Advances in Nuclear Science and Technology: Volume 24 (Advances in Nuclear Science & Technology)

Advances in

Nuclear Science And Technology VOLUME 24

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

Neuclear Science And Technology VOLUME 24

Edited by

Jeffery Lewins Cambridge University Cambridge, England

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

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PREFACE

Nuclear technology, which may be said to have started with devices like the Cockcroft-Walton machine of the thirties, or the later Lawrence Cyclotron, and blossomed with the development of nuclear reactors in the forties, is now well into its sixth decade. What future has it? A seminal turning point may be taken as the Chernobyl accident (in 1986) occurring at its half-century and we now enter the first decade beyond Chernobyl. Chernobyl may well be seen as a paradigm of the break-up of the Soviet Socialist Republic and while it would be foolish to think that this political upheaval was caused simply by the failure of the social as well as technical system evidenced at Chernobyl, it certainly marks a turning point in modern history. But if we are to look forward rather than back, as is our intention in this volume, we must nevertheless ask what sort of accident was Chernobyl? Certainly it raised substantial questions about the nature of nuclear electricity generating systems, the raison d’être of nuclear technology. What is the future of nuclear power? In the Occident we see a clear maturity of the technology. With the possible exception of France, which still has major overseas influence, the construction of power stations appears to have passed its peak. It is too soon to write off the operating industry, however; it accounts for 10% of production in the US, some 24% in the UK, to 80% in France. But nonetheless, the emphasis is passing from design through operating procedures, where fine tuning of the system raises profits, to consideration of long term questions of waste disposal—some of these being related to the war-time and cold-war military rather than civil programs. Matters in the Orient are different; Japan has a fully developed industry still committed to further construction; most of the young dragons have seen the necessity if not the virtue of nuclear power as an alternative to fossil fuels and have substantial construction programs. To contribute to the vision of the next century, this volume starts with the perceptive comparison and contrast drawn between the Chernobyl and the Bhopal accident that preceded it. In terms of immediate cost in human lives, Bhopal was far more serious; yet it is largely forgotten, certainly in the West. Nevertheless the common themes of both accidents spell out a lesson for the future, drawn for us by Malcolm Grimston. What are these lessons? That accidents happen is hardly new, and in the US nuclear scene was perhaps better established by Three Mile Island than Chernobyl. Robert Wright has painstakingly undertaken the forensic task of understanding the core melt-down that characterised this accident in 1979 so that we might be better prepared against further accidents. Certainly a better understanding of the nature of such a plant is necessary. Part of this understanding must also be directed to the fundamental physical nature of process. The paper by Russell Mosteller and his colleagues is directed to just such an analysis of our understanding of the interaction of neutrons with matter in their Monte Carlo assessment of the international data listing: ENDF/Bv

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PREFACE

VI. We are grateful for the careful evaluation this presents of the virtues and such continuing deficiencies of the international understanding of neutron interactions. If we state that nuclear power in the form of electrical generation is central, we should not ignore the other applications, particularly perhaps in the medical world for diagnostics and treatment. Here the problems of assessment of dose, etc., involve the acceptance of the variability, indeed randomness, of the distribution of matter—in things and in people. Gerry Pomraning has carried us forward by his review of particle transport theory in stochastically distributed systems. A particular lesson to draw from this work is the deficiency of deterministic methods based on mean values in assessing radiation distributions. Transport theory equations have to be solved and except for rare cases where analytical solutions can be obtained, valuable as they are for understanding the nature of their solutions, we turn to numerical computational methods for professional application. Benoist and Petrovic do more than review the theory that was established largely by the pioneering work of Pierre Benoist; they review the practical applications of the TIBERE models to explicit computation of leakage terms in realistic reactor geometry. We are delighted to have a presentation of this theory of such international stature. The final two papers our volume concerns systems more than fundamental elements. Modern computers could have prevented, perhaps, the Three Mile Island accident—if trusted and not overruled, they would certainly have avoided Chernobyl. But how shall such computers work to apply the maximum artificial intelligence to the problem? Pázsit and Kitamura are therefore to be thanked for surveying the rôle of neural networks in reactor control. But the final paper takes systems theory to the whole of nuclear power, and the work of Hansen and Golay in applying system dynamics to the broad problem of the development of nuclear power can, we hope, point the way forward to the coming century. They provide a technique with which to assess and quantify the broad issues of nuclear power: finance, risks and public perception. 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

Chernobyl and Bhopal Ten Years on Malcolm C. Grimston 1. 2. 3. 4. 5. 6.

Summary A Brief Outline of the Accidents at Chernobyl and Bhopal Causes of the Accidents Consequences of the Accidents Public Perceptions Conclusions

1 4 13 29 40 42

Bibliography

44

Transport Theory in Discrete Stochastic Mixtures G.C. Pomraning 1. Introduction 2. Mixing Statistics 3. Markovian Mixtures without Scattering 4. Markovian Mixtures with Scattering, Exact Results 5. Markovian Mixtures with Scattering, Low-order Models 6. Markovian Mixtures with Scattering, High-order Models 7. Non-Markovian Mixtures 8. Concluding Remarks

References

47 50 57 65 72 78 83 89 90

The Role of Neural Networks in Reactor Diagnostics and Control Imre Pázsit and Masaharu Kitamura 1. 2. 3. 4. 5. 6.

Introduction History and Principles of Artificial Neural Networks Survey of Applications in Reactor Diagnostics and Control Solving Inverse Problems with Neural Networks in Core Noise Diagnostics Case Study: Neutron Noise Diagnostics of Control Rod Vibrations Conclusions

95 97 107

References

127

116 119 126

Data Testing of ENDF/B-VI with MCNP: Critical Experiments, Thermal-Reactor Lattices and Time-of-Flight Measurements Russell D. Mosteller, Stephanie C. Frankle, and Phillip G. Young 1.

Introduction

131 vii

viii

2. 3. 4. 5. 6. 7.

CONTENTS

Data Testing: Critical Experiments Data Testing: Thermal Reactor Lattices Effect of ENDF/B-VI Release 3 for Data Testing: Time-of-Flight Measurements Conclusions and Recommendations Acknowledgements

137 153 163 169 190 192

References

192

System Dynamics: An Introduction and Applications to the Nuclear Industry K.F. Hansen and M.W. Golay 1. 2. 3. 4.

Introduction Background Application of System Dynamics to Nuclear Industry Summary

197 198 205 218

References

220

Theory: Advances and New Models for Neutron Leakage Calculations Ivan Petrovic and Pierre Benoist 1. 2. 3. 4. 5. 6. 7.

Introduction Homogeneous Theory Heterogeneous Theory Simplified Heterogeneous Models: TIBERE and TIBERE-2 Homogenization by an Equivalence Procedure Some Numerical Comparisons Conclusions

223 226 232 243 260 266 275

References

279

Current Status of Core Degradation and Melt Progression in Severe LWR Accidents Robert R. Wright 1. 2. 3. 4. 5. 6. 7. 8. 9.

Introduction Lessons from the TMI-2 Accident Core Degradation Early Phase Melt Progression Late Phase Melt Progression Melt-Water Interactions in the Lower Plenum Lower Head Failure Consequences of Reflooding a Severely Degraded Core Conclusions

283 284 288 288 295 301 304 310 312

References

313

INDEX

317

CONTENTS OF EARLIER VOLUMES1

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

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

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CONTENTS OF EARLIER VOLUMES

Vapor Explosion Phenomena with Respect to Nuclear Reactor Safety Assessment, A. W. Cronenberg and R. Benz 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

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CONTENTS OF VOLUME 17 A Methodology for the Design of Plant Analysers, T. H. E. Chambers and M. J. Whitmarsh- Everies 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

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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 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-Hermite 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 Information 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

CHERNOBYL AND BHOPAL TEN YEARS ON Comparisons and Contrasts

Malcolm C. Grimston, MA (Cantab.), BA (Open) Senior Research Fellow Centre for Environmental Technology Imperial College of Science Technology and Medicine London SW7 2PE

SUMMARY Within eighteen months in mid-1980s, two of the most serious industrial accidents in history occurred. At Bhopal, capital of Madhya Pradesh in central India, in the early hours of December 3, 1984, an explosion occurred at Union Carbide of India Ltd (UCIL)’s methyl isocyanate plant. Some 40 tonnes of a complex chemical mixture were released, causing several thousand deaths and hundreds of thousands of injuries. At Chernobyl in the north of Ukraine, then part of the USSR, in the early hours of April 26, 1986, an explosion at a state-owned nuclear power plant caused the release of some 6 tonnes (possibly more) of radioactive materials. The ‘immediate death toll was much lower than at Bhopal. The usually-quoted figure of 30, all of whom were on site in the immediate aftermath of the explosion, includes two who died on site from burns and falling masonry and 28 who died within the following few weeks from Acute Radiation Syndrome. (Another person on site died of a heart attack, which it is difficult to associate directly with the accident.) However, the effects of the accident were widespread, especially in Belarus, Ukraine and Russia, and theoretical calculations suggest that further illness and premature deaths are to be expected for several years to come. This paper compares and contrasts these two accidents. There are striking similarities. Both accidents involved highly technical industrial processes and plant which were set in relatively backward regions. Madhya Pradesh is one of the least industrialised states in India, manufacturing accounting for only 11% of the State Domestic Product in 1978/79, while Ukraine is a largely agrarian economy, with the exception of its five nuclear power stations. Both accidents occurred in the early morning, were blamed on ‘operator error’ and led to prosecutions of individuals. However, HOT (Human, Organisational, Technological) analysis reveals that deeper institutional and managerial weaknesses contributed significantly

Advances in Nuclear Science and Technology, Volume 24 Edited by Lewins and Becker, Plenum Press, New York, 1996

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to both accidents. For example, in the case of Chernobyl, experiments similar to those which led to the accident had been carried out on previous occasions, while at Bhopal several of the technical factors which were causes of the accident had previously been identified but not rectified. Indeed, if the term ‘operator error’ is used to mean actions by operators which were simple mistakes (inadvertent deviations from operating codes) then it is not clear that the phrase can be used in either case, with the exceptions of only one or two actions in each case. It seems that the operators in question generally knew precisely what they were doing. They broke safety ‘rules because of other institutional pressures, such as the unexpected need to generate electricity on the afternoon of April 25 1986 in Chernobyl, and general commercial pressures in a loss-making plant at Bhopal. In both cases technological factors were also implicated. Both plants had fundamental design flaws, such as the storage of MIC in large tanks and the back-fitted ‘jumper line’ in the pipework at Bhopal, and the positive void coefficient at Chernobyl which led to a positive power coefficient at low power outputs. Both plants also allowed operators to disable key safety equipment, contributing to the accidents. However, while there was considerable component failure at Bhopal (e.g. the failure of the pressure valve of tank E610), there was no apparent component failure at all at Chernobyl. Both accidents were exacerbated by a lack of information and immediate action on the part of the local authorities, which both reduced the effectiveness of measures which could have been taken earlier, and led to a loss of confidence in those authorities among the affected people. As a result, the detrimental effects of both accidents were wider than the somatic health problems caused by the toxic releases themselves, and included considerable problems related to stress. In both cases, different interested groups made widely different interpretations of the accidents and their effects. The States and operating organisations involved tended to concentrate on technical, quantifiable factors, while affected individuals and international activists tended to concentrate on the effects on humans and to rely on hearsay and anecdote as the source of information. The failure of these two approaches to address the concerns and interests of the others undoubtedly contributed further both to anxiety among those affected and to frustration on the part of the operators and regulators. Both accidents, to a greater or lesser extent, became ‘defining events’ for political activism against the chemical and nuclear industries respectively. Nonetheless, there are striking differences between the two accidents. At Bhopal, the morale of operators and supervisors was very low. The plant was losing money in a division which was regarded as a career ‘dead-end’ within Union Carbide. Highly skilled staff generally moved on rapidly. This was compounded by poor instrumentation (e.g. pressure gauges which could not be seen from the control desk, and in different and incompatible units; instructions in English which many operators did not speak) and poor maintenance. Chernobyl, by contrast, was the flagship plant of one of the Soviet system’s flagship technologies, and operators, who were well trained and well paid by Soviet standards and with high social status, appeared to believe it impossible for any major accident to occur, and therefore felt confident in ignoring operating instructions. The overall effect in both cases was remarkably alike; the human contributions to both accidents following similar courses. It would appear that there is a necessary ‘morale envelope’ to operating high technologies; operators must become neither too complacent nor too depressed. The demonstrable health effects of the material releases at Bhopal (3 800 deaths and 203 000 injuries by official 1990 estimates) considerably outweigh those of the material releases from Chernobyl (33 deaths - the original 30 plus three deaths from thyroid cancer, with perhaps one other among those who suffered from Acute Radiation Syndrome - and more than 600 cases of thyroid cancer, by late 1995).

CHERNOBYL AND BHOPAL TEN YEARS ON

3

The real financial cost of responding to the Chernobyl accident worldwide has been considerably higher than that for Bhopal. Radioactivity can be detected at much lower levels than chemicals such as methyl isocyanate, and is much more persistent in the environment. As a result evidence of the Chernobyl accident could be detected far afield (e.g. caesium contamination in fields in Cumbria and North Wales), while no such evidence was found for migration of the plume from the Bhopal accident. Hence countermeasures could be (and still are being) taken against the Chernobyl releases, even where it was felt that these were extremely unlikely to cause significant health problems, while similar countermeasures could not be taken against a potential health problem which could not be detected. Probably for the same reason, there has been far more research into the long-term effects of the Chernobyl accident than of Bhopal. As a result comparisons of the long-term health implications of the two accidents are difficult to make with any confidence, though it would seem that the overall health effects of Bhopal considerably outweigh those of Chernobyl. The place of the two accidents in public perceptions is rather different. Bhopal is largely forgotten by the general public and the media; there was for example, relatively little interest in the tenth anniversary of the accident, in 1994. Chernobyl, by contrast, still attracts considerable attention, the fifth anniversary being marked by three television documentaries in the UK, for example, and the incident continuing to attract constant, if low-level, media interest. The reasons for this would appear to be a mixture of factors contingent on the accidents themselves (especially the fact that Chernobyl happened on the edge of Europe while Bhopal occurred in the Third World), and differences in the perception of risks from radiation and from chemicals. Of particular relevance is the fact that fallout from Chernobyl can be detected at very low levels and so is perceived as a potential risk to everyone and to future generations, while it is assumed that the MIC released at Bhopal was hydrolysed relatively rapidly in the environment and hence does not represent a continuing and widespread threat. Accidents involving large numbers of acute deaths are far more frequent in developing countries and the former Soviet Union than in the developed world. Comparison of Bhopal and Chernobyl with incidents like the leak of aldicarb oxime in 1985 from the UCC plant at Institute, West Virginia, and with Three Mile Island, Pennsylvania (1979), cannot yield firm conclusions over why this should be the case, but certain themes do emerge, albeit tentatively. The level of human and organisational failings in the American accidents seem to be very similar to those at Bhopal and Chernobyl. For example, a very similar incident to that which led to the Three Mile Island accident had occurred at another Babcock and Wilcox nuclear plant two years previously, while an internal UCC report just three months before the Bhopal accident raised the possibility of a runaway reaction involving water at Bhopal. In neither case was the information passed on to the relevant plant management or operators. This would seem to point to superior technology in the developed world. For example, the emergency safety equipment at Institute stopped the leak of aldicarb oxime within 15 minutes. However, the severity of the Bhopal accident was clearly a function of the very high population density near the plant, compared e.g. to Institute. The month before Bhopal, for example, there was a release of an MIC/chloroform mixture from Institute which amounted to about one sixth of the release from Bhopal. Presumably had that occurred in an area as densely populated as Bhopal the death toll would have been about 500. Three Mile Island, by contrast, released about one millionth as much radioactive material as did Chernobyl, and remains the most serious incident in a Western nuclear power station. It may also be the case, then, that it is possible to design ‘failsafe’ nuclear stations while it is not possible to do the same with chemical installations (though there may well be other approaches to inherent safety in the chemical industry, e.g. reducing or eliminating storage of hazardous intermediates on the process site). However, more research would be necessary before any great confidence could be put in such conclusions.

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A BRIEF OUTLINE OF THE ACCIDENTS AT BHOPAL AND CHERNOBYL Bhopal Background Bhopal is the capital of the central Indian State of Madhya Pradesh. This is one of the least industrialised States in India, manufacturing accounting for just over 11% of the State Domestic Product in 1978/9 (Minocha, 1981) but the city was undergoing rapid expansion (from 102 000 in 1961 to 670 000 in 1981). In 1984 the Union Carbide Corporation (UCC) was Americas seventh largest chemical company, with both assets and sales of about $10 billion worldwide. It owned or operated companies in some 40 countries, producing a wide range of chemical products such as petrochemicals, industrial gases, pesticides, metals, carbon products, consumer products and technology services. However, there were questions over Union Carbides profitability, which was considerably lower than its major competitors such as Dow Chemical, Du Pont and Monsanto (Hiltzik, 1985). As a result, the late 1970s saw the company concentrate its investment in its most profitable lines such as industrial gases, consumer products, technology services and speciality products (UCC, 1984). Other, less profitable businesses were divested (Business Week, 1979). At the same time, economies associated with forward and backward integration of processes (e.g. reductions in transportation costs, economies of sales) were explored. One example was a move towards manufacturing component chemicals for various UCC products. Union Carbide’s Indian subsidiary had been formed in 1934 as the Ever Ready Company (India) Ltd, manufacturing batteries in Calcutta and Madras. Diversification in the mid-1950s led the company to change its name to Union Carbide (India) Ltd, UCIL, in 1959. By 1984 UCIL was India’s 21st largest company, with annual sales of around $170 million. 51% of UCIL’s shares were owned by UCC, 23% by the Indian Government, the rest by private investors and other institutions. Diversification into chemicals and plastics continued through the 1960s, and in 1969 a plant to formulate pesticides and to act as headquarters for the Agricultural Products Division was established in the north of Bhopal, close to the railway and bus stations (Figure 1). One of the operations in which the plant was engaged was the manufacture of pesticides. There was some concern that the potentially hazardous chemicals involved should be processed within a very densely populated part of the city, but as the plant was at first used for ‘formulation’ (the mixing of stable substances to make pesticides) it was accepted that it did not represent a major risk to local people. In 1974 UCIL was granted a license to manufacture pesticides, which involved reacting chemicals together to produce desired substances. By 1977 UCIL was producing more hazardous pesticides such as carbaryl, which was marketed under the name Sevin. Carbaryl is made from phosgene, naphthol and methylamine (aminomethane). There are two broad ways of carrying out the manufacturing process. Phosgene and naphthol can be reacted together, and the resulting compound reacted with methylamine. This route does not involve any highly dangerous intermediates (though of course phosgene itself is poisonous), and has been used e.g. by Bayer in Germany. Alternatively, phosgene and methylamine can be reacted together to form methyl isocyanate, MIC, which is then reacted with naphthol. This route involves the production and storage of MIC, which was known to be hazardous though relatively little work had been done on its long-term effects on human health, but it is also generally regarded as being more economic, especially if there is an external market for MIC. It was the latter method which was chosen for Bhopal.

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At this stage the component chemicals for carbaryl, one of which is methyl isocyanate, were imported in small quantities to Bhopal from what was then UCC’s only MIC manufacturing plant, at Institute, West Virginia. The policy of (backward) integration of processes led UCIL to decide, in 1979, to manufacture five component chemicals for the pesticides, including MIC, at Bhopal. The technology to be used was basically similar to that at Institute, though pressure from the local government to maximise employment led to the use of manual rather than computerised control systems. Surprisingly little was known of MIC’s possible consequences on human health - or at least surprisingly little was in the public domain. (There were suggestions both that UCC had carried out research which had not been made public, and that work done by various national Governments had been kept secret owing to the potential use of MIC in chemical weapons (Delhi Science Forum, 1985).) Though it is not a member of the cyanide family, it could be associated with a range of serious medical conditions, especially affecting the eyes, skin, respiratory tract and immune system (e.g. Smyth, 1980). The chemical is unstable, and has to be stored at low temperatures. Municipal authorities objected to the manufacture and use of dangerous chemicals in a very highly populated area, contrary to the plant’s original license for commercial and light industrial use (Bhopal Town and Country Planning Department, 1975). UCIL was a very powerful company in India and in Madhya Pradesh, however, and the state and central Governments overruled the city authorities and gave permission for the manufacture of MIC. There seemed to be a widespread belief among local managers and workers alike that MIC was ‘no worse than tear gas’, and that its effects could not be fatal. There was no known antidote to the gas. The early 1980s was a time of considerable overcapacity in the pesticides industry, coupled with reductions in demand. As a result, UCIL decided in July 1984 that the Bhopal plant, with the exception of the MIC plant, should be sold (US District Court, 1985).

Before the Accident MIC for use in manufacture of carbaryl (or, if contaminated, for reprocessing) was stored at Bhopal in three large underground tanks, designated E610, E611 and E619. This system was generally regarded as being potentially more dangerous than the alternatives, viz. smalldrum storage, or use of a manufacturing method which does not necessitate the production of MIC such as that used by Bayer. ‘UCC insisted on a process design requiring large MIC storage tanks over the objections of UCIL engineers’ (ICFTU, 1985). ‘The UCIL position was that only token storage of MIC was necessary’, but UCC ‘imposed the view and ultimately made to be built large bulk storage tanks patterned on the similar UCC facilities at Institute, West Virginia’ (Wall Street Journal Europe, 4.2.85). MIC was manufactured in batches in a refining still, and carried along a stainless steel pipe which branched off into the three tanks. When the MIC was required it was transferred out of the tanks under (very pure) nitrogen pressure. It then passed through a safety-valve to another stainless steel pipe, the ‘relief-valve header’, which led to the production reactor. This pipeline was about 7 metres above ground level. Another common pipe took rejected MIC back to the tanks for reprocessing, while contaminated MIC was taken to a vent-gas scrubber for neutralisation with caustic soda. Excess nitrogen could be released from the tanks through a ‘process pipe’ fitted with a blow-down valve. The relief-valve pipe and the process pipe, despite their very different functions, were connected by another pipe, the ‘jumper system’, fitted a year before the accident to simplify maintenance (Figure 2). The normal storage pressure in the tanks was or about 1 atmosphere (15 psi). Each storage tank had its own high temperature alarm, a level indicator and high- and low-level alarms. It was recommended that the level in each tank did not exceed about 50%

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of capacity. The safety valves between the tanks and the relief valve pipe were accompanied by a ‘rupture disk’ which kept in gas until it reached a certain pressure, then let it out. This could not be monitored from the central control point, requiring manual readings from a pressure indicator located between it and the safety valve. A batch of MIC had been manufactured between October 7 and October 22 1984. At the end of this period, tank E610 contained 42 tonnes of MIC, while E611 contained 20 tonnes. This represented a considerable inventory of the chemical. As noted by Dr Varadarajan of the Indian Council for Scientific and Industrial Research, and confirmed by UCIL manager S. Kumaraswami, UCIL stored ‘many times its own requirement’ at Bhopal in order to be able to supply other chemical firms in India, and to guard against the whole Sevin manufacturing process having to be closed down if there were a failure in MIC production (Everest, 1985). After the completion of production of this batch the flare tower was shut down so that a piece of corroded pipe could be replaced. The flare tower was used for burning normally vented gases from the MIC and other sections of the Bhopal plant; burning neutralised the toxicity of these gases before they reached the atmosphere. However, the flare tower was not designed to cope with large quantities of a chemical such as MIC, Unfortunately, other features of the plant’s safety systems were also shut down. The vent-gas scrubber received gases from the tanks, neutralised the gases with caustic soda (sodium hydroxide) solution, and released them into the atmosphere at a height of 30 metres or routed them to the flare tower. The scrubber had been turned off to a standby position a few weeks before the accident. The plant refrigeration unit was a 30 tonne unit which used the CFC Freon to chill salt water, the coolant for the MIC tanks. This system was shut down in June 1984 and drained of Freon, thus rendering it unavailable for use during an emergency (Diamond, 1985). There remained a set of water-spray pipes which could be used to control escaping gases, but these were to prove ineffective (see below). On October 21, the nitrogen pressure in tank E610 fell from the roughly normal 1.25 atmospheres to 0.25 atmospheres. This made it impossible to draw any MIC from this tank, so MIC for the reactor was drawn from tank E611. However, on November 30 tank E611 also failed to pressurise, owing to a defective valve. Attempts to pressurise E610 once again failed, however, and so the operators instead repaired the valve in tank E611.

The Night of December 2/3 1984 In the normal course of operation in the plant, small quantities of MIC and water react together in the pipes to produce a plastic-like substance which is a trimer of MIC (i.e. three MIC molecules attached together). Periodically the pipes were washed with water to flush out trimer that had built up on the pipe walls. Because MIC and water react together, albeit slowly if the MIC is quite pure, during flushing out the pipes were blocked off with a barrier disc known as a ‘slip blind’ to prevent water going into the storage tank. Such a flushing was ordered to be carried out on the night of December 2/3, and began at 2130. Responsibility for carrying out the flushing operation lay with the supervisor of the MIC plant. However, responsibility for fixing the slip blind lay with the maintenance supervisor. Several days earlier the post of second-shift maintenance supervisor had apparently been deleted, but nobody had been assigned the duty of inserting the slip blind. Several overflow devices (‘bleeder lines’) were clogged, and so water began to accumulate in the pipes. A valve used to isolate the various pipe lines was leaking (as were many others in the plant), and so water rose past that valve and into the relief-valve pipe. The operator noticed that no water was coming out of the bleeder lines, and shut off the flow with a view to investigating what was happening to this water. However, the MIC plant supervisor ordered him to resume the process (ICFTU, 1985). Water now flowed downhill

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from the relief-valve pipe towards tank E610. It then flowed through the jumper system into the process pipe (which is normally open), and then to the blow-down valve. This valve should have been closed. However, it is part of the system responsible for pressurising the tank with nitrogen. It is likely that a fault in this valve, or perhaps even that it had simply been left open, was the cause of the problem which had prevented the pressurisation of tank E610 for the previous six weeks, and had not been corrected. With the blow-down valve open, an estimated 500 kg of water flowed into tank E610 (through an ‘isolation valve’, which was normally left open), and started to react with the MIC. Because tank E610 had not been pressurised for six weeks, considerable quantities of contaminants were able to leak into the tank which would normally have met a pressure barrier. These contaminants included transition metal ions such as iron, chromium, nickel and cadmium which act as powerful catalysts for the reaction between water and MIC. (The iron probably originated in corroded carbon-steel pipes; it had originally been recommended that stainless steel be used, but this was not done, presumably for economic reasons.) As a result this reaction proceeded much more rapidly than would have been the case in the absence of these contaminants. At 23.00, after a change of shift, the new control-room operator, Suman Dey, noted that the pressure in tank E610 was about 0.7 atmospheres, well within the normal range of 0.1 to 1.7 atmospheres (2-25 psi) (but considerably above the 0.25 atmospheres at which the tank had remained for some weeks since the failure of the pressurising system). At 23.30 a leak of MIC and dirty water was detected near the scrubber. The mixture was coming out of a branch of the relief-valve pipe, downstream of the safety valve. Workers examining the situation found that another valve (a ‘process safety valve’) had been removed, and that the open end of the relief-valve pipe had not been sealed with a slip bind for flushing. By 00.15 on December 3 the pressure in tank E610 had reached 2 atmospheres, outside the normal range; by 00.30 the reading had gone off the top of the scale (just under 4 atmospheres, 55 psi). (It is indicative that some of the pressure dials in the plant were in psi, others in the incompatible units of kg per ) There was a hissing sound from safety valve of tank E610, indicating that it had opened. The local tank temperature and pressure gauges now showed values above their scale maxima of 25 °C and 4 atmospheres (55 psi). There was a loud rumbling and screeching from the tank. The operators tried to start the scrubber, but instruments showed the caustic soda was not circulating. Clouds of gas were now gushing from the plant chimney stack. The plant superintendent was summoned, and immediately suspended operation of the MIC plant. The toxic-gas alarm, designed to warn the community round the plant, was turned on. Bhopal was a city containing many shantytowns and slums, two of these large slum colonies containing several thousand people being literally across the road from the plant. However, a few minutes later the toxic-gas alarm was switched off, leaving just the plant siren to warn workers sounding. The water-sprays were turned on to douse the stack (as well as the tanks and the reliefvalve pipe). However, the water sprays did not reach the gases, which were being emitted at a height of 30 metres. The safety valve remained open for two hours. During that time, through it passed a complex mixture of gases, foam and liquid at a temperature of over 200 °C and a pressure of 15 atmospheres (UCC, 1985). It should have been possible to decant some of the MIC from tank E610 into E619, as one tank should always have been kept empty against such emergencies. However, E619 itself contained a considerable quantity of impure MIC, and there was some evidence that this was starting to react as well, though there were no releases.

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Chernobyl Background The Chernobyl Nuclear Power Plant is in the northern part of the now independent former Soviet Republic of Ukraine, near the borders with Belarus and Russia, in the eastern part of a large geographical region known as the Byelorusso-Ukrainian Poless (woodland), on the River Pripyat which flows into the Dnieper. It is 15 km away from the town that gives it its name, and some 130 km north of Kiev, in an isolated area of the country with low population density Apart from the company town, Pripyat (population 45,000, evacuated in 1986) 3 km from the station, there were no major centres of population near the plant (Figure 3). Ukraine is a largely agrarian economy (‘the breadbasket of the Russias’), with a population of some 52 million people, and an energy use of about 190 million tonnes of oil equivalent per year. This represents a slightly lower usage of energy per head than a country like the UK, though for a far lower industrial output. Electricity accounts for about a quarter of Ukraine’s energy use. Coal is the most important source of electricity providing 54% of the total, with nuclear power producing 38% from thirteen nuclear reactors at five power stations. Soviet interest in nuclear technology began during the Second World War, soon after nuclear fission was first described in 1939. The USSR became the second nation to test a nuclear device in August 1949, much to the surprise of the USA and the UK. Nuclear technology became one of the highest priorities for the Soviet regime. In June 1954, the small (5 MW) Soviet research reactor at Obninsk (which is still operated) became the first nuclear station in the world to generate electricity for use in an electricity grid. At that time a number of nations were at a similar stage in the development of nuclear power; the USA had made the first nuclear electricity in 1952 (lighting five electric light bulbs), while the UK was to become the first nation to operate a commercial-scale nuclear power station, at Calder Hall in 1956, the USA following in 1957. The Soviets launched the worlds first nuclear powered icebreaker in 1957. Soviet scientists were to develop two principal types of commercial nuclear station (as well as other systems, including two ‘fast reactors’). One, the VVER, is similar to the western Pressurised Water Reactor, and in its three versions (the 440-230 series, the 440-213 series and the 1000 series) has come to dominate electricity production in the former Soviet bloc (e.g. Bulgaria, Slovakia, the Czech Republic, Hungary, Slovenia, as well as republics of the former Soviet Union (FSU), and East Germany, where the reactors were closed on unification). However, there is another design, the RBMK, based on the Obninsk reactor. Naturally, all nuclear power stations in the FSU were built, owned and operated by the state. In 1989, just before the breakup of the USSR, it operated 46 nuclear power reactors with an installed capacity of 35.4 GW. Presently, there are fifteen commercial RBMKs operating, all in the FSU, at St Petersburg, Smolensk and Kursk in Russia, Ignalina in Lithuania and Chernobyl in Ukraine. In addition there are four reactors at Bilibino, Siberia, similar to the RBMK and used for district heating and power production, and RBMKs for weapons production at Krasnoyarsk and Tomsk. The RBMK has proved to be a relatively reliable producer of electricity with high availability. The RBMK is different in principle from any other commercial reactor system in the world, mixing water cooling with a fixed graphite moderator. (A moderator is necessary to slow down the neutrons which cause nuclear fission to take place in the reactor fuel.) Water passes through pipes which pass through the core where the nuclear fission takes place. This water boils in the pipes, and the steam produced goes to turbogenerators to create electricity. The ratio of liquid water to steam (‘void’) in the pipes is important. At very low power output the water in the pipes in the core would be almost entirely liquid. As liquid water is a

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good absorber of neutrons, this regime could only be sustained if the ‘control rods’ (which control the level of nuclear activity by absorbing neutrons themselves) were almost all removed from the core. If something should now happen to cause the water to boil, there would be an increase in the number of free neutrons, and hence in nuclear fission, and hence in heat production. This in turn would promote further boiling, and a ‘vicious circle’ would ensue. This is referred to as a ‘positive power coefficient’. The problem only affects operation at very low power (which was both forbidden in operating instructions, and prevented by the presence of an emergency core cooling system which would shut down the reactor at low steam levels), and the concept had been used in the early weapons production reactors in the USA. It was for example rejected by the British on safety grounds in 1947 (Arnold, 1992, p. 12), and has never been pursued outside the Former Soviet Union. The first commercial RBMK began operation at St Petersburg in 1974. The Chernobyl Nuclear Power Plant was the third site to use RBMKs (the first reactor of 1000 MW rating entering full service in 1978) and the first nuclear power station in Ukraine SSR. A second unit, which has been closed since a fire destroyed the roof of the turbine hall and instrumentation equipment in 1991, began operating in 1979, a third in 1982, and then unit 4 in 1984. It was unit 4 which exploded in 1986. Two other RBMK units on the site were under construction in 1986 but were abandoned soon afterwards.

Before the Accident Ironically, one of the potentially more serious events that can occur at a nuclear power station is a power cut. Though the nuclear fission processes themselves could be shut off immediately, the same could not be achieved for the residual heating caused by radioactive decay of fission products in the fuel. This heating would initially represent some 7% of normal operating power. If emergency safety (cooling) systems were unable to operate because of interruption in their power supplies, it is feasible that considerable damage could be done to the reactor. For this reason back-up emergency systems running on different energy sources such as diesel are fitted. Chernobyl unit 4 was scheduled to come out of service for routine maintenance on April 25, 1986. As the station came off line, it was proposed to carry out an experiment to determine whether even in the event of a power cut, or some other event which interrupted the flow of steam into the turbogenerators such as a major pipe failure, there would be enough energy in the turbogenerators as they ‘coasted down’ to operate the plant’s Emergency Core Cooling System (ECCS) for the 30 or 40 seconds required to get the diesel backup power supply working. In fact, the ECCS itself could not be used in this experiment, as it would shut down the reactor before the experiment could start. It was therefore decided to use the main circulating pumps and the feedwater pumps (which keep the coolant in the core circulating) to simulate the load on the turbogenerators which would be caused by the ECCS. The experiment was designed by an electrical engineer, who regarded it as something which would not affect the nuclear side of the plant (as it could have been done with the reactor switched off, using decay heat in the fuel). However, plant operators decided to keep the reactor operating at low power so that the experiment could be repeated in case of failure. Under such conditions, the effect of using circulating pumps and feedwater pumps to simulate the load of the ECCS was highly relevant to the nuclear side of the plant, as they would change the rate of flow and hence the temperature of water passing through the core, and thence the neutron absorption characteristics of that coolant.

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From 01.00 on April 25 the reactor’s power was reduced from its normal 3 200 MW (thermal), until by noon it was operating at half power. The experiment was to be carried out at 700-1 000 MW (thermal). Running the station on half power exacerbated one of the classic problems of nuclear power generation, ‘xenon poisoning’. The element xenon-135, which is created as a result of nuclear fission (both directly, and more importantly from radioactive decay of another fission product, iodine-135), is a very good absorber of neutrons. If too much of it builds up in the fuel, it can therefore prevent nuclear fission occurring. In normal reactor operation the amount of xenon remains constant, as the rate at which it is created is balanced by the rate at which it absorbs neutrons and turns into other elements. However, if the reactor is brought down from full power to half power, there will be a considerable period during which the rate at which xenon-135 is created is greater than that at which it is destroyed. This is because iodine-135, and hence the creation of xenon-135 through decay of this iodine, will be at full-power levels, but destruction of xenon-135 will be occurring at only half-power rates. The concentration of xenon-135 therefore increases. At 14.00 the reactor Emergency Core Cooling System (ECCS) was disconnected from the primary circuit. This system would have tripped (switched off) the reactor during the experiment and prevented a repetition if necessary. There was now an unexpected hitch in bringing a coal-powered station on line, and the Ukrainian electricity authority, Kievenergo, demanded that Chernobyl-4 remain on line for a further nine hours, until 23.10. The station was therefore run at half power for this length of time with the ECCS disengaged, contrary to operating instructions, though not directly relevant to the accident.

The Night of April 25/26,1986 At 23.10 the operators started to reduce power by inserting control rods (which absorb neutrons) into the core. However, at 00.28 an operator failed to reset the power regulation system to the desired 700-1 000 MW (th), and as a result the power kept falling. Power slumped to below 30 MW (th). This led to greatly increased levels of xenon in the fuel; xenon was still being produced (from iodine-135) at 1 600 MW rates, but hardly being burned up at all by the very low neutron levels in the fuel at the low power output. The increase in xenon poisoning made it much more difficult for the operators to reverse the mistake and bring power back up to what was required. The vast majority of the total of 211 control rods had to be removed, and even then the power could only be raised to 200 MW (th). At this relatively low power output most of the water in the pipes passing through the core was liquid. In this unstable (and forbidden) regime, several signals were sent from various parts of the station, including the steam pressure monitors, ordering a trip of the reactor, but these were overridden by the operators, who disabled that part of the automatic control system. At 01.19 the operators opened the main feed valve, thus increasing the levels of water in the system. As this cold water passed into the core steam production fell further, and power began to fall as more neutrons were absorbed by the water. To maintain power at 200 MW (th) it was necessary to remove all but the last 6 to 8 control rods from the core, something which was absolutely forbidden. At 01.21.50 the operator sharply reduced the supply of cold water, and steam began to be produced. Some automatic control rods started to reenter the core to compensate for reduced neutron absorption in the water. Nonetheless, there were still fewer than half of the design ‘safe’ minimum of control rods in the core, and less than a quarter of what was demanded in the operating instructions.

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At 01.23.04 the main steam valves to the turbogenerator were closed, simulating a failure in power supply and initiating the experiment. As already stated, the reactor could have been tripped at this stage, but was kept operating in case the experiment had to be repeated. The closing of these valves reduced the flow of main coolant and of feedwater, increasing the temperature in the coolant and hence steam production. The number of neutrons being absorbed by the coolant therefore fell, and an increase in reactor power was noted at 01.23.31. Automatic control rods entered the core but were unable to stop this increase in power. At 01.23.40 the shift manager attempted to shut off the fission process by inserting the manual control rods. However, these rods were inserted mechanically (rather than by gravity), and took about 10 seconds fully to insert. In fact, a design fault meant that the initial insertion of the rods actually increased the reactor power. Three seconds later the power had reached 530 MW (th). A condition known as ‘prompt criticality’ now ensued, and two explosions were heard. It is believed that the first resulted from an interaction between the fuel and the coolant once some of the fuel rods had been destroyed by the increase in power, while the second may have resulted from an explosion between hydrogen (formed when the very hot water came into contact with the graphite moderator) and air which had entered the reactor space. The second explosion is calculated to represent something like 480 times the normal operating power of the reactor, which had been running at 6% of normal power just five seconds earlier. At least 3% of the 190 tonnes of fuel in the core were ejected from the building, and over 30 fires were started. Highly radioactive material was strewn around the wreckage of the reactor building. Emissions from the stricken reactor continued until May 5, when in effect they ceased (Figures 5 and 6).

CAUSES OF THE ACCIDENTS It has been common practice, until recent years, to regard accidents as being caused either by ‘component failure’ or by ‘human error’. In fact, of course, these are not mutually exclusive categories. Simple component or system failure, for example, can often be traced back to human mistakes, for example in the original design of the plant, or in the maintenance and monitoring function. Similarly, it can be argued that a properly designed plant should be capable of ‘forgiving’ considerable deviation on the part of the operators. However, it is also clear that there is an intermediate class of errors between the two, which can be referred to as ‘organisational’. Organisational inadequacies might involve excess pressure on operators to ignore safety procedures (or indeed inadequate safety procedures in the first place); poor management or other factors leading to low morale; poor supervision of operation; poor flow of information through the operating utility etc. Such organisational factors can result in operators quite deliberately acting contrary to safety and other operating codes, or being unaware of them. In such cases the term human ‘error’ would not seem to be appropriate. Both accidents here under discussion exhibit many examples of this phenomenon. It is common now to carry out ‘HOT’ (Human, Operational and Technological) analyses of the causes of major accidents.

HOT Analysis of Bhopai Human Factors Though the Bhopal plant’ s technology was broadly similar to that used in UCC’s other MIC manufacturing plant at Institute, West Virginia, it relied much more heavily on manual, rather than computerised, control systems. This seems to have been as a result of the Indian government’s insistence that the plant should provide as much employment as possible. As a

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result, human involvement in the accident, both at operator and at manager level, was considerable. The plant, as discussed above, was losing money, and received a very low priority from the Union Carbide Corporation; indeed, apart from the MIC unit the whole plant was put up for sale in 1984. The Agricultural Products Division offered poor promotion prospects. These factors had obvious effects on employee morale. Some 80% of workers trained in MIC technology in the USA had left the plant in the four years before the accident, while many of those who remained had started their own businesses on the side to supplement their wages. Furthermore, cost-cutting led to reduction in staffing levels - between 1980 and 1984 the entire MIC unit work crew was cut from twelve to six, while the maintenance crew was cut from three to one. A 1982 safety audit revealed that there was a general carelessness and lack of safety consciousness in the plant’s operations at all levels (UCC, 1982). Examples cited include maintenance workers signing permits which they could not read (being written in English); workers in prohibited areas without the required permits; fire-watch attendants being called away to other duties. A number of specific unsafe practices can be cited as being relevant to the Bhopal accident. The position of maintenance engineer on the second and third shifts had been eliminated. This person had been responsible for fitting the slip blind which would isolate the pipes from the MIC tanks during flushing out, and the task had not been reassigned. Staff reductions had an inevitable effect on the level of human back-up to safety systems. Important instrumentation, e.g. the local pressure valves and dials for each MIC tank, were not accessible to the central control site, and required an operator physically going to the tank to check them. In addition, emergency communications within the plant, and from the plant to the local community, depended on messengers physically carrying messages from place to place. Levels of staff safety training were poor; in particular, workers had no training on how to deal with emergencies; indeed, there were no emergency plans. One manifestation of this was the early switching off of the site emergency alarm designed to warn the local community. Top managers seemed to know very little about the possible health consequences of MIC and other chemicals in use in the plant. When, several weeks before the accident, storage tank E610 suffered a considerable drop in pressure, nothing was done to rectify the situation. The lack of positive pressure in tank E610 allowed small amounts of water containing contaminants to leak into the tank and local pipes. This led to a blockage of those pipes with MIC trimer, necessitating the flushing operation on the night of December 2/3, and allowed catalysts to contaminate the MIC in the tank and so accelerate the reaction between water and MIC during the accident. When the worker who was carrying out the flushing procedure noticed that no water was coming out of the far end of the pipes, he stopped the procedure. However, he was ordered to recommence by the production supervisor who made no attempt to discover where the water was going. This supervisor had been transferred to the MIC plant from a UCIL battery plant a month before the accident and had little experience of the MIC plant and its hazards. Tank E610 was filled to 75-80% of its capacity, against the recommended 50%. Further, neither of the other tanks was empty to transfer MIC into if necessary. The Union Carbide Corporation claimed that the accident had been caused by deliberate sabotage by a worker (investigation carried out by Arthur D. Little Inc, May 10 1988, quoted in UCC, 1994) who attached a water pipe directly to the tank. Even if this were true, the above contributions to the accident remain relevant; the accident had involved simultaneous failures in plant, management and operation, and other relevant failures had occurred some weeks before the accident (Adler, 1985).

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Organisational Factors The relatively low importance of Bhopal to Union Carbide starved the plant of both resources and managerial attention. The whole of UCIL represented less than 2% of UCC’s worldwide sales, and Bhopal, one of thirteen plants owned by UCIL, had operated at less than 40% capacity for several years. (Indeed, the plant had been established just as the pesticides market began to decline and competition increased, and serious consideration was given to abandoning the plant even as it was being constructed.) In the fifteen years between the plant’s establishment and the accident there had been eight different senior managers, many coming from outside the chemical industry with little experience of the specific problems raised by the plant. This turnover of top management brought with it frequent changes in operational procedures and consequent uncertainty for operators. Technology policies, operating and safety manuals and maintenance procedures were all derived from UCC’s documents, but tended to be left to the local management for implementation, with occasional safety audits from the parent company. The last of these, in May 1982, detected inter alia the following weaknesses which were relevant to the accident: the potential for release of toxic materials in the MIC unit and storage areas because of equipment failure, operating problems or maintenance problems; lack of fixed water-spray protection in several areas of the plant; potential for contamination, excess pressure or overfilling of the MIC storage tanks; deficiencies in safety-valves and instrument maintenance programmes; problems created by high personnel turnover at the plant, particularly in operations. To these can be added the poor allocation of certain safety features, which allowed, for example, responsibility for flushing the pipes and responsibility for fitting the slip blind to rest with different functions within the plant. The local plant management did develop an action plan to deal with these problems (Mukund, 1982), but clearly this had not been entirely effected or effective. A further failure in the organisation/operational system allowed the earlier shut down of the refrigeration system. This could have cooled the tanks sufficiently to slow down the reaction between the impure MIC and water, or at least reduced the temperature so that fewer of the toxic by-products of the reaction would have been formed. There was no contingency plan to deal with major emergencies. Reference to the Indian Government’s enthusiasm for the plant to create as much employment as possible, which led to the installation of manual rather than computerised control systems, should also be made under this heading.

Technological Factors Broadly, technological contributions to an accident can be divided into those associated with plant design, and those associated with component failure. The preconditions for the Bhopal accident lay in the fact that large amounts of MIC were stored in underground tanks in an operating environment which relied on manual rather than computerised control systems, and hence had no reliable early warning system. Alternatives (small drum storage; constantflow production cycles which did not require storage of MIC) were available. The design did

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not include a backup system to divert escaping MIC into an effluent area for quick neutralisation. One of the other tanks should have been kept empty for diversion of material in an emergency, but at the time of the accident all three tanks contained significant amounts of MIC. Other relevant technological factors included: Failure of the pressurisation system in tank E610. This allowed ingress of water, containing contaminants that could catalyse the reaction between MIC and water, which would otherwise have met a pressure barrier. In the absence of these catalysts the 500 kg of water which entered the tank would have been insufficient to cause such a violent reaction in such a short period of time. Water approached the tank via the relief-valve pipe and the process pipe. This is strongly implied by the high level of sodium ions in the tanks. (Alkaline water from the scrubbers tended to accumulate in the relief-valve and process pipes.) Water could only enter the tank through the blow-down valve or the safety valve. The possibility of a malfunction in the blow-down valve was suggested by the failure in the pressurisation system of tank E610, but was not investigated. Water could only reach the blow-down valve from the process pipe, and could only reach the process pipe from the reliefvalve pipe through the jumper line, which had been retrofitted to the plant without due consideration of this possibility. The use of corrodible carbon steel rather than stainless steel for pipework was the probable cause of the high concentration of iron ions in the small quantities of water which entered the tank prior to the accident. Gases released from the storage tank could not be neutralised or contained. The scrubber was designed to deal with gases alone and not with a mixture of gases and liquids, and anyway failed to operate, having been put on stand-by. The flare tower was down for maintenance, though it is questionable as to whether it could have coped with the volumes of releases. Water sprinklers could not throw water high enough to neutralise the escaping gases. The general standard of maintenance in the plant was poor, though it had only been installed in 1981. Valves and pipes were rusted and leaking and instrumentation was unreliable.

HOT Analysis of Chernobyl Human Factors The Chernobyl accident is remarkable in at least two respects. The first is that there was no component failure of any description; all ‘technological’ factors apply purely to the design of the plant. The second is that, with the (very important) exception of the failure of the operator to reset the power regulation system to stabilise the reactor power at 700-1 000 MW(th) which caused a dramatic slump in power, none of the human actions can be described as ‘errors’. All of the other precipitating actions were deliberate violations of operating rules, presumably made within the context of a determination to carry out the experiment under all circumstances. This in turn may have derived from the fact that the experiment could only be carried out once a year, when the plant was coming off line for maintenance. The first set of human factors relevant to the accident concern the conception of the experiment itself. This experiment was designed by a consulting electrical engineer. He appears to have regarded the test as a purely electrical one, in which the reactor was not relevant, and indeed the reactor could have been tripped as soon as the experiment started, or even earner. However, presumably because the operators wished to be able to run the experiment a second time should the first attempt prove inconclusive, the reactor was kept running at low power. Now the importance of rates of water flow through the core became crucial to the operating parameters of the nuclear reactor.

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It is reported that the director of the station was unaware of the experiment, while the chief engineer and head of the reactor section seem to have approved the experiment without properly acquainting themselves with the details. The members of the state inspectorate, the Gosatomenergoadzor, had all gone to the local clinic for medical inspections on April 25, so nobody was on site to prevent breaches of the operating code (Ignatenko et al., 1989). The sloppy drawing up of the test programme, without agreement with the station physicists, the reactor builders, the RBMK designers or representatives of the inspectorate may appear bad enough. Yet the operators then deviated very significantly from this programme. When the Kievenergo control centre demanded that Chernobyl-4 be run for a further nine hours, the plant was run at half power with the Emergency Core Cooling System switched off, in contravention to strict operating instructions. Though of itself this was not relevant to the accident, it demonstrates the attitude towards operating instructions on the part of the operators. Of more relevance is the fact that the original test had been scheduled for the afternoon of April 25, but the test as carried out occurred in the early hours of the next morning, when most of the site’s professional scientists and engineers had left, and perhaps also when operators were not at the peak of alertness. After the dramatic dip in reactor power caused by the failure to enter a ‘hold power’ command at 00.28 (the ‘genuine error’ referred to above), operators fought to increase the reactor power, but could only raise 200 MW (th), against the 700-1 000 MW demanded by the test programme. To achieve this many more control rods were removed than allowed in the instructions. (Out of 211 control rods, the instructions recommend that an ‘Operating Reactivity Margin’, ORM, of an equivalent of at least 30 rods should be maintained at all times. The operator has discretion between an ORM of 30 and 15 rods, but below 15 rods, in the words of Valery Legasov, head of the Soviet delegation to the conference on the accident held in August 1986, ‘no-one in the whole world, including the President of the country, can allow operation. The reactor must be stopped.’ At the start of the experiment proper, 01.22.30, the ORM was between 6 and 8 rods. Legasov says ‘the staff were not stopped by this and began the experiment. This situation is difficult to understand’.) In this regime it was impossible to shut down the reactor quickly. The reactor was now operating at 7% of normal output, well below the 20% specified as a minimum in the operating instructions, and the 22-31% envisaged in the test programme. Under these conditions, a slight boiling in the coolant would reduce neutron absorption, causing an increase in nuclear activity, and hence of heat, which would further boil the coolant, resulting in a further reduction in neutron absorption, and so on. This is referred to as a ‘positive power coefficient’ - if power increases this causes a further increase in power. The reactor was operating at a far lower power than expected, but the flow of coolant through the core remained at levels demanded by the experiment programme. As a result there was even less steam in the mixture in the core, resulting in an even more unstable condition. The whole core was perhaps within 1 K of boiling, but no boiling was taking place. The operators overrode the many ‘trip’ commands coming from a variety of parts of the plant. Indeed, the whole process of keeping the reactor at power, in order to repeat the experiment if necessary, was not part of the test programme; the test could in fact have been carried out soon after switching the reactor off, using the decay heat of the fuel.

Organisational Factors The remarkable actions of the operators - Legasov said they ‘seemed to have lost all sense of danger’ - cannot be explained in terms of ‘error’, but must be seen against the institutional background of the plant’s operation.

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In many senses nuclear technology, along with the space programme, was the most prestigious of the industries of the USSR. Chernobyl had been described as the flagship of the Soviet nuclear fleet. Perhaps these factors contributed to the complacency about safety that was endemic in the station’s operations. For example, from 17 January 1986 until the day of the accident the reactor protection system had been taken out of service on six occasions without sufficient reasons. Between 1980 and 1986, 27 instances of equipment failure had not been investigated at all, being left without any appropriate appraisal of possible consequences being made (Ignatenko et al. (1989)). The operators thought that ‘no matter what you did with the reactor an explosion was impossible’ (Kovalenko, 1989). There were no RBMK control room simulators for operator training anywhere in the FSU. The kudos attached to nuclear power also affected the attitude of operators in a more subtle way. Operators were selected not only for their technical ability (typically it took a nuclear engineer seven years to get their first degree and four or five more for a PhD, in a highly competitive system), but also for their loyalty to the Party - it was not possible even to enter university without being an unswerving Party member. Officially, operating procedures, derived more from the plant design than from operating experience, were to be adhered to ‘by the book’; overtly to do otherwise would be to invite instant dismissal and a return to a 25-year waiting list for an apartment. In reality, though, operators were constantly being put into situations which conflicted with this imperative, e.g. the local mayor, a high Party official, demanding extra power during a cold spell, something which would be done if possible whatever ‘the book’ said. Thus the highly talented workforce was daily discouraged from using personal initiative and taking responsibility for it, while they were quite used to bending the rules covertly (Traves, 1995). The experiment had originally been proposed for the similar reactors at the Kursk station in Russia. However, there the plant manager, an experienced nuclear engineer, apparently appreciated the dangers and refused permission. Further, A. Kovalenko reports that a sequence of events similar to the early stages of the Chernobyl accident had occurred at the Leningrad (St Petersburg) RBMK plant in 1982. It is not clear whether these pieces of information reached the Chernobyl site management (Kovalenko believes the Leningrad papers had probably been read by the Chernobyl management, ‘but at the time of the tests they were all tucked up in bed’), but clearly there were considerable communication problems within the Ministry for Power and Electrification.

Technological Factors As mentioned above, the Chernobyl accident occurred without any of the plant’s components malfunctioning. In other words, though questions have been raised over the standard of construction and maintenance at the station, the accident was the result purely of poor plant design. The central flaw is a feature called ‘positive void coefficient’. If the proportion of steam in the steam/water mixture passing through the core should increase, this reduces the number of neutrons being absorbed by that mixture, and hence increases the amount of nuclear fission occurring. At normal operating temperatures this is outweighed by a ‘negative Döppler (fuel) coefficient’ - hot uranium is better at absorbing neutrons than is cold uranium. So if the power output of the reactor goes up, the number of neutrons absorbed by the coolant reduces, but the number of neutrons absorbed by the fuel increases by a greater factor, and the overall effect is to return the reactor power to a lower level. At very low power, however, the positive void coefficient is greater than the negative Döppler coefficient. Now, an increase in power output will result in an overall reduction in neutron absorption, hence an increase in nuclear fission, hence an increase in power output - a positive feedback loop or ‘vicious circle’ which could

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cause the reactor to run away with itself. This is the reason that very low power operation was forbidden. Stations such as Pressurised Water Reactors, Boiling Water Reactors, Advanced Gascooled Reactors and Magnox do not display a positive void coefficient, either because the water acts as the moderator and so loss of coolant results in the cessation of the fission processes (PWR, BWR), or because the reactor is gas-cooled and so no change of phase (‘voiding’) is possible (AGR, Magnox). The Emergency Core Cooling System (ECCS) served a number of safety purposes, one of which was to prevent operation at very low power. However, this system could be disabled manually, as could the various trip signals. The station did not have a fast shutdown system independent of the operational control rods. The emergency control rods entered the core mechanically, taking up to about 18 seconds, rather than by gravity. These rods were unable to enter the reactor at all during the accident, presumably because the initial release of power caused the control rod channels to buckle. A design fault in the control rods (the attachment of a graphite ‘rider’ on the bottom of the rods) meant that insertion of the rods could initially lead to an increase in reactivity, and hence in temperature production; this was especially important when there was considerable xenon poisoning in the fuel and hence very low levels of insertion of control rods. The station also lacked an ‘outer containment’ which is regarded as being necessary in water-cooled reactors in the West. It is likely that such containment would have significantly reduced the release of radioactive materials in the course of the accident.

Discussion The similarity between the two accidents was apparent almost immediately. For example, Academician Valery Legasov, head of the Soviet delegation to the Vienna conference on Chernobyl in August 1986, said (cited in Ignatenko et al., 1989): ‘Naturally, reactor design engineers studied all the accidents which have occurred at nuclear power stations and have, if necessary, adopted additional safety measures. But unfortunately they did not study accidents in other branches of industry. The train of events at Chernobyl NPS, which led to the tragedy, was in no way reminiscent of even one of the emergency situations at other nuclear power stations, but was very, very similar, right down to the last details, to what happened at the chemical works at Bhopal in 1984. ‘Right down to the last details. The Chernobyl accident occurred in the night from Friday into Saturday. The accident in India happened on a Sunday. At Chernobyl they switched off the emergency protection, in India they switched off the coolers and absorber which perform a protective function. In India there was a technical fault involving a gate valve, and passage of water resulting in an exothermic reaction, which developed exponentially, with the coolers switched off, whilst here there was an excess of steam and a rise in reactivity. The main thing was that both in India and here, the staff had been able (in spite of this being strictly forbidden) to switch off the protective devices. ‘If the reactor designers had drawn some conclusions from the Bhopal accident... but what use is there in talking about it now? To be fair I would just like to say that it was precisely after Bhopal that chemists knocked on the “reactor doors”, but such words as “methyl isocyanate”, “oxidation” and “chemical reactions” made the problem uninteresting for physicists. The lesson of Bhopal went unheeded.’ While Legasov may push the technical details of his comparison to the edge of usefulness, there is no doubt that a series of human, operational and technological failures were common to both accidents.

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Neither accident is characterised by ‘human error’ in the sense of inadvertent actions by operators. At Bhopal it is not clear that any action clearly fits into this category; even the failure to fit the slip blind which would have prevented water entering tank E610 arose from organisational weaknesses. At Chernobyl, the failure to set a new power level while bringing the plant power down in the early morning of April 26 was a clear ‘error’. But the rest of the human violations of operating instructions seem to be quite deliberate. This is probably typical of major accidents; Howard (1983) analysed five chemical industry accidents, and argued that only one could be assigned to human error in the sense defined above. ‘Management error’, which of course impinges on ‘organisational failure’, and general matters of worker attitudes were more salient features. In both accidents, there seems to have been a remarkably cavalier approach on the part of the operators to operating and safety instructions, insofar as these were available to them. A number of examples are outlined in the text above. Ironically, though, it seems that the causes of this laxity were almost diametrically opposed to each other. In the case of Bhopal, overall morale was very low. The plant was losing money and most of it had been put up for sale; there was a high turnover of top management and of US-trained operations staff, coupled with drastic staff cuts. The Agricultural Products Division was regarded as a career dead end. Against such a background it is unlikely that the highest quality operators would have been attracted to the plant. The general lack of awareness about the potential hazards of the plant’s operations, and the observation that the operating instructions were generally written in English which many of the operators did not speak or read are further evidence for the poor state of what might be called the ‘safety culture’. At Chernobyl the problem seems to stem from precisely the opposite vice, that of overconfidence. The operators were among the most highly skilled workers in the USSR, well paid and with good social standing. From management down there seemed to be a feeling that ‘no matter what you did with the reactor an explosion was impossible’. These observations imply that safe plant operation flourishes when operators are confident of their own value to the company, but are kept aware of potential hazards. There would appear to be a desirable ‘morale envelope’, within which the operators are neither too complacent about the possibility of serious accident, nor too depressed about their own futures. In this context the constant vacillation over the last ten years about the future of the Chernobyl plant is potentially very dangerous. It seems that every three months or so it is announced either that the station will close as soon as Western money is made available, or that refurbishment of the existing plant, including Unit 2, is planned which will allow operation of the plant for the rest of its planned lifetime (i.e. up to the year 2011) or even longer. The effects of such uncertainty on staff morale can be imagined, and a comprehensive and consistent approach to the future of the operators, including guarantees of alternative employment if closure should be pursued, should be regarded as an integral element in any attempt to improve safety. David Mosey (1990) categorises organisational factors as follows. 1. ‘Dominating production imperative’, institutional pressure to maintain production, and also to ‘get the job done on time’. At Bhopal, this pressure was considerable. The plant was losing money, and within a division which was losing money, against a background of considerable divestment by the parent company. It had been operating at 40% capacity for some time. Thus the flare tower had been dismantled during a break in MIC manufacture, despite the fact that it might still have been required in case of an MIC leak from the storage tanks; the Freon from the refrigerator had not been replaced; and as an operational example, the plant supervisor ordered that flushing of the pipes should continue before investigating why no water was coming out of the relief-valve pipe.

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At Chernobyl, it is difficult to understand why the operators should have continued with the test despite the fact that so many of the preconditions (especially the required 700-1 000 MW (th) output from the reactor) could not be met unless they were determined to get the job done whatever the circumstances. This would also explain why the reactor was kept running rather than shut off at or before the start of the experiment proper, thus allowing the possibility for a repeat of the experiment if it should not work first time. However, the motivation would seem to be different from that at Bhopal. There was no threat to the future of the Chernobyl plant itself, the flagship of one of the most prestigious industrial fleets in the USSR. It seems rather that loyalty to the Party, coupled with the above-mentioned belief that nothing could go seriously wrong in such a technology, drove the operators to ignore the most emphatic of safety instructions, if indeed they recognised the prohibition of operation below 25% of normal output as such. Legasov also refers to a number of examples where production considerations seem to have overridden quality concerns. In one case of a shoddily welded pipe, ‘they began to look into the documentation and found all the right signatures; the signature of the welder, who certified that he had properly welded the seam, and the signature of the radiographer who had inspected the seam - the seam that had never existed. All this had been done in the name of labour productivity.’ 2. ‘Failure to allocate adequate or appropriate resources’. This was a central issue at Bhopal. The generally run-down state of the plant, with several valve failures, clogging of bleeder lines, rusting pipes etc., was directly relevant to the accident. The retention of obsolete technology and control measures (e.g. the lack of a computerised control system) would not presumably have persisted had the plant been resourced more fully. Wage levels were low, meaning that many operators had to start their own businesses on the side, which will have contributed to fatigue in the early hours of the morning. Further, the dramatic reduction in staffing levels led directly, for example, to the failure to fit the slip blind to the relief-valve pipe which allowed water to enter the tank. It may also be noted that the Indian state provided very few resources for environmental management and monitoring of environmental and safety standards. In 1983 the central government Department of the Environment, established only in 1980, had a budget of just $650 000, with instructions to concentrate on deforestation and waste pollution issues (World Environment Centre, 1984). This factor is less relevant at Chernobyl, where there was no component failure as such. However, the RBMK design itself, and especially the positive void coefficient problem, was arrived at for reasons of economy, though it was recognised by the designers that under certain extreme operating conditions major failure could occur. One can also note the lack of any RBMK simulators for training anywhere in the USSR. 3. ‘Failure to acknowledge or recognise an unsatisfactory or deteriorating safety situation’. The 1982 safety audit at Bhopal identified ten serious weaknesses in the operating regime of the plant, at least five of which were relevant to the accident in 1984. However, there appears to have been little attempt by UCC to ensure that the recommended improvements had been made. Certain maintenance problems, such as that affecting nitrogen pressure in tank E610, seem simply to have been ignored. In September 1984 an internal UCC report warned that a runaway reaction between MIC and water could occur at UCC’s only other MIC plant, at Institute, West Virginia. UCC responded by increasing the level of sampling at Institute, but the warning seems never to have been passed on to the similar plant at Bhopal. At Chernobyl, revelations that the reactor had operated six times with the Emergency Core Cooling System uncoupled in the first four months of 1986, and that twenty seven component failures had been ignored from 1984 to the time of the accident, indicate a failure to recognise the flaunting of safety procedures.

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4. ‘Lack of appreciation of the technical safety envelope’. This was a major common factor in the two accidents. At Bhopal, there was almost no knowledge among the plant operators or supervisors about the potential toxic effects of MIC. (Research into the health effects of this chemical was relatively rudimentary at the time, but information was available in the academic and Governmental spheres which does not appear to have been made available to plant officials.) There were no emergency procedures in the case of a major release, and the disabling of many of the safety devices further suggests that operators were not alert to the possibility of major leaks even between batch manufacturing phases. The depressurisation of tank E610 some six weeks before the accident should have been addressed, but operators seemed entirely unaware of the consequent dangers of allowing contaminants to enter this tank. At Chernobyl, the ubiquitous impression among managers and operators alike was that safety rules were made covertly to be broken (though the official position was that they were adhered to literally). As well as the practice of simply crossing out sections of the instructions which appeared inconvenient, Legasov also quotes a site manager as saying, ‘What are you worrying about? A nuclear reactor is only a samovar’. 5. ‘Failure to define and/or assign responsibility for safety.’ The most striking example of this at Bhopal was the division of responsibility for carrying out the flushing and for fitting barriers such as the slip blind between the MIC supervisor and the maintenance supervisor. The abolition of the latter post some short time before the accident was not accompanied by a reassignation of this duty, so the flushing was carried out without it. Had responsibility for the whole process lay say with the MIC supervisor this would not have been possible. There seemed to be no allocation of responsibilities in the case of an emergency; it was not clear for example who was responsible for alerting the local community, as a result of which the off-site alarm was switched off soon after it was activated. At Chernobyl it is still not clear how it was possible for an electrical engineer with little experience of nuclear reactors in effect to take control of the plant during the experiment, without the site director even being aware of the test. The frequent violation of safety procedures in the period before the accident seems to have been done with the knowledge of the deputy chief engineer; to whom was he responsible? How was it possible for the test programme to be drawn up without any reference to any review or approval process? What powers did the state inspectorate have, and did they attempt to exercise them? (In an attempt to resolve such questions, the Soviets created a new Ministry of Nuclear Energy after the accident.) A further point arises. In his Memoirs, published after his suicide in 1988, Valery Legasov reports that ‘the level of preparation of serious documents for a nuclear power plant was such that someone could cross out something, and the operators could interpret, correctly or incorrectly, what was crossed out and perform arbitrary operations’. When taken alongside the comments by Traves (above), a worrying model of one organisational source of poor safety performance emerges. Operators at Chernobyl were under constant pressure to produce high output from the reactors. In the course of normal operation, then, one can postulate that operators, as they became more confident in operating the plant, began to identify sections of the operating instructions which were not essential to the production of power, and observance of which might well delay or reduce output. Some of these ‘short cuts’ may have been identified by accident during normal operation - an operator omitted a procedure, and discovered that on that occasion no apparent detriment resulted. These ‘operationally unnecessary’ procedures were presumably among those which Legasov discovered to be crossed out in the documentation.

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In any complex technology, there will undoubtedly be a number of procedures which, while desirable, can indeed be omitted with very low or zero risk of significant adverse consequences. Others, however, will be absolutely mandatory. Others will lie between the two extremes. In the case of operation of Chernobyl, one might perhaps categorise continuing to run the reactor at half power for nine hours as a ‘trivial’ breach, running the plant with the Emergency Core Cooling System disabled or over-riding trip commands as ‘serious’ breaches, and running the reactor at 6% of power with Operating Reactivity Margin of only six rodequivalents as ‘forbidden’. However, as Traves describes, the official position was that operators always went ‘by the book’, on pain of dismissal and disgrace. Hence it was presumably impossible for operators to discuss openly which corners could ‘safely’ be cut, and which could not under any circumstances. The Soviet system was undoubtedly an extreme example of such pressures, but one can assume that similar pressures must be felt by operators in all industries and societies; there is some evidence of similar practices in the chemical industry, for example (Jones, 1988). There would seem to be two possible ways of reducing the likelihood of such pressures leading to major problems. Either the regulatory and training regime is so strict that operators are not tempted to cut corners however sure they are that the measures in question are unnecessary or lead to inefficiencies; or operators are encouraged to discuss more openly any violations in operating procedures which they may have perpetrated, however involuntarily, and so to categorise them by seriousness. Clearly this latter course requires a great deal of confidence and the involvement of any external regulator to which the industry in question may have to report. However, unless the regulatory system and safety culture is so robust as to make the cutting of corners organisationally impossible, greater openness over the real practice of operation may be fruitful, and would certainly have been valuable in the case of Chernobyl. The technological causes of the accidents differ in one important respect. At Chernobyl there was no component failure, while at Bhopal at least one important component seems to have failed - the blow-down valve of tank E610 which caused the depressurisation of the tank some six weeks before the accident. This allowed ingress of contaminants into the stored MIC in the tank, and subsequently allowed the entrance of water which had flowed into the process pipe via the relief-valve pipe and jumper line. However, both plants displayed a number of design errors, outlined above, which made them especially vulnerable to violations of operational codes and procedures. Perhaps principal of these were the storage of large volumes of MIC in tanks and the backfitting of the jumper line at Bhopal, and the positive void coefficient of the Chernobyl reactor which created a positive power coefficient at very low power, and allowed the reactor to ‘run away with itself. In addition, though both plants had safety systems, these could be and were uncoupled by operators, either for maintenance or so that the safety experiment could be carried out. A contrast can be seen between the accidents at Bhopal and Chernobyl, and similar accidents in the West. In the first half of this century there were a number of major accidents in western countries involving chemicals. For example, explosions involving the fertiliser ammonium nitrate (which is also used as a commercial explosive) caused 560 deaths at Oppau, Germany in 1921, 200 deaths at Tessenderloo in Belgium in 1942 and 530 deaths in Texas City in 1947 (the worst industrial disaster in US history). However, the second half of the century has seen a reduction of major accidents involving chemicals in the West. Examples causing more than a hundred deaths are rare; Explosion of lorry (liquefied propene) Collapse of an off-shore oil rig Explosion of Piper Alpha oil rig

Los Alfaques, Spain Norway UK

216 deaths 123 deaths 167 deaths

1978 1980 1987

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MALCOLM C. GRIMSTON Examples are far more frequent in eastern Europe and in the developing world.

Explosion of liquefied natural gas Xilatopec, Mexico Oil explosion at power station Tacao, Venezuela Petrol explosion Sao Paulo, Brazil Liquefied petroleum gas reserves explosion Ixhuatepec, Mexico Release of MIC Bhopal, India Train sparking explosion from leaking gas pipeline Russia

100 deaths 145 deaths 508 deaths 452 deaths 3 800 deaths 500 deaths

1978 1982 1984 1984 1984 1989

(From Shrivastava, 1987.) (It should be noted that the demonstrable direct death toll from Chernobyl at the end of 1995 was 33-30 during the accident itself, plus 3 among thyroid cancer sufferers in Ukraine, Belarus and Russia - with perhaps one more among those who suffered from Acute Radiation Syndrome. However, more deaths are predicted owing to long-term effects of radiation.) The most obvious Western event which can be compared to Chernobyl is the Three Mile Island accident at Harrisburg, Pennsylvania, USA, in 1979. It is less easy to find a direct comparison for Bhopal, but an example might be the leak of aldicarb oxime from the Union Carbide plant at Institute, West Virginia in 1985.

Three Mile Island The 959 MW reactor Three Mile Island 2 was destroyed on March 28 1979, when a complete loss of feedwater led to severe core damage. Though it is not the intention of this paper to provide a detailed account of this incident, a HOT analysis emphasises a number of similarities between the causes of this accident and those of Chernobyl. The accident, which started at 04.00, is often described in terms of ‘operator error’. On the face of it, after the reactor had tripped for another reason (water entering a dry line through a check valve which had stuck open) the operators misread the information being presented to them (that a valve known as the Power Operated Relief Valve (PORV) had stuck open). In their attempts to understand and respond to the plant parameters they then made a number of errors, the most important of which were to throttle the high pressure injection flow machinery (which was providing cooling water to replace that which was escaping through the stuck valve), and then to turn off the main circulating pumps. The result was that the core boiled dry and about 40% of the fuel melted before the situation was recognised and controlled. However, further analysis once again indicates that these ‘errors’ should be seen against a less than perfect organisational and technological background, and indeed it seems harsh to blame the individual operators in any sense. For example, in 1977, at the Davis Besse plant, which like TMI had been built by Babcock and Wilcox, a PORV had stuck open, and operators again responded to rises in pressuriser water levels by throttling water injection. Because the plant was at low power (9%), and the valve closed after 20 minutes, no damage was done, but an internal Babcock and Wilcox analysis concluded that if the plant had been operating at full power ‘it is quite possible, perhaps probable, that core uncovery and possible fuel damage would have occurred’. In January 1978 a Nuclear Regulatory Commission (NRC) report concluded that if such a circumstance had arisen it was unlikely that the operators would have been able to analyse its causes and respond appropriately (Kemeny et al., 1979). Yet none of this information was passed on to other plant operators either by Babcock and Wilcox or by the NRC. (This is strikingly similar to the September 1984 internal UCC report on the dangers of a runaway reaction at Institute which was not passed on to Bhopal.) Indeed, the failure of a PORV valve was not one of the ‘analysed incidents’ with which operators were familiarised during training.

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The Kemeny Report also refers to a lack of staff and expertise in the area of nuclear plant operation; for example, review of technical information from other plants was carried out by people without nuclear backgrounds. This deficiency was felt both at TMI itself and throughout the organisation. In addition, the apportionment of responsibility for safety both within the operating utility and within the regulatory authority was not clear. One aspect of technological weakness was in the layout of the instrumentation. There was an indicator light associated with the PORV, and this light went out 12 seconds after the reactor was tripped. However, this merely reported that the signal to open the PORV had been cancelled; it did not actually confirm that the valve had closed. Just over three minutes later, the temperature of the water in the ‘reactor coolant drain tank’ rose dramatically, suggesting that the steam generators had boiled dry (and hence, indirectly, that the PORV was stuck open), but the relevant meter displaying this information was located on the back of the main control panel, out of the sight of the operators who were by now trying to cope with vast amounts of information. Similarly, though there was an alarm sound associated with this message it was undetectable under the audible alarms on the front panel, which were all sounding at once. This information was also sent to the alarm printer to be printed out, but by the time that signal was sent the printer was receiving over 100 other alarm messages per minute. It took several minutes for the printer to print it out, and then it was just one alarm among several hundred. It took two hours and 22 minutes to identify that the PORV was stuck open and to stop the loss of coolant from the circuit, but by then several other problems had arisen, and it took another 13 hours to stabilise the situation, by which time the reactor core had been destroyed. However, releases of radioactivity from the station were minimal; for example, they did not breach the plant’s weekly permitted discharge limit. The principle conclusion that can be drawn, then, is that although the Three Mile Island accident involved human and organisational weaknesses as severe as those at Chernobyl, and also involved component failure (which was absent at Chernobyl), key design features of the plant prevented radioactive leaks as a result of the incident which might have caused significant damage to the environment or to human health. Among these key design features include the physics of the plant which would cause it to trip under any exceptional conditions, and prevent a power surge under any circumstances. The containment of the plant was able to prevent releases of significant amounts of radioactivity into the environment. In other words, the design of the plant was ‘forgiving’, tolerating considerable violations by operators of operating and safety codes without causing major releases of hazardous materials.

Institute UCC operated an MIC plant at Institute, West Virginia. Immediately after the Bhopal accident this plant was closed, and $5 million spent on additional safety features. Public announcements were made that such a major accident could not occur in the USA. However, on March 28, 1985 there was a leak of mesityl oxide causing nausea to eight people, followed on August 11 1985 by a leak of aldicarb oxime which caused 135 injuries, 31 of whom were hospitalised. There is no consensus about the toxicity of aldicarb oxime. Dr Vernon Houk from the Centre for Disease Control in Atlanta, Georgia, said that it was ‘less toxic than MIC’, though he added ‘I am not saying this is an innocuous chemical - it is not’, while an unidentified health consultant in the Charleston Gazette (14.8.85) claimed it was seventy times more potent than MIC. In the same month, Rep. Waxman said that internal UCC documents indicated that aldicarb oxime was given the same hazard classification (Class 4) as MIC (Wilkins, 1987). Subsequent investigations revealed several occasions over the previous five years in which releases of chemicals had occurred, including one case a month before the Bhopal incident in which over six tonnes of an MIC/chloroform mixture had been released (US EPA, 1985).

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These incidents were caused by failures in operating procedures, equipment malfunctions, and human errors - the same factors that contributed to the Bhopal accident. ‘In the period leading up to the leak [on August 11] high-pressure alarms were repeatedly shut off and ignored. A high-temperature alarm was out of service. A level indicator in the tank that leaked and was known to be broken was not fixed. Meanwhile the unit’s computer, which silently recorded the rising problems for days, was never asked for the information by operators ... a total of 32 people in the sprawling plant had directed responsibility for the problem chemical unit in the days before the 11 August leak. Contrary to plant procedure, the workers never checked the tank and associated equipment before using it to make sure it was running properly, which it was not. In addition, the workers assumed that tank was empty because a pump being used to drain it had stopped a few days earlier. But the workers never verified that assumption. Such a check would have been difficult in any case because the tank’s level indicator was broken ...’ (New York Times, 24.8.85). There were delays in sounding the off-site alarm, and local people had little information about potential health hazards. In April 1986 the Occupational Health and Safety Administration (OHSA) imposed a record $1.4 million fine on UCC for 221 health and safety violations. Nonetheless, it was recognised that the safety systems operated quite efficiently, stopping the leak of aldicarb oxime after just fifteen minutes, and though there were significant health implications there were no direct deaths from the incident. It is of course difficult, if not futile, to draw too many firm conclusions from two case studies, or indeed from the relatively small number of major industrial accidents over the last half century. It would seem possible to come to a number of quite different, and perhaps even contradictory, conclusions from the available evidence. The first is that despite weaknesses, the organisational structure of industries in the West is generally more effective than of those in developing countries. Gladwin and Walter (1985) of New York University wrote, ‘a tentative reading of the publicly available evidence so far suggests that the Bhopal facility may have been operating quite independently of UCC. Much of this relative autonomy can probably be traced to the pattern of restrictive Indian regulations imposed on foreign investment and the importation of products, know-how and managerial and technical skills’, suggesting adverse local factors were relevant at least at Bhopal. However, the above comparisons suggest that human and organisational weaknesses are by no means limited to the developing world and the former eastern bloc, though safety cultures may be worse in the latter regions. The similarities between the organisational causes of Bhopal and Three Mile Island are especially stark. Perhaps, then, the undeniably greater success of Western nations in preventing major loss of life, especially among those off site, may derive from a more robust approach to technological design factors, which runs across all industries. Broadly, the engineered safety systems and fundamental design philosophies at Institute and Three Mile Island worked to prevent major health problems, while those at Bhopal and Chernobyl did not. However, both of these explanations assume that there is a qualitative difference between the course of accidents in the West and in the developing world. The relative frequency of major accidents in the two regions does lend support to this, but it must also be noted that the fact that no lives were lost as a result of the Institute incident was a function not only of the speed with which the emissions were stopped, but also of the low population density near the plant compared to Bhopal. If aldicarb oxime is indeed comparable in its toxicity to MIC then a release of Institute proportions into a population as dense as that of Bhopal would presumably have caused many deaths. Indeed, the month before the Bhopal accident there had been a release of over 6 tonnes of an MIC/chloroform mixture from Institute (US EPA, 1985), representing about a sixth of the Bhopal release. No deaths were caused, but a simple comparison would seem to suggest that the same release would have killed more than 500 people had it occurred in Bhopal.

