University of Tennessee, Knoxville TRACE: Tennessee Research and Creative Exchange Doctoral Dissertations Graduate School 12-2017 DETERMINATION OF CRITICAL EXPERIMENT CORRELATIONS VIA THE MONTE CARLO SAMPLING TECHNIQUE William Jay Marshall University of Tennessee, Knoxville, [email protected] Follow this and additional works at: https://trace.tennessee.edu/utk_graddiss Part of the Nuclear Engineering Commons Recommended Citation Marshall, William Jay, "DETERMINATION OF CRITICAL EXPERIMENT CORRELATIONS VIA THE MONTE CARLO SAMPLING TECHNIQUE. " PhD diss., University of Tennessee, 2017. https://trace.tennessee.edu/utk_graddiss/4750 This Dissertation is brought to you for free and open access by the Graduate School at TRACE: Tennessee Research and Creative Exchange. It has been accepted for inclusion in Doctoral Dissertations by an authorized administrator of TRACE: Tennessee Research and Creative Exchange. For more information, please contact [email protected]. To the Graduate Council: I am submitting herewith a dissertation written by William Jay Marshall entitled "DETERMINATION OF CRITICAL EXPERIMENT CORRELATIONS VIA THE MONTE CARLO SAMPLING TECHNIQUE." I have examined the final electronic copy of this dissertation for form and content and recommend that it be accepted in partial fulfillment of the equirr ements for the degree of Doctor of Philosophy, with a major in Nuclear Engineering. Ronald E. Pevey, Major Professor We have read this dissertation and recommend its acceptance: Steven E. Skutnik, Howard L. Hall, Vasilios Alexiades Accepted for the Council: Dixie L. Thompson Vice Provost and Dean of the Graduate School (Original signatures are on file with official studentecor r ds.) DETERMINATION OF CRITICAL EXPERIMENT CORRELATIONS VIA THE MONTE CARLO SAMPLING TECHNIQUE A Dissertation Presented for the Doctor of Philosophy Degree The University of Tennessee, Knoxville William Jay Marshall December 2017 Copyright © 2017 by William Jay Marshall All rights reserved. ii To my wife, Robyn Joy Marshall, without whom none of this would have been possible. To my son, Ian Jay Marshall, who has tolerated a tremendous loss of play time so this thing can be brought to a successful completion. And finally, to the 2016 Chicago Cubs, the ultimate low probability, high weight event. iii ACKNOWLEDGMENTS There are so many people to thank in the execution of a project and a document like this. I will endeavor to thank all the important ones, but I will inevitably forget some and for this I am truly sorry. First, thanks to Dr. Pevey. He willingly took me on as a part time doctoral student while I was (and still am) working full time at Oak Ridge National Laboratory, despite working with me more than 10 years prior when I was at UTK working on my Master of Science degree. He also allowed me the freedom to take classes and make progress on my research around my full time day (and night and weekend) job at the lab. Also, crucially, he reconsidered allowing me to use this project as my dissertation after initially rejecting it. Second, thanks also the other official members of my committee: Dr. Steven Skutnik, Dr. Howard Hall, and Dr. Vasilios Alexiades. They each dedicated time and effort to my work here despite having other things to do. I’m sure reviewing a dissertation isn’t high on anyone’s list of desired activities. Thanks to each of the committee members for their time, insights, and careful review. Third, thanks to the courtesy member of my committee Dr. Bradley Rearden. His support of my efforts in this area began in Stockholm, Sweden, in March, 2013, and have continued over the past 4 years. He saw the potential of this project as a doctoral dissertation despite my early protests, and supported me with discussion, some funding at the lab, and plenty of encouragement. He also supported my trips to Europe to discuss my ideas within the OECD/NEA Working Party on Nuclear Criticality Safety Expert Group on Uncertainty Analysis for Criticality Safety Assessment (WPNCS/UACSA). All of this has been invaluable to the pursuit and final conclusion of this effort. Fourth, thanks to Dr. Mike Dunn, Mr. Steve Bowman, and Dr. Doug Bowen at Oak Ridge National Laboratory for supporting my pursuit of a PhD while working for them full time at the lab. All three have provided support and encouragement despite this effort taking away from time I would have had to work on projects for them. I also need to thank the laboratory itself for paying the lion’s share of my tuition over the last 5 years as I have toiled along in this endeavor. Fifth, thanks to Elizabeth Jones, Emily Fenske, Jon Gill, and Dr. Adam Stratz for befriending an old guy like me. Having a few friends among the other graduate students has certainly made this easier and a whole lot