Advances in Nuclear Science and Technology: Volume 26 (Advances in Nuclear Science & Technology)

AdVANCES iN NUClEAR SCiENCE: ANd TECHNOlOGY VOLUME 26 AdVANCES iN NUClEAR SCiENCE ANd TECHNOlOGY Series Editors Jef...

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AdVANCES iN

NUClEAR SCiENCE: ANd TECHNOlOGY VOLUME 26

AdVANCES iN NUClEAR SCiENCE ANd TECHNOlOGY Series Editors

Jeffery Lewins Cambridge University, Cambridge, England

Martin Becker Oregon Graduate Institute of Science and Technology Portland, Oregon Editorial Board R. W. Albrecht Ernest J. Henley John D. McKean K. Oshima A. Sesonske H. B. Smets C. P. L. Zaleski

A Continuation Order Plan is available for this series. A continuation order will bring delivery of each new volume immediately upon publication. Volumes are billed only upon actual shipment. For further information please contact the publisher.

AdVANCES iN

NUClEAR SCiENCE ANd TECHNOlOGY VOLUME 26 Edited by

Jeffery Lewins Cambridge University Cambridge, England

and

Martin Becker Oregon Graduate Institute of Science and Technology Portland, Oregon

KIuwer Academic Publishers New York, Boston, Dordrecht, London, Moscow

eBook ISBN: Print ISBN:

0-306-47088-8 0-306-46110-2

©2002 Kluwer Academic Publishers New York, Boston, Dordrecht, London, Moscow Print ©1999 Kluwer Academic / Plenum Publishers New York All rights reserved No part of this eBook may be reproduced or transmitted in any form or by any means, electronic, mechanical, recording, or otherwise, without written consent from the Publisher Created in the United States of America Visit Kluwer Online at: and Kluwer's eBookstore at:

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PREFACE As we close on the Millennium we also approach a half-century of nuclear power. The major peaceful application centres on the large-scale production of electricity but international politics in the last year has shown that we have not left the origins in nuclear weapons behind. Can we assess the future of nuclear technology in the coming century? Electricity remains the dominant role for nuclear technology, though we may recognise the valuable contributions of radio-isotope work to biology, agriculture, disease control, medical diagnostics and therapy in particular, with ramifications in most industries and civilised life. The supply of electricity from nuclear power remains of the order of 10% in the USA but rises to one-third in the UK and on to some 80% in France. Far Eastern countries still have a major construction program that might lead to them rivalling such figures. Indeed if we are to accept the majority judgement of scientists that global warming through, principally, carbon dioxide is a realistic threat, it is to be hoped that nuclear power will indeed be maintained and increased to balance the almost inevitable expansion of the use of coal by China in particular. Claims that countries such as Sweden will reduce CO2 emission and simultaneously reduce nuclear power seem but a chimera, lacking all technical credibility. The down-side remains, however. Not only is the world still threatened with nuclear weapons, by repute or even testing, but no country has had the political will to manage the disposal of radio-active wastes, not just the Achilles heel but the foot, shin and leg of nuclear power. Many scientists and engineers will suppose that this is solely a political problem, claiming that the technical solutions are available. Unfortunately, it makes it no less a problem and renders the prospect of continued, let alone expanded nuclear power, no less vulnerable for being political. And who amongst the technicians is wise to say that the atavistic public perception of the threat of nuclear power is irrational when sabre rattling by emerging weapons powers echoes throughout the world? We see an interesting parallel in the modern development of genetically engineered plants and animals, to be compared with the last halfcentury of nuclear power. In the bright dawn of the post-war nuclear age, the public by-and-large accepted the claims made for the promise of the cheap production of electricity but have largely turned, or been turned by the media, against nuclear power. Do we see the same hubris mounting for those microbiologists who dismiss risks of genetic contamination of agriculture or dismiss the change in insurance attitudes when the threads of DNA predict our future like the threads of the Fates, those Spinners of the Years? Take note of the history of Nuclear Power and prepare for the backlash. If trust is to be restored in the civil use of nuclear power, rational and careful study must be made of its problems, and the results made available publicly, as instanced by the first article in this volume: Malcolm Grimston's careful assessment of the evidence that might link leukaemia with nuclear establishments in the U.K. We think this provides a useful complement to Janet Tawn's earlier assessment of Leukaemia around the Sellafield establishment in our last volume. Grimston's broader survey is followed by our Japanese contributor's development of the mathematics of non-linear stochastic models that might represent for us the risks and consequences of our operations. The substantial effort made in developing nuclear technology, from the rush into weapons to the more methodical but sustained efforts in understanding nuclear reactor physics, has led to developments in transport theory which, as seen in the third review, have applicability to microbial transport in the ground. In an era concerned with pollution, this of course has a significance for the restoration of areas such as Hanford where the U.S. authors are based, but

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vi we can, in passing, see an interesting application in the microbial recovery of inorganic elements, such as uranium and perhaps thorium, or indeed copper, in 'micro biological mining'. We see this paper as part of a steady progression in the review of transport theory, especially following articles in volume 24 by Williams and volume 25 by Pomraning. The Pressurised Water Reactor remains the leading contender for international development of nuclear power. It is a central tenet for the PWR that the failure of the pressure vessel is 'inconceivable' and that no back-up or redundancy is necessary. To hold by this belief demands an understanding therefore of crack failure and we are grateful, in returning to the UK for our fourth review, that modern ideas in crack arrest theory are brought before us by Weisner and Hayes from the Welding Institute, or TWI as it is now abbreviated. Closely associated with the prevention of catastrophic failure of the vessel is the concern for the behaviour of its contents in a severe accident. Our Swedish authors have therefore provided an assessment of the heat transfer processes in a meltdown, that allows a robust estimate of the circumstances. Much of this thinking arises from the Three Mile Island accident whose consequences over a period of twenty tears have been more pervasive than simply clearing up the now-frozen fuel melt. Finally we return to the topic of the optimum use of the fuel in power reactors, the focus of the benefit that nuclear power may give. Geoffrey Parks, who assessed optimisation methods for us in volume 21, is now joined with Paul Turinsky in surveying the application of new methods to the optimisation of both in-core and out-of-core fuel management. These new methods are driven in part by the challenge of increasing the economic value of the fuel passed through Light Water Reactors, and in part ❒indeed by the rapid increase in computing power that makes new numerical algorithms feasible. We suggest that the fresh ideas they are uncovering will find application in wider fields still. We are grateful to all our authors for contributing, each in their specialist way, to the forwarding of our discipline and it remains to commend them to their international audience, our readers. Jeffery Lewins Martin Becker

CONTENTS LEUKAEMIA AND NUCLEAR ESTABLISHMENTS: FIFTEEN YEARS OF RESEARCH Malcolm C. Grimston 1. 2. 3. 4. 5.

Summary .................................................................................................................................1 Leukaemia .................................................................................................................................3 Nuclear Establishments and Leukaemia ...........................................................................4 Possible Causes of Leukaemia Excesses Near Nuclear Establishments ................................8 Conclusions........................................................................................................................14 References

.........................................................................................................................15

NONLINEAR STOCHASTIC DYNAMICS AND INSTABILITY THEORY H. Konno 1. 2. 3. 4. 5. 6.

Introduction ..........................................................................................................................21 Basis for Nonlinear Stochastic Dynamics .......................................................................22 Nonlinear Effects in the Normal Steady State ...............................................................29 Space-Independent Stochastic Dynamics ......................................................................33 Space-Dependent Stochastic Dynamics..................................................................47 Concluding Remarks ..................................................................................................55 References ......................................................................................................................56

MODELING BACTERIAL TRANSPORT AND ACCUMULATION PROCESSES IN SATURATED POROUS MATERIAL: A REVIEW T.P. Clement, B.M. Peyton, T.R. Ginn and R.S. Skeen 1. 2. 3. 4. 5. 6.

Introduction...................................................................................................................59 Modeling Biomass in Porous Media..............................................................................61 Modeling Microbial Transport Kinetics .............................................................................66 Modeling Changes in Soil Properties Caused by Microbial Accumulation ..............69 Modeling Microbial Growth Kinetics ..............................................................................71 Summary and Conclusions .....................................................................................................72 References.............................................................................................................................74 vii

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CONTENTS

A REVIEW OF CRACK ARREST CONCEPTS FOR THE ASSESSMENT OF PRESSURE VESSEL INTEGRITY C.S. Wiesner and B. Hayes 1. 2. 3. 4. 5. 6. 7.

Introduction .........................................................................................................................79 Historical Context ...................................................................................................................................80 Measuring Crack Arrest Properties ............................................................................... 81 Analysis of Crack Arrest Behaviour ...........................................................................88 Crack Arrest Data for Pressure Vessel Steels .................................................................... 92 Application of Crack Arrest to Pressure Vessel Integrity Assessment...........................93 Conclusions . ......................................................................................................................96 References ...................................................................................................................... 98

HEAT TRANSFER PROCESSES IN REACTOR VESSEL LOWER PLENUM DURING LATE PHASE OF IN-VESSEL CORE MELT PROGRESSION B.R. Sehgal, V.A. Bui, T.N. Dinh and R.R. Nourgaliev 1. 2. 3. 4. 5. 6. 7.

Abstract.....................................................................................................................103 Introduction ........................................................................................................................104 Modeling of Natural Convection Heat Transfer in Core Melt Pool ......................................105 Validation of ECCM for Melt Pool Natural Convection Heat Transfer .............................113 Heat Transfer in Molten Metal Layer: Computational Fluid Dynamics (CFD) Analysis .................................................................121 Modelling of Heat Transfer in Metallic Layer by ECCM Approach .............................126 Assessments of the Thermal Loadings in Selected Severe Accident Scenarios .............128 Summary and Concluding Remarks ..................................................................................131 References......................................................................................................................133

ADVANCES IN NUCLEAR FUEL MANAGEMENT FOR LIGHT WATER REACTORS Paul J. Turinsky and Geoffrey T. Parks 1. 2. 3. 4. 5.

Introduction ..............................................................................................................137 Modern Fuel Management Practices ..................................................................................142 Computational Reactor Physics Methods ............................................................................148 Mathematical Optimization Techniques.................................................................................153 Future Developments .............................................................................................................162 References .................................................................................... ..............................162

INDEX ...................................................................................................................................167

CONTENTS OF EARLIER VOLUMES* CONTENTS OF VOLUME 10 Optimal Control Applications in Nuclear Reactor Design and Operations, W.B.Terney and D.C.Wade Extrapolation Lengths in Pulsed Neutron Diffusion Measurements, N.J.Sjsötrand Thermodynamic Developments, R.V.Hesketh Kinetics of Nuclear Systems: Solution Methods for the Space-Time Dependent Neutron Diffusion Equation, W. Werner Review of Existing Codes for Loss-of-Coolant Accident Analysis, Stanislav Fabic

CONTENTS OF VOLUME 11 Nuclear Physics Data for Reactor Kinetics, J.Walker and D.R.Weaver The Analysis of Reactor Noise: Measuring Statistical Fluctuations in Nuclear Systems, N.Pacilio, A.Colombina, R.Mosiello, F.Morelli and V.M.Jorio On-Line Computers in Nuclear Power Plants - A Review, M. W.Jervis Fuel for the SGHWR, D.O.Pickman, J.H.Gittus and K.M.Rose The Nuclear Safety Research Reactor (NSSR) in Japan, M.Ishikawa and T.Inabe Practical Usage of Plutonium in Power Reactor Systems, K.H.Peuchl Computer Assisted Learning in Nuclear Engineering, P.R.Smith Nuclear Energy Center, M.J.McKelly

CONTENTS OF VOLUME 12 Characteristic Ray Solutions of the Transport Equation, H.D.Brough and C.T.Chandler Heterogeneous Core Design for Liquid Metal Fast Breeder Reactors, P.W.Dickson and R.A.Doncals Liner Insulation for Gas Cooled Reactors, B.N.Furber and J.Davidson Outage Trends in Light Water Reactors, E.T.Burns, R.R.Pullwood and R.C.Erdman Synergetic Nuclear Energy Systems Concepts, A.A.Harms Vapor Explosion Phenomena with Respect to Nuclear Reactor Safety Assessment, A. W.Cronenberg and R.Benz

* Volumes 1-9 of the series were published by Academic Press.

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CONTENTS OF EARLIER VOLUMES CONTENTS OF VOLUME 13

Radioactive Waste Disposal, Horst Böhm and Klaus Kühn Response Matrix Methods, Sten-Oran Linkahe and Z.J.Weiss Finite Approximations to the Even-Parity Transport Equation, E.E.Lewis Advances in two-Phase Flow Instrumentation, R.T.Lahey and S.Benerjee Bayesian Methods in Risk Assessment, George Apostolakis

CONTENTS OF VOLUME 14 Introduction: Sensitivity and Uncertainty Analysis of Reactor Performance Parameters, C.R.Weisben Uncertainty in the Nuclear Data used for Reactor Calculations, R.W.Peeble Calculational Methodology and Associated Uncertainties, E.Kujawski and C.R. Weisben Integral Experiment Information for Fast Reactors, P.J.Collins Sensitivity Functions for Uncertainty Analysis, Ehud Greenspan Combination of Differential and Integral Data, J.H.Marable, C.P.Weisbin and G.de Saussure New Developments in Sensitivity Theory, Ehud Greenspan

CONTENTS OF VOLUME 15 Eigenvalue Problems for the Boltzmann Operator, V.Protopopescu The Definition and Computation of Average Neutron Lifetimes, Allen F.Henry Non-Linear Stochastic Theory, K.Saito Fusion Reactor Development: A Review, Weston M.Stacey, Jr. Streaming in Lattices, Ely M.Gelbard

CONTENTS OF VOLUME 16 Electrical Insulation and Fusion Reactors, H.M.Bamford Human Factors of CRT Displays for Nuclear Power Plant Control, M.M.Danchak Nuclear Pumped Lasers, R.T.Schneider and F.Hohl Fusion-Fission Hybrid Reactors, E.Greenspan Radiation Protection Standards: Their Development and Current Status, G.C.Roberts and G.N.Kelly

CONTENTS OF EARLIER VOLUMES

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CONTENTS OF VOLUME 17 A Methodology for the Design of Plant Analysers, T.H.E.Chambers and M.J.WhitmarshEveries Models and Simulation in Nuclear Power Station Design and Operation, M. W.Jervis Psychological Aspects of Simulation Design and Use, R.B.Stammers The Development of Full-Scope AGR Training Simulators within the C.E.G.B., C.R.Budd Parallel Processing for Nuclear Safety Simulation, A.Y.Allidina, M.C.Singh and B.Daniels Developments in Full-Scope, Real-Time Nuclear Plant Simulators, J. Wiltshire

CONTENTS OF VOLUME 18 Realistic Assessment of Postulated Accidents at Light Water Reactor Nuclear Power Plants, E.A. Warman Radioactive Source Term for Light Water Reactors, J.P.Hosemann and K.Hassman Multidimensional Two-Phase Flow Modelling and Simulation, M.Arai and N.Hirata Fast Breeder Reactors - The Point of View of French Safety Authorities, M.Laverie and M.Avenas Light Water Reactor Space-Dependent Core Dynamics Computer Programs, D.J.Diamond and M.Todosow

CONTENTS OF VOLUME 19 Festschrift to Eugene Wigner Eugene Wigner and Nuclear Energy, A.M. Weinberg The PIUS Principle and the SECURE Reactor Concepts, Kåre Hannerz PRISM An Innovative Inherently Safe Modular Sodium Cooled Breeder Reactor, P.H.Pluta, R.E.Tippets, R.E.Murata, C.E.Boardman. C.S.Schatmeier, A.E.Dubberley, D.M.Switick and W.Ewant Generalized Perturbation Theory (GPT) Methods; A Heuristic Approach, Augusto Gandini Some Recent Developments in Finite Element Methods for Neutron Transport, R.T.Ackroyd, J.K.Fletcher, A.J.H.Goddard, J.Issa, N.Riyait, M.M.R.Williams and J. Wood

CONTENTS OF VOLUME 20 The Three-Dimensional Time and Volume Averaged Conservation Equations of Two-Phase Flow, R.T.Lahey, Jr., and D.A.Drew Light Water Reactor Fuel Cycle Optimisation: Theory versus Practice, Thomas J.Downar and Alexander Sesonske The Integral Fast Reactor, Charles E.Till and Yoon I.Chang Indoor Radon, Maurice A.Robkin and David Bodansky

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CONTENTS OF EARLIER VOLUMES CONTENTS OF VOLUME 21

Nodal Methods in Transport Theory, Ahmed Badruzzaman Expert Systems and Their Use in Nuclear Power Plants, Robert E.Uhrig Health Effects of Low Level Radiation, Richard Doll and Sarah Darby Advances in Optimization and Their Applicability to Problems in the Field of Nuclear Science and Technology, Geoffrey T.Parks Radioactive Waste Storage and Disposal in the U.K., A.D.Johnson, P.R.Maul and F.H.Pasant

CONTENTS OF VOLUME 22 High Energy Electron Beam Irradiation of Water, Wastewater and Sludge, Charles N.Kurucz, Thomas D.Waite, William J.Cooper and Michael J.Nickelsen Photon Spectroscopy Calculations, Jorge F.Fernández and Vincenzo G.Molinari Monte Carlo Methods on Advanced Computer Architecture, William R.Martin The Wiener-Hemite Functional Method of Representing Random Noise and its Application to Point Reactor Kinetics Driven by Random Reactivity Fluctuations, K.Behringer

CONTENTS OF VOLUME 23 Contraction of Infomation and Its Inverse Problems in Reactor System Identification and Stochastic Diagnosis, K.Kishida Stochastic Perturbation Analysis Applied to Neutral Particle Transfers, Herbert Rieff Radionuclide Transport in Fractured Rock An Analogy with Neutron Transport, M.M.R.Williams

CONTENTS OF VOLUME 24 Chemobyl and Bhopal Ten Years on, Malcolm C. Grimston Transport Theory in Discrete Stochastic Mixtures, G.C. Pomraning The Role of Neural Networks in Reactor Diagnostics and Control, Imre Pázsit and Masaharu Kitamura Data Testing of ENDF/B-VI with MCNP: Critical Experiments, Themal-Reactor Lattices and Time-of-Flight Measurement, Russel D. Mosteller, Stephanie C. Frankl and Phillip G. Young System Dynamics: An Introduction and Application to the Nuclear Industry, K.F. Hansen and M.W. Gorlay BN Theory: Advances and New Models for Leakage Calculations, Ivan Petrovic and Pierre Benoist Current Status of Core Degradation and Melt Progression in Severe LWR Accidents, Robert R. Wright

CONTENTS OF EARLIER VOLUMES CONTENTS OF VOLUME 25 Childhood Leukaemia and Radiation: The Sellafield Judgement, E. Janet Tawn and Richard Wakeford Reactor Dynamics from Monte Carlo Calculations, Timothy E. Valentine Notes on a Simplified Tour: From the Fourier to the Wavelet Transform, Marzio Marseguerra Genetic Algorithms for Incore Fuel management and Other Recent Developments in Optimisation, Jonathan N. Carter The Computerization of Nuclear Power Plant Control Rooms, Bill K.H. Sun and Andrei N. Kossilov Consequences of Chemobyl: A View Ten Years On, A Borovoi and S. Bogatov Dynamic Reliability, Jacques Devooght

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LEUKAEMIA AND NUCLEAR ESTABLISHMENTS Fifteen Years of Research

Malcolm C. Grimston, MA (Cantab.), BA (Open) Senior Research Fellow T.H. Huxley School of Environment, Earth Science and Engineering Imperial College of Science Technology and Medicine London SW7 2PE

SUMMARY It is fifteen years since a television programme in the UK, called Windscale, the Nuclear Laundry, introduced the possibility that certain nuclear installations, notably BNFL’s Sellafield nuclear reprocessing site in Cumbria, England, may be associated with excesses of childhood leukaemia. Since then a great deal of research has been carried out to determine, first whether the association is a real one rather than one arising from chance, and secondly, ifit is a real one, what cause or causes might be implicated. Research into such phenomena tends to be of two kinds. Epidemiological investigations look for unusual patterns of the disease in a variety of populations, with a view to demonstrating crucial differences between populations which show a statistically high rate of the disease and those which do not and hence discovering possible causes. Biological investigations, by contrast, look at affected individuals in an attempt to determine directly what might induce the disease; animal studies are often involved. In the case of leukaemia, a generic name for a set of cancers of the bone marrow and lymphatic system which result in the production of large number of immature white blood cells and hence compromise the immune system, biological research has led to a recognition of the level of development at which the disease occurs, but has led to relatively little understanding of what agents, genetic, environmental or infective, may be implicated; the cause of more than 80% of leukaemias remains unknown, In particular, the biological studies, while confirming that high levels of radiation lead to increased risks of developing cancer, imply that the levels of radiation to be found near Sellafield and other relevant installations are too low, by at least two orders of magnitude, to explain the reported excesses of leukaemia. Further, if radiation were the cause, one would expect also to see excesses of other radiation-related diseases, and this is not the case.

Advances in Nuclear Science and Technology, Volume 26, edited by Lewins and Becker. Kluwer Academic / Plenum Publishers, New York 1599.

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This paper concentrates mainly on the epidemiological investigations which have been carried out. The first question to be addressed is whether the reported unusual patterns of the disease near certain nuclear establishments can be explained by chance. Extremely unlikely patterns of a variety of phenomena do occur randomly; if one notes an apparent excess of a disease in an area, say, and then carries out statistical analysis of the area, it is very likely that the analysis will confirm the excess. In the absence of a prior hypothesis, i.e. some reason for believing that an excess should occur in the area in question, the reason having emerged from other, independent, study, such an excess is difficult to interpret. It may be random, or it may be caused by some factor acting in the area. The excess of leukaemia in the Sellafield area, then, is perhaps best viewed as generating the hypothesis that some feature of ‘sites like Sellafield’ causes, or at least is associated with, increased risks of childhood leukaemia. The discovery of an excess of childhood leukaemia in the vicinity of the UK’s other nuclear reprocessing plant, at Dounreay in northern Scotland, and near the weapons research establishments at Aldermaston and Burghfield, coupled with more recent work suggesting an excess near the French reprocessing plant at Cap La Hague in Normandy, suggests strongly that these excesses are not chance occurrences, but are indeed associated with some feature of the plants themselves. In the UK the plants in question are nuclear establishments which were operating before 1955. Three possible causes for the excesses are considered: emissions of radioactive materials from the installations; genetic damage caused by irradiation of the father before conceiving the child (the ‘Gardner hypothesis’); and an infective aetiology for the disease, coupled with changed patterns of population mixing in the vicinity of the installations (the ‘Kinlen hypothesis’). Direct emissions of radioactivity as an explanation suffers from the fact that emissions, even making generous allowances for possible historical under-reporting, are far too low to account for the excesses, given our understanding of dose-response relationships involving radiation and cancer derived from several populations, notably the survivors of the atom bombs at Hiroshima and Nagasaki. The possibility that internal irradiation is more damaging than external has been considered, but seems unlikely to overcome the fundamental observation. The Gardner hypothesis, which suggests that high doses of radiation received by the father before conceiving the child increases the child’s risk of developing leukaemia, seems uniquely to apply to the town of Seascale near Sellafield; the association is not observed elsewhere in Cumbria, nor in Scotland, Denmark, Germany or Ontario, Canada. Interaction between paternal preconceptional irradiation and some other factor unique to Seascale but not present, say, in Egremont North, 11 km away, appears implausible. At present it seems possible that the Gardner correlation is one of chance. The Kinlen hypothesis holds that leukaemia is the rare result of a fairly common infective agent, perhaps a virus. Changes in patterns ofpopulation mixing, especially when an isolated geographical area, with a high proportion of people who have little experience of the virus and are hence unusually susceptible to it, receives an influx from more populated areas, leads to a spread of the virus and hence an outbreak of the disease. The wide range of populations which conform to the Kinlen hypothesis, including oil workers, wartime evacuees, military servicemen, the New Towns of the 1950s and those in occupations which involve regular contact with the public, implies that population mixing is at least part of the explanation. It is not clear at present whether the Kinlen hypothesis alone could explain the excesses, or whether some other factor must also be acting. The fact that, after such a major research programme, controversy remains is a reflection of the difficulty of determining the causes of rare diseases. It is believed that smoking is associated with some 100 000 early

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deaths per year in the UK, yet it took many years to establish the correlation, and even now one hears, correctly, that no causal link has been ‘proven’. The excesses of leukaemia near nuclear installations are running at one or two cases per year. Until, say, the virus associated with the illness is isolated, or an alternative method by which radioactive materials might induce leukaemia is characterised, the complete truth is unlikely to emerge. LEUKAEMIA Leukaemia is a generic name for certain cancers of the bone marrow and lymphatic system, and it manifests by the rapid multiplication of abnormal white blood cells in the marrow. The blood stream therefore becomes flooded by immature white cells which are not capable of the normal functioning of mature cells. There are several different types of leukaemia, and of the closely related diseases, lymphoma. In addition the excess cancerous tissue in the bone marrow affects the production of red blood cells and platelets, causing anaemia and bleeding. Fortunately leukaemia is relatively rare, affecting about 1 in 1800 live births by the age of 15, or about 500 children per year in the UK (Doll, 1989). Leukaemia kills more children between the ages of 2 and 15 than any other disease, but progressive improvements in treatment mean that nowadays the remission rate for childhood leukaemia is better than 60%. Childhood leukaemia accounts for about 10% of all leukaemias. Of the various forms of childhood leukaemia, acute lymphoid leukaemia is the most common. This is a rapidly progressive cancer of the lymphocytes, which are cells responsible for recognising infection. Apparent excesses of leukaemia over short periods of time in small areas have been reported for over 60 years, since before nuclear fission was discovered (e.g., Kellett, 1937). However, excesses do occur entirely by chance in any random process, and leukaemia does not seem to ‘cluster’ naturally (Smith, 1982; Heasman, 1986) - in other words, there is little evidence to suggest that excesses occur more often than would be expected by chance. However, leukaemia rates in different parts of the country do seem to vary, with Somerset in particular having rather higher levels than expected at all ages, and for most leukaemia types (Leukaemia Research Fund, 1990). In contrast to many childhood diseases, leukaemia seems to be especially associated with individuals of high socioeconomic status ( e.g., McWhirter, 1982; Alexander et al., 1990; Stiller and Boyle, 1996). This has led to leukaemia sometimes being called a ‘middle class disease’. There seems to be a remarkable (positive) correlation between risk of leukaemia and male employment rates (by county in England and Wales), and also between childhood leukaemia rates and parental employment (Wolff, 1991). Possible Causes It is believed that leukaemia can start with damage to the DNA of a single primitive blood cell in the bone marrow, although cells disrupted in this fashion are almost always dealt with by the body’s defence mechanisms. Several agents are known to cause damage to DNA, such as ionising radiation, ultraviolet light, chemicals such as benzene, and some viruses. It is believed that most of these can also cause some forms of leukaemia (Greaves, 1988). In addition, it is thought that hereditary factors may be involved in the development of leukaemia. People with Down’s Syndrome, for example, have a 20-fold increase in their risk of developing acute leukaemia (Miller, 1963). Also implicated in childhood leukaemia may be certain chemotherapeutic drugs, maternal smoking during pregnancy (Stjernfeldt et al., 1986), garden pesticides (Lowengart

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et al., 1987, and chickenpox (Frederick and Alberman, 1972) and influenza (Fine et al., 1985) during pregnancy, though some dispute remains in all of these cases. Ionising Radiation It is well established that large doses of ionising radiation to the body can increase the risk of developing leukaemia. This information derives from studies of radiologists early this century, from people treated with large doses of X-rays to the spine for ankylosing spondylitis; and from the survivors of the atom bombs at Hiroshima and Nagasaki, among whom there has been an excess of about 80 cases of leukaemia in a population of just over 90 000 (Preston and Pierce, 1987). It is also established that doses of radiation delivered to the foetus in utero can increase the subsequent risk of the child developing leukaemia (Mole, 1990), the excess risk being about 40% (Wakeford, 1995). In the UK the average annual dose of radiation is about 2.5 mSv (millisieverts). In Cornwall, owing to high levels of radon gas, the average dose is 7.8 mSv. A few thousand houses in the UK deliver annual doses ofmore than 50 mSv (Wrixon et al., 1988): a few tens may be delivering doses above 500 mSv (O’Riordan, 1990). It is not generally thought that high levels of radon are associated with leukaemia (Wood, 1960), although at least one study has suggested that there may be such a link (Henshaw et al., 1990). Between them, in utero X-rays and ionising radiation are unlikely to account for more than 8% of all childhood leukaemias (Doll, 1989), with another 3% associated with welldefined genetic abnormalities. Hence more than 80% of childhood leukaemia is unexplained. NUCLEAR ESTABLISHMENTS AND LEUKAEMIA In 1983 James Cutler, a journalist with Yorkshire Television, while researching a programme on the 1957 Windscale Fire, noticed an excessive number of cases of juvenile leukaemia in the town of Seascale, three miles south of the Sellafield nuclear complex. His findings, broadcast in the programme Windscale, the Nuclear Laundry, suggested that there had been seven cases of leukaemia in people under the age of 25 (five of these in children under 10) in Seascale between 1954 and 1983, when less than one case would have been expected by chance. The Sellafield complex includes the early Windscale military piles (one of which caught fire in 1957, the Calder Hall Power Station, and one of the world’s two largest reprocessing operations. Reprocessing is a chemical process which separates spent nuclear fuel into reusable uranium, plutonium, and waste products. Because of this operation Sellafield discharges more radioactive material into the environment than does a nuclear power station; historical discharges have been much higher than today’s. As a result of Cutler’s claims, a committee of enquiry was set up under Sir Douglas Black, a former president of the Royal College of Physicians of London. The Committee reported in 1984 (Black, 1984), confirming the excess in leukaemia rates in the area but saying that more research was necessary before any link with Sellafield could be confirmed or denied. At the Black Report’s recommendation, the Department of Health and Social Security set up the Committee on the Medical Aspects of Radiation in the Environment (COMARE), and further epidemiological studies were proposed. Epidemiology Epidemiology is the study of disease in relation to populations. Its main procedure is statistical analysis, for example observing rates of a disease near nuclear power stations and

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comparing them to the rates in a ‘control’ area which has a similar population but no nuclear power station. It tends to rely therefore on historical records rather than on designed experiment. The interpretation of statistical findings can be highly problematic and ambiguous, as strictly speaking, epidemiology does not prove causes, but demonstrates correlations. While it is true that epidemiological methods sometimes produce such overpowering evidence that deduction of causation is inescapable, as in the case of smoking and lung cancer, more often it is not clear whether the observed phenomenon arises through chance, or through some third (‘confounding’) factor. For example, cancer mortality rates in towns such as Eastbourne and Bournemouth are significantly higher than the national average. However, this cannot be taken to indicate some deadly environmental factor in these areas: seaside towns attract many senior citizens, among whom deaths from cancer are more common than among younger people. When dealing with a very small number of cases, the statistical picture is even more complicated. Chance becomes more important; the misdiagnosis of a single case of the disease can dramatically alter the apparent picture. If one area is more efficient at registering leukaemia, perhaps because it is an important issue among the local medical community, this can significantly distort the apparent incidence of the disease. It is also difficult to match the study group with a control group which has the right age profile, social group mixture and local environment to avoid the introduction of bias; in effect the choice of any control group is bound to reflect some hypothesis, if only that the population is similar to another in all respects except that under investigation. Epidemiological associations can be even more difficult to interpret if apparently abnormal distributions of disease (or other phenomenon) are noted without being predicted by a prior hypothesis. The likelihood of a particular pattern of numbers in the UK National Lottery is about 1 in 14 million. Yet it would be unjustified to assume that because such an unlikely pattern had occurred one week, there must be some bias towards that pattern If, on the other hand, the same, or a very similar, pattern were to occur twice running (i.e. that the first occurrence had been used to generate the hypothesis of such bias, the second observation being used to test the hypothesis) then one would be on firmer ground in claiming that the pattern was in some way favoured or caused. This point is relevant in the case of the Seascale leukaemias. An area with apparently high leukaemia rates was reported casually; when statistics were done it was found that leukaemia was indeed prevalent in the area, Of itself this could not demonstrate that the excess was caused by any factor, environmental or other. The observation should strictly have been used to generate the hypothesis that some feature of Sellafield, be it associated with radiation or not, was correlated with excesses of juvenile leukaemia, and other sites should then have been identified to test this hypothesis (or hypotheses). The response of elements of the media and of certain non-governmental organisations, however, was very different from this. Olsen et al. (1996) point out that excesses tend first to be noted in a small geographical area (e.g., Seascale). A local ‘cause’ (e.g., Sellafield) is identified. However, rather than then gathering data for other areas surrounding the ‘cause’, say a 10 km circle, attention still tends to focus on the original excess, which of its nature (the fact that it was noticed casually) is likely to be more significant. This weakness is prevalent in media reports of such events (“leukaemia rates ‘near Sellafield’ can be ‘up to’ ten times the national average”), though some commentators believe it can influence the research community as well (Kinlen et al., 1997). Olsen et al. likened this approach to a Texas sharpshooter, who would fire a bullet at a blank wall and then draw a target round the bullet mark. One cannot of course be sure that the sharpshooter would not have hit the bulls eye of a preexisting target (or that the apparent excess is not real and caused), but it is a less impressive demonstration of skill. The rest of this paper should be read with these statistical problems in mind.

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Nuclear installations operating in the UK before 1955 The definitive Office of Population Censuses and Surveys (OPCS) study into cancer near nuclear establishments m England and Wales in the years 1950 to 1984 (Cook-Mozaffari et al., 1987; Forman et al., 1987) established that overall cancer rates around nuclear installations were below average when compared to matching control areas, However, when discussing leukaemia the study drew a distinction between the pre-1955 sites, such as Sellafield, Dounreay, Aldermaston, Burghfield and Harwell, which did seem to display excess rates of childhood leukaemia, and the then CEGB (now Magnox Electric plc and British Energy plc) nuclear power stations, the fist two of which began operating in 1962, which did not. The first COMARE report (Bobrow et al., 1986) focused on the Sellafield area, and especially Seascale. The rates of stillbirth and of infant mortality in Seascale from 1950 to 1983 were significantly lower than the national average (Gardner et al., 1987), in all likelihood because the town had become the traditional home for many of the site’s senior management. However, the excess of leukaemia was of great concern, and it looked unlikely that it could be explained simply in terms of greater prevalence of leukaemia among higher socio-economic groups. Bobrow et al. took into account further discharges from Sellafield which had not been characterised in the Black Report, and pointed out that because monitoring practices before 1977 were less sophisticated than today’s it was difficult to estimate precise radiation doses frin the plant before that date. However, even taking revised estimates of radiation exposure, the report concluded that the leukaemia excess in Seascale was difficult to explain in terms of radioactive discharges; recent calculations suggest that environmental levels of radiation are two orders of magnitude below those necessary to cause such excesses (Simmonds et al., 1995). Further investigation revealed another case of leukaemia, in a child born in Seascale who had subsequently left the area, making a total of five fatal cases born in the village (Gardner et al., 1987). Four of these five cases occurred before 1970. The Seascale excess does not appear generally to extend to the rest of Copeland District Council or West Cumbria Health Authority (Draper et al., 1993). Craft et al. (1993) explored leukaemia and non-Hodgkin’s lymphoma rates in the 49 electoral wards in Copeland District, plus the neighbouring ward of Broughton in South Lakeland District. Excess rates ofacute lymphoblastic leukaemia were found in Egremont North and Broughton, and of acute myeloid leukaemia in Sandwith. However, while the Seascale ward possessed the most signibnt high rate of acute lymphoblastic leukaemia in the north of England and one of the most significant high rates of non-Hodgkin’s lymphoma, the significance of the other excesses was considerably lower. The second COMARE report (Bobrow et al., 1988) looked at leukaemia cases which occurred between 1968 and 1984 in the KW postcode area of Caithness, which includes the Dounreay nuclear establishment and the town of Thurso. Dounreay, the location of Britain’s research into the Fast Reactor, has on site a variety of facilities, including the 250 Mw Prototype Fast Reactor and its associated reprocessing plant. This reprocessing operation is much smaller than that at Sellafield. Between 1968 and 1984 there were 12 cases of leukaemia registered within the KW area: five within 12.5 km of Dounreay; one between 12.5 and 25 km; and six more than 25 km away. Furthermore, two cases which had been diagnosed as non-Hodgkin’s lymphoma, one Within 12.5 km, one between 12.5 and 25 km of the plant, were rediagnosed as leukaemia. Seven of the eight cases living within 25 km of Dounreay had been registered after 1978. The Dounreay pattern differs from the Sellafield one in that it seems to represent a temporal as well as spatial cluster; the excess is in a narrow time period as well as a small geographical area.

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COMARE 2 makes the point that to discover leukaemia excesses near both Britain's reprocessing plants makes it more likely that some common factor associated with these plants is implicated. In effect, ifthe Seascale cases generated the hypothesis that some feature of a plant like Sellafield was associated with leukaemia excesses, the Dounreay research prima facie confirmed this idea. The third COMARE report (Bobrow et al., 1989) found a small but significant increase in leukaemia registration rates within 10 km of Aldermaston and Burghfield. Practically all of the excess was found within 10 km of Burghfield (the discharges from which are minute) in an area which includes Reading, and in children under 5. The report also found a small, nonsignificant deficit in childhood leukaemias within 10 km of the UK Atomic Energy Authority site at Harwell, though Busby and Scott Cato (1997) report an excess in South Oxfordshire between 1981 and 1995. Barton et al. (1997) report no increased incidence of acute lymphoblastic leukaemia in west Berkshire since 1972. The UK nuclear power stations As noted above, Cook-Mozaffari et al. (1987), in the OPCS study, found no abnormal patterns of childhood leukaemia in the vicinity of the then CEGB nuclear power stations, the first two of which began operating in 1962. However, subsequent reanalysis of the OPCS data suggested that leukaemia rates in areas near to the CEGB stations seemed to be about 15% greater than expected (Cook-Mozaffari et al., 1989a). Discharges of radiation from these plants were (and are) extremely low when compared to natural levels of radiation. Further, it seemed that leukaemia rates near sites that the CEGB had considered for nuclear stations but subsequently rejected, such as Druridge Bay in Northumberland and Luxulyan in Comwall, were also higher than expected (Cook-Mozaffari et al., 1989b) - or to put it another way, if such areas had been used as the controls for the operating power stations no excess would have been observed. Hence there may be some other aspect of the type of area in which nuclear stations are built which causes a raised incidence of juvenile leukaemia, though case numbers in the sites at which stations were not built are small. The international picture , The only other large-scale reprocessing plant operating in the world is at COGEMA s Cap La Hague site in Nord-Cotentin, Normandy. Early studies showed slight (non-significant) deficits in cancer and childhood leukaemia rates in Nord-Cotentin, in a 35 km circle around Cap La Hague (Dousset, 1989; Viel and Richardson, 1990). This area also contains three other nuclear sites: the contiguous low level waste storage facility, a nuclear power station 16 km from La Hague, and a naval dockyard 19 km away. However, Viel et al. (1995) subsequently noted a small, non-significant excess in juvenile leukaemia from 1978 to 1993 in the electoral ward of Beaumont-Hague (4 cases against an expected 1.4) which contains the La Hague reprocessing plant. Pobel and Viel (1997) examined a number of possible associations, and found no links with, among other factors, parental occupation (including parental preconceptional irradiation), electromagnetic fields or characteristics of residence. However, in a recall study, mothers of children who developed leukaemia were significantly more likely to recall taking part m recreational activities on the local beach while pregnant than were mothers of the controls (relative risk 4.5). There was also a positive dose (frequency of visits)-response relationship. Children who recalled playing on the beach also seemed more at risk (relative risk 2.9). Children who recalled consuming fish and shellfish were also more likely to have developed leukaemia. A similar association with the beach, again involving a recall study, was noted by Urquhart et al. (1991) with respect to Dounreay, though not, for example, by Gardner et al. (1990) with respect to Sellafield, while Watson and Sumner ( 1996)

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showed that levels of radioactivity in affected and unaffected children near Dounreay were similar. Pobel and Viel concluded that some environmental pathway for radioactive materials, possibly marine, might be involved. However, the measured contamination on the beach is still far too low to account for such an excess (see below), and Clavel and Hémon (1997) showed that the size of the dose-response relationship noted by Pobel and Viel, coupled with the number of controls who recalled visiting the beach, was such that the overall level of leukaemia in Nord-Cotentin would be expected to be double the normal, rather than normal as is the case. In addition, the problems of recall bias should be borne in mind (Kneale and Stewart, 1980), especially in a study which involves births over a period of almost forty years, though Pobel and Viel argue that since La Hague was not associated with leukaemia before the study was carried out there is no reason why mothers of leukaemic children should selectively remember visits to the beach. Further investigation is under way. A study looking at cancer incidence near six French installations which began operation before 1973 found that leukaemia rates in these areas are slightly lower than would be expected by chance (Hill and Laplanche, 1990). Childhood cancer frequencies near nuclear installations in Germany show no abnormal patterns overall (Keller et al., 1992), but a cluster of leukaemias near the Krümmel nuclear plant in Lower Saxony has been documented (Lackland et al., 1997). The US equivalent of the OPCS study (Cook-Mozaffari et al., 1987), published by the American National Cancer Institute (NCI, 1990), failed to find any evidence of increased levels of sixteen different types of cancer, including leukaemia, in the 107 US counties surrounding 52 commercial nuclear power stations, nine Department of Energy research and weapons plants and one commercial fuel processing plant. Other studies m the USA have failed to show any excess of childhood leukaemia near, for example, Three Mile Island (Hatch et al., 1990); Hanford (Miham, 1988); Oak Ridge (Patrick, 1977). One study does suggest that leukaemia rates (all ages) near the plant at Pilgrim Massachusetts, may have been higher than expected in the early 1980s, but not subsequently (Massachusetts Department of Public Health, 1990). There have been several studies carried out on the health of the population near to Chernobyl in the wake of the 1986 accident. These have not resulted in any reliable evidence for increased leukaemia rates, but the poor state of record-keeping both before and after the accident makes interpretation difficult (WHO, 1995). In general, however, it did seem until recently that the apparent excess of leukaemia found near nuclear power plants was a British phenomenon which, for some reason, did not occur elsewhere in the world. This impression is now much less clear. POSSIBLE CAUSES ESTABLISHMENTS

OF

LEUKAEMIA

EXCESSES

NEAR

NUCLEAR

Radioactive discharges into the environment Perhaps the most superficially attractive explanation of these observations is that some radioactive substance, such as plutonium, released from the establishments is causing the leukaemias. However, this theory has significant weaknesses. First, and most obvious, are the very low levels of radioactive discharge compared to natural background radiation, and to the levels of radiation observed to be necessary to cause leukaemia among radiologists, spondylitics and atom bomb survivors. The National Radiological Protection Board (NRPB) calculated that discharges from Sellafield were too low by a factor of 300 to account for the five excess leukaemias, on current dose/response estimates (Stather et al., 1988a); more recent calculations confirm that radiation doses have been two orders of magnitude too low to account for the excess (Simmonds et al., 1995). Discharges cannot be much greater than thought, as the material involved would then have

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shown up in autopsies carried out in the area in far greater amounts than is the case (Doll, 1990). Similarly, the susceptibility of children to radiation is unlikely to be much greater than is believed at present, as natural radiation would then cause more leukaemia than actually occurs (Doll, 1990): there is no difference in radiation damage caused by natural and artificial radioactive materials. Bithell et al. (1994), examining the incidence of leukaemia in electoral wards of West Cumbria and South Lakeland reported by Croft et al. (1993), showed that, if the Seascale cases were excluded, there was no correlation between distance from the plant and incidence of leukaemia. For example, the fifth-furthest ward from the plant, Egremont North, had a significant excess of acute lymphoblastic leukaemia, while no cases were reported for the three wards ranking second to fourth-Mest. A distance-response relationship would be expected if plant discharges were a major cause. Recent revelations about an explosion in a waste shaft at Dounreay in 1977 (COMARE/RWMAC (Radioactive Waste Management Advisory Committee), 1995) have led to reviews of the estimated doses received by local residents. However, assays of the concentrations of radionuclides including plutonium-239, americium-241 and strontium-90 in people living near the plant revealed no significant differences in contamination between leukaemia cases, siblings, parents, matched local controls and controls living remote from reprocessing plants. It is unlikely, therefore, that the observed excess is due to the single factor of personal radioactive contamination from Dounreay (Watson and Sumner, 1996). The fallout from the atmospheric atom bomb tests carried out between 1958 and 1963 did not seem to cause any worldwide increase in leukaemia incidence. Yet these tests released the same materials, including plutonium, as are discharged from a reprocessing plant. Furthermore, the dose received because of the test fallout by anyone living in Thurso was about the same as that received because of discharges from Dounreay (Darby and Doll, 1987). So if these discharges are to blame, the whole world might have been expected to have suffered from dramatically raised leukaemia incidence in the early and mid 1960s, something which did not occur. The excesses near Sellafield, Dounreay, Aldermaston and Burghfleld all involve about five to fifteen cases, albeit in areas of significantly different population densities. Yet before the dramatic reductions of recent years, Sellafield’s discharges were about 15 times greater than Dounreay’s, which were in turn about 900 times greater than the minute discharges owing to Aldermaston and Burghfield (Lambert, 1990). Hence the effect seem to be independent of the amount, and type, of radioactive material released by the establishment in question. This is a very difficult observation to reconcile with present accepted theory. The third COMARE report (Bobrow et al., 1989), for example, states, ‘In our judgment, the authorised and accidental radioactive discharges from [Aldermaston and Burghfleld] are far too low to account for the observed increase in childhood cancer incidence in the area,’ and points out that people living 5 km from Aldermaston receive about twice as much radiation from the coal plant operating on the site at that period than they would have in the peak year of discharges from the weapons operations. The deficit, noted in the COMARE report, of leukaemias within 10 km of Harwell, which discharges rather more radioactive material than Aldermaston or Burghfield, was also puzzling, though there is subsequent evidence of an excess in South Oxfordshire between 1981 and 1995 (Busby and Scott Cato, 1997). If radioactive discharges into the environment are to blame, then, it would seem that they must involve an unusual radioactive substance with a specific tendency to cause childhood leukaemia; or there must be selective deposition in an organ in which childhood leukaemia is unusually easy to induce, presumably the bone marrow where most childhood leukaemias arise (Darby and Doll, 1987), though bone marrow develops relatively late in the human foetus, and in the early stages of foetal development blood cells are manufactured in a variety of sites, such as the liver and the thymus. No such pathways or substances have yet been identified.

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Paternal preconceptional radiation exposure As a further result of the Black Report recommendations, a study was launched, under the leadership of Professor Martin Gardner of the Medical Research Council’s Epidemiology Unit at the University of Southampton, to examine a collection of possibilities, including that of a link between juvenile leukaemia and paternal radiation exposure sustained before the child was conceived. The ‘Gardner Report’ (Gardner et al., 1990) revealed a statistical association between patemal occupation and leukaemia. Men working in the nuclear reprocessing industry in Cumbria apparently ran raised risks of fathering leukaemic children, as did men working in the iron and steel, chemicals and farming industries. The fathers of four of the five leukaemia cases born and diagnosed in Seascale were known to have worked at Sellafield, and had received cumulative radiation doses of at least 97 mSv prior to conceiving. Gardner also suggested that there is statistical connection between increased leukaemia incidence in children and paternal doses greater than 10 mSv in the six months prior to conception. The risk of fathering a leukaemic child in either of these exposure groups was said to be raised by a factor of about 6 to 8, i.e. about 1 in 300 children born to this group of men had developed leukaemia Gardner’s study showed no link between environmental radiation and leukaemia. For example, the report considered people who play on the beach, eat fish or shellfish, grow vegetables, etc., and could find no associated risk. The result caused considerable surprise within the academic community, as such a correlation had not been observed in the offspring of male survivors of the Hiroshima and Nagasaki bombs, who received average radiation doses of 492 mSv (Ishimaru et al., 1982); nor among men who received significant abdominal doses of X-rays (Kneale and Stewart, 1980). The absence of a significant excess of testicular cancer among Sellafield workers suggested no accumulation of radioactive material in the testes (Smith and Douglas, 1986). It was known that plutonium tends to deposit in the skeleton and liver: it was estimated that only about 0.035% of ingested plutonium concentrated in the testes (Newton et al., 1990). Further, it was known that acute doses of more than 4000 mSv were necessary to double the chromosomal mutation frequency in human sperm (Evans, 1990). The lack of any evidence of other inherited diseases in vicinity of Sellafield presented a further problem for interpreting Gardner’s findings (Jones and Wheater, 1989). It had also been suggested that any chromosomal alteration which would cause leukaemogenesis would be inconsistent with the viability of the early embryo, and so could not be inherited (Evans, 1990). As a result of the Gardner correlation, a series of other studies were concluded or set in train. It became clear (Health and Safety Executive, 1993; Parker et al., 1993; Wakeford et al., 1994) that the association between childhood cancer and paternal preconceptional radiation dose was confined to children born in Seascale, even though more than 90% of the offspring of exposed fathers were born in West Cumbria outside the village of Seascale. Kinlen (1993) demonstrated that paternal preconceptional exposure alone was insufficient to explain the Seascale excess of leukaemias and non-Hodgkin’s lymphoma between 1951 and 1991. The possibility that a multi-causal process might be in action, involving both paternal irradiation and something unique to Seascale, was suggested, but looked unlikely, for example because the second factor would have to be implausibly strong (Little et al., 1993) yet fail to operate e.g., in Egremont North just 11 km away from Seascale. The required initiating mutation rate would also be so high as to be incredible (Wakeford et al., 1994b). Doll et al. (1994) noted that the original association had not been confirmed in studies using independent data, and concluded that the Gardner correlation was ‘largely or wholly a chance finding’, a position also reached by UNSCEAR (1994), which said that preconceptional irradiation had ‘largely been discounted’, and by Neel (1994).

