Thermal and Reliability Criteria for Nuclear Fuel Safety (River Publishers Series in Chemical, Environmental, and Energy Engineering) 8770224013, 9788770224017

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Thermal and Reliability Criteria for Nuclear Fuel Safety

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Thermal and Reliability Criteria for Nuclear Fuel Safety

Maksym Maksymov Odessa National Polytechnic University, Ukraine

Svitlana Alyokhina A. Pidgorny Institute of Mechanical Engineering Problems of the National Academy of Sciences of Ukraine; V.N.Karazin Kharkiv National University, Ukraine

Oleksandr Brunetkin Odessa National Polytechnic University, Ukraine

River Publishers

Published, sold and distributed by: River Publishers Alsbjergvej 10 9260 Gistrup Denmark www.riverpublishers.com

ISBN: 978-87-7022-401-7 (Hardback) 978-87-7022-400-0 (Ebook) ©2021 River Publishers

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, mechanical, photocopying, recording or otherwise, without prior written permission of the publishers.

Contents

Afterword

ix

Preface

xi

List of Figures

xv

List of Tables

xxiii

List of Abbreviations

xxix

1

2

3

Physical Safety Basis of WWER Nuclear Fuel 1.1 Fuel Burn-Up as a Nuclear Safety Criterion . . . . . . . . . 1.2 Influence of the Reactor Operating Mode on the Efficiency of WWER-1000 Fuel Cycles . . . . . . . . . . . . . . . . . . 1.3 Design Constraints and Engineer Suitable Coefficients When Designing and Operating WWER Fuel Loads . . . . . . . . 1.4 Criteria and Methods of Nuclear Fuel Safety Evaluation Under Operation . . . . . . . . . . . . . . . . . . . . . . . Modern Approaches to the Heat Exchange Modeling in NPP Equipment 2.1 Abstract Models . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Generalization of the Mathematical Model . . . . . . . . . . 2.3 Simplified Method of the Numerical Solution of Nonstationary Heat Transfer Problem Through a Flat Wall . . . . . . . . . 2.4 Method of Approximate Analytical Solution of a Nonstationary Heat Transfer Problem Through a Flat Wall . . . . . . . . . Safety Criteria for WWER-1000 Fuel Assembly When Making a Decision About Its Dry Storage 3.1 Variable Modes of NPP Operation . . . . . . . . . . . . . .

v

1 1 6 10 12 29 29 48 55 73 93 94

vi Contents 3.2 3.3 3.4 4

5

6

Assessment of Emergency of Fuel Assemblies in Light Water Reactors (LWR) . . . . . . . . . . . . . . . . . . . . . . . 99 Qualitative Evaluation of WWER-1000 Fuel Assemblies . . 101 Performance Criteria of Fuel Elements . . . . . . . . . . . . 105

Effect of Reactor Capacity Cyclic Changes on Energy Accumulation of Irreversible Creep Deformations in Fuel Claddings 4.1 Simulation of Technological Parameters of the NPP with WWER-1000 Under Cyclic Loading . . . . . . . . . . . . . 4.2 Analysis of the Effect of Parameter Variations of Control Programs on the State of Fuel Assemblies During the Cyclic Change of the Reactor Capacity . . . . . . . . . . . . . . . 4.3 Uniformity of NPP Energy Release Parameters . . . . . . . 4.4 Simulation of Fuel Cladding Failure and Axial Offset in the Cyclic Mode of the NPP Operation . . . . . . . . . . . . . .

121 122

125 130 138

Analysis of WWER 1000 Fuel Cladding Failure 5.1 Initial Data of the Evaluation Model of the Probability of Fuel Cladding Depressurization . . . . . . . . . . . . . . . . . . 5.2 Simulation of Fuel Cladding Reliability . . . . . . . . . . . 5.3 Computational Method of Fuel Cladding Failure . . . . . . 5.4 Computational Analyses of Stress-strain State of the Fuel Cladding . . . . . . . . . . . . . . . . . . . . . . . . . . . .

151

Thermal Safety Criteria for Dry Storage of Spent Nuclear Fuel 6.1 Dry Storage of Spent Nuclear Fuel in Ukraine . . . . . . . . 6.2 Method of Thermal Safety Analysis of Dry Storage of Spent Nuclear Fuel . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Thermal Condition of Containers for Dry Storage . . . . . .

197 198

151 155 160 181

202 223

Index

257

About the Authors

259

Afterword

The trend in the development of commercial nuclear power indicates economic and technical attractiveness of the further application of light-water non-boiling pressure vessel reactors and the expansion of the application of such a technology for NPP power units. The expansion should be understood as two aspects: the extension of the plant life of operating units or the construction of new ones. One of the areas of commercialization is a significant reduction in the cost of technologies for SNF reprocessing or storing. The analysis has shown that even taking into consideration the variety of such technologies, only long-term dry storage of spent nuclear fuel from reactors of this class is common for fuel from different suppliers. This means that the issues of safe and reliable dry storage of spent nuclear fuel will be in the focus of attention of researchers for a long time. The authors of the book are convinced proponents of the idea that there is no alternative to the development of nuclear energy. Within the framework of the book, the issue of how to develop nuclear power in the future was not discussed; the attention was focused on the issues on which the development of nuclear power depends, namely, the criteria for the safe operation of spent nuclear fuel were discussed. The modern world is very dynamic in all its manifestations, but this dynamism is spasmodic; it is especially well manifested in the development of nuclear power, including the developed technologies for dry storage of fuel. The designers and manufacturers of this technology are changing, the geography of the introduction of new technology samples is expanding, but the basic technologies for removing energy from the irradiated fuel assembly and ensuring the strength of the cladding during its storage remain the same. Only the requirements for safety, reliability, and efficiency are constantly being reinforced. These requirements are applied not only to newly created and designed technologies, but also to those dry storage facilities that are in operation. Scientific research in the field of materials science, extension of knowledge of the physics of processes associated with dry storage, new approaches

vii

viii

Afterword

and methods for modeling processes occurring during long-term storage of spent nuclear fuel, as well as improving information and measuring systems and information processing facilities are the most important components of ensuring safe and reliable SNF storage. The potential for borrowing engineering ideas in the world practice of designing dry storage facilities is minimized with each project and has practically reached its limits. It is clear that the strategic search for innovations in order to improve dry storage technologies should become an important part of the complex task of innovation in nuclear energy. From the point of view of unsolved problems in the field of dry storage of spent nuclear fuel, the following direction presented in the book “Thermal and Reliability Criteria for Nuclear Fuel Safety” can be distinguished: the authors showed one of the possible ways of making a decision on the long-term storage of spent nuclear fuel in a dry storage facility and the possibility of its subsequent disposal. On the basis on the material presented in the book, a number of the following important conclusions can be drawn. First, compliance with the strength criteria for making a decision on dry storage of spent nuclear fuel guarantees the absence of cladding failure of the fuel element. Secondly, the cladding can be destroyed due to the violation of the removal of residual energy release through it. Both in the first and the second cases, the ongoing processes depend on the properties of the cladding material.

Preface

The authors of this book are united in their research activities by the desire to ensure the European level of operational safety of nuclear power plants (NPP) in Ukraine, namely in terms how to save the resource of operating nuclear fuel and the possibility of making a decision on its long-term dry storage, which will undoubtedly result in a significant reduction in the risks of nuclear incidents and reputational losses at the final stages of the nuclear fuel cycle. It should be noted that for a long time the authors have been cooperating with Studsvik, which is the moderator of a number of scientific and technical works as part of the Studsvik Cladding Integrity Project (SCIP). The cooperation is carried out on the basis of an international consortium, that includes Ukraine, which is represented by a number of research organizations. These are the results obtained in the framework of the SCIPIII and SCIP-IV projects that stimulated the desire to formalize the existing scientific groundwork on safety issues of the nuclear fuel for water-water energetic reactors (WWER) at the final stages of the nuclear fuel cycle in the form of a book. It is worth mentioning that it was the visual examination of irradiated nuclear fuel in the Studsvik laboratories, which the authors had the opportunity to observe; they influenced the final understanding of the paradigm that is presented in this book. The object-matter of the research in the book is the safety of nuclear fuel for WWER-1000 reactors under normal operating conditions at the final stages of the nuclear fuel cycle. The subject-matter of the research is the processes of thermal physics of nuclear fuel and the accumulation of failure to the cladding of nuclear fuel, which determine the model of its safe operation in the WWER-1000 reactor and in open dry container storage facilities for spent nuclear fuel under normal operation conditions. First of all, the book is based on the scientific achievements of the authors of the book, M. Maksymov, S. Alyokhina, and O. Brunetkin, who are the Doctors of Engineering Science. A number of the propositions that

ix

x

Preface

are included in the book were obtained by the authors in collaboration with PhD students, where the authors were scientific supervisors. There are six sections in the book. The first chapter “Physical Safety Basis of WWER Nuclear Fuel” is devoted to the safety of the SNF storage facility, which is necessary to create guaranteed conditions for thermal states throughout the entire operation time of the storage facility and it creates the possibility to control the sources of ionizing radiation. The second chapter «Modern Approaches to the Heat Exchange Modelling in NPP Equipment» reveals the methods of the development of mathematical models of nonstationary heat transfer of the technical system, which provides the heat transfer related to any state of nuclear fuel. It is necessary to generalize the principles of physical modelling in order to fold information, which gives opportunity to check it by means of a mathematical model, evaluation of alternative variants of the physical model under consideration, and the choice of the best one. The third chapter «Safety Criteria for WWER-1000 Fuel Assembly when Making a Decision about its Dry Storage» is devoted to the search for optimality criteria for the control of a NPP with WWER-1000 for which it is necessary to find efficiency criteria that would take into account the requirements of nuclear safety. This makes it possible to compare any methods of operating the reactor core, including power maneuvering. The fourth chapter «Effect of Reactor Capacity Cyclic Changes on Energy Accumulation of Irreversible Creep Deformations in Fuel Claddings» presents the modelling of the operation of nuclear fuel in cyclic modes; this is necessary to ensure compensation for power changes within the daily or weekly production scheduling of the power system requirements, which makes it possible to compare the considered control programs with the inherent specifics of each change of technological parameters, which has a significant effect on the interaction “the fuel pellet and the cladding» and leads to leakage. The fifth chapter «Analysis of WWER 1000 Fuel Cladding Failure» deals with the definition of the computed values of leakage probability of fuel element claddings, taking into the account the inhomogeneity of the distribution of energy release among fuel elements of fuel assemblies, which is necessary to control the properties of fuel elements; it also makes it possible to control the value of cladding failure and, therefore, at the same time, the predicted probability of depressurization of the cladding of fuel elements happens.

Preface

xi

The sixth chapter “Thermal Safety Criteria for Dry Storage of Spent Nuclear Fuel” is devoted to the development of the basis for the analysis of thermal regimes of SNF dry storage, which is necessary for the safe longterm operation of an interim SNF storage, as a result of which it is possible to make an informed decision about the possibility of subsequent reprocessing or disposal of nuclear fuel. The presented material in the chapters allowed the authors to formulate the following method, that ensures the safety of the nuclear fuel that is operated. The conservatism which lies in the design of the fuel elements can be used not only for an increase in the capacity of a nuclear power unit in excess of the design one or for operation in maneuverable modes, but also for ensuring long-term dry storage of spent nuclear fuel. It is generally accepted that mechanical failure of nuclear fuel according to the stress corrosion cracking model is completely excluded due to the limitation of the linear capacity and the rate of its increase, but this is not always the case, as it is possible to simultaneously impose technological operating conditions when such a previously excluded model starts affecting nuclear and thermal safety. To prevent such a possibility, it is constantly necessary to minimize or practically eliminate the following four processes during the operation of nuclear fuel in the reactor: — not to load any fuel assemblies with the first power leap immediately after refueling; — alternate switching on of main circulation pumps when gaining power, especially in the first 40 effective days; — if, after reloading of nuclear fuel, the unit operated at its nominal capacity for several days (there was no sufficient accumulation of cracks in the fuel pellets) and was unloaded or stopped, then its reloading must be carried out according to a special program, and not according to operating management recommendations; — not to allow an opposite change in the coolant temperature with changes in the current power in the upper and lower parts of the reactor core. If simultaneously any two of the stated processes occur, then they significantly reduce the resource and do not allow keeping the fuel in proper condition for its dry storage. Moreover, it will not be possible if at least three of any processes are superimposed in one time interval. It is very difficult to predict what will be the properties of the fuel if four processes coincide at the same time in the current time interval.

xii Preface As a rule, a probabilistic safety analysis is carried out in order to prevent emergency design conditions of severe accident conditions. Such an analysis does not objectively evaluate the situation, but the analysis method allows ultimately estimating the state of the fuel, which turns out to be erroneous in principle under current operation conditions. Due to the fact that all the options in the analysis were not estimated, during operation there is a high probability of making a wrong decision. As a result, the operations personnel choose “the best option from the worst one.” The best one of those, which were considered in the probabilistic safety analysis. If to apply this strategy when operating a power unit, then instead of objective assessing of the processes that occur, weighing all the pros and cons, the operator tries to evaluate the state of the nearest future. As a result, the decision made and the development of the situation keep the power unit from emergency modes, but at the same time the number of failures formed in the nuclear fuel increase. However, paradoxically, no one takes these failures into account in the future, and they usually begin to manifest themselves at the end of the period when the fuel is in the reactor core. The paradigm proposed by the authors is as it follows. To identify the current state of the fuel and the ongoing processes that affect the safety of the fuel, and then operate it so that the subsequent states of the fuel ensure its long-term operation. Anyone who cannot identify the current state of the nuclear fuel in the core simply does not know what is happening with the fuel at the moment.

List of Figures

Figure 1.1 Figure 1.2 Figure 1.3 Figure 1.4 Figure 1.5 Figure 1.6 Figure 2.1 Figure 2.2 Figure 2.3 Figure 2.4

Figure 2.5

Figure 2.6

k dependence on the burn-up for a standard UO2 nuclear fuel assembly. . . . . . . . . . . . . . . . . Dependence of the effective neutron fission coefficient for various calculation models [17]. . . . . . . Field of local fuel element loads with respect to the factor of margin depending on the burn-up. . . . . Fuel element linear load jumps as a result of refueling. . . . . . . . . . . . . . . . . . . . . . . Axial offset after the transient process at the end of a (1) three- and (2) four-year campaign. . . . . . . Computation scheme of thermotechnical reliability. Scheme of technical system operation and its interaction with the environment. . . . . . . . . . . Calculation model for one-dimensional discrete analogue. . . . . . . . . . . . . . . . . . . . . . . Minimal computational grid. . . . . . . . . . . . . Relative temperature Θ depending on the relative coordinate X and the relative moments of time Fo for analytical Θa and numerical Θn solutions. (a) Bi = 0.004 (0.008). (b) Bi = 0.5 (1.0). (c) Bi = 5 (10). (d) Bi = 50 (100) . . . . . . . . . . . . . . . Figure 2.5. Profiles of relative temperature Θ over relative thickness of the plate X depending on different Fo numbers. . . . . . . . . . . . . . . . . Temperature profile with asymmetric heating of the plate. (a) Examples of numerical calculation results of the relative temperature Θ depending on the relative coordinate X at different values of Fo. (b) Scheme for analytical calculations. . . . . . . . . .

xiii

3 4 12 12 18 19 30 56 61

67

69

81

xiv List of Figures Figure 2.7

Figure 4.1

Figure 4.2

Figure 4.3

Figure 4.4 Figure 4.5 Figure 4.6 Figure 4.7 Figure 4.8 Figure 4.9

Temperature changes Θ along the thickness of the plate X in nonstationary heat transfer at different times Fo. (a) The process of energy accumulation from the initial state Bi1 = 0, Bi2 = 100. (b) The process of energy accumulation from the initial state Bi1 = 1, Bi2 = 100. (c) Energy release from the initial state Bi1 = 4, Bi2 = 100. (d) computational scheme. . . . . . . . . . . . . . . . . . . . . . . . Static control programs for NPPs with WWER1000 under different control programs: 1 is the coolant temperature at the outlet of the reactor core, tout ; 2 is the average coolant temperature in the reactor core, tav ; 3 is the coolant temperature at the inlet to the reactor core, tin ; 4 is the temperature of the saturated steam in the second circuit, ts ; 5 is the pressure of the steam in the second circuit, pII . . . . Regulatory areas of the axial offset values depending on the capacity level of the reactor: 1 – recommended area; 2 – admissible area; 3 – non-recommended area; 4 – forbidden area. . . . . Schematic diagram of the NPP with WWER-1000 control, which implements the control program tav = const. . . . . . . . . . . . . . . . . . . . . . Simulation model of the control program when tav = const. . . . . . . . . . . . . . . . . . . . . . Schematic diagram of the NPP with WWER-1000 control when pII = const. . . . . . . . . . . . . . . Simulation model of the control program when pII = const. . . . . . . . . . . . . . . . . . . . . . Schematic diagram of the NPP WWER-1000 where tin = const. . . . . . . . . . . . . . . . . . . . . . Simulation model of the control program when tin = const. . . . . . . . . . . . . . . . . . . . . . . . Change of the position of the working group of the Control Rod Drive Mechanism in order to maintain the axial offset: 1 – high altitude position of the H working group the Control Rod Drive Mechanism, % from the lower part of the reactor core; 2 – axial offset, %. . . . . . . . . . . . . . . . . . . . . . .

84

123

129

133 135 135 136 137 138

142

List of Figures xv

Figure 5.1 Figure 5.2 Figure 5.3 Figure 5.4

Figure 5.5 Figure 5.6

Figure 5.7

Figure 5.8

Figure 5.9

Figure 5.10

Figure 5.11

Figure 5.12

Figure 5.13

Arrangement of the control group of the 10th group: (number) – number in the cell in the reactor core. . Change of WWER-1000 capacity depending on time. . . . . . . . . . . . . . . . . . . . . . . . . . Change of the position of the 10th group of the Control Rod Drive Mechanism depending on time. Scheme of the distribution of fuel assemblies in the cells of the sector of symmetry of the reactor core: I, II, III, and IV – first, second, third, and fourth year of campaign, correspondingly; (number) –number in the cell in the reactor core. . . . . . . . . . . . . Fuel element distribution in the groups I*,. . ., IV* in the rearrangement 5-30-10-43. . . . . . . . . . . Time-dependent failure in the rearrangement 5-3010-43 of algorithm 2 (four given groups of fuel elements). . . . . . . . . . . . . . . . . . . . . . . Time-dependent failure in the rearrangement 9-1120-1 of algorithm 2 (four given groups of fuel elements). . . . . . . . . . . . . . . . . . . . . . . Time-dependent failure in the rearrangement 3-2254-29 of algorithm 2 (four given groups of fuel elements). . . . . . . . . . . . . . . . . . . . . . . Time-dependent failure in the rearrangement 13-1921-42 of algorithm 2 (four given groups of fuel elements). . . . . . . . . . . . . . . . . . . . . . . Time-dependent failure in the rearrangement 231-18 of algorithm 2 (four given groups of fuel elements). . . . . . . . . . . . . . . . . . . . . . . Figure 5.11. Time-dependent failure in the rearrangement 55-41-12-6 of algorithm 2 (three given groups of fuel elements). . . . . . . . . . . . . . . Time-dependent failure in the rearrangement 4-3268-8 of algorithm 2 (four given groups of fuel elements). . . . . . . . . . . . . . . . . . . . . . . Time-dependent failure in the rearrangement 55-1118-43 of algorithm 6 (three given groups of fuel elements). . . . . . . . . . . . . . . . . . . . . . .

153 153 154

154 174

182

182

183

183

184

184

185

186

xvi

List of Figures

Figure 5.14

Figure 5.15

Figure 5.16

Figure 5.17

Figure 5.18

Figure 5.19

Figure 5.20 Figure 5.21

Figure 5.22 Figure 5.23

Figure 5.24 Figure 5.25

Figure 6.1 Figure 6.2 Figure 6.3 Figure 6.4

Time-dependent failure in the rearrangement 1332-20 of algorithm 6 (four given groups of fuel elements). . . . . . . . . . . . . . . . . . . . . . . Time-dependent failure in the rearrangement 3-3110-8 of algorithm 6 (four given groups of fuel elements). . . . . . . . . . . . . . . . . . . . . . . Time-dependent failure in the rearrangement 9-1968-42 of algorithm 6 (four given groups of fuel elements). . . . . . . . . . . . . . . . . . . . . . . Time-dependent failure in the rearrangement 4-4112-29 of algorithm 6 (four given groups of fuel elements). . . . . . . . . . . . . . . . . . . . . . . Time-dependent failure in the rearrangement 2-3021-6 of algorithm 6 (four given groups of fuel elements). . . . . . . . . . . . . . . . . . . . . . . Time-dependent failure in the rearrangement 5-2254-1 of algorithm 6 (four given groups of fuel elements). . . . . . . . . . . . . . . . . . . . . . . Time dependence σe /σ0 (t) for the given group IV* of fuel elements in the rearrangement 9-11-20-1. . . Time dependence σθ (t)/250 MPa for the given group IV* of fuel elements in the rearrangement 9-11-20-1. . . . . . . . . . . . . . . . . . . . . . . Time dependence σe /σ0 (t) for the given group IV* of fuel elements in the rearrangement 13-19-21-42. Time dependence σθ (t)/250 MPa for the given group IV* of fuel elements in the rearrangement 13-19-21-42. . . . . . . . . . . . . . . . . . . . . . Time dependence σe /σ0 (t) for the given group IV* of fuel elements in the rearrangement 3-22-54-29. . Time dependence σθ (t)/250 MPa for the given group IV* of fuel elements in the rearrangement 3-22-54-29. . . . . . . . . . . . . . . . . . . . . . Storage of spent nuclear fuel at the site of Zaporizhzhya NPP. . . . . . . . . . . . . . . . . . Calculation area for the group of containers. . . . . Geometric form of calculation area. . . . . . . . . Calculation area for the storage cask. . . . . . . . .

186

187

187

188

188

189 190

190 191

191 192

192 199 212 213 215

List of Figures

Figure 6.5

Figure 6.6 Figure 6.7

Figure 6.8 Figure 6.9 Figure 6.10 Figure 6.11 Figure 6.12

Figure 6.13

Figure 6.14 Figure 6.15 Figure 6.16

Calculation area for determining the thermal state of the spent fuel assembly: 1 – guide tube, 2 – helium, 3 – fuel element, 4 – burnable absorber rods. . . . . Temperature field of ventilation air. . . . . . . . . . Dependence of the value of equivalent thermal conductivity in the cask for spent nuclear fuel on its storage time. . . . . . . . . . . . . . . . . . . . Flux in the ventilation tract of the container. . . . . Density field of the convective heat flux. . . . . . . Temperature change along the axis of the storage container with respect to its height. . . . . . . . . . (a) Temperature field of the surface of the cask and (b) the outer surface of the container. . . . . . . . . (a) Temperature change on the surface of the cask and (b) on the surface of the container in the cross sections in the center of outlet ventilation ducts (1), between inlet and outlet ventilation ducts (2), between outlet channels, and on the axis which is perpendicular to the axis of the inlet ducts (3) with respect to the height. . . . . . . . . . . . . . . . . Heat transfer coefficient on the surface of the storage cask in the sections: in the middle of outlet vents (1); between inlet and outlet vents (2); between outlet vents and in the middle of inlet vents (3). . . . . . . . . . . . . . . . . . . . . . . . . . . Temperature field of the cask with spent fuel assemblies. . . . . . . . . . . . . . . . . . . . . . Numbers of fuel assemblies and surfaces for determining boundary conditions. . . . . . . . . . Temperature change on the surface of the guide tubes with respect to the height. (a) Fuel assembly 4. (b) Fuel assembly 5. (c) Fuel assembly 6. (d) Fuel assembly 12. (e) Fuel assembly 14. (f) Fuel assembly 16. (g) Fuel assembly 22. (h) Fuel assembly. 24. . . . . . . . . . . . . . . . . . . . .

xvii

217 225

227 227 228 229 230

231

232 233 234

235

xviii List of Figures Figure 6.17

Figure 6.18

Figure 6.19 Figure 6.20 Figure 6.21

Figure 6.22 Figure 6.23

Figure 6.24

Figure 6.25

Figure 6.26

Figure 6.27

Change of heat transfer coefficients on the surfaces which surround the fuel assembly with respect to the height. (a) Fuel assembly 4. (b) Fuel assembly 5. (c) Fuel assembly 6. (d) Fuel assembly 12. (e) Fuel assembly 14. (f) Fuel assembly 16. (g) Fuel assembly 22. (h) Fuel assembly 24. . . . . . . . . . Temperature field in the horizontal cross section of the storage cask for spent nuclear fuel at the level of their maximum temperatures when the atmospheric air temperature is 40◦ C. . . . . . . . . . . . . . . . Location of fuel elements with maximum temperature. . . . . . . . . . . . . . . . . . . . . . Options of wind direction (u – upper; l – lower ventilation ducts). . . . . . . . . . . . . . . . . . . Change of maximum temperature in the storage container in case of different options of the inward air flow with respect to its velocity. . . . . . . . . . Maximum temperature of fuel assemblies in case of inflow air (option A) by the speed. . . . . . . . . . Dependence of surface temperature of the container on the time of day at different heights. (a) North. (b) East. (c) South. (d) West. . . . . . . . . . . . . . . Time dependence of the concrete container temperature at different distances on the surface (western side). (a) 1.45 m from the ground. (b) 2.9 m from the ground. (c) 4.35 m from the ground. . . . . . . Time dependence of the concrete container temperature at different distances on the surface (northern side). (a) 1.45 m from the ground. (b) 2.9 m from the ground. (c) 4.35 m from the ground. . . . . . . Time dependence of the concrete container temperature at different distances on the surface (northern side) (eastern side). . . . . . . . . . . . . . . . . . Time dependence of the concrete container temperature at different distances on the surface (southern side). . . . . . . . . . . . . . . . . . . . . . . . . .

237

238 240 241

243 244

246

249

250

251

252

List of Tables

Table 1.1 Table 1.2 Table 1.3

Table 1.4

Table 2.1

Table 2.2

Table 2.3

Table 2.4

Table 2.5

Table 2.6

Technological and radiation criteria, which provide safe operation of NPP with WWER. . . . . . . . . . . Main axial offset perturbation actions. . . . . . . . . Computational results of the fuel element walls during the first 7 hours after the completion of the power maneuver. . . . . . . . . . . . . . . . . . . . . . . . Data on how to calculate the safety coefficient before the heat transfer crisis in the maximum loaded fuel assemblies. . . . . . . . . . . . . . . . . . . . . . . . Results of analytical and numerical calculations of the relative temperature Θ at symmetrical heating of an infinite plate. Magnitudes of errors of numerical calculations with respect to analytical calculations. . . Results of numerical calculation of the relative temperature Θ in nonstationary heat transfer through an infinite plate. The magnitudes of error of numerical calculation on a small computational grid. . . . . . . Results of numerical calculations of relative temperature Θ in the process of nonstationary heat transfer through an infinite plate at different calculation steps by time. . . . . . . . . . . . . . . . . . . . . . . . . Results of numerical calculations of the relative temperature Θ in the process of nonstationary heat transfer through an infinite plate at different combinations of the number of nodes in a calculation grid and the magnitude of the calculation steps by time. Comparison of the results of accurate [24] and approximated (2.119) calculations of dimensionless process termination time of heating the body. . . . . . Comparison of the results of minimum temperature calculations. . . . . . . . . . . . . . . . . . . . . . .

xix

13 17

20

21

68

70

71

72

80 83

xx List of Tables Table 2.7

Table 2.8

Table 2.9

Table 3.1

Table 3.2 Table 3.3

Table 3.4 Table 3.5 Table 3.6 Table 3.7 Table 4.1

Table 4.2

Table 4.3

Table 4.4

Table 4.5 Table 4.6

Comparison of the results on how to determine the position of the temperature “minimum” in the vertical section. . . . . . . . . . . . . . . . . . . . . . . . . . Comparison of the results on how to determine the temperatures on the inclined section of the trajectory of their “minimum.” . . . . . . . . . . . . . . . . . . Comparison of the results on how to determine the temperatures on the inclined section of the trajectory of their “minimum.” . . . . . . . . . . . . . . . . . . Change of maximum inter-cartridge gaps of fuel assemblies of alternative design in the process of the third–sixth fuel campaign. . . . . . . . . . . . . . . . Input data of a fuel pellet for the calculation according to [21]. . . . . . . . . . . . . . . . . . . . . . . . . . Computational results of cyclic loading capacity of fuel claddings and fuel claddings with Gd in cyclic modes. . . . . . . . . . . . . . . . . . . . . . . . . . Group of thermophysical criteria. . . . . . . . . . . . Group of corrosion criteria. . . . . . . . . . . . . . . Group of deformation criteria. . . . . . . . . . . . . . Group of strength criteria. . . . . . . . . . . . . . . . Values of the coolant temperature (◦ C) at the inlet to the reactor core for the considered static control programs in relation to the NPP capacity. . . . . . . . Values of the average coolant temperature (◦ C) in the reactor core for considered static control programs depending on the NPP capacity. . . . . . . . . . . . . Values of the coolant temperature (◦ C) at the outlet of the reactor core for considered static control programs depending on the NPP capacity. . . . . . . . . . . . . Values of the steam pressure (MPa) in the second circuit in front of the main steam valve (pII ) for the considered control programs depending on the NPP capacity. . . . . . . . . . . . . . . . . . . . . . . . . Analysis of the control programs for the NPP with WWER. . . . . . . . . . . . . . . . . . . . . . . . . Effect of changes in technological parameters on the change of the axial offset. . . . . . . . . . . . . . . .

88

89

90

102 103

104 114 114 114 115

126

127

128

129 139 141

List of Tables

Table 4.7 Table 4.8 Table 4.9 Table 4.10 Table 5.1 Table 5.2 Table 5.3

Table 5.4 Table 5.5 Table 5.6 Table 5.7 Table 5.8 Table 5.9 Table 5.10 Table 5.11 Table 5.12 Table 5.13 Table 5.14 Table 5.15

Results of axial offset computations during one cycle, %. . . . . . . . . . . . . . . . . . . . . . . . . Linear capacity of the NPP at 100% and 80% of the capacity for the considered control programs. . . . . . Fuel cladding failure for the considered control programs for every computational layer i. . . . . . . . Efficiency indicators of the control programs during the four-year campaign. . . . . . . . . . . . . . . . . Mode and design parameters of WWER-1000, fuel assemblies of alternative design, and fuel elements. . Parameters of the model for calculating the distribution of energy release in the fuel element. . . . . . . . . . Fuel cladding failure and burn-up in the axial segment of the sixth fuel element (accumulated work to material failure). . . . . . . . . . . . . . . . . . . . . Probability P j of fuel cladding depressurization in the jth algorithm. . . . . . . . . . . . . . . . . . . . . . . Probability of fuel cladding depressurization in the algorithm j, %. . . . . . . . . . . . . . . . . . . . . . Relative capacity in the computation cell (i, j) for the first group of fuel elements. . . . . . . . . . . . . . . Relative capacity in the computation cell (i, j) for the first group of fuel elements. . . . . . . . . . . . . . . Relative capacity in the computation cell (i, j) for the second group of fuel elements. . . . . . . . . . . . . Relative capacity in the computation cell (i, j) for the second group of fuel elements. . . . . . . . . . . . . Relative capacity in the computation cell (i, j) for the third group of fuel elements. . . . . . . . . . . . . . . Relative capacity in the computation cell (i, j) for the third group of fuel elements. . . . . . . . . . . . . . . Relative capacity in the computation cell (i, j) for the fourth group of fuel elements. . . . . . . . . . . . . . Relative capacity in the computation cell (i, j) for the fourth group of fuel elements. . . . . . . . . . . . . . Coefficients of relative energy release according to IP software. . . . . . . . . . . . . . . . . . . . . . . . . Coefficients of relative energy release in the axial segment 6 for fuel elements of four groups. . . . . . .

xxi

141 144 146 147 152 155

158 159 159 163 164 165 166 167 168 169 170 171 172

xxii

List of Tables

Table 5.16 Table 5.17 Table 5.18 Table 5.19 Table 5.20 Table 5.21 Table 5.22 Table 5.23 Table 5.24 Table 5.25 Table 5.26 Table 5.27 Table 5.28 Table 5.29 Table 5.30 Table 5.31 Table 5.32 Table 5.33 Table 5.34 Table 5.35 Table 5.36

Comparison of values kv,6,j (ANC − H) and kv,6,j (IP ). Relation of kv,6,j (ANC − H) to kv, 6,j (IP ). . . . . . Characteristics of algorithms 2 and 6. . . . . . . . . . Division of fuel elements into groups in the rearrangement 5-30-10-43 of algorithm 2. . . . . . . . I∗ , . . . , k IV∗ . . . . . Coefficients of energy release kv,i,j v, i,j Values ql, j,max for the groups of fuel elements I*, . . . , IV* in the rearrangement 5-30-10-43. . . . . . . . . . Axial distribution ki,j for the group I* in the rearrangement 5-30-10-43. . . . . . . . . . . . . . . Axial distributions ki,j for the group II* in the rearrangement 5-30-10-43. . . . . . . . . . . . . . . Axial distributions k i,j for the group II* in the rearrangement 5-30-10-43. . . . . . . . . . . . . . . Axial distributions ki,j for the group IV* in the rearrangement 5-30-10-43. . . . . . . . . . . . . . . Values ω for the given groups of fuel elements 5-3010-43. . . . . . . . . . . . . . . . . . . . . . . . . . Values ω for the given groups of fuel elements in the rearrangement 9-11-20-1. . . . . . . . . . . . . . . . Values ω for the given group of fuel elements in the rearrangement 3-22-54-29. . . . . . . . . . . . . . . Values ω for the given groups of fuel elements in the rearrangement 13-19-21-42. . . . . . . . . . . . . . . Values ω for the given groups of fuel elements in the rearrangement 2-31-18. . . . . . . . . . . . . . . . . Table 5.31. Values ω for the given groups of fuel elements in the rearrangement 55-41-12-6. . . . . . . Values ω for the given groups of fuel elements in the rearrangement 4-32-68-8. . . . . . . . . . . . . . . . Values ω for the given groups of fuel elements in the rearrangement 55-11-18-43. . . . . . . . . . . . . . . Values ω for the given groups of fuel elements in the rearrangement 13-32-20. . . . . . . . . . . . . . . . . Values ω for the given groups of fuel elements in the rearrangement 3-31-10-8. . . . . . . . . . . . . . . . Values ω for the given groups of fuel elements in the rearrangement 9-19-68-42. . . . . . . . . . . . . . .

172 172 173 173 175 175 176 176 177 177 178 178 178 179 179 179 179 179 179 180 180

List of Tables

Table 5.37 Values ω for the given groups of fuel elements in the rearrangement 4-41-12-29. . . . . . . . . . . . . . . Table 5.38 Values ω for the given groups of fuel elements in the rearrangement 2-30-21-6. . . . . . . . . . . . . . . . Table 5.39 Values ω for the given groups of fuel elements in the rearrangement 5-22-54-1. . . . . . . . . . . . . . . . Table 6.1 Data to calculate the volumes of the elements which are part of the storage cask. . . . . . . . . . . . . . . Table 6.2 Maximum temperature and flow of ventilation air in the storage container throughout the year. . . . . . . . Table 6.3 Maximum temperatures in the fuel assemblies. . . . . Table 6.4 Maximum temperatures in a separately located container is case of different values of velocity and directions of wind (Ta = 24◦ C). . . . . . . . . . . . . Table 6.5 Data for the definition of maximum temperatures in fuel assemblies. . . . . . . . . . . . . . . . . . . . .

xxiii

180 180 181 224 232 239

242 247

List of Abbreviations

BUC BWR CFD CRDM IAEA LOCA LWR MM NPP PWR SNF TDMA TS WWER

Burn-up credit boiling water reactor Computational Fluid Dynamic Control Rod Drive Mechanism International Atomic Energy Agency loss-of-coolant accident light water reactor mathematical model nuclear power plant pressurized water reactor spent nuclear fuel TriDiagonal-Matrix-Algorithm technical system water-water energetic reactor

xxv

1 Physical Safety Basis of WWER Nuclear Fuel

1.1 Fuel Burn-Up as a Nuclear Safety Criterion The safety of spent nuclear fuel (SNF) management is based on the implementation of the following criteria [1, 2, 3]: • non-exceedance of fuel element temperature limits due to residual energy release; • non-exceedance of the level of ionizing radiation effect on staff and the environment; • guaranteed subcriticality of the storage cask loading or transport cask of the spent nuclear fuel. The issue of ensuring the fuel cladding integrity as one of the physical safety barriers is a topical matter in the process of the development, implementation, and operation of spent nuclear fuel interim storage [4]. The system of thermal and strength criteria of the cladding integrity support has been internationally adopted. Herewith, the thermal criteria are established keeping in mind the necessity to ensure the strength of the fuel cladding. Consequently, the predictive validity of the cladding failure detection under various storage conditions can have a significant impact on the set permissible storage temperatures and, as a result, it can influence the economic factor of spent nuclear fuel dry storage projects [5]. It should be mentioned that the residual heat of each spent fuel assembly under production-line conditions is not currently controlled by standard methods. Instead, computational methods based on experimental dependencies obtained by calorimetric measurements in laboratory conditions are used. Slow kinetic processes cause the residual energy release, while fast kinetic processes are accompanied by the release of gamma radiation,

1

2

Physical Safety Basis of WWER Nuclear Fuel

which is not absorbed or recorded. The results of research establishing the dependence of the 137 Cs gamma radiation intensity and the power of heat formation in the fuel assembly are known [6–10]. However, there may be a significant difference between fuel assemblies, depending on their burn-up and operating conditions in the reactor core. In this regard, the process of forming a container loading cannot entirely rely on computations. In addition, we can achieve significant financial savings if accurate measurements of the residual energy release are established since each container is expensive; that is why we should make the best of its usage. Therefore, the nuclear fuel burn-up should be considered as one of the safety criteria when loading into the storage system. For its effective identification, the method for the experimental determination of the heat of residual SNF energy release by means of fast measurements of gamma radiation has been developed. When analyzing the safety of SNF management systems, the burn-up of specific fuel assemblies is not taken into account, that is, when making an estimate of nuclear safety parameters, all fuels are considered to operate under the same conditions and have some average characteristics. As a result, the calculated value of the subcriticality of the system is conservatively overestimated [11, 12]. This approach was initially due to the imperfection of the calculation programs for determining the reactivity of burned fuel systems and the eventuality of human errors. The development and improvement of computational methods in recent years allow reducing the conservatism of the computational results at the cost of the burn-up account of a specific fuel assembly, without sacrificing the required subcriticality (coefficient kef f ) of a system with a given geometry that takes into consideration neutron leakage and does not reduce its nuclear safety. Figure 1.1 shows an example of the k dependence (the subcriticality of the system with infinite geometric dimensions without taking into account neutron leakage) on the burn-up for a standard UO2 fuel assembly. For burnup of 40 MW-day/kg, the fission coefficient is approximately 30% less than that for fresh fuel [13]. When we consider burn-up as a nuclear safety parameter, the concept of nuclear safety maintenance can be used for all elements which provide the life cycle of spent nuclear fuel, spent fuel pool storage racks and central storage of the atomic nuclear plants, NSF dry storage cask, processing facilities, etc.