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A third possibility, then, would point to a difference between nuclear and chemical technology. It can be argued that there were equivalent human and organisational failures at Three Mile Island and Chernobyl; indeed, there was component failure at Three Mile Island but none at Chernobyl. Nonetheless, the radiological consequences of the two nuclear incidents were dramatically different. This may suggest that nuclear power technology can be made ‘fail safe’, or ‘forgiving’ of mistakes made by operators, while the considerable releases of chemicals from plants in the West as well as in the developing world would suggest that this is not possible, or at least has not been achieved, in the chemical case. This approach offers an explanation for the safety record of the nuclear power industry in the West, with no established examples of on-site accidents causing off-site deaths through the effects of radiation, in comparison to the safety record of the chemical industry in the West. However, it is less successful in explaining the difference in the major accident rate between the West and the developing world. Some combination of all three explanations may therefore be necessary to explain the various differences noted above. This would seem to be an area in which further research would be fruitful.

CONSEQUENCES OF THE ACCIDENTS Both Bhopal and Chernobyl were defining events in the world’s perception of the relevant industries. However, there are a number of key differences in those consequences. The immediate (‘early’) health effects of Bhopal were much more serious than those of Chernobyl. It is predicted that there will be considerable long-term (‘late’) effects as a result of Chernobyl, while understanding of the long-term effects of MIC is much less clear and so reliable predictions are more difficult to make. Chernobyl’s late effects could in principle affect people who were not born at the time of the accident but live in contaminated regions, as well as offspring of those directly affected by the accident, while at Bhopal it is likely that those affected will be limited to people alive at the time and their offspring. Fallout from Chernobyl is much easier to detect, and will be longer-lasting, than fallout from Bhopal. The releases from Chernobyl are quite well characterised, while those from Bhopal are far less so, owing to the complex chemistry of the precipitating event. Bhopal occurred in a densely populated city, Chernobyl in a sparsely populated rural area. Bhopal occurred in the developing world at a time when the West was quite used to images of third world suffering (e.g. soon after the drought which led to the ‘Band Aid’ project); Chernobyl happened on the edge of Western Europe.

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Health Effects (1) - Medical Bhopal Almost 40 tonnes of a complex chemical mixture were released in the course of the Bhopal. This mixture was calculated to be about two thirds MIC and about one third products of the reactions which occurred in tank E610 before release (UCC, 1985). As mentioned above, it is striking how little information was available in the public domain about the health effects of MIC, given the widespread use of the chemical in the pesticides industry. There are two broad theories about the health effects of MIC. The ‘pulmonary theory’ holds that, with the exception of irritation to the eyes, the other effects of MIC are all associated with its effects on the lungs, leading to oxygen deficiencies (hypoxemia) in other organs of the body. It seems unlikely that oxygen deficiency alone could account for the wide range of symptoms reported after the accident. Victims did suffer from breathlessness, dry coughs, chest pains, restrictive lung diseases and loss of lung capacity, dry eyes and photophobia (12 000 cases of cornea damage). The most serious and permanent damage was in the respiratory tract. Many victims died of oedema (fluid in the lungs). MIC also damaged mucus membranes, perforated tissue, inflamed the lungs, and caused secondary lung infections. Many survivors could not be employed because they suffered from bronchitis, pneumonia, asthma and fibrosis. These observations could be explained in terms of lung and eye disease only. However, there were many other problems, including fatigue, blurring of vision, muscle ache, headache, flatulence, anorexia, nausea, excessive lachrymation, tingling and numbness, loss of memory, anxiety and depression, impotence and shortening of the menstrual cycle (Medico-Friend Circle, 1985), as well as loss of appetite, vomiting, diarrhea, abdominal pains and suppression of lactation in nursing mothers, which could not so easily be assigned to lung and eye damage. The alternative ‘enlarged cyanogen pool theory’ holds that the effect of released gases on the patients was to increase the pool of cyanogenic chemicals in the body, leading to chronic cyanide-like poisoning. The cyanide controversy is referred to below. UCC reported that chronic-type, low level inhalation studies conducted in mice and rats in 1980 had “found no carcinogenicity” (Chemical and Engineering News 3.3.86). It is not known precisely how many people died as a result of the Bhopal accident. A few months after the accident the Indian Government officially put the death toll at 1 754, a figure derived from (incomplete) morgue records, to which were added figures reported from registered cremation and burial grounds and from out-of-town hospitals and cremation and burial grounds (eliminating names previously counted). A year later this figure was revised to 1 773, and to 2 200 a further eight months later. Newspapers put the total at 2 000 to 2 500, while others - social scientists, eye-witnesses, voluntary organisation, and figures gleaned from circumstantial evidence such as shrouds sold and cremation wood used produced estimates between 3 000 and 15 000. Such degrees of uncertainty concerning large numbers of deaths are not unusual in India. There was no systematic method to certify and count accurately the dead as they were brought to government hospitals or crematoria and burial grounds. Further, in the first few days after the accident all available medical personnel were engaged in treating the living. Though government officials stuck to the figure of 1 754 deaths, in private they would admit that they did not believe this number, and that counting errors, when rectified, could well raise the death count to around 3 000. In response, social activists claimed that there was a conspiracy between the Indian Government and Union Carbide to underestimate the death toll, and though there is little evidence that this is true, it did receive considerable currency in the

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months after the accident. It can certainly be argued that the Indian Government and UCC, by not providing better information in the immediate aftermath of the leak, lay themselves open to erosion of their credibility. In 1988 it was reported that two people a week in the city were still dying from the aftereffects of the accident (Jones, 1988). Within one year, it was reported, the number of stillbirths among 10 000 women exposed to MIC had doubled, and spontaneous abortions trebled (Kurzman, 1987). On November 16, 1990, the State Government of Madhya Pradesh submitted to the Supreme Court of India its completed categorisation of the claims of all of the victims. It determined that there had been 3 828 deaths. However, the immediate deaths were by no means the only health effects of the accident. Some 200 000 to 300 000 people suffered health problems of various degrees; the 1990 Madhya Pradesh State estimate was 203 500, the bulk of whom suffered only ‘temporary injury with no disability’. There were also suggestions of possible genetic effects of MIC. For example, Roy and Tripathi (1985) of Banaras Hindu University did tests on crops which they said ‘showed that MIC acted as a mutagen and changed the morphology and breeding behaviour of the plants’. They advocated destruction of all standing crops and keeping the land fallow until after the monsoons, though this was not done. The picture was considerably complicated by the circumstances of the accident. The release from tank E610 occurred at an estimated 250 °C and 15 bar pressure, and in the presence of several impurities, including chloroform. Under these circumstances a range of toxic chemicals were produced, including aminomethane, methylurea, 1,3-dimethylurea, trimethylurea, 1,3,5-trimethylbiuret, 1,1,3,5-tetramethylbiuret, ammonium chloride, dione, methyl-substituted amine hydrochlorides, MIC trimer, dimethylisocyanate, chloroform etc. (UCC, 1985). Each of these substances has its own potential health effects. It was also suspected that the mixture released on December 3 included hydrogen cyanide, which is produced when pure MIC is heated to 350 °C, and at lower temperatures if the MIC is contaminated with such chemicals as dimethylurea, dione or MIC trimer. Either of these routes could have been available during the accident. UCC claimed that the (average) temperature in the tank did not exceed 250 °C before the valves ruptured, but the possibility remained of localised ‘hot spots’ within the tank against a lower average temperature. This matter was of considerable importance; while MIC was not at that stage a recognised and well-understood poison, cyanide was. Proof of considerable releases of cyanide would have made UCC liable for much more serious damages. Furthermore, since there is an antidote for cyanide (sodium thiosulphate), unlike for MIC, it would have caused a considerable political scandal if state and central governments had not identified the substance at once and issued antidote to affected communities. There was evidence that cyanide may have been involved. Hydrogen cyanide was found in the vicinity of the storage tanks immediately after the accident (APPEN, 1986). Discolouration of organs, cherry-red blood and oedema of the lung and brain, all known symptoms of cyanide poisoning, were detected in some autopsies, and the ‘enlarged cyanogen pool’ theory of the health effects of MIC raised the possibility that cyanide could still be implicated in causing health problems even if it was not released in significant amounts in the accident. However, the issue was never resolved (Khandekar, 1985; Weisman, 1985), and administration of sodium thiosulphate was actively discouraged by the authorities, despite some apparently successful use of the substance. The lack of certain information concerning what was released during the Bhopal leak has made it extremely difficult to make predictions based on calculated doses received by individuals. Ongoing study of the health of individuals in Bhopal has led to the coining of a

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new phrase, ‘chemically-induced AIDS’, to describe gas-induced breakdown of the immune system, making victims especially susceptible to tuberculosis and respiratory problems. However, in 1995 the Indian government wound up its official monitoring programme in the city (Ghazi, 1994). It remains the case that little is known of the long-term effects of MIC exposure, or of exposure to many of the other chemicals which were formed and released in the course of the leak. Furthermore, there was no real attempt to track the released material in the environment around the plant. For example, Dr J. Kaplan, one of the two US Government doctors from the Centres for Disease Control who went to the area, said, ‘as far as can be determined, the cloud in India dissipated within two hours’ (JAMA, 12.4.85). The Indian Council on Medical Research was later to contradict this suggestion. Similarly, within a few days of the accident air and water were declared safe without any great degree of monitoring, but a report by the Nagrik Rahat aur Punerwas Committee (2.5.85) found a high level of thiocyanate in subsoil lakes and filtered water in Bhopal 100 days after the leak. Since the long-term effects of radiation dominate the concerns over the Chernobyl accident, this makes direct comparisons difficult.

Chernobyl About 6 tonnes of highly radioactive material was released from the core of the Chernobyl reactor. 30 people died as an ‘immediate’ result of the accident at Chernobyl, two from falling masonry, burns etc. during the accident itself, and a further 28 from Acute Radiation Syndrome in the following few weeks. All of these people were on site either at the time of the explosion or in the course of the firefighting operation which immediately followed. A further 106 cases of Acute Radiation Syndrome (ARS) were confirmed. All of these people recovered, though a further 14 had died by the end of 1995. The survivors have a greater likelihood of developing cancer in the future. There were no immediate health effects among people off site at the time of the accident. However, many people received measurable doses of radiation as a result of the event. It is known that high doses of radiation are associated with an increased risk of developing cancer, after a ‘latency period’ which differs for different forms of cancer. (Leukaemia has a latency period of 2 to 10 years, thyroid cancer of 5 years and above, most solid cancers of at least ten years.) This evidence arises from studies of a variety of irradiated populations, principally the survivors of the atom bombs at Hiroshima and Nagasaki. Below very high doses of radiation, it is assumed that this risk is independent of the rate at which radiation exposure is experienced, i.e. that radiation is proportionately just as dangerous in small amounts as it is in moderate. (There is no corresponding assumption concerning MIC. In practice there is a threshold for MIC workers of 0.02 parts per million, ‘the level to which it is believed that nearly all workers may be repeatedly exposed day after day without adverse effect’ (Kinnersly, 1973).) However, Fishbein (1981), considering carcinogenicity in general, says ‘since there are so few data and so many interpretations, the view is widely held that continuing arguments over thresholds are an exercise in futility’. This may be thought to be as applicable to radiation as it is to chemical exposures. In any case, concern over Chernobyl has focussed on the long-term health effects of the accident. The World Health Organisation has set up the International Project on the Health Effects of the Chernobyl Accident (IPHECA). The results of IPHECA’s pilot projects, reported at a WHO conference late in 1995, are as follows (WHO (1995)):

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There has been a very significant increase in thyroid cancer in the affected areas of Belarus, Ukraine and Russia. By the end of 1995 over 600 cases had been identified, three of whom had died. (Thyroid cancer, fortunately, is treatable with a high degree of success given appropriate medical facilities.) Further excesses are expected over the next few years. Dose reconstruction exercises have been carried out, taking into account weather conditions at the time of the accident and the observed iodine deficiency in the local diet which will have led an increase in uptake of radioactive iodine from the accident in the thyroids of affected individuals. These suggest that the levels of thyroid cancers being observed are consistent with previous understanding of the dose-response relationship for this disease. There is no evidence of increased levels of leukaemia among the population in contaminated areas. There is an apparent increase in mental retardation and behavioural problems among children who were in the womb at the time of the accident. This is difficult to evaluate, and attempts are being made to identify adequate cohorts of the relevant children for examination in the three Republics. There is no evidence of increases in solid cancers. IPHECA is now looking at the health of the ‘liquidators’, the estimated 800 000 people who were involved, in the months and years after the accident, in liquidating its aftermath, including shovelling highly radioactive graphite and spent fuel into the stricken reactor from the roof of the turbine hall, and also in decontaminating buildings and land near the reactor and burying the waste. There have been unconfirmed reports of deaths among this population owing to the accident, but as yet no evidence that the death rate is higher than would be expected after ten years in any similar population of this size. In the longer term, calculations based on (unproven) assumptions that radiation remains equally dangerous at low dose rates as at high suggest that cancer deaths in Europe over the next 70 years will be increased by 0.01% (Lynn et al., 1988), and in the northern hemisphere by 0.004% (Parmentier and Nénot, 1989). The United Nations Scientific Committee on the Effects of Atomic Radiation (UNSCEAR) has calculated that the total collective radiation dose to the population of the former Soviet Union is about 226 000 person-sieverts (UNSCEAR, 1988). Applying current estimates of the risk of radiation-induced cancer (ICRP, 1991), it can be estimated that 11 000 to 12 000 cancers may eventually result in the former Soviet Union (30 000 worldwide). With the exception of thyroid cancers discussed above, such numbers of extra cases would be undetectable against the ‘background rate’ of cancer deaths from other causes.

Health Effects (2) - Psychological A common feature of the two accidents is the level of public concern, and associated stress-related disease, which in the case of Chernobyl have come to be referred to as ‘radiophobia’. In both cases, there was an immediate failure on the part of the state governments, and at Bhopal on the part of UCIL, to give full information about the accidents and their possible health effects. It is not necessary to infer any sinister motivation for this, though a number of commentators have done so. The important point is the effect that this had on the local populations.

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Bhopal At Bhopal, the Indian Government was seen to be underestimating the death toll, not only among humans (see above), but also for example among animals (the official figure of 1 047 being well below that of 2 000 which was estimated by studies ‘on the ground’ (Indian Council of Agricultural Research, 1985)). The controversy over cyanide exacerbated the distrust that local people felt over the accuracy of government information. There were considerable pressures on the ‘government’, at both state and national level, in the immediate aftermath of the accident, complicated by the fact that elections were scheduled for just three weeks after the accident occurred. The ruling Congress (I) Party had to demonstrate to local people that it was their champion against UCIL and UCC, and hence distance itself from the companies. At the same time it had to avoid blame for allowing the accident to occur, both as a significant (23%) stockholder in UCIL, and as the regulator of industrial practice, including environmental and safety issues. However, Prime Minister Rajiv Gandhi, and Arjun Singh, Chief Minister of Madhya Pradesh State, also wished to attract multinationals into India and into Madhya Pradesh, and did not wish to take measures which would discourage such investment. Two tactics were used by the government to fulfil these needs. First, it took control of all information; at one stage, for example, the direction of the wind at the time of the accident was classified information (Nanda, 1985). In this way the answers to critical questions about blame for the accident, suitable punishments, future actions, the organisation of relief efforts and compensation, could be shaped to minimise blame attaching to government agencies. Secondly, a number of individuals were identified as being to blame, and punished accordingly. The state Minister of Labour was asked to resign for not ensuring action was taken after a previous accident at the plant in 1981; several officials were suspended pending investigation; operators and managers of the Bhopal plant were arrested. On the face of it these tactics were successful. In the elections, put off until late January 1985, the Congress (I) Party won all 239 seats in the state legislature, and 27 of the 35 seats for the national government elected by Madhya Pradesh. Presumably this also made it easier for the Indian Government to take powers to represent all the victims in March 1985. However, the tight control over information, and especially information about the health effects of MIC, made design of an appropriate response to the accident difficult. For example, the managers of the plant who had been arrested, though allowed to supervise making the plant safe, were prevented from talking to UCIL or to UCC, thus preventing outside expertise from coming to the aid of the situation. Union Carbide was less successful in avoiding public condemnation. As operator of the plant it could not distance itself from matters, and lack of information made its early statement to the media rather vague. Its concentration on technical, legal and financial issues made it appear heartless in the face of great human agony. In March 1985 the sabotage theory surfaced. This was never fully accepted even within the company - for example, Warren B. Anderson, Chairman and Chief Executive of UCC, said at a news conference, ‘The amount of water that got into this tank took a while to get in there. That is why we said that it might be deliberate. I can’t impugn malice here. I can’t say it’s an act of sabotage’ (United Press International, 20.3.85). However, it prompted UCC to say that it ‘accepted moral but not financial responsibility’ for the accident. The protracted legal battle (see below), in the course of which the company tried vigorously both to have cases tried in India, where compensation could be expected to be lower than in the USA, and to keep total compensation below its $240 million insurance ceiling, further gave the impression that the company was uninterested in the suffering it had ‘caused’. Many victims continued to view Government and UCIL as being in cahoots, and also to distrust traditionally respected groups such as doctors (who failed to distribute free milk, for

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example) and lawyers (from the US and India, who descended on the area in the hope of finding lucrative business representing victims). Shrivastava (1987) cites examples of vernacular songs and poetry expressing such cynicism over motives. One runs; What did Carbide do ? It murdered thousands of people. What did the government do ? It aided the murderers.

What did doctors do ? They did not give us milk. What did lawyers do ? They used us as pawns.

In addition, the government took steps to stifle public complaint, for example raiding the homes of activists the night before a major protest planned for June 25, 1985, and breaking the demonstration up when it went ahead. Distrust of the authorities was exacerbated by the slow progress, and poor targeting, of the government’s compensation programme. The overall effect was to erode trust in the small amount information that was coming from the authorities, and to add to the stress felt by people who perceived that nobody was taking their suffering seriously. Psychological and emotional symptoms included sleeplessness, anxiety, loss of libido, projection of guilt, increased family violence, and impairment of learning in children (Madhya Pradesh Chronicle, 1985). ‘People just won’t believe anybody anymore. They won’t believe the government. They won’t believe the doctors, and they won’t believe Union Carbide’ (Tatro, 1984). Another manifestation of distrust, which in itself added to the psychological damage done by the accident, was the ‘evacuation’ which accompanied ‘Operation Faith’, the neutralisation of remaining MIC in the plant. The problem of dealing with this material, in the other two tanks, was of some urgency, and after a review of the options it was decided that the best way of neutralising it was to convert it into Savin in the normal industrial process. Some 200 000 people had already fled during the accident, though there was no formal evacuation, it being assumed that the material would have dispersed or hydrolysed within a few hours. A further 400 000 left during Faith on December 18 and 19, despite assurances from UCIL and from the government that the operation did not represent a threat and that another accident was unlikely (‘Second evacuation’ Times of India 18.12.84). It is possible that the operation was interpreted by local people as ‘starting the plant up again’. Many of the evacuees pawned or sold all their belongings at distress prices. Chernobyl There was no corresponding division between ‘state’ and ‘operating utility’ in the case of Chernobyl, and the Soviet government therefore had even less success than the Indian in distancing itself from the accident and its consequences. After the accident there was an initial unwillingness on the part of the authorities to accept that such a major accident could have taken place. The reasons for this are presumably associated with the perception among operators, managers and government departments alike that the technology (and indeed the whole Soviet system) was incapable of such disaster. Against such a background, levels of distrust among the local populations when news of the accident did emerge were unsurprising. The first indication in the West that there had been a major incident was when fallout was detected in Sweden. The evacuation at Chernobyl, once started, was carried out systematically and efficiently. Evacuation of Pripyat, the ‘company town’ of 47 000 people adjacent to the plant, only began at noon on April 27 1986, 36 hours after the accident, when the wind changed. However, it was completed in less than four hours. By May 6 evacuation of a 30 km ‘exclusion zone’ around the plant had been undertaken. At least 116 000 people were evacuated in the course of this exercise.

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However, people were concerned at the lack of information about the accident, and the poor quality of what information was available. ‘Locals [in the 30 km exclusion zone] describe how young soldiers banged on their door and said, ‘Out! You have got one hour, there’s the bus, you can take only what you can carry and no pets’. All questions were ignored. Many towards the edge of the zone didn’t even know anything had happened. And when they arrived at their far destinations the inhabitants there didn’t know why they had come or from where. Things were different in Pripyat as everyone worked at the station, and in daytime on April 26 they had flocked down the road to a bridge 1 km from the reactor and stood watching the firework display for hours’ (Dr Eric Voice, personal communication, 1996). Evacuation itself caused problems. According to IPHECA: ‘When people are evacuated from their homes, they often suffer considerable stress because they do not have full information about what is going on, they undergo disruption in community infrastructure and social interaction, and they face uncertainty about housing and employment. Many evacuees who move to new settlements after the Chernobyl accident were particularly depressed in their new homes because of financial difficulties, fear of isolation, and concern for the health of their children. The tense situation caused considerable stress which, combined with the constant fear of health damage from the radioactive fallout, led to a rising number of health disorders being reported to local outpatient clinics. Although the countermeasures following the accident reduced radiation doses, they increased tension and the upheavals resulted in significant psychological stress in the affected population’ (WHO, 1995). Financial Effects The direct financial consequences of the Bhopal accident were largely limited to those associated with immediate damage. To an extent, this derives from the fact that MIC and the other emissions during the accident cannot be detected at very low levels in the environment. Estimates of business losses in the immediate area range from $8 million to $65 million (De Grazia, 1985). The closure of the plant itself, announced in April 1985, eliminated 650 permanent jobs and about the same number of temporary jobs. These jobs were especially important to the local economy because of UCIL’s relatively high wage policy. The redundant workers often found it very difficult to find alternative work, partly owing to superstitious prejudice against them. Family economies in the slum colonies were totally disrupted by the loss of income and the addition of financial burdens. Apart from the loss of principal wage-earners, after the accident many women and children had died or were incapacitated, forcing other members of the family to give up wage-earning occupations in order to do domestic work. The proliferation of local loan sharks added to the financial difficulties faced by local families. The Indian Government instituted a food relief scheme which by October 1985 had cost an estimated $13 million, though reports of corruption among officials administering the scheme raised questions as to how much reached the victims. Soon afterwards the programme was cut, to be replaced by a $27 million scheme to ‘beautify’ Bhopal. Most of this money was to be spent on general infrastructure, only 10% to go on health measures. These schemes formed part of a general relief effort, including medical treatment, job-training programmes and public works projects, which by March 1987 had cost an estimated $150 million. The issue of personal compensation claims was a difficult and frustrating one. Before the elections of February 1985 (which had originally been scheduled to take place in the month of

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the accident) the Government promised 10 000 Rupees ($800) to members of the families of people who had died, and 2 000 Rupees ($150) per survivor. After the election this latter sum was reduced to 1 500 Rupees, and was restricted to families who earned less than 500 Rupees ($40) a month. Many of these amounts were distributed as crossed cheques in an attempt to eliminate corruption among officials distributing the money. However, as many of those entitled to compensation did not have bank accounts, they had to cash these cheques with moneylenders at high discounts (Kurzman, 1987). Initial claims against the Union Carbide Corporation in the USA ran to some $ 100 billion, more than ten times the value of the company. Personal injury claims were handled by the Indian Government, which in March 1985 took powers through the Bhopal Gas Leak Disaster Act to represent all victims of the accident. There were also claims from UCC shareholders against the company for not informing them of the risks involved in doing business abroad, though these were dropped when the GAF Corporation made a hostile takeover bid for UCC in June 1985, boosting the share value. In estimating a financial ‘cost’ associated with Bhopal, it is necessary to place a financial value on a lost human life, and on disability caused in an industrial accident. In this sense the question of whether Bhopal was an ‘Indian’ or an ‘American’ accident became crucial. The price of a lost life was likely to be adjudged much higher in the USA. Though the Indian legal system was sophisticated (India had 230 000 lawyers, more than any other country apart from USA), the area of tort was not well codified. Punitive damages were almost unknown, and compensation tended to be very low. The estimated ‘cost of a life’ after the Gujerat dam collapse in 1979 had been about $250 (in money of the day), reaching $1 800 after a train crash killed 3 000 people in 1981 (Adler, 1985). Much higher sums could be expected under USA legal practice. A report by the Rand Corporation (Wall Street Journal Europe, 20.5.85) estimated that the ‘worth of a life’ in the USA was about $500 000. US payment to asbestos victims, who suffered a variety of diseases loosely comparable to those injured as a result of Bhopal, averaged $64 000. Against this, the equivalent figures for India were taken to be $8 500 for a lost life and $1 100 for serious injury. Assuming 3 800 deaths and 203 000 injuries, (the Madhya Pradesh State estimates from 1990) simple calculation yields figures of almost $15 billion for ‘human injury’ on US ‘values’ (1984 money), but little over $250 million on Indian ‘values’, the figures being dominated by the ‘value’ put on a personal injury rather than that on a death. (The 1995 Intergovernmental Panel on Climate Change report puts a value of $1.5 million on the life of a US citizen, and $100 000 on that of a life in the developing world (China).) The meaningfulness of such figures is not clear, and is probably very limited. Indeed, the very public discussions of such matters which continued for five years after the accident must have added considerably to the distress of affected families, whose perception was of being regarded as the subjects of awkward financial calculations, rather than as individual human victims of a tragedy. Such calculations led to protracted wrangling over where (which ‘forum’), and under what legal conditions, the claims should be heard. The Indian Government had in March 1985 taken on powers to act as the sole representative of the victims in their cases against Union Carbide. For the Indian Government there were several attractions to having the cases heard in the USA. UCC was a majority shareholder in UCIL, and the Bhopal plant had been designed in the USA and run by UStrained engineers. Much higher damages were likely to be awarded by a US court than in India. UCC internationally had assets of some $9 or $10 billion, while UCIL’s assets stood at less than $200 million, much reducing the scope of damages. Furthermore, US law would allow the Indian Government considerable powers of discovery of information from UCC concerning issues such as the known health effects of MIC and studies of the health of people

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after the accident. In addition, the portrayal of the accident as an ‘American’ one would assist the Indian Government to sidestep questions about its own role, both as regulator and as substantial stockholder in UCIL, which would be more awkward if the accident were to be perceived as a local one. In August 1985 UCC filed for dismissal of the suits in the USA, on the grounds that it was an inappropriate forum for the litigation. The event had occurred in India and most of the material evidence, witnesses and victims were in India. UCIL was an independent company, not controlled by UCC despite its majority stockholding. However, Judge John Keenan, appointed by the Reagan administration to rule on the matter, allowed partial discovery to the Indian Government before deciding the forum issue, as a result of which it was claimed that UCC, rather than UCIL, did indeed take major decisions relating to Bhopal plant, e.g. the decision to store MIC in large tanks. Vigorous attempts were made by both sides to reach an out of court settlement. In 1985 UCC offered first $100 million, then $230 million, a sum for which it carried insurance. Both offers were rejected by the Indian Government. The Indian Government also refused offers of interim payments from UCIL and UCC, e.g. $10 million in January 1986. This seems in part to have been because UCC demanded considerable information about the immediate health effects of the accident which the Indian Government would use in the trial, as a condition of making the money available. There may also have been a wish not to be seen to be ‘consorting with the devil’ at a time when for political reasons it was important for the Indian government to be seen as quite separate from Union Carbide. In February 1986 UCC raised its offer to $350 million. This offer was secretly accepted by US lawyers working on the case, but was again rejected by the Indian Government. Eventually, in May 1986, Judge Keenan decided that the cases should be heard in India. However, he imposed three conditions on UCC; that it must agree to be bound by Indian jurisdiction; that it must pay any damages decided by the Indian courts; and that UCC (but not the Indian Government) would provide pre-trial disclosure under the stricter American laws on the matter. In July 1986 UCC appealed against the last of these conditions, arguing that both sides should face the same laws of disclosure. Keenan’s decision represented a partial victory for both sides. For UCC, the case would not be heard in the US against the background of higher personal injury settlements. For the victims, all of the assets of UCC would be available for payment of damages (though UCC divested itself of considerable assets through 1985 and 1986), and full disclosure of information would help to establish the culpability of UCC as well as UCIL. By the time such matters were sorted out, almost five years had passed since the accident. In February 1989 the Supreme Court of India ordered UCC and UCIL to pay $470 million to the Indian Government as a final settlement for claims on behalf of the victims. UCC contributed a further $20 million towards the construction of a hospital in 1992, selling its shares in UCIL to finance it. Financial consequences arising from the health effects of the accident were limited to India. It was believed that MIC hydrolysed rapidly under atmospheric conditions, and so no attempts were made to monitor the spread of the chemical, or others released in the accident. In addition, little was (or is) known of the long-term effects of low-level exposure to such chemicals. The main financial effects of Bhopal elsewhere in the world arose from higher expenditure on plant safety and regulatory structures, obstacles to continued operation of existing plants and establishment of new ones, and increased insurance premiums. The perhaps ironic example of the extra $5 million spent on safety at UCC’s other MIC manufacturing plant at Institute has been mentioned above. In the USA two initiatives were launched by the Chemical Manufacturers’ Association; CAER (Community Awareness and Emergency Response) and the National Chemical Response and Information Centre (NCRIC). The Environmental

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Protection Agency instituted the Chemical Emergency Preparedness Programme. Another immediate effect of the Bhopal accident was the refusal of planning permission to UCC to build a $66 million plant to manufacture industrial gases in Livingston New Town, Lothian, in February 1985, despite unemployment rates in the town running at 26%. When a tantalum production plant in Phuket, Thailand, which used the highly corrosive chemical hydrogen fluoride and the radioactive compound thorium dioxide, was burned down in June 1986 it was reported that videos suggesting the plant could be another Bhopal or Chernobyl were circulating in the town beforehand (Asiaweek, 6.7.86). Available insurance coverage for toxic water sites was immediately reduced and premiums increased, with some sites threatened with closure (Nolan, 1984). However, even before Bhopal the chemical industry was facing difficulties in the area of insurance, especially in the USA. In the early 80s the sudden and accidental pollution insurance market had reduced individual companies’ coverage from $300 million to $50 million, with even this lower figure unavailable to large chemical companies, while the environmental-impact liability insurance market had all but evaporated, with only one major carrier, American International Group, offering such cover by February 1985 (Jones, 1988). This may have been in response to an increasing number of multi-million dollar verdicts against chemical companies (from 1 in 1962 to 251 in 1982 and 401 in 1984), and/or to the low pricing of such insurance cover in the past. Not only did prices rise, but it became increasingly difficult to get any cover for a number of risks, forcing many companies to set up their own insurance companies to avoid directors becoming personally liable for claims after adverse events.

Chernobyl The direct financial consequences of the Chernobyl were more significant. Within the FSU, it was reported that by 1990 over 200 billion roubles had been paid for direct and indirect material losses. These included the destruction of the reactor itself, the subsequent measures taken to make the site safe, including the construction of the ‘sarcophagus’ (Figure 4), and the abandonment of the partially completed Units 5 and 6 at Chernobyl and others around the Former Soviet Union. Another 2.5 billion roubles were spent on compensation and concessions to victims (OECD/NEA, 1994). However, the fact that radioactive materials, and especially caesium, can be detected in very low quantities resulted in a number of countries taking their own measures to reduce exposure. For example, in January 1996 in the United Kingdom some 300 000 lambs on 200 farms were still restricted owing to caesium levels in the flesh (MAFF and Welsh Office Press Releases, 1996). Compensation to farmers in Wales, where most of these farms were located, had reached almost £8 million ($12 million) by 1996 (Welsh Office, 15.2.96). There were restrictions on reindeer in Nordic countries, and on fish in Sweden. The USSR was not a signatory to the Vienna or Paris Conventions whereby all liability for a nuclear accident is channelled through the operator. As a result countries and companies in the West who were affected by the accident were unable to seek recompense from the USSR. This makes a direct comparison with the costs of Bhopal more difficult, but it is clear that the realised losses are much greater. However, it does not seem likely that the greater cost of the Chernobyl accident reflects a greater toll in human life or health or environmental damage. The above analysis demonstrates that the short-term effect of Bhopal were much more severe than those of Chernobyl, though the picture is less clear when it comes to examining long-term effects.

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PUBLIC PERCEPTIONS Despite their common role as ‘defining events’ in the campaigns against their respective industries, Bhopal has been largely forgotten among the population of the UK, while Chernobyl remains the subject of much public concern and media attention, for example prompting three TV documentaries in Britain in the week of the fifth anniversary in 1991. It is unlikely that this difference can be explained in terms of the objective seriousness of the two events. The demonstrable effects of Bhopal on human health are greater than those of Chernobyl, and though long-term consequences of Chernobyl are expected they can only be the subject of conjecture, as with certain very clear exceptions such as thyroid cancer they will be indistinguishable from background levels of the diseases, principally cancers, involved. However, the following factors may be relevant.

Location It is likely that the respective locations, and consequent media coverage, of the two accidents were important in forming the public response to the events. The Bhopal accident received a great deal of media attention in the USA. It was front page news in the New York Times for two weeks (Shrivastava, 1987), and developments were covered and analysis offered for some months, though coverage fell after about two months (Wilkins, 1987). It became the second biggest story of 1984 after the reelection of President Reagan. US Media coverage of the Bhopal accident tended to portray Indians, including doctors and others working to mitigate the effects, as helpless victims of the accident, while Americans who had gone out to help tended to be depicted as ‘powerful figures’ (people capable of doing something to improve the situation) more than twice as often as their Indian counterparts. ‘The overall effect - which was probably unintentional - was to portray Indians as something less than competent, responsive human beings’ (Wilkins, 1987). It was rare, for example, for the media to discuss the similarities between the Bhopal plant and its MIC-manufacturing sister in West Virginia. The coverage of the Chernobyl accident, on the other hand, tended to concentrate more on similarities between the technology and the level of industrial development of the USSR and the West. ‘Could it happen here’ was a more frequently asked question, and there was less (though still considerable) emphasis on comparing the technologies and organisational structures of the West and the country in question than there had been in Bhopal. It is unlikely that these observations alone could explain the continuing differences in perceptions of the two events. Most obviously, they cannot explain the fact that the media have been more interested in Chernobyl than in Bhopal for some years. A more promising suggestion derives from the old US media adage that ‘1 000 deaths in the Third World’ is equal to ‘100 deaths in Europe’ is equal to ‘one death in the USA’. Though the TV pictures of deaths at Bhopal were graphic and memorable (Wilkins, 1987), they occurred in a distant country, and of course to people of a different race to the majority in countries like USA and UK. Further, it is possible, though this can only be conjectured this long after the event, that the pictures of the Ethiopian drought and famine which had become headline news around the world only two months earlier, and had prompted the ‘Band Aid’ fund-raising movement, had to an extent induced ‘compassion fatigue’ among the populations of the West. Chernobyl, by contrast, happened in an ostensibly ‘developed’ country (though the largely peasant nature of the region in which the station was sited has been described above), and one relatively similar in racial background and in ‘customs and cultures’ to the West. The impression that something has happened to (and/or has been perpetrated by) ‘people like us’ presumably increases newsworthiness.

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Risk Perception A further likely clue to the difference is to be found in work which has been done on the factors which mediate between ‘real’ and ‘perceived’ risk, especially by the Oregon researchers (Slovic et al., 1980). The application of this work to these two accidents assumes that the key difference is that one was a nuclear accident while the other was a chemical accident. In essence, there are three relevant ‘factors’ in this mediation process. First, if a risk appears to be ‘new’, ‘unknown to those exposed’, ‘not observable’ and with ‘delayed effects’, it will be perceived as being more serious than a risk of equivalent magnitude which appears to be ‘old’, ‘known’, ‘observable’ and with ‘immediate effects’. This is often referred to as ‘fear of the unknown’. Both radiation and chemicals such as MIC are unfamiliar to the general public, and generally regarded as ‘new’ risks. The end-product of a nuclear power station - electricity is if anything more familiar than the end-product of the Bhopal plant (a pesticide). However, radiation cannot be detected by human senses, whereas it seems to be widely believed that toxic chemicals can be detected, especially by smell but also by sight. The smells of MIC and phosgene were a familiar feature of the Bhopal plant for workers and local people alike. The clouds which passed over Bhopal, and over Seweso nearly a decade earlier, could be seen, and the crystalline fallout of dioxins at Seweso was also very obvious (Fuller, 1977). (However, a number of toxic chemicals cannot be detected in such a simple way.) Fallout from Chernobyl, by contrast, was not observable without instruments. The fact that the fallout from a nuclear accident may be affecting people without their knowing it adds to the perception of radiation as a severe hazard. It is known (by researchers and by the public) that the health effects of radiation can often be delayed for a decade or more, while there is no clear evidence of carcinogenicity associated with MIC. However, Wilkins (1987, p. 121) reported that 72% of (US) respondents did believe MIC caused cancer, and so the ‘delayed effect’ parameter may be common to both accidents. The second factor refers to ‘controllability’, ‘voluntariness’ and also ‘catastrophe’. If a risk appears to be one which is run voluntarily by affected individuals, who can therefore control it to some degree, then it will be perceived as being less severe than a risk of a similar magnitude which is regarded as being uncontrollable and outside the choice of those involved. Again, both nuclear and toxic chemical industries score badly against this parameter. Decisions about the siting of plants, or even about whether such technologies should be developed or not, tend to be taken at a level far removed from the local community, or indeed sometimes from the wishes of the people of a nation as a whole. Factor 2, however, also includes such parameters as ‘dread’ (‘fate worse than death’), ‘global catastrophe’ and ‘risk to future generations’. (Slovic’s justification for grouping these parameters with the apparently unconnected ones of voluntariness and controllability is that their perceptual effects are well correlated.) Here there seem to be relevant differences between the nuclear and chemical industries. The safety record of the nuclear power industry worldwide has been impressive. Chernobyl is the only accident in a civil nuclear power station which has had demonstrable radiation-induced health effects among people who were not on site at the time. By contrast, on October 3 1985 the United States Government released a consultant’s report stating there had been at least 6 928 chemical accidents in the US since 1980, killing 135 and injuring 1 500 (Wilkins, 1987). However, the image of the atom bomb (much more closely linked with nuclear power in the public mind than in reality), and possibly of many films and other entertainments depicting ‘nuclear war’ and other hazards of radiation, imply that a major nuclear accident could be on a different scale from any other conceivable industrial accident. Once this idea had become established in people’s minds, it became difficult for them to reject it in the face of the evidence

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from the FSU showing that, with the exception of thyroid cancers, no radiation-induced health effects have been detected among the general population of the region. This presumably created a ready market among readers and viewers, and hence within the media themselves, for some of the more exaggerated claims about health effects of the accident. One example was the initial reports in the British press claiming that 2 000 deaths had been caused. ‘Seeking a comparison with events outside the nuclear industry [for Chernobyl], the press most often settled on a 1984 disaster at Bhopal, India, where a chemical cloud had escaped from a pesticide plant. That cloud had killed outright not a few dozen people but over two thousand, the long-term damage to the health of another ten thousand or so was not hypothetical but visible; yet to the press and most of the public, the Chernobyl accident seemed the more serious. That was largely because reactor worries centred on faint but widespread radioactivity, whereas in Bhopal the hypothetical long-term effects of dispersed chemicals for a wide population were hardly mentioned. As an emblem of contamination, the radioactive atom remained supreme. It was not pesticides but nuclear power that Newsweek (12.5.86) announced to be a “bargain with the devil’” (Weart, 1988, p 370). Though concerns about the use of chemicals certainly go back as far as the use of gas warfare in the First World War, and have been strengthened by books such as Rachel Carson’s Silent Spring (1962) and events like Seweso (1976), there does not seem to be the same image of a single catastrophic chemical event which could potentially threaten human life on earth. The fact that radiation is associated, at least in the popular mind, with the possibility of genetic effects (‘mutation’), while such beliefs do not seem to be as prevalent with environmental chemicals, brings the ‘risk to future generations’ into play. Slovic’s third factor reflects the number of people exposed to the hazard in question. The impression of the Bhopal accident, like car and mining accidents (much larger causes of death than either chemical or nuclear accidents), was that one knew who the victims were. It is believed that MIC hydrolyses quite rapidly in the environment, it cannot be detected at very low concentrations, and there is no clear evidence of its carcinogenicity in small amounts. Though relatively few people died at Chernobyl, the releases of radioactive material are still detectable over a large area, and though the likelihood of any individual, certainly outside of the FSU, developing cancer because of exposure to such materials is extremely small, it might affect anyone. Hence a person in the UK, say, will feel more at threat because of Chernobyl than because of Bhopal, though in practice they may not be in any real risk from either.

CONCLUSIONS Study of the causes and effects of the accidents at Bhopal and Chernobyl yields remarkably similar pictures. Both involved considerable weaknesses in human, organisational and technological weaknesses. Both occurred in plants which were especially susceptible to violation of operating codes. Both were exacerbated by a lack of credible information in the immediate aftermath. The exercise points to a number areas in which further research would be fruitful. The first refers to perceptions of the two events. Ten years after the events, the health effects of releases from Bhopal have been much more serious than those from Chernobyl. For example, Bhopal caused 3 800 deaths by official estimates, whereas so far the established death toll from Chernobyl would appear to be 33. Yet Chernobyl has had a far more long-lasting effect on public consciousness, both in the region and in the West. While part of the explanation for this may arise from the location of Chernobyl against that of Bhopal, part of it is undoubtedly because of psychological factors in the perception of different risks. It seems clear that the psychosocial effects of the

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Chernobyl accident vastly outweigh the effects caused directly by radiation; this was also the case at Three Mile Island. (This being said, the psychological detriment caused by Bhopal to the local population should not be underestimated.) A major question, then, is how can the psychological effects of such accidents be minimised? Some commentators (e.g. Shrivastava, 1987) have analysed major accidents in terms of a plurality of stakeholders. It is argued that ‘official’ bodies such as the operators of the installation involved and the State in its role as regulator, tend to concentrate on objective, statistical information, which is assessed rationally rather than emotionally. While such an approach may win the support of the scientific community and potentially lead to apparently ‘appropriate’ responses to prevailing conditions, it can also lead to the institutions involved appearing callous and unconcerned about the plight of individuals, viewing people as mere elements in the statistical analysis. In turn this can lead to the growth of mistrust of the official view among those most affected, and therefore make it more likely that such people will believe any scare story in circulation, or at least remain unreassured by information provided and remedial action taken by the authorities. Other stakeholders, principally the victims themselves and pressure groups (local, national and international) seeking to achieve policy changes as a result of the event in question, tend to approach events from a different frame of reference. Such groups invest greater credibility in personal experiences and anecdotes (which of their nature tend to be limited), and to seek to undermine trust in the official bodies. Clearly neither of these paradigms is adequate, and in conjunction both work to increase the level of psychological damage to the affected population. This in turn can lead to ‘overreaction’ on the part of the authorities, often after some time delay, creating further concern and mistrust. Ways should therefore be developed of synthesising these frames of reference to create a ‘statistics of the individual’ which values the widespread gathering of information while remaining sensitive of the needs of individual victims. An essential component of this would be meaningful and public dialogue between the two communities representing the two paradigms. Such dialogue was conspicuously absent at both Bhopal and Chernobyl, with both ‘sides’ having to bear some of the blame for this. Secondly, Chernobyl appears to represent an extreme case of a common human experience. Under the rigidity of the Communist regime, operators were officially expected to do everything ‘by the book’. In reality, however, pressure on them to ‘get the job done’ seems to have led to routine disregard for ‘the book’, to the extent that sections of the operating manual at Chernobyl were actually crossed out. Ideally one would expect operators always to follow instructions precisely. However, it seems likely that any experienced operator will become aware of possible ‘short cuts’ which will allow the required outcome to be achieved more quickly, more cheaply, etc. In part the curbing of the temptation to employ such short cuts can be achieved by a robust safety culture, as the success of the nuclear industry in the West in avoiding major incidents demonstrates. However, it would seem absolutely essential that any short cuts which are employed should be properly discussed with colleagues at all levels, and ideally with the safety regulators as well. The Soviet industrial structure seems to have made this impossible, with terminal consequences for the Chernobyl reactor. Consideration should be given to methods of quasi-institutionalising any such deviations from official procedures. Thirdly, the difference in severe accident rates both between the developed world and less developed countries including the former Communist bloc, and between the chemical and nuclear industries, needs to be explained. Considerable human and organisational failings were implicated in all four accidents considered above (Bhopal, Institute, Chernobyl and Three Mile Island), implying perhaps some technological explanation for any differences.

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Had leaks of the magnitude of those occurring at Institute happened in areas as densely populated as Bhopal it can be presumed that considerable numbers of deaths would have resulted. The same observation cannot be made when comparing Three Mile Island and Chernobyl. Does the key difference involve some difference between technological standards in the developed world and those elsewhere? Or is it possible to make nuclear technology ‘failsafe’ in a way which cannot be achieved in the chemical industry (unless a particular process can be carried out without the storage of significant quantities of hazardous intermediates)? Consideration of a small number of case studies alone can do no more than raise the questions.

BIBLIOGRAPHY Much of the information here presented about the course of the Bhopal accident has been derived from Shrivastava, P., 1987, Bhopal - Anatomy of a Crisis, Ballinger Publishing Company, Cambridge, Mass.. The course of the Chernobyl accident is outlined in Collier. J.G. and Davies, L.M., 1986, Chernobyl, CEGB, Gloucester, UK; and in Ignatenko, E.I., Voznyak, V. Ya, Kovalenko A.P. and Troitskii, S.N., 1989, Chernobyl - Events and Lessons (Questions and Answers), Moscow Political Literature Publishing House.

Other references. Adler, S.J., 1985, ‘Union Carbide plays hardball in court’, American Lawyer, November 1985. Asian Pacific Peoples’ Environment Network (APPEN), 1986, The Bhopal Tragedy - One Year After, an APPEN Report, Sahabat Alam, Penang, Malaysia. Arnold, L., 1992, Windscale 1957 - Anatomy of a Nuclear Accident, MacMillan, London. Business Week, 1979, ‘Union Carbide: its six business strategy is tight on chemicals’, 24.9.79. Central Water and Air Pollution Control Board, 1986, ‘Report of the Central Water and Air Pollution Control Board (Gas leak episode at Bhopal)’ in APPEN, The Bhopal Tragedy - One Year After (op. cit.). De Grazia, A., 1985, A cloud over Bhopal, Kalos Foundation, Bombay. Delhi Science Forum, 1985, Bhopal Gas Tragedy, Delhi Science Forum, Delhi. Diamond, S., 1985, ‘The Bhopal disaster: how it happened’, New York Times, 28.1.85. Environmental Protection Agency, 1985, Multi-Media Compliance Inspection Union Carbide Corporation, Institute, West Virginia, EPA-RIII, Environmental Services Division, Philadelphia. Everest, L., 1985, Behind the Poison Cloud: Union Carbide’s Bhopal Killing, Banner, Chicago. Fishbein, L., 1985, ‘Overview of some aspects of quantitative risk assessment’ in Occupational Cancer and Carcinogenesis, Hemisphere, Washington DC. Fuller, J.G., 1977, The Poison that Fell from the Sky, Random House, New York. Ghazi, P., 1994, ‘Bhopal struck by wave of “chemical AIDS’”, Observer, 20.11.94. Gladwin, T.J. and Walter, I., 1985, Wall Street Journal Europe, 21.1.85. Government of Madhya Pradesh, 1985, Bhopal Gas Tragedy, Relief and Rehabilitation - Current Status, Government of Madhya Pradesh, Bhopal. Hiltzik, M.A., 1985, ‘Carbide has a long history of difficulty’, Los Angeles Times, 19.8.85 Howard, H.B., 1983, ‘Efficient time use to achieve safety of processes or “How many angels can stand on the head of a pin?’” in Loss Prevention and Safety Promotion in the Process Industries vol. 1 ,Institution of Chemical Engineers Symposium Series, London. Indian Council of Agricultural Research, 1985, The Bhopal Disaster: Effect of MIC Gas on Crops, Animals and Fish, Indian Council of Agricultural Research, New Delhi. International Commission on Radiological Protection (ICRP), 1991, ‘1990 Recommendations of the International Commission on Radiological Protection’, Annals of the ICRP 21 no. 1-3. International Confederation of Free Trade Unions (ICFTU), 1985, The Trade Union Report on Bhopal, International Confederation of Free Trade Unions, Geneva. Jones, T., 1988, Corporate Killing: Bhopals Will Happen, Free Association Books, London. Khandekar, S., 1985, ‘Painful indecision’, India Today, 31.1.85.

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Kemeny, J.G. et al., 1981, Report of the President’s Commission on the Accident at Three Mile Island, Pergamon Press, New York. Kinnersly, P., 1973, The Hazards of Work: How to Fight Them, Pluto, London. Kovalenko, A., 1989, Soviet Weekly, 19.8.89. Kurzman, D., 1987, A Killing Wind: Inside Union Carbide and the Bhopal Catastrophe, McGraw-Hill, New York. Lynn, R. et al., 1988, ‘Global impact of the Chernobyl reactor accident’, Science 242. Madhya Pradesh Chronicle, 1985, ‘Psychological effects on gas victims?’, 21.5.85. Medico-Friends Circle, 1985, The Bhopal Accident Aftermath: An Epidemiological and Socio-Medical Survey, Medico-Friends Circle, Bangalore, India. Minocha A.C., 1981, ‘Changing Industrial Structure of Madhya Pradesh: 1960-1975’, Margin 4 no. 1, pp 4661. Mosey, D., 1990, Reactor Accidents: Nuclear Safety and the Role of Institutional Failure, Nuclear Engineering International Special Publications, Sutton, Surrey, UK. Mukund, J., 1982, Action Plan - Operational Safety Survey May 1982, Union Carbide (India) Ltd., Bhopal. Nanda, M., 1985, ‘Secrecy was Bhopal’s real disaster’, Science for the People 17 no. 6. NEA/OECD, 1994, Liability and Compensation for Nuclear Damage: An International Overview, OECD, Paris. Nolan, J., 1984, ‘Bhopal likely to alter dramatically how insurance is written abroad’, Journal of Commerce, 31.12.84. Parmentier, N. and Nénot, J-C., 1989, ‘Radiation damage aspects of the Chernobyl accident’, Atmospheric Environment 23. Roy, S.K. and Tripathi, D.S., 1985, Madhya Pradesh Chronicle, 16.3.85. SFEN, 1991, Proceedings of the International Conference ‘Nuclear Accidents and the Future of Energy’, SFEN, Paris. Slovic, P., Fischhoff, B. and Lichtenstein, S., 1980, ‘Facts and Fears; Understanding Perceived Risk’ in ed. Shwing, R.C. and Al Albers, W., Societal Risk Assessment: How Safe Is Safe Enough?, Plenum, New York. Smyth, H.F., 1980, Current state of Knowledge about the Toxicity of Methyl Isocyanate, unpublished paper, Mellon Institute, Carnegie-Mellon University, Pittsburgh. Town and Country Planning Department, 1975, Bhopal Development Plan, Municipal Corporation, Bhopal. Tatro, E.F., 1984, ‘Life in gas-stricken city gradually returns to normal’, Associated Press 24.12.84. Traves, A. 1995, University of Birmingham, personal communication. Union Carbide Corporation, 1982, Operating Safety Survey CO/MIC/Sevin Units, Union Carbide India Ltd, Bhopal Plant, Union Carbide Corporation, Danebury, Conn.. Union Carbide Corporation, 1984, Annual Report, Union Carbide Corporation, Danebury, Conn.. Union Carbide Corporation, 1985, Bhopal Methyl Isocyanate Investigation Team Report, Union Carbide Corporation, Danebury, Conn.. Union Carbide Corporation, 1994, Bhopal Chronology, Union Carbide Corporation, Danebury, Conn.. United Nations Scientific Committee on the Effects of Atomic Radiation (UNSCEAR), 1988, Sources, Effects and Risks of Ionising Radiation, United Nations, New York. U.S. District Court, Southern District of New York, 1985, Memorandum of law in opposition to Union Carbide Corporation‘s Motion to dismiss these actions on the grounds of forum and non convenience, MDL Docket no. 626 Misc., No. 21-38 (JFK) 85 Civ. 2696 (JFK). Varadarajan, S., et al., 1985, Report on Scientific Studies in the Factors Related to Bhopal Toxic Gas Leakage, Council of Scientific and Industrial Research, New Delhi. Weart, S.R., 1988, Nuclear Fear - a History of Images, Harvard University Press. Weisman, S., 1985, ‘Doctors in India disagree on drug’, New York Times, 10.4.85. Wilkins, L., 1987, Shared Vulnerability - the Media and American Perceptions of the Bhopal Disaster, Greenwood Press, Westport, Conn.. World Environment Centre, 1984, The World Environment Handbook, World Environment Centre, New York. World Health Organisation, 1995, Health Consequences of the Chernobyl Accident - results of the IPHECA pilot projects and related national programmes (Summary Report), World Health Organisation, Geneva.

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TRANSPORT THEORY IN DISCRETE STOCHASTIC MIXTURES G. C. Pomraning School of Engineering and Applied Science University of California, Los Angeles Los Angeles, CA 90095-1597

INTRODUCTION The flow of neutral particles which interact with a background material but not with themselves is described in some generality by a linear kinetic, or transport, equation. This equation, while algebraically complex, has a very simple physical content; it is simply the mathematical statement of particle conservation in phase space. Applications of such transport descriptions are numerous. They include neutron migration in nuclear reactors, radiative transfer (thermal photon flow), neutrino flow in astrophysical problems, neutral particle transport in plasmas, gamma ray transport in shielding considerations, and Knudsen flow arising in the kinetic theory of gases. A vast literature exists on the formulation and solution methods, both analytical and numerical, of such transport problems, but generally only in the nonstochastic area. We use the terms stochastic and nonstochastic throughout this article in a special sense. That is, particle transport is in itself a stochastic process, but this is not the stochasticity or lack thereof that we mean when we use the terms “stochastic transport” and “nonstochastic transport.” In our use of the word, nonstochastic means that the properties, as functions of space and time, of the background material with which the particles interact are either specified or can conceptually be computed in a deterministic fashion. During the last decade, there has been a renewed interest in formulating linear kinetic theory and particle transport descriptions when the properties of the background material, as functions of space and time, are only known in a statistical sense. In our terminology, this would be called stochastic transport theory. For example, the background material with which the particles interact may consist of a turbulent mixture of two immiscible fluids. At any point in space and time, one does not know with certainty which of the two fluids is present; one only knows the probability of one or the other fluid being present. The goal in such a statistical setting is to find the ensemble average (mean), and perhaps higher statistical moments such as the variance, of the particle intensity in phase space. These averages are computed over all possible physical realizations of the underlying statistical description of the background

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material. Conceptually, the solution to such stochastic transport problems is straightforward. One generates from the presumed known statistics of the background material, either deterministically or by a Monte Carlo procedure, a certain physical realization of the statistics. For this realization, the transport equation describes a deterministic transport problem and the solution is in principle computed, either analytically or numerically. One repeats this process for all possible statistical realizations, and than averages the corresponding transport solutions to obtain the ensemble-averaged solution. This averaging process is performed with proper account taken of the probability of occurrence of each of the physical realizations of the statistics. In general the number of realizations is infinite, and hence this straightforward solution method for the ensemble-averaged result involves a very large number of deterministic transport calculations to obtain an accurate estimate of this ensemble average. A zero error computation of the ensemble average would require an infinite number of deterministic transport calculations. A highly desirable alternative to this computationally intensive process is to derive a relatively small and simple set of deterministic equations which contains the ensemble-averaged intensity directly as one of its unknowns. In principle, this is also a conceptually straightforward process. One forms appropriate statistical moments of the transport equation, leading to an infinite set of coupled equations for the ensemble-averaged intensity, as well as higher moments (1-4). This can be done using either the integro-differential (1-4) or the purely integral (1) form of the transport equation. One then truncates this infinite set to a finite number of equations by introducing an approximate closure. The accuracy of the resulting finite set of equations, and in particular the corresponding prediction of the ensemble-averaged intensity, obviously depends upon the accuracy of the closure. A substantial fraction of this review article deals with various closures that have been suggested (1-8). The coupling of transport theory and stochasticity is a quite old subject, going back at least to the Manhattan project of the 1940’s. Three books, within the context of nuclear reactors, have been written summarizing much of this work. These are due to Thie (9) in 1963, Uhrig (10) in 1970, and Williams (11) in 1974. These books deal with a wide range of stochastic problems within the context of reactor physics and transport theory. In this review article, we adopt a much more modest goal, treating only a very restricted class of stochastic transport. This class corresponds to neutral particle transport through a background material consisting of two (or more) randomly mixed immiscible materials. That is, at any point in space and time one or the other component of the mixture is present in its pure state according to some prescribed statistics of mixing; we do not allow these two components to mix at the atomic level. Thus the physical picture that emerges is that of a grainy background material made up of randomly distributed chunks of random sizes and shapes of the two (or more) components of the mixture. For any given physical realization of the statistics, the particle flow is described by the deterministic transport equation. As mentioned earlier, the goal in this statistical setting is to develop a simple formalism to compute the ensemble-averaged intensity in phase space, as well as higher statistical moments that may be of interest, such as the variance. To further simplify the discussion, we treat only time-independent particle transport. The linear kinetic, or transport, equation we shall deal with is then (12, 13), with full display of all arguments,

Equation (1) is written in standard neutronic notation, with

denoting the

TRANSPORT THEORY IN DISCRETE STOCHASTIC MIXTURES

49

angular flux at space point r, energy E, and direction The quantities and denote the macroscopic total cross section and double differential scattering kernel, respectively, and represents any internal source of particles. The only significant assumption made in writing Eq. (1) is that the material through which the particles flow is isotropic, so that is independent of direction and depends upon and through the single variable the cosine of the scattering angle. Aside from this commonly assumed isotropy restriction, Eq. (1) is a quite general linear transport equation. The stochasticity in this problem is embodied in the parameters and S in Eq. (1). For a binary stochastic mixture of immiscible materials, these parameters are two-state discrete random variables which assume, at any spatial point r, one of two triplets of values associated with each of the two materials constituting the mixture. We will identify the two materials by an index with and 1, and in the ith material we denote these three parameters by and As a particle traverses the mixture in any direction it encounters alternating segments of the two materials, each of which has known deterministic values of and S. The statistical nature of the problem enters through the statistics of the material mixing, i.e., through the probabilistic knowledge as to which material is present at each space point r. Since and S are random variables, the solution of the transport equation is a (continuous) random variable, and we let denote the ensemble average (mean) of this random variable. The goal of any stochastic transport model is to develop simple and accurate equations for and perhaps higher statistical moments as well. The problem of describing particle transport through such a binary stochastic mixture was first addressed at the Ninth International Conference on Transport Theory, held in 1985 in Montecatini, Italy. This early and rudimentary treatment was summarized in the proceedings of that conference (14), as well as in a more complete journal article (15). During the next few years, substantial progress was made in this area of research, as reported in the 1991 book by the present author (16). Our purpose in this review article is to briefly summarize the highlights of this book, and to describe recent results which have been published between 1991 and the present. A brief outline of this article is as follows. The next section discusses fundamental statistical concepts which are needed to formulate transport theory in a binary stochastic mixture. The following section considers the simplest transport problem, corresponding to no scattering Under the assumption of Markovian mixing, exact and simple equations for the ensemble-averaged intensity, as well as higher-order statistical moments, are easily derived in this purely absorbing case. In the following section, retaining the assumption of Markovian mixing, an exact formalism is developed in the presence of the scattering interaction, but only for a very restricted class of problems. In the next section, various low-order closures to the infinite set of statistical moment equations are discussed, leading to two equations (for a binary mixture) describing the ensemble-averaged intensity, again for Markovian mixing. Following this, higher-order closures are discussed, which lead to four equations describing in the Markovian case. The penultimate section considers the stochastic transport formulation for mixing statistics more general than Markovian. The final section of this article is devoted to a few concluding remarks. We close this introduction by pointing out the importance, as well as a few applications, of having an understanding and usable formulation of transport theory for a stochastic mixture of two (or more) immiscible materials. All materials in nature have a heterogeneous and stochastic character to some extent, and it is important to understand in any given application if this stochasticity must be accounted for in particle flow calculations. For example, in radiation protection calculations involving neutron and

50

G. C. POMRANING

gamma ray transport through a concrete shield, it may be important to account for the random heterogeneities of the concrete. Neutron transport in a boiling water reactor involves a stochastic background material, namely a mixture of liquid water and vapor. Light transport through sooty air and murky water clearly involves photon transport through stochastic media. The treatment of randomly dispersed burnable poison grains in a nuclear fuel element has been treated by the formalism to be described here (17). In the area of inertially confined fusion, a proper treatment of radiative transfer is crucial to the prediction of the performance of the fusion pellets. Since various hydrodynamic instabilities are likely to be excited at material interfaces in the implosion and subsequent explosion of such a pellet, one must be able to treat thermal photon transport in turbulent fluid mixtures which could arise in this context (18). It was this fusion arena which supplied the original motivation for the formulation of a transport theory formalism in a binary random medium in the mid-1980’s (15). Another important, and generally accepted, application is within the context of general circulation models of the atmosphere. The interaction of solar and earth-reflected radiation with clouds is very important in modeling climate change. Radiative transfer through a partially cloudy sky is treated as thermal photon flow through a binary stochastic mixture consisting of clouds and clear atmosphere (19-26). Applications have also been made in the astrophysics context to photon transport through interstellar molecular clouds and solar prominences (27,28). Finally, it has recently been suggested that the small-scale heterogeneities of the human body may be well modeled as a two (or more) component stochastic binary mixture. This might then lead to improved planning algorithms for electron and photon beam cancer treatments (29-32). In short, there appears to be a large number of potential, and some already realized, applications for a theory of particle transport through a stochastic background material. Hopefully the material summarized in this article, and the references supplied, will be useful to persons who must deal with stochasticity in various applications of transport theory.

MIXING STATISTICS In this section, we summarize certain statistical considerations we shall subsequently use to formulate transport theory in a binary stochastic mixture of immiscible materials. Specifically, we consider certain general statistical results concerned with the infinite real line populated statistically with alternating segments (materials), which we label as 0 and 1, of random lengths. The material in this section is purely statistical in nature, and is completely independent of its subsequent application to transport theory. These binary results are available in the literature (15,16,24,33), and some generalizations to mixtures with more than two components have also been reported (34). On the infinite real line, we consider two points and with and further consider the closed interval We define seven probabilities, namely

TRANSPORT THEORY IN DISCRETE STOCHASTIC MIXTURES

51

where is the index indicating the segment type. We note that and have similar, but not identical, definitions. That is, specifies that the left boundary of the interval, is an interface between two segments, with segment type to the right, whereas states that is anywhere in segment type Similarly, specifies that the right boundary of the interval, is an interface between two segments, with segment type to the left, whereas states that is anywhere in segment type Within this general statistical setting, it has been shown (16,33) that is the fundamental probability, with the other six probabilities defined by Eqs. (3) through (8) being simply related to These relationships are

As a special case of these results, we consider homogeneous statistics, by which we mean that all points on the line have identical statistical properties. This implies that and depend upon the single displacement argument and the are constant, independent of position on the line. If we define we can simplify the notation in this homogeneous case by writing

Using similar simplified notation for and (11) become, in the case of homogeneous statistics,

we find that Eqs. (9) through

where the prime in Eq. (15) indicates differentiation. We now consider a restricted class of homogeneous statistics as follows. Let be the probability density function for the length of a segment of material defined such that

The function is independent of the position on the line; all segments of material are chosen from the same density function. Thus the statistics are homogeneous. It is clear from the definitions of and that

The mean chord length

in material is given by

52

G. C. POMRANING

Integrating Eq. (18) by parts shows that to

can be expressed in terms of

according

Further, the probabilities which are independent of position on the line, are clearly related to these mean chord lengths according to

We now solve the differential equation given by Eq. (15) subject to the boundary condition that This gives

Setting

in Eq. (21) and using and hence we have

Using Eq. (22) in Eq. (14), and once again using

as well as Eq. (19), establishes that

gives

Thus we see that all statistical quantities of interest are simply related to the chord length distribution function since depends simply upon according to Eq. (17). A special case of this class of homogeneous statistics is Markovian statistics, in which case the chord lengths are exponentially distributed according to (15)

which has a mean

In this case, Eqs. (17), (22), and (23) give the simple results

Equations (24) through (26) give complete statistical results for a homogeneous Markov process. We now consider a more general Markov (no memory) process, in which we do not restrict our considerations to homogeneous statistics. Specifically, we consider the following statistical situation. If the line consists of material at position then the probability of position being in the other material is given by

where here is a given, nonnegative, function of It is the dependence of one or both of the that makes these statistics inhomogeneous. According to Eq. (27), the Markov process is completely characterized by the two Markov transition lengths and We now show that the Markov process can equivalently be characterized

TRANSPORT THEORY IN DISCRETE STOCHASTIC MIXTURES

53

by the unconditional probability of finding point in material and a correlation length to be introduced shortly. We define as the conditional probability that position is in material given that position is in material That is

The

satisfy the forward form of the Chapman-Kolmogorov equations, given

by (35)

with boundary conditions at

given by

We seek a solution to Eqs. (29) through (31) in terms of functions according to

and

where

and

In view of the way and enter into Eq. (32), the function must be interpreted as the unconditional probability of point being in material and the function is reasonably called a correlation length. It is clear that Eq. (32) satisfies Eq. (31). It can be shown by direct substitution that Eq. (32) is a solution of Eqs. (29) and (30) if and are related to the according to

where the prime indicates differentiation. We note that since implies

Eq. (35)

as a physical realizability condition for inhomogeneous Markovian statistics. Equation (35) represents two equations, which can be solved on for the and as

with

in Eq. (37) defined by

54 We now compute statistics. We write

G. C. POMRANING as defined by Eq. (5) for these inhomogeneous Markovian

The left-hand side of this equation is the probability that the interval is in material given that the point is in material The right-hand side is the product of two factors, namely which is the probability that the interval is in material given that the point is in material and which is the probability of no transition out of material in the distance Equation (40) leads to the differential equation

which must be solved subject to the boundary condition

The solution of Eqs. (41) and (42) is simply

Using Eq. (35) for

one finds that Eq. (43) can be written equivalently as

From Eqs. (9) and (10) it then follows that

and Eq. (10) for

with

Finally, using Eq. (45) to evaluate

gives

and

according to Eq. (11), we find

with and as just given by Eqs. (44) and (46). As a special case of these considerations we consider homogeneous Markovian statistics. By this we mean that the Markov transition lengths and hence the and are independent of position on the line. Then Eqs. (44) through (47) reduce to

Further, in this homogeneous case one can simply relate the segment lengths in materials 0 and 1, as we now show.

and

to the mean

TRANSPORT THEORY IN DISCRETE STOCHASTIC MIXTURES

55

Using Eq. (49) for in Eq. (17) with and differentiating establishes the fact that homogeneous Markovian statistics correspond to choosing the segment lengths from an exponential distribution given by

The mean

of this distribution is given by

Further, Eq. (35) reduces to, in the case of homogeneous statistics ( position),

independent of

A comparison of Eqs. (51) and (52) shows that the mean chord lengths are precisely equal to the Markov transition lengths as defined by Eq. (27). A solution of Eq. (52) for the and yields the simple results

These last two results also follow directly from Eqs. (37) and (38). We note that using Eq. (52) in Eq. (50), with reproduces Eq. (24), the exponential distribution stated earlier as corresponding to homogeneous Markovian statistics. In the general case of inhomogeneous Markovian statistics, Eqs. (37) and (38) still relate the and to the transition lengths but in this general case one cannot identify the with mean segment lengths. There is an additional result for homogeneous Markovian statistics which we will find useful in the transport theory context. Let denote a parameter associated with all segments of material We then define between any two points a distance apart, say and as

In the transport theory context, is the total cross section associated with material and is the optical depth corresponding to a geometric distance However, we have no need for these physical interpretations in the present development. Now, since is a random variable, so is We seek the probability density function for the random variable given a distance Since we shall restrict our attention to statistics that are independent of position, the point is irrelevant; the statistics of the random variable are independent of Because there are only two states, 0 and 1, and assuming and to be independent of position on the line, we have

To obtain the distribution of the total track lengths through materials 0 and 1 in a distance we make use of a problem discussed by Lindley (36). We define the random variable as the total track length in material 1 in a distance given that the point is in material 0. Similarly, we define the random

G. C. POMRANING

56

variable as the total track length of material 0 in a distance given that the point is in material 1. Then, according to Eq. (56), the optical depth as a function of the distance is given by one of two expressions, namely

or Equation (57) occurs with probability and Eq. (58) with probability The cumulative distribution function is the probability that is less than a value given a geometric distance In view of Eqs. (57) and (58), we have

Simple rearrangement gives

If we label the materials such that

Eq. (60) immediately gives the result, since

which is just the statement that in a distance the minimum value for is and the maximum is The key to finding for is to use the facts that the segment lengths are exponentially distributed for homogeneous Markovian statistics [see Eqs. (24) and (50)], and that the sum of identically distributed exponential random variables with a parameter is given by a gamma distribution with parameters and The details are available in the literature (15,16), and the result is

where

is the modified Bessel function, and we have defined

Equations (61) and (62) give the cumulative distribution function for all values of with the geometric distance simply a parameter in this distribution. The probability density function corresponding to follows in the usual way as

Since is discontinuous at and it follows that contains Dirac delta functions at these two discrete points. We conclude this section by pointing out two nonexponential (non-Markovian) chord length distributions which have been reported within the context of stochastic particle

TRANSPORT THEORY IN DISCRETE STOCHASTIC MIXTURES

57

transport. The first of these is due to Vanderhaegen (38) who considered a uniform and parallel beam of particles incident upon a sphere of radius . He showed that the probability density function for a chord length through the sphere is given by

This distribution has a mean

given by

which is identical to the mean chord length for a sphere placed in an isotropic particle field, which follows from the well known formula (39). This result was subsequently generalized to ellipsoids (24), with the result also given by Eq. (65) with in this equation replaced by given by

This result was obtained within the climatology context, where the ellipsoid represents an idealized cloud shape, with the vertical semi-axis and the horizontal semiaxes. A parallel beam of radiation impinges upon this cloud at an angle measured with respect to the vertical axis. The same reference (24) considers the chord length distribution for the spacing between clouds. In this case, no exact analytic results are possible, but an approximate expression for this cloud spacing probability density function agrees quite well with benchmark Monte Carlo results.

MARKOVIAN MIXTURES WITHOUT SCATTERING We consider Eq. (1) in the absence of scattering. Then the only relevant independent variable is the path length in the direction of travel of the particle. Thus the transport equation is, at each energy E,

where is the directional derivative in the direction As assumed throughout this article, and are discrete binary random variables. At we assign the simple boundary condition

where is the prescribed boundary data which we take as nonstochastic (i.e., it is the same for each realization of the statistics). We seek a solution for the ensemble-averaged angular flux In general, we let denote the ensemble average of any random variable , and we let denote the deviation of from Thus we have

With this notation, an ensemble-averaging of Eq. (68) gives

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G. C. POMRANING

It is clear from Eq. (71) that one needs to calculate (or approximate) the cross correlation term in terms of to obtain a closed formulation for the quantity of interest, namely To do this, assuming such a closure is possible, one obviously needs a knowledge of the statistics of the random variables and S. The simplest closure (approximation) is to simply neglect this cross correlation term, i.e., set On physical grounds, this simplest of treatments approaches exactness as the chord lengths of one or both materials in the mixture become vanishingly small. In general, however, this very simple closure will underestimate This follows from the fact that as the cross section becomes larger (smaller), the solution becomes smaller (larger). Hence is negative for each realization of the statistics, and accordingly Thus this term, when transposed to the other side of Eq. (71), acts as a positive source terms for the equation. The result is that this the exact result, is larger than that computed by neglecting this cross correlation source term. An alternate way to proceed to obtain is to consider the integral, rather than the differential, formulation of particle transport. In this simple case of no scattering, the integral formulation of particle transport is, in fact, a solution for the angular flux. That is, the solution of Eqs. (68) and (69) for a given realization of the statistics is

Given this solution, one can obtain by ensemble averaging, over all physical realizations of the statistics, the right-hand side of Eq. (72). It is clear once again that in order to perform this average one must know the details of the statistics. In particular, we see from Eq. (72) the joint occurrence of and S, indicating a need for a knowledge of the correlation between these two random variables. Vanderhaegen (40) pointed out that Eq. (71) can be closed exactly, under the assumption of Markovian statistics as defined by Eq. (27), by invoking the Liouville master equation (41,42). We begin by giving a brief discussion of this master equation in a general setting, without regard to particle transport. This equation is generally associated with initial value problems in time. Such a problem is Markovian in the sense that if the solution is known at any time the solution is uniquely defined for all times We consider, then, a dynamical system which at any time can be in one of two states. We label these states with indices 0 and 1. The specification of which state is present at any time is taken as a statistical process, and is assumed to be Markovian. Specifically, if the system is in state at time the probability of the system being in state at time is assumed to be given by [see Eq. (27)]

where the are prescribed functions of time. For any realization of the statistics, the dynamics of the system is taken as described by

where is the dependent variable, and F is a prescribed, nonstochastic function of its arguments, provided that the state of the system is specified. We let be this function corresponding to state Since a nonstochastic initial value problem is Markovian, and we have prescribed the transitions between states as Markovian, we have a joint Markov process. Associated with any joint Markov process as defined by

TRANSPORT THEORY IN DISCRETE STOCHASTIC MIXTURES

59

Eqs. (73) and (74) is the Liouville master equation (41,42). This equation, in a discrete state context, describes the joint probability density defined such that is the probability that the system is in state at time and that the stochastic solution lies between and In our particular case of two discrete states, we have the two coupled equations

The ensemble average of any function of the solution, say

, is given by

Thus the key to obtaining complete results for this statistical problem is finding the solution for the A numerical procedure for solving Eq. (75) for general functions has recently been reported in the literature (43). Vanderhaegen (40) introduced this Liouville master equation in the particle transport context. He observed that the time-independent, no-scattering transport equation can be thought of as an initial value problem, with the space coordinate playing the role of time If the mixing is taken as Markovian, the stochastic transport problem is then a joint Markov process, and the Liouville master equation applies. Specifically, if Eq. (68) is the underlying transport equation, the function F in Eq. (74) is, with and replaced with and respectively,

and Eq. (75) becomes

where

and are the cross section and source associated with material Here is the probability of finding material at position and having the stochastic solution lie between and The initial condition on Eq. (78) is [see Eq. (69)]

which expresses the certainty of the solution at position being in material and is related to

Here is the probability of according to

The integration range does not include negative values of since negative angular fluxes do not exist. Integration of Eq. (78) over gives simple differential equations relating the to the namely

Equation (81) also follows from Eqs. (29) and (30) by noting that

G. C. POMRANING

60

Further, using Eq. (38) for in Eq. (35) also reproduces Eq. (81). If we define as the conditional ensemble average of conditioned upon position being in material the definition of gives

as the relationship between average of the angular flux,

and The overall, unconditional, ensemble is given in terms of the as

We note that in the transport setting, since is linear in as shown in Eq. (77), it is not necessary to solve the partial differential equations given by Eqs. (78) and (79) in order to obtain the One can obtain equations for the directly. If one multiplies Eq. (78) by and integrates over making use of Eqs. (80) and (83), one finds the two coupled equations (after integrating by parts) given by

These two equations for and are subject to the boundary conditions found by multiplying Eq. (79) by and integrating over namely

Equations (84) through (86) give a complete and exact description for the ensembleaveraged intensity for time-independent, no-scattering transport in a binary Markovian mixture, with arbitrary spatial dependencies for the and An alternate, but equivalent, form of these results can be found by changing dependent variables from and to and according to

Then Eqs. (85) and (86) become

Here we have defined the parameters

with given by Eq. (38). The advantage of this form of the equations is that the quantity of interest, namely is one of the two unknowns, with being the (irrelevant) other. A comparison of Eq. (88) with Eq. (71) shows that is just the cross correlation term in Eq. (71).

TRANSPORT THEORY IN DISCRETE STOCHASTIC MIXTURES

61

These results for purely absorbing transport problems can be obtained in several other, but more complex, ways. In fact, they were first obtained (14-16) by using the method of smoothing (44-47). In this treatment, one uses the ensemble-averaged equation given by Eq. (71), together with the equation for the fluctuating component of the angular flux, given by

Equation (94) follows from using and in Eq. (68), and subtracting Eq. (71) from the result. Equation (94) is formally solved for in the form of an infinite Neumann series. This result is then multiplied by and ensemble averaged to obtain a result for which serves to close Eq. (71). This result for involves with correlations involving and In the case of Markovian statistics, all of these correlations can be computed, and the infinite sum performed. The result is identical to the predictions of the Liouville master equation approach as described here. In two separate developments, Sahni (48,49) showed that these purely absorbing transport results for Markovian mixtures can be obtained in two other ways, namely by using the techniques of nuclear reactor noise analysis (48) and by assuming that each particle track is independent of prior tracks (49). Finally, stochastic balance methods (1,2) have also been shown to reproduce the results given here, as we discuss in some detail later in this article. One advantage, in addition to its simplicity, of the master equation approach is that it allows one to obtain simple equations for the higher moments of the random variable If we define for as the conditional ensemble average of conditioned upon the position being in material we then have, from the interpretation of the relationship

If one now multiplies Eq. (78) by equations

and integrates over all

one finds the two coupled

For Eq. (96) reduces to our previous result given by Eq. (85), since in our earlier notation. The higher moments, corresponding to an index can be seen to satisfy two coupled linear equations, with the solutions providing inhomogeneous terms in the equations with index Thus these equations can be solved sequentially, beginning with there is no closure problem. The boundary conditions on Eq. (96) follow from Eq. (79) as

The overall, unconditional, ensemble average for powers of is related to the according to

Of particular interest is the quadratic V defined as

which we denote by

moment since it enters into the variance

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G. C. POMRANING

The variance gives an indication of the magnitude of the spread of the stochastic solution about the mean A large variance implies a large spread, and in this case a knowledge of alone may not be enough to fully understand a stochastic particle transport problem. If we consider homogeneous statistics so that the and hence the are independent of position and additionally assume that the and are independent of one can obtain a simple and complete analytic solution for In this case, the parameters and are all constant, independent of and Eqs. (88) and (89), subject to the boundary conditions given by Eq. (90), are easily solved. The result for is (15,16)

with

given by

We emphasize that Eq. (100) is an exact solution for the problem under consideration, namely homogeneous Markovian statistics for a purely absorbing binary mixture with constant component properties and sources. As was stated at the beginning of this section, another way to obtain the solution for is to ensemble average the solution as given by Eq. (72). We carry out the details of this averaging for the source-free problem (15,16). In this case, Eq. (72) is simply where

is the optical depth given by

Ensemble averaging this pure exponential, we have

where is the probability density function for the optical depth random variable with a parameter in this distribution function. An integration of Eq. (104) by parts introduces the cumulative distribution function For homogeneous binary Markovian mixtures, is given by Eqs. (61) and (62) of the last section. We thus have

with in Eq. (105) given by Eq. (62). To evaluate the right-hand side of Eq. (105), we introduce transform of Transforming Eq. (105) gives

as the Laplace

We change integration variables in Eq. (106) from to where and are defined by Eq. (63). The double integral in Eq. (106) then becomes a double integral

TRANSPORT THEORY IN DISCRETE STOCHASTIC MIXTURES over the first quadrant of

space. Inserting Eq. (62) for

where we have defined the function

63

we then have

as

Using Eq. (108) in Eq. (107), one must evaluate a triple integral over can be done by appropriately interchanging orders of integration (first finally ), and the result is the simple expression (15,16)

and then

This and

with and defined by Eqs. (92) and (93). Laplace inversion of Eq. (109) then gives the exact result for in a source-free, homogeneous, two-material Markovian mixture (with constant) as

with once again given by Eq. (101). We see that Eq. (110) agrees with our previous Liouville master equation result, Eq. (100), in the source-free case. The final item we consider in this section is the possibility of obtaining a renormalized transport equation, accounting approximately for stochastic effects, of the form

Here and are nonstochastic “effective” quantities which in some sense account for the statistical nature of the problem. That is, the definition of these two effective quantities provides the recipe for homogenizing the binary mixture into a uniform, effective background material. If appropriate definitions for these two homogenized material properties can be found, the nonstochastic solution methods already developed for the usual transport equation can be applied directly to Eq. (111) since this equation is of the same form as the classic nonstochastic equation [see Eq. (68)]. Here we give one strategy for defining and for our binary Markovian mixture, with and taken as independent of space, leading to and which are also independent of space. We first note that no definitions for these two effective quantities will allow Eq. (111) to predict exact results for For example, in the source-free problem, the exact solution is the sum of two decaying exponentials [see Eq. (110)], whereas Eq. (111) predicts a single decaying exponential according to

with again denoting the boundary value. Thus the definitions of and are necessarily approximate and hence somewhat arbitrary. In the present treatment, the best we can do is to choose and to preserve two characteristics of the exact solution.