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A report concerning the Caithness leukaemias failed to find any link between preconceptual paternal irradiation exposure and childhood leukaemia (Urquhart et al., 1991); Kinlen et al. (1993a) subsequently studied every case of childhood leukaemia in Scotland between 1958-1990, showing that in no case had the father received a measurable preconceptional dose of radiation, Other populations have also now been studied. The phenomenon is absent from the survivors of the Hiroshima and Nagasaki atom bombs (Neel and Schull, 1991). However, this population received mainly external doses of radiation, so the question remained as to whether internal dose of radiation was more important. Study of Danish Thorotrast patients (Andersson et al., 1994; Little et al., 1996) makes this highly unlikely. Danish neurosurgical patients were given injections of the thorium-bearing diagnostic contrast medium Thorotrast for cerebral arteriography. (Thorium is known to metabolise like plutonium in most respects - ICRP 1990, 1995.) This resulted in very high gonadal doses (up to 1.5 Sv), almost entirely from alpha-particle irradiation from internally incorporated radionuclides. No excess mortality or cancer rates were observed among the offspring of these patients. McLaughlin et al. (1993), in a case-control study of childhood leukaemia cases born to mothers residing near nuclear installations in Ontario between 1950 and 1988, showed no evidence of an increased risk due to paternal preconceptional radiation. Similarly a historical cohort study of children born to fathers working in the West German nuclear industry (Michaelis et al., 1994) found no evidence for an increased risk of either childhood cancer or leukaemia due to paternal preconceptional irradiation. Evidence supporting Gardner is weak. Roman et al. (1993), in a case-control study, found an association with fathers being monitored for external radiation dose in the preconceptional period at the Aldermaston and Burghfield nuclear weapons establishments in Berkshire and subsequent childhood leukaemia, but the recorded doses were low (less than 5 mSv) and there was no significant dose-response relationship. There seemed to be a mild excess of leukaemia among the children of male radiographers who received pre-conceptual doses ofradiation (McKinney et al., 1991), but as no dose records were available for these men the observation was difficult to interpret; this study also found apparent associations with preconceptional exposure to benzene and to wood dust and raised incidence of childhood leukaemia. Shu et al. (1988, 1994b) carried out retrospective case-control studies on parents of leukaemic children in Shanghai and North America, each seeming to demonstrate a link with paternal preconceptional irradiation (diagnostic X-rays). However, these studies depended on interview with parents, and indeed it tended to be mothers who were asked about doses received by the fathers. Recall bias is a notorious problem in such circumstances (Kneale and Stewart, 1980), and the results of these studies are therefore regarded as less reliable than those of studies involving more objective measures of dose. A follow-up to the first Shu study (Shu 1994a) failed to confirm the initial finding. Animal studies have been inconclusive, but offer little evidence to support the Gardner hypothesis. For example, Cosgrove (1993) found that no excess leukaemias or cancers developed in the offspring of mice after preconceptional irradiation. In October 1993, after a case which lasted for a year, Judgment was given in the High Court, London, on two test cases of childhood cancer mortalities brought against British Nuclear Fuels plc, the owners and operators of Sellafield. The Gardner hypothesis formed the most important part of the Plaintiffs’ case, and the Judge concluded that ‘on the evidence before me, the scales tilt decisively in favour of the Defendants’. A full account of the Sellafield court cases has been published in Advances in Nuclear Science and Technology Volume 25 (Tawn and Wakeford, 1997).

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Draper et al. (1997) brought together the findings of several studies in the field, using data from the National Registry for Radiation Workers, the National Registry of Childhood Tumours (Stiller et al., 1995), the Oxford Survey of Childhood Cancers and the Scottish casecontrol study (Kinlen et al., 1993). A total of 35 949 British children diagnosed as having cancer, together with matched controls, was involved. The analysis confirmed that fathers of children with leukaemia or non-Hodgkin’s lymphoma (excluding the Gardner cases) were significantly more likely to have been radiation workers than were fathers of control (relative risk 1.77, 95% confidence interval 1.03 to 3.03). However, there was no dose-response relationship - indeed, the highest excess was found among those receiving doses below the level of detection. No increased risk was found for fathers with a lifetime preconceptional dose of 100 mSv or more, or with a dose in the six months prior to conception of 10 mSv or more, nor was there any increased risk for the group of other childhood cancers. Ironically, this study did suggest a possible raised risk of leukaemia among offspring of mothers who had received preconceptional doses of radiation, something not observed in other studies. Other site factors The possibility remains that leukaemia may be caused by some chemical released by the plant, or the chemical behaviour of a rare radioactive substance, and some research has been carried out into possibilities (Little, 1990; Little 1991). For example, Lowengart et al. (1987) found a marginally significant correlation between paternal preconceptional exposure to chlorinated solvents in the year prior to conception and childhood leukaemia, but this study, as most others of its type, relied on exposure reported at interview or through questionnaires rather than objective records and is therefore subject to recall bias. At present there is no reliable reason for believing that such a factor is at work. It has also been suggested that stress because of the very presence of the plant might be causing suppression of the immune system. However, there is little direct evidence of this, and anyway it seems an unlikely cause in the UK. There is no reason to believe that local people feel such stress because of the establishments, nor is there any suggestion of raised leukaemia levels near other ‘stressful’ plants outside the nuclear industry (though few studies have been done on such establishments). Viral or other infective causes At least one virus (Human T-cell Lymphotropic Virus type 1, and possibly type 2) is known to cause rare forms of leukaemia in man (Hoover et al., 1982), though no viral association with acute lymphoblastic leukaemia has been demonstrated. It is also known that a virus can cause leukaemia in cats (Hoover et al., 1982) and cattle. However, the observation that leukaemia in man does not seem to cluster more than would be expected by chance militates against the theory that leukaemia is a common response to a rare viral infection. There remains the possibility that leukaemia is a rare response to a common virus, against which people living in stable communities will normally develop immunity (Smith, 1982). The link between the hepatitis B virus and hepatocellular carcinoma is an example of such a disease (Whittle et al., 1983), but such links are difficult to establish when the supposed agent is unidentified and its commonest manifestation (if any) is unrecognised. If this is the case, excesses of leukaemia might be expected when people from isolated communities come together into areas with high population densities and hence experience greater levels of population mixing, as this would presumably facilitate the transmission of this virus. In a series of papers, Professor Leo Kinlen of the Cancer Research Campaign and coworkers have identified many such populations and investigated the patterns of juvenile leukaemia within them.

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In his early work, Kinlen examined the New Towns of the 1940s, such as Glenrothes, Corby, Peterlee, Aycliffe and Cwmbran (Kinlen, 1988; Kinlen et al., 1990), and discovered significant excesses of childhood leukaemia. These excesses persisted for about the first ten years after the populations of the towns began to undergo significant expansion. The very peculiar nature of the population mixing at Seascale and Thurso when the plants at Sellafield and Dounreay were established would lead one to expect a similar effect to be observed there. Subsequent work by Kinlen has addressed some of the early apparent anomalies which the hypothesis faced. For example, Aldermaston and Burghfield were established in highlypopulated Berkshire, but Kinlen et al. (1991) proposed that the key factor was changing patterns of population mixing rather than the level of social isolation of the site per se. The Thurso cases of 1980-1984 came rather too late in the lifetime of the Dounreay site (established 1950), but may be associated with the influx ofoil workers into the area in the late 1970s (Kinlen et al., 1993b). The continuing nature of the excess in Seascale between 1984 and 1992 (COMARE 1996) is similarly at odds with the data from the New Towns, but Seascale’s peculiarly high rate of population turnover may result in a continuing supply of individuals who may be susceptible to the proposed virus (Kinlen et al., 1997 - see consideration of the fourth COMARE report below). Studies have also been carried out on other populations which have experienced considerable changes in patterns ofpopulation mixing. These include, in the UK, the wartime evacuation of urban children to rural districts (Kinlen and Johns, 1994); rural concentrations of national servicemen during the period of compulsory national military service from 1950 to 1963 (Kinlen and Hudson, 1991); populations experiencing large increases in commuting (Kinlen and Stiller, 1993); the North Sea oil industry in northern Scotland (Kinlen et al., 1993b); and rural areas affected by other, non-nuclear, construction projects (Kinlen et al., 1993c). In each case a statistically significant excess was found. Similar associations have been identified in Greece (Petridou et al., 1995), Italy (Kinlen and Petridou, 1995) and Hong Kong (Alexander et al., 1997). Most recently, Kinlen (1997) has detected an increased risk of leukaemia among children of fathers whose jobs involve unusual levels of contact with the public, including teachers, construction and transport workers. The strength of this work, in epidemiological terms, is that the epidemiology has been used to test a prior hypothesis (that childhood leukaemia is associated with a relatively common infective agent and is likely to be promoted by increases or other changes in population mixing patterns) which had been derived from other observations (the leukaemia excess in Seascale). Indeed, most ofthe excesses which have been identified in pursuit of this hypothesis had not previously been noted, being hidden in the statistics of larger areas. This is clearly more satisfactory than the ‘Texas sharpshooter’ approach which has been applied by some commentators with respect to the Seascale cluster. The fourth COMARE report In 1996 COMARE published its follow-up report on the Sellafield excess, summarising studies which had reported since the first COMARE report in 1986. The report confirmed that radioactive discharges from Sellafield could not account for the excess (Simmonds et al., 1995), and also dismissed the Gardner hypothesis. The role of environmental chemicals was considered and discounted. The Committee accepted that Kinlen’s hypothesis of population mixing (Kinlen, 1995) could play a role in the excess of childhood leukaemia in Seascale, saying that ‘many scientists working in the field now believe that an infective aetiology is likely for at least some leukaemias and that population mixing plays some role’, but concluded that the phenomenon was too marked for the viral hypothesis alone to offer a full explanation. The possibility of

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synergy between two or more factors, perhaps including radiation, was considered, but no conclusions were drawn in view of the implausibility that such an interaction could be unique to Seascale (3 km from Sellafield) while not operating e.g., in Egremont North just 8 km from the plant. Kinlen et al. (1997) pointed out that the COMARE report, though claiming to consider the ‘incidence of cancer and leukaemia in young people in the vicinity of the Sellafield site’, in fact effectively restricted its interest to Seascale, hence falling into one of the pitfalls outlined by Olsen et al. (1996) and other commentators (e.g. Elliott et al., 1992; (US) National Conference on Clustering of Health Events, 1990) on the interpretation of apparent localised excesses of disease. In fact, if a 10 km circle of the Sellafield plant is taken, even after excluding Seascale, the excess m the 1984-1993 period is similar to that near large non-nuclear construction sites in rural areas such as the Drax coal-fired station in south Yorkshire (Kinlen et al., 1993c), while the excess in the 1950-1983 period is almost identical. Even if population mixing is a convincing explanation of the excesses near the site excluding Seascale, the excess within Seascale itselfremains rather higher than those normally associated with population mixing studies. As remarked above (e.g. with respect to Olsen et al., 1996), it is difficult to determine whether this is a ‘real’ phenomenon, or one which has arisen through chance but become important initially through casual observation. However, Kinlen et al. (1997) argue that in fact Seascale is atypical, and that its unique nature could lead to a magnification of population-mixing effects when compared to populations such as the New Towns, evacuees, national servicemen and others. For example, Seascale is ‘a cul-de-sac in an already isolated area’, and among rural parishes of comparable size it has had the highest proportion of households in Social Class I in England and Wales, four times greater than in any other rural ‘growth’ parish (COMARE, 1996). It is established that rates of childhood lymphoblastic leukaemia are higher in higher socioeconomic classes (e.g. McWhirter, 1982; Alexander et al., 1990). Furthermore, Seascale has seen an unusually high turnover of families, presumably ensuring a steady ‘replenishment’ of susceptible individuals. The ratio of non-Cumbrian to Cumbrian fathers in Seascale is almost twice as great as that in the rest of West Cumbria (HSE, 1993). COMARE (1996) does note that ‘Seascale is extreme in terms of the length of time in which population mixing has occurred’. Over the 40 years of operation of the site Sellafield has had the largest number of construction workers of any site in the UK. Kinlen et al. (1997) argue that it is therefore not surprising that sites with fewer construction workers should exhibit lower rates of leukaemia, it being postulated that construction workers are one important source of exposure to the proposed virus (Kinlen et al., 1993c). Determining the extent to which these factors might promote a (postulated) infective aetiology for leukaemia in Seascale, and hence whether population mixing alone could account for the Seascale excess as it appears to be able to account for excesses elsewhere near the plant, will undoubtedly provide a focus for research in the near future. CONCLUSIONS Behind the human tragedy represented by every case of childhood leukaemia lie several important lessons for the investigation of the causes of rare diseases. It is vital not to jump to the most obvious conclusion about the aetiology of such diseases, especially on the basis of epidemiological evidence, as an apparent cause ( e.g., emissions of radioactive substances from a plant) may be a surrogate for another factor (nuclear establishments tend to be rather isolated and hence to involve unusual patterns of population mixing). Failure to recognise this can lead to unnecessary waste of resources, and also to considerable anguish on the part of the families

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who are suffering if they are led to believe that a cause has been found but that science is unable to bring the perceived perpetrators to justice. Whether the media, and certain non-governmental organisations, have learned these lessons, or even wish to do so, is highly doubtful. The debate has been dominated by a determination on the part of these agents to focus attention on a single possible explanation radiation from the installations - despite its growing scientific implausibility, even though more promising lines of enquiry, especially the infective hypothesis, have emerged. The failure, so farm to demonstrate unequivocally the cause of the leukaemia excesses near nuclear installations has not been the result of a shortage of effort or resources. Several millions of pounds have been spent investigating the phenomenon. The emergence of the Kinlen hypothesis, which appears to be a genuine advance in our understanding of the disease, is a considerable reward for this effort. However, in the course of those 15 years there have been many thousands of cases of childhood leukaemia in the UK, the vast majority nowhere near nuclear installations and hence of less interest to those for whom demonstrating the link with radiation has been an important part of a wider political campaign. It is not possible to predict what progress could have been made in the fight against all leukaemias had the resources committed to the few tens of cases near nuclear installations instead been dedicated to detecting leukaemia anywhere and identifying possible causes, but it is at least likely that the overall outcome would have been more beneficial. It is to be hoped that the message of the last fifteen years will be heeded next time attempts are made to demonstrate a simplistic environmental cause for a complex medical phenomenon. REFERENCES Alexander F.E., Ricketts T.J., McKinney P.A. and Cartwright R.A., 1990, ‘Community lifestyle characteristics and risk of acute lymphoblastic leukaemia in children’, Lancet 336 pp 1461-1465. Alexander F.E., Chan L.C., Lam T.H., Yuen P., Leung N.K, Ha S.Y., Yuen H.L., Li C.K, Lau Y.L. and Greaves M.F., 1997, ‘Clustering of childhood leukaemia in Hong Kong: association with the childhood peak and common acute lymphoblastic leukaemia and with population mixing’, B. J. Cancer, 75 pp 457-463. Andersson M., Juel K., Ishikawa Y. and Storm H.H., 1994, ‘Effects of preconceptional radiation on mortality and cancer incidence in the offspring of patients given injections of Thorotrast’, J. Nat. Cancer Inst. 86 pp 1866-1870. Barton C., Ryder H. and Sargent M., 1997, ‘Case-control studies have been done in Britain’, BMJ 314 p 1554. Bithell J.F., Dutton S.J., Draper G.J. and Neary N.M., 1994, ‘Distribution of childhood leukaemias and nonHodgkin’s lymphomas near nuclear installations in England and Wales’, BMJ 309 pp 501-505. Black D., 1984 Investigation of the Possible Increased Incidence of Cancer in West Cumbria: Report of the Independent Advisory Group, HMSO: London. Bobrow M. et al. (First COMARE Report), 1986, The Implications of the New Data on the Releases from Sellafield in the 1950s for the Conclusions of the Report on the Investigation of the Possible Increased Incidence of Cancer in West Cumbria, HMSO: London. Bobrow M. et al. (Second COMARE Report), 1988, Investigation of the possible increased incidence of leukaemia in young people near Dounreay Nuclear Establishment, Caithness, Scotland, HMSO: London. Bobrow M. et al. (Third COMARE Report), 1989, Report on the Incidence of Childhood Cancer in the West Berkshire and North Hampshire area, in which are situated the Atomic Weapons Research Establishment, Aldermaston, and the Royal Ordnance Factory, Burghfield, HMSO: London. Busby C. and Scott Cato, M., 1997, ‘Death rates from leukaemia are higher around nuclear sites in Berkshire and Oxfordshire’, BMJ 315 p 309. Clavel J. and Hémon, D., 1997, ‘Bias could have been introduced into study’, BMJ 314 p 1553. COMARE/RWMAC, 1995, Potential Health Effects and Possible Sources of Radioactive Particles Found in the Vicinity of the Dounreay Nuclear Establishment, HMSO: London. COMARE, 1996, The Incidence of Cancer and Leukaemia in Young People in the Vicinity of the Sellafield Site, West Cumbria: Further Studies and an Update of the Situation since the Publication of the BIack Advisory Group in 1984 (Fourth COMARE Report), HMSO: London.

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Cook-Mozaffari P.J., Ashwood F.L., Vincent T., Forman D. and Alderson M., 1987, Cancer incidence and mortality in the vicinity of nuclear installations, England and Wales 1959-80, OPCS: London. Cook-Mozaffari P.J., Darby S.C., Doll R, Forman D., Herman C. and Pike M.C., 1989a ‘Geographical Variation in mortality from leukaemia and other cancers in England and Wales in relation to proximity to nuclear installations, 1969-78’, B. J. Cancer 59 pp 476-485. Cook-Mozaffari P.J., Darby S.C. and Doll R., 1989b ‘Cancer near potential sites of nuclear installations’, Lancet ii pp 1145-47. Cosgrove G.E., Selby P.B., Upton A.C., Mitchell T.J., Steele M.H. and Russell W.L., 1993, ‘Lifespan and autopsy findings in the first-generation offspring of X-irradiated male mice’, Mutat. Res. 319 pp 7179. Craft A.W., Parker L., Openshaw S., Charlton M., Newell J., Birch J.M. and Blair V., 1993, ‘Cancer in young people in the north of England 1968-1985: analysis by census wards’, J. Epidemiol. Commun. Health 47 pp 109-115 Darby S.C. and Doll R., 1987, ‘Fallout, radiation doses near Dounreay, and childhood leukaemia’, BMJ 294 pp 603-607. Doll R., 1989, ‘The epidemiology of childhood leukaemia’, J. Royal Statistical Society 152 pp 341-351. Doll R, 1990, ‘The effects of low-level radiation - current epidemiology’, Nuclear Energy 29 pp 13-18. Doll R, Evans H.J. and Darby S.C., 1994, ‘Paternal exposure not to blame’, Nature 367 pp 678-680. Draper G.J., Stiller C.A., Cartwright RA., Craft A.W. and Vincent T.J., 1993, ‘Cancer in Cumbria and in the vicinity of the Sellafield nuclear installation 1963-90’, BMJ 306 pp 89-94. Draper G.J., Little M.P., Sorahan, T., Kinlen L.J., Bunch LJ., Conquest, A.J., Kendall G.M., Kneale G.W., Lancashire R.J., Muirhead C.R., O’Connor C.M., Vincent T.J., Thomas J.M., Goodill A.A., Vokes J. and Haylock R.G.E., 1997, Cancer in the Offspring of Radiation Workers - a Record Linkage Study, NRPB-R298, HMSO: London. Dousset M., 1989, ‘Cancer mortality around La Hague nuclear facilities’, Health Physics 56 pp 875-884. Evans H.J., 1990, ‘Leukaemia and radiation’, Nature 345 pp 16-17. Elliott P., Cuzick J., English D. and Stem R, ed., 1992, Geographical and Environmental Epidemiology: Methods for Small Area Studies, Oxford University Press: Oxford. Fine P.E.M., Adelstein A.M., Snowman J., Clarkson J.A. and Evans S.M., 1985, ‘Long term effects of exposure to viral infections in utero’, BMJ 290 pp 509-511. Forman D., Cook-MozafFari P.J., Darby S.C., Davy G., Stratton I. and Doll R, 1987, ‘Cancer near nuclear installations’, Nature 329 pp 499-505. Frederick J. and Alberman E.A., 1972, ‘Reported influenza in pregnancy and subsequent cancer in the child’, BMJ II pp 485-488. Gardner M.J., Hall A.J., Downes S. and Terrell J.D., 1987, ‘Follow-up of children born to mothers resident in Seascale, West Cumbria’, BMJ 295 pp 822-827. Gardner M.J., Hall A.J., Snee M.P., Downes S., Powell C.A. and Terrell J.D., 1990, ‘Results of a case-control study of leukaemia and lymphoma among young people near Sellafield nuclear plant in West Cumbria’, BMJ 300 pp 423-429. Greaves M.J., 1988, ‘Speculations on the causes of childhood acute leukaemia’, Leukaemia 2 pp 120-126. Hatch M.C., Beyea J., Nieves J.W. and Susser M., 1990, ‘Cancer near the Three Mile Island nuclear plant: radiation emissions’, A. Epidemiology 132 pp 397-412. Health and Safety Executive, 1993, HSE Investigation of Leukaemia and Other Cancers in Children of Male Workers at Sellafield, HSE books: Sudbury. Heasman M.A., 1986, ‘Geographical distribution of leukaemia in young persons in Scotland 1968-83’, Lancet 266 p 385. Henshaw D.L., Eatough J.P. and Richardson RB., 1990, ‘Radon as a causative factor in induction of myeloid leukaemia and other cancers’, Lancet 335 pp 1008-1012. Hill C. and Laplanche A., 1990, ‘Overall mortality and cancer mortality around French nuclear sites’, Nature 347 pp 755-757. Hoover E.A., Olsen R.G., Hardy W.D. jr, Schaller J.P. and Mathes L.E., 1982, ‘Feline leukaemia virus infection: age-related variation in response of cats to experimental infection’, J. Nat. Cancer Inst. 69 pp 333-337. ICRP, 1990, ‘Age-dependent doses to members of the public from intake of radionuclides: Part 1’, Annals of the ICRP 20 (2), Pergamon: Oxford. ICRP, 1995, ‘Age-dependent doses to members of the public from intake of radionuclides: Part 3, Ingestion dose coefficients’, Annals of the ICRP 25 (1), Pergamon: Oxford. Ishimaru T., Ichimaru M. and Mikami M., 1982, Leukaemia incidence among individuals exposed in utero, children of atomic bomb survivors and their controls, Hiroshima and Nagasaki 1945-79, RERF TR 11-81.

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Jones K.P. and Wheater A.W., 1989, ‘Obstetric outcomes in West Cumberland Hospital - is there a risk from Sellafield?’, J. Royal Soc. Medicine 82 pp 524-527. Keller B., Haff G., Kaatsch P., and Michaelis J., 1992, Study of the Frequency of Cancer Cases in Childhood in the Vicinity of W. German Nuclear Plants, Institute for Medical Statistics and Documentation, University of Maine: Maine, Germany. Kellett C.E., 1937, ‘Acute myeloid leukaemia in one of identical twins’, Arch. Dis. Child 12 pp 239-252. Kinlen L.J., 1988, ‘Evidence for an infective cause of childhood leukaemia: comparison of a Scottish new town with nuclear reprocessing sites in Britain’, Lancet II pp 1323-1327. Kinlen L.J., Clarke K. and Hudson C., 1990, ‘Evidence from population mixing in British New Towns 1946-85 of an infective basis for childhood leukaemia’, Lancet 336 II pp 577-582. Kinlen L.J. and Hudson C.M., 1991, ‘Childhood leukaemia and poliomyelitis in relation to military encampments in England and Wales in the period ofnational military service, 1950-63’, BMJ 303 pp 1357-1362. Kinlen L.J., Hudson C.M. and Stiller C.A., 1991, ‘Contacts between adults as evidence for an infective origin of childhood leukaemia: an explanation for the excess near nuclear establishments in West Berkshire?’, B.J. Cancer 64 pp 549-554. Kinlen L.J., 1993, ‘Can paternal preconceptional radiation account for the increase of leukaemia and nonHodgkin’s lymphoma in Seascale?’, BMJ 306 pp 1718-1721. Kinlen L.J. and Stiller C., 1993, ‘Rural population mixing and excess of childhood leukaemia’, BMJ 306 p 930. Kinlen L.J., Clarke K. and Balkwill A., 1993a ‘Paternal preconceptional radiation exposure in the nuclear industry and leukaemia and non-Hodgkin’s lymphoma in young people in Scotland, BMJ 307 pp 1153-1158. Kinlen L.J., O’Brien F, Clarke K., Balkwill A. and Matthews F., 1993b, ‘Rural population mixing and childhood leukaemia: effects of the North Sea oil industry in Scotland, including the area near Dounreay nuclear site’, BMJ 306 pp 743-748. Kinlen L.J., Dickson M. and Stiller C.A., 1993c, ‘Childhood leukaemia and non-Hodgkin’s lymphoma near large rural construction sites, with a comparison with Sellafield nuclear site’, BMJ 310 pp 763-768. Kinlen L.J. and John, S.M., 1994, ‘Wartime evacuation and mortality from childhood leukaemia in England and Wales in 1945-9’, BMJ 309 pp 1197-2102. Kinlen L.J., 1995, ‘Epidemiological evidence for an infective basis in childhood leukaemia’, B. J. Cancer 71 pp 1-5. Kinlen L.J. and Petridou E., 1995, ‘Childhood leukaemia and rural population movements: Greece, Italy, and other countries’, Cancer Causes and Control 6 pp 445-450. Kinlen L.J., Craft A.W. and Parker L., 1997, ‘The excess of childhood leukaemia near Sellafield a commentary on the fourth COMARE report’, J. Radiol. Prot. 17 pp 63-71. Kinlen L.J., 1997, ‘Contact paternal occupations, infection and childhood leukaemia: five studies of unusual population mixing in adults’, B.J Cancer 76 pp 1539-1545. Kneale G.W. and Stewart A.M., 1980, ‘Re-conception X-rays and childhood cancers’, B. J. Cancer 41 pp 222226. Lackland D., Grosche B., Mohr L., Dunbar J., Burkart W. and Hoel D., 1997, ‘Leukaemia in the vicinity of two tritiwn releasing nuclear facilities: a comparison of the Kruemmel site, Germany, and the Savannah River site, South Carolina, Proceedings of the Conference ‘Health effects of low dose radiation’ 1114 May 1997, BNES: London. Lambert B.E., 1990, How safe is safe? Unwin: London. Langford I., 1991, ‘Childhood leukaemia mortality and population change in England and Wales 1969-73’, Soc. Sci. Med. 33 pp 434-440. Leukaemia Research Fund, 1990, Leukaemia and lymphoma - an atlas of distributions within areas of England and Wales 1984-1988. Little M.P., 1990, ‘A comparison between the risks of childhood leukaemia from parental exposure to radiation in the Sellafield workforce and those displayed among the Japanese bomb survivors’, J. Radiol. Prot. 10 pp 185-198. Little M.P., 1991, ‘A comparison of the risks of childhood leukaemia from parental pre-conception exposure to radiation in the Sellafield and Dounreay workforces and the Japanese bomb survivors’, J. Radiol. Prot. 11 pp 231-240. Little M.P., Wakeford R and Charles M.W., 1994, ‘A comparison of the risks of leukaemia in the offspring of the Sellafield workforce born in Seascale and those born elsewhere in West Cumbria with the risks in the offspring of the Ontario and Scottish workforces and the Japanese bomb survivors’, J. Radiol. Prot. 14 pp 187-201. Little M.P., Wakeford R., Charles M.W. and Andersson M., 1996, ‘A comparison of the risks of leukaemia and non-Hodgkin’s lymphoma in the first generation offspring (F1) of the Danish Thorotrast patients with those observed in other studies of parental preconception irradiation’, J. Radiol. Prot., 16 pp 25-36.

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Lowengart RA., Peters J.M., Cicioni C., Butley J., Bernstein L., Preston-Martin S. and Rappaport E., 1987, ‘Childhood leukaemia and parents’ occupational and home exposures’, J Nat. Cancer Inst. 79 pp 3946. Massachusetts Department of Public Health, 1990, Investigation of Leukaemia Incidence in 22 Massachusetts Communities, 1978-1986 Massachusetts. McKinney P.A., Alexander F.E., Cartwright RA. and Parker L., 1991, ‘Paternal occupations of children with leukaemia in west Cumbria, northHumberside, and Gateshead‘, BMJ 302 pp 681-686. McLaughlin J.R., King W.D., Anderson T.W., Clarke E.A. and Ashmore J.P., 1993, ‘Paternal radiation exposure and leukaemia in offspring: the Ontario casecontrol study‘, BMJ 307 pp 959-966. McWhirter W.R., 1982, ‘The relationship of incidence of childhood lymphoblastic leukaemia to social class’, B. J. Cancer 46 pp 640-645. Michaelis J., Kaatsch P. and Zöllner I., 1994, ‘Querschnittsuntersuchumg zur Häufigkeit von Krebserkrankungen bei Kindern von beruflich strahlenexponierten Beschäftigten in west-deutschen kerntechnischen Anlagen’, Arbeitsmed Sozialmed Unweltmed 29 pp 324-355. Milham S., 1988, private communication to the Department of Social and Health Services, Olympia, Washington. Miller R.W., 1963, ‘Down’s syndrome (mongolism), other congenital malformations and cancers among sibs of leukaemic children’, New England Journal of Medicine 268 pp 393-401. Mole R.H., 1990, ‘Childhood cancer after prenatal exposure to diagnostic X-ray examination in Britain’, B. J. Cancer 62 pp 152-168. National Cancer Institute (US), 1990, Cancer Mortality around Nuclear Power Plants: Bethesda, Maryland. National Conference on Clustering of Health Events, 1990, Am. J. Epidemiol. 132 suppl. Neel J.V. and Schull W.J. (ed.), 1991, The Children of Atomic Bomb Survivors: a Genetic Study, National Academic Press: Washington DC. Neel J.V., 1994, ‘Problem of “false positive” conclusions in genetic epidemiology lessons from the leukaemia cluster near the Sellafield nuclear installation’, Genet. Epidemiol. 11 pp 213-233. Newton D., Warner A. and Talbot R, 1990, ‘Leukaemia risk and plutonium’, Nature 347 p 521. Olsen S.F., Martuzzi M. and Elliott P., 1996, ‘Cluster analysis and disease mapping - why, when and how? A step by step guide’, BMJ 313 pp 863-866. O’Riordan M.C., NRPB, 1990, personal communication. Parker L., Craft A.W., Smith J., Dickinson H., Wakeford R., Binks K., McElvenny D.M., Scott L.E. and Slovak A,, 1993, ‘Geographical distribution of preconceptional radiation doses to fathers employed at the Sellafield nuclear installation, West Cumbria’, BMJ 307 pp 966-971. Patrick C.H., 1977, ‘Trends in public health in the population near nuclear facilities: a Critical assessment’, Nuclear Safety 18 pp 647-662. Petridon E., Revinthi K., Alexander F., Haidas S., Koliouskas D., Kosmidis H., Piperopoulou F., Tzortzatou F. and Trichopoulos D., 1996, ‘Space-time clustering of childhood leukaemia in Greece: Evidence supporting a viral aetiology’, B. J. Cancer, 73, pp 1278-1283. Pobel D., and Viel J-F., 1997, ‘Case-confro1 study of leukaemia among young people near La Hague nuclear reprocessing plant the environmental hypothesis revisited‘, BMJ 314 pp 101-106. Preston D.L. and Pierce D.A., 1987, The Effect of Changes in Dosimetry on Cancer Mortality Risk Estimates in the Atomic Bomb Survivors, RERF-TR9-87. Roman E., Watson A., Baal V., Buckle S., Bull D., Baker K. and Ryder H., 1993, ‘Case-control study of leukaemia and non-Hodgkin’s lymphoma among children aged 0-4 years old living in West Berkshire and North Hampshire health districts’, BMJ 306 pp 615-621. Shu X-O., Gao Y.T., Brinton LA., Linet M.S., Tu J.T., Zheng, W. and Fraumeni J.F., 1988, ‘A populationbased case-control study of childhood leukaemia in Shaughai’, Cancer 62 pp 635-644. Shu X-O., Jin F., Linet M.S., Zheng W., Clemens J., Mills J. and Gao Y.T., 1994a, ‘Diagnostic X-ray and ultrasound exposure and risk of childhood cancer’, B. J. Cancer 70 pp 53 1-536. Shu X-O., Reaman G.H., Lampkin B., Stather H.N., Pendergrass T.W. and Robison L.L., 1994b, ‘Association of paternal diagnostic X-ray exposure with risk of infant leukaemia’, Cancer Epidemiol. Biomarkers Prev. 3 pp 645653. Simmonds J.R., Robinson, C.A., Phipps A.W., Muirhead, C.R. and Fry, F.A., Risks of Leukaemia and Other Cancers in Seascale from all Sources of Ionising Radiation Exposure, Report NRPB-R276, HMSO: London. Smith P.G., 1982, ‘Spatial and temporal clustering’ in ed. Schottenfield D. and Fraumeni J.R., Cancer Epidemiology and Prevention, Saunders: Philadelphia Smith P.G. and Douglas A., 1986, ‘Mortality of workers at the Sellafield plant of British Nuclear Fuels’, BMJ 293 pp 845-854. Stather J.W., Clarke R.H. and Duncan K.P., 1988a, The risk of childhood leukaemia near nuclear establishments, NRPB-R215, HMSO: London.

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Stather J.W., Dionian J., Brown J., Fell T.P. and Muirhead C.R., 1988b, ‘The risk of leukaemia in Seascale from radiation exposure’, Health Physics 55 pp 471-481. Stiller C.A. and Boyle, P.J., 1996, ‘Effects of population mixing and socioeconomic status in England and Wales, 1979-85, on lymphoblastic leukaemia in children’, BMJ 313 pp 1297-1300. Stjernfeldt M., Berglund K., Lindsten and Ludvigsson J., 1986, ‘Maternal smoking during pregnancy and risk of childhood cancer’, Lancet 266 pp 1350-1352. Stiller C.A., Allen, M.B. and Eatock E.M., 1995, ‘Childhood cancer in Britain: the National Registry of Childhood Tumours and incidence rates 1978-1987’, Eur. J. Cancer 31A pp 2028-2034. Tawn E.J. and Wakeford R., ‘Childhood Leukaemia and Radiation: The Sellafield Judgment’, in ed. Lewins J. and Becker M., Advances in Nuclear Science and Technology Volume 25, Plenum: New York UNSCEAR (UN Scientific Committee on the Effects of Atomic Radiation), 1994, Sources and Effects of Ionising Radiation, UN: New York. Urquhart J.D., Black R.J., Muirhead M.J., Sharp L., Maxwell L., Eden O.B. and Adams Jones D., 1991, ‘Casecontrol study of leukaemia and non-Hodgkin’s lymphoma in children in Caithness near the Dounreay nuclear installation’, BMJ 302 pp 687-692. Viel J-F. and Richardson S.T., 1990, ‘Childhood leukaemia around the La Hague nuclear waste reprocessing plant’, BMJ 300 pp 580-581. Viel J-F., Pobel D. and Carre A., 1995, ‘Incidence of leukaemia in young people around the La Hague nuclear waste reprocessing plant: a sensitivity analysis, Stat. Med. 14 pp 2459-2472. Wakeford R., Tawn E.J., McElvenny D.M., Scott L.E., Binks K., Parker L., Dickinson H. and Smith J., 1994a, ‘The descriptive statistics and health implications of occupational radiation doses received by men at the Sellafield nuclear installation before the conception of their children’, J. Radiol. Prot. 14 pp 3-16. Wakeford R., Tawn E.J., McElvenny D.M., Binks K., Scott L.E. and Parker L., 1994b, ‘The Seascale childhood leukaemia cases - the mutation rates implied by paternal preconceptional radiation doses’, J. Radiol. Prot. 14 pp 17-24. Wakeford, R., 1995, ‘The risk of childhood cancer from intrauterine and preconceptional exposure to ionizing radiation’, Environmental Health Perspectives 103 1018-1025. Watson W.S. and Sumner D.J., ‘The measurement of radioactivity in people living near the Dounreay Nuclear Establishment in Caithness, Scotland’, Int. J. Radiat. Biol. 70 117-130. Whittle H.C., Bradley A.K., McLoughlan K., Ajdukiewicz A.B., Howard C.R., Zuckerman, A.J. and McGregor I., 1983, ‘Hepatitis B Virus infection in two Gambian villages’, Lancet I pp 1203-1206. Wolff, S.P., 1991, ‘Leukaemia and unemployment’, Nature 350 p 268. Wood E.E., 1960, ‘A survey of leukaemia in Cornwall, 1948-1959’, BMJ I pp 1760-1764. World Health Organisation, 1995, Health consequences of the Chernobyl accident; Results of the IPHECA pilot projects and related national programme - summary report, WHO: Geneva. Wrixon A.D., Green B.M.R., Lomas P., Miles G.C., Francis E.A., Driscoll C.O., James A.C. and O’Riordan, M.C., 1988, Natural Radiation Exposure in UK Dwellings, NRPB-R190, HMSO: London.

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NONLINEAR STOCHASTIC DYNAMICS AND INSTABILITY THEORY

H. Konno Institute of Information Sciences and Electronics University of Tsukuba Tsukuba, Ibaraki 305, Japan 1.

INTRODUCTION

The Boltzmann equation for low-density gases has supplied various useful concepts for statistical mechanics of transport and relaxation phenomenon; Bogoliubov (1946) and Kirkwood (1946, 1947). Ehrenfest (1911) pointed out the importance of the coarse-graining in µ space. Twenty years ago, Tokuyama and Mori (1976) proposed a statistical mechanical theory and derived the Boltzmann equation without the use of any assumption like Bogoliubov’s functional postulate and Kirkwood’s time smoothing and the product assumption for the spatially coarse-grained particle density in µ space (1) where c = N/Ω , Ω being the volume of the system, pi(t) and ri(t) are the momentum and the position of the i-th particle at time t, respectively. ∆(r) is the coarse-grained δ function (2) where qc = l/b and Σq' is the sum over wave vectors q whose magnitudes are smaller than the cutoff qc. The projection-operator method in the Heisenberg picture leads to the following nonlinear evolution: (3) where Gpr(t) is a fluctuating force. Jpr(A(t); t) denotes a generalization of the collision term which differs from that of the memory-function type evolution equation (Mori, 1965). Since the neutron density is dilute enough even in the case of power reactors, the collision term Jpr in the Boltzmann equation (3) can be neglected. This fact is the basis of the linear

Advances in Nuclear Science and Technology, Volume 26, edited by Lewins and Becker. Kluwer Academic / Plenum Publishers, New York 1999.

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theory of zero-power reactor noise which has been published by Uhrig (1970), Thie (1963), Williams (1974) and Saito (1974). The importance of nonlinearity in nuclear power reactors is deduced from the fact that the absorption Σ a(r,t) and the fission Σ f(r,t) cross section is proportional to the fuel temperature Tf (r, t), the coolant temperature Tc (r, t) and others: (4) and

(5) The spatio-temporal variation of the state variables such as temperatures, coolant velocity, voids and others become significant noise sources of neutron fluctuations. In power reactors, the strength of noise sources are greater than that of the noise source due to the neutron branching process. Therefore, the dynamics of these state variables gives rise to various nonlinear effects in nuclear power reactors. What is the significance of the introduction of nonlinearity in the model of power-reactor dynamics? First, it is important for the stability of the dynamics even in the steady state. Second, the nonlinearity is also contributing to reduce the neutronic variance of fluctuation. Third, when a limit cycle oscillation takes place, its saturated amplitude is determined by the strength of the nonlinearities. As far as the linear theory of transition rates of neutron fission and neutron capture are concerned in zero-power reactors, the neutron number N fluctuation in the steady state is subjected to Poissonian or super-Poissonian statistics, i.e., (6) On the other hand, the magnitude of practical observations of neutron fluctuations in power reactors at the steady state are estimated within a few percentage of the total power in PWRs; it takes larger values in BWRs than that of PWRs though. The neutron number fluctuation is anyway subjected to the highly sub-Poissonian statistics, i.e., (7) To explain noise suppression, the role of nonlinearity must be accounted for in addition to other averaging effects such as spatial diffusion and neutron counting processes. Also to explain the change of characteristics of neutron fluctuation especially at the critical point of instabilities (bifurcations), one must take into account nonlinearity-noise interplay from the viewpoint of nonlinear stochastic dynamics under the influence of multiplicative (parametric) noises. We see more reasons for incorporation of nonlinearity to give explanation of various features of space-dependent neutron fluctuations in nuclear power reactors: There exist complex bifurcations and associated spatio-temporal non-uniform noise sources due to nuclear, thermal and hydraulic fluctuations. Namely, there exist spatio-temporal high-dimensional chaotic movements, turbulence and complexities. This space-dependent nonlinearity-noise interplay in conjunction with multiplicative (parametric) noise is the key concept to understand the spatio-temporal patterns; the phenomenon of (a) noise rise and that of (b) noise suppression, in nuclear power reactors as complex systems. In spite of the complexity of the fluctuation phenomena in a nuclear power reactor, one conventionally adopts homogenized and/or averaged models for simplicity. Therefore, one might find in this review many challenging academic problems which must be solved, along the line of studying fluctuation phenomena in stochastic nonlinear dynamics in nuclear power reactors. 2. BASIS FOR NONLINEAR STOCHASTC DYNAMICS A. The Slaving Principle (Stochastic center manifold)

NONLINEAR STOCHASTIC DYNAMICS

23

The theory of stochastic processes is based on the idea that complex phenomena can be analyzed and/or interpreted by a relatively small number of (spatial and/or temporal) modes. This attitude saves time and money if compared to the amount that is required in taking a rigorous attitude to solve the problem by the first principle calculations, e.g. simulations with the use of finite difference scheme (FDS) and/or the finite element method (FEM) for the partial differential equations (PDEs). To reduce the complex governing equations to simple equation(s), there are systematic way of thinking: one of them is the idea of the slaving principle (Haken, 1980). The principle states that one can take only the slowly varying state variables since the fast variables slave to the slow ones (modes). The fast variables can be sometimes regarded as the random forces with Gaussian-white nature. The concept of the center manifold theorem in nonlinear dynamics, states that there exists a central region wherein the motion of a relevant system is bounded. The concept of a center manifold does not hold in stochastic systems in the rigorous sense. The slaving principle (or the stochastic center manifold) is a more general concept than the center manifold (cf. Haken, 1983). Center manifold equations are derived by (I) the projection operator method (Mori, 1965; Grabert, 1982), (II) the direct elimination of state variables (Haken, 1980), (III) the reductive perturbation method (Taniuchi, 1974), (IV) the Galerkin method, and so forth. The projection operator method (I) teaches us (a) how a kinematical equation is reduced to a stochastic equation; (b) what is the relation of random force and memory function. The method takes the projection operator as PX = 〈 X A *〉 〈 A A *〉 -1 A (where * denotes the complex conjugate and 〈 ...〉 is the statistical average) leading the Liouville equation A (t) = iLA(t) to the memory-function type Langevin equation for the state vector A as (8) where iΩ = 〈 AA*〉 〈 AA *〉 -1 is the collective frequency and the random force f(t) is written in terms of (1 - P)iL as f(t) = exp{(1− P)iLt}(1 − P)iLA (0) . The memory function Φ(t) is expressed in terms of the random force (9) The causality 〈 f(t)A*(0)〉 = 0 ( t > 0 ), holds between the random force f(t) and the state variable A. This physically means that the random force f(t) is not affected by the initial state A(0) . One should note that there is another type of reduced equation, i.e., the convolution-less type Langevin equation derived by Tokuyama and Mori (1976) with the use of a projection operator identity. Further, introducing PX = Σ n 〈Xφ n({A})〉φ n ({A}), φ n ({A}) being a nonlinear function, one can derive nonlinear Langevin equations (Zwanzing, 1960; Kawasaki, 1973). The adiabatic elimination method (II) (Haken, 1980), the direct elimination (the formal integration) of fast varying state variables, leads to non-Markovian or a higher-order Markovian SDE. Examples can be seen in the later sections. The basic idea of this method and physical validity of it are also based on the slaving principle. The reductive perturbation method (III) (Taniuchi, 1974; Nayfeh and Mook, 1979) is a systematic method for simplifying a set of nonlinear coupled PDEs (or ODEs) into Schrodinger type or complex Ginzburg-Landau type PDE (or a ODE). B. Nonlinear Transformations Nonlinear transformations (NTRs) have been used to find exact solutions for ordinary and partial differential equations in physics and engineering problems. The most well known example of a topic in physics is the “soliton” solution (see for example, Ablowitz and Segur, 1981; Hirota, 1992). The general method for finding exact solutions via NTRs in engineering are described in the book by Ames (1965).

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H. KONNO

A theory of NTR for stochastic partial differential equations (PDEs) has not been developed, in contrast to the advancement of the theory of nonlinear transformation in nonlinear PDEs. A few examples are given in the article by Fox (1978). Exact solutions for stochastic KdV equation (Wadati, 1983) and stochastic Burgers equation (Konno, 1995a) under the influence of spatially uniform Gaussian-white noise are obtained with the aid of a NTR. Now let us take the Logistic-Verhulst equation in population growth which is equivalent to the simplest model for power reactor dynamics in nuclear reactors (10) where Λ is the effective neutron lifetime, ρ is the reactivity that is inserted to keep the reactor power constant and γ is the feedback coefficient. The method of NTR is useful (I) to find an exact solution; (II) to find approximate solutions with good accuracy; and (III) to find an excellent state space for applying numerical Langevin simulation with high accuracy. To exhibit the usefulness (I), let us consider the case ρ = ρ 0 = const; the steady state solutions are Ns ,0 = 0 and Ns = ρo/γ. If the state variable is scaled by the non-trivial steady state Ns, viz., n = N/Ns , the dynamical equation reduces to (11) where a = ρ 0 /Λ. This is a Riccati-type ordinary differential equation. Therefore, if one apply (a) a NTR n = 1/x, one obtains the linear equation (12) The solution of x is easily given by (13) When we take (b) a NTR n = exp(y) without the external perturbations, we have the equation with the exponential (Toda) potential ( V(y) = exp(y) – y – 1 ) (14) This is also integrable since separation of variables (y and t) is possible. Apparently, one get the same result as the above. Then if we take (c) a NTR zm = nm , we have (15) This is equivalent to (16)

(17)

The formal solution is given by Z(t) = exp(Lt)Z(0). One can contrive other types of NTRs depending on various specified purposes in practical applications. Then, to show the usefulness (II), let us incorporate a parametric noise ρ(t) = ρ0 +ρ1(t) , into eq.(10). When the normalized fluctuating force η(t) = ρ1(t)/ ρ0 , has Gaussian-white

NONLINEAR STOCHASTIC DYNAMICS

25

nature 〈 η(t)〉 = 0 and 〈 η(t)η(0)〉 = Dδ(t) , the corresponding Fokker-Planck (FP) equation becomes (18) The steady state solution for eq.(18) is given by (19) On the other hand, the NTR (a) n = 1/ x leads eq.(10) to the following linear SDE with parametric noise (20) The FP equation for the Langevin equation (20) takes the form (21) The exact analytical solution P(x, t) can also be found. When the neutron fluctuation becomes large, the effect of parametric noise becomes important. In this case, it is convenient to take the NTR (b) n = exp(y ) which leads to the form of a SDE with an additive noise as (22) The corresponding FP equation becomes (23) When the assumption of small fluctuation, viz., the expansion exp(y) ≈ 1 + y (the logarithmic linearization) is insufficient to describe the system, it is convenient to perform Langevin simulations in the state space y for the reason that an accurate numerical algorithm for simulating a Langevin equation can be applied. Excellent numerical algorithm have not been well developed for simulating nonlinear unstable systems under the influence of parametric colored noises (cf. Kolden and Platen, 1992). C. Bifurcation Theory The theory of nonlinear oscillations shows that the phase change of nonlinear systems is associated with bifurcations (see Chow and Hale, 1982; Gukenheimer and Holmes, 1983). The theory teaches us what kind of changes in dynamical behaviours can appear if a set of relevant non-linear equations is assumed and/or derived from theoretical as well as experimental points of view. Consider a kinematic equation (24) where f[U;R] is a nonlinear function and R is a parameter. Suppose that U0 is a fixed point of eq.(24): (25) (26) where L is the Jacobian derivative of f evaluated at the fixed point Uo : (27) If all the eigenvalues of the matrix L are negative, δU(t) decays at long times and U0 is said to be linearly stable.