1.1 Fuel Burn-Up as a Nuclear Safety Criterion

3

Figure 1.1 k dependence on the burn-up for a standard UO2 nuclear fuel assembly.

Nowadays, the conditions for the promotion of nuclear power plant (NPP) competitiveness require bringing up average burn-up to 60–65 MW-day/kg, which, in its turn, puts a limit on the initial enrichment value of 235 U 4.8%–5.1% for reactors with the capacity of 1000 MW [14]. Under the specified enrichment values, the transportation of SNF in the current transportation cask without the account of burn-up is not possible. This problem has already been encountered when using fuel with enrichment of 4.4% in WWER-440 reactors. We can increase the enrichment value for the CASTOR-V/52 transportation cask from 4% up to 4.6% keeping track of burn-up. It is reported that a possible increase of transportation cask capacity is between 10% and 100% [15]. The allowable increase of capacity depends on the initial enrichment and the minimal guaranteed burn-up. The researchers [13] found out that maximum permissible enrichment of 230 nuclear fuel assemblies, which are located simultaneously in the plant interim storage facility of La Hague, can be increased from 3.3% to 4% on condition that the burn-up is not less than 10 MW-day/kg. The use of burn-up as a nuclear safety parameter faces quite complex problems; the main problems are [16]: • which isotopes must be considered when determining the fission coefficient;

4

Physical Safety Basis of WWER Nuclear Fuel

• what burn-up value should be taken into account since fuel assemblies have different burn-up profiles. To solve the first problem, i.e., when choosing isotopes, three main approaches are used [17]: • accounting only for the depletion of primary fissile material; • additional accounting for actinoids with large atomic masses formed during the operation of the reactor; • additional accounting for fission products, which have a high neutronabsorption cross section. It is relatively simple to bring the first scheme into action because most of the calculation programs and evaluated nuclear data file are verified on a large amount of experimental data [18–20]. Any additional analysis of fuel compositions with respect to actinoids with large atomic masses formed during the operation of the reactor requires a higher level of experience with software codes and evaluated nuclear data file. Complete account of neutron absorption by fission products is one of the most complicated tasks, especially for high level of burn-up. This is largely due to abundance of isotopes that should be taken into account. Most of the corresponding computational programs and evaluated nuclear data file are currently being under implementation and validation. Figure 1.2 shows the dependence of the effective neutron fission coefficient for various calculation models [17].

Figure 1.2 Dependence of the effective neutron fission coefficient for various calculation models [17].

1.1 Fuel Burn-Up as a Nuclear Safety Criterion

5

It is worthwhile noting that the analysis of the use of burn-up as a safety criterion requires more calculations than the standard analysis of criticality, as it requires the calculation of the SNF isotopic composition. As it is pointed out in [21], typical program codes (OECD/NEA) used for reactor computations may not be suitable to use Burn-Up Credit (BUC) as a safety criterion. This is due to the fact that complex models are used in the calculations of the reactor core and special requirements are imposed on the initial data. Therefore, the codes and data are closely related. The purpose of computations of the reactor is its efficiency. When we use codes for nonreactor zone facilities (for example, an SNF transport container, SNF dry storage container, etc.), the purpose of the computations is maximum safety of their operation. It should also be taken into account that these facilities can contain fuel with different production history and which was produced by different manufacturers. It is noted in [22] that the use of BUC involves knowledge of the exposure time, burn-up, initial enrichment, and isotope distribution. For example, the practical application of this approach in France requires the fulfillment of the following criteria: • burn-up value is based on the least irradiated 50 cm of the active length of the nuclear fuel assembly; • actual burn-up value must be checked by measuring each nuclear fuel assembly. The type of measurements, whether qualitative or quantitative, depends on specific conditions, expected burn-up, and initial enrichment. For example, if an expected burn-up is less than 5.6 MW-day/kg and enrichment is less than 3.3%, qualitative measurements are enough, while higher burn-up and enrichment values require quantitative measurements. The concept of burn-up usage as a nuclear safety parameter is not a modification of the basic safety principles or an attempt to define new safety principles [23]. From this point of view, real-time burn-up definition allows ensuring the principle of safety priority directly in the process of SNF overload while improving the economic performance of nuclear power plants. To provide the implementation of these alternatives, it is necessary to consider methods and means how to control nuclear materials, determine the nuclear fuel burn-up, as well as to find technical solutions that allow real-time burn-up measurement.

6

Physical Safety Basis of WWER Nuclear Fuel

1.2 Influence of the Reactor Operating Mode on the Efficiency of WWER-1000 Fuel Cycles The main issues of the economics of a fuel cycle are presented in the following works [24–28] and others. Even without taking into account macroeconomic aspects, they are extremely complex and belong to the class of optimization tasks. The traditional approach is based on two main principles: • the power unit is to operate on rated capacity between refueling; • the ratio of the plant unit downtime to its on-time is to be minimal. Hence, we have the task of reducing the duration of preventive maintenance and developing a program to increase the duration of fuel lifetime. Due to the fact that a clear, comprehensive criterion for the efficiency of nuclear power plants is still unknown, many aspects are excluded from consideration. For example, in [29], the authors show that even the regular use of the operation mode of the WWER-1000 power unit with partial use of negative reactivity effects leads to a decrease in the average power level per a campaign. On the other hand, it allows increasing the yearly average power production or decreasing the cost value of the fuel component of supplied electricity without the reduction of the yearly average power production. The operation time between refueling depends not only on the commercial efficiency of the cycle options but also on such conditions as ensuring preventative and predictive maintenance at any given time (for example, operation out of the bounds of the autumn and winter peak of electricity consumption, operation of other units of the given nuclear power plant, and the reliability of the equipment). Thus, the duration of the campaign can be significantly reduced, regardless of the type of the used fuel cycle. The introduction of cycles with reduced neutron leakage, as it was implemented at the Khmelnitsky NPP (KhNPP), makes it possible to form a wide range of loadings within the framework of the limitations mentioned above. In this case, the fuel campaign becomes, on average, 10% shorter than the project provides. From the point of view of the traditional approach, these cycles are less efficient than the projects due to the proportional increase in the constant constituent of the nuclear generating cost. But in such a cycle, the neutron flux on the inner surface of the reactor vessel is reduced by 25%–40%, which creates the prerequisites to a proportional increase of the life of the reactor, and, thereby, it practically proportionally reduces the constant constituent of the cost value. In addition, such a cycle makes it

1.2 Influence of the Reactor Operating Mode

7

possible to obtain significant savings in the fuel factor of the cost value and to reduce the specific amount of SNF per unit of supplied energy, as well as more frequent performance of preventative and predictive maintenance additionally increases the reliability of the nuclear power plant operation during the campaign. The influence of reactor core layout arrangements on the resource of the reactor vessel is so high that it actually makes it possible to operate and control it [30]. Under actual operating conditions, the average reactor capacity during the campaign is lower than the rated capacity. This can be caused by many factors: partial equipment malfunctioning, which requires a reduction in power, power line capacity, operation on the power reactivity effect, etc. When estimating the efficiency of such a unit, the balance of contributions from different criteria changes. There are data about the possibility of operation of WWER-1000 power units at the so-called “daily and weekly” load schedule [28, 31, 32]. This mode of operation of power units is a promising one, as the market value of such energy increases by 1.5–2 times and requires a corresponding feasibility study. Computational and experimental researches [28, 31, 32] show rather stable behavior of the WWER-1000 reactor core in transient modes under the appropriate choice of control actions. As a result, operating conditions of the power unit which can be constituents of estimation criteria of its operation (average capacity per a campaign, calendar and effective loading work time, duration of preventative and predictive maintenance, depth of burn-up of upload SNF fuel, average integral density of neutron flux on the reactor vessel, etc.) can be described by a system of dependencies. Constraint functions in these dependencies are non-linear and are determined starting with the characteristics of the reactor core that range from the simplest one (e.g., the number of reactor fuel assemblies) and ending with quite complex, empirically determined connections. The example can be a link between the fuel make-up nomenclature, the reduction of neutron leakage from the reactor core, and the value of the reactivity coefficient according to the coolant temperature at a minimum controlled power level. The type and amount of fuel assemblies depend on the reliability characteristics of the equipment as well as the relation between increasing the depth of fuel burn-up and severization of requirements for reactor control quality in transient modes [33–37]. In order to choose the type and amount of fuel assemblies, it is necessary to use a complex approach: at the first stage, based on the experience of steady-state fuel cycle formation, it is necessary to analyze the layouts

8

Physical Safety Basis of WWER Nuclear Fuel

of specific loads as perturbations of standard cycles. The second stage is to ensure that the phenomena manifested in the accumulated operating experience and subject to systematization are taken into account. At the third stage, the stochastic element must be taken into account, which can be done using the theory of optimal processes. After the second and third stages, it is necessary to adjust the results of the previous stages each time. After a series of iterations, the problem of optimal control of the entire fuel cycle of nuclear power plants is to be solved. Some fuel loading layout problems are discussed in more detail below. The study of fuel cycles in “ideal” conditions of operation of the reactor excludes from consideration all the parameters except the characteristics of the reactor core and gives a schematic representation of fuel consumption effectiveness. Therefore, in the future, an analysis of the impact of the operation of the power unit in maneuverable mode on fuel consumption will be given. It can be shown how the actual operating conditions influence the properties of irradiated fuel by taking into account the average power level of the reactor plant per campaign. The consideration of the average capacity level in a number of regarded parameters allows including two reactivity effects. Both of them are related to the fact that when operating at reduced power, the reactivity margin for fuel burn-up in the reactor is higher than when operating at the nominal level. The first effect is the most significant, and the power level at which the reactor operates at the end of the load plays a crucial role here. Reduced power provides the possibility of longer calendar work as well as longer effective work. Although this leads to a decrease in the average reactor capacity per campaign, the average annual energy production and plant capacity coefficient can increase. A detailed discussion of this phenomenon is given below. The second effect is not of such importance, but with an accurate evaluation of the operational efficiency of the reactor, it should be determined and taken into account. It is caused by the fact that with a decrease in the full capacity of the reactor, the redistribution of energy release between the fuel cells in the reactor core occurs. As a result, the neutron leakage value outside the reactor core changes, i.e., inefficient losses of reactivity margin as well as the distribution of nuclear fuel burnup rate over the reactor core take place. The capacity level of operation and the duration of its operation per a campaign are of great importance for this effect. In this case, the advantage is in the effective duration of the campaign as well as the depth of fuel burn-up in the unloaded part of the reactor core.

1.2 Influence of the Reactor Operating Mode

9

The example of taking into account the decrease in power when analyzing the cycles is given in [38]; it demonstrates one of the transition methods to the consideration of the reactor within the conditions of a real operation. It is shown that for an additional evaluation of the effectiveness of fuel cycle options, it is possible to analyze the degree of their sensitiveness, from the point of view of reducing efficiency, and from reducing the reactor capacity during operation as well. The control of axial offset is one of the tasks of reactor safety protection, the quality of its operation in case of the efficiency increase of fuel utilization, and the use of the established capacity level. In addition, the control of axial offset is one of the two main components of the problem, which is linked to the adaptation of WWER-1000 power units to operate in the maneuverable mode [39–41]. The studies undertaken allow identifying several main, conditionally independent, possible components of cost advantages: • • • •

the effect of the operation in the maneuverable mode; the effect which is related to the nuclear fuel reliability growth; the increase of the installed capacity utilization factor; the effect of the operation in the mode of the higher burn-up.

The first component of the effect of maneuvering is determined by the fact that in the European energy market, the electricity generated by the power units participating in the regulation of the power system frequency is paid at a higher rate than the electricity generated by the power units, which provide the standard component of the capacity of the system. The effect can be estimated at the level of 50% of the cost of electricity generated at nuclear power plants, allowing for about 7% of losses from generation reduction during night unloading, as well as increased cost of advanced fuel and equipment, which will be determined by suppliers and, apparently, can be estimated at half of the expected effect. The total value of the effect from the entire complex of works can reach up to 20%–22% of the cost of the energy generated by the NPP. If using advanced control algorithms of WWER-1000 control, the second part of the effect of maneuvering allows the nuclear power unit No. 1 of KhNPP to operate without preschedule reactor fuel assembly unload due to the leakage of fuel cladding, with coolant activity in the primary circuit that makes it possible to operate the reactor core without annual control of fuel assembly leak resistance. Thus, this component is close to the cost of all prematurely unloaded leaking fuel assemblies. The average number of prescheduled fuel assembly unload can be estimated at the level of two fuel

10 Physical Safety Basis of WWER Nuclear Fuel assemblies for one power unit per year; the increase of fuel factor of the cost of energy caused by their replacement is about 1%–2% [42]. The third part of the effect of maneuvering allows reconsidering the approaches to the base-load operation condition. Here the effect of inefficient financial resources appears; in this case, the power unit has to operate at reduced parameters due to constraint violations set for the power distribution in the reactor core caused by the xenon transient process. This value can be evaluated by the 30-hour of load decrease of 25%. The fourth component of the effect of maneuvering requires a deeper understanding, which is given below. In addition to the studied components of the cost advantages, there are other aspects, the quantitative assessment of which has not been performed by the authors. For example, the reliability growth of nuclear fuel leads to a decrease in reactor coolant activity, a decrease in the activity of gases in the ventilation system of a power unit, and a reduction of the personnel radiation doses during scheduled maintenance. It can also lead to a reduction in these maintenances and an additional increase in the installed capacity utilization factor. If handling with the fuel is on the critical path of the preventative and predictive maintenance conduction, the absence of necessity to change leaking reactor fuel assemblies or conduction of additional control of the leak resistance leads to the reduction of the maintenance period and to the additional increase of the installed capacity utilization factor [33–35].

1.3 Design Constraints and Engineer Suitable Coefficients When Designing and Operating WWER Fuel Loads Operational limits or design limits under standard operation are values of parameters and characteristics of the system state and nuclear power plants as a whole, which are set by the project for normal operation [43]. The promotion of the nuclear energy competitiveness and nuclear fuel competitiveness in the global market requires the introduction of new, more efficient fuel cycles [44]. New fuel cycles include an increase in the burn-up depth, profiling of enrichment, introduction of burnable absorbers in fuel assemblies, and an increase of the capacity in a power unit New fuel cycles make it necessary to review the existing set of operational limits, namely:

1.3 Design Constraints and Engineer Suitable Coefficients

• • • •

11

introduction of new constraints; exclusion of duplication; physical “transparentness”; exclusion of unreasonable conservatism.

Operational limits lie at the heart of the concept of safety and its constituents, from which the safety criteria follow. Necessary and sufficient conditions of safety criteria are given in Table 1.1. Table 1.1 includes the requirements for the operation of nuclear fuel that comply with the IAEA recommendations. The principle is invariability, maintenance of safety criteria such as input data for operational limit development. The accomplishment of operational limits is a purpose which can be achieved by controlling other parameters on the basis of the in-core control system. Below, there are operational limits of the WWER-440 (B-213), which, under the operation of the reactor, are used when choosing loadings [45, 46]: • reactor thermal capacity can exceed the nominal value of 1375 MW not more than 4%; • coolant pressure at the output of fuel assemblies with six operating main circulation pumps (MCP) may differ from the nominal value (12.26 MPa) at the most 0.2 MPa. • coolant inlet flow when there are six working MCPs is not less than 39,000 m3 /h. • average coolant temperature at the inlet of the reactor core is to be in range of 265◦ C–270◦. • maximum capacity of the fuel element and fuel element with Gd is 54.5 kW if pin-to-pin spacing in an assembly is 12.2 mm and maximum capacity 56.6 if pin-to-pin spacing in an assembly is 12.3 mm. • marginal linear load and changes (jumps) in the linear load of a fuel element and fuel element with Gd depending on the burn-up is in accordance with the graphs shown in Figures 1.3 and 1.4. • subcriticality in case of shutdown is 1%, and in case of refueling it is 2%.

12 Physical Safety Basis of WWER Nuclear Fuel

Figure 1.3 Field of local fuel element loads with respect to the factor of margin depending on the burn-up.

Figure 1.4

Fuel element linear load jumps as a result of refueling.

1.4 Criteria and Methods of Nuclear Fuel Safety Evaluation Under Operation When performing a maneuver with a power pressurized water reactor, the operator faces the problem as to how to control the power density field because of xenon transient process occurrence and axial offset oscillations, which are the result of this occurrence. The axial offset is determined by a dependence

Provision of NPP safety due to the operation and maintenance reliability of barrier efficiency as well as provision of personnel with technical means and organizational measures necessary for NO mode.

Purpose:

I Level – Normal operation (NO).

Protection Level

Technological and radiation criteria, which provide safe operation of NPP with WWER. Technological Criteria Fuel Matrix Fuel Coolant System of Radiation Criteria Cladding Circulation Leak-Tight Circuit Enclosure No fuel NonUnder NO, The amount Limits of occupational exposure: melting. exceedance the amount of of leakage The yield of of the uncontrolled from the 1. The limit of a personal dose of radiologically operational leakage from containment external and internal exposure is 20 hazardous limit of fuel the primary is not more mSv/yr. fission element coolant than 0.1% of Design values of a dose rate fractions failure: circuit is not the corresponding to the given limit with from the fuel more than containment respect to twofold design suitable factor matrix is not • defect of gas 100 l/h. volume per according to the job characteristics are more than leakage is not The pressure day. used to develop the protection against 0.3% of the more than is less than or ionizing radiation. total amount. 0.2% of fuel equal to the assemblies; operating 2. A limit on the collective dose of pressure. In personnel during routine maintenance is • close contact case of set. The routine maintenance is related of nuclear emergency, to the dose consumption (preventative fuel with the there can be a and predictive maintenance), refueling coolant is not pressure – 0.5 mSv/yr. more than boost up to 0.02% of fuel 1.15 from the Dose limits of public exposure: assemblies; operating Individual exposure – 0.1 mSv/yr. pressure. The limit refers to the annual effective equivalent dose for the critical group of population caused by NPP operation, taking into account direct and indirect pathways of radiation exposure.

Table 1.1

1.4 Criteria and Methods of Nuclear Fuel Safety Evaluation Under Operation

13

Purpose: Provision of NPP safety due to the restriction of NO to the extent to its shutdown and maintenance of barrier efficiency as well as personnel activities with technical means including safety systems.

II Level – Abnormal operation (AO).

Protection Level

Fuel Matrix

Table 1.1 Continued Technological Criteria Fuel Coolant System of Cladding Circulation Leak-Tight Circuit Enclosure • No boiling crisis, suitable factor before crisis is 1.2 ÷ 1.3 (1+2σ), where σ is a root-mean square error of the used correlation In a similar way to NO The limit corresponds to 10% of basic dose limits established for population by the safety radiation level.

Radiation Criteria

14 Physical Safety Basis of WWER Nuclear Fuel

Purpose: Provision of NPP safety due to the safe decommissioning by means of the safety system.

III Level – Design Basis Accident (DBA)

Protection Level

• local oxidation rate of fuel element claddings is not more than 18% from initial wall thickness;

Table 1.1 Continued Technological Criteria Fuel Matrix Fuel Coolant System of Cladding Circulation Leak-Tight Circuit Enclosure No fuel NonLoss of The amount melting. exceedance primary of leakage of maximum coolant can from the Fuel enthalpy design lead to containment under the damage limit short-term is not more reactivity for fuel dryout of the than 0.1% of effect is not elements: core. the more than containment 145 cal/g. • temperature Pressure does volume per of fuel not exceed day. element 1.15 Pop . claddings is not more than 1200◦; The limits relate to the integral doses received by the personnel during the accident and the rectification of its consequences for external and internal exposure due to inhalation.

It is installed in the design to reliably ensure the permanent stay of personnel in the main control room and emergency control room.

The limit of individual effective equivalent dose in the main control room and emergency control room is – 25 mSv/yr.

The limit of a personal dose of external exposure is 40 ÷ 80 mSv/yr.

Planned special exposure of personnel.

Radiation Criteria

1.4 Criteria and Methods of Nuclear Fuel Safety Evaluation Under Operation

15

Protection Level

Fuel Matrix

Table 1.1 Continued Technological Criteria Fuel Coolant System of Cladding Circulation Leak-Tight Circuit Enclosure • the proportion of reacted zirconium is not more than 1% of its mass in the claddings of fuel elements.

Exposure effect on the public caused by accidental releases of radioactive substances into the environment during design accidents and/or external effects do not require the introduction of protective measures for the public.

Individual total body dose is 5 mSv/yr.

Dose limits of public exposure:

Radiation Criteria

16 Physical Safety Basis of WWER Nuclear Fuel

1.4 Criteria and Methods of Nuclear Fuel Safety Evaluation Under Operation

17

N−N × 100 %, (1.1) N+N where NB and NH are the capacity of core in upper and lower packages, respectively. Obviously, almost any change in the parameters of the WWER-1000 core (capacity, temperature, coolant flux rate, position of the regulating mechanisms, concentration of the integral absorber, etc.) can lead to xenon oscillations of the axial offset [47, 48]. As a rule, a power maneuver is planned and carried out as a certain sequence of relatively fast transitions between power levels at which the reactor operates for a rather long time. In this case, the task of controlling the power density field centers on maintaining the axial offset current value in proximity of the given value [49]. The analysis of the qualitative dependences of variations in axial offsets with respect to changes of the WWER-1000 main operational parameters shows that any exposure on the reactor installation leads to an ambiguous change in the axial offset (Table 1.2). In this regard, it is of interest to evaluate the thermal technological reliability of the WWER-1000 core during transient processes. Since the existing WWER-1000 in-core control system does not allow measuring any local parameters of the power density field, a numerical simulation was used to obtain the power density field parameters in the WWER-1000 core. It was carried out for a reactor that was in the steadystate fueling mode with a three- and four-year campaign. The computer code BIPR-7A was used as a modeling tool [50]. For both campaigns, a AO =

Table 1.2 Main axial offset perturbation actions. Action Direction Change of reactor thermal capacity N⇑ N⇓ Change of k-group position of regulating Hk ⇑ mechanisms of control system and protection in the Hk ⇓ upper half of the core Flow variation G⇑ G⇓ Boric acids concentration change in a heat pump of Cb ⇑ the primary coolant circuit Cb ⇓ Temperature change of a heat pump at the core inlet Tin ⇑ Tin ⇓

Result AO ⇑ AO ⇓ AO ⇓ AO ⇑ AO ⇓ AO ⇑ AO ⇓ AO ⇑ AO ⇓ AO ⇑

18 Physical Safety Basis of WWER Nuclear Fuel hypothetical xenon transient process was considered both at the beginning and at the end of the campaign. A three-hour decrease of the thermal power in the reactor to 50% and its increase again up to 100% was simulated. This mode was chosen as the most logical and the most cost-effective one with a possible maneuverable operation cycle. The analysis of the table data showed that the largest axial offset oscillations occur during a power maneuver at the end of the fuel campaign during a four-year fuel cycle (Figure 1.5). At the beginning of the campaign, with a large margin of reactivity, the arising axial offset oscillations tend to decay, but at the end of the fuel campaign, the opposite effect is observed. Let us consider the external temperature of the fuel element claddings and the safety flux before the heat transfer crisis in the most loaded fuel assemblies at the upper and lower maximums of the power density. An unambiguous relation between axial offset oscillations and local values of the power density field is shown in the computation and experimental work [51]. The rate of heat transfer from the fuel elements to the coolant determines the temperature regime of the fuel element cladding. The value of the heat transfer coefficient varies significantly depending on the hydrodynamic structure of the coolant flow and its state of aggregation. When calculating nuclear power reactors with water coolant in a general case, it is possible to consider convective heat transfer, heat transfer with surface boiling, and heat transfer with developed volume boiling of coolant.

Figure 1.5 campaign.

Axial offset after the transient process at the end of a (1) three- and (2) four-year

1.4 Criteria and Methods of Nuclear Fuel Safety Evaluation Under Operation

19

The heat transfer coefficient for convective heat transfer conditions, as is the case for WWER reactors, can be calculated using the well-known Mikheev empirical formula [52]   λ ρω dΓ 0,8 α = 0, 021 Pr0,43 (1.2) dΓ µ where λ, µ, and Pr are correspondingly the heat conductivity coefficient, dynamic coefficient of viscosity, and Prandtl number for coolant in a design sector of the reactor fuel assembly; ρω and dr are a mass flow of the coolant and a hydraulic diameter of a fuel element. To determine the temperature of the fuel element wall, we shall apply the following equation: qs · KV + Tcool (1.3) Twall = α where T cool is the coolant temperature in the corresponding reactor core elementary volume; KV is the radial power peaking coefficient; qs is the graded density of fuel element thermal radiation. The determination of thermal characteristics of the reactor core is possible according to the following scheme (Figure 1.6), taking into account the fact that the source data for it are the output files of the BIPR-7A program.

Figure 1.6 Computation scheme of thermotechnical reliability.

20 Physical Safety Basis of WWER Nuclear Fuel Table 1.3 Computational results of the fuel element walls during the first 7 hours after the completion of the power maneuver. Time after the Maximum Loaded Upper Maximum Loaded Lower Beginning of the Section of the Fuel Section of the Fuel Transient Process, Assembly Assembly Hours T cool , ◦ C KV T wall , ◦ C T cool , ◦ C KV T wall , ◦ C 3 291.47 1.071 308.52 323.33 1.681 350.10 4 291.36 1.051 308.10 323.45 1.68 350.20 5 291.36 1.058 308.20 323.58 1.658 349.98 6 291.48 1.089 308.82 323.74 1.619 349.51 7 291.7 1.143 309.90 323.91 1.565 348.83

The definition of thermal characteristics of the core is possible according to the following scheme (Figure 1.6), taking into account the fact that the input data for it are the output files of the BIPR-7A program. Such calculations allow running some versions of programs such as RELAP, DIN3D, as well as their combination. The results of the calculations of T cool and KV , as well as the temperature of the fuel element walls for the upper, most loaded part of the fuel assembly and its lower section, which is symmetrical to it relatively to the horizontal plane of the core, are given in Table 1.3. In order to calculate the margin of power before the heat transfer crisis (usually expressed as the ratio of the critical heat flux to the nominal K = qcr /qS ), it is necessary to define the so-called heat flux critical density qcr . of the fuel assemblies. There are several empirical formulae to define fuel assembly qcr which are based on the research of flow models for various types of reactors, various thermohydraulic parameters, and states [53]. For this analysis, we used the equation that allows evaluating qcrI in a fuel assembly with an error limit of 11% qcrI = 0, 0274 · (ρω)0,505 · (1 − E)1,965 · (1, 3 − 9, 4 · 10−4 · P )

(1.4)

where P is the coolant pressure in the primary circuit; ρω is the coolant mass flux; and x is the flow quality. This dependence is applicable to the pressure of WWER-1000 (P = 16.7 MPa) reactors under conditions of uniform heating of the rod bundles. In addition, in order to calculate the heat transfer crisis in fuel assemblies of WWER-1000 reactors, experimental design bureau “Hydropress” recommends the formula obtained under conditions as close as possible to the operating conditions of this reactor [53]

1.4 Criteria and Methods of Nuclear Fuel Safety Evaluation Under Operation

21

Table 1.4 Data on how to calculate the safety coefficient before the heat transfer crisis in the maximum loaded fuel assemblies. Time after the Beginning of qcr I , qcr II , qs , the Transient Process, Hours MW.m−2 MW.m−2 MW.m−2 Csf 1 Csf 2 3 3.163 5.139 0.940 3.37 5.470 4 3.160 5.126 0.939 3.37 5.459 5 3.155 5.111 0.927 3.40 5.516 6 3.150 5.094 0.905 3.48 5.629 7 3.145 5.075 0.875 3.60 5.802

qcrII = 0, 795 · (1 − E)n · (ρω)m · (1 − 0.0185 · P )

(1.5)

where n = 0.105.P = 0.5, and m = 0.184 − 0.311.x; the other parameters are as in the formula for qcrI . The behavior of the analyzed parameters allows concluding that during the first hours after the beginning of the transient process, there are favorable conditions to operate the reactor according to safety criteria (Table 1.4). We can observe an increase in the temperature of the fuel element cladding at the bottom of the reactor core due to a shift in the maximum power density exactly there, which, in fact, is not dangerous, because the temperatures at the bottom of the core are initially 20◦ C−25◦ C lower than at the top. Further, the temperature begins to fluctuate because of the redistribution of the neutron field due to the temperature effect of reactivity. The temperatures in the lower and upper parts of the reactor core fluctuate in antiphase. If to analyze the temperature behavior in the upper part of the fuel assemblies, it can be noted that during the first 7 hours of the transient process, there is an opportunity to control the state of the reactor as, at this time, the lower harmonic of temperature fluctuations is observed. Fluctuations in the temperature of the containment reach dangerous values (350◦ C) (from the point of view of the storage before the beginning of surface boiling) which is especially evident at the end of the fuel campaign. The methodology for evaluating the heat and technology reliability of the reactor core, its calculation, and experimental justification are given in [54] and [55], where the authors show that under such conditions, any temperature fluctuations (fluctuations in coolant or cladding of fuel elements) are practically absent, i.e., they are within the estimated error. The formulae available in the literature and used here to calculate the critical heat flux qcr and corresponding safety coefficient before the crisis approximately equally describe the behavior of the reactor core and its heat

22 Physical Safety Basis of WWER Nuclear Fuel and technical reliability but differ significantly in the numerical values of the safety coefficient. The WWER-1000 reactor is designed on a conservative approach, including a significant reserve of thermal and technical characteristics of fuel assemblies during operation in transient modes. The presence of a significant reserve before the heat transfer crisis in fuel assemblies creates prerequisites for transferring the WWER-1000 reactor to a maneuvering mode of operation, which is extremely important for the current state of the energy industry in many countries. Problems before the heat transfer crisis in fuel assemblies belong to the class of unsteady-state heat transfer problems and are currently solved exclusively by numerical methods. The problems of unsteady-state heat transfer in bodies that have the form of geometric primitives (an endless plate, a cylinder, and a ball) belong to classical problems. The results of their analytical solution are given in monographs and many textbooks on heat transfer. In well-known sources, the results of solving such problems are presented in a dimensionless form for generality. As a result, the functional dependence of the temperature of the bodies on time, for example, for the body central points, is presented in a two-criteria form: depending on the Fourier number (dimensionless time) and the Biot criterion. In a graphical form, it corresponds to a set of curves for each considered point. For each of the given bodies, its own solution and, accordingly, an individual set of curves are designed. But for the engineering application of such functional dependencies, the understanding of the adequacy of the mathematical model to a real unsteady-state heat transfer process is necessary. One of the methods that allow adjusting the adequacy is the ability to bring the model to an automodeling form. The theory of similarity is closely connected with this method. In the given field, there is a basic π-theorem (in English literature − Buckingham theorem; in French – Vaschy theorem), fixing the possible number of dimensionless values in transformable models. On the way to the further development of dimensionless methods to solve problems of unsteady-state heat conduction, a certain success has been achieved. But the methods used by researchers are the result of the intuition of their developers. As for the scientific approach, it is necessary to consider the method as a coherent logical system which provides the basis for further work in this direction and indicates the relevance of research. Therefore, in the future, it is advisable to consider new approaches for modeling unsteadystate heat transfer processes in nuclear power plant equipment as well as nuclear fuel.

References

23

References [1] Odessa National Polytechnic University. Report on Study of selfradiation fields of nuclear fuel and methods of their registration in order to create systems for monitoring the state of nuclear fuel in real time at all stages of NPP operation. Dec. 2002. [2] Odessa National Polytechnic University. Report on Study of the distribution of fission products of nuclear fuel in fuel assemblies of light water reactors in real time in order to verify the reliability of the programs for calculating the core and loading containers of a dry storage facility for spent nuclear fuel. Dec. 2005. [3] Design of fuel handling and storage systems for nuclear power plants: safety guide. IAEA, 2003. NS-G-1.4–STI/PUB/1156. [4] Pavlov, S.V., Smirnov, V.P., Mytarev, A.V., Vlasenko, N.I. and Biley, D.V. (2003). “Methods for WWER-1000 fuel testing under dry storage conditions,” in Proceedings of an International Conference on Storage of Spent Fuel from Power Reactors. 2003 (Vienna: IAEA), 541–551. [5] Nuclear technology review 2006. 2006. IAEA. IAEA/NTR/2006. [6] Maslov, O. V. (2001). Radiation-technological monitoring system for spent fuel from light water nuclear power plants. Ph.D. thesis. Odessa National Polytechnic University. Odessa. [7] Odessa National Polytechnic University. Report on Study of fields of nuclear fuel self-radiation and methods of their registration in order to create systems for monitoring the state of nuclear fuel in real time at all stages of NPP operation. Dec. 2002. [8] Odessa National Polytechnic University. Report on Study of the distribution of fission products of nuclear fuel in fuel assemblies of light-water reactors in real time in order to verify the reliability of the programs for calculating the core and loading containers of a dry storage facility for spent nuclear fuel. Dec. 2005. [9] Uppsala University. Report on Gamma-ray measurements of spent PWR fuel and determination of residual power. Dec. 1997. [10] Jansson, P. (2000) Studies of Nuclear Fuel by means of Nuclear Spectroscopy Methods. Licentiate thesis. [11] Olejnik, S. G., Maslov, M. V., Maksymov, M. V. (2002). Equipment and methodology for monitoring the distribution of FP in the SNF management technology in Proceedings of VIII Russian Scientific Conference Radiation protection and radiation safety in nuclear technologies. Obninsk, Russian Federation.

24 Physical Safety Basis of WWER Nuclear Fuel [12] U.S. Department of Energy, Office of Civilian Radioactive Waste Management. Topical Report on Actinide-Only Burn-up Credit for PWR Spent Nuclear Fuel Packages. April 1997. [13] Gesellschaftfür Anlagen- und Reaktorsicherheit (GRS) GmbH. Consideration of Burnup Rates in the Analysis of the Safety of the Nuclear Fuel Cycle. Oct. 1997. [14] Güldner R, Burtak F. Contribution of Advanced Fuel Technologies to Improved Nuclear Power Plant Operation. 1999. The Uranium Institute. London [15] Wagner JC, Parks CV. A Critical Review of the Practice of Equating the Reactivity of Spent Fuel to Fresh Fuel in Burn-up Credit Criticality Safety Analyses for PWR Spent Fuel Pool Storage. NUREG. 2000; 230: 44. [16] Gesellschaftfür Anlagen- und Reaktorsicherheit (GRS) GmbH. Report on the Analysis of the Safety of the Nuclear Fuel Cycle. Oct. 1997. [17] Parks CV, DeHart MD, Wagner JC. Phenomena and parameters important to burnup credit. IAEA Technical Committee Meeting on the Evaluation and Review of the Implementation of Burn-up Credit in Spent Fuel Management Systems. 2000. IAEA. Vienna, Austria. [18] Parks CV, Broadhead BL, DeHart MD, Gauld IC. Validation Issues for Depletion and Criticality Analysis in Burnup Credit. IAEA Technical Committee Meeting on the Evaluation and Review of the Implementation of Burnup Credit in Spent Fuel Management Systems. 2000. IAEA. Vienna, Austria. [19] DeHart, M. D., Hermann, O.W., Parks, C.V. (1995) “Validation of a method for prediction of isotopic concentrations in burn-up credit applications” in Proceedings of ICNC’95 Fifth International Conference on Nuclear Criticality Safety. Albuquerque, NM (US). [20] Wagner J.C. Computational Benchmark for Estimation of Reactivity Margin from Fission Products and Minor Actinides in PWR Burnup Credit. 2001. ORNL. ORNL/TM-2000/306. [21] Brady MC, Takano M, DeHart MD, Okuno H, Nouri A, Sartori E. Findings of the OECD/NEA study on burn-up credit. 1998. Nuclear energy agency organization for economic cooperation and development. [22] Proceedings of the Twenty-Seventh Water Reactor Safety Information Meeting. 1998. NUREG. Bethesda, Maryland (US). [23] Spent Fuel Project Office Interim Staff Guidance – 8, Rev. 1 – Limited Burnup Credit. 1999. US Nuclear Regulatory Commission.

