 Fast Reactor Lattices ...... 94 4.4 Thermal Reactor Lattices ...... 98 The Four Factor Formula ...... 99 Pressurized Water Reactor Example ...... 108

5 Reactor Kinetics ...... 115 5.1 Introduction ...... 115 5.2 Neutron Balance Equations ...... 116 Infinite Medium Nonmultiplying Systems ...... 116 Infinite Medium Multiplying Systems ...... 117 Finite Multiplying Systems ...... 119 5.3 Multiplying Systems Behavior ...... 120 5.4 Delayed Neutron Kinetics ...... 123 Kinetics Equations ...... 124 Reactivity Formulation ...... 126 Job Name: 209838t

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5.5 Step Reactivity Changes ...... 126 Reactor Period ...... 127 Prompt Jump Approximation ...... 131 5.6 Prologue to Reactor Dynamics ...... 133

6 Spatial Diffusion of Neutrons ...... 139 6.1 Introduction ...... 139 6.2 The Neutron Diffusion Equation ...... 140 Spatial Neutron Balance ...... 140 Diffusion Approximation ...... 142 6.3 Nonmultiplying Systems—Plane Geometry ...... 143 Source Free Example ...... 143 Uniform Source Example ...... 144 6.4 Boundary Conditions ...... 145 Vacuum Boundaries ...... 146 Reflected Boundaries ...... 147 Surface Sources and Albedos ...... 147 Interface Conditions ...... 148 Boundary Conditions in Other Geometries ...... 149 6.5 Nonmultiplying Systems—Spherical Geometry ...... 149 Point Source Example ...... 150 Two Region Example ...... 151 6.6 Diffusion Approximation Validity ...... 153 Diffusion Length ...... 154 Uncollided Flux Revisited ...... 155 6.7 Multiplying Systems ...... 157 Subcritical Assemblies ...... 157 The Critical Reactor ...... 160

7 Neutron Distributions in Reactors ...... 167 7.1 Introduction ...... 167 7.2 The Time-Independent Diffusion Equation ...... 167 7.3 Uniform Reactors ...... 169 Finite Cylindrical Core ...... 170 Reactor Power ...... 172 7.4 Neutron Leakage ...... 174 Two Group Approximation ...... 174 Migration Length ...... 178 Leakage and Design ...... 179 7.5 Reflected Reactors ...... 180 Axial Reflector Example ...... 181 Reflector Savings and Flux Flattening ...... 184 Job Name: 209838t

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7.6 Control Poisons ...... 186 Reactivity Worth ...... 186 Partially Inserted Control Rod ...... 188 Control Rod Bank Insertion ...... 190

8 Energy Transport ...... 199 8.1 Introduction ...... 199 8.2 Core Power Distribution ...... 199 Finite Cylindrical Core ...... 200 Uniform Cylindrical Core Example ...... 203 8.3 Heat Transport ...... 204 Heat Source Characterization ...... 204 Steady State Temperatures ...... 205 Pressurized Water Reactor Example ...... 209 8.4 Thermal Transients ...... 211 Fuel Temperature Transient Examples ...... 212 Coolant Temperature Transients ...... 213

9 Reactivity Feedback ...... 221 9.1 Introduction ...... 221 9.2 Reactivity Coefficients ...... 221 Fuel Temperature Coefficient ...... 223 Moderator Temperature Coefficient ...... 225 Fast Reactor Temperature Coefficients ...... 227 9.3 Composite Coefficients ...... 227 Prompt Coefficient ...... 228 Isothermal Temperature Coefficient ...... 228 Power Coefficient ...... 229 Temperature and Power Defects ...... 230 9.4 Excess Reactivity and Shutdown Margin ...... 230 9.5 Reactor Transients ...... 232 Reactor Dynamics Model ...... 233 Transient Analysis ...... 234

10 Long-Term Core Behavior ...... 243 10.1 Introduction ...... 243 10.2 Reactivity Control ...... 243 10.3 Fission Product Buildup and Decay ...... 245 Xenon Poisoning ...... 247 Samarium Poisoning ...... 250 10.4 Fuel Depletion ...... 252 Fissionable Nuclide Concentrations ...... 252 Burnable Poisons ...... 255 10.5 Fission Product and Actinide Inventories ...... 257 Job Name: 209838t

Contents xi

Appendices ...... 263 A Useful Mathematical Relationships ...... 265 Derivatives and Integrals ...... 265 Definite Integrals ...... 265 Hyperbolic Functions ...... 266 Expansions ...... 266 Integration by Parts ...... 266 Derivative of an Integral ...... 267 First-Order Differential Equations ...... 267 Second-Order Differential Equations ...... 268 r2 and dV in Various Coordinate Systems ...... 268 B Bessel’s Equation and Functions ...... 269 C Derivation of Neutron Diffusion Properties ...... 273 D Fuel Element Heat Transfer ...... 279 E Nuclear Data ...... 283

