Data Bank ISBN 978-92-64-99056-2 NEA/DB/DOC(2008)1 Analytical Benchmarks for Nuclear Engineering Applications Case Studies in Neutron Transport Theory B.D. Ganapol Department of Aerospace and Mechanical Engineering University of Arizona with Forewords by Paul F. Zweifel Richard Sanchez and Norman J. McCormick OECD © 2008 NEA No. 6292 NUCLEAR ENERGY AGENCY ORGANISATION FOR ECONOMIC CO-OPERATION AND DEVELOPMENT ORGANISATION FOR ECONOMIC CO-OPERATION AND DEVELOPMENT The OECD is a unique forum where the governments of 30 democracies work together to address the economic, social and environmental challenges of globalisation. The OECD is also at the forefront of efforts to understand and to help governments respond to new developments and concerns, such as corporate governance, the information economy and the challenges of an ageing population. 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In these and related tasks, the NEA works in close collaboration with the International Atomic Energy Agency in Vienna, with which it has a Co-operation Agreement, as well as with other international organisations in the nuclear field. © OECD 2008 OECD freely authorises the use, including the photocopy, of this material for private, non-commercial purposes. Permission to photocopy portions of this material for any public use or commercial purpose may be obtained from the Copyright Clearance Center (CCC) at [email protected] or the Centre français d’exploitation du droit de copie (CFC) [email protected]. All copies must retain the copyright and other proprietary notices in their original forms. All requests for other public or commercial uses of this material or for translation rights should be submitted to [email protected]. Foreword FOREWORD P.F. Zweifel Forty years ago, Kenneth Case and I wrote our treatise on transport theory1 which dealt with a very idealized set of problems, basically one-speed transport in one-dimension, models that were amenable to exact solution. We gave a rationale for treating such idealized models on page 2 of our book: 1) In some special situations these [models] may be a good approximation to physical reality. [Let us admit that these special situations are indeed few and far between.] 2) The results are of (exploratory) mathematical interest in that knowledge is gained of the structure of solutions of linear transport boundary-value problems. [This aim was indeed fulfilled, as exemplified by an explosion of papers in linear transport in the years following book’s publication, not to mention the birth, in 1969, of the bi-annual series of International Conferences in Transport Theory still very much alive, a series in which Professor Ganapol has been an active and important participant. The aim of this series, as stated in the manifesto of its founders, Charles Siewert, Robert Erdmann and myself, was to bring together researchers in the several fields of engineering and science who used similar – or even identical – mathematical methods in their studies, meaning those which evolved from the methods described in Ref. 1.] 3) Of greatest importance2 is probably the fact that these exact solutions provide a sample set of problems against which approximate methods may be tested… This last item is precisely that which Professor Ganapol has seized upon in his search for benchmark solutions to nuclear reactor physics problems. He has been a true pioneer in the exploitation of exact solutions as benchmarks. Although some tentative, earlier work had been carried out in the days immediately after the publication of Ref. 1 (by Dale Metcalf) an intensive program never got underway until Professor Ganapol began his studies, in the 1970s and 80s. His benchmarks have gone far beyond the simplified problems treated in Ref. 1; for example he has considered multi-speed (aka multigroup) models of transport as well as two-dimensional and even three-dimensional transport. He has also studied zero-dimensional transport, i.e. slowing down problems using methods originated in 19553 for calculating the neutron spectrum in water-moderated reactors with the aim of evaluating few-group cross sections by spectral averaging. The Ganapol treatise which follows gives derivations of the benchmark equations which are then studied numerically, so it is complete in itself and could be the text for a graduate course in nuclear engineering. Or, if I may be permitted an anagram, in unclear engineering, since Professor Ganapol has his own unique methods in transport theory; for example, he is (in)famous as the creator of a multiple collision solution of the collisionless Vlasov equation. But all roads lead to Rome, and the material contained in this book offers a wealth of useful information and data to the practical engineer who wants to be assured that the numerical approximations being used have some correspondence to physical reality. 1 Case, K.M., P.F. Zweifel, Linear Transport Theory, Addison-Wesley, Reading, MA (1967). 2 Italics not in the original. 3 Hurwitz, H., Jr., P.F. Zweifel, J. Appl. Phys., 26, 923 (1955). iii Foreword FOREWORD Richard Sanchez and N.J. McCormick In this book on analytical benchmarks Professor Ganapol presents twelve reference calculations in transport theory. The material in the book is presented in order of progressive complexity, from problems in only the energy or only the spatial variables to problems with both variables. Chapter 1 contains a detailed introduction to the different forms of the transport equation that are solved. Chapter 2 covers slowing down and thermalization in an infinite medium, Chapter 3 deals with one-group space dependent problems, including the classical Green’s function, albedo and Milne problems, and Chapter 4 presents two multigroup space-dependent applications. True, the problems that are treated are mostly for homogeneous media in an infinite space, a half-space and the one- dimensional geometries. However, as one of us told Professor Ganapol a long time ago, it is much more difficult to accurately solve a simple transport problem than to carry out a transport calculation in a complicated geometry for reactor physics applications. The treatment of the problems presented in this book excels by the rigor of both the mathematical and the numerical analysis. To solve these problems, the author has applied and further developed a number of classical transport methods, including Caseology (i.e. singular eigenfunction expansions), H-functions and invariant embedding, the FN method. In addition, he develops Green’s functions and the mathematical relationships between solutions in elementary one-dimensional geometries, via pseudo fluxes. However, it is not enough to write down an analytical or semi-analytical explicit solution for a transport problem, there is still the problem of obtaining an accurate numerical solution. Professor Ganapol has