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[24] The Economics of the Nuclear Fuel Cycle. 1994. Nuclear Energy Agency. Paris. [25] Abagian AA, Matveev AA, Ignatenko EI, Pshechenkova TV. Improvement of the criteria for assessing the efficiency of operation of NPP with VVER. Electric stations. 1983; 10: 15–18. [26] Massachusetts Institute of Technology. Report on The Future of Nuclear Power. An interdisciplinary MIT study. 2003. [27] Shevelev YaV. Application of discounted costs to assess the effectiveness of economic activities in nuclear power. Economic and mathematical methods. 1984; 20(6): 1103–1112. [28] Kramerov AJa, Shevelev JaV. Engineering calculations for nuclear reactors. 1984. Energoatomizdat. Moscow. [29] Korennoi AA, Nedelin OV, Pismennyi EN, Vajner LG. Optimization of the operating time of VVER-type power units in the campaign extension mode. Industrial Heat Engineering. 2000; 22(5–6): 82–88. [30] Bukanov VN, Vasiljeva E.G, Nedelin OV. Method for determining the radiation load of a VVER-1000 reactor vessel. Nuclear and radiation safety. 2000; 3(3): 32–40. [31] Baskskov VE, Maksymov MV, Maslov OV. Algorithm for the operation of a power unit with a VVER to maintain the daily balance of the power system. Trudy Odesskogo politehnicheskogo universiteta. 2007. 2(28): 56–59. [32] Maksymov MV, Pelykh SN, Maslov OV, Baskskov VE. Method for evaluating the efficiency of the maneuvering algorithm power of a power unit with a VVER-type reactor. Nuclear Energy. 2008; 4: 128–139. [33] Maksymov MV, Fridman NA, Maslov OV. Determination of the efficiency criterion for the operation of NPP with VVER in the variable part of the electrical load graph / Trudy Odesskogo politehnicheskogo universiteta. 2001; 2(14): 86–89. [34] Maksymov MV, Fridman NA, Maslov OV. NPP Operation Efficiency Analysis. Trudy Odesskogo politehnicheskogo universiteta. 2001; 4(16): 42–45. [35] Maksymov MV, Fridman NA, Maslov OV. Evaluation of the efficiency of NPPs with VVER-1000 reactors. Trudy Odesskogo politehnicheskogo universiteta. 2002; 1(17): 70–75. [36] National Research Center Kurchatov Institute. Report on Development of an improved algorithm for controlling the power and energy distribution of the core of the serial VVER-1000, taking into account

26 Physical Safety Basis of WWER Nuclear Fuel

[37]

[38]

[39]

[40]

[41]

[42]

[43] [44]

[45]

[46]

[47] [48]

the results of experimental power maneuvers at the 5th unit of the Zaporizhzhya NPP. Dec. 1998. National Research Center Kurchatov Institute . Report on Adaptation of advanced power control optimization algorithms for the first unit of the Rostov NPP. Dec. 2000. Korennoi AA, Fridman NA. Improvement of efficiency criteria for the operation of fuel loads for VVER-1000 reactors. Zbirnik naukovih prac Institutu yadernih doslidzhen. 2002; 2(8): 85–88. Reshetnikov FG, Bibilashvili YuK, Golovnin IS, Platonov PA, Reshetnikov NG. Developing fuel elements of the water-moderated VVÉR-1000 reactions intended for working under the conditions of maneuvered NPP and increased depletion. Soviet Atomic Energy. 1988; 64(4): 258–266. Gorokhov AK. Method for analyzing axial xenon oscillations and modes of their suppression in VVER-1000 reactors and some results of its application. Problems of Atomic Sciences and Technology. 2006; 15: 13–30 Gorokhov AK. Limiting the axial offset in VVER-1000 reactors when performing power maneuvers. Problems of Atomic Sciences and Technology. 2006; 15: 31–44 Aver’yanova SP, Semchenkov YuM, Filimonov PE, Gorokhov AK, Korennoi AA, Makeev VP. Adoption of improved algorithms for controlling the energy release of a VVER-1000 core at the Khmel’nitskii nuclear power plant. Atomic Energy. 2005; 98(6): 414–421. Tevelin SA. Nuclear power plants with VVER-1000 reactors. Textbook. 2008. Izdatelskij dom MEI. Moscow. Shystov V, Petrov Yu, Shmelkov S, Malyshev S. Alarm signaling and radiation environment monitoring systems. Contemporary Technologies in Automation. 2000; 2: 42–49 Nosik L, Sobakar T, Kondrychin E. BARS emergency monitor: features and practical application. Contemporary Technologies in Automation. 2001; 3: 52–57 Chebyshov SB. Construction of functionally integrated information and measurement systems for radiation monitoring. Nuclear measurement and information technologies. 2006; 4(20): 37–45 Ovchinnikov FJa, Semenov VV. Operating modes of pressurized water power reactors. 1988. Energoatomizdat. Moscow. Anikanov SS, Dunaev VG, Mitin VI. VVÉR-1000 power distribution during power level changes. Atomic Energy. 1993; 75(1): 3–8.

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[49] Novikov AN, Pshenin VV, Lizorkin MP. Code package for WWER cores analysis and some aspects of fuel cycles improving. // Problems of Atomic Science and Technology. 1992; 1: 3–10. [50] National Research Center Kurchatov Institute. Report on The complex of programs for neutron-physical calculations of the RRC KI. Complex of programs CASCADE. BIPR-7A program. Description of the algorithm. Application Description. Dec. 2002. [51] Aver’yanova SP, Lunin GP, Proselkov VN. Monitoring the linear local power (LLP) of the fuel elements in the WWER-1000 core by means of the offset-power diagram. Atomic Energy. 2002; 93(1): 13–18. [52] Kirillov PL, Yur’ev ES, Bobkov VP. Thermal Hydraulic Calculations Handbook: Nuclear Reactors, Heat Exchangers, Steam Generators. 1984. Energoatomizdat. Moscow. [53] Maksimov MV, Maslov OV, Pysklova T.S. Estimation of error in calculating the depth of fuel burn-out with reactor VVER -1000 model. Trudy Odesskogo politehnicheskogo universiteta. 2005; 1(23): 34–39. [54] Korennoi AA, Titov SN, Litus VA, Nedelin OV. Control of the axial distribution of the energy-release field in the VVÉR-1000 core during transient processes. Atomic Energy. 2000; 88(4): 257–262. [55] Korennoi AA, Nedelin OV. A model for controlling the axial distribution of the energy release of a nuclear reactor with a physically large core. Problems of Atomic Science and Technology. 2001; 3: 15–22.

2 Modern Approaches to the Heat Exchange Modeling in NPP Equipment

A comprehensive investigation of any process is closely related to modeling and models. The variety of fields of science and technology in which modeling is used, as well as the desire to strive for the fact that the model could meet the special features of the task, generates numerous specific models and types of modeling. The general definition can cover all the variety of models; however, for the practical use of this powerful research tool, it is important to know the basic principles of construction and evaluation of models. This is particularly true for complex systems such as nuclear power reactors, temporary storage systems, and operated storage facilities, as the study of processes that happen there and that are accomplished with the help of mathematical modeling is a necessary component of ensuring the safety of their operation. The purpose of this section is to develop a scientific basis for building a mathematical model (MM) of a technical system (TS). Under an MM, we mean any nuclear power plant operation in a shift-based load mode where well-known laws of heat transfer and mass transfer take place. We are going to generalize it in order to compress information, test it with the help of a physical model, and evaluate alternatives to MM and choose the best one.

2.1 Abstract Models Schematically, the TS, the environment, and the interactions between them, which control the actions and reactions of the TS, can be presented as shown in Figure 2.1.

29

30 Modern Approaches to the Heat Exchange Modeling in NPP Equipment

Figure 2.1 Scheme of technical system operation and its interaction with the environment.

The TS is controlled by the vector ~u, exposed to the influence of the environment described by the vectors f~ and ~v . The reaction of the TS to the effects is presented as the vector ~y . The environment is divided into two parts: the first one is not sensitive to the system response; that is why f~ does not depend on ~y , but the behavior of the second environment depends on the TS. An example of “the environment-1” is the pressure in the atmosphere, which does not depend on the influence of the TS. An example of “the environment2” can be its temperature in a closed space, where the TS is located, and which depends on the process of heat transfer. Interacting with the environment, the TS is to operate to meet a number of requirements regulated by the technical requirements for the facility. In addition, the technical requirements determine the environmental conditions on the basis of which the boundary conditions of the MMs of a TS are formed; technical requirements determine the parameter regions of the TS and the environment as well. All the information on technical requirements related to the TS operation can be divided into two groups: (a) conditions and values of environmental parameters and ~u under which the TS is to operate; (b) requirements for the TS operation.

2.1 Abstract Models

31

Let us assume that j-a is a technical requirement condition, where the TS is to operate according to a predicate Φj . Then the conditions which are given by the technical requirements can be written as follows: ¯j ) ¯D Φj : (pi j ∈ Dj ) ∧ (pi j ∈

∀j ∈ Jm ,

∀i ∈ Jn ,

where pij is the value of the ith parameter according to the jth condition; Dj is the acceptable region of parameters; n is the number of parameters, which are specified by technical requirements; m is the number of conditions. Let us assume that kth requirement to TS operation is the predicate Rk , and ~y is an initiating signal of the TS. Then the satisfaction of TS conditions can be written as: ∃~y ∈ Y : R1 (~y ) ∧ . . . ∧ Rk (~y ) ∧ . . . ∧ Rl (~y ) = 1

∀k ∈ Jl .

Conclusively, the requirements and TS requirements can be written as follows: ¯ j )] ¯D ∃~y ∈ Y : [R1 (~y ) ∧ . . . ∧ Rl (~y ) = 1] ∧ [(pi j ∈ Dj ) ∧ (pi j ∈ ∀i ∈ Jn ∀j ∈ Jm .

(2.1)

The design of the MM of the TS is accomplished by means of TS decomposition according to the elements with their further description. Each element is chosen in such a way that it should be uniform in the physical space S = X1 ⊗ X2 ⊗ X 3 ⊗ T , (2.2) where X 1 , X 2 , and X 3 are independent one-dimensional geometric spaces and T is the time space. If the process is described in dynamics, then there are elements which are homogeneous in space, for example, there is a transient process and a stationary mode. The process in the system in space X is distinguished by the features of homogeneity of the working fluid; for example, there are liquid, gaseous, and solid phases. Further decomposition allows obtaining a new level of the TS description. In a mass of the working fluid, which is homogeneous in nature, it is possible to detect areas homogeneous in the nature of the processes which occur in them. For example, in the study of hydrodynamics and heat transfer in the fluid flow, the core and the boundary layer can be distinguished; in a solid body, it can be the depth of a heat wave penetration. In its turn, a laminar sublayer, etc., can be found in the boundary layer.

32 Modern Approaches to the Heat Exchange Modeling in NPP Equipment Thus, consistently carrying out the TS decomposition on the basis of homogeneity of the description of its elements, it is possible to represent the structure of an MM, in which as the level decreases, the details of the TS description increase. In order to study the structure of the MM and the relation between its elements, a generic MM is designed where the degree of abstraction is higher than that in the MM relative to the TS. Each equation, formula, and boundary condition can be represented as the sum of operators over variables that are equal to zero. Next to the predicates, there are various restrictions and conditions. In view of this, the generic MM shall be written as follows: X * * * Ai j (~y , ~s , ~z , c , k , w ) = 0, j ∈ Jm , i ∈ Jn j , (2.3) i * * *

Lq (~y , ~s , ~z , c , k , w ) , q ∈ Jnq , (2.4) where Aij is the operator of the ith term in the jth equation or by a formula of a combined equation, formulae, and boundary conditions; nj is a number of terms in the jth equation or a formula; m is a number of equations, formulae, and boundary conditions which belong to the MM; ~y is the vector of the TS outcoming signal;~s is the coordinate vector of physical space; ~z is the TS status vector; ~c is a sequence of working fluid properties; ~k is the sequence of the properties of the system and its elements; w ~ is the sequence of other values that are included in the MM; Lq is the qth predicate. The variables of the physical space are independent: ~s ∈ S ,

Def

~s = (s1 , s2 , s3 , s4 ) = (x1 , x2 , x3 , t) .

(2.5)

Physical constants of working fluid properties such as viscosity, heating capacity, density, etc., depend on the TS status parameters: ~c = c(~z ) .

(2.6)

The properties of the systems and their elements (e.g., vorticity diffusion and viscosity, spring hardness, etc.) depend on the variables of the physical space and the TS state parameters: ~k = k (~s , ~z ) .

(2.7)

TS state parameters in a general case depend on variables of the physical space: ~z = z (~s ) . (2.8)

2.1 Abstract Models

33

Thus, in a general case, the values ~c , ~k , ~z are dependent on the variables of the physical space or, in other words, the model of the TS is distributed in the coordinate space. The dependence (2.6) causes nonlinearity of the MM, and (2.7) is the nonlinearity of the system of equations, boundary conditions, and variability of MM coefficients in geometric space and time. However, for some tasks, the dependences (2.6) and (2.7) in some areas or in the whole space of the TS are weak and have insignificant influence on the result of the solution. In this case, in order to simplify the MM in the relevant areas and with respect to the appropriate coordinates of the physical space, the properties of the TS and the working fluid, as well as the parameters of the TS process are averaged. The average-integrated values are used when averaging some properties of the distributed model; that is why, thereafter, we will use the term “integral” talking about such models. Therefore, we can say that at the upper level of the MM structure, there are integrated models based on the phenomenological conservation laws of thermodynamics. They are represented as stationary and lumped models and are used to find the range of values of process parameters and make evaluations that are important at the stage of the problem analysis and design. At the lower level of the MM structure, we can imagine a differential model that takes into account that parameters and properties are differentiated and taken into account by dependences (2.6) and (2.7) (nonstationary and nonlinear model with distributed parameters and variables) at each point in the physical space of the TS. Consequently, it is possible to make the MM structures, which differ in the differentiation of parameters and properties in the physical space and to arrange them in accordance with the completeness of the recording of changes in the properties of substances and systems. In the case of a joint variation of a number of system parameters, its properties begin to depend on a set of parameters. In thermodynamics, this is described by means of cross-coefficients [1]. The introduction of crossconnections significantly complicates the MM process and its solution. As a rule, however, a part of the epiphenomena influence on the process is different. It gives a fundamental opportunity to make a ranked structure of the MM, where the factors which influence the process from a higher level to a lower level are taken into account more and more. Thus, the MM ordered series, which differs in the degree of adequacy and simplicity, provides a fundamental opportunity to choose the best MM with the help of a criterion that includes both of the above-mentioned factors: adequacy and simplicity of the model.

34 Modern Approaches to the Heat Exchange Modeling in NPP Equipment The scheme of a construction of a number of alternative MMs can be as follows: a complete differential MM and an integral MM, which represent the poles in the description of the process and define the boundaries of many alternative models. Intermediate MMs obtained by simplifying a complete MM on the way to an integral one are created on the basis of two different procedures: reduction of factors irrelevant to the problem and simplification of the type of dependencies with the same number of factors. In order to perform the first procedure, you need to evaluate the difference in solving the problem with two or more MMs with different number of factors taken into account and address the admissibility of MM simplification and error increase that occurs in such a case. Basically, this can be done by adopting a certain procedure to establish the adequacy (equality) of the results to the nearest certain predetermined value of ε: Ad

~y(n) (p1 , . . . , pnp ) = ~y(k ) (p1 , . . . , pk ) , pn ⊇ D ∗

ε

pk ⊇ D ∗

∀pi ∈ pn

pk ⊂ pn ,

(2.9)

where pi is the ith factor, which is taken into account in the MM; np and k are full and light set of factors; D* is a range space of factors which are defined by technical requirements for the TS. The procedure for adequacy establishment of the compared objects in a number of research works is performed in more than one way. As it has been mentioned before, we can proceed from a complicated model to a simple one only in exceptional cases when no other verification of the model is possible, for example, when solving problems of ecology and environmental protection, where an experiment is either physically impossible or can lead to irreversible catastrophic consequences and therefore is impossible due to the high risk [2–4]. If there is an opportunity to test the MM experimentally, it is necessary to reduce the factors taken into account according to the prediction of the MM solution. In this case, the decrease in the reliability of the obtained result must be counterbalanced by testing a simple MM in another reliable way, for example, experimentally. The prediction of a solution is possible by different methods, but all of them are based on simplified solutions or solutions of simple MMs. In order to develop a scientific basis for such a procedure, it is necessary to analyze the known options and perform their development in terms of solving the problem of the procedure of MM simplification by reducing the number of factors taken into account.

2.1 Abstract Models

35

If the solution of the problem depends on the complex, in which the factors are included in the form of the sum of the components, it is possible to estimate the contribution of each of them in relation to the largest of the additive components. If the relation of pk factor to the basic p1 order is ε  1( ppk1 = 0(ε)), then the influence of pk factor for the solution can be neglected. The relation pi /p1 ∀ i ∈ Jn ranges the factors according to their influence on the MM solution in the simplest way: p2 p3 pk pn 1> > > ··· > > ··· > . p1 p1 p1 p1 P In this case, due to their limited number and if nj=k pp1i  1, especially for different signs of the components, the reduction of the list of factors up to n − k does not significantly influence the result of the solution of the problem. This technique is well known and effective, but a group of MM competing factors is not always represented in the right form. The results that follow from the application of the theory of similarity and dimension are the development of these ideas. In such a case, the comparison is performed in relation to the complexes of factors that characterize separate phenomena. The theory of similarity and dimension allows not only simplifying a MM but also significantly generalizing it. Its value and results take a special place in all studies related to modeling, especially in the study of safety criteria for nuclear fuel. The fundamental theorem for the dimensional analysis is π-theorem [5]. Its main point is as follows. If there is a relation between n-dimensional values, it shall take the form: f (p1 , . . . , pn ) = 0,

(2.10)

among which k values have independent dimensions, then by means of transformations, the dependence (2.10) can be represented in the form F (π1 , . . . , πn−k ) = 0, where π1 , . . . , πn−k are dimensionless combinations of dimensional qualities. It is important to note that each dimensionless complex has the form of an exponential monomial [6]. Thus, on the basis of π-theorem, it is shown that the space of dimensional variables n by transformation to a dimensionless form is reduced to dimensionality n − k. There are plenty of dimensional values which are part of dimensionless complexes that correspond to each set πi , i = 1, . . . , n−k. There are many separate problems, which are determined by a large number of a single-valuedness condition and are reduced to a set of dimensionless complexes. This shows that the transformation to a

36 Modern Approaches to the Heat Exchange Modeling in NPP Equipment dimensionless form is a generalization, that is, one MM can correspond to an infinite number of specific implementations of parameters ∃pj : πi − idem ∀i ∈ Jn−k . According to the experimental design theory, the number of studies is an indicative function of the number of factors that influence the process [7, 8]. Thus, the number of studies N of a complete factorial is determined by the formula: N = q · ml , where q is the number of repetitions; m is the number of levels of every factor changes; and l is the number of variable factors. If the number of variable factors in the dimensional form of the MM is lp , then according to π-theorem after reducing the MM to a dimensionless form, they are lπ = lp − k. Thus, the number of studies when we use dimensionless complexes in the experimental study is mk times less (if all else were equal) than if we used dimensional factors. The results of the theory of similarity are widely used in the physical modeling of processes. According to the classical theory for two similar processes, the relation of the homogeneous values at similar points in the system is a constant value [9]: pi = µi = const, ∀i ∈ Jn , (2.11) p∗i where µi is the scale of the ith dimensional value; * is the notation of values which characterize the physical original. The procedure of reduction to a dimensionless form and establishment of similarity between two processes is performed by authors in different ways and has different results. The following reduction procedure can be considered as a classic one. For each dimensional value, a relative unit of measurement is chosen, which is, as a rule, equal to its value in the space−time continuum of the system. Dividing and multiplying each value by its relative value, we shall obtain the product of a unit of quantity by its dimensionless representation: pi Def = pei · p¯i , (2.12) pei pei pei = const, where pei is a relative unit of the ith value; the dash indicates the normalized value, which is transformed to dimensionless. If we convert each value of the equation (boundary condition) and take all constants out the signs of the operators, then we shall obtain coefficients of all

2.1 Abstract Models

37

its terms. According to the Fourier rule, their dimension is the same. Dividing all the coefficients by one of them, the equation (boundary condition) is reduced to a dimensionless form. If we reduce all the equations and boundary conditions of the MM to the dimensionless form, then it will have a form identical to the initial one but with dimensionless variables and criteria. In some cases, dimensionless complexes can be combined with variables and, consequently, move to generalized dimensionless variables (also called similarity numbers [11]). Similarity conditions of the two phenomena can also be established using the scales of quantities. Substituting all variables in the MM by means of relation (2.11) on variables of the original, taking quantity scale out of the operator symbols, and having divided all the coefficients by one of them, we shall obtain the form of an MM in variables of the original. Under the similarity condition, the form of both MMs must be identical; from here, the conditions of unit equality of all coefficients of equations in the form of power monomials follow [10, 12] Y αij µi = 1, i ∈ Jn , (2.13) i

where αij are the indicators of the power of the ith value in the jth coefficient. The procedure of bringing an MM to a dimensionless form allows determining the criteria that represent the ratio of factors which act on the described process. The comparison using the ratio allows ranking them with further simplification of the MM. These procedures, different in detail, are called maintenance [4], inspection [13], and fractional [5] analyses. Every rth variable is transformed in a dimensionless form: Def

p¯r =

pr − pr min , prmax − pr min

p¯r ∈ [0, 1] ∀r ∈ Jn ,

(2.14)

where pr min , prmax are the least and the largest values of the rth variable. After the substitution of Equation (2.14) in the MM, constant values that form the coefficients are taken out of operator symbols. Their values reflect the contribution of the relevant factor to the process described by the MM. The effectiveness of this procedure lies in the fact that simultaneously with the transformation of the space of variables into dimensionless, it becomes homogeneous and normalized. This transformation is called normalization [5].

38 Modern Approaches to the Heat Exchange Modeling in NPP Equipment Here in after, it is assumed that all the derivatives included in the MM are equal to 0. ∂ q pr Def = 0(1) k, e, l ∈ Jn . ∂pe · . . . · ∂pl {z } | q

Then all the equations, formulae, and boundary conditions of the MM represent the sum of terms with variables and their derivatives of the order of unit and coefficients different in value: XX ∂ q pr =0 ∀j ∈ Jm . πij ∂pe · . . . · ∂pl r i

If the coefficients are ranked in each equation, formula, and boundary condition of the MM, then, neglecting the small coefficients, we shall simplify the MM: πkj :

πkj Def = 0(ε) ⇒ πkj = 0 πjmax

∀k ∈ Jnj ,

where πjmax is the least coefficient. The method of smooth estimators, which is the core of fractional analysis, is essentially a simplified solution of the MM, in which all the dependences of the variables are represented by a linear form: p¯k = p¯r. but herewith:

∀¯ pk , p¯r ∈ [0, 1] ⇒

∂ p¯k = 1, ∂ p¯r

∂ 2 p¯k ∂ p¯k =1 ⇒ 6= 1, ∂ p¯r ∂ p¯2r ∂ p¯k ∂ 2 p¯k =1 ⇒ 6= 1. ∂ p¯2r ∂ p¯r

This inherent contradiction of the method of smooth estimators significantly limits the method. Satisfactory simulation results are obtained when the MM is written in first-order derivatives [6]. The inherent contradictions of the method of normalization and smooth estimators are the reason for other limitations of its application. For example, in the case of a two-dimensional problem and the ratio of the characteristic dimensions of the region in which the process ll12 − 0(ε) develops, one term of the equation can be neglected, according to the method of smooth estimators. But the equation of continuity disappears, while it must always be fulfilled

2.1 Abstract Models

39

as the law of conservation. The method is not very effective in the case of significant nonlinearities. The procedure of normalization meets insuperable problems in some tasks, as for a number of dependent variable values pr min and prmax are unknown in advance. Despite these disadvantages, the procedure of normalization in combination with the method of smooth estimators on a restricted class of problems can be successfully used, and it can be the basis for improving the method of evaluation. In order to expand the scope of applications of the method of smooth estimators to problems with nonhomogeneous change of parameters with respect to domains, Klain [5] suggests a method of zonal estimates. It can be used in problems where it is important to take into account the boundary layer (such problems include heat transfer problems in a nuclear reactor, steam generator, regenerative heaters, etc.). The method involves the selection of an area within which the coordinates of space are normalized separately, and the evaluation is designed using the exponential function. When applying this method, labor costs for the evaluation increase significantly compared to the method of smooth estimators. The procedure of zonal estimates is written in general terms; in the context of the details of the transformations, the success depends on the qualifications of the researcher, which reduces the effectiveness of the method even within a selected class of problems. The numbers of similarity of homochrony “Ho” represent a special group of criteria, the analysis of which can predict the solution of the MM. The analysis shows that for linear systems with fixed coefficients, the value of this number characterizes the completion of the process in time and space. This is due to the fact that the numbers of homochrony belong to the category of dimensionless variables, where the time variable refers to a complex with the dimension of time and is measured in “natural coordinates” [5] or a “distinctive scale” [14, 15]. If the complex with the dimension of time is correct, then it characterizes the dynamics of the object; that is why the ratio of the current time to this complex in the “Ho” numbers characterizes the duration of the transient process. The analysis of homochrony numbers allows making generalizations that are beneficial to evaluate the dynamics of the process and to develop rules for weighted idealization of the MM. Processes of heat and mass transfers in a nuclear reactor, filling facilities with liquids and gases (for example, transportation casks or storage of spent nuclear fuel), elastic loading of solid, liquid, and gaseous bodies, movement of a body in a force field (gravitational, magnetic, etc.), change of free surface

40 Modern Approaches to the Heat Exchange Modeling in NPP Equipment of phases, and a number of other phenomena have similar relations between the properties that determine these phenomena; so they can be described by one MM and studied according to its solution. The main property of these processes is the ability to accumulate (discharge) mass or energy. This property can be measured by the capacity “u.” The intensity of the process depends on the flux of substance or energy ˙ The capacity of physical objects is a bounded quantity. For example, the Q. heat capacity is limited by melting or evaporation of the body; the load is limited by the strength properties of the materials. Therefore, in a real TS, the change “u” is limited to a certain value umax . It is possible to impose restrictions on the process, for example, by terminating the action on the TS at time t1 t1 : u(t1 ) = umax ; Q˙ = 0 ∀ t > t1 . An idealized influence shall be written down with constraints: ( Q˙ 0 − const; 0 ≤ t ≤ t1 ; Q˙ = 0; t > t1 ,

(2.15)

and without constraints: ( Q˙ 0 − const; Q˙ = 0;

∀t ≥ 0; t < 0.

(2.16)

The process in the TS shall be written in the form of the model du ˙ = Q, u(0) = u0 (2.17) dt which represents the equation of an ideal integrating factor [16]. The solutions (2.15) and (2.17) in criterion variables take the form:   Ho < 0 ; 0 ; u ¯ = Ho ; 0 ≤ Ho ≤ 1 ; (2.18)   1; 1 < Ho , where

t u − u0 ; Ho = . umax − u0 t1 For the case of influence (2.16) the solution has the following form: ( 0; Ho < 0 ; u¯ = Ho ; Ho ≥ 0 . u ¯=

(2.19)

2.1 Abstract Models

41

The influence of the type (2.17) is an idealized connection of the TS with the environment. In a general case [1], the specific flux that changes the j parameter of the TS is determined as follows: qj =

n X

γjk Yk ,

(2.20)

k=1

where γjk is the transport coefficient and Yk is the generalized thermodynamic force in accordance with the “k”th parameter. Taking into account Equation (2.2), the flux in the TS can be determined by the scalar product of the specific flux vector by the vector normal to the elementary surface area of the TS through which the flux enters: Z X n n X −−−→ ˙ qj = γjk Yk Qj = γjk Yk · d ~s. (2.21) Sj k=1

k=1

In the linear theory, the transport coefficients do not depend on the TS parameters. In many cases, only main transfer effect is practically taken into account when determining the flows. This effect corresponds to the equality k = j in the dependence (2.20). Bearing in mind the comments above, the integrated flux and the specific flux shall be defined as follows: Q˙ j = Bj · ∆yj , qj = βj · ∆yj ,

(2.22)

grad y s is the integral coefficient of the where Bj = − sj βj |grad y| · d ~ transportation of the jth flux and ∆yj is the pressure of the jth potential. When researching the TS properties, not so much its capacity is measured, but the parameter “y” associated with the property of capacity Z Z u= c · d ζ · d V, (2.23)

R

VE

y0

where V T S is the TS volume and c is a coefficient, which, in general case, depends on “y” and the TS space coordinate. For a homogeneous TS relative to “c,” there is a relation: Z y c · dζ, (2.24) u = VTS y0

and in a linear approximation, it is Def

u = VTS · c(y − y0 ) = C(y − y0 ),

(2.25)

42 Modern Approaches to the Heat Exchange Modeling in NPP Equipment where C is the coefficient, which characterizes the TS capacity. Let us assume that the external influence of the generalized environment on the TS has the form: yge − y0 = a · E(t),

(2.26)

where a is invariable and E(t) is Heaviside unit step function. Then taking into consideration the dependencies (2.22), (2.25), and (2.26), the MM of the TS process can be represented as follows: T

dy = yge − y ; dt

y(0) = y0 ;

yge − y0 = a · E(t).

(2.27)

After the transformation of the variable “y” according to the formula y* = y − y0 , the MM shall be written in the following form: T

d y∗ ∗ = yge − y0 − y ∗ = yge − y∗ ; dt

y ∗ (0) = 0 ;

∗ yge = a · E(t) , (2.28)

C where T = B is the time constant. Thus, the time constant that characterizes the dynamics of the process in the TS represents the ratio of the coefficients of capacity and conductivity for the corresponding flux. Here in after, as it is conventional to the theory of automatic control, let us assume that at t < 0, the TS and the environment are in equilibrium, and the initial conditions of the TS are equal to zero. The * sign can be omitted. The Aij is (2.15), (2.17), and (2.28) are easily generalized. Then

y/yge = 0(ε)

⇒

∆y ≈ ygec

∆y = (yge − y) = yge (1 − ε) ∀y : y/yge = 0(ε) .

(2.29)

Substituting Equations (2.22) and (2.25) in Equations (2.15) and (2.17) in approximated (2.29), it is easy to find that the MMs (2.15) and (2.17) are the idealization of the MM (2.28) for the case (2.29). In other words, at slight relative deviations, the potential of the initial state of the feedback has little effect on the TS. The solution of the MM (2.27) under normal variables is # " −Ho  , (2.30) y¯ = 1 − exp T t∆ where yˆ =

y−y0 a−y0 ; Ho

=

t M tM ; t

is the time scale, which shall be equal to T.

2.1 Abstract Models

43

Thus, for processes that have the MMs (2.28) and (2.27), the number of homochrony is determined by the ratio: Ho =

t . T

(2.31)

Generalizing the obtained results, we can formulate the rule to evaluate the dynamics of a single-site system where the processes of mass or energy exchange described in the linear approximation take place. Having the value C , of the number Ho = Tt , which corresponds to the relation (2.31), and T = B the dynamic of the process is characterized as follows:   − nonequilibrium process 0(ε) (2.32) Ho = 0(1) − transient process   0 (1/ε) − equilibrium process. The evaluation of the TS dynamic properties and the environment with lumped parameters can be generalized to the case with distributed parameters. A feature of the latter is that in addition to the dependence of parameters of the space coordinates, there is the finite value of mass and energy flux resistance during their transmission inside a spacious TS and the environment. In macroscopic systems, changes in parameters in space are described by piecewise differentiable functions with a finite number of discontinuities of the first kind. Thus, choosing the area of the TS elements between the parameter discontinuities and reducing the characteristic size of the area l, we can select the TS element in which the parameter variations do not exceed a predetermined value of ε and consider it as a lumped value. In this case, the conservation equation with respect to the average integral values of the TS parameters of the ith element shall be written as follows:   X Z X −−−→ X d y˜ij γjk Yk · d ~s + = J˙j i m , (2.33) cij dt sj m j

k

i

where J˙j i m is the mth source of the ith element of the jth substance or energy. Finding the TS element and the description of the processes in a group of joined elements is the concept of the theory of finite elements, which has found a wide application is different disciplines that deal with

44 Modern Approaches to the Heat Exchange Modeling in NPP Equipment continuous medium. Thus, Equation (2.33) shall be written in the following form: ! P PPP d y˜ji cji d t = sj p q · βj k p q · ∆yk p q + J˙j i m ; (2.34) m q p k i
∆yk p q = y˜k (i+p−2) − y˜k (i+p−1) q , (p = 1, 2) , where ∆yk p q is the pressure of the kth potential in the direction q via p−y surface of joint elements; p-indexes: 1 is the input signal, 2 is the output signal, and ∼ is the notion of the average integral value. In a dimensionless form, Equation (2.34) can be written as follows: ! P P P si p q · βj k p q d y¯ ˜j i = ∆¯ yk p q dHoj q i q p k sj 1 q · βj 1 q i P ˙ Jj i m m + , (2.35) (sj 1 q · βj 1 q )i (˜ yj ooc − y˜j i 0 ) where Hojq i =

t·sj 1 q ·βj 1 q ; cj i

y¯ ˜j i =

y˜j i −˜ yj i 0 y˜j ooc −˜ yj i 0 ;

∆yk p q =

(y˜k (i+p−2) −˜yk (i+p−1) )q y˜j

yj i 0 ooc −˜

The transmission coefficient β can be represented by virtue of the coefficient of transmission χ and the characteristic dimension ˜l of the space, where the transposition takes place: βj p q =

χj q . ˜lp q

(2.36)

Relying on the dependence (2.36), the number of homochrony has the form: Hoj q i =

t · sj 1 q · χj q . cj i · ˜lp q

(2.37)

The indexes in Equations (2.33)–(2.35) and expressions (2.36) and (2.37) represent the features of a distributed TS in comparison with a lumped one, in which the direction of transfer and the dimensions of the element are significant. The specific features of the distributed TS are reflected in the block diagram as well. Between the conjugate elements of the TS, there are direct and feedback links that characterize the transfer problem as a boundary one. The parameters of the process include the dimensions of the surfaces

.

2.1 Abstract Models

45

through which the transfer of mass and energy takes place. It reflects the impact on the process of the geometric shape of the TS. In a general case, the environment can have anisotropy of conductivity, which is taken into account by the block diagram and expressions (2.33)–(2.35). In order to evaluate the dynamics of the TS and its environment, we can divide it into a small number of elements. It is of great importance to take into account the physics of processes and specific features of the distributed system when distinguishing the area of elements. In the general case, under the appropriate boundary conditions, the solution of Equation (2.35) cannot be represented as it is done for the lumped TS. Therefore, to illustrate the effectiveness of estimates for the values of the HO numbers, in the following sections, we shall consider heat exchange problems, which have an explicit solution. The presence of nonlinear relations in the MM is its very important characteristic. The admissibility of the corresponding idealization can be characterized by two criteria. The first criterion is a qualitative one. As we know, the linear models cannot describe some features of highly nonlinear systems in principle. Thus, it is impossible to obtain a solution of the autooscillation process in linear systems [16]. That is why the fundamental impossibility of nonlinear effect description serves as a qualitative criterion for the inadmissibility of MM linearization. The second criterion that characterizes nonlinearity is a quantitative criterion. It can be based on the importance of the output signal deviation of the linear model from the similar − a nonlinear one [17]. In this case, it is necessary to have a solution of a nonlinear MM or to have the result of experimental testing of the corresponding object. It has already been mentioned that in order to solve the issue of the factor or effects relevance, which are included in the MM, it is rational to use a simplified solution of the MM. The method of integrated coefficients, which is described below, can be used as such a technique. In a closed MM, the differential parameters could be converted to the average integral values that in a number of cases significantly simplifies MM and do not decrease the results informativeness.. In order to achieve these purposes, the author [2] developed the averaging method with the help of which a number of difficult tasks (such as turbulent transfer, flux of twophase media, and processes in porous media) has been solved. However, the averaging method is not always easy for evaluations, and obtaining of integrated values requires knowledge of the dependences of variables from coordinates and time.

46 Modern Approaches to the Heat Exchange Modeling in NPP Equipment The normalization of variables allows obtaining useful generalizations on the basis of which it is easier to find analogies and make evaluations based on the known results of similar tasks as well as to intuitively evaluate their value. In order to use this, normalized values are introduced; they relate the average integral variables to the scales of these values and the coordinates of the space in which the process is studied. When describing the process, the effective area Sef is chosen in the general case of four-dimensional space “time−geometric space,” where the change is important. Another space is uninformative. An analogue of the effective space is the depth of penetration in integrated methods [18, 19]. Let us assume that Z Z 1 ... δpi ds1 . . . dsn , (2.38) δ p˜i = Vn ef S1 ef Sn ef where δpi = pi − pi min is the fluctuation of the variable, δ p˜i is the average integrated fluctuation of the variable in the n-dimensional space, Vnef is the volume of effective n-space, and Sj is the coordinate of space of independent variables. Let us assume that n Y Vnef = ξVnef · s∆ (2.39) jef j=1

where ξVn ef is the completeness coefficient Sef in the parallelotope Φef = {sj ∈ Sj ef : sj min ef ≤ sj ≤ sj max ef ∀j ∈ jn } , Q where volume is equal to nj=1 s∆ jef In order to relate δ˜ pi with the scale of the ith value p∆ i = pimax − pimin , an integral coefficient is introduced ˜i Def δ p p∆ i

ζpi Vn ef =

∈ [0, 1]

δ p¯˜i = ζpi Vn ef .