26

H. KONNO In the spatially extended system, the deviation with the wave number q is expressed as (28)

If an initially localized structure around x 0 grows larger, U0(x) is called absolutely unstable. On the other hand, if the solution grows in amplitude but moves away from x 0 such that its value at any fixed spatial position eventually decays to zero, it is convectively unstable (Landau and Lifshitz, 1959, p. 111). The simple types of bifurcations for one state variable u are classified as follows: 1. saddle-node : (29) 2. transcritical : (30) 3. pitchfork : (31) 4. Hopf : (32) and (33) When a new complex state variable (34) is introduced, the set of equation (32) and (33) reduces to the complex normal form : (35) 5. imperfect : In the case of asymmetric system due to the existence of a source term and the second-order nonlinear term, the imperfect bifurcation takes place: (36) In any case, the above nonlinear systems loose stability during the change of the value of R changing from negative to positive. The transcritical and the pitchfork bifurcations have been discussed in connection with nonlinear dynamics of nuclear power reactors by Karmeshu (1978, 1981b). From the practical point of view, the Hopf bifurcation and the imperfect bifurcation become relevant to practical power reactors as will be shown in the subsequent sections. The simplest nonlinear model in eq.(10) can have a transcritical bifurcation. The stability of the fixed points in the simple normal form equation are summarized in Table I. When the eigenvalue λ of the system takes a real positive number, one call this (a) soft mode instability. On the other hand, when the eigenvalue takes a positive real part and non-zero imaginary part, one call this (b) hard mode instability (limit-cycle oscillation). It is important to note the fact that in the reactor model (10) [or (30)] one fixed point u0 = 0 corresponds to the zero-power state; the other one u0 = R/g corresponds to the at-power state. After inserting the constant reactivity ρ = ρ o , the zero power state u0 = 0 becomes unstable and u0 = R/g becomes stable. The stability of the two fixed points alters at the transcritical bifurcation point.

NONLINEAR STOCHASTIC DYNAMICS Table I

27

Classification of the stability of the fixed points

1. saddle-node:

2. transcritical:

3. pitchfork:

4. Hopf:

5. imperfect:

In terms of conventional reactor physics, this transition point u0 = 0 is “the just critical state” where the generation of neutrons and the loss of neutrons is balanced in the zero power state. On the other hand, the steady state in the sense of nonlinear dynamics is called “the critical state”. D. Wong-Zakai Noise Correction Consider a nonlinear SDE for a state variable X with multiplicative (parametric) noise (37) where F(X, t) and Gi(X, t) (i = 1, 2, .., m) are, in general, well-behaved nonlinear functions with respect to X. The nature of the noise is not necessarily Gaussian white in general. From the point of view of mathematical tractability, one usually avoids non-Gaussian and non-white noises. So let us restrict that the noises {ξi(t)} ( i = 1, 2,..,m) have Gaussian-white nature. It is known under this assumption that the noise correction term is needed in the case of a nonlinear SDE with the parametric noise in eq.(37). Namely, one must interpret the SDE in eq.(37) in the Stratonovich sense and the Wong-Zakai noise correction (1965) is required as (38) where {xi(t)} (i = 1, 2, .., m) are independent Gaussian white noises with null mean: (39) Consider a power reactor model (Sako, 1980; Williams, 1983), (40) The WZ noise correction terms which must be added to (40) are (41)

28

H. KONNO

Figure 1: Classification of bifurcations for nonlinear dynamical systems.

NONLINEAR STOCHASTIC DYNAMICS

29

3. NONLINEAR EFFECTS IN THE NORMAL STEADY STATE A. One Component Model with Feedback 1. Model Consider a nonlinear stochastic power reactor model for the neutron density N(t) without precursors with an additive noise FN (t) and a parametric noise ρ (t) : (42)

8

where ρ0 is the reactivity inserted and Λ is the neutron lifetime. In a zero-power reactor, the reactivity fluctuation ρ(t) may be assumed to be a stationary fluctuation around ρ 0 = 0. Note here that if there is no feedback term and assuming that 〈ρ(t)〉 = 0, 〈ρ(t) ρ(t’)〉 = σ2δ (t–t’) , results contradictory to practical observations are obtained (a) the steady state does not exist; (b) the neutronic variance diverges as t → though ρ0 = 0 (cf. Williams, 1974a, Chapter 5.7). Williams (1974b) shows that nonlinear feedback can suppress the divergence. Actually, in a power reactor with feedback, ρ (t) represents the net influence of feedback so that one must, for example (Dutre et al., 1977), put (43) where γ is the net feedback coefficient to the neutron field. This model oversimplifies the practical negative feedback in the form of coupled stochastic differential equations from the fuel and the coolant temperatures and others. But this simplification helps readers to understand clearly the main consequences of nonlinear feedback effects and the crucial differences of the results predicted in power reactors from those in the zero-power reactors. Now let us assume that the additive noise FN (t) and the parametric noise k(t) are independent and Gaussian white with null mean : (44) (45) 2. Fokker-Planck equation Without the noise sources, viz. FN(t) = k(t) = 0, the neutron density N 0 takes a constant value, which is given by (46) The Fokker-Planck (FP) equation corresponding to eq.(42) after the WZ noise correction is given by (47) where

. The steady state solution of the FP equation is (48)

where P0 is the normalization constant and (49) When a > 0, a reactor is operated with a constant power. When a < 0 is realized, the state of a reactor comes into the state of zero-power after the noise-induced transition (Konno, 1983).

30

H. KONNO

3. Without additive noise When the influence of the additive noise FN (t) can be neglected compared with that of the parametric noise K(t), viz. DNN ≡ 0, the steady state pdf becomes where P0 is the normalization constant and a and b are defined in eq.(49). The exact analytical expressions for the mean and the variance σ N N (= 〈 N 2 〉 - 〈N 〉 2 ) are (51) Note that 〈 N 〉 = N 0 , independent of Dkk . This means physically that ∆ρ induced by k(t) (the WZ term) is compensated by the effect of suppression due to the nonlinearity. The variance-to-mean ratio R becomes (52) which is independent of the power level N 0. 4. Preservation of Gaussianity and sub-Poissonian statistics One should note that the most probable value N m of the pdf in eq.(50) is estimated from (53) 〈N 〉 = N 0 . It seems that a non-Gaussian pdf in eq.(50) contradicts Notice that N m practical observations in power reactor at the normal steady state. The key concept to solve the question is the sub-Poissonian statistics (54) Under the condition, the non-Gaussian pdf in eq.(50) reduces to the Gaussian distribution. Actually, putting N = N m + n and using the asymptotic expansion, ln for 1 , we obtain the Gaussian distribution for the fluctuation n around the most probable value N m : (55) ∼σ

where NN = ∆ NNm. How is the reactor-power dependence of the neutronic variance to be validated ? Since our model in eq.(42) with the feedback in eq.(43) is equivalent to the one that the effect of temperature fluctuation is adiabatically eliminated, the square 〈 N〉 2 power dependence of σ NN is expected as has been pointed out by Oka et al. (1972) and others. The physical reason for the reduction of the reactor-power dependence of the neutronic variance is partially due to the renormalization from the nonlinear term. According to the exact FP analysis, one finds that neutronic fluctuation may be written as (56) where

and (57)

The important point is that the neutronic damping Γ 0 depends on the power level N 0. This is the reason that 〈 N〉 2-dependence is reduced to 〈 N〉 -dependence in the variance in eq.(51).

NONLINEAR STOCHASTIC DYNAMICS

31

B. Inclusion of Delayed Neutrons 1. Model Next let us account for the effect of precursor C(t) (Konno et al., 1990) : (58) (59) where β, λ are the fraction of precursors and its decay constant. To obtain the exact expressions of the mean values 〈 N 〉 and 〈C 〉 of the system is not a trivial problem due to nonlinearity. So we will take a naive attitude that their values at the steady state are equal to the values of N 0 and C0 without the noise source, viz. FN (t) = FC(t) = k(t) = 0 making use of the result in the previous section that the WongZakai term may compensate the nonlinear term (60) 2. Linearized Langevin eauation The equation for the fluctuation steady state (N 0 , C0 ) becomee

around the expected (61)

where (62) and (63) The Boltzmann operator B in eq.(61) is different from the conventional one based on the linear theory (cf. Saito, 1974; Otsuka, 1972): (64) The coefficient takes a negative value though k 0 > 1, which comes from the nonlinear negative feedback term. In the linear zero-power model, the fluctuation diverges for k 0 > 1 . The noise source matrix takes the form: (65) 3. Eigenvalue The characteristic equation for fluctuations in eq.(61) becomes (66) where (67) The solutions of the characteristic equation are (68) The most important point different from the zero-power reactor's case without feedback is that the eigenvalues s± depends on the steady state value, viz. N 0 ( or ρ 0 ). When ρ0 = 0, we have s+ = 0 and s– = β /Λ. In this case, the eigenvalues calculated form BL in eq.(64) takes the same values as the ones from B in eq.(62). However, this

32

H. KONNO

is not the case for power reactor since N 0 = 0 means that the reactor power is zero from eq.(60). The eigenvalues for k 0 > 1 are approximately estimated as (69) where Γ 0 is same as the relaxation constant of neutron fluctuation without delayed neutrons in eq.(57), i.e., For the zero-power steady state p0 = 0, the eigenvalues becomes No-independent quantities. 4. Power spectral density The power spectral density (PSD) of neutron fluctuation becomes (70) The PSD takes the same form as the linear Brownian oscillator with the resonant frequency and the damping constant C1. The break frequency wb is approximated by |s_ | which shifts toward higher frequency region as the reactor power increases. 5. Reactor-power dependence of neutronic variance The variance of neutron fluctuation σ NN becomes (71) where . This means that when k 0 > 1, C2 > 0; the stable steady state is realized under the constant reactivity insertion. The variance diverges as C2 → +O. Namely, C2 = 0 is the just critical point of the reactor. The divergence of variance known in linear theory is suppressed by the nonlinear negative feedback. Since C1 > 0 always hold even for k 0 > 1, the hard mode instability does not take place when the strength of the parametric noise of Gaussian white nature is small. When a reactor is operated at high power level, so that all the additive noise may be neglected, viz. DNN ≡ DNC ≡ DCC ≡ 0, the above expression reduces to (72) The effect of precursors contribute to reducing the variance of the neutrons, as is seen in the subcritical zero-power reactor (Otsuka, 1972). But its reduction rate is small even in the limit of large N 0 since the factor is 1/(1 +β). In power reactors with nonlinear negative feedback by which the steady state is maintained, the effect of precursors does not play a crucial role as far as the magnitude of neutron fluctuation is concerned. It can be shown that a critical fluctuation with a non-exponential decay can appear near the critical point C2 = 0 in the presence of parametric noise (Konno, et al., 1990). C. Inclusion of Temperature Feedback Consider next the effect of a temperature feedback (Debosscher et al., 1979a, 1979b; Karmeshu, 1981b; Konno, 1983, 1984) based on a two-variables model ; (73) where the neutron density n and the reactor temperature θ are normalized by the steady state density Ns and the temperature Ts as (74)

NONLINEAR STOCHASTIC DYNAMICS

33

Also the parameters (a, b) and the normalized random force η(t) are expressed by the noise-corrected reactivity ρ~0 (= ρ 0 + ∆ ρ), the effective neutron lifetime Λ, the heat transfer coefficient h and the heat capacity of the reactor cR as (75) The contraction of the state variable θ leads to (76) where the dot (.) denotes the time derivative (d/dt) for simplicity, and the effective potential becomes (77) The NTR n = exp(x) leads eqs.(73) to an oscillator with Toda potential : (78) where (79) Approximating exp(x) ≈ 1 + x + x 2/2 + x 3/6 + x 4/24, one obtains a stochastic Duffing type oscillator with a second- and a third-order nonlinearity. A higher-order approximation of the adiabatic elimination of the temperature T leads also to a same type of equation classified into eq.(36) (cf. Konno, 1983 and 1984). It is clear that the collective frequency and the damping constant of eq.(78) are (80) Since n is the normalized neutron density defined by eq.(74), one can write n = 1 + y and (81) for small fluctuation y (|y| << 1). By noticing the resonant frequency w 1 = one can readily write down the decay ratio γ D of the system as (82) for ab – b2 /4 > 0. On the other hand, for ab – b2/4 < 0, the decay ratio is not defined due to monotonous damping. Note that the constants a and b are multiplied to the strength of the noise source η(t) in the presence of feedback. As far as this model in eqs.(73) is concerned, the temperature feedback works to stabilize the system, and there is no element to destabilize the system. 4. SPACE-INDEPENDENT STOCHASTIC DYNAMICS A. Stochastic Power Oscillation in the NSRR 1. Delayed feedback model Power oscillations have been observed in the Nuclear Safety Research Reactor (NSRR), a TRIGA-type pulsed reactor, at JAERI, Japan. The power oscillation was first recognized in 1983 when the reactor was operated in a stationary power mode at about 300 kW before its recent modification. The feature of power oscillations was firstly reported by Hayashi (1984). The origins of the power oscillations have been examined (Konno et al., 1990, 1992; Suzudo et al., 1994a, 1994b) from experimental and theoretical points of view.

34

H. KONNO

The structure of the NSRR inside the reactor pool and the layout of the reactor core and the locations of the sensors have been reported (Hayashi et al., 1984). It has been recognized that there is no apparent experimental indication of void formation when the power oscillation was observed. However, weak power oscillations (the amplitude is about 4 percent to the total power) have been observed in cases when (a) the reactor was operated at a stationary high power level; (b) the diffuser pumps are turned on or off; and (c) the temperature of the water in the reactor pool is relatively high. In addition, various types of oscillations have been observed. The oscillations from the neutron detectors can be classified into three types: (i) sinusoidal, (ii) rectangular and (iii) sawtooth, which are schematically illustrated in Fig. 2.

Figure 2: Features of oscillations observed in the NSRR: (a) Sinusoidal Type, (b) Rectangular Type and (c) Sawtooth Type. In the automatic control system, there was a dead band, a limiter and a time delay of the control signal to the mechanical driving unit (the control rod). Therefore, it has been considered that an element which gives rise to an instability exists in the automatic control system. To explain the observed variations of the damping ratio and of oscillations, i.e. rectangular type (0.12 Hz), sawtooth type (0.04 Hz) and sinusoidal type (0.12 Hz), the effect of finite delay time τ is examined on account of the physical situation in the NSRR; the characteristics of the driving mechanism of the control rod, the existence of the dead band and the limiter in the control svstem: (83) where the detailed position and the movement of the control rod is completely neglected. Due to the nonlinear feedback, the neutron density N for τ = 0 can be kept constant and the steady state value takes N 0 (= ρ0 / γ) for ρ0 > 0 . When τ is relatively small, N(t – Τ) can be expanded around τ; (84)

NONLINEAR STOCHASTIC DYNAMICS

35

and truncating the terms up to the second order, one can approximate this equation by (85) where α2 = 2/ τ 2 . This equation explains (a) why the delay τ contributes to the negative damping, (b) why the power N 0 = ρ/γ plays the role of a source to sustain the oscillation and (c) the system becomes more unstable when γN and/or τ become large. 2. Characteristic equation The characteristic equation of the linearized equation for the model (83) is given by (86) or in terms of p = τs and a = (ρ0 /Λ) as (87) This is a transcendental equation which takes the form of infinite order polynomial equation if one expands exp(p) in the power series of p. One can easily find the exact solutions for equation (87): (a) the pure imaginary solutions, (88) (b) the real solution, (89) When the solutions are complex numbers, viz. p = pR + ipI , the real and the imaginary part must satisfy the relations: (90)

Figure 3: The principal solution of the transcendental equation (87) as a function of a τ.

Figure 3 shows the principal solution (eigenvalue) r1 of the characteristic equation (87) as a function of the parameter aτ(= ρ0 τ/Λ). When at < exp(–1), the only one real solution with a negative value exists On the contrary, finite values of imaginary part of the root appear when (91) However, as far as (92)

36

H. KONNO

the real part of the root takes the negative value, viz. Re.(p1 ) < 0. So the system behaves like a damped oscillator in this region. When aτ becomes large enough to have the values (93) the real part of the solution becomes positive, viz. Re.(p1) > 0, the fluctuation will grow and the effect of nonlinear suppression becomes important. 3. Temporal oscillation without noise The above analysis indicates why the reactivity feedback by the delayed insertion of the control rod behaves like a damped harmonic oscillator which has been experimentally estimated. Note that we can explain the variation of the resonant frequency ω 0 and the damping ratio γD depending upon the value of the parameter aτ according to the model (83). The scale transformations of time and the state variable via (94) in eq.(83) leads to the integral equation for y(ξ): (95) As is expected easily from the expression (87), the system with the time-delay τ is equivalent to a system with infinite degrees of freedom. There exists infinite number of eigenvalues, and the memory effect becomes important when aτ becomes large. Examination of the behavior of oscillation under the initial distribution, y(ξ) = A sin(πξ), leads to the conclusion (Konno et al., 1990) that: (a) when aτ > π/2, stationary oscillations take place; when a τ < π/2, damped oscillations appear. Concerning the profile of the oscillations, we have obtained the facts that; (b) when aτ >> π/2, the amplitude of oscillation does not depend on the initial distribution; when aτ << π/2, the amplitude of oscillation is proportional to the initial magnitude of perturbation; (c) The profile of oscillations near the unstable threshold (aτ)c (i.e., Re.(p1) = 0) is well characterized by the principal eigenvalue p1, viz. sinusoidal. 4. Approximate analytic solution of limit cycle oscillation There are a number of methods to obtain an excellent approximate solution for nonlinear ordinary differential equations: (a) equivalent linearization method; (b) averaging method of Krylov-Bogoliubov-Mitropolsky (KBM) (or Harmonic balance method); (c) embedding method and others. To obtain an analytic expression of saturated amplitude of the scaled variable y(ξ ) (cf. Konno et al., 1992), let us put (96) Under the NTR, we can rewrite (83) as (97) Following the method of KBM, let us put (98) where A(ξ) , B(ξ) and C(ξ) are assumed to be functions slowly varying with time. Substituting eq.(98) into eq.(97) and balancing the coefficients of cosw 0 ξ, sin w 0ξ and constant terms, one obtains (99) (100)

NONLINEAR STOCHASTIC DYNAMICS

37 (101) (102) (103)

2 1/2

and R = (A + B ) be determined from 2

. The steady-state amplitude of oscillation

+

can (104) (105)

and (106) Since R s ≠ 0, we have When have the first order approximation for the saturated amplitude:

(107) Under this situation, we

(108) Note that the amplitude of the normalized neutron density around the steady-state value N 0 is given by (109) The second order approximation is obtained by accounting for the R 4 term in X 0 (R) and X 1 (R): (110) It is important to note the fact that the higher-order harmonics are incorporated in the analytical solution by virtue of the NTR of the state variable via eq.(96). Availability of nonlinear transformations for nonlinear coupled ODEs is demonstrated in an article (Konno and Sawahata, 1995). 5. Incorporation of noise source Consider the situation that a parametric noise ρ (ξ) is incorporated into the reactivity, viz. ρ0 → ρ0 + ρ(ξ). Under the approximation of small delay time τ, the introduction of scaled variables y = N(ξ)/N 0 - 1 and 1/(1 + y) ≈ 1 - y + y 2 leads (83) to a van der Pol type oscillator with additive noise as (111) where (112) Let us assume the form of the solution for eq.(111) as (113) where R(ξ) and f(ξ) are slowly varying variables. Following the method of KBM, one can put (114)

38

H. KONNO

Eqs.(113) and (114) lead to (115) and (116) Inserting eq.(113) into (116) in noticing that mω

2 0

= 1, we obtain (117)

Substituting (113) and (114) into (117), we obtain

(118) Averaging the cosine terms in the coefficients of R, R 2 and R 3 over one cycle [0,2π] of oscillation, one obtains (119) where (120) Similarly, application of the above averaging procedure to the equation of the phase, (121) leads to (122) where (123) 6. Experimental verification (a) complex normal form In the previous section, we have derived a van der Pol type equation with additive noise (111) for a small amplitude limit cycle oscillation near the onset of Hopf bifurcation. But, for the purpose of nonlinear system identification, it is more convenient to simplify it by applying the averaging method in the previous section or the rotational wave approximation (Haken, 1970) and transforming (Konno et al., 1994) it into the SDE in the complex normal form in eqs.(119) and (122), viz., (124) where A(= x + iy) is the state variable of complex number, F(t)(= Fx(t) + iFy(t)) is the fluctuating force of complex number, ω 0 is the frequency of oscillation and the constants g and h are assumed to take real values. The strength of noise sources in both the real Fx and the imaginary Fy part of F(t) are assumed to take the same value D, viz., áFx(t)Fx(0)ñ = Dδ(t) and 〈 Fy(t)Fy(0)〉 = Dδ (t) . One should note that the second moment 〈 |A| 2 〉 has a special physical meaning. Namely, it is proportional to the angular momentum in the 2D phase space, as was pointed out by Tomita et al. (1974) (cf. also Kishida et al., 1976). He called the fluctuation associated with this angular momentum the irreversible circulation of fluctuation (ICF). To get an analytical expression for the ICF near the instability point is quite difficult due to complicated nonlinearity-noise interplay. Konno et al. (1994) obtained the analytic expression of ICF A for the model (124) below and above the onset point (g = 0) of Hopf bifurcation:

NONLINEAR STOCHASTIC DYNAMICS

39 (125)

where erf(x) is the Error function. Figure 4 shows that the ICF takes the maximum value at the bifurcation point (g = 0) and it decreases as g increases. This also means that nonlinearity works to suppress the fluctuation.

Figure 4: The variation of ICF A as a function of g in the vicinity of Hopf bifurcation. There is one more reason for taking this equation; we must reconstruct, at least, 2D phase space (N(t), N(t)) only from time series {N(t)} with the use of an embedding method. The length. scales of constructed 2D space are same order even if the original length scales of N and N are different. So canonical variables such as A should be taken for the analysis (Konno et al., 1994; Konno and Hayashi, 1996). The stochastic model in eq.(124) provides the steady-state pdf of the amplitude R (= |A|) and the phase as (126) where the amplitude distribution W s (R) = W 0 exp(–2 V(R)/D) is expressed in terms of the effective potential and the phase distribution W s (φ) = is uniform over the range [–π, π]. The amplitude R and the phase f motions are decoupled by virtue of the high symmetry of the potential function V(R) and the uniform distribution of the phase in the stochastic model in eq.(124). (b) identification of pdf The amplitude and phase pdfs obtained from the power oscillation data in the phase-III reactor noise experiment in the NSRR after reconstruction of 2D phase space by applying the embedding method due to Takens (1981) to the time-series data of power oscillation measured in the NSRR are depicted by the solid line with the triangle mark in Figs. 5 and 6. The mark GX indicates the strength of gain-level of feedback to the automatic control system which may increase delay-time in the control system. Namely, larger number X corresponds to larger gain-level GX, i.e., G1 < G2 < G3 < G5 . The experimental amplitude pdf for G5 is identified by the theoretical amplitude pdf: (127) where W 0 is the normalization constant, a and b are fitting parameters. We get a good fit by taking a = 10.0 and b = 6.5 as seen in the dashed line shown in Fig. 5(a). Though the

40

H. KONNO

experimental phase pdf in Fig. 6(a) for G5 is almost uniform, a weak localization of phase is observed. So we try to fit by taking the theoretical phase pdf: (128) where W 1 is the normalization constant. By choosing the fitting parameter c = 0.3, we get a good fit as shown in the dash-dot line in Fig. 6(a). This weak localization of phase indicates that we must account for the terms g'A * and h'(A|2A*, where g' and h' are constants and * denotes the complex conjugate. It seems from a previous qualitative analysis that the amplitude pdf for G3 is identified by the theoretical amplitude pdf (127). However, we cannot get a good fit by using this distribution as shown in the dashed line in Fig. 5(b). Therefore, we will adopt a new theoretical amplitude pdf: (129) where W 2 is the normalization constant. This amplitude pdf (129) provides a good fit to the experimental one as shown in the dash-dot line in Fig. 5(b). Also this amplitude pdf gives good fits in the other experimental ones for G2 and G1 as seen in Figs. 5(c) and (d). On the other hand, the phase pdf (128) is available for all the cases through G5 to G1 to get good fits to the experimental ones. However, the case of G0 show (Konno et al., 1994) both the amplitude and phase pdfs do not accept a good fit to the above theoretical pdfs in eqs.(127) and (129), as was expected before analysis due to its strong intermittent oscillation. Table II Estimated parameter values from the phase-III experimental data

The parameters that give good fits for the cases of G5, G3, G2 and G1 are summarized in Table II. In order to get the parameters, we have used conveniently the most probable value Rm in terms of a and b for the amplitude pdf. The most probable value rm for the amplitude pdfs (127) and (129) take the same expression as (130) For the phase pdf, a phase-shift ∆φ must be introduced in replacing φ by φ+ ∆φ in eq.(128) to fit experimental data. The values of the phase-shift are not described in Table II since the values are not important for physical interpretation. (c) modification of complex normal form Now let us discuss the physical situation for the profile change in the identified amplitude and hase pdfs in the previous section. When the symmetry breaking terms g'A * and h'|A|2 A* (g' and h' are assumed to be constants of real value here) must be introduced as was explained in introduction, the amplitude and phase motions cannot be decoupled in the rigorous sense. Actually, the equations of the amplitude R (= |A|) and the phase are given by (131) and (132)

NONLINEAR STOCHASTIC DYNAMICS

41

Figure 5: Variations of the amplitude pdfs in the NSRR (the Phase-III experiment) as a function of feedback gain GX (G5 > G3 > G2 > G1) to the control system. Experimental amplitude pdf is indicated by solid line with triangle; the theoretical amplitude pdf (127) is shown in dashed line; the theoretical amplitude pdf (129) is shown in dash-dot line.

42

H. KONNO

Figure 6: Variations of the phase pdfs in the NSRR (the Phase-III experiment) as a function of feedback gain GX (G5 > G3 > G2 > G1) to the control system. Experimental phase pdf is indicated by solid line with triangle; the theoretical amplitude pdf (128) is shown in dash-dot line.

NONLINEAR STOCHASTIC DYNAMICS

43

where FR (t) and Fφ(t) are random forces for the amplitude and the phase, respectively. The amplitude motion in eq.(131) gets a parametric influence from the phase fluctuation. The phase motion in eq.(132) gets also a parametric influence from the amplitude motion. When h' can be ignored the phase motion becomes independent of the amplitude motion. However, the amplitude motion is still under the parametric influence from the phase motion. To discuss the pdfs identified experimentally in the previous section, let us derive approximate analytical expressions for the amplitude and phase pdfs at the steady-state. Let us begin with the phase pdf. Under the mean field approximation in which the term R 2 is replaced by the ensemble average 〈 R 2 〉 , one gets eq.(128) with (133) Note that when h' = 0, the phase motion is independent of the amplitude without the mean field approximation. Direct numerical simulation supports the validity of these approximations. The relevance of the nonlinear term h'|A | 2 A* is not easily determined only from the phase distribution. Consider next the amplitude pdf. The application of mean field theory to the amplitude equation, viz., the cosine term cos2φ is replaced by 〈 cos 2φ〉, oversimplifies the coupling between the amplitude and the phase. Mean field theory is not appropriate to provide the amplitude pdf (129). So let us introduce a new stochastic amplitude equation (134) where η(t) = cos2f. Assuming the strength of the parametric noise, i.e., 〈 g’η(t)g’η(0)〉 = Qδ(t), we get an approximate amplitude pdf after the WZ noise-correction as (135) where W 3 is the normalization constant, (136) When D = 0, the expression in eq.(135) reduces to eq.(129). Thus it becomes clear what are the necessary terms to provides the form of amplitude and phase pdfs to explain experimental results. One should note here that (a) the interference between nonlinear terms and noises change the strength of fluctuations; (b) the strengthes of noises Q and D do not take constant values which are dependent on the system parameters since they are expressing the intrinsic noise sources in a complex nonlinear feedback system. B. Stochastic Power Oscillation in PWRs The dynamics of PWRs may be characterized by coupled equations which take into account the feedback from the fuel and coolant temperatures. Theoretical power reactor noise analysis based on the linearized treatment accounting for two temperatures and the effect of mechanical vibrations has been extensively studied as seen in Saito (1973, 1974), Kitada (1974) (see also Williams, 1974a). Approach along this line have been performed by Wood and Perez (1991). On the other hand, historically, purely kinematic studies on the stability boundary and the profile of limit cycle oscillations in a two temperature feedback model has been performed by many authors (Shotkin, 1964; Torlin, 1966; Akcasu and Shotkin, 1967; Devooght and Smets, 1967; Schmidt and Hetrick, 1967; Shotkin et al., 1970; Vreeke and Sandquist, 1970; Ash, 1970, Poddar et al., 1983; Nomura et al., 1986, 1988). Consider the following stochastic two-temperature feedback model: (137)

44

H. KONNO (138)

and (139) where the physical meaning of the parameters are listed as follows: γ f = temperature coefficient of reactivity due to fuel; γ c = temperature coefficient of reactivity due to coolant; Cf = thermal capacity of fuel; Cc = thermal capacity of coolant; µf = fuel-to-coolant heat transfer constant; W c = mass flow rate of coolant through core; M c = mass of coolant in core at any instant; T c = transit time of coolant through core; γ m = 2W c Cc /M c = 2Cc /T c ; P0 = qN s = steady-state power level; Tf = average temperature of fuel; Tc = average temperature of coolant; T ( ci ) = inlet coolant temperature The effect of heat removal due to coolant fluid is incorporated although the explicit treatment of the Navier-Stokes equation coupled with the heat conduction equation is not incorporated. Instead, its effect are introduced phenomenologically through the term γ m T c in the equation (139) for coolant temperature T c . In the analysis described below, we will assume that the physical parameters other than γ m are fixed. The physical consequence associated with the variation of γ m will be shown below. The steady state solutions become

and (140) The equation of fluctuation becomes

= (δN, δTf , δT c )T around the steady state

= (N s , Tf s ,T c s ) (141)

_ where F(t) = (FN (t),Ff (t),Fc (t))T and

(142) The characteristic equation becomes (143) where (144) and

(145) According to the Routh-Hurwitz stability criteria, the system is stable when C1 > 0, C2 > 0, C3 > 0 and C1C2 > C3. It is clear that C1 > 0, C2 > 0 and C3 > 0. Therefore, the soft mode instability does not take place. Let us examine the last inequality C1C2 > C3 which is relevant to the hard mode instability (the limit cycle oscillation) by setting S=C1C2–C3.

(146)

NONLINEAR STOCHASTIC DYNAMICS

45

Since (147) the necessary condition for the occurrence of a limit cycle oscillation is (148) Otherwise, the limit cycle oscillation does not take place. The equation of S is a secondorder algebraic equation for γ m. Under this condition the parabola intersects the line S = 0 as shown in Fig. 7. Therefore, S takes negative values in the range (149) where (150) Apparently, γ m = 0 is the most unstable state under the necessary condition of instability (148) described above. If one neglects the effect of coolant temperature Tc in eqs.(137) and (138), the limit cycle oscillation does not take place except the cases that (i) a delay time is introduced; or (ii) an oscillatory colored noise is introduced in the external force (t). Though the model is a homogeneous one, the physical processes of an inhomogeneous nuclear reactor are incorporated phenomenologically. Namely, the heat generated in the fuel region transfers to the coolant fluid; the movement of coolant lowers its temperature. The delay time associated with the process is the origin of onset of power oscillation. The unstable region in (γ m , S) space and in (γ m, P0) space are illustrated by the hatched lines in Fig. 7.

Figure 7: Unstable regions in (a) (S,γ m) and (b) (P0,γ m) parameter spaces.

The variance Σ and the ICF A below the Hopf bifurcation can be expressed in terms of C1, C2 and C3 via the power spectral density matrix of fluctuation P(w) = G[iw]DG[-iw] T (where G[iw] = [iwI + B] −1 is the Green's function matrix for eq.(141) and D is the noise source matrix for as (151) and (152)

46

H. KONNO

Therefore, it is easy to understand that the values of the variance Σ and the ICF A increase as the system approaches to the Hopf bifurcation point (viz., C1C2 = C3). Kishida et al. (1976) studied these features numerically after an asymptotic evaluation. The mathematical divergence is suppressed by the nonlinearities, which results in the finite amplitude of oscillation. C. Stochastic Power Oscillation in BWRs Power oscillations in BWRs have been extensively studied experimentally and theoretically from the viewpoint of reactor safety. Historically, a power oscillation in the BORAX-2 (Thie, 1963) was described by Rice (Wax, 1954) and Akcasu (1961). Akcasu described phenomenologically its beating oscillation by (153) where Z(t) is Gaussian white noise and κ1 (t) is Gaussian (not necessarily white) noise. When κ1 (t) is white noise, the characteristic equation after the WZ noise correction becomes (cf. also Arnold, 1974, Chapter 11) (154) This means that the noise κ1 (t) works to destabilize the system. Analog simulations showed that non-white noise plays a crucial role to produce oscillating packets (cf. Ackasu, 1961 and Williams, 1974). Since then, BWRs have been operated under condition such that the oscillation does not occur. Due to the accident of Lasalle-2 in 1988 (Murphy, 1988) and the development of a new type of fuel that improves the fuel thermal margin, much work has been done to demonstrate the main features of the occurrence of power oscillations in BWRs. A BWR model accounts for the void-reactivity feedback since the effect is considered to be the main origin of oscillation: (155) (156) and (157) Konno (1985) assumed and u = 0 and Fα (t) = 0 provided that the void-reactivity feedback is characterized by the slow drift motion of a colored noise η(t). On the other hand, March-Leuba et al. (1986) take a damped oscillator model by setting K(p, η(t)) = a1 (> 0) . By applying the reductive perturbation method (Taniuchi, 1974; Nyfeh and Mook, 1979) to a multiple-variable BWR model (155)-(157) and incorporating a complex fluctuating force F(t), one obtains (cf. e.g., Munoz-Cobo and Verdu, 1991 and 1992 for the case without noise sources) . which is rewritten by setting A = Rexp(iφ) as

(158)

(159) and (160)

NONLINEAR STOCHASTIC DYNAMICS

47

These stochastic equations express a stochastic center manifold of a nuclear reactor near the onset of a spatially uniform limit cycle oscillation. Since the symmetry of the equation is broken due to (de + b|A| 2 )A*, the motion of amplitude and the phase are not decoupled as seen in eqs.(131)-(132) in 4.A.6. 5. SPACE-DEPENDENT STOCHASTIC DYNAMICS A. Space-Dependent Theory of Nonlinear BWR Dynamics Numerical simulations in 3-D space, especially, the computer codes such as RETRAN (Araya et al., 1991), STANDY (Yokomizo et al., 1987; Muto et al., 1990), EPA (Wullf, 1992), TOSDYN (Tsunoyama et al., 1984; Takigawa et al., 1987; Takeuchi et al., 1991, 1992a, 1992b, 1994a, 1994b), RAMONA (Moberg et al., 1979) and LAPUR (Blakemann et al., 1989), have been performed to manifest the essential mechanism of core-wide (uniform or in-phase) and regional (out of phase) oscillations which have been observed in BWRs shown in Table III. The zeros of the Bessel function and the corresponding spatial modes in the cylindrical geometry (r, θ, z) are summarized in Table III. Table III Zeros of the Bessel Function Jl(z l m) = 0 Mode (l, m)

zl m

Fundamental mode (0,0)

2.40482

First azimuthal mode (0,1)

3.8371

Observation LaSalle-2 (USA) Peach Bottom (USA) Vermont Yankee (USA) KBR-B/C (Germany) Caorso (Italy) Leibstadt (Switzerland) TVO-2 (Sweden) Ringhals (Sweden) Gundermmingen (Germany)

The BWR instabilities are complicated due to the complicated non-linear couplings in conjunction with the nuclear, thermal and hydraulic feedback including liquid-gas two-phase fluid flow. It is known that there are many instabilities associated with the liquid-gas twophase flow (Boure et al., 1973; Lahey et al., 1980). Roughly speaking, they are classified into two elementary types: the static one called (I-a) “the Ledinegg instability” due to the appearance of negative pressure drop in flow channel; the dynamic one called (I-b) “the density wave oscillation (DWO)” due to the result of delayed feedback among fluid flux, fluid density and pressure. There are also many combined instabilities such as (I-c) “the pressure drop oscillation”, (I-d) “the chugging instability” and so on. BWR instability is one of the combined ones resulting from the coupling with the void-reactivity (VR) feedback. By virtue of many experimental and theoretical works done up to now, the essential mechanism of power oscillation in BWRs are becoming clear. However, there are many unsolved questions, e.g., (q-1) why the beating oscillations take place; (q-2) what is the origin of the appearance of higher harmonics; (q-3) what is the governing law for the oscillation pattern selection.

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H. KONNO

1. Density wave oscillation To understand BWR instability deeply, let us first review the mechanism of (I-a) the DWO which is regarded as the main origin of power oscillation in BWRs, by taking the 1D single channel model for two-phase flow without the VR feedback to the neutron filed. Fundamental (mass ρ, energy h and momentum G) equations and constitutive equations which account for the correlations, (i) void fraction correlation (Zuber-Friendlay's correlation); (ii) flow quality correlation in the subcooled region (Levy formula with Saha-Zuber's and (iv) two-phase subcooled boiling initiation); (iii) two-phase friction multiplier local loss multiplier (cf. Yokomizo, 1990), are written down as follows: (161) (162) (163) where xF is the flow quality, q is the rate of heating per unit area and unit time, α is the void fraction, (164) is the density of fluid and the pressure drop R due to the flow-channel friction is defined as m

(165) The equilibrium quality x and the flow quality xf are related by (166) In the single-phase region, xF = 0 and hl(x). On the other hand, in the bulk boiling region, In the subcooled boiling region, xF and h l are independent. Now let us seek solutions of the coupled equations by applying the method of KBM after assuming that (167) (168) (169) After averaging the phase ς = ωt over [0,2p] by multiplying by sin{wt + φx(z)}, sin{wt + and sin{wt+φP} , one obtains the coupled equations for x 0(z), G0(z), P0(z), Ax(z), A G (z), A P, φx(z), φG(z) and φP as shown in Table IV. Table IV ODEs for x 0(z), G0(z), P0(z), Ax (z), A G (z), A P, φ x (z), φG (z) and φ P in eqs.(167)-(169). (170) (171) (172) (173)

NONLl NEAR STOCHASTIC DYNAMICS

49 (174) (175) (176) (177) (178)

The functions f c , x 1 , ξ and Φ in Table IV are defined as (179) (180) The parameters like A i j are abbreviations for A i - A j . The above equations in Table IV teach us that (a) the average mass flow is constant along the flow path, (181) (b) the average quality at the stable steady state x 0 is proportional to z along the flow path, (182) and (c) Po depends on the feature of void fraction. Hence the physical parameters we must give are: τ = (pl g H/G0 ) (the coolant transit time through the core), ∆x = (qmH/h fgG0) (the difference of the equilibrium qualities between the inlet and exit), η = (pg/pl), G0 , ξ, and fc. H denotes the length of the flow path. For a given angular frequency w and a set of boundary conditions at the channel inlet, numerical solutions of AG(z), φG (z), x 0(z), Ax(z) and φx(z) give averaged pressure drop ∆P0(z), amplitude AP and its phase φP (Yokomizo, 1990). The channel thermal hydraulic impedance is given by (183) This Z[iω] determines the stability of the channel for the DWO. Namely, the characteristic equation Z[s] = 0 has a root δ + iω 0 and the decay ratio is expressed as (184) When γD > 1, the system is unstable. The above analysis shows that a limit cycle oscillation can appear without the VR feedback, which is the main reason that Konno (1985, 1989, 1991) proposed a BWR model with the use of a stochastic Van der Pol oscillator as VR fluctuation in eq.(157). One should also note that the above theoretical analysis is based on the 1D spacedependent model and on the KBM averaging method, a limit cycle oscillation with a sinusoidal wave form is due to the approximation adopted. Dorning et al. (1988), Lahey et al. (1989, 1990) and Takenaka et al. (1991a, 1991b) have shown theoretically that periodic and chaotic oscillations can be extracted from a boiling channel model. Ozawa (1996) has shown experimentally that a pressure drop oscillation (I-c) can be described by a modified

50

H. KONNO

Van der Pol model which exhibits chaos. 2. Detailed linear stability analysis BWR instability is characterized by a combination of instabilities due to (a) the VR loop and (b) the DWO loop. Do the two feedback loops enhance or suppress the power oscillation? The feature of feedback loops is illustrated in Fig. 8. It is clear that the sub-loop works to enhance the instability since the loop is, so to speak, auto-catalytic. In order to confirm the fact analytically, one can perform complicated linear stability analysis with the use of various non-dimensional numbers such as the phase-change number N p, the Froude number Fr, the Jacob number Ja and the fuel time-constant τ M , as was studied by Ran et al.(1995). The results are summarized as follows: (a) the Ledinegg instability (I-a) will not takes place under the strong influence of the VR feedback; (b) (Np, Ja, Fr, τM ) are the key nondimensional parameters determining the DWO instability, viz. (i) Fr has a much stronger effect on the stability of the system when the VR feedback is working; (ii) the lower the Fr the less stable the system; (iii) the fuel-time-constant τ M exhibits clearly the stability boundary in the presence of the VR feedback loop. Wang et al. (1994) and Hagen et al. (1997) have been studying experimentally and theoretically the boundary of instability with the use of the useful nondimensional numbers, i.e. the Zuber number and the subcooling number, for a natural circulation BWR. Hagen et al. (1997) have examined the availability of their numbers in conjunction with the decay ratio estimation.

Figure 8: The main feedback loops in BWRs.

3. Space-dependent theory The main feedback loops in BWRs are illustrated in Fig. 8. The space-independent model in eqs.(155)-(157) accounts for the VR feedback. On the other hand, the DWO described with the use of eqs.(161)-(166) does not account the VR feedback. Recently various reduced order BWR models (March-Leuba et al., 1991; Takeuchi et al., 1992; Hashimoto et al., 1993; Takeuchi et al., 1994a, 1994b; Munoz-Cobo et al., 1996; Torlin et al., 1997) have been introduced in conjunction with the origin of the regional (out of phase) oscillation. Consider the space-dependent neutronic equation and heat diffusion equation as (185)

NONLINEAR STOCHASTIC DYNAMICS

51

and (186) where is the destruction operator and is the production operator, is the heat production rate of the reactor and kf is the thermal conductivity. The effect of delayed neutrons is neglected here, for brevity. Expanding the neutron field with the use of the λ-modes of the steady state problem (Takeuchi et al., 1992; Hashimoto, 1993), (187) where L 0 is a destruction and M 0 is a production operator at the steady state, Munoz-Cobo et al. (1996) have derived a set of neutronic and temperature equations: (188) (189) and (190) where nm is the m-th mode, θr is the temperature of the r-region of the core, (191) (192) is the qm,r is the coefficient of heat production for the m-th mode in the r-region, averaged temperature feedback coefficient in the r-region due to the combination of the (m, n)-mode. Combining the two-phase flow model in eq.( 161)-(166), linearizing them, introducing the void fluctuation δα into eq.(191) and accounting for the symmetry of the fundamental and the first mode, (193) they have arrived at a reduced order model: (194) (195) (196) and (197) where θav = (1/2)(θ1 +θ 2 ) is the averaged temperature and θd i = θ 1 –θ 2 is the temperature difference of the two regions of the core. When the fundamental mode is dominant, the above model reduces to the space-independent model due to March-Leuba et al. (1986). On the other hand, when the second mode is dominant, one must account for all coupled equations. They also demonstrated with a few numerical examples that the reduced-order model in eqs.(194)-(197) is available for both the core-wide oscillation and the regional oscillation.

52

H. KONNO B. Stochastic Center Manifold Dynamics near the Hopf Bifurcation

1. Experimental noise analysis in BWR In spite of a space-dependent DWO analysis (5.A.1) and an unified description of corewide and regional oscillations (5.A.3), the set of reduced order equations in eqs.(155)-(157) and eqs.(194)-(197) are expressing center manifolds, viz. mean field equations. They have neglected fluctuations associated with the means. According to experimental data in the cases of power oscillation which have been observed in BWRs such as BORAX-2 (Thie, 1963, 1981; Akcasu, 1961; Williams, 1994a), FORSMARK-1 (Bergdahl et al., 1989; Oguma, 1996) and Dodewaard (Hagen et al., 1994; 1997), there appeared modulations or beating oscillations. It is known that the neutron fluctuations δN associated with them take mostly Gaussian-like profiles, (198) This seems .to be due to the averaging effect of LPRM, a consequence of the central limit theorem. But, a striking fact is that the flow fluctuations take also mostly Gaussian-like profiles. This fact suggests that (a) local fluctuations and global (coherent) fluctuations have a similar nature; (b) the degrees of freedom might be large. The theoretical models in 5. A. 1 and 5. A. 3 provide the amplitude pdf giving the inverse sine law, (199) Rmax being the maximum amplitude. On the other hand, the power oscillation observed in the NSRR takes the cylindrical distribution in eq.(126). 2. Space-dependent stochastic center manifold Now consider the stochastic space-dependent center manifold with the spatial diffusion term as (200) where ∆ 2 denotes Laplacian, 〈 Fj(r,t)〉 = 0 (j = x and y ) and (201) where E(r, r') is the strength of the noise source F(r, t). Here we take the coefficients of the model in general complex values like (202) Positive definite values of the coefficients (203) are required for preserving global stability of the system. Let us consider the space-dependent center manifold (200) with complex coefficients in relation to nuclear systems and clarify the spatial effects associated with the complex coefficients. When gI is eliminated (this is always possible) and if hI = DI = 0, the following potential function is defined: Φ = When all the circulating components exist, viz. gI, hI and DI take non-zero values, the potential function Φ does not exist. These circulating components are relevant in nonequilibrium open systems. We have shown an illustrative example, the two-group neutron fields with heat diffusion and the two-phase flow field. This example shows how the imaginary part of the coefficients are originated in multiple-variables coupled nonlinear systems. Other examples can be seen in plasma (Taniuchi, 1974) and fluid mechanical systems (Thompson, 1984).

