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The Consistency of Differential and Integral

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The Consistency of Differential and Integral THE CONSISTENCY OF DIFFERENTIAL AND INTEGRAL THERMONUCLEAR NEUTRONICS DATA A DISSERTATION Presented to The Faculty of the Division of Graduate Studies By William Albert Reupke In Partial Fulfillment of the Requi rements for the Degree Doctor of Philosophy in the School of Nuclear Engineering Georgia Institute of Technology December, 1977 THE CONSISTENCY OF DIFFERENTIAL AND INTEGRAL THERMONUCLEAR NEUTRONICS DATA Approved: n_, / /y ^2- )'. Narl Davidson, Chairman Dougla^l. Muir _^2_ -> JoseftJf D. Clement ^ M( Jonathan.Hairm^hari. Haire ^/ Jeifinyfr.."Postcn ^ Date approved by Chairman/u^r. /? /77/ TABLE OF CONTENTS ACKNOWLEDGMENTS ..... .............. LIST OF TABLES . ... ... LIST OF ILLUSTRATIONS . Chapter I. INTRODUCTION Neutronic Analysis of Fusion Reactor Blankets and Shields Quantitative Consistency Analysis in Nuclear Design II. THEORY OF QUANTITATIVE CONSISTENCY ANALYSIS . Methodological Considerations Algorithms Used in the Present Work Considerations in the Use of the Algorithms III. DEVELOPMENT OF A COMPUTER PROGRAM FOR CONSISTENCY ANALYSIS Rationale for a New Code Sensitivity Model Programming of Consistency Algorithms Programming of Sensitivity Model ALVIN2 Graphics Features Program Architecture Code Validation IV. COMPUTATIONAL PROCEDURE. ............. Method of Attack Description of Integral Data: The Wyman Experiment Consistency of the Data Prior to Adjustment Consistency of the Data Subsequent to Adjustment Summary of Computational Procedure V. RESULTS AND DISCUSSION OF RESULTS Chi-Square of System Multigroup Cross Sections: Results Multigroup Dispersion Matrices: Results Multigroup Cross Sections and Dispersion Matrices: Discussion Tritium Production Distribution and Dispersion Matrix Tritium Breeding Ratio Correlation Coefficient Matrix VI. CONCLUSIONS AND RECOMMENDATIONS , . Conclusions Recommendations APPENDIX I. DATA PROCESSING CONSIDERATIONS. ........ APPENDIX IT. METHOD OF ESTIMATING RESPONSE FUNCTION VARIANCE DUE TO SOURCE ANISOTROPY. APPENDIX HI. VALIDATION OF SENSITIVITY CALCULATIONS APPENDIX IV. LEAKAGE SPECTRA FROM PULSED SPHERES IV ACKNOWLEDGMENTS Among the community of people throughout the world who recognize the research enterprise, I wish to thank the citizens of the State of Georgia for making possible the preparatory phases of this research and the citizens of the United States of America for their support, particu­ larly during the latter stages of this work. My friends in the Federal Republic of Germany, who - during turbulent times in the States - spurred on my interest in resuming the formal quest for knowledge, must be thanked tenfold. Amongst those who maintain the pathways of research, the Division of Graduate Studies and the School of Nuclear Engineering at the Georgia Institute of Technology, and the Los Alamos Scientific Laboratory, oper­ ated for the U.S. Department of Energy by the University of California, have provided essential support, with coordination provided by the As­ sociated Western Universities. Here I wish to acknowledge the central role of my adviser at Georgia Tech, J. Narl Davidson. In addition, I must acknowledge the equally crucial, invisible network maintained by my fellow students and by others who have so ably served on short notice. Finally, I wish to thank the handful who accompanied me to the threshold: the members of the Applied Nuclear Data Group at Los Alamos, who provided a congenial atmosphere for research; my adviser at Los Ala­ mos, Douglas W. Muir, for his constant advice and counsel; my late mother, Mrs. Albert Reupke, for her infinite patience; and finally, my wife Karen, for her constant support and assistance with the typing. V LIST OF TABLES e Page Hierarchy of Observation Statements 27 Consistency Analysis Codes and their Applications 67 Consistency Analysis Criteria: Wyman Experiment. ...... 98 Sources of Error in Tritium Production Data 118 Partial Reactions of Principal Interest . .......... 124 Sensitivity of Tritium Production in 15 cm Li Sample . .133 Evaluated Multigroup Errors of the D(n92n) Reaction 137 Evaluated Errors of the 6Li(n,t) Reaction ... 138 Equivalent Group Cross Section Observations 7 for the Li(n9xt) Reaction. 144 Development of Successive Approximations 152 Present Results in Context of Typical Fitting Results .... 159 Contributions to Chi-Square •• . 161 Tritium Breeding Ratio Before and After Consistency Analysis . 132 VI LIST OF ILLUSTRATIONS re Page ALVIN Flowchart , 87 Deuterium Cross Sections. .. j07 Lithfum-6 Cross Sections. .... 108 Lithium-7 Cross Sections. ..... 109 Radial Distribution of Tritium Production, Before Consistency Analysis . 113 Tritium Production Dispersion Matrix, Before Consistency Analysis . 