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G. C. POMRANING

To obtain we demand that the average distance to collision (the mean free path) for a particle impinging upon a source-free medium be given correctly by Eq. (112). This mean free path is given by the weighted average of the distance traveled, with the weight function in this averaging process being the fraction of particles which collide at That is

Using Eq. (112) in Eq. (113) gives the simple result

If we use the exact result for

as given by Eq. (110) in Eq. (113), we find

where the last equality in this equation makes use of the expressions for given by Eq. (101). If we now equate Eqs. (114) and (115), and use Eqs. (92) and (93) for and as well as make use of Eq. (53), we find

This result can be written in terms of the obtain

and

by using Eqs. (53) and (54). We

To determine we demand that the transport equation involving and namely Eq. (111), give the exact solution in an infinite medium. From Eq. (111), this result is clearly given by

whereas the exact result follows from Eq. (100) by setting

Equating Eqs. (118) and (119) and using Eq. (116) for

This result is

gives

Making use of Eqs. (53), (54) and (91) through (93), this can be written as

or equivalently,

We see from Eqs. (116), (117), (121), and (122) that

TRANSPORT THEORY IN DISCRETE STOCHASTIC MIXTURES

65

and thus this homogenization procedure is robust, predicting in all instances a nonnegative cross section and source. We conclude this discussion by emphasizing again that these results for and apply only to particle transport in a binary mixture obeying homogeneous Markovian statistics, and with the properties of each component of the mixture, namely and being constant, independent of position. In later sections, we will relax these restrictions of homogeneous Markovian statistics and constant properties. Further, we will include the scattering interaction. Non-Markovian statistics are treated using integral transport theory and the methods of renewal theory (1,16,33,38,50-53), and effective properties, with scattering and inhomogeneities, are obtained as an asymptotic limit (16,54-57) of a general, but approximate, model of particle transport in a binary stochastic mixture (2,16,48,49,58-61). Prior to these considerations, however, we consider in the next section certain exact results which can be obtained for a restricted class of problems when the scattering interaction is extant (16,62-66).

MARKOVIAN MIXTURES WITH SCATTERING, EXACT RESULTS We now consider the stochastic transport problem in the presence of the scattering, i.e., when Eq. (1) is the underlying transport equation. In this case, the analysis is much more complex, and in general no simple and exact formulations for can be found. This is discussed in some detail by Sanchez (1). The problem is not the inclusion of the energy variable. Even in a monoenergetic setting, and even in planar (or, even simpler, rod) geometry with isotropic scattering, it is the presence of the scattering interaction which causes the complications. The method of smoothing (14-16,44-47) discussed briefly in the last section is, in principle, still applicable, but the algebraic details are too complex to obtain useful and general results. The approaches of Sahni (48,49) cannot be carried through without introducing approximations, as we point out in the next section. Stochastic balance methods (1-4) require an approximate closure, as we also discuss in the next section. The Liouville master equation approach (16,4043) which proved so useful in the purely absorbing context, is not strictly applicable when scattering is extant. Let us discuss in some detail why this is so. In the last section we observed that, in the absence of scattering, the transport problem can be thought of as an initial value problem, with the spatial coordinate in the particle flight direction playing the role of time. Since the mixing of the background material was assumed to be Markovian, the stochastic transport problem is then a joint Markov process. As such, the Liouville master equation applies, and gives exact equations for the joint probability of a given spatial position being in material and simultaneously exhibiting a solution for the intensity lying between and With the inclusion of scattering, however, the situation is radically different, as first pointed out by Vanderhaegen and Deutsch (62). The transport problem in this case can no longer be put in analogy to an initial value problem. Rather, the transport problem with scattering is a true boundary value problem. For example, in planar geometry one must assign boundary conditions to the transport equation for incoming directions at each edge of the planar system. The scattering interaction mixes, through the integral scattering operator, the transport equation for different directions. Thus, in contrast to no-scattering problems, one cannot solve for the intensity in a given direction independently of the solution in other directions. This means that, for a given direction, the solution is dependent upon the boundary conditions for all other directions. Further, as we pointed out earlier, these boundary conditions are applied at

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two different positions, namely the two edges of a planar system. Thus, with scattering, one cannot interpret the transport equation as an initial value problem; the solution is dependent upon more than just an initial value. Another observation is that backscattering allows a particle to traverse the same material more than once, and this too is inconsistent with a Markov process. The conclusion based upon all of these observations is that, with scattering included, the transport equation does not describe a Markov process and hence, when coupled with the mixing process, be it Markovian or not, the joint process is not Markovian. Thus the Liouville master equation cannot be used to obtain an exact formulation for particle transport in a Markovian mixture when scattering is present. Our overall conclusion is that none of the techniques which produced simple, exact, and general results for the purely absorbing Markovian problem will yield simple, exact, and general results when scattering is extant, even if the mixing is taken as Markovian. However, an exact formalism is available (16,62-66) for treating a restricted class of statistical transport problem including the scattering interaction. This formalism treats planar geometry problems (so that the stochasticity consists of alternating slabs of random thicknesses of the two materials), and then Eq. (1) is written

where µ, is the cosine of the angle between the direction and the axis. We assume that Eq. (124) holds on the finite interval We assign nonstochastic boundary conditions to Eq. (124) given by

The class of planar geometry problems we can treat consists of those for which the total cross section the scattering kernel and the internal source S obey common statistics, in a sense to be made precise shortly. However, these common statistics of the mixture in which the particles transport are allowed to be completely general. Explicit results can be obtained in the case of a binary mixture obeying homogeneous Markovian statistics. The key to the analysis is the introduction of an optical depth variable defined by In terms of

where

Eq. (124) becomes

TRANSPORT THEORY IN DISCRETE STOCHASTIC MIXTURES

Equation (128) holds on the interval optical depth of the planar system given by

67

where T is the energy dependent

Equations (125) and (126) transform to

which give the boundary conditions on Eq. (128). To proceed, we now assume that and all obey the same statistics in the sense that and are nonstochastic. Then the transport problem described by Eqs. (128), (133), and (134) is only stochastic through the optical depth variable and the optical size of the system T. The statistics of the problem, then, are entirely described by the joint probability density function defined such that is the probability that, for a given geometric position and a given geometric system thickness the position corresponds to an optical depth lying between and and the thickness corresponds to an optical depth lying between T and The ensemble average of the intensity which we denote by is thus given by

In writing in Eq. (135), we have indicated that it depends upon the optical thickness of the system, T, as well as the usual independent variables and µ. The above formulation for is quite general in that it treats arbitrary statistics (given that and are nonstochastic). All of the statistical information is embodied in the joint probability density function From the transport point of view, this joint probability density function is assumed known. In the special case of binary homogeneous Markovian statistics, one has the separable form, if we denote the two materials making up the binary mixture by subscripts 0 and 1,

This form for arises from the no-memory character of Markovian statistics. Here represents the probability of finding material at any point in the system. In terms of the mean slab thicknesses, of the alternating slabs of the two materials making up the planar system, we have [see Eq. (53)]

For homogeneous Markovian statistics, these slab thicknesses are exponentially distributed, with mean for material and the Markovian assumption is equivalent to a classical Poisson process [see Eq. (24) with The function in Eq. (136) is the probability density function defined such that is the probability that the planar system has an optical depth lying between

68 and material

G. C. POMRANING given that the geometric depth is It is given in the usual way by

and given that the point

lies in

where is the cumulative distribution function corresponding to For the case of homogeneous binary Markovian statistics being discussed, the functions are known in closed analytic form (16,64). They are related to the cumulative distribution function for the optical depth, as defined above Eq. (59), according to

With it is clear that for and for [see Eq. (61)]. For a comparison of Eqs. (62) and (139) establishes the expressions for the Thus in the special case of binary homogeneous Markovian statistics, Eq. (135) becomes

and the are explicitly known as discussed above. Equation (140) simplifies considerably if the problem under consideration is a halfspace In this case, the optical thickness of the system, T, is infinite for each realization of the statistics, and at any is the same for each realization. That is, for halfspace problems the solution in optical depth space, which we now denote by depends only upon the single random variable Thus in Eq. (140) can be taken outside of the inner integration, and the remaining integration over T yields unity, since is a probability density function, normalized to a unit integral. Thus for a halfspace problem, Eq. (140) reduces to

where This function is the probability density function for the optical depth for a given geometric distance For homogeneous binary Markovian statistics, an explicit result for has been given in the second section of this article [see Eqs. (61) through (64)]. Equation (141) was applied (16,64-66) to several classical monoenergetic halfspace transport problems with isotropic scattering, both in planar geometry as in Eq. (124) and in rod geometry (67), where particles are constrained to move along the axis. These problems were the Milne, albedo, and emissivity (a constant source halfspace) problems, with the mean number of secondaries, the same for each component of the mixture, as required in Eq. (141). The emissivity problem involves a source given by where B is nonstochastic. Thus a nonstochastic quantity, again as required in Eq. (141). This form of the source is appropriate in the radiative transfer context, where B is the Planck function. The infinite medium Green’s function problem, with an isotropic source, was also treated by Eq. (141), since this problem can be considered as two adjoining halfspace problems. In all of these problems, the solution is a linear combination of the discrete and continuum Case modes (68) (plus a constant in the emissivity problem), which are exponentials in Thus the key to

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69

obtaining solutions for these problems is to find the ensemble average of an exponential in Such an average is easily computed for homogeneous binary Markovian statistics. We consider a pure exponential in optical depth space, with being a constant, and ask for the ensemble average (expected value) of this exponential at a geometric point (16,64). This analysis generalizes the analysis given earlier [see Eqs. (104) through (110)], where was taken as minus one. If we denote the ensemble average of by we have

where is the same function as in Eqs. (141) and (142). An integration of Eq. (143) by parts gives

with given by Eq. (62). We now define variable, of We find

as the Laplace transform, in the

To proceed, we change integration variables from to according to Eq. (63) with replaced by and interchange orders of integration [see the discussion under Eq. (106)]. In an analogous manner that allowed Eq. (106) to be written as Eq. (109), we find that Eq. (145) reduces to the simple result

where

with

and

are given by Eq. (92), and

is defined as

given by Eq. (38). The Laplace inversion of Eq. (146) yields

where

and the

are given by

For Eq. (148) agrees with our earlier result, Eq. (110) with It is easily shown that both and are real for real. Additionally, for one can show that both and are negative. For is always positive, but the sign of is the same as the sign of Thus we see that a single exponential in optical depth space bifurcates into two exponentials in physical space. Since the solutions of the halfspace problems under consideration are linear combinations of exponentials in space, the corresponding solutions for the ensemble-averaged intensities are, according

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to Eqs. (141), (143), and (148), simply sums of exponentials in space, with each exponential in yielding two exponentials in The formalism in this section has also been applied (63) to the purely scattering, monoenergetic, finite rod geometry albedo problem, defined on the interval by the equations,

Here is the intensity in the positive direction, and is the intensity in the negative direction. In optical depth space, Eqs. (151) and (152) become

We seek the transmission through the rod, which for each physical realization of the statistics is given by The solution of Eqs. (153) and (154) is simply

To find the ensemble-averaged transmission, it is clear that we need to average the right-hand side of Eq. (155) over the probability density function for the optical thickness T, given a fixed geometric distance Although this is apriori clear, we can also show it from our general result given by Eq. (140). If, as a special case of Eq. (140), we consider the point in this equation, we then must have

which expresses the fact that at the point the optical depth variable is, with certainty, equal to the optical depth variable T. Setting and using Eq. (156) in Eq. (140), the integration over is trivial, and we find, writing

with

again given by Eq. (142). For homogeneous binary Markovian statistics, is known explicitly and is given by Eqs. (61) through (64). We use Eq. (157) in the monoenergetic rod geometry setting to obtain the ensemble-averaged transmission, for the problem under consideration. That is, making use of Eq. (155) for we have

To evaluate this integral, we note that Eq. (158) can be written as

where

is the ensemble average of

i.e.,

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71

The reason for rewriting in this way is that is known and simple. The ensemble average indicated in Eq. (160) is given by Eq. (148) with and If we write and change integration variables in Eq. (159) according to one finds

which is a form convenient for numerical integration. Equation (161) was used (63) to compute representative numerical results, and these results were found to be in complete agreement with independent invariant imbedding (67) results of Vanderhaegen and Deutsch (62). This invariant imbedding approach converts the boundary value transport problem for the intensity into an initial value problem for the transmission Since the statistics of the binary mixture are taken as Markovian, one then has a joint Markov process in this invariant imbedding setting. Thus the Liouville master equation is valid, and this was the method used by Vanderhaegen and Deutsch (62) to solve this stochastic rod geometry transmission problem. For small and large values of one can evaluate Eq. (161) analytically (63). One finds for small

and for large

Thus for both short and long rods, the ensemble-averaged transmission depends only upon the ensemble-averaged cross section; all other aspects of the statistics, such as the correlation length, are irrelevant. This prompted Vanderhaegen and Deutsch (62) to define an effective cross section, for this problem by the equation

Now, the Schwartz inequality gives

and Eqs. (132) and (155) yield

We thus deduce that Since for and [compare Eqs. (162) and (163) with Eq. (164)], either for all or goes through a minimum for one or more values of It is the latter of these two alternatives that is the case, as has been established by numerical results (62,63). The fact that goes through a pronounced minimum for a single value of has been described as a transmission window which may have significance in certain physical applications (62). We will use the existence of this transmission window as a guide in the development of a general Markovian stochastic model later in this article.

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MARKOVIAN MIXTURES WITH SCATTERING, LOW-ORDER MODELS

In this section, we summarize available results concerning low-order Markovian models for in a general setting with scattering present. By this we mean in a setting beyond the restricted class of scattering problems discussed in the last section. Thus, we consider general three-dimensional transport in a binary Markovian mixture with and all independent parameters. In particular, and are allowed to be stochastic (dependent upon the index ), in contrast to the treatment in the last section. By low-order models, we mean models which consist of two coupled transport equations for the two defined as the conditional ensemble average of conditioned upon position r being in material The overall ensemble average of the intensity, is then given by a weighted average of the according to Eq. (84), with the variable in this equation replaced by the triplet We have argued in some detail in the last section that the Liouville master equation is not valid as a treatment of stochastic transport in a Markovian mixture when scattering is present. Nonetheless, it has been suggested (2,16,58-61) that the use of the master equation, while not rigorous, should produce a useful and simple, albeit approximate, model of particle transport in stochastic mixtures. As we shall see, this resulting model gives correct results in all known limiting cases and is robust away from these limits. The approximation made in applying the master equation in the scattering problem is that the inscattering term in Eq. (1), namely

can be treated in the same way as the internal source term This is clearly an approximation since the inscattering and internal source terms are completely different in character. The source S is a two-state discrete random variable, assuming a known value if the system is in material at space point r. Although the scattering kernel in Eq. (168) is a similar two-state discrete random variable, the intensity is not. Given that the system is in material at a given r, the intensity is a continuous random variable and not deterministic as is Thus the approximation in using the master equation in the presence of scattering is to replace the continuous random variable in Eq. (168) with a purely discrete two-state random variable. These two states are reasonably taken as and Thus the approximation is to write the inscattering term, given that position r is in material as

We can then write the Liouville master equation for the joint probability of finding material at position r, and simultaneously finding a solution for the intensity lying between and Multiplying this equation by and integrating over just as in the purely absorbing treatment, we then find two coupled equations for the two given by

Here and have the same meaning as earlier; is the probability of point r being in material and is the Markov transition length from material to material By

TRANSPORT THEORY IN DISCRETE STOCHASTIC MIXTURES

73

introducing the change of dependent variables given by Eq. (87), we find that Eq. (170) can be written equivalently as

Here and have been previously defined by Eqs. (91) through (93), and the three new parameters in Eqs. (171) and (172) related to the scattering process are given by

As pointed out earlier, the stochastic transport model given by Eq. (170), or equivalently Eqs. (171) and (172), is an approximate model because of the presence of the scattering interaction. The approximation used is embodied in Eq. (169). We now discuss a second derivation of this model, based on stochastic balance methods (1-4), which gives an alternate viewpoint concerning the approximate nature of this result. Following Adams et al. (2), we introduce the characteristic function defined as

We multiply Eq. (1) by integrate the result over an arbitrary volume, and let this volume shrink to zero. After some algebra, the result is (2)

where

It is clear that is simply the probability of finding material at position r, and is the ensemble-averaged value of the intensity, given that position r is in material The other two quantities in Eq. (177) are and The intensity is also a conditional ensemble average of the intensity, conditioned upon the position r lying at an interface between two materials, with material to the left (the vector points from left to right). The quantity is a purely geometric/statistical quantity defined by

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G. C. POMRANING

where V is an arbitrary volume surrounding the point r, and denotes the portion of the interfaces within V for which where is a local normal unit vector pointing out of material At this point in the development, no approximations have been made. Equation (177) is exact for arbitrary binary statistics (in particular, it is not restricted to Markovian statistics). However, a comparison of Eq. (177) with the master equation model, Eq. (170), meant as an approximate treatment of transport with scattering in a Markovian mixture, shows that these two equations have very nearly the same structure. Also, Eq. (177), while exact, is clearly a formal result in that it represents two equations for the four unknowns and A closure is needed to make this formalism useful. We recall that is an ensemble average of the intensity at interface points for which To close Eq. (177), we approximate this average by using an analogue of upwind differencing in the numerical analysis of hyperbolic equations. Specifically, we replace in Eq. (177) with the ensemble-averaged intensity within material With this approximation, Eq. (177) becomes a closed set of two equations for the two unknowns and given by

If one identifies the geometric quantities in Eq. (181) with the Markov transition lengths in Eq. (170), these two equations are identical. Thus straightforward particle balance considerations have led to an alternate derivation of the master equation model. In this alternate treatment, the sole approximation made is very clear, namely

which in words says that a restricted ensemble average is approximated by an unrestricted (except that r be in material ) ensemble average. We also note that this master equation/balance model of stochastic particle transport can be obtained in yet two additional ways, one by using the techniques of reactor noise analysis (48), and the other by assuming that each particle track is independent of prior tracks (49). Both of these derivations, due to Sahni, were restricted to Markovian mixing statistics, as is the master equation considerations, but the present derivation shows that this model, while approximate, applies to arbitrary binary mixing statistics. The details of the statistics are embodied in the evaluation of the contained in Eq. (181), and defined by Eq. (180). We also note that this model is readily extended to a immiscible mixture containing an arbitrary number of components (55). Finally, we mention the work of Titov and colleagues (20,25), who developed a model for monoenergetic stochastic transport in a binary Markovian mixture with the framework of integral transport theory. To obtain a usable result, a closure was introduced into certain stochastic balance equations. This closure is exact for the purely absorbing problem, but approximate when scattering is present. The resulting integral equation model has been shown (22,23) to be entirely equivalent to the model discussed here, and given by Eq. (170). We point out that Eq. (181) is known to be exact for purely absorbing binary mixtures obeying Markovian statistics. It is clear from the master equation considerations that this must be the case since, with no scattering, particle transport in a Markovian mixture is a joint Markov process and thus the master equation applies with no approximations. It is also clear within the context of the present derivation why this

TRANSPORT THEORY IN DISCRETE STOCHASTIC MIXTURES

75

is so. The approximation of replacing the interface ensemble-averaged intensity with the volumetric-averaged intensity, i.e., setting is in the case of no scattering not an approximation; it is an exact replacement. This observation follows from the fact that for purely absorbing transport the solution of the transport equation at any spatial point depends only upon the optical depths from the point in question to the system boundary and source points. For Markovian statistics, the ensemble-averaged optical depth between a boundary or source point and the field point is independent of whether the average is taken over all realizations for which the field point is an interface, or if the average is taken over all realizations for which the field point is not an interface. This is related to the equalities given by Eq. (47) which show that for general inhomogeneous Markovian statistics certain probabilities are independent of whether a point is identified as an interface point or not. Further, in the Markov case the geometric quantity defined by Eq. (180) must, in fact, be identical to the Markov transition length The argument leading to this conclusion is that this must be so in the no-scattering case, since in this instance the present model is exact and must coincide with the exact master equation treatment. But since and are both purely geometric quantities dependent only upon the material mixing, the equality of and is independent of the scattering, or lack thereof, of the transport process. Thus for Markovian mixing, Eq. (180) must predict

where the the Markov transition lengths, are defined in the usual way such that is the probability of transition from material to material in a distance Finally, for spatially (but not necessarily angularly) homogeneous statistics, the parameter is simply the mean (average) chord length in material in the direction under consideration for all statistics, not just Markovian (2,16). Let us now address the accuracy of this model of stochastic transport. It is easily argued that the correct behavior is exhibited in all simple limits. If both functions in Eq. (181) [or in Eq. (170)] increase without bound, then the coupling terms vanish, and the equations for and decouple as they should. If one of the (or ), say is vanishingly small and the other, is nonvanishing, we then have and hence Thus and addition of the two equations in Eq. (181) shows that satisfies the nonstochastic transport equation characteristic of material 1. Finally, it is easily shown (16,54,56), using asymptotics, that if both and are vanishingly small, then one obtains a nonstochastic equation for characterized by ensemble-averaged parameters and This is the correct limit on physical grounds. Thus in these limiting cases, Eq. (181) predicts correct results for all statistics. Away from these cases, comparisons with benchmark calculations shows that this model is robust, and predicts to an accuracy of the order of ten percent (2,53,69,70 ) in the case of homogeneous Markovian statistics. The benchmark results, for monoenergetic, finite thickness albedo problems in both rod and planar geometry, were obtained by using Monte Carlo methods to generate a physical realization of the statistics (exponentially distributed chord lengths), solving the resulting transport problem numerically, and averaging over a large number of such calculations. Concerning accuracy, we should also point out that the Markovian model given by Eq. (170) is, in fact, exact for one class of problem including scattering. This class consists of problems with a very peaked, in the forward direction, scattering kernel in a thin (as measured in transport mean free paths) system. Under these conditons, backscattering can be ignored as negligibly small, and thus the particles cannot traverse the same statistical distribution of materials more than once. The

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neglect of backscattering also converts the boundary value transport problem to an initial value problem (with space playing the role of time). The intensity is essentially zero in the backward directions, and thus the entire process, the Markovian mixing and the particle transport, is a no-memory process with a unique time arrow (again with space playing the role of time). This class of problems arises in charged particle and photon transport in beam physics associated with cancer therapy (29-32). In an attempt to obtain even simpler models of stochastic transport in discrete mixtures, the model described here by Eqs. (170) and (181) has been subjected to further simplying analyses. These analyses include the development of several asymptotic limits (26,30-32,54-57,71-74) as well as and diffusive descriptions (75,76) and flux-limiting algorithms (77,78). Most of this analysis was done in the monoenergetic, isotropic scattering setting, with isotropy assumed for both the sources and the Markov transition lengths (both are independent of ). In this case, Eq. (170) is written, with full display of all arguments,

where is the macroscopic scattering cross section for material In one simple piece of asymptotic analysis (54,56), the in Eq. (184) were assumed vanishingly small, which led to the expected asymptotic limit

A second asymptotic limit considered in these same papers (54,56) is that corresponding to a small amount of large cross sections, large source, material admixed with a large amount of small cross sections, small source, material. In this case, one finds the renormalized transport equation given by

where the effective source and cross sections are given by

Here and are defined by Eqs. (91) through (93) and (173) through (175). Equation (186) is robust in the sense that all three effective parameters defined by Eqs. (187) through (189) are always nonnegative. Equation (188) for and Eq. (187) for in the absence of scattering agree with our previous purely absorbing results [see Eqs. (117) and (122)] which were obtained in a completely different manner. The accuracy of Eqs. (170), (185), and (186), as well as the accuracy of other models of stochastic transport in a homogeneous binary Markovian mixture, has been addressed by representative numerical calculations (69).

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The analysis leading to Eq. (186), involving effective parameters, has been generalized to an entire family of reduced transport descriptions for the case of an arbitrary number of components in the immiscible mixture (55,57). A generalization for binary mixtures accounting for anisotropic scattering as well as anisotropic statistics ( dependent upon ) has also been reported (26). Another generalization of the analysis leading to Eq. (186) has been published for highly peaked scattering (32). In this case, the differential Fokker-Planck operator is a valid description of the scattering process. Within this Fokker-Planck framework, in the arena of beam transport, this same asymptotic limit was considered, namely the limit of a small amount of opaque material admixed with a large amount of transparent material. A third asymptotic limit has been considered for problems involving the FokkerPlanck scattering operator. This is the small correlation length limit (30-32), which gives first order corrections in to Eq. (185). Two of these results are in planar geometry, one for monoenergetic, purely scattering transport (30), and the other including absorption and energy dependence (31). The third contribution (32) is again for monoenergetic transport in a purely scattering medium, but in full three-dimensional generality. This same asymptotic limit of a small correlation length has been considered for anisotropic scattering described by the integral scattering operator (26). Several papers are available (54,56,71-74) describing still another asymptotic limit, that which leads to a diffusive description of stochastic transport. The first of these (54,56) scales and in Eq. (184) as where is a formal smallness parameter, and scales the gradient term as The are unscaled, and hence taken as O(1). For nonstochastic transport, it is well known that such scalings reduce the transport equation to a diffusion equation. In the stochastic setting (54,56), one obtains two coupled diffusion equations for and the scalar fluxes given as the integrals over solid angle of and Subsequent work (71) generalized this analysis to include various scalings of the Depending upon the assumed scaling, one obtains either two coupled diffusion equations for the or a single diffusion equation for the ensemble-averaged scalar flux. This analysis was generalized still further (72) by considering a stochastic mixture consisting of an arbitrary number of components. Finally, for binary mixtures, asymptotic diffusive descriptions have been obtained in the presence of anisotropic statistics, and independent scalings for the two (73,74). Other diffusive, but not arising from asymptotic analyses, descriptions corresponding to Eq. (170) have been reported (75-78). These are the and spherical harmonic approximations and an asymptotic (in the sense of Case discrete modes, and not in the sense of asymptotic analysis) treatment (75,76), and flux-limited diffusion theories (77,78). Flux-limiting means, in neutron transport theory terminology, that the diffusion coefficient in Fick’s law of diffusion is nonlinear, in just such a way that the magnitude of the current can never exceed the scalar flux, no matter how steep the spatial gradients. We close this section by pointing out recent work (5,6) which suggests two alternate closures to the simple upwind strategy given by Eq. (182). These are, within the context of monoenergetic transport with isotropic scattering,

with

and

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with Here and all other symbols are as previously defined. The qualitative difference between these two closures is that Eq. (192) mixes the various directions, through the scalar flux, term. Such mixing is not present in Eq. (190). We note that both of these closures are exact in the absence of scattering. They both reduce to Eq. (182) which is an exact closure for The genesis of these closures is the demand that the deep-in behavior for certain rod geometry halfspace albedo problems be predicted exactly by the models arising from using either Eq. (190) or Eq. (192) as the closure for Eq. (177). For homogeneous Markovian statistics with the and independent of space, this exact deep-in behavior for the ensemble-averaged intensity in the two rod geometry directions, is given by decaying exponentials, i.e.,

where the and are constants. The parameter in two cases. The first of these is when section. In this case we have (6,63)

is known in simple analytic form the case treated in the last

where is the nonstochastic mean number of secondaries per collision. The second class of problems for which is known is the small correlation length limit for arbitrary and This result is (5)

Equation (196) is a generalization of a result first given by Vanderhaegen et al. (79), in which but That is, the two materials have a common total cross section, but are allowed a different scattering cross section. The generalization given by Eq. (196) relaxes the Vanderhaegen et al. constraint that The accuracy of these two closures has been discussed in connection with representative numerical calculations (5,6). The overall conclusion is that the closures given by Eqs. (190) and (192) are generally not inferior to the classic closure given by Eq. (182), and in some cases are significantly better. It was further concluded that it is difficult to choose between Eqs. (190) and (192). Both closures seem to yield about the same accuracy, over a wide range of problems. MARKOVIAN MIXTURES WITH SCATTERING, HIGHER-ORDER MODELS

In the last section, we discussed three low-order models describing stochastic transport in a binary Markovian mixture. These three models were based upon the stochastic balance equation given by Eq. (177), combined with three different closures as given by Eqs. (182), (190) and (192). These models are referred to as low order in that they

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are based upon introducing a closure for into the lowest order stochastic balance equation given by Eq. (177). Recent work (3,4,7,8) has suggested that better accuracy may be achieved by going to higher order. Specifically, Eq. (177) is retained in its exact form, with no closure introduced. A stochastic balance equation for the interface intensity is then derived. This equation contains, in addition to still higher (more restricted) stochastic ensemble averages, and a closure is introduced for these terms. This then yields a model consisting of four coupled transport equations for the four unknowns and The quantity of interest, the ensemble-averaged intensity is still given by All reported results (3,4,7,8) have been within the context of monoenergetic transport with isotropic scattering, in either rod or planar geometry. In this case, the exact stochastic balance results given by Eq. (177) is written, with

Here is the scalar flux corresponding to the angular flux and the streaming operator is written in rod geometry and in planar geometry. Two higher order models have been proposed (3,4,7,8). The first of these (3,4) combines Eq. (198) with a closed set of equations for the given by

Here H is the collision operator, written in planar geometry as

with an analogous expression in rod geometry. Equation (199) was derived by introducing a closure for the higher statistical averages which occur in the balance equation for The arguments leading to this closure are available in the literature (3,4), and will not be given here since they are somewhat lengthy. Equations (198) and (199), which have been referred to as the interface model, represent a closed system of four equations for the four unknowns and Since Eq. (199) is only an approximate description for the interface average intensity the model taken as a whole is generally an approximation for computing However, in the special case of it is easily seen from Eqs. (199) and (200) that we have Thus this interface model is exact for purely absorbing problems. This interface model has been numerically tested in both rod geometry (3) and planar geometry (53). A remarkable feature of this model is that it consistently predicts very accurate reflection and transmission results for the finite geometry albedo problem, even while giving inaccurate and qualitatively wrong spatial intensity profiles (8,53). The reason for such accurate estimates of transmission and reflection results has been attributed to the fact that the interface model has the capability of predicting the transmission window for purely scattering problems (8). This transmission window phenomenon has been discussed earlier in this article [see the discussion starting with Eq. (151)]. In particular, it was shown that as defined by Eq. (164) approaches as (the rod length) approaches zero and as increases without bound. Between and goes through a pronounced minimum. The interface model

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exhibits this same qualitative behavior. It has been shown (63) that the classic loworder model arising from using Eq. (182) as the closure for Eq. (198) predicts for and decreases monotonically as increases. Further, the other two loworder models discussed in the last section, arising from using Eqs. (190) and (192) as closures, also do not exhibit this transmission window. This follows from the fact that both models predict, for purely scattering problems, and it then follows that for all values at Thus the conclusion is that none of the three low-order models predict the transmission window, but that the interface model does. From this observation, it is reasonable to assert that among these four models, the interface model is the most accurate for dealing with purely scattering problems (8). Since the interface model reduces to the classic closure model based upon Eq. (182) for purely absorbing problems, and this model is exact for such problems, it is reasonable to expect that the interface model will predict accurate reflection and transmission results for problems which are dominated by either absorption or scattering. Based upon these arguments, it has been suggested (8) that any good model of stochastic transport in binary Markovian mixtures should possess two characteristics. First, it must be an exact description for purely absorbing problems, thus having a good possibility of being accurate for absorption dominated problems. Second, it should be able to predict, at least qualitatively, the transmission window for purely scattering problems, thus having a good possibility of being accurate for scattering dominated problems. Only models possessing these two features can be expected to do well for general problems, neither absorption nor scattering dominated. Based upon this reasoning, a modification to the interface model has very recently been suggested (8). The genesis of this model is a qualitative argument as to why the interface model is not robust. This lack of robustness manifests itself in its prediction of unphysical spatial intensity profiles, especially when one of the two materials is thick and purely scattering (8,53). In simple terms, this unphysical behavior can be explained as follows. Let us denote by the coupling terms in the exact balance equation for as given by Eq. (198). We have

In the interface model, the are computed from Eq. (199) and may involve considerable errors since Eq. (199) is only an approximation for the interface intensity After is approximated by Eq. (199), the equation for as given by Eq. (198) is totally decoupled from the equation for with If we further decompose as the sum of its exact value, and its error, i.e.,

then the accuracy of is determined solely by which may be large compared with When the angular integration of is nonzero, acts as an additional source (positive or negative) of particles in the equation. If the problem under consideration is such that it is very sensitive to source terms, as in a thick purely scattering material, it will also be sensitive to Such sensitivity of to the errors in is not observed in the low-order models discussed in the last section, where the equations for and are coupled. This coupling reduces the effects of on the accuracy of the by the following argument. The preferred way to improve the interface model is to develop new equations for the such that the are significantly smaller. No suggestions have been reported in the literature in this regard. A second way to proceed is to make the model less

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81

sensitive to the by coupling the equations for and When and are defined by two coupled transport equations, the error in depends upon both and which have opposite signs according to Eqs. (201) and (202). Thus, the effects of the errors and may cancel each other to some extent in the computation of If so, the solution for becomes less sensitive to the errors Based upon this argument, the closure given by

has been proposed (7,8), with given by Eqs. (199) and (200) with replaced by in these equations. To completely specify this model, one must specify and We first note that Eq. (203) reduces to the classic closure given by Eq. (182) by setting and We also note that the interface model is recovered by setting and In this sense, the closure given by Eq. (203) is a linear combination of the classic closure, which is exact for purely absorbing problems, and the interface model, which is reasonably accurate for purely scattering problems. To specify and in general, the small correlation length limit of this model was considered for the halfspace rod geometry albedo problem (7,8). For homogeneous statistics with the and independent of position, it was found that is given by, to first order in where the

and

are constants, with

given by

One relationship determining the closure parameters and was obtained by equating Eq. (205) to the exact result for in this small correlation length limit, as given earlier by Eq. (196). This yields

Obviously, a second equation involving and is needed to uniquely determine these two parameters. The proposers of this model were unable to find a usable second constraint on and and thus they proceeded in a physically motivated way to specify the parameters and solely from Eq. (206). To do this, they examined Eq. (206) in two limiting cases. Consider first purely absorbing mixtures for which In this case, the right-hand side of Eq. (206) is unity, and this leads to It is easily seen from Eqs. (198) and (199) [with replaced with in Eq. (199)] that and then the closure given by Eq. (203) reduces to the classic closure which is exact in this case, as long as Thus, the closure given by Eq. (203) is always exact for purely absorbing problems provided that and satisfy Eq. (206). That is, a second equation for and is unnecessary and the separate determination of and is irrelevant in this limiting case. Consider now the purely scattering problem for which In this case the right-hand side of Eq. (206) is zero. This leads to with undefined. As discussed earlier, the interface model, corresponding to and is better than any of the low-order models discussed in the last section for purely scattering problems, in that it qualitatively predicts the transmission window. Thus, one would want the closure given by Eq. (203) to reduce to the interface model in this case. That is, one would desire an expression for α which predicts for purely scattering problems is predicted by Eq. (206) for purely scattering problems].

82

G. C. POMRANING With this in mind, Eq. (206) can be rewritten

which suggests (7,8)

Then the parameters

and

follow as

The use of Eq. (209) in Eq. (203) explicitly gives the closure defining this model of stochastic transport in a binary Markovian mixture. The justification for Eq. (209), which must be described as adhoc, is two-fold. First, it leads to very simple and thus appealing expressions for the closure parameters; probably Eq. (209) is one of the simplest definitions for and which satisfy Eq. (206). It is easily seen that both and are positive and range from zero to unity. Thus the closure given by Eq. (203) interpolates between the classic low-order model based upon Eq. (182) and the interface model. Second, this closure has the correct limiting values. On the one hand, Eq. (209) yields and for purely absorbing problems, and thus the present closure reduces to Eq. (182), the classic exact closure, in this case. On the other hand, Eq. (209) predicts and for purely scattering problems, and thus the present closure reduces to the interface model, which is reasonably accurate for this class of problems. Hence the closure parameters given by Eq. (209) should predict quite good results for problems close to these two limiting cases (pure absorption and pure scattering). A final point to be made is that, in addition to the limiting cases of no scattering and no absorption, the present closure parameters given by Eq. (209) simplify for one additional class of problems. Specifically, this higher-order closure reduces to a loworder closure we discussed in the last section for problems with a common cross section, i.e., In this case Eq. (209) predicts

and the closure given by Eq. (203) reduces to

with given by the simple radical in Eq. (210). Equation (211) is identical to Eq. (190) of the last section, since for Eq. (191) also predicts as given by Eq. (210). Thus for this class of problems this higher-order model reduces to the low-order model given by Eqs. (190) and (191). This gives further confidence in this higher-order closure since Eqs. (190) and (191) have been shown to be very accurate when (5,6,79). Representative numerical calculations (7,8) have demonstrated that the closure given by Eq. (203), with and defined by Eq. (209), is quite accurate over a wide range of problems. In particular, this model of stochastic transport in a binary Markovian mixture corrects the unphysical behavior (8,53) associated with the interface model, and is generally more accurate than any of the three low-order models.

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83

NON-MARKOVIAN MIXTURES All of the discussion thus far in this article has assumed that the statistics of the mixing is described by a Markov process. In this section, we summarize the few results that are available for more general statistics. This analysis uses the integral, rather than the integro-differential, equation as the underlying description of particle transport. The stochastic formalism used is the theory of alternating renewal processes. In particular, an exact model involving four integral renewal equations is derived which describes the ensemble-averaged intensity for particle transport in a purely absorbing medium composed of two randomly mixed immiscible materials. The use of renewal theory to describe stochastic transport in a binary mixture was first suggested by Vanderhaegen (38), and was subsequently considered by other authors (1,16,33,50-53,70). We begin by considering the purely absorbing problem (33) with a nonstochastic boundary condition as described by Eqs. (68) and (69). The three probabilities that enter into the renewal analysis are the probability that a point on the line is in material the probability that the interval is in material given that is the right-hand boundary of material and the probability that the interval is in material given that is anywhere in material The probabilities have been defined and discussed earlier [see Eqs. (4), (6), and (8)]. Another quantity that enters into the renewal analysis is the nonstochastic solution of Eq. (68). If we let denote the solution of Eq. (68) for at position given the boundary condition that the solution is at position and given that the entire interval consists of material we have [see Eq. (72)],

Here

is the optical depth between points

and given by

To derive the renewal equations, we first define the ensemble average of for a subset of all physical realizations of the statistics on the line. This subset corresponds to those realizations such that the point is an interface between two materials. Specifically, we define as the ensemble average of for this subset at an interface with material lying to the left and material lying to the right. This quantity is the same as introduced earlier in Eq. (177). We now consider the subset of all possible physical realizations of the statistics such that the point of interest is an interface between materials, with material to the left of this interface and material to the right. Within this subset, all physical realizations can be placed in one of two categories, namely: (1) the interval is made up entirely of material and (2) one or more material interfaces are extant in the interval Considering first category 1, the solution for for each statistical realization within this category is simply given by where the function is given by Eq. (212). Since this solution is nonstochastic, the ensemble average of this solution is clearly given by the same expression. Further, from the definition of as given by Eq. (8), it is clear that the probability of category 1 is simply Thus if we define as the contribution of category 1 to the ensemble-averaged solution at given that the position is an interface between materials with material to the left of this interface,

84

G. C. POMRANING

we have We next consider category 2 and take the first material interface to the left of position to occur at position The solution for each such physical realization of the statistics is given by We need to ensemble average this result over all statistical realizations with the first interface to the left of occurring at Since in is the only stochastic quantity in this expression, and since it occurs linearly [see Eq. (212)], the ensemble average of this quantity is with A subtlety in this argument has been discussed in the literature (1,51), having to do with the class of statistics for which this argument holds. This class is that for which the material distribution to the left of any point is independent of that to the right. Statistics defined by material lengths chosen from a distribution as in Eq. (16) are in this class. From the definition of as given by Eq. (8), the probability of the first interface to the left of position occurring at position is given by

Thus, if we define to be the contribution of category 2 to the ensemble-averaged solution at given that is an interface between materials with material to the left, and given that is the first interface to the left of we have

Since these two categories of the physical realizations of the statistics are exhaustive, the ensemble-averaged solution at position denoted by since we are considering to be an interface position with material to the left, is the sum of these contributions. The sum over the various possibilities for in Eq. (216) takes the form of an integral, and to this integral we must add Eq. (214). Thus, we are led to the renewal equations

Equation (217), for and 1, represents a closed set of two integral equations which, in principal, can be solved for the two unknowns once the statistics, as represented by the functions have been specified. However, the unknown corresponds to the ensemble-averaged solution for a subset of all of the physical realizations of the statistics, namely that subset for which the position is the right-hand endpoint of material To obtain the ensemble average over the entire set of physical realizations, we need to derive additional renewal equations involving the defined as the ensembleaveraged intensity averaged over all realizations such that the point is anywhere in material To do this, we consider the point to be in material and place, as before, all physical realizations with position anywhere in material into one of two categories, namely: (1) the interval is made up entirely of material and (2) one or more material interfaces are extant in the interval In this case, the relevant statistical quantity is as defined by Eq. (6), since this probability is conditioned only upon position being in material Considering first category 1, we use identical arguments that led to Eq. (214) to find in this case

TRANSPORT THEORY IN DISCRETE STOCHASTIC MIXTURES

85

where we have defined as the contribution of category 1 to the ensemble-averaged solution at given that position is anywhere in material Considering now category 2, we take the first material interface to the left of position to occur at position The solution for any physical realization of the statistics under these conditions is given by Ensemble averaging this solution over all statistical realizations with the first interface to the left of occurring at gives for The ensemble average with occurs in this expression since the position has been designated as an interface with material to the left, and the ensemble average of at such a position has previously been defined as From the definition of as given by Eq. (6), the probability of the first interface to the left of position occurring at position is given by

Thus we have

where we have defined as the contribution of category 2 to the ensembleaveraged solution at given that is anywhere in material and that is the first interface to the left of Summing the contributions to as given by Eqs. (218) and (220), we have the renewal equations.

for

and 1. Once the are obtained from the solution of Eq. (216), these solutions can be simply inserted into Eq. (221) and, in principle, the right-hand side of this equation evaluated to determine the The expected value (ensemble average) of the random variable which we denote by is then found by multiplying the by the probability that position is in material and forming the sum over namely

with defined by Eq. (4). All of the statistical information needed in this renewal analysis, namely the and is related to the fundamental statistical quantity as discussed in the second section of this article [see Eqs. (2) and (9) through (11)]. For Markov statistics, we have previously shown that [see Eq. (47)]

Then a comparison of Eqs. (217) and (221) establishes

a result we used earlier [see Eq. (182) and the subsequent discussion] as a closure, and noted its exactness in the absence of scattering. We note that the use of Eq. (212) for in Eqs. (217) and (221) leads to terms involving double integrals, over and By interchanging orders of integration in these terms, performing the integration over first, and then relabeling the remaining integration over as an integration over

86

G. C. POMRANING

one finds that Eqs. (217) and (221) can be written in an equivalent, but somewhat simpler and more explicit, form. We find

Renewal theory as described here for purely absorbing systems has been used to analyze transmission through a binary stochastic mixture described by homogeneous statistics of the form given by Eq. (16) for various functions (50). Specifically, numerical results were obtained for several unimodal distributions, and it was concluded that the mean and variance of the chord length distributions for the two materials constitute sufficient information from the statistics to obtain a good approximation for whereas the mean alone is not sufficient information. It was emphasized that this conclusion was based upon a study of unimodal distributions, and may not hold for chord length distributions with more structure. Analytic analyses of these purely absorbing renewal equations, for homogeneous statistics have also been reported (52), using both singular eigenfunction and integral equation methods. Benchmark results, in rod and planar geometry and including the scattering interaction, are also available in the literature (53,70). These numerical results were obtained by sampling from various chord length distributions as defined by Eq. (16) to obtain a physical realization of the statistics, numerically solving the transport equation for this realization, and averaging the results over a large number of such calculations. Finally, we mention an attempt to use variational methods to obtain accurate estimates of the ensembleaveraged intensity for non-Markovian binary mixtures (80). This endeavor was only modestly successful. A very general and simple procedure has been suggested (50) to introduce nonMarkovian mixing effects into the Liouville master equation model for Markovian statistics as given by Eqs. (171) and (172). The parameter in Eq. (172) is defined as [see Eq. (93)]

where is the correlation length for the Markovian statistics, given by Eq. (38). The suggestion is to replace in Eq. (227) by an effective correlation length, to account for non-Markovian statistics. The starting point in the analysis to define is the purely absorbing renewal equations given by Eqs. (225) and (226) in the special case of homogeneous statistics and with cross sections which are spatially independent. We then have

In writing these equations, we have used the fact that for homogeneous statistics the functions and depend only upon the displacement argument Further, in subsequent considerations we take and to be given by Eqs. (17) and (23). Equations (228) and (229) are easily solved by Laplace transformation since the integral terms are of the convolution form. If we consider the source-free case,

TRANSPORT THEORY IN DISCRETE STOCHASTIC MIXTURES and define as the Laplace transform of any function Eq. (23), written for the variable, gives

87

we find that transforming

where is the mean chord length in material i as given by Eq. (19). Then, transforming Eqs. (228) and (229) with using Eq. (230) to eliminate and forming the ensemble average gives

where the argument of

in Eq. (231) is

and we have defined

as

In writing Eq. (231), we have assumed a unit incoming intensity, i.e., Knowing the statistics, one could in principle compute the and then a Laplace inversion of Eq. (231) would give the ensemble-averaged intensity We choose here to use Eq. (231) in another way. Specifically, if we compute the average distance to collision for a particle impinging upon a halfspace, we have [see Eq. (113) with ],

Setting

in Eq. (231) gives

where we have defined

We now approximate the exact solution for exponential according to

for

by a single decaying

and define such that Eq. (236) gives the correct mean distance to collision. Use of Eq. (236) in Eq. (233) gives and equating this to Eq. (234) we find

It has been shown (50) that for all chord length distributions, and hence is always nonnegative. Equation (237) is the same result obtained by Vanderhaegen (38) from diffusion theory considerations. In the case of Markovian statistics given by Eq. (26), one has [see the discussion under Eq. (52)]. Then a comparison of Eqs. (54) and (232) establishes that Finally, in this case Eq. (235) gives

88

G. C. POMRANING

and Eq. (237) reduces to our earlier result given by Eq. (117), as well as the asymptotic result given by Eq. (188). To obtain a corresponding expression for the effective source, we take the in Eqs. (228) and (229) to be constant, independent of and look for the deep-in solution This solution will be a constant, independent of Omitting the straightforward algebraic detail, we find

with and once again given by Eqs. (232) and (235). We now define an effective source, by the equation

Since is nonnegative, and from the physics so is we conclude that is nonnegative for all chord length distribution functions. The motivation for this definition of is that a classic purely absorbing transport equation involving and (to be written shortly) will give the correct deep-in solution in the case of homogeneous non-Markovian statistics when and are spatially independent. From Eqs. (237) through (239), we then deduce

We note the similarity of Eqs. (237) and (240). In particular, if the sources and cross sections are related according to we then find

In the radiative transfer context, B is the Planck function. For the Markov case defined by Eq. (26), we have and as noted earlier, and thus Eq. (240) reduces to our earlier Markovian result given by Eq. (122), and the no-scattering limit of Eq. (187). In terms of the effective quantities and given by Eqs. (237) and (240), the equation

is the simplest description that incorporates non-Markovian statistical effects into a standard, purely absorbing, transport equation. The dependences of and in Eq. (242) arise from using local values of the quantities on the right-hand sides of Eqs. (237) and (240). Equation (242) is a robust equation in that both and are nonnegative. Further, Eq. (242) has the property of predicting the correct mean distance to collision as well as the correct deep-in solution for homogeneous nonMarkovian statistics when the and are spatially independent. We note that as tends to zero for both and 1, Eqs. (237) and (240) predict

which are the expected results. When scattering is present in the transport problem, it has been suggested (50) to use these purely absorbing considerations to modify the low-order Markovian master

TRANSPORT THEORY IN DISCRETE STOCHASTIC MIXTURES

89

equation model given by Eqs. (171) and (172) to account for non-Markovian statistics. Specifically, the proposal is that Eqs. (171) and (172) [with the parameters and given by Eqs. (91) through (93) and (173) through (175)], will (approximately) describe stochastic transport in a non-Markovian binary mixture if the correlation length in the definition of as given by Eq. (93) is replaced by an effective correlation length. That is, Eq. (93) is replaced by

where

is defined by

with and given by Eqs. (232) and (235). An equivalent model results from replacing the in Eq. (170) by This model for non-Markovian mixing described by chord length distributions [see Eq. (16)] is obviously exact in the special case of purely absorbing transport in a statistically homogeneous Markovian mixture. Further, all of the simplifications that were discussed for the Markovian master equation model, starting with the paragraph containing Eq. (184), apply equally well to this non-Markovian model. In particular, the renormalized monoenergetic transport description given by Eqs. (186) through (189) describes non-Markovian transport in the asymptotic limit described above Eq. (186) if in Eqs. (187) through (189) is given by Eq. (245).

CONCLUDING REMARKS In this article we have attempted to summarize the research which has been reported during the last ten years describing particle transport through a background medium consisting of two (or more) randomly mixed immiscible materials. Following an introduction and a section dealing with purely statistical considerations, four sections were devoted to such transport in the special case of Markovian mixing statistics. This was followed by one section which discussed particle transport in the presence of non-Markovian statistics. This emphasis on Markovian statistics in our review reflects the corresponding emphasis in the literature. For this special class of statistics, many relatively simple results can be obtained which are not possible in the non-Markovian case. For example, in the purely absorbing context, a coupled set of two differential equations describes exactly the ensemble-averaged intensity [see Eq. (85)]. The corresponding result for non-Markovian statistics is a set of four coupled integral equations [see Eqs. (217) and (221)]. Additionally, in the no-scattering case, a simple set of two coupled differential equations can be obtained for each of the higher statistical moments, such as the variance [see Eq. (96)]. No similar results, even in the form of coupled integral equations, have been reported for non-Markovian statistics. When scattering is present in the underlying transport process, exact stochastic transport results are only available for a very restricted set of problems. The most widely used and studied general approximate model, with scattering and Markovian mixing, is the first low-order model we discussed in the fifth section of this article [see Eq. (170)]. This two-equation model is simple, robust, and reasonably accurate over a wide range of problems. Further, it is easily generalized to a mixture of more than two materials, and can be modified to include non-Markovian effects by introducing an effective correlation length, or equivalently effective Markov transition lengths [see Eq. (246) and the discussion immediately thereafter]. Many limiting cases and approximations to this model have been studied, including various asymptotic limits

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G. C. POMRANING

and several diffusive descriptions. What appears to be an improved, and at this time the best, general model for Markovian statistics is the higher-order modified interface model [see Eqs. (198) and (203)]. This model is more complex than the two-equation low-order model in that it consists of four coupled transport equations. It must also be stressed that this description of stochastic transport is very recent, and it has not been tested or studied to any great extent. It appears to be robust and accurate, but more experience with this model is required to make this assertion with full confidence. As mentioned in the introduction to this article, the potential applications of a linear transport theory in the presence of a stochastic binary mixture of immiscible materials are many. Several of these are currently being pursued in various research communities. Hopefully, this review article may be of some help in guiding other researchers potentially interested in this class of stochastic transport to the literature currently available on the subject. Acknowledgment

The preparation of this article was partially supported by the U.S. Department of Energy under grant DE-FG03-93ER14355. REFERENCES 1. R. Sanchez, “Linear kinetic theory in stochastic media,” J. Math. Phys. 30: 2498 (1989). 2. M.L. Adams, E.W. Larsen, and G.C. Pomraning, “Benchmark results for particle transport in a binary Markov statistical medium,” J. Quant. Spectrosc. Radiat. Transfer 42: 253 (1989). 3. G.C. Pomraning, “A model for interface intensities in stochastic particle transport,” J. Quant. Spectrosc. Radiat. Transfer 46: 221 (1991). 4. G.C. Pomraning, “Simple balance methods for transport in stochastic mixtures,” Proc. Workshop on Nonlinear Kinetic Theory and Mathematical Aspects of Hyperbolic Systems, V.C. Boffi et al., eds., World Scientific, Singapore, pp. 204-211 (1992). 5. B. Su and G.C. Pomraning, “Limiting correlation length solutions in stochastic radiative transfer,” J. Quant. Spectrosc. Radiat. Transfer 51: 893 (1994). 6. G.C. Pomraning and B. Su, “A closure for stochastic transport equations,” Proc. International Conference on Reactor Physics and Reactor Computations, Ben-Gurion University of the Negev Press, Israel, pp. 672-679 (1994). 7. G.C. Pomraning and B. Su, “A new higher order closure for stochastic transport equations,” Proc. International Conference on Mathematics and Computations, Reactor Physics, and Environmental Analyses, American Nuclear Society Topical Meeting, Portland, OR, pp. 537-545 (1995). 8. B. Su and G.C. Pomraning, “Modification to a previous higher order model for particle transport in binary stochastic media,” J. Quant. Spectrosc. Radiat. Transfer 54: 779 (1995). 9. J. Thie. Reactor Noise, Rowman and Littlefield, New York (1963). 10. R.E. Uhrig, Random Noise Techniques in Nuclear Reactor Systems, Ronald Press, New York (1970). 11. M.M.R. Williams. Random Processes in Nuclear Reactors, Pergamon Press, Oxford (1974). 12. G.I. Bell and S. Glasstone. Nuclear Reactor Theory, Van Nostrand Reinhold, New York (1970). 13. J.J. Duderstadt and W.R. Martin. Transport Theory, Wiley-Interscience, New York (1978). 14. G.C. Pomraning, “Transport and diffusion in a statistical medium,” Transport Th. Statist. Phys. 15: 773 (1986). 15. C.D. Levermore, G.C. Pomraning, D.L. Sanzo, and J. Wong, “Linear transport theory in a random medium,” J. Math. Phys. 27: 2526 (1986). 16. G.C. Pomraning. Linear Kinetic Theory and Particle Transport in Stochastic Mixtures, World Scientific, Singapore (1991).

TRANSPORT THEORY IN DISCRETE STOCHASTIC MIXTURES 17. R. Sanchez and G.C. Pomraning, “A statistical analysis of the double heterogeneity problem,” Ann. Nucl. Energy 18: 371 (1991). 18. G.C. Pomraning, “Radiative transfer in Rayleigh-Taylor unstable ICF pellets,” Laser Particle Beams 8: 741 (1990). 19. G.L. Stephens, “The parameterization of radiation for numerical weather prediction and climate models,” Mon. Weather Rev. 112: 826 (1984). 20. G.A. Titov, “Statistical description of radiation transfer in clouds,” J. Atmos. Sci. 47: 24 (1990). 21. G.L. Stephens, P.M. Gabriel, and S-C Tsay, “Statistical radiative transport in one-dimensional media and its application to the terrestrial atmosphere,” Transport Th. Statist. Phys. 20: 139 (1991). 22. F. Malvagi, R.N. Byrne, G.C. Pomraning, and R.C.J. Somerville, “Stochastic radiative transfer in a partially cloudy atmosphere,” J. Atmos. Sci. 50: 2146 (1993). 23. F. Malvagi and G.C. Pomraning, “Stochastic atmospheric radiative transfer,” Atmos. Oceanic Optics 6: 610 (1993). 24. B. Su and G.C. Pomraning, “A stochastic description of a broken cloud field,” J. Atmos. Sci. 51: 1969 (1994). 25. V.E. Zuev and G.A. Titov, “Radiative transfer in cloud fields with random geometry,” J. Atmos. Sci. 51: 176 (1995). 26. G.C. Pomraning, “Effective radiative transfer properties for partially cloudy atmospheres,” Atmos. Oceanic Optics 9: 7 (1996). 27. P. Boissé “Radiative transfer inside clumpy media: The penetration of UV photons inside molecular clouds,” Astron. Astrophys. 228: 483 (1990). 28. S. Audic and H. Frisch, “Monte-Carlo simulation of a radiative transfer problem in a random medium: Application to a binary mixture,” J. Quant. Spectrosc. Radiat. Transfer 50: 127 (1993). 29. G.C. Pomraning and A.K. Prinja, “On the propagation of a charged particle beam in a random medium. II: Discrete binary statistics,” Transport Th. Statist. Phys. 24: 565 (1995). 30. G.C. Pomraning, “Small correlation length solutions for planar symmetry beam transport in a stochastic medium,” Ann. Nucl. Energy 23: 843 (1996). 31. G.C. Pomraning, “The planar symmetry beam problem in stochastic media,” J. Quant. Spectrosc. Radiat. Transfer 55: 771 (1996). 32. B. Su and G.C. Pomraning, “The Fermi-Eyges beam analysis for a heterogeneous absorbing stochastic medium,” Ann. Nucl. Energy 23: 1153 (1996). 33. G.C. Pomraning, “Statistics, renewal theory, and particle transport,” J. Quant. Spectrosc. Radiat. Transfer 42: 279 (1989). 34. R. Sanchez, O. Zuchuat, F. Malvagi, and I. Zmijarevic, “Symmetry and translations in multimaterial line statistics,” J. Quant. Spectrosc. Radiat. Transfer 51: 801 (1994). 35. E. Parzan, Stochastic Processes, Holden-Day, San Francisco (1962). 36. D.V. Lindley. Introduction to Probability and Statistics from a Bayesian Viewpoint, Cambridge University Press, Cambridge (1980). 37. J. Medhi. Stochastic Processes, Wiley, New York (1982). 38. D. Vanderhaegan, “Impact of a mixing structure on radiative transfer in random media,” J. Quant. Spectrosc. Radiat. Transfer 39: 333 (1988). 39. K.M. Case, F. deHoffmann, and G. Placzek. Introduction to the Theory of Neutron Diffusion, Vol. 1, Los Alamos Scientific Laboratory, Los Alamos, NM (1953). 40. D. Vanderhaegen, “Radiative transfer in statistically heterogeneous mixtures,” J. Quant. Spectrosc. Radiat. Transfer 36: 557 (1986). 41. R. Kubo, “Stochastic Liouville equations,” J. Math. Phys. 4: 174 (1963). 42. N.G. van Kampen. Stochastic Processes in Physics and Chemistry, North Holland, Amsterdam (1981). 43. V.C. Boffi, F. Malvagi, and G.C. Pomraning, “Solution methods for discrete-state Markovian initial value problems,” J. Statist. Phys. 60: 445 (1990). 44. U. Frisch, “Wave propagation in random media,” Probabilistic Methods in Applied Mathematics, Academic Press, New York (1968). 45. J.B. Keller, “Wave propagation in random media,” Proc. Symp. Appl. Math. 13, Amer. Math. Soc., Providence, RI (1962). 46. J.B Keller, “Stochastic equations and wave propagation in random media,” Proc. Symp. Appl. Math. 16, Amer. Math. Soc., Providence, RI (1964). 47. J.B. Keller, “Effective behavior of heterogeneous media,” Statistical Mechanics and Statistical Methods in Theory and Application, Plenum Press, New York (1977).

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48. D.C. Sahni, “An application of reactor noise techniques to neutron transport problems in a random medium,” Ann. Nucl. Energy 16: 397 (1989). 49. D.C. Sahni, “Equivalence of generic equation method and the phenomenological model for linear transport problems in a two-state random scattering medium,” J. Math. Phys. 30: 1554 (1989). 50. C.D. Levermore, G.C. Pomraning, and J. Wong, “Renewal theory for transport processes in binary statistical mixtures,” J. Math. Phys. 29: 995 (1988). 51. G.C. Pomraning and R. Sanchez, “The use of renewal theory for stochastic transport,” J. Quant. Spectrosc. Radiat. Transfer 43: 267 (1990). 52. H. Frisch, G.C. Pomraning, and P.F. Zweifel, “An exact analytic solution of a radiative transfer problem in a binary mixture,” J. Quant. Spectrosc. Radiat. Transfer 43: 271 (1990). 53. O. Zuchuat, R. Sanchez, I. Zmijarevic, and F. Malvagi, “Transport in renewal statistical media: Benchmarking and comparison with models,” J. Quant. Spectrosc. Radiat. Transfer 51: 689 (1994). 54. F. Malvagi, C.D. Levermore, and G.C. Pomraning, “Asymptotic limits of a statistical transport description, ” Transport Th. Statist. Phys. 18: 287 (1989). 55. F. Malvagi and G.C. Pomraning, “Renormalized equations for linear transport in stochastic media,” J. Math. Phys. 31: 892 (1990). 56. F. Malvagi, C.D. Levermore, and G.C. Pomraning, “Asymptotic limits of a statistical transport description,” Operator Theory: Advances and Applications 51: 200, W. Greenberg and P. Polewczak, eds., Birkhauser Verlag, Basel (1991). 57. F. Malvagi and G.C. Pomraning, “Renormalized equations for linear transport in stochastic media,” Proc. International Conf. on Advances in Mathematics, Computations, and Reactor Physics, American Nuclear Society Topical Meeting, Pittsburgh, PA, Vol 3: p. 13.1, 2-1 (1991). 58. G.C. Pomraning, C.D. Levermore, and J. Wong, “Transport theory in binary statistical mixtures,” Lecture Notes in Pure and Applied Mathematics 115: 1, P. Nelson et al., eds., Marcel Dekker, New York (1989). 59. G.C. Pomraning, “Particle transport in random media,” Proc. Venice I: Symposium on Applied and Industrial Mathematics, R. Spigler, ed., Kluwer, Dordrecht, The Netherlands, pp. 309-315 (1991). 60. G.C. Pomraning, “Linear transport and kinetic theory in stochastic media,” Proc. Vth International Conf. on Waves and Stability in Continuous Media, S. Rionero, ed., World Scientific, Singapore, pp. 349-359 (1991). 61. G.C. Pomraning, “Transport theory in stochastic mixtures,” Trans. Am. Nucl. Soc. 64: 286 (1991). 62. D. Vanderhaegen and C. Deutsch, “Linear radiation transport in randomly distributed binary mixtures: A one-dimensional and exact treatment for the scattering case,” J. Statist. Phys. 54: 331 (1989). 63. G.C. Pomraning, “Radiative transfer in random media with scattering,” J. Quant. Spectrosc. Radiat. Transfer 40: 479 (1988). 64. G.C. Pomraning, “Classic transport problems in binary homogeneous Markov statistical mixtures,” Transport Th. Statist. Phys. 17: 595 (1988). 65. F. Malvagi and G.C. Pomraning, “A class of transport problems in statistical mixtures,” J. de Physique, Coll. C7 49: 321 (1988). 66. G.C. Pomraning, “The Milne problem in a statistical medium,” J. Quant. Spectrosc. Radiat. Transfer 41: 103 (1989). 67. G.M. Wing. An Introduction to Transport Theory, Wiley, New York (1962). 68. K.M. Case and P.F. Zweifel. Linear Transport Theory, Addison-Wesley, Reading, MA (1967). 69. F. Malvagi and G.C. Pomraning, “A comparison of models for particle transport through stochastic mixtures,” Nucl. Sci. Eng. 111: 215 (1992). 70. B. Su and G.C. Pomraning, “Benchmark results for particle transport in binary non-Markovian mixtures,” J. Quant. Spectrosc. Radiat. Transfer 50: 211 (1993). 71. M. Sammartino, F. Malvagi, and G.C. Pomraning, “Diffusive limits for particle transport in stochastic mixtures,” J. Math. Phys. 33: 1480 (1992). 72. F. Malvagi, G.C. Pomraning, and M. Sammartino, “Asymptotic diffusive limits for transport in Markovian mixtures,” Nucl. Sci. Eng. 112: 199 (1992). 73. G.C. Pomraning, “Diffusive transport in binary anisotropic stochastic mixtures,” Ann. Nucl. Energy 19: 737 (1992).

TRANSPORT THEORY IN DISCRETE STOCHASTIC MIXTURES 74. G.C. Pomraning, “Anisotropic diffusive limit for particle transport in stochastic mixtures,” Proc. 18th Rarefied Gas Dynamics Symposium, Progress in Astronautics and Aeronautics, Vol. 159, B.D. Shizgal and D.P. Weaver, eds., AIAA, Washington, DC, pp. 653-661 (1994). and asymptotic approximations for stochastic transport,” 75. B. Su and G.C. Pomraning, Nucl. Sci. Eng. 120: 75 (1995). diffusive approximation to a stochastic transport description,” 76. B. Su and G.C. Pomraning, Proc. International Conf. on Mathematics and Computations, Reactor Physics, and Environmental Analyses, American Nuclear Society Topical Meeting, Portland, OR, pp. 564-573 (1995). 77. M. Sammartino and G.C. Pomraning, “Flux-limiting in stochastic transport,” J. Quant. Spectrosc. Radiat. Transfer 46: 237 (1991). 78. G.C. Pomraning, “Flux-limiting in three-dimensional stochastic radiative transfer,” J. Quant. Spectrosc. Radiat. Transfer 54: 637 (1995). 79. D. Vanderhaegen, C. Deutsch, and P. Boissé, “Radiative transfer in a one-dimensional fluctuating albedo mixture,” J. Quant. Spectrosc. Radiat. Transfer 48: 409 (1992). 80. B. Su and G.C. Pomraning, “A variational approach to stochastic transport,” J. Quant. Spectrosc. Radiat. Transfer 51: 467 (1994).

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THE ROLE OF NEURAL NETWORKS IN REACTOR DIAGNOSTICS AND CONTROL

Imre Pázsit1 and Masaharu Kitamura2 1

Department of Reactor Physics, Chalmers University of Technology, S-412 96 Göteborg, Sweden 2 Department of Quantum Science and Energy Engineering, Tohoku University, Sendai, 980-77 Japan

1. INTRODUCTION Reactor diagnostics and core diagnostics in particular, is an inverse task, just as most other diagnostics. One measures some physical parameter at some position, or the fluctuation thereof, which is given rise by the fluctuation of another parameter, presumably at a different position. In neutron noise diagnostics, the measured quantity is the neutron noise, whereas the cause, fluctuations of the core material, is called the “noise source”. The relationship between the cause (noise source) and the induced noise (effect) is determined by the physics of the process and can usually be described by a theory. This means that the direct task, calculation of the noise from the noise source, can always be achieved. In diagnostics, however, the process starts from the back end, i.e. one observes the effect of some cause. The task is to infer the cause (noise source) from the effect (induced noise), which is an inverse task (sometimes also called unfolding). The situation is very much similar even regarding control problems. The system state is described by a vector in the parameter space, When it deviates from the desired one, the control parameters need to be changed to bring the system into the desired state. The computation of the change in the state vector, due to a change in the control vector, can be calculated with no difficulty in principle. The control requires however the inverse task to be performed, i.e. to determine which changes of the control vector would bring the state vector into the desired state. The solution of inverse problems is usually not unambiguous. Besides, there are no standard methods to perform the inversion or unfolding, rather in each case one has to devise a way of solving the particular problem at hand. The chances for a successful inversion are the highest if an analytical solution of the direct task is available, because then even the inversion may be attempted analytically. However, in complicated systems like a reactor

Advances in Nuclear Science and Technology, Volume 24 Edited by Lewins and Becker, Plenum Press, New York, 1996

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core, there is not even a chance of achieving an analytical solution of the direct problem unless it is drastically simplified. Most neutron noise methods up to date followed this path of action, i.e. using an extremely simple core model. If an analytical solution is not possible, then another strategy can be to make a large “database” of the solutions of the direct problem covering as many cases as possible. The input parameters (i.e. source parameters) for each such solution are known. The parameters of a new observation or measurement are then found by comparing the measurement with the database and picking the most similar one. The parameters of that “baseline” function will constitute the solution of the inverse problem. Such methods have extensively been used, e.g. in geophysics, i.e. in geoelectric measurements. They require of course a method of identifying the “most similar” solution from the data base, which may be a quite complicated task. Sometimes, instead of calculating a large data base in advance and then searching in it, prior knowledge of the process is used to try to generate a solution of the direct task, by way of direct trial and error, that will be similar to the observation. Artificial neural network (ANN) algorithms provide a very powerful alternative for the solution of inverse problems. A neural network is a computing paradigm which performs a non-linear mapping of a multi-dimensional input space into an output space. The mapping is performed in one or several steps such that in each step, the input values may be operated on in parallel, so the algorithm is fast. The key point however is that there are explicit algorithms with the help of which the network can be “trained” to yield the desired set of outputs to a given set of inputs. There is no need to have an explicit rule of mapping available for the training, only numerical examples of input-output pairs. An observation can be given as input and the searched parameters (the results of the inverse task) as its output. Thus the network can be trained to give the solution of an inverse task, whereas the data for its training may be produced by solving the direct task only. Once trained, the network will be able to yield the solution of the indirect task with rather little computational effort. Of course if the inversion of the direct task is not unique (i.e. if several sources can lead to the same noise), the network may fail to give the correct answer. In most cases, however, a sufficiently high success rate or precision can be achieved. The above description of ANNs may appear similar to the “database” method above in that in both cases a large set of direct solutions need to be provided in advance. However, in complicated systems, the computational effort in the training of the network consists basically of the calculation of the training set. Thus on the whole, use of an ANN requires much less effort than the traditional method since the identification of the “matching solution” to any new sample pattern is nearly automatic and effortless. This ability of ANNs to perform pattern recognition to solve inverse problems in a transparent and very fast way is extremely useful. In addition, since the network constitutes a distributed memory, there is the additional advantage of being “fault tolerant” regarding errors in both the network itself and in the input data (i.e. faulty or noisy sensors). Thus neural network technology has seen an explosion of feasibility studies and applications in the past few years. Applications are seen in all branches of engineering sciences, process control, economy, medicine, weather prediction, optimization, character recognition etc. Not the least is this true in nuclear engineering where there are several important inverse tasks in diagnostics and control, and where any real-world problems can only be treated numerically and by an extensive calculation. This chapter gives an account on the potentials of the use of neural networks in reactor diagnostics and control. The principles and properties of various ANN paradigms will be described. One example of application in reactor diagnostics, namely the localisation of vibrating core components with neural network techniques, will be described in detail. This latter will also give an illustration of several features and advantages of ANNs.

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2. HISTORY AND PRINCIPLES OF ARTIFICIAL NEURAL NETWORKS There exist excellent textbooks on neural networks, covering also the history and basic principles of ANNs (Rumelhart and McClelland, 1986; Grossberg 1988;Wasserman, 1989; Eberhart and Dobbins, 1990; Freeman and Skapura,1991; Hertz et al., 1991; Kosko, 1992; Churchland and Sejnowski, 1992; Haykin, 1994). Our intention here is to give enough material to make this chapter a self-contained reading.

2.1 History The idea of ANN systems, and in particular its basic computing element, the neuron, arose out of studies of the central nervous system. In the brain there exist a large number of nerve cells or neurons, which communicate with each other via synapses. In the human brain there are about neurons. Each neuron is connected to about other neurons, resulting in approximately synapses or connections. Communication means receiving, processing and passing on signals to other neurons. The synapses act as connections as well as linear amplifiers whereas the neurons as multipleinput single-output non-linear amplifiers. The synapses are flexible in the sense that the amplification of a synapse is increased in case of frequent communication between two neurons and vice versa. Artificial neural networks are computing structures whose building stones are based on similar principles. The first works on the development of models of the nerve cells that performed logical functions date back to the 1940’s (McCulloch and Pitts, 1943). Major steps of development are marked by the following works: the general rule of updating the synaptic weights in the learning process was formulated (Hebb, 1949); the perceptron, a twolayered feedforward network, was developed (Rosenblatt, 1958); the research in neural networks started finding applications in and contributions from the engineering sciences, shown e.g. by the construction of the gradient descent method of training (Widrow and Hoff, 1960). The perceptron was capable of solving relatively complicated tasks. However, it had concurrently important drawbacks; for instance it could not solve the “exclusive or” (XOR) problem. Expert systems, a competing technology, could on the other hand handle that problem with ease. This shortcoming was taken up in the technical literature (Minsky and Papert, 1969) and as a consequence, the interest in ANNs diminished very rapidly during the 70’s. Thereafter the developments in machine or artificial intelligence were concentrated on expert systems instead. The research on ANNs was, however, not abandoned completely. It was gradually understood that the shortcomings of the perceptron, among others its inability to solve the XOR problem, could be tracked down to a lack of non-linearity. Namely, a perceptron can only separate two sets in the N-dimensional output space if the sets can be separated by an N-1 dimensional hyperplane (linear surface). Sets that are separated by nonplanar surfaces require a certain non-linearity that the perceptron does not possess. The simplest way to furnish an ANN with the required non-linear classification property is to augment it with one (or more) intermediate or hidden layer of neurons between the input and output layers. Such ANNs are indeed capable to tackle very complicated tasks that the perceptron could not handle. The multi-layered perceptron emerged, and the backpropagation algorithm was developed for the training of such networks (Rumelhart and McClelland 1986). Also, new types of network structures and algorithms were developed such as feedback (recurrent) networks (Hopfield, 1982, 1984) and self-organizing maps (Kohonen 1972).

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This development has led to a fast revival of interest in neural networks in the early 80’s. There is intensive research and development work going on, comprising both theory development, feasibility studies of potential applications and real applications. This can be witnessed by the number of publications, books, and journals and conferences devoted to the subject. There are many proven applications and several commercial products regarding both software and dedicated equipment. However, it must be noted that the number of concrete field applications is much less than the number of studies of potential applications. This is valid also concerning the nuclear engineering applications.

2.2 Basic principles The basic computing element of the ANN is one neuron, also called a node or a processing element. A node has one or several inputs, and computes an output from the weighted sum of inputs. This signal is then passed on either to other neurons or to the network output. One can distinguish between two basic types of neurons: binary and continuous. The first of these can have only two outputs, zero and unity or ±1. The first Hopfield models and the Boltzmann machine fall into this category. The second type uses continuous values between certain limits, the limits being usually (0,1) or (-1,1). The backpropagation algorithm is based on such continuous node values. Since this is the type of networks we shall be dealing with most, this type of processing element will be assumed unless stated otherwise. A typical node in an intermediate or hidden layer looks like shown in Fig. 1. The

neuron number i in layer preceding layer node value

receives signals

from the neurons of the

through the synapses with gains or weights

The

before activation, will thus be a weighted sum of all signals

to which the node is connected, plus an offset or threshold

of layer

as

Its output, or activation value, will lie in the interval (-1, 1) or (0,1) and is calculated by a non-linear mapping or activation function, usually a sigmoid function, as

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Here T is a parameter describing the sensitivity of the non-linear transfer, i.e. that of the output to the input. Its role can be associated to that of temperature in statistical systems. The function g in eqn (2) is called a sigmoid function and is depicted in Fig. 2 for two dif-

ferent values of the scaling parameter T. However, in the following we shall take T = 1 for simplicity.

2.3 Network structures and terminology The taxonomy of networks with respect to architecture, training type, etc. is not completely unique, but we try to give a classification as structured as possible. Thus we will distinguish between two basic types of networks, one being the feedforward, the other the recurrent or feedback type networks. A feedforward (FFW) network is (multi)-layered, and the signal propagation is always one-way, from input to output. Correspondingly, there exist clearly distinct input and output layers. The perceptron, the first network of this kind, is a two-layered network with input and output layers or nodes only. The multi-layered perceptron, or simply the feed-forward network, is the most frequently encountered type to date. Its structure is shown in Fig. 3. It consists of an input layer, an output layer, and one or more so-called hidden layers. The network receives input through the nodes (receptors) in the input layer, from which the signals propagate forward to the nodes of the consecutive layers by the rules given in (1) and (2). The output is calculated directly in one sweep from the input data. Recurrent or feedback networks show a larger variety of architectures and are thus more difficult to classify. Also, their functioning is more complicated. The common feature of such networks is that the signal can propagate between certain, or all, nodes in both directions. In addition, they can be both layered or non-layered. For instance, in the network in Fig. 3, one may keep the same layered structure, but allow the signal to propagate between certain nodes in both directions. For such networks the output cannot be calculated in one sweep, since the node values in a given layer are affected by signals from not only the preceding (lower) but also from the subsequent (higher) layers. This means that, in contrast to the feedforward networks, a second sweep will lead to a different result and so on. In such networks the calculation must be repeated (iterated) a number of times until a stable output is attained. As mentioned above, this type of network is not restricted to a layered structure. A completely recurrent network, consisting of one layer only (or, better to say, without a

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layered structure), is shown in Fig. 4. In this network the nodes are initiated with the values of an input pattern, after which the network is iterated until it reaches an equilibrium. Thus in this case the input and output nodes are the same. It is also possible to divide such a nonlayered network into input and output nodes, and even hidden nodes, without changing the non-layered character of the network. The recurrent or feedback-type networks are also called Hopfield networks after their inventor (Hopfield, 1982). Such networks are especially powerful regarding optimization problems and pattern recognition, which require so-called hetero-associative memory. By this latter one means an associative memory that can reconstruct images from distorted data. (The terminology is, however, not unambiguous on this point, and other uses and definitions of the terms auto- and hetero-associative are also in use).