NONLINEAR STOCHASTIC DYNAMICS

53

Putting A(r, t) = R(r, t) exp(iΘ(r,t)), the model (200) is rewritten in terms of the amplitude R(r, t) and the phase Θ(r, t) as (204)

(205) It is readily seen that the natural frequency ω 0 = gI is reduced due to the existence of the nonlinear term hIR2 . This phenomenon is called “non-linear pulling” of the natural frequency. One should also note here that: (i) the amplitude and the phase motion cannot be decoupled due to the diffusion term; (ii) their motions have parametric influence on each other. Stochastic Navier-Stokes equation for phase When the amplitude takes an almost constant value, viz., R = R 0(t), the above coupled equation reduces to the equation of phase Θ(r, t) as (206) where equation:

. Putting v(r, t) =

t) , one obtains the stochastic Navier-Stokes (207)

If the fluctuating force FΘ(r, t) does not exist, the equation (207) can be linearized with the use of the Hopf-Cole transformation, (208) so that we have the diffusion equation. However, the existence of non-zero FΘ (r, t) leads (206) to the parametrically driven stochastic diffusion equation: (209) 3. Linear stability of uniform solution One should note here that the model (200) has the special spatially uniform solution for the case of positive gR(> 0): (210) Now let us take the model (200) and consider the stability of the uniform spatial limit cycle oscillation (210). Let us put (211) and derive linearized equations for ρ (r, t) and υ/ (r, t): (212) and (213) Putting (214)

54

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we have the characteristic equation : (215) where Note here that C1 > 0 always hold. Therefore, the Routh-Hurwitz stability criteria teaches us that when C2 > 0 , the system is stable. On the other hand, when C2 < 0, the system becomes unstable. Actually, the solution of eq. (215) is (216) There exists a parameter region where the uniform oscillation (210) becomes unstable when C2 < 0, viz. (217) Since DR > 0 and hR > 0, the uniform limit cycle oscillation becomes unstable ( gR > 0) when (218) A new spatial non-uniform oscillation appears, where the wave number k 2 region of unstable mode is such that (219) Since the necessary conditions for taking the limit cycle oscillation are DR > 0 and hR > 0, either hI or DI should be negative, not both of them, for the appearance of the spatially inhomogeneous state. Table V Condition of stability of phase, frequency shift

C. Discussion (i) non-stationarity, noise-induced limit cycle Traditional linear stochastic theory in science and engineering is based on weak stationarity: the correlation function C(t, t') can be expressed by time difference, viz. C(t, t') = C(t - t'). This property guarantees the availability of the Fourier-Laplace transform. However, as is suggested by stochastic models with multiplicative noise due to Akcasu (1961) and Konno (1985, 1990), their associated correlation functions can not be described by time differences. Consider, for example, the model in eq.(153) due to Akcasu (1961). Under the conditions that (a) the rotational wave approximation and (b) ω 0 >> κ0 are valid, it reduces to (220) Here we assume an exponential correlation (colored noise) with the Gaussian nature for the parametric noise κ1(t). By virtue of the causality between the random force f(t) and the state variable A, 〈 f(t)A*(0)〉 = 0 and Bloch's theorem, 〈 exp(x)〉 = exp(〈 x 2〉 /2) for a Gaussian random variable x, we obtain (Konno and Kanemoto, 1998) (221)

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It is clear that C(t, t') ¹ C(t – t'), viz. the condition of the weak stationarity is broken. Also, random switching (annihilation and creation) between stable and unstable state takes place when the strength of the parametric noise κ1(t) becomes large, even if the system is in the stable state in the sense of mean field theory. This phenomenon is a noise-induced limit cycle oscillation. (ii) global-local interference The global-local concepts (Behringer et al., 1977) have been proposed to interpret the noise characteristics in BWRs. The importance of effects of spatial higher harmonics on the PSD has been exhibited by Analytis (1978), Konno and Saito (1982a,1982b) and others within the framework of the two-energy group diffusion equations. The concept states that infinite summation of spatial higher harmonic modes are relevant even at the steady state. But one should note that the exact decomposition into the global and local components is feasible for a system without feedback loops; the rigorous decomposition under the existence of feedback loops is not possible (Konno and Saito, 1982b). When one spatial mode becomes unstable, the remaining stable modes are governed by the unstable one due to the slaving principle. So it seems that a description with a few degrees of freedom works well. Though the reduction of degrees of freedom in crossing from stable to unstable state near the Hopf bifurcation is observed (Takeuchi and Miyamoto, 1994), the degrees of freedom are not small. Theoretically, the local flow noise associated with the DWO seems random locally and behaves coherent globally. The same is true for the local and global neutron fluctuations. We see that the global and the local component can not be decoupled due to strong feedback, global-local interference. (iii) weak turbulence We have demonstrated with the use of the model (200) that at least near the onset of Hopf bifurcation, there exist a region wherein fluctuations become important and infinitely many degrees of freedom are vital, viz. the description with a few degrees of freedom is not appropriate. In other words, even when the amplitude of oscillation is almost constant, the spatio-temporal variation of phase associated with it becomes important. This is a sort of weak turbulence. (iv) preservation of Gaussianity, critical spatial dimensionarity Finally, we must comment on the preservation of the Gaussianity for fluctuations and the central limit theorem (CLT). The CLT states that even if the local fluctuations have nonGaussian nature, the averaged fluctuation tends to have a Gaussian nature. As far as neutron fluctuations is concerned, the observed neutron fluctuation via LPRMs tends to have a Gaussian nature even if the local neutron fluctuation is chaotic and/or turbulent driven by the spatially extended chaotic elements with a non-Gaussian pdf. With the use of the coupled map lattice approach, Konno, Kozma and Kitamura (1996) demonstrate that the CLT are relevant and suggest that there exists a critical spatial dimensionarity which separates the Gaussian and the non-Gaussian pdf. 6. CONCLUDING REMARKS We have reviewed nonlinear stochastic dynamics and instability theory as applied to nuclear reactors. Control, temperature and voidage instabilities are discussed within the framework of “nonlinear stochastic center manifold”. Xenon instability is not described in this paper (see Hitchcock, 1960; Hetrick, 1989). The key concept to understand power reactor noise is the multiplicative (or parametric) couplings among global and local state variables. These global-local couplings are the origin of complex fluctuations in nuclear reactors leading to the phenomenon of noise rise and of

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noise suppression which are related to current scientific terms: (1) amplitude modulations; (2) non-stationarity; (3) switching between stable and unstable state; (4) noise-induced spatio-temporal order; (5) noise-induced limit cycles; (6) stochastic resonance; (7) weak turbulence; (8) preservation of Gaussianity. Nuclear reactors are huge and complex engineering systems. There are many disciplinary problems which require profound academic insights and technical developments both theoretically and experimentally. The efforts to solve these problems might feedback favorably to other complex science and engineering problems. Acknowledgements I would like to thank Drs J. D. Lewins and M. Becker for their invaluable assistance in writing this paper. I would also like to thank Dr K. Kishida for his continual advices and friendship. 7. REFERENCES Ablowitz M. J. and Segur H. (1981) Soliton and Inverse Scattering Transform, SIAM. Akcasu A. Z. (1961) Nucl. Sci. Eng., 10, 337. Akcasu A. Z. and Shotkin L. M. (1967) Nucl. Sci. Eng., 28, 72. Ames W. F. (1965) Non-linear Partial Differential Equations in Engineering, Academic Press, NY. Analytis G. Th. (1978) Ann. Nucl. Energy, 5, 597. Araya F., Yoshida K., Hirano M. and Yabushita Y. (1991) Nucl. Technol., 93, 82. Arnold L. (1974) Stochastic Differential Equations: Theory and Applications, Wiley, NY. Ash M. (1970) Nuclear Reactor Kinetics, Second ed., McGraw-Hill, NY. Behringer K., Kosaly G. and Kostic Lj. (1977) Nucl. Sci. Eng., 63, 306. Bergdahl B. G., et al. (1989) Ann. Nucl. Energy, 16, 509. Blakeman E.D. and March-Leuba J. (1989) 7th Power Plant Dynamics, Control and Testing Symp., Knoxville, Tennessee, USA. Bogoliubov N. N. (1946) J. of Physics, (U.S.S.R.) 10, 265. Boure, J. A., Bergles, A. E. and Tong, L. S. (1973) Nucl. Eng. Des., 25, 165-192. Chow S.-N. and Hale J. K. (1982) Methods of Bifurcation Theory, Springer, Berlin. Debosscher A. F. and Durte W. L. (1979a) Nucl. Sci. Eng., 69, 347. Debosscher A. F. (1979b) Nucl. Sci. Eng., 69, 354. Devoot J. and Smets H. B. (1967) Nucl. Sci. Eng., 28, 226. Dorning J. J., et al. (1988) Nucl. Sci. Eng., 100, 393. Durte W. L. and Debosscher A. F. (1977) Nucl. Sci. Eng., 62, 355. Ehrenfest P. und T., (1911) Encyklopadie der mathematischen, Wissenschaften, Vol. IV, Pt. 32, Leipzig-Berlin. Fox R. F. (1978) Gaussian Stochastic Process in Physics, Phys. Rep., 48, 179. Grabert H. (1982) Projection Opemtor Techniques in Nonequilibrium Statistical Mechanics, Springer, Berlin. Gukenheimer J. and Holmes P. (1983) Nonlinear Oscillations, Dynamical Systems ond Bifurcations of Vector Fields, Springer, Berlin. Haken H. (1970) Laser Theory, Springer, Berlin. Haken H. (1980) Synergetics, 2nd Ed., Springer, Berlin. Haken H. (1983) Advanced Syergetics, Springer, Berlii. Hashimoto K. (1993) Ann. Nucl. Energy, 20, 789. Hayashi K. et al. (1984) JAERI-M 84-056; JAERI-M 84-137 (in Japanese). Hirota R. (1992) Mathematical Structure of Soliton, Iwanami, Tokyo (in Japanese) . Hitchcock A. (1960) Nuclear Reactor Stability, Temple Press, London. Hetrick D. L. (1989) Nonlinear Xenon Oscillations in Noise and Nonlinear Phenomena in Nuclear

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MODELING BACTERIAL TRANSPORT AND ACCUMULATION PROCESSES IN SATURATED POROUS MEDIA: A REVIEW

T.P. Clement1, B.M. Peyton2,T.R. Ginn3, and R.S. Skeen2 1 2 3

Battelle Pacific Northwest National Laboratory, Richland, WA 99352. Department of Chemical Engineering, Washington State University, WA 99164. Department of Civil Engineering, University of California, Davis, CA 95616.

1. INTRODUCTION During the past few decades, leaking underground fuel storage tanks have created soil and groundwater contamination problem throughout the United States. In addition, defense nuclear production by the U.S. Department of Energy (DOE), its predecessor agencies, and its contractors has generated large volumes of hazardous and radioactive wastes. Discharge of these wastes to soils has resulted in extensive contamination around DOE sites. Harnessing the potential of in situ microorganisms to degrade these chemical contaminants in the subsurface has been widely studied in the past two decades. Soil microbiologists have been investigating the role of shallow, root-zone microbes in recycling nutrients since the early twentieth century. However, only during the early 1970s did it become evident that subsurface microbes have a potential to degrade contaminants in much deeper subsurface zones. Developments in subsurface aseptic sediment sampling methods played a major role in providing conclusive evidences for the presence of microbes in these zones (Dunlap et al., 1977). Later experimental studies conducted under controlled laboratory conditions have also showed that these organisms can mineralize many man-made chemicals (Vogel and Grbic-Galic, 1986). In the early 1980s, researchers started to develop methods to harness these natural degradation potential of microbes to cleanup subsurface contaminants. Currently, two types of bioremediation approaches are being considered for field-scale cleanup: intrinsic bioremediation and active bioremediation. The intrinsic remedial approach is a plume management strategy that relies on the inherent degradative capacity of native subsurface microbes (Chapelle et al., 1994; Semprini et al., 1995; Wiedemeier et al., 1995; Rice et al., 1995). In contrast, active bioremediation is an accelerated clean-up strategy that uses various nutrient-delivery strategies to enhance growth and degradative capacity of native microbes (Semprini et al., 1991; Truex, 1995). As a part of the active bioremediation strategy, a few researchers have also explored the possibility of introducing non-indigenous cultured microbial strains into the subsurface to remove specific Contaminants (Mayotte et al., 1996; Duba et al., 1996).

Advances in Nuclear Science and Technology, Volume 26, edited by Lewins and Becker. Kluwer Academic / Plenum Publishers, New York 1999.

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On reviewing the history of bioremediation processes, it is evident that the application of the technology has run ahead of the basic science required to resolve the fundamental issues. Therefore, although field-scale applications have been used at several sites, a design basis for implementing bioremediation technology is yet to be formulated. Development of mathematical descriptions for predicting spatial and temporal changes in microbes, nutrients, and contaminants in the subsurface is a first major step in rationally understanding this technology. Advection, dispersion, and reaction are the three basic processes that control transport in subsurface porous media. Use of deterministic, single species, advection-dispersion models for analyzing simple, first-order reactions in porous media has been well documented in the literature. These models are relatively simple and, in some special cases, are amenable to analytical solutions (Bear, 1979). Several public-domain computer codes have been developed for multidimensional modeling of single species reactive transport in natural porous media (MOC by Konikow and Bredehoeft, 1978; SUTRA by Voss, 1984; and MT3D by Zheng, 1990). However, none of these models can solve multi-species, reactive transport equations required for modeling bioremediation. The general macroscopic equation describing the fate of aqueous- and solid-phase (attached or sorbed) species, respectively, in saturated porous media are (1)

(2)

where ϕ is the porosity, c is the aqueous-phase macroscopic averaged concentration of the transported species [ML-3], Dh is the hydrodynamic dispersion tensor, q is the Darcy flux [LT-1], Rreact is the biochemical reaction rate that describes the mass of the species removed or produced per unit volume per unit time [ML-3T-1], Ratt and Rdet, respectively, are attachment or adsorption and detachment or desorption rates that describe inter-phase exchange of the transported species [ML-3T-1], and all three reaction rates can be functions of transported and/or attached species concentrations; m is the total number of aqueous-phase species, n is the total number of solidphase species, and in equation (2) the tilde symbol is used for representing solid-phase reactions and concentrations. The Darcy flux vector, q , which describes fluid motion in porous media, is computed by solving the saturated groundwater flow equations: (3)

(4) where h is hydraulic head [L], Ss is the specific storage coefficient [L-1], w is the sink/source term, and K is hydraulic conductivity tensor [LT-1]. The hydraulic conductivity values of biologically-active porous media can have both spatial and temporal changes depending on the solid-phase biomass levels. Hence, flow equations (3) and (4) are coupled to the transport equations (1) and (2) through the hydraulic conductivity tensor.

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Detailed examples illustrating the use of transport equations of the form (1) and (2) to model coupled solute and microbial transport problems in saturated porous media are described in Taylor and Jaffe (1990c), Zysset et al. (1994), and Clement et al. (1996a, 1997b, 1997c). However, the scope of this work is restricted to a review of modeling issues involved in describing microbial transport and growth. Hence, the paper reviews the application of equations (1) and (2) only when the transported species of interest are microorganisms.

2. MODELING BIOMASS IN POROUS MEDIA Figure 1 shows a schematic representation of microorganisms in porous media and the relevant processes that govern their transport. As shown in the figure, in saturated porous media microorganisms exist and grow, as suspended cells or as attached cells. Field-scale experiments indicate that, under natural conditions, most bacteria reside on the solid phase (Harvey, 1984). Relative changes in the biomass concentration at solid and aqueous phases are functions of kinetic processes that dictate microbial growth, attachment, and detachment. The rates of these processes are functions of the concentration of biomass present at each phase. Therefore, precise estimates of biomass at both aqueous and attached phases are essential for modeling the overall fate and transport of microbes in porous media. Usually, for modeling purposes, aqueous-phase cells are assumed to transport in a manner similar to dissolved solutes; hence, they are simply quantified as liquid-phase concentrations (Taylor and Jaffe, 1990c; Zysset et al., 1994). However, for solid-phase biomass, three different conceptual modeling approaches have been used in the literature to represent their accumulation patterns and to quantify their spatially variable concentrations (Baveye and Valocchi, 1989). These approaches describe solid-phase biomass as either biofilms or micro-colonies or macroscopic averaged biomass without any specific structure. In the following sections all three conceptual approaches are reviewed. 2.1 Biofilm approach Biofilm-based models were originally used for subsurface applications because environmental researchers were experienced with this formulation in biological reactor systems. This approach assumes a continuous and uniform biomass coating on the exposed surface of each soil particle (Fig. 2). The amount of biomass in the system is typically quantified in terms of biofilm thickness, volumetric density, and available surface area. Biofilms are usually classified as fully-penetrated or diffusion-limited. A fully-penetrated biofilm has non-rate limiting concentrations of growth substrates throughout its entire thickness. In contrast, in a diffusion-limited biofilm, reaction rates are limited by diffusion of one or more of substrates through the biofilm structure. Mass-transport limitations across a thick, diffusion-limited films are modeled using an effectiveness factor (Rittmann, 1993; Zysset et al., 1994). Several researchers have used the biofilm approach to model microbially mediated transport of an electron donor and mixed-electron acceptors in one-dimensional soil columns (Bouwer and Cobb, 1986; Taylor and Jaffe, 1990a; Taylor and Jaffe, 1990c; Odencrantz et al., 1990). Rittmann (1993) reviewed the significance of several biofilm concepts for modeling porous media transport applications. More recently, Dykaar and Kitanidis (1996) used a biofilm model within a structured porous medium to study macro-scale transport. They evaluated the macro-scale transport by up-scaling the basic equations used to predict diffusionlimited mass transfer and biochemical kinetics at the pore scale.

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Figure 1. Schematic of Microbial Transport Processes in Porous Media

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Figure 2. Schematic Diagram Illustrating the Microcolony and Biofilm approaches used for Modeling Biomass in Saturated Porous Media

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2.2 Microcolony approach In the microcolony approach, microorganisms are assumed to grow as scattered, discrete colonies (Fig. 2). Several laboratory and field experimental observations that support this concept have shown that attached solid-phase bacteria may not always form continuous biofilms; rather they tend to aggregate in microcolonies (Campbell, 1977; Harvey et al., 1984). In a field study conducted at an unconfined aquifer at Cape Cod, Massachusetts, Harvey et al. (1984) observed microcolonies containing 10 to 100 bacteria on soil particle surfaces. Vandevivere and Baveye (1992 a, b, and c) conducted soil column studies and published scanning electron micrographs to show the sparse and heterogeneous bacterial colonization inside pore spaces. Molz et al. (1986) developed a mathematical framework to model bioreactive transport using the microcolony approach. Transport equations were derived to describe microbial growth dynamics coupled to nutrient and oxygen transport in one-dimensional porous media. Widdowson et al. (1988) extended this model to simulate oxygen- and nitrate-based respiration linked to substrate and nutrient availability in porous media. Chen et al. (1992) used a similar approach to model the transport and biodegradation of benzene and toluene in soil columns and compared model results against experimental data. Although the microcolony approach seems to provide a reasonable approximation for describing subsurface microbes, the model predictions of microbial pattern are difficult to verify because the complex growth patterns of microcolonies are tedious to measure on a routine basis. 2.3 Macroscopic approach In the macroscopic approach, bacterial cells are assumed to be uniformly distributed through the Representative Elementary Volume (REV) of porous media. Microscopic details regarding the growth patterns are not necessary because the model makes no assumption about the spatial structure of biomass. The biomass levels are quantified in terms of macroscopic averaged values that describe the mass (wet or dry) of biomass per unit volume or mass of porous media. Many researchers have successfully used this approach for a variety of applications (Corapcioglu and Haridas, 1984 and 1985; Borden and Bedient, 1986; Kindred and Celia, 1989). In formulating models to simulate column experiments, MacQuarrie et al. (1990) used a similar approach and hypothesized biomass as a volumeless dissolved species undergoing transport; inter-phase biomass transport were represented by equilibrium partitioning. In intermediate-scale experiments of biodegradation under flow parallel to geologic layering, Wood et al. (1994) used an unstructured approach in which biomass was assumed to be fully attached to the solid phase. Kinzelbach et al. (1991) presented a numerical model using a variation of the macroscopic approach in which biomass is unstructured but associated with adiffusion limitation to describe natural and enhanceddenitrification processes in groundwater environment. Zysset et al. (1994) present averaged biofilm-based transport equations that are similar to macroscopic-model based transport equations. Clement et al. (1996b) demonstrate how Zysset’s (1994) model can also be written purely from the macroscopic viewpoint. Clement et al. (1996a) used a macroscopic approach to analyze radial flow and reactive transport near nutrient delivery wells used for in situ biostimulation. 2.4 Comparison and critical analysis of biomass modeling approaches The conceptual differences and similarities between microcolony and biofilm models at different scales of observations are illustrated in Fig. 2. As shown in the figure, both biofilm and microcolony approaches are based on a fixed pore-scale growth pattern for biomass

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accumulation. However, these pore-scale details need to be finally averaged over an REV to describe the overall Darcy-scale, macroscopic transport. Unfortunately, rigorous mathematical methods for performing this averaging process are not available for any type of pore-scale biological model. Baveye and Valocchi (1989) were the first to point out the conceptual disparity in the ambivalent use of pore-scale biomass models along with macro-scale transport equations to describe biorective transport in porous media. Recently, Dykaar and Kitanidis (1996) attempted to use arigorous scaling effort to derive macroscopic equations from a single, pore-scale biofilm model using the periodic-media arguments. However, this effort assumes continuous uniform biofilm of known density. It is also subject to assumptions that specify an idealized pore geometry which would limit extensions of this approach to realistic porous media. Baveye and Valocchi (1989) evaluated the validity of various assumptions used in the three biomass modeling approaches and identified the similarities among them. Presently available experimental evidences are insufficient to determine the significance of the differences between biofilm and microcolony models. For example, the microscopic features of biomass, such as biofilm density, and microcolony density are difficult to measure with currently available experimental techniques. In addition, the values of biofilm density depends on several factors, such as type of bacteria, vicinity of bacteria to the aqueous phase, history of substrate supply, and amount of shear force on the biofilm. Peyton and Characklis (1995) report that published biofilm density values range from 10 to 130 kg /m3 (dry mass/ wet volume). Such uncertainties in biofilm density values will severely affect the results of biofilmbased models. Hence, lack of sound experimental data limits verification of most pore-scale modeling efforts. In biologically active natural soils, microbes may not grow exclusively as either microcolonies or biofilms. Microbes that initially start growing in discrete colonies can expand, if required nutrients are available to form continuous biofilms. Hence, use of a fixed type of pore-scale model may not be adequate for modeling natural systems. Baveye et al. (1992) state that, although biofilm approaches may be appropriate in coarse materials (like 2-mm glass bead medium), the assumption of complete continuous coverage of the solid particles by bacteria and their associated extracellular polymer seems inappropriate in natural fine textured porous media. Rittmann (1993) noted that at low nutrient levels, typical of natural groundwater conditions, distinguishing continuous from discontinuous biofilms is not so important when the goal of modeling is to predict substrate removal. Hence, for practical large-scale contaminant transport applications, accurately modeling the microscopic details of biomass distribution may be a futile effort. However, the distinction between the biomass patterns may be critical when the modeling goal is to predict not only substrate or contaminant degradation, but also the spatial distribution of biomass and biologically induced loss of permeability (Rittmann, 1993). The current literature does not provide clear guidelines for using a specific conceptual model for a particular situation. Disagreements regarding the use of a particular model are apparent in the rather interesting exchanges between various groups of researchers with conflicting preferences (Widdowson, 1991; Baveye and Valocchi, 1991; Baveye et al., 1992; Jaffe and Taylor, 1992). Overall, to choose among the three available biomass modeling alternatives, the macroscopic approach seems to provide the most practical and consistent approximation for describing large-scale contaminant degradation and transport problems. The microcolony and biofilm approaches may be required only for more detailed calculations involving biomassinduced reductions of permeability or porosity.

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3. MODELING MICROBIAL TRANSPORT KINETICS Experimental studies with actively growing cells in porous media have verified that bacteria can exist and grow simultaneously on both solid and aqueous phases (Taylor and Jaffe, 1990a; Lundman, 1992; Jennings, 1994; Peyton et al., 1995). Several published biologicallyreactive transport models do not explicitly consider the presence of cells in the two distinct phases or the kinetic exchange of cells between the phases (Bouwer and Cobb, 1986; Molz et al., 1986; Widdowson et al., 1988; Kindred and Celia, 1989, Kinzelbach et al., 1991). Corapacioglu and Haridas (1984) were the first to perform theoretical investigations of the effects of kinetic exchange of biomass between aqueous and solid phases in porous media. Using model simulations combined with experimental data, Taylor and Jaffe (1990c) demonstrated the importance of considering biomass exchange kinetics. These studies have identified that biomass exchange in porous media is controlled by two distinct kinetically controlled mechanisms, namely, attachment and detachment. The attachment and detachment processes are represented in the transport equations (1) and (2) through appropriate kinetic expressions for the terms Ratt and Rdet, respectively. Recently, Reddy and Ford (1996) compared a similar kinetic modeling approach against an equilibrium modeling approach and concluded that, for analyzing realistic systems with active bacterial growth, kinetic descriptions for bacterial attachment and detachment rates are more appropriate. 3.1 Microbial attachment Microbial attachment refers to all processes by which aqueous-phase biomass is transferred to the solid phase. Biomass in porous media can directly attach to the exposed surface of soil particles or to an existing biomass layer. Biomass attachment can occur via sorption, straining, or filtration. Sorption of biomass may take place in two steps: 1) initial fast and reversible adsorption of cells to the solid surfaces; 2) permanent attachment, possibly by aid of extracelluar polymeric substances (Characklis et al., 1990). The rate of sorption is also a function of solid surface properties. Straining and physical filtration also remove microbes from solution by physical forces. Straining is the trapping of microbes in pore throats that are too small to allow passage; this process is exclusively a function of pore geometry. Sakthivadivel (1969) and Herzig et al. (1970) describe methods for predicting mass removal by straining; these methods are based on geometric relations between the effective diameter of colloids and the diameter and packing (coordination number) of grains forming the porous media. Researchers have found microbial straining to be an insignificant phenomenon in idealized packed beds (porous media made up of identical spherical grains) where the colloid diameter is less than 5% of the porous media grain diameter (Herzig et al., 1970; Corapcioglu and Haridas, 1984; McDowell-Boyer et al., 1986). This result has been invoked in analysis of subsurface bacterial transport (e.g., Matthess et al., 1988; Harvey and Garabedian, 1991). More general results on straining in granular media with distributed grain sizes are discussed in Sherard et al. (1984 a and b). Physical filtration is the removal of particle mass from solution by collision with and fixation to the porous media. Physical forces involved in this process are gravity, particleparticle and particle-solvent collisions (Brownian forces), electrostatic interaction potentials between the particle and the porous media, van der Waals attractive potentials, particle inertia, and pore-water hydrodynamic forces (Yao et al., 1971; McDowell-Boyer et al., 1986). Filtration due to gravity is termed “sedimentation”, and its rate depends on particle buoyancy (McDowell-Boyer et al., 1986). Since many bacteria are neutrally buoyant, sedimentation effects are often negligible. However, Gerba et al. (1975) report significant sedimentation for some bacteria; in such cases, the gravitational velocity expressed by Yao et al. (1971) can serve as a measure of sedimentation.

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The dominant mechanisms for removing bacteria involve filtration due to the remaining physical forces such as Brownian, electrostatic, van der Waals, and pore-water hydrodynamic forces. The effects resulting from these forces are modeled by up-scaling the pore-scale interaction potentials and hydrodynamic forces of a chemically inert spherical particle transported through a uniformly packed homogeneous bed of spherical grains (Herzig et al., 1970; Tien et al., 1979). The mass removal due to filtration is described in terms of pore-water velocity, viscosity, density, grain size, concentration, and porosity. The resulting "filtration theory" relations have been applied to microbial transport since Harvey et al. (1989) introduced the idea, and was later applied it to a field problem as reported in Harvey and Garabedian (1991). The effects of changes in solute ionic strength on physical filtration mass removal and the physical effects of changes in the direction of pore-water flow are generally ignored in applications of filtration theories. This is because all the filtration models represent irreversible deposition under conditions of uniform flow direction and fixed solution chemistry. Recent research has demonstrated that physical filtration is influenced by solute ionic strength, pH, and mineralogy (Scholl et al., 1990, Tan et al., 1994). Shonnard et al. (1994) found that increasing ionic strength of solution in coarse sands significantly increased adsorption of a trichloroethylene degrading bacterium. Scholl et al. (1990) found that iron hydroxide-coated sands filtered more bacteria than clean sand. Mills et al. (1994) observed similar results when they examined the combined effects of ionic strength variations in the aqueous phase and iron oxide presence in the solid phase. Primarily quartzitic materials have negatively charged surfaces, as do most bacteria; thus, bacterial immobilization can occur only when the hydrodynamic and attractive van der Waals forces overcome the repulsive electrostatic forces. The positive surface charges on iron hydroxide-coated sand grains reverse the electrostatic force from repulsive to attractive and increases the likelihood of microbial attachment. Ryan and Gschwend (1990) and McCaulou et al. (1994) investigated the mechanism of hydrophobic sorption by measuring the hydrophobicity of the colloid particles on porous media with a homogeneous distribution of organic matter. They observed that even small amounts of organic matter on porous media may enhance transfer of colloids from the aqueous phase to the solid phase. Numerous researchers have experimentally investigated attachment processes that influence the transport of non-growing bacterial cells in one-dimensional porous media columns. A large part of the literature is dedicated to the development and application of filtration theory-based representations of attachment processes examined in such laboratory column experiments. The modeling approaches used to predict these experimental observations are mostly phenomenological; an advection-dispersion equation is used with first-order kinetic or equilibrium terms to account for bacterial attachment and detachment processes. The attachment coefficients are either estimated from a filtration theory (e.g., Bales et al., 1991) or evaluated by curve fitting a phenomenological model to the experimental data. Estimation of effective first-order kinetic attachment coefficients using filtration theories has met with mixed success, partly because of the non-ideal and distributed characteristics of natural granular media and partly because of limitations of the theory (for example, original filtration theory-based attachment is unlimited and irreversible). Fontes et al. (1991) performed experiments to study the influence of various physiochemical factors on the transport of non-growing bacterial cells through one-dimensional, saturated soil columns. Homberger et al. (1992) analyzed the data of Fontes et al. (1991) and determined the values of attachment coefficients by fitting an advection-dispersion model and by using the filtration theory of Tien et al. (1979). The fitted coefficients varied within an order of magnitude of those predicted on the basis of the filtration theory. Other researchers have also usedfiltration theories to predict bacterial attachment in porous media (Harvey and Garabedian, 1991; Martin et al., 1992; Johnson et al., 1995). Logan et al. (1995) evaluated the use of various filtration theories for predicting colloid removal in one-dimensional porous media and

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clarified several aspects of its applications. Tan et al. (1994) and Lindqvist et al. (1994) augment the first-order kinetic attachment model with a nonlinear governing factor intended to represent the attachment-limiting effect of site-saturation. Saiers et al. (1994) also proposed almost an identical model for describing colloid transport. Many researchers have included attachment kinetics within their models used for predicting biologically-active transport in one-dimensional soil columns (Taylor and Jaffe, 1990a; Lundman, 1992; Jennings, 1994; Peyton et al., 1995; Clement et al., 1997a). In all these models, a constant value for attachment coefficient is estimated using a curve-fitting procedure or using a simple filtration model. As reviewed in Peyton and Characklis (1995), only a few biofilm studies have explored the variations of bacterial attachment coefficients under active growth conditions. We have found no studies that measure the rate at which suspended cells attach to preexisting porous media biofilms; however, Gunawan (1991) attempted to study this process in an annular biofilm reactor. While the past research has focussed on the attachment of non-growing cells onto clean porous media, future research efforts in this area need to focus on quantifying the attachment kinetics of growing cells on to actively growing microbial films. 3.2 Microbial detachment process Detachment refers to the loss of biomass from the solid phase and its re-entrainment into the aqueous phase. Detachment is mainly caused by fluid shear forces, but the rate is controlled by other physio-chemical and biological factors (Peyton and Characklis, 1993). Microbial detachment processes can be categorized as erosion, abrasion, and sloughing (Characklis et al., 1990). Erosion is continuous loss of small quantities of biomass. Abrasion is the loss of biomass due to repeated collision between the substratum particles; it occurs mainly in fluidized beds where biofilms are subjected to abrasion by moving sand particles. Sloughing is a sudden transient process in which massive detachment of biomass occurs instantaneously. Sloughing may be initiated by gradual loosening of biomass due to shear forces generated by the moving liquid; it can also be due to gradual accumulation of toxic substances, dead bacteria, and gaseous metabolic products at the biomass-solid interface. Some early models of bacterial transport, based on the intrinsically irreversible filtration theory, incorporated no detachment at all. Support for this approach arose when early column studies performed under non-growth conditions indicated that attachment was not generally reversible (Wollum and Cassel, 1978; Smith et al., 1985). Extensions of this concept to include additional equilibrium attachment processes were used by Matthess et al. (1988). However, observations of bacterial transport in natural media in soil columns and field experiments showed that the attachment processes are reversible not only at column-scale transport (Scholl et al., 1990), but also at larger time and space scales (Harvey and Garabedian, 1991). The evidential significance of detachment processes in experimental studies has led to increased reliance on a kinetic detachment term, as espoused by Corapcioglu and Haridas (1984). The resulting model for non-growth associated detachment is the "two-site” reactive transport model, which incorporated irreversible kinetic reactions for attachment at one type of reaction site (in accordance with filtration theory) and either equilibrium or kinetic reversible terms for attachment and detachment at a second type of reaction site (Harvey and Garabedian, 1991; Bales et al., 1991; Lindqvist and Bengtsson, 1991; and McCaulou et al., 1994). The recent reliance on kinetic detachment is not actually new; dual attachment-detachment mechanisms were modeled similarly in the particulate “clogging/ declogging” studies by Sakthivadivel and Irmay (1969) and Herzig et al. (1970). In comparison to attachment, predicting non-growth related detachment is a more complex process since it includes sloughing, an inherently uncertain process. Hence, detachment coefficients are usually evaluated by curve fitting phenomenological models to column data (Hornberger et al., 1992; Saiers and Homberger, 1994; Tan et al., 1994; Lindqvist et al., 1994; Johnson et al., 1995).

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Detachment rates in biofilm reactors have been observed to increase with increasing growth rates (Peyton and Characklis, 1993, Speitel and DiGiano, 1987). Only few studies have experimentally investigated detachment of microbes in porous media under growth conditions (Lundman, R.S., 1992; Jennings, 1994). Peyton et al. (1995) used an advection-dispersion model to analyze Lundman (1992) data and observed the detachment coefficient to increase with increases in substrate concentration and hence with growth rate. Clement et al. (1997a) used first-order detachment and attachment models coupled to advection-dispersion equations to analyze Jennings (1994) experimental data. The detachment coefficient values estimated for their actively-growing, denitrifying population were higher than the literature values of nongrowing bacterial detachment coefficients. Clement et al. (1997a) also re-evaluated some of Lundman’s (1992) data and observed that detachment values for aerobic microbes were higher than the values for denitrifying microbes. They therefore hypothesized that similar to reactorgrown biofilms, detachment in porous media may increase with microbial growth rate. However, because the amount of data was limited, they did not prescribe models for predicting the growth rate dependence. Since the overall understanding of microbial attachment and detachment processes in natural porous media is in its infancy, only a few researchers have attempted to study microbial transport at field sites. Harvey et al. (1989) and Harvey and Garabedian (1991) performed field scale, forced- and natural-gradient tests to study the transport of non-growing bacterial cells in a sandy aquifer. They used filtration models to describe bacterial attachment and observed several uncertainties in applying filtration theories to predict field-scale bacterial transport. So far, no one has attempted to describe field-scale bacterial transport processes under active growth conditions. This will be a formidable task because growing cells will reduce the size of pores causing additional pore-scale heterogeneities over existing, natural, field-scale heterogeneities. The resulting flow field will be more complex to predict because additional models may be required to predict the changes in soil properties due to microbial accumulation within pore spaces.

4. MODELING CHANGES IN SOIL PROPERTIES CAUSED BY MICROBIAL ACCUMULATION 4.1 Permeability and porosity changes In porous media, actively growing microbial cells clog pores thereby reducing soil porosity and permeability values. Reduced porosity will affect the contaminant transport equation (1) and reduced permeability will affect the groundwater flow equation (4), thus leading to a coupled nonlinear transport problem. Subsurface bacterial growth and its role in reducing well permeability values were first observed and reported in the petroleum literature (Carlson et al., 1961). Kalish et al. (1964) studied the permeability of sandstone cores to learn whether bacterial accumulation can adversely affect brine injectivity during secondary recovery or disposal operations. More recently, bacterial clogging has become a major concern in the bioremediation industry because microbial accumulation on well screens and in pore spaces reduce the permeability of nutrient injection wells. Typical bioremediation systems commonly use injection-extraction well pairs to deliver and recirculate growth stimulating nutrients (Semprini et al., 1991, Truex, 1995). When the required nutrients are injected continuously into the subsurface, rapid bacterial growth can plug pores near the well screen and thereby increase the well pressure needed to inject the flow (Taylor and Jaffe, 1991). Pulsed nutrient delivery techniques have been proposed to alleviate these injection-well clogging problems (Taylor and Jaffe, 1991; Semprini et al., 1991; Shouche et al., 1993; Peyton, 1996). When two-well recirculation systems are used, design of nutrient pulsing strategies requires a careful

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a priori analysis because insufficient opportunity for pulse mixing can cause nutrients to break through and mix at or near the extraction well; this would lead to clogging near the extraction well and could pose severe operational problems. Since extraction-well flow rates are always limited by available drawdown, partially clogged extraction wells could run dry at low extraction rates and could render remediation activities impossible. Even though bacterial clogging has severe detrimental effects, a few researchers have devised ingenious methods to use it for beneficial purposes. Lappin-Scott et al. (1988) proposed an innovative approach for enhancing oil recovery by injecting starved cells into sandstone cores and resuscitating them with appropriate nutrients to selectively plug highpermeability rock strata that have already been drained of oil. Environmental restoration engineers are currently exploring the feasibility of novel technologies that stimulate in situ bacterial growth to create clogged barriers (called as “biobarriers”) for controlling plume migration. A comprehensive mathematical model that can predict the effects of bacterial growth on porous media properties is required to assess the potential ill- or useful-effects associated with pore clogging. However, developing such a model is a formidable task because it involves describing several microscopic details of microbial growth relative to the pore structure. Moreover, aside from bacteria cells, plugging can also be caused by fungi or protozoa (Okubo and Matsumoto, 1983), production of extracellular polymers (Vandevivere and Baveye, 1992 a and b), gas bubbles released during biological processes (Sanchez de Lozada et al., 1994), precipitation of metals by sulfate-reducing bacteria (van Beek and van der Kooij, 1982), precipitation of metal hydroxides by iron-reducing bacteria (van Beek, 1984), and accumulation of fines and colloidal material in the pore spaces (Khilar and Fogler, 1983). Despite these complexities, a few researchers have attempted to use the biofilm approach to develop analytical expressions for modeling the clogging process. Okubo and Matsumoto (1979) conceptualized porous media as a bundle of capillary tubes and used the Hagen-Poiseuille equation to model the changes in the capillary diameter caused by biofilm accumulation. Taylor et al. (1990) modified several pore structure and permeability models to predict how biofilms affect the properties of porous media. Cunningham et al. (1991) modified the Ergun equation to describe the effects of biofilm on porous media hydrodynamics. Vandevivere and Baveye (1992 a,b,c) investigated the reduction of saturated hydraulic conductivity caused by the growth of aerobic bacteria and observed that presence of a biofilm may not be necessary for pore clogging to occur. These authors report that reductions in soil permeability can be caused by the presence of large aggregates (microcolonies) of cells that form plugs in pore throats. To describe these observations, Vandevivere et al. (1995) proposed a simple clogging model assuming formation of bacterial plugs between uniform, cylindrical capillaries. Vandevivere et al. (1995) also tested the performance of the Taylor et al. (1990), Okubo and Matsumoto (1979), and their own simple bacterial-plug models by comparing the model predictions against published experimental data. They concluded that the currently available clogging models do not satisfactorily predict saturated hydraulic conductivity reductions that were experimentally observed in fine-textured materials. They also suggested that removing biofilm assumptions and accounting for bacterial plugs in the interstices of clogged porous media allows prediction of more severe clogging effects. In this respect, models based on a microcolony approach have the greatest potential to influence permeability with a least amount of biomass. Based on the macroscopic approach, Clement et al. (1996b) derived analytical equations to model changes in porous media properties caused by biomass accumulation. They demonstrated that, for uniform porous media, the biofilm-based models of Taylor et al. (1990) are identical to their macroscopic models. Presently available analytical models may only yield approximate estimates for biomassaffected porous media properties. However, this information can be useful for prioritizing pore

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clogging processes and their overall implications on flow and transport. Most clogging-related modeling or experimental studies reported in the literature are restricted to one-dimensional systems where the entire porous media is assumed to be sampled by the flow field. But, in realistic, multi-dimensional problems, exclusion of flow from clogged zones is expected be a major effect; significance of this issue, especially in multi-dimensional systems, has not yet been addressed in the literature. 4.2 Size exclusion effects Unlike molecular-scale solutes which more thoroughly sample the entire pore size and velocity distribution, bacterial cells, by virtue of their size, are excluded from micropores and the regions of low velocity that occur closer to the solid surface. Consequently, within porous media, effective transport velocities encountered by the cells are higher, and thus cells move faster than tracer ions. Results from hydrodynamic chromatography indicate that, in idealized porous media, the occurrence of exclusion requires the colloid diameter to be less than 1% of the media grain diameter (DeMarsily, 1986). Thus, exclusion cannot be ruled out in bacterial transport experiments in porous media with grain sizes up to two orders of magnitude larger than bacterial cells. Size exclusion effects cause bacterial cells to breakthrough well in advance of bromide tracer in transport experiments. Harvey et al. (1989) observed this behavior in their field-scale transport studies. In more controlled, laboratory-scale, soil-column experiments Homberger et al. (1992) and Mayotte et al. (1996) observed break through of relatively smaller quantities of bacterial cells well before the tracer breakthrough. Homberger et al. (1992) hypothesized that size exclusion effects are the probable cause for the observed results. The exclusion effects can be amplified in the presence of ionic forces (this is anionexclusion, as opposed to size-exclusion). When the electrostatic forces between the media and colloid are repulsive, as is the case with negatively charged microbes in negatively charged quartzitic media, the force field tends to channel the microbes closer to the pore throat centerline and away from the walls and the slower fluid velocities (DeMarsily, 1986). The effect is more pronounced at larger observation scales (Enfield and Bengtsson, 1988; Harvey et al., 1989; Shonnard et al. 1994). Modeling bacterial cell exclusion processes for actively growing microbes under field-scale heterogeneous conditions poses a considerable amount of challenge to the modeling community. 5. MODELING MICROBIAL GROWTH KINETICS In biologically-active subsurface environments, modeling microbial growth rate is critical in predicting the rates of reaction terms shown in the transport equations (1) and (2). Growth of microorganisms under natural conditions depends on environmental variables, such as pH, temperature, presence of inhibitory or toxic materials, and the types of organisms that are present. In addition, growth rate can be also a function of other transported nutrients and substrate concentrations. Advanced structured models may be used to simulate the effects of all these variables on cell-level metabolic processes (Williams, 1967). However, to keep the mathematical descriptions simple, semi-empirical unstructured models are often used. The Monod model is the most commonly used unstructured models for environmental applications (Monod, 1949). Different forms of Monod-based kinetic models are available for predicting various types of subsurface microbial metabolisms (Molz et al., 1986; Widdowson et al., 1988; Kindred and Celia, 1989; Kinzelbach et al., 1991; Wood et al., 1994; Hooker et al., 1994; Skeen et al., 1995, Clement et al., 1996 a and c). Most of these models assume that microbial kinetics are independent of the phase where the bacterial cells are located; i.e. growth and other

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kinetic constants for attached and suspended bacteria are assumed to be identical. This may not be an adequate assumption in all cases because cells attached to the solid phase may behave differently from the cells suspended in the aqueous phase. Van Loosdrecht et al. (1990), reviewed the influence of solid surfaces on microbial activity and concluded that solid surfaces may positively or negatively (or not at all) affect microbial substrate utilization rates. The review suggests that the available evidences were insufficient to conclusively support any particular hypothesis. Harms and Zehnder (1994) studied the degradation of contaminant by attached and suspended bacteria. They observed maximum specific activities of cells in the two systems to be the same. However, the half-maximum uptake rate constants were different. They described this difference using a model that was based on the assumption that the intrinsic half-maximum uptake rate-associated concentration was unchanged and that the observed deviations resulted from stereometry and hydrodynamics around the cells. Doong and Wu (1995) observed that the addition of inert glass beads into their microcosm bottles, to facilitate attached growth, substantially enhanced microbial accumulation and carbon tetrachloride destruction rates. Harms (1996) investigated the influence of substrate diffusion across various matrices on bacterial growth in a heterogeneous system. The mathematical model used in this study combined Monod growth kinetics extended by a term for culture maintenance and substrate diffusion to explain the observed bacterial growth curves on distant naphthalene diffusing through water, air, and water-saturated and non-saturated porous media. Information concerning the fate and bioavailability of sorbed contaminant to bacterial cells is also necessary for choosing an appropriate model structure for predicting bacterial growth and contaminant degradation. Gorden and Millero (1985) compared biodegradation of organic nutrients by attached and free-living systems and concluded that the issue of whether bacteria can metabolize adsorbed organics directly or if they can only utilize the desorbed organics is not clear. However, most experimental efforts reported in the literature seem to indicate that bacteria can only utilize contaminant in the dissolved state and hence desorption rate of contaminants usually limit the rate of its biodegradation (Wodzinski and Coyle, 1974; Stucki and Alexander, 1987; Robinson et al., 1990). Further modeling studies supported by experimental evidences are needed before general conclusions can be made about the bioavailability of sorbed contaminants, and the differences between liquid and solid phase growth/contaminant utilization rates. A few studies, particularly field-scale modeling studies, have assumed subsurface microbial reactions to be instantaneous (Rifai et al., 1988; Chiang et al., 1989). These studies assumed oxygen to react instantaneously with a benzene (or other hydrocarbon contaminant) plume, leading to complete mineralization of a stoichiometric equivalent of the hydrocarbon. Subsurface aerobes were assumed to mediate this instantaneous reaction; however, their presence was not explicitly considered. The most commonly used numerical model, BIOPLUME II (Rifai et al., 1987) is based on this instantaneous reaction concept. Another approximate approach using simple, plume-scale, first-order degradation rates to assess intrinsic biodegradation of plumes has been detailed in Wiedemeier et al. (1995). Chapelle et al. (1996) and Wiedemeier et al. (1996) proposed field and laboratory methods to measure these apparent first-order field-scale decay rates for BTEX plumes. Although widely used in field-scale applications, such lumped first-order reaction models or instantaneous reaction models may not be appropriate for modeling several slowly degrading environmental contaminants. 6. SUMMARY AND CONCLUSIONS A comprehensive mathematical model for describing microbial transport and accumulation processes in saturated porous media should include an appropriate conceptual model for describing porous media biomass and models for microbial attachment, detachment,

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and biomass growth kinetics. In addition to these fundamental microbiological process descriptions, auxiliary models may also be required to describe secondary effects such as poreclogging, diffusion limitation, and/or size exclusion effects. Although several conceptual models are available to describe detailed growth patterns of porous media biomass, accurate modeling of microscopic biomass patterns may not be necessary for practical large-scale contaminant transport applications. Modeling biomass structure may be critical when the goal is not only to predict substrate or contaminant degradation, but also to predict spatial distribution of biomass and biologically induced permeability losses. Hence the choice of a conceptual model to represent porous media biomass will depend on the modeling objective. Among the three available biomass modeling approaches, namely, biofilm, microcolony and macroscopic approaches, the macroscopic approach seems toprovide the most practical andconsistent approximation for describing largescale contaminant transport problems, at least at the current level of fundamental knowledge. However, the microcolony or biofilm model may be required for more detailed calculations involving biomass induced permeability and porosity reductions. Lumped, first-order kinetic expressions are the only available models for predicting microbial attachment and detachment processes. Estimating the value of first-order attachment coefficients based on filtration theories has met with mixed success, partly because of the nonideal and distributed characteristics of natural granular media and partly because of limitations of filtration theories. More experimental evidence is needed to refine these models to account for the growth dependent attachment and detachment. Unstructured Monod-type models to describe subsurface microbial growth rates seem to provide an adequate approximation for predicting biomass growth in subsurface environments under simplifying assumptions on biomass structure. The influence of porous media surfaces on microbial ecology, growth rate, and contaminant degradation rate is not clear. Further modeling studies supported by experimental evidences are needed before general conclusions can be made about the bioavailability of sorbed contaminants to attached or freeliving microbes, and the differences between these microbes growth/contaminant utilization rates. In summary, even though modeling microbial transport in subsurface systems has several inherent uncertainties, it is important to realize that modeling is the only rational approach to develop a fundamental understanding of subsurface transport processes. Even simple, approximate models can be useful for examining the sensitivity of different process formulations and for prioritizing processes relative to their implications on overall flow and transport. Progress in understanding the basic processes of subsurface microbial transport may be best reached through iteration between modeling analysis of existing data and further experimental investigations of process uncertainties that are prioritized through modeling studies. Hence, field or laboratory experimental investigations of microbial transport can be more productive if they are substantiated with a modeling framework. ACKNOWLEDGMENTS We would like to thank Dr. Jeffery Lewins for this help and suggestions in the preparation of this manuscript. This work was supported by the U.S. Department of Energy Office of Science and Technology. The Pacific Northwest National Laboratory is operated for the U.S. Department of Energy by Battelle Memorial Institute.

