Fractional-order Modeling of Nuclear Reactor: From Subdiffusive Neutron Transport to Control-oriented Models (eBook)

Vishwesh A. Vyawahare is a faculty in the Department of Electronics Engineering at Ramrao Adik Institute of Technology, Nerul, Navi Mumbai, India. He received his Master of Engineering degree in Control Systems from the Government College of Engineering, Pune, India in 2004, followed by a PhD in Systems and Control Engineering from the Indian Institute of Technology Bombay, Mumbai, India, in 2012. His doctoral work focused on the fractional-order modeling of nuclear reactors. His current research areas include modeling and control using fractional-order, complex-order and variable-order calculus. Paluri S. V. Nataraj is a faculty in the Systems and Control Engineering Group at the Indian Institute of Technology Bombay (IIT Bombay), Mumbai, India. He received his PhD in Process Dynamics and Control from the Indian Institute of Technology Madras, Chennai, India in 1987. He subsequently worked at the CAD Centre at IIT Bombay for one and a half years before joining the Systems and Control Engineering Group at IIT Bombay in 1988, where he has been involved in teaching and research for the past 28 years. His current research interests are in the areas of fractional-order modeling and control, global optimization, parallel computing, reliable computing, and robust control.

Preface 6
Acknowledgements 9
Contents 11
About the Authors 14
Acronyms 15
1 Fractional Calculus 16
1.1 Introduction 16
1.2 Special Functions in Fractional Calculus 18
1.2.1 Gamma Function 18
1.2.2 Mittag-Leffler Function 18
1.3 Fractional-order Integrals and Derivatives: Definitions 20
1.4 Fractional-order Differential Equations 23
1.5 Fractional-order Systems 24
1.6 Chapter Summary 25
2 Introduction to Nuclear Reactor Modeling 26
2.1 Introduction 26
2.2 Nuclear Reactor Theory 27
2.3 Slab Reactor 28
2.4 Mathematical Modeling of Nuclear Reactor 29
2.4.1 Modeling of Neutron Transport 29
2.4.2 Point Reactor Kinetics Model 32
2.4.3 Modeling of Large Commercial Reactors 34
2.4.4 Modeling Neutron Transport as Random Walk 34
2.5 Anomalous Diffusion 35
2.5.1 Continuous-Time Random Walk 36
2.6 Fractional Calculus Applications in Nuclear Reactor Theory 37
2.6.1 Analysis of FO Neutron Transport Equation 38
2.6.2 FO Modeling of Neutron Transport and Analysis of Nuclear Reactor 38
2.6.3 Development and Analysis of FO Point Reactor Kinetics Model 39
2.6.4 Design of FO Controller for Nuclear Reactor 40
2.7 Chapter Summary 41
3 Development and Analysis of Fractional-order Neutron Telegraph Equation 42
3.1 Introduction 42
3.2 Motivation 44
3.3 Derivation of FO Neutron Telegraph Equation Model 46
3.4 Analysis of Mean-Squared Displacement 49
3.4.1 General Diffusion Case 50
3.4.2 IO Neutron Diffusion Equation (INDE) 52
3.4.3 IO Neutron Telegraph Equation (INTE) 52
3.4.4 FO Neutron Diffusion Equation (FNDE) 53
3.4.5 FO Neutron Telegraph Equation (FNTE) 57
3.4.6 FO Neutron Telegraph Equation by Paredes (PNTE) 61
3.5 Solution Using Separation of Variables Method 63
3.5.1 Solution of IO Neutron Diffusion Equation 65
3.5.2 Solution of IO Neutron Telegraph Equation 67
3.5.3 Solution of FO Neutron Diffusion Equation 71
3.5.4 Solution of FO Neutron Telegraph Equation 76
3.6 Chapter Summary 86
4 Development and Analysis of Fractional-order Point Reactor Kinetics Model 87
4.1 Introduction 87
4.2 Point Reactor Kinetics Model 88
4.2.1 Survey of FPRK Models 89
4.2.2 Steps for Development of Point Reactor Kinetics Model 89
4.3 Derivation of FPRK Model 90
4.3.1 Observations 90
4.3.2 Separation of Variables Method for FO Neutron Diffusion Equation 90
4.3.3 Longtime Behavior 92
4.3.4 Derivation 94
4.3.5 Discussion 97
4.4 Solution of FPRK Model with One Effective Delayed Group 97
4.5 Chapter Summary 105
5 Further Developments Using Fractional-order Point Reactor Kinetics Model 106
5.1 Introduction 106
5.2 Fractional Inhour Equation 107
5.3 Inverse FPRK Model 112
5.3.1 Reactivity Insertion for Exponential Rise of Power 114
5.3.2 Reactivity Insertion for Sinusoidal Power Variation 116
5.3.3 Reactivity After a Positive Power Transient 120
5.4 Constant Delayed Neutron Production Rate Approximation 122
5.5 Prompt Jump Approximation 127
5.6 Zero Power Transfer Function of the Reactor 129
5.6.1 Derivation of ZPFTF Using the Small Amplitude Approximation 130
5.6.2 Analysis of ZPFTF with One Effective Delayed Group 132
5.7 Chapter Summary 135
6 Development and Analysis of Fractional-order Point Reactor Kinetics Models with Reactivity Feedback 137
6.1 Introduction 137
6.2 Modeling of Reactivity Feedback in a Reactor 139
6.2.1 Reactivity Feedback Mechanism 139
6.2.2 Models of Temperature Feedback of Reactivity 141
6.3 Fractional-order Nordheim--Fuchs Model 143
6.4 FPRK Model with Reactivity Feedback (Below Prompt Critical) 152
6.4.1 Step Reactivity Insertion 154
6.4.2 Sinusoidal Reactivity Insertion 158
6.5 Linearized FO Model with Reactivity Feedback 162
6.6 Chapter Summary 176
7 Development and Analysis of Fractional-order Two-Group Models and Fractional-order Nodal Model 179
7.1 Introduction 179
7.2 IO Two-Group Diffusion Model 182
7.3 Fractional-order Two-Group Telegraph-Subdiffusion Model 184
7.3.1 Motivation 184
7.3.2 Derivation 185
7.4 Fractional-order Two-Group Subdiffusion Model 188
7.5 Fractional-order Nodal Model 189
7.6 Chapter Summary 192
Appendix Fractional Second-order Adams--Bashforth--Moulton (ABM) Method 194
References 196
Index 209