A network, either feedforward or recurrent, can be fully or partially connected. For multi-layered feedforward networks this refers only to the connections between neighbouring layers. If all possible connections between the nodes exist, the network is fully connected. Thus a multi-layered feedforward network can be fully connected although there may be no direct connections between the input and output layers.

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2.4 Training techniques The most important feature of ANNs is the possibility to “tune” them, i.e. to change the input-output mapping. This process is called training or learning. Training can be performed in two distinct ways, corresponding to two distinct uses of ANNs. For obvious reasons these are called supervised and unsupervised training, respectively. The type of training used overwhelmingly up to date is the supervised training. Its purpose is to tune the network such that a prescribed set of outputs (targets) be obtained for a given input set. This can be achieved without an explicit rule of the input-output mapping being available, only a number of numerical realisations of the mapping, i.e. a sufficiently large set of associated input-output pairs. Unsupervised training, on the other hand, uses only input data, without a known or desired output (target) being available. Such networks perform a grouping or clustering of the input data set, and this clustering requires that the network performs an interactive training of itself. One can say that in unsupervised training, the desired output is not known, but some implicit information is given to the network on the rules of classification. The output of a network, either feedforward or recurrent, to a given input, depends on the architecture (connection topology), the weights of the synapses (see (1) and Fig. 1) and the form of the function g (u) in (2). In principle all of these can be changed and thus could be used for training purposes. These changes are though not completely independent of each other. For instance the change of the parameter T in the activation function (2) is equivalent to a change of the weights and the learning rate in eqn (5) below (van der Hagen, 1994). At any rate, for various practical and principal reasons, in practice training is most often achieved by changing only the connection weights, whereas all other parameters and factors are fixed once for all at the setup. In some works, however, the number of nodes is also changed during learning. In the following the methods used for tuning the connection weights are described. Supervised learning: Feedforward networks. Without doubt, these are the most extensively used network architectures. Their training and usage is also conceptually the simplest. The concrete example that we will discuss in Section 5 is also based on a feedforward network. The training is performed according to the following scheme. One defines a set containing N input - output pairs or vectors and p = 1, 2, ...N, respectively. Here, for simplicity of notations, the sets of input and output node values are denoted as vectors (of different dimensions in general). The network is then initialised by randomly chosen weights. Once these are selected, the network response to any input vector can be calculated. In general, the output to an input vector will differ from the desired one, i.e. the target vector To minimise the deviation, one defines an error function E

Then one adjusts the weights such that E is minimised. It is seen from (1) and (2) that E is an analytical function of all the weights. Thus a condition for E to be minimum can be formulated by requiring the partial derivatives of E w.r.t. the vanish. This condition will lead to a non-linear equation system which cannot be solved analytically in general. In practice therefore an iterative procedure is used to find the roots of the equation system

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Such root searching procedures are based on a multi-variate version of the Newton-Raphson method. The most widely spread method is called the generalized delta rule, and it is a variant of the gradient descent method (Rumelhart et al., 1986; Werbos, 1990; Freeman and Skapura, 1991). The essence of the method is that the weight matrix of a given layer is changed, in a number of iteration cycles, along the gradient of the scalar function as Here the parameter determines the length of the step along the gradient and thus the speed of convergence towards the minimum. This parameter is called the learning rate. Its choice is optional, and contributes to the several degrees of freedom one has when performing the training. Proper choice of requires some knowledge of the physical process one simulates with the network, and it depends also on the initialisation of the network, etc. Its value can even be adaptively decreased when approaching the minimum. The rule (5) will secure that the change of will tend to a minimum, although there is no guarantee that one ends up in the global minimum of the system. Making sure that one gets an optimal or nearly optimal solution to (4) is a large problem area in itself and we refer again to the literature for more information. Here we only mention one method that may help to avoid oscillations in the iteration. It consists of adding one more term to the r.h.s. of (5) as

where is called the momentum rate, and is the weight update matrix from the previous iteration (Freeman and Skapura, 1991). Again, as with the learning rate, the choice of the momentum rate is to a large extent arbitrary. As already mentioned, the gradient in (5) can be expressed explicitly from (1) - (3). Because of the nested functional dependence of on the weights, the gradient can be expressed by using the chain rule of derivation. The gradients w.r.t. the weights between the last hidden layer and the output layer can be expressed in terms of the same weights. The gradients w.r.t. to weights between preceding layers, on the other hand, will depend also on the values of the connection weights of the subsequent (higher) layers, due to the forward signal propagation. Therefore, the lower that two layers lie behind the output, the lengthier the formulae for the weight updates become. For orientation we give here the update formulae for the last and the last but one layer of connections. For simplicity of notation we choose M = 1 in (3), drop notation on (p) , and also neglect the momentum rate term as well. For the last layer of connections, i.e. between the output and the last hidden layer, one will have

where

This leads to the updating rule of the weights

as

For the connections between the layers one step below the last connection layer the same procedure will result in the updating rule

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As (10) shows, the updating of the layers becomes more and more involved for the earlier layers, due to the fact that the weights of all later layers enter the formula. For this reason, this type of training is usually called backward error propagation, or just backpropagation algorithm. This terminology is though somewhat incorrect, because the update of any layer can be made independently of (and thus concurrently with) any other layers, even higher ones, at least as long as only the old weights are used in (10) throughout (which is usually the case). Multilayer perceptrons with the above learning method are called feedforward networks with backward error propagation, or just backpropagation networks (BPNs). The way of iterating the gradient descent method via (9) and (10), etc., until a minimum of E is reached, is also somewhat arbitrary. One could select e.g. M = N in (3), i.e. use all available training vectors at a time, and iterate on the same set. The generally followed procedure, however, is to select M = 1 and use different input-target pairs in sequence. After having used all training vectors, the same set can be gone through again, preferably in another (random) order, such that one training vector pair is used several times. The reason for selecting M = 1, i.e. only one pattern at a time is that in general, in the updating formulae (7) - (10) a summation from 1 to M is involved (not denoted here since we took M = 1). For large M values this imposes a large storage requirement (memory allocation or heavy disk read-write) during the calculations which is impractical to handle. It is because of the above, i.e. using M = 1 and calculating the gradient for different patterns in sequence that the iteration procedure is called the generalized delta rule and not the gradient descent method. The difference between the two is that in the gradient descent method the same function is iterated on, whereas in the method used here, in each iteration a different function is used. Using a different input-output pattern pair in calculating the error function during the iteration means that one uses slightly different error functions in each iteration. Thus the weight update method described above is not strictly a gradient descent method, hence the name generalized delta rule. The reason for the fact that the pattern vectors need preferably to be used in a random sequence such that patterns belonging to different classes must be “mixed” has its explanation in how the memory of the network is built up during training. If all patterns constituting a certain class are used first and then only other classes are used, then the network will gradually “forget” the former as the training progresses. This is a manifestation of the Hebbian learning rule which is embedded in the above backpropagation training algorithm. It is also seen from the above that the training of a network contains several moments of empirical character. Such a moment is, e.g., the selection of the learning rate In many approaches one uses a learning rate that is decreasing when the network error gets closer to the minimum, to avoid oscillations around the minimum with no convergence. The selection of all such tunable parameters is usually made by way of “trial-and-error”. Another point is that the output of any node after activation, including the output nodes, will lie between (0,1) or (-1,1). Hence, applying the network to any problem requires that the physical parameters (target component values) be normalized such that they will lie, say between 0.1 and 0.9 or -0.9 and 0.9, respectively. Achieving this is a non-trivial problem, since sometimes the range of the physical parameters is not completely known for all possible patterns. The training is normally terminated when the network error for a given training set is less than a prescribed (again, user selected) value. The network can be used thereafter in recall mode. If the error is small enough, the network will give the correct output to all input that is part of the training set. By presenting the network any new input vector that lies within

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the range of the training set but is not identical to any of the patterns used during the training, the network is likely to produce the correct output. Whether or not this is the case will depend on partly whether the solution of the inverse task is unique, and whether or not the training led to the global minimum of the network error function or only to a local minimum. Such feedforward networks with backpropagation are effective in producing numerical output to non-linear multivariate problems. In such a case their functioning is similar to a multidimensional fitting procedure. Generally, the network performs much better in interpolation than in extrapolation; i.e. it will more likely yield the correct output for input vectors that lie within the multidimensional convex hull spanned by the input training vectors in the input hyperspace. Hence it is important that the training covers all significant cases of interest sufficiently well. This is where knowledge on the physics of the process to be identified or the parameter to be determined becomes important. Feedforward networks can also perform pattern identification, both in an auto-associative (i.e. recognition of incomplete patterns) and a hetero-associative (recognition of distorted or noisy patterns) way. They usually perform better with the former type, whereas the latter, identification of distorted patterns, is performed better by recurrent (Hopfield type) networks. Supervised learning: feedback/recurrent networks. For simplicity, we shall describe here the so-called Hopfield model (Hopfield 1982, 1984), a fully connected one-layer network. In this network all nodes are both input and output nodes concurrently. Another type of recurrent network with input-output nodes and a hidden layer with stochastically chosen node values, a so-called Boltzmann machine, is described in Rumelhart and McClelland (1986) and Marseguerra and Zio (1994). In the Hopfield model the output to a given input is obtained by iteration. That is, the nodes are initiated with an input vector and the network is iterated with unchanged input until it reaches an equilibrium state. In most cases the node values assume only the values ±1. The updating of the node values is performed through the rule

Such networks are mostly used for recognition of distorted images, i.e. for pattern recognition, or in more general, for structure classification. The patterns or structures to be recognized are stored in the network, and this initial storage is achieved through training. With the network trained to store a number of patterns or images, for any further input vector the network will converge to an output which is in some sense the “most similar”. The principle of operation, which determines also the training algorithm, is based on an “energy function”

An iteration of the type (11) will lead to the minimum of the energy function (12). This means that for a given pattern to be a minimum of E, the weights must be appropriately chosen. If there are M patterns chosen according to

p = 1, 2, …M to be stored, then the weights can be

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Thus the training of the Hopfield network is not iterative, but is performed in one step. On the other hand, identification via a Hopfield network is made through iterations. It is to be mentioned that there is an upper limit on the number of the images or patterns that can be stored in a recurrent network, which is 0.14N, N being the number of nodes or neurons in the net. The Hopfield network gives an opportunity to illustrate a concept which is used with respect to both training of and identification with ANNs. It is the fact that whenever the minimum of a multivalued function is searched by iteration, either that of the error function (3) or the energy function (12), there is no guarantee that the iteration will end up in the global minimum. The iteration procedure of the Hopfield network by (11) may well lead to a local minimum, which does not necessarily correspond to the searched pattern. Especially when using a Hopfield network for a global optimization problem, such as that of the travelling salesman, a global minimum of the whole system is searched for. Similar problems may be encountered during the training of feedforward networks. To prevent the iteration converging to a local minimum, a method borrowed from statistical physics, the so-called simulated annealing or simulated cooling can be used. The simulated annealing is based on the idea that in a statistical system with thermal disordered motion there will be a statistical distribution of system state occupancy, and not only the state with the lowest energy is occupied. Such a system can, under disturbance (“noise”) perform “hill climbing”, i.e. can get out of local minima. If such a system is cooled slowly, in the end it will occupy the global energy minimum. The probability of this occurring is only asymptotically unity, i.e. if the cooling is infinitely slow. In reality only a finite time can be spent at any given temperature, thus it is not absolutely certain that the global minimum will be reached, but the probability of this happening can be made sufficiently high. The essence of the method, illustrated on a network with binary nodes, is as follows. A control parameter T, analogue to the temperature is selected. The network is then initialized by input data without iteration. The energy function E of this initial state is calculated via (12). Then one node is picked randomly, and the value of the node is reversed. Without further iteration, the energy function of the modified configuration, and then the energy difference is calculated. The modified configuration will be accepted with the probability

and rejected with probability 1 – P. This procedure is repeated a sufficient number of times until “thermal equilibrium” is attained. The parameter T is then decreased and the whole procedure repeated. In the limit the global minimum of the system will be attained. Structural learning with forgetting. A particular training algorithm, called the structural learning algorithm with forgetting, is worth mentioning here due to its potential of bridging the gap between knowledge engineering and neuro-computing. One criticism of neural network technology is that it functions as a black-box. We have to accept the trained ANN as it is without obtaining explanatory information about the mechanism behind. This is sometimes regarded as a prohibitive difficulty, particularly in applications to highreliability, high-potential-hazard systems like a nuclear power plant where the mechanism understanding and assurance of correct behaviour are extremely important. The structural learning algorithm allows one to alleviate this difficulty since it prunes the unimportant links in the ANN during training and thus results in a skeletonized ANN structure with a small number of links remaining.

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The skeletonized ANN is far easier to understand and interpret. Furthermore, it is often possible to derive a set of rule-like pieces of knowledge from the ANN. Thus it can also alleviate the serious difficulty called knowledge acquisition bottleneck, commonly experienced when developing knowledge-base systems. The structural learning algorithm is basically a modification of the backpropagation algorithm. The non-standard part is to modify the connection weights as follows: Here, is the change of the ij-th weight based on the standard backpropagation. The parameter is called the forgetting rate. The second term on the right hand side of the above expression represents the continuous forcing term to decrease the connection weights. Because of this term, each weight tends to decay unless it is reinforced by the standard backpropagation correction. The above modification is equivalent to a modification of the cost function E as

with Here, E is defined by eqn (3), and is the learning rate in the standard backpropagation, cf. eqn (5). After successful training, the reduction in network complexity is generally quite significant. The possibility of understanding the functionality of the ANN, and of deriving symbolic knowledge representations as well, is far better in the resultant network. Theoretical basis and case studies of this method are given in Ishikawa (1995). Unsupervised learning (self-organising maps). A new type of network architecture, based on unsupervised training, was suggested by Kohonen (1972). This type of network is non-layered, but consists of an input and an output field of nodes. As the name also suggests, there are no known output samples or targets (i.e. correct answers) to the inputs with which the network could be trained. The task is rather to extract existing structure (clustering) from the input data set. Thus, there are no numerical examples of the inputoutput mapping available, but there is an implicit information on the mapping available in that it is assumed that a clustering, i.e. based on proximity, is applicable. This way the network performs functions similar to the classical K-means algorithm of pattern recognition (Tou and Gonzalez, 1974), except that the number of clusters need not be known in advance. Such a task is performed by a network through self-organization. The Kohonen networks perform an adaptive clustering of the input data. Identification is achieved through learning, thus these two functions, performed separately in networks with supervised training, are intermingled here. The algorithm consists of an updating of the system, and the updating strategy can be either competitive or collective in character. Such networks may be useful in dividing an input data set into a smaller number of classes or clusters. For a more detailed description of the actual algorithms and architecture the reader is referred to the literature. Besides the above basic network architectures, there is a large variety of sub-versions of the main ones, and further even more exotic types. In achieving a certain identification or fault detection task in a complex system, combinations of the different algorithms can also be effective. At any rate, the dedicated literature on neural networks abounds in different varieties of paradigms. For a somewhat off-putting review of the depth and breadth of the jargon and parlance of neural network research the reader is referred to the book of Kosko (1992).

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2.5 Main usage forms and applications of neural networks This subsection is a short summary of the usage types that were mentioned scattered in the previous subsections. A more detailed list of concrete applications will follow in the next section. Parameter estimation. One important class of applications is to assign values of a uni- or multi-variate function to an input vector. The network output can represent interpolation values of a multivariate function as well as a solution of a direct or indirect problem. For such purposes feedforward networks are used with supervised training and backpropagation. Classification/pattern recognition/fault detection. In many cases no detailed numerical answers or solutions are searched, only class identity of the input need to be determined. The classes may or may not be known in advance. In cases when the distribution of the classes is known in advance, the training can be performed in a supervised way, i.e. via training patterns. The training can be executed by both feedforward and recurrent algorithms. However, as is often the case with pattern recognition, the distribution of the classes may not be known in advance; thus part of the pattern recognition process is to perform a clustering of the data set. Clustering may refer both to the initial (input) data or the mapped (output) data. Clustering can also be performed by neural networks. Such networks execute an adaptive version of either the K-means algorithm, or the determination of the data distribution. These networks use unsupervised learning (Kohonen nets). Sensor validation and process estimation. Sensor validation is an area with clear safety relevance. Whether a change in signal characteristics is due to degradation in the process (equipment to be monitored) or degradation in the sensors is important to know. The issue has received increased attention after the Three Mile Island (TMI-2) accident. There are classical methods of state estimation in use, based on e.g. redundancy in the measured process values. Again, any underlying process and sensor models are non-linear and nonanalytical. ANNs can be effectively used for sensor validation.

3. SURVEY OF APPLICATIONS IN REACTOR DIAGNOSTICS AND CONTROL Typical applications of neural network technology in reactor diagnostics and control are reviewed in this section. Although optimization is also one of the rich areas of ANN applications, this topics will not be discussed in this article since our main interest is in diagnostics and control. Application of neural networks to nuclear engineering gained popularity in the early 90s. No ANN-relevant paper can be seen in the 1988 ANS topical meeting on “AI and other Innovative Computer Applications in Nuclear Industry” (Majumdar et al., 1988). Only one paper (Uhrig, 1990) addressed the issue of ANN applications in the 1990 ANS topical meeting on Advances in Human Factors Research on Man/Computer Interactions: Nuclear and Beyond (Proceedings, 1990). The statistics was drastically changed in a short period of time. Twelve out of ninetyfive papers were dealing with ANN applications in the 1991 Conference “AI’91, Frontiers in Innovative Computing for the Nuclear Industry” (Proceedings, 1991). The statistics is roughly similar since then, as typically observed in the 1993 and 1996 ANS Topical Meetings on NPIC and MMIT. This steady development is somewhat in contrast with the

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once popular expert system technology. It would not be too prejudiced to conclude that the ANN technology has established a firm base in nuclear industry, particularly in instrumentation and control applications. It is to be added, however, that most of the applications concern feasibility studies or proof-of-principle investigations. This is good to have in mind even if most applications treat concrete practical problems, and the networks used are trained and/or tested on either measured data, or data obtained in a detailed and realistic simulation, i.e. from a plant simulator. In this respect the combination of training on specific simulated data and using the trained network on measured data must be considered as especially effective. Nevertheless, we need to keep in mind that those cases count as off-line applications, and that the number of cases when ANNs are an integrated part of plant diagnostics or control systems, working on-line, is still not too high. In this section an extensive set of such applications or corresponding feasibility studies will be listed, grouped around the main types of uses given in Section 2.5. We also refer to a review article by Uhrig (1991) where several applications of ANNs in nuclear engineering are described. In this latter reference a few specialised cases are also described that will not be mentioned here, Further material can be found in recent conference proceedings (Proceedings 1995a, 1995b).

3.1 Parameter estimation The term parameter estimation will be used in a wide sense here. The conventional meaning is to determine the numerical value of a (continuous or discrete) physical parameter. We will however include here also localisation through measurement of neutron flux (an inverse transport problem), and the estimation of certain discrete parameters that could be just as well called classification or pattern recognition. In addition, fault detection is often based on threshold-passing of the numerical value of a parameter. Hence, the distinction between parameter estimation and fault detection is not sharp. Since parameter estimation is based on estimating the mapping between some measured (input) variables and the parameter of interest which is a function of the former, parameter estimation is most conveniently performed by multilayered BPN algorithms. Unless otherwise stated, the applications in this subsection use this network type. Moderator temperature coefficient. A representative application is estimation of moderator temperature coefficient (MTC), which has been studied by applying various methods of noise analysis. Efforts to develop a method to estimate MTC by means of noise analysis were made by many researchers (Shieh et al., 1987; Herr and Thomas, 1989; Thomas et al., 1991, 1994). The essential idea is to estimate the neutron power-coolant temperature cross power spectral density and the coolant temperature auto power spectral density

The MTC is usually estimated using the relationship

where is the transfer function from reactivity to neutronic fluctuation. The estimation of the ratio is called the frequency response function between neutron flux fluctuation and coolant temperature fluctuation. The above expression is nevertheless only an approximation, for several reasons. One reason is that the measured parameters whose spectra are used in (19) are not equal to the core-averaged parameters which are used in the definition of the MTC. Second, the above model is based on linear theory, whereas the core neutronics and thermal hydraulics are

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basically non-linear. At the best, it can be assumed that the above expression is proportional to the MTC, but the scaling is not known. Besides, the method is sensitive to the choice of frequency band. The scaling may depend on burnup, etc., so that the relationship between measurements and the MTC is quite complex. If the r.h.s. of (19) is treated as a frequency dependent function, it may contain enough information such that the correct MTC can be recovered. In other words, the MTC is a functional of the function This relationship is nevertheless rather complicated and not known. On the other hand, if a few calibration points are available (associated pairs of the MTC and an ANN can be trained to learn the mapping of the MTC from time or frequency signals and be used to interpolate between the calibrations (training patterns). This possibility was investigated by Upadhyaya and Glöckler (1991) which is the pioneering work in the field. Training data were generated from a model calculation, simulating a power plant. The trained network was able to give a correct estimation of the MTC as long as the recall patterns lie within the training domain. Boiling water reactor stability. Stability of BWRs under start-up is usually monitored by determining the so-called decay ratio (DR), which is a measure of the decay of the oscillating autocorrelation function (Upadhyaya and Kitamura, 1980). The decay ratio is usually determined by calculating the autocorrelation function from in-core neutron detector signals and then taking the ratio of two consecutive maxima. This gives a correct value if the system is a pure second-order one and without a dominating background noise. In practice, these conditions are seldom fulfilled, thus among others the ratio of consecutive maxima is not constant. Several tricks are known to cope with these difficulties. An alternative possibility is to use an ANN which receives the values of the autocorrelation function at different values, and yields the decay ratio and the resonant frequency as output (van der Hagen, 1995). The training can be made on synthetic ACFs, where also different types of disturbances (e.g. departure from second order system) can be simulated. It is found that this approach is slower than model based methods of determining the DR, but it is more robust against noise and deviations from second order in the system dynamics. Estimation of heat rate performance. The heat rate performance is an operational parameter with economic rather than safety relevance. It is defined as the reciprocal of the thermodynamic efficiency of the plant. Estimation of this parameter requires modelling of the thermodynamic behaviour of the plant. This modelling can be executed by a neural network approach using operational plant data. In one particular study a hybrid ANN approach was used (Guo and Uhrig, 1992). In the first step, the original data are rearranged into several classes or clusters by a self-organizing network. Then these clusters are used as input (training patterns) to a BPN algorithm. This hybrid approach reduces both training time and error. After training, a sensitivity analysis can be performed to determine which operational parameters have the largest influence on the heat rate performance. Due to the speed of the recall part of the trained algorithm, the method can be used on-line, thus being able to improve plant economy. Prediction of thermal power. Plant thermal power is measured by calorimetric methods. On the other hand, the safety instrumentation is based on ex-core neutron detectors. These latter should be calibrated to the thermal power and the calibration updated during the fuel cycle. This can be achieved by an ANN. Such an approach is described e.g. in Roh et al. (1991) where a conventional BPN is used and signal preprocessing is performed before the data are submitted to the ANN input.

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Reactivity surveillance. A knowledge of net reactivity, deduced from core and plant operational data, would make it possible to determine the required control rod position in reactors where the control is achieved by rods. The dependence of the net reactivity on temperature, flow, etc. data is rather complicated and non-linear. However, it can be modelled by an ANN. Training can be performed by data from the reactor. This strategy was tested on the Fast Breeder Test Reactor (Arul 1994). The resulting trained network can actually be used for reactor control as well. Localisation of sources. Two different cases will be mentioned here. The first is a static problem, and it consists of locating the position of a radioactive source in a known medium (Wacholder et al. 1995). This is a typical problem in inverse transport theory. Both feedforward and recurrent networks can be used, and the training can be made by simulated data. A Hopfield-type network was found to be more effective in the above reference. The main incentive in using a recurrent network lies in the fact that it requires much fewer training patterns than a feedforward network. The difference can be quite significant when good spatial resolution is required. The second localisation problem we mention here is that of vibrating control rods from neutron noise measurements. This case will be discussed in detail in Section 5. As was shown in earlier papers (Pázsit and Glöckler, 1983, 1984, 1988), if the reactor physics model is kept simple, a closed form solution for the direct problem can be obtained, from which a direct inversion or unfolding of rod position is possible. However, this traditional localisation is cumbersome and has certain limitations. The localisation can be performed much more effectively with a neural network, trained by simulated noise data. Monitoring thermal margins. One of the key safety parameters in PWRs is the departure from nucleate boiling ratio (DNBR), which is used to estimate thermal margins. Traditionally, this parameter is monitored either by on-line measurements using precalculated safety boundaries (corresponding only to a generic case), or by on-line calculations that need to be simplified and thus made overly conservative. ANNs provide the possibility of making fast determination of the DNBR possible in both stationary and transient state. In a study by Kim et al. (1993) the dependence of the DNBR on five important plant parameters, namely the core inlet temperature, core power (or flux), the enthalpy rise peaking factor core inlet flow rate, and the system pressure, was modelled. The training was based on simulated data provided by the subchannel analysis code COBRA-IV. An interesting feature of the training consisted of a strategy for “randomizing” the input sequences by so-called Latin Hypercube Sampling to secure most effective training with a given pattern set. The trained network estimated the DNBR correctly under both stationary and transient conditions. Analysis of core vibrations. Vibrations of various core components such as the core barrel and core barrel support are usually monitored by ex-core neutron detectors in several PWRs. Neural networks can be used to monitor such vibrations, and verify the integrity of the components (Alguindigue and Uhrig, 1991; Uhrig, 1991). The algorithm identifies both the modes of vibration and further can detect and identify mechanical failures. Test were made using data from 28 PWRs in France with a correct diagnosis in approximately 98% of the cases.

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3.2 Classification, pattern recognition and fault detection As in the case of parameter estimation, this category is not sharply defined either. In general, classification or pattern recognition means the process of establishing the class identity of a sample pattern. The classes are then further divided into normal or healthy and abnormal or faulty, with a possible further sub-classification of the faulty states into different fault types or fault severity. However, sometimes the possibility of detecting faults is based on the observation that certain selected parameters (“features”) show clustering properties in parameter space, with a reliable separation between the normal and abnormal state clusters. Especially with unknown (new) types of failure, the relationship between clustering and fault development or fault type may not be know in advance. In such cases it is also the task of the pattern recognition system to establish the existence and properties of the clusters, after which the fault detection becomes possible. As may be apparent from the above, networks with both supervised learning (both feedforward and recurrent) as well as with unsupervised learning (self-organising maps) are suitable for fault detection. Indeed, it is in this category that we find the largest diversity of neural network architectures and training algorithms. Identification of special faults. One task that often appears is to detect whether or not a certain failure of known type has occurred. Again, as in all diagnostic cases, the task is to find out the failure from indirect measurements. One such example is boiling detection in a nuclear reactor which has been realized using a feed-forward ANN architecture (Kozma et al.,1996). The specific feature of the method applied is the structural learning with forgetting as it allows to infer causal relationships from the measured data. Another example is the detection of a pipe break in the auxiliary feedwater system (Marseguerra and Zio, 1994). A recurrent type of network, a Boltzmann machine, was used in this case with one visible and one hidden layer. The former was further divided into input and output nodes. Training is supervised and is made from simulated data from a plant simulator. In addition, simulated annealing was used to attain a global minimum of the energy (or error) function. A Boltzmann machine was found to be effective for solving the task with high success ratio, but this network algorithm is relatively slow due to its size and the number of iterations required. Applications with handling several fault or transient types. Such works are reported in several papers. The general strategy is to use a large number of plant variables (both from the core, primary circuit and the containment) as input, and have a few expected or known types of failures (e.g. loss-of-coolant, pipe break, loss of power etc.) as output. Feedforward networks are used with backpropagation. The training data can be generated e.g. by a plant simulator. Several versions of BPN can be employed. Bartlett and Uhrig (1992) use a BP network with stochastic optimization where, among others, the size of the network (number of nodes) is also changed dynamically during the training period to achieve optimum network size. Vibration diagnostics of rotating machinery is one case when operational faults can be identified by pattern recognition. Experience shows that operational data cluster around certain centres, one normal and several abnormal clusters corresponding to various faults. ANNs can perform both an adaptive clustering as well as consecutive state identification based on operational data. Thus a combination of two different network types is used sequentially to perform the fault detection.

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ANNs, expert systems and fuzzy logic. Expert systems (rule based algorithms) and fuzzy logic have been used in fault detection and state classification for a long time. In certain cases significant advantages can be achieved by combining them with neural techniques. Neural networks can for instance be used to set up a knowledge base for expert systems. Such constructions are called connectionist expert systems (Cheon and Chang, 1993). Likewise, neural nets can be employed to transform measurement data to fuzzy logical parameters that can be used in a fuzzy system for state identification (Ikonomopoulos et al., 1993) and for the identification of anomalies (Kozma et al., 1995a, 1995 b).

3.3 State estimation, sensor or signal validation Estimation of the validity of signals, i.e. the integrity of sensors monitoring a process is an important part of any diagnostic system. We shall refer to it as signal validation. The interest in signal validation as an autonomous part of process monitoring and diagnostics became very much enhanced after the TMI-2 accident. The accident showed clearly that the possibility of faulty signals can complicate diagnostics immensely. The traditional techniques for signal validation are mostly based on methods of estimation of state variables by means of a Kalman filter or Luenberger observer. The algorithms used are based on the assumption that the system behaviour, i.e. that of the process and the sensor, is governed by the equations

and where X, Y, and U are vectors corresponding to state variables, observed (measured) variables and control variables with dimensions n, m and respectively. The coefficient matrices A , B , C and D are assumed to be time-invariant and have proper dimensions (n, n) , (m, n) and The vectors V and W represent system noise and observation noise. The essential idea of signal validation is to estimate the state variable X by various independent mechanisms functioning with mutually exclusive subsets of Y as input data. Formally, one can estimate by means of the estimation mechanism with input subset where contains one or more components of Y. The only restriction is that the members of should not coincide with the members of The identification of the faulty sensor is performed according to the following rules: a) Define an error limit and an error variable as b) If If

then the sensors included in and are normal; then the sensors included only in or are likely to be faulty. By applying the above rules to various estimations, one can gradually identify the most doubtful set of sensors. As the simplest case, one can define as a single member set, containing only signal i. In this case the principle of identification is easy to interpret. Signal validation techniques with the above or similar schemes were applied to nuclear power plants by several researchers (Kitamura, 1980, 1983; Clark and Campbell, 1982; Tylee, 1982, 1983; Upadhyaya, 1985) with reasonable success as far as the cases tested are concerned. The main obstacle to practical application is that the above approach requires the existence of precisely known linear and time-invariant models (behaviour) of the object. In most practical cases non-linear models must be used which are not known analytically. This

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fact motivates the application of neural networks as models for signal validation. Actually, signal validation is one of the earlier topics of the application of neural networks in the nuclear engineering field (Eryurek and Upadhyaya, 1990; Upadhyaya et al. 1990; Eryurek et al. 1991; Nabeshima et al. 1995). Several architectures of neural networks have been used for this purpose, but the dominating one is again the standard feedforward network with backpropagation. Other architectures that have been used for signal validation is the auto-associative network (Fantoni and Mazzola, 1995). Signal validation, as any item in the broader field called system identification, includes both structural identification and parameter identification. Due to the massively parallel computational regime of neural networks, structural identification may often be nontransparent. In this respect the structural learning algorithm, described in Section 2.4, may be applied more effectively.

3.4 Reactor control Reactor control is a typical inverse task where speed of solution is important. The general aim is to determine the desired value of some control parameter(s) that assure a certain desired state of the system. In such a case the task to be performed is very much the same as in the case of parameter estimation, Section 3.1. Also, the tools applied (network architecture, training) are very much the same too. Application of ANNs to control of nuclear plant or its subsystems is conceptually straightforward, similarly to other types of applications. However, despite the principal simplicity of the application, the number of such cases is astonishingly small in this area. The reason is that unlike diagnostics, control is an intrusive or active process, where much conservatism is encountered due to demanding safety requirements. However, it is no doubt worth evaluating the feasibility of neural network based control of nuclear plant because all advantages of neural paradigms mentioned earlier exist also in this area. The early studies (Carre, 1991; Parlos, 1992) addressed the applicability of ANNs to control problems, but the demonstrations remained at the stage of empirical model identification. Below we only give a short list of such applications that are known in this field. Reactivity control. The method of estimating net reactivity from plant parameters, described in Section 3.1, is also suitable to be used for reactor control. The value of net reactivity, determined by the output of the network, can be used to control the rod axial elevation. Optimal reactor temperature controller. To achieve good performance of a plant such that large variations in power can be accommodated requires handling of non-linear dynamics. Neural nets are capable of handling such problems well. This was illustrated e.g. in case of an optimal temperature controller (Ku et al., 1991). A backpropagation network was used with the power level, control rod gain and desired output as network inputs, and the optimal state feedback gains as the network output. The learning algorithm was based on minimizing the least-squares error between the plant and the reference model over a time interval. This way the number of training cycles could be significantly reduced. ANNs as an aid in identifying control goals. One special type of control is safety control, although it is not control in the conventional sense. ANNs can be useful to set up guidelines for decision support by analysing different fault trees of various accident scenarios and the

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probabilities of occurrence, detection and consequence. Such networks use specialised methods such as the Sutton-Barto algorithm (Jouse and Williams, 1993). Autonomous plant project. A more advanced and broader control application can be found within the autonomous plant project (Ugolini et al., 1994, 1995) which is conducted by a group of several Japanese research institutes under the initiative of the Japanese Science and Technology Agency. One of the goals of the project is to develop fault-tolerant, self-evolving control techniques. Neural networks are one of the tools being considered for this purpose. Two basic schemes of neuro-control are illustrated in Figs 5 and 6. The main

idea is to train the neural network to emulate the input operation to minimize the mismatch between the plant output and the desired output. In short, the neural network is trained to generate inverse dynamics of the objective plant. However, this training is not so easy since the inverse mapping is not one-to-one. In order to make the problem tractable, various modifications of the basic scheme have been studied (Kitamura, 1992).

The basic concept needs to be further modified if one wants to cover the change in system dynamics along with the operational conditions. In other words, the model of the plant to be used in the control system needs to be dependent on operational condition and history. The incorporation of an adaptive scheme is a natural solution to fulfil this requirement. The proposed control scheme becomes more involved, as shown in Fig. 7. The aim of this control system is to manipulate the steam temperature at the outlet of the

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evaporator of a fast breeder reactor (FBR). Two neural networks are incorporated in the scheme: one for system identification and the other for control action generation. As shown in Fig. 8, the neural network employed for system identification is a semi-autoassociative network with seven inputs, ten intermediate units and four outputs. All the output variables are included in the input variables as well. The identification ANN is trained to minimize the error between plant output and output of its own. The information about the weights update is conveyed to the controller ANN, which is supposed to generate the control action similar to the reference controller designed to provide a pre-defined optimal performance. The control scheme is thus called model reference adaptive control (MRAC) system (Narendra and Annaswamy, 1989; Ugolini et al., 1995).

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The performance of the MRAC system has been tested with the FBR evaporator with several transients in the inlet sodium temperature, leading to fairly good performance. However, because of the innovative nature of the MRAC controller, additional modification of the control scheme was required to prevent any unexpected behaviour of the controller while maintaining the high performance. To this end an aggregated controller scheme was introduced as illustrated in Fig. 9. The supervisor (decision maker) unit selects the best controller action among a set of candidates including conventional PI (proportional and integral) controller and LQG (linear quadratic Gaussian) controller. Although the neural controller is supposed to show sufficiently high performance, other controllers are standing by to substitute it whenever judged necessary. By having the traditional PI controller within the scheme, scepticism and concern on the applicability of the advanced controller was significantly calmed down. System integration of this kind would always prove to be useful prior to the final acceptance of a novel technique.

4. SOLVING INVERSE PROBLEMS WITH NEURAL NETWORKS IN CORE NOISE DIAGNOSTICS Diagnostics of power reactor cores with neutron noise methods is based on the fact that fluctuations in reactor material, given rise to by various technological processes, lead to the fluctuations of the neutron flux (neutron noise) that are specific to the inducing phenomena. Specifically, any perturbation or deviation from the normal state, such as vibrations, boiling and two-phase flow transport, constitute a space-time-dependent fluctuation of the local macroscopic cross sections around their nominal values. The effect of these fluctuations on the neutron yield can be calculated through the time-dependent transport or diffusion

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equations. Thus for any perturbation or noise source, the induced neutron noise can be determined. The strategy of neutron noise diagnostics is described in a few books and review papers (Williams, 1974) but we will briefly summarize it here for later use. For simplicity of notation one-group diffusion theory will be used throughout. The main principles remain the same in more complicated reactor physics models. One starts with a critical system and the static equations for the flux and precursor density as

(23) (24) Here,

and all symbols have their usual meaning. For simplicity all material constants are assumed space-independent here. The occurrence of a perturbation (noise source) will then be associated with the change of the cross sections in the form of a space-dependent fluctuation around the static (critical) value as The effect of this perturbation will be that the neutron flux will also show fluctuations around the critical flux: The fluctuations in both the perturbation and the neutron noise are assumed to be first-order small quantities. The relationship between them can be derived in first-order theory as follows. One starts with the time-dependent equations

Putting (26) and (27) into the above, neglecting the second order term and making use of the critical equations, the remaining terms can be Fourier transformed in time. Eliminating the delayed neutron precursors leads to the following equation for the neutron noise in the frequency domain: where

and

In (31),

is the well-known zero power reactor transfer function.

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Since (30) is an inhomogeneous equation, its solution can be given by the Green’s function technique. Let us define the Green’s function or system transfer function as the solution of

Then the solution of (30) is given as

Eqn (34) is the basis of core diagnostics with neutron noise methods. It shows that the effect of a perturbation can be factorized into a noise source term and a system transfer As (33) shows, the system transfer is only dependent on the parameters of the unperturbed system, which is a consequence of using linear theory. The transfer function can thus be calculated for a reactor independently of the type of perturbation. Hence, can be assumed to be known in a diagnostic task. The task of diagnostics is to quantify the noise source from the measured neutron noise using the knowledge of the transfer function. At a first sight it may appear that this is a direct task if eqn (30) is used. In this latter the transfer function does not need to be known, only the frequency dependent buckling If is known from measurement, then it may seem that all that needed is to apply the operator on to obtain the noise source. The fact that this is both impossible and impractical is, however, obvious. First, is measured only in a few discrete points, thus the Laplace operation cannot be performed (or only in a too coarse way to be useful). In addition, even if this was possible, this operation gives the noise source only locally (in the detector position and nowhere else). This is very seldom needed or useful. One usually diagnoses a noise source by measuring the neutron noise at some distance away. This is the very essence of noise diagnostics in that through the spatial propagation of the neutron noise, a detector collects information over a large area a distance away from the detector. Thus (30) cannot be used. Rather, (34) must be employed since it displays how values of the noise source and the neutron noise in different points are related through the transfer function. Determining from through (34) is an inverse task. Formally, if and are known, (34) is a Fredholm-type integral equation for the unknown which may be inverted. However, the difficulty that has already been mentioned arises even here in that is only known in a few discrete points, i.e. in the detector positions. If nothing is known about inversion of (34) has only a chance if a relatively large number of detectors is available at a time, which is normally not the case in practice. Thus, usually another approach is selected. One assumes a certain type of anomaly or perturbation, such as two-phase flow transport, vibrations etc. Such assumptions are usually made by studying the frequency content of the measured noise, or the type of noise source may be apparent from other, independent measurements. A given noise source is then represented by a simple functional form, in which only a few parameters remain as unknowns. In the next Section such a case will be discussed. Then, it may be possible to determine those few parameters by using a few neutron detectors. Reducing the arbitrariness of the noise source form into a given function with a few unknown parameters, is thus essential if spatial reconstruction of the noise source is attempted. However, this is not necessarily sufficient for elaborating a method or algorithm for unfolding those parameters from the neutron noise if at the same time the neutron

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physical transfer is not sufficiently simple. If can only be calculated numerically, then any analytical or functional relationship between the noise source and the neutron noise is lost too. The inversion or unfolding procedure is then, in general, prohibitively complicated. This fact too will be returned to in the next Section. Due to the above circumstances, in the overwhelming majority of neutron noise diagnostic methods, extremely simple neutron physical models have been used such that the transfer function could be given in a closed form or at least in form of an expansion of orthogonal eigenfunctions of the Helmholtz operator. Then there will be an analytical or functional dependence of the neutron noise on the searched noise source parameters. This gives both insight into the character of the noise induced by a specific noise source, and a possibility for inversion. Much of the known diagnostic work stayed at this level. The most complicated neutron noise model ever used in a diagnostic investigation until recently was still based on homogeneous reflected system. Only very recently has the possibility of nodal noise models been considered (Pázsit 1992; Makai 1994; Holstein 1995). The reasons for this lie both in the difficulties in calculating the complex, frequency-dependent function in a realistic core as well as the difficulty of using such a transfer function in an inverse problem. It is at this point that the neural nets may bring about a breakthrough. An ANN can be trained to learn the inverse mapping between two function sets if training patterns, calculated from the direct task, are available in sheer numerical form. When neural techniques are used for the solution of the inverse task, the complexity of the transfer function does not play any role. This, in turn, may give some extra impetus for elaborating codes for the nodal calculation of the transfer function in realistic (i.e. non-homogeneous) cores. The impact of ANN techniques is thus twofold: they may help the transition of noise methods from model problems to realistic problems, and they may expedite the development of advanced core transfer function calculation methods.

5. CASE STUDY: NEUTRON NOISE DIAGNOSTICS OF CONTROL ROD VIBRATIONS 5.1 Theory of vibration induced noise and the traditional localisation algorithm In this Section the use of neural networks will be illustrated in a diagnostic problem. The problem is the localisation of a vibrating control rod by neutron noise methods. This problem was solved earlier with conventional methods (Pázsit and Glöckler, 1984). It will serve as an example to illustrate noise source unfolding in general, and the advantages provided by the neural approach. The history of neutron noise induced by two-dimensional (2-D) vibrations of control rods is described in earlier publications (Pázsit and Glöckler, 1983). The fact that a vibrating control rod leads to detectable neutron noise is known from the ORR and HFIR reactors (Fry, 1971). This meant that at least the occurrence of stronger than normal vibrations could be detected by neutron noise measurements. Later it turned out that control rod vibrations can occur even in power reactors (Lucia et al., 1973). Since excessive vibration is always a sign of beginning malfunction, detecting and locating vibrations by neutron noise methods became an interesting task. To this end, first the neutron noise, induced by control rod vibrations had to be solved. A solution, based on Green’s function techniques and assuming a weak absorber, was employed in the traditional localisation algorithm, and this solution will be reviewed here.

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The key point is the construction of a model describing the vibrating rod with a simple analytical form. A two-dimensional model will be used throughout, since the rod will be assumed to vibrate only laterally without showing any axial dependence. Thus dependence on the axial variable will be completely neglected and a two-dimensional cylindrical reactor model will be used. The flux will be treated in a horizontal cross-section of the reactor only, with the coordinate being specified in either Cartesian or planar polar co-ordinates. An axial control rod, being perpendicular to this plane, will be characterised by its position on this plane. Since the rod diameter is much smaller than system size (core diameter), it is convenient to assume that the rod can be treated as a point on this plane. Its contribution to the macroscopic cross sections will be described by a 2-D Dirac function. Thus, the steady (not vibrating) control rod at is described by

where is the rod equilibrium position and is the so-called Galanin’s constant, which characterizes the rod strength. When vibrating, the rod will move on a two-dimensional stochastic trajectory around the equilibrium position such that its momentary position will be given by where and is the core radius. Then the perturbation represented by the vibrations is given as Eqn (37) shows that the vibrating rod is characterized by the parameters and The latter describe the amplitude and the frequency content of the vibrations. The purpose of the diagnostics is to determine these parameters. From (32), (34) and (37) everything is available for a formal solution except that is not explicitly available since eqn (37) cannot be Fourier-transformed directly. However, performing the spatial integral in (34) first, then utilizing the smallness of the vibration amplitude through a one-term Taylor expansion in this latter appears explicitly and can be Fourier transformed. One then obtains for the neutron noise in the frequency domain

where

and similarly for It is implicit in the above that both the static flux and the Green’s function are calculated in a homogeneous reactor without the static rod being present. That is, they are calculated from (23) and (33), respectively. This leads to simplifications in the calculation of the flux and the Green’s function. This approximation corresponds to neglecting higher orders of the rod strength in (38) and thus it is called the weak absorber approximation. It is applicable in most practical cases. Eqn (38) shows that the vibration components and appear explicitly in the noise expression, whereas the rod position is contained implicitly. One can thus eliminate the former and attempt to determine the latter first. This process is called localisation. The procedure is as follows. One selects three detectors at positions with an arrangement as e.g. in Fig. 10. Denoting the detector signals by one has

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for all three signals, with Using two equations of the type (40), the displacement components can be expressed by, say, and Using them in the third equation results in an expression of the form Here,

etc. Eqn (42) is called the localisation equation, and it constitutes the formal solution of the localisation problem. In this equation all quantities are known from measurement or calculation, except the argument i.e. the searched rod position. This parameter is determined as a root of the equation. The localisation in a given case is thus performed as finding the root of the complex equation (42). In practice, however, the procedure is somewhat more complicated. Namely, for stationary random processes, it is not the Fourier transform of the signals that is used, rather the auto- and cross-spectra of the signals. This means that the auto- and cross-power spectra of i.e. and respectively, are used. Likewise, instead of and the auto- ana cross-spectra and of the displacement components need to be used as input source. With the application of the Wiener-Khinchin theorem the auto- and cross-spectra of the detector signals can be expressed from (40) as

In the above form, account was taken of the fact that both the Green’s function and the displacement cross-spectra are real in the simplified model used in this paper. The localisation procedure means eliminating the vibration spectra etc., and deriving an equation similar to (42) in which only the transfer functions and the neutron noise spectra are present. Such an equation can be formally written down with the help of the spectral and transfer matrices in the form

where

is the neutron noise spectral matrix, and

is the transfer function matrix.

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Taken component-wise and in different combination of the equalities, eqn (46) is actually several independent equations. They are the spectral equivalent of (42). As before, the rod position is given as the joint root of all equations. As described in (Pázsit and Glöckler, 1984), the roots of any single localisation equation lie along a complete “localisation curve” on the plane; the sought rod position is determined by the joint intersection in one point of all localisation curves. The above described procedure was investigated in numerical tests and it was also applied in one case at operating plant successfully. We shall describe the numerical tests first. In these, noise data were generated by selecting a rod position and the vibration spectral properties and With these the neutron detector auto- and cross-spectra can be calculated through (44) and (45) and then these were used in (46) to reconstruct the known rod position. The purpose was to see whether the inversion is unique, to test the sensitivity to (simulated) background noise, and to check the calculational load of the procedure. For the concrete calculations, a reactor model had to be chosen such that and G can be calculated, and realistic values of the displacement spectra had to be selected. Regarding the first, (39) shows that the derivative of the Green’s function in the second spatial argument is needed. Thus a model that allows a solution that is analytical in the second argument is advantageous. For this and other reasons, a bare homogeneous 2-D cylindrical core was chosen. In addition, the so-called power reactor approximation was used, according to which one can set in (33). This converts it from a Helmholtz equation into a Poisson one, which, with vacuum boundary conditions, leads to a closed analytical solution in the form

Regarding the vibration spectra, in the simplest non-trivial random model these can be parametrized by two variables, an ellipticity (anisotropy) parameter and the preferred direction of the vibration as

The above core transfer and vibration model, expressed through eqs. (50)-(53) will also be used in this paper for the ANN-based localisation. Especially, the parametrization of the vibration spectra by and will prove very useful. Some sample simulation tests are shown in Fig. 10 for two different rod positions and vibration patterns. In these simulations background noise was added to the calculated signals before performing the localisation. As a result, the multiple intersection of all curves in one point, which would indicate the true rod position, is split up into pair-wise intersections that lie over a certain area instead of constituting one point. Thus in certain cases several areas can be suspected, as in the case shown. This may make the localisation process ambiguous in certain cases. Despite this difficulty, the method was applied with success once in a real case at a PWR plant during operation. This application is described in detail in Pázsit and Glöckler (1988) and will be returned to below.

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For a comparison with the ANN-based version of the localisation procedure, which will be described in the next Section, we review in sequence how the traditional localisation procedure is applied. a) select one equation from (46); b) select an value and calculate the corresponding which yields a root of the equation; c) repeat b) with several other values such that a whole localisation curve is calculated; d) repeat a) - c) with several different equations from (46) such that all localisation curves are calculated; e) plot all curves and pick the highest order intersection point. The computational load of this procedure is now clear. Since the root searching is performed numerically, step b) alone requires several iterations, i.e. calculations of six different transfer functions Step c), yielding one complete localisation curve, incurs repeating the above a few hundred times, bringing the total number of transfer function calculations in the range of Then the whole procedure is repeated a few times to get several different localisation curves. The calculation of just one function is more complicated than that of the static flux since G is both complex and has more arguments. In a realistic reactor model performing the above procedure can easily lead to several days computing time, prohibiting on-line applications. Of course the computational burden could be severely reduced if, instead of searching on the whole two-dimensional plane, the search could be confined to the few positions of the control rods in the reactor. However, the above algorithm does not make this possible. A second difficulty is the selection of the highest-order intersection of the localisation curves, which can only be made by a subjective decision. The third inconvenience is that if more than three detectors should be available simultaneously, the redundancy represented by the extra detectors cannot be utilized in an effective way. It could only lead to more localisation curves which would hardly improve the readability of the plots. As we shall see below, all these problems can be alleviated or eliminated by the use of neural network methods.

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5.2 Rod localisation via a feedforward backpropagation network The principle is the following. A three-layered feedforward network can be chosen for a given reactor configuration. The number of input nodes is equal to the number of detector auto- and cross-spectra, and the number of output nodes is equal to the number of control rods in the core. The network can then be trained such that for a given set of input spectra, it

identifies one rod as the vibrating one. This latter is made by assigning an output equal to 1 to the node corresponding to the suspected rod, and assigning zero to the others. In the case described here a configuration with seven rods and 3 or 4 detectors will be discussed (Fig.l 1). The corresponding network architecture in the 3-detector case is shown in Fig. 12.

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The training is done by using training patterns, i.e. associated rod position/vibration data as well as neutron spectra. These are generated by (44) and (45). One has to generate a sufficient number of training patterns such that all possible cases are sufficiently covered. In this procedure it is convenient that the vibration spectra of a given rod can be parametrized by only two variables, and of (51) - (53). The entire variety of possible vibrations can be covered by varying the control rod number (identity) as well as and With 7 control rods, this means a few thousand training patterns. The above number is actually significantly less than the number of corresponding calculations one needs to perform in one single localisation in the traditional method. Further, since the training is performed off-line, there is no constraint on the speed of the calculation. Thus here we see immediately an advantage of the ANN approach in that the calculationally demanding part of the procedure can be performed off-line, and that it is still much less burdensome than the calculational load of the traditional algorithm. Once the training patterns are available, the training is a rather routine process and thus will not be discussed much in detail here. The adjustable parameters, such as the number of training patterns, number of nodes in the hidden layer, as well as the learning and momentum rates, need to be fine-tuned for optimum performance. This was performed with both 3 and 4 detectors. To indicate the increased efficiency of the 4-detector case over the 3-detector one, the dependence of the network error on the progress of the training is shown in Fig. 13.

When the network is trained, performing one localisation (the recall mode of the network) requires only one input-output mapping, which is done in a very short time as usual with feedforward networks. Thus in the recall phase the difference in computing time between the traditional and ANN methods is indeed tremendous. Besides, the running time of the network in the recall mode is independent of the level of complexity which is used in producing the training patterns. This complexity only affects the training time, or rather the time it takes to produce the training patterns. As is obvious, 4 detectors at a time can be used without difficulty. The computation time for producing the training patterns will be larger in this case, but the number of training cycles needed to reach a certain network error will be decreased due to the redundancy in the detector signals. Moreover, the “success ratio”, i.e. the percentage of correct identifications, will be higher than in the 3-detector case. Numerical tests (Pázsit et al., 1996) show that a

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success ratio of about 99.6% can be achieved with a reasonable training time using 4 detectors. The identification process in this method is completely transparent, explicit and objective. There is no need for a subjective decision since the output of the network is directly the identity of the suspected rod. The ANN-based localization procedure was tested on real data, taken in a PWR with an excessively vibrating control rod. The algorithm made a correct identification of the faulty rod. Details of this ANN-based identification are described in (Pázsit et al., 1996). Here again the successful application of an ANN, trained on simulated data, to a real case, is seen.

6. CONCLUSIONS It has been seen that the ANN-based algorithm is more effective, simpler and faster in every respect than the conventional method. In our view, the main advantage of the use of ANNs is the fact that they do not require the use of analytical or simple core models, they work just as well with any numerical model. Thus they make it possible for neutron noise based core diagnostic methods to move out from the stagnation point where they have been in the past decade, and may lead to a renewed effort in the development of advanced methods for transfer function calculations. In the more direct or more empirical diagnostic and control application where there is no need to elaborate an underlying calculational model because either the training patterns can be obtained from measurements or an existing plant simulator, the applications have already gone a long way. With the further development of both software and hardware, there are still more applications to come. However, it would be fair to note here a related difficulty in the diagnostic application of ANNs. This is the fact itself that there is in general a lack of training data from real world. Because it is unlikely to have access to enough plant data, covering sufficient number of anomalies, one has to rely on numerical data from simulation codes. Therefore, ANN applications are always subject to severe scepticism on their dependability. Of course, the same criticism could apply to other methodologies as well. Any other diagnostic technique, e.g. expert system technology, is subject to the same criticism insofar as realistic anomaly data are not available. A reasonable claim is that more research effort should be invested to improve the fidelity of numerical simulations. More significantly, however, it seems necessary to establish a technique to provide a modestly reliable solution and the credibility (self-confidence) about the solution even for inexperienced (untrained) input data. To allow the neural network to output “I am not certain”, or “my guess of the failure is the mode-X, but only with moderate confidence” is of crucial importance in making the network credible, robust and user-friendly. Also, it is imperative to have a learning or data updating capability whenever the network meets the inexperienced events. Though a lot of issues remain to be solved, the hybrid neuro-fuzzy paradigm could be one of the promising ways to meet the need. One of the benefits of employing neural network techniques, and fuzzy logic as well, is the ease of practical implementation. The value of this benefit has been underestimated for a long period of time. Rather, the method has often been criticized since it lacks theoretical foundations, completeness, etc. This criticism is one-sided in the sense that any of the theoretically rigid methods is also subject to the similar defects in applications to complex, large scale system. The theoretical method in its standard form usually requires a mathematical model of the objective systems. The model is, by definition, an abstract and simplified subclone of the objective system with many simplifying assumptions. For

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successful implementation of model-based diagnosis or control, introduction of significant simplifications is inevitable. Therefore, the diagnosis/control schemes can be rigorous within the simulated regime, but not in the real world. On the contrary, neural computing can work without simplifying assumptions. The only requirement is availability of sufficient data to train the network. More precisely, the system performance can be improved evolutionally along with the increase in real world experiences. We are tempted to conclude that the development of diagnosis/control technique based on neural network is more practical and realistic in this respect. The importance of hybrid computational paradigms such as combination of neuroknowledge base, neuro-fuzzy, fuzzy-neuro and other methods should be stressed since we cannot rely on a single paradigm to create or simulate intelligent behaviour of the designed system. This recognition can be well endorsed by recent trends in diagnosis/control applications, where the usage of multiple paradigms is becoming a common practice. We do not suggest, however, just a blind search for multiple paradigms with naive expectation of improved performance. A proper allocation of functions and sub-functions to allocate the various building blocks with individual computational discipline would appear to be a promising path towards reliable machine intelligence to support humans in case of real problems.

ACKNOWLEDGEMENT The authors are indebted to Dr. O. Glöckler of Ontario Hydro, Dr. N.S. Garis of Chalmers University of Technology and Dr. R. Kozma of Tohoku University for many stimulating discussions and joint work in the field. Thanks are due to Dr. Garis for helping with the literature survey and to Mr. S. Takei for technical help.

7. REFERENCES Alguindigue, I.A., Uhrig, R.R., 1991, Vibration monitoring with artificial neural networks, Proceedings of SMORN-VI, Gatlinburg, 59.01 Arul, A.J., 1994, Reactivity surveillance on a nuclear reactor by using a layered artificial neural network, Nucl. Sci. Engng 117:186 Bartlett, E.B., and Uhrig, R.E., 1992, Nuclear power plant status diagnostics using an artificial neural network, Nucl. Technology 97:272 Carre, J.C. and Martinez, J.M., 1991, Approach to identification and advanced control of PWR using neural networks, Proceedings of SMORN-Vl, Gatlinburg, 56.01 Clark, R.N. and Campbell, P., 1982, Instrument fault detection in a pressurized water reactor pressurizer, Nucl. Technology 56:23 Cheon, S.W., and Chang, S.H, 1993, Application of neural networks to a connectionist expert system for transient identification in nuclear power plants, Nucl. Technology 102:177 Churchland, P.S., and Sejnowski, T., 1992, The Computational Brain, MIT Press Eberhart, R.C., and Dobbins, R.W., 1990, Neural Network PC Tools, A Practical Guide, Academic Press, Inc., N.Y. Eryurek, E. and Upadhyaya, B.R., 1990, Sensor validation for power plants using adaptive backpropagation neural networks, IEEE Trans. on Nuclear Sci. Symp.. San Francisco 37:1040 Eryurek E, and Türkcan, E., 1991, Neural networks for sensor validation and plant-wide monitoring, Int. Rep. ECN-RX--91-057, June 1991

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Fantoni, F.P., and Mazzola, A., 1995, Transient and steady signal validation in nuclear power plants using autoassociative neural networks and pattern recognition, Proc. SMORN-VII, 19-23 June 1995, Avignon, France 13.5 Freeman, A. and Skapura, D.M., 1991, Neural Networks, Algorithms, Applications and Programming Techniques, Addison-Wesley Publishing Co., NY. Fry, D.N., 1971, Experience in reactor malfunction diagnosis using on-line noise analysis, Nucl. Technol. 10:273 Grossberg, S., 1988, Neural Networks and Natural Intelligence, The MIT Press, Cambridge, Massachusetts Quo, Zh., and Unrig, R.E., 1992, Use of artificial neural networks to analyze nuclear power plant performance, Nucl. Technology 99:36 Haykin, S., 1994, Neural Networks: A Comprehensive Foundation, Maxmillan/IEEE Press Hebb, D.O., 1949, The Organization of Behaviour, John Wiley, New York. Herr, J.D. and Thomas, J.R.,1989, Low frequency coolant temperature and neutron flux perturbations, Proc. 7th Power Plant Dynamics, Control and Testing Symposium, Knoxville, Tennessee 50.01 Hertz, J., Krogh, A., and Palmer, R.G., 1991, Introduction to the Theory of Neural Computation. AddisonWesley, N.Y. Hollstein F., Meyer K., 1995, Calculation of neutron noise due to control rod vibrations using nodal methods for hexagonal-Z-geometry, Proc. SMORN-VII, 19-23 June 1995, Avignon, France 12.1 Hopfield, J.J., 1982, Neural networks and physical systems with emergent collective computational abilities. Proc. Natl. Acad. Sci. 79:2554 Hopfield, J.J., 1984 Neurons with graded have collective computational properties like those of two-state neurons. Proc. Natl Acad. Sci. 81:3088 Ikonomopoulos, A., Tsoukalas, L.H., and Uhrig, R.E., 1993, Integration of neural networks with fuzzy reasoning for measuring operational parameters in a nuclear reactor, Nucl. Technology 104:1 Ishikawa, M., 1995, Learning of modular structured networks, Artificial Intelligence, 75: 51 Jouse, W.C:, and Williams, J.G., 1993, Safety control of nuclear power operation using self-programming neural networks, Nucl. Sci. Engng 114:42 Kim, H.K., Lee, S.H., and Chang, S.H., 1993, Neural network model for estimating departure from nucleate boiling performance of a pressurized water reactor core, Nucl. Technology 101:111 Kitamura, M., 1980, Detection of sensor failures in nuclear plant using analytic redundancy, Trans. Am.Nucl. Soc. 34:351 Kitamura, M., 1983, Use of analytic redundancy for surveillance and diagnosis of nuclear power plants, Proc. 5th Power Plant Dynamics, Control and Testing Symposium, Knoxville, Tennessee 33.01 Kitamura, S., 1992, Neural networks and inverse problems in measurement and control, Proc. IMACS/SICE International Symposium on Robotics, Mechatronics and Manufacturing Systems. Sep. 16-20,1992, Kobe, 205 Kohonen, T., 1972, Correlation matrix memories I.E.E.E. Transactions on Computers C 21(4):353 Kosko, B., 1992, Neural Networks and Fuzzy Systems, Prentice Hall, NJ. Kozma, R., Yokoyama.Y., and Kitamura, M., 1995a, Intelligent monitoring of NPP anomalies by an adaptive neuro-fuzzy signal processing system, Proc. Topical Meeting on Computer-Based Human Support Systems: Technology, Methods, and Future, June 25-29, 1995, Philadelphia, 449. Kozma, R., Sakuma, M., Sato, S., and Kitamura, M., 1995b, An adaptive neuro-fuzzy signal processing method by using structural learning with forgetting, Intelligent Automation and Soft Computing, 1:389 Kozma, R., Kitamura, M., and Sato, S., 1996, Monitoring of NPP state using structural adaptation in a neural signal processing system, Proc. 1996 ANS International Topical Meeting on Nuclear Plant Instrumentation, Control, and Human-Machine Interface Technologies, May 6-9, 1996, The Pennsylvania State University, 273 Kim, K., and Bartlett, E.B., 1994, Error prediction for a nuclear power plant fault-diagnostic advisor using neural networks, Nucl. Technology 108:283 Kohonen, T., 1984, Self-Organization and Associative memory, Springer-Verlag, Berlin

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Ku, C.C., Lee K.Y., and Edwards R.M., 1991, Neural network for adapting nuclear power plant control for wide-range operation. Trans. Am. Nucl. Soc. 63:114 Lucia, A., Ohlmer, E., and Schwalm, D., 1973, Correlation between neutron noise and fuel element oscillations in the ECO reactor, Atomkernenergie, 22:6 Majumdar, M.C., Majumdar, D and Sacket, J.I. (ed.), 1988, Proceedings of Topical Meeting on AI and other Innovative Computer Applications in Nuclear Industry, Plenum Press. Marseguerra, M., and Zio, E., 1994, Fault diagnosis via neural networks: the Boltzmann machine, Nucl. Sci. Engng, 117:194 Makai M., 1994, Nodewise analytical calculation of the transfer function, Ann. nucl. Energy 21:519 McCulloch, W.C, and Pitts, W., 1943, A logical calculus of the ideas immanent in nervous activity. Bulletin of Math. Biophysics, 5:115 Michal, R.A., 1994, Before the year 2000: Artificial neural networks may set the standard, Nucl. News July 1994:29 Minsky, M.L., and Papert, S., 1969, Perceptrons, MIT Press, Cambridge, MA. Nabeshima K., Suzuki K., and Türkcan E., 1995, Neural network with an expert system for real-time nuclear power plant monitoring. Proc. SMORN-VII, 19-23 June 1995, Avignon, France 4.1 Narendra, K.S., and Annaswamy, S., 1989, Stable Adaptive Systems, Prentice Hall, Inc., Englewood Cliffs, New Jersey. Ohga, Y., and Seki, H., 1993, Abnormal event identification in nuclear power plants using a neural network and knowledge processing, Nucl. Technology 101:159 Parlos A.G., Atiya, A.F., Chong, K.T., and Tsai, W.K., 1992, Nonlinear identification of process dynamics using neural networks, Nucl. Technology 97:79 Pázsit I., and Glöckler O., 1983, On the neutron noise diagnostics of PWR control rod vibrations I. Periodic vibrations. Nucl. Sci. Engng. 85:167 Pázsit I., and Glöckler O., 1984, On the neutron noise diagnostics of PWR control rod vibrations II. Stochastic vibrations. Nucl. Sci. Engng. 88:77 Pázsit I., 1992, Dynamic transfer function calculations for core diagnostics. Ann. nucl. Energy 19:303 Pázsit I., and Glöckler O., 1988, On the neutron noise diagnostics of PWR control rod vibrations III. Application at power plant. Nucl. Sci. Engng. 99:313 Pázsit I., Garis N.S., and Glöckler O., 1996, On the neutron noise diagnostics of PWR control rod vibrations IV. Application of neural networks. To appear in Nucl. Sci. Engng. Proceedings, 1990, Proceedings of the Topical Meeting On Advances In Human Factors Research On Man/ Computer Interactions: Nuclear and Beyond, Nashville, Tennessee. Proceedings, 1991, Proceedings of the AI91, Frontiers in Innovative Computing for the Nuclear Industry, Jackson Lake, Wyoming. Proceedings, 1993, Proceedings of the Topical Meeting on Nuclear Plant Instrumentation, Control and ManMachine Interface Technologies, Oak Ridge, Tennessee. Proceedings, 1995a Proc. 9th Power Plant Dynamics, Control and Testing Symposium, Knoxville, Tennessee Proceedings, 1995b, Proc. SMORN-VII, 19-23 June 1995, Avignon, France Roh, M.S., Cheon, S.W., and Chang, S.h., 1991, Power prediction in nuclear power plants using a back-propagation learning neural network, Nucl. Technology 94:270 Rosenblatt, F., 1958, The perceptron: a probabilistic model for information storage and organization on the brain. Psychological Review, 65:386 Rumelhart, D.E:, and McClelland, J.L., 1986, Parallel Distributed Processing, Explorations in the Microstructure of Cognition, Vol. I: Foundations. MIT Press, Cambridge, MA Rumelhart, D.E., Hinton, G.E., and Williams, R.J., 1986, Learning internal representations by error propagation, in: Parallel Distributed Processing, MIT Press, Cambridge, MA. Shieh, D.J., Upadhyaya, B.R.and Sweeney, F.J, 1987, Application of noise analysis technique for monitoring the moderator temperature coefficient of reactivity in pressurized water reactors, Nucl.Sci. and Engng. 95:14

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Thomas, J.R. and Clem, A.W, 1991, PWR moderator temperature coefficient via noise analysis: time series methods, Proceedings of SMORN-VI, Gatlinburg, 34.01 Thomas, J.R. and Adams, J.T., 1994, Noise Analysis Method for Monitoring the Moderator Temperature Coefficient of Pressurized Water Reactors: Neural Network Calibration, Nucl. Technology 107:236 Tou, J.T., and Gonzalez, R.C., 1974, Pattern Recognition Principles, Addison-Wesley Publishing Co., Inc., Reading, Massachusetts. Tylee, J.L., 1982, A generalized likelihood ratio approach to detecting and identifying failures in pressurizer instrumentation, Nucl. Technology 56:484. Tylee, J.L., 1983, On-line failure detection in nuclear power plant instrumentation, IEEE Trans, on Automatic Control, AC-28:406 Ugolini, D., Yoshikawa, S. and Endou, A., 1994, Implementation of a model reference adaptive control system using neural network to control a fast breeder reactor evaporator, Proc. 2nd Specialists' Meeting on Application of Artificial Intelligence and Robotics to Nuclear Plants, AIR'94, JAERI, Tokai, Japan, 157 Ugolini, D., Yoshikawa, S. and Ozaki, K., 1995, Improved neural network based MRAC system to control an FBR evaporator, Proc. 9th Power Plant Dynamics, Control and Testing Symposium, Knoxville, Tennessee 52.01 Uhrig, R.E., 1990, Use of artificial intelligence in nuclear power plants, Proceedings of the Topical Meeting on Advances in Human Factors Research on Man/Computer Interactions: Nuclear and Beyond, Nashville, Tennessee, 210 Uhrig R.E., 1991, Potential application of neural networks to the operations of nuclear power plants, Nuclear Safety, 32:68 Upadhyaya, B.R., and Kitamura, M., 1980, Stability monitoring of boiling water reactors by time series analysis of neutron noise, Nucl. Sci. Engng. 77:480. Upadhyaya, B.R., 1985, Sensor failure detection and estimation, Nuclear Safety, 26:32 Upadhyaya B.R., Eryurek E., and Mathai G., 1990, Neural networks for sensor validation and plant monitoring, Proc. Int. Fast Reactor Meeting, 12-16 August 1990 Snowbird, Utah Upadhyaya, B.R., Glöckler, O., 1991, Estimation of feedback parameters in pressurized water reactors using neural networks. Proc. SMORN-VI, May 19-24 1991, Gatlinburg, Tennessee, USA 58.01 van der Hagen, T.H.J.J., 1994, The scaling parameter of the sigmoid function in artificial neural networks, Nucl. Technology 106:135 van der Hagen, T.H.J.J., 1995, Artificial neural networks versus conventional methods for boiling water reactor stability monitoring, Nucl. Technology, 109:286 Vitela, X.E., and Reifman, J., 1994 Accelerating learning of neural networks with conjugate gradients for nuclear power plant applications, Nucl. Technology 106:225 Wacholder, E., Elias, E., and Merlis, Y., 1995 Artificial neural networks optimization method for radioactive source localization, Nucl. Technology 110:228 Wasserman, Ph. D., 1989, Neural Computing: Theory and Practice, van Nostrand Reinhold, NY. Werbos, P., 1990, Backpropagation through time; what it does and how to do it? Proc. IEEE 78:1550 Widrow B., and Hoff, M.E., 1960 Adaptive switching circuits. 1960 IRE WESCON Convention Record: Part 4, Computers: Man-Machine Systems. Los Angeles, 96-104 Williams, M.M.R., 1974, Random Processes in Nuclear Reactors, Pergamon Press, Oxford

DATA TESTING OF ENDF/B-VI WITH MCNP: CRITICAL EXPERIMENTS, THERMAL-REACTOR LATTICES, AND TIME-OF-FLIGHT MEASUREMENTS

Russell D. Mosteller,1 Stephanie C. Frankle,2 and Phillip G. Young3 1

Technology and Safety Assessment Division Applied Theoretical and Computational Physics Division 3 Theoretical Division Los Alamos National Laboratory Los Alamos, NM 87545

2

INTRODUCTION The United States evaluated nuclear database, ENDF/B, is organized and implemented by the Cross Section Evaluation Working Group (CSEWG), which is a cooperative industrial/governmental activity involving, at its peak, some 20 different organizations. Since its inception in 1966, CSEWG systematically improved the evaluated database as new experimental and theoretical information became available, periodically issuing new versions of the ENDF/B library. The ENDF/B-VI file initially was issued in 1989–1990 and has been followed by three updated releases, with a fourth to follow soon. The purpose of this paper is to review the status of the ENDF/B-VI data file and to describe recent data testing results obtained with the Monte Carlo N-Particle (MCNP) code.1

Deficiencies in ENDF/B-V The previous version of the database is referred to as ENDF/B-V.2, which was originally issued2 in 1979 but was followed by significant updating (Release 2)3 in 1981. Thorough reviews of the fission-reactor-related cross-section data in ENDF/B-V.2 were published4 in 1984. While Version V represented a significant improvement over previous versions, it was soon apparent that limitations in the formats made substantive improvement of the data at lower energies (resonance region) difficult and made adaptation of the file for higher energy applications (above 20 MeV) virtually impossible. Furthermore, at the time of issuance of ENDF/B-V, it was only possible to include neutron-induced data in the

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official ENDF/B file, even though the need for evaluated charged-particle-induced data for fusion reactors and other applications was apparent. Finally, in the period following the issuance of ENDF/B-V.2, several important differential measurements were completed, and advances in nuclear theory and evaluation methods occurred that made updating of the database desirable. A number of deficiencies were identified in both content and breadth of coverage that provided further impetus for improving the database.

Objectives for ENDF/B-VI Prior to ENDF/B-VI, the only resonance-parameter formalisms allowed were singlelevel Breit-Wigner,5 multi-level Breit Wigner,6 and Adler-Adler7 formalisms. With only these possibilities available and with the procedures in effect at that time, it was not possible to adequately represent the high-resolution data that were becoming available, particularly from the Oak Ridge Electron Linear Accelerator (ORELA). Additionally, many of the resolved resonance parameter evaluations only extended to relatively low incident neutron energy, well below the energy at which modern, high-resolution measurements were possible. For example, the ENDF/B-V.2 resolved resonance parameter evaluations for and only extend to 82 and 301 eV, respectively. Beginning around 1985, there was increased interest in expanding evaluated data files to higher incident energies for both neutrons and charged particles, initially for national defense applications and later for accelerator-based transmutation technologies and accelerator-shielding applications. The most serious limitation in the ENDF/B format for higher energy evaluations was the inability to represent energy-angle correlated spectral distributions of emitted neutrons and charged particles. The formats at that time did permit a form of energy-angle correlations for emitted neutrons (only) to be given, but the method was cumbersome and inadequate. Most importantly, processing codes did not exist to handle this option, because it was generally felt that improvements to the format were needed before any processing-code development was merited. Serious deficiencies in energy balance were noted in ENDF/B-V by MacFarlane,8 which led to questions as to the efficacy of the file for damage and kerma calculations. Some of the evaluations identified as having energy-conservation problems were quite important for either fusion or fission applications (or both), particularly for several important structural materials Additionally, many evaluations did not contain covariance information, and there were serious deficits in both the coverage and the scope of fission-product evaluations. Finally, the concept of simultaneous evaluations had not yet materialized, and, for example, evaluations of and data for and were done independently. While data-testing results indicated that the quality of ENDF/B-V evaluations was high,9 it was still possible for small systematic shifts or biases to be present, and it was felt desirable to analyze simultaneously as many of the important reactions as possible. Efforts to modernize and update the ENDF/B-V.2 file began in the mid-1980s. The primary objectives were to (1) modernize and generalize the ENDF/B format for wider scope and applicability, especially in the resonance region and for higher energy evaluations; (2) perform a state-of-the-art simultaneous evaluation of the standard crosssection reactions, including all other reactions with significant absolute cross-section databases and with ratio links to the standards; (3) correct the most serious energy-balance problems, particularly for structural materials, utilizing isotopic data evaluations; (4) significantly update fission-product yield and decay data files; (5) incorporate new highresolution total, scattering, and (n,f) cross-section data in high-quality resonance parameter analyses; and (6) extend evaluations to higher energies and to include energyangle correlated neutron and charged-particle emission data utilizing the new format developments.

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Status of ENDF/B-VI Data File Four separate distributions of ENDF/B-VI data files have occurred thus far, and a new distribution is in preparation. The initial distribution (Release 0) occurred in 1990, which included the extensively revised standards cross-section materials.10,11 This distribution was followed by a largely corrective Release 1 in 1991. Additional distributions occurred in 1993 (Release 2) and 1995 (Release 3), and a new distribution (Release 4) is planned for 1996. A summary of the major new ENDF/B-VI evaluations that have been released to date is given in Table 1. Not included in Table 1 are a number of substantive revisions of the evaluated data for key fission products, primarily in the resonance region. Summary documentation for Releases 1–3 is given in Ref. 12. Standards Cross Sections. The methodology used in the simultaneous standards evaluation13 is especially important for fission-reactor applications because it permits inclusion of several important control and actinide materials in the analysis. In addition to data for the usual standards cross sections (which include the and reactions), absolute cross-section measurements and connecting ratio data for the and reactions were incorporated in the simultaneous analysis. While this technique improves absolute cross sections, it is especially effective in determining relative cross sections and covariances, hopefully eliminating some of the bias that has been evident in the past between uranium and plutonium critical systems. Light-Element Cross Sections. Significantly improved evaluations are included in ENDF/B-VI for several light elements. Most important are the and evaluations, which include the important standards cross sections. The basis for these evaluations (and for the and evaluations below the inelastic threshold) are coupled-channel Rmatrix analyses that facilitate accurate analysis of comprehensive experimental databases and, at the same time, permit inclusion of very accurate charged-particle and neutron total cross-section data. Important new evaluations also are included for and C. In the case of a complete new evaluation was performed with special emphasis on accurately specifying the energy-angle correlated neutron emission spectra from (n,2n) reactions.10 The new evaluation is primarily based on improved experimental data, with the most significant improvement occurring in the neutron total cross section, which in ENDF/B-V.2 is in error by as much as 40%. Structural Materials. Fe, Ni, and Cr are among the most important structural materials for fission reactors. Separate evaluations were performed for all stable isotopes of each of these elements. The evaluations are based on analyses of experimental data and extensive new theoretical studies in the MeV region.14 Additionally, new multichannel resonance parameter analyses were performed15 for and using high-resolution transmission, capture, and scattering data from ORELA. A similar analysis of transmission and capture also is included in the evaluation. Finally, substantially improved new evaluations of V, and are included in ENDF/B-VI. Major Actinides. New evaluations of resolved resonance parameters16 using the Reich-Moore multilevel formalism17 are included in the ENDF/B-VI evaluations for and Each of these resonance-parameter evaluations covers a considerably larger energy range than does the previous version of ENDF/B and thereby reduces difficulties and uncertainties in calculating self-shielding effects. The resolved resonance region extends to 2.25 keV, to 10 keV, and to 1 keV. The simultaneous standards evaluation provides significantly improved (n,f) cross sections above neutron energies of ~ 10 keV for and and improved cross sections in

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the same range for Additionally, the new or updated theoretical analyses of the elastic scattering, (n,2n), and (n,3n) reactions for the major actinides provide significantly improved data for outgoing neutron cross sections, angular distributions, and emission spectra. Yield, Decay, and Delayed-Neutron Data. Substantially expanded fission-product yield data files are provided in Version VI of ENDF/B. New or updated evaluations of independent, cumulative, and mass chain yields with uncertainties are given for 36 fissioning nuclides at one or more incident neutron energies.18 Evaluated yields for spontaneously fissioning nuclides also are included, so that a total of 60 nuclide-energy combinations is available, resulting in the formation of approximately 1100 different fission products. Decay data (average decay energies, decay spectra, and, to a lesser extent, cross sections) for many of the present 979 fission-product and actinide nuclides were improved for Version VI using new experimental measurements and nuclear-model calculations.19 Finally, delayed-neutron data are vastly improved in ENDF/B-VI as compared to ENDF/B-V.2. In particular, individual decay spectra and emission probabilities are included for some 271 delayed-neutron precursors, with temporal 6-group data provided for most of the fissionable nuclides.