120 Multigroup Sensitivity Matrix, D(d92n) Reaction, Total Cross Section Constant. ........ 128 Multigroup Sensitivity Matrix, 6Li(n,t) Reaction, Total Cross Section Variable 131 Multigroup Sensitivity Matrix, 7Li(n,xt) Reaction, Total Cross Section Constant. , 135 7Li(n,xt) Reaction Multigroup Dispersion Matrix, Before Consistency Analysis . ..*•• . -j47 . Flow Diagram of Consistency Analysis ' 151 7Li(n,xt) Reaction Group Cross Section Before and After Consistency Analysis . 154 7Li(n,xt) Reaction Multigroup Dispersion Matrix, After Consistency Analysis 155 Comparison of 14 MeV 7Li(n,xt) Reaction Cross Section Data 171 Radial Distribution of Tritium Production Before and After Consistency Analysis ... 177 Tritium Production Dispersion Matrix, After Consistency Analysis 180 Cross-Type Correlation Coefficient Matrix, After Consistency Analysis. •.- 185 1 CHAPTER I INTRODUCTION To increase the accuracy of the neutronics analysis of nuclear reactors, physicists and engineers have employed a variety of tech­ niques, including the adjustment of multigroup differential data to improve consistency with integral data. Of the various adjustment strategies, a generalized least-squares procedure which adjusts the combined differential and integral data can significantly improve the accuracy of neutronics ca-lculations compared to calculations employ­ ing only differential data. This investigation analyzes 14 MeV neutron-driven integral experiments, using a more extensively developed methodology and a newly developed computer code, to extend the domain of adjustment from the energy range of fission reactors to the energy range of fusion reactors. Neutronics Analysis of Fusion Reactor Blankets and Shields A classical test of engineering design analysis is the agreement of calculation and experiment within prescribed limits of error. The present state-of-the-art in calculation methods and integral experi­ ments, .as applied to fusion reactor neutronics design, is reviewed here. In a subsequent section of this chapter the principles of con­ sistency-analysis and data adjustment are introduced. 2 State-of-the-Art in Calculation .Methods Cornerstones of neutronics calculations are evaluated nuclear data and methods of solving the neutron transport equation, together with the sensitivity calculation technique of relating changes in nuclear data to changes in the results of neutron transport calculations. This subsection reviews the state of the present art in these basic com­ ponents of neutronics calculation. Status of Evaluated Nuclear Data. In the earliest controlled fusion reactor designs investigators at Princeton performed their own evaluations even on the most basic nuclear data. In subsequent years, nuclear data initially evaluated for weapons applications became more widely disseminated and was combined with evaluations undertaken for fission reactor applications to permit controlled fusion reactor calcula­ tions on a somewhat firmer data base. The emergence of organized nuclear data evaluation for controlled fusion reactors may be traced to the 1970 International Atomic Energy Agency Helsinki conference on nuclear data, where both British and Soviet reviewers noted continuing serious defi- . 2 ciencies. For example, it was noted by V. S. Crocker et_ aj_ that the important Li(n,xt) reaction remained 25% uncertain throughout the range of interest. In some energy regions., namely 8-13 MeV, no measurements were available. At the most recent critical review of evaluated nuclear data needs for the near-term requirements of the United1States' fusion 3 reactor program, the Neutronics Working Group concluded that new 3 experimental data and more accurate representations of secondary neutron spectra for Li should be incorporated in the ENDF/B-V evaluation. A number of other improvements were suggested, including the organized evaluation of cross section uncertainties. This introduction emphasizes that the utility of the existing data is limited in most cases by the continuing absence of quantitative uncertainty estimates. Sensitivity theory has, in large measure, pro­ vided new incentive for organized cross section error evaluation. How­ ever, in most- cases, and in the present work, the designer must still provide his own evaluation, just as in former years the designer was faced with evaluation of the cross sections proper. This remark applies particularly to secondary energy and angular distribution uncertainties. Since the wealth of possible cross section data far exceeds the data of immediate interest, the evaluation of uncertainties must be guided by the larger context of engineering significance. Since this same con­ text also guides the selection and design of integral experiments, one may expect an appropriate base of evaluated cross section uncertainty data will evolve in parallel with the requirements of
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