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A REVIEW OF CRACK ARREST CONCEPTS FOR THE ASSESSMENT OF PRESSURE VESSEL INTEGRITY C S Wiesner and B Hayes TWI (The Welding Institute) Abington Hall, Abington, Cambridge, UK [email protected] : [email protected] 1. INTRODUCTION The failure prevention concept based on the avoidance of fracture initiation is now applied world-wide during design, operation and evaluation of structures. The crack arrest philosophy provides a complementary and, in some circumstances, alternative approach which is particularly useful where there exists the possibility that a specific combination of circumstances might promote initiation of a flaw, but where such initiation is not threatening to the overall structural integrity. For example, a weld flaw in a locally embrittled region subject to an unusual combination of loading, possibly as a result of an accident, might initiate but then arrest as the propagating defect emerges from the critical region. Such an approach can provide additional confidence in the safety of a structure, especially under accidental conditions, and it may also be used for justifying life extensions. The crack arrest approach to structural integrity is simple in its philosophy if not always in its application. The basic concept is that the material used will arrest brittle, fast propagating cracks, initiated in regions of low toughness and/or high stress, when they emerge from the critical zone. It is an approach which is particularly useful for nuclear applications where additional assurance of integrity is required and localised in-service embrittlement may occur. An advantage of considering crack arrest is that attention can be redirected from the microscale (i.e. from the local brittle zone and the local stress concentration) to the intermediate scale (i.e. the properties of the parent plate, weld metal, or heat affected zone, and the nominal applied stress). Much of the early work on crack arrest was carried out in naval laboratories in the UK and in the USA. Other industries were also interested in crack arrest (storage tanks, nuclear pressure vessels and pipelines), but the fracture initiation approach to structural integrity was also being developed and became dominant. Two primary reasons for this were that fracture initiation conditions were inherently simpler to define, and there was a significant costpenalty associated with steels capable of arresting fast running cracks. For more modern steel structures, designing for crack arrest has become an economically viable option. Questions still remain, however, as to the best practical method of determining crack arrest properties and how to analyse the applied stress conditions in a structure in the presence of a running crack. The present paper is based on a review of crack arrest models carried out for the UK Health and Safety Executive (HSE) (Wiesner and Hayes, 1996).

Advances in Nuclear Science and Technology, Volume 26, edited by Lewins and Becker. Kluwer Academic / Plenum Publishers, New York 1999.

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As background to this review, a brief chronology of developments is given below, further details and references will be given in the relevant sections of the main text: 1950s The stress-temperature dependence of brittle crack propagation and arrest and the concept of a crack arrest temperature (CAT) is established. The nil-ductility transition temperature (NDTT) is defined as the upper temperature limit for crack propagation under purely elastic loads. The large-scale ESSO (named after the oil and gas company by which it was developed) test is introduced to measure crack arrest properties in storage tank steels, The double tension test, in which the crack is initiated under slow loading, is developed in Japan. 1960s Pellini proposes the Fracture Analysis Diagram (FAD) in which the conditions for fracture initiation (flaw-size dependent) and crack arrest are plotted in terms of stress as a function of temperature. A fracture mechanics treatment of crack arrest is introduced by Irwin. 1970s The fracture toughness reference curve is included in the ASME nuclear pressure vessel code (ASME III Appendix G). The KIR curve is the lower bound of relevant KIc, KId and KIa data over a range of temperatures, where the temperature is expressed relative to the NDT temperature. A number of large research projects supported by the US Nuclear Regulatory Commission (NRC) and the Electric Power Research Institute (EPRI) investigate crack arrest in thermal shock and pressurised thermal shock (PTS) circumstances in nuclear pressure vessels. Small-scale fracture mechanics tests for crack arrest are developed and scrutinised in the ASTM Co-operative Test Program. The issue of static or dynamic analysis is subject to debate. Direct measurements of KI at the tip of running and arresting cracks are made by the optical method of shadow caustics. Two ASTM Symposia on crack propagation and arrest are held in the USA (Hahn and Kanninen, 1977,1980) 1980s Moment modified compact tension (MMCT) specimen for measurement of KIa in tough materials subjected to a rising stress field (PTS simulation) is developed. Analysis and modelling of PTS crack arrest tests are developed in EPRI sponsored projects. Publication of final report on ASTM Co-operative Test Program. The statically determined KIa value measured in wedge-loaded compact crack test is found to give an useful estimate of dynamically calculated KI at arrest. Standard test method for KIa determination is published by ASTM (E1221-88). Short crack arrest (SCA) wide plate test for welded joints developed. 'Structurally relevant' large scale tests are used to establish correlations between large and smallscale crack arrest tests for modern steels. Japan Welding Research Institute runs a co-operative programme to evaluate KIa test for Japanese reactor pressure vessel steels. 1990s Full thickness Ka quality control test proposed. Publication of Canadian Standard for steel offshore structures (CAN/CSA-S473-92) which includes crack arrest requirements based on drop-weight NDT temperature. HSE/TWI seminar on crack arrest concepts for failure prevention and life extension (Wiesner, 1996a).

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MEASURING CRACK ARREST PROPERTIES

Crack arrest tests may be divided in three groups: (i) large-scale, structurally representative tests, (ii) small-scale tests which can be correlated empirically with crack arrest properties and (iii) fracture mechanics based crack arrest toughness tests. Large-scale crack arrest simulation Large-scale experiments are carried out to model the geometry and loading of the actual structure as closely as possible. If the test set-up is such that the conditions in the actual structure can be modelled, the experiment is described as 'structurally representative'. Two types of large-scale crack arrest tests are used: i) Component testing such as thermal shock experiments on pressure vessels or on rotating cylinders; ii) Wide plate testing, e.g. Robertson, ESSO, double tension and short crack arrest (SCA) tests plus large-scale single edge notch tension (SENT) or centre crack tension (CCT) panels. Component testing is the most realistic type of test. For nuclear applications, thermal shock experiments (TSEs) or pressurised thermal shock experiments (PTSEs) aim to simulate hypothetical loss of coolant accidents (LOCA) or steam line break accidents in pressurised water reactor vessels. In such events, a low temperature coolant is injected into the vessel, passing over the hot inner surface of the vessel and giving rise to the following effects (Cheverton et al., 1980; Pugh et al., 1991a): i) steep temperature gradient through the pressure vessel wall thickness; ii) high thermal tensile stresses at the inner surface (decreasing through the wall), and associated stress intensity factors KI for pre-existing flaws. For pressurised vessels, the additional membrane stresses can lead to a rising stress intensity factor with increasing crack depth; iii) reduced fracture toughness of the inner part of the vessel due to the reduced temperature; and iv) a positive fracture toughness gradient through the wall due to increasing temperature and decreasing irradiation damage. The above effects are shown in Fig.1. A pre-existing flaw may initiate in a brittle manner (applied KI > KIc). However, because of the positive fracture toughness gradient, the applied stress intensity factor falls below the crack arrest fracture toughness at some point within the vessel wall and the crack arrests (KI ≤ KIa ). Roos and Griesinger (1987) presented an experimental method for crack arrest characterisation which simulates the stress conditions in pressure vessels by employing rotating discs. The applied stresses in pressure vessels and rotating discs or cylinders are similar and in both geometries the crack driving force is maintained fully during the crack propagation phase. A recent spinning cylinder experiment has been carried out within the Network for Evaluating Steel Components programme (Hurst et al., 1996) and will be subject to crack arrest analysis. Full-scale component testing, albeit the most realistic way of assessing a particular structure, is often neither an economic nor practical option. An alternative is to carry out large-scale wide plate tests. Stress waves which emanate from the propagating crack tip may be reflected,

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however, at the test specimen loading boundaries back to the crack tip region and so influence the crack arrest event. Therefore, to ensure structurally representative conditions, the crack arrest event should occur prior to the time, ts, which is needed for the stress waves to travel to the specimen boundaries and back. An estimation of this time has been given by Willoughby and Wood (1984):

Fig.1 Schematic through-thickness distribution of temperature, stress applied, stress intensity factor and initiation (KIc) and crack arrest toughness (KIa) for a pressure vessel subject to thermal shock. From Cheverton et al. (1980). ts ≤ 2 (distance to nearest loaded boundary)/stress wave velocity

[1]

The first wide plate crack arrest test was developed by Robertson (1953). The test plate is cooled to the temperature of interest and subjected to the maximum design stress. A running crack is then initiated by impacting the notched 'ear' of the test plate. The test result is simply a statement whether, at a given combination of applied stress and temperature, a running crack is arrested ('no go') or not ('go'). The lowest temperature at which arrest occurs is termed crack arrest temperature (CAT). A second widely used large-scale crack arrest test is the ESSO test, which was developed in the 1950s (Feely et al., 1955) as a modification of the Robertson test. In most recent studies, the test is analysed not only in terms of a crack arrest temperature for a given applied stress but also of the applied stress intensity factor at arrest, i.e. the crack arrest toughness, Ka. From an analysis of a number of large-scale crack arrest tests Wiesner and Hayes (1996) concluded that the CAT corresponds to the temperature where Ka is about 100 to 200 MPa m½. The double tension test, developed in Japan by Yoshiki and Kanazawa (1958), avoids the complication of the impact blow which is necessary to initiate the brittle crack in Robertson and ESSO tests. Instead, the crack is initiated by applying a subsidiary load to the edge of the plate via a secondary loading tab. Crack initiation may be assisted by local embrittlement and/or cooling. An isothermal test will lead to a 'go'/'no-go' result for the applied temperature.

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The applied stress intensity factor at the moment of arrest can also be calculated (Wiesner et al, 1994). Strictly speaking, a dynamic analysis of the test is required, however, the difference between static and dynamic stress intensity factors is less than about 15% during the crack propagation phase (Curr and Turner, 1987). A static analysis is appropriate as long as the run/arrest event is not influenced by unloading stress waves. A variation of the double tension test, the short crack arrest test was developed by Tanaka et al. (1982) to improve the accuracy of locating the propagating crack in a welded wide plate specimen and to take into account fully the transverse welding residual stresses which are highest in the centre of the welded panel. This is especially desirable for assessing the structural relevance of local brittle zones (LBZs) which can be present in weld heat affected zones (HAZ). A vital part of the US Nuclear Regulatory Commission (NRC) sponsored Heavy Section Steel Technology (HSST) programme was a series of single edge notched wide plate crack arrest tests on nuclear pressure vessel steels. The test panels were loaded in monotonically increasing tension until crack initiation occurred. As a thermal gradient was imposed on the specimens along the crack path, crack propagation took place into a rising temperature and fracture toughness region. Small-scale empirical methods As opposed to large-scale, structurally representative tests, the predictive capabilities of small-scale tests rely on empirical correlations based on past experience. However, such empirical correlations exhibit a degree of inherent uncertainty due to the following factors (Fearnehough, 1973): i) the small-scale specimens do not exhibit the same constraint as the structure of concern (due to differences in dimensions and stress state). This can cause general yielding of small-scale specimens at stresses when structural behaviour can be still described by LEFM; ii) the loading and strain rates in small-scale tests cannot always simulate the structural condition; and iii) the combination of small-scale specimen compliance and testing machine stiffness is not normally representative of the structure. This means that developed correlations have to be re-established for new materials. Nevertheless, because of the expense involved in carrying out large-scale tests, small-scale tests and correlations with structurally representative tests continue to be sought, as recently summarised by Wiesner (1996b). The most Widespread small-scale fracture test is the Charpy test. The energy absorbed by a Charpy specimen or its fracture appearance is a measure of both fracture initiation and propagation processes. Therefore, these properties are not directly compatible with crack arrest behaviour and existing correlations are purely empirical. Correlations were reported in the 1960s by Nichols (1965), Feamehough and Vaughan (1963) and Cowan et al. (1967), between the crack arrest temperature (CAT) and the Charpy fracture appearance transition temperature (FATT). CAT tends to increase with increasing FATT although there is considerable scatter. Smedley (1989) collected a considerable amount of data and calculated CATS from Charpy 50% FATTs using semi-empirical relations developed in Japan; he found increased scatter below transition temperatures of about –13 °C which was attributed to the development of modern grain refined steel developments which have led to lower Charpy

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changes. Wiesner et al. (1993) have compared different Charpy transition temperatures of modern steels and their weldments with structurally representative crack arrest temperatures for applied stresses of 2/3 the yield strength. Whilst marked scatter makes the use of conventional Charpy results difficult, the ‘best’ correlation was found using the 50% FATT. A summary of the empirical reactions between measured CAT and 50% FATT is shown in Fig.2. This confirms that rough estimates of CAT can be made from 50% FATT and that increasing scatter is observed for more modern steel.

Fig.2 Measured large-scale crack arrest temperatures (CAT) versus Charpy 50% fracture appearance transition temperature (FATT). The drop-weight test (often referred to as 'Pellini' test) was developed as a simple method to determine the nil-ductility transition temperature (NDTT). The NDTT was defined in the 1950s (Puzak et al., 1952) as the test temperature in explosion bulge tests at which the plate remained flat at fracture, that is crack propagation occurred in the presence of elastic strains only. The drop-weight test was developed to simplify the determination of the NDTT. The importance of the NDTT as a reference temperature for the ductile to brittle transition temperature of ferritic steels was established from studies of service failures (Puzak et al., 1954 and 1958) and structurally representative crack arrest tests (Pellini and Puzak, 1963). In the UK, comparisons between isothermal CATs, as determined in Robertson tests, and the NDTT were made for pressure vessel steels by Feamehough and Vaughan (1963), Nichols (1965) and Lessels and Leggett (1971). Graville (1989) established a database of more than 250 large-scale test results to compare the fracture performance of structural steels with their NDTT, and Wiesner et al. (1993) reported results from 1990s steels and their weldments. Smedley (1989) collected data from 1960s and 1970s steels and developed correlation between CAT and NDTT containing empirical corrections for thickness, B, and stress levels, σ. The correlation was further developed by Wiesner (1996b) and Wiesner et al (1997) to give: CAT = [NDTT + 10] + [1n(σ)/0.046-105] + [153 (B - 5)1/13-190] (CAT and NDTT in °C, σ in MPa and B in mm)

[2]

All data collected by Smedley (1989), Graville (1989) and Wiesner (1993) were re-analysed and the measured CATs are compared in Fig.3 with the predicted values from NDTT, see Eq. [2].

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Instrumented Charpy tests allow the initiation and propagation stages of the test to be separated, see Fig.4. Fearnehough (1973) used the post brittle fracture energy as a measure of crack arrest, which gave markedly improved correlations with CATs, as compared to conventional Charpy transition temperatures. Similar findings were reported by Hagedorn (1983). He correlated thickness corrected CAT with the temperature at which the brittle crack extension in an instrumented Charpy test caused the load to drop to zero.

Fig.3 Crack arrest temperature calculated from NDTT as a function of measured CAT. From Wiesner (1996).

Fig.4 Typical load-time trace from instrumented Charpy test in the transition region of a C-Mn pressure vessel steel.

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A number of German researchers (Vasoukis, 1971; Berger et al 1979 and Gillot, 1988) have developed an NDTT criterion which is based on force at the arrest of the brittle crack in instrumented Charpy tests (see Fig.4). Tests carried out by Berger et al. (1979), Planman et al. (1995) and recently by TWI established that the NDTT correlates well with the temperature where the force at arrest corresponds to 5kN for a variety of steels see Fig.5. This good correlation between NDTT and the 5kN transition temperature from instrumented Charpy tests suggest that predictions of structural crack arrest behaviour (see Eq. [2]) can be made using instrumented Charpy tests only.

Fig.5 Nil-ductility transition temperature (NDTT) versus temperature in instrumented Charpy tests where the arrest load equals 5kN, T5kN. Fracture mechanics based crack arrest toughness tests In fracture mechanics crack arrest testing, propagation/arrest events are produced in relatively small, inexpensive specimens and fracture mechanics principles are used to calculate the applied stress intensity factor at arrest, i.e., the crack arrest toughness of the material. The measured crack arrest toughness can be used in the analyses of structures and components to predict whether a particular combination of applied stress and crack length will give conditions under which an initiated crack will arrest after some propagation or will continue to propagate. The most widely used geometries for fracture mechanics based crack arrest tests are double cantilever beam (DCB), tapered double cantilever beam (TDCB), compact crack arrest (CCA), and moment modified compact tension (MMCT) specimens. Of these tests, the CCA test has been standardised by ASTM (Specification E1221-88). Some work also been carried out using single edge notched bend (SENB) and single edge notched tension (SENT) specimens. DCB and CCA specimens are loaded by forcing a wedge between the loading pins. Blunt (sometimes embrittled) notches cause the applied stress intensity factor at initiation to be greater than the KIc of the material and so, as soon as a crack initiates, it becomes unstable and propagates into the specimen. Due to the basically constant displacement loading

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arrangement, the applied stress intensity factor decreases with increasing crack length and crack arrest is likely to occur. The CCA specimen has the advantage over the DCB specimen of lower machining costs and greater crack path stability. The crack arrest toughness can be calculated using static relationships between the crack mouth opening displacement and the stress intensity factor (Crosley and Ripling, 1969, 1971, 1977). Alternatively, a dynamic or kinetic approach, which takes into account the crack propagation process and inertial effects prior to arrest, has been developed (Hahn et al., 1973; Kanninen, 1974; Hoagland et al., 1977; Kanninen et al., 1977). The validity of the static approach versus the dynamic approach was somewhat disputed ( e.g. Crosley and Ripling (1980) and written discussion to that paper). Caustic and photoelastic analyses published by Kalthoff et al. (1977) and Kobayashi and Dally (1980) have provided evidence that kinetic effects are important for rectangular DCB specimens, see Fig.6. When dynamic effects are small, as in the CCA specimen (Fig.6), the simpler static approach can be used to calculate the static K at arrest Ka (or KIa if plane strain conditions are met).

Fig.6 Stress intensity factor - time plots for rectangular DCB and CCA specimens. From Kalthoff et al. (1980). The statistical variation of KIa test results was studied by Rosenfield et al. (1984) and found to be generally smaller than that observed in initiation fracture toughness tests (see also Crosley

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and Ripling, 1969) This has been attributed to the fact that crack arrest is governed by the bulk material properties whilst the initiation toughness is controlled by local microstructural events. Most crack arrest tests are carried out at temperatures below RT NDT + 60 °C (where RT NDT is the ASME reference nil-ductility transition temperature). If crack arrest data at higher temperatures are required (thus approaching the onset of upper shelf behaviour), special experimental problems occur. Such data are especially relevant to thermal shock and pressurised thermal shock scenarios. One approach for obtaining compact crack arrest data at upper shelf temperatures is the use of increasingly large specimen dimensions (Kussmaul and Gillot 1987, 1991). Alternatively, Rosenfield et al. (1988) have proposed a compact specimen geometry with a gradually increasing sidegroove depth to produce an increasing stress intensity as the crack propagates into the test material. Marshall et al. (1983a) have compared full size CCA test results (50 mm thick by 200 mm square) with those obtained on specimens down to 6.4mm thickness. They concluded that similar crack arrest toughness values are obtained provided that the measured crack opening displacement used to calculate KIa does not include a significant plastic component. The successful use of small specimens has important implications for surveillance specimens testing. Marshall et al. (1986) have also proposed reconstituted CCA specimens for crack arrest testing of irradiated material. An additional small scale test configuration which has been used to characterise crack arrest behaviour is the moment modified compact tension (MMCT) specimen. The specimen was developed (Schonenberg and Noms, 1986) to model pressurised thermal shock conditions in nuclear pressure vessels, i.e. thermal stress and toughness gradients. This is achieved by creating a temperature gradient in the specimen and by applying a secondary load via the moment arms to impose an increasing stress and stress intensity factor field (Ayres et al., 1988). Crack arrest toughness values at temperatures up to T-RT NDT = 70 °C can be readily measured and Fabi et al. (1988) have developed a simplified analysis method which reconciles the test results of MMCT and large and full-scale tests results. MMCT tests have been further analysed by Kobayashi and Giovanola (1989a, 1989b) and Giovanola and Kobayashi (1989) using crack opening displacement measurements and post test fracture surface topography analysis. 4.

ANALYSES OF CRACK ARREST BEHAVIOUR

Comparison of static with dynamic analyses The fracture mechanics analysis of crack arrest was the subject of much debate throughout the 1970s when the basic concepts were developed. One view point suggested that crack arrest may be characterised as the reverse of crack initiation (Irwin and Wells, 1965): the knowledge of the arrested crack length, aa , and the applied stress enables one to calculate a fracture toughness value which characterises the crack arrest capability, KIa, of the material under consideration. Since no input from the crack propagation phase is required, the calculation is carried out employing relations developed for static loading conditions, Thus, the approach can be expressed as (Kanninen and Popelar, 1985). KI = KIc for initiation and KI ≤ KIa for arrest

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where K I is the applied stress intensity factor (calculated statically) and KIc and KI a are assumed to be temperature dependent material properties. In the dynamic or kinetic approach, which takes into account the effects of kinetic energy in the cracked body (Kanninen and Popelar, 1985), crack propagation takes place as long as the crack driving force is sufficient to continually extend the crack K I = K ID (v, T) where KI is the applied stress intensity factor calculated dynamically and K ID is the dynamic propagating fracture toughness which is a function of crack speed and temperature. Crack arrest occurs when the driving force falls below the minimum value of K ID , normally termed KI A : K I ≤ K I A (Note that K IA is also frequently written as K I D m , K I m , K IM or K ldm .) Experimental evidence (see Fig.6) and numerical computations have shown that dynamic or kinetic effects do exist during crack propagation and arrest events. However, static crack arrest toughness values can be readily determined experimentally and, when the conditions of short crack jump lengths and moderate crack speeds are met, K I a is a very good estimate of K IA . In practice, the above pre-requisites for K I a approaching K IA values are frequently met as exemplified by the successful static analysis of thermal shock scenarios. Prior to the return of reflected stress waves, the difference between dynamic and static stress intensity factor is a function of crack speed. The relationship between dynamic and static K I has been obtained analytically (Freund, 1972; Broberg, 1960; and Baker, 1962) and is plotted versus the crack speed, v, normalised by the Rayleigh surface wave speed, cR, (about 3000 m/s for steel), in Fig.7 for tensile loading configurations in an infinite body. It can be seen how the ratio of dynamic to static stress intensity factor decreases with increasing crack speed. For common brittle crack propagation speeds in steel, 200 ≤ v ≤ 2000 m/s (i.e. 0.1 ≤ dyn v/cR ≤ 0.3), it follows from Fig.6 that 0.8 ≤ K I / K Is t a t £ 0.95. This means the difference between static and dynamic stress intensity factor is about 5 to 20% for typical crack speeds in steel. The dynamic stress intensity factor depends on the (static) stress field which exists prior to crack propagation as this has little time to change given the rapid crack growth. This is the , basis of the reflectionless K described by Smith (1992). ,

Depending on the loading configuration of the specimen studied, there are two possibilities with respect to the development of the applied stress intensity factor after crack arrest. In homogeneous DCB specimens, which exhibit decreasing stress intensity factors with increasing crack lengths, the dynamic crack arrest toughness (i.e. the stress intensity factor at dyn decreases and arrest) can be higher than that calculated statically. Following arrest, K I oscillates around the statically calculated stress intensity at arrest with decreasing amplitude, see Fig.6. This behaviour is referred to as 'ring-down' or 'down swing' (Kobayashi and Giovanola, 1989b).

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Fig.7 Ratio of dynamic to static stress intensity factor in infinite body as a function of crack speed, normalised by the Rayleigh surface wave speed. However, in the more common case, crack arrest occurs because a region of increased toughness is encountered during a phase where the dynamic value lies below the static one. The dynamic stress intensity factor rises following the arrest event, see Fig.8, and again oscillates around the statically calculated value. This behaviour is known as 'ring-up' or 'up swing'. Whether or not these dynamic variations will cause further crack extension is governed by the dynamic initiation fracture toughness of the material. Sumpter and Curr (1996) have proposed that sustained crack arrest in a structure can be predicted using the statically determined crack arrest toughness from small-scale tests if: K dpeak structure ≤ K

peak specimen d

where K dpeak is the maximum dynamic K arising after arrest.

Fig.8 Normalised dynamic stress intensity factor in Araldit duplex CCA specimen. From Kobayashi and Giovanola (1989b).

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Plasticity and three-dimensional effects Experiments have demonstrated clearly that extensive plasticity and crack tunnelling can take place during run/arrest events. The analytical or numerical analysis (using viscoplastic models) of these effects has provided valuable information as to how the crack during force and thus the crack arrest toughness is affected by these effects (Kanninen et al., 1986; Bass, et al., 1987; Schwartz and Bass, 1987; Bass et al., 1988a, 1988b; Pugh et al., 1991b; Curr and Turner, 1992). Since these more complicated analyses generally lead to a reduction of calculated crack driving force, their use will be considered for situations where the elimination of over-conservatism is of great importance (e.g. life extension programmes). Analysis of large-scale tests The analysis of the HSST wide plate test programme compared KI values calculated at crack arrest using both static and dynamic methods (generation mode) and constant load and constant displacement boundary conditions (Pugh et al., 1991b; Bass et al., 1988a). In a dynamic generation mode analysis, the crack extends incrementally according to a prescribed crack tip position-time history, as measured in the experiments, and the dynamic stress intensity factor is calculated as a function of time (or crack extension). The effect of the different analysis methods and boundary conditions is demonstrated in Fig.9 in which the crack arrest toughness -temperature curve is plotted for one test of the series (WP-2.4), which exhibited seven run-arrest events. Figure 9 shows that the difference between the various analyses increases with increasing temperature, i.e. for increasing crack length and elapsed time in the experiment as this was a thermal gradient test.

Fig.9 Comparison between various analysis methods (and boundary conditions) for crack arrest toughness in a SENT wide plate test. Further indications as to which analysis method and boundary conditions best describe the crack arrest behaviour in SENT specimens were obtained from tests and analyses at the MPA in Stuttgart, Germany. Gradient temperature SENT tests on two sizes of wide plates were carried out (Kussmaul et al., 1991; Elenz, 1992). The tests were analysed using static and generation mode dynamic FEA with both constant load and constant displacement boundary

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conditions. In addition, the experimentally determined load-time history at the load application points was used as a time dependent boundary condition. Elenz (1992) compared the experimental strain gauge readings with both static and dynamic FEA predictions from the smaller wide plate specimens (which fractured completely) and concluded that the dynamic analysis using the experimentally determined force-time boundary conditions gave the most accurate predictions. The difference between these and those from the static method with constant displacement boundary conditions was at most 30%. Dynamic analysis of TWI double tension tests (Curr and Turner, 1987) shows similar results, but with a difference of only about 15%. 5.

CRACK ARREST DATA FOR PRESSURE VESSEL STEELS

The United States Heavy Section Steel Technology (HSST) programme produced initiation fracture toughness, KIc, dynamic initiation fracture toughness, KId and crack arrest fracture toughness, KIa, data on A533 B-1, A508-2 and A508-3 steels. An outcome of the programme was the development of the ASME boiler and pressure vessel code KIR reference curve (Anon., 1972), which is taken to represent a lower bound to fracture toughness data. The equation of the KIR curve is given relative to RT NDT as: KIR = 42.5 exp[0.0261 (T - RTNDT + 88.9)] + 930.57 (KIR in N/mm3/2, T and RTNDT in °C) Structurally representative crack arrest toughness values for ASTM A508 and A533B steels were calculated from TSEs and PTSEs (Cheverton et al., 1983, Pellissier-Tanon et al., 1983; Pugh et al., 1986; Bryan et al., 1986 and Pennell, 1992). All measured values (calculated using static LEFM) lie above the ASME KIR reference toughness curve and some exceed by far the ‘upper shelf limit of 220MPa m1/3 given in ASME. This is also the case with the HSST Program SENT wide plate tests results calculated with dynamic FEA. All of these results are plotted together in Fig.10.

Fig.10 Crack arrest toughness values deduced from TSEs, PTSEs and SENT wide plates

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Kussmaul and Gillot (1987), however, measured crack arrest toughness at temperatures of up to T NDT + 180 °C on German designation 17 MoV 8 4 and 22Ni Mo Cr 3 7 (mod) steels and found that the data exhibited a smaller temperature dependence than the ASME reference curve, see Fig. 11. For the two steels, the KIR curve is intersected at temperatures between 50 and 100 °C above T NDT. It should be noted that one of the steels exceeds the strength validity limits for the K IR reference curve.

Fig.11 Crack arrest toughness value for three pressure vessel steels determined at temperatures up to T = TNDT + 150 °C. From Kussmaul and Gillot (1987). Marschall et al. (1983b) found good agreement between compact crack arrest data and crack arrest toughness values inferred from full-scale thermal shock experiments for temperatures up to RTNDT + 60 °C. Agreement between lower bound CCA and large-scale test results was also found in a major study instigated by the Japan Welding Engineering Society (Sakai et al., 1986). This study included almost 400 crack arrest tests on A533 Type B Class 1 plates, A508 Class 3 forgings, plus weld metal and HAZ tests. The results are shown in Fig.12. An EPRI research programme investigated the temperature shift in materials properties which results from neutron irradiation (Anon., 1984). Parent plate and weld metal with low and high Cu and P contents were studied. An interesting finding of the study, but one which is not discussed in the work cited, was that several results, including both as-received and irradiated specimens from a low-copper weld, fell below the KI R curve. A similar phenomenon was reported by Hahn et al. (1980a) who carried out a test programme using six different heats of A533 Type B and A508 Class 2 pressure vessel steels, plus a submerged-arc weldment. Some KI a data, but not the corresponding dynamically calculated arrest toughnesses, fell below the K IR curve. The results of the Sixth Series of the HSSI (Heavy-section steel irradiation) described by Iskander et al. (1992) did not include points lower than the reference curve. All of the data referred to above plus some from other sources are brought together in Fig. 13. 6.

APPLICATION OF CRACK ARREST TO PRESSURE VESSEL INTEGRITY ASSESSMENT

Crack arrest analyses have been applied to assess the significance of hypothetical loss of coolant accidents (LOCAs) or steam line break accidents in pressurised water reactor vessels

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Fig.12 Crack arrest fracture toughness results versus T-RTNDT from Japanese Welding Engineering Society test programme. Adapted from Sakai et al. (1986): a) Parent plates; b) Weld metal and heat affected zone. (Marston et al., 1979, 1980). Events during LOCA transient conditions can be illustrated by critical crack depth curves, initially proposed by Ball et al. (1983), which are deduced from plots of the applied KI ,KI c and KI a curves against a/W, see Fig. 1. The intersection points of KI with KI c and KIa curves are plotted against time to give the critical depth curve shown in Fig.14. When KI = KI c , crack initiation occurs and the crack propagates until it intersects the crack arrest toughness curve, KI = K la, where it arrests. Re-initiation is possible at later times during the transient. In Fig.14, there are two re-initiation events at times of about 3.3 and 5.8 minutes. This sequence of events is stopped when the crack propagation 'path' intersects the warm prestress (WPS) curve, which indicates the time at which the maximum applied KI is reached throughout the wall thickness. Following this, the applied KI decreases with time and previous work has shown that crack initiation does not occur when dK/dt<0 (for more details of the WPS effect, see for instance Chell et al. (1981)). Critical crack depth curves have been used to analyse actual PTS transients. Examples were given by Blauel and Nagel (1985), Cheverton and Ball (1986), Fabi et al. (1988) and Pennell(1992).

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Fig.13 Summary of collected crack arrest toughness data for pressure vessel steels.

Fig.14 Critical crack depth-time curves for a hypothetical LOCA accident situation similar to that resulting in the applied conditions shown in Fig.1. From Cheverton and Ball (1986). Analyses of TSEs and PTSEs have demonstrated that, for the small crack jump lengths generally observed in these simulation tests, static assessments are appropriate (e.g. Cheverton et al., 1980; Kanninen et al., 1985; Brickstad and Nilsson, 1986a, 1986b). This may not be the case, however, for experiments or structural situations where longer crack jumps are experienced. Elenz (1992) analysed the fracture behaviour of 21 mm thick pressure vessels into which round inserts of a brittle material were welded. For a test in which crack arrest, the applied dynamic KI value was less than the KI a . value of the vessel material as the crack emerged from the brittle insert. The arrest behaviour could not have been predicted by

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the applied static KI which exceeded the arrest toughness value of the vessel material, This illustrates that static analyses can be over-conservative for structural integrity arrest assessments of pressure vessels containing long flaws. More recent work at ORNL (Dickson et al., 1989) has shown that under PTS circumstances the limiting factor for pressure vessel integrity is tearing stability of arrested cracks rather than the arrest of cracks under high driving force conditions. The statistical analysis of a number of PTS transients indicated that crack arrest has little effect on failure probability. For non-pressurised thermal shock conditions, crack arrest does, however, contribute to a reduction in failure probability (Dickson, 1995). Recent work (Sumpter, 1991; Sumpter and Hayes, 1992; Wiesner and Pisarski, 1996) has shown that for small regions which may be susceptible to cleavage fracture (‘pop-in’ type fracture events), it is possible to complement an initiation-based structural integrity assessment with crack arrest arguments. In recent years, there has been some consideration of the relationship between crack arrest temperature (CAT) and the onset of upper shelf temperature (OUST) for initiation toughness (Cottrell, 1996; Lidbury et al., 1996; Smith, 1996). If OUST lies above CAT, then basing a structural integrity assessment method on OUST will subsume crack arrest. However, recent TWI work (Hayes and Wiesner, 1998) has shown that for a successful use of such an argument, appropriate definitions of CAT and OUST have to be established. Further, Smith (1996) argues that if OUST based on initiation properties rather than crack arrest properties is used to define operational temperature envelopes, then a temperature margin should be applied as brittle crack propagation at temperature exceeding OUST has been observed experimentally. The relationship between CAT and OUST for a particular material will be dependent on its strain-rate sensitivity. For instance, if CAT is taken to be the transition temperature of the dynamic toughness of a running crack, then CAT will probably be close to the static initiation toughness transition temperature (and hence OUST) for materials with low strain-rate sensitivity. On the other hand, if the material is highly strain-rate sensitive, then CAT could exceed considerably OUST based on slow rate tests. Finally, in the UK, efforts are now underway to develop Appendix 15 of the R6 structural integrity assessment procedure for the evaluation of structures using crack arrest concepts (Conners, 1997), and the applicability of crack arrest concepts to pressure vessels has been reviewed by the UK Technical Advisory Group on Structural Integrity, TAGSI, (Burdekin et al, 1998). 7.

• •

CONCLUSIONS There is a clear role for arrest concepts to support initiation based assessments, even if comprehensive crack arrest arguments are not possible. Crack arrest can only be effective if there are gradients in stress (reducing) or fracture toughness (increasing due to either microstructural effects or temperature) away from the initiation location. In the vast majority of structural situations, if crack arrest occurs, it takes place because the crack emerges from a local low toughness region, and propagates into a higher toughness area.

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The application of a validated crack arrest assessment procedure will give additional confidence in safety during the operation of some welded components and efforts are underway in the UK to formalise the crack arrest approach as Appendix 15 of the R6 structural integrity assessment procedures. The fracture mechanics approach to crack arrest is consistent with the crack arrest temperature (CAT) in that the CAT corresponds to the temperature for a certain crack arrest toughness value (typically 100 to 200 MPa m½ depending on the relevant yield strength). It may be possible to use other proxies for defining different positions on the overall crack arrest toughness transition curve (which incluses the CAT), such as the ‘Pellini’ dropweight test nil-ductility transition temperature (NDTT), appropriate transition temperatures determined from instrumented Charpy tests, or the temperature corresponding to a given value of the fracture initiation toughness. The application of such proxies must take into account corrections for component thickness and applied stress. A dynamic analysis is required for an accurate crack arrest analysis of the applied conditions during propagation and arrest. However, for propagating cracks, the dynamic applied crack driving force is lower than the value calculated statically for the vast majority of structural situations. (Only in cases where the effect of reflected stress waves becomes dominant, such as in double cantilever beam crack arrest test specimen, can the static crack driving force be smaller than the dynamic value). When crack arrest occurs, there is an increase in the dynamic applied crack driving force (‘overshoot’ effect) in most structurally relevant situations. A static analysis will therefore result in conservative assessments for short crack jumps (with respect to the component dimensions) when dynamic effects, and hence the overshoot effect are small or negligible. Expressed in terms of fracture mechanics, sustained crack arrest will take place if the applied crack driving force falls below the crack arrest toughness and if the dynamic crack driving force peak shortly after arrest does not exceed the dynamic initiation toughness. For accurate structural integrity assessments, it is necessary to determine the peak value of the dynamic crack driving force in the component shortly after arrest and compare this with the experimentally determined crack arrest toughness value (which is also a lower bound for dynamic initiation toughness); this will assess whether sustained crack arrest will take place, i.e. the crack will not re-initiate. For a conservative assessment, the dynamic crack driving force peak can be assumed to be 1.5 to 1.7 times the statically calculated stress intensity factor value for the arrested crack length. (From a review of dynamic crack driving forces during propagation and arrest (figures in Wiesner and Hayes, 1996), it has been concluded that most peaks are between 1.05 and 1.3 times the static crack driving force value at arrest.) There is still discussion as to the most appropriate test to determine crack arrest toughness experimentally, but it is generally agreed that standardised compact crack arrest (CCA) tests give conservative estimates for both crack arrest and dynamic initiation toughness. Although the broad micromechanical aspects of the processes leading to crack arrest have been established, a detailed predictive model based on this understanding, including effects of strain rate and work hardening, is not yet available. The relation between the onset of upper shelf temperature (OUST) for initiation and the CAT depends strongly on the exact definition of OUST and CAT and needs to be assessed on a case-by-case basis, but data do exist for which CAT can be subsumed by OUST.

ACKNOWLEDGEMENTS Financial support from the UK HSE is acknowledged gratefully. The authors would also like to thank Drs A A Willoughby and J D G Sumpter for helpful discussions.

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REFERENCES Anon.: Pressure Vessel Research Committee Recommendations on Toughness Requirements for Ferritic Materials. WRC Bull. No.175, (August 1972), 1-24. Anon.: Westinghouse Electric Corporation and Battelle Columbus Laboratories: Development of a crack arrest toughness data bank for irradiated reactor pressure vessel materials. Volume 2: Detailed Technical Description of the Work. EPRI Report No.NP-3616, July 1984. ASTM E208-91: Standard test method for conducting drop-weight test to determine NDTT of ferritic steels. American Society for Testing and Materials, 1993. ASTM E1221-88: Standard test method for determining plane-strain crack arrest fracture toughness, Kla. of ferritic steels. American Society for Testing and Materials, 1988. ASME: Pressure vessel and boiler code, Part III: Rules for construction of nuclear power plant components. American Society of Mechanical Engineers, New York, 1989. Ayres D J, Fabi R J, Schonenberg R Y and Noms D M: Comparison of analysis and experimental data for unique crack arrest specimen. STP 969, 724-751, American Society for Testing and Materials, 1988. Baker B R: Dynamic stresses created by a moving crack. J. Appl. Mechs, 29 (1962), 449-458. Ball D G, Cheverton R D, Brake J B and Iskander S K: Report No. NUREG/CR-3491, Oak Ridge National Laboratory, 1983. Bass B R, Pugh C E, Keeney-Walker J: Elastodynamic fracture analyses of large crack arrest experiments. Nucl. Eng. Des. 98 (1987) 157-169. Bass B R, Pugh C E, Merkle J G, Naus D J and Keeney-Walker J: Fracture analysis of heavy section steel technology wide plate crack arrest experiments. STP 969, 691-723, American Society for Testing and Materials, 1988a. Bass B R, Pugh C E, Keeney-Walker J, Schwartz C W: Late event viscoplasticity in wideplate crack arrest tests. Int. J. Press. Ves. & Piping, 31 (1988b), 325-348. Berger C, Ewald J, Wiemann W and Wojaczyk H.-G Ermittlung von Sprodbruch und bruchmechanischen Kennwerten mittels Kerbschlagproben. Proc. Conf. Sitzung DVM AK Bruchvorgänge, 182-195, DVM, 1979 (in German). Blauel J G and Nagel G: Crack arrest safety analysis for a reactor pressure vessel under emergency cooling conditions. Proc. Conf. 8th Struct. Mech. React. Technol. (SMIRT), Vol.C/D, 607-614, North Holland, 1985. Brickstad B and Nilsson F: A dynamic analysis of crack propagation and arrest in PTS experiments. Eng. Fract. Mechs. 23 (1986a), 99-102. Brickstad B and Nilsson F: Dynamic analysis of crack growth and arrest in a pressure vessel subjected to thermal and pressure loading. Eng. Fract. Mechs. 23 (1986b), 61-70. Broberg K B: The propagation of a brittle crack. Arkiv for Fysik, 18 (1960), 159-192. Bryan R H, Bass B R, Merkle J G, Pugh C E, Robinson G C and Whitman G D: The heavy section steel technology pressurised thermal shock experiment PTSE-I. Eng. Fract. Eng., 23 (1986), 81-97. Burdekin F M, Knott J F, Sumpter J D G and Sherry A H: TAGSI view on aspects of crack arrest philosophies for pressure vessels with thicknesses up to 100 mm. Submitted to Int. Journal of Press. Vess & Piping, 1998. CAN/CSA - S473-92: Steel structures (offshore structures). Canadian Standards Association, 1992. Chell G G, Haigh J R and Vitek V: A theory of Warm prestressing: experimental validation and the implications for elasticplastic failure criteria. Int. J. Fract., 17 (1981), 61-81. Cheverton R D, Gehlen P C, Hahn G T and Iskander S K: Application of crack arrest theory to a thermal shock experimental. STP 711, 392-421, American Society for Testing and Materials, 1980.

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Cheverton R D, Canonico D A, Isakander S K, Bolt S E, Holz P P, Nanstad R U and Stelzman W J: Fracture mechanics data deduced from thermal shock and related experiments with LWR

pressure vessel material. J. Press. Vess. Techn., 105 (1983), 102-110. Cheverton R D and Ball D G: The role of crack arrest in the evaluation of PWR pressure vessel integrity during PTS transients. Eng. Fract. Mechs., 23 (1986), 71-80. Conners D C: Assessment using crack arrest concepts, R6, Draft Appendix 15, August 1997. Cottrell AH: Fifty years on the shelf: Int. J. Pres. Ves. & Piping, 64, (1995), 171-174. Cowan A, Fearnehough G D, Gilles J and Lees G M: Irradiation damage in mild steel: a comparison of crack arrest tests and Charpy criteria and influence of neutron spectrum. STP 426, 107 -134, 1967. Crosley P B and Ripling E J: Dynamic fracture toughness testing of A533 steel. J. Basic Engn., Trans. ASME D, 91 (1969), 525. Crosley P B and Ripling E J: Crack arrest toughness ofpressure vessel steels. Nucl. Eng. Des., 17 (1971), 32-45. Crosley P B and Ripling E J: Guidelines for measuring KIa with a compact specimen. Presented to the Dynamic Initiation and Crack Arrest Task Group, American Society for Testing and Materials, 23 March 1977. Crosley P B and Ripling E J: Comparison of crack test methodologies. STP 711, 211-227, American Society for Testing and Materials, 1980a. Curr R M and Turner C E: The numerical analysis of crack propagation and crack arrest in wideplates of C -Mn steel. TWI Report 5554/10/87, Appendix A, TWI, November 1987. Curr R M and Turner C E: Numerical modelling of crack tunnelling in dynamic fracture of medium -strength steel. Report for the UK Science and Engineering Research Council (SERC), Imperial College of Science, Technology and Medicine, London, 1992. deWit R, Low S R (111) and Fields R J: Wide plate crack arrest testing: Evolution of experimental procedures. STP 969,679-690, American Society for Testing and Materials, 1988. Dickson T L: Crack arrest effects on the conditional probability of through-wall cracking. EPRI Pressurized Thermal Shock Workshop, Scottsdale, Arizona, Sept 1995. Dickson T L, Cheverton R D and Shum D K: Inclusion of unstable tearing and extrapolated crack- arrest toughness data in PWR vessel integrity assessment. USNRC Report NUREG/CR5473 (ORNL/TM-11450), 1989. Elenz T: Experimentelle und numerische Untersuchungen von instabiler Rissausbreitung und Risstop beim schnellen Bruch von zugbelasteten Platten. Techn.- wissenschaftl. Berichte MPA Stuttgart, No.92-03, 1992 (in German). Fabi R J, Ayres D J and Noms D M: A simplified method for calculating the stress intensity factor at crack arrest. Int. J. Pres. Vess. & Piping, 34 (1988), 207 -225. Fearnehough G D: The small -scale test and its application to fracture propagation problems. Proc. Conf. Dynamic Crack Propagation, 77-1 02, Noordhoff Publishing, 1973. Fearnehough G D and Vaughan H G: Comparisons between drop-weight and crack arrest tests for the estimation of the brittle fracture transition of steel. Weld. J. (May 1963), 202-s to 204-s. Feely F J (Jr), Northup M S, Kleppe S R and Gensamer M: Studies on the brittle failure of tankage steel plates. Weld Journ., 34 (1955), 596s -607s. Freund L B: Crack propagation in an elastic solid subjected to general loading - Part I: Constant rate of extension. Part II: Non uniform rate of extension. J. Mech. Phys. Jol. 20, (1972), 129-140 and 141-152. Giovanola J H and Kobayashi T: Crack opening displacement measurement methods for crack arrest experiments. Proc. Conf. ICF7, Vol.V, 35 17-3524, Pergamon Press, 1989. Gillot R: Experimentelle und numerische Untersuchungen zum Risstopverhalten von Stählen und Gusseisenwerkstoffen. Techn. Wissenschaftl, Berichte, MPA, Stuttgart, No.88-03 1988 (in German). Graville B A: Crack arrest requirements for arctic structures. Graville Associates Inc. Report, March 1989.