Overview of MCNP Development of Monte Carlo methods began at Los Alamos Scientific Laboratory, later renamed Los Alamos National Laboratory (LANL), during World War II. MCNP is the product of more than 50 years of research and more than 400 person-years of development. Code Capabilities. MCNP is a general-purpose Monte Carlo code that can be used to perform a variety of types of calculations involving neutrons, photons, and/or electrons in one-, two-, and three-dimensional geometries. For the studies discussed herein, however, it was used only to calculate eigenvalues, neutron reaction rates, and fluxes. MCNP employs three different types of estimators for collision, absorption, and track-length. It then combines those estimators to produce in such a way that the associated standard deviation is minimized. All of the values reported herein for are based on the combined collision/absorption/track-length estimator. Continuous-Energy Libraries. One of the principal advantages of MCNP is its ability to work with continuous-energy neutron libraries. A group-wise library requires preprocessing on a case-by-case basis to incorporate the effects of self-shielding, group-togroup scattering, and/or geometry (viz., Dancoff factors). A continuous-energy library, in contrast, eliminates the need for such preprocessing and all associated approximations. Continuous-energy MCNP libraries derived from ENDF/B-V and ENDF/B-VI specifications are available for a large number of isotopes and for some naturally occurring elements.20 The ENDF/B-VI isotopic libraries are current through ENDF/B-VI Release 2 (ENDF/B-VI.2). In addition, a library based on the evaluation for in ENDF/B-VI Release 3 (ENDF/B-VI.3) has been generated for internal use at LANL, and calculations with that library are compared with results from ENDF/B-V and ENDF/B-VI.2 later in this paper. All of the libraries employed in this study are based on a temperature of 300 K. The ENDF/B-V.2 and ENDF/B-VI.2 isotopic libraries for MCNP hereafter will be designated simply as the ENDF/B-V and ENDF/B-VI libraries unless otherwise noted.

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DATA TESTING: CRITICAL EXPERIMENTS Data testing has been performed for a large number of critical experiments. Those experiments have been selected to encompass most combinations of materials that are of interest in criticality safety. Where feasible, similar experiments have been included to ensure that the results are not distorted by some unique feature of a single experiment. In addition, in many cases more than one experiment from a particular set of experiments has been included so that the effect of variations within such a system can be evaluated. Specifications for all of the experiments discussed herein have been approved either by the CSEWG or by the working group for the International Criticality Safety Benchmark Evaluation Project (ICSBEP). The detailed specifications for the CSEWG benchmarks are given in their benchmark-specifications report,21 while the ICSBEP specifications are given in the International Handbook of Evaluated Criticality Safety Benchmark Experiments.22 (In some cases, the specifications were approved subsequent to the last formal publication of those documents and therefore have not yet appeared in them.) Specifications for several of the experiments are given in both documents, and in such cases the ICSBEP specifications usually have been chosen. In most of those cases, the differences in the specifications are minor. However, the ICSBEP specifications provide an uncertainty for the benchmark value of whereas the CSEWG specifications usually do not. The specific CSEWG and ICSBEP identifiers for each benchmark are given in Appendix A. Except as noted, all of the MCNP calculations for these critical experiments utilized 440 generations of neutrons with 2500 neutrons per generation, and the first 40 generations were excluded from the statistics. Consequently, the quoted results are based on 1,000,000 active neutron histories.

Uranium Critical Experiments The uranium critical experiments that have been used for data testing are summarized in Table 2. They include highly enriched uranium (HEU) and metal systems, lowenriched uranium (LEU) lattices, and HEU, LEU, and solutions. Fast neutrons produce most of the fissions in the metal systems, while thermal neutrons produce most of the fissions in the lattices and the solutions. The benchmark values for for most of these experiments are unity, but some of them are slightly higher or slightly lower. There are two reasons for such differences. First, some of the benchmark specifications include idealizations that were made to simplify the representation of the experiment, and the benchmark was adjusted to account for the reactivity effect of those idealizations. Second, the configuration of some of the experiments was slightly supercritical when the measurement was made. Metal Systems. The metallic uranium systems include experiments with HEU and with The experiments with HEU include a bare sphere, two spheres that were reflected with normal uranium, two stacks of alternating platters of HEU and normal uranium, and a sphere and a cube of HEU that were reflected by water. The experiments with include a bare sphere and a sphere reflected by normal uranium. All of these experiments were performed at LANL, mostly during the 1950s. Godiva23,24 is a bare sphere of HEU enriched to 93.71 wt.% in It has a radius of 8.7407 cm and a density of Both the Topsy25 and the FLATTOP-2523,26 configurations contain a sphere of HEU enclosed in a sphere of normal uranium. The inner Topsy sphere is slightly smaller and slightly less dense than the inner FLATTOP-25 sphere, but its enrichment is slightly higher. The Topsy reflector is slightly less dense than its FLATTOP-25 counterpart, but it is thicker. A summary of the specifications for the two configurations is given in Table 3.

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The two Jemima experiments27 have circular disks of HEU and normal uranium stacked to form a cylinder. The cylinder of disks is supported by a steel platform. The Jemima pairs experiment alternates disks of HEU and normal uranium, while the triplets experiment has two disks of normal uranium between successive disks of HEU. In the ICSBEP idealizations of the Jemima experiments, all of the disks have an outer radius of 13.335 cm. The thicknesses of the idealized disks of HEU and normal uranium are 0.804 cm and 0.604 cm, respectively. The average enrichment of the HEU disks is 93.4 wt.% for the pairs experiment and 93.5 wt.% for the triplets experiment, while their average density is for the pairs experiment and for the triplets experiment. The average density for the disks of normal uranium is for both experiments. The stack of idealized disks is 15.648 cm high for the pairs experiment and 24.973 cm high for the triplets experiment Both the water-reflected sphere28,29 and the water-reflected cube30 of HEU are enclosed in a neutronically infinite medium of water. The sphere is more highly enriched than the cube (97.7 wt.% versus 94.0 wt.%), and it is very slightly more dense ( versus ). Consequently, the difference between the critical mass of the sphere (22.16 kg) and that of the cube (24.00 kg) is slightly larger than it would be because of just the difference in shape. The radius of the sphere is 6.5537 cm, and each side of the cube is 10.863 cm long. Most of the fissions for both of these cases are caused by fast neutrons, but the water sufficiently thermalizes enough neutrons that approximately 15% of the fissions are caused by thermal neutrons. Jezebel-23323 is a bare sphere of uranium enriched to 98.1 wt.% in It has a radius of 5.9838 cm and a density of FLATTOP-2323,26 is similar to FLATTOP-25 except that the inner sphere is enriched in rather than The inner sphere has the same density and enrichment as Jezebel-233, but its radius is 4.2058 cm. The outer sphere of normal uranium has a density of and an outer radius of 24.1194 cm. The results of the MCNP calculations for the metallic uranium benchmarks are summarized in Table 4. For most of these cases, ENDF/B-V and ENDF/B-VI produce good agreement with the benchmark values of However, the ENDF/B-VI results are in significantly better agreement with the benchmark for the two Jemima cases than are the corresponding ENDF/B-V results. Furthermore, with the exception of the water-reflected cases, the ENDF/B-VI result is consistently lower than the corresponding ENDF/B-V result. (The ENDF/B-VI sample means are lower than the corresponding ENDF/B-V sample means for the water-reflected cases too, but the difference is not statistically significant.) Both ENDF/B-V and ENDF/B-VI slightly underpredict the benchmark for Godiva, although the ENDF/B-VI difference is about twice as large as the ENDF/B-V difference. Both libraries predict values of for the Topsy sphere and FLATTOP-25 that are within

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a single standard deviation of the corresponding benchmark value. This behavior produces a significant swing between the values of for the bare Godiva sphere and the reflected spheres (approximately for both ENDF/B-V and ENDF/B-VI). Neutron leakage from Godiva is nearly 60%, whereas the leakage from the reflected cases is only about 30%. A possible explanation is that both libraries tend to overpredict neutron leakage for high-leakage configurations. Although the neutron leakage for both Jemima cases exceeds 50%, ENDF/B-VI produces values of for both cases that are in excellent agreement with the benchmark values. In contrast, ENDF/B-V produces a significant overestimate for for both cases. Sensitivity studies indicate that the better agreement for the ENDF/B-VI case is due primarily to changes in the fast cross sections for and, to a lesser extent, steel. The changes to the cross sections appear to have little or no impact on reactivity. The two libraries produce results for the water-reflected cases that are statistically indistinguishable. Both libraries underpredict for the water-reflected sphere by approximately and they both overpredict for the water-reflected cube by about the same amount. Consequently, on average, they produce very good agreement with the benchmarks for these two cases. ENDF/B-V and ENDF/B-VI produce very similar results for Jezebel-233 and FLATTOP-23. This behavior is not surprising, because the differences between the ENDF/B-V and ENDF/B-VI evaluations for are very minor. The same pattern that was observed for Godiva and FLATTOP-25 persists for Jezebel-233 and FLATTOP-23, although it is more exaggerated. The predicted value for for the bare sphere is low relative to the benchmark value, but the predicted value for the reflected configuration is in excellent agreement with the benchmark Specifically, the reactivity swing between the bare and reflected configurations is approximately for ENDF/B-V and approximately 0.0085 for ENDF/B-VI. The pattern of neutron leakage also is similar: neutron leakage is slightly more than 60% for Jezebel-233, while it is approximately 30% for FLATTOP-23. The low values for Jezebel-233 strongly suggest that the fast cross sections for need to be improved. In general, both the ENDF/B-V and the ENDF/B-VI libraries produce good agreement with metallic uranium benchmarks. The notable exceptions are Jezebel-233 and, for ENDF/B-V, the two Jemima cases. As noted previously, there are only very slight differences between the ENDF/B-V and ENDF/B-VI evaluations for and so the two results for that case would be expected to be very similar. ENDF/B-V not only overestimates for the two Jemima cases, but it overestimates it by a significandy larger margin for the triplets case than for the pairs case. In marked contrast, ENDF/B-VI matches the benchmark very well, and the ENDF/B-VI values for the two cases are statistically indistinguishable. Two additional patterns can be observed for these cases. First, ENDF/B-VI tends to produce lower values of for these cases than ENDF/B-V does, although the differences usually are small. Second, both libraries tend to underestimate for bare spheres of fissile material, but they both predict very accurately for reflected systems. This behavior may be due to leakage-related spectral effects. The neutron leakage for Godiva and Jezebel-233 is approximately 60%, but the leakage for the corresponding cases reflected by normal uranium is only about 30%. Lattices. The three lattices of fuel pins are based on experiments31 that were performed at Babcock & Wilcox’s Lynchburg Research Center in 1970 and 1971. This series of experiments was designated as Core XI, and the individual experiments were characterized as different "loadings." The entire series of 17 experiments appears as ICSBEP benchmarks, and the 3 loadings that are discussed herein also have been accepted

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as benchmarks by the ad hoc committee on reactor physics benchmarks,32 which is part of the Reactor Physics Division of the American Nuclear Society. The experiments were performed inside a large aluminum tank that contained borated water and thousands of fuel pins. The water height for each loading was exactly 145 cm, and the soluble boron concentration in the water was adjusted until the configuration was slightly supercritical, corresponding to The standard deviation in the measured soluble boron concentration is ±3 parts per million (PPM), by weight. For the three cases of interest, the central region of the configuration closely resembles a 3 x 3 array of pressurized-water-reactor (PWR) fuel assemblies. Each of these assemblies contains 225 lattice locations (i.e., in PWR parlance, they are "15 x 15" assemblies.) The 9 assemblies are surrounded by a buffer of fuel pins, and the buffer in turn is surrounded by a water reflector region. The buffer contains 2,936 pins for each of these cases, and the core as a whole contains more than 4,800 pins. A schematic of this arrangement is shown in Figure 1. Each of the fuel pins is clad in aluminum and has an enrichment of 2.459 wt.%. The pins in both the central region and the buffer are arranged on a grid with a pitch of 1.63576 cm. In loading 1, fuel pins occupy all of the lattice positions in the central 9 fuel assemblies. Consequently, there is no distinction between those assemblies and the buffer, and the lattice is uniform. In loading 2, the fuel rods have been removed from 17 positions in each assembly. The location of those positions within the assembly corresponds to the location of the water holes in a normal 15 x 15 assembly. In loading 8, Pyrex rods have been inserted in 16 of the open positions in each assembly, while the central position remains vacant. The progression from loading 1 to loading 2 to loading 8 therefore corresponds to a transition from a uniform lattice to assemblies with water holes to assemblies heavily loaded with discrete burnable absorbers.

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Because pin-by-pin fission distributions had been measured in the central assembly of loadings 2 and 8, the number of histories in the MCNP calculations for these three cases was increased to produce standard deviations of approximately 0.1% in the predicted pinwise fission rates. Specifically, each MCNP calculation employed 1,050 generations with 4,000 neutrons per generation, and the first 50 generations were excluded from the statistics. Consequently, the results for each of these cases are based on 4,000,000 active histories. The normalized pin-by-pin fission distributions predicted for the central assembly in loadings 2 and 8 are compared with the measured distributions in Figures 2 and 3, respectively. The ENDF/B-V distributions for these 2 loadings both produce a slightly better match with the measured distributions than do the corresponding ENDF/B-VI distributions. For the complete set of loadings, however, the two libraries produce comparable results.33 The root-mean-square (RMS) differences between the ENDF/B-V and ENDF/B-VI fission distributions for loadings 2 and 8 are both 0.016, which indicates that the fission distributions predicted by ENDF/B-V and ENDF/B-VI agree slightly better with each other than either of them does with the measured distributions. The MCNP eigenvalues for these cases are presented in Table 5. Both ENDF/B-V and ENDF/B-VI produce good agreement with the benchmark values for although they underpredict it slightly. On average, ENDF/B-V is low by approximately and ENDF/B-VI is low by approximately Both libraries tend to produce an RMS variation of about 2% relative to the measured pin-by-pin fission rates, which is quite acceptable. These differences are representative of those obtained for other loadings as well.33 The reactivity difference between the ENDF/B-V and ENDF/B-VI results is due primarily

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to changes in the cross sections for and Relative to ENDF/B-V, the ENDF/B-VI cross sections for tend to increase reactivity, while the cross sections for tend to decrease it. The net effect of these tendencies depends upon the enrichment of the fuel and the neutron spectrum, as will be shown in the discussion of data testing for reactor lattices. In this particular case, the net effect is that ENDF/B-VI consistently underpredicts ENDF/B-V by approximately It has been reported34 recently that the MCNP ENDF/B-V library produces a bias of approximately in the calculated in thermal lattices with LEU. If that bias is applied here, ENDF/B-V produces excellent agreement with the benchmark value of (the difference is one standard deviation or less for all 3 cases), and the reactivity difference between ENDF/B-V and ENDF/B-VI increases to approximately Solutions. The uranium solutions include six HEU solutions, one LEU solution, and five solutions of Five of the HEU solutions are spheres of uranyl nitrate in light water, and the sixth is a sphere of uranyl fluoride in heavy water. The LEU solution is an annular cylinder of uranyl fluoride in light water, and the solutions of are spheres of uranyl nitrate in light water. The sphere of HEU uranyl nitrate in heavy water is reflected by heavy water, but none of the other solutions are reflected. The experiments35,36 with the spheres of HEU uranyl nitrate were performed at Oak Ridge National Laboratory (ORNL) in the 1950s and are designated as ORNL-1, ORNL-2, ORNL-3, ORNL-4, and ORNL-10. The enrichment for all five cases is 93.2 wt.%. ORNL-1, ORNL-2, ORNL-3, and ORNL-4 have an outer radius of 34.595 cm, while ORNL-10 has an outer radius of 61.011 cm. The CSEWG benchmarks for these spheres represent them as completely bare with no container. ORNL-1, ORNL-2, ORNL-3, and

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ORNL-4 contain successively increasing amounts of uranyl nitrate, while the uranyl nitrate in ORNL-10 is more dilute than in ORNL-1. ORNL-2, ORNL-3, and ORNL-4 contain dilute amounts of boron (on the order of 20 to 50 PPM relative to the water), but ORNL-1 and ORNL-10 do not. The leakage in ORNL-1 is approximately 20%, while the leakage in the other three spheres with the same radius is approximately 17%. Although the leakage is lower in the latter three cases, the neutron spectra actually are harder because of the higher concentration of fissile material and the presence of boron. The spectrum for ORNL-10 is softer than that for any of the smaller spheres, because it has nearly three times the volume, contains no boron, and its fissile material is more dilute. Leakage for ORNL-10 is less than 7%. The CSEWG specifications, unfortunately, do not include any uncertainty in for the ORNL HEU spheres. However, uncertainties of for ORNL-1 and for ORNL-2, ORNL-3, and ORNL-4 recently have been established,37 and it is likely that the actual uncertainty for ORNL-10 is about the same size. The experiment38 with the sphere of uranyl fluoride in heavy water was performed at LANL in the 1950s. The mixture of uranyl fluoride and heavy water is contained inside a spherical shell of stainless steel, and its heavy-water reflector is contained inside a concentric second spherical shell, also made of stainless steel. The inner and outer radii of the inner shell are 22.211 and 22.313 cm, respectively, while the inner and outer radii of the outer shell are 44.411 and 44.665 cm, respectively. The enrichment of the uranium is 93.65 wt.%. The SHEBA-II experiment39,40 was performed at LANL in the 1990s. It contains a uranyl-fluoride solution inside an annular cylinder of 304L stainless steel. The uranyl fluoride contains uranium with an enrichment of approximately 5 wt.%. The stainless-steel container is slightly less than 80 cm tall, and the hollow central column is 5.08 cm wide. The inner radius of the outer wall is approximately 24.4 cm, and the critical height of the solution is 44.8 cm. The inner wall of the container is 0.635 cm thick, while the outer wall is slightly less than 1 cm thick. The MCNP results for uranium solutions are summarized in Table 6. ENDF/B-V produces generally good agreement with the benchmark values for for the ORNL HEU spheres, although it has a tendency to underpredict it slightly. ENDF/B-VI tends to underpredict ENDF/B-V by approximately Sensitivity studies indicate that this reduction in is due primarily to differences between the ENDF/B-V and ENDF/B-VI evaluations for and, to lesser extent, for Unfortunately, the uncertainty for the benchmark value of for the reflected sphere of uranyl fluoride is so large that no definitive conclusions can be drawn about the relative accuracy of ENDF/B-V and ENDF/B-VI for this experiment. (The ICSBEP benchmarks include other spheres as well as cylinders in the same series of experiments. However, the uncertainties for those other experiments are comparable to that for the one discussed herein.) Nonetheless, the MCNP results demonstrate that ENDF/B-V and ENDF/B-VI produce significantly different results for the combination of uranyl fluoride and heavy water. In particular, the ENDF/B-VI for this case is approximately 0.007 higher than the corresponding ENDF/B-V value. Sensitivity studies indicate that differences between the ENDF/B-V and ENDF/B-VI cross sections for and Fe all contribute significantly to the reactivity difference. Both ENDF/B-V and ENDF/B-VI overpredict the reactivity for SHEBA-II, although ENDF/B-VI does so by a substantially smaller margin. The difference in reactivity is due primarily to differences in cross sections for and, to a lesser extent, for Overall, ENDF/B-V tends to underpredict slightly for the HEU uranyl-nitrate solutions, and ENDF/B-VI tends to underpredict it by a slightly larger margin. On average, ENDF/B-V produces a of approximately 0.997 for the ORNL HEU spheres, while ENDF/B-VI produces a value of approximately 0.995. In contrast, both ENDF/B-V and

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ENDF/B-VI overpredict the reactivity for the LEU uranyl-fluoride solution in SHEBA-II. ENDF/B-VI continues to underpredict ENDF/B-V, but the difference has increased to approximately The reactivity difference for the reflected sphere of uranyl fluoride in heavy water reverses the pattern between the two libraries, however: ENDF/B-VI produces a value for that is approximately higher than the corresponding ENDF/B-V value. Unfortunately, the large uncertainty associated with this last benchmark precludes a conclusive determination of which library produces the better result. The pattern reversal between the case with uranyl fluoride in heavy water and the other cases may be related to the neutron spectrum. Even though the former case is reflected, its leakage is substantially higher than that of any of the other solutions. Leakage for that case is slightly more than 40%, whereas the leakage for SHEBA-II is slightly less than 25%, and the leakage for all of the ORNL HEU spheres is less than 20%. Furthermore, the moderator and reflector for the uranyl-fluoride sphere are heavy water, whereas the moderator for the ORNL HEU spheres and for SHEBA-II is light water. Consequently, the neutron spectrum for the uranyl-fluoride sphere is substantially harder than the spectra for the other solutions. The experiments35,36 with uranyl-nitrate solutions were performed at ORNL in the 1950s and are designated ORNL-5, ORNL-6, ORNL-7, ORNL-8, and ORNL-9. The uranium was enriched to 97.7 wt.% in All five of the solutions are enclosed inside a thin, spherical shell of aluminum-1100. The shell has an inner radius of 34.595 cm and is 0.32 cm thick. (It is worth noting that the outer radius of the uranyl nitrate solutions in ORNL-5 through ORNL-9 is the same as that of the HEU uranyl-nitrate solutions for ORNL-1 through ORNL-4.) ORNL-5 contains no boron, while ORNL-6 through ORNL-9 contain successively increasing amounts of boron. The concentration of uranyl nitrate, although relatively dilute for all five cases, increases from one case to the next to offset the negative reactivity introduced by the increase in boron. However, the uranyl nitrate remains sufficiently dilute that the leakage from these spheres is only about 2%. The results for ORNL-5 through ORNL-9 are included in Table 6. Generally speaking, both the ENDF/B-V and the ENDF/B-VI values for fall within a single standard deviation of the benchmark However, a pattern is evident: the ENDF/B-VI values are consistently about lower than the ENDF/B-V values. As was mentioned earlier, the ENDF/B-VI evaluation for differs only slightly from the corresponding ENDF/B-V evaluation. Therefore, the difference in reactivity is not attributable to differences in the cross sections for Sensitivity studies demonstrated that differences between the cross sections for are responsible for essentially the entire reactivity difference. However, the mechanism for this effect is not obvious. The leakage is too low for differences in the total cross section at high energies to have much effect, and is too weak an absorber to produce that reactivity change directly. The difference may be due to interactions between and other isotopes, most likely

Plutonium Critical Experiments The plutonium critical experiments are summarized in Table 7. They include metal spheres, mixed-oxide (MOX) lattices, and plutonium-nitrate solutions. Fast neutrons produce most of the fissions in the metal systems, while thermal neutrons produce most of the fissions in the lattices and the solutions. The metal and solution systems were critical, but the MOX lattice experiments all were slightly supercritical. However, the benchmark values for for four of the MOX experiments are less than unity, because they incorporate adjustments, given in the benchmark specifications, to account for particle effects.

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Metal Spheres. The metallic plutonium systems include two bare spheres, a sphere reflected by normal uranium, and a sphere immersed in water. The distinction between the two bare spheres is the amount of that they contain. All of these experiments were performed at LANL during the 1950s and 1960s. Jezebel23,41,42 is a bare sphere of delta-phase plutonium that contains 4.5 at.% It has a radius of 6.3849 cm and a density of Jezebel-24042 also is a bare sphere of delta-phase plutonium, but it contains 20.1 at.% It has a radius of 6.6595 cm and a density of FLATTOP-Pu23,26 is a sphere of delta-phase plutonium encased inside a spherical shell of normal uranium. The plutonium sphere contains 4.83 at.% It has an outer radius of 4.533 cm and a density of and the uranium shell has an outer radius of 24.13 cm and a density of The water-reflected plutonium sphere43 contains alpha-phase plutonium with 5.20 at.% It has an outer radius of 4.1217 cm and a density of The water that surrounds it is neutronically infinite. Although some of the fissions that occur in the sphere are produced by thermal neutrons, the majority of the fissions is produced by fast neutrons. The results for the metallic plutonium spheres are presented in Table 8. ENDF/B-V and ENDF/B-VI produce statistically identical values for for the first three spheres, and the values for the water-reflected sphere differ by only Furthermore, the results are in excellent agreement with the benchmark values for All of the calculated values are within two standard deviations of the benchmark value, and four of them are within one standard deviation. Sensitivity studies demonstrated that the difference between the ENDF/B-V and ENDF/B-VI results for the water-reflected sphere are due not to the plutonium isotopes but rather to differences in the cross-section libraries for This behavior appears to arise from interactions between plutonium and oxygen, however, because there was no corresponding difference in the results for the water-reflected HEU sphere or cube. Lattices. The MOX lattice experiments44,45 were performed by Battelle Pacific Northwest Laboratory (PNL) during the 1970s. They contain MOX fuel pins arranged in a uniform lattice with a square pitch, in a roughly cylindrical arrangement. The fuel pins contain 2 wt.% and the plutonium nominally contains 8 at.% The benchmark specifications, however, correspond to 7.73 at.% The active length of the fuel pins is 91.44 cm, and they have an outer radius of 0.6414 cm. The pins are clad in zirconium with an outer radius of 0.7176 cm, and they are supported by an aluminum platform. A lead radiation shield rests on top of the pins. The water in which the pins are immersed is neutronically infinite on the bottom and sides of the lattice, and the water height above the radiation shield ranges from 2.29 to 15.24 cm, depending on the particular experiment. The six cases involve three different pitches, and there is a borated and (essentially) unborated case at each pitch. More details about these cases are presented in Table 9. The CSEWG specifications for these cases do not include a stated uncertainty. It is assumed, therefore, that the uncertainty in is small. The results from these cases are summarized in Table 10. Three trends are observable: (1) ENDF/B-VI consistently underpredicts ENDF/B-V, by about (2) the values of for the borated cases are consistently higher than those for the unborated cases at the same pitch, and (3) the value of increases as the pitch increases. Overall, ENDF/B-VI produces marginally better agreement with these benchmarks than ENDF/B-V does. Relative to the benchmark the sample means for the ENDF/B-V and ENDF/B-VI results are given by, respectively,

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and

where is the benchmark value for Although the bias for ENDF/B-VI is smaller than that for ENDF/B-V, the standard deviation for both of them is quite large. Sensitivity studies indicate that the consistent difference between the ENDF/B-V and ENDF/B-VI results is due primarily to differences in the cross sections for and Differences in the cross sections for tend to increase slightly for ENDF/B-VI relative to ENDF/B-V, while differences in the cross sections for and tend to decrease it slightly. However, these differences tend to offset each other, and, in any case, the impact of the cross-section differences on for any one of these three isotopes is only about The differences between the calculated values of for borated and unborated cases at the same pitch are about for both ENDF/B-V and ENDF/B-VI. The leakage for these experiments is very low (approximately 2% for the unborated cases and less than 1% for the borated cases). Because the presence of the boron tends to harden the spectrum, the conversion ratio increases by about 10% for a borated case relative to the corresponding unborated case. This pattern suggests that spectral effects may be responsible for the observed differences. On the other hand, this trend also could be explained by a systematic underestimate of the boron level in the borated cases. The trend of increasing with pitch also could be a spectral effect, because the spectrum becomes increasingly softer as the pitch (and hence the moderator-to-fuel ratio) increases. In particular, the conversion ratio decreases by more than 40% in going from the tightest pitch to the loosest. However, the variation of with pitch is essentially the same for ENDF/B-VI as it is for ENDF/B-V: the value of for the loosest pitch is approximately higher than that for the tightest pitch for both borated and unborated cases. This behavior suggests either that a deficiency in the ENDF/B-V cross sections also exists in their ENDF/B VI counterparts or that there is a problem with the benchmark specifications. The only definitive conclusion that can be drawn from these results is that ENDF/B-VI consistently produces a value of for MOX lattices that is approximately lower than the corresponding ENDF/B-V value. Solutions. Five unreflected and three water-reflected spheres of plutonium nitrate were studied. All of the experiments46,47 upon which these benchmarks are based were performed by PNL during the 1960s. The CSEWG benchmark specifications for PNL-1 through PNL-5 represent them simply as bare spheres of plutonium nitrate. However, the experiments upon which PNL-3 and PNL-4 are based recently have been evaluated as part of the ICSBEP, and it has been suggested48 that those two CSEWG benchmarks be revised to conform to the ICSBEP specifications. Consequently, the ICSBEP specifications have been used herein for PNL-3 and PNL-4. The principal difference between the two sets of specifications is that the ICSBEP benchmark retains the stainless steel sphere that encloses the solution and its cadmium cover. The unreflected spheres differ primarily with respect to the concentration of plutonium nitrate, although there also are variations in size and content. The outer radius of the bare spheres for PNL-1 and PNL-2 is 19.509 cm, and the plutonium contains 4.58 at.% PNL-1 contains approximately 38.8 g/l of plutonium, while PNL-2 contains approximately 171.7 g/l. The outer radius of the sphere for PNL-3 and PNL-4 is 22.7 cm, and the plutonium contains 4.18 at.% As noted above, the plutonium-nitrate solution is contained in a stainless steel shell that is enclosed by a cadmium cover. Both the stainless steel shell and

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the cadmium cover are quite thin, however; the shell is approximately 0.13 cm thick and the cover is only .05 cm thick. PNL-3 contains approximately 22.4 g/l of plutonium, while PNL-4 contains approximately 27.5 g/l. Both of these concentrations are significantly more dilute than those in PNL-1 and (especially) PNL-2. The outer radius of the bare sphere for PNL-5 is 20.1265 cm, and the plutonium contains 4.15 at% The plutonium density in PNL-5 is approximately 43.2 g/l, which is only about 11% higher than that for PNL-1. The three water-reflected spheres all have the same radii but different concentrations of plutonium. In addition, one of them contains a different fraction of than the other two. The outer radius of the plutonium-nitrate solution is 17.79 cm, and the thickness of the stainless-steel shell that contains it is less than 0.13 cm. The water reflector is neutronically infinite for all three cases. The first water-reflected plutonium-nitrate sphere has a plutonium density of approximately 29.57 g/l, and the plutonium contains 3.12 at.% The second sphere has the same fraction as the first, but its plutonium density is 39.38 g/l. The third sphere has a slightly lower plutonium density (26.27 g/l) than the first, but its plutonium contains only 0.54 at.% Therefore, relative to the first case, the second case contains significantly more plutonium, while the third case contains significantly less As the results in Table 11 demonstrate, ENDF/B-V tends to overestimate the value of substantially; the computed is high for all of the cases, and the difference exceeds 0.005 for seven of the eight. This pattern also has been observed for earlier versions of ENDF/B. However, ENDF/B-VI produces a striking improvement: its values for are 0.006 to lower than those from ENDF/B-V and are within of unity for seven of the eight cases. (A criterion of seems reasonable for these cases, because that is the approximate size of the uncertainty given for the benchmark value of in the ICSBEP specifications for PNL-3 and PNL-4. Although the CSEWG specifications do not include uncertainties, it is likely that the actual uncertainties would be about the same as those for PNL-3 and PNL-4 because of the similarity of the cases.) Sensitivity studies indicate that the improvement is due primarily to differences between the ENDF/B-V and ENDF/B-VI cross sections for and, to a lesser extent,

and Fe. The pattern that was observed for MOX lattices continues to hold for these plutoniumnitrate solutions, but it is more extreme; ENDF/B-VI consistently produces a value for that is approximately lower than its ENDF/B-V counterpart. For these solutions, it is quite clear that ENDF/B-VI produces results that agree much better with the benchmark values of than do those produced by ENDF/B-V.

DATA TESTING: THERMAL-REACTOR LATTICES ENDF/B-V generally has predicted the neutronic behavior of light-water-reactor (LWR) lattices very accurately. Consequently, comparisons between ENDF/B-V and ENDF/B-VI results for LWR lattices may provide a good indication of how successfully ENDF/B-VI will predict the neutronic behavior of such lattices. Comparisons also have been made for lattices from a variety of other thermal-reactor types. In addition to PWRs and boiling water reactors (BWRs), the reactor types include Russian RBMK reactors, modular high-temperature gas-cooled reactors (MHTGRs), CANDU heavy-water reactors, and heavy-water production reactors (HWPRs). The cases are summarized in Table 12. All of these cases employ reflective boundary conditions, and therefore they have no leakage. Furthermore, they all are axially uniform. Consequently, they are effectively twodimensional. As was noted previously, ENDF/B-VI libraries for MCNP currently are available only at 300 K. Consequently, the comparisons were limited to "cold" conditions (i.e., room

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temperature and pressure). However, some parameters of considerable importance for reactor safety are determined at cold conditions (e.g., cold shutdown margin). Furthermore, for several of the lattices, the only significant difference between hot and cold conditions is the additional Doppler broadening of resonances. Even for LWRs, the two main distinguishing features between cold and hot conditions are the differences in Doppler feedback and water density. No correction has been applied to any of these results to account for the recent report34 that the MCNP ENDF/B-V library systematically underestimates reactivity for thermal LEU systems. If such a correction were included, it is anticipated that the ENDF/B-V values for would increase by approximately for LEU and normal-uranium lattices. Furthermore, it is possible that it would increase even more for thermal lattices with significant amounts of plutonium, because the harder spectrum in such lattices produces higher fluxes in the in the energy range where low-lying epithermal capture resonances occur.

PWR Lattices Comparisons were made for infinite lattices of both once-through and MOX pin cells. Five different lattices and three different MOX lattices were studied. Lattices. Three of these lattices are "numerical benchmarks" that have been defined elsewhere.49,50 These numerical benchmarks are of interest because they represent a range of spectra that more than spans the range normally seen in LWRs. The other two lattices are based on actual PWR designs. Numerical benchmark 1 (NB-1) is an idealization of case T6 in the TRX critical experiments.51 It corresponds to an infinite hexagonal lattice of identical fuel pins clad in aluminum and immersed in water. The uranium is only slightly enriched (1.3 wt.%), and

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the fuel pin has an outer radius of 0.4864 cm. The cladding is 0.0889 cm thick, and the cell pitch is 1.5578 cm. The combination of low fuel enrichment and a relatively high moderator-to-fuel ratio produces a spectrum that is significantly softer than that for a typical PWR pin cell. NB-4 is an idealization of a "typical" PWR pin cell, although the water that surrounds the fuel pin contains no boron. NB-4 corresponds to an infinite square lattice of identical fuel pins clad in zirconium. The enrichment of the uranium is 2.75 wt.%, and the outer radius of the fuel pin is 0.5080 cm. The cladding is 0.08674 cm thick, and the cell pitch is 1.4605 cm. NB-5 is an idealization of the HiC-13 critical experiment.52 It corresponds to an infinite hexagonal lattice of fuel pins clad in aluminum and immersed in water. The enrichment of the uranium is 3.05 wt.%, and the outer radius of the fuel is 0.4675 cm. The thickness of the cladding is 0.0615 cm, and the cell pitch is 1.166 cm. This very tight pitch produces a low moderator-to-fuel ratio, and the spectrum therefore is significantly harder than that for a typical PWR pin cell. The results for NB-1, NB-4, and NB-5 are presented in Table 13, where the reactivity difference is defined as

Although on average the agreement between ENDF/B-V and ENDF/B-VI is quite good, there is a trend to the difference. The ENDF/B-VI value for is significantly lower than the corresponding ENDF/B-V value for NB-1, but the reverse is true for NB-4 and NB-5. This pattern could be attributable either to the difference in enrichment or to the difference in moderator-to-fuel ratio. To resolve that question, further studies were performed for two sets of PWR pin cells. Within each set, the dimensions were held fixed, but the enrichment was changed. The enrichments ranged from as low as 0.711 wt.% (normal uranium) to as high as 3.9 wt.%. The first set of pin cells is based on a numerical benchmark for the Doppler coefficient of reactivity,53 although the atomic number densities were increased to reflect the change from hot to cold conditions. These pin cells are based on an "optimized" fuel assembly (OFA) design that has been used in both initial and reload cycles of several PWRs. The outer radius of the fuel rod is 0.39306 cm, the cladding is 0.06496 cm thick, and the cell pitch is 1.26209 cm. This fuel-pin radius is somewhat smaller than that for the corresponding conventional design, and it produces a higher moderator-to-fuel ratio and therefore a somewhat softer spectrum. The fuel pin is clad in zirconium and is immersed in water that contains 1400 PPM of soluble boron.

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The second set of pin cells is the same as the first, except that the outer radius of the fuel pin and the thickness of the cladding have been changed to 0.41169 cm and 0.06420 cm, respectively. These changes produce a pin cell that corresponds very closely to a standard PWR design. The results for these cases are presented in Table 14. The reactivity difference between the ENDF/B-V and ENDF/B-VI results follows the same pattern for both sets of pins: the difference becomes increasingly more positive as the enrichment increases. The moderator-to-fuel ratio does have a small effect, however, because the reactivity difference at a given enrichment is consistently more positive for the standard design than for the OFA design. Sensitivity studies indicate that the reactivity difference is due primarily to competition between and The ENDF/B-VI evaluation for produces a more positive reactivity contribution for these lattices than does the corresponding ENDF/B-V evaluation, while the ENDF/B-VI evaluation for produces a more negative reactivity contribution than ENDF/B-V. As a result, ENDF/B-VI tends to predict lower reactivity than ENDF/B-V for normal-uranium or slightly enriched fuel cells, but that difference becomes less negative as the enrichment increases and eventually becomes positive. MOX Lattices. Three different infinite lattices of MOX pin cells were studied. One of them has been defined elsewhere49,50 as a "numerical benchmark," while the other two lattices are based on the same OFA and standard designs that were used for the lattices discussed previously. NB-2 is an idealization of the PNL-33 critical experiment44,45 that was described earlier. Apart from the fact that NB-2 is two-dimensional and PNL-33 is threedimensional, the principal difference between them is that the former corresponds to an infinite lattice of fuel cells, while the latter contains a finite number of cells. The other two MOX lattices54 simply replace the fuel pin in the corresponding lattice with a MOX pin. A significant distinction between NB-2 and the other two MOX lattices is the isotopic composition of the plutonium. In NB-2, the plutonium is primarily accounts for slightly less than 8 at.% of the plutonium, and there are only trace amounts of and The plutonium in the OFA and standard MOX lattices, in contrast, corresponds closely to that in discharged fuel and contains 45 at.% 30 at.% 15 at.% and 10 at.%

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The results for the MOX pin cells are presented in Table 15. They appear to be independent of the moderator-to-fuel ratio as well as the plutonium isotopics. ENDF/B-VI produces essentially the same reactivity as ENDF/B-V for the PWR cases with 2 wt.% but it produces significantly lower reactivity for the PWR cases with 1 wt.% and for NB-2. Sensitivity studies demonstrated that this behavior is not due to differences in the plutonium cross sections. Instead, it is due primarily to spectral variations and differences between the evaluations for

BWR Lattice Calculations were performed for an infinite lattice of identical BWR fuel bundles at three different conditions: unrodded at beginning of life (BoL), rodded at BoL, and unrodded at a bundle-averaged uranium burnup of 25,000 kWd/kg (MWD/MTU). The bundle is an idealization of bundle type 4 for Peach Bottom Unit 2, which was an 8 x 8 design with one internal water hole and 5 gadolinia-loaded fuel pins.55 The average enrichment for the bundle at BoL is 2.74 wt.%. In the idealization, the fuel pins are clad in zirconium, and the hollow rod in the water hole also is made of zirconium. The rodded case is identical to the unrodded case except that a control rod with rodlets clad in stainless steel has been inserted in the wide water gaps. The isotopics for the depleted bundle were taken from a study56 of boron-retention requirements following a severe accident in a BWR. It should be noted, however, that in that study fission products were not represented explicitly. Instead, they were represented by two pseudoisotopes with fixed microscopic cross sections, and the MCNP calculation for the depleted BWR lattice also employed that representation. Consequently, the results of the calculations for this bundle do not involve actual ENDF/B-V or ENDF/B-VI fission products. A schematic of the bundle is shown in Figure 4, and the geometry of the rodded bundle is shown in Figure 5. The results from the calculations are presented in Table 16. The ENDF/B-VI values for for the unrodded and rodded bundles at BoL are statistically identical to the

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corresponding ENDF/B-V values. However, the ENDF/B-VI value for the depleted bundle is slightly but significantly lower than the ENDF/B-V value. At a uranium burnup of 25,000 kWd/kg, accounts for less than 50% of the fissions in this bundle, and sensitivity studies demonstrated that the small reduction in reactivity is attributable primarily to differences between the evaluations for At BoL, the differences between the cross-section libraries compensate for the differences between the libraries, but at 25,000 kWd/kg the reduced content cannot completely offset the effects of the differences. The sensitivity studies also demonstrated that differences between the ENDF/B-V and ENDF/B-VI evaluations for the plutonium isotopes do not produce a significant reactivity change for the depleted bundle.

RBMK Lattice The idealized RBMK lattice corresponds to an infinite lattice of fuel cells for an RBMK reactor at BoL.57 The central region contains 18 fuel pins with an enrichment of 2.4 wt.%. The fuel pins are clad in stainless steel. The assembly is cooled by water flowing through the central tube in the lattice, and it is moderated by the surrounding block of graphite. The geometry of the fuel cell is shown in Figure 6. Results for the RBMK fuel cell are summarized in Table 17. The ENDF/B-VI value for is only slightly lower than the ENDF/B-V value, but the difference is statistically significant. The reactivity difference is about the same as that observed for the OFA pin cell with the same enrichment. Sensitivity studies indicated that, once again, the difference is due to competing effects from and

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MHTGR Lattice The MHTGR lattice corresponds to an infinite array of standard core blocks for the MHTGR design that was developed as part of the now-discontinued New Production Reactor program.58 Small spheres of HEU are contained in blocks of graphite moderator and are cooled by helium gas that flows through small channels bored in the graphite. A diagram of this standard cell block is shown in Figure 7. Results for the MHTGR lattice are presented in Table 18. ENDF/B-VI produces a value for that is significantly higher than the ENDF/B-V value. Sensitivity studies demonstrated that this reactivity difference is due primarily to differences between the evaluations for For the LEU cases discussed previously, differences between the evaluations tended to compensate for the differences between the evaluations. Such compensation does not occur here, however, because the fuel is highly enriched.

CANDU Cluster The CANDU cluster contains 37 fuel pins surrounded by a heavy-water coolant and a heavy-water moderator. The pins contain natural uranium in the form of and they are clad in a zirconium alloy. The outer radius of the fuel is 0.25 cm, and the cladding is 0.15 cm thick. As Figure 8 illustrates, the fuel is arranged in rings of 1, 6, 12, and 18 pins.

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The fuel pins and their surrounding coolant are contained inside a zirconium-alloy pressure tube, which in turn is contained inside a zirconium-alloy calandria tube. The calandria tube is surrounded by the heavy-water moderator. This case is based on the "sample cluster case" from WIMS-AECL user’s manual.59 The results for the CANDU cluster are summarized in Table 19. The ENDF/B-VI value for is slightly but significantly lower than the ENDF/B-V value. Although this pattern is consistent with that previously seen for PWR lattices, its cause is not entirely the same. The reduction in reactivity is due almost entirely to differences between the ENDF/B-V and ENDF/B-VI cross sections for and for the reactivity effect of the cross-section differences for is negligible. The reactivity change due to is approximately the same as the reactivity change due to Although reactivity changes due to were observed previously for critical experiments with significant neutron leakage (including one with heavy water), there is no leakage for the CANDU cluster because it is represented as an infinite lattice. The neutron capture by oxygen is about 20% higher in the ENDF/B-VI calculation than in the ENDF/B-V calculation, but oxygen is such a weak absorber that the increase in capture accounts for less than half of the observed reactivity difference. Consequently, the remainder of that difference is presumed to result from changes in the capture rate of other isotopes due to spectral changes induced by differences between the ENDF/B-V and ENDF/B-VI evaluations for

HWPR Cell and Supercell These cases mimic the hexagonal fuel cell and the repeating lattice of supercells that formed the Mark 22 design for the Savannah River K-Reactor.60 Inside a fuel assembly, rings of HEU are surrounded by flow channels for the heavy-water coolant and by rings

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of absorber material containing boron. The heavy water that is external to the fuel assembly acts as the moderator. Six hexagonal fuel cells and the hexagonal control cell that they surround form a supercell. The geometry of the supercell is shown in Figure 9. Results from calculations for the HWPR cell and supercell are presented in Table 20. The ENDF/B-VI values for are only marginally higher than their ENDF/B-V counterparts, even though the uranium is HEU. However, the coolant and moderator are heavy water. The differences between the ENDF/B-VI and ENDF/B-V evaluations for do produce a small increase in reactivity, but the differences between the evaluations tend to compensate for it. Consequently, the patterns observed previously for HEU fuel and for heavy water continue to hold, but they largely offset each other.

EFFECT OF ENDF/B-VI RELEASE 3 FOR As was noted previously, the ENDF/B-VI continuous-energy libraries for MCNP were generated from the evaluations that were current through ENDF/B-VI.2. However, ENDF/B-VI.3 included an updated evaluation61 for The revisions to that evaluation are limited to the energy range below 900 eV and principally affect the range below 110 eV.62,63 The changes increase the capture resonance integral and slightly reduce the fission resonance integral. This combination increases the epithermal capture-to-fission

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ratio and thereby improves the agreement of that parameter with measured values. Thermal data also were revised slightly to increase reactivity and thereby to partially offset the epithermal reduction in reactivity. "Unthinned" continuous-energy libraries for were generated at LANL based on the ENDF/B-VI.2 and ENDF/B-VI.3 evaluations so that the effects of differences between the revisions could be investigated without being masked by approximations introduced during the processing of those evaluations into cross-section libraries. The unthinned libraries are several times larger than the libraries normally employed because, as the term suggests, they employ a much finer energy grid. Consequently, they constitute a more rigorous representation of the evaluation because approximations that are introduced during processing are minimized. The calculations for the critical-experiment benchmarks with bare spheres of HEU uranyl nitrate and with lattices of fuel pins were repeated with the unthinned ENDF/B-VI.2 and ENDF/B-VI.3 libraries. The values obtained for are compared to those obtained previously in Tables 21 and 22. These results suggest that the reactivity differences between the thinned and unthinned ENDF/B-VI.2 libraries are negligible for the lattices. However, the results for the ORNL spheres demonstrate that differences in the processing of the thinned and unthinned libraries can produce effects that are comparable in magnitude to those caused by differences in the ENDF/B-VI.2 and ENDF/B-VI.3 evaluations. Consequently, comparisons between those two evaluations hereafter will be based exclusively on the unthinned ENDF/B-VI.2 library. The ENDF/B-V results will continue to be based on a thinned library, however, because an unthinned library for MCNP was not available. These results also suggest that the reactivity changes produced by ENDF/B-VI.3 are likely to be small. For that reason, the scope of the comparisons will be extended to include spectral indices as well as These spectral indices are and the conversion ratio (CR). is the ratio of fast fissions to thermal fissions in while is the ratio of fissions in to fissions in (summed over all energies). Similarly, is the ratio of fast to thermal captures in while is the ratio of fast to thermal captures in By convention, the breakpoint between the fast and thermal ranges is taken to be 0.625 eV. CR is the ratio of the production of fissile isotopes to the destruction of fissile isotopes; in the systems examined here, it is simply the ratio of

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captures in to absorptions in Because all the cases studied employed 1,000,000 active neutron histories, the computed uncertainties in the spectral indices typically are less than 1% (and often substantially so.) Consequently, those uncertainties will be ignored in the discussion that follows.

Lattices Comparisons were made for six different lattices with fuel: NB-1, NB-5, the OFA PWR cell with 1.6 wt.% enrichment, the standard PWR cell with 3.9 wt.% enrichment, the CANDU cluster, and the RBMK fuel cell. The first four lattices are moderated by light water, while the other two are moderated by heavy water and by graphite, respectively. As was noted earlier, NB-1 has a low enrichment (1.3 wt.%) and a relatively soft spectrum. The results for NB-1 are shown in Table 23. The ENDF/B-VI.3 result for increases by nearly 8% relative to that for ENDF/B-VI.2 (and by more than 3% relative to ENDF/B-V). The increase in epithermal capture reduces slightly, but it has relatively little impact on the other spectral indices. With the notable exception of the results from ENDF/B-VI.3 tend to accentuate the differences between ENDF/B-V and ENDF/B-VI rather than to reduce them. NB-5 has a very tight lattice and consequently a relatively hard spectrum, as a comparison of values from Table 24 with those from Table 23 demonstrates. Because the largest differences between the ENDF/B-VI.2 and ENDF/B-VI.3 evaluations occur in the epithermal range, the harder spectrum in NB-5 produces larger differences between the spectral indices than occurred for NB-1. still increases by essentially the same amount (nearly 8%), but because of the harder spectrum the increase in epithermal capture has a larger impact on which decreases by 0.0044 ± 0.0007 relative to ENDF/B-VI.2. Furthermore, the difference between the ENDF/B-VI.3 and ENDF/-V values for is more than 6%, which is double the difference for NB-1. In addition, the further hardening of the spectrum from the increase in epithermal capture increases not only but also and (at least marginally). The increased capture in more than compensates for the increased capture in from the hardening of the spectrum, and therefore CR decreases.

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The results for the OFA pin cell with an enrichment of 1.6 wt.% are shown in Table 25. The pattern of those results is very similar to that for NB-1. The only parameter that changes substantially from ENDF/B-VI.2 to ENDF/B-VI.3 is which again increases by nearly 8%. To the extent that the other parameters do change, they tend to accentuate the differences between ENDF/B-VI and ENDF/B-V rather than to mitigate them. The similarity between the results from NB-1 and this case is not surprising, because both of them contain slightly enriched fuel and have a relatively soft spectrum. The results for the standard PWR pin cell with an enrichment of 3.9 wt.% are shown in Table 26. Those results are intermediate between those for the OFA cell and NB-5. This behavior is not surprising, because the spectrum for this case is intermediate between the relatively hard spectrum of NB-5 and the relatively soft spectrum of the OFA cell. Once again, the ENDF/B-VI.3 value for is nearly 8% higher than the ENDF/B-VI.2 value, and the reactivity decreases (although by an intermediate amount, 0.0016 ± 0.0005 ). The changes in the other spectral indices are quite small. The ENDF/B-VI.3 value for for the CANDU cluster also increases by nearly 8% relative to the corresponding ENDF/B-VI.2 value, but, as shown in Table 27, the spectrum for that case is so soft that it has very little impact on or the other spectral indices. (For example, the CANDU value for is only about 40% as large as that for NB-1, which has the softest spectra of any of the previous four cases.) However, it is noteworthy that, even for this case, the ENDF/B-VI.3 changes to the thermal data for produce only a marginal increase in reactivity.

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The RBMK fuel cell also produces a very thermal spectrum, although it is slightly harder than that for the CANDU cluster. As Table 28 demonstrates, the patterns for the CANDU cluster also characterize the RBMK cell. Although increases by slightly more than 8% for this case, it has only marginal impact on and the other spectral indices.

Solutions Comparisons were made for four different solutions, two with HEU uranyl nitrate (ORNL-4 and ORNL-10), one with LEU uranyl fluoride (SHEBA-II), and one with HEU uranyl fluoride (sphere of uranyl fluoride in heavy water). ORNL-4 has the hardest spectrum of any of the ORNL HEU uranium-nitrate spheres because it contains the most boron and has the highest concentration of fissile material. Nonetheless, its spectrum is more thermal than that of any of the lattices discussed above except the CANDU cluster. As Table 29 illustrates, the value of increases by slightly less than 8% for ENDF/B-VI.3 relative to ENDF/B-VI.2, but the changes in and are essentially negligible. The reduction in most likely is due to competition between and rather than to a spectral effect, because the increase in epithermal capture should harden the spectrum rather than soften it. Similarly, the reduction in CR is due primarily to the small overall increase in absorption. ORNL-10 has the softest spectrum of any of the ORNL HEU uranyl-nitrate spheres because it has the lowest leakage, contains no boron, and has the most dilute concentration of fissile material. Its spectrum is sufficiently thermal that ENDF/B-VI.3 predicts an increase in reactivity relative to ENDF/B-VI.2, even though its value for is nearly 8% higher. This pattern is expected because of the differences between the ENDF/B-VI.3 and ENDF/B-VI.2 evaluations for in the deep thermal range. As shown in Table 30, the changes in the other spectral indices are essentially negligible. SHEBA-II has a harder spectrum than any of the ORNL HEU spheres, because its leakage is higher and, even though it contains LEU rather than HEU, its concentration of fissile material is nearly double that of the ORNL sphere with the highest fissile concentration. Nonetheless, its overall spectrum is intermediate between those of the lattice

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cases (softer than NB-5 and the 3.9 wt.% pin cell but harder than the others). As the results in Table 31 illustrate, the ENDF/B-VI.3 value for is marginally higher than that for ENDF/B-VI.2, even though the ENDF/B-VI.3 value for is approximately 8% higher than its ENDF/B-VI.2 counterpart. The small decrease in probably is due to increased competition for epithermal neutrons between capture and fission. The differences between the ENDF/B-VI.3 and ENDF/B-VI.2 results for and CR are negligible. The reflected sphere of uranyl fluoride in heavy water has the hardest spectrum of any of the solutions studied because it contains HEU, is moderated and reflected by heavy water, and has the highest leakage of any of the uranium solution experiments. Consequently, the differences in the results from ENDF/B-VI.3 and ENDF/B-VI.2 are more pronounced than for the other solutions, as Table 32 demonstrates. increases by nearly 10%, and drops by more than The effects on the other spectral indices also are more pronounced than in previous cases, although the actual changes remain relatively small. CR decreases by approximately 1.5% percent, and and each change by between 0.5% and 1%. The small changes to these spectral indices most likely reflect increased competition between and for epithermal neutrons rather than a softening of the spectrum. Overall, the reactivity differences between ENDF/B-VI.2 and ENDF/B-VI.3 are essentially negligible for most of these lattices and solutions. However, as the results for NB-5 and the sphere of uranyl fluoride in heavy water demonstrate, ENDF/B-VI.3 does produce a lower reactivity than ENDF/B-VI.2 for the cases with harder spectra.

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DATA TESTING: TIME-OF-FLIGHT MEASUREMENTS LLNL Pulsed-Sphere Experiments The pulsed-sphere benchmark experiments were performed at Lawrence Livermore National Laboratory (LLNL) beginning in the 1960s and continuing through the 1980s.64-69 The primary purpose of these benchmark experiments was to address the need for detailed neutron transport measurements that were sufficiently simple to calculate, yet complex enough to test some of the more sophisticated features of the transport codes and crosssection data. These experiments used an almost isotropic 14-MeV neutron source created at the center of a sphere of target material from the reaction at 400 keV. The neutron emission spectrum was measured using time-of-flight techniques over an energy range from 10 eV to 14 MeV. For the low-energy data between 10 eV and 1 MeV, a glass scintillator was used, while a Pilot B or NE213 scintillator was used for the high-energy measurements from 2 to 15 MeV. These detectors were placed at angles of 26°, 30°, and 120° with respect to the incident d-beam direction, and they had flight paths ranging from 750 to 975 cm. A representative experimental setup is shown in Figure 10. The thickness of the target materials ranged from 0.5 to 5.0 mean free paths (mfp) for the 14-MeV neutrons. The target materials included nuclides from to as well as composite materials such as concrete. Later experiments from the 1980s also included photon production measurements for the various targets. This discussion will focus on the high-energy range from the earlier neutron-transmission measurements.66-69 These earlier experiments included at least 49 measurements for 21 target materials. There were three types of target assemblies: (1) bare spheres of materials such as carbon, (2) clad spheres of materials such as the stainless-steel-clad lead sphere, and (3) spherical stainless-steel dewars for materials such as light water. Figures 11, 12, and 13 illustrate these spherical geometries for the 2.9 mfp carbon sphere, the 1.4 mfp lead sphere, and the 1.9 mfp lightwater dewar, respectively. Region 1 in Figure 12 is lead, and region 2 is stainless steel; region 1 in Figure 13 contains water, while regions 2 and 3 are stainless steel. The measured data for each spherical assembly ("target in") was normalized to the total flux measured with the material of interest removed from the spherical assembly ("target out").

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LLNL Pulsed-Sphere Benchmarks The high-energy pulsed-sphere benchmarks were first implemented for MCNP3B in the late 1970s70 and later for MCNP4A.71,72 We recently made revisions to these earlier implementations of the benchmarks, as well as implementing benchmarks for additional materials. Currently, we have implemented 36 benchmarks for 20 target materials, including 17 nuclides and 5 composite materials (light and heavy water, polyethylene, teflon, and concrete) as indicated in Tables 33 and 34, respectively. With the exception of Mg, Al, Si, Ti and W, these nuclides represent new evaluations for ENDF/B-VI.

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Revisions to the previously reported results71,72 include small geometry, source, and material corrections, improvements in the detector efficiencies used in the MCNP tallies, changes in some of the density specifications, and a more accurate conversion from time of flight to energy. In earlier implementations, the density was altered to reflect the total mass reported in various publications. In the revised benchmarks, densities reported in later publications were used for those materials.64,65 The improved relativistic conversion between the time of flight and the neutron energy is66

where E is the energy in MeV, L is the length of the flight path, c is the speed of light, and t is the time of flight. The source specification for the reaction includes the appropriate energy and angle distributions with respect to the incident d-beam direction, and it also includes the appropriate Gaussian time distribution for each experiment.67,70 For the lighter materials, the actual target assembly for the neutron source does not need to be included in the simulation. Ring detectors were used to calculate the flux at a given detector location. The corresponding detector efficiency was folded into the MCNP tally using the DE (detector energy) and DF (dose function) cards for the Pilot B and NE213 detectors.66,67,69 The detector tallies were binned in 2-ns time bins. Simulations were performed for both the target-in and target-out measurements. The target-in results were then normalized to the total flux of the target-out results for comparison to the reported experimental data. A more detailed description of these benchmarks will be reported at a later date.73

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The results may be compared graphically or by tabulating the integrated flux ratios within specified energy bins. Figures 14 through 49 graphically compare the experimental results with results calculated using MCNP and its continuous-energy ENDF/B-V and ENDF/B-VI libraries. The time scale employed for these Figures is shakes, where one shake is 10 ns. Table 35 presents the same comparisons over the neutron energy ranges of 2-3, 3-5, 5-10, 10-13, and 13-16 MeV. The standard deviations in the values shown in Table 35 are less than the number of significant figures given for those values. Consequently, the standard deviations will be ignored.

Pulsed-Sphere Benchmark Results As these results demonstrate, there is good agreement with the measurements for the light and fissionable nuclides and for the composite materials. However, the agreement for the larger spheres of the heavier, non-fissionable nuclides is not as good. The results show an underestimate of the flux over the energy region between 7.3 and 11.6 MeV (160200 ns) but an overestimate of the flux at lower energies. The results for are in much better agreement, particularly for the larger sphere. The ENDF/B-VI results for Be produce much better agreement with the measurements than do those from ENDF/B-V, although both underestimate the flux at energies below 2.5 MeV. The ENDF/B-VI results for C, however, are not as good as those from ENDF/B-V, although both underestimate the flux at the lowest energies. The ENDF/B-VI results for N produce improvement similar to that seen for Be, although both ENDF/B-VI and ENDF/B-V overestimate the flux at lower energies for the larger sphere. The results

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for O still show poor agreement with measurement, although ENDF/B-VI shows improvement at the lower energies. The ENDF/B-VI results for Pb indicate a significant improvement over the corresponding ENDF/B-V results. (ENDF/B-VI has an evaluation for each isotope of Pb, while ENDF/B-V contained only a single evaluation for the element.) However, the flux for Pb still is somewhat underestimated at lower energies. The results shown here for Fe are not conclusive. However, other studies74 have indicated that the ENDF/B-VI evaluation is an improvement over ENDF/B-V. The results for Al, Mg, Ti, and W are equivalent because the ENDF/B-V evaluations for those elements have been retained through ENDF/B-VI.2. Mg shows reasonable agreement for the smaller spheres, but the agreement is poor for the larger sphere. The Al results indicate reasonable agreement over the entire energy range, while the Ti results show very poor agreement. The W results indicate a significant underestimate of the flux at energies below 3.7 MeV. Both ENDF/B-V and ENDF/B-VI produce good agreement with the measured results for the fissionable nuclides. In particular, ENDF/B-VI produces improved results for in the energy range between 7.6 and 13 MeV. The results for show an overestimate of the flux from 10 to 13 MeV. In addition, ENDF/B-VI produces a slight increase in the flux over the energy range from 4 to 6 MeV relative to ENDF/B-V. The results for show no significant differences between ENDF/B-V and ENDF/B-VI. There is good agreement for the light- and heavy-water spheres and the polyethylene spheres for both data libraries, although the ENDF/B-VI data do show some improvement over ENDF/B-V. There also is improvement in the teflon spheres for ENDF/B-VI, particularly for the thicker sphere. The concrete spheres were composed primarily of O, H, and Si, and they show good agreement for both sets of evaluations.

CONCLUSIONS AND RECOMMENDATIONS ENDF/B-VI produces good to excellent agreement with most of the benchmark measurements studied herein. It also produces results that are very similar to those from ENDF/B-V for a majority of them. However, it produces dramatically better results for a few cases and slightly worse results for a few others.

Improvements in ENDF/B-VI The most obvious success for ENDF/B-VI is the dramatic improvement in the agreement with the benchmark values of for plutonium nitrate solutions, an area in which previous versions of ENDF/B had not performed well. In addition, ENDF/B-VI improves the agreement with benchmark values of for some other critical configurations relative to ENDF/B-V, including the Jemima experiments and SHEBA-II. For most of the reactor lattices and solutions examined in this study, ENDF/B-VI.3 produces negligible to marginal changes in reactivity relative to ENDF/B-VI.2. However, the results for NB-5 and the sphere of uranyl fluoride in heavy water demonstrate that ENDF/B-VI.3 decreases the reactivity significantly for lattices and solutions with harder spectra. ENDF/B-VI produces significantly better agreement than ENDF/B-V with time-offlight measurements for several materials, including Be, N, Pb, and heavy water. The improvement for Be and N extends throughout the energy range between 2 and 16 MeV. However, the improvement for Pb and heavy water occurs primarily in the range between 2 and 5 MeV, while the improvement for occurs primarily in the range between 7.6 and 13 MeV.

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Deficiencies in ENDF/B-VI Although ENDF/B-VI agrees as well as or better than ENDF/B-V with most of the benchmarks, there a few for which it produces marginally worse agreement. In particular, it underestimates by a larger margin than ENDF/B-V for Godiva, the B&W lattices, and the solutions of HEU uranyl nitrate. In general, ENDF/B-VI appears to increase neutron leakage relative to ENDF/B-V. Consequently it tends to produce slightly better agreement than ENDF/B-V for those benchmarks where ENDF/B-V overpredicts but to produce slightly worse agreement for those where ENDF/B-V underpredicts (The notable exception to this generalization is the sphere of uranyl fluoride in heavy water, which has a spectrum that is significantly harder than those of the other solutions.) ENDF/B-VI produces slightly worse agreement with the time-of-flight benchmarks for C than ENDF/B-V does. The deterioration occurs throughout most of the energy range between 2 and 16 MeV, but it is particularly noticeable at the lowest energies. Although the ENDF/B-VI and ENDF/B-V evaluations for O produce significantly different results for the time-of-flight benchmarks, neither of them produce good agreement. ENDF/B-VI produces reasonable agreement above 10 MeV, but the agreement with measurement deteriorates at lower energies. Additional deficiencies in ENDF/B-VI are carryovers from ENDF/B-V. In particular, the cross sections for produce a significant underestimate of for the Jemima-233 benchmark, and Ti shows poor agreement with the time-of-flight benchmarks. In addition, the tendency to overestimate leakage for bare spheres of uranium is common to both ENDF/B-V and ENDF/B-VI.

Other Observations There are some other differences between results from ENDF/B-VI and ENDF/B-V, but at this point it is not clear whether those differences represent improvements. Specifically, ENDF/B-VI produces a more pronounced variation in reactivity with enrichment for LEU lattices than ENDF/B-V does. Furthermore, relative to ENDF/B-V, produces small but significant reactivity differences for many of the cases with thermal spectra. This behavior improves the agreement with plutonium solution benchmarks, but it worsens the agreement for spheres of uranium nitrate.

Recommendations Further revisions to the ENDF/B-VI evaluation for already are under way. However, based on the results obtained herein, there are other areas that should be improved as well. Spheres of HEU and both exhibit significant reactivity swings between bare and reflected configurations. This behavior suggests that the current effort to revise the evaluation for should be extended to include the total cross section at high energies (~1 MeV). In addition, the cross sections for at high energies need to be revised to improve agreement with the Jezebel-233 benchmark and to reduce the reactivity swing between bare and reflected configurations. The ENDF/B-V evaluation for Ti was carried forward to ENDF/B-VI, but the time-offlight benchmarks clearly indicate that it needs to be revised. The evaluation for needs to be revisited as well. As the comparison with time-offlight benchmarks indicates, the ENDF/B-VI cross sections for are deficient between 2 and 10 MeV. In addition, the ENDF/B-VI evaluation for tends to produce more leakage in critical configurations than its predecessor did, and it also reduces reactivity slightly in cases with thermal spectra.

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ACKNOWLEDGMENTS We would like to thank a number of our colleagues at LANL for their contributions, without which much of this study could not have been performed. Robert E. MacFarlane prepared the unthinned ENDF/B-VI libraries for MCNP. Joseph L. Sapir, Raphael J. LaBauve, and Roger W. Brewer provided us with benchmark specifications for several of the critical experiments prior to the formal publication of those specifications. Denise B. Pelowitz gave us an MCNP input deck for the (very) complicated model of the MHTGR lattice. Guy P. Estes, Ronald C. Brockhoff, and J. David Court provided information on the previous implementation of the LLNL pulsed-sphere benchmarks, and Robert C. Little provided many helpful discussions on the results.

REFERENCES 1. Judith F. Briesmeister, ed., "MCNP—a general Monte Carlo n-particle transport code, version 4a," LA-12625-M, Los Alamos National Laboratory (November 1993). 2. R. Kinsey, "ENDF-201: ENDF/B summary documentation," BNL-NCS-17541, Brookhaven National Laboratory (1979). 3. B. A. Magurno and P. G. Young, "ENDF-201 supplement 1: ENDF/B-V.2 summary documentation," BNL-NCS-17541, Brookhaven National Laboratory (1985). 4. See S. Pearlstein, "Nuclear data for nuclear reactor analyses — an overview," Prog. Nucl. En., 13:75 (1984) and subsequent articles in that volume. 5. G. Breit and E. Wigner, "Capture of Slow Neutrons," Phys. Rev. 49:519 (1936); see also, George I. Bell and Samuel Glasstone, Nuclear Reactor Theory, pp. 391-398, Van Nostrand Reinhold Company (1970). 6. P. F. Rose and C. L. Dunford, "ENDF-102: data formats and procedures for the evaluated nuclear data file ENDF-6," BNL-NCS-44945, Brookhaven National Laboratory (1991). 7. F. T. Adler and D. B. Adler, "Interpretations of neutron cross sections of the fissionable materials for the resolved resonance region," P. B. Hemmig, Ed., Proc. Conf. on Neutron Cross Section Technology, CONF-660303, Vol. II, pp. 873-894 (1967). 8. R. E. MacFarlane, "Energy balance of ENDF/B-V," Trans. Am. Nucl. Soc. 33:681 (1979). 9. C. R. Weisbin, R. D. McKnight, J. Hardy, Jr., R. W. Roussin, R. E. Schenter, and B. A. Magurno, "Benchmark Data Testing of ENDF/B-V," BNL-NCS-31531, Brookhaven National Laboratory (1982). 10. P. F. Rose, "ENDF-201: ENDF/B-VI summary documentation," BNL-NCS-17541, Brookhaven National Laboratory (1991). 11. A. D. Carlson, W. P. Poenitz, G. M. Hale, R. W. Peele, D. C. Dodder, C. Y. Fu, and W. Mannhart, "The ENDF/B-VI neutron measurement standards," NISTIR 5177 (ENDF-351), National Institute of Standards and Technology (1993). 12. V. McLane, "ENDF-201 supplement: ENDF/B-VI summary documentation," BNL-NCS-17541, supplement 1, Brookhaven National Laboratory (1996). 13. S. T. Perkins, E. F. Plechaty, and R. J. Howerton, "A reevaluation of the reaction and its effect on neutron multiplication in fusion blanket applications," Nucl. Sci. Eng. 90:83 (1985). 14. For example, see D. M. Hetrick, C. Y. Fu, and D. C. Larson, "Calculated neutron-induced cross sections for 58,60Ni from 1 to 20 MeV and comparisons with experiments," ORNL/TM-10219, Oak Ridge National Laboratory (1987). 15. C. M. Perey, F. G. Perey, J. A. Harvey, N. W. Hill, N. M. Larson, and R. L. Macklin, transmission, differential elastic scattering, and capture measurements and analysis from 5 to 813 keV," ORNL/TM-10841, Oak Ridge National Laboratory (1988). 16. For example, see H. Derrien, G. De Saussure, R. B. Perez, N. M. Larson, and R. L. Macklin, "Rmatrix analysis of the cross sections up to 1 kev," ORNL/TM-10098, Oak Ridge National Laboratory (1986). 17. C. W. Reich and M. S. Moore, "Multilevel formula for the fission process," Phys. Rev., 111:929 (1968). 18. T. R. England and B. F. Rider, "Evaluation and compilation of fission product yields — 1993," LA-UR-94-3106 (ENDF-349), Los Alamos National Laboratory (1994). 19. T. R. England, J. Katakura, F. M. Mann, C. W. Reich, R. E. Schemer, and W. B. Wilson, "Decay data evaluation for ENDF/B-VI," Proc. Sym. on Nuclear Data Evaluation Methodology, Brookhaven National Laboratory, Upton, NY, 12-16 October 1992, p. 611.

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20. S. C. Frankle, "A revised table 1 for appendix g of the MCNP4A manual," XTM:96-146, Los Alamos National Laboratory (1996). 21. "Cross section evaluation working group benchmark specifications," BNL-19302 (ENDF-202), Brookhaven National Laboratory (November 1974). 22. "International handbook of evaluated criticality safety benchmark experiments," NEA/NSC/DOC(95)03, OECD Nuclear Energy Agency (April 1995). 23. G. E. Hansen and H. C. Paxton, "Re-evaluated critical specifications of some Los Alamos fastneutron systems," LA-4208, Los Alamos Scientific Laboratory (June 1969). 24. R. E. Peterson, "Lady Godiva: an unreflected uranium-235 critical assembly," LA-1614, Los Alamos Scientific Laboratory (September 1953). 25. V. Josephson, "Critical-mass measurements on oralloy in tuballoy and WC tampers," LA-1114, Los Alamos Scientific Laboratory (May 1950). 26. H. C. Paxton, "Los Alamos Critical-Mass Data," LA-3067-MS, Rev., Los Alamos Scientific Laboratory (December 1975). " LA-1671, Los 27. H. C. Paxton, "Bare critical assemblies of oralloy at intermediate concentrations of Alamos Scientific Laboratory (July 1954). 28. Cleo C. Byers, Jerry J. Koelling, Gordon E. Hansen, David R. Smith, and Howard R. Dyer, "Critical measurements of a water-reflected enriched uranium sphere," Trans. Am. Nucl. Soc. 27:412 (1977). 29. R. G. Taylor, "Fabrication and isotopic data for the water reflected U(97.67) metal sphere critical experiment," Y/DD-622, Oak Ridge Y-12 Plant (November 1993). 30. J. C. Hoogterp, "Critical masses of oralloy lattices immersed in water," LA-2026, Los Alamos Scientific Laboratory (March 1957). 31. M. N. Baldwin and M. E. Stern, "Physics verification program, part III, task 4, summary report," BAW-3647-20, Babcock & Wilcox (March 1971). 32. D. J. Diamond, R. D. Mosteller, and J. C. Gehin, "Ad hoc committee on reactor physics benchmarks," Trans. Am. Nucl. Soc., 74:285 (1996). 33. Russell D. Mosteller, "Benchmarking of MCNP against B&W LRC core XI critical experiments," Trans. Am. Nucl. Soc., 73:444 (1995). 34. F. Rahnema and H. N. M. Gheorghiu, "ENDF/B-VI benchmark calculations for the Doppler coefficient of reactivity," Ann. Nucl. Energy, 23:1011 (1996). and for critical experiments," Nucl. Sci. Eng. 12:364 35. R. Gwin and D. W. Magnuson, "Eta of (1962). 36. Alan Staub, D. R. Harris, and Mark Goldsmith, "Analysis of a set of critical homogeneous spheres," Nucl. Sci. Eng. 34:263 (1968). 37. Michelle Pitts, Farzad Rahnema, T. G. Williamson, and Fitz Trumble, "Benchmark description for unreflected 174-1 spheres containing uranyl nitrate," Trans. Am. Nucl. Soc., 74:227 (1996). 38. Richard N. Olcott, "Homogeneous heavy water moderated critical assemblies. part 1. experimental," Nucl. Sci. Eng. 1:327 (1956). 39. Kenneth B. Butterfield, "The SHEBA experiment," Trans. Am. Nucl. Soc. 70:199 (1994). 40. R. J. LaBauve and Joseph L. Sapir, "SHEBA-II as a criticality safety benchmark," Proceedings of the Fifth International Conference on Nuclear Criticality Safety (September 1995). 41. G. A. Jarvis, G. A. Linenberger, D. D. Orndoff, and H. C. Paxton, "Two plutonium-metal critical assemblies," Nucl. Sci. Eng. 8:525 (1960). 42. L. B. Engle, G. E. Hansen, and H. C. Paxton, "Reactivity contributions of various materials in Topsy, Godiva, and Jezebel," Nucl. Sci. Eng. 8:543 (1960). 43. David R. Smith and William U. Geer, "Critical mass of a water-reflected plutonium sphere," Nucl. Appl. Technol. 7:405 (1969). 44. R. I. Smith and G. J. Konzek, "Clean critical experiment benchmarks for plutonium recycle in LWR's," NP-196, Vol. I, Electric Power Research Institute (April 1976). 45. R. Sher and S. Fiarman, "Analysis of some uranium oxide and mixed-oxide lattice measurements," NP-691, Electric Power Research Institute (February 1978). 46. R. C. Lloyd, C. R. Richey, E. D. Clayton, and D. R. Skeen, "Criticality studies with plutonium solutions," Nucl. Sci. Eng. 25:165 (1966). 47. C. R. Richey, "Theoretical analyses of homogeneous plutonium critical experiments," Nucl. Sci. Eng. 31:32 (1968). 48. R. Q. Wright, Oak Ridge National Laboratory, private communication (May 1995). 49. M. L. Williams, R. Q. Wright, J. Barhen, W. Rothenstein, and B. Toney, "Benchmarking of epithennal methods in the lattice physics code EPRI-CELL," in: Proceedings: Thermal Reactor Benchmark Calculations, Techniques, Results, and Applications, P. F. Rose, ed., NP-2855, Electric Power Research Institute (February 1983).

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50. M. L. Williams, R. Q. Wright, and J. Barhen, "Improvements in EPRI-CELL methods and benchmarking of the ENDF/B-V cross-section library," NP-2416, Electric Power Research Institute (June 1982). 51. R. D. Mosteller, L. D. Eisenhart, R. C. Little, W. J. Eich, and J. Chao, "Benchmark calculations for the Doppler coefficient of reactivity," Nucl. Sci. Eng., 107:265 (March 1991). 52. J. R. Brown, et al., "Kinetic and buckling measurements of lattices of slightly enriched uranium or rods in light water," WAPD-176, Bettis Atomic Power Laboratory (January 1958). 53. A. R. Boynton, Q. L. Baird, K. E. Plumlee, W. C. Redman, W. R. Robinson, and G. S. Stanford, "High conversion critical experiments," ANL-7203, Argonne National Laboratory (January 1967). 54. R. D. Mosteller, J. T. Holly, and L. A. Mott, "Benchmark calculations for the Doppler coefficient of reactivity in mixed-oxide fuel," Proceedings of the International Topical Meeting on Advances in Mathematics, Computations, and Reactor Physics, CONF-910414 (April 1991). 55. N. H. Larsen, "Core design and operating data for cycles 1 and 2 of peach bottom 2," NP-563, Electric Power Research Institute (June 1978). 56. R. D. Mosteller and F. J. Rahn, "Monte Carlo calculations for recriticality during the reflood phase of a severe accident in a boiling water reactor," Nucl. Technol., 110:168 (1995). 57. "Russian RBMK reactor design information," PNL-8937, Pacific Northwest Laboratory (November 1993). 58. Denise B. Pelowitz, Joseph L. Sapir, and Janet E. Wing, "Analysis of the NP-MHTGR concept: a comparison of reactor physics methods," Proceedings of the 1992 Topical Meeting on Advances in Reactor Physics (March 1992). 59. J. Griffiths, "WIMS-AECL users manual," RC-1176, COG-94-52, Chalk River Laboratories (March 1994). 60. R. T. Perry, R. D. Mosteller, J. R. Streetman, J. L. Sapir, and R. J. J. Stamm’ler, "Preliminary benchmarking of the HELIOS code for hexagonal lattice reactors," Trans. Am. Nucl. Soc., 66:513 (1992). 61. C. R. Lubitz, "A modification to ENDF/B-VI to increase epithermal alpha and " Proceedings of the International Conference on Nuclear Data for Science and Technology, CONF-940507 (May 1994). 62. A. C. Kahler, "Homogeneous critical Monte Carlo eigenvalue calculations with revised ENDF/B-VI data sets," Trans. Am. Nucl. Soc., 72:384 (1995). 63. M. L. Williams, R. Q. Wright, and M. Asgari, "ENDF/B-VI performance for thermal reactor analysis," Trans. Am. Nucl. Soc., 73:420 (1995). 64. E. Goldberg, L. F. Hansen, T. T. Komoto, B. A. Pohl, R. J. Howerton, R. E. Dye, E. F. Plechaty, and W. E. Warren, "Neutron and gamma-ray spectra from a variety of materials bombarded with 14-MeV neutrons," Nucl. Sci. Eng. 105:319 (1990). 65. L. F. Hansen, et al., "Updated summary of measurements and calculations of neutron and gamma-ray emission spectra from spheres pulsed with 14-MeV neutrons," UCID-19604, Rev. 1, Lawrence Livermore National Laboratory (1989). 66. C. Wong, et al., "Livermore pulsed sphere program: program summary through July 1971," UCRL-51144, Rev. 1, Lawrence Livermore National Laboratory (1972). 67. W. M. Webster and P. S. Brown, "Low energy time-of-flight spectra from spheres pulsed by 14-MeV neutrons," UCID-17223, Lawrence Livermore National Laboratory (1976). 68. W. M. Webster, et al., "Measurements of the neutron emission spectra from spheres of N, O, W, and pulsed by 14-MeV neutrons," UCID-17332, Lawrence Livermore National Laboratory (1976). 69. E. F. Plechaty and R. J. Howerton, "Calculational model for LLL pulsed spheres (CSEWG shielding benchmark collection no. STD 10)," UCID-16372, Lawrence Livermore National Laboratory (1973). 70. G. P. Estes, "Pulsed sphere calculations with ENDF/B-V cross sections (for transmittal to CSEWG)," Los Alamos National Laboratory memoranda (October 24, 1979 and May 9, 1980). 71. Daniel J. Whalen, David A. Cardon, Jennifer L. Uhle, and John S. Hendricks, "MCNP: neutron benchmark problems," LA-12212, Los Alamos National Laboratory (1991). 72. John D. Court, Ronald C. Brockhoff, and John S. Hendricks, "Lawrence livermore pulsed sphere benchmark analysis of MCNP ENDF/B-VI," LA-12885, Los Alamos National Laboratory (1994). 73. S. C. Frankle, "Revised LLNL pulsed sphere benchmarks for MCNP," to be published by Los Alamos National Laboratory. 74. John D. Court and John S. Hendricks, "Benchmark analysis of MCNP ENDF/B-VI iron," LA-12884, Los Alamos National Laboratory (1994).