⇒

(2.40)

Using the definitions (2.38)–(2.40), the integral of the variable variation in the space Sef can be represented by the integrated coefficient, the coefficient of the completeness of the space, and scales of the value and coordinates: ζpi Vn ef ·

p∆ i

· ξVn ef ·

n Y j

s∆ j

Z =

Z ...

s1 ef

δpi ds1 . . . dsn . s1 ef

(2.41)

2.1 Abstract Models

47

The obtained result can be extended to a variation function of one or more variables. The integrated coefficient can be determined by calculating the quadrature with an approximation of the variable variation by means of the testing function δpi = fi (s1 , . . . , sn , k1 , . . . , kl ) to an accuracy up to the constant kj , which describes its behavior. The integrated coefficient can be determined at an intuitive level when performing evaluations. The integrated coefficient can also be determined by analogues which are known from a source of literature or from specialpurpose experiments. All conservation laws for the denoted space Sef can be represented in the integral form with the operator of full-time differentiation over the integral of geometric space [2]. The remaining MM equation can also be represented by integral and algebraic equations. Thus, taking into consideration the transformation (2.41), the MM can come to an equivalent system of standard differential and algebraic equations, which is much easier to solve than a boundary value problem with partial derivatives. The solution of the equivalent system of equations is performed after the similarity transformation, which allows generalizing the result and using additional information on similar or analogous phenomena. If closing relations of the MM are not enough for its single-valued solution, then it is possible to obtain the solution with the accuracy up to the constant kj which are identified by the results obtained in the literature or specially set experiments. The generalization of normalization and the introduction of normalized integrated coefficients and completeness coefficients of the effective space make the method of integrated coefficients more flexible and universal than the method of averaging [2] and the method of linear estimates [5], without excluding the possibility of using all the results of these methods. Every process significantly depends on the interaction between the TS elements or the environment and the TS as a whole, as well as the state of the TS elements, such as the development of transfer phenomena, the completion of the process over time, and parameter variations in space. In addition, the dependence of the properties of the involved working bodies on the process parameters or other factors can be significant. For example, other factors can include definitely unknown composition of the working body. Each of the characteristics mentioned above may be the result of the solution of a complex MM or long-term experimental studies. However, it is easy to establish upper and lower estimates of these characteristics in many cases. Thus, the process in time is limited to completely nonequilibrium and

48 Modern Approaches to the Heat Exchange Modeling in NPP Equipment equilibrium states, the parameter variation in space is limited to values from the initial to the boundary ones, and the process of transfer is confined from the molecular level to the molar turbulent or convective one. The properties of working bodies can be evaluated in the same way. The general MM of the process can be represented in the form of blocks with direct directorial links: X Aij (~y , p~, ~s) = 0; i ∈ jn j , j ∈ (Jm /Jk ) ; (2.42) i

X

Aij (p1 , . . . , pl1 , ~q, ~s) = 0;

j ∈ (Jk /Jd ) ;

(2.43)

Aij (q1 , . . . , ql2 , ~r, ~s) = 0;

j ∈ (Jd /Jl ) ,

(2.44)

i

X i

where ~y is the TS output signal, p1 , . . . , pl1 are the characteristics of the first level of the process in the TS, q1 , . . . , ql2 are the characteristics of the second level of the process in the TS, p~, ~q, ~r are the dependent variables of the MM, and ~s are independent variables of the MM. The MM block (2.42) represents its core. According to the scheme (2.14), after normalization, the MM core in its general form can be solved up to the characteristics p1 , . . . , pl1 , every one of which can be represented by the completeness coefficients or the completion coefficient of the process p¯k = ηk ∈ [0, 1] .

(2.45)

The introduced coefficient can be estimated intuitively, determined by selfmodeling areas, set experimentally, and calculated from the solution of the MM block (2.43). The latter corresponds to the MM solution at the second level of completeness of the description of the process in contrast to the first one, when the completeness coefficients and completeness of subprocesses are estimated. It is possible to further precise the MM solution due to the solution of the block (2.57) and subsequent blocks.

2.2 Generalization of the Mathematical Model Let the dimensional value qi be determined by the measure function (numerical value) pi and the unit of measurement ei , which belongs to the ith class of comparison Qi . The values qi and qj belong to the corresponding classes of comparison qi ∈ Qi and qj ∈ Qj , and when the units

2.2 Generalization of the Mathematical Model

49

of measurement change, the corresponding value of the measure function is determined by the scale of change of the units µ. For two different classes, there is no scale to set up a correspondence between measure functions in these classes: ∀qi , ∀qi ,

qi ∈ Qi , qi = pi · ei , qi0 = T p0i · e0i ∃µ : p0i = µi · pi , ¯ : p0 = µi · pi . qj : qi ∈ Qi , qj ∈ Qj Qi Qj = ∅∃µ i

(2.46)

In [6], it is shown that in the physical laws, which do not depend on the choice of the units of measurements, the ratio between scales of dimensional values when changing the units of measurements has the following form: Y αij µi = µj , i, j ∈ Ju , (2.47) j

where αij is the correspondent degree indicator and u is the number of dimensional values. The value of the exponents in (2.47) is determined by the structure of the MM. It is useful to establish a relation between the ratios of scales and measure functions of dimensional values, which are determined by the MM structure. If the scales (u) ofQall dimensional values, which are included in the MM, are α in the ratio µi = j µj ij , ∀i, j ∈ Ju , then measures of these values are in a similar relation: Y αij pi = pj , ∀i, j ∈ Ju . j

Q α It follows from pi = ⇒ µi = j µj ij , which is determined by the homogeneity of the initial dependence on the scale [6]Q and is checked by α the direct substitution p0i = µi · pi . The assertion µi = j µj ij ⇒ pi = Q αij Q α is proved by contradiction. Let us assume that µi = j µj ij ⇒ j pj Q β Q α Q β pi = j pj ij , but then we have j pj ij = j pj ij . The comparison of the quantities pi and pj can be accomplished only in their class; hence, αij = β ij . Consequently, Y αij Y αij µi = µj ⇔ pi = pj . (2.48) Q

αij j pj

j

j

Q αij According to the Fourier rule in the formula pi = p dimensions Q j αjij of values in right and left parts are equal to Ei = that allows j Ej Q αij . dimensionless complexes j pj pi together with the results (2.48) to transform the MM space.

50 Modern Approaches to the Heat Exchange Modeling in NPP Equipment The MM shall be written as follows: i=n Pj

    

*

Aij ( p ) = 0;

j=1

*

(2.49)

p ∈ P, P = Y ⊗ S ⊗ W,   *  * * y ∈ Y, s ∈ S, w∈ W,

where Aij is the operator of the i-term in the jth equation or the MM formula; * p = (p1 , p, . . . , pn ) is the finite sequence of all dimensional values which * are part of the MM; y = (y1 , y, . . . , yny ) is the finite sequence of all *

values which define the behavior of the system; s = (s1 , s2 , s3 , s4 ) is the finite sequence which define the coordinates of the geometric space and time; * * * w= (w1 , w2 , . . . , ynw ) is the finite sequence which adds to s and y till * p ; nj is a number of members in the jth equation or the MM formula; m is a number of equations and formulae in the MM. According to the theory of similarity, the establishment of the conditions of physical modeling is performed on the MM, which is reduced to a dimensionless form. This can be done in different ways. However, the most effective procedure is performed in a general form. To do this, let us normalize all dimensional variables. In this case, the quantities, by which the normalization is performed, are not yet determined. This is convenient to do later to get the desired results. After the normalization of all dimensional values, the normalized ones are factored out from the sign of the operator. The product of values, which are normalized, and other constants form coefficients, the dimension of which, according to the Fourier rule, in each equation or formula is the same. The divisions of all others by any coefficient form dimensionless complexes. These complexes and normalized variables make up a list of values that determine the process described by the original MM. The formalized record of the procedure of reduction to a dimensionless form and obtaining dimensionless variables and complexes shall be represented as follows: i=nj

ϕ:

X

i=nj

*

Aij ( p ) = 0 →

X

(2.50)

i=1

i=1 * ¯

y = (¯ y1 , . . . , y¯ny ), yl y¯l = ∆ , yl

* ¯ * ¯ * Aij ( y , s , π ) = 0,

skl s¯k = ∆ , sk

* ¯

s = (¯ s1 , . . . , s¯ns ) πi j =

q=u Y q=1

α

pq i j q

(2.51)

2.2 Generalization of the Mathematical Model

51

where u is a number of dimensional values in the MM and ∆ is the designation of the variate in standard measure. Some of π ij are identically equal to one or other numbers. Let us exclude them from consideration as the same for all the systems described by the original MM, and, therefore, they do not add information within the considered class of the MM. Let each pair (i, j), which corresponds to a dimensionless complex π ij , have one and only one element of a natural series where π ij is not equal to one or another number: (i, j) ↔ h,

h ∈ Jt ,

(2.52)

where t is the number of dimensionless complexes. Now each complex shall be represented as: πh =

q=u Y

α

pq nq .

(2.53)

q=1

Thus, as a result of the transformation (2.50), the MM is reduced to a dimensionless form, the solution of which, if it exists, shall be written as follows: * ¯ * * y = f ( ¯s , π ), (2.54) *

where π = (π1 , π2 , . . . , πt ). It is known that a necessary and sufficient condition for the similarity of the two processes G is the equality between all like criteria and dimensionless variables: G:

¯ ∀sk ∈ S,

πh ∈

⊃ D∗ : (¯ sk = s¯∗k ) ∧ (πh = πh∗ )

(2.55)

where * is the designation of a membership of a natural physical system and D is the range of values of a certain technical task for the system. In Equation (2.55), the condition of similarity is specified by the fact that the values of the criteria must lie in the area denoted by the technical task for a pilot physical system. If the MM of the original and its model after the procedure (2.50) meets the condition (2.55), then they become indiscernible, and the identity of the operators of the MM original and the model determines the transparency of the solutions (2.54) for the original and the model. In our further research, we should definitely try finding transformations that reduce the dimensionality of the space in which the problem is solved. If

52 Modern Approaches to the Heat Exchange Modeling in NPP Equipment we make a mutually unique transformation Ψ between a great number of dimensional values of the full-scale system and a great number of values of the model quantities, then, with the account of the transformation Ψ, the simulation conditions shall be written as follows: ∃pq : (Ψ : p∗q ↔ pq ) ∧ G,

∀p∗q q ∈ Ju .

(2.56)

Let the transformation Ψ have the following form: pq = µq · p∗q ,

∀q ∈ Ju .

(2.57)

In this case, the first numbers of the natural series are assigned to the elements * * of the finite sequence y , the next ones to s , and other dimensional values of the MM. The latter is the parameters included in the boundary conditions and the physical constants of the working bodies involved in the process. Taking into account the relation (2.52), the substitution of the expression (2.57) in the formula (2.51) allows obtaining the following result: q=u Y

α

µq hq = 1,

∀h ∈ Jt .

(2.58)

q=1

The logarithmization of the expression (2.57) allows obtaining a system of linear homogeneous algebraic equations



~ = 0, A1 · M (2.59)  . . . α1u · · · · · ·  is the matrix of αnq degree indicators . . . αtu

α11 α12 whereA1 =  · · · · · · αt1 αt2 and ~ = [ln(µ1 ), ln(µ2 ), . . . , ln(µu )]T is the column-vector of a scale log M of all dimensional values. Using the Gauss−Jordan method, the matrix A1 can be converted to the form:   . A1 → E .. B , (2.60) where E is the unity matrix of the size (r × r), r − rank [A1 ]. Lines with linearly dependent elements are removed from the matrix A1 , and B is the matrix of exponents βhv of the size [r × (u − r)].

2.2 Generalization of the Mathematical Model

53

In a general form, the matrix shall be represented as: p1 , . . . , pq , . . . , pr, pr+1 , . . . , pu y1 , . . . , yny , s1 , . . . , sns , z1 , . . . , znz , zΩ1 , . . . , zΩn , c1 , . . . , cnc   1 ··· 0 k β11 · · · β1(u−r) π1  ···  k   πn  1 · · · k · · · βnv ···  · · ·  .   ··· k πr 0 ··· 1 k βr1 · · · βr(u−r) (2.61) Above the matrix, there is the finite sequence of all dimensional values of the MM, the powers of which in complexes πh constitute the matrix A1. Here, z is process parameters, c is physical constants of working bodies, and Ω is quantities included in the boundary conditions of the problem.  .. Using the matrix E . B , it is easy to write down the solution of the system (2.58) and analyze the results. The solution of the system (2.59) for the qth scale shall have the form: µq =

v=u−r Y

−βqv

µv

∀q ∈ Jr .

,

(2.62)

v=1

Now the normalizing values (2.47) will take the form: pM q =

v=u−r Y

−βqv

pv

, ∀q ∈ Jr ,

(2.63)

.

(2.64)

v=1

and normalized ones: p¯q =

pq v=u−r Q

−βqv

pv

v=1

After transformations, dimensionless complexes πh shall take the form according to the formula: πh0 =

q=r Y

p∆ q

q=1

Y
αnq q=u

α

(pq hq ).

q=r+1

Substituting Equation (2.63) in Equation (2.65), we can get: !α q=r Y Y −βqv nq q=u Y αhq πh0 = pv (pq ). q=1

v=1

q=r+1

(2.65)

54 Modern Approaches to the Heat Exchange Modeling in NPP Equipment We can draw two conclusions from the results of the transformations: [1] If the rank A1 < ny + ns , then such MMs are reduced to an automodeling form. This follows from the fact that if rank A1 = r, then ny + ns − nr of independent variables will be included in the right part of formula (2.63), and nr of the normalizing values will be expressed through these variables and other u − (ny + ns ) dimensional values. The space of independent variables after the transformations (2.64) will be reduced, and the MM in the new space will have an auto-modeling form. [2] If rank A1 < t + ny + ns , then the number of dimensionless complexes will be less than that followed from π-theorem. From the statement of π-theorem, it follows that the number of dimensionless complexes is equal to n – k, where k is a number of dimensional values with independent dimensions by definition [31]. In this case, the structural features of the MM are not taken into account, and, as a result, n − k ≥ t + ny + ns . After the transformations (2.64) and (2.65), the number of dimensionless values will be nr , where nr ≤ rank A1 . Hence, taking into account the initial condition, we have nr < n − k. In other words, after the transformations (2.64) and (2.65), the number of dimensionless complexes will be less than that followed from π-theorem. The performed transformations make it possible to obtain all known results of reducing the dimensionality of space up to bringing the MM to an automodeling form, as well as to obtain new results due to the deep penetration in the link of the similarity theory with the structure of the MM. The developed algorithm of transformations allows widely using reduction of space dimensionality of the MM in various researches. The dimensions of the matrix A1 of complex processes are larger; it causes significant difficulties in transformations to a dimensionless form if, in accordance with the procedure described above, the elements of the matrix are considered as rational fractions. In order to simplify the computer implementation of the algorithm in accordance with the formula (2.60), we suggest at, the initial stage, replacing rational fractions with their decimal values. After performing all the calculations at the final stage, the results in the form of decimal fractions will be replaced by rational fractions, for example, according to the Euclidean algorithm [22]. This makes it possible to perform all subsequent transformations (2.64) and (2.65).

2.3 Simplified Method of the Numerical Solution

55

2.3 Simplified Method of the Numerical Solution of Nonstationary Heat Transfer Problem Through a Flat Wall The nonstationary heat transfer problem can be solved using numerical methods. A large number of similar practically important processes can be described using relatively simple one-dimensional models. Processes such as thermal conductivity in the fuel pellet and heat transfer from the fuel pellet through the fuel rod cladding to the coolant medium (coolant in any phase or in the gaseous medium) can be distinguished here. The problem is the discrepancy between the simplicity of the model and the complexity of the proposed numerical method for its solution. When using simple models, no additional opportunities to generalize the results of the solution by presenting the model in a dimensionless form are used. Therefore, it is necessary to develop a simplified discrete specific analogue of the real thermophysical process to calculate the process of nonstationary heat transfer through an infinite plate (one-dimensional model). This will make it possible: [1] to represent a discrete analogue and, as a consequence, the results of the calculations in a dimensionless form with the possibility of their appropriate generalization; [2] to obtain a discrete analogue that makes it possible to carry out calculations with the required accuracy on rough computational grids (with a small number of calculation nodes) and with large time steps with maintaining the stability of numerical calculations. In some studies, when using the method of control volumes, the emphasis is on creating universal two- and three-dimensional analogues. The example of one-dimensional analogue only explains the principle of their construction but does not consider the possibility of how to solve a one-dimensional version of the problem on the basis of dimensionless values. The universalism of the simplified discrete specific analogue is manifested in the fact that even in the one-dimensional case, there is a source term that takes into account the sources or heat fluxes inside the heat exchange surfaces. In the case of heat transfer through the fuel rod cladding, the sources (fuel pellet) and the energy flux (heat carrier or gas) are outside the cladding through which the heat exchange takes place. To understand the algorithm, the term that describes the sources was excluded from consideration. Within the framework of the method of control volumes, the examined space, as well as the finite element method, is divided into separate small

56 Modern Approaches to the Heat Exchange Modeling in NPP Equipment

Figure 2.2

Calculation model for one-dimensional discrete analogue.

elements – control volumes. In a three-dimensional orthogonal coordinate system, these are “orthogonal” elements, in the Cartesian reference system, they take the form of parallelepipeds, and in the one-dimensional model, they are layers (Figure 2.2). Inside these layers, there are nodes, which, similar to the method of finite differences, are assigned with the value of calculated quantities. The geometrical characteristics of the calculated region presented in this way are: δx are the distances between grid nodes; ∆x is the dimension of the layers (elements) into which the region is split. The position of the nodes inside the different layers may differ. The very dimensions of layers can also differ. That is why, in a general (nonstationary) case, both δxi and ∆xi differ, and, in the general case, δx 6= ∆x. In the case of uniformity of thermophysical properties in the heat transfer surface, in order to simplify the discrete analogue, let us assume that: δxi = const;

∆xi = const;

δx = ∆x.

(2.66)

In the method of control volumes, the nodes of the computational grid (points “1” and “N” in Figure 2.2) are situated on the boundaries of the calculated region. They are surrounded by noncomplete boundary control volumes. Taking into account Equation (2.66), their value in this case is ∆x/2. It must be considered when recording a discrete analogue for nodes at the boundaries of the calculated region. In addition, for these nodes, in contrast to the internal ones, it is necessary to take into account the boundary conditions on the corresponding surfaces. Thus, discrete analogues for internal and boundary nodes differ, but they must have a general character of their record for the possibility of their regular solution.

2.3 Simplified Method of the Numerical Solution

57

Let us consider a discrete analogue for internal points. Based on the general constructions from [21] for the internal point “Pi ” and taking into account the parameters at the points “Wi ” and “Ei ” (Figure 2.2), the calculation of temperatures can be written as follows: aP · TP = aE · TE + aW · TW + b ,

(2.67)

where ∆x 0 ·T . ∆t P (2.68) Here, ρ, c, and λ are the density, heat capacity, and the thermal conductivity of the material of the heat exchange surface, respectively; ∆t is the calculation step by time; TP , TE , and TW – (current) temperatures in specific points; TP0 is the value of the temperature at point P from the previous calculation step by time. The value from the initial condition (initial temperature profile) is represented at the first step of calculation. Substituting Equation (2.68) in Equation (2.67), let us divide all terms of the equation by λ/δx. As a result, taking into account (2.66), we shall obtain   1 (∆x)2 1 (∆x)2 · TP = TE + TW + · · TP0 , (2.69) 2+ · a ∆t a ∆t aP = aE + aW + ρ · c ·

∆x ; ∆t

aE =

λ ; δx

aW =

λ ; δx

b = ρ·c·

where a = λ/(ρ · c) is the temperature conductivity coefficient. The expression (2.69) is an expression of a simplified discrete specific analogue in a dimensional form to solve the problem of nonstationary thermal conductivity. In order to present it in a dimensionless form, let us set the thickness of the heat exchange surface equal to 2l. Let us multiply the second term of sum in square brackets by (2l)2/(2l)2 = 1 and denote in this expression ∆x/2l = δx/2l = ∆. It is a dimensionless (relative) layer thickness of a discrete analogue. Taking into account further transformations, we shall receive: 1 (∆x)2 (2l)2 1 · = · (∆)2 = · (∆)2 . a ∆t a · ∆t ∆(F o)

(2.70)

Here, ∆(Fo) is the dimensionless calculation step by time, that is, the change step of Fourier number. Having performed similar transformations for the last addend in the right part of the expression (2.69), we shall obtain:   1 1 2 2+ · (∆) · TP = TE + TW + · (∆)2 · TP0 . (2.71) ∆(F o) ∆(F o)

58 Modern Approaches to the Heat Exchange Modeling in NPP Equipment The expression (2.71) is a dimensionless discrete analogue for the internal points of the calculated region. Let us consider a discrete analogue for the node of the calculation grid at the boundary of the calculated region, to be definite at point “1” on the left boundary (Figure 2.2). In order to calculate the temperature at point P1 , it is necessary to take into account the boundary conditions to the left of it, as well as the values at point E1 . As boundary conditions, let us consider the conditions of the third kind as the most general ones. Based on the general constructions from [21], for the boundary point “1,” it can be written aP · TP = aE · TE + b ,

(2.72)

where aP = aE + α1 + ρc

λ ∆x 1 0 ∆x ; aE = ; b = α1 Tav1 + ρc · T . (2.73) ∆t δx 2 ∆t P

Here, α1 , T av1 is the heat-exchange coefficient and the temperature of environment from the corresponding side of the heat exchange surface. As in the previous case, let us substitute Equation (2.73) in Equation (2.72) and divide all terms of the equation by λ/δx. As a result, taking into account Equation (2.66), we shall obtain   δx 1 (∆x)2 1 1 + α1 · + · · · TP λ a 2 ∆t = TE + α1 ·

δx 1 (∆x)2 1 · Tav1 + · · · T0. λ a 2 ∆t P

(2.74)

In the expression (2.74), let us multiply the terms containing δx by (2l)/(2l) = 1. Those terms which contain (∆x)2, we multiply by (2l)2/(2l) 2 = 1. We shall also take into account that ∆x/2l = δx/2l = ∆. Let us denote α1λ·2l = Bi1 − Biot criterion from the side of the node “1” of the computational grid. As a result, we shall obtain:   1 1 2 1 + Bi1 · (∆) + · · (∆) · TP 2 ∆(Fo) = TE + Bi1 · Tav1 · (∆) +

1 1 · · (∆)2 · TP0 . 2 ∆(Fo)

(2.75)

The expression (2.75) is a dimensionless discrete analogue for the left boundary point of the calculated region (Figure 2.2).

2.3 Simplified Method of the Numerical Solution

59

Having performed transformations similar to Equations (2.72)–(2.75), but for the right boundary (node “N,” Figure 2.2), we shall obtain a dimensionless discrete analogue for the right boundary point of the calculated region:   1 1 2 1 + Bi2 · (∆) + · · (∆) · TP 2 ∆(Fo) = TW + Bi2 · Tav2 · (∆) +

1 1 · · (∆)2 · TP0 . 2 ∆(Fo)

(2.76)

Here, Bi2 , Tav2 is the Biot criterion and the ambient temperature from the side of the “N” node of the computational grid. The expressions (2.71), (2.75), and (2.76), taken together, represent a dimensionless simplified discrete specific analogue for the calculation of nonstationary heat transfer through a flat plate under boundary conditions of the third kind. The solution algorithm based on such a discrete analogue can be implemented by means of TriDiagonal-Matrix-Algorithm (TDMA). The computational results are obtained in one “sweep,” without iterations, which simplifies the calculation. In order to simplify the implementation of the solution algorithm, let us write a discrete analogue (2.71), (2.75), and (2.76) in the index form, where the indices are calculated from the left to the right boundary of the calculated region (Figure 2.2): • for the internal points: ai · Ti = bi · Ti+1 + ci · Ti−1 + di , where ai = 2 +

1 ∆(Fo)

· (∆)2 ;

bi = 1;

ci = 1;

di =

(2.77) 1 ∆(Fo)

· (∆)2 · TP0 ;

• for the left boundary: a1 · T1 = b1 · T2 + d1 ,

(2.78)

where a1 = 1 + Bi1 · (∆) + d1 = Bi1 · Tav1 ·

1 2 ∆(Fo) · (∆) ; 1 (∆) + 21 · ∆(Fo) · 1 2

·

b1 = 1; (∆)2 · TP0 ;

• for the right boundary: aN · TN = cN · TN −1 + dN ,

(2.79)

60 Modern Approaches to the Heat Exchange Modeling in NPP Equipment where aN = 1 + Bi2 · (∆) + dN = Bi2 · Tav2 ·

1 2

1 ∆(Fo) (∆) + 21

·

· (∆)2 ; ·

1 ∆(Fo)

cN = 1; · (∆)2 · TP0 .

The solution algorithm is set up on the ground of [21] as follows. Direct course of computation – auxiliary coefficients P and Q are calculated. First, from (2.78) P1 and  1 b1   = P1 =   1 1  a1  1 + Bi1 · (∆) + · · (∆)2    2 ∆(Fo)   , (2.80) Q1 1 1   Bi1 · Tav1 · (∆) + · · (∆)2 · TP0   d1 2 ∆(Fo)   Q1 = =   1 1 a1   1 + Bi1 · (∆) + · · (∆)2  2 ∆(Fo) and then from Equations (2.77) and (2.80) Pi and  1 bi   = Pi =   1  ai − ci · Pi−1  2+ · (∆)2 − Pi−1    ∆(Fo)   Qi 1   · (∆)2 · TP0 + Qi−1   d + c · Q ∆(Fo) i i i−1   = . Qi =   1 a − c · P  i i i−1  2+ · (∆)2 − Pi−1  ∆(Fo)

(2.81)

Inverse course of computation − for the right boundary bi = 0. Thus, PN = 0.3 (2.81): 1 1 · · (∆)2 · TP0 + QN −1 dN + Qi−1 2 ∆(Fo) QN = = . 1 1 aN − Pi−1 · (∆)2 − PN −1 1 + Bi2 · (∆) + · 2 ∆(Fo) (2.82) Let us read from Equation (2.82), Bi2 · Tav2 · (∆) +

TN = QN

(2.83)

2.3 Simplified Method of the Numerical Solution

61

and further in the reverse order: Ti = Pi · Ti+1 + Qi .

(2.84)

The algorithm must qualitatively and quantitatively reflect the examined processes. Therefore, he evaluation of the physicality and adequacy of the developed discrete analogue can be carried out in two ways: [1] after appropriate transformation by comparing the analogue with the exact solution in the extreme case of stationary heat transfer through an infinite flat wall (evaluation of physicality is a qualitatively correct representation of the studied process); [2] by comparing the results of numerical calculations with the results of analytical solutions in a particular case of nonstationary process of symmetrical heating (cooling) of an infinite plate (evaluation of the accuracy of the quantitative representation of the studied process). For both of these cases, there are exact analytical solutions. When solving the problem of stationary heat transfer through a flat wall, the heat flux through both wall surfaces is assumed to be identical. To be definite, let us assume (Figure 2.3) that the ambient temperature to the left of the wall is higher than the temperature to the right and the heat flux is directed from left to right. This can be written as: • q1 = α1 · (Tamb1 − Tw1 ) – conditionally for the left side of the wall; • q2 = α2 · (Tw2 − Tamb2 ) – conditionally for the right side of the wall.

Figure 2.3 Minimal computational grid.

62 Modern Approaches to the Heat Exchange Modeling in NPP Equipment Here, q1 , q2 is respectively the heat flux, which comes from the environment from left to the wall and comes from the wall to the right into the environment; α1 , α2 are heat transfer coefficients on the left and right sides of the wall, respectively; Tw1 , Tw2 are the temperatures of the left and right sides of the wall, respectively; T amb1 , T amb2 are the ambient temperatures from the left and right sides of the plate, respectively. With respect to the set condition q1 = q2 , we shall obtain: α1 · (Tamb1 − Tw1 ) = α2 · (Tw2 − Tamb2 ) or

(Tamb1 − Tw1 ) α2 Bi2 = = . (2.85) (Tw2 − Tamb2 ) α1 Bi1 Here, Bi1 and Bi2 are Biot criteria for the left and right sides of the wall. The discrete analogue of the form (2.82)–(2.84) is obtained for the case of nonstationary heat transfer. But at constant ambient temperatures on the left and right sides of the wall and quite a long heat transfer process, a properly constructed analogue (2.82)–(2.84) must yield a result similar to Equation (2.85). In addition, in this case, the profile of the temperature change from Tw1 to Tw2 (Figure 2.3) inside the wall must be linear, and a properly constructed discrete analogue also must reflect it. Let us transform the analogue (2.82)–(2.84) to test the feasibility of these requirements. Let us split the considered flat wall into three layers (Figure 2.3): two of which are wall boundary layers and one inner layer. In this case, the temperatures T 1 and T 3 of the analogue correspond to the temperatures Tw1 and Tw2 on the surface of the wall. The temperature T 2 is the temperature in the central layer. According to Figure 2.2, the wall boundary layers have half valve thickness. The choice of only three layers for partition is due to the following considerations. The discrete analogue for the temperatures Ti in the central units (2.82) is set up using the temperatures in the adjacent units Ti − 1 and Ti + 1 . Three layers allow constructing a similar analogue for the temperature T 2 with respect to the temperatures Tw1 and Tw2 . The wall boundary layers allow using analogues (2.83) and (2.84) for the left and right boundaries of the plate. Thus, three layers for calculation are enough to evaluate the accuracy of the representation of the process of stationary heat transfer using the developed discrete analogue, and an increase in the number of inner layers in case of need of partition adds nothing fundamentally new to the calculation. In the analogue (2.82)–(2.84), the values Tp0i , Tp01 , Tp0n coefficients di , d1 , and dn , and represent the temperatures in respective nodes of the

2.3 Simplified Method of the Numerical Solution

63

computational grid from the previous calculation step by time. To calculate the heat transfer process for a long period of time and entering the stationary mode, the temperatures from the previous calculation step by time must be equal to the temperatures calculated at the current time at the appropriate points. For the case of three layers, we have: Tp02 = T2 ;

Tp01 = T1 ;

Tp03 = T3 .

(2.86)

After the substitution of the expressions from Equation (2.82) for all the coefficients, we shall obtain: 2 · T2 +

1 1 · (∆)2 · T2 = T3 + T1 + · (∆)2 · TP02 . ∆(Fo) ∆(Fo)

(2.87)

With respect to Equation (2.86) of the equality Tp02 = T2 , we shall receive the equality of terms from the left and right parts of the expression (2.87): 1 1 · (∆)2 · T2 = · (∆)2 · TP02 ∆(Fo) ∆(Fo) and after their reduction, we have: T3 + T1 (2.88) 2 Having performed similar transformations for analogues of the left (2.83) and right (2.84) boundaries, we shall obtain: 2 · T2 = T3 + T1

or T2 =

• for the left boundary T1 + Bi1 · (∆) · T1 = T2 + Bi1 · (∆) · Tamb1 ;

(2.89)

• for the right boundary T3 + Bi2 · (∆) · T3 = T2 + Bi2 · (∆) · Tamb2 .

(2.90)

Let us substitute the expression for T2 from (2.88) to (2.89) and (2.90); accomplish transformations, and we shall obtain: • for the left boundary T3 − T1 = Bi1 · (∆) · T1 − Bi1 · (∆) · Tamb1 ; 2 • for the right boundary T3 − T1 = Bi2 · (∆) · Tamb2 − Bi2 · (∆) · T3 . 2

(2.91)

(2.92)

64 Modern Approaches to the Heat Exchange Modeling in NPP Equipment In the expression (2.91) and (2.92), left parts are equal. Let us set right parts equal and reduce them by ∆: Bi1 · (T1 − Tamb1 ) = Bi2 · (Tamb2 − T3 ), or Bi1 · (Tamb1 − T1 ) = Bi2 · (T3 − Tamb2 ).

(2.93)

As a result, we have: Bi2 (Tamb1 − T1 ) α2 . = = (T3 − Tamb2 ) α1 Bi1

(2.94)

The comparison of the expressions (2.94) and (2.85) shows their coincidence, which confirms the accuracy of the construction of the analogue (2.82)–(2.84) in this part. Now let us analyze the second part of the test − the representation using the proposed discrete analogue of the form of the temperature profile inside a flat wall. To do this, let us determine the tangents of the inclination angles of the temperature profiles in the regions (T1 − T2 ) and (T2 − T3 ): • from Figure 2.3 for the region between the temperatures T1 and T3 with respect to Equation (2.89), we have: tg(φ1,2 ) =

(T1 − T2 ) = Bi1 · (Tamb1 − T1 ); ∆

(2.95)

• from Figure 2.3 for the region between the temperatures (T2 − T3 ) with respect to Equation (2.90), we have: tg(φ2,3 ) =

(T2 − T3 ) = Bi2 · (T3 − Tamb2 ). ∆

(2.96)

From Equation (2.93) follows the equality of the right parts in expressions (5.31) and (5.31). Hence, left parts are equal as well: tg(φ1,2 ) = tg(φ2,3 ).

(2.97)

Thus, in the process of stationary heat transfer, the sections of the temperature profiles provided by the analogue (2.82)–(2.84) have a common point T 2 and equal inclination angles (2.97). Therefore, they represent one straight line. This corresponds to the analytical solution and confirms the accuracy of the construction of the analogue (2.82)–(2.84) in this part as well.

2.3 Simplified Method of the Numerical Solution

65

The accuracy of numerical calculations using the proposed discrete analogue can be evaluated by comparing their results with the available analytical solutions. One of the few available is the case of unilateral heating of the fuel rod cladding from the fuel pellet. Let us analyze the case when, at the initial time (t = 0), the initial temperature in the plate is evenly distributed. Under these conditions, the proposed analytical solution has the form of the sum of a series. In [21], it is stated that when Fo ≥ 0.3, the series starts converging so rapidly that the temperature distribution is quite accurately determined by the first term of the series in the form: 2 · sin µ1 Θ=1− · cos(µ1 · X) · exp(−µ21 Fo). (2.98) µ1 + sin µ1 · cos µ1 Here, Θ is the relative (dimensionless) temperature of the plate; X is the relative (dimensionless) coordinate of the considered point, which is calculated from the center of the plate to its surface; and Fo = a·t is the δ2 Fourier number, where a is the coefficient of thermal conductivity and δ is the half-thickness of the plate. The relative temperature of the plate Θ is determined by the relationship Θ=

T (X) − T0 , Tamb − T0

(2.99)

where T (X) is the current temperature in the corresponding point of the plate; T0 is the initial temperature which is uniformly distributed in the plate; and Tamb is the ambient temperature up to which the plate can be heated. The relative temperature of the plate changes in the range Θ ∈ [0. . .1]. The relative coordinate is determined from the relationship X = x/δ and varies in range X ∈ [0. . .1]. Here, x is the absolute coordinate calculated from the center of the plate to its surface. The value µ in Equation (2.98) is the root of the transcendental equation ctg(µ) = µ/Bi. Equation (2.98) is the first term of the series. That is why µ1 , which is the first positive root of the transcendental equation, is used here. In order to compare the results of analytical and numerical calculations, it should be noted that the Biot criterion in the analytical solution is calculated for the half-thickness of the plate (due to its symmetry). However, in the numerical calculation, taking into account the possibility to solve nonsymmetric problems, it is determined for the full thickness of the plate. Therefore, it is necessary to use the relationship (Bi1 )s = (Bi2 )s = 2 · Bi0 .

(2.100)

66 Modern Approaches to the Heat Exchange Modeling in NPP Equipment Here (Bi1 )s and (Bi2 )s are Biot criteria for the respective sides of the plate in a discrete analogue (numerical calculation) for the case of symmetrical heating, and (Bi)a is the Biot criterion in analytical calculations. The Fourier number also differs for the cases of analytical and numerical solutions. The comparison of the results corresponding to the identical moments of dimensionless time should be carried out according to the ratio: 4 · Fon = Foa . (2.101) Here, Foa is the Fourier number for analytical calculations, and Fon is the Fourier number corresponding to the numerical calculations. In Figure 2.4, the results of the calculation based on the analytical expression (2.98) and the discrete analogue (2.77)–(2.89) are presented. Since the analytical calculations are performed for the dimensionless temperature (2.99), then the numerical solutions are obtained on its basis. Figure 2.4 shows the results covering a wide range of changes of the Bio numbers. Its values for numerical calculations based on Equation (2.100) are indicated in parentheses. Each figure shows the results for two moments in time. The values of Fo numbers corresponding to them are bound by the relation (2.101). For every time moment, the maximum relative error ε of numerical results from all calculated points on the plate thickness is given. In total, we examined 21 computational points. The relative error was determined in relation to the range of temperature changes Θ ∈ [0. . .1]. The comparison of the results of numerical and analytical calculations shows a close agreement. However, other numerical methods, if they have a large number of calculation points, can also give similar accuracy. The method of control volumes is distinguished by the implementation of the conservation laws on the calculation grids of any accuracy. In order to evaluate the influence of the number of calculation points on the computational error, we performed calculations using maximally small grids – where there are only three nodes (Figure 2.3). The obtained results and their comparison with analytical and numerical calculations on a large grid (21 nodes) are given in Table 2.1. Here, the values of Bi and Fo related to numerical calculations are given in parentheses as well as in Figure 2.4. The values of Bi and Fo in analytical and numerical calculations, as in the previous case, are related by the formulae (2.100) and (2.101). The first three lines show the values of relative temperatures for Fo = 0.4 (0.1). In this case: • the first line shows the temperatures in the analytical calculations;

2.3 Simplified Method of the Numerical Solution

67

Figure 2.4 Relative temperature Θ depending on the relative coordinate X and the relative moments of time Fo for analytical Θa and numerical Θn solutions. (a) Bi = 0.004 (0.008). (b) Bi = 0.5 (1.0). (c) Bi = 5 (10). (d) Bi = 50 (100) .

• the second line shows the temperatures in numerical calculations on a grid with 21 nodes; • the third line shows the temperatures in numerical calculations on a grid with three nodes.

68 Modern Approaches to the Heat Exchange Modeling in NPP Equipment Table 2.1 Results of analytical and numerical calculations of the relative temperature Θ at symmetrical heating of an infinite plate. Magnitudes of errors of numerical calculations with respect to analytical calculations. X Fo/ε Bi 0.0 0.5 1.0 0.4 0.842 0.378 0.842 (0.1)21 0.837 0.366 0.837 (0.1)3 0.873 0.393 0.873 ε21 0.005 0.012 0.005 ε3 0.031 0.015 0.031 5 ε21−3 0.036 0.027 0.036 (10) 1.0 0.944 0.779 0.944 (0.25)21 0.941 0.766 0.941 (0.25)3 0.955 0.765 0.955 ε21 0.003 0.013 0.003 ε3 0.011 0.014 0.011 ε21−3 0.014 0.001 0.014 0.4 0.985 0.507 0.985 (0.1)21 0.984 0.487 0.984 (0.1)3 0.991 0.528 0.991 ε21 0.001 0.020 0.001 ε3 0.006 0.020 0.006 50 ε21−3 0.007 0.041 0.007 (100) 1.0 0.996 0.881 0.996 (0.25)21 0.996 0.868 0.996 (0.25)3 0.997 0.851 0.997 ε21 , (5.1) where < q1 > is average linear capacity in the fuel element over the reactor core, W/cm. Denoting the maximal lineal capacity in the j-cell of the reactor core as qi,j,max , and denoting the coefficient of relative capacity of (i, j)-cell as ki,j , the expression (5.1) was used to define the linear capacity in the central point of (i, j)-cell as follows: < ql,i,j >= ql,j,max · ki,j .

