MIT-3944- & MITNE-109 A SINGLE-ELEMENT METHOD FOR HETEROGENEOUS NUCLEAR REACTORS by S.S. Seth, M.J.Driscoll, 1. Kaplan, T.J. Thompson and D.D. Lanning May 1970 Contract AT(30-1)3944 U.S.Atomic Energy Commission MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Nuclear Engineering Cambridge, Massachusetts 02139 A S - MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Nuclear Engineering Cambridge, Massachusetts A SINGLE-ELEMENT METHOD FOR HETEROGENEOUS NUCLEAR REACTORS by S. S. Seth, M. J. Driscoll, I. Kaplan, T. J. Thompson and D. D. Lanning May, 1970 MIT - 3944 - 3 MITNE - 109 AEC Research and Development Report UC - 34 Physics (TID-4500, 47th Edition) Contract AT(30-1) 3944 U. S. Atomic Anergy Commission ............ DISTRIBUTION MIT-3944-3 MITNE-109 AEC Research and Development Report UC-34 Physics 1-4. U. S. Atomic Energy Commission, Headquarters Division of Reactor Development and Technology Reactor Physics Branch (2 copies) Core Design Branch (1 copy) Water Reactor Branch (1 copy) Washington, D. C. 20545 5. U. S. Atomic Energy Commission Savannah River Laboratory Attn: B. C. Busche Aiken, South Carolina 29801 6. AEqL Chalk River Laboratory Attn: C. Millar Sheridan Park, Ontario, Canada 7. H. S. Potter, NY Patent Group U. S. Atomic Anergy Commission Brookhaven Office Upton, New York 11973 8-9. U. S. Atomic Energy Commission Cambridge Office IT I'll 2 ABSTRACT A "single-element" method is described for the experimental determination of the parameters P, n and A which characterize the neutronic properties of a fuel element in heterogeneous reactor theory. This method requires the use of only one fuel element located at the center of a tank of moderator in an exponential facility. The measurements are made outside this single fuel ele- ment and include the following quantities to which the hetero- geneous fuel parameters are related: the radial distance to the thermal neutron flux peak, the inverse relaxation length of the axial flux, the cadmium ratio of gold at a given radial distance, and the ratio of the epicadmium activities (per unit isotopic weight) of gold-197 and molybdenum-98 irradiated on the fuel surface. The single-element method was applied to 19 and 31 rod clusters of plutonium containing fuel. The reactor physics parameters of uniform lattices composed of these clusters, calculated from the measured values of P, n and A, show good agreement with the results of full-lattice studies of the same fuel at the Savannah River Laboratory. It is concluded that the proposed method should increase the efficacy of heterogeneous reactor theory and make possible the evaluation of new, promising and scarce nuclear reactor fuels at very low cost. 3 ACKNOWLEDGEMENTS The success of the M.I.T. Reactor Physics Project is due to the support of the 'U. S. Atomic Energy Commission and to the contributions of a number of individuals. The work described in this report has been performed primarily by the principal author, Shivaji S. Seth, who has submitted substantially the same report in partial fulfillment of the requirements for the degree of Doctor of Science at M.I.T. Overall direction of the project has been shared by Professors M. J. Driscoll, I. Kaplan, D. D. Lanning and T. J. Thompson (now on leave of absence). Mr. A. T. Supple, Jr., has provided great assistance in the experimental work. The contributions of Mr. E. McFarland and Mr. T. C. Leung are specially acknowledged. The staffs of the M.I.T. Reactor, the Reactor Machine Shop, and the Reactor Electronics Shop have provided advice and assistance. All computer calculations were done on the IBM-360 at the M.I.T. Information Processing Service Center. Special thanks are due to Miss Marcia Clear for her unfailing patiende, good humor and skill in preparing this manuscript. 4 TABLE OF CONTENTS Page Chapter 1. Introduction 10 1.1 Foreword 10 1.2 M.I.T. Reactor Physics Project 11. 