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SYSTEM DYNAMICS: AN INTRODUCTION & APPLICATIONS TO THE NUCLEAR INDUSTRY

K. F. Hansen and M. W. Golay Nuclear Engineering Department Massachusetts Institute of Technology Cambridge, MA 02139

1. INTRODUCTION The field of nuclear technology has been developing for over 50 years and has moved from the laboratory into a very large commercial industry. The growth in the underlying science and engineering has been remarkable both in its breadth and depth. The ability to design, analyze, and understand the behavior of nuclear plants is firmly established. There remain many challenging technical problems, but success of the industry is not contingent upon solving those technical problems. Rather, the success of the industry will be determined by a wider array of concerns than pure technology. For instance, nuclear plants in the future will have to compete economically against efficient, versatile, and reliable fossil technologies. In addition, potential users must be assured that the indirect costs, such as those of environmental effects and waste disposal, are acceptable. Finally, public perceptions about risks must somehow be allayed, if not resolved. The objective of this paper is to provide an introduction to a tool that may be useful to the industry in addressing the types of issues suggested above. The tool we discuss is system dynamics. It has been used with considerable success in many other fields in ways that are similar to the needs of the nuclear field. In the next section we provide some background on the system dynamics method and illustrate how system dynamics models are constructed. In section 3 we discuss two applications in the nuclear field, the first relating to construction of nuclear plants and the second in the operation of a nuclear utility in the social/political environment of today in the United States. We conclude with some summary comments.

Advances in Nuclear Science and Technology, Volume 24 Edited by Lewins and Becker, Plenum Press, New York, 1996

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2. BACKGROUND System dynamics is a methodology involving the application of engineering systems analysis to business, economic, and public policy problems. Originally developed at MIT in the 1960s, it is currently being used by many public and private sector organizations to help managers understand and forecast the results of their actions. System dynamics is used to analyze organizational, market, or other behavior which arises from the dynamic interaction, over time, of many interrelated variables. It simulates system behavior using explicit models of the internal cause-and-effect interactions of the system components. These models are recursive and deterministic. They are primarily distinguished from other types of models by their rich treatment of non-linear relationships, including feedback loops (Forrester, 1961; Lyneis, 1980; Pugh, 1986; Roberts, 1978). The application of system dynamics models to project management began with the analysis of complex research and development projects (Roberts, 1964; Weil, Bergan, and Roberts, 1973). Subsequent use included many different major naval and civil shipbuilding programs (Cooper, 1980). Over the past ten years, the range of applications has widened; system dynamics models have been used to analyze, design, and build projects in the aerospace, electronics, civil construction, nuclear power, and software industries (AbdelHamid and Madnick, 1989). Over sixty projects involving more than $40 billion in total cost have been modeled. The first system dynamics project models allowed managers to evaluate alternate prospective management policies and initiatives on ongoing projects. Later, system dynamics was established as a powerful means of determining responsibility for past project performance problems, by quantitatively answering the question "What would have happened if...?" (Weil and Etherton, 1990). It has been used many times to assist settlement of complex contractual disputes. Newer applications to project management have included cost forecasting to support new project bids; comprehensive project risk analysis and contingency planning; diagnosis of performance problems; tests of possible management initiatives and mitigating actions; analysis of the behavior of competitors and project manager training (similar conceptually to use of aircraft flight simulators). The ideas and concepts are most easily understood by a simple example. The development of a system dynamics model is usually preceded by the development of a causal loop diagram in which an analyst attempts to illustrate the cause-effect relations being modeled. The construction of a causal loop is very similar to the flow charts used in other modeling methods. Once the analyst has captured the structure of a problem in a causal loop, it is then necessary to define quantifiable variables to represent the systems. This stage of the process leads to a model diagram where variables of various types become defined and classified according to the role the variable plays in the model.

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Figure 1. Flow of ncutrons into and out of Group 1.

In order to illustrate both stages of development, we chose as an example the familiar few-group neutron difusion problem. For purposes of the example, we will represent a three-group problem with the following plausible characteristics: all fission neutrons are born fast slowing down is only between adjacent groups there is no up scatter out of the thermal group These assumptions are made to keep the model simple. The system dynamics technique is sufficiently versatile that any degree of complexity in the physics of the problem could be represented. For ease of illustrating the technique, we chose a simple, yet realistic, reactor physics model. We denote the fast neutrons as group 1, the intermediate neutrons as group 2, and the thermal neutrons as group 3. A causal loop diagram can be evolved starting anywhere in the system. We begin by asking how neutrons in group 1 are created and lost. Figure 1 represents the processes by which neutrons are produced, i.e. fissions, and lost via leakage, absorption, and down scattering. Note that there is one feedback loop apparent in the fast fission term. We then couple the fast group to the intermediate group in Figure 2. Neutrons in group 2 are lost via leakage, absorption, and slowing down. Neutrons enter group 2 only by slowing down from group 1. Note the model now has 2 feedback loops. The causal loop is completed in Figure 3 where the group 3 neutrons are represented.

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The next step in the model development is to convert the causal loop into a quantifiable model. It is clear from the context that the group 1, 2, and 3 fluxes are variables that satisfy a conservation principle; hence they are levels or stocks in the system dynamics jargon. It is also clear that the loss mechanisms for each group represent rates of decrease, i.e. outflows. Figure 4 illustrates the outflows as three different paths. The leakage path leads to a neutron sink which is represented as a cloud. The valve controlling the flow is represented by a parameter, and the level itself. These two combine to determine the rate of neutron leakage. The absorption path is similar, except that the control valve is governed by the level and the parameter The slowing down rate is determined by the group 1 flux and the transfer cross section The flow into the group 1 flux is determined by the fission rate. Part of the production comes from fast fission, i.e., by combining the group 1 flux and The fast fission rate is represented as an auxiliary quantity that will be combined with intermediate and thermal fission rates to determine the total group 1 production rate.

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Figure 5 shows the addition of the intermediate group to the model. The group 2 flux is a level. The outflows due to leakage and absorption are combined as a single flow path. The fission rate is an auxiliary quantity that combines with the fast fission rate to produce the flow into group 1. The inflow into the group 2 flux is via the slowing down rate from the fast group. Finally, Figure 6 shows the full representation of the entire three-group model. There are very sophisticated software packages (ITHINK, 1994; VENSIM, 1995) that assist analysts in producing figures such as 6. Further, the software packages guide the user

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in constructing the equations that are used to represent the model. The software solves the dynamic equations in discrete (user selected) time steps. For instance, each level equation must be of the form:

For group 1 the equation would be of the form

The inflow rate will be Inflow rate = Fast fission rate + Intermediate fission rate + Thermal fission rate.

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The software will guide the user in making each fission rate a function of the indicated variables. Likewise, the outflow rates will be a function of the group flux and the parameters and

3. APPLICATION OF SYSTEM DYNAMICS TO NUCLEAR INDUSTRY We illustrate how system dynamics can be used in the nuclear industry via two largescale examples. The first is a model of nuclear power plant construction. The model evolved from earlier project management models used in the shipbuilding industry (Cooper, 1980). The second example is a recent development from an MIT research program on nuclear safety. The model attempts to integrate nuclear plant operations with the external social/political/regulatory environment (Hansen, Turek, and Eubanks, 1995).

3.1 Nuclear Power Plant Construction Many of the 109 United States nuclear power plants experienced large cost increases during construction. In an effort to understand their causes, an analysis was undertaken of the plant construction process. This work was guided by interviews with many engineers and managers who had been involved in past nuclear power plant projects. They consistently indicated that a major difficulty of completing a project successfully was the unstable decision-making climate in which a project was developed. Our analysis investigated the implications for costs of such an environment, where instability resulted from an unending stream of changes in the project's requirements, constraints and resources. Many of these changes arose within the nuclear safety regulatory system, but many others arose within the organizations pursuing the projects. In modeling the work of designing and building a nuclear power plant it was necessary to take into account the resources needed to accomplish the work to be done, and also the factors summarized in the causal loop of Figure 7. Many of these factors are related to the degree that the project is able to maintain the rate of progress needed for all of the interacting work tasks to mesh harmoniously. From people having extensive experience in such projects it is consistently reported that the quality of the work being performed (i.e., the fraction of the work being accomplished which is actually useful in completing the project) is degraded as the project falls behind schedule. This can happen for many reasons. Among the most important in the United States experience are the following: Changed performance requirements render obsolete some of the previously accomplished work. Recently-imposed project changes interfere with project coordination and distract the workers and their managers from their tasks.

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The demands of the world outside the project demand the attention of the project managers at a higher priority than the project itself. A more complete listing of factors affecting both productivity and work quality in the work presented here is listed in Table 1.

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In modeling a United States nuclear power plant project it is necessary to model both the design and construction phases of work -- which ultimately results in construction of a completed power plant. The structure of the work flow paths is the same in both phases, as illustrated in Figures 7 and 8. The flow of work obeys the following sequence: The existence of a work backlog (e.g., drawings to be made, power plant sections to be built) triggers the assignment of workers to perform the needed work. The assigned workers produce completed work products at a rate equal to the product of their population and their productivity. The fraction of completed work products flowing into the category of accepted work products is equal to the work quality and that flowing into the category of rework products is equal to (1- work quality). Previously accepted work products can be rendered obsolete by new changes to the project. New rework gives rise to new work to be performed. As the project proceeds, workers and managers are added and shed from the project according to the populations needed to complete the work backlogs and rework inventories by the planned completion date. As workers are added, they take some time to become fully capable; as they leave the job they take their experience and capabilities with them. All of these processes have their respective non-productive time delays which must be experienced before they can be completed. These delays can cause serious project disruptions if they have not been planned for. Particularly, those associated with finding previously existing, but undiscovered, rework can seriously impair project progress and drive up costs. The longest rework time delays involve engineering errors discovered during construction. If the number of such errors is large, the project can be affected very seriously. The project costs mount as funds are expended for work, materials, and parts. Should insufficient funds be available, of if restrictive policies are imposed, the populations of workers, the rates of work progress, and the rates of supplies being purchased can be restricted. Doing this often costs far more than the short term savings accomplished.

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Thus, the effect of a new change to the project is felt in several areas simultaneously -in terms of new work which must be performed at all levels of the project organization, in decreased worker productivity and product quality, and in some previously completed work being rendered obsolete. These individual influences are felt directly and also in an amplified form arising from their mutual interactions. When we simulate the construction experience of the 1970s, utilizing model parameter data typical of electric utility and power plant design organizations of that time, we obtain the power plant construction experience summarized in Figure 9. We see in a stable decision-making environment that the power plant is constructed in approximately eight years, with a peak engineering work force of 220 and a peak construction work force of 1150. The cumulative labor hours required for engineering and construction, respectively, are 1.13 and 7.69 million (Bespoloka, et al., 1994). Then, we take into account the unstable decision-making environment of the 1970s. This is done by considering the perturbations to the project by the new regulations issued by the US Nuclear Regulatory Commission (NRC) and its predecessor the Atomic Energy Commission. When we treat each new regulation as having the project disrupting effects described above, and when we apply the new regulations to the project at the rate at which they were issued during the 1970s, we obtain the results shown in Figure 10. We see that the duration of the project is extended from 8 to nearly 10 years, and the peak engineering and construction workforces grow from 220 to 400 and from 1150 to 1550, respectively. In subsequent studies exploring the effects of various management practices, we find that the combination of maintaining an accelerated project completion schedule in combination with improved work practices (i.e., improved basic work productivity and quality) in the presence of these chronic changes reduces the cumulative labor hours to 3.25 million for engineering and 10.71 for construction, with a project completion duration of 7 years. From this we can see that the effects of unanticipated changes can be very disruptive to a project; however, good management practices can help to control them. System dynamics analysis can help in identifying these practices.

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3.2 Nuclear Utility Model A second application of system dynamics to the nuclear industry is in the operation of a nuclear plant. The safety and performance of the plant are functions of many technical factors such as initial design, service and maintenance programs, and staff training. Safety and performance are also influenced by the social/political environment in which the plant operates. Thus, exaggerated safety concerns by the public could cause safety regulators to impose constraints that reduce resources available at the plant. The nuclear power plant model described below attempts to model those complex interactions. The overall model is shown in Figure 11. The nuclear utility sector represents the internal functions of the utility and the nuclear plant. The government sector represents the regulation of the industry, primarily through actions of the Nuclear Regulatory Commission (NRC). The public sector models the behavior of the local and national publics, the actions of the intervenor groups, the role played by the media in influencing public opinion, and the effects of industry/utility public relations programs. The financial resources sector models the flow of income and expenses in operating the plant, as well as related factors such as Public Utility Commission (PUC) rate decisions. The final sector, safety information, models the flow of industry-wide experience into improved practices and procedures. The overall model is very complex and incorporates about 1500 relations. We illlustrate a typical representation of the utility sector in the following. Figure 12 is a causal

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loop of the plant capacity determination. The "production" term represents the electric energy generation. As the plant produces energy, the equipment functions and is therefore subject to "wear and tear". The more wear and tear, the greater the breakdown of equipment. The more "equipment broken down", the less will be the amount of "functional equipment". Finally, the "plant capacity", and its rated capacity, determine the energy "production". This simple loop is modified by several other loops as seen in Figure 13. Broken equipment is repaired by corrective maintenance. Some equipment may be removed from service via preventive maintenance and inspections. Conversely, preventive maintenance and inspections will reduce the amount of broken equipment. The inspections made may be mandated by the NRC, or they may be discretionary. The work to repair, maintain, or inspect equipinent is done by the plant staff. Figure 14 illustrates the coupling between the plant capacity loops and the staffing loops. Taken together the corrective maintenance, the preventive maintenance, and the inspections

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determine the amount of work to be done. This total, combined with a desired schedule, determines the staff level required to do the work. The actual work accomplished is determined by the available staff. Any discrepancies between required and available staff determine the need to hire or fire workers. Limits on staff size come through budget allocations. The connections to other sectors are easy to imagine. Thus, production feeds to the financial sector where it combines with allowed rates to generate revenue. The financial sector feeds back information to the utility sector for the budgetary allocations to staff levels and for discretionary maintenance work. The information sector provides information to be analyzed by the utility staff for discretionary inspections. This sector also influences the learning curves that affect workers' performances and productivities in conducting their work. Finally, efforts by the utility to process and learn from industry experience are used as part of the NRC evaluation of utility performance. The government sector couples through the NRC actions regarding mandatory inspections and requests for information. NRC actions are affected by plant events and events elsewhere in the industry. In addition, information requests to the NRC from Congress may generate regulatory requests to the utilities. Plant performance affects the attitudes of the local and national public. Events at the plant can create public concern which, in turn, leads to greater resources being generated for intervenors. The intervenor groups influence the media, Congress, and the NRC. The impact is felt in the utility by increasing NRC requests for information, or increased levels of regulation.

3.3 Some Typical Results The model can be used to analyze a host of utility policy issues. For example, what are the benefits and costs of an aggressive preventive maintenance program? Another problem

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of interest is the value of public relations efforts and/or reducing events such as forced outages at the plant. We present below some results that analyze the long-term impacts of a large accident, such as TMI, at a plant other than the utility being modeled. The problem illustrates the "connectedness" between all nuclear utilities. The model assumes that the industry is operating in a near steady state environment until week 150, at which time a large accident occurs at a different utility. Figure 15 is a representation of the media activity and the state of public concern. Initially, there is little media interest in nuclear power. Public concern is a vague concept which we represent in terms of the fraction of the population that is willing to provide financial support to intervenor groups. In the simulation about 15% of the population is represented as being "concerned" prior to the event. Immediately after the accident, media reports pour out and these induce a rapid increase in public concern . After a few weeks, the rate of production of reports declines rapidly. (The data for the rate of decline was obtained from analyzing the actual post-TMI record.) The small upsurge after week 260 results from declining nuclear plant performance to be discussed below. Public concern remains high for a much longer time than media interest. This reflects the much longer memory of the public. Figure 16 represents the NRC response to the accident. Before the event, the NRC conducts a steady level of investigations which produce a slowly rising number of regulations. After the accident, the NRC begins a large number of investigations, which place heavy burdens upon the utilities. The investigations ultimately decline as studies are completed. However, completed studies produce a near doubling of regulations over the next 3 years. The results are similar to what actually happened after TMI. Finally, we assume there is a slow obsolescence of regulations over the long-term.

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K. F. HANSEN AND M. W. GOLAY The response at the plant is presented in Figure 17. Graph 1 is the plant capacity factor.

Immediately after the accident, plant staff are diverted to respond to NRC investigations. The maintenance program suffers and the plant capacity drops. The revenues then decline. Costs increase rapidly due to overtime, and, with some delay, due to increasing the staff size with added work. The plant gets back to full capacity after nearly a year. However, the increase in regulations then begins to drop plant capacity, which in turn decreases revenue. The need to increase the staff drives up costs. The decline in performance around week 260 is noted by the media and leads to an upsurge in media reports. Ultimately the plant gets back to full capacity. However, the staff required is now nearly 50% larger because of the cumulative effect of new NRC regulations. Further, unless the PUC will allow a rate increase, the cost of operation would exceed the revenue from sales. The simulation results presented are not based upon calibrating the model against a specific nuclear plant. As such, the results presented must be viewed as suggestive of the types of analyses that can be performed with the model. Other simulations have been used to study the complex interactions between operations, maintenance, resource allocation, and external events such as an accident. One of the most interesting results is the value of preventative maintenance (PM). Our results suggest that there are direct and indirect benefits to preventative maintenance. Direct benefits include better plant performance and reduced risk of events that could create media attention and thus increased public concern. An indirect benefit is that a plant with a preventative maintenance staff is better able to respond to the influence of an accident elsewhere. The PM staff represents a surge capacity to deal with NRC requests for information without an immediate, negative impact on capacity. Our simulations show that the utilities with an adequate PM staff suffer less capacity loss, and lower overall lost revenue, in responding to an incident. We reiterate that the model is still relatively new and cannot be considered validated against real data. It is presented here for the purpose of showing how system dynamics can be used to couple the technical aspects of nuclear power to the external social/political environment.

3.4 What is Useful in a Model In addition to the obvious use of a model to simulate scenarios of interest, a system dynamics model can have many valuable uses. Among the most important are the following: Improved understanding and group consensus of the system structure, important internal causal relationships gained through the process of building causal loop, and process flow diagrams.

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Improved group consensus concerning the nature of the problems which can be attacked to improve the performance of the system, and of the effects of the external world upon the system. Improved understanding of the system's performance during the past, gained through the process of gathering data for comparison to simulations of past performance. Improved insights into the system's sensitivities and vulnerabilities under a range of internal policies and external conditions. Improved policies for optimizing system performance and for reducing vulnerabilities to potential external circumstances. Potentially more successful negotiation among stakeholders in a common problem involving the system by using a model to simulate the outcomes expected to result from various proposed problem solutions and policies (and establishment of a record of the information provided in advance by these analyses). Thus, it is evident that building a system dynamics model can be beneficial even when the model development effort is not carried to the point of completion, or even of quantitative model development. Since the early stages of model development are relatively less costly, engaging in causal loop development can especially be one of the most beneficial parts of the model development process.

3.5 Problems of Using a Model Using a system dynamics model requires art as well as knowledge. This is because the approximations which are inevitably part of a model introduce a requirement for interpretation and review of the results to ensure that they are realistic. Also, it is important to have a disciplined process for ensuring that the definition of each variable of the model is kept invariant during the model's use. This requirement arises because the variables are defined using a natural language, and the terms of such languages are often ambiguous, particularly in describing aspects of social systems. Once problems of model realism and consistency have been addressed, one still must deal with problems of model comprehensives, correctness, and veracity. The problem of comprehensives, or ensuring that all of the variables which are important in the problem are included in the model, cannot be solved definitively, at least in modeling the behavior of social systems. This is because there is no known way to ensure that the important causal interactions within the system have been identified. The process of identification is pursued by an iterative direct search, with the model which is being developed being compared repeatedly to one's knowledge of the actual system and then refined. Because our evidence of the system's behavior is restricted to that of past experience, our knowledge of the actual non-linear system structure will always be imperfect, at least in the sense that we cannot be sure that the model does not contain behavior modes which have not yet been revealed.

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Then with a given state of knowledge of the system's structure and causal relationships, we still can have only limited confidence in our model's correctness. The causal relationships must be stated in explicit mathematical terms, when we can often only guess at plausible forms for these relationships. Usually the data needed for confident quantification are absent, but the qualitative nature of the causal relationship will be known. Finally, once a quantitative model has been specified, based upon our best knowledge, there remains the problem of building an operating computational model and verifying that it satisfies the specifications. With many computational models this is done by requiring the model to reproduce the results of analytical test cases or, failing that, to produce the output results known from an oracle to correspond to a set of input data cases. Because the social systems to which system dynamics models are often applied are non-linear and their past behavior is poorly documented, neither approach for verification is usually successful. Rather, the best that one can do is usually limited to requiring the model to reproduce the data of past performance which can be found. The process of reproduction is usually an iterative one, requiring considerable adjustment of the model's free parameter values. Doing this does not establish model verification, but rather provides a basis for plausible belief. Thus, with system dynamics models one cannot typically be sure of the model's correctness at the structural, quantitative, and executable levels. Nevertheless, system dynamics models can be very useful as a means of informing us of potential system behavior which would otherwise be very difficult to anticipate. We must always be careful to ensure that this information is trustworthy. However, with a good model we are very often better off with it than without it. In addition to these factors, system dynamics models also provide opportunities for including in their results the types of errors common to all numerical models, including those of truncation, stability, and correct input formulation.

4. SUMMARY We have attempted to introduce a tool that extends the range and nature of nuclear industry analysis beyond the conventional technical analysis. Our goal has been twofold: to illustrate the simplicity and generality of the system dynamics technique, as well as to provide examples of how the technique can be used to understand and assess complicated problems. The examples we have chosen illustrate the diverse problems of nuclear construction in an environment of regulatory change, and plant operations in a social/political context. We believe the technique could be used in a number of ways in the future. In the context of internal utility planning and organization we can imagine several uses, for example, in resource allocation between corrective and preventative maintenance. The

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model described in Section 3.2 can be used to study the safety and economic impacts of various allocation schemes. Similarly, the costs and benefits of investment in lessons learned for the worldwide industry can be assessed. Another potential application could be the analysis of various management structures and organizations. The dynamic behavior of a managerial system is dependent upon the distribution of authority and resources, as well as the flow of information throughout the system. Progress in modeling such problems is being made in other fields (Senge, 1990) and should be applicable to nuclear utilities. Finally, we believe that system dynamics models can be a valuable source in training and educating personnel. Operator simulators are highly advanced in the nuclear industry as well as other industries. There are emerging numerous training tools for managers and others using system dynamics models (Ledet, 1996). Of particular interest is the opportunity for managers to gain greater insight into the impacts of various policies on many aspects of plant performance, for instance, the long-term impacts of resource allocation. On a much broader scale, we foresee that system dynamics models could be used by safety or economic regulators to better understand the impacts of regulatory changes. The nuclear plant construction model indicated how very costly changes are. Generally, the direct costs of change are a small fraction of the total cost because of the delays and disruptions that change produces. In a similar way, a new regulation that might increase mandatory inspection rates could be very expensive to a utility with a "lean and mean" organization. In general, lean and mean implies an organization fully occupied without any capacity to increase the work load. Any change would have a highly nonlinear impact on the organization's productivity and efficiency. In fact, we believe the costs of maintaining surplus staff can be justified to economic regulators on the grounds of lower long-term costs and better, safer plant performance. At the most abstract level, we believe system dynamics models can be used in reaching a social consensus on the future of nuclear power. In order for society to develop appropriate control of the technology, it is vital that there be a common understanding of the complex relationships between plant performance -- including safety -- and regulations. There is no such common understanding today -- and very little progress to date in reaching that goal. System dynamics is a tool for developing such understanding based upon creating a collective understanding of cause-effect relations, a collective recognition and representation of feedback pathways, and a collective set of quantifiable variables with which to express the relations. Use of such a system can lead to a shared understanding of how behavior responds to various policies and what are the unexpected and unintended consequences of various policies. We may hope that a sufficiency of reasonable people can come together to formulate a coherent policy on nuclear power.

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5. REFFERENCES Abdel-Hamid, T. K., and S. E. Madnick, 1989. Software Project Management, Englewood Cliffs, NJ: Prentice-Hall. Beckjord, E., M. Golay, E. Gyftopoulos, K. Hansen and R. Lester, 1987. "International Comparison of LWR Performance." MIT Energy Laboratory Report No. MIT-EL 87-004. Bespolka, C., W. Dalton, M. Golay, K. Hansen, and H. Weil, 1994. "Competition and the Success of Complex Projects." Presented at the 1994 International Systems Dynamics Conference, August 1994. Cooper, K. G., 1980. "Naval Ship Production: A Claim Settled and a Framework Built." Interfaces, Vol. 10, No. 6, December 1980. Forrester, J. W., 1961. Industrial Dynamics. Cambridge, MA: The MIT Press. Hansen, K. F., M. G. Turek, and C. K. Eubanks, 1995. "System Dynamics Modeling of Social/Political Factors in Nuclear Power Plant Operations." Proceedings International Conference on Mathematics and Computations. Reactor Physics, and Environmental Analyses, April-May 1995, pp. 715-724. "ITHINK", 1994. High Performance Systems Inc., Hanover, NH. Institute for Nuclear Power Operations, 1992. "Annual Reports of U.S. Nuclear Power Industry Performance Indicators." Personal Communication, 1996. Kingwood, TX.

Winston P. Ledet, "The Manufacturing Game."

Lyneis, J. M., 1980. Corporate Planning and Policy Design. Cambridge, MA: The MIT Press. Now available from Pugh-Roberts Associates. Pugh, A. L., 1986. Professional DYNAMO Plus Reference Manual. Cambridge, MA: Pugh-Roberts Associates. Roberts, E. B., 1964. The Dynamics of Research and Development. Cambridge, MA: The MIT Press.

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Roberts, E. B., ed., 1978. Managerial Applications of System Dynamics. Cambridge, MA: The MIT Press. Senge, P. M., 1990. The Fifth Discipline. New York, NY: Doubleday/Currency. "VENSIM", 1995. Ventana Systems, Inc., Belmont, MA. Weil, H. B., T. A. Bergan, and E. B. Roberts, 1973. "The Dynamics of R&D Strategy." Proceedings of the 1973 Summer Computer Simulation Conference. Also published in Roberts, E. B., ed., Managerial Applications of System Dynamics. Cambridge, MA: The MIT Press, 1978. Weil, H. B., and R. I. Etherton, 1990. "System Dynamics in Dispute Resolution." In D. F. Anderson, G. P. Richardson, and J. D. Sterman, eds., System Dynamics '90. Cambridge, MA: Sloan School of Management, Massachusetts Institute of Technology.

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BN THEORY: ADVANCES AND NEW MODELS FOR NEUTRON LEAKAGE

CALCULATION

Ivan Petrovic1 and Pierre Benoist2 1

Department of Nuclear Engineering, University of California at Berkeley, Berkeley, CA 94720, USA 2 32 rue du Montparnasse, 75006 Paris, France

INTRODUCTION Nuclear reactor physics design and analysis require broad knowledge of parameters affecting reactor operation. Power distributions, control rod worth, shut down margins, isotopic depletion rates, etc., have to be determined throughout the reactor cycle. The efficiency of predicting these quantities depends strongly on models used to determine neutron density in space, direction and energy. When thermal-hydraulic characteristics of the reactor and nuclear data are available it seems as if one should be able to solve the three-dimensional transport equation. Unfortunately this is not true. The use of direct methods for solving the three-dimensional transport equation is limited because of the complexity in explicit modeling of every fuel pin, control rod, burnable poison rod and coolant channel. Although considerable progress has been made in recent years to increase the capabilities of digital computers, the magnitude of the computational problem created by explicit modeling is such that even most sophisticated computers are still not capable of evaluating reactor parameters to a satisfactory degree of accuracy. In order to overcome this computational problem of explicit geometrical modeling, most reactor analysis methods are based on coupling geometrically simple and energy dense calculations with geometrically complex and energy scarce calculations by way of spatial homogenization and energy group collapsing. In other words the neutronic calculations for a nuclear reactor are separated into two global steps. First, the fine multigroup transport calculation of assemblies or cells is carried out, with 60 to 100 energy groups, or more, where each cell consists of fuel pin, clad, coolant and/or moderator. This calculation is followed by a space homogenization and an energy group condensation of cross sections using the neutron spectrum obtained for these assemblies or cells. Secondly, the few group diffusion, less often transport, calculation of the whole reactor composed of homogenized zones with 2 to 6 energy groups is performed. These homogenized zones are usually homogenized assemblies or cells.

Advances in Nuclear Science and Technology, Volume 24 Edited by Lewins and Becker, Plenum Press, New York, 1996

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The homogenization of heterogeneous lattice of assemblies or cells has been a subject of much work because of its significance to whole-core calculations. Many authors have investigated and proposed different homogenized diffusion theory parameters for a homogenized heterogeneous cell, which are to be implemented in a whole-core calculation. The main question is which flux should be used in order to obtain homogenized parameters as well as how the diffusion coefficient should be defined. The difficulty arises from the fact that the diffusion coefficient, which may be interpreted as the leakage cross section per unit buckling, is not a quantity appearing in the primary form of the transport equation, but it seems that it needs to be defined via the balance equation in such a way as to conserve the neutron balance. Consequently, the definition could come out as a function of cross sections and that would depend on the finite size of the considered medium. From the point of view of practical reactor concerns the terms not dealing with leakage seem to be the more important ones since they define important reaction rates. At the same time these reaction rates themselves depend on the neutron leakage from one region to another and therefore the overall leakage effects influence considerably global integral quantities as well as reactivity. In order to understand the meaning of the diffusion coefficient it would be useful to perform an analysis. If it is assumed that the reaction rates are the quantities of primary importance and since they depend on angle integrated flux the angular distribution can be ignored to a certain extent. Therefore the integrodifferential transport equation can be integrated over the whole solid angle. But this integration does not eliminate the angular dependence since the resulting equation contains two angular moments of the flux: the zeroth moment being the scalar flux and the first moment being the vector current, with the latter occurring exactly in the leakage term. To relate these two moments another equation is required and the most direct relationship would be the one which resembles Fick’s law in which the diffusion coefficient occurs as a proportionality constant relating the current to the negative of the gradient of the scalar flux. Here, we have said that the relation which resembles Fick’s law has been used. Looking strictly, when Fick’s law is used within diffusion theory it results in the diffusion coefficient being equal to the well known one third of the mean free path of the considered medium. In transport theory the diffusion coefficient does not have this value except in the case of the homogeneous theory when dimensions of a reactor are great compared to the mean free path of the medium and if the development for the diffusion coefficient is limited to the first order of buckling. This is the reason why we will from now on when working within transport theory use the name leakage coefficient instead of the diffusion coefficient. Another reason is that by multiplying the leakage coefficient with the square buckling the leakage cross section is obtained. However, many authors use the name diffusion coefficient in both cases. Moreover, after the homogenized parameters for the diffusion calculation have been obtained there is still a choice for the user according to his experience to decide which coefficient to introduce into the diffusion equation, since there is no straightforward prescription concerning the choice. When performing the transport calculation of an assembly or cell it is usually assumed that we deal with an infinite lattice of identical symmetric motives (assemblies or cells). This regular infinitely large periodic lattice is equivalent to the specular reflecting (mirror effect) of neutrons on the motive boundary. Of course, this is an idealized model since reactor lattices are finite and in the case of modern reactors not regular. But, most of the theory behind the calculation of homogenized parameters stipulates such idealized lattices, since at this beginning step of reactor calculation one has no information about what is going on in the neighboring assemblies or cells. Other types of boundary conditions can be also considered, for example isotropic reflection (the white boundary condition).

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However, the finite size of a reactor is reality and it should influence the lattice calculation. If the finite system is such that there exists, in its interior, a large region where the spatial variation of flux is the same for all neutron energies there emerges the concept of the fundamental mode with the buckling B as a measure of the finite size, and the effects induced by finiteness become synonymous with buckling-dependent effects. This is the basic idea of theory. But one should be aware that there exists also additional transients in the vicinity of the boundary of the system. These transient modes seem to be very difficult to take into account in homogenization theory. That is why, in the following, we shall deal only with the homogenized parameters corresponding to the fundamental mode. As a further step, one extends the lattice with an analytical continuation to cover the whole space. This extension is equivalent to the well-known "method of image pile" which removes the effect of boundary transients and the solution obtained is the deep-interior spatially asymptotic one. This is equivalent to considering the finite medium solution as a harmonic of an infinite lattice and is analogous to considering spatial harmonics of finite media. In the early days of reactor theory when reactor lattices were regular and modeled as homogeneous, by the simple flux weighting homogenization of cross sections, bucklingdependent effects were basically identified as leakage effects. A diffusion equation was set up for homogeneous media where the leakage coefficient figured in place of the diffusion coefficient and also as a measure of the leakage when multiplied by the buckling. The leakage coefficients were calculated for an infinite homogeneous lattice, which is of course an idealization, on the bases of existence of a fundamental mode. This represents the concept of homogeneous theory. The buckling does not only include leakage, but also introduces attendant change in the neutron spectrum with the result that the average reaction rates are also modified. These changes become also a component of buckling-dependent effects. The desire for improved modeling and the need to extend diffusion theory to heterogeneous media gave rise to the problem of defining the average leakage coefficient for a heterogeneous periodic lattice. In these heterogeneous lattices the streaming of neutrons, which is the straight propagation of neutrons, is different along different directions of the lattice, as it is the case with leakage. This why the use of heterogeneous theory becomes inevitable. First, we shall be concerned with the derivation of homogeneous theory. Limiting to the zeroth and then to the first order of the expansion of scattering law we will obtain the homogeneous and models which are even today most often used for practical leakage coefficient evaluation in heterogeneous transport calculations. In fact, the leakage is calculated for a homogenized heterogeneous motive and one obtains a homogeneous leakage cross section. It is obvious that this cross section is not consistent with other cross sections, which vary throughout the heterogeneous motive. Then we present heterogeneous theory which has been used because of its complexity only for some very simple geometries. Further, we discuss several definitions and approaches of homogenized directional leakage coefficients based on the first-order expansion of the solution of the zeroth order of this theory, which is also called the small buckling approximation, and we compare them with an asymptotic method. At the price of a few approximations introduced in the first order of the heterogeneous theory we derive two simplified consistent heterogeneous models which enable one to compute space-dependent directional leakage coefficients. If these coefficients are multiplied by corresponding directional bucklings one obtains directional space-dependent leakage cross sections which have the same energy and space dependence as all other cross sections.

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It seems that after performing the transport calculation of a heterogeneous motive using either homogeneous or heterogeneous theory one has everything that is needed to evaluate homogenized parameters which are to be used in the whole-core calculation. But it has been proved that simple flux weighting of cross sections does not satisfy the equivalence between the reference calculation (the heterogeneous multigroup transport calculation of an assembly or cell) and the macro-calculation (the diffusion or transport calculation accomplished for an assembly or cell composed of macro-regions, with cross sections homogenized at the scale of each macro-region and collapsed in macro-groups). Hence, the problem of equivalence consists of defining in a homogenized macro-region and in a collapsed macro-group a cross section for a reaction in such a way that using an approximate computational procedure one obtains the same reaction rate as in the reference calculation. This is why we will also briefly present a method which solves this problem and uses directly and consistently results of the simplified heterogeneous models. This superhomogenization method is based on conservation of reaction and leakage rates in each macro-region of a considered assembly. The idea of fundamental mode computations followed by a homogenization by an equivalence procedure is the procedure generally used in France, while for example in USA and Germany the method of discontinuity factors is used instead. Some connections exist between these two approaches, although they are not immediately obvious and should merit further investigations. In the case of modern reactors which have heterogeneous and even irregular lattices the concept of a fundamental mode is difficult to justify, since in the corresponding assemblies or cells this mode is perturbed by transients. However, in conjunction with a homogenization by an equivalence procedure, the buckling can be considered as a criticality parameter. Some numerical comparisons between the homogeneous and the two simplified heterogeneous models are given as well. Moreover, comparisons of these models with an experiment and a Monte Carlo prediction is also illustrated. HOMOGENEOUS

THEORY

The simplest way to calculate a finite lattice of heterogeneous assemblies representing a reactor consists in replacing this lattice by a homogeneous medium and extending it to infinity. Moreover, the idea that in a reactor there exists a fundamental mode which is the same for all neutron energies allows the spatial dependence of the over-all flux to be approximated by an exponential term unique for all energies. Therefore the basic hypothesis of the homogeneous theory assumes factorization of the physical flux of this homogeneous medium as a product of the exponential term independent of neutron energy and a function depending on solid angle and energy. Considering a multigroup decomposition of the neutron energy spectrum the proposed factorization can be written as1:

where is the buckling familiar from the elementary reactor theory2, 3. This factorization is possible whatever the orientation of vector is, but on the other hand, it is possible only for a certain value being its modulus. It is known that is the lowest eigenvalue of the Helmholtz equation with the boundary condition of zero flux at the extrapolated boundary of the system. The solution of this equation in plane geometry is a cosine, in cylindrical geometry a Bessel function, etc. All these solutions, as well as the solution for a parallelepiped, can be expanded in a sum of exponential modes.

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For the sake of simplicity, the energy group indices are dropped and all values, including fluxes and currents, are considered to be multigroup vectors or matrices. Let us write the time-independent integrodifferential form of the transport equation with an isotropic fission source:

with the integral physical flux being

and

where is the total, the scattering and the fission cross section, is the number of neutrons emitted by fission, is the contribution of the fission spectrum and is the infinite multiplication factor. The angular dependence of the scattering cross section may be expanded as the sum of a series of Legendre polynomials, i.e.,

where given by

are the Legendre polynomials and the

are the expansion coefficients

Since we are dealing with an infinite homogeneous medium the space variable r will be dropped. Inserting the factorization of the physical flux (Eq.(1)) and the expansion of the scattering cross section (Eq.(5)) into the integrodifferential transport equation (Eq.(2)) one obtains:

where is the effective multiplication factor emerging from introduction of buckling. To proceed with the derivation one would need to use the addition theorem of spherical harmonics functions which would, in this case, lead to rather complicated expressions. Instead of using directly Eq.(7) it will be first considerably simplified. Since we are only searching for the energy shape in a homogeneous medium and as it has already been noted that the final solution does not depend on orientation of the vector , we may choose the direction of to suit our convenience. Accordingly, we make parallel to the axis in the three-dimensional system of coordinates. Consequently, this reasoning permits use of the simpler Eq.(7) and implies that we are looking for the solution that is independent of the azimuthal angle of the solid angle Thus, Eq.(7) becomes

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where is the cosine of the angle between the initial and final direction of the scattered neutron and is the cosine of the angle between and the z axis. Expanding in Eq.(8) the angular fluxes and in a series of Legendre polynomials and applying the addition theorem, as well as multiplying this new equation by and integrating it over the solid angle one would obtain the corresponding theory. But, without using any approximation for the angular fluxes one obtains the homogeneous theory. Hence, first the addition theorem of spherical harmonics is applied on in Eq.(8) and one obtains

Further Eq.(9) is divided by and then multiplied by the solid angle to obtain a set of equations for

and integrated over

where

It is evident that the flux appearing in Eq.(l0) and defined by Eq.(4) corresponds to the flux defined by Eq.(11) for with The set of coupled Eqs.(l0) can be solved numerically for provided the sum on the right hand side is truncated. Neglecting all the components of the scattering cross section for and solving the resulting N + 1 equations for the first N + 1 components one obtains the homogeneous approximation. This approximation has a property that its solution is final for a given order of N, while this not true for the corresponding approximation. To illustrate this point it may be noted that if the is truncated for and the first N + 1 components have been computed, computation of components for does not change the value of already computed for Hence, neglecting the for is justifiable since the first N + 1 components are computed exactly. On the contrary, this is not the case with the corresponding approximation where if the vanish for evaluating values of the higher components by including more equations alters the components found by a lower order computations Usually, for practical applications the expansion for the scattering cross section is limited to the zeroth or the first order, leading to the homogeneous or approximation respectively. To illustrate some properties of the vectors involved when deriving these two approximations, a slightly different derivation procedure will be applied, i.e., the azimuthal dependence will not be neglected. Homogeneous

Approximation

If the scattering is isotropic the expansion for the scattering cross section will have only the zeroth term In this case, since Eq.(5) becomes

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The energy group index has been introduced to avoid misleading conclusions. Dividing Eq.(7) by and integrating it over the solid angle one obtains, after some algebra, the equation for the scalar flux

which corresponds to Eq.(l0) for and represents the homogeneous approximation. Here we have defined When the effective multiplication factor is equal to 1, the value which is the smallest one on the real positive axis satisfying Eq. (13) is the critical buckling of this medium. Then, the scalar flux is the critical spectrum corresponding to the fundamental mode. Let us write the general definition for the vector current

which, in this case, corresponds to Eq.( 11) for where (or ). Proceeding in the similar way as when deriving Eq.(13) one obtains the expression for the vector current

corresponding to Eq.(l0) for and From Eq.(15) it is obvious that the vector current and the buckling vector are colinear. Since in an infinite homogeneous medium the scalar flux and the vector current depend only on energy, they can be related via a coefficient, also depending on energy, in the following way:

and we can write

where is the leakage coefficient in group Eq.(16) resembles the Fick’s law used in the diffusion theory. Thus we shall see that the leakage coefficient appearing in Eq.(16) differs from the diffusion coefficient obtained in the diffusion theory. Inserting Eq.(13) and Eq.(15) into Eq.(17) one obtains the expression for the leakage coefficient

When the critical buckling is small compared to the total cross section i.e., when dimensions of a reactor are big with respect to the total mean free path in the group using the expansion of for the small argument Eq.( 18) is transformed into

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If the expansion for the leakage coefficient, given by Eq.(19), is limited to the first order one obtains the diffusion coefficient of diffusion theory. This is why we did not consider strictly Eq.(16) as Fick’s law. The buckling of a supercritical system is real and one can use directly Eq.(18). On the contrary, the buckling of a subcritical system is a purely imaginary value and one has to work with the modulus . Using the transformation for when the argument is an imaginary value Eq.(18) becomes

We shall discuss the physical situation leading to an imaginary buckling later on when analyzing the heterogeneous approximation. Let us write the expression for the vector current corresponding to the angular physical flux given by Eq.(l)

The justification for existence of the leakage coefficient is to evaluate the leakage rate per unit volume which can be given as

The term plays a role of a leakage cross section. The leakage coefficient in the energy group depends only on the total cross section in this same group and not on cross sections belonging to other groups. Hence, in the homogeneous approximation the leakages in two different groups are not correlated.

Homogeneous

Approximation

If the scattering is assumed to be linearly anisotropic the expansion for the scattering cross section will have two terms Accordingly, Eq.(5) becomes

since Proceeding in the same way as when deriving Eqs.(13) and (15), one obtains two equations, one for the scalar flux, corresponding to Eq.(l0) with and and the other for the vector current corresponding to Eq.(10) with and

These two equations represent the homogeneous approximation. It can easily be shown that if by introducing in Eqs.(24) and (25) the expansion for

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limited to the first order, these equations reduce to the corresponding equations. Inserting Eqs.(24) and (25) into Eq.(17) one obtains the equation for the leakage coefficient

with

where is the leakage coefficient obtained in the case of isotropic scattering. Moreover, one can see from Eq.(26) that the scattering anisotropic term induces coupling of leakages belonging to different energy groups.

Calculation of Critical Conditions of an Assembly or Cell Homogeneous and approximations are very often used for practical and routine computations of leakages. Almost all transport computer codes give a very fine description of the multigroup flux map and hence the reaction rates of a heterogeneous assembly or cell, but the leakages are calculated for a motive (a homogenized assembly or cell). The habitual procedure is as follows. Once the multigroup flux map of a reflected motive is obtained using a numerical method to solve the transport equation (Eq.(2)), one can define a homogeneous medium, equivalent to the considered motive, by weighting cross sections for all reactions with the flux. The homogeneous medium is not critical and in order to make it critical it is necessary to introduce the leakage cross section calculated as shown previously. This critical buckling is obviously not the fundamental buckling of the actual heterogeneous motive because the homogenized medium has been obtained by weighting cross sections with the flux map where leakages have not been taken into account. This is why another calculation of the heterogeneous motive has to be accomplished with these leakages being represented by an additional volumetric absorption term Solving with this new cross sections the transport equation one obtains a new flux map which allows to determine a new equivalent homogeneous medium that is still not critical. The evaluation of a new and the corresponding is then repeated. Further, the calculation of the heterogeneous motive including the new is carried out and this continues until the convergence is attained. Hence, the described procedure leads to the critical flux map in a heterogeneous motive, the critical buckling and the leakage coefficient of the equivalent homogenized motive. The quantity represents the volumetric leakage rate of the motive in the group with the representing the leakage cross section (being independent of r). The drawback of this procedure lies in the fact that the leakages are calculated for a homogenized motive (assembly or cell), and not for the actual motive, i.e., the leakage cross section should depend on r. This imperfect representation of leakages of a heterogeneous motive tends to underestimate actual leakages. In particular, the procedure presented here does not take into account the anisotropy of leakage coefficients, in other words the fact that, due to heterogeneity of the assembly or cell, the axial coefficient differs from the transverse coefficient.

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It has been shown that to represent the global influence of a reactor core on an assembly or cell calculation, one most frequently uses a procedure that approximately takes into account in a heterogeneous situation the equivalent absorption due to the macroscopic structure of the physical flux. This absorption is introduced as the product of the leakage coefficient and the buckling, obtained for an equivalent homogeneous medium using homogeneous theory. Hence the influence of lattice heterogeneity on neutron leakage seems to be somewhat neglected. However, the errors introduced by this procedure are small under normal operating conditions in pressurized water reactors, PWRs (which are not so heterogeneous), and even in Canada deuterium uranium (CANDU) reactors. But, in case of a loss-of-coolant accident, when these reactors include voided or almost voided regions the streaming effect becomes important. Since the streaming effect is a result of the straight propagation of neutrons the errors introduced appear particularly visible. In these cases, the homogeneous leakage model significantly underestimates the actual leakages. This effect also appears even under normal operating conditions in the upper parts of boiling water reactors (BWRs), in fast reactors (LMFRs) due to the presence of sodium, which has a long mean free path, as well as in gas-cooled reactors (gas-graphite, HTGRs), since the gas-coolant is very transparent for neutrons. The streaming effect always produces a decrease of reactivity, except in over-moderated media where coolant is a moderator. However, the streaming effect is just one of the components of the global void effect. In particular, the spectrum variation effect, may according to the case lead to a decrease or an increase of reactivity, with an increase being possible in reactors containing plutonium. The influence of lattice heterogeneity on neutron leakage appears in another way as one of the components of the temperature effect of a reactor with a liquid moderator or coolant. Indeed, an increase of temperature of the liquid media induces a decrease of their density, which provokes an increase of neutron leakage and a loss of reactivity. As the global void effect is an algebraic sum of several effects, some being positive and others negative, one can understand why the emphasis on the streaming effect calculations is important. In addition, the power distribution is influenced by the effect of lattice heterogeneity on the neutron leakage. Considering current computing capabilities many theories of neutron streaming through heterogeneous media as well as corresponding definitions of the homogenized leakage coefficient were proposed in the past. After this pioneering period it seemed that reactor physicists entered into a period of doubt and hesitation, at least of interrogations, about foundations of the theory which appeared to be not as simple as one could have believed. Not infrequently, the leakage coefficient definitions that seem to be totally different have turned out to be equivalent. Some definitions emerged from diffusion and the others from transport theory. There existed some ambiguity in the definition of the leakage coefficient leading to troubling questions. On the contrary, it was shown that the migration area is defined without any ambiguity, but its relation with the group leakage coefficient was not absolutely clear. It seems, after lots of time and effort had been invested, that the answer depends on the weighting functions one uses for averaging reaction and leakage rates. The ambiguity of the leakage coefficient definitions result from the choice of weighting with the only condition being that the balance of neutrons is conserved. This point will be discussed later on. The influence of lattice heterogeneity on neutron leakage has been studied and taken into account in the calculations of old type gas-graphite reactor lattices. One of the first important

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studies about the theory of the leakage coefficient in lattices, although not explicitely based on heterogeneous theory, was published by Behrens4. This very influential paper discusses a particularly severe type of lattice composed of a homogeneous material medium pierced with cavities of any shape. In spite of several drawbacks and limitations, shown by Schaefer and Parkyn5 comparing with the experimental results and with Monte Carlo calculations, and also by Laletin6 using analytic methods, this work has been the starting point of several theories of the leakage coefficient. The theory of Behrens was extended and generalized by Benoist 7-10 . Although comprising ideas from earlier work, the approach of Benoist to streaming computations was founded on heterogeneous theory. The computational procedures he developed were convenient, fast and accurate for applications. Let us consider a reflected heterogeneous assembly; or in other words, an infinite lattice of identical and symmetrical assemblies. The basic hypothesis of the heterogeneous theory assumes factorization of the physical flux of a reflected assembly (or cell) in the form of a product of two functions. One is the periodic fine structure flux with the period equal to that of the lattice, describing fast variations of the flux inside the assembly and the other is the macroscopic function describing global variations of the lattice flux. The local flux variations are determined by variations of cross sections at the scale of assembly, while the macroscopic flux is defined by material balance at the larger scale. These ideas are transposed from the notion of the fundamental flux, solidly established in an infinite homogeneous medium, and generalized in a heterogeneous situation. Of course, this idealized situation does not correspond to an actual reactor, where the nonrepetitivity of assemblies introduces transients in the neighborhood of the assembly boundaries. Anyway, the proposed factorization is a practical tool used for homogenization techniques. Let us remind that in fact this factorization corresponds to Bloch’s theorem in solid state physics, regarding the effect of the periodic structure of a lattice on the free-particle wave function. Hence, one can write

For the sake of simplicity the group indices will be dropped again. The first use of the above factorization is attributed to Brissenden by Leslie11, although no details are available regarding how Brissenden came to this outcome, apart from the fact that he argued in terms of the integral transport equation. At the same time this factorization was used by Benoist9 in the integral and by Leslie12 in the integrodifferential transport equation, while a detailed mathematical basis was worked out by Deniz13. Because of convenience the integrodifferential transport equation will be written using operators and corresponding to the left and the right side of Eq.(2) respectively. Thus, we can write

Inserting Eq.(28) into Eq.(2) one obtains

with the only difference that the form12, 13

becomes

. The solution of Eq.(30) can be sought under

Substituting the solution given by Eq.(31) into Eq.(30) and separating the real from the imaginary part one obtains a system of two real equations

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We can also write

As the system of Eqs.(32) and (33) is linear, the solution of the second equation can be written as

Inserting Eqs.(34) and (35) into Eq.(33), one obtains three equations (for following type

) of the

where is the angular flux corresponding to directions The existence of solution of Eq.(36) is established by Fredholm’s theorem. Hence Eq.( 31) can be written as

where the flux vector The physical angular flux having a physical meaning,

is introduced via its components

and

is now, considering the real part which is the only one

The first term of Eq.(38) is simply a product of the macroscopic flux and the microscopic flux which depends on the buckling vector . The second term is proportional to the gradient of the macroscopic flux, and the flux vector reflects the variation of the microscopic flux induced by the presence of the gradient of the macroscopic flux. Thus, the fluxes and are the solutions of the system of equations

To obtain a general three dimensional form of the heterogeneous theory one would be forced to use spherical harmonics expansions for the scattering cross section appearing on the right side of Eqs.(39) and (40). This would lead to rather ugly equations and we do not see the point writing them here. The angular fluxes and are not expanded in a series because this would lead to the corresponding approximation and prevent a

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straightforward transformation of the integrodifferential equations into their integral form which can be solved by a very simple numerical method. In heterogeneous theory when all components of the critical buckling vector are real values the end point of the vector moves on a closed surface defined by the following critical relation:

where is the buckling-dependent migration area in the direction In the small buckling approximation, that will be discussed latter, the migration area is independent of and the closed surface reduces to an ellipsoid. In homogeneous theory the migration area depends only on the modulus of the vector but not on its orientation and the closed surface represents a sphere. The only simple example of the structure taken by the set of Eqs.(39) and (40) occurs for the almost lowest order, the equations. Moreover, these equations can have practical application and will allow us to perform an analysis of symmetry properties of the functions involved. Thus, assuming that the scattering is linearly anisotropic (Eq.(23)), we can write Eqs.(39) and (40) in a more explicit way

where

The scalar fluxes and currents are defined as follows

As it is the case with the homogeneous approximation, the heterogeneous approximation has also the property of finality. If, for example, and (the zeroth moments of angular fluxes) have been calculated for the expansion of being truncated for computation of and (the first moments of angular fluxes) does not change the values of and calculated for the zeroth order of the scattering cross section expansion.. This is not the case with the corresponding approximation. When analyzing the practical application of both the homogeneous and the heterogeneous approximation we are confronted with the range of buckling values for which the proposed factorizations are valid. In the reactor core under normal operating conditions these factorizations are possible. In a reflector the assumption of a separable flux is difficult to

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justify for any buckling value, since there are no fission sources and thus no fundamental mode; the slowing-down spectrum varies with the space variable. Another case where there exists no exponential fundamental mode is when a subcritical assembly contains infinite voided channels or ducts. More precisely, in a subcritical infinite lattice, with an imaginary buckling in one direction, there exists generally a fundamental mode. But if this lattice contains a one-dimensional channel or a two-dimensional duct in the direction where the buckling component is imaginary no fundamental mode can exist. This condition appearing in the space-decay problems is similar to the well known Corngold14,15 condition in the time-decay problems: the space-decay rate cannot be faster than the smallest total cross section (in other terms, as Corngold said: "The graduate school class cannot proceed faster than the slowest pupil"). If this cross section is zero, no exponential fundamental space-decay mode can exist. This comes from the fact that there is a competition between two interfering neutron populations each having different decay rates: in the solid material the decay rate tends to be exponential while in one-dimensional voided channels or two-dimensional voided ducts the decay rate is geometrical because of the decrease of the solid angle. However, to simplify the evaluation of homogenized parameters the assumption of a separable flux is still adopted even in the regions that are quite subcritical, under the condition that there are no infinite voids. In such cases the value of the buckling is imaginary. Thus, it appears that for practical calculations the separability of the flux depends, not on the sign of the buckling, but only on its magnitude. For better understanding of the heterogeneous system of equations it would be useful to perform an analysis of symmetry properties of the fluxes and currents involved. The angular flux is very anisotropic due to a very particular source This source sends positive neutrons in the direction forming an acute angle with the direction and negative neutrons in the direction forming an obtuse angle with the direction For apprehension of the meaning of a symmetry, which will be often used in the following, we define it as a symmetry with respect to the axis Changing into means that the effect of this symmetry is to change the sign of the associated variable and of the vector components along that variable without changing the sign of the variable. For instance, the symmetry changes and into and respectively. Thus, and are antisymmetric with respect to Since the assembly and the boundary conditions are symmetric, it follows that the angular flux solution of Eq.(30) must be symmetric also because a symmetry does not change the scalar product Then, according to the decomposition in Eq.(37), it follows that and are symmetric. This means that is symmetric for all values of , and this regardless of the symmetry properties of the The latter cannot be inferred from this argument since we only know that the scalar product must be symmetric. However, since the definition of the component does not break the symmetry (due to the source which is antisymmetric with respect to ), it follows that is symmetric, and therefore that is symmetric with respect to symmetries and antisymmetric with respect to the symmetry. Consequently, the angular flux is unchanged with respect to the change of into whatever is On the contrary the angular flux changes sign when is changed into but unchanged when is changed into for

Since the real flux component

of the proposed solution given in Eq.(37) is

symmetric with respect to the subscript has been attributed, while the subscript has been attributed to the imaginary flux component due to its antisymmetric behavior with

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respect to Corresponding currents, regardless their symmetry properties, have a subscript according to the flux. Symmetry properties for integral fluxes and current components follow easily the precedent discussion. The integral flux is symmetric with respect to while the current in the direction corresponding to the flux is an antisymmetric function with respect to The integral flux is antisymmetric with respect to but symmetric with respect to The current in the direction corresponding to the flux in the direction is an antisymmetric function with respect to for but a symmetric function with respect to for Another consequence of these symmetry and antisymmetry characters is that according to Eqs.(42) and (43), the functions and are symmetric with respect to the median plane perpendicular to any symmetry axis of the assembly; on the contrary the functions and (for ) are antisymmetric with respect to this plane. Moreover, and are symmetric with respect to the two other orthogonal planes, while (for ) is antisymmetric with respect to both median planes orthogonal to and and symmetric with respect to the third one. Let us now consider, as a particular case, a homogeneous reflected assembly, which is equivalent to an infinite homogeneous medium. It can easily be shown that the angular flux solution of Eq.(42), is uniform and isotropic. Thus, the integral flux is uniform, while the current component is zero. It can also be shown that the angular flux solution of Eq.(43), is proportional to Therefore, the integral flux is zero. The current component for is also zero since is invariant with respect to On the contrary, the current for is not zero due to the integral of over the solid angle. If in a heterogeneous, symmetrical and reflected assembly, cross sections are independent of the direction (equivalent to homogeneity in the direction), we have a twodimensional problem, which is the one most frequently used in reactor calculations. In this case the foregoing discussion can be summarized by the following symmetry relations

The symmetry properties have been analyzed partially or from different points of view by Benoist7, 9, 10, Leslie12, Deniz16, 17 and Williams18.

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Small Buckling Approximation If

i.e., when becomes great compared to the size of an assembly, then and Eq.(42) becomes

with being the limit of This is exactly the transport equation for an reflected assembly without buckling. Since is assumed to be small we can replace in Eq.(43) by and we have

where becomes Eq.(37) changes into

Hence, at the limit when

the solution given by

It is evident that Eq.(50) contains the first two terms of an expansion (in powers of ) for the small buckling value. Such an expansion was obtained by Benoist7, 9 and Leslie12. Several leakage coefficient definitions are based or related to the solution expansion for the small buckling. This is why it would be useful to review them and to sketch their derivation. Although the work of Behrens4 did not emerge directly from small buckling approximation it can be interpreted via this approximation. The theory assumed that the diffusion area could be calculated from the sum of the neutron mean squares of elementary paths, that is by using in a lattice a property valid only in a homogeneous medium (with isotropic scattering). Instead of that, the diffusion area should be computed from the mean square distance that a neutron crosses from its source to its absorption. This mean square is composed of the sum of the mean squares of the elementary paths, plus the sum of the mean scalar products of two paths separated by collisions. In other terms, the Behrens theory in spite of its great merit at the time of issue is incorrect since it does not take into account the angular correlation terms. The meaning of the angular correlation terms can be explained in a very simple way. Consider a lattice composed of voids rather spaced with respect to the mean free path of the material medium. A neutron having just crossed a cavity has a better chance to traverse a long path after a collision if it is scattered backwards than if it is scattered forwards. The Behrens theory, which does not consider this effect, gives an exact value for the axial leakage coefficient but overestimates the transverse leakage coefficient; the bigger the cavity is compared to the mean free path of the material medium the larger the overestimation is. An extensive work on analyzing and defining the leakage coefficient has been done by Benoist7-10. As a result of his detailed theoretical treatment two definitions emerged, known conventionally as the “uncorrected”7, 8 and the “corrected”9, 10 coefficients. Both definitions take leakage anisotropy into account, the second definition being a genuine correction with respect to the first, enabling to evaluate the leakage effect in the actual finite system. The “uncorrected” leakage coefficient is proportional to the ratio of integrals over the assembly of the microscopic current and flux. This corresponds to weighting of cross sections by the microscopic flux (the fine structure flux). The flux having the periodicity of the lattice

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implies that homogenized cross sections and leakage coefficients are independent of how the assembly or cell is defined but the leakage coefficients do not represent leakage effects from the true flux with its associated macroscopic flux curvature. The “corrected” leakage coefficient corresponds to an equivalence between true reaction and leakage rates of a cell or assembly and those of the homogenized motive. It gives the same importance to all neutrons of the motive. But, as pointed out by Gelbard19, the drawback of this definition lies in the fact that because of the weighting of cross sections by the physical flux which has not the periodicity of the lattice, cross sections and leakage coefficients defined in this way depend on how the motive is defined. In other terms, if the cell is shifted by half a pitch, the values of the homogenized cross sections and leakage coefficients will change although physically nothing has changed, not least that the effective multiplication factor is unchanged. Hence, the terms of the neutron balance equation are all double-valued and they compensate each other providing that the neutron balance remains unchanged. This drawback makes the “corrected” leakage coefficient not too convenient in practice. But, Leslie11 mentioned that the equivalence principle of Selengut20 leads to a different normalization than the one used by Benoist. Deniz21, 22 wisely pointed out that there is no reason to give the same importance to all neutrons in a cell or assembly. Thus, one can weight the numerator of homogenized parameters by one complex function and the denominator by another. Although there is a total arbitrariness in the choice of the weighting function, with the only condition being that the balance of neutrons is conserved, some choices seem to be more logical than others. Deniz has shown that different leakage coefficient definitions although not derived explicitly from small buckling approximation can be interpreted via this approximation. However, the “uncorrected” leakage coefficient has been widely used in practice. In the interest of efficiency, Benoist has defined directional collision probabilities that he has used to developed a simple computational procedure. There is another feature indicated by Benoist9 which is that there exists an essential inconvenience inherent in the definition of the finite buckling leakage coefficient. In reality there is never a unique separable fundamental mode in a reactor since reactor lattices are not uniform and regular, and there are also additional transients near the boundary. Even if there is a fundamental mode, the flux is not a unique mode but a linear combination of separable modes. In the small buckling approximation, the directional leakage coefficients, according to their definition, are independent of the buckling. Therefore, the directional leakage coefficients provide a convenient and probably sufficient tool for practical use. Leslie12 as well as Benoist has shown that the directional (anisotropic) leakage coefficient can be interpreted as the product of the square directional diffusion area and the average absorption cross section. The square directional diffusion area is defined as one half the mean square of the directional component of the vector linking the point of birth of a neutron to the point where it is absorbed in the infinite lattice. This interpretation becomes in a homogeneous medium the usual diffusion area defined as one sixth of the mean square distance between the birth and the absorption of a neutron. In a channeled reactor with gas-coolant the mean square distance of travel would be larger for the axial direction than for the transversal direction. Leslie has shown that the averaged axial leakage coefficient is obtained by weighting the local diffusion coefficient with the flux being the zeroth term of the expansion, while the averaged radial leakage coefficient is derived from the flux being the first term of the expansion for the flux solution. There is thus a migration area anisotropy, even in the absence of voids. The analysis he performed for a cell consisting of a circular hole surrounded by moderator led to a formula already deduced by Carter23, that was in good agreement with available evidence.

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The leakage coefficient of Bonalumi24 is somewhat related to the “corrected” Benoist’s definition. In arguing for the use of the cell-edge flux normalization, instead of the cellaverage flux normalization used by Benoist, he asserts that the leakage coefficient thus obtained satisfies Selengut’s equivalence principle20, which states that if corresponding cells from a lattice and its equivalent homogeneous medium are interchanged the overall condition of each medium is unchanged. As the “corrected” Benoist’s coefficient, this coefficient has the double-valued character when the definition of the cell is changed since the weighting physical flux does not have the periodicity of the lattice. Based on the Benoist’s analysis, Deniz16, 17 aimed his attention at the eigencoefficients. He tried to determine the change in the effective multiplication factor induced by the existence of the macroscopic dependence in an infinite lattice. Realizing that the expression obtained was implicit he made a perturbation between the buckled and the unbuckled lattice using the weighting with the adjoint function of the zero-buckling problem. Thus, the formal solution of the eigenproblem was rendered explicit; the decay-constant eigenproblem as well as the effective multiplication factor eigenproblem were treated and interpreted. The bucklingindependent directional migration area so obtained, which is an energy-integrated quantity (independent of the energy group) measures the reactivity change due to all the bucklingdependent effects such as neutron leakage but also modification of the fine structure and of the spectrum caused by the macroscopic curvature. Upon a dimensional transformation of the directional migration area Deniz derived, Gelbard19 generalized it to a multigroup form and deduced the leakage coefficient which became known as the Deniz-Gelbard coefficient. Using the Deniz approach buckling-independent homogenized group parameters were obtained, i.e., weighting is carried out with the zeroth term of the flux expansion. This is done by calculating the net importance of the energy dependent cell leakage, leading to homogenized leakage coefficients defined in such way that for each energy group the net importance is the same in the homogenized medium as in the lattice. A similar leakage coefficient had been suggested earlier by Williams18 who started from the same definition of the directional leakage coefficient as Benoist, although later in his work making different approximations leading to different computational procedures. The leakage coefficient of Larsen25, 26 results from an entirely different approach which does not use, at least explicitly, the concept of buckling. Larsen develops an asymptotic theory based on a multiscale method. The method requires that the physical medium is nearly critical and that the mean free path is small compared to the diameter of the medium of interest. Defining a small dimensionless parameter as the ratio of these two lengths, the solution of the transport equation which is asymptotic with respect to the small parameter is formally constructed. The asymptotic solution is obtained using two length scales, one being microscopic, describing spatial variations on the order of the mean free path and the other, being macroscopic, describing spatial variations on the order of the entire medium The leading term of the asymptotic solution is a product of two functions: one being spatially periodic and determined by a cell calculation, and the other determined by a diffusion equation whose coefficients depend on the results of the cell calculation. This asymptotic method, like the Deniz-Gelbard method, does not lead directly to leakage coefficients. An additional condition of some sort should be supplied before leakage coefficients can be defined. The leakage coefficient thus obtained is rather similar to those of Deniz-Gelbard and Williams, and differs by a weighting factor, while homogenized cross sections are buckling-independent. A contribution to the asymptotic theory development was also given by Chiang and Dorning27. Deniz21, 22 has proposed another leakage coefficient different from the Deniz-Gelbard coefficient. The new one results from a perturbational equivalence, earlier proposed by

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Marchuk28, between the buckled lattice and the equivalent homogeneous medium having the same eigenvalue (the effective multiplication factor) and the same buckling vector. The resultant definition is thus an importance weighted definition where the angle integrated buckling dependent adjoint flux of the homogeneous medium appears. The new definition of Deniz reduces in an exact manner to the “uncorrected” Benoist’s definition in the case of a one-energy group problem. This equivalence remains if the energy is introduced as a continuous variable, but if the energy is treated by a multigroup scheme the definition of Deniz reduces to the “uncorrected” Benoist’s definition in an approximate manner for well thermalized systems. Köhler29 has proposed leakage coefficients that have been obtained using the leastsquare fit to the reaction rates throughout the lattice. In fact, this is a compromise between obtaining reaction rates correctly in one cell but not in another. This formulation leads to the unambiguous homogenized leakage coefficients, independent of the definition of a cell, and is closely related to the “uncorrected” Benoist’s definition. Each of the foregoing leakage coefficient definitions has been derived in its own particular way. Using a one-dimensional symmetric slab lattice with monoenergetic neutrons Hughes30 has shown that they can be derived from a single approach which illustrates the differences and the advantages of the various definitions. Another comparison has been proposed by Duracz31. Interpreting the small parameter from the asymptotic theory, Larsen and Hughes32 came to a conclusion that one of the possible representations could be the absolute value of the buckling vector. This interpretation enabled them to establish an equivalence between the asymptotic method and small buckling transport theory treatment. The present authors have proposed a model33, also based on the small buckling approximation, for the calculation of homogenized directional leakage coefficients. It uses directional first-flight collision probabilities and is able to evaluate, via the directional migration area, the reactivity difference between isotropic and anisotropic treatment of leakages in a reactor lattice. As it has been noted the anisotropic leakage coefficients defined by Benoist are those often used for practical calculations. But when these formulas are applied to PWR assemblies containing cavities that have a form of plane slits, or voided ducts situated between fuel rods, enabling a neutron to escape to infinity (in a two-dimensional space), they diverge for the directions parallel to the slit or duct. This is due to the analytic divergence of an underlying integral and was pointed out initially by Behrens4. If instead of void there is a medium having a very long mean free path, as in sodium-cooled reactors, there is no analytic divergence, but the leakage coefficients are very overestimated. However, these facts have been somewhat forgotten because most practical calculations assumed cylindrical cells and therefore removed this difficulty artificially. The method of Köhler and Ligou34, which combines analytic and numerical techniques, was the first to solve the problem and to point out its significance. They treated a fast reactor lattice at the finite buckling which led to a buckling dependent leakage coefficient. Limiting the expansion to the lowest order in powers of buckling, which is a logarithmic term, they arrived at a slow divergence for an infinite lattice providing good values for the leakage coefficients. Aside from numerical approximations this method may be regarded as the finite-buckling version of the method of Benoist. Benoist35, 36 solved the problem of planar voids by an analytical method aimed at recovering the use of his formulas, though in a modified form. This was done not by limiting to the zeroth order in powers of buckling which is impossible in this case, but at the lowest order which is logarithmic, neglecting terms of higher order. Due to the relatively small

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heterogeneity of the fast reactor, this approximation appears to be sufficient in practice, with a correction taking into account higher order terms calculated for the homogenized lattice. Previously, Benoist37, 38 had performed an in-depth analysis of the angular correlation terms, which introduced energy coupling, both for isotropic and linearly anisotropic scattering, and came to the conclusion that in many cases the influence of these terms is negligible in a bidimensional lattice of reactors concerned here. This fact considerably simplifies calculations. It has also been shown that the Wigner-Seitz model, calculates badly directional leakage coefficients and that a bi-dimensional treatment is required. On the contrary, cylindrization of a cell can be considered as a good model for calculations of angular correlation terms since it does not violate fundamentally the physical reality considered. Regarding the matter of infinite planar voids we would like to underline also studies by Eisemann39, Gelbard and Lell40, Gelbard et al.41, Köhler42 and Duracz43. Gelbard and Hughes44 reexamined the earlier work of Gelbard and Lell40 which was based on first-order perturbation theory used to obtain relations between mean-square chord lengths and lattice eigenvalue for given buckling. In the improved model the mean-square chord length is redefined by inserting the missing terms in square buckling. Moreover, Gelbard40, 41, 45 has developed several Monte Carlo methods for leakage treatment, i.e., for the computation of an eigenvalue, as a function of buckling. In the foregoing discussions it was assumed that a considered cell or assembly was symmetric. Yet, the problem of nonsymmetric cells has been also studied revealing that complex buckling modes could occur, thus suggesting that homogenization of asymmetric cells is likely to be troublesome. Gelbard and Lell40 have shown on an asymmetric cell that, in the presence of buckling, the effective multiplication factor is not real. Moreover, in a slab reactor formed of cells being not symmetric the overall flux is not a cosine. Also Larsen and Williams46 have examined the case of an asymmetric cell by the asymptotic method. Using the buckling approach Hughes47 has noticed that in this case a complex eigenvalue is obtained for all real bucklings, but that a real eigenvalue is obtained for certain complex bucklings. He has concluded that in a lattice formed of asymmetric cells the physical flux will be factorized with a complex buckling. Further investigations have been carried out by Larsen and Hughes32 and Duracz48. However, it seems that a definitive issue has not yet been found. In the preceding comments on various definitions of anisotropic leakage coefficients we have focused on those arising from transport theory mainly in terms of the small buckling approximation. We did not enter into a detailed explanation of each or comparison between them since it has already been done extensively and carefully by Gelbard49 and by Deniz50, although certain assertions may be regarded as somewhat contradictory. Moreover, Benoist51 analyzed also different definitions of leakage coefficients.

Heterogeneous

and

Models

Taking advantage of increased capabilities of computers further progress has been made in improving knowledge of the influence of lattice heterogeneity on neutron leakage. Therefore, studies and approaches via the concept of the small buckling approximation have been abandoned and less approximate models, although based on previous experience, have been proposed. Using heterogeneous theory Lam-Hime52 has studied the case of a one-dimensional slab lattice in a one-speed problem. In his work he did not introduce the physical flux factorization (Eq.(28)) into the integrodifferential form of the transport equation but into its

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integral form. After discretization by first-flight collision probability method he obtained a rather complicated form of these probabilities, since they are buckling dependent, and was forced to compute various power series expansions depending on buckling. These expansions converged satisfactory for a slab lattice containing only solid media, but diverged for a lattice containing a void slab. In the case of an almost voided slab these expansions had a poor convergence. Another attempt, representing an extension of the work of Lam-Hime, has been carried out by Roy et al.53 avoiding previous difficulties since no expansions in terms of buckling have been used. Moreover, conditions of existence of a fundamental mode have been discussed. They treated also a one-dimensional slab lattice in one-energy group. Introducing in the integral transport equation spherical harmonics expansions of sources, anisotropic buckling-dependent collision probabilities have been obtained. These probabilities, which are moments of integral transport kernels weighted by spherical harmonics, have been calculated using cyclic tracking method54 with double angular quadratures. Calculations with isotropic and linearly anisotropic scattering have been performed and analyzed. The one-group numerical results show the variation of the critical buckling, and the parallel and the perpendicular leakage coefficient versus the orientation of the buckling vector. The presence of voided media does not lead to convergence problems. However, for complex geometries heterogeneous and models would require very time-consuming calculations and thus would be rather inappropriate for practical or routine calculations.

SIMPLIFIED HETEROGENEOUS

MODELS: TIBERE AND TIBERE-2

Benoist and Petrovic55 have tried to find an efficient approximation to heterogeneous theory, which would use numerical solution methods already existing in available neutron transport codes without increasing considerably computing time compared to ordinary calculations of multigroup flux maps. As a result of this research two simplified heterogeneous models named TIBERE55 - 57 and TIBERE-258, 59 come out. These new models take into account the streaming anisotropy and heterogeneity in a consistent manner, within a fine flux transport calculation. Both the TIBERE and the TIBERE-2 model are founded on a few common approximations, assuming that the lattice is regular and symmetric. The bases on which these approximations are developed are the isotropic and the anti-isotropic behavior of angular fluxes, as well as the symmetrical and the antisymmetrical shapes of integral fluxes and currents. In addition, two corrections are introduced; one that guarantees the exact solution at the homogeneous medium limit, and the other that guarantees the exact solution for a homogenized heterogeneous medium. Thus, one obtains an iterative scheme to simultaneously compute coupled multigroup scalar fluxes and directional currents. This scheme yields directional space-dependent leakage coefficients, and one can define space-dependent leakage cross sections having the same type of space-dependence as all other cross sections. Consequently, perfectly consistent reaction and leakage rates are obtained, and can be used in an equivalence procedure that evaluates homogenized cross sections and leakage coefficients required for a whole-core calculation. Since there is a straightforward link between these new leakage models and the homogenization by an equivalence procedure, it will be described later on. In fact, we may say that they are unseparable in the sense of obtaining homogenized parameters for the overall reactor calculation. The main difference between the TIBERE and the TIBERE-2 model is with regard to

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the boundary conditions at the limit of a heterogeneous, symmetric and reflected assembly or cell. The TIBERE model55 - 57 assumes that a neutron arriving at the boundary is reflected in a specular way (mirror effect), that is equivalent to an infinite lattice of identical assemblies or cells. Such an assembly or cell represents a reference situation for a homogenization by an equivalence calculation. This model requires that the classical and directional first-flight collision probability be calculated. Although considerable progress has been made with respect to the homogeneous leakage model, which is still the one most frequently used, and also with respect to previously described cell-homogenized directional leakage coefficient definitions, two difficulties appear with the TIBERE model. Both difficulties arise from the chosen boundary condition, that is the specular reflection of neutrons. In fact, this type of reflection extends neutron paths to infinity. Therefore, the TIBERE model calculations may become rather time consuming for production calculations and even unnecessary long in the case of a lattice where coolant-voiding appears in small and closed zones and not in large zones. At the same time, in the case when the periphery of an assembly or the whole assembly is coolant-voided, this model leads to infinite leakage coefficients. This well-known divergence, that we have already discussed earlier, is caused by the fact that because of the specularly reflecting boundary condition here, a neutron can escape to infinity without suffering a collision. Calculations of coolant-voiding in a reactor are important for safety analysis. For these reasons, another simplified heterogeneous leakage model, named TIBERE258, 59, has been proposed. This model assumes that a neutron arriving at the assembly boundary is reflected in a quasi-isotropic way. The adopted boundary condition is of course approximate, but no more so than the specular boundary condition in actual core situation. The resulting assembly or cell represents a new reference situation for a homogenization by an equivalence procedure. The TIBERE-2 model, apart from the classical and directional first-flight collision probabilities, requires the calculation of the classical and directional escape and transmission probabilities. But the classical and directional collision probabilities have to be calculated only for an open assembly. Consequently, calculations with this model circumvent the divergence described earlier and are much less time consuming than those with the TIBERE model since the neutron paths in the classical and directional first-flight collision probabilites are now limited to just one assembly or cell.