(5.2)

The methods of analysis of the probability of fuel cladding depressurization use verified software reactor simulator [11] and FEMAXI according to the energy version of the creep theory [12, 13]. The software IP was used to calculate the change of the linear capacity of the (i, j)-cell of the WWER-1000 reactor core and for the control of axial offset stability during the NPP power maneuver [1–4, 6, 8, 9, 11]. The FEMAXI software was used to calculate the evolution of stresses and strains in the fuel claddings [10, 12–14].

5.2 Simulation of Fuel Cladding Reliability The calculated value of the probability of fuel cladding depressurization without taking into account the heterogeneity of energy distribution among the fuel elements in fuel assemblies can be used to predict the heterogeneity WWER-1000 fuel cladding reliability. If we consider the average fuel

156

Analysis of WWER 1000 Fuel Cladding Failure

element with respect to the fuel assembly, the probability of fuel cladding depressurization is based on the accepted maximum fuel cladding failure, which allows predicting the fuel cladding reliability [1–4]. The method of modeling the fuel cladding reliability uses a criterion model of the efficiency on how to control the properties of fuel elements, which allows taking into account the requirements for both safety and efficiency of fuel element operation. According to the criterion model, the maximum efficiency Eff of the fuel element property control is determined by the criterion, the most general form of which is given in [1, 8, 9] and [14]. The method of predicting the fuel cladding reliability is the development of the method on how to control the properties of fuel elements at the stages of the WWER design and operation [14] because when controlling the amount of fuel cladding failures, the change of predicted probability of fuel cladding depressurization immediately occurs. The method of prediction of fuel cladding reliability includes [1]:] 1) the application of the algorithm on how to control the properties of fuel elements on the basis of the method of the energy version of the creep theory and the organization of iterative calculation of sets of determining factors, which are characterized by the largest values of the criterion Eff [8, 14]; 2) the calculation of the probability of fuel cladding depressurization for sets of determining factors with the largest Eff values; 3) the choice of the best sets of determining factors based on the condition of ensuring a minimum probability of fuel cladding depressurization. The following assumptions were made: 1) averaged fuel element with respect to the WWER-1000 fuel assembly is examined; the material of the fuel cladding is zircaloy-4; 2) the “alternative” algorithm of daily maneuver capacity of the reactor unit is 100% → 80% → 100% Nnom , tin = const [15]; 3) the control method of fuel assembly rearrangements in the reactor core is used to predict the fuel cladding reliability of WWER fuel elements [14]; 4) the model of fuel assembly rearrangement is taken in accordance with Figure 5.4. According to the control method of fuel assembly rearrangements, if to examine the fuel assemblies used in a certain algorithm j, the controlled parameters are maximum ωjmax and average < ω >j failure damage, minimal value of burn-up Bjmin , whereas the algorithm of fuel assembly

5.2 Simulation of Fuel Cladding Reliability

157

rearrangement is the varied discrete factor [1–3, 4, 8]. The cells in the selected sector of the WWER-1000 reactor core were randomly chosen by the “rand” function of the MATLAB software [16]. When applying the model of fuel assembly rearrangement and taking into account the displacement of the control group, ω(τ ) and the depth of fuel burn-up B(τ ) in the most intense axial segment were calculated in the following sequence: 1) coefficients ki,j were calculated for all axial segments in all the cells of the sector of symmetry of the reactor core applying the software IP when N = 80 and 100% Nnom [11]; 2) the most stressed axial segment was defined; for this segment, ω(τ ) and B(τ ) were calculated applying the software FEMAXI for five algorithms of fuel assembly rearrangement when τ = 0...1460 days. 3) The criterion of the energy version of the creep theory was used when A0 = 30MJ/m3 (ultimate work to material failure), the corrosion rate of the fuel cladding was set according to MATPRO-A model [12–14, 17]. The algorithms 17 and 18, which were practically used at the NPP, as well as three random algorithms (2, 3, and 6) are represented in Table 5.3. The calculations of effectivity Eff for five algorithms, which are given in Table 5.3, have shown that the aim of the control of fuel assembly rearrangement is accomplished in the algorithms 3, 6, and 18 (in the order of Eff decreasing) [1, 9, 14]. In order to determine the probability of fuel cladding depressurization, averaged fuel cladding failure with respect to the fuel assembly for the jth algorithm of the fuel assembly rearrangement was considered as a random variable value rand distributed according to the normal law in the interval j [< ωjrand > −∆ωjrand ; < ωjrand > +∆ωjrand ], j = 2; 3; 6; 17; 18. Taking into account the three sigma rules of the normal distribution and using the data in Table. 5.3, standard deviations σ(ω rand ) of the random value ωjrand j were found. For the jth algorithm of the rearrangement the probability P j of the depressurization of the averaged fuel cladding with respect to the fuel assemblies was calculated according to the expression (Table 5.4)   (ωjrand −)2 Z ωmax exp − · dωjrand 2[σ(ωjrand )]2 j √ . (5.3) Pj = σ(ωjrand ) 2π ω lim

158

Analysis of WWER 1000 Fuel Cladding Failure

Table 5.3 Fuel cladding failure and burn-up in the axial segment of the sixth fuel element (accumulated work to material failure). Algorithm Rearrangement A, MJ/m3 ω(τ ) = A/A0 , % V, MW·day/kg 5-30-10-43 1.838 6.127 63.04 9-11-20-1 1.443 4.81 57.26 3-22-54-29 1.843 6.143 63.89 13-19-21-42 2.652 8.84 68.13 2 2-31-18 1.209 4.03 47.61 55-41-12-6 1.955 6.517 59.1 4-32-68-8 1.368 4.56 57.02 9-19-21-8 2.253 7.51 62.49 5-41-68-43 1.391 4.637 60.47 55-22-10 2.167 7.223 54.67 13-11-20-6 1.421 4.737 56.8 3 3-30-54-1 1.387 4.623 55.04 4-32-18-42 1.722 5.74 62.69 2-31-12-29 1.976 6.587 63.88 55-11-18-43 1.568 5.227 63.84 13-32-20 2.019 6.73 54.19 3-31-10-8 1.816 6.053 59.65 9-19-68-42 2.054 6.847 65.55 6 4-41-12-29 1.935 6.45 64.93 2-30-21-6 1.522 5.073 54.82 5-22-54-1 1.238 4.127 53.05 2-22-12-6 1.463 4.877 54.35 3-41-29 1.184 3.947 48.8 4-11-68-43 1.078 3.593 60.63 5-19-10-8 1.498 4.993 57.18 17 9-30-20-1 2.058 6.86 59.39 13-32-21-42 2.667 8.89 68.23 55-31-54-18 2.437 8.123 67.45 2-22-21-6 1.55 5.167 54.86 3-41-68 1.18 3.933 48.83 4-11-29-18 1.159 3.863 60.84 5-19-20-1 1.449 4.83 54.55 18 9-32-12-42 2.586 8.62 67.86 13-30-10-43 2.551 8.503 67.73 55-31-54-8 1.982 6.607 61.37

The expression (5.3) is characterized by the error, which is related to the fact that < ωjrand > 6= < ωj > (see Tables 5.3 and 5.4): < ωjrand > − < ωj > max{ } = 10 %. < ωj >

5.2 Simulation of Fuel Cladding Reliability

159

Table 5.4 Probability P j of fuel cladding depressurization in the jth algorithm. j ω lim , % < ωjrand >, % 2 · ∆ωjrand , % ωjmin , % ωjmax , % σ(ωjrand ), % Pj 2 6.435 4.81 4.03 8.84 0.8017 0.0035 3 6.067 2.887 4.623 7.51 0.4812 0 6 5.487 2.72 4.127 6.847 0.4533 0 8.5 17 6.242 5.3 3.593 8.89 0.8833 0.0039 18 6.242 4.757 3.863 8.62 0.7928 0.00085

It is possible to improve the accuracy of the calculation of the probability of fuel cladding depressurization by modifying the expression (5.3) with the help of the combination of truncated normal distributions [1]. Since six fuel assemblies of the fourth year campaign are used in the algorithm of the fuel assembly rearrangement within the sector of symmetry of the reactor core, the total number of fuel elements in the fuel assemblies of the fourth year is 3126 = 1872. Knowing the probability P j of depressurization according to Bernoulli distribution, it is possible to calculate the probability of depressurization of k fuel elements where n = 1872 fuel elements which are contained in six fuel assemblies, and which have operated for four years, k Pj,1872 (k) = C1872 · (Pj )k · (1 − Pj )1872−k ,

(5.4)

1872! k where C1872 = k· (1872−k) !. If we examine six identical sectors of symmetry of the reactor core, the event “fuel cladding depressurization” in any sector of the reactor core means simultaneous fuel cladding depressurization of the corresponding fuel element in the other five sectors; so the expression (5.4) implies the expression for the probability of the depressurization of 6 k fuel elements where n = 11, 232 fuel elements which are contained in 36 fuel assemblies operating for four years in all the sectors of the reactor core (Table 5.5): k Pj,11232 (6 · k) = C1872 · (Pj )k · (1 − Pj )1872−k .

(5.5)

Table 5.5 Probability of fuel cladding depressurization in the algorithm j, %. Number of depressurized fuel claddings 6 · k (k = 0, 1, 2, . . . , 12) j 0 6 12 18 24 30 36 42 48 54 60 66 72 Probability of depressurization of 6 · k fuel claddings, % 2 0.14 0.93 3 6.55 10.9 14.3 15.7 14.7 12 8.76 5.72 3.41 1.85 3 100 0 6 100 0 17 0.07 0.49 1.79 4.36 7.97 11.6 14.2 14.8 13.5 11 8 5.3 3.22 18 20.4 32.4 25.8 13.7 5.44 1.72 0.46 0.1 0

160

Analysis of WWER 1000 Fuel Cladding Failure

Algorithms 3 and 6 dominate all other variants of the fuel assembly rearrangement, including practically used algorithms 17 and 18 because they have zero probability of fuel cladding depressurization at the accepted maximum allowable value of the fuel cladding failure ω lim = 8.5%. The probability of depressurization for practically used rearrangement algorithms 17 and 18 is less than 18 fuel elements, and it is 2.4% and 78.6%, respectively. The probability of depressurization of 18−72 fuel elements for algorithms 17 and 18 is 94% and 21.4%, respectively. The probability of depressurization of more than 72 fuel elements for algorithms 17 and 18 is 3.6% and 0%, respectively. In the light of the condition of ensuring a minimum probability of fuel cladding depressurization, algorithms 3 and 6 have the best sets of discrete factors. Thus, examining the averaged fuel element with respect to fuel assemblies, a method for calculating the probability of fuel cladding depressurization depending on the sequence of sets of factors that determine the fuel element failure was developed. The possibility of predicting the fuel cladding reliability by controlling the factors that define its failure is shown. Taking into account all fuel assemblies which have operated in the reactor core for four years and assuming ω lim = 8.5% as the maximum allowable value of the fuel element failure, the probabilities of depressurization of more than 18 fuel elements for practically used algorithms 17 and 18 are 97.6% and 21.4 %, respectively. Algorithms 3 and 6 of fuel assembly rearrangement which have zero probability of fuel element depressurization at the accepted ω lim are found. As it is shown in the analysis, it is possible to significantly increase the accuracy of the calculation of the probability of fuel cladding depressurization if to take into account the non-uniformity of the distribution of energy release through the fuel elements inside the fuel assemblies [1].

5.3 Computational Method of Fuel Cladding Failure The calculation of fuel cladding failure was performed on the basis of the following assumptions taking into account the non-uniformity of the distribution of the energy release among the fuel elements of fuel assemblies: • the control program where tin = const is used for the capacity control of the reactor core; • the scheme of the control group placement in the reactor core which is accepted for tin = const is used; only the 10th group of the Control Rod Drive Mechanism is used to control the capacity in the reactor core;

5.3 Computational Method of Fuel Cladding Failure

161

• when maneuvering capacity, the control group moves with an amplitude of 0.04 from their effective length; the coolant temperature at the inlet to the reactor core is set tin = 287◦ C; • the model of fuel assembly rearrangement of the four-year cycle is used; it is shown in Figure 5.4; • the length of the fuel element is conditionally divided into eight equal axial segments, which are numbered from the bottom edge of the lowest fuel pellet; • the energy release in fuel assemblies is evaluated calculating kv,i,j in eight axial segments (i = 1, . . . , 8) for the cells of the specified segment of the reactor core, which are shown in Figure 5.1 (j = 2, . . . , 68); • the non-uniformity of energy release distribution in the fuel elements of fuel assemblies of alternative design is considered by conditional allocation of four groups of fuel elements, each of which is characterized by the averaged coefficients kv,i,j ; • mode and design parameters of the fuel element and the reactor core were set in accordance with the data in Table 5.1; • computations were accomplished for the beginning of the campaign; • load graphs and position of the 10th group correspond to Figures 5.2 and 5.3; • the calculation of relative energy release coefficients for four conditionally selected groups of fuel elements was performed with a three-dimensional and two-group diffusion nodal software ANC-H (Advanced Nodal Code-Hexagonal) [3, 4]; • the energy release of each fuel element was identified by correction coefficients to homogenized sections; • a layout, according to which 1 node in the radial direction and 24 nodes in the axial direction, was used in the computational model; these nodes were then averaged in eight axial layers. Layout methods for fuel elements in fuel assemblies into groups: For each fuel assembly, the layout of fuel elements into four groups was performed according to the value in the maximum energy intensive axial segment of the fuel element (axial segment 4 or axial segment 5) on the basis of the following conditions: I group: kv,4−5,j ∈ [0, 4...1]; II group: kv,4−5,j ∈ [1...1, 2];

162

Analysis of WWER 1000 Fuel Cladding Failure

III group: kv,4−5,j ∈ [1, 2...1, 4]; IV group: kv,4−5,j ∈ [1, 4...1, 7]. The computational values of the relative capacity in the cells (i, j) for four conditionally selected groups of fuel elements in all the cells of the sector of symmetry of the reactor core at capacity levels which are equal to 100% and 80% are given in Tables 5.6–5.13. The coefficients in the axial segment 6 were also calculated by means of the program IP for the case when N = 80% and 100% of Nnom for all the cells of the sector of symmetry of the reactor core (Table 5.14). On the basis of the data in Tables 5.6–5.13, the coefficients of relative energy release in the axial segment 6 for fuel elements of four groups (group number m = 1, . . . , 4) are recorded (Table. 5.15). Using the data in Table 5.15 for the fuel assemblies 2–5, 9, 13, and 55 of the first year of operation, the values kv,6,j averaged over four groups of fuel elements are calculated according to the following expression: kv6,j (ANC − H) =

4 X m=1

kv,6,j,m

nm . 312

(5.6)

The value kv,6,j (ANC − H) obtained according to expression (5.6) was compared with the value kv,6,j (IP) given in Table 5.14, obtained according to the program IP, Table 5.16. The value of the relation of kv,6,j (ANC − H) to kv,6,j (IP ) is given in Table 5.17. The relation of kv,6,j (ANC − H) to kv,6,j (IP ) is equal to ≈1.07 when N = 100% and 80% which is averaged for seven fuel assemblies; this indicates an acceptable value of the systematic error of the method used for the division of the fuel elements into four conditionally selecte groups. Methods of obtaining the given groups of fuel elements: Let us examine the algorithms of fuel assembly rearrangement 2 and 6 as an example; their characteristics are given in Table 5.18. In the rearrangement 5-30-10-43 of algorithm 2, the division of fuel elements with conditionally selected groups is given in Table 5.19.

6 6 6

0.240 0.379 0.427 0.451 0.456 0.437 0.388 0.245

80

100 55 0.236 0.359 0.397 0.419 0.427 0.417 0.374 0.238

80

0.230 0.353 0.394 0.419 0.426 0.406 0.359 0.234

8

100 13 0.323 0.501 0.557 0.581 0.584 0.562 0.499 0.314

0.371 0.608 0.687 0.721 0.725 0.693 0.610 0.370 106

80

6

5 0.380 0.616 0.687 0.713 0.714 0.684 0.601 0.362 108 30

100

0.251 0.396 0.446 0.470 0.476 0.456 0.405 0.256

0.396 0.658 0.742 0.777 0.779 0.743 0.649 0.378 63

80

80

4 0.410 0.673 0.748 0.774 0.773 0.738 0.644 0.380 67 22

100

—

—

—

—

—

—

—

—

—

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—

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—

—

—

—

—

I kv, i,j , i = 1, . . ., 8

—

—

—

—

—

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—

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n

0.583 0.802 0.889 0.945 0.963 0.86 0.814 0.586 2

41 0.598 0.814 0.892 0.940 0.960 0.937 0.846 0.589 2

32

31

0.396 0.657 0.741 0.755 0.778 0.742 0.650 0.388 61

80

6

3 0.414 0.678 0.753 0.780 0.778 0.743 0.649 0.384 67 19

100

9 0.255 0.399 0.443 0.462 0.466 0.448 0.398 0.249

0.374 0.613 0.691 0.725 0.729 0.698 0.615 0.374 98

80

100

2 0.386 0.625 0.696 0.722 0.723 0.692 0.609 0.367 102 11

100

j

j

N,%

n

Relative capacity in the computation cell (i, j) for the first group of fuel elements.

I kv, i,j , i = 1, . . . , 8

Table 5.6

5.3 Computational Method of Fuel Cladding Failure

163

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80

100 21

80

100 54

80

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—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

n

6 0.212 0.308 0.344 0.362 0.367 0.357 0.323 0.224 312

0.241 0.352 0.398 0.422 0.429 0.415 0.375 0.263 312

1 0.245 0.355 0.397 0.417 0.422 0.410 0.369 0.257 312

80

—

— —

— —

—

—

—

—

—

—

—

—

—

—

—

—

—

0.376 0.532 0.596 0.635 0.647 0.628 0.572 0.409 308

0.585 0.790 0.879 0.936 0.955 0.902 0.787 0.594 216

—

0.466 0.633 0.707 0.754 0.771 0.743 0.679 0.509 305

0.583 0.784 0.871 0.929 0.949 0.897 0.784 0.581 225

— 43 0.599 0.797 0.876 0.927 0.949 0.929 0.833 0.589 226

—

— 42 0.601 0.804 0.884 0.934 0.956 0.934 0.838 0.602 218

—

— 29

100 68 0.480 0.646 0.714 0.757 0.777 0.764 0.705 0.520 305

—

0.572 0.823 0.921 0.976 0.991 0.955 0.864 0.604 15

100 20

80

—

—

—

—

I kv, i,j , i = 1, . . . , 8

8 0.385 0.539 0.598 0.629 0.640 0.622 0.566 0.400 310

—

80

—

—

—

j

100 18 0.590 0.839 0.928 0.972 0.982 0.951 0.858 0.593 31

—

100 12

—

—

n

0.208 0.305 0.344 0.366 0.372 0.360 0.327 0.229 312

—

80

—

i = 1, . . . , 8

—

—

j

100 10

N,%

Relative capacity in the computation cell (i, j) for the first group of fuel elements.

I kv, i,j ,

Table 5.7

164 Analysis of WWER 1000 Fuel Cladding Failure

—

—

100 55

80

—

—

—

—

—

— —

—

1.123

80

0.605 0.945 1.062 1.115

1.138

j

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

II kv, i,j , i = 1, . . . , 8

—

—

—

—

—

—

—

—

—

—

—

—

1.088 0.963 0.611 19 31

1.046 0.922 0.565 46

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

— —

— —

—

0.639 0.971 1.073 1.139 1.157 1.071 0.899 0.548

2

3

—

—

1

2

—

—

—

—

—

—

n

—

0.692 0.940 1.040 1.107 1.129 1.075 0.956 0.699 175

— 41 0.710 0.956 1.045 1.103 1.127 1.101 0.994 0.702 181

1.076 0.956 0.612 27

1.092 0.967 0.611 39 32 0.663 0.997 1.088 1.143 1.161 1.128 0.989 0.569

—

—

0.645 0.977 1.079 1.145 1.163 1.097 0.947 0.583

1.046 0.919 0.599 54 30 0.670 1.009 1.101 1.155 1.174 1.139 0.990 0.564

1.048 0.915 0.548 31

1.037 0.903 0.536 28 22

1.033 0.904 0.542 28

1.045 0.911 0.541 28 19

1.063 0.936 0.575 49

1.055 0.926 0.565 50 11

n

0.615 0.955 1.070 1.125 1.134134 1.088 0.968 0.624 13

1.134

1.094

1.094

1.102

1.089

1.086

1.097

1.112

1.102

100 13 0.638 0.987 1.093 1.136

80

0.640 0.986 1.090 1.132

100

9

0.570 0.923 1.039 1.089

80

0.594 0.951 1.056 1.094

100

5

0.568 0.937 1.053 1.100

80

0.587 0.956 1.059 1.093

100

4

0.561 0.923 1.038 1.084

80

0.593 0.963 1.067 1.101

3

100

0.600 0.959 1.064 1.102

0.580 0.939 1.057 1.107

2

II kv, i,j , i = 1, . . ., 8

Relative capacity in the computation cell (i, j) for the second group of fuel elements.

80

100

N, % j

Table 5.8

5.3 Computational Method of Fuel Cladding Failure

165

0.661 0.973 1.087 1.145 1.157 1.108 0.989 0.666 11

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

II kv, i,j , i = 1, . . ., 8

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

n

4 5 4 6 6 7 7

0.689 0.966 1.071 1.134 1.151 1.080 0.916 0.649

100 21 0.702 0.979 1.072 1.123 1.138 1.101 0.964 0.631

0.684 0.961 1.065 1.128 1.144 1.076 0.911 0.635

80

100 54 0.598 0.961 1.054 1.107 1.123 1.088 0.966 0.551

0.582 0.946 1.050 1.111 1.124 1.065 0.933 0.546

80

80

100 68 0.626 0.869 0.953 1.005 1.027 1.005 0.917 0.644

80

0.609 0.853 0.946 1.006 1.024 0.982 0.886 0.636

5

0.660 0.941 1.051 1.113 1.129 1.087 0.981 0.689 228

100 20 0.707 0.984 1.077 1.128 1.142 1.106 0.972 0.649

80

0.630 0.858 0.953 1.014 1.034 0.960 0.777 0.565 87

43 0.646 0.871 0.956 1.010 1.033 1.008 0.873 0.572 86

0.643 0.877 0.972 1.035 1.054 0.982 0.810 0.600 96

42 0.660 0.890 0.977 1.030 1.052 1.026 0.895 0.605 94

0.687 0.957 1.059 1.122 1.140 1.088 0.971 0.687 4

29 0.726 0.993 1.084 1.138 1.155 1.122 1.009 0.701 7

0.588 0.850 0.950 1.006 1.021 0.985 0.890 0.618 4

100 18 0.686 0.965 1.063 1.113 1.126 1.090 0.982 0.680 245 8 0.608 0.868 0.960 1.005 1.016 0.983 0.885 0.607 2

80

—

100 12 0.698 1.018 1.121 1.167 1.174 1.129 1.007 0.668 44

6

—

0.656 0.958 1.069 1.126 1.139 1.090 0.973 0.662

—

6

80

1

100 10 0.694 1.011 1.112 1.158 1.164 1.120 1.000 0.665 17

i = 1, . . ., 8 j

j

n

N,%

Relative capacity in the computation cell (i, j) for the second group of fuel elements.

II kv, i,j ,

Table 5.9

166 Analysis of WWER 1000 Fuel Cladding Failure

III kv, i,j , i = 1, . . . , 8

0.682 1.139 1.269 1.331 1.338 1.267 1.101 0.620

0.740 1.136 1.256 1.327 1.341 1.277 1.133 0.706

6

6

5

0.721 1.107 1.229 1.296 1.308 1.246 1.103 0.684

6

0.813 1.141 1.259 1.336 1.357 1.277 1.096 0.762 49

0.709 1.105 1.236 1.297 1.305 1.249 1.109 0.709 61

0.790 1.122 1.243 1.317 1.335 1.166 0.564 0.387 190

9 0.735 1.130 1.247 1.293 1.294 1.240 1.097 0.694 62 31 0.813 1.142 1.250 1.311 1.330 1.288 0.947 0.389 210

0.685 1.105 1.242 1.301 1.306 1.246 1.097 0.673 67

5 0.720 1.145 1.269 1.314 1.312 1.254 1.101 0.671 75 30 0.842 1.166 1.270 1.335 1.357 1.318 1.163 0.766 61

0.688 1.124 1.262 1.318 1.319 1.255 1.096 0.661 41

4 0.712 1.147 1.269 1.309 1.304 1.242 1.083 0.648 50 22 0.803 1.174 1.287 1.343 1.353 1.305 1.165 0.754 16

0.679 1.111 1.247 1.301 1.303 1.239 1.084 0.653 40

3 0.713 1.149 1.271 1.311 1.305 1.242 1.084 0.648 45 19 0.754 1.145 1.252 1.304 1.312 1.263 1.124 0.684

0.690 1.113 1.250 1.308 1.314 1.254 1.105 0.680 57

6

n

0.695 1.083 1.214 1.274 1.282 1.226 1.089 0.696 60

0.807 1.127 1.245 1.322 1.345 1.265 1.086 0.757 76

80

0.755 1.127 1.251 1.323 1.339 1.271 1.126 0.750 202

0.771 1.062 1.174 1.247 1.271 1.209 1.073 0.711 126

100 55 0.777 1.147 1.259 1.320 1.340 1.301 1.166 0.758 206 41 0.793 1.082 1.182 1.244 1.267 1.235 1.112 0.770 124

80

100 13 0.731 1.125 1.244 1.291 1.292 1.238 1.096 0.693 55 32 0.832 1.151 1.255 1.320 1.342 1.305 1.152 0.757 87

80

100

80

100

80

100

80

100

80

2 0.720 1.147 1.270 1.314 1.311 1.254 1.100 0.671 68 11 0.697 1.161 1.276 1.321 1.319 1.258 1.093 0.599

j

100

n

j

N,%

III kv, i,j , i = 1, . . ., 8

Table 5.10 Relative capacity in the computation cell (i, j) for the third group of fuel elements.

5.3 Computational Method of Fuel Cladding Failure

167

j

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

III kv, i,j , i = 1, . . ., 8

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

n

—

—

80

—

—

100 68

80

—

—

—

—

—

— —

— —

— —

— —

—

—

—

0.781 1.119 1.238 1.312 1.332 1.269 1.131 0.782 249

—

100 54 0.804 1.139 1.246 1.310 1.333 1.297 1.170 0.789 258 43

0.788 1.096 1.216 1.288 1.306 1.231 1.056 0.745 279

80

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

0.798 1.103 1.219 1.293 1.314 1.256 1.126 0.806 268 —

0.792 1.101 1.221 1.293 1.312 1.237 1.067 0.774 277

100 21 0.806 1.110 1.216 1.274 1.289 1.248 1.098 0.733 288 42

80

100 20 0.810 1.114 1.220 1.278 1.293 1.252 1.106 0.759 286 29 0.817 1.119 1.221 1.281 1.299 1.262 1.137 0.791 276

0.729 1.032 1.149 1.215 1.233 1.183 1.065 0.749 69

—

100 18 0.761 1.063 1.167 1.220 1.233 1.192 1.072 0.742 36

80

—

0.738 1.063 1.186 1.251 1.266 1.214 1.084 0.745 301

80 8

—

100 12 0.758 1.076 1.183 1.234 1.244 1.200 1.071 0.726 268 6

0.749 1.077 1.200 1.267 1.282 1.230 1.100 0.761 305

—

n

80

III kv, i,j , i = 1, . . ., 8

—

j

100 10 0.765 1.086 1.194 1.245 1.254 1.211 1.083 0.736 295 1

N,%

Table 5.11 Relative capacity in the computation cell (i, j) for the third group of fuel elements.

168 Analysis of WWER 1000 Fuel Cladding Failure

IV kv, i,j , i = 1, . . ., 8

n

j

IV kv, i,j , i = 1, . . ., 8

n

0.811 1.293 1.449 1.517 1.524 1.455 1.283 0.797 108

0.926 1.381 1.534 1.610 1.621 1.541 1.359 0.895 307

0.869 1.393 1.559 1.628 1.631 1.550 1.359 0.835 183

0.925 1.338 1.482 1.562 1.580 1.506 1.340 0.914 306

0.867 1.391 1.557 1.627 1.629 1.548 1.355 0.832 177

0.900 1.303 1.446 1.525 1.542 1.470 1.304 0.884 306

0.795 1.269 1.424 1.491 1.497 1.429 1.260 0.781 93

0.928 1.314 1.450 1.534 1.555 1.462 1.255 0.868 262

0.921 1.427 1.591 1.666 1.674 1.597 1.415 0.903 232

0.851 1.209 1.339 1.418 1.436 1.253 0.601 0.411 122

0.858 1.199 1.325 1.404 1.427 1.357 1.203 0.850

9

0.808 1.202 1.334 1.411 1.427 1.352 1.191 0.794 104

80

0.915 1.296 1.432 1.515 1.536 1.444 1.235 0.849 234 5

0.893 1.3863 1.548 1.622 1.629 1.554 1.375 0.876 219

100 55 0.831 1.223 1.342 1.407 1.428 1.385 1.236 0.802 100 41 0.885 1.228 1.340 1.407 1.429 1.387 1.243 0.846

80

100 13 0.915 1.403 1.545 1.600 1.599 1.530 1.350 0.849 210 32 0.939 1.317 1.435 1.503 1.522 1.473 1.295 0.841 222

80

100 9 0.942 1.442 1.585 1.640 1.638 1.566 1.382 0.781 225 31 0.875 1.228 1.343 1.408 1.426 1.379 1.010 0.410 102

80

100 5 0.822 1.293 1.431 1.482 1.481 1.415 1.245 0.765 75 30 0.952 1.334 1.453 1.521 1.539 1.490 1.312 0.858 249

80

100 4 0.889 1.409 1.554 1.603 1.596 1.521 1.328 0.805 167 22 0.921 1.318 1.443 1.505 1.515 1.461 1.300 0.861 296

80

100 3 0.892 1.411 1.556 1.605 1.598 1.552 1.330 0.808 172 19 0.943 1.348 1.473 1.535 1.546 1.489 1.328 0.884 3 306

80

100 2 0.835 1.315 1.453 1.503 1.501 1.436 1.262 0.777 92 11 0.943 1.389 1.521 1.577 1.579 1.512 1.335 0.861 306

N,% j

Table 5.12 Relative capacity in the computation cell (i, j) for the fourth group of fuel elements.

5.3 Computational Method of Fuel Cladding Failure

169

j

—

—

—

—

—

IV kv, i,j , i = 1, . . . , 8

—

—

—

—

—

80

100 18

80

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

— 8

—

— 6

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

IV kv, i,j , i = 1, . . . , 8

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

n

—

—

80

—

—

100 68

80

—

—

—

—

—

— —

— —

— —

— —

—

—

—

0.832 1.194 1.321 1.400 1.422 1.352 1.201 0.829 57

—

100 54 0.856 1.216 1.330 1.398 1.422 1.384 1.245 0.836 48 43

0.852 1.221 1.358 1.434 1.450 1.378 1.214 0.836 29

80

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

0.867 1.227 1.360 1.438 1.457 1.392 1.246 0.873 40 —

0.855 1.220 1.355 1.430 1.448 1.377 1.220 0.861 31

100 21 0.880 1.247 1.368 1.428 1.439 1.386 1.228 0.822 19 42

80

100 20 0.880 1.243 1.363 1.422 1.433 1.383 1.232 0.843 21 29 0.894 1.253 1.370 1.432 1.448 1.400 1.257 0.859 29

—

j

— 1

n

0.839 1.199 1332 1.403 1.417 1.357 1.216 0.845 1

—

100 12

80

100 10

N, %

Table 5.13 Relative capacity in the computation cell (i, j) for the fourth group of fuel elements.

170 Analysis of WWER 1000 Fuel Cladding Failure

171

5.3 Computational Method of Fuel Cladding Failure Table 5.14 Coefficients of relative energy release according to IP software. Fuel

Fuel

Fuel

Fuel

N, % Assembly kv, 6,j (IP) Assembly kv, 6,j (IP) Assembly kv, 6,j (IP) Assembly kv, 6,j (IP) 100

2

80 100

3

4

5

9

80

22

0.98

1.325

13

1.315

30

1.36

12

1.39

1.39

31

1.42

18

1.395

20

1.4

1.365 1.41

6

0.995

1.165

21

1.195

8

1.055 1.08

68

0.56 0.58

29

1 1.035

42

1.24 54

0.345 0.36

1.2

1.445 41

1.16

0.37 0.38

1.03

1.47 32

1

1.2

1.435

1.375 55

1.385

1.16 1.205

1.445

1.38

80 100

1.12

10

1.445

1.02

80 100

19

1.165

80 100

1.125

1.235 1.285

1.17

80 100

11

1.03

80 100

0.99

0.975 1

43

0.98 1.005

1.005 1.03

Thus, for I conditional group: nIj=5 = 108; nIj=30 = 0; nIj=10 = 0; nIj=43 = 226.Here,nIj=5 6= nIj=30 6= nIj=43 . For II conditional group: II II II II II II II nII 5 = 54; n30 = 2; n10 = 17; n43 = 86, nj=5 6= nj=30 6= nj=10 6= nj=43 .

For III conditional group: III III III III III III III nIII 5 = 75; n30 = 61; n10 = 295; n43 = 0, nj=5 6= nj=30 6= nj=10 6= nj=43 .

172

Analysis of WWER 1000 Fuel Cladding Failure

Table 5.15 Coefficients of relative energy release in the axial segment 6 for fuel elements of four groups. N, % Cell kv, 6,j,m (nm ) 100 2 0.692 (102nfe) 1.055 (50nfe) 1.254 (68nfe) 1.436 (92nfe) 80 0.698 (98nfe) 1.063 (49nfe) 1.254 (57nfe) 1.455 (108nfe) 100 3 0.743 (67nfe) 1.045 (28nfe) 1.242 (45nfe) 1.552 (172nfe) 80 0.742 (61nfe) 1.033 (28nfe) 1.239 (40nfe) 1.550 (183nfe) 100 4 0.738 (67nfe) 1.037 (28nfe) 1.242 (50nfe) 1.521 (167nfe) 80 0.743 (63nfe) 1.048 (31nfe) 1.255 (41nfe) 1.548 (177nfe) 100 5 0.684 (108nfe) 1.046 (54nfe) 1.254 (75nfe) 1.415 (75nfe) 80 0.693 (106nfe) 1.046 (46nfe) 1.246 (67nfe) 1.429 (93nfe) 100 9 0.448 (6nfe) 1.088 (19nfe) 1.240 (62nfe) 1.566 (225nfe) 80 0.456 (6nfe) 1.088 (13nfe) 1.249 (61nfe) 1.597 (232nfe) 100 13 0.562 (8nfe) 1.092 (39nfe) 1.238 (55nfe) 1.530 (210nfe) 80 0.437 (6nfe) 1.076 (27nfe) 1.226 (60nfe) 1.554 (219nfe) 100 55 0.417 (6nfe) — 1.301 (206nfe) 1.385 (100nfe) 80 0.406 (6nfe) — 1.271 (202nfe) 1.352 (104nfe) *nfe: number of fuel elements.

Table 5.16 Comparison of values kv,6,j (ANC − H) and kv,6,j (IP ). N, % Fuel Assembly kv, 6,j (ANC − H) kv, 6,j (I) 100 2 1.092 0.99 80 1.119 1.03 100 3 1.288 1.125 80 1.306 1.17 100 4 1.265 1.12 80 1.297 1.165 100 5 1.059 0.98 80 1.083 1.02 100 9 1.451 1.325 80 1.486 1.38 100 13 1.399 1.315 80 1.428 1.375 100 55 1.311 1.36 80 1.281 1.4

Table 5.17 Relation of kv,6,j (ANC − H) to kv, 6,j (IP ). Fuel assembly 2 3 4 5 9 13 55 N, % 100 80 100 80 100 80 100 80 100 80 100 80 100 80 kv, 6,j (ANC−H) 1.10 1.09 1.14 1.12 1.13 1.11 1.08 1.06 1.10 1.08 1.06 1.04 0.964 0.915 kv, 6,j ( )

5.3 Computational Method of Fuel Cladding Failure

173

Table 5.18 Characteristics of algorithms 2 and 6. Algorithm Rearrangement A, MJ/m3 ω(τ ) = A/A0 , %(A0 = 55 MW · day/kg−U ) 2 5-30-10-43 1.838 3.342 9-11-20-1 1.443 2.624 3-22-54-29 1.843 3.351 13-19-21-42 2.652 4.822 2-31-18 1.209 2.198 55-41-12-6 1.955 3.555 4-32-68-8 1.368 2.487 6 55-11-18-43 1.568 2.851 13-32-20 2.019 3.671 3-31-10-8 1.816 3.302 9-19-68-42 2.054 3.735 4-41-12-29 1.935 3.518 2-30-21-6 1.522 2.767 5-22-54-1 1.238 2.251 Table 5.19 Division of fuel elements into groups in the rearrangement 5-30-10-43 of algorithm 2. Group Cell 5 Cell 30 Cell 10 Cell 43 I

108

0

0

226

II

54

2

17

86

III

75

61

295

0

IV

75

249

0

0

For IV conditional group: IV IV IV IV IV IV nIV 5 = 75; n30 = 249; n10 = 0; n43 = 0, nj=5 6= nj=30 6= nj=10 .