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Hagedom K E: Comparison of transition temperatures from instrumented impact tests with the Robertson crack -arrest temperature. Arch. Eisenhüttenwes., 54 (1983), 239 -244 (in German). Hahn G T, Hoagland R G, Kanninen M F and Rosenfield A R: A preliminary study of fast fracture and arrest in the DCB specimen. Proc. Conf. Dynamic Crack Propagation, 679 - 662, Noordhoff, 1973. Hahn G T, Hoagland R G, Lereim J, Markworth A J and Rosenfield A R Fast fracture toughness and crack arrest toughness of reactor pressure vessel steel. STP 71 1, 289 - 320, American Society for Testing and Materials, 1980. Hayes B and Wiesner C S: Evaluation of the potential for crack arrest using the UK definition for upper shelf toughness behaviour, to be published in Proc. of Fourth Conf. on Eng. Structural Integrity assessment, AEAT, 1998. Hoagland R G, Rosenfield A R, Gehlen P C and Hahn C T: A crack arrest measuring procedure for KIM, KID and K I a properties. STP 627, 177 -202, American Society for Testing and Materials, 1977. Hurst R C, Hemsworth B and Wintle J B: NESC: the network for evaluating steel components. International Conference on Nuclear Engineering, 1, Part A, ASME, 1996. Irwin G R and Wells A A: A continuum mechanics view of crackpropagation. Metallurgical Reviews, 91 (1965), 223 - 270. Iskander S K, Corwin W R and Nanstad R K: Effects of irradiation on crack arrest toughness of two high - copper welds. STP 1125,251 - 269, American Society for Testing and Materials, 1992. Kalthoff J F, Beinert J and Winkler S: Measurements of dynamic stress intensity factors for fast running and arresting cracks in DCB specimens. STP 627, 161 - 176, American Society for Testing and Materials, 1977. Kanninen M F: A dynamic analysis of unstable crack propagation and arrest in the DCB test specimen. Int. J. Fract. 10 (1974), 415 - 430. Kanninen M F, Popelar C H and Gehlen P C: Dynamic analysis of crack arrest in contoured and straight- sided DCB test specimens for wedge loading and machine load condition. STP 627, 1938, American Society for Testing and Materials, 1977. Kanninen M F, Ahmad J, Papaspyropoulos V, Hudak S J, FitzGerald J H: An assessment of dynamic fracture mechanics for the analysis of crack arrest in a pressurized thermal shock event. Electric Power Research Institute Report EPRI NP - 4043s, May 1985. Kanninen M P and Popelar C H: Advanced fracture mechanics. Oxford University Press, 1985. Kanninen M F, Dexter R J, Polch E Z and Popelar C H: The analysis of crackpropagation and arrest in welded LNG storage tanks. GRI Report GRI- 86/0006, 1986. Kobayashi T and Dally J W: Dynamic photoelastic delamination of the a- K relation for 4340 alloy steel. STP 711, 189- 210, American Society for Testing and Materials, 1980. Kobayashi T and Giovanola J H: Crack opening profile observations for dynamic cleavage crack propagation and arrest. J. Mech. Phys. Solids., 37 (1989a), 759-777. Kobayashi T and Giovanola J H: Dynamic crack behaviour in a field of rising stress intensity and increasing toughness. Proc. Conf. Dynamic Fracture Mechanics, ASME PVP 160, 55- 63, American Society of Mechanical Engineers, 1989b. Kussmaul K and Gillot R: Riβstopp-Untersuchungen bei hohen Temperaturen an keilbelasteten Compact Proben. Materialprüfung, 29 (1987), 340- 345 (in German). Kussmaul K and Gillot R: Determination of crack arrest toughness at high temperatures using compact specimens. EGF/ESIS8, 381- 399, Mechanical Eng. Publ, London, 1991. Kussmaul K, Gillot R and Elenz T: Full thickness crack arrest investigations on compact specimens and a heavy section wide plate. Proc. Conf. 17th MPA Seminar, MPA Stuttgart, Germany, 1991. Lessels J and Leggett J: Crack arrest properties of carbon manganese structural steels: Part I and II. Metal Constr. and Brit. Weld. J. (May, July 1971), 103- 197 and 266-268. Lidbury D P G, Druce S G and Tomins B: Crack arrest: Some comments on microscopic and macroscopic aspects in relation to the assurance of structural integrity. Seminar on Crack

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Arrest Concepts for Failure Prevention and Life Extension, Abington, Cambridge, Sept 1995, Abington Publishing, 1996. Marschall C W, Rosenfield A R, Landow M P, Mager T R and Tagart S W (Jr): Radiation effects on crack-arrest toughness . Proc. Conf. Environmental Degradation of Materials in Nuclear Power Systems - Water Reactors, 384- 394, 1983a. Marschall C W, Mincer P N and Rosenfield A R: Subsize specimens for crack arrest testing. STP 791, Vol.II, 295- 319, American Society for Testing and Materials, 1983b. Marschall C W, Rosenfield A R and Landow M P:Specimen size considerations in crack arrest testing of irradiated RPV steels. STP 888, 339- 352, American Society for Testing and Materials, Philadelphia, PA, 1986. Marston T U, Stahlkopf K E and Smith E:An overview of crack arrest as it applies to reactor vessel integrity . Proc. Conf. 5th Structural Mech. React. Technol. (SMIRT), 6 , Paper G216, North Holland, 1979. Marston T U, Smith E and Stahlkopf K E:Crack arrest in water cooled reactor pressure vessels during loss-ofcoolant accident conditions . STP 71 1, 422- 431, American Society for Testing and Materials, 1980. Nichols R W: Fast and brittle fracture studies related to steelpressure vessels. Proc. Roy. Soc., 285A (1965), 104- 119. Pellini W S and Puzak P P:Fracture analysis diagram procedures for the fracture-safe design of steel structures. Welding Res. Counc. Bull. No.88, May 1963. Pellissier-Tanon T, Sollogoub P and Houssin B:Crack initiation and arrest in a SA 508 Class 3 cylinder under liquid nitrogen thermal shock experiment. Proc. Conf. 7th SMIRT, Vol.G, Paper GlF18, 137- 142, August 1983. Pennell W E: Heavy section steel technology program: Fracture issues. Journal of Pressure Vessel Technology, 114 (August 1992), 255- 264. Planman T, Wallin K and Rintamaa R:Comparison of some indirect measure of crack arrest. Proc. Seminar Crack Arrest Concepts for Failure Prevention and Life Extension, Cambridge, Sept 1995, Abington Publishing, 1996. Pugh C E, Corwin W R, Bryan R H and Bass B R:Some advances in fracture studies under the heavy-section steel technology program . Nucl. Eng. Des.,96 (1986), 297- 312. Pugh C E, Bryan R H, Bass B R and Nanstadt R K: Evaluation of fracture models through pressurised thermal shock testing. EGFESIS 8, 283-306, Mechanical Engineering Publications, London, 1991a. Pugh C E, Naus D J, Bass B R and Keeney - Walker J: Crack arrest behaviour of reactor pressure vessel steels at high temperatures. EGFESIS 8,357-380, MEP, London, 1991b. Puzak P P, Eschbacher E W and Pellini W S:Initiation and propagation of brittle fracture in structural steels. Weld. J (Dec. 1952), 561-s to 581- s. Puzak P P, Schuster M E and Pellini W S:Crack starter tests of ship fracture and project steels. Weld. J. (October 1954), 481-s to 496-s. Puzak P P, Babecki A J and Pellini W S:Correlations of brittle fracture service failures with laboratory notch-ductility tests. Weld. J (Sept 1958), 391- s to 410- s. Robertson T S: Propagation of brittle fracture in steel. Joum. of the Iron and Steel Institute,175 (1953), 361 - 374. Roos E and Griesinger H: Crack arrest investigations on rotating discs. Int. J. Pres. Vess. & Piping, 30 (1987), 23-36. Rosenfield A R: Validation of compact specimen crack arrest data. Journ. Eng. Mat. Techn,106 (1984), 207 - 208. Rosenfield A R, Mincer P N, Marschall C W: High temperature crack arrest toughness measurements using compact specimens. STP 945, 73-85, American Society for Testing and Materials, Philadelphia, PA, 1988.

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Sakai Y, Miya K, Takahashi I and Ando Y:Properties of heavy section steel plates and forgings for nuclear vessels produced in Japan - Part 2. Fracture Toughness. Int. J. Pres. Ves. & Piping, 22 (1986), 79- 109. Schonenberg R Y and Norris D M: Moment modified compact tension specimen for measuring crack arrest toughness. Nucl. Eng. Des., 96 (1986), 277-286. Schwartz C W and Bass B R: Evaluation of the presence of constraint in crack run/arrest events. Proc. Conf. 15th Water Reactor Safety Information Meeting, Report No. NUREG/CP0091, 113 -122, 1987. Smedley G P: Prediction and specification of crack arrest properties of steel plate. Int. J. Pres. Vess. & Piping, 40 (1989), 279- 302. Smith E: Use of the reflectionless stress intensity factor procedure to predict crack arrest during a hypothetical PTSE: The effect of initial cracksize . Int. J. Pres. Vess. & Piping, 49 (1992), 317325. Smith E: Relationship between the normal operation temperature for a nuclear reactor steel pressure vessel and the onset of upper shelf temperature. Int. J. Pres. Ves. & Piping, 68, (1996), pp45-52. Sumpter J D G: A strategy for assessing weld metal pop-in, Inst. J. Offshore and Polar Engineering, 1 (1 991), 208-2 11. Sumpter J D G and Curr R M: Crack arrest concepts for structural safety justification. Proc. Seminar Crack Arrest Concepts for Failure Prevention and Life Extension, Cambridge, Sept 1995, Abington Publishing, 1996. Sumpter J D G and Hayes B: Demonstration of crack arrest capability for welds showing low toughnesspop-in , Eng. Fract. Mechs. 43 (1992), 855- 862. Tanaka K, Sato M and Ishikawa T: Fatigue COD and short crack arrest tests. Proc. Conf. Fracture Toughness Testing: Methods, Interpretation and Applications, paper 18, The Welding Institute, 1982. Vasoukis G: Fraktographie und Analyse des Kerbschlagbiegeversuchs . Techn. Wissenschafth, Berichte, MPA, Stuttgart, No.71- 01, 1971 (in German) Wiesner C S, Hayes B and Willoughby A A. Crack arrest in modern steels and their weldments -comparison between small and large-scale experiments. Int. J. Pres. Ves. & Piping, 56 (1993), 369 - 385. Wiesner C S, Hayes B, Smith S D and Willoughby A A: Investigations into the mechanics of crack arrest in large plates of 1.5%Ni TMCP steel . Fat. Fract. Eng. Mat. Struct.17 (1994), 221233. Wiesner C S and Pisarski H G: The significance of pop-ins during fracture toughness tests 3R International, 35 (1996), 638-643. Wiesner C S and Hayes B: A review of crack arrest tests, models and applications, Proc. Seminar Crack Arrest Concepts for Failure Prevention and Life Extension, Paper 3, Abington Publishing Ltd, Cambridge, UK, 1996. Wiesner C S (Technical Director): Crack Arrest Concepts for Failure Prevention and Life Extension , Proc. Seminar, Abington Publishing Ltd, Cambridge, UK, 1996a. Wiesner C S: Predicting structural crack arrest behaviour using small-scale material characterisation tests, Int. J for Pressure Vessel and Piping, 69 (1996b), 185-196. Wiesner C S, Garwood S J and Denham B: The specification of crack arrest properties for storage tanks background and recommendations, ASTM STP 1337, Symposia on Design Criteria to assure Structural Inetegrity, New Orleans, May 1997. Willoughby A A and Wood A M: Crack initiation and arrest in 9%Ni steel weldments at - 118, liquefied natural gas temperature . Proc. Conf. Transport and storage of LPG and LNG, 111 Koninklijke Vlaamse Ingenieurs vereniging, Antwerpen (Belgium), 1984. Yoshiki M and Kanazawa T: On the mechanism of propagation of brittle fracture in mild steel. J. S oc. Nav. Arch. Japan, 102 , (1958), p.39 (in Japanese).

HEAT TRANSFER PROCESSES IN REACTOR VESSEL LOWER PLENUM DURING LATE PHASE OF IN-VESSEL CORE MELT PROGRESSION B.R. Sehgal, V.A. Bui, T.N. Dinh, and R.R. Nourgaliev Division of Nuclear Power Safety Royal Institute of Technology Brinellvagen 60, S-10044 Stockholm, Sweden

ABSTRACT Late phase in-vessel core melt progression during the course of a hypothetical severe (meltdown) accident in a light water reactor (LWR) is considered, with particular emphasis on thermal processes occurring in the lower plenum of the reactor pressure vessel (RPV). The formation of a melt pool, from the initial state of a uniform composition, dried out, debris bed, is investigated. The objective of this work is to analyze the effects of various heat transfer phenomena, which are of potential importance to the vessel thermal loading. In particular, numerical investigation of heat transfer in a metallic layer was performed for cases with different boundary conditions. A two-dimensional model was developed to simulate heat transfer in naturallyconvected, internally-heated corium melt pool and in metallic layer heated from below. The model solves the energy conservation equation in general curvilinear coordinates, while taking into account homogeneous-orthotropic anisotropic heat conduction. A method is developed to describe buoyancy-induced convective and turbulent heat transport by means of effective heat, conduction and pseudo-convective (effective velocity) terms. Extensive validation was performed with analyses of simulant experiments and good agreement was obtained between calculated parameters (energy splitting, transient and steady temperature fields, local heat. flux distributions) and measured data. These included cases with different boundary conditions, and geometries, and high Rayleigh numbers. This method provides a sufficiently mechanistic modeling in an integrated manner. The method was applied, in two-dimensions, to analyze the heat transfer and phase change processes in the RPV lower plenum in several selected scenarios of core melt-down accident progression. It was found that the multi-dimensional representation of heat conduction in, and melting of, the vessel wall significantly reduce the ’focussing effect’ of a thin metallic layer resident on top of an oxidic melt pool, compared to the earlier predicions of thermal loadings at the vessel wall obtained with 1-D representation. Advances in Nuclear Science and Technology, Volume 26, edited by Lewins and Becker. Kluwer Academic /Plenum Publishers, New York 1999.

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1

INTRODUCTION

During the course of a hypothetical severe (core melt-down) accident in a LWR, molten core materials may be relocated to the RPV lower plenum in large quantities. Prediction of the survival and integrity of the RPV lower head during core melt-vessel interactions has been the subject of several experimental and numerical investigations, for both pressurized water reactors (PWRs) and boiling water reactors (BWRs) reactors. A review of physical phenomena involved in, and results of previous studies related to, the assessment of RPV thermal loading and vessel failure, can be found in the work by Rempe et al. (1993) [1]. The present work focuses only on the last phase of in-vessel core melt progression, i.e. a debris of core materials is assumed to be present in the lower plenum. A thorough analysis of in-vessel coolability and retention of core melt was performed by Theofanous et al. (1995) for the AP-600 reactor design, with external vessel cooling [2]. The vessel thermal loading was examined for the terminal steady-state configuration of a melt pool, with a set of heat transfer correlations, which were obtained from simulant experiments. Essentially, a one-dimensional approach was employed with an energy-balance formulation. It was determined that external vessel cooling, as a severe accident management strategy, provides sufficient thermal margins against the thermal loading imposed. The current study is motivated by the desire to analyze a range of in-vessel core melt progression scenarios, instead of the terminal steady state, for different reactor designs. In particular, assessments of severe accident management schemes, proposed for existing PWRs and BWRs, and for advanced LWRs, need to evaluate the timing and modes of the reactor vessel failure. In addition, recently, the in-vessel debris coolability has been advocated as a possible mechanism for in-vessel retention of core debris. Evaluation of these accident management applications necessitate the development of an integrated-mechanistic-modeling approach to describe and predict in-vessel debris coolability and retention. The most-known models for melt pool formation are those incorporated in the codes MAAP [3] and SCDAP/RELAP [4]. The MAAP model employs the lumped-parameter approach with transient pool development controlled by energy balances. It employs chemical heat generation and provides a model for the hypothesized in-vessel coolability. The pool natural convection heat transfer at its boundaries is obtained through selected heat transfer correlations. A model named COUPLE has been added to SCDAP-RELAP code, which performs a two-dimensional finite element calculation for the heat conduction and the natural convection processes. Since COUPLE is strictly a heat conduction code, the convective heat flux at the phase change interface is simulated by means of effective conductivities. Natural convection is modeled by assigning a large value for thermal conductivity to all liquefied elements, combined with an effective conductivity to the phase change elements at the boundary of the liquefied pool. It is worth noting that the thickness of the phase-change (mushy) region may vary in time and, in the case of a fine computational grid, it is difficult to allocate a unique mushy node to specify the effective heat conductivity. Although interrelationship between different heat transfer mechanisms has been accounted for in such codes and models, they remain restricted to quasi-steady-state situations, based on a priori configurations of core debris in the lower plenum. Another aspect of previous modeling efforts is that they directly employ empirical heat transfer correlations obtained from simulant and/or small-scale experiments. This approach has uncertainties, when extended to reactor scale prototypic situations; see, for example, Dinh et al. (1996) [5] for a review and analysis of non-prototypic effects on core melt pool heat transfer calculations.

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A research program was initiated and developed at the Royal Institute of Technology (RIT, Stockholm) to provide new knowledge regarding the relevant heat transfer rates and to support modeling activities. First, multi-dimensional computational fluid dynamics (CFD) methods are employed to gain insights into the melt natural convection and heat transfer processes; see e.g, [6] [7] [8]. These insights formed the basis for developing mechanistic models (such as the effective conductivity-convectivity model of melt pool heat transfer [9] and debris heating and melting model [10]), which maintain major dependencies, and enable coupling with models for other processes. Relatively fast-running two-dimensional (2D) models of separate processes are then implemented in an integrated model for assessments of reactor accidents. Such a model, named MVITA1 , was developed, validated and employed for reactor predictions at, RIT [11]. The objectives of the present paper are to describe the modeling approach, the MVITA model development, validation and application. Notably, the effects of the possible presence of a metallic layer above the oxidic pool, or debris, have also been incorporated in the MVITA model. Such a metallic layer was found to have a profound ’focussing’ effect [2] to enhance thermal loading in the vessel wall region adjacent to the metallic layer.

2

MODELING OF NATURAL CONVECTION HEAT TRANSFER IN CORE MELT POOL

2.1 Background Natural convection heat transfer in a core melt pool has been the major focus of a number of studies. These studies have been supported by experimental programs conducted at various facilities. The COPO experiments, conducted in Finland, have provided data for heat fluxes as functions of the azimuthal angle for a two-dimensional slice representation of the torospherical lower head of the Loviisa reactor [12]. The UCLA experiments [13] have employed a three-dimensional vessel head at considerably smaller scale than the COPO facility. A program named BALI [14] has been in progress in France, using a full scale slice geometry representation of a the spherical lower head of the French PWRs. These experimental programs have used simulant materials to represent the corium melt and the vessel wall. An OECD/Russian experimental program, named RASPLAV, using prototypic materials for core melt and vessel is also in progress at the Kurchatov Institute, Moscow. Heat removal in the large decay-heated core melt pool, resident in the lower head of the vessel of a LWR, is through natural circulation flows at high Rayleigh number (1014 – 1016). The flow regime is turbulent and the imposed heat flux on the vessel wall varies greatly over the azimuthal angle. A significant fraction of the heat generated in the pool has to be removed through its upper cooled surface. It has been observed that the flow in the upper pool portion is unsteady, with countercurrent flows of ascending plumes and falling cooled blobs. Heat transfer to the side wall is controlled by the boundary layers developed in the flows moving downwards along the cooled wall, thereby varying the imposed heat flux on the vessel wall along the azimuthal angle. The COPO data [12], measured at Rayleigh numbers of 1014 – 1015, have shown that the heat flux is extremely low at the bottom part of the head, but, rises to substantial values in the upper elevations of the head; and along the vertical wall of the vessel. This measured behavior is actually beneficial, since the heat removal capability, due to the boiling process on the outside surface of the vessel wall, also varies in the same fashion; and a good matching may be obtained. MVITA stands for Melt Vessel Interaction - Thermal Analysis.

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Prediction of the heat fluxes, in prototypical situations [2], have relied primarily on measured correlations ([12] [23] [27] [28]). Calculations based on representation of the turbulence in the high-Rayleigh-number melt pool, through the standard (k– ε) models, have not been successful, since analyses of the COPO and the UCLA experiments, performed with these models, have shown large discrepancies between the measured and the calculated results. It has been found that advanced turbulence models, would be needed to correctly predict the flow field, and the heat transfer characteristics, of the unstably-stratified region of highRayleigh-number melt pools [6],[7]. Studies have been performed to develop and validate a theoretical model of natural convection heat transfer in heat-generating pools, in which only the heat conduction equation is solved over the pool domain; and the effect of natural convection flows is accounted for by effective directional heat conductivities. This effective diffusivity model was successfully used to describe the natural convection heat transfer in fluid layers and pools with simple boundary conditions (one cooled surface) [15] [16]. However, its validation for more prototypic boundary conditions is not yet available. In the next section, a novel simulation method, based on an effective conductivity-convectivity approach, is introduced.

2.2

The Effective Conductivity-Convectivity Model (ECCM)

The mathematical formulation of the heat transfer problem is based on the energy conservation equation:

(1) In this study, the temperature distribution T in the pool is assumed to be axisymmetric, twodimensional, and governed by the energy conservation equation, while taking into account the transient homogeneous-orthotropic anisotropic heat conduction process [17].

(2) with qx = – kx.gradT, qy = –ky .gradT. qx, qy, kx, and ky are the heat fluxes and (effective) heat conductivities on the vertical (x) and horizontal (y) directions, respectively; U, V are the (effective) vertical and horizontal velocities; n has the value 0 in the Cartesian coordinate system and 1 in the axisymmetric, cylindrical coordinate system. This equation is then rewritten for a general curvilinear coordinate system [x = x( ξ, η), y = y( ξ, η)] in order to account for the geometry of liquid (melt) pools. Such complex geometry pools are divided into several computational domains, which are ’sewn’ together by a special computational treatment. Numerical method: The resulting energy conservation equation in a general curvilinear coordinate system (ξ , η) has the following form [18]

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(3) (4) (5) (6) (7)

Figure 1: A computational control volume. The energy conservation equation is solved iteratively in all computational domains, using control-volume methodology [19]; Fig.1. The values of temperature on the inter-domain boundaries are redefined after each iteration, and are used as the domain boundary conditions for the next iteration. The convergence of iterative procedure is controlled by the energy balance. The modeling approach: The major features of the modeling approach are based on the observed flow fields. It can be stated as follows. The heat transfer inside the naturalconvection liquid (melt) pool with a volumetric energy source, is assumed to be driven by two mechanisms: (a) the vertical upward movement of plumes delivering heat to the upper boundary; and (b) the horizontal heat transfer to the cooled side wall through the liquid boundary layer developing downwards along the cooled curved wall. In the present work, the first heat transfer mechanism is modeled using a new method (named the effective convectivity approach), in which the convective heat transfer is defined through the heat transfer coefficients on the boundaries of the melt pool. For decay-heated melt pools, the horizontal velocity is neglected and the effective upward velocity is estimated by the simple heat balance equation. The second mechanism is modeled by means of the effective diffusivity approach, in which the horizontal heat conduction coefficient kx for a vertical position y is obtained from the heat transfer coefficient as a function of boundary-layer development length, counted from the upper edge of the pool side wall, expressed as an Eckert-type correlation. Thus, the horizontal velocity component V is neglected and the vertical heat conductivity is assumed unchanged ky = k.

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2.3

Model derivation and basic validation

In this section expressions for effective velocity U and effective conductivity kx are derived, based on consideration of the three simple cases of internally heated pools, i.e., fluid layers. 2.3.1

Fluid layer cooled from top (effective vertical velocity in the unstablystratified region)

Figure 2: A fluid layer cooled from the top. The natural convection in a heat generating fluid layer cooled from its top (Fig.2) is characterized by an unstably stratified flow pattern, in which the average flow moves mainly upwards and carries most of the heat generated within the volume to the isothermally-cooled boundary. Simultaneously, cooled liquid tongues fall down in a chaotic manner, carrying liquid mass back to the lower region of the fluid layer. In this a fluid layer, the time-averaged velocity is zero. The time-averaged temperature distribution within the fluid layer is quite uniform, except for the very top part near the cooled surface. In such a case the heat may be assumed to be removed from the fluid layer, only, by the average flow with upward velocity U, that may be defined from the heat balance equation pcU ∆T = QvL. Thus, turbulent upward heat transport is represented by an effective velocity U. Expressing the relation between heat flux QvL and temperature difference ∆T through a Nusselt number, we obtain the following expression for U: (8) where Nuup may be determined from the experimental correlation by Kulacki et al. [20] [21]: (9) It should be noted here that the effective upward velocity U has the same order of magnitude as the turbulent fluctuation velocity; value of U is of the order of 10–4 m/s for Ra ~– 1015. Kulacki et al. [20] ,[21] reported measurements of transient temperature distribution across volumetrically heated horizontal fluid layers, which were cooled from top and thermally insulated at the bottom. Measurements were performed for wide ranges of Rayleigh numbers and layer depths, L. The calculated results of the steady-state and transient temperature profiles (lines) with ECCM, presented in Figs.3-5, are in reasonably-good agreement with the experimental data (filled points).

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Figure 3: Steady-state temperature distribution in fluid layers cooled from top.

Figure 4: Transient temperature profiles (Ra = 1.18 × 1010) in a fluid layer cooled from top.

2.3.2 Fluid layer cooled from top and bottom (effective vertical velocity in stably-stratified region) In this case, the fluid layer can be divided into two sublayers with distinctive effective velocity for each sublayer. The upper sublayer is unstably stratified and the generated heat is removed by the effective velocity Uup to the upper cooled surface, eq.(10). The lower sublayer is stably stratified, with heat removed upwards by the effective velocity Ulow and downwards by heat conduction; eq.(11). The heat balance in both parts of the layer can be represented as follows: (10) (11)

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Figure 5: Transient temperature profiles (Ra = 5.59 × 104) in a fluid layer cooled from top.

Figure 6: A fluid layer cooled from top and bottom. Thus, the effective velocities can be calculated from the following correlations (12) (13)

(14) (15) Calculations were performed for layers of different sizes, with wide ranges of Rayleigh numbers, and their results are compared to experimental data obtained by Kulacki and Gold-

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stein [24]. The calculated fraction of energy transported to the lower boundary is presented in Fig.7 and is found to be in good agreement with experimental data. Fig.8 shows the temperature profiles obtained by calculations (lines) in comparison with experimental data (points) for different Rayleigh numbers. The discrepancies in these comparisons are within the uncertainty range of the measurements, and of the experimental correlations used in the present work.

Figure 7: Fraction of heat down in fluid layers cooled from top and bottom.

Figure 8: Temperature profiles in fluid layers cooled from top and bottom. 2.3.3

Side-cooled liquid cavity (effective horizontal heat conductivity)

In the case of cavities cooled from the side, heat transfer to the walls depends on the characteristics of the boundary layer developing along the vertical wall (Fig.9). The heat transfer regime might change from laminar to turbulent if the local Rayleigh number exceeds a critical value. Because such a boundary layer development occurs only in a very narrow region next to the cooled boundaries, and the heat transfer in the horizontal direction is dominant, the proposed approach, based on average (effective) velocities, may not be applicable. In this

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Figure 9: Square cavity cooled from side walls. case the effective diffusivity approach [15] [16] may be used. The heat conductivity in the horizontal direction, k x is defined from the following correlation: (16) where T is the cavity, horizontally-averaged temperature for a vertical position. Therefore, (17) where is the local Nusselt number. There exist two options to determine the local heat transfer coefficient. In the first approach, experimentally obtained (for specific pool geometry) correlations like Nu = f(Ra) and shape functions of the local heat flux distribution can be utilized. However, this would significantly reduce the prediction capabilities of the simulation model for different geometry pools. In the present work, heat transfer coefficient h is determined from the boundary layer concept. This second approach is based on the assumption that a boundary layer of the descending flow is developing along the cooled sidewall surface. The flow in the boundary layer may be laminar or turbulent, depending on the local Rayleigh number Ray, which is based on the boundary-layer development length from the leading edge of the cooled surface, y', and the temperature difference across the boundary layer, ∆Tw : (18) In the simplest case of laminar natural convection heat transfer from an inclined surface, the Eckert-type correlation proposed by Chawla and Chan [25] can be applied to define Nu y as follows: (19) giving Calculations were performed for a cavity cooled from its sides with the conditions listed in the experiments performed by Steinberner and Reineke [23]. The size of the cavity is 0.8 m x 0.8 m. The calculated results are presented in Fig.10, together with the experimental data. The calculated local distributions of Nus(y) exactly reproduce the experimental dependency eq.(19). It can be seen that reasonable agreement between the calculated and measured

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Figure 10: Local distribution of Nus at the cooled vertical side wall. Nusselt numbers is achieved for the upper part of the sidewall surface (y’/L ≤ 0.75), while differences are observed for the lowermost portion of the vertical surface. In cases with a laminar boundary layer along the vertical surface (according to ref.[23]), the measured data at the lowermost portion of the vertical surface are lower than the calculated values. This is due to the fact that interaction between the boundary layer and the cavity’s bottom surface, as well as with the stagnant liquid layer at the lower part of cavity was not represented in the present model, since Nuy presents the heat transfer across the one-dimensional free boundary layer along an infinite heat-exchange surface. In addition, the laminar boundary layer model can not adequately describe the heat flux distribution in case of transition to turbulence (Ra = 3.71 x 1013 [23]). For Ray > Ray,transition, the turbulent heat transfer correlation (Nuy ~ Ray1/ 3 ), developed for turbulent boundary layer [25], should be applied. In the turbulent part of the vertical surface, the local heat flux distribution is uniform, i.e. independent of y'.

3

VALIDATION OF ECCM FOR MELT POOL NATURAL CONVECTION HEAT TRANSFER

In this section the model is applied to calculate heat transfer in internally-heated liquid pools with varied boundary conditions and complex geometry. Simulant material experiments available in the literature are analyzed, with particular emphasis on validation of the model.

3.1 Fluid layer with one spherical boundary The following analysis pertains to the experimental study by Min and Kulacki [26] on convective heat transfer in liquid pools with uniformly distributed volumetric energy sources, in which measurements of the transient temperature distribution along the centerline of the pool were performed. The pool was bounded from below by the segment of a spherical shell, and from the side, by a cylindrical wall both maintained at zero heat flux (adiabatic), and from above by a horizontal surface maintained at constant temperature. In Fig.11 the calculated transient temperature profiles along the pool centerline are compared to the experimental data. The model provides good agreement for the steady-state temperature distribution,

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while overpredicting the transient from an initial uniform temperature with a step change in Rayleigh number. This may be related to the effect of the bottom surface curvature and the small aspect ratio on the two-dimensionality of the flow pattern under relatively low Rayleigh number conditions, which were present in the experiment.

Figure 11: Comparison with experimental data of Min and Kulacki (1978), Run 148.

3.2 Square cavities with all walls cooled The experiments of Steinberner and Reineke [23] in a Joule-heated, water filled, rectangular cavity, with all walls cooled, were chosen for validation of the model. Water was the working fluid. For quasi-steady state conditions, the internally-generated heat has to be removed through the isothermal vertical and horizontal boundaries. For this case, both the concepts of effective velocity, and effective heat conductivity, have to be employed in order to describe the heat transport in the cavity. Calculations were performed for three cases for which data was obtained (Ra = 1012; 2.15. 1013 and 3.52. 1013). Holographic observations showed that the boundary layers along the vertical cooled wall in the first two cases are laminar, while in the third case transition to turbulence may have occurred in the region near the bottom surface. In the present analysis, the model of laminar boundary layer (along the vertical wall) is applied to calculate the effective horizontal heat conductivity kx. Comparison of the calculated results with the experimental data shows that the model is capable of providing the distributions of the heat removed through the top, bottom, and side walls of the cavity (see Fig.12). In addition, the local distribution of Nus at the vertical wall can also be described (Fig.13) by the model.

3.3 Complex geometry pools with all walls cooled In this section, three sets of experiments, on (i) semicircular cavity by Jahn and Reineke .. .. [27]; (ii) hemispherical pool by Asfia and Dhir [13]; and elliptical cavity (COPO) by Kymalainen et al. [12] are analyzed. The common feature of these experiments is that the test sections include a curved cooled bottom surface, a characteristic of a molten corium pool inside the

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Figure 12: Ratios of heat removed to top, bottom and side walls.

Figure 13: Local distribution of Nus number at the vertical wall (all walls cooled). reactor pressure vessel lower head (Fig.14). The UCLA and COPO experiments are also distinguished by high Rayleigh numbers (1011-1014 in the UCLA experiments and 1.34. 10141.61 . 1015 in the COPO experiments) achieved. In order to apply the proposed modeling approach, we rewrite eq.(16) for the heat flux q(z). perpendicular to the wall at vertical location z, into a more general form as follows:

(20) Here γ(z) is the angle of inclination of the wall from vertical direction at the vertical location z. Also, the gravitational acceleration g in eqs.(18-19) should he defined as g- = g cosγ (z). It should he noted that most of correlations for Nuy [like eq.(19)] are valid for γ ≤ 300, so we assume that only kx changes along the cooled wall and takes its ordinary value (k x = k) for γ ≤ 60° . Then kx has the form

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Figure 14: Semicircular and hemispherical pools.

(21) Fig.15 shows a typical decomposition of computational domains and nodalization used in the 2D simulation of heat transfer in hemispherical geometry pools. Sensitivity studies performed showed that the calculated results are sufficiently robust, and independent of grid resolution and domain decomposition technique, providing non-zero Jacobian [eq.(4)] for each subdomain.

Figure 15: Decomposition of computational domains and grid. 3.3.1

Semicircular cavity experiments

Calculations were performed for a semicircular cavity of 75 mm radius with Rayleigh is number ranging from 1.4 × 108 to 1.4 × 1013. The calculated distribution of depicted in Fig.16 and compared with experimental data obtained by Jahn and Reineke [27].

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The average Nuup and Nudn received from calculations (points) are shown in Fig.17 and agree well with the experimental results (lines).

Figure 16: Local distribution of Nusselt number on the curved surface of the semicircular cavity.

Figure 17: Average Nusselt numbers (semicircular cavity). 3.3.2

Hemispherical pool experiments

In the UCLA experiments [13], the pool had hemispherical form and contained Freon-113, which was volumetrically heated using microwave energy. For comparison with the UCLA experimental data, calculations were performed for hemispherical pools cooled both from the upper flat surface and from the curved surface with geometrical ratio H / R set equal to 1.0 and 0.43, respectively. The calculated results for Nudn are presented in Fig.18 and compared with Asfia and Dhir’s experimental data [13]. As reported in Asfia and Dhir’s study, as well as in studies by other authors, large variation in heat transfer coefficient along the pool curved surface has been observed. The heat transfer coefficient is lowest at the stagnation point and increases almost linearly along the curved surface. The calculated results for the heat transfer ratio are presented in Fig.19 and compared with the measured data [13]. The angular variation in heat transfer is well represented by the model for hemispherical pool geometry.

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Figure 18: Comparison of the calculational results with other analytical and experimental results (hemispherical pools).

Figure 19: Ratio of the local to average Nusselt numbers on the bottom curved wall of the hemispherical pool. 3.3.3

Torospherical cavity experiments

The COPO experiments [12] used a two-dimensional ’slice’ of the Loviisa PWR lower head (including a portion of the cylindrical vessel wall) and ZnSO 4 − H2 O solution, as simulant for the melt. The pool was heated by electrical current between flat wall-electrodes. The calculations were performed for the conditions corresponding to some of the COPO tests: runs No.32f and No.29a with 80 cm-depth pool, and runs No.40a and No.42c with 60 cmdepth pool. The calculated results for Nuup, Nusd, and NuC,dn are presented in Figs.20-22 together with the COPO data, and the experimental correlations by Steinberner and Reineke [23] for the upper surface Nusselt number Nuup, and the sidewall vertical surface Nusselt number Nusd, i.e.,

(22)

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and (23) and with the experimental correlation of Mayinger et al. [28] for downward heat transfer, i.e.,

(24) where H is the pool depth and R is the radius of curvature of the segment. In general, good agreement between the predicted and measured data was obtained. The underprediction of Nusselt number on the upper pool surface (Fig.20) is related to the use of Kulacki correlation for Ra > 1015.

Figure 20: Variation of upward heat fluxes with Rayleigh number (COPO torospherical slice).

Figure 21: Variation of the heat fluxes on the vertical boundary of the COPO torospherical slice with Rayleigh number. Since the Rayleigh numbers (Ra and Ray ) in the COPO experiments were high, the model of turbulent boundary layer is used in determining the effective heat conductivity Kx.

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The predicted heat flux on the side wall (vertical portion) is almost uniform, as confirmed by COPO experimental data. The distribution of the calculated heat flux on the curved portion of the pool is shown in Fig.23 and is in reasonably good agreement with the COPO experimental data.

Figure 22: Variation of downward heat fluxes with Rayleigh number (COPO torospherical slice).

Figure 23: Distribution of heat flux on the lower curved pool boundary (COPO torospherical slice).

It is seen that the proposed effective convectivity-diffusivity model is capable of describing major heat transfer characteristics of internally-heated liquid (melt) pools for all the cases investigated. The approach can provide reasonable temperature distributions in the pools during steady and transient conditions. The energy splitting (fraction of heat removed through different cooled surfaces) and the local distribution of heat flux on the vertical and curved boundaries can also be predicted by the model. More importantly, the simulation method developed has been adopted to include modeling of heat, transfer and phase change processes in a multiphase (liquid, mushy, and solid) domain, using a fixed grid, temperature-based, enthalpy method [11]. New knowledge on natural convection heat transfer, should it become available from experiments or analyses, can be easily introduced in the present model.

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4.1

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HEAT TRANSFER IN MOLTEN METAL LAYER: COMPUTATIONAL FLUID DYNAMICS (CFD) ANALYSIS Background and problem formulation

It was demonstrated by Theofanous et al. in [2] that during the late phase of core melt progression in the AP-600 reactor pressure vessel, with external vessel cooling, the core configuration that will produce the most thermal loading includes ” all the core debris, as either solid oxidic debris or as a molten oxidic pool in the lower head” , and ” a molten steel layer on top of the oxidic pool, made up of all the ’light’ components of the core region, the reflector and core support plate, and perhaps, a portion of the core barrel, and lower (plenum) internal structures” . Schematically, this is shown in Fig.24.

Figure 24: Schematic of in-vessel melt retention phenomenology.

A comprehensive analysis of heat transfer in the metallic layer and the assessment of its impact on the in-vessel melt retention thermal margin were performed in [2]. Two heat transfer mechanisms, namely Rayleigh-Benárd natural convection [ N U RB = f( RaR B )] in a fluid layer heated from below and natural convection heat transfer to an isothermal, cooled, side vertical wall [ N uvw = f(Ravw )] of the fluid layer, were accounted. The experimental correlation of Globe and Dropkin [30] was applied to describe the former mechanism, while Churchill's and Chu's correlation [31] was used for the latter. It should be noted that the above correlations for the surface-averaged heat transfer coefficients were obtained from separate experiments. In order to examine the use of the above correlations for the mixed convection problem of the molten steel layer, MELAD experiments were designed and performed [2] (MELAD stands for MEtal LAyer Demonstration experiments). The MELAD experimental data were used to demonstrate the validity of the assessment method developed to determine the energy splitting to the upper and side surfaces of the fluid layer, heated from below, and cooled from top and sides. However, no detailed information about the local heat flux distribution on the vertical wall was determined. Also, the effects of the fluid layer's aspect ratio. and of the boundary conditions were not investigated. In the present study, the method of direct numerical simulation is employed to investigate natural convection heat transfer in fluid layers heated from below. The AEAT-CFX 3D computer code [32] is used. Details of the direct numerical simulation method are provided in [7]. Both Rayleigh-Benárd natural convection, and mixed convection problems, are investigated. The former is included here for validation purposes. The formulation of the two convection modes is shown in Fig.25.

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Figure 25: The formulation and boundary conditions employed in the CFD simulations.

4.2

Rayleigh-Benárd natural convection heat transfer: validation

Both two- and three-dimensional direct numerical simulations of Rayleigh-Benárd natural convection heat transfer were performed. Validation of these computations against the O'Toole & Silveston's experimental correlations [33] for the range of RaRB from 4000 to 3.109 is provided in Fig.26, which shows the calculated (spatially and temporally) averaged Nusselt numbers on the top surface. It was found that the flow structure in the layer varies significantly over the wide range of Rayleigh numbers selected. It should be noted that for RaRB > 105, when turbulent convection is characterized by a disordered, highly transient flow pattern, the dependence of Nu RB on RaRB is in accordance with the 3_1 -power-law [33]. = 0.104

Pr

0 . 084 .

The fluid Prandtl number has only a minor effect in eq.(25). It was found that a 3D simulation is required for the prediction of Rayleigh-Benárd natural convection heat transfer for Rayleigh number RaRB > 105. Several boundary condition configurations were tested for the fluid layers. It was determined that quite similar heat transfer coefficients were obtained for both isothermal and constant heat flux boundary conditions at the top and bottom surfaces. Furthermore, application of either periodic or adiabatic conditions, at the vertical sides of the computational domain yields essentially equal upward heat transfer coefficients. 4.3

Mixed Natural Convection Heat Transfer

Direct numerical simulations were also performed for mixed convection, with isothermal vertical side walls T uw and isothermal upper wall Tup . Equal temperature was assumed for all cases presented in this section, i.e. Tuw = Tup . Since we define

(25)

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Figure 26: Rayleigh-Benárd natural convection heat transfer. f(Pr) = 1, if RaRB < 105 and f(Pr) = Pr 0.085 otherwise.

and Tb it can be shown that isothermal upper and vertical walls.

= 2 for the cases with the equal-temperature

Heat transfer rates at the upper surface agree well with O’Toole & Silveston’s correlations [33] and with data from the MELAD-A series [2] and from the Kirkpatrick & Bohn’s experiments [34]; as shown in Fig.27.

Figure 27: Upward heat transfer in a cavity with mixed boundary conditions. f ( Pr ) = 1. if RaRB < 105 and f(Pr) = Pr 0.085 otherwise (curves are O’Toole’s and Silveston’s correlations).

The sideward heat transfer rate is found to be in reasonable agreement with Churchill & Chu’s correlations for Ravw ≥ 106 [31], and with Chawla & Chan’s turbulent correlation [25] for surface-averaged Nusselt number Nuvw; eq.(26). Also, Fig.28 depicts reasonable agreement between the calculated results and the data of the (vertical-wall) surface-average heat transfer coefficient obtained from the MELAD-A experiments.

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(26)

Figure 28: Comparison of surface- and time-averaged values of Nuvw number against correlation of Chawla & Chan.

4.4

Parametric investigations

Due to the wide range of mixed convection heat transfer configurations, a complete discussion of all aspects here is impossible. Effects of the major parameters of importance to reactor safety assessments are presented below. 4.4.1 Local heat flux distribution Fig.29 depicts calculated dimensionless heat fluxes q* (normalized to the surface-average heat q— flux — v w ) over cooled vertical walls in mixed convection cases, for different Ravw numbers. A distribution of the heat flux, determined from the laminar boundary-layer solution by Chawla and Chan [25] is also presented in this figure for comparison. It can be seen that the calculated local heat fluxes are approaching zero at the upper corner since it is the junction between the two equal temperature isothermal walls. With increase of Rayleigh number (i.e. with enhanced mixing level in the fluid layers), the heat fluxes at the upper region of the vertical wall also increase. At the region near the bottom surface, where a hot surface is joined with the vertical cooled surface, infinitely large heat fluxes are calculated. This prediction is unreal, since the heat transfer process at the lower corner of the fluid layer (near the vertical cooled wall) would smear out such a singularity. However, at (very) low RaVW numbers, the peaking of heat flux near and at the lower corner of the fluid layer remains abnormally high.

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Figure 29: Local distribution of dimensionless heat flux over vertical cooled wall. Cavity with mixed boundary conditions.

4.4.2

Aspect ratio

For isothermal upper and vertical walls (of equal temperature), the aspect ratio is not an important parameter. Indeed, it was found from numerical analysis for a number of cases when RaRB and Ravw numbers are comparable, Rayleigh-Benard cellular structures exist at distances 2.H from the vertical walls, where H is the depth of the layer. Thus, the physical picture of heat transfer in the bulk fluid remains Rayleigh-Benárd natural convection, while a natural convection boundary layer is present at the vertical cooled walls (driven by the temperature difference ∆Tb = Tb - Tvw. However, for the cases with different temperatures of upper and vertical walls (and, hence. different governing temperature differences), in particular when Ravw >> RaRB, the cellular structure of Rayleigh-Benárd natural convection may be destroyed by flow circulation associated with strong descending flows at the vertical walls. Thus, heat transfer to the upper surface may be larger than that for the pure Rayleigh-Benárd convection. 4.4.3 Radiative vs. isothermal boundary conditions A smaller temperature difference for Rayleigh-Benárd convection, than that for the boundary layer convection on the cooled vertical wall, may occur for a thin metallic layer, with radiative heat flux boundary condition on the upper surface. Temporal and spatial variations of heat flux on the upper surface were calculated to be much less for radiative (free) boundary conditions than for isothermal walls. 4.4.4

Reactor-scale predictions

Calculations were performed for reactor-scale conditions. In these cases, crust formation was simulated by changing the local boundary conditions from free and radiative to rigid and isothermal. A uniform heat flux was assigned for the lower surface of the metallic layer. This heat flux was estimated from the upward heat flux of an oxidic melt pool (9.3 m3, Rapool = 3 . 1015, q 500 kW/m2). The calculated surface average heat transfer rates are shown in Figs.27-28.

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Calculated local heat flux distributions along the vessel wall for thin and thick metallic layers of height H are presented in Fig.30. As it can be seen, the local heat flux distributions on the vertical wall for cases with H = 0.02 and 0.2 m are similar to those shown in Fig.29. A somewhat different distribution was obtained for the thick metallic layer ( H = 1 m). An increase in the heat flux distribution in the region 0.05 < y /H < 0.3 of the thick metallic 1010) is attributed to a transition-to-turbulence in the vertical-wall boundary layer (Ravw layer.

Figure 30: Reactor-scale predictions of local heat flux distribution. Steel layers of H = 2 cm, 20 cm and 1 m. In general, it was found that the local peak heat flux occurs at the edge (upper and lower) regions of the vertical wall. The peaking, however, appears not to be significant in the reactor-scale situations (generally, less than 50%, except for the lowermost and uppermost regions; see discussion above on the local heat flux distribution). The assessment method, proposed in [2], is found to be valid for determining the energy splitting (to the upper and side surfaces) of the metallic layer, which is uniformly heated from below. The correlations of O'Toole and Silveston, eq.(25) [33], and the correlations by Chawla and Chan, eq.(26)[25], are recommended for determining the heat transfer coefficients on the upper and sidewall surfaces, respectively.

5

MODELING OF HEAT TRANSFER IN METALLIC LAYER BY ECCM APPROACH

In this section, the ECCM approach is extended to simulate natural convection heat transfer in liquid layers heated from below (Qu = 0). From experimental observations and CFD analysis, it is seen that the liquid layer, heated from below and cooled from above, is symmetric with respect to the boundaries. A characteristic velocity for turbulent heat transport, i.e. the effective velocity in the MVITA model, can be calculated as:

(27)

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Figure 31: Temperature field in the fluid layers heated from below. MVITA calculation.

NURB of Rayleigh-Benárd natural convection can be determined from O’Toole’s & Silveston’s correlations [33]. Fig.31 depicts the temperature field of liquid layers heated from below, calculated with the MVITA model. It agrees qualitatively and quantitatively with measured temperatures. In particular, it can be seen that the central region of the fluid layer is well mixed.

Figure 32: Heat transfer rate of Rayleigh-Benárd natural convection and of natural convection in cavity with mixed boundary conditions. MVITA calculations.

Thus, the general correlations of O’Toole & Silveston and of Chawla & Chan (for the local heat transfer coefficient) are implemented into the MVITA model for simulation of natural convection heat transfer in liquid layers heated from below and cooled from the top and side walls. Fig.32 depicts the calculated surface average heat transfer coefficients and for different Rayleigh numbers. It is seen that the measured dependences =f (RaRB) and = f (RaVW) were also reproduced by the MVITA code. For cases with side-cooled walls, the MVITA model, which includes both molecular heat conduction and effective conductivity, is capable of describing the heat flux distribution. Both limiting cases of pure conduction and high Rayleigh number (RaVW) convection can he calculated; as seen in Fig.33.

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Figure 33: Local heat flux distribution on the vertical wall of cavities with mixed boundary conditions. MVITA calculations. Also, see Fig.29.

ASSESSMENTS OF THE THERMAL LOADINGS IN SELECTED SEVERE ACCIDENT SCENARIOS

6

In this section, results of the MVITA model calculations for prototypic reactor accidents will be discussed. A large quantity of core materials is assumed to present in the lower plenum of the AP-600 reactor vessel. Geometrical characteristics of the RPV and physical properties of the oxidic and metallic phases were taken from [2]. External vessel cooling is employed as the accident management strategy, so that an isothermal boundary condition is set for the external vessel surface ( T 390K).

Figure 34: Computational mesh. Case C (thick metallic layer).