1.3 The Heterogeneous Reactor Method 1 1.3.1 A Review 12 1.3.2 The Source-sink Method 13 1.3.3 Advantages 16 1.3.4 A Major Limitation 17 1.4 The Single-element Method 19 1.4.1 Outline 19 1.4.2 Highlights 23 1.5 Organization of the Report 25 Chapter 2. Theory 26 2.1 Heterogeneous Fuel Element Parameters 26 2.1.1 Thermal Constant, r 26 2.1.2 Fast Neutron Yield, n 29 2.1.3 Epithermal Absorption Parameter, A 30 2.2 Experimental Parameters 32 2.3 Determination of r 34 2.3.1 Thermal Neutron Flux Distribution 34 2.3.2 Expression for r 41 2.3.3 Physical Basis of the Expression for r 44 5 2.3.4 The Slowing-down Density, q(r,z,T) 44 2.3.4.1 Effect of Epithermal Absorption on q0 ,r(rT) 49 2.3.5 The Integrals I and I 51 2.3.6 Transport Correction to X 52 2.4 Determination of n 53 2.4.1 The Determination of n from the Axial Buckling 54 2.4.2 The Determination of ti from the Gold Cadmium Ratio, R 55 2.4.3 Comments Regarding the Determination of r and n 58 2.5 Determination of A 60 2.5.1 Motivation for the Method 60 2.5.2 Exact Relationship between fc and A 62 2.5.3 Normalization of fc 65 2.5.4 Summary Procedure to, Evaluarte A 68 2.5.5 Effects of Spectral Differences 68 2.6 Summary 72 Chapter 3. Experiments, Analysis and Results 75 3.1 Introduction 75 3.2 M.I.T. Exponential Facility 75 3.3 Single Elements 77 3.3.1 Some Features of the Cluster Design 79 3.3.2 SRL and MIT Cluster Differences 81 3.4 Determination of X 81 3.4.1 Foil Irradiation 82 6 Page 3.4.2 Activity Measurement 82 3.4.3 Curve-fitting for X 85 3.5 Determination of y 86 3.5.1 Foil Irradiation 86 3.5.2 Curve-fitting for y 88 3.6 Determination of R 88 3.6.1 Foil Irradiation 89 3.6.2 Height Correction 89 3.7 Determination of F and fe 90 3.7.1 Foil-packet Irradiation 90 3.7.2 Activity Measurement 91 3.7.3 Relating F to f 93 3.8 Moderator Parameters 95 3.9 Calculation of ', n, A 96 3.10 Sensitivity of the Heterogeneous Parameters 99 Chapter 4. Application to Uniform Lattices 104 4.1 Introduction 104 4.2 Thermal Utilization, f 105 4.3 Fast Neutron Yield, q 108 4.3.1 Summation of the Age Kernel 110 4.3.2 Summation of the Uncollidpd Flux Kernel 113 4.4 Resonance Escape Probability 118 4.4.1 Relation between AL and A 119 4.4.2 Cal4ulation of 8 122 4.5 Infinite Medium Multiplication Factor 125 7 Page 4.6 Material Buckling, B2 128 4.7 Accuracy of Values of Material Buckling 135 Chapter 5. Conclusion 139 5.1 Summary 139 5.2 Suggestions for Future Work 140 Appendix A. Perturbation in Radial Buckling 146 Appendix B. Uncollided Flux Kernel 149 Appendix C. Correction of Detector Resonance Integrals 151 Appendix D. Sample Data 154 Appendix E. r of Natural Uranium Rod 161 Appendix F. ERI238 and A of Single-elements 162 Appendix G. Concentration of Nuclides in the UO -PuO Fuel Clusters 2 2 164 Appendix H. Nomenclature 165 Appendix I. Bibliography of Publications on Heterogeneous Reactor Theory 169 Appendix J. References 173 8 LIST OF FIGURES Fig. No. Page 1.1 Fuel Element Arrangement 21 2.1 Thermal Neutron Flux Distribution around the Single Fuel Element 28 2.2 The Single-element Model 33 2.3 Comparison of Slowing-down Age Kernels 50 2.4 Epithermal Flux Ratio (f ) Versus Epithermal Absorption Parameter (A) C 64 2.5 Comparison of Epithermal Spectra 70 2.6 Cumulative U-238 Resonance Absorptions in a Lattice and around a Single Fuel Element versus Energy 73 3.1 Vertical Section of the Subcritical-Assembly 76 3.2 Schematic Side-view 80 (a) Fuel Rod (Type B) (b) Fuel Cluster 3.3 Position of Radial Foil Holderp 83 3.4 Schematic Set Up of Counting Equipment 84 3.5 Position of Vertical Fail Holders 87 3.6 Mounting of Foils for Measurement of R 87 3.7 Sensitivity of P 100 3.8 Sensitivity of n 101 3.9 Sensitivity of A 102 4.1 Lattice Array for Age Kernel Summation 111 4.2 Lattice Array for Uncollided Flux Kernal Summation 114 4.3 Buckling of 1.0 in. Diameter, Natural Uranium Rods 132 in D20 4.4 Sensitivity of B2 136 9 LIST OF TABLES Table No. Page 1.1 Isotopic Composition of Simulated Burned Fuel used 22 in 19 and 31 Rod Clusters 3. 1 Description of Fuel and Cladding 78 3.2 Housing Tube Differences between SRL and MIT Clusters 81 3.3 Properties of Au-197 and Mo-98 92 3.4 Values of the Experimental Parameters 94 3.5 Moderator Properties 97 3.6 Heterogeneous Fuel Parameters 98 4.1 Results for Thermal Utilization, f 107 4.2 Values of the Fast and Epithermal Nuclear Constants 116 4.3 Results for the Fast Neutron Yield, n L 117 4.4 Values of the Advantage Factor, S 126 4.5 Results for Resonance Escape Probability, p 127 4.6 Diffusion and Slowing-down Areas 130 4.7 Comparison of Single Element and Lattice Results for D20 Moderated and Cooled, Plutonium Containing Fuel Clusters 134 D.1 Typical Gold Foil Activities for Radial Traverses 155 D.2 Values of X (cm) 156 D.3 Typical Corrected Gold Foil Activities for Axial Traverses 157 D.4 Values of y2 (x'10 6 -2 158 D.5 Values of R 159 and f 160 D.6 Values of F e F.1 Calculated Values of ERI 28and A 163 10 Chapter 1 INTRODUCTION 1.1 FOREWORD Heterogeneous nuclear reactors moderated by heavy water or graphite have been important since the early days of nuclear tech- nology.