The non-uniformity of the distribution of the fuel elements of I, . . . ,IV groups in the cells of the reactor core requires a special algorithm which shall take into account the distribution of fuel elements in the groups. Let us introduce the notion of the given groups of fuel elements I*, ..., IV* according to such conditions: 1) the number of fuel elements of the given groups I*,. . ., IV* in the cells of the reactor core of each fuel assembly rearrangement is constant; I∗ , . . . , k IV∗ for 2) the coefficients of specific volumetric energy release kv, i,j v, i,j the given groups of fuel elements I*,. . .,IV* are calculated according to the weight of each conditionally specified group I,. . ., IV in the particular cell of the reactor core, which is determined according to the condition: vacancies in each of the groups I*„. . ., IV* are filled with

174

Analysis of WWER 1000 Fuel Cladding Failure

fuel elements on the basis of the conservative principle, the main point of which is that vacancies in each given group are first filled with fuel elements from a conditionally specified group with a higher number (with higher energy intensity) for each cell of the reactor core. Applying the conservative principle of filling vacancies in these groups, which is based on the data in Table 5.19, it is possible to write down the distribution of fuel elements in the groups I*„. . ., IV* placed consecutively in the cells 5, 30, 10, and 43 during the first, second, third, and fourth years of the fuel cycle in the fuel assemblies, respectively (Figure 5.5). The proposed approach assumes that for four groups I*,. . ., IV* the equations are: I∗ I∗ I∗ nI∗ j=5 = nj=30 = nj=10 = nj=43 = 108; II∗ II∗ II∗ nII∗ j=5 = nj=30 = nj=10 = nj=43 = 54; III∗ III∗ III∗ nIII∗ j=5 = nj=30 = nj=10 = nj=43 = 75; IV∗ IV∗ IV∗ nIV∗ j=5 = nj=30 = nj=10 = nj=43 = 75.

According to the distribution of fuel elements in the groups I*, . . . , IV* in the rearrangement 5-30-10-43, we shall obtain the coefficients of specific I∗ , ..., k IV∗ for the given groups of fuel elements volumetric energy release kv, i,j v, i,j I*, . . . , IV*, respectively (Table 5.20).

Figure 5.5 Fuel element distribution in the groups I*,. . ., IV* in the rearrangement 5-30-10-43.

5.3 Computational Method of Fuel Cladding Failure

175

I∗ IV∗ Table 5.20 Coefficients of energy release kv,i,j , . . . , kv, i,j .

j=5

j = 30

j = 10

j = 43

Group I*, nI∗ = 108 I kv, i,5

II III IV kv, i,30 ·2+kv, i,30 ·61+kv, i,30 ·45

II III kv, i,10 ·17+kv, i,10 ·91

nI∗

nI∗

I kv, i,43

Group II*, nII∗ = 154 II kv, i,5

III∗

Group III*, n III kv, i,5

IV kv, i,30

I kv, i,43

III kv, i,10

IV kv, i,30

= 75

III kv, i,10

II I kv, i,43 ·11+kv, i,43 ·64

nIII∗

Group IV*, nIV ∗ = 75 IV kv, i,5

IV kv, i,30

III kv, i,10

II kv, i,43

Table 5.21 Values ql, j,max for the groups of fuel elements I*, . . . , IV* in the rearrangement 5-30-10-43. Parameter ql, j,max , W/cm Cell 5 30 10 43 I* 100% 120.3 240.8 208.9 159.9 80% 97.73 193.6 169.7 127.9 II* 100% 184.3 259.3 211.3 159.9 80% 147.5 209.6 172.8 127.9 III* 100% 221.4 259.3 211.3 172.0 80% 176.0 209.6 172.8 137.8 IV* 100% 249.7 259.3 211.3 174.1 80% 201.8 209.6 172.8 139.4

The values ql, j,max obtained on the basis of the data presented in Tables 5.6–5.13 and 5.20 for the given groups of fuel element rearrangements 5-3010-43 according to algorithm 2 are presented in Table 5.21. Axial distributions ki,j for the given groups of the fuel element rearrangement 5-30-10-43 of algorithm 2 are presented in Tables 5.22–5.25. It was accomplished on the basis of the data in Tables 5.6–5.13 and 5.20. The values ω (A0 = 55 MJ/m3 ) based on the data from Tables 5.22–5.25 for the given groups of fuel element rearrangement 5-30-10-43 according to algorithm 2 are presented in Table 5.26. The value of the failure damage for the group IV* in the rearrangement 5-30-10-43 of algorithm 2 is 5.889%, which is much higher than the value ω = 3.342% for the averaged fuel element with respect to the fuel assemblies, i.e., in the case of a one-group model.

176

Analysis of WWER 1000 Fuel Cladding Failure

Table 5.22 Axial distribution ki,j for the group I* in the rearrangement 5-30-10-43. ki,j Given Group N, % i j = 5 j = 30 j = 10 j = 43 I* 100 1 0.532 0.619 0.608 0.631 2 0.863 0.863 0.866 0.840 3 0.963 0.940 0.953 0.923 4 0.998 0.986 0.993 0.977 5 1.000 1.000 1.000 1.000 6 0.958 0.970 0.966 0.979 7 0.842 0.855 0.863 0.878 8 0.507 0.561 0.585 0.621 80 1 0.512 0.598 0.583 0.614 2 0.839 0.842 0.840 0.826 3 0.948 0.930 0.936 0.918 4 0.994 0.985 0.989 0.979 5 1.000 1.000 1.000 1.000 6 0.956 0.941 0.959 0.945 7 0.841 0.808 0.858 0.826 8 0.510 0.559 0.592 0.612

Table 5.23 Axial distributions ki,j for the group II* in the rearrangement 5-30-10-43. ki,j Given Group N, % i j = 5 j = 30 j = 10 j = 43 II* 100 1 0.543 0.619 0.610 0.631 2 0.869 0.867 0.866 0.840 3 0.965 0.944 0.952 0.923 4 1.000 0.988 0.993 0.977 5 1.000 1.000 1.000 1.000 6 0.957 0.968 0.966 0.979 7 0.840 0.853 0.864 0.878 8 0.547 0.558 0.587 0.621 80 1 0.521 0.597 0.584 0.614 2 0.843 0.845 0.840 0.826 3 0.950 0.933 0.936 0.918 4 0.995 0.987 0.988 0.979 5 1.000 1.000 1.000 1.000 6 0.956 0.940 0.959 0.945 7 0.843 0.807 0.858 0.826 8 0.516 0.558 0.594 0.612

5.3 Computational Method of Fuel Cladding Failure

177

Table 5.24 Axial distributions ki,j for the group II* in the rearrangement 5-30-10-43. ki,j Given Group N, % i j = 5 j = 30 j = 10 j = 43 III* 100 1 0.548 0.619 0.610 0.626 2 0.871 0.867 0.866 0.842 3 0.966 0.944 0.952 0.925 4 1.000 0.988 0.993 0.978 5 0.999 1.000 1.000 1.000 6 0.954 0.968 0.966 0.976 7 0.838 0.853 0.864 0.849 8 0.511 0.558 0.587 0.562 80 1 0.525 0.597 0.584 0.609 2 0.847 0.845 0.840 0.829 3 0.951 0.933 0.936 0.920 4 0.997 0.987 0.988 0.980 5 1.000 1.000 1.000 1.000 6 0.955 0.940 0.959 0.930 7 0.840 0.807 0.858 0.761 8 0.515 0.558 0.594 0.555

Table 5.25 Axial distributions ki,j for the group IV* in the rearrangement 5-30-10-43. ki,j Given Group N, % i j = 5 j = 30 j = 10 j = 43 IV* 100 1 0.555 0.619 0.610 0.626 2 0.873 0.867 0.866 0.843 3 0.966 0.944 0.952 0.925 4 1.000 0.988 0.993 0.978 5 0.999 1.000 1.000 1.000 6 0.955 0.968 0.966 0.975 7 0.840 0.853 0.864 0.845 8 0.516 0.558 0.587 0.554 80 1 0.531 0.597 0.584 0.609 2 0.848 0.845 0.840 0.830 3 0.951 0.933 0.936 0.922 4 0.996 0.987 0.988 0.981 5 1.000 1.000 1.000 1.000 6 0.954 0.940 0.959 0.928 7 0.841 0.807 0.858 0.751 8 0.522 0.558 0.594 0.546

178

Analysis of WWER 1000 Fuel Cladding Failure

The values ω (A0 = 55 MJ/m3 ) for the given groups of fuel elements in the rearrangement 9-11-20-1 of algorithm 2 are presented in Table 5.27. The values ω (A0 = 55 MJ/m3 ) for the given groups of fuel elements in the rearrangement 3-22-54-29 of algorithm 2 are presented in Table 5.28. The values ω (A0 = 55 MJ/m3 ) for the given groups of fuel elements in the rearrangement 19-21-42 of algorithm 2 are presented in Table 5.29. The values ω (A0 = 55 MJ/m3 ) for the given groups of fuel elements in the rearrangement 2-31-18 of algorithm 2 are presented in Table 5.30. The values ω (A0 = 55 MJ/m3 ) for the given groups of fuel elements in the rearrangement 55-41-12-6 of algorithm 2 are presented in Table 5.31. The values ω (A0 = 55 MJ/m3 ) for the given groups of fuel elements in the rearrangement 4-32-68-8 of algorithm 2 are presented in Table 5.32. The values ω (A0 = 55 MJ/m3 ) for the given groups of fuel elements in the rearrangement 55-11-18-43 of algorithm 6 are presented in Table 5.33. The values ω (A0 = 55 MJ/m3 ) for the given groups of fuel elements in the rearrangement 13-32-20 of algorithm 6 are presented in Table 5.34. Table 5.26 Values ω for the given groups of fuel elements 5-30-10-43. Group Number of Fuel Elements ω, % ω for One-Group Model, % I* 108 2.154 II* 54 3.599 3.342 III* 75 4.613 IV* 75 5.889 Table 5.27 Values ω for the given groups of fuel elements in the rearrangement 9-11-20-1. Group Number of Fuel Elements ω, % ω for One-Group Model, % I* 6 0.797 II* 19 3.850 2.624 III* 62 4.609 IV* 225 7.965 Table 5.28 Values ω for the given group of fuel elements in the rearrangement 3-22-54-29. Group Number of Fuel Elements ω, % ω for One-Group Model, % I* 67 4.147 II* 28 5.785 3.351 III* 45 6.812 IV* 172 10.8

5.3 Computational Method of Fuel Cladding Failure

179

Table 5.29 Values ω for the given groups of fuel elements in the rearrangement 13-19-21-42. Group Number of Fuel Elements ω, % ω for One-Group Model, % I* 8 1.397 II* 39 4.013 4.822 III* 55 4.713 IV* 210 7.75 Table 5.30 Values ω for the given groups of fuel elements in the rearrangement 2-31-18. Group Number of Fuel Elements ω, % ω for One-Group Model, % I* 102 0.737 II* 50 1.082 2.198 III* 68 1.631 IV* 92 2.907 Table 5.31 Table 5.31. Values ω for the given groups of fuel elements in the rearrangement 55-41-12-6. Group Number of Fuel Elements ω, % ω for One-Group Model, % I* 6 0.845 II* 0 — 3.555 III* 206 1.375 IV* 100 2.256 Table 5.32 Values ω for the given groups of fuel elements in the rearrangement 4-32-68-8. Group Number of Fuel Elements ω, % ω for One-Group Model, % I* 67 0.718 II* 28 0.831 2.487 III* 50 2.353 IV* 167 3.956 Table 5.33 Values ω for the given groups of fuel elements in the rearrangement 55-11-18-43. Group Number of Fuel Elements ω, % ω for One-Group Model, % I* 6 0.749 II* 0 — 2.851 III* 206 4.258 IV* 100 5.401 Table 5.34 Values ω for the given groups of fuel elements in the rearrangement 13-32-20. Group Number of Fuel Elements ω, % ω for One-Group Model, % I* 8 0.782 II* 39 2.042 3.671 III* 55 2.587 IV* 210 6.161

180

Analysis of WWER 1000 Fuel Cladding Failure

The values ω (A0 = 55 MJ/m3 ) for the given groups of fuel elements in the rearrangement 3-31-10-8 of algorithm 6 are presented in Table 5.35. The values ω (A0 = 55 MJ/m3 ) for the given groups of fuel elements in the rearrangement 9-19-68-42 of algorithm 6 are given in Table 5.36. The values ω (A0 = 55 MJ/m3 ) for the given groups of fuel elements in the rearrangement 4-41-12-29 of algorithm 6 are presented in Table 5.37. The values ω (A0 = 55 MJ/m3 ) for the given groups of fuel elements in the rearrangement 2-30-21-6 of algorithm 6 are presented in Table 5.38. The values ω (A0 = 55 MJ/m3 ) for the given groups of fuel elements in the rearrangement 5-22-54-1 of algorithm 6 are presented in Table 5.39. Table 5.35 Values ω for the given groups of fuel elements in the rearrangement 3-31-10-8. Group Number of Fuel Elements ω, % ω for One-Group Model, % I* 67 1.205 II* 28 1.766 3.302 III* 45 2.195 IV* 172 3.998 Table 5.36 Values ω for the given groups of fuel elements in the rearrangement 9-19-68-42. Group Number of Fuel Elements ω, % ω for One-Group Model, % I* 6 0.746 II* 19 2.527 3.735 III* 62 3.016 IV* 225 5.455 Table 5.37 Values ω for the given groups of fuel elements in the rearrangement 4-41-12-29. Group Number of Fuel Elements ω, % ω for One-Group Model, % I* 67 2.042 II* 28 2.672 3.518 III* 50 3.076 IV* 167 5.443 Table 5.38 Values ω for the given groups of fuel elements in the rearrangement 2-30-21-6. Group Number of Fuel Elements ω, % ω for One-Group Model, % I* 102 1.838 II* 50 3.288 2.767 III* 68 4.068 IV* 92 5.685

5.4 Computational Analyses of Stress-strain State of the Fuel Cladding

181

Table 5.39 Values ω for the given groups of fuel elements in the rearrangement 5-22-54-1. Group Number of Fuel Elements ω, % ω for One-Group Model, % I* 108 2.437 II* 54 3.381 2.251 III* 75 4.127 IV* 75 5.926

In the case of one-group models of energy distribution over fuel elements in fuel assemblies (without consideration of the non-uniformity of energy distribution in fuel elements), the values of fuel cladding strain failure in the axial segment 6 in the rearrangement algorithms 2 and 6 are in the range of 2.2% ... 4.82% and 2.25% ... 3.74%, respectively. However, taking into account the non-uniformity of the energy release distribution in the fuel elements of the fuel assemblies, the values of strain failure of the fuel cladding in the axial segment 6 in the rearrangement algorithms 2 and 6 are in the range of 0.72% ... 10.8% and 0.75% ... 6.16%, respectively. Therefore, according to the proposed method on how to take into account the non-uniformity of energy distribution in fuel elements of the fuel assemblies, the maximum fuel cladding failure, reached after a four-year fuel cycle, for the algorithms 2 and 6 increased in 2.2 and 1.6 times in comparison with the one-group model, respectively.

5.4 Computational Analyses of Stress-strain State of the Fuel Cladding As a result of the implementation of the given computation algorithm, we obtained the time-dependent strain fuel cladding failure for four given groups of fuel elements in the sixth segment in the rearrangement algorithms 2 and 6 (Figures 5.5–5.18). The time-dependent failure in the rearrangement 5-30-10-43 of algorithm 2 is presented in Figure 5.6. The maximum allowable value of the strain failure is ω lim = 10 %, which is defined by the norm safety coefficient K = 10; in the rearrangement 5-3010-43 of algorithm 2, it is not reached. The time-dependent failure in the rearrangement 9-11-20-1 of algorithm 2 is given in Figure 5.7. The maximum allowable value of the strain failure for the rearrangement 9-11-20-1 of algorithm 2 is not reached.

182

Analysis of WWER 1000 Fuel Cladding Failure

Figure 5.6 Time-dependent failure in the rearrangement 5-30-10-43 of algorithm 2 (four given groups of fuel elements).

Figure 5.7 Time-dependent failure in the rearrangement 9-11-20-1 of algorithm 2 (four given groups of fuel elements).

The time-dependent failure in the rearrangement 3-22-54-29 of algorithm 2 is represented in Figure 5.8. In the rearrangement 3-22-54-29 of algorithm 2, the maximum allowable value of the strain failure ω lim = 10 % exceeds for the IV-th group. The time-dependent failure in the rearrangement 13-19-21-42 of algorithm 2 is given in Figure 5.9.

5.4 Computational Analyses of Stress-strain State of the Fuel Cladding

183

Figure 5.8 Time-dependent failure in the rearrangement 3-22-54-29 of algorithm 2 (four given groups of fuel elements).

Figure 5.9 Time-dependent failure in the rearrangement 13-19-21-42 of algorithm 2 (four given groups of fuel elements).

The maximum allowable value ω lim of the strain failure in the rearrangement 13-19-21-42 of algorithm 2 is not reached. The time-dependent failure in the rearrangement 2-31-18 of algorithm 2 is given in Figure 5.10. The maximum allowable value of the strain failure ω lim = 10 % in the rearrangement 2-31-18 of algorithm 2 is not reached.

184

Analysis of WWER 1000 Fuel Cladding Failure

Figure 5.10 Time-dependent failure in the rearrangement 2-31-18 of algorithm 2 (four given groups of fuel elements).

Figure 5.11 Time-dependent failure in the rearrangement 55-41-12-6 of algorithm 2 (three given groups of fuel elements).

The time-dependent failure in the rearrangement 55-41-12-6 of algorithm 2 is given in Figure 5.11. The maximum allowable value of the strain failure ω lim = 10 % in the rearrangement 55-41-12-6 of algorithm 2 is not reached. The time-dependent failure in the rearrangement 4-32-68-8 of algorithm 2 is given in Figure 5.12.

5.4 Computational Analyses of Stress-strain State of the Fuel Cladding

185

Figure 5.12 Time-dependent failure in the rearrangement 4-32-68-8 of algorithm 2 (four given groups of fuel elements).

The maximum allowable value of the strain failure ω lim = 10% in the rearrangement 4-32-68-8 of algorithm 2 is not reached. Since there are 172 fuel elements (see Table 5.28) in the case of the rearrangement 3-22-54-29 in the given group IV* of fuel elements, and in all other rearrangements of algorithm 2, the maximum allowable value of the strain failure is not reached (Figures 5.9–5.12), the predicted number of depressurized fuel claddings in the sector of symmetry of the reactor core with WWER-1000 in the case of implementation of algorithm 2 will be equal to 172. Taking into account six identical sectors of symmetry in the reactor core of WWER-1000, the total number of depressurized fuel elements in the case of implementation of the rearrangement algorithm of fuel assemblies No. 2 are equal to 1032, which exceed the limit of safe operation (there cannot be more than 1% of total number of fuel elements with the defect of “gas leak” type in the reactor core, i.e., not more than 508 fuel elements) [18]. The time-dependent failure in the rearrangement 55-11-18-43 of algorithm 6 is given in Figures 5.13–5.19. The maximum allowable value of the strain failure ω lim = 10 % in the rearrangement 55-11-18-43 of algorithm 6 is not reached. The time-dependent failure in the rearrangement 13-32-20 of algorithm 6 is given in Figure 5.14.

186

Analysis of WWER 1000 Fuel Cladding Failure

Figure 5.13 Time-dependent failure in the rearrangement 55-11-18-43 of algorithm 6 (three given groups of fuel elements).

Figure 5.14 Time-dependent failure in the rearrangement 13-32-20 of algorithm 6 (four given groups of fuel elements).

The maximum allowable value of the strain failure ω lim = 10 % in the rearrangement 13-32-20 of algorithm 6 is not reached. The time-dependent failure in the rearrangement 3-31-10-8 of algorithm 6 is given in Figure 5.15. The maximum allowable value of the strain failure ω lim = 10 % in the rearrangement 3-31-10-8 of algorithm 6 is not reached.

5.4 Computational Analyses of Stress-strain State of the Fuel Cladding

187

Figure 5.15 Time-dependent failure in the rearrangement 3-31-10-8 of algorithm 6 (four given groups of fuel elements).

Figure 5.16 Time-dependent failure in the rearrangement 9-19-68-42 of algorithm 6 (four given groups of fuel elements).

The time-dependent failure in the rearrangement 9-19-68-42 of algorithm 6 is given in Figure 5.16. The maximum allowable value of the strain failure ω lim = 10 % in the rearrangement 9-19-68-42 of algorithm 6 is not reached. The time-dependent failure in the rearrangement 4-41-12-29 of algorithm 6 is given in Figure 5.17.

188

Analysis of WWER 1000 Fuel Cladding Failure

Figure 5.17 Time-dependent failure in the rearrangement 4-41-12-29 of algorithm 6 (four given groups of fuel elements).

Figure 5.18 Time-dependent failure in the rearrangement 2-30-21-6 of algorithm 6 (four given groups of fuel elements).

The maximum allowable value of the strain failure ω lim = 10 % in the rearrangement 4-41-12-29 of algorithm 6 is not reached. The time-dependent failure in the rearrangement 2-30-21-6 of algorithm 6 is given in Figure 5.18. The maximum allowable value of the strain failure ω lim = 10 % in the rearrangement 2-30-21-6 of algorithm 6 is not reached.

5.4 Computational Analyses of Stress-strain State of the Fuel Cladding

189

The time-dependent failure in the rearrangement 5-22-54-1 of algorithm 6 is given in Figure 5.19. The maximum allowable value of the strain failure ω lim = 10 % in the rearrangement 5-22-54-1 of algorithm 6 is not reached. Since there is no exceedance of ω lim = 10 % in any rearrangement of the fuel assemblies in algorithm 6, it can be concluded that algorithm 6 is admissible from the point of view of non-exceedance of the boundary of safe operation for the value of stress–strain failure in contrast to algorithm 2. Changes in the values of equivalent and tangential stresses for the given group IV* of fuel elements of fuel assembly rearrangements with the largest calculated values of failure are shown in Figures 5.20–5.25. The dependence σe /σ0 (t) in the rearrangement 9-11-20-1 is given in Figure 5.20. The value σe /σ0 (t) for the given group IV* of fuel elements in the rearrangement 9-11-20-1 does not exceed 41%. The dependence σθ (t)/250 MPa in the rearrangement 9-11-20-1 is given in Figure 5.21. The value σθ (t)/250 MPa for the given group IV* of fuel elements in the rearrangement 9-11-20-1 does not exceed 11%. The dependence σe /σ0 (t) in the rearrangement 13-19-21-42 is given in Figure 5.22.

Figure 5.19 Time-dependent failure in the rearrangement 5-22-54-1 of algorithm 6 (four given groups of fuel elements).

190

Analysis of WWER 1000 Fuel Cladding Failure

Figure 5.20 Time dependence σe /σ0 (t) for the given group IV* of fuel elements in the rearrangement 9-11-20-1.

Figure 5.21 Time dependence σθ (t)/250 MPa for the given group IV* of fuel elements in the rearrangement 9-11-20-1.

The value σe /σ0 (t) for the given group IV* of fuel elements in the rearrangement 13-19-21-42 does not exceed 38%. The dependence σθ (t)/250 MPa in the rearrangement 13-19-21-42 is given in Figure 5.23. The value σθ (t)/250 MPa for the given group IV* of fuel elements in the rearrangement 13-19-21-42 does not exceed 10%. Dependence σe /σ0 (t) in the rearrangement 3-22-54-29 is given in Figure 5.24.

5.4 Computational Analyses of Stress-strain State of the Fuel Cladding

191

Figure 5.22 Time dependence σe /σ0 (t) for the given group IV* of fuel elements in the rearrangement 13-19-21-42.

Figure 5.23 Time dependence σθ (t)/250 MPa for the given group IV* of fuel elements in the rearrangement 13-19-21-42.

The value σ/ σ0 (t) for the fuel elements of the group IV* in the rearrangement 3-22-54-29 does not exceed 48%. The dependence σθ (t)/250 MPa in the rearrangement 3-22-54-29 is given in Figure 5.25. The value σθ (t)/250 MPa for fuel elements of the IV* group in the rearrangement 3-22-54-29 does not exceed 15%.

192

Analysis of WWER 1000 Fuel Cladding Failure

Figure 5.24 Time dependence σe /σ0 (t) for the given group IV* of fuel elements in the rearrangement 3-22-54-29.

Figure 5.25 Time dependence σθ (t)/250 MPa for the given group IV* of fuel elements in the rearrangement 3-22-54-29.

Despite the fact that the exceedance of ω lim = 10 % is reached for the fuel elements of the IV* group in the rearrangement 3-22-54-29, the exceedance of the maximum allowable values for the equivalent stress (100% of the creep limit if the storage coefficient for SC2 K = 1) and tangential stress (83% of 250 MPa if the storage coefficient for SC1 K = 1.2 [19]) is not observed. This conclusion contradicts the widespread opinion (for example, [20]) that the value of the strain does not play a significant limiting role in the calculated assessment of the boundary state of the fuel element cladding.

5.4 Computational Analyses of Stress-strain State of the Fuel Cladding

193

Thus, the proposed method for assessing fuel cladding failure of fuel elements with the account of the non-uniformity of the energy distribution in the fuel elements of WWER-1000 fuel assemblies is as follows: 1) Setting the initial data of the model in order to calculate the probability of fuel cladding depressurization, namely: design parameters of fuel elements, fuel assemblies, and the reactor core; the algorithm for regulating the capacity of the reactor unit, the scheme of the placement of the control group of the Control Rod Drive Mechanism in the reactor unit; the programs of the capacity control of the reactor unit, including the programs of the capacity change in the reactor unit, the position of the control group of the Control Rod Drive Mechanism, as well as the coolant temperature to the inlet of the reactor core, the duration of the fuel cycle and the model of the fuel assembly rearrangements, and slag distribution in the reactor unit at the beginning of the campaign. 2) The choice of the number of conditionally selected groups and the degree of discretization of the fuel element distribution in the fuel assemblies according to the coefficient of volumetric non-uniformity of energy release, which is in a certain interval for each conditional group. 3) The choice of the degree of the discretization of the fuel element length, the assessment of the length of one axial segment, and the formation of the dimension of the matrix of coefficients, where i is the number of the axial segments, j is the cell number of the reactor core, and (i, j) is the computation cell. 4) The division of fuel elements of each fuel assembly into conditionally selected groups I, . . . , IV on the basis of the value in the maximum energy-stressed axial segment of the fuel element taking into account the boundary values in each group. 5) Accounting for the non-uniformity of the fuel element distribution of I, . . . , IV groups in the cells of the reactor core in each rearrangement of fuel assemblies by calculating the composition of the given groups I*, . . . , IV* of fuel elements based on the conservative principle of filling vacancies in each of the given groups with fuel elements from conditionally selected groups; in this case, we shall take into account parts of fuel elements of each of the groups I, . . . , IV in the formation of the given groups I*, . . . , IV* to ensure the consistency of the number of fuel elements in each given group I*, . . . , IV * in all the cells of the reactor core in each rearrangement of fuel assemblies and accounting for the worst possible load scenarios for fuel elements.

194

Analysis of WWER 1000 Fuel Cladding Failure

6) The calculation of the parameter of fuel cladding failure ω according to the method of creep energy theory in the maximally loaded axial segment of the fuel cladding for the given groups of fuel elements in fuel assembly rearrangement algorithms, checking the non-exceedance of the maximum allowable value of the failure parameter. 7) The conclusion about the predicted number of depressurized fuel claddings in the reactor core in the case of the implementation of this algorithm in the fuel assembly rearrangement can be accomplished in the case of the exceedance of fuel elements in any fuel assembly rearrangement and in any rearrangement algorithm for any of the given groups, taking into account the number of fuel elements in these groups and the presence of six identical sectors of symmetry in the WWER1000 reactor core. The conclusion about the admissibility of this fuel assembly rearrangement algorithm based on the normative boundary of safe operation of fuel elements is made. Further, it is expedient to use the developed method for estimating fuel cladding failure which is based on four groups of fuel elements by nonuniformity of the energy release in fuel assemblies depending on the linear capacity and the coolant temperature to the inlet, which most affect the fuel cladding failure, to substantiate the storage of fuel assemblies in the dry waste storage. In this case, the value of the stress–strain failure of the fuel claddings plays a limiting role in determining their boundary state. This requires taking into account the non-uniformity of energy distribution in the fuel elements according to the uncertainty of the input parameters of the calculation model ω(τ ) and using the method of predicting the reliability of fuel claddings during the storage of spent fuel in a dry storage, which will determine the probability of fuel cladding depressurization based on the available range of ω values during the storage.

References [1] Pelykh SM, Maksimov MV, Nikolsky MV. A Method for VVER Fuel Element Cladding Reliability Prediction. Nuclear Physics and Atomic Energy. 2014; 15(1): 50–58. [2] Pelykh S, Maksimov M, Nikolsky M. A method for minimization of cladding failure parameter accumulation probability in VVER fuel elements. Problems of Atomic Science and Technology. 2014; 4: 108– 116.

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[3] Pelykh SN, Nikolsky MV, Riabchikov SD. Method for limiting the probability of damage accumulation in VVER fuel-element cladding. Tr. Odes. Politehn. un-ta. 2014; 2(44): 82–87. [4] Pelykh, S.N., Maksimov, M.V., Nikolsky, M.V., Riabchikov, S.D. (2015). “Method for minimizing the probability of accumulation of damage to cladding of VVER-1000 fuel elements, taking into account the unevenness of energy release in SFA” in Proceedings of the XXII annual scientific conference of the Institute for Nuclear Research of the National Academy of Sciences of Ukraine, Kyiv, Ukraine. [5] Shmelev VD, Dragunov YG, Denisov VP. VVER Cores for Nuclear Power Plants. 2004. Academkniga, Moscow. [6] Pelykh S, Maksimov M, Baskakov V. Grounds of VVER-1000 fuel cladding life control. Annals of Nuclear Energy. 2013; 58: 188–197. [7] Nikolsky MV. Axial offset as a measure of the stability of a lightwater nuclear reactor with a daily maneuver of power. Avtomatizaciya tehnolog. ta biznes-procesiv. 2014; 6(4): 65–72. [8] Pelykh SN, Maksimov MV. A method of Fuel Rearrangement Control Considering Fuel Element Cladding Damage and Burnup. Problems of Atomic Science and Technology. 2013; 5(87): 84–90. [9] Pelykh S, Maksimov M, Parks G. A method for VVER-1000 fuel rearrangement optimization taking into account both fuel cladding durability and burnup. Nuclear Engineering and Design. 2013; 257(4): 53–60. [10] Pelykh S, Maksimov M. Cladding rupture life control methods for a power-cycling WWER-1000 nuclear unit. Nuclear Engineering and Design. 2011; 241(8): 2956–2963. [11] Filimonov PE, Mamichev VV, Averianova SP. The “simulator reactor” to simulate the maneuvering modes of VVER-1000. Atomic Energy. 1998; 84(6): 560–563. [12] Suzuki M. Simulation of the behavior of a fuel element in a light-water reactor under various loading conditions. 2010. Astroprint, Odessa. [13] Suzuki M. Light water reactor fuel analysis code FEMAXI–V (Ver. 1). 2000. Tokai: Japan atomic energy research institute. [14] Pelykh S. Fundamentals of VVER Fuel Rod Properties Control. 2013. Saarbrücken: Palmarium Academic Publishing. [15] Todorcev YK, Kokol EA, Nikolsky MV. Estimation of the mass of the coolant in the reactor plant with a complete loss of feed water. Tehnologicheskij audit i rezervy proizvodstva. 2013; 1(14): 26–29.

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[16] MATLAB Version 7.10.0. Natick. Massachusetts: The MathWorks Inc., 2010. [17] Vorobiev RY. Albums of neutron-physical characteristics of the core of the power unit No. 5 ZNPP, campaign 20-23. 2011. Zaporizhska NPP, Energodar. [18] Nuclear Safety Rules for Reactor Installations of Nuclear Power Plants NP-082-07. 2008. Federalnaya sluzhba po ekolog., tehnolog. i atom. nadzoru, Moscow. [19] Alekseev, E. E. (2008). Development of methods for calculating the performance of VVER fuel elements in a probabilistic and deterministic formulation. Ph.D. thesis, National Research Centre ”Kurchatov Institute”, Moscow. [20] Novikov VV, Medvedev AV, Bogatyr SM. Ensuring the operability of nuclear fuel in maneuverable modes. 2005. KhAES. Khmelnitsky.

6 Thermal Safety Criteria for Dry Storage of Spent Nuclear Fuel

The decision that there is no failure in fuel and its condition is suitable for long-term dry storage is based on the methods described in detail in the previous chapters. Thus, the analysis of the data presented in the fourth and fifth chapters demonstrated that cladding fatigue which was caused by cyclic loads during the entire fuel operation time is not a critical factor when it is estimated according to stress amplitude since there is a large boundary for material fatigue failure. However, when assessing the creep deformation of the fuel cladding, the upper safe limit is within the range of values quickly achievable during operation in the reactor. This is due to the fact that with an increase of the irradiation period, the fuel pellet swells, and when the gap between the fuel pellet and the fuel cladding is closed, the latter experiences “outward” creep deformation. For the operating conditions of WWER-1000 fuel assemblies, the value of the accumulated creep strain is about 2% and is within the second stage of creep, the so-called steady-state creep. When it comes to the stage of the accelerated growth of the accumulated creep strain, its value begins to grow rapidly and exceeds 10%. Due to the fact that there is no energy load in the dry storage mode, it is possible to admit variable effects for the fuel cladding in the areas of steady creep and at the beginning of the stage of accelerated growth, where the values of the accumulated creep strain are small. It follows from the above-mentioned that the strength criteria are not critical for assessing the durability of safe dry storage of spent nuclear fuel. The fulfillment of the strength criteria does not guarantee the absence of fuel cladding failure due to the violation of the removal of its residual energy release. In other words, the fuel cladding can be reliable in terms of strength, but it can melt due to the melting overtemperature of its material. In this regard, for the safe dry storage of spent nuclear fuel, it is advisable to use

197

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thermal safety criteria in addition to the existing reactivity criteria, which were discussed in the first chapter.

6.1 Dry Storage of Spent Nuclear Fuel in Ukraine Ukraine is a country with considerable experience in dry storage of spent nuclear fuel. Nowadays, the long-term storage of spent nuclear fuel in Ukraine is actually carried out only at the largest NPP in Zaporizhzhya. The storage of spent nuclear fuel at the station is designed to store more than 9000 spent fuel assemblies of six WWER-1000 reactors at Zaporizhzhya NPP and it anticipates the operation for about 50 years, i.e., during the design life of the NPP [1], [2]. Having studied various storage technologies, it was concluded that the most practical, efficient, and economical storage system for the specific conditions at Zaporizhzhya NPP is a storage system in ventilated concrete containers VSC-24, developed by Sierra Nuclear Corporation (USA) [3], [4]. This technology is now used as well (Figure 6.1). Like any object of the nuclear power complex of the dry storage of spent nuclear fuel, Zaporizhzhya NPP requires a comprehensive study of its safe operation. According to the International Atomic Energy Agency (IAEA) Glossary on Safety [5], the term “safety” primarily refers to control of radiation sources. In IAEA safety regulations, the term “safety” means “nuclear safety,” unless otherwise specified. In a broader sense, the term “protection and safety” (i.e., radiation protection and nuclear safety) is often used to refer to the safety of nuclear facilities, radiation safety, safety of radioactive waste management, and safety of transportation of radioactive materials; however, this concept does not include aspects unrelated to radiation safety. Therefore, if there is the need to consider such non-radiation safety aspects, then other terms should be used. For example, measures to protect nuclear material source from unauthorized access are referred to as “physical protection” (or, in some cases, “physical safety”; see, for example, [5]). Unfortunately, if there is the need to examine compliance with certain temperature regimes of the radiation source, the corresponding term in the Glossary and other regulations is not recorded. The safety of operation of any storage facility of spent nuclear fuel is a complex concept (see, for example, [5]−[7]) and, in addition to nuclear safety and radiation protection measures, it includes the provision of appropriate conditions for thermal regimes during the operational life of

6.1 Dry Storage of Spent Nuclear Fuel in Ukraine

Figure 6.1

199

Storage of spent nuclear fuel at the site of Zaporizhzhya NPP.

the storage facility. In addition, during the whole life of such a facility, it is necessary to keep under steady watch the temperature of both the stored spent nuclear fuel and the main storage equipment. That is to say that there is permanent control of the source of radiation (in this case, this source is spent nuclear fuel).