Basic Assumptions The approximations that enable a simple way of solving the heterogeneous system of equations as well as the corrections leading to exact solutions at certain limiting cases follow. Approximation of Isotropy of and Anti-isotropy of The proposed approximation55 - 57, that will be partially corrected later, concerns the real flux component appearing in the third term on the right side of the equation for the imaginary flux component, i.e., Eq.(43), and the imaginary flux component appearing in the third term on the right side of the equation for the real flux component, i.e., Eq.(42). It is admitted that the referred angular flux that is symmetric with respect to direction is replaced by its angularly averaged value, that according to Eq.(44) leads to

and represents the isotropic approximation. Further, it admitted also that the referred angular flux that is antisymmetric with respect to due to the source is replaced by a quantity proportional to

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i.e., Upon calculating A(r) from the definition for the current

given by Eq.(47) one obtains

which represents the anti-isotropic approximation. Approximation of Neglecting Angular Correlation Terms. The second proposed approximation55 - 57 concerns the antisymmetrical structure of the angular correlation terms. Let us remind the sense of the angular correlation terms. A neutron having crossed a voided cavity, being great compared to the mean free path of the surrounding medium, has better chance to traverse a long path after a collision if it is scattered backwards than if it is scattered forwards. This effect tends to decrease the diffusion area and consequently the leakage coefficient since the cosine of an obtuse angle is negative. However, this effect, although important in old type gas-graphite reactors, is not important in PWRs, CANDU reactors, BWRs and probably in HTGRs since in voided tight lattices a neutron having crossed a long path in void and then after being scattered in a fuel pin has a small chance to cross another long path in whatever direction. Supposing that the previously described approximations have been introduced into Eqs. (42) and (43) we will comment on the angular correlation terms. The angular correlation terms are: the second term on the right side of Eq.(42) and the first term on the right side of Eq.(43), as well as the directional cross terms appearing in the third term on the right side of Eq.(42) and the second term on the right side of Eq.(43). The approximation consists of replacing the flux and the currents and for appearing in these terms and which are as shown previously antisymmetric with respect to by their average values at the scale of a heterogeneous and symmetrical assembly or cell. These average values of antisymmetrical functions are obvious zero. We would like to recall that these antsymmetrical functions are identically zero in a homogeneous medium, as well as in axial direction in a two dimensional calculation. We could have inverted the order of presenting these two approximations without any consequence for the final result. In that case we would have considered also the angular flux which is antisymmetric with respect to and would have replaced it with its averaged value being equal to zero. Moreover, the angular correlation terms have been studied in the past by Benoist37, 38, and one has good reasons to think that regarding the results obtained, these terms do not have a considerable role for the actual types of assemblies or cells. This would not be the case in old-type gas-graphite cells with large channels, where these terms become important in radial direction. In any case, the approximation adopted considerably simplifies calculations since the angular correlation terms introduce coupling between neutron paths, as well as between energy groups, even if the collision is isotropic. To better understand the angular correlation terms we will briefly discuss their physical sense. We consider here a heterogeneous, symmetrical and reflected assembly. For the sake of simplicity, we suppose that the scattering is isotropic. The integral form of the transport equation governing the flux and corresponding to Eq.(43) where the approximation of the isotropy of the flux has been inserted, is written as where K is the integral transport operator. According to the definition of Eq.(47), the corresponding current is

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We may say that the nondiagonal terms of the current tensor for are certainly small because they reflect the influence of leakages in one direction on leakages in another direction. The angular correlation term that is the most important is without doubt the one involving the antisymmetric flux The solution of Eq.(54) can be written in the form of a Liouville-Neuman expansion, i.e., a multicollision expansion, or in other words, in powers of as follows:

Substituting the solution given by Eq.(56) into the equation for the directional current defined by Eq.(55), one obtains

The first term of the expansion of Eq.(57) represents the contribution to the current of neutrons that did not suffer any collision since the source the second term represents the contribution of neutrons that suffered one collision; the term represents the contribution of neutrons that suffered collisions. Consequently, as we shall see later, the first term introduces the first-flight directional collision probability, and the other terms introduce directional collision probabilities for neutrons that suffered from one to collisions. The directional collision probability, concerning a neutron that suffered collisions, takes into account angular correlation between two straight paths of the neutron separated by collisions. The approximation adopted here consists in conserving just the first term. This assumption is justified by the fact that the following terms, that is the angular correlation terms, result from the source that is antisymmetrical with respect to This hypothesis means in fact that we admit that this source can be reasonably replaced by its average value at the scale of the assembly, i.e., the flat flux approximation; this average value is obviously zero. The importance of the angular correlation terms with respect to the principal term depends on the degree of heterogeneity of the lattice. If the lattice is not very heterogeneous one may think that the angular correlation terms will not be too important and it will be possible to neglect them, at least as a first approximation. In the case of very heterogeneous lattices, these terms play an important role. In a multigroup scheme, the angular correlation terms introduce coupling between energy groups, even when the collision is isotropic. This means that these terms, belonging to one group of energy depend on cross sections of another group. Inserting the described approximations into Eqs.(42) and (43) one obtains the following system of equations

Fitting on Homogeneous model. It is possible to correct the first approximation in order that Eqs.(58) and (59) fit exactly the homogeneous model at the homogeneous limit55 - 57. Consider a homogeneous medium. By integrating Eqs.(58) and (59) over the solid angle, where Eq.(59) is first multiplied by the following two equations are obtained

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with

where all variables are space independent. In a homogeneous medium the directional current does not depend on the direction As a matter of fact, Eqs.(60) and (61) correspond to a homogeneous approximation. To get the homogeneous system of equations, Eq.(61), which corresponds to Fick’s relation, should be somewhat revised, while Eq.(60) being the conservation relation, which is rigorous, remains unchanged. Moreover, in a homogeneous medium Eq.(30) reduces to

Using Eq.(37) one can easily obtain the following relation for the flux:

while for the current one obtains

After applying the symmetry properties discussed earlier, Eq.(64) reduces to while Eq.(65) becomes Integrating Eq.(63) over the solid angle and inserting the last two relations one obtains Eq.(60), i.e., the conservation relation. To obtain the second equation of the homogeneous system one has to divide Eq.(63) by to introduce Eqs.(66) and (67), and to integrate it over the solid angle. After some algebra an equation similar to Eq.(61) is derived

with the fitting correction defined as

This fitting correction, which is generally close to unity, guarantees the exact homogeneous solution. When we have that We propose to accomplish the same operation with the imaginary flux component given by Eq.(59), regarding a heterogeneous and reflecting assembly. Hence, multiplying the right side of Eq.(59) with the fitting correction, one obtains

The system of Eqs.(58) and (70) gives the exact solution in the case when the homogeneous medium is considered. However, these equations are written for a heterogeneous medium while the fitting correction assumes a homogeneous medium. To obtain an equivalent homogeneous medium the total cross section, appearing in the fitting correction, should be homogenized by flux weighting and it becomes (where H means the homogenization by flux weighting).

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Fitting on Homogenized model. The fitting correction is not rigorously valid when a heterogeneous medium is homogenized, since the simple flux mapping technique involved is insufficient to correctly take into account directional streaming effects. This is why another consistent correction58, 59 is added that will result in the exact solution also in the case of the equivalent homogenized medium. Consider first the corresponding solution for a homogeneous medium given by Eqs.(60) and (61). In the case of a homogenized medium, after integrating Eqs.(58) and (59) over the solid angle, where Eq.(59) was first multiplied by (where H will mean flux homogenization and h directional current homogenization), one obtains

Here and are the usual flux homogenized cross sections, and are the directional current homogenized cross sections, while and are the volume averaged flux and directional currents. Now compare the homogeneous with the homogenized medium, Eqs.(60) and (61) with Eqs.(71) and (72). The equivalence between Eq.(60) and Eq.(71) is obvious since the last term on the right side of Eq.(71) can be written as where satisfies the following relation

and represents a buckling averaged homogenized current. In order to obtain an equivalence between Eq.(61) and Eq.(72) some manipulations are required. Multiplying Eq.(72) by and after summing it over all directions one obtains

with

and where is defined by replacing with in Eq.(75). The cross sections and represent buckling averaged current homogenized cross sections. One can see that Eq.(74) is similar to Eq.(61). However, the total cross sections appearing in the homogenized equations (Eq.(71) and Eq.(74)) are not identical, contrary to the homogeneous medium case(Eqs.(60) and (61)). The reason is that they are homogenized differently, one by the flux and the other by the directional currents. To restore a uniquely defined total cross section in Eqs.(71) and (74), we choose the flux homogenized cross section, since it will appear, later on, in the fitting correction For this reason can be added to each side of Eq.(74), which is now written as follows with A direct equivalence not only between Eqs.(60) and (71), and Eqs.(61) and (76), but also between the two systems of equations has now been obtained. At the same time a new cross section defined by Eq.(77), which is in fact a corrected cross section, has also been obtained.

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This correction will be applied to the heterogeneous case. Consider Eq.(70) without the fitting correction The cross section should be corrected by adding to it the term and to ensure conservation of neutrons, one should add the equivalent term, namely, to the left side of Eq.(70). It can be easily shown that starting from the transformed form of Eq.(70) one directly obtains Eq.(61) in the case of a homogeneous medium and Eq.(76) in the case of a homogenized medium. Now, by multiplying the right side of the transformed Eq.(70) with the fitting correction one obtains

Finally, after rearranging terms in Eq.(78) and after defining

one obtains the new Eq.(70), and we can write the system of equations to be solved as

What is important with this additional fitting correction is that the buckling obtained by the homogeneous model, for a medium homogenized in the manner presented here, is identical to the sum of directional bucklings obtained by Eqs.(80) and (81) which comprise the additional consistent correction. This is not the case when only flux mapping homogenization is carried out, i.e., when only the fitting on the homogeneous model is used. The additional fitting correction (fitting on the homogenized model) practically does not change the final result compared to the previous fitting correction (fitting on the homogeneous model), however it has the advantage that consistency of the model is assured. It is important to note that for both types of corrections the coefficient remains close to unity. Space Dependent Directional Leakage Coefficients. The second term on the right side of Eq.(80) can be written in the following form

with being the definition of the space-dependent directional leakage coefficient55 - 57. The product represents an additional absorption cross section, having the same space-dependence as all other cross sections and therefore being completely consistent with them. This was not the case with previous leakage coefficient definitions. Integral Form. To apply the collision probability technique as a solution method60 - 66 for the system of transport equations given by Eqs.(80) and (81), the integrodifferential form will be replaced by its corresponding integral form

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with where is on the assembly or cell boundary. Here is the neutron optical path between two points.

TIBERE Model So far, practically no comments have been made regarding the type of reflection considered on the assembly or cell boundary. Let us assume that a neutron arriving at the boundary is reflected in a specular way (mirror effect). This is equivalent to the neutron spreading throughout an infinite lattice of identical assemblies. In this case, tends to infinity, and therefore, the first term on the right side of Eqs.(84) and (85) vanishes. The resulting equations represent the TIBERE model 55-57 . Thus, the adopted boundary condition for an assembly or cell has been chosen to represent a reference situation for the homogenization by an equivalence procedure, which provides cross sections for a whole-core calculation. Real Flux Component. The angular flux obeys the specular boundary condition, i.e., a neutron hitting a boundary is remitted with a reflection angle equal to its incident angle,

Imaginary Flux Component. The angular flux obeys the antispecular boundary condition, i.e., a neutron hitting a boundary orthogonal to the axis is remitted as a negative neutron with its angle equal to its incident angle (for a neutron hitting a boundary parallel to the axes the flux obeys the specular boundary condition),

Numerical Discretization. Proceeding now with the usual discretization, where the flat flux approximation is used, one can obtain two equations corresponding to the real and the imaginary flux component, Eqs.(84) and (85) without their first term that has vanished due to the boundary condition adopted. The first equation is obtained by integrating Eq.(84) over the solid angle, and then introducing it is integrated over the volume. The second equation is obtained applying the same procedure to Eq.(85), but with the only difference that it is first multiplied by Let us write the resulting discretized equations of the TIBERE model:

where is the first-flight collision probability, i.e., the probability for a neutron uniformly and isotropically emitted in the volume to suffer its first collision in the volume having the total cross section and defined as

while the definition of

being the directional first-flight collision probability is

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It can easily be proved that the classical and the directional collision probability satisfy the following reciprocity relations

as well as the following conservation relations

Moreover, regarding the fact that the following relation between classical and directional collision probabilities can be established

More details regarding basic assumptions of the TIBERE model as well as a slightly different analysis of symmetry properties of the real and the imaginary flux components have been given by Petrovic56.

TIBERE-2 Model Another type of reflecting on the assembly or cell boundary may be chosen as well and will, in fact, represent another reference situation for the homogenization by an equivalence calculation. This boundary condition may be referred as a quasi-isotropic reflecting boundary condition since it will be shown that there exists for the real flux component a nonisotropic (or under certain assumptions isotropic) and for the imaginary flux component an reflecting boundary condition. Applying this type of reflection on the system of Eqs.(84) and (85) the TIBERE-258, 59 emerges. For practical calculations, when using homogeneous model, very often instead of the specular boundary condition, the isotropic boundary condition is imposed for the calculation of the heterogeneous flux which is used for homogenization of cross sections. Note that the specular boundary condition, being equivalent to the infinite lattice of identical assemblies, is one of the main assumptions of the fundamental mode theory. These homogenized cross sections represent the equivalent homogeneous medium on which homogeneous model is applied. The isotropic boundary condition is used in the heterogeneous calculation since especially if the collision probability method is used computations with the specular boundary condition are rather time-consuming. This comes from the fact that the specular reflection extends neutron paths to infinity, while the isotropic reflection limits them just to the assembly or cell being considered. It is obvious that the isotropic boundary condition does not correspond rigorously to the fundamental mode theory, but is closer to reality, although in most cases it almost does not change the final result compared to the option when the specular boundary condition is used. In a similar way the TIBERE-2 model may be regarded compared to the TIBERE model. But, one should be aware of the fact that in reactors, particularly modern reactors, lattices of identical assemblies or cells do not exist and that some types of reactors do not have regular

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lattices. Therefore, although not respecting, in certain sense, one of the main assumptions of the fundamental mode theory, the TIBERE-2 model, due to its quasi-isotropic reflection, corresponds better to the reality since neighboring assemblies or cells are different from the one being considered. This is why in this case the buckling could be considered only as a criticality parameter. Hence, the TIBERE-2 model enables to calculate in a more realistic manner a voided assembly surrounded by non-voided assemblies. In the case of several voided assemblies one could consider them as one multiassembly motive within the TIBERE-2 model calculation. But, this kind of modeling would lead to rather time-consuming calculations. If in this case one would consider just one assembly the computation would certainly underestimate the streaming effect, since neutron paths would be artificially truncated at the limit of the assembly. This is why a study is under way that will still allow to treat one assembly, instead of all voided assemblies, but where neutron paths would be extended in an approximate way and introduced as a simple correction. As it will be shown the TIBERE-2 model introduces incoming and outgoing currents and currents of momentum at the surface of a considered assembly or cell. Although at this starting step of calculation one has no information about what is happening in the neighboring assemblies or cells, these currents and currents of momentum could allow a better coupling, in an iterative procedure, between the considered and the neighboring assemblies or cells. But, this iterative procedure would be rather time-consuming. However, the following presentation will still be limited exclusively to the quasi-isotropic reflection on an assembly or cell boundary. Real Flux Component. The first idea which comes to mind is to consider a uniform and isotropic reflecting (the white) boundary condition for the real flux component. However, this is not completely correct as will be shown on a simple example. In a homogeneous and infinite medium Eq.(80) reduces to k

which proves that the angular flux is independent of r, but depends on in all cases except when where a special case is when all Hence, an isotropic reflecting boundary condition, i.e. an isotropic incoming flux, for the real flux component is a priori incorrect, since even in a homogeneous medium such a flux can hardly exist. However, based on Eq.(98), one can assume that the incoming real flux component is proportional to the real angular flux component existing in a homogenized medium. This means that neutrons are reflected on the boundary in a uniform and non isotropic manner

where H and denote homogenized values as previously described when fitting on the homogenized model. The proportionality constant A is determined by normalizing the total incoming current to one neutron

where n_ denotes the incoming unit normal at the surface S. To simplify writing we have assumed that there is only one surface. Thus, the incoming angular flux is

THEORY where the shape parameter9, 10

253 is defined as

satisfying the following relation

This parameter depends on the geometry and on the direction If the boundaries are two infinite plane surfaces, if is perpendicular to the planes and if is parallel to the planes. If the boundary is an infinite cylinder of whatever section, if is parallel to the generatrices. If the cylinder is of a square section and if is perpendicular to one of the sides, or if the cylinder is circular and any direction perpendicular to the generatrices on the average. However, as it will be shown, the incoming angular flux given by Eq.(101) may complicate practical calculations and can decrease the efficiency of the numerical model. This is why another assumption can be made. In production calculations the differences between the directional bucklings are generally small. Hence, in order to take advantage of an isotropic incoming flux it will be assumed that the directional bucklings are identical. Even if in an actual calculation this is not the case, this assumption should have only a slight effect on the final result. Consequently, the second fraction in Eq.( 101), defining the non isotropic incoming angular flux, becomes unity and the well known uniform and isotropic form is obtained

which implies an isotropic reflecting boundary condition for the real flux component. Imaginary Flux Component. Before considering Eq.(85), it should be useful to perform another analysis. The integration of the equation for the real flux component, i.e., Eq.(80), over the solid angle leads to the conservation relation. On the contrary, one cannot obtain anything useful by integrating the equation for the imaginary flux component, i.e., Eq.(81), over the solid angle. But, by weighting Eq.(81) by and then integrating over the solid angle, one obtains the conservation relation of the total momentum of neutrons in the direction

where another type of current vector appears, defined as9, 10

Since the angular flux is antisymmetric with respect to the direction (due to the antisymmetrical source term), it will be assumed that the incoming angular flux is uniform and proportional to where the proportionality constant A will be evaluated upon normalizing the total incoming current of momentum in the direction to one neutron

Hence, the incoming angular flux is

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Numerical Discretization. Proceeding in the usual way67, 68 when discretizing the equation for the real flux component, i.e., Eq.(84), including the corresponding adopted boundary condition, i.e., Eq.( 101), and using the flat flux approximation, one obtains three corresponding equations. The first equation is obtained by integrating Eq.(84) over the solid angle, and then by introducing and it is integrated over the volume. To obtain the equation for the outgoing angular flux, we impose first in Eq.(84). Multiplying the resulting equation by where denotes the outgoing unit normal at the surface, and integrating it over the solid angle and over the surface yields the second equation. The third equation represents the conservation of the incoming and outgoing total currents. Finally, for an assembly or a cell, having one surface (for the sake of simplicity), the following system of equations can be written

with

The quantities and are the classical and the directional first-flight collision probabilities already defined, but calculated now for an open assembly, i.e., the neutron paths are limited inside it. Moreover, and are the well known classical escape, penetration and transmission probabilities defined as

Moreover, and abilities, defined as9, 10

are the directional escape, penetration and transmission prob-

These directional probabilities differ from the classical probabilities by the presence of ( and ) or ( and ). The classical and the directional probabilities satisfy the following conservation relations

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as well as the reciprocity relations

The reciprocity relations given earlier for the classical and the directional first-flight collision probabilities are valid here as well. After summing each type of directional probability over all directions the following relations are obtained

To discretize the equation for the imaginary flux component, i.e., Eq.(85), with its corresponding adopted boundary condition, i.e., Eq.(109), and to obtain corresponding equations, one proceeds in the same way as in the case of Eq.(84), but with the exception, that the integration over the solid angle is carried out with the weight Now, three discretized equations corresponding to the equation for the imaginary flux component (assuming one surface for the sake of simplicity) can be written

where the third equation represents the equality between the entering and leaving total current of momentum in the direction Final Form. It has been shown that by introducing the quasi-isotropic reflecting boundary conditions, the non isotropic for the real flux component and the reflecting boundary condition for the imaginary flux component, two systems of equations have been obtained. Consequently, classical and directional collision probabilities are calculated for an open assembly or cell, which limits neutron paths inside it. In fact, due to the quasi-isotropic reflecting of neutrons on the boundary, in the TIBERE-2 model no infinite neutron paths are used for the calculation of these collision probabilities, while such paths are required by the TIBERE model. This has two advantages. First, the divergence appearing in the TIBERE model when an assembly is completely or peripherally voided, induced by the specular reflecting of neutrons on the boundary, is in the TIBERE-2 model eliminated. The second advantage is the decrease in the computational time of the TIBERE-2 model, compared to the TIBERE model, since the

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neutron paths used for the computation of classical and directional collision probabilities are shorter (they are limited to one assembly or cell). Eliminating the currents and from the system of Eqs.(110),(111)and(112), and the currents of momentum and from Eqs.(131), (132) and(133), the system of equations representing the discretized form of the TIBERE-2 model is obtained

with modified collision probabilities defined as

and with

The modified collision probability satisfies the conservation and the reciprocity relations, while and satisfy only the conservation relation. Since in practical calculations directional bucklings are similar to each other, we can assume that they are identical. This assumption smoothes the path from Eq.(101) to Eq.(104) and considerably simplifies the calculation procedure. Hence, the terms in brackets multiplying thecurrent J_ in Eq.( 110) reduce to andinEq.(111)they reduce to Consequently, one obtains another definition of the probability P given as follows It is obvious that in the case of the non isotropic reflecting boundary condition the probability P has a more complicated form than in the case of the isotropic boundary condition. Moreover, it can be easily shown that with the probability P defined by Eq.(140) the modified collision probabilities and satisfy the conservation and the reciprocity relations. But, this is not true for which only satisfies the conservation relation; while the reciprocity relation cannot be satisfied.

An Alternative to Models TIBERE and TIBERE-2 Based on a previous work37 Benoist has proposed an amelioration69 compared to the TIBERE model consisting in replacing the approximation of isotropy of the real flux component and anti-isotropy of the imaginary flux component with an estimation of the smallest order of in Eqs.(42) and (43) having the specular boundary condition. In other words, this a kind of iterative use of the small buckling approximation within the heterogeneous theory. Let us write the integral form of Eqs.(42) and (43), including the specular boundary condition, where only the approximation neglecting angular correlation terms has been introduced. Thus, one obtains

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Limiting the system of Eqs.(141) and (142) to the zeroth order of and inserting in the equation for the real flux component into the equation for the imaginary flux component, the following system is obtained

with and Here and are neutron optical paths between two points. Integrating Eqs.(141) and (142) over the solid angle, where Eq. (142) is first multiplied by and the over the volume in the same way as in the case of the TIBERE model, obtains equations for scalar flux and the directional current Then, inserting into these equations Eqs.(143) and (144) and after several integrations by parts one obtains the following system of equations

In the system of Eqs.(145) and (146) higher orders in powers of have been neglected. Moreover, there appear the neutron mean free path as well as the square mean free path To take them into account, in an approximate manner, but sufficient for practical calculations, coefficients depending on B will be introduced in this system of equations. These coefficients will be calculated in such way as to obtain from Eqs.(145) and (146) the exact homogeneous system in a homogeneous medium. In a homogeneous medium the system of Eqs.(145) and (146) reduces to

Proceeding similarly as shown when fitting on the homogeneous coefficients

model, one obtains two

where should multiply the first two terms of Eq.(147), while the third term of Eq.(147) and both terms of Eq.(148) leading to the exact solution at the homogeneous medium limit. Performing multiplication of the same terms in Eqs.(145) and (146) and after discretizing these equations as shown earlier, the following equations are obtained

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where H means that the equivalent homogeneous cross section has been obtained by flux weighting. In Eqs.(151) and (152) appears a new type of directional collision probabilities defined as follows

These new directional collision probabilities and satisfy the usual reciprocity relations, but they do not satisfy the conservation relations. It is obvious that these new directional collision probabilities, as they are defined, are not dimensionless. It is important to note that this improved model compared to the actual TIBERE model is an asymptotic approximation of Eqs.(42) and (43) at the limit of small buckling. However, it requires the calculation of new, not typical, directional collision probabilities that are not normalized to unity. An advantage of this model, compared to the TIBERE model (Eqs.(88) and (89)), is that one needs not to calculate collision probabilities in the case when the region or both regions and are void or close to void since these probabilities are multiplied by a cross section that is zero or negligible. Since in the TIBERE model this is not the case, to solve numerical problems that appear, one has to use expansions. The improved model should be used in cases when one expects that the angular fluxes and are far from being isotropic and anti-isotropic respectively. We will not elaborate further the considered improvement, but fitting on a homogenized model can be performed as shown earlier. Moreover, this asymptotic approximation could be extended also to the corresponding equations with quasi-isotropic boundary conditions and obtain an improvement with respect to the TIBERE-2 model.

Practical Utilization of Models TIBERE and TIBERE-2 in Transport Codes Currently the TIBERE and the TIBERE-2 model are working within several transport theory codes. At the time being their application is restricted to two-dimensional geometry. One of the advantages of the models TIBERE and TIBERE-2 lies in the fact that in codes already using the classical collision probabilities one has only to add computation of the directional collision probabilities. Moreover, the iterative scheme of the eigenvalue evaluation using these models is very similar to the one used for standard eigenvalue calculations (assembly flux map calculations). Before starting an iterative procedure the classical and directional collision probabilities are calculated once and for all. If the nonisotropic reflecting boundary condition is used for the real flux component, within the TIBERE-2 model, the modified collision probabilities and should be recalculated before each outer iteration. This is done because the probability P contains homogenized values evaluated by flux and directional currents mapping arising from the previous iteration. This fact can decrease the efficiency (fast calculation) of the model since the inversion of the matrices included in the matrices P should be performed in each outer iteration and hence, may become time consuming. On the opposite, if the isotropic reflecting boundary condition is adopted, the modified collision probabilities and are calculated once and for all. Since 1992 the TIBERE model, for geometry, is in operation within the APOLLO-2 transport theory code70. The integration of the exponential functions appearing in the definitions of probabilities is performed using Bickley-Naylor functions71 (which are except for the sign successive primitives of Bessel function obtained by the analytical integration in the axial direction, while the numerical integration is carried out in plane. For the classical

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collision probabilities the following Bickley-Naylor functions are used: (when there is no voided media), (when the medium or is void) and (when both media and are void). The directional collision probabilities involve the same functions but with a different order. Since it has been assumed that the considered assembly or cell is symmetric the radial collision probability averaged over all transverse directions, has been introduced. The function appearing in the classical probabilities has to be replaced by in the radial and by in the axial collision probability. Moreover, Eq.(97) becomes Consequently, one has to compute only and while is evaluated from Eq.(155). The exact evaluation of Bickley-Naylor functions is very time consuming. This is why they are tabulated for equidistant arguments, with the step being small enough to obtain the desired precision, and a linear interpolation is used to evaluate these functions. Consequently, in the problems that are considered, because the medium and/or may be void or almost void (fog), numerical difficulties arise in the calculation of collision probabilities. These difficulties can also appear when calculating the difference of two functions where arguments are very near one to another. Hence, the problem of small arguments (optical paths) and difference effects can spoil the accuracy of the calculation or even lead to results without significance. To overcome these problems, additional analysis were performed, and MacLaurin’s and Taylor’s expansions of Bickley-Naylor functions were introduced for small optical paths and small differences of optical paths, respectively. It should be emphasized that in the classical collision probability computations used in the calculations of an assembly flux map, these difficulties were not apparent because in the case when either or both and are void or almost void, classical collision probabilities are multiplied by the cross section that is equal to zero or negligible. The system of equations representing the TIBERE model is solved by simultaneously computing multigroup scalar fluxes and directional currents in a heterogeneous geometry using successive inner and outer iterations. Before each outer iteration, the coefficient is evaluated, where is calculated by homogenizing the heterogeneous medium with the flux map last obtained. The buckling B is obtained via the critical neutron balance relation using the scalar flux and the directional current maps last obtained. By performing the homogeneous calculation in each outer iteration, using the homogenization described within fitting on the homogenized model, it is possible to reduce the number of outer iterations (already accelerated by standard methods) by ~ 70 %. This is done by renormalizing, before each outer iteration, the last flux and directional current maps by the last critical homogeneous flux and current obtained. In this way the flux and directional current source terms of heterogeneous equations are approached in advance, by the homogeneous critical calculation (rapidly carried out), to the criticality of heterogeneous equations. Further on, supposing that two directional bucklings are known, the third is obtained from the critical neutron balance relation, and is used in the next iteration. Once, the system of equations has converged, one obtains the directional leakage coefficients, in each calculation zone and in each group, using the converged scalar flux and directional current maps. The calculation of the classical first-flight collision probability in an assembly in geometry represents the major part of the total computation time of the flux map and consequently of the homogeneous leakage model calculation. In other words, if one uses a fine mesh numerical integration (a small step between and a certain number of angles for ray tracking), allowing good numerical precision, the computational time of the classical collision probabilities will be ~ 70 % of the total computational time. In the framework of the TIBERE model, apart from the classical collision probabilities, one also calculates directly radial colli-

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sion probabilities (axial are evaluated from these two, Eq.(155)). Therefore, the computation time of collision probabilities (classical and radial) increases by ~ 70 % (but not by 100 % because the tracking is the same for the two calculations). The use of expansions for small optical paths somewhat increases the computational time of collision probabilities. Consequently, the computational time of collision probabilities in the framework of the TIBERE model is ~ 80 % longer, and we may conclude that the TIBERE model computation procedure is also ~ 80 % longer than the procedure of homogeneous leakage model. More details regarding the TIBERE model itself as well as within the APOLLO-2 code are given by Petrovic56. In 1994 the TIBERE-2 model, as well as the TIBERE model, have been implemented in the DRAGON transport theory code72. It is worth mentioning that due to the adopted boundary condition the TIBERE-2 model, as it is developed for geometry, can equally be applied to geometry. Apart for few exceptions, almost all we have said about the TIBERE model within the APOLLO-2 code is valid for the models TIBERE and TIBERE-2 within the DRAGON code. The DRAGON code uses a parabolic interpolation with adding logarithmic terms of the expansion for small arguments for the tabulated Bickley-Naylor functions and thus the expansions for small optical paths and differences are not necessary. When using the specular reflection of neutrons as the boundary condition, this code also offers the possibility of calculating collision probabilities using the cyclic tracking method, that has been used in the characteristics formulation of the transport equation54. The TIBERE model has been introduced in such a manner as to take advantage of this fact. This technique, which uses the numerical integration of exponential functions in all directions, allows the specular reflection of neutrons to be represented by reentering the leaving neutrons on the appropriate track, therefore cycling to infinity. The foregoing conclusions of comparisons between the TIBERE and the homogeneous model are valid also for the comparisons of the TIBERE-2, with the isotropic boundary condition for the real flux component, and the homogeneous model under one condition which is that the heterogeneous flux map used for the homogenization, that is needed for the homogeneous model, is calculated also with the isotropic reflecting boundary condition. The TIBERE-2 model has been also introduced in the transport code MAGGENTA73 which is based in part on the existing GTRAN2 code methodology74. This version of the TIBERE-2 model uses expansions for small optical paths. The implementation of the TIBERE model in this code is foreseen as well. Recently the TIBERE-2 model has been implemented75 in the APOLLO-2 code.

HOMOGENIZATION BY AN EQUIVALENCE PROCEDURE As a reactor core is composed of different assemblies, the only exact calculation would consist of solving the multigroup transport equation for the whole reactor, with its reflector. Despite very fast progress in supercomputing, this is still not possible or would be extremely expensive. Therefore, one is led to decompose the reactor in small pieces, cells or assemblies, that are heterogeneous but will be represented in the whole core calculation as equivalent homogenized media. There are two concepts of homogenization for pressurized water reactors. One is the heterogeneous diffusion approach, which relies on finite difference (low-order) discretization and uses pin-by-pin homogenized nuclear properties (a fuel rod surrounded by its coolant). These pin-by-pin properties are obtained using the assembly flux map. This concept enables accurate prediction of on-site measurements. The other concept is the homogeneous diffusion approach based on nodal or finite element (high-order) discretization and uses homogeneous nuclear properties defined at the scale of each assembly. Similar homogenizations are

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performed for other types of reactors.

Homogenization and Homogeneous Leakage Coefficients In the past, when the homogeneous leakage model has been used, first the critical buckling, the assembly multigroup leakage coefficient and the corresponding multigroup flux map are calculated separately for each reflected assembly. Using the multigroup flux map one can define homogenized parameters at the scale of each macro-region (cell or assembly). Simultaneously, a condensation of energy groups is accomplished giving homogenized parameters collapsed in a small number of macro-groups76. Thus, for the reaction one obtains the reaction rate in a macro-region M and in the macro-group G

while the leakage rate

in the macro-region M and macro-group G is

The average flux in the macro-region M and macro-group G is

and the obtained reaction and leakage rates, as well as the average flux will be regarded as the reference values. Using the reaction and the leakage rates one can define the average cross section for the reaction in each macro-region M and in each macro-group G

as well as the average leakage coefficient

The leakage coefficients differ from one macro-region M and macro-group G to another within the same assembly. This is obtained in an artificial manner since the leakage coefficient is calculated for a homogenized assembly and is unique for the whole assembly. In a one-group problem, the leakage coefficient will evidently be the same in all macro-regions, which proves that in a spectral problem the deviation of the leakage coefficient from the spatial uniformity arises only from the spectral variation from one macro-region to another. In any case, this effect is small. Hence, it is obvious that the macro-region leakage coefficient is obtained in a completely artificial way because of spectrum weighting of the leakage coefficient at the scale of each macro-region.

Homogenization and Directional Space-Dependent Leakage Coefficients However, if the model TIBERE or the model TIBERE-2 is used one will obtain completely consistent homogenized cross sections and leakage coefficients, i.e., leakage cross sections. Since the complex angular flux given by Eq.(37), is the solution of the transport equation, Eq.(30), the reference reaction and leakage rates can be defined using earlier discussed symmetry properties. Thus, the reaction rate in the macro-region M and the macro-group G is

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while the corresponding volumetric integral flux is

The expressions for the reference reaction rate and the volumetric integral flux are the same as those in the classical homogenization. Let us now define the reference reaction rate for the volumetric leakage reaction in the direction in the macro-region M and in the macro-group G. Thus, the reaction rate corresponding to the cross section appearing in Eq.(30) is

Consequently, one can define the average cross section for reaction

as well as the leakage cross section in the direction

where is the average leakage coefficient. One can see that the homogenized cross section for the reaction and for the leakage in direction have been obtained in the same way since their origins have the same space dependence. As shown, this is not the case when leakages are calculated using the homogeneous model.

Equivalence It has been proved that the simple flux weighting does not satisfy the equivalence between the reference calculation and the macro-calculation, i.e., the diffusion or less often transport calculation (used if the whole core calculation is carried out applying the transport theory instead of the diffusion theory), accomplished for a reflected assembly, composed of homogenized macro-regions and condensed in macro-groups. More precisely, the equivalence between reaction rates was not satisfied in the case of the macro-calculation using the heterogeneous diffusion approach. But, in the case of the macro-calculation using the homogeneous diffusion approach, although this equivalence was satisfied, the homogeneous diffusion approach itself failed to reproduce the control rod worth with sufficient accuracy compared to the measured values. Apart from the early works related to the equivalence theory that have been performed by Selengut20 and Bonalumi77, many important contributions to the homogenization theory were presented at the IAEA conference at Lugano in 1978. Kavenoky78 proposed SPH (superhomogénéisation) factors as an attempt to generate SPH-corrected cross sections for the heterogeneous diffusion analysis of an irregular lattice. This procedure involves a renormalization of the homogenized cross sections and fluxes to preserve all the reaction rates and interface currents in a simplified assembly calculation. The simplified calculation, referred to as the macro-calculation, is carried out with homogenized pin cells and uniform interface currents, and is defined over a coarse energy grid. The number of SPH factors in a coarse energy group is equal to the number of distinct pin cells in the assembly. This approach is consistent only in the case when the number of interface currents is equal to the number of SPH factors and is therefore limited to situations in which each pin cell is surrounded by a unique isotropic interface current. Mondot79 introduced the idea also using SPH factors as a mean for correcting the errors due to both Fick’s law and to low-order discretization of the diffusion equation. At

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this conference contributions by Koebke80 and by Henry et al.81 set up the bases of the discontinuity factor method. The problem of boundary conditions for cylindrical cells, which is of interest for homogenization, has been discussed by Altiparmakov82 in the context of the calculation of disadvantage factors. Later, a detailed analysis of homogenization techniques for light water reactor has been performed by Smith83, with an emphasis on the discontinuity factor method. Hébert84 - 86 has developed several SPH algorithms for the transport-transport and the transport-diffusion equivalence techniques, preserving two main ideas of Kavenoky’s work. They are that an SPH technique never needs the solution of an eigenvalue (K or buckling search) problem in the adjustment phase (the distribution of the fission and diffusion sources is always known from the initial flux calculation) and a unique equivalence factor per pin cell always gives enough degrees of freedom to preserve all the reaction and leakage rates. It should be noted that the equivalence calculation is a step between the transport and the diffusion (or transport) calculation, and that no modification of reactor diffusion (or transport) calculation is required. The aim of an equivalence calculation consists of defining in a homogenized macroregion M and in a macro-group G the equivalent cross section for the reaction in such a way that using an approximate computation procedure one finds the same reaction rates as in the reference calculation, The same holds for the leakage cross section, i.e., it is necessary to find the directional leakage cross section or the directional leakage coefficient that satisfies (or in the case of the homogeneous leakage model for ). The SPH method gives as the solution the SPH factor that satisfies the equivalence of reaction and leakage rates in such a way that equivalent cross sections are defined as and

where consequently one can write

while in the case of the homogeneous

leakage model

It is obvious, that the factor does not depend on the type of reaction, but just on space and energy. Moreover, this factor, which is obtained in a nonlinear procedure, depends also on the chosen type of macro-calculation, i.e., diffusion or transport, as well as on the degree of discretization, i.e., space and energy discretization. As the consequence of conservation of rates and because the flux does not depend on the type of reaction, one can write the corresponding equivalent flux as follows

Using a generalized perturbation theory to evaluate SPH factors Hébert84 noticed that an infinite number of SPH factor sets can be the solution of the problem and satisfy the reaction rates. This brings an extra degree of freedom that is subject to a normalization condition in order to obtain a unique set of SPH factors. Since the choice of the additional condition is arbitrary, there are several possibilities. To illustrate different normalizations in a simple way we will limit the following presentation to a one-group case where the macro-calculation is performed using the homogeneous diffusion approach (the whole assembly is homogenized). However,

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this presentation can easily be extended to a multigroup case and to the heterogeneous diffusion approach, but with more complicated relations. In the following, when writing the real integral flux component we will also drop the subscript s, since as it has already been indicated, it is equal to the integral flux being the solution of the transport equation (the imaginary integral flux component has been neglected because of its antisymmetric properties). The simplest normalization condition is the one in which the reference average flux in an assembly is preserved in the equivalence calculation, i.e., that the equivalent flux is equal to the average flux in the volume of an assembly Obviously, this normalization leads to the homogenization by simple weighting of cross sections with the reference flux and one has a unique SPH factor equal to unity. It has been shown that the cross sections obtained by this homogenization, i.e., the whole assembly homogenization, lead in the whole core calculation to results having not sufficient precision compared to experiments. This is particularly apparent in the case of the antireactivity evaluation of a pressurized water reactor control assembly. However, if the heterogeneous diffusion approach (homogenization per cell within an assembly) is used in the macro-calcualtion one would obtain a SPH factor for each cell. Using the cross sections corrected with these factors one obtains in the whole core calculation results that are quite better than in the previous case. Another choice consists in assuming that the equivalent flux is equal to the average value of the reference flux obtained not over the volume but over the assembly surface. The reason to use this normalization is to satisfy the principle of Selengut20 that has been already proposed for a cylindrical cell within the scope of the diffusion-diffusion equivalence. This principle admits that the homogenization of a reflected system should not perturb the flux in its environment. But, this principle cannot be absolutely satisfied except by the diffusion-diffusion equivalence. If the reference calculation is the transport calculation, this principle is not satisfied rigorously because of the existence of transport transients between the heterogeneous and the homogenized zone. Still, the conservation of the average surface flux satisfies this principle sufficiently, since there is only one adjustment parameter. The Selengut’s concept has been first used by Bonalumi77. Using the proposed normalization, where is the average value of the reference flux at the assembly surface, one can write

where the SPH factor µ, is defined as Thus, the equivalent cross section of a homogenized system is defined as where has been obtained by the simple flux weighting. A normalization close to the previous one, but still different, and which has been proposed in the original SPH method by Kavenoky78 for a cylindrical cell, has been extended for a square assembly by Hébert85, 86. He also established a procedure so that the generalized perturbation theory is no longer used. This new algorithm leads to the SPH factors that are similar to the inverse of heterogeneous factors used in the simplified equivalence theory of Koebke87 or in the simplified discontinuity factor theory of Tanker and Henry88. It is clear that in a reflected homogeneous medium the flux is uniform and isotropic. If is the incoming current (or the outgoing current, since they are equal) and S the surface of the system, one can write

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The adopted choice in the original SPH method, for a cell78, and later for an assembly85, 86, consists not in preserving the average flux at the surface as previously, but in preserving the total incoming current in the system. If is the total incoming current given by the reference transport calculation we can write and consequently

where The corresponding equivalent cross section is obtained by Eq.(174). The value of µ is slightly different from the one obtained by the previous normalization, since the relation

is valid only for a uniform and isotropic flux which is not the case with the reference calculation of a heterogeneous reflected assembly. In fact, if is the angular flux, one can write

while where n_ is the incoming unit normal at the surface S. Some intuitive reasons are in favor of this normalization instead of the previous one. In both cases Selengut’s principle is satisfied (at the limit of the only one degree of freedom), but the flux in the environment of a system is probably conditioned more by the current J_, i.e., by the number of neutrons leaving the system, than by the average flux at the surface. Moreover, the current is more convenient than the flux for calculation by the collision probability method, since it can be computed directly (using the collision probability conservation and reciprocity relations) if in the reference calculation it is assumed that neutrons are reflected at the boundary following a uniform and isotropic flux. Thus, using the considered normalization the principle of Selengut implies that the transparency of an assembly (or a cell) is preserved in the process of homogenization for a neutron incoming in a uniform and isotropic manner. This SPH factor, and consequently the equivalent cross sections are slightly different than those obtained by the previous normalization. However, if one uses the diffusion-diffusion equivalence these two normalizations lead to perfectly identical results, because in the reference diffusion calculation one obtains Eq.(179). It should be noted that the flux obtained in the whole core calculation, using equivalent cross sections obtained by the foregoing three normalizations, is continuous at all assembly (or cell) interfaces. The foregoing normalizations can be easily generalized to the heterogeneous diffusion approach (pin-by-pin homogenization within an assembly), but its presentation is more complicated. Moreover, these three normalizations are used in the homogenization procedure in France. Let us return to the first normalization, the one in which the reference average flux in an assembly is preserved in the equivalence calculation (µ = 1). In the homogenization procedure the reaction rates are preserved, but the principle of Selengut is not satisfied since the average flux at the surface or the incoming current are not preserved. Still, one can satisfy Selengut’s principle by means of an assumption which consists in adopting that by preserving the average flux in the volume one preserves equally the average flux at the surface. Since, in a homogenized system the flux is uniform, and since

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in the reference calculation there emerges an artificial discontinuity between the volumetric and the surface flux. Thus, in a homogenized system we have

where

If all the assemblies are identical, f does not vary from one assembly to another and writing has no consequence. But, in a whole core calculation belonging to different assemblies I will be different and there appears a discontinuity of the flux between two adjacent assemblies. The whole core calculation is performed by imposing at the interface of the two assemblies 1 and 2 the condition of equality of the average surface fluxes, which is an approximation resulting from the unique degree of freedom. Hence, the discontinuity condition of the fluxes is as follows

This is the scheme, based on works of Koebke80 and Henry81, which is used in the diffusion theory in Germany and USA. Regarding reactor reactivity calculation it is equivalent to the second normalization proposed here and close to the third normalization at least in the version transport-diffusion. Note that for all foregoing normalizations the leakage coefficient (both homogeneous and directional), appearing in the leakage cross section, is evaluated in each case without ambiguity, but that the diffusion coefficient, i.e., the coefficient of the operator appearing in the diffusion equation of the transport-diffusion equivalence, is completely arbitrary. The factor µ does not depend on it since the assembly is reflected. In a whole reactor calculation the diffusion coefficient affects the flux shape in each assembly.

SOME NUMERICAL COMPARISONS Several sets of two-dimensional x-y calculations56 - 59 performed with the transport codes APOLLO-270 and DRAGON72 using the homogeneous and the TIBERE leakage model, and the homogeneous and the leakage models TIBERE and TIBERE-2 have been considered respectively. The APOLLO-2 code was used for calculations of a PWR assembly. It was also used in a computation scheme, that includes the homogenization by equivalence and the diffusion code CRONOS89, to perform comparisons with an experiment belonging to the experimental program EPICURE90. The DRAGON code was used to calculate a PWR assembly and a CANDU reactor cell. A comparison91, 92 of results obtained by the DRAGON and by the MCNP code76 in the case of a CANDU cell has also been performed. Currently, a comparison94 between the models TIBERE and TIBERE-2 within the transport code MAGGENTA73 and the MCNP code for BWR and HTGR assemblies is under way.

APOLLO-2 Code PWR Assembly. The infinite multiplication factor, obtained by the APOLLO-2 code70, of a symmetric mixed-oxide fuel PWR assembly with 17×17 cells, the central cell being a

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water hole, is model is buckling) is

The critical buckling obtained by the homogeneous leakage Assuming that the axial buckling (the geometrical axial the TIBERE model gives the critical radial buckling or the total critical buckling A simple calculation shows that the homogeneous leakage procedure overestimates the reactivity by i.e., the leakages are slightly underestimated. The flux and directional current maps, obtained by the TIBERE model, are not uniform but the effect is much more important in the following case56, 57. If the central part of 7 × 7 cells of the same assembly water is substituted by void, the infinite multiplication factor is The critical buckling obtained by the homogeneous leakage model is Assuming the same axial buckling as in the previous case, the TIBERE procedure gives the critical radial buckling i.e., the total buckling The decrease of the critical buckling in the case of the TIBERE model (compared to the homogeneous treatment) implies that the dimensions of the considered reactor should be increased to obtain criticality. The increase of the reactor dimensions compensates for the loss of neutrons induced by the increase of leakages and the change of spectrum due to the loss of moderator. Again, the simple calculation shows that the homogeneous leakage model overestimates the reactivity by By imposing in the homogeneous leakage calculation the critical buckling given by the TIBERE model one obtains the same overestimated reactivity value.

Probably more interesting than the reactivities are the maps of the scalar flux and the radial and the axial currents obtained by the TIBERE model56, 57. Since the

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calculation was carried out with fine mesh and in 99 energy groups, flux and current maps were homogenized at the scale of each cell, while energy groups were condensed in two groups (the limit between groups being 0.625 eV). Thus, one obtains flux and current maps, as functions of cell positions on the half-median, in the fast and the thermal group. These maps are given in Fig. 1. Note that in the zone of the assembly where water is voided, fuel rods remain (except for the central cell that was a water hole and is now void). In the fast group, the flux is practically uniform, but the currents have an important peak in the voided zone. According to the definition of the directional leakage coefficients it is obvious that they have also an important peak in this zone. In the thermal group, the flux is strongly depressed in the voided zone because of the absorption in fuel, but there is practically no thermal source since there is no water. However, the directional currents still have an important peak in this zone, and consequently the directional leakage coefficients. One can notice that in the voided zone, particularly in the central cell (which is a complete void) the axial current is greater than the radial one. This is the consequence of the fact that a neutron has a longer streaming (without suffering a collision) moving in the axial than in the radial direction since in the radial direction the voided zone is limited. In the part of the assembly that is not voided these streaming lengths are almost the same. Hence, there is a variation of radial and axial leakage coefficients through the assembly. The homogeneous leakage procedure would lead in both groups to uniform leakage coefficients in the assembly because is proportional to and is independent of radial or axial direction (except for a small variation due to the spectral condensation). The results presented here for a regular lattice of identical assemblies indicate that the effect can be important, but it is probable that in the case of a real loss-of-coolant accident, the voiding of water will not occur in the center of all assemblies at the same time. Thus, the equivalence method is of considerable importance because it allows more realistic cases to be treated. EPICURE Experiment. To improve the knowledge of the influence of lattice heterogeneity on neutron leakages, at about the same time, the TIBERE model has been developed at CE Saclay, while at CBN Cadarache the experimental program EPICURE (carried out on the EOLE experimental reactor) has been completed by Mondot et al. 90. A part of this experimental program was devoted to the investigation of two-dimensional local void effects in a lattice. The aim of these experiments was to measure the perturbation caused by progressive removal of water in the central part of a regular core in terms of reactivity and pinwise fission rate distributions. First, a critical reference core made of a regular lattice has been constructed. It has been carefully characterized by buckling measurements derived from a number of different reaction rate distributions. At the center of this core, a zone of 7 × 7 pins has been reserved for special geometrical changes. Only the total removal of water will be anticipated here. For safety reasons, aluminum is used instead of void to replace evacuated water. This is justified by the value of the mean free path of aluminum which is big enough. The region of 7 × 7 pins is replaced by a solid block of aluminum containing 49 channels in which fuel pins are inserted. Criticality has been obtained by using a compensating boric acid adjustment in the moderator, and radial pinwise fission rate distributions have been obtained from integral gamma scanning on the pins themselves. It has been observed that in the reference case (0 % of aluminum), the radial fission rate distribution in the core is very near to the fundamental mode in cylindrical geometry except for edges of core where the existing perturbation is caused by the presence of the water reflector. In the case with aluminum, there is a depression of radial fission rates in the central part of the core, but this

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perturbation vanishes rather quickly and the fundamental mode is recovered a few pins from the voided-unvoided zone interface, except for the core edges as explained earlier. As the experimental results indicate, in the reference case there exists not too near the boundaries of the reactor a fundamental mode where the ratio of the fast and the thermal flux does not depend on the space variable. This circumstance allows the simplification of the calculation by adopting an approximate approach56, 57. The core with its reflector is not calculated, but the core without the reflector is calculated, since the existence of the fundamental mode defines the flux extrapolated vanishing point. Moreover, because in the case of the aluminum core one can observe in the same way the existence of a fundamentalmode zone between the voided zone and the reflector (transients from the voided zone do not overlap), a hypothesis is adopted assuming that the flux extrapolated vanishing point is the same here as in the reference (non-voided) case. The extrapolated vanishing point was found in cylindrical geometry and an equivalent core lattice in geometry (without the reflector) was defined, being increased compared to the original core with its reflector. To be able to calculate the equivalent reactor core by diffusion theory, a square assembly of 21 × 21 cells has been defined containing in its center the aluminum zone in such a way that the assembly boundary lies inside the fundamental-mode zone. This means that the dimensions of the assembly are defined in such a way that the perturbation caused by the presence of aluminum vanishes inside the limits of the assembly. The calculation of the assembly has been carried out in 99 energy groups by the APOLLO-2 code using the homogeneous and the TIBERE leakage model respectively. Once transport reaction and leakage rates of the cells belonging to the assembly have been calculated and condensed in two energy groups by using the SPH transport-diffusion equivalence procedure (with the heterogeneous diffusion approach being applied on the diffusion calculation), the equivalent cross sections were obtained. The cells situated around the assembly were homogenized by simple flux weighting. The heterogeneous diffusion calculation (homogenization per each cell) of the whole equivalent core has been performed with the diffusion code CRONOS89 where the diffusion coefficients have been assumed to be equal to the leakage coefficients. Note that the zone with aluminum represents 2.13 % of the volume of the equivalent core. The comparison56, 57 of the core reactivity obtained by the diffusion code CRONOS, where in one case the leakages were calculated with the homogeneous leakage model within the APOLLO-2 code and in the other with the TIBERE model (within the APOLLO-2 code) show a decrease of reactivity being in the case of the TIBERE model. This decrease results from the increase of leakages computed by the TIBERE model compared with the homogeneous model. The homogeneous model underestimates leakages. The most interesting comparison concerns the fission rate distribution in the EPICURE core. The relative discrepancies of fission rates between the experiment and the calculations with the diffusion code CRONOS where the leakages were calculated with the homogeneous leakage model and with the TIBERE model are considered, and they are as follows. In the case where leakages are calculated with the homogeneous model an underestimation of the depression in the aluminum zone is observed being at the average ~ 3 %, while in the case of the TIBERE model a good agreement with the experiment is obtained being within the experimental error which is ~ 1.5 %. The relative discrepancies are even more interesting in the central part of the aluminum zone being 3 to 4 % and less than 0.5 % respectively. Thus, compared with the uniform leakage coefficient resulting from the homogeneous model, the TIBERE model gives in the aluminum zone bigger leakage coefficients and, hence, bigger additional equivalent absorption. Another interesting parameter is the core maximum-to-minimum fission rate ratio. This value is 2.447 for the experiment, 2.369 for

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the calculation of leakages with the homogeneous model (representing the relative error of 3.19% with respect to the experiment), and 2.435 for the TIBERE model leakage calculation model (representing the relative error of 0.49 % with respect to the experiment).

For the fast and the thermal group, Fig. 2 gives the curves of the leakage coefficient D, obtained by the homogeneous leakage model, and the radial and the axial leakage coefficients obtained by the TIBERE model, as functions of the abscissa along the half-diagonal of the assembly. These curves result from the condensation of the 99group APOLLO-2 calculations of fluxes and currents into two groups. Note that directional leakage coefficients are calculated for a chosen discrete number of regions in each cell of the assembly, as it is the case with fluxes and directional currents, and they are afterward homogenized at the scale of each cell. Because of the spectrum condensation, the leakage coefficient D obtained by the homogeneous leakage model is not absolutely uniform as it was before the group-condensation since the spectrum is space dependent. On the contrary, the directional leakage coefficients given by the TIBERE model, are strongly space dependent because of streaming in aluminum. It is obvious that the homogeneous leakage model considerably underestimates the leakage coefficients in the aluminum zone; this effect is even more important in the thermal group. Comments given earlier regarding directional currents for a PWR assembly are equally valid for directional leakage coefficients in Fig. 2. A more detailed description of comparisons between the experimental and the numerical results has been given by Petrovic56.

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DRAGON Code PWR Assembly. Since both the TIBERE and the TIBERE-2 model have been introduced into the DRAGON code72 the emphasis will be given on the comparison of these two models58, 59. Calculations of another symmetric 17 × 17 PWR assembly, where the central 7 × 7 cells were coolant-voided, and where the central cell does not contain fuel has been carried out. A 69-group cross section library was used in these calculations. For this assembly the infinite multiplication factor is The critical bucklings obtained by the homogeneous procedure and by the TIBERE-2 model are and respectively. The critical buckling obtained using the TIBERE model is After performing a two-group condensation (the limit being at 0.625 eV) and a homogenization at the scale of each cell, the comparison of directional leakage coefficients has been made. In both groups the relative discrepancy between radial leakage coefficients, obtained by the two models, is about 0.1 %. The relative discrepancy between axial leakage coefficients is, in both groups, less than 0.4 %, except in the central cell (which is the one without fuel) where the discrepancy in the fast group is 2.4 % and in the thermal 3.4 %. The increase of this discrepancy in this cell could be attributed to the following fact. As has already been indicated, when using the TIBERE model within the DRAGON code, the numerical integration of exponential functions is carried out both for the axial and the radial direction (the cyclic tracking method). When using the TIBERE-2 model, the integration is carried out analytically in the axial direction, by introducing the Bickley-Naylor functions, while the numerical integration is performed only in plane. Since the central cell is completely voided and since the relative difference between the axial and the radial leakage coefficients is at its maximum in this very cell, especially in the thermal group, the error in the numerical integration in the axial direction is also at its maximum in this cell. However, it could be said that the numerical differences are negligible. On the contrary, there is a very important difference in the computing time. For the same numerical integration precision in both calculations, the TIBERE-2 model is about ten times faster. A series of calculations regarding gradual peripheral voiding of an assembly has been performed as well. The aim of this progressive decrease in material density is to illustrate the convergence of directional leakage coefficients from the non-voided to the voided case, as well as to illustrate the fact that using the TIBERE-2 model, with its quasi-isotropic reflecting of neutrons on the boundary, one can evaluate leakages for an assembly or cell with peripheral voiding. Such a task is impossible to accomplish with the TIBERE model because of its specular reflecting boundary condition, where a neutron can escape to infinity without suffering a collision. CANDU Cell. Calculations of a 69-group CANDU reactor heavy-water-cooled cell with 37 fuel rods distributed in 4 rings (1, 6, 12 and 18 fuel rods), under normal operating conditions and in the case of coolant-voiding have been carried out58, 59. In the non-voided case The critical buckling obtained by the homogeneous procedure is and by the TIBERE-2 model In the voided case while and were obtained by the homogeneous and by the TIBERE-2 model respectively. As the homogenized and condensed leakage coefficient maps illustrate the global behavior of leakages, the detailed leakage coefficient maps over half of the cell for two characteristic energy bins (groups), both for the normal, Figs. 3 and 4, and for the coolant-voided cell, Figs. 5 and 6, are presented. These are the values obtained directly by the

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code, without any further condensation or homogenization. A bin belonging to the very fast part of the spectrum (1.35 to 2.23 MeV) and a bin at the beginning of the thermal spectrum (0.63 to 0.78 eV) have been chosen. The regions correspond, from left to right, to fuel, clad, coolant, clad, fuel, clad, coolant,...., coolant, pressure tube, void, calandria tube and several moderator regions. In fact, these are the values as they were calculated. Here, the leakage coefficients D obtained by the homogeneous model are completely uniform reproducing the fact that this model gives one leakage coefficient per group. Directional leakage coefficients and (obtained by the TIBERE-2 model) have a minimum in fuel rods, while elsewhere they increase and see a very important maximum in the void between the pressure and the calandria tube. The differences are much more important in the coolant-voided case. In the moderator, directional leakage coefficients (as well as currents) are very similar, except for the bin belonging to the very fast part of the spectrum where they are rather different, and where the radial coefficient is more important. This effect is the inverse to the one existing in the thermal group of the centrally voided PWR assembly, considered previously, where the axial leakage coefficient is more important. The thermal neutrons in a PWR assembly, are produced in the non-voided region and absorbed everywhere in fuel, resulting in an important decrease of the flux towards the assembly center. At the same time in the voided region the axial leakage coefficients (and the axial currents) are more important than the radial coefficients, since in this region a neutron can cross a longer path, in the axial than in the radial direction without, suffering a collision. For the very fast part of the spectrum in a CANDU reactor cell, fast neutrons are produced in the central part, containing fuel, and disappear by scattering everywhere but mostly in the moderator. This induces an important decrease of the fast flux towards the cell edge. Here, a neutron has a greater probability leaving the cell, without suffering a collision, by moving in the radial than in the axial direction (the radial

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current being more important), since it crosses less material in this direction. It is evident that the TIBERE-2 model yields a better representation of the distribution of the leakage coefficients than with the homogeneous model. Calculations for this cell with the TIBERE model were also performed. The relative discrepancy, with respect to the TIBERE-2 model, is even less important than in the case of the PWR centrally voided assembly, considered previously. This could be explained by the fact that it seems that the streaming effect, due to voiding, is less important in a CANDU cell than it is in a PWR assembly. However, the streaming effect exists even under normal operating conditions between the pressure and the calandria tube. In fact, in a CANDU reactor cell the density of fuel rods and its non-uniform distribution, in the pressure tube, is considerably different from the one existing in a PWR assembly. As a result, in a CANDU reactor cell, a neutron cannot cross in any direction from one edge of the coolant-voided region to the other without suffering a collision in a fuel rod, as is the case in a PWR assembly. Moreover, the computing time of the leakage calculation with the TIBERE-2 model is here about seven times shorter than with the TIBERE model. There is another indication implying that in a CANDU cell the streaming effect seems to be relatively weak. It is the fact that the relative difference between the critical bucklings obtained by the homogeneous model and by the TIBERE-2 model does not change considerably in this cell whether or not the coolant is voided. Recently, based on the Monte Carlo technique, Milgram91 has developed a method for calculating of the axial leakage coefficient using the MCNP code93. Compared to other cleverly employed Monte Carlo sampling, as those developed by Gelbard40,41,45 that depend on simple models and approximations, Milgram’s method allows both the observation and verification of the approximations inherent in the standard methodology, with no more

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approximation than are associated with the usual Monte Carlo technique. In the case of a CANDU reactor heavy-water-cooled and voided channel cell, Marleau and Milgram92 have compared the axial leakage coefficients generated by the TIBERE-2 leakage model implemented in the DRAGON code with those generated by Monte Carlo MCNP code using the Milgram’s method91. Both for the case of the non-voided and the voided fuel channel of the cell the agreement between the axial leakage coefficients obtained by the TIBERE-2 model and the Milgram’s method is very good. Such similar results have been obtained although the MCNP simulations use a finite three-dimensional cell, while the DRAGON calculations (comprising the TIBERE-2 model) are performed using a two-dimensional model for this cell.

MAGGENTA Code BWR and HTGR Latticess. It has been shown that the streaming effect, as a result of a straight propagation of neutrons, is rather important in voided or almost voided zones of an assembly, i.e., under the loss-of-coolant accident conditions in a PWR or CANDU reactor. However, even under normal operating conditions almost voided zones exist in the upper parts of BWRs, where there is steam, and in HTGRs, since the gas-coolant is very transparent for neutrons. In BWRs the streaming effect should be more pronounced in the case of the loss-of-coolant accident. Since the TIBERE-2 model has been introduced in the transport code MAGGENTA73 investigations of the streaming effect in BWR and HTGR lattices, under normal and accidental conditions, are under way and the results obtained will be compared with Milgram’s method using the MCNP code. We expect some interesting results to emerge94.

CONCLUSIONS Since a reactor core, as a whole, is still not calculated in a single computation step, the aim of theory is to couple in a reasonably approximate way the fine multigroup transport calculation of assemblies or cells and the few group diffusion calculation of the whole core composed of homogenized regions, i.e., assemblies or cells. Homogenization and condensation of cross sections is done using the multigroup flux map obtained by the transport calculation. But there remains a question of how this homogenization should be accomplished. The concept of a fundamental mode, which is the basic idea of homogeneous theory, assumes that in a reactor core there exists a large zone, not near the core boundary, where the spatial variation of flux is the same for all neutron energies. Although idealized and not accounting for transients in the vicinity of the core boundary, this idea that corresponds to a homogeneous lattice is extended to a heterogeneous core lattice. This is done by factorizing the physical flux in a homogenized medium as a product of an exponential term corresponding to the fundamental mode and a function depending on solid angle and energy. The equivalent homogenized medium is obtained by flux weighting of cross sections of a perfectly reflected assembly or cell, i.e., of an infinite lattice of identical assemblies or cells. Choosing a perfectly reflected (mirror effect) assembly or cell one removes the effect of boundary and the solution is the deep-interior spatially asymptotic one. Thus the factorization described enables to introduce the influence of a core as a whole, in an approximate way via the global flux shape in a reactor core, on the microscopic flux in a homogenized assembly or cell. As a result one obtains the leakage cross section of a homogenized medium, i.e., an

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additional absorption cross section, that is defined as the product of buckling and the leakage coefficient. While flux and reaction rates are calculated for a heterogeneous assembly or cell, the leakage coefficient is calculated for a flux-weighted homogenized one. This is a rough representation of actual leakages of a heterogeneous assembly or cell and leads to an underestimation of leakages. Moreover, this imperfection becomes important when a reactor itself or certain reactor assemblies contain voided zones or zones with low density media, i.e., media with large mean free path. Resulting from a large mean path a straight propagation of neutrons occurs, known as the streaming effect, and which alters reactivity. These voided zones existing under normal operating conditions in old-type gas-graphite reactors (gas can be in practice approximated by void) and in more modern HTGRs, may also appear in PWRs, BWRs, sodium-cooled and CANDU reactors as a consequence of a loss-of-coolant accident. Even in normal operating in the upper part of BWRs there exists steam that has a low density. It is the same with sodium-cooled reactors because sodium is very transparent to neutrons due to their long mean path in this medium. Leakage calculations are very important in a loss-of-coolant accident particularly for cores containing plutonium. In these cores there can exist a competition between a positive reactivity effect induced by the decrease of absorption and a negative reactivity effect which is the consequence of leakages. Still this method or more precisely homogeneous model is currently most often used for leakage calculations. The influence of assembly or cell heterogeneity on the neutron leakage was first studied and taken into account in the calculations of the old-type gas-graphite reactor lattices. This study was based on heterogeneous theory that enables one to superimpose the macroscopic character of the flux, which is at the scale of the core, on to the microscopic flux, which is periodic and reflects fine variations in a perfectly reflected assembly or cell. In fact, the superposition is achieved by factorizing the physical flux in the product of the macroscopic and the microscopic flux. Further the model used in this study was limited to the first-order expansion of the solution of heterogeneous theory, which is also called the small buckling approximation. This model made it possible to define buckling-independent cell-homogenized directional leakage coefficients using directional currents. It has been shown that directional probabilities are a well suited numerical method for calculation of these directional currents. Also, the effects of the first-order scattering anisotropy on the neutron leakage have been investigated. Extensive researches have been dedicated to defining anisotropic leakage coefficients based on small buckling approximation, or on other approximations, as well as means how to obtain homogenized parameters for the whole-core calculation. Also asymptotic approaches have been proposed which can be interpreted, under certain conditions, as equivalent to the small buckling expansion. It should be noted that heterogeneous theory has been studied, without any approximation, for low density ducts as well as for a slab lattice. In addition, heterogeneous theory has been investigated for a slab lattice. However, for complex geometries heterogeneous or theories would require very time consuming calculations. The aim of heterogeneous theory, i.e., in practice the model, is to provide a critical transport calculation for the situation of an infinite lattice of identical assemblies or cells. This idealized situation may not correspond to the situation of an actual reactor but is used to determine homogenized parameters to be used in the reactor calculation. This assembly-by-assembly or cell-by-cell modeling greatly simplifies the calculation, compared with very time-consuming (or even impossible) direct deterministic or stochastic simulation of the whole reactor. However, this modeling is possible only when homogenized parameters do not depend strongly on the environment of a considered assembly or cell. That is the case in normal operating situation but is not the case when a reactor contains voids or low-density

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ducts extending over a long distance. When such voids are present, a direct long-range interaction between two assemblies or cells situated far from another can exist because of the straight-line propagation of neutrons. This interaction makes the assembly-by-assembly or cell-by-cell modeling more difficult. In particular, if heterogeneous formalism is used without approximation in an infinite lattice of identical assemblies or cells containing two-dimensional infinite void ducts, it leads to finite leakage coefficients, but with a weak convergence (i.e., due to the existence of positive and negative phases of the macroscopic flux) which is somewhat artificial. For this reason, the heterogeneous model strictly speaking, is maybe not the most suitable choice for a reference situation for assembly or cell modeling when infinite voids are present. It can easily be seen that the difficulties come from the specular reflecting boundary condition (mirror effect), which is equivalent to an infinite lattice of identical assemblies or cells. But the choice of such a boundary condition, as a reference situation, is not at all an obligation. Another possible simple choice is, for instance, a generalization of a white boundary condition. This boundary condition is of course approximate but not more than the specular one in actual core situation. Let us consider, for example, the case of a water-voided pressurized water reactor assembly surrounded with non-voided assemblies. The white boundary condition is in this case certainly much closer to reality than the specular one. In the case where the considered assembly is not voided, both boundary conditions are reasonably justified. Thus, the calculation of an assembly with a white boundary condition can be taken as a reference for the evaluation of homogenized parameters of this assembly (at the scale of each cell or the whole assembly). Not that this boundary condition avoids the artificial difficulties mentioned earlier that are induced by the specular reflecting boundary condition; in particular, the treatment of a completely voided assembly becomes possible. Moreover, with the generalized white boundary condition it is possible to treat the case where the considered voided assembly is surrounded by any number of voided assemblies; no divergence will appear, but a certain underestimation of leakages will arise because the long straight paths will be cut at the edge of each assembly and the correlation between the paths in different assemblies will be neglected. To overcome this drawback, a solution could be to treat not a single assembly but a set of voided assemblies (superassembly), with a white boundary condition at its border. But this solution would be rather time consuming. Another option could be to treat only one voided assembly but to introduce a correction based on an approximate extension of neutron paths. To consider the streaming anisotropy and heterogeneity within a fine flux transport calculation in a consistent manner and to use numerical solution methods already existing in available neutron transport codes, without increasing considerably computing time compared to ordinary calculations of multigroup flux maps, two simplified heterogeneous models, named TIBERE and TIBERE-2, have been derived. The two models, based on a few common approximations assuming that the considered lattice is regular and symmetric, yield an iterative scheme to simultaneously compute coupled multigroup scalar fluxes and directional currents. Thus a definition of the directional space-dependent leakage coefficient emerges and one can define directional space-dependent leakage cross sections having the same type of space-dependence as all other cross sections. Therefore, perfectly consistent reaction and leakage rates are obtained, and can be used for evaluation of homogenized parameters required for a whole-core calculation. The main difference between the TIBERE and the TIBERE-2 model concerns boundary conditions at the limit of a heterogeneous, symmetric and reflected assembly or cell. The TIBERE model assumes that a neutron arriving at the boundary is reflected in a specular way (mirror effect), that is equivalent to the infinite lattice of identical assemblies or cells. Such an assembly or cell represents a reference situation

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for homogenization. This model requires evaluation of the classical and directional collision probabilities. The TIBERE-2 model assumes that a neutron arriving at the boundary is reflected in a quasi-isotropic manner (generalized white boundary condition). This represents another reference situation for homogenization. Apart from the classical and directional collision probabilities, but now calculated for an open assembly or cell, the TIBERE-2 model requires the evaluation of the classical and directional escape and transmission probabilities. While the TIBERE model can treat only one-dimensional infinite void channels, since for two-dimensional ones this model gives infinite leakage coefficients mentioned earlier, the TIBERE-2 model, due to its boundary condition, can treat also two-dimensional infinite void channels without having any problems with the divergence. Moreover, the calculations with the TIBERE-2 model are much less time consuming than those with the TIBERE model since the neutron paths in the classical and directional first-flight collision probabilities are now limited to just one assembly or cell. Although the models TIBERE and TIBERE-2 give perfectly consistent reaction and leakage rates it has been noticed that, as in the case of homogeneous model, the simple flux weighting of cross section does not satisfy the equivalence of reaction rates between the reference calculation and the macro-calculation (diffusion calculation accomplished for a reflected assembly composed of homogenized macro-regions and condensed in macrogroups). However, using an equivalence method to obtain homogenized parameters for the whole-core diffusion calculation one can satisfy the equality of reaction and leakage rates obtained by the reference and the macro-calculation. As it has already been noticed the aim of theory is to link in an approximate manner the fine multigroup transport calculation of assemblies or cells and the few group diffusion calculation of the whole core composed of homogenized regions. Therefore, to obtain homogenized parameters for a whole-core calculation, the TIBERE and the TIBERE-2 model are unseparable from homogenization by an equivalence method. Another approach for linking the fine transport and the global diffusion calculation is the method of discontinuity factors. The simplified heterogeneous models TIBERE and TIBERE-2 have been introduced in a few transport codes. Numerical comparisons between the models TIBERE and TIBERE2 with the homogeneous model indicate that the simplified heterogeneous models give a much better representation of leakage when there exist voided or almost voided zones in a PWR assembly and in a CANDU reactor cell. A comparison of fission rates obtained using the TIBERE model for leakage calculation with those obtained by an experiment regarding voiding effect show that the calculation using the TIBERE model is within the experimental error which is not the case when leakages are calculated by the homogeneous model. Moreover, the calculations with the TIBERE-2 model are much faster than those with the TIBERE model without almost any effect on the accuracy of computation. Investigations of streaming effect in BWR and HTGR lattices are under way. Solution methods for neutron transport have evolved from the earliest days of interest in nuclear reactors. There was a close link between the development of improved numerical techniques for the transport equation solution and the evolution of computers. Rather accurate transport calculations are currently available. Hence, it appears that some of earlier and elegant techniques for neutron leakage evaluation and corresponding homogenization approaches now have a small role for practical calculations. With the fast progress of computers probably in future a fine calculation of a whole reactor core would be possible in one step, although time-consuming, and might loose its significance. However, for practical and production calculations the computation of reactors in two steps will probably remain for a longer time and thus the need for theory.

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IVAN PETROVIC AND PIERRE BENOIST K. Koebke, A new approach to homogenization and group condensation, IAEA Proc. Spec. Mtg. Homog. Meth. React. Phys., Lugano, Switzerland, November 13-15, 1978, 303:322, TECDOC-231, International Atomic Energy Agency (1980) A. F. Henry, B. A. Worley and A. A. Morshed, Spatial homogenization of diffusion theory parameters, IAEA Proc. Spec. Mtg. Homog. Meth. React. Phys., Lugano, Switzerland, November 13-15, 1978, 275:302, TECDOC-231, International Atomic Energy Agency (1980) D. V. Altiparmakov, On the boundary conditions in cylindrical cell approximation, IAEA Proc. Spec. Mtg. Homog. Meth. React. Phys., Lugano, Switzerland, November 13-15, 1978, 27:42, TECDOC-231, International Atomic Energy Agency (1980) K. S. Smith, Assembly homogenization techniques for light water reactor analysis, Prog. Nucl. Ener., 17, 303:335(1986) A. Hébert, Développement de la methode SPH: Homogénéisation de cellules dans un réseau non uniform et calcul des paramètres de réflecteur, Note CEA-N-2209, Commissariat à l’Energie Atomique, Saclay (1981) A. Hébert and P. Benoist, A consistent technique for the global homogenization of a pressurized water reactor assembly, Nuc. Sci. Eng., 109, 360:372 (1991) A. Hébert and G. Mathonnière, Development of a third-generation Superhomogénéisation method for the homogenization of a pressurized water reactor assembly, Nuc. Sci. Eng., 115, 129:141 (1993) K. Koebke, Advances in homogenization and dehomogenization, ANS/ENS Proc. Int Topic. Mtg. Advanc. Math. Meth. Solut. Nucl. Engin. Prob., Munich, Germany, April 27-29, 1981, Vol. 2, 59:73, Kernforschungszentrum Karlsruhe (1981) E. Tanker and A. F. Henry, Finite difference group-diffusion theory parameters that reproduce reference results, Trans. Am. Nuc. Soc., 50, 280 (1985) J. J. Lautard, S. Loubiere and C. Fedon-Magnaud, Three dimensional pin by pin core diffusion calculation, Proc. ANS/ENS Int. Top. Mtg. Advanc. Math. Comput. React. Phys., Pittsburgh, Pennsylvania, U.S.A., April 28-May 2, 1991, Vol. 2, 6.11-1:6.11-10, American Nuclear Society (1991) J. Mondot, J. C. Gauthier, P. Chaucheprat, J. P. Chauvin, C. Garzenne, J. C. Lefebvre and A. Vallée, EPICURE: An experimental program devoted to the validation of the calculational schemes for plutonium recycling in PWRs, Proc. ENS Int. Conf. Phys. React. Operation, Design and Computation (PHYSOR’90), Marseilles, France, April 23-26, 1990, Vol. 1, VI-53:VI-64, European Nuclear Society (1990) M. S. Milgram, Estimation of axial diffusion processes by analog Monte Carlo: theory, tests and examples, Ann. Nucl. Ener., to be published in summer or autumn 1996 G. Marleau and M. S. Milgram, A DRAGON-MCNP comparison of axial diffusion subcoefficients, Trans. Am. Nuc. Soc.,72, 163:164 (1995) J. Briesmeister, Ed., MCNP - A general Monte Carlo code for neutron and photon transport, Version 4, Report LA-7396-M, Rev. 2, Los Alamos National Laboratory (1991) I. Petrovic, J. Vujic and P. Benoist, Validation of the heterogeneous leakage models TIBERE and TIBERE-2 on BWR and HTGR lattices, being submitted to Nuc. Sci. Eng.

CURRENT STATUS OF CORE DEGRADATION AND MELT PROGRESSION IN SEVERE LWR ACCIDENTS Robert W. Wright Consultant 14400 Homecrest Road, #220 Silver Spring, MD 20906 (301) 598-1406

INTRODUCTION This paper is a summary of the current state of technical understanding and the principal remaining technical uncertainties regarding the processes of core degradation and core melt progression in severe Light Water Reactor (LWR) accidents. It is to be emphasized initially that technical resolution of all these uncertainties is not required for making safety assessments of different issues in the regulatory process, nor is such resolution probably feasible. Different regulatory issues require knowledge of different aspects of meltprogression technology and at different levels of accuracy. Regulatory considerations will not be further addressed in this paper, which treats the technology of core degradation and core melt progression. In-vessel core degradation and core melt progression describe the state of the reactor core in severe LWR accidents from core uncovery to reactor vessel failure, or to temperature and geometry stabilization in accidents which are recovered by core reflooding. Melt progression provides the characteristics of the melt released from the damaged core and possibly later from the reactor vessel. These characteristics are the melt mass, the melt composition (in particular the metal fraction), the melt temperature, and the rate of melt release. They provide the initial conditions for assessing the loads on the reactor vessel and possibly later on the containment in severe LWR accidents. The uncertainties in these initial conditions often provide the largest of the uncertainties in assessing the integrity of the vessel lower head and the integrity of the containment. Melt progression also provides the in-vessel hydrogen generation, the core conditions that determine the in-vessel fission-product release, transport, deposition, and revaporization, and also the core conditions for assessing the consequences of accident-recovery actions, core reflooding and vessel depressurization in particular. The sequences of core degradation and melt progression in severe LWR accidents consist of a number of distinct stages. These are: Core degradation (initial intact rod geometry); Early-phase melt progression (metallic melts);

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Late-phase melt progression (ceramic melts and solid ceramic debris relocation); Lower plenum melt-water interactions, both non-explosive and explosive, and melt and debris cooling; Lower head failure, threshold and mode of failure; The consequences of reflooding a severely degraded core. At the same point in time, different regions of the core will generally be in different stages of melt progression. The initial core degradation takes place in essentially the initial intact rodded core geometry. This stage, at local temperatures less than about 1500 K, involves: coolant boil down and the heat up of the uncovered part of the core by fission-product decay, clad ballooning and rupture (fuel failure), and the failure of Ag-In-Cd Pressurized Water Reactor (PWR) control rods and Boiling Water Reactor (BWR) control blades (both stainlesssteel clad).1,2 This stage also includes the beginning of the transient rapid oxidation of the core Zircaloy by the boil-off steam, with accompanying local heating and hydrogen generation. The early phase of melt progression includes most of the rapid transient heating of the uncovered part of the core from Zircaloy oxidation by the steam and at local temperatures between about 1500 K and 2800 K. This heating melts the upper portion of the remaining unoxidized Zircaloy cladding and the control-rod materials, with downward relocation of the melt.1,2 This relocated metallic melt freezes to form a blocked core in accidents like the one at Three Mile Island, Unit 2 (TMI-2).3,4 The late phase of melt progression involves ceramic melts at local temperatures above about 2800 K, the sintering behavior and relocation of solid, ceramic core debris, and, in blocked-core accidents, the formation, growth, and later melt-through from the core of an essentially ceramic melt pool.4 A major question here is the mass of the ceramic melt that is released from the core on melt-through, and this is determined by the threshold and the location of pool melt-through from the pool-containing ceramic crust.4 When either ceramic melts or metallic melts fall into the lower plenum water, the interaction between the melt and the water becomes significant. There is a relatively slow breakup of the melt with non-explosive steam generation (and hydrogen generation with metallic melts) along with melt cooling.5 Under some conditions, an explosive thermal interaction may also occur with the generation of a high pressure shock.5 This is commonly known as a steam explosion. The threshold of failure of the reactor-vessel lower head determines whether or not the reactor vessel fails for given melt conditions and vessel pressure. Such failure is of major significance. The threshold and the mode of failure under melt attack determine the mass and the rate of release of the melt into the containment. Such melt attack on the internallyflooded vessel lower head did occur in the TMI-2 accident, but the lower head did not fail.4 The consequences of reflooding a severely degraded core at different stages of core degradation are also of LWR safety significance. Of primary concern is whether or not, at a given stage of core damage, reflooding will promptly terminate the accident, as it did not do at TMI-2.4 The large amount of generated hydrogen (with local core heating) from Zircaloy oxidation by the reflood steam, the amount of reflood steam, and the possibility of a large steam explosion are also of reactor safety interest.