Index ...... 287 Job Name: 209838t Job Name: 209838t

Preface

This text is intended as a first course in the physics of nuclear reactors. It is designed to be appropriate as an introduction to reactor theory within an undergraduate nuclear engineering curriculum, as well as for a stand-alone course that can be taken by undergraduates in mechanical, electrical, or other fields of engineering who have not had a previous background in nuclear energy. Likewise, it is planned to be useful to practicing engineers from a variety of disciplines whose professional responsibilities call for familiarity with the physics of nuclear reactors. Why a new book on reactor physics when a number of legacy texts are still in print? The better of these are well written, and since the fundamentals of the subject were already worked out at the time of their publication, they remain useful today. My conviction, however, is that for today’s undergraduates and practicing engineers an introduction to reactor physics is better presented through both reorganizing and refocusing the material of earlier texts, and in doing that emphasizing the characteristics of modern power reactors. Earlier textbooks most commonly have begun with the relevant nuclear physics and neutron interactions, and then presented a detailed treatment of neutron slowing down and diffusion in homo- geneous mixtures of materials. Only in the latter parts of such texts does the analysis of the all-important time-dependent behavior of fissionable systems appear, and the dependence of criticality on lat- tice structures of reactor cores typically is late in receiving attention. To some extent such a progression is necessary for the logical development of the subject. However, both in teaching undergraduates and in offering continuing education instruction for practicing engineers, I have found it advantageous to present a quantitative but more general overview early in the text, while deferring where possible more detailed analysis, and also the more advanced mathematics that accompanies it, to later. Thus I have moved the treatment of reactor kinetics forward, attempting to inculcate an understanding of the time-dependent behavior of chain reactions, before undertaking the detailed treatment of spatial power distributions, reflector saving,

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xiv Preface and other topics dependent on the solution of the neutron diffusion equation. Likewise, the compositions of power reactor cores are incorporated into the discussion early on, emphasizing the interdis- ciplinary nature of neutronic and thermal design. My intent in this text is to emphasize physical phenomena, rather than the techniques necessary to obtain highly accurate results through advanced numerical simulation. At the time the legacy texts appeared, computers were emerging as powerful tools for reactor analysis. As a result, the pedagogy in teaching reactor theory often emphasized the programming of finite differencing techniques, matrix solution algorithms, and other numerical methods in parallel with the analysis of the physical behavior of reactors, and thus extended the range of solutions beyond what could be obtained with paper and pencil. Now, however, high level programming languages, such as MathcadÒ or MATLABÒ, allow students to solve transcendental or linear systems of equations, to integrate differential equations, and to perform other operations needed to solve the preponderance of problems encountered in an introductory course without programming the algorithms themselves. Concomitantly, the numerical simulation of reactor behavior has become a highly sophisticated enterprise; one to which I have devoted much of my career. But I believe that the numerical techniques are better left to more advanced courses, after a basic understanding of the physical behavior of reactors has been gained. Otherwise the attempt to incor- porate the numerical approaches employed in reactor design dilutes the emphasis on the physical phenomena that must first be understood. By reorganizing and refocusing the materials along these lines, I hope to have broadened the audience for whom the text may be useful, in particular those advancing their professional development, even while they may not be taking a formal reactor physics course in a university setting. My goal is that the book may be read and physical insight gained, even though for lack of time or background some of the more mathematical sections are read and the results accepted without following each step of the development in detail. With this in mind, many of the results are presented in graphical as well as analytical form, and where possible I have included represen- tative parameters for the major classes of power reactors in order to provide the student with some feel for the numbers. Examples are integrated into the text’s narrative, and a selection of problems is provided at the end of each chapter. The problems both enforce the concepts that have been covered and in some cases expand the scope of the material. The majority of the problems can be solved analytically, or with the use of a pocket calculator. In some cases, where multiple solutions or graphical results are called for, use TM of the formula menu of a spreadsheet program, such as Excel , Job Name: 209838t

Preface xv removes any drudgery that might otherwise be entailed. Selected problems require the use of one of the earlier mentioned high level computing languages for the solution of transcendental or differential equations. These are marked with an asterisk. The preparation of this text would have been immensely more difficult if not impossible without the help and encouragement of many friends, colleagues, and students. Advice and assistance from the staff of the Nuclear Engineering Division of Argonne National Laboratory have been invaluable in the text’s preparation. Won Sik Yang, in particular, has provided advice, reactor parameters, graphical illustrations, and more as well—taking the time to proofread the draft manuscript in its entirety. Roger N. Blomquist, Taek K. Kim, Chang-ho Lee, Giuseppe Palmiotti, Micheal A. Smith, Temitope Taiwo, and several others have also pitched in. Bruce M. Bingman and his colleagues at the Naval Reactors Program have also provided much appre- ciated help. Finally the feedback of Northwestern University students has been most helpful in evolving a set of class notes into this text. Most of all, Ann, my wife, has endured yet another book with grace and encouragement, while covering for me in carrying much more than her share of our family’s responsibilities.