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It is of great importance to ensure the safe storage of spent nuclear fuel in order to prevent the destruction of structural elements of spent fuel assemblies, especially of fuel claddings, as they provide the integrity and mechanical strength. The thermal effect can be high enough to have an adverse impact on the structural stability of spent nuclear fuel and high-level radioactive waste. Corrosion or leaching processes can also be accelerated at increased temperatures. The effect of these phenomena can be significant and should therefore be taken into account when assessing safety. The decision on the maximum allowable temperature level of system components and heat release from radioactive waste and spent nuclear fuel should be made with respect to the specific conditions of the site and the design of the storage. When changing the thermal conditions of operation of fuel elements, the main parameter that characterizes its integrity is the temperature of the fuel cladding. The destruction of the fuel cladding starts when the stresses (including thermal stress) exceed the strength limit. The values of temperatures used as emergency criteria are determined for certain conditions of this type of fuel elements taking into account the design of the storage. Nowadays, the following values are used as thermal safety criteria for the equipment for the dry storage of spent nuclear fuel [4]: a) the maximum allowable temperature of fuel claddings for spent nuclear assemblies of the WWER-1000 reactor at a normal mode of long-term dry storage in the helium medium is 350◦ C; b) for short-term modes of the effect of extreme weather conditions and at the time of transport and technological operations, the temperature must not exceed 450◦ C; c) the permissible residual heat release of the fuel to be stored must not exceed 0.99 kW for each spent fuel assembly in the system of the ventilated storage container of the WWER-1000 reactor; d) under normal operation and at abnormal ambient temperatures (40 C), the average temperature of the concrete body of the container must not exceed 66 C, the maximum temperature must be 107◦ C, and in the conditions of design accidents, these values are 93 C and 177 C, respectively. It should be noted that the external thermal effects on the dry storage of spent nuclear fuel are primarily determined by weather conditions. The change of the latter over time is characterized not only by evident seasonal and daily fluctuations but also by the presence of numerous random factors; as a result,

6.1 Dry Storage of Spent Nuclear Fuel in Ukraine

201

their exact consideration in forecasting the thermal state of the dry storage of spent nuclear fuel is impossible. Therefore, when formulating the abovementioned criteria, the following averaged data of meteorological values were taken. The normal long-term mode of operation is equal to the average daily air temperature 24◦ C in the absence of solar load, which with some assumptions (including conservative ones) corresponds to the averaging of these parameters in the summer period. The atmospheric air temperature of 38◦ C at maximum solar load or 40◦ C in the absence of solar load are considered as short-term effects of extreme weather conditions, which, again with some assumptions, correspond to the maximum daily temperature parameters in the hottest period. Moreover, due to global warming, climatic conditions in Ukraine have changed. Thus, in Energodar, the maximum daily air temperature of 37◦ C−40◦ C can be reached for several weeks in a row, which is a cyclical short-term extreme effect, but it was not foreseen at the design stage of the dry storage of spent nuclear fuel. Finally, under a design accident, we mean a situation where the ambient temperature reaches 52◦ C with the maximum solar load. In all cases, we assume calm weather conditions, i.e., the wind velocity is 0 m/s. As a result of the analysis of the boundary values of thermal and hydraulic parameters of other system components, it was concluded that the above values are determinative (for ASME SA516 steel which is used in the cask, allowable temperatures exceed 538◦ C and for RX-277 material used in the protective cover, it is 177◦ C). In order to monitor the thermal mode of the dry storage of spent nuclear fuel, the thermal control of the ventilated storage container is carried out [12]. Temperature measurements are recorded at temperature control points at the outlets of the ventilation ducts of the storage container. The temperature control is carried out using resistance temperature detectors, which are installed on the axis of the by-pass ventilation ducts at a distance of 150−200 mm from the protective grid. According to the safety analysis report [4], the temperature of the by-pass ventilation air must not exceed the air temperature by more than 61◦ C. The purpose of this restriction is to control the heating of the spent fuel assemblies and, as a consequence, the maximum temperature of fuel claddings, as well as to detect a significant blockage of the ventilation path of the WWER ventilation containers during the operation. It is obvious that using this method of control, it is impossible to obtain complete information about the temperature state of spent fuel assemblies stored in the container. Therefore, additional studies were conducted to

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confirm the safety of the storage of spent nuclear fuel at the site of the nuclear spent fuel dry storage when changing various external influences. The method of analysis described below was used for this purpose.

6.2 Method of Thermal Safety Analysis of Dry Storage of Spent Nuclear Fuel The mathematical modeling apparatus is of great importance in order to determine the thermal state of spent fuel assemblies both at the design stage of a new storage facility and at the operation stage or in case of the modernization of the existing one. It should be noted that the system “spent fuel assembly − helium − storage cask − ventilation air − container body − environment” is quite complex from the thermal point of view. It involves all three mechanisms of heat transfer: conduction, convection, and radiation. All these processes occur in an object of a rather complex spatial form in the presence of numerous factors that result in internal and external effects on this system: initial power of residual heat release of spent fuel assemblies when they are sent for storage, capacity change of residual heat release during storage, and various effects of weather conditions (change in ambient temperature of the wind, wind load, solar radiation, and precipitations). Therefore, for a comprehensive solution of the problem of thermal state of spent fuel assemblies, the tools of mathematical modeling were introduced, in particular, to conjugate heat transfer problems. The definition of the temperature field in the solid body under consideration and the fields of pressure, velocities, and temperatures of the surrounding gas by solving a system of differential equations under known conditions of unambiguity is the subject of solving the direct conjugate heat transfer problem. In the general case, the system of differential equations that determines the direct conjugate heat transfer problem consists of [13], [14]: • continuity equation ∂ρ + div (ρv) = 0, (6.1) ∂t where ρ is the density of the moving medium, t is time, and v is the velocity vector; • Navier–Stokes equation     dv 2 ρ = −grad p + µef div v + 2div µef S˙ + FV , (6.2) ∂t 3

6.2 Method of Thermal Safety Analysis of Dry Storage of Spent Nuclear Fuel 203

where p is the pressure of the moving medium; µef is the effective (with respect to the turbulent component µt ) dynamic coefficient of viscosity; S˙ is the rate-of-strain tensor; and FV is the vector of bulk force; • energy equation ρ

dU = −pdiv(v) + µef Φ(v) + q, dt

(6.3)

where U is the specific internal energy; q is the amount of heat, which is reduced to a unit of mass per a time unit; F(v) is the dissipative function that characterizes the part of mechanical energy that is converted into heat as a result of overcoming viscosity forces during the motion of the moving medium. If a compressible medium is examined, then this system of differential equations is closed, as a rule, by the thermal equation of state. In addition to the equation of state, other algebraic relations can be used to close the original equations. If there is a turbulent flow, in order to calculate the turbulent components of thermophysical constants which are part of the equations of motion and energy, some physical values are also taken into account (for example, turbulent kinetic energy and its dissipation rate) and differential equations describing storage laws for these values are introduced into the system of equations along with additional algebraic closures, which describe the components of viscosity and thermal conductivity. All these equations form the so-called turbulence model [16]. In case of conjugate heat transfer problem, the single-valued conditions include: • geometric information about the system under consideration; • thermophysical properties of a solid body and liquid (gas); • initial conditions, which consist of the fields of temperature, pressure, and velocity and possibly other physical parameters in a solid body and liquid (gas) at the initial moment of time; • information on internal sources, heat sink, and mass; • boundary conditions. In this case, the thermal boundary conditions for a solid body are set only on that part of its surface which does not come into contact with the liquid or gas under consideration. In the part of the boundary of the remaining area, the boundary conditions for liquid (gas) are set, which, in addition

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to thermal conditions, include conditions for other physical fields (pressure, velocities, etc.). The thermophysical interaction of a solid body and liquid (gas) that flows around can take place both in the presence of a phase transformation on the boundary of a solid body and a moving medium and in its absence. In the latter case, the conditions of equality of heat fluxes and the conditions of no mass flux are simply set at the contact boundary. In the presence of a phase transformation (Stefan problem), a more complex mathematical model is examined, which takes into account the heat of the phase transition and the velocity of the phase front. The presence of a multiphase or multicomponent moving medium also introduces additional differential and algebraic equations into the mathematical model. In this case, the equations of continuity and energy must be examined separately for each phase (or component), taking into account its mass part. A non-zero term that characterizes the source or mass flow as a result of the transition from one phase to another one or chemical reactions between the components may appear in the equation of continuity in the right part. In the case when the moving medium is transparent and separates surfaces with different temperatures, heat between them is transferred by radiation. If the bodies involved in radiant heat exchange are gray (i.e., absorb the same part of the incident radiation over the entire wavelength range), the heat flux transmitted from one body to another can be expressed as [13], [17]:
Q12 = σ0 ε T14 − T24 H12 , (6.4) where σ 0 = 5672.10−8 W/(m2 .K4 ) is the Stefan−Boltzmann constant;  is the reduced power of blackness; T 1 is the temperature of the surface of one (more heated) body; T 2 is the temperature of the surface of the other (less heated) body; H 12 is the area of the mutual irradiation surface that can be found from the equation Z Z cos θ1 cos θ2 H12 = dF1 dF2 , 2 πR12 F1

F2

where F 1 and F 2 are the areas of surfaces, which are involved in the heat exchange; θ1 and θ2 are the angles between the line that connects the centers of elementary areas dF 1 and dF 2 , and appropriate normals to these areas; R12 is the distance between these elementary areas. The reduced power of blackness can be defined as ε = [1 + (1/ε1 − 1)ϕ12 + (1/ε2 − 1)ϕ21 ]−1 ,

6.2 Method of Thermal Safety Analysis of Dry Storage of Spent Nuclear Fuel 205

where ε1 and ε2 are the powers of surface 1 blackness and surface 2 blackness; ϕ12 = H 12 /F 1 and ϕ21 = H 12 /F 2 are the radiation coefficients or angular radiation coefficients indicating which part of the radiation emitted by one body falls on the other one. If several surfaces are separated by a transparent medium, their pairwise thermal interaction by radiation should be taken into account. In this case, the expression (6.4) is considered for each pair of surfaces. Formula (6.4) is written in the assumption of the isothermality of the surfaces involved in heat transfer. Otherwise, the surfaces must be divided into elementary sites, and we shall examine Equation (6.4) for each pair of elementary areas. As it can be seen, in order to solve the direct conjugate heat transfer problem, it is necessary to solve a problem, the mathematical model of which includes a system of non-linear differential equations in partial derivatives and algebraic equations. Previously, analytical methods were used to solve these problems [14], the development of which was finally limited to simple one-dimensional problems. The emergence of high-speed computers with large amounts of RAM and the development of modern computational methods (finite-element method, finite difference method, etc.) have made it possible to solve conjugate heat transfer problems for rather complex devices. Among numerous methods, we can highlight the method of control (or in other terminology finite) volume, which is one of the most effective methods. It has been widely used in solving problems in hydrogas dynamics, including conjugate heat transfer problems. This method belongs to the group of methods of finite differences and allows building conservative difference schemes of the second order. It is based on the integral formulation of conservation laws (mass, momentum, energy, etc.) for each elementary volume, into which the calculation area is divided. The advantage of the method of control volume is the ability to comply with the laws of conservation on both structured and unstructured grids; its disadvantage is a relatively complex form of recording for the derivatives of the second order. The implementation of this method in solving numerical problem is simple; so the use of this method is substantiated both in existing software and in those programs that are being developed now [15]. The equations of the mathematical model of the conjugate heat transfer problem given above are written in a general form. Let us dwell in more detail on the features of the mathematical model when examining thermophysical processes that occur during the dry storage of spent nuclear fuel in ventilated storage containers.

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In this case, the gravitation force is the volume force FV , which, due to Archimedean forces, causes both the air movement in the ventilation ducts and the circulation of helium in a sealed cask. Therefore, Equation (6.2) takes the form     dv 2 ρ = −grad p + µef div v + 2div µef S˙ + ρg, (6.5) ∂t 3 where g is free fall acceleration. Air and helium are the moving media in the dry storage of spent nuclear fuel at relatively low pressures and temperatures, which allowed considering them as ideal gases, i.e., to accept the dependence for specific internal energy U = cv T, as well as to use Mendeleev–Clapeyron equation as the equation of state ρ=

p , RT

(6.6)

where cv is the specific heat at constant volume; T is the gas temperature; R is the gas constant 8314 J/(K·mol). In addition, the conversion of kinetic energy into thermal energy during gas movement is so insignificant due to low viscosities and velocities that the dissipative function can be neglected. Since there are no heat sources in the air and helium, the energy equation (6.3) is written as cp ρ

dp dT − = div (λef gradT ) , dt dt

(6.7)

where cp is the specific heat capacity under constant pressure; λef is the effective (taking into account the turbulent component λt ) heat capacity. For the solid bodies (concrete shell of the container, metal body of the cask, spent fuel assemblies, etc.), the special case of the energy equation is the equation of thermal conductivity: cp ρ

∂T = div (λgradT ) + qV . ∂t

(6.8)

The k-ε-model can be used as the model of turbulence, which has proven itself to be stable and economical; this model also allows obtaining acceptable accuracy in the process of modeling turbulent flows in case of heat transfer in many technical devices. The model includes two differential equations of

6.2 Method of Thermal Safety Analysis of Dry Storage of Spent Nuclear Fuel 207

transfer: one is for turbulent kinetic energy k and the second one is for the rate of its dissipation ε:  ∂ (ρk) + div (ρkv) = div µ + ∂t  ∂ (ρε) + div (ρεv) = div µ + ∂t

  µt gradk + Gk + Gb − ρε, (6.9) σk   µt ε ε2 gradε + C1ε (Gk + C3ε Gb ) − C3ε ρ , σε k k (6.10)

where Gk is the term that describes the generation of the turbulent kinetic energy, which occurs due to the velocity gradient (according to Boussinesq’s hypothesis Gk = µT S2 ); Gb is the term that describes the generation of turbulent kinetic energy caused by the action of Archimedean forces (for an ideal gas):   ∂p ∂p ∂p + gy ∂y + gz ∂z ); C1 , C2 , and C3 are the constants Gb = − ρ µPrT T gx ∂x of the models which are equal to 1.44, 1.92, and 0.09, respectively; σ k = 1.0 is the turbulent Prandtl number for k; σ = 1.3 is the turbulent Prandtl number for ε. In Equations (6.9) and (6.10), the term responsible for the dissipation of destruction due to large Mach numbers is omitted. The turbulent component of the dynamic viscosity is determined by the expression k2 (6.11) µT = 0, 09ρ , ε and the turbulent component of the thermal conductivity of the moving medium is cp µT , (6.12) λT = PrT where PrT = 0.85 is the turbulent Prandtl number. Using the Reynolds criterion, the nature of air and helium flows was assessed. Studies have shown that the flow of helium inside the cask (in the channels between the spent fuel assemblies, partitions, and the wall of the cask inside the spent fuel assemblies) should be considered as laminar (Re ∼ 100 ÷ 500 < 2300), and the air flow in the ventilation channels should be considered as a developed turbulent flow. Quite often, when conducting applied research, the procedure of solving a direct conjugate heat transfer problem requires a significant amount of computer resources (time and memory). First of all, this is due to the need to build a very detailed grid, the application of which will allow the chosen

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numerical method to take into account all the details of thermophysical processes. All this makes it impossible to solve repeatedly the direct problem in the process of minimizing the target functionality of the inverse problem. In order to solve this problem, an iterative approach was developed; here, models of different levels of complexity are used. A low-level model is built for the entire object. In this model, the geometry of its components and physical processes are simplified as much as possible, which makes it possible to use a computational grid with a relatively small number of nodes and obtain a solution of the direct problem of conjugate heat transfer for the whole object using a numerical method in a short time. Definitely, such a solution will have quite a low accuracy. In order to make it more precise, high-level mathematical models are used; they are built separately for all components into which the object under study is divided. Each model takes into account the detailed geometry of the corresponding part of the object and describes in detail the thermophysical processes which are modeled. In this case, the step between the nodes in the computational grid or the size of its cells is chosen smaller than the corresponding value in the low-level model, which allows obtaining a more accurate solution of the direct problem of conjugate heat transfer for the selected part of the object. In order to build the high-level model for a part of the object, certain boundary conditions were set. Some of these boundary conditions were taken from the boundary conditions of the original problem, namely, for those sections of the boundary of the part of the object that coincide with the boundary of the whole object. In those boundary parts of the selected part of the object that are inside it, the boundary data are calculated according to the temperature field (and, if necessary, other physical fields), which was obtained by solving the problem of the low-level model. The method of their calculation will be described below. When using the low-level model, as many simplifications as possible are usually made in order to describe physical processes in each of its components. For example, a part of the object in which both solid bodies and a moving medium are present can be replaced by an isotropic solid body with equivalent thermophysical properties. At this stage, this allows solving the problem of thermal conductivity in this part of the object instead of the conjugate heat transfer problem. The choice of values that ensure maximum approximation of physical fields calculated by means of the high-level model with respect to physical fields which correspond to the conjugate statement of the problem in this part of the object refers to the equivalence. Equivalent thermophysical properties can be calculated as a result of the solution of the

6.2 Method of Thermal Safety Analysis of Dry Storage of Spent Nuclear Fuel 209

inverse identification problem of thermal conductivity for a specified part of the object. Here, in contrast to the classical inverse identification problem, the temperature values (and, if necessary, values of other physical fields), which are obtained by means of the high-level model are used instead of temperature and other measurements. Thus, the low-level model made it possible to obtain boundary conditions for high-level models, which, in its turn, allow determining the equivalent thermophysical properties of the corresponding components for the lowlevel model. The algorithm for calculating the initial conjugate heat transfer problem is based on the iterative process of alternating use of low- and high-level models during which the data exchanged in these models are specified. Let us point out that this approach can be extended to more levels of mathematical models. The calculation of boundary data in those boundary parts of the specified part of the object, which are inside it, is performed based on the physical sense of the boundary conditions. Since each boundary condition characterizes the physical interaction of the bodies in the calculation area with the environment, the boundary data in such areas can be easily calculated from the values of the corresponding physical value at points on both sides of this boundary. These values are taken from the corresponding physical field, which was obtained as a result of solving the direct problem of conjugate heat transfer for the whole object by means of the low-level model. For example, in case of thermal boundary conditions, it is quite easy to calculate the normal component of the heat flux for such a boundary in the absence of a moving medium near it, using the finite-difference approximation of Fourier law, which is constructed according to the points of the found temperature field. If there is a flow on one or both sides of this section of the boundary in the finite-difference approximation of Fourier’s law, it is necessary to take into account conductive heat fluxes and convective heat fluxes for the calculation of which, along with the temperature field, the velocity field is used. The equivalent thermophysical properties were determined as it follows. The selected part of the object is considered as an isotropic solid body. The internal inverse problem to find its thermal conductivity and heat capacity (in case if it is a non-stationary problem) is set. Instead of temperature measurements, the temperature values obtained for this part of the object as a result of solving a direct conjugate heat transfer problem of the highlevel model are taken as initial values. The objective functional, which

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Thermal Safety Criteria for Dry Storage of Spent Nuclear Fuel

is minimized in the process of solving such an inverse problem, is the discrepancy between the temperature obtained using the high-level model and the temperature which, at each iteration of the minimizing algorithm, is calculated using the low-level model for current trial values of the required equivalent thermophysical properties. Let us note that in this case, the lowlevel model is not applied to the entire object under study but only to the part for which the equivalent thermal conductivity and heat capacity are identified. The purpose of creating the following method for determining the thermal state of spent nuclear fuel is to reduce the amount of computing resources that are usually required to define the thermal state of containers for spent nuclear fuel by dividing the calculation area into separate components. Therefore, the calculation algorithm is as it follows. At the first stage, the thermal state of the container for spent nuclear fuel is determined, provided that the cask with spent fuel assemblies is considered as a solid body with certain thermophysical properties. Next, the heat transfer coefficients are determined on the surface of this body, and they serve as boundary conditions for the calculation area at the following stage that contains only a storage cask with certain details, and the fuel assemblies are represented as a solid body. The next stage, by analogy with the previous one, determines the boundary conditions for each of the fuel assemblies. Then, the fuel assembly and, finally, each fuel element of the fuel assembly are examined in detail. Thus, in contrast to the conservative approach, it is possible to determine actual temperatures of fuel claddings, which are a criterion for safe dry storage of spent nuclear fuel. As it was mentioned above, the feature of the proposed method is a combination of the solutions of direct and inverse heat transfer problems. The solutions of direct conjugate heat transfer problems are used to determine the thermal state of individual elements in the calculation area, whereas the solution of inverse problems is used to determine the boundary conditions and equivalent properties of the bodies that replace certain structural elements. In order to study the inverse problems associated with the identification of conditions of unambiguity, the operator equation of the first kind is used as a convenient notation of the mathematical model Aϕ = f, ϕ ∈ Φ, f ∈ F,

(6.13)

where ϕ and f, respectively, denote the sought and observed characteristics of the object, which belong to certain metric spaces ϕ and F. The operator A, which acts on the basis of Φ in F, formalizing the set of the operations defined by the original mathematical model of the phenomenon and the conditions

6.2 Method of Thermal Safety Analysis of Dry Storage of Spent Nuclear Fuel 211

of unambiguity, establishes causal relations between the sought and input (observed) values. In this case, the problem in the form of operational equation (6.13) is reduced to the variational problem of minimizing the objective functional J(φ) = kAφ − f kF .

(6.14)

In practice, the functional (6.14) is the mismatch between the measured and calculated temperatures. According to the general strategy of extreme methods for solving inverse problems, the minimization of the objective functional J (ϕ) involves the search for unknown conditions of unambiguity in the process of multiple numerical modeling of the thermophysical process in the object under consideration. Thus, when solving the inverse problem by means of the extreme method, it is necessary to solve a number of direct conjugate problems of thermal conductivity at “trial” values of the required parameters. An approximate mathematical model obtained by approximation (for example, finite-difference, common-volume, and common-elementary) of the original mathematical model is used. As it has already been mentioned, the procedure for solving direct conjugate problems of thermal conductivity is quite time-consuming. Therefore, it is advisable to apply an approach that has perfectly proven itself in classic inverse problems of heat conductivity, namely, the usage of ready-made software complexes to solve direct problems of heat and mass transfer or so-called CFD-packages (FLOWORKS, ANSYS, FLUENT, PHOENICS, STAR-CD, etc.). These complexes are universal, i.e., aimed at solving a wide range of problems, including conjugate heat transfer problems. They use well-established and verified algorithms for the numerical solution of the considered systems of differential equations. In modern software packages for solving problems of heat and mass transfer, the problem can be fully formulated in the source file by means of the data description language. Therefore, it is possible to automate the procedure of changing the information in the source data and use ready-made Computational Fluid Dynamic packages (CFD-packages) when solving inverse problems. The following calculation areas were used to solve direct conjugate heat transfer problems. The calculation area for the group of containers is presented in Figure 6.2. When constructing the calculation area for the group of WWER ventilated storage containers, the distance between the containers in groups, the distance between the groups of containers, as well as the distance to the radiation

212

Thermal Safety Criteria for Dry Storage of Spent Nuclear Fuel

Figure 6.2 Calculation area for the group of containers.

protection structure were taken into account. The number of containers in the calculation area was chosen according to the problem to be solved. The boundaries B2 and B4 were placed at a distance from the axes of the containers, which is equal to half the distance between the axes of the containers in transverse rows. During the computation study, the boundary conditions were as follows: • on the boundaries B1−1 , B2 , B3−1 , B4 , and B5 , the atmospheric air temperature and atmospheric pressure are – p|B i = pa , – T|B i = Ta ; • on the boundary B6 (the surface of the concrete base of the storage site), there is no heat flux; • the temperature value and speed of ventilation air are set in the sections of the boundaries Bki , which correspond to the ventilation ducts at the inlets and outlets of WWER ventilated storage containers; • in other sections of the boundaries Bki , i = 1, ..., n (the surfaces of the containers), the absence of mass and heat flux is set (since the percentage

6.2 Method of Thermal Safety Analysis of Dry Storage of Spent Nuclear Fuel 213

of heat removed from the spent nuclear fuel through the concrete body of the container is insignificant in comparison with the heat removal due to air cooling in the ventilation channel); • on the boundaries B1−2 and B3−2 , which correspond to the radiation protective wall, the absence of mass and heat flows is set. The studies of the thermal state of one container for spent nuclear fuel were performed using the calculation area, the shape of which is shown in Figure 6.3.

Figure 6.3 Geometric form of calculation area.

214

Thermal Safety Criteria for Dry Storage of Spent Nuclear Fuel

When designing the calculation area, such assumptions were set: 1. The storage cask was considered as a group of homogeneous bodies (zone of heat release, head and bottom nozzle of fuel assemblies, and cask cover) with equivalent thermal conductivity. 2. The ambient medium was restricted by planes located at a distance from the center of the container, and which are equal to half the distance between two adjacent containers. The following boundary conditions were set for computational research: • on the boundaries B1 −B5 , the temperature and pressure of the atmosphere are p|Γi = pa ; T|Γi = Ta ; • on the boundary B6 (the surface of the concrete foundation of the storage site), there is no heat flux because the percentage of heat dissipated through concrete slabs into the ground is insignificant compared to heat dissipated into the atmosphere by air which passes through the ventilation system of WWER ventilated storage containers. The calculation area for the storage cask has the form shown in Figure 6.4. The numbers indicate the metal body and guide tubes (1), fuel assemblies (2), the cask cover (3), helium (4), the area of the assembly heads (5), the fuel zone (6), and the area of the bottom nozzle (7). The following boundary conditions were set on the surfaces of the calculation area: • heat-transfer coefficients and the temperature of the cooling air are on the surfaces S1 and S2; • heat flux, which corresponds to the amount of heat carried away in the concrete container through the lower surface of the cask, is on the surface S3. When carrying out the research, the spent fuel assemblies were considered as a set of homogeneous bodies (head, fuel zone, and bottom nozzle) with an equivalent value of thermal conductivity [4] of 1.04 W/(m·K). The metal parts (cask shell and guide tubes) had thermal conductivity of 45.00 W/(m·K), and the multilayer cask cover was considered as a homogeneous body with equivalent thermal conductivity of 3.06 W/(m·K). Thermophysical characteristics of helium in the cask with respect to the temperature were presented in the form of step and polynomial functions

6.2 Method of Thermal Safety Analysis of Dry Storage of Spent Nuclear Fuel 215

Figure 6.4 Calculation area for the storage cask.

216

Thermal Safety Criteria for Dry Storage of Spent Nuclear Fuel

obtained by approximating the tabular data [19] in the temperature range from 90◦ C to 400◦ C: • density (when the pressure is p = 0.1 MPa) ρ = 9.0 · 10−10 · (T + 273.15)3 + 2.0 · 10−6 · (T + 273.15)2 −1.3 · 10−3 · (T + 273.15) + 0.4086; • kinematic viscosity v = 4.253 · 10−7 · (T + 273.15)0.6732 ; • thermal conductivity λ = −5.0 · 10−10 · (T + 273.15)3 + 6.0 · 10−7 · (T + 273.15)2 +1.0 · 10−4 · (T + 273.15) + 0.0771; • heat capacity Cp = 5204 J/(kg·K). The calculation area for determining the thermal state of the spent fuel assembly in the storage cask is presented in Figure 6.5. The boundary conditions (heat transfer coefficients) are set according to the location of the fuel assembly in the cask on the surfaces S1−S6. The fuel elements were considered as solid heat-emitting elements. The heat release capacity of each fuel element was set as 3.2 W, given that total heat release capacity of one fuel assembly when loaded into the storage container must not exceed 1 kW [4]. As it has been mentioned above, the complexity of approaches that use simplified solid bodies with equivalent thermophysical properties lies in the definition of these thermophysical properties. For the problem under consideration, the value of the equivalent heat transfer coefficient of the fuel elements is fundamental. Many scientists have made serious work determining the equivalent coefficient of the thermal conductivity of fuel elements [20], [21]. Therefore, based on the information presented in the literature, the thermal conductivity coefficient was 5.5 W/(m·K) for fuel elements, 22 W/(m·K) for burnable absorber rods, and 45 W/(m·K) for metal in guide tubes. The thermal conductivity of helium was represented by the polynomial dependence on the temperature T according to the formula (6.15). The addition of the equation of radiant heat transfer to the mathematical model when solving the problem of determining the thermal state of a single container on the site of the dry storage of spent nuclear fuel requires substantiation.

6.2 Method of Thermal Safety Analysis of Dry Storage of Spent Nuclear Fuel 217

Figure 6.5 Calculation area for determining the thermal state of the spent fuel assembly: 1 – guide tube, 2 – helium, 3 – fuel element, 4 – burnable absorber rods.

Using the specified geometric form of the calculation area by solving the related problems of heat transfer, the assessment of the necessity to take into account the heat transfer by radiation was performed. In order to accomplish this, the heat flux from the storage cask to the container body was estimated using the formulae [13], [17] Qr = ϕεσ0 (T14 − T24 ), Qr = qr F,

(6.16) (6.17)

where qr is the density of the radioactive heat flux; σ0 = 5672·10−8 W/(m2 ·K4 ) is Stefan−Boltzmann constant; T 1 is the temperature of the surface of the solid body, which irradiates heat (in our case, it is the surface of the cask); T 2 is the temperature of the surface of the solid body, which absorbs heat (inner surface of the body); Qr is the heat capacity transmitted by radiation; F is the area of the body surface, which takes part in the heat

218

Thermal Safety Criteria for Dry Storage of Spent Nuclear Fuel

exchange by radiation; ε is the surface emissivity factor; and ϕ is the angular radiation coefficient. In order to determine the values T 1 and T 2 , a computation study was performed without taking into account radiation heat transfer. It was found that at the ambient air temperature Ta = 24◦ C, the temperature of the outer surface of multiplace sealed cask at the height of the cask first increases from 31.4◦ C to 121.4◦ C (in the central part of the cask) and then decreases to 75.7◦ C. The temperature of the inner surface of the concrete container increases evenly over the height from 24.0◦ C to 56.5◦ C, according to the change in air temperature in the annular duct between the storage cask and the container. At the ambient temperature Ta = 40◦ C, these ranges take the values of 47.2◦ C, 138.6◦ C, 92.9◦ C, 40.0◦ C, and 72.3◦ C, respectively. Since the width of the annular duct (70 mm) and the width of the gap between the multiplace sealed cask and the cover of the WWER ventilated storage containers (10 mm) are small in comparison with the height (4973 mm) and diameter (1715 mm) of the multiplace sealed cask, when estimating heat transfer by radiation, we shall neglect the propagation of heat flux in the direction, other than normal one, to the wall or to the cover of the cask. Therefore, as the temperatures T 1 and T 2 in expression (6.16), we shall take the temperatures at the points on the outer surface of the cask and the inner surface of the container, located opposite to each other. In this regard, ϕ = 1.0. In order to obtain the evaluation of the heat flux under conservative storage conditions (with a margin), let us assume that the outer surface of the multiplace sealed cask and the inner surface of WWER ventilated storage containers are absolutely black bodies, i.e., ε = 1.0. As a result, it is obtained that the heat flux density transferred from the outer surface of the cask to the inner surface of the container due to heat transfer radiation with the increase of the height first jumps from 45.7 to 804.3 W/m2 and then decreases to 170.2 W/m2 when Ta = 24◦ C. At Ta = 40◦ C, these values are 51.9, 936.6, and 210.6 W/m2 , respectively. These values were compared with the average integral heat flux dissipated from the storage cask by all heat transfer mechanisms, which can be generally calculated as q = Q/F, where Q = 24 kW is total heat release capacity of the spent fuel assemblies placed in the multiplace sealed cask, F = 29.08 m2 is the total area of the side surface and top outer surface of the cask. Thus, q = Q/F = 825 W/ m2 . It should be noted that the usage of calculation results, which took into account only natural convection but not heat transfer by radiation, leads to the overestimation of the density of heat flux. This is due to the fact that considering the heat transfer by radiation in the mathematical model,

6.2 Method of Thermal Safety Analysis of Dry Storage of Spent Nuclear Fuel 219

the temperature of the outer surface of the storage cask is lower, and the temperature of the inner surface of the container is higher than the temperatures T 1 and T 2 which have already been used. In addition, as it has been noted above, the obtained results are also overestimated for another reason − the surfaces involved in heat transfer by radiation were considered as completely black bodies with ε = 1.0. However, this estimate shows that the amount of heat dissipated from the multiplace sealed cask by radiation heat exchange can have the same order as the amount of heat dissipated by convection, i.e., the mathematical model must include an equation that takes into account heat transfer by radiation when studying the thermal state of a single spent fuel storage container at the storage site. Radioactive heat transfer occurs between the outer surface of the cask and the inner surface of the container in addition to convective and conductive heat transfer due to air movement in the ventilation tract, which, according to the research above, must be taken into account. Similarly, using the same approach, the assessment on whether we should consider the heat transfer radiation inside the cask or not was made. For this purpose, it was necessary to examine the problem where the heat exchange inside the storage cask with the fuel assemblies installed in it was modeled. On the surface of the cask, averaged boundary conditions of the third kind were set: the heat transfer coefficient on the surface is α = 11.95 W/ (m2 ·K) and medium temperature is Tm = 46.4◦ C; these values correspond to the averaged values based on the results of the numerical experiment. The ducts through which helium flows are formed by the side wall, guide tubes, and fuel assemblies; their width is small compared to the height, and that is why the propagation of radiant heat flux in a direction other than horizontal one can be neglected when assessing heat transfer radiation. In addition, the strongest intensity of heat fluxes will obviously be observed at the level of the zone of heat release in the fuel assemblies. Therefore, let us denote the temperatures at points on the surfaces of the fuel assemblies, guide tubes, and the frame of multiplace sealed cask, located opposite each other at the level of the zone of heat release in the fuel assemblies as the temperatures T 1 and T 2 in the expression. As a result of the numerical solution of the model problem, it was found that the surface temperatures of solid bodies emitting heat vary within the range of T 1 = 379 ÷ 540 K, with the temperature difference between the radiating and irradiated bodies T 1 − T 2 = 4 ÷ 39◦ C. Using the formula (6.16), we shall obtain that the heat flux density transferred between the fuel assemblies, the guide tubes, and the cask frame

220

Thermal Safety Criteria for Dry Storage of Spent Nuclear Fuel

varies in the range of 109 ÷ 571 W/m2 due to the heat exchange by radiation. Let us compare these values with the average integral density of the heat flux removed from the fuel assembly surface by all heat transfer mechanisms, which can be defined as q = Q/F, where Q = 1 kW is the capacity of heat release in one fuel assembly; F = 3.7 m2 is the area of the lateral surface of the fuel assembly (we shall neglect heat transfer at the end faces of the fuel assemblies due to their small values). Thus, the density of the heat flux is q = Q/F = 270 W/m2 . This estimation shows that the amount of heat transferred inside the cask by radiation heat exchange has the same order as the amount of heat dissipated by convection, i.e., the mathematical model of thermophysical processes inside the cask must include an equation that takes into account heat transfer by radiation. With respect to the proposed method of separation of calculation area into the parts, several types of boundary conditions will be used: [1] pressure and temperature of atmospheric air for the calculation area of the first level; [2] temperature and capacity rate of ventilation air for the calculation area of the second level; [3] heat transfer coefficients and temperature of the moving medium for the calculated areas of the third, fourth, and fifth levels. It should be noted that due to some features of storage cask cooling of the spent fuel assembly and fuel element in the mathematical models of the third, fourth, and fifth levels, we shall consider the boundary conditions of the third kind which describe convective heat transfer: q|B = −α(T|B − Tm ),

(6.18)

where q is the density of the heat flux; α is the heat transfer coefficient; T is the cask temperature; Tm is the temperature of the medium (air in the ventilation duct); index B means that the values that it refers to are examined at the boundary of the calculation area coinciding with the surface of the cask. In order to describe the heat transfer on the surface of the cask by means of Equation (6.18) as accurately as possible, we shall examine local heat transfer coefficients, i.e., such coefficients where the value varies with respect to the coordinates of the cask surface. The method of determining the local heat transfer coefficients is as follows. Having calculated the temperature field by means of the

6.2 Method of Thermal Safety Analysis of Dry Storage of Spent Nuclear Fuel 221

mathematical model of the first level for each ith elementary section of the cask surface, heat flux Qi through this section and its average temperature Ti are found. In order to calculate the value Q, the same finite-difference approximation is used as in the calculation of the temperature field, i.e., already found values of temperatures in the nodes of the difference grid are applied. After that, taking into account Equation (6.18), the heat transfer coefficient on the ith elementary section can be calculated as αi =

Qi , Fi (TC − T i )

(6.19)

where Fi is the area of the surface of the ith elementary section. It should be noted that the air temperature in the ventilation duct increases as the air passes by the cask surface and is heated by it. If we consider not only the local heat transfer coefficients as a boundary condition (6.18) but also the change of the air temperature TC in special coordinates, it will only complicate the mathematical model but will not improve the accuracy of the calculation, as it is preferable to calculate the temperature field of the cask q|B , which is already taken into account in the local heat transfer coefficients. Therefore, we shall consider the air temperature at the entrance to the ventilation tract as TM , i.e., we shall relate the local heat transfer coefficients to the atmospheric air temperature. The basic equipment of the dry storage of spent nuclear fuel (container and storage cask) has a rather complex structure, which causes some difficulties in modeling the thermal processes occurring at the storage site. Since detailed consideration of the structure of all the elements of the storage system is not possible due to limited computing resources, it is advisable to use the above presented methodology, which is based on the use of calculation areas of different levels of particularization. As it has been mentioned above, under this approach, it is necessary to use the equivalent thermophysical properties of those bodies that replace the part of the object with a complex structure. The method of determining the equivalent thermal conductivity is based on the solution of the inverse thermal conductivity problem [18], [22]. In contrast to the case when the equivalent thermal conductivity of composite solids is found and classical inverse heat conduction problem is used, due to the presence of the moving medium (helium and air), this problem must be considered in a conjugate formulation and must be solved as an inverse conjugate heat transfer problem [23].

222

Thermal Safety Criteria for Dry Storage of Spent Nuclear Fuel

The proposed method is as follows. Under the same external thermal influences, two problems are considered, one of which uses a detailed geometric model that takes into account the internal structure of the object under examination (a storage cask), and the second one is simplified, i.e., the object under study is replaced by a homogeneous isotropic body with equivalent thermal conductivity λe . The latter is determined in the process of multiple solutions of the problem with simplified geometry by a trial and error method in order to minimize the standard deviation between the values of temperatures obtained by solving problems with detailed and simplified geometry: v uN
uP d s 2 u T − T i i t → min, σ(λe ) = i=1 N where N is the number of the so-called control points in which the temperature deviation is considered; Ti d is the temperature in the ith control point which is obtained as a result of the solution of the problem with a detailed geometric model of the object under examination; Ti s is the temperature in the ith control point which is obtained as a result of the solution of the problem with a simplified geometric model. This approach allows obtaining the value of equivalent thermal conductivity that gives the temperature distribution in the object with simplified geometry (which is considered) as close as possible to the temperature distribution in the object with detailed geometry. In addition to the thermal conductivity of the simulated elements, for nonstationary research, it is necessary to know their heat capacity (it is equivalent heat capacity in case of the simplification of the object). Unfortunately, the review of the literature has shown that there are no works dedicated to equivalent heat capacity. Usually, non-stationary problems are considered without geometric simplifications and heat capacity is set for individual elements made of materials with previously known thermophysical properties [24]–[30]. In all studies on this subject, heat capacity is given either by a constant or in the form of a functional dependence on temperature, but always for separate elements of the storage cask for spent nuclear assemblies. The density ρ and the specific heat Cp are included in the equation of the thermal conductivity as a product. In view of this, we shall also determine the equivalent product of these values for the cask. That is, it is necessary to calculate the equivalent volumetric heat capacity of the storage cask Ce : Ce = Ce ρe .