In this study, the thermal margin of the in-vessel melt retention scheme was assessed for three configurations, namely ”A” without metallic layer; ”B” with a thin layer (H = 0.1 m) and ”C” with a thick layer (H = 0.8 m) of molten metals above the oxidic debris. Fig.34 shows the computational mesh as a multi-block domain 2. See refs.[11] [10] for more discussion on numerical aspects of phase change modeling and solution of eq.(1)

2

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Figure 35: Steady state configuration of in-vessel melt retention (Case "A", without upper metallic layer, qv = 1.4 MW/m3).

Initial conditions of 2000 K for the oxidic debris and 1550 K for the metallic layer were employed in the calculations, whose results are presented here. It is worth noting that the initial conditions influence the time taken for the debris heat-up and remelting, but they are of minor importance in the assessment of in-vessel melt retention. This is because the lower head thermal loading reaches maximum value when the debris pool achieves steady thermal state as a slightly superheated melt pool (see also [2]). Both oxidic pool and metallic layers were modeled as multicomponent materials. Such systems feature complex phase diagrams; and phase change occurs in a temperature range between the liquidus and solidus points, or in the so-called mushy zone. The temperature differences over the mushy zone depend on mixture composition and were chosen to be 70 K and 40 K for the oxide pool and metallic layer, respectively.

Figure 36: Steady state configuration of in-vessel melt retention (Case "B", with thin upper metallic layer, qv = 1.4 MW/m3). in a complex domain including the conjugated regions of the oxidic pool, crust, molten steel and the vessel

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Figure 37: Steady state configuration of in-vessel melt retention (Case ”C ”, with thick upper metallic layer, qv = 1.4 MW/m3).

The effect of radiative heat transfer from the upper surface of the debris (or the metallic layer) was investigated. In this work, the emissivity coefficients for oxide and metallic layer surface were chosen 0.75 and 0.45, respectively. In the report by Theofanous et al. [2] it was assumed that the energy radiated from the surrounding cavity to the layer is negligible. This assumption holds for the situations when the surface temperature of the surrounding structure 4 ). It was T∞ is lower than the surface temperature of the metallic layer T up (then T up4 >> Τ ∞ also found from the calculation results for case ” C ” , that the temperature of the upper surface of the metallic layer is only slightly higher than the melting point of the metallic phase. At the same time, the surface of the surrounding structure, which is exposed to the metallic layer, is also nearly at the steel melting-point temperature. For this case (Tup T∞ ), the resulting radiative heat removal from the upper surface is limited, so that the vessel thermal loading increases. Figs.35-37 depict the calculated temperature fields in the three chosen cases. The metal melting point is assigned for T ∞ (T ∞ = 1600 K) in order to be conservative. It can be seen that, some vessel melting occurs at the corner region of the debris pool 3 . The vessel ablation in case ” B” is significantly deeper than that in case ” C ” (Fig.37). Previous assessments of the impact of the metallic layer, e.g. [2], indicated that the thin metallic layer (10 cm) may threaten the vessel thermal margins with respect to critical heat flux at the external cooled surface. The MVITA calculations confirmed that the sideward heat fluxes of the metallic layer to the vessel wall are significantly higher than the maximum heat, fluxes occurring at the molten oxidic pool corner. Significant vessel ablation at the oxidic pool corner and in the location next to the thin metallic layer is predicted. However, it should be noted that the thick, highly-conductive, steel vessel wall acts as a heat distributor. The multi-dimensional heat, diffusion significantly reduces the effect of the local hot spots occurring on the inner face of the vessel wall, e.g. the heat flux peaking in the thick metallic layers, or the high heat, flux imposed by the thin metallic layers. 3 The role of natural convection in the vertical layer of molten vessel steel was identified and analyzed in [8] and [11].

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A comparison of one-dimensional and two-dimensional representation of the heat diffusion in, and the melting of, the vessel wall was performed in order to relate the results obtained here to those obtained in the previous studies [2]. Fig.38 shows this comparison; it is seen that the heat flux peaking on the external vessel wall, next to the thin metallic layer, obtained in the 2-D representation is significantly lower than that obtained in the 1-D representation. The case of heat generation of 1.4 MW/m3 shows a small thermal margin to CHF, with 2-D representation, while with the 1-D representation the CHF limit is substantially exceeded.

Figure 38: Thermal margins of in-vessel core melt retention by external vessel cooling, with thin and thick upper metallic layers above the oxidic debris (qv = 1.4 MW/m3). The dotted line results from the one-dimensional vessel melting modeling for the thin metallic layer case.

7

SUMMARY AND CONCLUDING REMARKS

In this paper, heat transfer in internally heated liquid pools is described with the MVITA model. Particular attention is paid to the validation of the developed MVITA model for various geometries and for different boundary conditions. It was shown, by comparing the calculated results with the available experimental data, that the method is capable of describing (1) the fractions of heat transported to the cooled surfaces of the pools, (2) the temperature fields within the pools for steady-state and transient conditions, and (3) local heat flux distributions on the vertical and bottom curved surfaces. The model is based on solution of the two-dimensional energy conservation equation, with provision for anisotropic heat conduction. Effects of natural convection are modeled by means of ” pseudo-convective” terms and effective diffusivity coefficients. The heat-driven effective velocities are calculated using a heat-balance treatment and the experimental correlations for heat transfer coefficients on the upper horizontal cooled boundaries. The heat transported to the cooled curved and side walls is represented by an effective heat conductivity, which is determined by using the Eckert-type heat transfer correlations for the laminar and turbulent naturally-convected boundary layers along those walls. It should be noted that, although the MVITA is able to portray almost all significant features of natural convection heat transfer in heat-generating liquid pools, it is bound by the uncertainties of the experimental heat transfer correlations employed. The MVITA model provides a good description of the core melt pool formation, progression and of the thermal loads imposed on the RPV lower head. The characteristics of molten pool during its formation,

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namely the pool temperature field and the heat fluxes imposed on pool boundaries, can be properly modeled by this robust and efficient modeling approach. Such information is needed for evaluating the mode and timing of vessel failure as well as the characteristics of melt, discharged from reactor vessel, upon its failure. In the paper, natural convection heat transfer in metallic layers was analyzed with the CFD direct numerical simulation method. It was found that the time-averaged characteristics of natural convection flow and heat transfer in metallic layers (heated from below) are insensitive to either 'radiative' or isothermal boundary conditions at the upper surface. The presence of side cooled (isothermal) walls, however, is able to significantly modify the flow pattern, in which side-wall boundary-layer descending flows tend to overpower the Rayleigh-Benárd natural convection cell structure. Nevertheless, time- and surface-averaged heat transfer characteristics may be predicted by a technique proposed in [2]. It was found that the ECCM (effective conductivity-convectivity model), developed for core melt pool heat transfer and melt pool formation, can be successfully extended to the more complex debris configuration, which includes an overlying metallic layer. The local heat flux distribution on the side wall of the metallic layer was found to depend on RaVW, i.e. the input heat flux and the layer thickness. However, the results from the integrated analysis (MVITA calculations) showed that the focusing effect of the metallic layer is significantly diminished in a reactor configuration. This is due to the multi-dimensionality of heat diffusion into, and melting of, the highly conductive steel wall of the reactor pressure vessel. In general it was found that the 2D conjugated formulation employed here is necessary for more realistic predictions of the prototypic reactor accident conditions. This is particularly true when the interactions between distinct regions ( metal layer, pool, crust, vessel steel) have substantial influence on the heat transfer process, and temperature distributions.

Nomenclature Arabic c Fo g

H h k L Nu Pr q q* R Ra RaVW T t U, V U x y

Specific heat capacity, J/(kg.K) Fourier number, Fo = Gravitational acceleration, m/s 2 Height of melt pool or fluid layer, m Heat transfer coefficient, W/ (m.K) Thermal conductivity, W/(m2.K) Characteristic length, m Nusselt number, Nu = Prandtl number, Pr = v/α Heat flux, W / m2 Dimensionless heat flux normalized to surface-average heat flux Pool radius, radius of curvature, m Rayleigh number, Ra = gβ∆ TH 3 /αv Rayleigh number , Temperature, K Time, s Vertical and horizontal velocity components, m/s Velocity vector ( U, V ) Horizontal coordinate, m Vertical coordinate, or Boundary layer development length, m

HEAT TRANSFER IN REACTOR LOWER PLENUM Greek a β κ µ ν ρ θ∗ ∆T

Thermal diffusivity, m 2/s Thermal expansion coefficient, 1/K Heat Conductivity, W/m . K Dynamic viscosity, Pa.s Kinematic viscosity, m 2 /s Density, kg/m3 Dimensionless temperature, Temperature difference, K

Subscripts b c dn int low s.sd up v w x y x, y

Bulk Curved wall Down, bottom surface Interface Lower surface Sideward, Vertical surface Upper, upward, top surface Volume Wall Horizontal direction; Local Vertical direction; Local Horizontal and vertical space directions

RB VW

Rayleigh-Benárd Vertical Wall

Abbreviations CFD ECCM

Computational Fluid Dynamics Effective Conductivity-Convectivity Modelling (Method)

Acknowledgement. The constructive comments of Dr. J. Green (RIT) are gratefully acknowledged. This work was supported by the European Union, Swedish Nuclear Power Inspectorate (SKI), US Nuclear Regulatory Commission (US NRC), the Swiss Nuclear Inspectorate (HSK), and the Swedish and Finnish Power Companies.

REFERENCES [1] J.L. Remple et al., “Light Water Reactor Lower Head Failure Analysis”. NUREG/CR-5642. EGG-2618 (October 1993). [2] T.G. Theofanous et al., “In-Vessel Coolability and Retention of a Core Melt”, DOE/ID-10460 (July 1995). [3] “MAAP 4 Users Manual”, V01.2, Fauske Associated Inc., 1994.

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B. R. SEHGAL ET AL. Damage Progression Model Theory”.

[5] T.N. Dinh, R.R. Nourgaliev, and B.R. Sehgal, ”On Heat Transfer Characteristics of Real and Simulant Melt Pool Experiments”, Intern. J. Nuclear Engineering and Design, Special volume on ”In-vessel Melt Retention and Coolability”, Vol. 169, pp.151-164, 1997. [6] T.N. Dinh and R.R. Nourgaliev, ”Turbulence Modeling in Large Volumetrically Heated Liquid Pools”, Intern. J. Nuclear Engineering and Design, Special Volume on ”In-vessel Melt Retention and Coolability”, Vo1.169, pp.131-150, 1997. [7] R.R. Nourgaliev and T.N. Dinh, ”An Investigation of Turbulence Characteristics in Internally Heated Unstably Stratified Fluid Layers”, Nucl. Eng. & Des., 1997 (Also, ANS Proc. of 1996 National Heat Transfer Conference, Houston, Texas, 1996, HTC-Vol.9, pp.357-367). [8] T.N. Dinh, V.A. Bui, R.R. Nourgaliev, and B.R. Sehgal, ”Crust Dynamics under PWR InVessel Melt Retention Conditions”, ANS Proc. of 1996 National Heat Transfer Conference, Texas, 1996, HTC-Vol.9, pp.368-375. [9] V.A. Bui and T.N. Dinh, ”Modeling of Heat Transfer in Heat-Generating Liquid Pools by an Effective Diffusivity-Convectivity Approach”, Proc. of 2nd European Thermal-Sciences Conf., Rome, Italy, 1996, pp.1365-1372. [10] V.A. Bui, T.N. Dinh and B.R. Sehgal, ”Numerical Modeling of Heating and Melting Processes in Internally-Heated Debris Beds in a Reactor Vessel Lower Plenum”, Proc. of 4th Intern. Conf. ”Advanced Computational Methods in Heat Transfer”, 1996, Udine, Italy. [11] V.A. Bui, T.N. Dinh, and B.R. Sehgal, ”In-Vessel Core Melt Pool Formation during Severe Accidents”, ANS Proc. of 1996 National Heat Transfer Conference, Houston, Texas, 1996, HTC-Vol.9, pp.86-94. [12] O. Kymäläinen, et al. ”Heat Flux Distribution from a Volumetrically Heated Pool with High Rayleigh Number”, Proceedings of the 6th Int. Topical Meeting on Nuclear Reactor Thermal Hydraulics, NURETH-6, Grenoble, France, October 1993, pp.47-53. [13] F.J. Asfia and V.K. Dhir, ”Natural Convection Heat Transfer in Volumetrically Heated Spherical Pools”, Proceedings of the OECD/CSNI/NEA Workshop on Large Molten Pool Heat Transfer, Grenoble, France, March 9-11, 1994. [14] J.M. Bonnet, S. Rouge and J.M. Seiler, ”Large Scale Experiments for Core Melt Retention”, Proceedings of the OECD/CSNI/NEA Workshop on Large Molten Pool Heat Transfer, Grenoble, France, March 9-11, 1994. [15] F.B. Cheung, et al. ”Modeling of Heat Transfer in a Horizontal Heat-Generating Layer by an Effective Diffusivity Approach”. ASME HTD-Vol.192, pp.55-62, 1992. [16] M.J. Tan, D.H. Cho, and F.B. Cheung, ”Thermal Analysis of Heat Generating Pool Bounded From Below by Curved Surfaces”. ASME J. Heat Transfer, Vo1.116, pp.127-135, 1994. [17] H.S. Carslaw and J.C. Jaeger, ”Conduction of Heat in Solids”. Oxford Press, 1959. [18] C.J. Kim, S.T. Ro, and J.S. Lee, ”A Efficient Computational Technique to Solve the Moving Boundary Problems in the Axisymmetric Geometries”. Int.J. Heat Mass Transfer, Vol.36(15), pp.3759-3764, 1993. [19] C.J. Kim and M. Kaviany, “A Numerical Method for Phase-change Problems with Convection and Diffusion”. Int. J. Heat Mass Transfer, Vol.35(2), pp.457-467, 1992.

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[20] F.A. Kulacki and A.A. Emara, ”Steady and Transient Thermal Convection in a Fluid Layer with Uniform Volumetric Energy Sources”. J. Fluid Mech. Vo1.83, part 2, pp.375-395, 1977. [21] F.A. Kulacki and M.E. Nagle, ”Natural Convection in a Horizontal Fluid Layer With Volumetric Energy Sources”. ASME J. Heat Transfer, Vol.97, pp.204-211, May 1975. [22] V.S. Arpaci, ”Buoyant Turbulent Flow Driven by Internal Energy Generation”. Int. J. Heat Mass Transfer, Vo1.38(15), pp.2761-2770, 1995. [23] U. Steinberner and H.H. Reineke, ”Turbulent Buoyancy Convection Heat Transfer with Internal Heat Sources”. Proceedings of the 6th Int. Heat Transfer Conference, Toronto, Canada, Vo1.2, pp.305-310, 1978. [24] F.A. Kulacki and R.J. Goldstein, ”Thermal Convection in a Horizontal Fluid Layer with Uniform Volumetric Energy Sources”. J. Fluid Mech. Vo1.55, part 2, pp.271-287, 1972. [25] T.C. Chawla and S.H. Chan, ”Heat Transfer From Vertical/Inclined Boundaries of HeatGenerating Boiling Pools”, J. of Heat Transfer, Transactions of the ASME, Vo1.104, pp.465473, 1982. [26] J.H. Min and F.A. Kulacki, ”An Experimental Study of Thermal Convection with Volumetric Energy Sources in a Fluid Layer Bounded From Below by a Segment of a Sphere.” Proceedings of the 6th Int. Heat Transfer Conference, Toronto, Canada, paper NR-27, 1978. [27] M. Jahn and H.H. Reineke, ”Free Convection Heat Transfer with Internal Heat Sources: Calculations and Measurements.” Proceedings of the 5th Int. Heat Transfer Conference, Tokyo, Japan, Vo1.3, paper NC-2.8, 1974 [28] F. Mayinger, M. Jahn, H.H. Reineke, and V. Steinberner, ”Examination of Thermalhydraulic Processes and Heat Transfer in a Core Melt”. BMFT RS 48/1, Institut fur Verfahrenstechnik der T.U., Hanover FRG, 1976. [29] R.R. Nourgaliev, T.N. Dinh, and B.R. Sehgal, ”Effect of Fluid Prandtl Number on Heat Transfer Characteristics in Internally Heated Liquid Pools with Rayleigh Numbers up to 1012”, Intern. J. Nuclear Engineering and Design, Special Volume on ”In-vessel Melt Retention and Coolability”, Vol. 169, pp.165-184,1997. [30] S. Globe and D. Dropkin, ”Natural-Convection Heat Transfer in Liquids Confined by Two Horizontal Plates and Heated From Below”, J. Heat Transfer, Transactions of the ASME, February 1959, pp.24-28. [31] S.W. Churchill and H.S. Chu, ”Correlating Equations for Laminar and Turbulent Free Convection from a Vertical Plate”, Int. J. Heat Mass Transfer, Vo1.18, pp.1323-1329, 1975. [32] CFX-F3D. Version 4.1 User Manual. AEA Technology, England, October 1995. [33] O‘Toole, J.L. and Silveston, P.L., ”Correlations of Convective Heat Transfer in Confined Horizontal Layers”, AIChE. Chem. Engng. Progr. Symp. Ser., 57(32), pp.81-86, 1961. [34] A.T. Kirkpatrick and M. Bohn, ”An Experimental Investigation of Mixed Cavity Natural Convection in High Rayleigh Number Regime”, Int. J. Heat Mass Transfer, Vol.29, 1, pp.6982, 1986.

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ADVANCES IN NUCLEAR FUEL MANAGEMENT FOR LIGHT WATER REACTORS

Paul J. Turinsky,1 and Geoffrey T. Parks2 1

Department of Nuclear Engineering North Carolina State University Raleigh, NC 27695-7909 U.S.A.

2

Cambridge University Engineering Department Trumpington Street Cambridge CB2 1PZ U.K.

INTRODUCTION Nuclear fuel management entails making decisions that influence how a reactor core's reactivity and flux, power and burnup spatial distributions vary. These spatial distributions impact the thermal-hydraulic and material properties of the core, which are constrained by various performance limits. The objective of nuclear fuel management is to minimize the cost of nuclear power generated electricity while satisfying all constraints imposed. Our focus is on Light Water Reactors (LWRs), in particular Boiling Water Reactors (BWRs) and Pressurized Water Reactors (PWRs). LWRs are characterized by requiring shutdown to refuel, and when refueling, replacing a fraction of the total fuel inventory. Decisions required include the design of the fresh fuel lattices and assemblies (bundles), the specification of the number of fresh assemblies and identity of partially burned assemblies to be reinserted, determination of the core loading pattern (LP), and management of core reactivity via control materials. Lattice design needs to address issues such as fuel enrichment and burnable poison (BP) radial loadings within a bundle, in addition to mechanical configuration. Bundle design involves the utilization of various lattices over different elevation spans. The core LP specifies the location and orientation of fresh and burned assemblies. Reactivity management addresses the need for BPs, and for BWRs control rod positions as a function of burnup and power level. All these decisions interact with each other, making fuel management a very complex optimization problem. Traditionally nuclear fuel management has been subdivided into two topics, out-ofcore and in-core nuclear fuel management, though, clearly this is an artificial separation due to the strong coupling between out-of-core and in-core decisions.

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Out-Of-Core Nuclear Fuel Management Out-of-core nuclear fuel management focuses on answering the questions “What to purchase?” and “What to reinsert?” over a multi-cycle planning horizon. It involves making decisions about cycle energy production requirements, utilization of stretch-out operation, the number and composition of fresh assemblies, and which burned assemblies to reinsert. The objective is to minimize the levelized energy cost of the electrical grid, where the levelized energy cost is defined as the time invariant cost that would generate the same present worth revenues that the actual time varying cost would generate over some specified time span. This implies that when considering a single nuclear unit, replacement power costs from other units must be factored in. Given cycle energy production requirements for a specified nuclear unit, the objective is to minimize the unit’s levelized fuel cycle cost. Multicycles must be considered since fresh fuel loaded into the reactor this cycle will impact what fresh fuel must be loaded in later cycles, since it will reside in the core for several cycles. In our discussion, a region of fuel is characterized as being loaded into the core as fresh fuel at the same time. A batch of fuel denotes a subset of the assemblies making up a region, characterized by being identical when fresh, i.e. the same bundle design, and experiencing the same irradiation history, i.e. irradiated in the same cycles. The most important constraints for out-of-core nuclear fuel management are the cycle energy production requirements, the maximum fuel enrichment allowed, various discharge burnup limits, and reactivity hold-down limits. The cycle energy production requirement, which, given an operating capacity factor, equates to the days of operation between refueling outages, mainly determines the average enrichment of the fresh assemblies, and, via reactivity and/or discharge burnup limit constraints, the number of fresh assemblies. The maximum enrichment allowed is normally constrained by criticality considerations at the fuel fabrication plant. Discharge burnup limits impose constraints on fuel rod, fuel bundle, batch and region burnups. For out-of-core fuel management, the focus is on batch and region burnup limits, which determine the minimum number of fresh assemblies that are needed in each cycle. For a BWR, the major concern in regard to reactivity hold-down limits is assuring that the core is adequately subcritical with control rods inserted at cold conditions at any time during the cycle. Since BWRs are under-moderated, the core is most reactive in the cold condition, when all the voids are collapsed. It must be assumed that one of the control rods is stuck out, giving rise to the so called “N-1” stuck rod configuration. The associated negative reactivity at cold zero power (CZP) with the limiting N-1 stuck rod configuration is referred to as the shutdown margin (SDM). Without the presence of soluble boron, only the core LP and BP loading are available to ensure an adequate SDM in a BWR. Assuming a fixed LP, the total BP loading is mainly dictated by the SDM constraint. For BWRs, an integral BP design based upon gadolinia (Gd2O3) is used. Via the weight percent loading in the fuel pellet, advantage can be taken of the thermal self-shielding effect to control the rate of depletion of gadolinia. Thus, by a prudent combination of total core gadolinia loading in fresh fuel and weight percent gadolinia loading, which can vary spatially, the hold-down reactivity of gadolinia can be adjusted such that the SDM requirement can be satisfied at all cycle exposures. In addition, the flexibility of gadolinia loading can be used for local and global power distribution shaping. PWR cold SDM requirements are easily met by the addition of soluble boron during a controlled cooldown. For an uncontrolled cooldown, as would occur during a steam-line break accident, the core must be sufficiently subcritical after trip that the associated cooldown transient can be tolerated either without any actions or with the automatic injection of an emergency, high concentration soluble boron solution. Thus the SDM for PWRs is specified at hot zero power (HZP) conditions, again assuming an N-1 stuck rod configuration. This imposes a limit on the core LP but not the total BP loading required.

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From a reactivity viewpoint, the BP hold-down reactivity loading is dictated by the requirement for a negative moderator temperature coefficient (MTC) at higher powers. With the soluble boron concentration increasing with cycle energy production requirement, in order to offset the more reactive fuel loading, a positive MTC is to be expected, as the MTC increases with boron concentration. Indeed for cycle lengths of about eighteen months or greater, a positive MTC can occur. To reduce the soluble boron concentration, and hence make the MTC negative, BPs are introduced. They are loaded in the fresh assemblies so as to flatten the global and local power distributions. PWRs utilize both integral and discrete BP designs. In addition to gadolinia, one finds erbium oxide (Er2O3) also being blended into the pellet. An alternative integral BP design is the Integral Fuel Burnable Absorber (IFBA) design, which utilizes a thin coating of a boron containing compound on the fuel pellet surface. These integral BP designs, via total and weight percent loadings, can tailor the reactivity hold-down as a function of exposure. Discrete BPs, which can be located within the guide tubes found in the lattice, are also utilized in PWRs, sometimes in combination with integral BPs. Whether used in a BWR or PWR, there is a residual reactivity hold-down effect of BPs at end of cycle (EOC), which necessitates a higher feed enrichment in order to satisfy the cycle energy production requirement. The associated fuel cycle cost increase, in addition to the direct manufacturing cost of burnable absorbers, motivates one to utilize the minimum amount possible. Additional constraints that may be imposed in out-of-core fuel management are that feed enrichments and BP weight loadings must be selected from a palette of specified values. This restriction is imposed to reduce fuel fabrication costs. Clearly in most cases this will result in the utilization of split feed enrichments; that is, a region being split into two batches with differing average bundle enrichments, so that the fresh assemblies’ average enrichment satisfies the cycle energy production requirement while utilizing batch enrichments from the palette of enrichments available. For cases where all the fuel in a region is not discharged simultaneously, split enrichments are normally utilized anyway since one desires to load more reactivity into the assemblies that will be going on for one additional cycle of irradiation, in order to carry reactivity forward thereby reducing the feed enrichment requirement in the additional cycle of irradiation. In-Core Nuclear Fuel Management Having decided what fresh and partially burned assemblies to load in the core for a specific cycle, in-core fuel management is concerned with answering the question “Where to position?” these assemblies and BPs within the core. Thus, the decision variables for this problem are the placement and orientation of the assemblies, including their BP content, and for a BWR the position of control rods as a function of cycle burnup, i.e. the control rod program (CRP). Clearly in a BWR there is a strong interaction between the LP and CRP The objectives for in-core nuclear fuel management are many and often conflicting. From an economic viewpoint, one wishes to minimize the fuel cycle cost. For a fixed fresh and burned fuel inventory, one may wish to make decisions that either maximize the cycle energy production or maximize the burnup of assemblies to be discharged at the end of the cycle. The objective of maximizing cycle energy production, i.e. maximizing core reactivity near EOC, is usually achieved at the expense of future cycles. This is because maximizing cycle energy production is mainly achieved by low leakage LPs, where less reactive fuel is located out-board and fresh fuel is located in-board. This causes the fresh fuel to operate at higher powers and receive a higher burnup, which increases the feed enrichments required in subsequent cycles. By contrast, recognizing that the objective of maximizing the discharge burnup is equivalent to maximizing the product of cycle energy production and average relative power of the assemblies to be discharged at the end of the cycle, this objective tends to lead to less aggressive burning of the fresh fuel assemblies. These

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Figure 1. PWR core (1/4 symmetry) LPs with maximized EOC keff and discharge burnup.1

differences in LP associated with maximizing cycle energy production and maximizing region average discharge burnup can be seen by contrasting the two LPs shown in Figure 1, determined by the FORMOSA-P in-core optimization code. 1- 2 Note that increasing region number equates with increasing region reactivity and decreasing region burnup, with Region 4 denoting the feed region. Another potential objective related to economics is minimization of the fresh assemblies cost, with the cycle energy production requirement imposed as a constraint. For a PWR, this can equate with minimizing the average feed enrichment. For a BWR, where bundle designs are often selected from available designs, this equates to loading those bundles whose total cost is minimized. An alternative is to minimize the total number of fresh assemblies loaded, letting feed enrichment or bundle design vary to ensure the cycle energy production requirement is satisfied. However, the region average discharge burnup limit will impose a lower limit on this number, implying multi-cycles must be considered when using this objective. Other in-core nuclear fuel management objectives are mainly concerned with maximizing thermal margins from limits, which will presently be discussed under the topic of constraints. Since there are multiple objectives, in recent years there has been an interest in understanding the trade-off of objectives against each other. Indeed, one can quantify the cost of thermal margin using multiobjective optimization methods, a topic we will address later. In-core nuclear fuel management constraints are many and often tightly constraining. Indeed the trend is for more and more constraints to be imposed as new fuel degradation modes are recognized, making finding a feasible LP more difficult. There are numerous physical constraints that the nuclear reload design engineer almost unconsciously imposes, but which need to be articulated mathematically when employing formal optimization techniques. Examples of such constraints are: one assembly per core location; fuel inventory available; orientation mix imbalance; and for PWRs no discrete BPs under control rod locations. The orientation mix imbalance constraint is introduced so that radial quadrant power tilts in earlier cycles do not grow in subsequent cycles due to the associated radial quadrant burnup tilt. In addition, one may want to impose constraints on the position or orientation of certain assemblies in the core, either fixing assemblies in or excluding assemblies from certain core locations. These constraints are useful in imposing restrictions of highly burned PWR assemblies and mixed oxide (MOX) assemblies from control rod

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locations. Physical constraints are desirable because they limit the search space. Other constraints require the assessment of the core attributes to determine whether they are satisfied. This implies solution of the few-group neutron diffusion equation over the cycle, which is computationally demanding. For BWRs, the additional computational burden of solving the two-phase flow equations is encountered. Common to BWRs and PWRs are constraints on: maximum feed enrichment; minimum SDM; maximum pellet, pin and node power densities; maximum pin, assembly, batch and region average discharge burnup limits; and for integral BPs maximum weight percent loading. Power density constraints originate because of such things as the loss of coolant accident (LOCA) peak clad temperature limit, the critical heat flux (CHF) limit, and the clad strain limit. Burnup limits are imposed to minimize internal rod pressure, assembly bow, and number of fuel rods whose clad integrity has been breached, i.e. “leakers”. In addition, constraints on maximum octant power tilt and maximum/minimum batch and region power sharing may be imposed. The batch/region power sharing limits are a convenient way to ensure that fuel to be discharged is highly burned and the high enrichment batch of a split feed enrichment region carries reactivity forward for its extra cycle of irradiation. Unique to PWRs are constraints on maximum soluble boron concentration, maximum MTC, and target EOC soluble boron concentration (a way of stating the cycle energy production requirement). The maximum soluble boron concentration constraint is imposed to limit the pH of the coolant which affects the clad corrosion rate. Unique to BWRs are constraints on minimum critical power ratio and target EOC core flow rate (also a way of stating the cycle energy production requirement). For PWRs, in-core fuel management entails determining the LP, including BP loading, such that the objective is maximized/minimized within constraints. Control rod insertions, if any, are not constrained by the criticality constraint since soluble boron is used to satisfy this condition. Thus, during determination of the LP, one can position the control rods at their known, desired insertions. In contrast, BWRs must determine the LP and CRP simultaneously, since without soluble boron, control rods must be positioned to satisfy the criticality constraint. Clearly there are many control rod positions that ensure criticality for the desired, known core flow rate, implying a very large optimization problem when the LP and CRP are joint decisions. If BWRs were to operate with a Haling power distribution3, i.e. position the control rods to hold the power shape constant throughout the cycle while retaining core criticality, the LP and CRP decisions would be separable. Unfortunately, it is known that BWRs do not operate with Haling power distributions for a variety of reasons, one of which is that the premise of why the Haling power distribution is the preferred power distribution is invalidated when gadolinia integral BPs, which display non-monotonic assembly reactivity versus burnup behavior, are utilized.4 Thus for BWRs, one must assess the optimum CRP for each potential LP. To limit the freedom in CRP decision possibilities, heuristic rules have been employed, which will be described in a later section. Thus, the in-core nuclear fuel management problem is characterized by a very large search space, possibly exceeding 10100 different LPs, integer or mixed-integer decision variables, and active constraints. For BWRs there is the added complexity of where to position control rods to ensure criticality at the desired core flow rate, i.e. core average void fraction. Lattice & Bundle Design Clearly some estimate of lattice and bundle designs must be known to complete out-ofcore nuclear fuel management, and detailed designs are required to complete in-core nuclear

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fuel management. To achieve a flat within-lattice power distribution, multiple enrichments may be utilized. This is common practice in MOX and BWR UO2 lattices to overcome local thermal flux leakage and moderation effects, respectively. The degree of flexibility of BP lattice loading depends upon BP type. For discrete BPs used in some PWRs, their positions are limited to the guide tubes within the lattice, and then only allowed in bundles not under control rod locations. For integral BPs, whether blended into or coated onto the fuel pellet, greater within lattice loading flexibility exists. In either case, one is attempting via number and weight percent of BPs to tailor the reactivity profile of the lattice with burnup. In addition, via the selection of radial position within the lattice, the within-lattice power distribution can be flattened. In regard to bundle design, the placement of various lattice designs spanning different elevations is employed to produce the desired axial zoning of fuel enrichments and BPs. This is done mainly to shape as desired the axial power and flux distributions. In addition, the use of natural uranium blankets at the top and bottom of the bundle reduces the enriched uranium requirements, thereby decreasing the fuel cycle cost. Computational Considerations Recognizing that the few-group neutron diffusion equations and thermal-hydraulics equations applicable to the core must be solved as a function of cycle exposure to evaluate the objective and constraints, it becomes clear that both the out-of-core and in-core nuclear fuel management problems, when searching for optimum decision variables, impose a very large computational burden. This reality has until recently limited the utilization of formal mathematical optimization methods in solving real-world nuclear fuel management problems. MODERN FUEL MANAGEMENT PRACTICES Heuristics Extensive experience in making out-of-core and in-core nuclear fuel management decisions has led to the widespread employment of heuristic rules to guide the process. However, the results of applying formal mathematical optimization techniques to real-world nuclear fuel management problems have recently begun to cast doubt on the validity of some heuristic rules. Note that sometimes heuristic rules are employed within formal mathematical optimization methods, with the advantage of increased computational efficiency via a prior limitation of the search space and the disadvantage of exclusion from the search space of areas of potential benefit. Out-Of-Core Nuclear Fuel Management We have described much of the current practice in out-of-core fuel management in the Introduction section. To minimize the levelized fuel cycle cost, one maximizes the number of regions by minimizing the number of fresh assemblies loaded each cycle in the planning horizon. For non-equilibrium cycles, this implies that multi-cycle predictions are required to ensure that the batch and region average discharge burnup limits are satisfied. Having set the number of fresh assemblies, the remaining core positions are filled with partially burned assemblies. Clearly their reinsertion is limited by lifetime and discharge burnup limits. Of the available burned assemblies, the most reactive are normally selected for reinsertion to minimize the fresh assemblies’ average enrichment. These burned assemblies may have been utilized in the previous cycle or be available from the spent fuel pool. For transition cycles due to changing cycle energy production requirements, there are situations for which

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not using the most reactive burned assemblies may be economically desirable. This occurs since more energy can be extracted within burnup constraints if some of the more reactive burned assemblies are placed in the spent fuel pool for a cycle and reinserted in the subsequent cycle. Clearly multi-cycle analysis is required to ascertain whether there is a benefit from this strategy. Table 1 shows a cycling scheme for a PWR transitioning from 12 month to 18 month cycles, the cycling scheme determined via application of a formal mathematical optimization technique, which displays the behavior just described.5 The challenge of out-of-core nuclear fuel management is to complete the multi-cycle analysis initially using highly efficient computational methods with adequate fidelity for the intended application, without a detailed knowledge of the core LPs. Having filled the core with assemblies, some estimate of the feed enrichment and total BP loading requirement can be made. As noted earlier, for BWRs the minimum total BP loading is dictated by the cold SDM requirement, which may be approximately related to the hot full power (HFP) unrodded excess core reactivity, thereby avoiding the need for cold SDM calculations. For PWRs, the estimate of the BP loading is based upon the maximum MTC limit, which may Table 1. PWR cycling scheme for a 12-18 month cycle length transition using random backfill for burned fuel.5

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be approximately related to the HFP critical soluble boron concentration except when using Er2O3 burnable absorbers, thus avoiding the need for MTC calculations. Clearly the average feed enrichment will increase if BPs are required due to their residual reactivity worth at end-of-cycle (EOC), implying a connection between total BP loading and average feed enrichment as constrained by the cycle energy production requirement. Once a cycling scheme has been determined for all the cycles in the planning horizon, the levelized fuel cycle cost can be calculated. This calculation requires following all fuel loaded during the planning horizon to final discharge, which implies that cycles beyond the planning horizon enter the analysis. It is common practice to assume these cycles are equilibrium cycles. One can now make perturbations to the cycling scheme decisions to determine if the levelized fuel cycle cost can be further reduced. One can also vary cycle energy production requirements, factoring in replacement power costs, to see if there are economic benefits in changing these requirements. For partial MOX cores, the additional decisions of the number of fresh assemblies to load with MOX and their weight percent content must be addressed. These additional decisions are constrained by numerous considerations, including SDM, and for PWRs boron worth and reactor vessel fluence. The use of about one-third MOX assemblies is common practice, a value that correspond to the number of MOX assemblies available from self-generated plutonium recycle in an equilibrium situation. In-Core Nuclear Fuel Management – Loading Pattern Design With feed assembly number and composition, including BP content, fixed, in-core nuclear fuel management refines the cycling scheme by determining the LP, and for BWRs the CRP. Heuristic rules for LPs have evolved greatly over the years, much of this evolution occurring due to changing objectives. In the past, a greater emphasis was placed on thermal margin versus fuel cycle cost. In addition, BP technology was less developed. This led to out-in scatter LPs. In these patterns, fresh fuel is loaded outboard in the core to offset leakage effects, giving rise to a flatter power distribution. Burned fuel with differing number of cycles of exposure is checker-boarded inboard, the higher reactivity fuel driving the lower reactivity fuel via neutron leakage. This is desirable from both power flattening and discharge burnup perspectives. For yearly reload PWRs, the utilization of BPs could be totally avoided since neither MTC nor power peaking limits required their use. Current LPs are low leakage to maximize neutron economy, and hence minimize feed enrichment and therefore fuel cycle cost, and have the additional advantage of reducing pressure vessel irradiation levels. Low leakage patterns load less reactive fuel outboard in the core, particularly in core locations with two surfaces adjacent to the reflector. Fresh fuel can be loaded one core location inboard from the edge, producing the so called “ring of fire” and/or checker-boarded more uniformly in the interior with burned fuel assemblies. The left-hand figure in Figure 1 illustrates a ring of fire PWR LP. Since the fresh fuel is located inboard, it is normally necessary to use BPs to reduce the power peaking. For longer cycle energy production requirements in PWRs and always in BWRs, BPs are required to address reactivity related constraints, providing more than enough BPs for power peaking suppression. Given that fresh and burned assemblies have differing neutronic properties due to differences in enrichments and BP loadings, and for burned assemblies irradiation history, the determination of the LP is much more complex than implied above. With partial MOX cores, restrictions on placing MOX assemblies in control rod positions, in order to maximize the SDM, and outboard locations, in order to minimize reactor vessel fluence, also come into play. To maximize the SDM, it would seem desirable to place fresh fuel under control rod locations; however, one also needs to avoid putting fresh fuel in the potential worst stuck rod core locations. Similar considerations enter for the PWR ejected rod and BWR

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Figure 2. BWR conventional core LP (1/8 symmetry).6

dropped rod accidents. In-Core Nuclear Fuel Management _ Control Road Programming For BWRs the LP and CRP decisions are tightly coupled. Utilizing a low leakage LP as just described, the so called “conventional core”, heuristic rules have been developed to guide the selection of the CRP. Figure 2 illustrates a conventional core LP.6 Recognizing that some control rods must be deeply inserted for some time to suppress excess fuel reactivity, high power peaking upon control blade withdrawal is a concern with fresh fuel located next to inserted blades. This fresh fuel will have a lower burnup, particularly in fuel pins adjacent to the control blade, causing it to be relatively more reactive when the blade is withdrawn to maintain core criticality. This effect is referred to as control rod burnup shadowing. By operating with a low core flow, hence high core average moderator void, one can reduce the excess fuel reactivity and thereby minimize deep control rod insertions. This has the added benefit of producing more fissile plutonium via the harder neutron spectrum and fewer wasteful neutron captures in control materials. Later in the cycle, e.g. in the last 20% of the cycle, the core flow rate can be increased, thereby increasing core reactivity and plutonium consumption. Such spectral shift operation is common practice in BWRs thanks to the flexibility of recirculation flow rate made possible by the Critical Heat Flux (CHF) margin. Even with spectral shift operation, control rod burnup shadowing is a problem. To minimize this effect, the practice is about every six weeks to withdraw the control rods inserted and insert other control rods. In this way, burnup shadowing does not accumulate very much before the core condition is changed, such that any accumulation that has occurred can be corrected by burning the effected assemblies a little harder after the rod swap. General Electric has identified four groupings of control rods that maintain radial core symmetry, referred to as Al, A2, B1 and B2, that are alternately sequenced into and out of the core as just described and illustrated in Figure 3. To further avoid axial rod burnup shadowing effects associated with partial control rod insertions, a deep-shallow insertion pattern is employed. The upper illustration in Figure 4 presents this, where “D” and “S” denote deep and shallow respectively, for a designated subgroup of control rods within the

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Figure 3. BWR control rod groups for a conventional core (1/8 symmetry).6

group. The drawback of the sequence exchange approach is increased operator burden and the need to complete the sequence exchanges at reduced power, adversely impacting operating capacity factor. The reduced power operation is required to satisfy fuel mechanical performance constraints. For these reasons, the control cell core, shown in Figure 5, has been developed.6 In this approach, the core LP is similar to conventional cores, except that less reactive fuel is

Figure 4. BWR control rod programs for a conventional and control cell core.6

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Figure 5. BWR control cell core LP (1/8 symmetry).6

located about selected control rod locations, thereby deviating from the checker-board pattern at these locations. This cluster of four assemblies around selected control rod locations is the origin of the name control cell core. The associated control rod worths and assembly powers of the control cells are lower than in a conventional core. This minimizes the control rod burnup shadowing effect. The net result is that no control rod sequences are required, with a single control rod group, e.g. A2, being used to control core reactivity at HFP over the entire cycle via infrequent deep-shallow exchanges within the group, occurring at an only slightly reduced power level. The lower illustration in Figure 4 shows this. The disadvantages are a reduced flexibility in LP designs and, perhaps, a slightly higher total BP loading. Lattice & Bundle Design The complexity of lattice and bundle designs has grown by leaps and bounds as fuel vendors compete to squeeze out the last ounce of improved economic performance. The radial loadings of fuel enrichment and BP within the lattice are chosen so as to minimize the within-lattice power peaking and to obtain the desired reactivity profile with burnup, an attribute clearly coupled to the bundle design and LP via the core wide power distribution. Axial heterogeneities are introduced into the bundle design through the utilization of different lattices over different elevation spans. For PWRs, the degree of axial heterogeneities is not that great. Axial blankets of natural or low enriched uranium are introduced at the top and bottom of the fuel, thereby minimizing axial leakage and the total fresh fuel enriched uranium requirement. Cut-back BPs are also introduced, leading to a flatter axial power distribution in particular in the fresh fuel, which reduces axial power peaking and total BP loading. In BWRs, the degree of axial heterogeneities is much greater. This is required to offset the negative reactivity effect of higher moderator voiding at upper core elevations. Without axial heterogeneities in the bundle design, the flux and power distributions would be strongly bottom peaked, particularly so if low core flow is being used for spectral shift operation. The bottom entry control rods could be partially inserted in an attempt to balance the axial reactivity profile, but with the adverse effect of control rod burnup shadowing at partial insertions. An alternative approach is to decrease the average void at higher bundle

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elevations. This is done via water rods whose radius increases at higher elevations, facilitated by the employment of some partial height fuel rods. Even with the introduction of this mechanically more complicated bundle design, axially dependent fuel enrichment and BP loadings are required. To offset the void induced axial reactivity profile, one wishes to deploy relatively more reactive lattices at higher elevations. This can be done via fuel enrichment and BP loading; however, one must do this in such a manner that the SDM requirement is satisfied. If the bundle is designed to be relatively more reactive at higher elevation, very strong top peaking will occur at cold conditions. Given that spectral shift operation will preferentially produce plutonium at higher elevations and that the fuel’s burnup will be lower at higher elevations due to the HFP axial power shape, the core’s reactivity will increase substantially as the flux redistributes at cold conditions to the higher worth upper core spatial region. This redistribution effect places constraints on the axial zoning of bundles, where at very high core elevations a relatively less reactive lattice must be utilized. This is done just below the upper axial blanket, since upper and lower axial blankets are also utilized in BWRs. BWR lattice and bundle designs are so important and complex that fuel vendors treat the design details as proprietary information. At present, lattice and bundle designs are almost exclusively the preserve of the expert designer working from experience. To assist in this task graphical user interface (GUI) packages are being developed which enable the expert to set up and analyze individual designs more easily.7-8 The first attempts at applying formal mathematical optimization methods to these tasks are also being made.9-12 COMPUTATIONAL REACTOR PHYSICS METHODS Whether performing out-of-core or in-core nuclear fuel management studies, one requires the values of few-group, spatially homogenized, neutron cross-sections. Conceptually, one starts with many group calculations and fine spatial detail for a limited spatial domain, e.g. a lattice, so that the computational burden is acceptable. One then collapses to few-group cross-sections that have been properly spatially homogenized. These cross-sections are then utilized in coarse mesh (nodal) core simulators to determine core reactivity and flux, power and burnup distributions. Lattice Physics Codes The many-group, fine spatial domain calculations are referred to as lattice physics calculations, since their spatial domain is normally an octant or quadrant of the lattice. As implied, two-dimensional calculations are completed, assuming properties to be homogenous in the axial direction. Since the lattice is highly heterogeneous, transport theory must be employed. Given the limited spatial domain and complex geometry, integral transport theory is very well suited for lattice physics calculations. Monte Carlo techniques have been employed, but recognizing that the flux distribution is desired versus some localized interaction rate, integral transport theory proves more computationally efficient. Codes such as APOLLO13, CPM14, CASMO15, HELIOS16, PHOENIX17 and WIMS18 are used to complete lattice physics calculations, frequently offering multiple solution method options, e.g. Monte Carlo, integral transport theory and discrete ordinates (Sn ). How the integral transport equation is numerically solved varies from code to code, with the collision probability, interface current, current coupling collision probability 19, and method of characteristics20 approaches being alternatives. Within these lattice physics codes are imbedded various techniques for treating energy and spatial resonance self-shielding, ranging from equivalence theory to multi-band/broad group methods.21 The past practice of energy group collapsing and spatial homogenization

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over successively larger spatial domains found in older lattice physics codes seems to be disappearing in the newer codes thanks to increased computational speed and more efficient numerical methods. In addition, the need for more accurate modeling of the highly heterogeneous lattice designs of today requires more spatial detail to be retained. The isotopic depletion equations are, of course, solved within lattice physics codes so that the lattice can be depleted. The output of such codes are few-group, spatially homogenized cross-sections, both macroscopic and microscopic. In addition, within lattice flux and power distributions are available for later use in pin-power reconstruction. Finally, assembly discontinuity factors are produced, 22-23 a necessary input to modern nodal codes in order to preserve node leakages when utilizing a coarse mesh core simulator. To account for the dependence of the few-group parameters on the instantaneous effects due to moderator density, fuel temperature, and control rod state, numerous lattice physics branch cases at various instantaneous states are completed at specified burnup values. This is necessary since the value of the instantaneous state variables are only known later when the core simulation is completed. By characterizing cross-sections to these quantities, they can be appropriately determined by the core simulator. Core Simulators for Out-Of-Core Nuclear Fuel Management With out-of-core nuclear fuel management decisions requiring assessment over the many cycles associated with the planning horizon, very efficient core simulators are required to reduce the computational burden. Once some initial estimates of the cycling scheme are developed, higher fidelity core simulators can be employed for near-term cycles to refine the cycling scheme. Given the uncertainties in future cycles’ actual energy productions, fuel component costs, e.g. the price of uranium ore, and availability of assemblies to reinsert due to potential fuel damage, one can make an argument that high fidelity core simulators are unwarranted for multi-cycles studies. A wide range of core simulators historically have been employed, trading off computational efficiency with fidelity. An example of a simple model is the Linear Reactivity Model (LRM)24, where one energy group is employed, batches are assumed to be homogenous, and each assembly within the batch is assumed to be surrounded by critical assemblies with core average properties. These assumptions allow the batch power shares, i.e. fraction of total core power produced in a batch, Pi to be determined without solving the spatially coupled neutron diffusion equation using the following simple expression: (1) where (2) and Ni and NTotal denote the number of assemblies in batch i and the total core, respectively, h the assembly pitch, M 2 the migration area, and ρi the batch i reactivity. Equation (2) can be refined to account for spatially dependent leakage effects. Given the batch power shares, the core reactivity can be approximated by batch power share weighting of the batch reactivities. Batch burnups can also be determined using the batch power shares. In this manner the batch burnups and core reactivity can be determined as a function of cycle burnup, allowing the feed enrichment needed to satisfy the cycle energy production requirement and batch and region average discharge burnups to be determined. An alternative to the LRM with similar fidelity and computational burden is the Radial