LESSONS FROM THE TMI-2 ACCIDENT Much of the current information base on the severe accident behavior of LWRs, particularly PWRs, has come from the accident at the Three Mile Island Unit 2 reactor and

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from the TMI-2 post-accident core examination.3,6 This holds particularly true for the late phase of melt progression (ceramic melts) in blocked-core LWR accidents and for the governing phenomena involved in potential lower head failure under melt attack, and it is also true for the more general system behavior in severe accidents in PWRs. The TMI-2 core was reflooded in the course of the accident, but the general melt-progression phenomenology in the TMI-2 accident appears also to be applicable to unreflooded blocked-core accidents.4 This is because most of the sequence of core degradation and melt progression at TMI-2, including the initial formation of the ceramic melt pool in the core, occurred before the core was reflooded by high-pressure-injection-system water.4 The TMI-2 results are quite consistent with the results of the earlier Severe Fuel Damage tests in the Power Burst Facility (PBF) test reactor and experiments in the CORA fuel-damage test facility.1,2 About 25 minutes prior to core reflooding in the TMI-2 Accident, there was a brief injection of water into the hot, degraded core by a flow transient which was terminated after about 15 seconds by pump cavitation.3 This water injection appears to have fragmented the hot, upper declad portion of the core, from which most of the molten metal had previously relocated downward. 1,4 The bundle reflooding in the CORA tests, in the PBF Scoping Test, and in the FP-2 test in the Loss Of Fluid Test (LOFT) facility produced strong local heating in the core and much hydrogen from Zircaloy oxidation by the reflood steam.1,2,7 It appears that similar extensive oxidation also occurred during the brief flow transient in the TMI-2 accident that was terminated by pump cavitation and, to some extent, later during core reflooding by the high-pressure-injection system.4 The core-degradation and melt-progression phenomena that occurred during the TMI-2 accident are as follows:4 coolant boil down and core uncovery; Ag-In-Cd control-rod internal melting and stainless-steel control-rod-cladding failure; clad ballooning and fuel-rod failure; the rapid transient oxidation of the Zircaloy clad by boil-off steam with resultant local core heating, Zircaloy melting, and hydrogen generation; metallic melt relocation by gravity and refreezing to form a metallic core blockage or crust; the formation of a particulate, mostly ceramic debris bed in the remaining free-standing array of fuel pellets and shards, possibly by the water from the short flow transient; growth of a ceramic melt pool in the debris bed; reflooding of the hot, highly degraded core; ceramic pool melt-through (under water and out the side of the core); melt drainage through the lower plenum water onto the vessel lower head, with non-explosive melt-water interactions and transient melt cooling; and the combined thermal and pressure attack of the melt and solid debris on the integrity of the water-flooded vessel lower head. The early development of metallic core blockages similar to the one in the TMI-2 accident occurred in the PBF, CORA, ACRR, and LOFT FP-2 tests, and in other severe LWR accident tests.4 From the TMI-2 post-accident core examination and thermal analysis, it was estimated that about 50 percent of the TMI-2 core mass was in the ceramic melt pool at the time of pool melt-through.6,8 Because the melt through from the core was from the side and not the bottom, the melt pool drained only partially, and only about 20 percent of the core mass, 20 tonne (te), drained into the vessel lower plenum.4 Readings from in-core selfpowered neutron detectors indicate that drainage of the ceramic melt into the lower plenum took about 100 seconds.4 The measured metal content of the in-core ceramic melt pool was less than about one percent by mass.6 The underwater release into the lower plenum water at 12 MPa vessel ambient pressure of about 20 te of ceramic melt at temperatures above 2800 K did not produce a steam explosion. The breakup and cooling of the ceramic melt stream did produce a quasi-static "steam spike" of about 3 MPa magnitude and 15 minutes duration.9 The total hydrogen generated in the TMI-2 accident corresponds to the oxidation of 45 percent of the core Zircaloy.4 The vessel lower head did not fail under the combined

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thermal load of 20 te of ceramic melt under water and a pressure load on the vessel lower head that varied from 10 MPa to 14 MPa 9 Core reflooding 26 minutes before pool melt-through from the core obviously did not stop the continuing growth of the ceramic melt pool in the core and the later melt-through from the core.4 Thus, reflooding did not promptly terminate the TMI-2 accident. The water not only did not penetrate into the melt pool, but it did not penetrate far enough into the hotter regions of the core debris to stop the continuing growth of the melt pool. Reflooding did, however, terminate the downward migration in the core of the melting and reforming metallic crust (blockage) that had occurred during coolant boil down.4 This termination may have caused the melt-through of the ceramic melt pool out the side of the core (rather than downward) with the resultant partial drainage of the melt pool onto the vessel lower head. At the time of the accident, the TMI-2 operators did not realize that they were dealing with a small break loss-of-coolant accident (LOCA).10 The accident was initiated by loss of feed water to the steam generators. After the reactor was scrammed and the steam generators had boiled dry, the continuing fission-product-decay heating of the core water caused the Power Operated Relief Valve (PORV) on the pressurizer to open on overpressure. The PORV stuck open causing vessel depressurization, coolant boil off, and core uncovery, and the operators had also mistakenly turned off the high-pressure-waterinjection system. After the accident proper, fear of an in-vessel explosion occurred when it was realized from measurements in the containment that the large gas volume existing in the vessel (called a "bubble") must have contained hydrogen.10 It was incorrectly feared that the "mysterious" hydrogen came from radiolytic decomposition of the vessel water. This would have created a stochiometric hydrogen-oxygen mixture in the vessel that is explosive. The actual source of the hydrogen was steam oxidation of the Zircaloy fuel cladding, and this does not generate oxygen. It also was not realized, however, that earlier in the accident, about 20 te of ceramic melt at about 3000 K had relocated (under water) onto the vessel lower head, and that the margin to failure of the lower head under the combined thermal and pressure loads could not have been large.9 A drawing is shown in Figure 1 of the end-state configuration of the TMI-2 core and reactor vessel that is based on the results of the post-accident TMI-2 core examination.6 This drawing illustrates the melt-progression phenomena discussed in this paper. At the top of the vessel are the upper grid plate and the light stainless-steel upper-plenum structure, which are only slightly damaged. At the top of the core is the cavity produced by the debris relocation (slumping) that followed the densification of the rubblized upper core debris upon melting, and also by the partial drainage of the ceramic melt pool from the core into the lower plenum. (The void fractions of the intact rodded core and the rubblized core debris are both about 50%, so there is no gross compaction and slumping in the fragmentation process.) In the center of the core are the solidified remnants of the ceramic melt pool and its surrounding crust following the partial drainage of the melt pool into the lower plenum. The drawing does not differentiate between the ceramic crust that surrounded and contained the ceramic melt pool and the metallic crust below the pool that originally blocked the core. (There was always cooling water below the metallic crust during the accident. The minimum water level before core reflooding was at about 30% of the core height.) The TMI-2 post-accident core examination and the results of the MP-2 experiment in the ACRR test reactor indicate that the expanding ceramic crust around the growing ceramic melt pool in the debris bed had grown into the original metallic core blockage (crust) which had been axially stabilized by core reflooding.6,11 At the bottom of the core are the undamaged sections of the fuel assemblies, most of which were always under water. At the bottom of the Figure are the nearly undamaged core support plate and lower plenum structure and the frozen relocated ceramic melt on the vessel lower head. Also shown in Figure 1 is the melt-drainage pathway from high at the side of the melt pool and down the side of the core.

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CORE DEGRADATION The initial core degradation in LWR core-melt accidents starts upon core uncovery and continues during coolant boil down under fission-product decay heating. This initial core degradation phase in intact rod geometry extends up to peak local temperatures of about 1500 K. The significant phenomena involved are: clad ballooning and rupture (fuel failure); the initial liquifaction and later failure of the Ag-In-Cd PWR control rods and the BWR control blades, both of which are stainless clad; and the beginning of significant transient oxidation heating of the clad Zircaloy by the boil-off steam with accompanying hydrogen generation.

Most Significant Results Clad ballooning and rupture occur at relatively low temperature, about 1150 K at low pressure.12,13 At high pressure, rupture occurs without clad ballooning and at somewhat higher temperature. The fission-product inventory in the cladding gap is released upon cladding rupture. Significant oxidation of the Zircaloy cladding by boil-off steam starts at about 1300 K and produces a rapid local core-heating transient.12,13 Validated rate laws with a strong Arrenhius temperature dependence have been obtained for steam oxidation of the Zircaloy cladding with accompanying hydrogen generation.12,13 The parabolic reaction rate is diffusion limited by the thickening surface oxide film on the cladding.12,13 Melting of the Ag-In-Cd control-rod alloy occurs at about 1100 K, but failure of PWR control rods and BWR control blades, both stainless clad, does not occur until about 1500K. 12,13 Current information on the core-degradation processes during core uncovery is summarized in the CSNI State of the Art Report "In-Vessel Core Degradation in LWR Severe Accidents."13 This information has also been incorporated into the current severe accident codes SCDAP/RELAP5,14 ICARE2,15 ATHLET/CD,16 MELCOR,17 and others. A comparison of the modeling in these different codes and an assessment with current data are given by Brockmeier.18

Principal Remaining Uncertainties For LWR core degradation in the initial intact rod geometry, the experimental data base and the models are generally adequate and the remaining uncertainties are generally of low reactor-safety significance. There are still uncertainties, however, on the effects of grid spacers on fuel-failure thresholds. There are uncertainties regarding the governing mechanisms in two-sided (outside and inside) oxidation of the Zircaloy cladding following ballooning and cladding failure, and on the rate of such two-sided oxidation with hydrogen generation. There are also uncertainties regarding the effects of irradiated fuel on fuel failure and core degradation.

EARLY-PHASE MELT PROGRESSION The early phase of melt progression includes most of the rapid transient oxidation of the Zircaloy cladding in the uncovered portion of the core by the boil-off steam with rapid local heating and hydrogen generation as local temperatures increase above about 1500 K. This oxidation, the rate of which increases rapidly with increasing temperature, produces a rapid local high temperature transient at rates in the 10 K/s range and local melting of the cladding Zircaloy as temperatures reach about 2200 K.1,2 There is significant partial dissolution of the

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fuel pellets in the molten Zircaloy.1,2,19 When the shell fails by partial dissolution in the molten Zircaloy, the Zircaloy melt is released and relocates downward by gravity and may freeze. This starts the buildup of a metallic blockage in blocked-core accidents like TMI-2. Control materials precede the clad Zircaloy in the relocation process.2 The melting temperatures of the reactor core materials and the liquifaction temperatures of interacting materials (eutectics) are of the greatest importance in melt progression. Figure 2, which is based on the work of Hofmann et al., gives the significant melting and liquifaction temperatures.19

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Most Significant Results The downward relocation by gravity of the metallic melt separates the melt (with some dissolved ) from the ceramic fuel-rod pellets that remain in place in the core.20 This physical separation of the molten metallic core material from the solid ceramic material has major permanent consequences in severe accident sequences.20 The melts released from the core and later, after possible failure, from the reactor vessel are almost purely ceramic (as at TMI-2),6 or almost purely metallic (as hypothesized for US BWR accidents with depressurization but without core reflooding),21 but not mixtures of metal and ceramic of a composition similar to the average initial (or partially oxidized) core composition. The melt composition is incorrectly assumed to be the initial (or partially oxidized) core composition in some severe accident analyses and codes. It was found in the DF-4 BWR test in the Annular Core Research Reactor (ACRR) and also in tests with BWR fuel assemblies in the CORA fuel-damage test facility that the melt of control-blade materials ( and stainless steel) at about 1500 K rapidly melts through the Zircaloy channel boxes that contain the fuel rods by alloying with and liquifying the Zircaloy.2,22 This opens up the originally channeled BWR core to cross flow of gas and also of the melt alloy.22 Metallic melt relocation is by a mixture of rivulet flow and droplet flow, not the uniform film flow down the rods that is sometimes assumed in severe accident models and codes.2 The in-core grid spacers have a major effect on metallic melt relocation as well as on the original cladding failure.2 The grid spacers also form preferred sites for the formation of metallic blockages. During the boil-down process in the core, the partial metallic blockages that have formed are melted-out by incoming metallic melt and reform below in a repetitive process.2

Principal Remaining Uncertainties There is a major question as to whether or not all risk-significant severe accidents follow a blocked-core pathway as at TMI-2, or whether there are some sequences in which the metallic melt drains from the core (and BWR core-plate structure) as it forms. Which pathway is followed has a major effect on the characteristics of the melt released later from the reactor vessel and therefore on the accident consequences. The melt mass and composition (metal fraction) are of particular importance, but the melt temperature is also significant. The metallic melt-drainage or core-blockage branch point is illustrated in Figure 3, which shows the accident sequence up to the branch point and the sequences for each of the two alternative pathways beyond the branch point. The left side of the figure shows the pathway for a TMI-2-like blocked-core PWR accident. The right side shows the pathway for metallic melt drainage which has been assumed to occur in a dry-core BWR accident. In this latter accident in US BWRs, manual depressurization without reflooding lowers the water level by flashing to below the core plate, and the core heats up slowly in a low steam-flow, "dry-core" environment with a low oxidation rate. There may also be dry-core PWR accident sequences in which this question is significant, in particular the pump-seal LOCA with loop-seal clearing. The vertical lines at each side of the figure illustrate qualitatively the rate of oxidation and hydrogen generation, with a dotted line indicating a low rate. The dry-core BWR metallic melt-drainage sequence shown is that indicated by the analysis of Hodge, et al., at Oak Ridge National Laboratory (ORNL).21 In this drainage sequence, resolidified particulate metallic melts and later ceramic melts accumulate in layers in the lower plenum water in order of the time of melting and melt drainage or of solid debris

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slumping. The bottom layers are metallic (with some dissolved ) and the top layers are fuel-containing ceramic. There may also be some relocated, mostly ceramic solid debris in the top layer. Following dry out of the lower plenum, re-melting will occur in layers depending on the debris liquifaction temperatures, the specific power (decay heat) in each layer, and the heat transfer between layers. On vessel failure, the released melt will be mostly if not entirely metallic and will have a much lower temperature than the ceramic melt released onto the TMI-2 lower head. There is a major question regarding whether the metallic meltdrainage pathway can occur in any risk-significant severe accident sequences, even under drycore BWR accident conditions. TMI-2 and wet-core fuel-degradation tests (PBF, CORA, ACRR, LOFT FP-2) developed incipient or complete metallic blockages. CORA-33, the only dry-core BWR fuel-degradation test that has been performed, provided information on core degradation and metallic melt behavior under dry-core conditions, but the necessary prototypic conditions could not be maintained to the time of incipient blockage formation.23

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A complementary ex-reactor experiment, XR2-1, has recently been performed at Sandia National Laboratories (SNL) that provides the necessary highly prototypic conditions at the time of potential incipient blockage in dry-core BWR accidents, including the appropriate pre-oxidation of the Zircaloy surfaces.24 The results of this experiment and possible future experiments should provide definitive results on the question of metallic melt drainage or core blockage under these conditions. The XR2-1 experiment had a full scale radial section of the lower quarter of a BWR core and core-plate structure, with prototypic components and a prototypic axial temperature distribution. Real time x-ray imaging of melt behavior and blockage development was provided. In an accident, any blockage would occur in the lower quarter of a BWR core where the axial temperature distribution decreases strongly downward, or below the core in the core-plate structure. In the XR2-1 experiment, metallic melt of prototypic control-blade and Zircaloy composition, temperature, and pour rate (dribble), was poured into the top of the test assembly. This melt represented the metallic melt that relocates downward from the upper three-quarters of the core in a BWR dry-core accident. Internal heating of the metallic melt was neither relevant nor needed. The initial conditions for the experiment were obtained from analysis at ORNL of the unrecovered station blackout accident in a BWR with manual vessel depressurization without reflooding and from the CORA-33 results.23 The relocation pathways in the lower fuel canisters and the structure in the region of the core plate are very complex. In addition, materials interactions between the Zircaloy and the stainless-steel control-blade materials (and also the fuel) change the liquifaction and the solidification temperatures and provide significant chemical heats of reaction. These effects can produce large changes in the geometry as well as in the solidification process itself. Thus metallic melt drainage or core blockage is not a simple thermal-hydraulic melt-freezing problem, and an experiment that includes both the complex geometry and the materials interactions is required for its resolution.

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The test-assembly cross section in the core region for the XR2-1 experiment is shown in Figure 4.24 The test assembly was a full scale radial section of a BWR core with core-plateregion components below. The cross section included fueled regions (Zircaloy-clad fuel rods) and the channel-box walls with the gap between the walls. The gap contained both a control-bladed section and an unbladed section. A schematic drawing is shown in Figure 5 of a BWR core plate region, which is an important region for metallic blockage development. As shown in the figure, the terminology "core plate" is somewhat misleading in that two-thirds of the "plate" is open structure. There are relatively open flow paths for both the "eutectic" melts of control-blade and channel-box Zircaloy, at about 1500 K, and the later Zircaloy melts from the cladding and the remaining channel box Zircaloy at about 2200 K. Earlier simplified XR1 developmental tests showed a continuing process of freezing and subsequent melt-out of the temporary blockages that were formed in the gap, with the melt refreezing below.24 Later the channel-box walls were liquified by the control-blade melt, which leads to melt spreading among the fuel rods in actual BWR accidents. The XR2-1 experiment, with the high degree of prototypicality required for the results on the metallic melt-drainage or core-blockage question to be meaningful, proved to be very difficult to develop and to perform. There were major problems in developing the necessary low-feed-rate melt-delivery system for the required prototypic materials and high temperatures. XR2-1 was recently performed successfully. The preliminary results indicate that about half of the metallic melt penetrated the core and core-plate regions in the experiment into a "lower plenum" volume below the core plate.25 This result would be in agreement with the melt drainage pathway in Figure 3.

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A porous-media model of Melt Relocation In Reactor Structure, MERIS, has been developed that treats the relocation of metallic melt, starting in the initial, intact rodded core geometry, and proceeds to melt freezing and potential blockage formation.45 MERIS uses an anisotropic permeability to account for the initial rod geometry of the core and does not attempt to describe the degrading core geometry in detail. MERIS treats melting, freezing, and melt relocation in the anisotropic porous media and the more important of the materials interactions, including the changes in both the liquifaction temperatures and the heats of reaction. MERIS was developed from the somewhat similar DEBRIS porous-media model of late-phase melt progression that treats the behavior of ceramic melt pools and crusts in particulate ceramic debris beds.44 MERIS has shown reasonable agreement with the results of the simplified XR1 developmental tests that addressed the behavior of control-blade melts in the gap between BWR channel-box walls.39 Another principal remaining uncertainty in early-phase melt progression is the threshold for molten metallic Zircaloy relocation following failure of the shells that are formed during cladding oxidation.26 This threshold is sometimes mistakenly called the "fuel-failure threshold". At this threshold, the rapid local oxidation temperature transient is essentially terminated by the removal of the Zircaloy "chemical fuel", and this has a major effect on the subsequent temperatures within the core. It has been found that this is not a simple temperature threshold.2 Some severe accident codes currently use a two-parameter representation of this threshold. Molten Zircaloy release through the oxide cladding shells is assumed to occur for a temperature Tzirc>Tth if the oxide-shell thickness (before dissolution thinning by molten Zircaloy) is less than a critical value. Currently SCDAP/RELAP5 uses 2400 K for Tth and 60% of the initial clad thickness for the critical thickness, although indications are that a higher temperature would better fit the data.14,26 Mechanistic modeling of the failure process is currently underway.26 The governing mechanisms in metallic blockage formation under the flow conditions that exist in severe LWR accidents are uncertain because of the effects of materials interactions and, in BWRs, of complex geometries. Not only does chemical alloying (eutectic formation) often reduce the liquifaction temperatures of the metallic melts, but the chemical heats of alloying usually exceed the latent heat of freezing of the pure metals. Experiments with socalled "simulant materials" may give very misleading results regarding phenomena in reactor safety in which materials interactions are important. Materials interactions and rate limitations are particularly important to the question of metallic melt drainage or core blockage under BWR dry-core accident conditions that were addressed in the XR2-1 experiment.24 There are significant uncertainties regarding the dissolution rates of fuel pellets in molten Zircaloy. This is a very significant phenomenon. The materials interactions between the core metals, their oxides, steam, and hydrogen, and the rate limitations on these interactions, are of great importance throughout the melt-progression process. Because the usual assumptions in reactor-safety analysis of parabolic rate laws and the limited available data that are mostly for pairs of materials, Veshchunov and Hofmann are performing multilayer diffusion analysis of the significant interactions. One purpose of this work is to determine the accuracy and the range of validity of the commonly used parabolic rate laws. These multi-layer diffusion models are included in the SVECHA package of mechanistic modules that have been validated against the available experimental data and are being implemented in the SCDAP/RELAP5, the ICARE 2, and the ATHLET-CD severe accident codes.27 There are also very large uncertainties regarding the rate limitations on the oxidation of relocating and relocated Zircaloy melts during both coolant boildown and reflood. Major

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uncertainties exist regarding the cracking of existing oxide films and the extent to which new surfaces on Zircaloy melts generate protective oxide films. These uncertainties are reflected in the very poor agreement of existing codes with the available data on the oxidation rates during reflood.18 Substantial swelling without melting of high-burnup irradiated fuel can occur at temperatures below 2500 K in the reducing environment down stream from the onset of steam starvation in the core. This behavior was observed in the ST (Source Term) experiments in ACRR with high-burnup Zircaloy-clad fuel rods in flowing hydrogen where this swelling (foaming) filled the gaps between the fuel rods.28 Similar swelling without melting was also observed in the post-test examination of individual high-burnup fuel rods in severe-fuel-damage tests in the NRU reactor.29 There is significant uncertainty regarding the mechanisms involved in this large fuel swelling without melting, in the range of severe accident conditions under which this swelling occurs, and on its significance to severe accident behavior. With the very limited existing data base, there are also questions as to whether there are significant but as yet undiscovered irradiated-fuel effects on core degradation and early-phase melt progression. There are also remaining quantitative uncertainties regarding the failure and the failure threshold of the hot-leg nozzles and the steam-generator tubes from natural convection of the primary-system steam and hydrogen coming from the hot PWR core during the early phase of melt progression. Counter-current natural-convection gas flow in the hot leg and in the steam-generator tubes provides this heating. Steam-generator-tube failure provides bypass of the containment into the atmosphere for the fission-product inventory in the primary system gas. Either of these failures depressurizes the primary system, and this depressurization strongly affects the remainder of the severe accident sequence. Phenomena affected are: the transient cooling of the core during the post-failure blow-down; the lowering of the water level below the core, unless the core is subsequently reflooded, to produce dry-core conditions during the subsequent core heat up by continuing fission-product decay as the melt-progression sequence is reentered; and, finally, the effective elimination of the contribution of high pressure loads to lower head failure with subsequent melt ejection and direct containment heating (DCH). A related potential failure mode that leads to primary system depressurization in stationblackout accidents and in some other sequences is the failure of the pressurizer line because of hot gas flow out the PORV and the safety relief valves. If the pressurizer line or the hotleg nozzle fails first, this depressurizes the primary system and removes the pressure load on the steam-generator tubes and the potential for steam-generator-tube failure that can have serious radiological consequences. With the failure of any one of the three, the pressurizer line, the hot-leg nozzle, or the steam-generator tubes, the primary system is depressurized so that failure of the other two is effectively precluded. Analysis with SCDAP/RELAP5 has indicated that unintentional depressurization of the primary system by failure of the pressurizer line or a hot-leg nozzle is likely in most PWR accidents, and that, as expected, this depressurization would occur well before lower head failure from melt attack.30 There are quantitative uncertainties remaining here involving the actual state of the core during heatup, the natural-convection process itself, and details of the accident sequence.

LATE-PHASE MELT PROGRESSION Late-phase melt progression involves ceramic melt behavior at temperatures above about 2800 K, and also, at somewhat lower temperatures, the pre-melting behavior of solid ceramic debris, including fuel swelling and foaming, fuel fragmentation, fuel sintering, and fuel slumping. The mass and the other characteristics of the ceramic melt released from the core

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into the lower plenum water in blocked-core accidents like TMI-2 are determined by latephase melt-progression processes, in particular the mechanism and threshold of the meltthrough of the ceramic melt pool from the core and also the location of melt-through around the pool boundary (side or bottom melt-through). Also of significance to late-phase melt progression as well as later to lower head failure are natural-convection thermal-hydraulic effects from the internal heating of the ceramic melt pool at high Rayleigh numbers.

Most Significant Results The current information base on the behavior of ceramic melt pools in blocked-core accidents has come primarily from the TMI-2 post-accident core examination.6 More detailed information on the basic mechanisms involved has come from the MP-1 and the MP2 (Melt Progression) experiments in the ACRR test reactor.11,31 More information on ceramic melt-pool behavior will be obtained in the Phebus FP (Fission Products) tests that are currently underway.32 The Phebus FP tests, except for the initial scoping test FPT-O, use high-burnup irradiated fuel. In blocked-core accidents, as TMI-2 demonstrated, a mostly ceramic melt pool grows in the pre-existing particulate ceramic debris bed under fission-product-decay heating. This bed is formed by fragmentation of the free-standing pellet stacks and oxidized cladding shards (mechanism unknown) that remain after the molten unoxidized cladding Zircaloy has drained and frozen lower in the core.20 The growing ceramic melt pool is contained in the ceramic debris bed by a growing ceramic crust that is constantly melting and reforming in the cooler debris bed that surrounds the melt pool.20 The MP experiments demonstrated the process of melting and reformation of the ceramic crust as the melt pool grows, and also demonstrated the growth of the ceramic crust into the supporting metallic crust without ceramic crust failure.11,31 The essentially one-dimensional MP experiments involved a growing ceramic melt pool in a pre-formed particulate ceramic debris bed that was supported by a pre-cast metallic crust across 32 clad fresh fuel-rod stubs, all in an inert helium environment.11,31 A schematic drawing of the MP-2 test assembly, shown in Figure 6, illustrates the geometry of the MP experiments. Fission heating by the ACRR test-reactor neutrons provided internal heat generation in the debris bed, the fuel rod stubs, and the dissolved in the metallic crust. On-line characterization of internal temperatures was provided by an array of special high-temperature thermocouples, but most of the experimental information was obtained from post-test examination of the test assemblies. MP-1 had an eutectic metallic crust with a 2200 K liquifaction temperature. MP-2 had a metallic crust of prototypic TMI-2 composition that included Zircaloy, control rod materials, and and and had a measured slump (not liquifaction) temperature of 1800 K. In MP-2, peak ceramic melt-pool temperatures of about 3400 K (inferred from indirect measurements) were maintained for ten minutes.11 The MP-1 and MP-2 experiments showed that, in a blocked-core accident, the growing ceramic melt pool is contained by a growing ceramic crust in the ceramic particulate debris bed with local crust melting and refreezing occurring in the debris bed as the pool and the crust grow. In MP-2, the original metallic blockage melted away and relocated downward between the fuel-rod stubs, but some oxidized remnants of the original metallic crust remained in place. The ceramic crust that actually contained the melt pool and that migrated into the fuel-rod stubs, however, did not fail. (In TMI-2, the fixed water level after reflooding prevented movement and melting away of the original metallic blockage, and the ceramic crust migrated into the fixed metallic crust as the melt pool grew, a process that was observed in the MP-2

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experiment.) Ceramic melting started as melting at 2900 K, but later dissolution occurred to give eutectic melting with a liquifaction temperature of 2800 K. Gauntt has hypothesized that pool melt-through from the core occurs when the growing melt pool reaches a boundary that has insufficient material and heat sink to support reformation of the crust.33 An x-radiograph of the post-test MP-2 test assembly is shown in Figure 7 along with metallographs of an axial section of the test assembly. The metallographs show details of the interaction of the ceramic melt pool and the crust. They show clearly that the ceramic melt pool expanded downward into the original metallic crust across the lower fuel-rod stubs and replaced the metallic crust with the ceramic crust, and that the advancing ceramic crust successfully contained the ceramic melt pool at all times without drainage of the melt pool. The voids in the refrozen MP-2 ceramic melt pool in Figure 7 are an artifact of the MP-2 experiment, and such voids should not be present in a ceramic melt pool in a severe LWR accident.25 They were caused by helium capsule fill-gas displaced by the final melt-pool relocation that filled the region formerly occupied by the metallic crust. Upon detection of this change of melt position, the ACRR was immediately scrammed and the melt-pool containing the voids quickly froze.

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The melt-progression process described above produces a very low fraction of unoxidized metal in the melt pool because of prior metallic melt relocation, and this is very significant for the subsequent in-vessel and ex-vessel parts of the accident sequence. The measured metal content of the frozen melt in the TMI-2 lower plenum was less than about one percent by mass.6 The partial or full drainage of the ceramic melt pool into the lower plenum water after melt-through is quite rapid (100 seconds in TMI-2).4 The thermal hydraulics of ceramic melt pools with the internal heat generation that causes pool convection is important to both melt progression in the reactor core and to lower head failure. The significant thermal-hydraulic characteristics are the heat flux distribution around the melt-pool boundary (particularly the up-to-down heat-flux ratio) and the flow start-up time constant. The thermal resistances of a possible gap above the pool surface, of the intervening ceramic crust, and of the core debris out to a convection and/or radiation-cooled boundary must be considered along with the Nusselt-number distribution around the pool boundary in order to determine the actual heat-flux distribution around the boundary, and also the derivative up-to-down heat-flux ratio. The Nusselt number distribution may thus not be the primary determinant of the actual distribution of the heat flux from the pool boundary. Modified Rayleigh numbers for a growing in-core ceramic melt pool in a PWR core-melt accident range up to and the convective flow at the higher Rayleigh numbers is turbulent. Experimental data and correlations on the thermal hydraulics of melt pools have indicated that the equilibrium upward Nusselt number is several times the downward Nusselt number, and that the distribution is peaked around the upper edge of the pool.34,35,36,37 This enhances sideways pool growth. Most of these data were obtained with electrically heated water pools with plane-parallel side walls ("slice geometry") and with low Rayleigh number laminar flow, although very limited data do exist in the Rayleigh number range with turbulent flow. Further experiments are underway. Analytical models for the melt-pool thermal hydraulics that use the formulation for the turbulent kinetic energy (k) and the turbulent dissipation applied to natural convection flow have been developed and assessed against the available data.41,42 These models agree reasonably well with the available data, that, however, are sparse in the high Rayleigh number range. One model includes buoyancy-driven turbulence and predicts that, at Rayleigh numbers above (turbulent flow), there is aperiodic unsteady flow in the pool.41 Available data and analysis indicate that the flow start-up time constant for both in-core and lower-plenum ceramic melt pools in PWR blocked-core accidents is on the order of 40 minutes.38,39,40 Thus the common usage of steady-state values for the up-to-down Nusselt number ratio for such periods is clearly incorrect. Nearly all the existing results on core degradation and early-phase melt progression have been obtained from tests with fresh fuel or trace-irradiated fuel, so little information exists for high-burnup irradiated fuel. The PBF SFD 1-4 test and the partially defective SFD 1-3 test, however, did use high-burnup irradiated fuel, individual irradiated fuel rods were included in the FLHT-4 and FLHT-5 full-length fuel-rod tests in the NRU reactor, and the small ACRR ST-1 and ST-2 fission-product-release experiments used irradiated fuel. The primary difference observed from the effect of the high-burnup fuel in the PBF SFD 1-4 test was that more extensive fragmentation of the remaining fuel pellets occurred following relocation of the molten metallic Zircaloy cladding, although this might have been caused by the higher peak test temperatures in SFD 1-4.43 The ACRR ST-1 and ST-2 experiments had reducing conditions with hydrogen flow and Zircaloy-clad fuel. As discussed under early-phase melt progression, after Zircaloy melting and relocation in these experiments, large fuel swelling (foaming) closed the gaps between the high burnup fuel pellets without fuel melting and at temperatures below 2500 K.28 Similar local fuel swelling in the hydrogen-containing steam-

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starved sections of the test fuel bundles were found in the post-irradiation examination (PIE) of the NRU FLHT-4 and FLHT-5 tests.29 A porous media model of late-phase melt progression has been developed that treats the melting, relocation, and freezing dynamics of a melt pool and crusts in a particulate ceramic debris bed.44 This DEBRIS model is a predecessor of the somewhat similar MERIS earlyphase porous-media model of metallic melt relocation and freezing in reactor structure.45 Porous-media modeling with relatively fine nodding, as opposed to uncertain geometryprescriptive modeling, provides a mechanistic treatment of the melt-pool growth in blockedcore accidents as well as a mechanistic treatment of the formation, relocation, and failure of both the ceramic crust that contains the ceramic melt pool and the original metallic core blockage. DEBRIS agrees well with the MP-1 and the MP-2 results.46 Porous-media latephase melt-progression modules based on DEBRIS have been added to the ICARE2 and KESS severe accident codes in order to provide mechanistic modeling of late-phase melt progression.15,47 A scoping sensitivity analysis has been performed by Schmidt and Humphries at Sandia National Laboratories on the effects of the different processes involved in the growth of the ceramic melt pool in a particulate ceramic debris bed in a PWR blocked- core accident, and particularly on whether the pool growth in the core is predominately sideways or downward.48 The analysis used idealized PWR core-debris configurations and included a comparison of the results of calculations with DEBRIS, that includes melt relocation, with those of the TAC2D thermal analysis code that does not. It was found that the most significant parameters that affect the direction of pool growth are the initial power distribution in the core and melt-relocation effects. The core boundary conditions had a relatively small effect because of the low thermal diffusivity of ceramic core debris. Predominately sideways pool growth with melt-through out the side of the core was generally predicted, and this results in only partial drainage of the melt pool. A peaked radial power distribution, however, promoted downward melt-pool growth, so that a blocked-core BWR accident would be much more susceptible to bottom failure with full melt-pool drainage than an otherwise similar PWR accident.

Principal Remaining Uncertainties The major remaining uncertainty regarding late-phase melt progression is the phenomenology of the melt-through of the ceramic melt pool from the core in blocked-core accidents that provides melt relocation into the lower plenum. Important are the threshold and the location of melt-through that determine the released melt mass. Also important is the capability for reasonably accurate predictive modeling of the phenomenology for use in accident analysis codes. The DEBRIS porous-media model with possible modifications may prove to be able to provide such predictive capability.44 More data on the relevant processes are badly needed, and some should be provided by the Phebus FP tests.32 One test, FPT-4, will use debris-bed geometry like the MP experiments, but the others will start from cladfuel-rod geometry.11 There is very little information available on the mechanism and the time in the accident sequence at which the local transition occurs from declad-rod geometry into particulatedebris-bed geometry. This transition is assumed to be a parametric temperature in current severe accident codes. It is not really known whether the existence of a particulate ceramic debris bed is necessary for the formation of a ceramic melt pool, or whether such a pool can form directly from free-standing declad fuel rods. In the PBF SFD 1-4 test with high-burnup irradiated fuel, the fragmentation of the relatively free-standing pellet stacks was somewhat greater and the debris appeared to be more "debris bed like" than in previous tests with trace-

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irradiated fuel and at lower peak temperatures.43 There is also some indication that dissolution by the molten Zircaloy may rubblize the fuel pellets to produce a debris bed.25 It is generally assumed that the extensive fragmentation in the TMI-2 upper core was the result of the brief flow transient in the partially uncovered, hot core that preceded the later core reflooding.3 Thermal analysis, however, indicates that a small, growing ceramic melt pool already existed in the core at the time of the flow transient.8 Future Phebus FP tests that have irradiated fuel and no reflooding may provide definitive information on these fragmentation questions.32 Also not known is the extent to which the dissolved in the metallic Zircaloy melt and further oxidation of the melt Zircaloy by steam contribute to the mass of the ceramic melt pool in the core. Uncertainties remain regarding the data and correlations and the processes involved in the thermal hydraulics of the internally-heated ceramic melt pool in the reactor core, along with the flow start-up time constant. It is not clear, however, that these uncertainties are really significant to the mass and the other characteristics of the melt released from the core into the lower plenum. Data at high Rayleigh numbers and for hemispheric geometry are limited, but further experiments in these areas are under way. More directly applicable data on the flow start-up time constants in the relevant Rayleigh number range and data in hemispheric geometry would be useful. Data that include two-phase (solidus-liquidus) and phase-change effects in the boundary layer for growing and for freezing melt pools are currently nonexistent and would be useful. As mentioned previously under early-phase melt progression, there are large uncertainties regarding the process of irradiated fuel swelling (foaming) under reducing conditions and at temperatures below 2500 K, on the consequences of such swelling on severe accident sequences, and on accident consequences.28,29 Because the TMI-2 accident did not provide such information (low-burnup fuel), there is also large uncertainty about whether or not there are any other significant irradiated-fuel effects on late-phase melt progression. It appears that the numerical density of fission gas bubbles in the ceramic melt pool would be too low to affect significantly the melt-pool natural convection and thermal hydraulics.49 There are, of course, significant uncertainties in the modeling of late-phase meltprogression behavior in severe accidents with porous-media models or with other models. These uncertainties occur, in large part, because of the very limited data base on late-phase melt progression and the limited range of conditions for which data exist. The forthcoming Phebus FP tests should provide much of this data for both improved phenomenological understanding and for model development and assessment.32

MELT-WATER INTERACTIONS IN THE LOWER PLENUM When either ceramic melts or metallic melts drain from the core into lower plenum water, melt-water interactions occur that are significant to the continued course of the accident. Upon entry into water, a relatively slow breakup of the melt jet occurs that is accompanied by non-explosive steam generation (and hydrogen generation with metallic melts) and by cooling of the melt.5 An explosive thermal interaction with a high pressure shock may also occur following the initial melt breakup and the mixing of the melt with water and the generated steam.5 This latter interaction is commonly called a steam explosion. Of reactor-safety interest are the breakup and cooling of the melt in transit through the lower plenum water to the vessel lower head, the amount of the generated steam and hydrogen, and also the possible occurrence and consequences of a steam explosion. Considered here are only in-vessel meltwater interactions in the lower plenum. Not considered are potentially destructive ex-vessel steam explosions in the reactor cavity, although the explosive and non-explosive melt-water

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interaction processes are the same in-vessel and ex-vessel, except for differing initial and boundary conditions.

Most Significant Results Historically, there has been a great deal of reactor-safety interest in the possible generation of a large containment-failing missile by a sequence of events initiated by a steam explosion in the lower plenum water. This sequence was indicated by events that occurred in the SL-1 accident.50 In this hypothetical LWR sequence, a lower plenum melt-water steam explosion accelerates an overlying slug of water and core material into impact upon the vessel upper head. This impact may break the head (or a major part of it) away from the reactor vessel and may expel it as a large mass missile through the containment. This sequence of events was labeled the alpha containment-failure mode in the Reactor Safety Study, WASH 1400, (a name that has persisted), and it has continued to be of LWR safety interest.51 A review group was convened by the NRC in 1985 to examine the risk from the alpha containment-failure mode, and it was concluded that this process is not risk significant.52 A second NRC review group is currently reassessing this conclusion. An explosive melt-coolant interaction involves rapid and fine fragmentation of the melt with very rapid heat transfer to the water (or other cold liquid) and gives shock pressures that have been observed sometimes to exceed the water critical pressure.53 The governing processes, however, long remained unknown. In 1972 Board and Hall proposed that meltwater thermal explosions were actually thermal detonations in analogy to chemical detonations.54 The thermal detonation model, as developed further, includes the following stages: 1) a pre-explosion melt breakup and mixing of the melt with the water and insulating steam to develop a thermally-detonable mixture; 2) a triggering pressure event (natural, or artificial in some experiments); 3) escalation in the mixture of the trigger event into a shock front by rapid fragmentation of the melt droplets and mixing with and heat transfer to the water with very rapid steam generation; 4) passage of the developed and self-sustaining detonation shock through the interacting mixture zone; and 5) expansion work by the postdetonation high pressure mixture against system constraints.5 The initial mixing (and melt cooling) phase with non-explosive steam generation occurs to some extent in almost all cases of melt pours into water. Only if a region of thermally-detonable mixture develops along with a sufficient trigger and escalation does an explosive interaction (steam explosion) actually occur. There is a question, however, regarding whether or not the steam explosions that have actually been observed are fully developed detonations in the sense of the Chapman-Jouquet conditions. Vapor-blanketed mixtures of high temperature melt and cold coolant form a metastable system, and it is known that explosive interactions can occur in at least two other modes than the pouring mode of contact. Explosive surface interactions have been observed between quasi-static layers of melt and coolant separated by a vapor layer, and this process might be relevant to some severe LWR accident situations.55 Because of the limited melt-coolant mixing in surface explosions, however, these explosions are less energetic. In the SL-1 super-prompt-critical power-excursion accident, fuel vaporization in the center of the uranium-aluminum fuel plates fragmented and dispersed molten fuel from the plates into the core water to produce the steam explosion that destroyed the reactor.53 This fuel-vapordriven fragmentation and mixing of the molten fuel with coolant has also been of interest in hypothesized super-prompt-critical accidents in sodium-cooled fast reactors. Many mostly small scale experiments in which melt is dropped into water have been performed over the years with non-reactor materials (simulants).56 These experiments have provided general information on the processes involved in melt-coolant interactions, both non-explosive and explosive. Thermite mixtures have often been used to "simulate" ceramic

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melts, and these mixtures must contain metal unless the metallic and ceramic components have been separated by gravity before pouring. There have also been separate-effects experiments and modeling directed to the individual processes involved in steam explosions.56 Most of these have been concerned with the physical limits to the preexplosion mixing of masses of melt and water that provide a limit to the steam explosion energetics. Data on melt-jet breakup in water have also been obtained with thermite simulation of ceramic melt jets.57 More recently, large scale experiments with well defined geometries in which furnaceheated ceramic melts are poured into water have been performed in the FARO and KROTOS facilities in the Commission of the European Communities (CEC) laboratory at Ispra, Italy.58,59 Tests on melt-jet breakup and cooling have been performed in the FARO facility with 150 kg melts, both with and without some metallic zirconium in the melt. These tests were pressurized, in part, to suppress steam explosion triggering and have yielded quantitative data. Zirconium-metal addition produced finer breakup because of metal oxidation with hydrogen generation.58 KROTOS is a well defined and well instrumented one-dimensional ambient-pressure facility in shock-tube geometry for the detailed study of steam explosions. Tests with 1.5 kg melts, both with and without external triggering, produced very energetic explosions with measured shock pressures of up to 65 MPa. On the other hand, no explosions occurred in similar tests with 4 kg melts (the same volume), even though strong external triggers were applied.59 The key to the very significant difference in the behavior of these two different ceramic melts appears to be that the void fraction in the mixing zone is about three times greater with the non-explosive melt than with the explosive melt at the time during mixing that the spontaneous explosion occurs with the melt.59 The reason for this difference that appears to cause the observed non-explosivity with melts is not understood and is the subject of high priority continuing research. The one-dimensional TEXAS steam-explosion code has given good agreement with the measured propagating steam-explosion pressures in the KROTOS experiments.60 As mentioned previously, a steam explosion did not occur at 10 MPa ambient pressure when 20 te of ceramic melt relocated under water from the TMI-2 core, through the lower plenum, and onto the lower vessel head.3 There was, however, a non-explosive 3 MPa "steam spike" of about 15 minutes duration from the breakup and cooling of the melt jet. The non-explosive melt-coolant interaction in the TMI-2 accident has been analyzed by Epstein and Fauske.61 There are also indications of possible pressure suppression of either steam-explosion triggering or escalation in the non-occurrence of a steam explosion in the FARO tests with 150 kg melt jets. These tests were mostly performed at 5 MPa ambient pressure, but with one test at 2 MPa.58 In intermediate scale experiments with 1 to 5 kg thermite melts, spontaneous explosions occurred at 0.1 MPa ambient pressure but external triggers were required to produce explosions at 1.0 MPa.62 Other small scale experiments have also indicated that a pressure threshold does exist above which spontaneously triggered (i.e. natural) steam explosions do not occur. There are considerable data and several models for the initial stage melt breakup and mixing with water that are accompanied by melt cooling and non-explosive steam generation.5 There are few data and models for triggering and escalation into a thermal detonation, including a possible ambient-pressure threshold.63 There are some data and models for the thermal detonation itself, the shock impulse, and the expansion work.64

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Sandia National Laboratories has developed an Integrated Fuel-Coolant Interaction code (IFCI) that models all of the stages of a steam explosion with the exception that a parametric trigger is required.65 IFCI is a 2D, 3-field compressible hydrodynamics code, and it also includes oxidation of metallic melt and the resulting hydrogen generation. IFCI is currently undergoing validation assessment with the available experimental data and peer review.

Principal Remaining Uncertainties The remaining uncertainty of overriding reactor-safety importance regarding melt-water interactions is whether or not energetic steam explosions can ever occur with melts, and, if they can, for what range of conditions. The ongoing KROTOS program at CEC Ispra is addressing this question on a priority basis. A second major uncertainty is whether there is an ambient-pressure threshold above which naturally-triggered steam explosions do not occur, and, if so, what that pressure threshold is. A third major uncertainty is whether or not substantial chemical augmentation of the thermal explosion energetics (pressures) can occur with metallic Zircaloy melts and water. Such augmentation of the explosion energetics by factors of several times has been observed with aluminum-melt pours into water.66 Experiments have been started at Argonne National Laboratory (ANL) to determine whether such chemical augmentation of the explosion energetics does occur in pours into water of melts of metallic Zircaloy and melts of mixtures.67 These well instrumented experiments are performed in a well defined, one-dimensional shock-tube geometry, with integral measurements of the generated hydrogen. There are also significant uncertainties regarding melt-jet break up and melt mixing with water, on the triggering process, and on the spectrum of naturally-occurring triggers in LWR core-melt accidents.

LOWER HEAD FAILURE The integrity of the reactor lower head is a major barrier to the further progression of a severe LWR accident. Lower head failure releases core melt and solid debris with their fission-product inventories into the containment, generates loads that are potentially threatening to containment integrity, and opens a pathway from the primary system into the containment for fission-product transport. The threshold and mode of failure of the lower head under melt attack are major factors in determining the mass, the temperature, and the rate of release of the melt and solid debris into the containment. In the TMI-2 accident, 20 te of ceramic melt were released onto the water-flooded vessel lower head at high pressure (10 MPa), but the lower head did not fail.3

Most Significant Results After all the core debris had been removed from the TMI-2 reactor vessel, a remote video examination was made of the lower head, the lower plenum structure, and the debris. The solidified lower head melt and solid debris were then broken up, removed, and examined. The lower head surface and penetration remnants were examined, and samples were taken from the vessel lower head for metallurgical examination to obtain information on the peak temperatures that had occurred within the lower head steel.9 The international Organization for Economic Cooperation and Development (OECD) then established the TMI-2 Vessel Investigation Project (VIP) to perform metallurgical examinations of these samples and to analyze the results in terms of failure modes and the margin to failure of the vessel lower head (that turned out to be negative).68

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The composition of the solidified melt on the lower head was found to be about 80 w% and about 20 w% with less than 1 w% structural stainless steel and Inconel and less than 0.5 w% Ag-In-Cd control-rod alloy.9 There was little melting of the structural steel by the melt-relocation transient because of the self-insulating ceramic crusts formed on the steel surfaces during relocation, a phenoma that had previously been observed and modeled. From the metallurgical examination, it was determined that a local "hot spot" of about one meter diameter and with a maximum inner surface temperature of about 1350 K had existed for about 30 minutes at the interface of the A522B ferritic vessel steel and the 5 mm thick 308 L stainless-steel weld cladding on the inner surface of the lower head.9 The hot spot was not caused by jet impingement because a ceramic melt jet forms an insulating ceramic crust at the site of impingement on metal and because the hot spot was located away from the center of the head and opposite the side where the ceramic melt relocated down the side of the core and into the lower plenum. A drawing of the lower plenum region that includes the hot-spot geometry is shown in Figure 8.9 This drawing also shows the degraded instrumentation nozzles, but the holes in the core support plate and the flow diverter plate through which the ceramic melt relocated are not shown. The metallurgical examination of the lower head samples showed that, away from the hot spot, the temperatures in the lower head did not exceed the ferritic to austenitic transition temperature of about 1000 K.9 Global creep-rupture failure of the lower head was assessed

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using Larson-Miller parameters constructed from creep and tensile-strength data for the A533 B steel used in the vessel that were obtained as part of the TMI-2 VIP program.9 This analysis was performed for the 14.4 MPa pressure that existed in the vessel from 2 hours after the time of melt relocation to 4 hours after relocation, following vessel repressurization. The analysis indicated that, for reasonable temperature gradients through the lower head, global creep-rupture failure would be expected at 0.43 hours after relocation at an assumed inner wall temperature of 1000 K and at 7.5 hours after relocation at an inner wall temperature of 900 K.68 A margin-to-failure analysis was then performed for the lower head under the measured time-varying pressure loading, with calculated lower head heating by the core debris and with global creep-rupture failure analysis of the lower head.68 The thermal analysis used measurements of the mass and the composition of the core debris, assessments of the decayheat source in the debris and the debris temperature on relocation, and assumptions regarding the debris-lower-head and the debris-water heat transfer and the outward heat loss from the lower head. Both a nominal (best estimate) case and a "lower-bound" case were considered for the heating of the lower head. It was found in the nominal case that lower head failure occurred 1.7 hours after melt relocation. This was soon after block-valve closure at 1.6 hours after melt relocation raised the primary system pressure from 8.2 to 9.2 MPa and when the increasing inner wall temperature had reached 1100 K.68 In the lower-bound case, lower head failure occurred at 2.3 hours after melt relocation when the pressure had risen to 14.4 MPa and, with the decreased lower head heating, the inner wall temperature had reached 1000 K.68 Since the lower head did not fail in the TMI-2 accident, and since the calculated inner vessel wall temperatures in this analysis were still rising at the time of failure for both cases considered, it was postulated in the TMI-2 VIP Margin-to-Failure report that "enhanced cooling mechanisms" for the debris must have been present that prevented lower head failure.68 The first of these postulated mechanisms is slow cooling of the debris by assumed convective water flow through cracks in the debris. The second is rapid cooling by an assumed convective water flow in the gap between the lower head and the debris. Thermal analysis of the cooling to be obtained from each of these mechanisms was performed for some assumed geometries and used both natural and forced-convection heat-transfer correlations for single phase water flow (taken for "conservatism"). It was concluded that single phase water cooling through channels containing less than 1% of the debris volume and through a debris-to-wall gap thickness of 1 mm would provide the additional cooling necessary to have prevented lower head failure and would be consistent with the metallurgical examination of the lower head samples, and that both of the postulated rapid and slow cooling mechanisms are required to prevent failure.68 It was also concluded that, in order to prevent lower head failure, the additional cooling of the lower head and the debris from that used in the margin-to-failure calculations would have had to be effective during the first two hours after melt relocation and well before the vessel was repressurized to 14.4 MPa. There are physical reasons for questioning the "enhanced cooling mechanisms" that the TMI-2 VIP report states are required to account for the non-failure of the TMI-2 lower head. The entry of liquid water into fine cooling cracks in newly frozen ceramic debris under a natural convection head for the slow cooling appears to be particularly questionable as a cooling mechanism. The ceramic melt in a pool on the lower head would freeze at about 2800 K, and, in this model, would soon need to undergo cooling cracking in order to provide operating time for the postulated enhanced cooling. If a temperature for cracking of about 2000 K is assumed as reasonable, the initial wall temperature is several times the 647 K thermodynamic critical temperature of water, the temperature above which water cannot exist as a liquid and there is no latent heat of vaporization. This effectively leads to

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convective gas cooling. This restriction is less severe for liquid-water cooling in the debristo-head gap because the initial head temperature is at saturation (584 K), which is below the water critical temperature, and the opposite insulating ceramic surface crust on the melt has previously been cooled by passage through the lower plenum water. Nevertheless, the continuing entry of liquid water into the gap to provide boiling heat transfer would appear difficult to achieve, particularly as the steel surface heats up. On the other hand, the freezing of the surface at the leading edge and bottom of the "lava-like" ceramic melt as it flows across the lower vessel head would appear to leave a large, very irregular gap with a very low gap conductance to the lower head. Evidence of the nature of this surface was destroyed in the necessary breaking-up of the frozen melt pool during the remote de-fueling of the debris from the lower head. It does appear possible, however, that some water might have remained in the gap during relocation and that this water might possibly have been replenished by convection to provide the rapid "enhanced-cooling" mode of the TMI-2 VIP report. The debris-to-head gap did widen on-the-order of because of thermal expansion and creep as the lower head heated to near-failure temperatures (assuming that the ceramic crust did not fail). Research on this potential mode of gap cooling is very significant, and experiments in this area are planned. A sensitivity analysis was performed as part of the margin-to-failure analysis in the TMI-2 VIP report that included the gap conductance and also the exterior lower head heat-transfer coefficient.68 Further examination of the possible range for these parameters would appear to be useful, however, particularly with the very large uncertainties that exist regarding the configurational and the thermal-hydraulic conditions that existed. The margin-to-failure analysis showed that the local and short time (30 minutes) hot spot alone would not have produced vessel failure so long as the global background temperatures in the lower head remained relatively low (600 K). The effect of the hot spot on global creep-rupture failure of the lower head was found to decrease the time between relocation and calculated failure for the "lower-bound" case from 2.3 hours to 1.9 hours, which is before the time of repressurization of the vessel to 14.4 MPa.68 There was extensive melting of a few of the instrumentation nozzles above the lower head surface, as shown in Figure 8, which also includes the maximum temperature contours of the hot spot.9 The nozzles, which penetrated the lower head near the hot spot, were melted off near the lower head, but, away from the hot spot, the nozzles extended higher above the head surface indicating the existence there of the insulating ceramic crust at the bottom of the ceramic melt pool.9 Calculations indicated that ceramic melt would not penetrate through the instrumentation tubes in the lower head, thus eliminating the possibility of ex-vessel tube rupture.9 Calculations were performed that indicated that there was a large margin to weld failure and instrument-tube ejection.9 The thermal hydraulics of the internally-heated ceramic melt pool on the water-flooded lower head is significant to lower head failure as well as to late-phase melt progression in the reactor core. As previously discussed under late-phase melt progression, the thermal resistance from the pool and the ceramic crusts to the water above and through the gap to the lower head below must be combined in lower head heating analysis. The up-to-down heatflux ratio at the boundary of the melt pool is significant here. The up-to-down Nusseltnumber ratio is generally taken from the available steady-state experimental data.34,35,36,37 The pool-cooling transient starts as conduction upon pool formation (equal up and down Nusselt numbers), and evolves towards steady state with the flow-start up time constant. This time constant is on the order of 40 minutes for the ceramic melt pools in LWR accidents, so the flow start-up transient is significant.38,39,40 In addition, as the pool freezes at the top and bottom during cooling, the ratio of the up-to-down pool Nusselt numbers decreases to one roughly linearly with the decrease in pool depth (decreasing Rayleigh number), but with modification by the flow change (flow start-up) time constant.

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The TMI-2 VIP report suggests two accident management actions that would appear to be effective in preventing lower head failure under core-melt attack.9 The first is vessel depressurization to a nominal level, if core-melt conditions do occur, in order to remove most of the pressure loading on the lower head. At TMI-2, repressurization of the vessel from 8.2 MPa to 14.4 MPa about 2 hours after melt relocation onto the lower head increased the lower head stress by 75% at a time when the lower head was still near its peak global temperatures.68 The second proposed accident management action is provision for flooding the reactor cavity to provide direct ex-vessel cooling of the lower head and the lower part of the reactor vessel. Either of these actions would have eliminated the threat of lower head failure in the TMI-2 accident.68

Principal Remaining Uncertainties From the negative margin to failure in the analysis in the TMI-2 VIP report (that predicted failure of the TMI-2 lower head), the most significant uncertainty regarding lower head failure appears to be the heating of the lower head under melt attack. The TMI-2 VIP report emphasizes debris cooling rather than the more directly significant lower head heating.68 The lower head heating rate depends on the thermal resistance of the large and irregular gap between the irregular surface of the frozen ceramic melt crust and the lower head steel surface as well as on the heat source in and the cooling of the melt. The possibility of continuing penetration of liquid water into the gap by convection is the second of the proposed "enhanced-cooling" mechanisms proposed in the TMI-2 VIP report, the rapid cooling mechanism, and this clearly needs to be investigated.68 This mechanism does appear to be questionable, however, because, as discussed previously, the temperatures of the surfaces of the melt crust and (except initially) of the lower head steel are above the water critical temperature, and this make the entry of liquid water into the gap difficult to achieve with convective driving pressures. This mechanism, if operative, could provide a very substantial reduction in the calculated lower head temperatures for given core-melt conditions. The first of the "enhanced-cooling" mechanisms proposed in the TMI-2 VIP report, slow cooling by convective water flow through cracks through the body of the solidified part of the melt, appears to be far less viable physically for the reasons previously discussed. Several experiments are being started on the potential for cooling in the gap between the ceramic melt crust and the lower head by convective liquid water flow into the gap. As part of the FARO program of experiments at CEC, Ispra, on the breakup and cooling of a 150 kg ceramic melt jet in water, measurements are being made of the heat transfer from the ceramic debris to a metal plate at the bottom of the test vessel. These are transient experiments, of course, and do not involve continued internal heating of the ceramic melt. They should, however, provide much of the needed information on the effective gap conductance and on potential convective cooling effects in the gap, at least for the initial stages of lower head heating. The cause of the local hot spot on the TMI-2 lower head and its general significance to lower head failure are not currently known. Thus it is not known whether local hot-spot effects need to be included in lower head failure assessments, and, if so, how they should be modeled. No planned research in this area is know to the author. There are significant uncertainties regarding the threshold and the mode of lower head failure under combined pressure and thermal loads at high head temperature. The mode of failure determines the melt-ejection rate into the containment. There are also significant remaining uncertainties regarding the properties of reactor-vessel steel at temperatures in the 1000 K range.68 Experiments have been started at Sandia National Laboratories at one-fifth scale to measure the threshold and the mode of lower head failure under known combined

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thermal and pressure loads. Radiation heating is used to provide a specified heat-flux distribution on the surface of the pressurized lower head as the pressure is slowly increased to failure. The first of these experiments was recently performed successfully. There are uncertainties remaining regarding the behavior of the internally-heated ceramic melt pool on the water-flooded vessel lower head. Most of these were discussed previously in the section on late-phase melt progression. For lower head failure, there are additional uncertainties regarding the thermal-hydraulic behavior of slowly freezing melt pools with internal heat generation, including phase-change and two-phase effects in the boundary layer and the increasing and decreasing flow transients. How significant these uncertainties are is not clear. The OECD and Russia are sponsoring a large comprehensive research program at the Kurchatov Institute on the thermal and chemical interaction of internally-heated core melt with the vessel lower head in order to address the conditions under which a molten core can be successfully retained inside the reactor vessel. This program is named RASPLAV, and includes integral experiments of medium scale (~20 kg melts) and large scale (~ 200 kg melts), small and large scale materials-interaction experiments, and analytical modeling.69 The experiments use melts of and metallic Zr, in different proportions, or salt as a melt simulant. Emphasis in the experiments and modeling thus far has been on the thermal hydraulics of internally-heated melt pools, for which 2D and 3D models have been developed as RASPLAV/LHF.42 These models are somewhat similar to the Kelkar model,41 and have been validated against experimental data with reasonable agreement, particularly at the lower laminar-flow Rayleigh numbers.42 Most of the experiments are performed in "slice" geometry with either electrical-resistance heating of the melt across the flats or with a ribbon electrical-resistance heater in the melt. Much new information on melt-pool thermal hydraulics should be obtained from these experiments, including information on crust behavior and on melt interaction with lower head structure. Data for models of boundarylayer behavior in melt pools that include two-phase phenomena in crust melting should also be obtained. Most of these experiments are performed with reactor materials and at coremelt temperatures. Experiments in the RASPLAV program are also planned on the thermal hydraulics of stratified ceramic-metallic melt pools with internal heating of the ceramic melt. Except for small metallic additions to pre-existing ceramic melt pools, however, such mixed ceramicmetallic melt pools do not appear to exist in realistic melt-progression sequences as long as there is water in the vessel lower plenum. This is because of the separation in the core of the ceramic and the metallic materials during melt progression, as previously discussed. There is a very large difference in the liquifaction temperatures of metallic and ceramic melts, which, depending on the specific composition of the metallic melts, is roughly 1000 K. This produces a large difference in the times in the accident sequence at which metallic melts and ceramic melts are formed, in the times at which they relocate, and in their post-relocation behavior. If provisions have been made to provide external cooling of the reactor vessel by flooding the reactor cavity in an attempt to prevent vessel failure, a blocked core PWR accident may progress to lower plenum dry out without vessel failure. In such an accident, radiation heating of the vessel wall would add substantial steel melt to the surface of the ceramic melt pool, and the high-conductivity metallic layer might challenge the integrity of the reactor vessel. Such accidents are not further considered here. Several large scale integral tests (800 kg melts) on core-melt attack on an LWR vessel lower head including penetrations and structures are underway at the Paul Scherrer Institute in the international CORVIS program.70 These experiments use thermite melts that are separated by gravity into iron and alumina melts. These melts are then individually poured into the test assembly with heating of the ceramic melt by an electric arc to sustain the melt temperature against heat losses. Results of early preparatory tests confirm previous

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results that transient ceramic-melt flow does not melt steel surfaces because of the formation of an insulating ceramic crust on the steel. In one test, early failure (6 seconds) of a simulated BWR drainline occurred upon contact with hot flowing metallic melt. Further tests should give more information on these interaction processes.

CONSEQUENCES OF REFLOODING A SEVERELY DEGRADED CORE The consequences of reflooding a severely degraded core are also significant for LWR severe accident assessment. The primary interest here is in determining the maximum state of core degradation for which reflooding will promptly terminate the accident. Such prompt termination did not occur upon reflooding in the TMI-2 accident.4 Also of interest are the quantities of steam and hydrogen generated by the core reflooding and the possibility and consequences of an explosive melt-water interaction (steam explosion).

Most Significant Results The most significant information on reflooding a degraded core has come from the LOFT FP-2 test, the PBF Scoping Test, and from the non-occurrence of prompt termination of the TMI-2 accident by core reflooding.1,3,4,7 In the LOFT FP-2 test there was strong local heating, fuel-bundle degradation, and hydrogen generation from oxidation by the reflood steam of the metallic Zircaloy that remained in the previously steam-starved upper portion of the test fuel bundle.7 Most of the hydrogen generated in the FP-2 test was generated during the bundle reflooding.7 The regions of the test fuel bundles that had increased temperatures and core damage from steam oxidation during reflooding were eventually quenched by the rising water level, but, at TMI-2, uncertain temperature measurements of greater than 1000 K persisted in the core for several days, even though forced cooling was reestablished by one of the four pumps at about 12 hours after melt relocation onto the vessel lower head.3 There were two reflood-type events in the TMI-2 accident. About 25 minutes before the actual core reflooding, there was a brief injection of water into the partially uncovered and hot degraded core by a brief flow transient that was terminated after about 15 seconds by pump cavitation.3 There is no measurement available of the hydrogen generation rate at TMI-2, but it is generally considered that: Zircaloy oxidation resulting from the flowtransient steam generated most of the hydrogen in the accident; that this steam also produced strong local heating in the core and some melting by oxidation of the Zircaloy that remained in the hot, uncovered portion of the core; and that the hot gas flow from the core produced most of the observed surface melting in the stainless-steel upper fuel assembly grid plate above the core.71 The total hydrogen generated in the TMI-2 accident corresponds to the oxidation of about 45% of the total core Zircaloy. It has also been hypothesized that the initial thermal shock from the cold water injected into the core during the short flow transient fragmented the declad, free-standing fuel-pellet stacks (as had been observed in the PBF SFD tests) to form the particulate ceramic debris bed in which the ceramic melt pool grew.1,3 The TMI-2 core was reflooded by the high pressure injection system starting 200 minutes after scram, and the reactor vessel refilled in about 7 minutes.3 While the outer region of the debris bed was slowly quenched, water did not penetrate to the surface of the melt pool (~2800 K). Since the melt pool continued to grow, less power was removed from the pool by cooling than was generated in the pool by decay heating. At 226 minutes after scram when the melt pool had grown to about 50% of the core mass, the pool melted through the growing and reforming ceramic crust in which it was contained and out the side of the core.3,4,20 Nearly half of the melt pool (20% of the core mass) drained under water through

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the lower plenum and onto the lower vessel head. Thus it is seen that core reflooding did not promptly terminate the TMI-2 accident. At the time of reflooding in the LOFT FP-2 test, most of the rapid oxidation transient had already occurred in the uncovered part of the fuel-rod bundle along with the melting and relocation of most of the remaining unoxidized cladding Zircaloy.7 The rapid and extensive oxidation and hydrogen generation produced by the reflood steam showed, however, that a significant fraction of the unoxidized cladding Zircaloy remained in the declad upper region of the hot core on the fuel pellet surface and/or in cracks in the pellets because of wetting by the molten Zircaloy. The reflooding of the severely damaged core in the TMI-2 accident at a time when about 50% of the core mass was in the ceramic melt pool within the core did not produce a steam explosion. The reflood water was excluded from the region of the pool by the very hot particulate debris exterior to the hot ceramic crust that contained the melt pool. The drainage 26 minutes later of about 20% of the core mass (20 te) from the ceramic melt pool under water into the lower plenum at 12 MPa ambient pressure also did not produce a steam explosion. The melt-water interaction was non-explosive and produced a 3 MPa "steam spike" of about 15 minutes duration.4 It is the personal opinion of the author that an explosive interaction of ceramic melt and water (a steam explosion) is extremely unlikely during core reflooding because of the difficulty in achieving water penetration through the very hot debris that surrounds the ceramic melt pool. Such a conclusion is far less certain, however, for any metallic melt pools that have an open surface. Steam explosions did not occur, however, in a series of experiments at SNL where 10 kg aluminum melts were flooded with water.72 Potential steam explosions cannot currently be excluded, however, for all accidents with ceramic melt drainage under or into water, as in the TMI-2 accident, but when, unlike TMI-2, the reactor vessel is at low ambient pressure.

Principal Remaining Uncertainties The principal remaining uncertainty regarding the consequences of reflooding a severely degraded core is the determination of the boundary of degraded core conditions beyond which reflooding will not promptly terminate the accident. Here prompt accident termination is defined as achieving decreasing temperatures or melt fractions and stable geometry at all points within the core and the reactor vessel after reflooding has been completed. It is clear that prompt termination of the TMI-2 accident upon reflooding did not occur. The accident was only successfully terminated several hours after reflooding and the subsequent massive relocation under water of ceramic melt onto the vessel lower head. This relocation increased the surface area and the heat sink for cooling the melt mass, which was now in two parts, and also the decay-heat power continued to decrease. At this time, the temperatures in the lower head were all decreasing and stable geometry was assured (assuming that the reactor vessel does not fail and that there is a continuing supply of cooling water). A major part of this problem is uncertainty, under core reflood conditions, in the rate of water entry into regions of hot particulate debris and the rate of steam cooling of that debris. A program to determine the boundary beyond which reflooding will not promptly terminate the accident is certainly feasible with current knowledge, but, as far as the author knows, none is currently underway. Major uncertainties regarding the consequences of reflooding a severely degraded core are the magnitude and the rate of the oxidation by reflood steam of the retained Zircaloy in the hot, declad region of the core and the resultant local heating and further core degradation and hydrogen generation. Such oxidation occurred during reflood in both the LOFT FP-2 test and the TMI-2 accident and also in some CORA experiments. It is not predicted at all well by current codes.18 Much of this uncertainty comes from the very large uncertainty in the fraction of the unoxidized molten Zircaloy cladding that is retained in the hot, declad

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region of the core. During the relocation of the metallic Zircaloy melt, this molten Zircaloy is retained on fuel pellet surfaces and/or in fuel cracks by wetting. This retained Zircaloy provides the "chemical fuel" for oxidation by the steam generated during reflooding, and results in local heating with further local core degradation and hydrogen generation. This fraction of the retained molten Zircaloy on the hot, declad fuel rods is not known, nor is there a quantitative model of the retention process. A related uncertainty is the governing ratelimiting process on the oxidation of this retained core Zircaloy. A major program of experiments and analysis on the consequences of reflooding a degraded core, the QUENCH program, has been started at Forschungszentrum Karlsruhe (FZK).73 This research, which is follow-on to the recently completed CORA program, should resolve much of the uncertainty regarding the consequences of reflooding a degraded core through core degradation and early-phase melt progression, including questions of cooling. The significant remaining uncertainties regarding the processes of both non-explosive and explosive melt-water interactions were discussed in the previous section, and these also apply directly to melt-water interactions under reflood conditions. An energetic steam explosion is precluded under severe accident conditions unless significant masses of melt (on the order of kilograms) and water can mix. Such mixing of ceramic melts and water does not appear possible during reflood in LWR core-melt accidents because of the intervening crust, debris, and temperature difference. Thus an energetic water-ceramic-melt steam explosion would not appear to be possible during reflooding, in agreement with the TMI-2 accident. An energetic steam explosion might occur later upon ceramic pool melt-through and drainage into the lower plenum water, however, if the reactor vessel is at low pressure. For metallic melts, it appears difficult to preclude with confidence the access of significant masses of liquid water and molten Zircaloy to each other. Thus the possibility of an energetic moltenZircaloy-water steam explosion is more difficult to preclude, even though no steam explosions occurred in a small number of water reflood tests with aluminum melts.72

CONCLUSIONS This paper is a summary of the current state of technical knowledge and the principal remaining technical uncertainties regarding the processes of core degradation and in-vessel core-melt progression extending up to vessel failure in LWR core-melt accidents, including the consequences of core reflooding. The yard stick for considering the significance of the remaining technical uncertainties has been estimates of whether or not the range of the current uncertainty might affect, by about 10% or more, the mass or other characteristics of the melt released from the core and possibly later from the reactor vessel, the hydrogen generated, or the core characteristics for determining the fission product release and primary system retention. Reducing all the uncertainties given here to this level would provide an exceedingly high level of technical certainty to the understanding of core degradation and melt progression. It is found that only in the core degradation phase of the LWR loss-ofcoolant accident (T<1500 K) does current knowledge appear to be sufficient to meet this yard stick. It again needs to be emphasized most strongly that resolution of all the technical uncertainties, particularly to this level, is not necessary for safety assessments of different issues in the regulatory process. Resolution of all of these uncertainties may not be technically feasible, and certainly would not meet reasonable cost-benefit criteria. In addition, different regulatory issues require knowledge of different aspects of melt progression technology, and with different permissible levels of uncertainty. What this paper does is to provide an awareness and a listing of the currently existing technical uncertainties.

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ACKNOWLEDGMENTS The author would like to express his appreciation to these of his colleagues for their valuable comments on the manuscript: Dr. Ali Behbahani, U.S. Nuclear Regulatory Commission; Dr. Dae H. Cho, Argonne National Laboratory; and Drs. Randall O. Gauntt, Kenneth O. Reil, and Rodney C. Schmidt, Sandia National Laboratories.

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INDEX

Accidents, major, 25–26 psychological effect, 43 Acute Radiation Syndrome, 1, 2, 32 Adaptive learning, 115 model reference control, 115–116 Adjoint: see Importance Adler–Adler, 132 AIDS, chemically induced, 32 Albedo problem, 79 ANN (Artificial Neural Networks), 96 learning & training, 101 APOLLO-2 code, 266, 269 Asymptotic analysis, 76 ATHLET/CD code, 288 Autonomous plant: see Plant Back scattering, 75 Balance model, 74, 111 Beginning of life (BoL), 158, 159 Behren’s theory, 238 Bickley–Naylor function, 258 Bloch’s theorem, 233 Boiling Water Reactor (BWR), 153, 232 Boron cross section, 142, 152, 167 Boundary condition, 250 BPN (Back Propagation Network), 103, 109, 111 Breit–Wigner, 132 Buckling, neutron, 225 et seq. small approximation, 238, 276 BWR lattices, 158, 275 Caesium contamination, 3 Cancer leukaemia, 33 thyroid, 32–33 CANDU cell, 275 cluster, 163 reactor, 232 Carbaryl, 4 Causal loops, 198 Chapman– Jouquet conditions, 302 Kolmogorov equations, 53 Clad ballooning, 288 Closure, mathematical, 48, 58, 76, 81–82 Collision probability, 248, 252–259 Competition, 197 Construction, plant, 208

Containment, 21 Control automatic, 12 reactor, 113 rod, 284, 288 Conversion ratio (CR), 164–168 Coolant-voiding, 244 CORA fuel damage test facility, 285, 291, 311 Core-blockage, 292 Corngold condition, 236 Correlation angular, 245, 259 CORVIS program, 302 Creep rupture, 305 Critical buckling, 229, 267 conditions, 231 Criticality, prompt, 13 CRONOS code, 266, 269 Cross Section Evaluation Working Group (CSEWG), 131 Cyanogen pool theory, 30 Dancoff factor, 136 DEBRIS code, 300 Deniz–Gelbard coefficient, 240 Departure from nucleate boiling (DNB), 110 Diffusion, neutrons, 199 coefficient, 224et seq. parameters, 224 Direct Containment Heating (DCH), 295 Discontinuity factor, 226 Döppler coefficient, 20, 155, 156 DRAGON code, 259, 266, 271 Embedding, invariant: see Invariant Emergency Core Cooling System (ECCS), 11, 19 plans, 17, 24 Enhanced cooling mechanism, 306 Environmental effects, 197 EPICURE program, 266, 268 Equivalence procedure, 226, 239, 262, 278 Eutectics, 289, 293 Evacuation, 35–36 Expert systems, 113 Explosion melt-water, 302 steam, 284

317

318 Failure component, 29 management, 22 organisational, 22 FARO program, 308 Fick’s law, 7, 141, 224, 229, 262 Finality: see Closure Fission product heating, 288 release, 299 Flattop, 136–137, 149 Flooding core, 286, 310 progression (FP), 300 Fluctuations, 116 Focker–Planck, 77 FSU (former Soviet Union), 9 Fuel-failure threshold, 294 Fuzzy logic, 113, 126 Galanin’s constant, 120 Gas reactors gas-graphite, 232 high temperature (HTGR), 232, 276 Godiva, 136, 141 Green’s functions, 119–122 GTRAN2 code methodology, 259 Heat rate, 109 Hebbian rule, 103 Helium fill-gas, 298 Helmholtz equation, 122 operator, 119 HiC experiment, 156 Hopfield model, 98, 104 network, 100, 105 HOT (Human, Organisational, Technological), 1, 13, 18 HTGR: see Gas reactor HWPR cell, 162 Hydrogen explosion, 13 generation, 285, 299, 310 ICARE2 code, 288 ICSBEP, 137 Image pile method, 225 Importance weighting, 241 Institute, West Virginia, 27 Insurance, 38–39 Integrated Fuel–Coolant Interaction (IFCI) code, 304

INDEX Interface model, 80, 90 International Project on the Health Effects of the Chernobyl Accident (IPHECA), 32–33 Invariant embedding, 71 Inverse tasks, 95 et seq. ITHINK program, 201 Jemima, 137, 190, 191 Jezebel, 141, 149, 191 Kalman filter, 112 turbulence modelling, 299 Keenan, Judge John, 38 Kemeny report, 26–27 KESS code, 300 Kohonen networks, 106 K-Reactor, 162 KROTOS experiment, 304 LANL(LASL), 136, 137 Larson–Miller parameters, 306 Leakage coefficient, 224 et seq. cross section, 230 Legasov, Valery, 21 Leuneberger observer, 112 Leukaemia, 33 Life, worth of, 37 Light Water Reactor (LWR), 283 Liouville equation, 58, 63, 72 -Neumann expansion, 246 Liquid Metal Fast Reactor (LMFR), 232 LLNL, 170 et seq. Loss of coolant Accident (LOCA), 286 Loss of Fluid Test (LOFT) facility, 285, 311 MAGGENTA code, 259, 266, 275 Maintenance, 18, 212–4 preventative, 213 Master equation: see Liouville MCNP, 136 et seq., 266 Media, 211–216 MELCOR code, 288 Melt jet, 303–305 Progression (MP), 283 et seq. Relocation in Reactor Structure (MERIS), 294 Melt-drainage, 290 Methyl icocyanite (MIC), 1 et seq. health effects, 30 MHTGR lattice, 191 Migration area, 233, 235 Milgram’s method, 275 Milne problem, 68

INDEX Mixtures, binary stochastic, 49 Modelling problems, 217 Monte Carlo, 48, 57, 233, 242, 273 Morale, 2, 22 Motive (cell), 224 MOX lattices, 149, 157 MTC: see Temperature coefficient Multigroup modelling, 223 Multiplication factor (K), 227 Neumann series, 61 Neuron, artificial, 97 Neutron leakage, 224 New Production Reactor, 161 Nuclear Regulatory Commission (NRC), 209 et seq. Nusselt number, 299 up-to-down, 307 Optical depth, 62, 66–69, 83 ORM: see Reactivity Perceptron, 97 Perturbation theory generalised, 263 Planck function, 88 Plant, autonomous project, 114 nuclear, 208 approximation, 76, 228 et seq. Poisson equation, 122 PORV: see Valve Power coefficient, 11 Operated Relief Valve (PORV): see Valve plant, 210 plant construction, 208 Pressurised water reactor, 2560; see also PWR Pripyat, 9 Productivity, 206 Project management, 198 Public concern, 212 Utility Commission (PUC), 210 Pulsed-sphere benchmark, 174 PWR lattices (assemblies), 153, 232, 271– 273 QUENCH program, 312 Radiophobia, 33 RASPLAV program, 309 Rayleigh number, 296, 299 RBMK (graphite-water reactor), 9 et seq. lattice, 153, 160

319 Reactivity control, 113 Operating Margin (ORM), 19, 25 surveillance, 110 Reactor lower head, 304 Reciprocity relation, 253 Reflection, 248, 275 quasi-isotropic, 249 RELAP code, 288 Renewal theory, 86 Rod geometry, 70 Scattering isotropic, 228 linearly anisotropic, 243 SCDAP code, 288 Selengut’s principle, 264 Self-organising maps, 106 Sevin: see Carbaryl Seweso accident, 41–42 SHEBA-II, 146–148, 190 Simulators, lack of, 20, 23 SL-l reactor accident, 302 Smoothing method, 65 Social systems, 219 Source Term (ST), 295 Specular reflection, 248 SPH (superhomogénéisation), 262–265 Stability BWR, 109 Statistics, homogeneous, 51 et seq. Steam explosion, 302, 304, 312 reflood, 284 spike, 285 Structural learning algorithm, 105 Symmetry properties, 236 et seq. Temperature coefficients, 108 control, 113 TEXAS code, 303 Three Mile Island (TMI II, TMI-2), 3, 26, 107, 215, 284 et seq. TIBERE model, 243 et seq. Training, 16, 219 patterns, 125 Transfer function, 117 Transport equation, 223 stochastic, 47 Turbulent mixtures, 50 TV accident coverage, 40 UCC (Union Carbide Corporation–US), 3, 4, 23 UCIL (Union Carbide Company of India), 1 Uranyl flouride, 168, 191

320 Valve blow-down, 5 isolation, 8 power operated relief (PORV), 26, 286, 295 relief, 5 safety, 8, 17 VENSIM program, 201 Vibrations, 110 Voids, 9, 277 coefficient, 20 VVER (water reactor), 9

INDEX WASH-1400 study, 302 Waste disposal, 197 Wigner–Seitz model, 242 Window, transmission, 71, 81 Work quality, 205 Xenon poisoning, 12 XOR problem, 97 XR2–1 experiment, 292 Zircaloy, 288 et seq.