(6.20)

6.3 Thermal Condition of Containers for Dry Storage

223

Let us consider the physical meaning of volumetric heat capacity. The given value characterizes the ability of the volume unit V of a particular substance to increase its energy Q with a change in temperature T. Based on this, let us write the equation: QΣ = Ce VΣ ∆T, (6.21) where QΣ and VΣ are the energy and volume of the whole storage cask. Similarly, let us write down the formula for a specific nth element of the storage cask: Qn = Cn Vn ∆T. (6.22) Let us rewrite relation (6.21) using formula (6.22): QΣ =

n X

Ci Vi ∆T .

(6.23)

i=1

According to Equations (6.21) and (6.23), we shall find the expression for the equivalent volumetric heat capacity of the storage cask: Pn i=1 Ci Vi . (6.24) Ce = P n i=1 Vi From the final formula (6.24), it is seen that in order to find the equivalent volumetric heat capacity Ce , it is enough to determine the volume of each substance that is part of the storage cask and its heat capacity. Using data on the geometric dimensions of the storage cask and its basic elements, we can find the volume occupied by each of them. In order to do this, we shall fix the necessary geometric parameters of each element and perform trivial calculations to find the volume. All data and results are summarized in Table 6.1. According to the obtained data (see Table 6.1) and using the formula (6.24), let us determine the value of the equivalent volumetric heat capacity kJ for the storage cask as Ce = 1620.5( K·m 3 ). This value will be used in the future when conducting non-stationary studies of the thermal state of spent nuclear fuel using the calculation area of the second level.

6.3 Thermal Condition of Containers for Dry Storage of WWER-1000 Spent Nuclear Fuel Let us consider the operation of containers for the storage of spent nuclear fuel under normal conditions, i.e., those that are provided by the report on the

224

Thermal Safety Criteria for Dry Storage of Spent Nuclear Fuel

Zr

Steel RX-277 Steel ASME SA516

Fuel cell

7.53

Fuel cladding Cladding of the fuel assembly head Cladding of the fuel-assembly bottom nozzle

7.7

3530

7488 1,176,522,265 1,176,522,265

9.1

3837 425

7488 24

532,278,939 144,933,656

270

24

92,075,499.4

50.8

1

109,543,860

10,954,386

4973

1

672,738,812

2,714,082,978

Neutron shield 1657.4

769,288,095

Shell of multiplace sealed cask

1664

Force cover

1651

76.2

1

16,304,923.8

Protective cover

1657.4

190.5

1

410,789,473

Block of hexagonal tubes

1587.5 1625.6

4320

1

415,147,686

249.6

4320

24

1,199,102,083

425 270

24 24

338,178,532 214,842,832

553,021,363

6,179,343,710

6,179,343,710

Guide tubes Steel AISI 321

Head Bottom nozzle

He2

Other space of multiplace sealed cask

1715

Number in multiplace sealed

Height, h, mm

UO2

Outer diameter D, mm

Diameter d, mm

Table 6.1 Data to calculate the volumes of the elements which are part of the storage cask. Volume Total Material Elements of multiplace V, mm3 volume ΣV, sealed cask mm3

259.1

safety analysis of the dry storage of spent nuclear fuel at Zaporizhzhya NPP [4], [31]. In order to do this, let us apply the iterative methodology outlined above. According to the methodology of the study of the thermal state of spent nuclear fuel, we shall start with the analysis of a group of containers, then a separate container, a storage cask, a fuel assembly, and a fuel element. Let us consider each of these levels of the model one by one.

6.3 Thermal Condition of Containers for Dry Storage

225

The model of the first level is a group of containers. The purpose of the calculations at the first level is to obtain the parameters of air at the inlet to the cooling system. These parameters will serve as a kind of “indicator” of whether there is mutual influence of containers in normal operating conditions or not. If the parameters of air at the inlet to the cooling system of the container do not differ from those conditions that are accepted as normal operating conditions in the analysis of spent nuclear fuel storage safety [4], we can say that there is no mutual influence of storage containers. The temperature field of ventilation air for two containers from the group under no-wind conditions is shown in Figure 6.6. The computation model takes into account the real structure of the ventilation ducts of storage containers, which allows estimating the real thermal state of the containers on the site of the dry storage of spent nuclear fuel. Figure 6.6 shows that ventilation air evenly comes out of the containers and rises. The neighboring containers do not influence each other – ventilation air is not mixed and does not come into a ventilating tract of the next container. When analyzing a group consisting of a larger number of containers, the results are similar − the containers do not affect each other. At the next stage, let us investigate the thermal state of one container with spent nuclear fuel under normal operating conditions. Due to the lack of mutual thermal influence, we can assume that the container is located

Figure 6.6

Temperature field of ventilation air.

226

Thermal Safety Criteria for Dry Storage of Spent Nuclear Fuel

separately (wide apart) from other containers. It should be noted that the calculation area has an accurate description of the ventilation system without simplifications, which allows obtaining more complete picture of thermal processes that occur during the storage of spent nuclear fuel under normal operating conditions of the storage container. Using the model of the second level (according to the method described above), let us study the thermal state of a separate container. To do this, it is necessary to determine the equivalent thermal conductivity of the cask with spent nuclear fuel. In order to determine the equivalent thermal conductivity, control points were selected; they took into account the temperature used in the root mean square mis-tie. The points were selected in the most critical places from the thermal point of view: on the axes of the fuel assemblies, i.e., at the points where the temperature reaches its highest values. We considered 80 control points (for a quarter of the cask), selected on the axes of eight spent fuel assemblies (No. 1, 2, 3, 8, 10, 12, 18, and 20), which were uniformly located at 10 levels. Sixty control points on the guide tubes and 30 points on the cask body were added to them. They were selected at the same levels with the points on the axes of a spent nuclear assembly, i.e., the points located on three vertical lines passing through the cask shell and the points which were located on six vertical lines passing through the walls of the guide tubes. In addition, 14 control points were selected on the cover and the bottom of the cask. Obviously, when changing the storage cask to a solid body with corresponding equivalent thermophysical properties, it will be important to take into account changes in the heat release capacity of fuel assemblies over time because the thermal conductivity of the elements in the cask depends on the level of temperature, which, in its turn, depends on the heat capacity of spent nuclear fuel. Due to the weak change in the heat release capacity of the cask for spent nuclear fuel over time, the problem of determining the equivalent thermal conductivity of the cask taking into account its storage time can be considered in a quasi-stationary formulation. The change in the equivalent thermal conductivity of the storage cask arranged by the years is presented in Figure 6.7. It is seen that the value of equivalent thermal conductivity decreases during its storage time. The nature of this change is similar to the nature of the change of the heat release capacity of spent nuclear fuel. The structure of the ventilation air movement in the storage container has the form shown in Figure 6.8 [32]. It is seen that the flux of ventilation air

6.3 Thermal Condition of Containers for Dry Storage

227

Figure 6.7 Dependence of the value of equivalent thermal conductivity in the cask for spent nuclear fuel on its storage time.

Figure 6.8 Flux in the ventilation tract of the container.

in the ventilation tract has zones of vorticity, which affect the nature of the cooling in the cask for spent nuclear fuel. In the outlet ducts, there are also zones of vorticity at the corners formed by the main flow of exhaust air. The density of convective heat flux, which is due to the intensity of the ventilation air in the annular channel on the surface of the storage cask, is

228

Thermal Safety Criteria for Dry Storage of Spent Nuclear Fuel

Figure 6.9

Density field of the convective heat flux.

shown in Figure 6.9. In the part of the cask where the speed of ventilation air is higher, convective heat exchange is more intensive, whereas it is lower in stagnant zones (zones where vortices are formed). Positive and negative values of heat flux density indicate the release or absorption of heat by the surface. Thus, in the zone of heat release where the flow of cooling air is quite intensive and slightly heated, heat is dissipated, whereas in the upper part, the flow of ventilation air is already significantly heated and part of its heat is given to the cask for spent nuclear fuel. The maximum temperature during normal operation of the storage container under no-wind conditions at the air temperature of 24◦ C reaches about 335◦ C, which is acceptable according to the thermal safety criterion. The change in temperature along the axis of the container is shown in Figure 6.10. The point of maximum temperature is located at a distance of 2.7 m from the bottom of the container, which is 0.107 m higher than the geometric center of the zone of heat release. This arrangement is apparently caused by convective heat release in the vertical ventilation duct of the container. The non-uniform flow of cooling air in the annular channel affects not only the maximum temperature inside the cask for spent nuclear fuel. The temperature field of the surface of the cask and the surface of the

6.3 Thermal Condition of Containers for Dry Storage

229

Figure 6.10 Temperature change along the axis of the storage container with respect to its height.

container have a corresponding structure (Figure 6.11); higher temperatures are observed in those areas where cooling is not intensive. The temperature field of the surface of the container is more uniform than the temperature field of the surface of the cask due to the convection of ambient air. The surface temperature of the cask and the surface temperature of the container in three cross sections are shown in Figure 6.12. The temperature gradient on the cask surface reaches 128.1◦ C in the cross section through the middle of the outlets; maximum temperature is at a height of 2.7 m from the bottom of the container. The surface temperature of the container in the cross section through the outlet can be divided into two parts: from the bottom to the outlet and from the outlet to the top of the container. The highest temperature in the second part is observed due to heating of the surface by exhausting ventilation air and reaches 42.5◦ C. In the first part, the highest temperature is 41.2◦ C and is located at a height of 2.7 m from the bottom of the container. The temperature gradient on the surface of the container is not more than 14.5◦ C. The distribution of heat transfer coefficients on the surface of the storage cask is presented in Figure 6.13. In the zone of vorticity, the heat transfer

230

Thermal Safety Criteria for Dry Storage of Spent Nuclear Fuel

Figure 6.11 (a) Temperature field of the surface of the cask and (b) the outer surface of the container.

coefficient is lower; however, in the areas where the ventilation air moves faster, it is higher. The decrease in the heat transfer coefficient occurs at the top of the storage cask due to the fact that the heated ventilation air hardly cools the storage cask. Heat transfer coefficients on the surface of the cask are necessary in order to determine the thermal state of fuel assemblies in the storage cask.

6.3 Thermal Condition of Containers for Dry Storage

231

Figure 6.12 (a) Temperature change on the surface of the cask and (b) on the surface of the container in the cross sections in the center of outlet ventilation ducts (1), between inlet and outlet ventilation ducts (2), between outlet channels, and on the axis which is perpendicular to the axis of the inlet ducts (3) with respect to the height.

232

Thermal Safety Criteria for Dry Storage of Spent Nuclear Fuel

Figure 6.13 Heat transfer coefficient on the surface of the storage cask in the sections: in the middle of outlet vents (1); between inlet and outlet vents (2); between outlet vents and in the middle of inlet vents (3).

Since, as has been mentioned above, the thermal state of fuel is affected by the temperature of the atmospheric air, let us determine the maximum temperature in the storage cask for the average monthly temperature during the year. Weather factors such as rain, snow, wind, etc., were not taken into account as they have an irregular and short-term effect on storage containers. Maximum temperatures in the fuel cask and mass flux of ventilation air through the ventilation system at different air temperatures during the year are presented in Table 6.2. When the ambient temperature decreases, the maximum temperature decreases as well and vice versa, and when it increases, the maximum temperature does the same. The flow of ventilation air through the ventilation tract fluctuates around 0.3 kg/s. Throughout the year, the criterion of thermal safety was not exceeded, which indicates the safety of spent fuel storage. Table 6.2 Maximum temperature and flow of ventilation air in the storage container throughout the year. Month Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Ta.av . ,◦ C −3.5 −2.6 2.0 10.1 16.4 20.2 22.4 21.4 16.2 9.6 3.5 −1.1 T max , ◦ C 312.5 313.5 317.7 325.4 330.7 334.3 335.7 335.4 330.9 324.7 318.6 314.6 G, kg/s 0.318 0.320 0.309 0.299 0.292 0.289 0.286 0.287 0.295 0.302 0.310 0.319

6.3 Thermal Condition of Containers for Dry Storage

233

Figure 6.14 Temperature field of the cask with spent fuel assemblies.

At the third stage of the application of the iterative method, how to determine the thermal state of spent nuclear fuel during its dry storage, the temperature field of the cask with assemblies which have the same heat release, was determined (Figure 6.14). The following temperatures were observed in the assemblies: assembly 4◦ C−314◦ C, assembly 5◦ C−287.2◦ C, assembly 6◦ C−232.7◦ C, assembly 12◦ C−304.9◦ C, assembly 14◦ C−291.6◦ C, assembly 16◦ C−250.9◦ C, assembly 22◦ C−259.8◦ C, and assembly 24◦ C−236.43◦ C. The level of maximum temperature is slightly lower compared to the results obtained by

234

Thermal Safety Criteria for Dry Storage of Spent Nuclear Fuel

the model of the second level because it takes into account convective heat transfer in the middle of the storage cask. The conditions of heat exchange on the surfaces of the guide tubes were determined for the next stage of the research. The difficulty is that, in contrast to the situation when the boundary conditions were determined on the surface of the cask, here there is no surface for spent nuclear assemblies that would be common to a simplified and detailed object. In this case, it is proposed to examine not only the object of study but also the part of the environment where it is located. Thus, the boundary conditions should be determined on the outer surface of the guide tubes for each assembly. Taking into account the symmetry of the location of spent fuel assemblies in the storage cask, let us examine only those fuel assemblies that are located in a quarter of the cask as shown in Figure 6.15. The numbering of the surfaces of the guide tubes for each of the assemblies was performed clockwise looking from above. According to the results of the calculation, the temperature distributions on each of the surfaces of the guide tube for each spent fuel assembly were also determined (Figure 6.16). The lowest temperatures are recorded in the area of the bottom nozzle, the heating of this part of the guide tube for each of the assemblies is almost the same, and the temperature gradient is about 50◦ C. The highest temperatures are in the zone of heat release at a distance of about 3 m from the bottom of the guide tube. The surface S2 of the assembly No. 4 located in the center of the cask is the most heated; the surface S5 of the assembly No. 6 is the least heated. All guide tubes of each of the assemblies are characterized by an increase in temperature in the areas from the bottom nozzle to the head and a temperature decline in the zone of the head, where heat is not released. This nature of the cooling of the assemblies determines the distribution of heat transfer coefficients on the surfaces of the guide tubes.

Figure 6.15 Numbers of fuel assemblies and surfaces for determining boundary conditions.

6.3 Thermal Condition of Containers for Dry Storage

235

Figure 6.16 Temperature change on the surface of the guide tubes with respect to the height. (a) Fuel assembly 4. (b) Fuel assembly 5. (c) Fuel assembly 6. (d) Fuel assembly 12. (e) Fuel assembly 14. (f) Fuel assembly 16. (g) Fuel assembly 22. (h) Fuel assembly. 24.

236

Thermal Safety Criteria for Dry Storage of Spent Nuclear Fuel

Figure 6.17 represents the distribution of heat transfer coefficients on the boundary surfaces for each of the assemblies. Negative or positive values of heat transfer coefficients indicate the direction of heat flux: in case of negative values, heat flux is directed outward from the calculation area, and in case of positive values, it is directed inward. All surfaces of the guide tubes are characterized by jumps of the values of heat transfer coefficients in the zones of the bottom nozzle and head. This nature of the distribution of heat transfer coefficients is due to the lack of uniform heating on that part of the spent nuclear assemblies, the surfaces of which are heated by heat transfer from the middle part of the guide tube or by radiation heat transfer. As it can be seen from Figure 6.17, the heat transfer coefficients on the surfaces which separate the assemblies and are not adjacent to the wide or narrow ducts have the lowest values due to the weak movement of helium in the medium near the assembly with approximately the same temperature and the fact that the surface is located next to another fuel assembly. More intensive heat transfer occurs on the surfaces, which are located closer to the edge of the cask, thus proving higher level of heat transfer coefficients. When the fuel assemblies were presented as solid bodies, their detailed temperature fields were determined, using the computational results. Boundary conditions were set on the outer surfaces of the guide tubes according to the results of calculation of the thermal state of the spent fuel assemblies, which were obtained at the previous stage of work. The boundary conditions were determined in a horizontal section located at the level of maximum temperatures in the fuel assembly. This allowed obtaining the most important information (from the point of view of the development of thermal safety criteria) about the temperatures in the fuel elements of each fuel assembly. The temperature fields of the fuel elements, guide tubes, and helium that fill the space between the fuel elements and the guide tubes are shown in Figure 6.18. The figure shows that the highest temperatures are observed in the assembly which is the closest to the center of the cask – assembly No. 3. The lowest temperatures are in the assembly No. 18, which is the closest to the wall of the cask and is cooled by helium in narrow and wide ducts. The applied approach allowed determining the maximum temperatures in each of the assemblies during their dry storage. The results are presented in Table 6.3. Such data correspond well with the data on maximum temperatures in the storage cask which were obtained at the previous stages of work, when the storage cask was considered as a solid body and in modeling a cask with

6.3 Thermal Condition of Containers for Dry Storage

237

Figure 6.17 Change of heat transfer coefficients on the surfaces which surround the fuel assembly with respect to the height. (a) Fuel assembly 4. (b) Fuel assembly 5. (c) Fuel assembly 6. (d) Fuel assembly 12. (e) Fuel assembly 14. (f) Fuel assembly 16. (g) Fuel assembly 22. (h) Fuel assembly 24.

238

Thermal Safety Criteria for Dry Storage of Spent Nuclear Fuel

Figure 6.18 Temperature field in the horizontal cross section of the storage cask for spent nuclear fuel at the level of their maximum temperatures when the atmospheric air temperature is 40◦ C.

spent fuel assemblies, when the assemblies were considered as solid bodies with equivalent thermophysical properties. The obtained data are important, first of all, for the development of thermal safety criteria for dry storage of spent nuclear fuel. In order to do this, it is necessary to have information about the location of the fuel elements with the maximum temperature. In Figure 6.19, fuel elements with the highest temperatures are marked in black.

6.3 Thermal Condition of Containers for Dry Storage

Number of fuel assembly T max , ◦ C

Table 6.3 1 262.9

Maximum temperatures in the fuel assemblies. 2 3 8 10 12 296.6

310.5

272.4

299.6

305.4

239

18

20

260.2

271.2

As we can see from Figure 6.19, the thermally stressed fuel elements are located in the central part of the fuel assembly, where, apparently, the level of radiant heat flux from the surface of the fuel elements is much lower than that in the fuel elements, which are located on the periphery of the fuel assembly, and convective heat transfer plays a minor role in their cooling. However, it should be noted that the fuel elements with the highest temperatures are not located in the center of the fuel assembly but closer to the wall of the guide tube, which is located closer to the center of the cask and borders on a hotter fuel assembly. Since the operation of containers with spent nuclear fuel occurs in an open storage facility, the containers are exposed to a variety of factors. In particular, the most significant factors are wind, temperature change during the day, and insolation. Let us consider in detail the influence of wind in the three main directions with respect to the ventilation ducts of the container (Figure 6.20) [33], [34]. The following boundary conditions were set during the computational studies (see Figure 6.3). 1. Option A: • the velocity vector and the temperature of the main air on the boundaries B1 and B5 are: vx |B1 = vx |B5 = c, vy |B1 = vy |B5 = 0 m/s, vz |B1 = vz |B5 = 0 m/s, T|B1 = T|B5 = Ta ; • on the boundary B3 , the atmospheric pressure and ambient temperature are: p|B3 = 101, 300 Pa, T |B3 = Ta ; • there are no mass and heat fluxes on the boundaries B2 and B4 .

240

Thermal Safety Criteria for Dry Storage of Spent Nuclear Fuel

Figure 6.19

Location of fuel elements with maximum temperature.

2. Option B: • the velocity vector and the temperature of the main air on the boundaries B4 and B5 are: vx |B4 = vx |B5 = 0 m/s, vy |B4 = vy |B5 = c, vz |B4 = vz |B5 = 0 m/s,

6.3 Thermal Condition of Containers for Dry Storage

Figure 6.20

241

Options of wind direction (u – upper; l – lower ventilation ducts).

T|B4 = T|B5 = Ta ; • the atmospheric pressure and ambient temperature on the boundary B2 are: p|B2 = 101, 300 Pa, T|B2 = Ta ; • there are no mass and heat fluxes on the boundaries B1 and B3 . 3. Option C: • the velocity vector and the temperature of the main air on the boundaries B1 , B4 , and B5 are: √ vx |B1 = vx |B4 = vx |B5 = 0.5c, √ vy |B1 = vy |B4 = vy |B5 = 0.5c, vz |B1 = vz |B4 = vz |B5 = 0 m/s, T|B4 = T|B5 = Ta ; • the velocity vector and the temperature of the main air on the boundaries B2 and B3 are: p|B2 = p|B3 = 101, 300 Pa, T|B2 = T|B3 = Ta . In the above boundary conditions, the following notations were used: c is the wind velocity in the undisturbed flux, m/s; v is the air velocity on the boundaries of the calculation area, m/s; Ta is the air temperature in the undisturbed flux, ◦ C.

242

Thermal Safety Criteria for Dry Storage of Spent Nuclear Fuel

Table 6.4 Maximum temperatures in a separately located container is case of different values of velocity and directions of wind (Ta = 24◦ C). Wind velocity Maximum temperature in the storage cask, ◦ C c, m/s Direction of wind, Direction of wind, Direction of wind, option A option B option C 0 335.1 1 337.9 338.2 339.3 2 341.7 340.9 340.5 3 347.5 341.1 334.9 5 355.9 345.9 312.0 6 358.2 341.8 297.9 8 356.6 328.5 282.3 9 355.2 327.4 272.9 10 336.0 317.5 260.4 12 320.7 292.2 260.3

In all these options, the heat removal on the boundary B6 (the surface of the base plate – concrete base of the dry storage of the spent nuclear fuel) is not taken into account. The temperature of atmospheric air is Ta = 24◦ C. Therefore, using the calculation area (Figure 6.3), more detailed studies of the effect of wind on the thermal state of spent nuclear fuel in a ventilated storage container were conducted. The maximum temperature for each of the options of inflow for different air velocities is presented in Table 6.4. As it can be seen from the computation results, the wind increases the temperature in the storage container, but the further increase in wind velocity leads to a decrease in the level of the temperature of spent nuclear fuel. These results basically coincide with the previously obtained data: high air velocity results in the entry of cold atmospheric air into the storage container and the redistribution of the structure of the ventilation air, which, in its turn, reduces the temperature of the cask for spent nuclear fuel. The data listed in Table 6.4 perfectly reflect the influence of wind on the state of spent nuclear fuel, if to place the points on the graph (Figure 6.21). It is seen that there is a tendency for each of the options of wind flow. The highest temperature level is observed when the wind blows according to option A, and the lowest one is when the wind blows according to option C. In order to use the obtained results during the analysis of the safety of dry storage of spent nuclear fuel in ventilated storage containers more effectively, statistical processing of the presented data was performed and regression dependences of maximum temperature on wind velocity were determined for each of the considered directions (they are represented by a dashed line in Figure 6.21).

6.3 Thermal Condition of Containers for Dry Storage

243

Figure 6.21 Change of maximum temperature in the storage container in case of different options of the inward air flow with respect to its velocity.

When the inward wind flow is according to option A, the dependence takes the following form: Tmax = −0.0642c3 + 0.2958c2 + 4.4092c + 334.03,

(6.25)

where T max is the maximum temperature in the storage container, ◦ C; c is the velocity of the inward air flow, m/s. The maximum temperature T max (6.561) = 357.56◦ C is reached when the wind velocity is 6.561 m/s. When the inward wind flow is according to option B, the dependence takes the following form: Tmax = −0.017c3 − 0.4532c2 + 4.3772c + 334.54.

(6.26)

The maximum temperature T max (3.951) = 343.711◦ C is reached when the wind velocity is 3.951 m/s. When the inward wind flow is according to option C, the dependence takes the following form: Tmax = 0.1562c3 − 2.921c2 + 6.242c + 336.3.

(6.27)

244

Thermal Safety Criteria for Dry Storage of Spent Nuclear Fuel

Figure 6.22 the speed.

Maximum temperature of fuel assemblies in case of inflow air (option A) by

The maximum temperature T max (1.18) = 339.855◦ C is reached when the wind velocity is 1.18 m/s. Since the thermal criterion for storage safety is the non-exceedance of a certain temperature level, it is advisable to consider the thermal state of fuel assemblies in the option with the highest temperatures, that is, in option A. Using the calculation area of the third level, the thermal state of the fuel assemblies in the storage cask was examined. The maximum temperatures in each of the assemblies are presented in Figure 6.22. It can be seen that from the side from which the wind flux flows (from the side where the ventilation ducts are blocked and the ventilation air does not come out freely from the container), the fuel assemblies have a higher temperature. The fuel assemblies located in the middle of the cask have the highest temperature; therefore, the wind effect is not significant enough to change the location of the hottest fuel assembly. It should also be noted that the temperature difference in the fuel assemblies located on the windward and leeward sides do not exceed 5 degrees. This indicates a slight effect of wind on the thermal state of fuel assemblies.

6.3 Thermal Condition of Containers for Dry Storage

245

Now let us examine the effect of daily fluctuations in atmospheric air temperature on the thermal state of containers for spent nuclear fuel by means of the model of the third level (the one where fuel assemblies are represented by solid bodies) in order to determine the temperatures in each of the fuel assemblies [35]. The change of maximum temperatures in the storage cask at different daily fluctuations in atmospheric air temperature is shown in Figure 6.23. Fluctuations in atmospheric air temperature were taken as a sinusoid Ta (τ ) = Tair aver + Cair · sin(2π

τ ), 24

(6.28)

where T air aver is the average daily temperature of atmospheric air, ◦ C; Cair is the daily amplitude of the temperature of atmospheric pressure, ◦ C; τ is the storage time, which is counted off from the midday, h. During the computation, the value T air aver = 24◦ C was chosen, which prevails on the grounds of Zaporizhzhya. The functional dependence (6.28) of the atmospheric air temperature from time perfectly coincides with the results of meteorological observations. As part of the study, different oscillation strengths were observed: Cair = 10◦ C, Cair = 7◦ C, Cair = 5◦ C. The analysis of the calculation results has shown that the amplitude of the change in the maximum temperature of the spent nuclear fuel does not exceed 0.14◦ C in case when Cair = 10◦ C and decreases to 0.06◦ C when Cair = 5◦ C. The peaks of maximum temperatures in each fuel assembly have a time shift relative to the peaks of the atmospheric air temperature. The maximum “delay” is 13.3 hours for the fuel assembly located in the center of the cask. Statistical processing of the computation results has shown that the change in the maximum temperature in each fuel assembly depending on the storage time can be represented by a regression dependence:   τ − τshift , Tasm (τ ) = Taver asm + Casm · sin 2π 24 where T aver asm is a time-averaging operation T asm ; Casm is the amplitude of daily oscillations of the temperature T asm ; τshift is the time shift of daily oscillation change T asm . The regression parameters in accordance with the parameters of atmospheric air temperature fluctuations are given in Table 6.5.

246

Thermal Safety Criteria for Dry Storage of Spent Nuclear Fuel

a

b

c

d Figure 6.23 Dependence of surface temperature of the container on the time of day at different heights. (a) North. (b) East. (c) South. (d) West.

6.3 Thermal Condition of Containers for Dry Storage

247

Table 6.5 Data for the definition of maximum temperatures in fuel assemblies. No of Average temperature of the fuel Amplitude Time shift fuel assembly (T aver asm ) (Casm ) (τshift ) assembly Cair = Cair = Cair = Cair = Cair = Cair = ±5◦ ±7◦ ±10◦ ±5◦ ±7◦ ±10◦ 1 205.53 205.53 205.48 0.36 0.53 0.71 6.30 2 253.39 253.36 253.33 0.12 0.20 0.26 10.13 3 273.60 273.55 273.54 0.06 0.10 0.13 13.17 8 227.71 227.68 227.64 0.21 0.33 0.43 6.00 10 260.19 260.13 260.11 0.10 0.18 0.23 10.48 12 268.16 268.10 268.08 0.07 0.13 0.18 12.52 18 210.39 210.33 210.30 0.27 0.44 0.61 5.92 20 228.79 228.72 228.69 0.13 0.26 0.37 8.43

The results of the computations have shown that the daily fluctuations of the sinusoidal air temperature lead to similar and variables in time fluctuations of the temperatures of the fuel assemblies. Temperature fluctuations in the fuel assemblies occur during the same daily period but with a phase shift and with insignificant amplitude. In order to determine the level of sun exposure on the container and the fuel assemblies located in it, a separate container was studied. This approach can be considered conservative, as there is no shading from neighboring containers and protective structures and the surface that receives solar radiation (and therefore the amount of heat absorbed by the container due to insolation) is greater than that in the case of a group of containers. A secondlevel model was used for this type of study, i.e., the cask was considered as a solid body. The change in temperature at the corresponding points on the surface of the container at different heights from the surface of the plate on which it is located is shown in Figure 6.23. The lowest values of the surface temperature are observed on the northern side of the container, where the sunlight does not fall. This situation is typical for the northern hemisphere and the latitude of the storage site. The eastern and western sides of the storage container are exposed to solar radiation of slightly lower intensity than the southern side, but during the day, the maximum surface heating is achieved from the western side. This situation is due to the fact that after the night decrease of temperature, the western side is heated in the first half of daylight, when the sun does not shine on it; it is heated up due to the rise of temperature of the atmospheric air. In the afternoon, it is heated by insolation. Thus, at the beginning of the absorption of solar radiation, the eastern side has a lower

248

Thermal Safety Criteria for Dry Storage of Spent Nuclear Fuel

temperature than the western side. And since during the day both western and eastern sides receive the same amount of heat due to solar radiation, the temperature of the western side will rise to a higher level than the eastern one. The southern side in this sense occupies an intermediate position because its pre-heating due to the air temperature rise is much lower than that on the western side. The values of the surface temperature of the container are different in height (Figure 6.23) due to the influence of external factors. Thus, the lowest values of the temperature are in the upper part of the container because they are under the action of intensive convective fluxes, and the highest ones are in the lower part. The higher temperatures in the lower part of the container are most likely due to the stagnant zone of atmospheric air at the foot of the container, which prevents heat dissipation from its surface by convection. Figures 6.24–6.27 illustrate the change in temperature of a concrete container over time, taking into account the effects of solar radiation. The figures show temperature fluctuations at the points located on the surface and inside the concrete body at different distances from the surface of the container (0, 0.07, 0.17, and 0.34 m) and at different levels (1.45, 2.90, and 4.35 m) from the concrete base of the container. It is seen that the surface of the container is exposed to the greatest heat during the day, and the amplitude of temperature fluctuations in the middle of the concrete body decreases with the distance from its surface. The highest temperature fluctuations are observed on the western side of the container and reach the depth of 17 cm; the least temperature fluctuations are observed on the northern side; their depth is not more than 10 cm from the surface of the container. At greater distances from the surface of the container, the daily temperature fluctuations are extremely insignificant and almost completely disappear at a depth of more than 30 cm. Thus, it can be concluded that insolation does not directly affect the spent nuclear fuel assemblies located in the storage container; however, it affects the temperature of the concrete storage container. The proposed method for determining the maximum temperature of fuel assemblies allowed analyzing the safety of dry storage of spent fuel by the criterion of maximum temperature. This method allows significantly reducing the level of conservatism, as not just the maximum temperature in the storage cask is determined but also the maximum temperature of each fuel element, taking into account the effects of external factors.

6.3 Thermal Condition of Containers for Dry Storage

249

(a)

(b)

(c) Figure 6.24 Time dependence of the concrete container temperature at different distances on the surface (western side). (a) 1.45 m from the ground. (b) 2.9 m from the ground. (c) 4.35 m from the ground.

250

Thermal Safety Criteria for Dry Storage of Spent Nuclear Fuel

(a)

(b)

(c) Figure 6.25 Time dependence of the concrete container temperature at different distances on the surface (northern side). (a) 1.45 m from the ground. (b) 2.9 m from the ground. (c) 4.35 m from the ground.

6.3 Thermal Condition of Containers for Dry Storage

251

(a)

(b)

(c) Figure 6.26 Time dependence of the concrete container temperature at different distances on the surface (northern side) (eastern side).

252

Thermal Safety Criteria for Dry Storage of Spent Nuclear Fuel

(a)

(b)

(c) Figure 6.27 Time dependence of the concrete container temperature at different distances on the surface (southern side).

References

253

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Index

Destruction of the fuel cladding, 111, 200 Dry storage, 1, 2, 5, 23, 93, 94, 116, 121, 194, 200–202, 238, 242, 248, 253–255

A

axial offset, 9, 12, 17, 18, 26, 105, 145, 147, 148, 151, 155, 195 B

Biot criteria, 62, 66, 87 Burn-up, 1–5, 7–12, 24, 98, 103, 105–114, 142–144, 156–158 Burn-Up Credit, 5, 24

E

equivalent thermal conductivity, 210, 214, 221, 222, 226, 227, 254

C

F

Conjugate heat transfer, 202, 203, 205, 207–211, 221 control program, 123–130, 132– 138, 141, 142, 145, 147, 148, 151, 160 control rod, 103, 124, 125, 130, 133, 134, 136, 138–143, 145, 151, 154, 155, 160, 193 Control Rod Drive Mechanism, 103, 124, 125, 130, 134, 136, 138–143, 145, 151, 155, 160 coolant temperature, 7, 11, 19, 91, 106, 122–128, 130–134, 136– 138, 140, 141, 193, 194 Critical heat flux, 20, 21, 106 Criticality, 5, 24, 107, 108

fatigue, 103–105, 108, 113, 115, 197 Fission, 2–4, 13, 23, 24, 105, 109, 112, 121 Fourier number, 22, 57, 65, 66, 78, 79 Fuel assembly, 1–3, 5, 9, 19, 20, 93, 108, 159–162, 172, 173, 189, 193, 194, 200, 202, 210, 216, 217, 220, 224, 234–237, 239, 244, 245 fuel campaign, 6, 18, 21, 101–103, 107, 108, 141 fuel cladding integrity, 1 Fuel enrichment, 108 Fuel fragmentation, 105, 111

D

I

Deformation, 97, 98, 108, 113–115, 117, 197

inverse heat conduction problem, 221

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loss-of-coolant accident, 111, 118 N

nonstationary heat transfer, 55, 59, 62, 69–75, 78, 83–86, 88, 89 O

Oxidation, 15, 100, 105, 106, 109, 111, 112 P

pressurized water reactor, 12 R

reactivity coefficient, 7, 96, 106, 107 reactor capacity, 7–9, 102, 103, 121, 125, 138, 141 reactor core, 2, 5, 7–11, 19, 21, 90, 91, 96–98, 101, 103, 106, 107, 122–128, 130–132, 134, 136– 138, 140–143, 145, 151–157, 159–162, 173, 174, 185, 193, 194 relative temperature, 65, 67–72, 81

reliability, 6, 7, 9, 10, 13, 17, 19, 21–23, 34, 93, 98, 100, 101, 103, 113, 117, 155, 156, 160, 194 residual heat, 1, 200, 202 S

safety, 1–3, 5, 9, 11–15, 18, 21– 25, 29, 35, 80, 83, 91–101, 105, 106, 108, 110, 112, 115– 118, 130, 133, 142, 143, 148, 150, 156, 181, 196–198, 200– 202, 224, 225, 228, 232, 236, 238, 242, 244, 248, 253, 255 spent nuclear fuel, 1, 2, 23, 24, 39, 94, 116, 121, 197–202, 205, 206, 210, 213, 216, 221, 223– 228, 233, 238, 239, 242, 245, 248, 253–255 strain, 98, 103, 104, 108–110, 113, 115, 118, 181–189, 192, 194, 197, 203 subcriticality, 1, 2, 11, 107 T

technical system, 29, 30

About the Authors

Maksymov Maksym was born in June 9, 1964 in Odessa. He graduated from Odessa Polytechnic Institute in 1986 with a degree in nuclear power plants. In 1990 he defended his PhD dissertation related to the dynamic modes of operation of nuclear power plants with WWER-1000. After defending his dissertation doctor of science in 2000, which was related to the automation of refuelling of nuclear fuel at nuclear power plants with WWER-1000, he was elected professor at the Department of Nuclear Power Plants. He currently holds the position of Head of the Department of Computer Automation Technologies. Scientific interest is devoted to models and methods that provide automated control of the reliability and efficiency of operation of a nuclear power plant depending on the modes of its operation. Supervised of training four doctors of sciences and twelve PhD. Alyokhina Svitlana was born in July 31, 1981 in Kharkiv (Ukraine). After graduating from the Kharkiv National University of Radio Electronics in 2003, in A.Podgorny Institute of Mechanical Engineering Problems of the National Academy of Sciences of Ukraine she studied thermal physical processes in energy equipment of power plants and defended her PhD in 2008. Defended her doctor of science dissertation: “Scientific Bases of the Thermal Safety of the Spent Nuclear Fuel Dry Storage” in 2019. Svitlana Alyokhina is senior scientific researcher in A.Podgorny Institute of Mechanical Engineering Problems of the National Academy of Sciences of Ukraine and professor in V.N. Karazin Kharkiv National University. She has more than 15 years of scientific experience in spent nuclear fuel management. Her research covers thermal aspects of safe interim storage of the spent nuclear fuel from power reactors. Brunetkin Oleksandr was born in January 21, 1958 in Chelyabinsk. After graduating from the Chelyabinsk Polytechnic Institute in 1981, he studied

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the dynamics of the thermophysical properties of chemically reacting gas mixtures, water hammer in pipes in an alternating field of mass forces and defended his PhD in 1990. Further scientific research was continued at the Odessa National Polytechnic University. The range of scientific interests is determined by the problems of the theory of dimension lessness and similarity, as well as unsteady heat transfer. Methods for solving nonlinear differential equations for solving control problems are investigated. Defended his doctor of science dissertation: “Models and methods of mathematical support for automated systems controlling the process of using variable composition fuels” in 2018. Occupies the post of professor of the Department of Computer Automation Technologies.