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Ring Model. In this the core is radially homogenized within radial rings. In addition, for PWRs axial homogenization can also be completed. This allows for PWRs the onedimensional (r) and for BWRs the two-dimensional (r, z), one-group neutron diffusion equation to be solved. Again core reactivity and flux, power and burnup distributions can be obtained. An improvement can be obtained by using crude nodal methods, similar to the original implementation in the FLARE code.25 The one-group diffusion equation is now solved with FLARE-type derived spatial coupling coefficients. Nodalization can be as coarse as a batch, when homogeneity of a batch is assumed, producing the following equation26: (3) where I denotes the total number of batches, k is the core multiplication factor, v i average number of neutrons released per fission reaction in batch i, κi is the average energy released per fission reaction in batch i, (4) and (5) where Wi is the average absorption probability in batch i, fi j the fraction of assembly edges of batch i adjacent to batch j (j = α ⇒ adjacent to the core exterior), and α i the average radial albedo for batch i. This leads to a very small total number of nodes, so the approach is very computationally efficient and more accurate than the LRM, which employs similar nodalization. One could also select an assembly as a node, gaining increased fidelity at the cost of a substantially greater computational burden. This would require a more detailed estimate of the core LP in order to define node neutronic properties (equation (3) only requires edge data for the core LP). Finally, one could use modern nodal methods, as employed in in-core nuclear fuel management studies, which will be discussed presently. For PWRs, this may be done in two-dimensions via axial collapsing with the retention of sufficient fidelity, unless axial heterogeneities in the bundle designs cannot be adequately treated. The use of modern nodal methods in out-of-core nuclear fuel management studies has so far been limited to near-term future cycles and the refining of the cycling scheme established by cruder methods. Perturbation theory to date has played a limited role in outof-core nuclear fuel management analysis. With the exception of modern nodal methods, macroscopic cross-sections models are employed, implying the isotopics as a function of burnup are based upon the lattice physics code’s predictions. This point is also relevant for fuel cycle cost evaluations, normally completed as a part of out-of-core studies, since both batch energy production and isotopics as a function time, i.e. cycle burnup, are required. Core Simulators for In-Core Nuclear Fuel Management In-core nuclear fuel management, which refines the cycling scheme and determines very spatially detailed core attributes, e.g. pellet and pin powers and burnups, requires the use of high fidelity core simulators. Fortunately one is concerned with assessing only the next reload cycle in great detail, and the subsequent few reload cycles in less detail. Modern nodal methods are the staple of core simulators for in-core studies.27 Modern nodal methods utilize a consistent derivation originating from the few-group neutron diffusion equation,

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given by (6) where the notation is standard. It is interesting to note that diffusion theory usually works fairly well despite the strong fuel heterogeneities at the lattice level. This is because the strong spatial heterogeneities have been treated using transport theory in the lattice physics code, and thus been homogenized away by the time the core simulator is utilized. All nodal methods start by integrating equation (6) over the volume of node l to generate a local neutron balance equation in terms of the face-averaged net leakages and the node volume average flux:

(7) where the face-averaged net leakages are defined in terms of the average surface currents, for example in the x direction by:

(8)

To obtain the average surface currents, most modern nodal codes employ the transverse integral method, where one-dimensional diffusion equations are obtained in each direction by integrating equation (6) across the area transverse to the direction of interest to obtain, for example in the x direction: (9) where the average transverse leakage is defined, for example in the y direction, by:

(10)

The solution of the coupled nodal balance and transverse integral equations completes the nodal solution. To solve the transverse integral equations, several approaches have been employed. Since only node average surface leakages are known from a nodal solution, the transverse leakage spatial dependence needs to be estimated. This is commonly done using a quadratic polynomial, preserving the node average surface leakages of the node in question and its adjacent nodes along the direction of interest. The solution of the one-dimensional neutron diffusion equations can be accomplished analytically or using spatial trial functions. The analytic solution gives rise to the Analytic Nodal Method (ANM). The use of trial functions gives rise to the Nodal Expansion Method (NEM) or the Semi-Analytic Nodal Method (SNM). NEM employs polynomials, normally

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up to quartic, as trial functions. SNM supplements the NEM polynomials with hyperbolic functions to more closely match the analytic solution. This is important in the thermal group, where a step change in flux gradient can occur near assembly interfaces due to heterogeneity across the interfaces. The advantages of SNM over ANM are moderately improved computational efficiency and derivation simplicity. PANBOX28, SIMULATE 29 , ANC30, NEM031, PANTHER32, MICROBURN33 and NESTLE34 are just some of the many modern nodal codes in use today. The transverse integral method approach is, however, questionable for partial MOX cores with heavily loaded MOX assemblies. When a MOX assembly is positioned adjacent to a UO2 assembly, a very high thermal flux spatial gradient occurs since the MOX assembly is relatively black to thermal neutrons. This causes the intra-nodal flux shape to deviate substantially from that predicted by the lattice physics code, introducing errors in homogenized cross-sections due to errors in both the intra-nodal flux and burnup distributions. In addition, with large transverse leakage in the radial directions, the simple quadratic approximation given above becomes questionable. For these reasons nodal methods not based upon the transverse integral method have recently been developed. Now spatially non-separable (in the radial directions) trial functions are employed to approximate the intra-nodal flux.35 Via a non-linear iterative method, some radial de-coupling can be introduced to reduce the computational burden. With the current interest in weapon’s grade plutonium disposition in commercial LWRs, characterized by very heavy fissile plutonium loadings, more wide-spread application of non-separable trial function based nodal methods may occur. Both macroscopic and microscopic cross-section models are employed in modern nodal codes. When using a microscopic model, the spatially dependent isotopic depletion equations must be solved for the isotopes being individually tracked, increasing the computational burden. Microscopic models are employed to overcome weaknesses introduced at the lattice physics code level. One weakness is the assumption of zero current boundary conditions, implying neighboring assemblies’ energy spectral effects are not accounted for. This introduces errors in the predicted isotopics, particularly for the outer rows of fuel rods. Microscopic models can keep track of node average volume and surface isotopics, allowing intra-nodal cross-section behavior to be approximated by low order polynomials. In addition, the isotopics of a node not only depend upon burnup, but also moderator density, fuel temperature, for a PWR soluble boron, and for a BWR control rod insertion history effects. These are not known at the time lattice physics calculations are completed, but are determined by the core simulator. The microscopic model can capture most of these effects. The macroscopic model requires macroscopic cross-section values to be characterized as functions of these history effects, this characterization being achieved by completing multiple depletions for different history effects using the lattice physics code. Indeed multiple depletions may also be required when using a microscopic model to characterize the few-group microscopic cross-sections’ history dependences, since the flux energy spectrum used in group collapsing has history dependences. However, this characterization can be cruder since the isotopic number densities dominate history spectral effects on macroscopic cross-sections. Even using a microscopic model, for partially loaded MOX cores inadequate fidelity may be obtained when using a one- or two-group formulation because of the transitional energy spectra about MOX-UO2 assembly interfaces. Since pellet and pin detail are required to ensure that power and burnup constraints are not violated, one must recover this detail from the coarse mesh nodal solution by utilizing the so called “dehomogenization” process. The miracle of modern nodal methods is that this can be done with high fidelity for most situations. It is accomplished by combining information from the nodal and lattice physics solutions.36 The nodal solution provides the node volume and surface average fluxes and power distributions. The lattice physics

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solution provides the pin-by-pin flux and power distributions within the lattice, referred to as form factors when properly normalized. The zero current boundary condition assumption is normally employed in the lattice physics calculations. For a homogeneous node with a zero current boundary condition, a flat flux profile would exist, i.e. the intra-nodal flux is constant. The nodal solution is based upon homogeneous nodes, but with the actual node boundary condition, which implies a spatially varying intra-nodal flux. If the intra-nodal flux distribution can be determined, then the pin-by-pin flux distribution can, in turn, be determined as the product of the intra-nodal flux and form factor. How the intra-nodal flux distribution is obtained from the nodal solution distinguishes different approaches. If the transverse integration method is employed, the surface average currents and fluxes, and node average flux are known from the nodal solution. This is insufficient information to determine the intra-nodal flux with adequate fidelity. If the node comer-point fluxes were also known, then that provides enough information to fit intra-nodal trial functions, selected from a combination of non-spatially separable polynomials and hyperbolic functions, so that all these nodal properties are preserved. The determination of the comer-point fluxes involves using neighboring node flux values available from the nodal solution. How this is completed varies substantially from one approach to another, with none of the approaches being particularly satisfying from a mathematical viewpoint. Recent evidence has indicated that the comer-point approaches being commonly employed have inadequate fidelity for partial MOX cores with heavy plutonium loadings. The advantage of using non-separable trial functions in the original nodal solution, thereby avoiding the need to use the transverse integration method, is that the solution so obtained is the intra-nodal flux. Recognizing that modern nodal method fidelity is required for in-core nuclear fuel management analysis, attention has been given to solving the associated equations as efficiently as possible to minimize the computational burden. Since these equations form large, sparse matrix systems, they are solved using iterative methods with nested iterations. The computational reactor physics community has taken advantage of advances in applied mathematics to solve such matrix systems efficiently, utilizing Krylov subspace and multilevel iterative methods.37-38 A distinguishing feature of in-core nuclear fuel management is that many LPs, and for BWRs many CRPs, must be examined if a nearoptimum set of decisions is to be determined. This implies that a substantial computational effort can be expended up-front, if the result of that effort increases the computational efficiency of the solution of the nodal equations. This is why Generalized Perturbation Theory (GPT)1-2 and Neural Networks (NNs)39 are being employed within in-core nuclear fuel management optimization codes. GPT has the overhead of determining the generalized adjoint fluxes for the responses of interest, e.g. nodal power distribution, but is extremely efficient in evaluating the responses of interest once the adjoint information is determined. Indeed, when the response of interest is the nodal flux distribution, GPT can be cast into an iterative method with extremely rapid convergence properties. NNs require the network to be trained, which therefore entails solving the nodal equations multiple times to generate the training data set. Once trained, the NN can rapidly determine the responses of interest. GPT fidelity has been shown to be acceptable for in-core nuclear fuel management. NN fidelity is not yet adequate, but progress is rapidly being made towards this goal. With its current state of fidelity, a NN may be useful in pre-screening candidate LPs prior to assessment by a higher fidelity, more computationally intensive method. MATHEMATICAL OPTIMIZATION TECHNIQUES It is fair to say that currently the most widely used mathematical optimization technique for nuclear fuel management is the nuclear reload design engineer. Using

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experience, which equates to heuristic rules and informed trial-and-error, a good nuclear reload design engineer can make fuel management decisions that provide feasible solutions which are satisfactory if not near-optimum. Through the utilization of visual user interfaces on powerful workstations,40-42 the nuclear reload design engineer’s productivity and decision making can be enhanced. The Holy Grail of nuclear fuel management is to automate nuclear fuel management decision making using formal mathematical optimization tools. The motivation is to further enhance the nuclear reload design engineer’s productivity and determine decisions that produce near-optimum solutions. Conceptualizing nuclear fuel management decision making as an optimization problem, it can be characterized by its mathematical attributes. Both the out-of-core and incore problems present very large decision spaces. Assuming quarter core symmetry for a PWR, over 10100 LPs are possible when the number of assemblies with distinct nuclear characteristics due to initial fuel enrichment, BP loading and irradiation history are taken into consideration. Determining the optimum LP by articulation, i.e. exhaustive search, is not computationally feasible since to assess the objective function and constraints, solution of the few-group neutron diffusion equations over the cycle for each LP is required. Another attribute of the nuclear fuel management optimization problem is that it is highly constrained, with a number of the constraints active. This causes difficulties in using gradient search techniques to determine the optimum decisions. In addition, gradient information on the objective function and constraints is not readily available. For the out-of-core problem, the decision variables are fresh assembly number and enrichment, potentially burned assemblies to reinsert if heuristic rules are not employed to make this decision, and total BP loading. For the in-core problem, the decision variables are fuel location and orientation, BP location and loading, and for a BWR the CRP and potentially feed bundle selection, and for a PWR potentially the feed enrichment. Thus the decision variables are either integer or mixed-integer and often combinatorial. Finally, the objective and constraints are highly nonlinear functions of the decision variables, implying that a linearized or quadratic representation to facilitate the use of linear or quadratic programming methods is likely to result in the identification of a local, rather than global, optimum. A recent study43 has shown that the search space for a typical PWR reload core design is very highly multimodal, i.e. there are many local minima. In combination these attributes describe an extremely difficult optimization problem, which can be expected to defeat the vast majority of (if not all) deterministic methods, and therefore preference has been given to employing stochastic optimization techniques to solve nuclear fuel management optimization problems. Simulated Annealing Advances in mathematical optimization techniques utilizing stochastic approaches have been rapid over the past two decades. The principal tool for much of this work has been the statistical mechanics algorithm due to Metropolis44, which was first introduced as a means of finding the equilibrium configuration of a collection of atoms at a given temperature. Its major advantage over other methods is an ability to avoid becoming trapped at metastable states (local minima). The algorithm employs a random search which not only accepts changes that decrease the system energy but also some changes that increase it. In the original problem, the latter were accepted with a probability given by the Boltzmann factor: (11) where δE is the change in the system energy, k is the Boltzmann constant and T is the

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absolute system temperature. Over many random steps the computer simulation of the system tends towards thermal equilibrium with its macroscopic parameters fluctuating about their mean values with a Boltzmann distribution appropriate to the temperature. The connection between the Metropolis algorithm and minimization was first noted by Pincus45, but it was Kirkpatrick46 who proposed that it form the basis of an optimization technique for combinatorial problems, in which a set of candidate solutions to minimize objective function F is generated by random trial moves (perturbations in the decision variables). Moves that decrease F are, of course, accepted, while moves which increase F by δF are accepted with probability: (12) where T is a control parameter, which by analogy is known as the system “temperature” irrespective of the objective function involved. As T is usually decreased over time in an “annealing schedule”, the technique is now commonly called optimization by Simulated Annealing (SA). At high temperatures many moves which degrade the objective function are accepted, whereas at lower temperatures few are. If the temperature is lowered too quickly, the search may get trapped in a local minimum; if too slowly, the search will be inefficient. Fortunately, by characterizing the behavior of the objective function values an appropriate annealing schedule is readily determined.47 Constraints can be handled in SA as “hard” limits, meaning that any move which violates them is rejected, but the disadvantage of this approach is that trapping in local minima can occur, particularly if the decision variables’ search space is composed of disjoint regions separated by infeasible regions. The alternative is to define an augmented objective function: (13) where i denotes a specific constraint, e.g. pin power peak, λi is a positive valued penalty coefficient and Θi is a function quantifying the magnitude of any violation of that constraint. The values of the penalty coefficients are usually increased as the search progresses, biasing it progressively more heavily towards feasible space as it proceeds. SA and other stochastic optimization techniques have the desirable feature of identifying a family of near-optimum decision variables, due to the wide-ranging search pattern. For many optimization problems, it can be difficult to mathematically state all the considerations that go into decision making. Given a family of near-optimum solutions from which to choose, the user can factor in these other considerations in making a final decision. SA has proven to be extremely robust in determining families of near-optimum decision variables, but the algorithm can require many iterations and hence be very computationally expensive. Genetic Algorithms Genetic Algorithms (GAs), first formalized as an optimization method by Holland48 and described more recently by Goldberg49, are search routines modeled on the Darwinian theory of evolution. The algorithm is based upon an analog of the “survival of the fittest” principle, where fitness is measured by the value of the (augmented) objective function. GAs differ from most optimization techniques by searching from one group (or population) of solutions to another, rather than from one solution to another. The GA selects the fittest members of the current population as parents for the next generation (population). The GA

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seeks to construct better solutions by combining the features of these good ones through an analogue of biological breeding. Using a procedure known as crossover the “chromosomal” information (an appropriate encoding of the decision variables) of parents is combined to produce offspring. To introduce and maintain diversity in the population, mutations (the analogue of gene copying errors) are introduced with some low frequency. (The mutation operator often corresponds to the operator used to perturb solutions in SA implementations.) Generation after generation is selectively bred in this manner, with the attributes of the best solutions surviving. Clearly the selection of the appropriate crossover and mutation operators are key to the performance of a GA. GAs have a number of advantages over SA. First, GAs appear to do a much better job of searching the decision space and locating the vicinity of the global minimum.50 This can be attributed to the fact that the crossover operator allows major perturbation in the decision variables while still carrying forward desirable attributes. The perturbations made in SA are random. If large perturbations were used, SA would revert to a random, undirected search and be highly inefficient. Second, GAs have the inherent ability to complete multi-objective optimization, where one is interested in determining the trade-off surface of one objective against the other objectives.51 This information is most useful in assessing the “cost of margin”, the amount by which an objective could be improved for a given relaxation in a constraint limit. Finally, GAs are embarrassingly parallel since breeding and solution evaluation can be done in parallel, allowing efficient implementation on multi-processor computers. The disadvantage of GAs is that it has inferior performance to SA for local searches in the vicinity of the global minimum, because the disruptive nature of the crossover operator makes it difficult for GAs to make small changes. This weakness can be addressed by not using the crossover operator once the vicinity of the global minimum has been determined, only using the mutation operator to complete the search, which is essentially the same as using SA. Out-Of-Core Nuclear Fuel Management There have, thus far, been many fewer attempts to apply formal, mathematical optimization techniques to the out-of-core nuclear fuel management problem for LWRs than Table 2. PWR cycling schemes – fresh fuel loadings for a 12-18 month cycle length transition using most reactive backfill for burned fuel.5

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to the in-core problem. If equilibrium cycles are of interest, mathematical optimization to make out-of-core decisions is not warranted since sufficient knowledge captured in heuristic rules should produce optimum decisions. For non-equilibrium cycles, mathematical optimization can probably improve upon the cycling schemes the nuclear reload design engineer can produce using heuristic rules, and at a minimum, produce equal quality cycling schemes in less time via computer automation of the decision making process. What is found when employing a stochastic optimization technique is that the family of nearoptimum cycling schemes nearly have identical levelized fuel cycle costs, their costs differing by less than 0.3% for the top fifty schemes. This is illustrated in Table 1, which presents a sample of the top 50 cycling schemes for a PWR transitioning from 12 month to 18 month cycles found by the OCEON code.5 Indeed, the cycling scheme shown in Table 1 corresponds to the third best scheme identified in Table 1. This enables the nuclear reload design engineer to factor in other considerations in selecting the cycling scheme to employ from this near-optimum family, since economics, at least as measured in terms of levelized fuel cycle cost, does not provide a strong differentiation of cycling schemes. In-Core Nuclear Fuel Management The PWR in-core nuclear fuel management problem has seen fairly wide-spread employment of stochastic optimization techniques, mainly SA and GAs. This has led to the development of software packages that are now being routinely utilized to solve this problem. For PWRs, FORMOSA-P 1-2, 52-53 , PANTHER 50, SIMAN 54-55, ROSA 56 and ALPS57 are examples of software packages that solve the LP optimization problem using SA and, in the case of the first two, GAs. Figure 6 to Figure 10 present FORMOSA-P results for a minimization of feed enrichment objective utilizing SA, with MTC and power peaking factor constraints imposed. This performance is typical of current in-core fuel management optimization packages employing SA. Figure 6 shows the initial LP from which the SA based optimization began and the optimum LP identified. Figure 7 and Figure 8 show the objective function and power peaking factor behaviors versus SA step for all LPs; while Figure 9 and

Figure 6. PWR core (1/4 symmetry) initial LP and LP optimized for minimum feed enrichment.52

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PAUL J. TURINSKY AND GEOFFREY T. PARKS

Figure 7. PWR feed enrichment values for all histories.52

Figure 8. PWR radial power peaking factor values for all histories.52

Figure 9. PWR feed enrichment values for all accepted histories.52

Figure 10. PWR radial power peaking factor values for all accepted histories.52

Figure 10 show the same thing for only SA accepted LPs. The horizontal line in Figure 8 and Figure 10 denotes the power peaking limit. Note the much greater “scatter” in the values of the objective function and power peaking when all LPs are considered, particularly for the power peaking, a very sensitive attribute to which a single perturbation can have a great effect, Figure 7 shows clearly how the SA search escapes from local minima in its quest to locate the vicinity of the global minimum. Figure 10 shows how the decrease in the SA temperature and the increase in the penalty function multiplier as the search progresses make it very unlikely that a sampled LP with a constraint violation will be accepted later in the search. Researchers are continuing to explore the potential of GAs51, 58-63 and Evolution Strategies64 for solving the PWR in-core nuclear fuel management problem. (Evolution Strategies65 are similar to GAs but emphasize mutation rather than crossover as the primary means of development.) There is also some interest in automating the application of heuristic howled e either through an expert system66 or in combination with a stochastic search method54,62. How much these methods improve upon the value of the objective function that a nuclear reload design engineer would determine using heuristic rules cannot be definitively stated, since the complexity of the problem, experience of the nuclear reload design engineer, and time available to solve the optimization problem all impact on the improvement realized. It is, however, clear that one should be skeptical of claims of vast improvements in the objective function value, since such a claim may be based upon an

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initial LP that any experienced nuclear reload design engineer can improve upon via manual optimization. Much less progress has been made in BWR in-core nuclear fuel management optimization, This is undoubtedly due to the increased complexity of BWR optimization because of the complicated interactions between lattice design, bundle design, loading pattern and control rod programming. In addition, a three-dimensional core simulator must be used in order to capture properly all the axial heterogeneity effects. (In contrast, PWR LP optimization can usually be completed using a two-dimensional core simulator.) Most of the optimization work on BWRs has addressed control rod programming, given an assumed LP. This work relies heavily on heuristic rules to direct the optimization rather than the use formal mathematical optimization rnethods.67-70 Recently, the capability to optimize BWR LPs, given an inventory of available bundle designs and the CRP, using a high fidelity core simulator and SA has been demonstrated by the FORMOSA-B code71. Figure 11 to Figure 19 and Table 3 present some results from employing FORMOSA-B to optimize an LP with the objective of maximizing the EOC reactivity. Figure 11 presents the initial LP from which the SA based optimization began and the optimum LP identified. Table 3 gives the values of the objective function (EOC keff ) and constraints for the initial and optimized LPs and the associated limits. Maximum Fraction of Limiting Power Density (MFLPD), Maximum Fraction of Average Planar Linear Heat Generation RATe (MAPRAT) and Maximum Fraction of Limiting Critical Power Ratio (MFLCPR) refer to the thermal limits, and the other limits are self explanatory. SDM was not imposed as a constraint in this particular problem. To limit the search space, bundle exchanges were restricted such that the associated reactivity change ∆ρ ≤ 10% at any time

Figure

11. BWR core (1/8 symmetry) initial LP and LP optimized for maximum EOC reactivity.

72

160

PAUL J. TURINSKY AND GEOFFREY T. PARKS Table 3. BWR LP optimization with multiple active constraints (no SDM).72 Core Attribute

Initial LP value

0.99997 EOC keff MFLPD 1.0703 MAPRAT 1.0635 MFLCPR 0.9599 Peak node discharge burnup <44000 (MWD/STU**) Peak assembly discharge burnup < 34500 (MWD/STU) Region 1 average discharge burnup 30751 (MWD/STU) Power sharing, Region 1/Batch 2 0.579 Power sharing, Region 3/Batch 2 1.374

Limit

Active in Active in Optimized survey?* search? LP value

– 50.98 S0.98 £0.93

– Yes Yes Yes

– Yes Yes Yes

0.99464 0.9766 0.9742 0.9225

≤ 44000

Yes

Yes

<44000

≤ 34500

Yes

Yes

34933

≤ 31000

Yes

Yes

30816

≥ 0.650 ≤ 1.200

Yes Yes

Yes Yes

0.672 1.209

* Survey period refers to the penalty function multiplier and temperature initialization period ** 1 MTU= 1.1023113 STU

during the cycle. It will be noted that a number of the constraints are violated by the initial LP. This is a highly constrained problem, and hence difficult for a formal, mathematical optimization method to tackle. As shown in Table 3, all the constraints are satisfied by the optimized LP except those on peak assembly average discharge burnup and maximum power sharing, which both exhibit small violations. It has been shown that these remaining constraint violations can be removed if the 10% reactivity change restriction on bundle exchanges is lifted, which implies that the number of feasible LPs in the associated subspace is small or even zero.72 Note that the EOC core reactivity actually decreased, i.e. the objective function was made worse, in going from the initial to the optimized LP, but the degree of constraint violation was, of course, greatly improved. Figure 12 and Figure 13 show the augmented and true objective function behaviors. Note again how SA is capable of climbing out of local minima in its search for the global minimum. The behavior of the constraints are shown in Figure 14 to Figure 19, with the

Figure 12. BWR augmented objective function behavior for all accepted histories.72

Figure 13. BWR EOC reactivity for all accepted histories.72

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Figure 14. BWR cycle limiting MFLPD values for all accepted histories.72

161

Figure 15. BWR cycle limiting MAPRAT values for all accepted histories.72

limits indicated by a horizontal line. During the search, the constraint on peak assembly discharge burnup is satisfied at times, but the thermal margin constraints are violated for the associated LPs. These results clearly demonstrate the difficulty of solving via formal, mathematical optimization techniques a problem which is highly constrained.

Figure 16. BWR cycle limiting MFLCPR values for all accepted histories.72

Figure 17. BWR core peak node discharge burnup for all accepted histories.72

Figure 18. BWR peak fuel assembly average discharge burnup for all accepted histories. 72

Figure 19. BWR region 1 average discharge burnup for all accepted histories.72

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FUTURE DEVELOPMENTS There is still plenty of scope for improving the effectiveness of the optimization methods applied to the out-of-core nuclear fuel management problem. With the continuing advances in the performance of core simulators and the inexorable increase in the speed of workstations, it is now realistic to contemplate coupling a GA, say, with a three-dimensional core simulator. The solution of the PWR in-core nuclear fuel management has now reached a level of maturity with sophisticated optimization methods, as described here, being routinely used by utilities. Further advances in this area are likel to be made through the introduction of new optimization methods58, such as Tabu Search 73 or Genetic Programming74-75. The near-term need for BWR in-core nuclear fuel management optimization is to complete the LP and CRP optimization together. The ultimate in-core nuclear fuel management optimization problem, that is optimizing lattice design, bundle design, loading pattern, and in the case of BWRs control rod program, together remains as a future objective. The grand challenge of nuclear fuel management optimization is to remove the artificial separation of out-of-core and in-core nuclear fuel management decision making by solving the multi-cycle nuclear fuel management optimization problem. References 1. D.J. Kropaczek and P.J. Turinsky, In-core nuclear fuel management optimization for PWRs utilizing Simulated Annealing, Nucl. Technol. 95:9 (1991). 2. G.I. Maldonado and P.J. Turinsky, Application of non-linear nodal diffusion Generalized Perturbation Theory to nuclear fuel reload optimization, Nucl. Technol. 110:198 (1995). 3. R.K. Haling, Operational strategy for maintaining an optimum power distribution throughout life, Nuclear Performance of Power Reactor Cores, TID-7672, U.S. Atomic Energy Commission (1964). 4. S. Sun, D.J. Kropaczek and P.J. Turinsky, Haling principle: true or false?, Trans. Am. Nucl. Soc. 68: 482 (1993). 5. S.A. Comes and P.J. Turinsky, Out-of-core fuel cycle optimization for non-equilibrium cycles, Nucl. Technol. 83:31 (1988). 6. W.P. Gassmann, An improved low leakage method of Boiling Water Reactor core design and core management, Proc. Topl. Mtg. Advances in Nuclear Fuel Management II, Myrtle Beach, ANS, 16-37 (1997). 7. C. Vidal, PYROS, a front end for CASMO, Proc. Topl. Mtg. Advances in Nuclear Fuel Management II, Myrtle Beach, ANS, 2-35 (1997). 8. K. Adielson, Hot-Bird, a tool for automated optimization of fuel enrichment design, Proc. Topl. Mtg. Advances in Nuclear Fuel Management II, Myrtle Beach, ANS, 17-31 (1997). 9. K. Hida, Burnup shape optimization for BWR cores by enrichment and gadolinia zoning, Proc. Topl. Mtg. Advances in Reactor Physics, Knoxville, ANS, Vol. III, 233 (1994). 10. Y. Hirano, K. Hida, K. Sakurada, M. Yamamoto, Optimization of fuel rod enrichment distribution for BWR fuel element, Proc. Int. Conf. Reactor Physics PHYSOR96 – Breakthrough of Nuclear Energy by Reactor Physics, Mito, JAERI, B-45 (1996). 11. G.I. Maldonado and T. Guo, Penalty-based constraints applied to within-bundle loading optimization, Trans. Am. Nucl. Soc. 78: 235 (1998). 12. G.I. Maldonado, T. Guo and P. Engrand, Dual-objective Simulated Annealing applied to

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within-lattice loading optimization, Trans. Am. Nucl. Soc. 78: 236 (1998). 13. D. Belhaffaf, G. Mathonniere, S. Mengelle, A. Nicolas and M. Soldevila, Reference calculations performed with APOLLO-2, Proc. Topl. Mtg. Advances in Reactor Physics, Knoxville, ANS, Vol. II, 103 (1994). 14. R.L. Grow, M.L. Williams, D.B. Jones, J.R. Fisher, S.P. Baker and E.L. Fuller, CPM-3, improved lattice physics computer code, Proc. Topl. Mtg. Advances in Reactor Physics, Knoxville, ANS, Vol. I, 29 (1994). 15. M. Edenius, K. Ekberg, B.H. Forssen and D. Knott, CASMO-4: A Fuel Assembly Burnup Program, User’s Manual, SOA-95/1, Studsvik of America (1995). 16. J.J. Casal, R.J.J. Stammler, E.A. Villarino and A.A. Ferri, HELIOS: geometric capabilities of a new fuel-assembly program, Proc. Advances in Mathematics, Computations, and Reactor Physics, Pittsburgh, ANS, 10.2.1-1 (1991). 17. T.Q. Nguyen, K.C. Hoskins, M.M. Weber, S. Srinilta, E. Saji, R.K. Disney and C. Durston, Qualification of the PHOENIX-P/ANC Nuclear Design System for Pressurized Water Reactor Cores, WCAP-11596-P-A, Westinghouse Electric Corporation (1988). 18. M.J. Halsall, WIMS7, an overview, Proc. Int. Conf Reactor Physics PHYSOR96 – Breakthrough of Nuclear Energy by Reactor Physics, Mito, JAEIU, B-1 (1996). 19. E. Fuentes and P.J. Turinsky, Consistent fidelity comparison of integral transport methods, Trans. Am. Nucl. Soc. 71:270 (1994). 20. M. J. Halsall, CACTUS , A Characteristics Solution to the Neutron Transport Equation in Complicated Geometries, AEEW-R 1291, AEEW (1980). 21. D.E. Cullen, Nuclear cross section preparation, in: Handbook of Nuclear Reactors Calculations, Y. Ronen, ed., CRC Press, Boca Raton (1986). 22. K. Koebke, A new approach to homogenization and group condensation, Proc. Specialists’ Mtg. Homogenization Methods in Reactor Physics, Lugano, IAEA, 303 (1978). 23. K.S. Smith, Spatial Homogenization Methods for Light Water Reactor Analysis, PhD Thesis, Massachusetts Institute of Technology (1980). 24. M.J. Driscoll, T.J. Downar and E.E. Pilat, The Linear Reactivity Model for Nuclear Fuel Management, ANS, La Grange Park (1990). 25. D.L. Delp, D.L. Fisher, J.M. Harriman and M.J. Stedwell, FLARE – A ThreeDimensional Boiling Water Reactor Simulator, GEAP-4598, General Electric Co. (1964). 26. C.L. Beard Jr., An Improved Long Range Fuel Management Program, MSc Thesis, Massachusetts Institute of Technology (1978). 27. R.D. Lawrence, Progress in nodal methods for the solution of the neutron diffusion and transport equations, Prog. Nucl. Energy 17:271 (1986). 28. R. Boer, R. Bohm, H. Finnemann and R. Muller, The coupled neutronics and thermalhydraulics code system PANBOX for PWR safety analysis, Kerntechnik 57:49 (1992). 29. J.T. Cronin, K.S. Smith and D.M. ver Planck, SIMULATE-3 Methodology, SOA-95/18, Studsvik of America (1995). 30. Y.A. Chao and Y.A. Shatilla, The theory of ANC-H: a hexagonal nodal diffusion code using conformal mapping, Proc. Topl. Mtg. Advances in Reactor Physics, Knoxville, ANS, Vol. II, 324 (1994). 31. G.H. Hobson and R.C. Aigle, Nodal code developments at FRAMATOME/BWFC, Proc. Topl. Mtg. Advances in Reactor Physics, Knoxville, ANS, Vol. I, 2 (1994). 32. P.K. Hutt, N. Gaines, M.J. Halsall, M. McEllin and R.J. White, The UK core performance code package, Nucl. Energy 30:291 (1991). 33. H. Moon, B. Burdick and D.H. Timmons, Recent improvements in MICROBURN-B Boiling Water Reactor simulation method, Proc. Int. Conf. Mathematics and Computations, Reactor Physics, and Environmental Analyses, Portland, ANS, 39

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(1995). 34. P.J. Turinsky, R.M.K. Al-Chalabi, P. Engrand, H.N. Sarsour, F.X. Faure and W. Guo, NESTLE: A Few-Group Neutron Diffusion Equation Solver Utilizing the Nodal Expansion Method for Eigenvalue, Adjoint, Fixed-Source Steady State and Transient Problems, EGG-NRE-11406, INEL (1994). 35. J.M. Noh and N.Z. Cho, A new diffusion nodal method based upon analytic basis function expansion, Nucl. Sci. Eng. 116:165 (1994). 36. K.R. Rempe, K.S. Smith and A.F. Henry, SIMULATE-3 pin power reconstruction: methodology and benchmarking, Nucl. Sci. Eng. 103:334 (1989). 37. H.G. Joo and T. Downar, An incomplete domain decomposition preconditioning method for nonlinear nodal kinetics calculations, Nucl. Sci. Eng. 123:403 (1996). 38. R.M. Al-Chalabi and P.J. Turinsky, Multigrid method applied to the solution of an elliptic, generalized eigenvalue problem, Proc. Conf. Iterative Methods, Copper Mountain, SIAM, Vol. II, Session I, Paper 1 (1996). 39. H. Noda, A. Yamamoto, Y. Nagasawa, H. Murao and S. Kitamura, Core burnup calculations using neural networks, Proc. Topl. Mtg. Advances in Nuclear Fuel Management II, Myrtle Beach, ANS, 20-39 (1997). 40. J.G. Stevens and K.R. Rempe, Recent enhancements in the X-IMAGEISIMAN core design environment, Proc. Topl. Mtg. Advances in Nuclear Fuel Management II, Myrtle Beach, ANS, 8-19 (1997). 41. D. Hodges and R.R. Rose, Interactive BWR core physics workstation, Proc. Topl. Mtg. Advances in Nuclear Fuel Management II, Myrtle Beach, ANS, 2-1 (1997). 42. G. Wiksell, OpenJack – a tool for optimizing core shuffle scheme, Proc. Topl. Mtg. Advances in Nuclear Fuel Management II, Myrtle Beach, ANS, 2-11 (1997). 43. A. Galperin, Exploration of the search space of the in-core fuel management problem by knowledge-based techniques, Nucl. Sci. Eng. 119:144 (1995). 44. N. Metropolis, A.W. Rosenbluth, M.N. Rosenbluth, A.H. Teller and E. Teller, Equation of state calculations by fast computing machines, J. Chem. Phys. 21:1087 (1953). 45. M. Pincus, A Monte Carlo method for the approximate solution of certain types of constrained optimization problems, Oper. Res. 18: 1225 (1970). 46. S. Kirkpatrick, C.D. Gerlatt Jr. and M.P. Vecchi, Optimization by Simulated Annealing, Science 220:671 (1983). 47. P.J.M. van Laarhoven and E.H.L. Aarts, Simulated Annealing: Theory and Practices, D. Reidel, Dordrecht (1987). 48. J.H. Holland, Adaptation in Natural and Artificial Systems, University of Michigan Press, Ann Arbor (1975). 49. D.E. Goldberg, Genetic Algorithms in Search, Optimization and Machine Learning, Addison Wesley, Reading (1989). 50. G.T. Parks and M.P. Knight, (A comparison of) PANTHER loading pattern optimization search options, Proc. Topl. Mtg. Advances in Nuclear Fuel Management II, Myrtle Beach, ANS, 8-67 (1997). 51. G.T. Parks, Multiobjective PWR reload core design by nondominated Genetic Algorithm search, Nucl. Sci. Eng. 124:178 (1996). 52. G.I. Maldonado, P.J. Turinsky, D.J. Kropaczek and G.T. Parks, Employing nodal Generalized Perturbation Theory for the minimization of feed enrichment during Pressurized Water Reactor in-core nuclear fuel management optimization, Nucl. Sci. Eng. 121:312 (1995). 53. P.M. Keller and P.J. Turinsky, Recent developments in FORMOSA-P, Proc. Topl. Mtg. Advances in Nuclear Fuel Management II, Myrtle Beach, ANS, 8-39 (1997). 54. J.G. Stevens, K.S. Smith and K.R. Rempe, Optimization of Pressurized Water Reactor shuffling by Simulated Annealing with heuristics, Nucl. Sci. Eng. 121:67 (1995). 55. J.G. Stevens and K.R. Rempe, Recent enhancements in the X-IMAGEISIMAN core

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design environment, Proc. Topl. Mtg. Advances in Nuclear Fuel Management II, Myrtle Beach, ANS, 8-19 (1997). 56. F.C.M. Verhagen, M. van der Schaar, W.J.M. de Kruijf, T.F.H. van de Wetering and R.D. Jones, ROSA, a utility tool for loading pattern optimization, Proc. Topl. Mtg. Advances in Nuclear Fuel Management II, Myrtle Beach, ANS, 8-31 (1997). 57. J.L. Bradfute, Y.A. Shatilla and B.J. Johansen, Recent developments in Westinghouse automated fuel management code, ALPS, Proc. Topl. Mtg. Advances in Nuclear Fuel Management II, Myrtle Beach, ANS, 8-49 (1997). 58. J.N. Carter, Genetic algorithms for incore fuel management and other recent developments in optimisation, Adv. Nucl. Sci. Technol. 25: 113 (1997). 59. P.W. Poon and G.T. Parks, Application of Genetic Algorithms to in-core nuclear fuel management pptimization, Proc. Joint Int. Conf. Mathematical Methods and Supercomputing in Nuclear Applications, Karlsruhe, Vol. 1, 777 (1993). 60. E. Tanker and A.Z. Tanker, Application of a Genetic Algorithm to core reload pattern optimization, in: Reactor Physics and Reactor Computations, Y. Ronen and E. Elias, eds., Ben Gurion University of the Negev Press (1994). 61. M.D. DeChaine and M.A. Feltus, Nuclear fuel management optimization using Genetic Algorithms, Nucl. Technol. 111: 109 (1995). 62. M.D. DeChaine and M.A. Feltus, Fuel management optimization using Genetic Algorithms and expert knowledge, Nucl. Sci. Eng. 124:188 (1996). 63. T. Bäck, J. Heistermann, C. Kappler and M. Zamparelli, Evolutionary algorithms support refueling of Pressurized Water Reactors, Proc. 3rd IEEE Conf. Evolutionary Computation, IEEE Press, 104 (1996). 64. J.K. Axmann, Parallel adaptive evolutionary algorithms for Pressurized Water Reactor reload pattern optimizations, Nucl. Technol. 119:276 (1997). 65. G. Rudolph, Evolution Strategies, in: Handbook of Evolutionary Computation, T. Bäck, D.B. Fogel and Z. Michalewicz, eds., IOP Publishing Ltd. and Oxford University Press (1997). 66. A. Galperin and Y. Kimhy, Application of knowledge-based methods in in-core fuel management, Nucl. Sci. Eng. 109:103 (1991). 67. T. Fukuzaki, K. Yoshida, Y. Kobayashi, H. Matsuura and K. Hoshi, Knowledge based system for control rod programming of BWRs, J. Nucl. Sci. Technol. 25: 120 (1988). 68. C. Lin, An automatic control rod programming method for a Boiling Water Reactor, Nucl. Technol. 92:118 (1990). 69. L.S. Lin and C. Lin, A rule-based expert system for automatic control rod pattern generation for Boiling Water Reactors, Nucl. Technol. 95:1 (1991). 70. M.S. Taner, S.H. Levine and M-Y. Hsiao, A two-step method for developing a control rod program for Boiling Water Reactors, Nucl. Technol. 97:27 (1992). 71. B.R. Moore and P.J. Turinsky, FORMOSA-B: a BWR incore fuel management optimization package, Proc. Topl. Mtg. Advances in Nuclear Fuel Management II, Myrtle Beach, ANS, 17-21 (1997). 72. B.R. Moore, Higher Order Perturbation Theory for BWR In-Core Nuclear Fuel Management Optimization, PhD Thesis, North Carolina State University (1996). 73. F. Glover and M. Laguna, Tabu Search, Kluwer Academic Publishers, Boston (1997). 74. J.R. Koza, Genetic Programming, MIT Press, Cambridge (1992). 75. J.R. Koza, Genetic Programming II, MIT Press, Cambridge (1994).

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INDEX AEAT-CFX code, 121 Aldermaston,2 ALPS code,157 ANC code, 152 ANM (Analytic Nodal Method),1511 AP-600 design,104,121,128 APOLLO code, 148 ASME (American Society of Mechanical Engineers US) code,80,92 ASTM (American Society for Testing Materials) program,80 attachment, microbial,66-68 coefficients,68,73 BALI program,105 Bifurcation,22,26 biofilm,61 biomass,61 accumulation,68 exchange kinetics,66 bioremediation,59 Black Report,4 Boltzmann equation,21 operator,31 BORAX-2 reactor,52 BP (Burnable Poison),137 BTEX plume,72 Burghfield,2 BWR (Boiling Water Reactor),137 oscillations,46

control cell core,146 COPO facility (Finland),105,114,118-120 corium,105 COUPLE code model, 104 CPM code,148 crack brittle,80 running,79 CRP (Control Rod Program),139 Cutler, James,4 CZP (Cold Zero Power),138 Darcy flux,60 DCB (Double Cantilever Beam) geometry, 86-87 debris pool,129 decay heat, 105 detachment, microbial,68-69 diffusivity, effective,106,13 1 discontinuity factors, 149 DNA damage,3 DOE (Department of Energy US),59 Dounreay,2,6,9 double tension test,82 Down's syndrome,3 'down-swing',89 DWO (Density Wave Oscillation),47-50

ECCM (Effective Conductivity-Convectivity Model),106,126,132 Eckert correlations,107,112 Cap la Hague,2,7 EOC (End of Cycle),139 CASMO code,148 Epidemiological studies, 1,4 CAT (Crack Arrest Temperature),80,92-960 EPRI (Electrical Power Research Institute US) CCA (Compact Crack Arrest) geometry,86, study, 80 87,97 erbium oxide, 139 CCT (Centre Crack Tension),81 ESSO test,80-82 center manifold,23 Evolution Strategy,158 CFD (Computational Fluid Dynamics),121,132 FAD (Fracture Analysis Diagram),80 failure prevention (crack),79 Charpy test,83-86 Chernobyl FATT (Fracture Appearance Transition and leukaemia,8 Temperature),83-84 CHF (Critical Heat Flux), 141 feedback, 29 et seq. filtration theory,66-67 CLT (Central Limit Theorem),52,55 FLARE code, 150 COMARE (Committee on the Medical Aspects of Radiation in the flaws,79 Environment UK),4 FORMOSA-B code,140,157,159

167

168 FP (Fokker-Planck) equation,25,29 GA (Genetic Algorithms),155-157 Gardner hypothesis,2 gadolinia,138-139 Gaussian noise,24 et seq. Genetic Programming, 162 GPT (Generalised Perturbation Theory), 153 GUI (Graphical User Interface) uses,148 Haling principle, 141 harmonic balance,36 HAZ (Heat Affected Zone),83,87 Heisenberg,21 HELIOS code,148 heuristics, 142 Hopf bifurcation point,27,38,45,52 HSE (Health and Safety Executive UK),79 HSST (Heavy Section Steel Technology), 83,96 hydraulic conductivity,61 et seq. HZP (Hot Zero Power),138 ICF (Irreversible Circulation of Fluctuation),38 IFBA (Integral Fuel Burnable Absorber), 139 inheritance leukaemiac disposition, 10 instability, chugging,47 hard and soft mode,26 Ledinegg,47 Jacob number,50 Jacobian,25 JAERI (Japanes Atomic Energy Research Institute),33 KBM

(Krylov-Boguliubov-Mitropolsky) averaging,36 k-ε models,106 et seq. Kinlen hypothesis,2 La Hague - se Cap la Hague Langevin equation,23 linearised,31 lattice physics,148 LBZ (Local Brittle Zones),83 Ledinegg instability,46,50 LEFM (Local Embrittlement Failure Mode),83,92 LOCA (Loss Of Coolant Accident),83, 93-95,141 Loviisa reactor, 105,118 LP (Loading Patterns),137 LRM (Linear Reactivity Model),149-150 LWR (Light Water Reactor),103

INDEX MELAD (MEtal LAyer Demonstration) experiment,121,123 Metropolis algorithm,154 microbial kinetics,7 1-72 MICROBURN code,152 microcolony,64 MMCT (Moment Modified Compact Tension) specimen,80,86,88 Monod growth,71,73 Monte Carlo methods,148 MOX (Mixed OXide) assemblies,140,152-153 MTC (Moderator Temperature Coefficient), 139,157 MVITA (Melt Vessel Interaction - Thermal Analysis) code model, 105,126-128, 130- 132 Navier-Stokes equation,53 NDTT (Nil-Ductility Transition Temperature), 80,84,97 NEM (Nodal Expansion Method), 151 NEMO code, 152 NESTLE code, 152 NN (Neural Network), 153 nodal methods, 150 et seq. Noise, reactor,22 rise and supression,22,55 source,37 Wong-Zakai,27 NRC (Nuclear Regulatory Commission US), 80 NRPB (National Radiological Protection Board UK),8 NSRR (Nuclear Safety Research Reactor Japan),33 NTR (Nonlinear TRanformation),23 nutrient delivery,69 OCEON code, 157 OPCS (Office of Population Census and Surveys UK),6 ORNL (Oak Ridge National Laboratory),96 oscillation, density wave 48 pressure drop,47 oscillator, damped harmonic,36 OUST (Onset of Upper-Shelf Temperature),96-97 PANBOX code, 152 PANTHER code, 152,157 paternal exposure to radiation,10 Pellini test,84,97 PHOENIX code,148 plasticity,91 plutonium and leukaemia,8 Poissonian statistics,22 power reactor theory,22

INDEX PTSE (Pressurised Thermal Shock Experiment),80,92,96 PWR (Pressurised Water Reactor),141 oscillations,43 Radial Ring Model,149-150 Rayleigh-Benárd convection, 123,125,127,132 reactor-scale, 125-126 RELAP code, 104 REV (Representative Elementary Volume),64 Riccati equation,24 'ring-down',89 'ring-up',90 ring of fire,144 RIT (Royal Institute of Technology Sweden),105 Robertson test,82 ROSA code, 157 Routh-Hunvitz criterion,43,54 RPV (Reactor Pressure Vessel),103 et seq. 131 RWMAC (Radioactive Waste Management Advisory Committee UK),9 SA (Simulated Annealing),155,157 SCA (Short Crack Arrest) plate test,80,83 SCDAP code, 104 SDM (ShutDown Margin), 138,143-144, 159 sediment, aseptic,59 Sellafield, 1 et seq. SENB (Single Edge Notched Bend) test,86 SENT (Single Edge Notch Tension) test,81, 86,91 SIMAN code,157 SIMULATE code, 152 Slaving Principle,22 smoking, disease related,2

169 soliton solution,23 spectral shift operation,145,147 stochastic power oscillations,43 stress intensity factor,86-87 waves,83,89 Tabu Search,162 TDCB (Tapered Double Cantilevered Beam) geometry,86 Texas sharpshooter,5,13 thermal loading, 103 Thorotrast,11 Three Mile Island and leukaemia,8 Toda potential,24 toughness test,86 TRIGA reactor (Japan),33 TSE (Thermal Shock Experiment),81,92,95 TWI (formerly The Welding Institute),80,96 UCLA (University of California Los Angeles) heat transfer facility,l05,115,117 'up- swing',90 van der Pol equations,38,49 velocities, effective,109 viral cause leukaemia, 12 VR (void-reactivity),47 waves, stress,83,89 wide plate test,82 WIMS code,148 Windscale, 1 and leukaemia,8 WZ see noise