more enjoyable. Sixth, thanks to the members of WPNCS/UACSA for generating a benchmark on this topic at a convenient time for me. Thanks also for thoughtful discussions during meetings, especially in Paris in 2015 and 2016 and in Garching in 2016. A special thank you to Dr. Maik Stuke, Dr. Fabian Sommer, and Dr. Elisabeth Peters at the Gesellschaft für Analgen- und Reaktorseicherheit (GRS), who have also spent a considerable amount of time struggling with critical experiment correlations. iv Seventh, thanks to Rob Lefebrve, Will Wieselquist, and Brandon Langley. Each of them implemented some sort of improvements somewhere in the tool chain of Sampler based on my experiences. All of the changes, except the “plus_x” thing, were improvements. Many of them, like speeding Sampler by about a factor of 50, were tremendously helpful in completing all the work presented here. Eighth, thanks to my parents for providing a loan to make sure I could take enough coins to the ivory tower when required. Their financial and moral support for all my educational and extracurricular activities is greatly appreciated, and I’ll pay that loan back soon. Promise. Ninth, finally, and most importantly, I owe an incredible debt of gratitude to my wife Robyn and my son Ian. They both put up with the lost time of me going to class, working on research, and writing this dissertation. Ian’s been as tolerant of my ignoring him at times as any 7 – 12 year old can be. Robyn has provided the right mix of support, encouragement, nagging, sympathy, and patience somehow at all the right times. She has also cooked, cleaned, raised a child, run a household, taught preschool, and earned a master’s degree and teaching certification in the meantime. There’s no way that my degree or this document could have been completed without her. v ABSTRACT Critical benchmark experiments are the foundation of validation of the computational codes used in criticality safety analyses because they provide a basis for comparison between the calculated results and the physical world. These experiments are often performed in series varying a limited number of parameters to isolate the effect of the independent parameter. The use of common materials, geometries, machines, procedures, detectors, or other shared features can create correlations among the resulting experiments. Most validation techniques used in criticality safety practice do not treat these correlations explicitly, and the effect of this is unclear as the correlations themselves are not well known. Generalized linear least squares methods used for advanced validation or in data adjustment studies also rely on correlation coefficients to constrain the adjustments allowed in critical experiment results. The purpose of this dissertation is to develop a methodology for the calculation of critical experiment correlations using a Monte Carlo sampling technique. The use of this technique allows for the determination of the uncertainty in each individual experiment, and identical perturbations applied to shared parameters provide estimates of the covariance between the experiments. The correlation coefficient is then calculated by dividing the covariance between any pair of experiments by the product of the individual experiment standard deviations. This technique is applied to high-enriched uranium solution experiments and low-enriched uranium pin lattice experiments to determine correlation coefficients for these types of systems. The important parameters governing the correlation coefficients are determined, and the results are compared with correlation coefficients in the literature determined using other methods at other institutions. The general method for the determination of the correlation coefficients is presented along with other conclusions and recommendations for further study in this area. vi TABLE OF CONTENTS Chapter I: Introduction ....................................................................................................................... 1 Chapter II: Literature Review ............................................................................................................. 7 Review of Statistics .......................................................................................................................... 7 Correlation Coefficient ............................................................................................................... 7 Assessment of Non-zero Correlation Coefficient.................................................................. 8 Confidence Interval Estimation for Correlation Coefficients .............................................. 9 Current Validation Guidance Documents ................................................................................