Modelling of Nuclear Reactor Multi-physics: From Local Balance Equations to Macroscopic Models in Neutronics and Thermal-Hydraulics

Description

Modelling of Nuclear Reactor Multiphysics: From Local Balance Equations to Macroscopic Models in Neutronics and Thermal-Hydraulics is an accessible guide to the advanced methods used to model nuclear reactor systems. The book addresses the frontier discipline of neutronic/thermal-hydraulic modelling of nuclear reactor cores, presenting the main techniques in a generic manner and for practical reactor calculations. The modelling of nuclear reactor systems is one of the most challenging tasks in complex system modelling, due to the many different scales and intertwined physical phenomena involved. The nuclear industry as well as the research institutes and universities heavily rely on the use of complex numerical codes. All the commercial codes are based on using different numerical tools for resolving the various physical fields, and to some extent the different scales, whereas the latest research platforms attempt to adopt a more integrated approach in resolving multiple scales and fields of physics. The book presents the main algorithms used in such codes for neutronic and thermal-hydraulic modelling, providing the details of the underlying methods, together with their assumptions and limitations. Because of the rapidly expanding use of coupled calculations for performing safety analyses, the analysists should be equally knowledgeable in all fields (i.e. neutron transport, fluid dynamics, heat transfer). The first chapter introduces the book's subject matter and explains how to use its digital resources and interactive features. The following chapter derives the governing equations for neutron transport, fluid transport, and heat transfer, so that readers not familiar with any of these fields can comprehend the book without difficulty. The book thereafter examines the peculiarities of nuclear reactor systems and provides an overview of the relevant modelling strategies. Computational methods for neutron transport, first at the cell and assembly lev Front Cover Modelling of Nuclear Reactor Multi-physics Modelling of Nuclear Reactor Multi-physics: From Local Balance Equations to Macroscopic Models in Neutronics and Thermal-Hydraulics Copyright Dedication Contents List of Abbreviations 1 - Introduction 1.1 Topics covered in the book 1.2 Structure of the book 1.2.a Contents 1.2.b Pedagogical approach 1.3 Notations and conventions used in the book 1.4 Reminder about some useful mathematical concepts 1.4.a Calculus on scalars, vectors and tensors 1.4.b Spherical coordinates and solid angles 1.4.c Gauss divergence theorems References 2 - Transport phenomena in nuclear reactors 2.1 Nuclear reactors as multi-physics and multi-scale systems 2.1.a Multi-physics aspects 2.1.b Multi-scale aspects 2.2 Neutron transport 2.2.a Introduction Elementary concepts in neutron physics Useful quantities in reactor physics Physical significance of the neutron current density vector 2.2.b Derivation of the neutron transport equation Integro-differential form Integral form Characteristic form 2.3 Fluid dynamics 2.3.a Mathematical formalism Temporal and spatial derivatives Leibnitz rules 2.3.b Generic differential conservation laws 2.3.c Mass and momentum differential conservation equations Mass conservation equation Momentum conservation equation 2.4 Heat transfer 2.4.a Heat transfer by conduction 2.4.b Heat transfer by convection 2.5 Overview of the modelling strategies 2.6 Deterministic and macroscopic modelling of nuclear systems 2.6.a Equations governing the neutron flux 2.6.b Equations governing the temperature and flow fields 2.6.c Coupling between the neutron kinetic and thermal-hydraulic modellings 2.7 Conclusions References 3 - Neutron transport calculations at the cell and assembly levels 3.1 Representation of the energy dependence 3.1.a Multi-group formalism 3.1.b Nuclear data libraries 3.2 Treatment of resonances 3.2.a Introduction 3.2.b Neutron slowing-down without absorption 3.2.c Neutron slowing-down with absorption Resonance modelling in homogeneous media Resonance modelling in heterogeneous media using the equivalence method Resonance modelling in heterogeneous media using the subgroup method 3.3 Resolving the energy dependence 3.4 One-dimensional micro-group pin cell calculations 3.4.a Introduction 3.4.b Transport correction Transport correction in the mono-energetic case Transport correction in the poly-energetic case 3.4.c Method of collision probabilities 3.4.d Properties of the probabilities 3.4.e Application of the method of collision probabilities 3.4.f Rational approximation 3.5 Two-dimensional macro-group lattice calculations 3.5.a Introduction 3.5.b Method of characteristics 3.5.c Discrete ordinates (SN) method 3.5.d Interface current method 3.5.e Acceleration methods Acceleration by coarse-mesh re-balancing Diffusion synthetic acceleration 3.6 Criticality spectrum calculations 3.6.a Introduction 3.6.b Properties of integral operators in infinite and homogeneous media 3.6.c Integral operators in critical systems 3.6.d Homogeneous B1 method 3.6.e Homogeneous P1 method 3.6.f Fundamental mode method 3.7 Cross-section homogenization and condensation 3.8 Depletion calculations 3.9 Cross-section preparation for core calculations 3.10 Conclusions References 4 - Neutron transport calculations at the core level 4.1 Angular discretization of the neutron transport equation 4.1.a Spherical harmonics (PN) method 4.1.b Diffusion theory 4.1.c Simplified PN method (SPN) 4.1.d Boundary conditions Free surface boundary condition Reflective boundary condition Albedo boundary condition 4.2 Spatial discretization of the neutron transport equation 4.2.a Introduction 4.2.b Finite difference methods 4.2.c Nodal methods 4.2.d Finite elements 4.3 Determination of the steady-state core-wise solution 4.3.a Introduction 4.3.b Direct methods 4.3.c Iterative methods Stationary methods Krylov subspace methods 4.4 Determination of the non-steady-state core-wise solution 4.4.a Introduction 4.4.b Analysis of the balance equations with respect to the prompt neutrons 4.4.c Analysis of the balance equations with respect to the delayed neutrons 4.5 Conclusions References 5 - One-/two-phase flow transport and heat transfer 5.1 Tools required for flow transport modelling 5.1.a Introduction 5.1.b Two-phase flow regimes 5.1.c Mathematical tools Descriptors in two-phase flow dynamics Time-averaging properties Space-averaging properties Space-time-averaging properties 5.2 Derivation of the space- and time-averaged conservation equations for flow transport 5.2.a Introduction 5.2.b Space-averaging of the local conservation equations 5.2.c Time-averaging of the space-averaged conservation equations 5.2.d Equations to be solved 5.3 Flow models 5.3.a Two-fluid model 5.3.b Mixture models with specified drift velocities 5.3.c Homogeneous equilibrium model 5.4 Spatial and temporal discretizations of the flow models 5.5 Modelling of heat conduction in solid structures 5.6 Conclusions References 6 - Neutronic/thermal-hydraulic coupling 6.1 Introduction 6.2 Modelling of the dependencies of the nuclear material data 6.2.a Introduction 6.2.b Data functionalization on base and partial values 6.2.c Tree-leaf representation 6.2.d Polynomial fitting 6.3 Spatial coupling 6.3.a Thermal-hydraulic to neutronic coupling 6.3.b Neutronic to thermal-hydraulic coupling 6.3.c Coupling coefficients 6.4 Temporal coupling 6.4.a Introduction 6.4.b Operator Splitting approaches 6.4.c Integrated approaches Principle Pre-conditioning Initialization and globalization Perturbation ε 6.5 Conclusions References 7 - Conclusions 7.1 Summary 7.2 Outlook References Index A B C D E F G H I J K L M N O P Q R S T U V W Back Cover