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# Modelling of Nuclear Reactor Multi-physics

## From Local Balance Equations to Macroscopic Models in Neutronics and Thermal-Hydraulics

- 1st Edition - November 18, 2019
- Author: Christophe Demazière
- Language: English
- Paperback ISBN:9 7 8 - 0 - 1 2 - 8 1 5 0 6 9 - 6
- eBook ISBN:9 7 8 - 0 - 1 2 - 8 1 5 0 7 0 - 2

Modelling of Nuclear Reactor Multiphysics: From Local Balance Equations to Macroscopic Models in Neutronics and Thermal-Hydraulics is an accessible guide to the advanced method… Read more

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Request a sales quote*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 levels, then at the core level, and for one-/two-phase flow transport and heat transfer are treated in depth in respective chapters. The coupling between neutron transport solvers and thermal-hydraulic solvers for coarse mesh macroscopic models is given particular attention in a dedicated chapter. The final chapter summarizes the main techniques presented in the book and their interrelation, then explores beyond state-of-the-art modelling techniques relying on more integrated approaches.

- Covers neutron transport, fluid dynamics, and heat transfer, and their interdependence, in one reference
- Analyses the emerging area of multi-physics and multi-scale reactor modelling
- Contains 71 short videos explaining the key concepts and 77 interactive quizzes allowing the readers to test their understanding

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 2.2.b Derivation of the neutron transport equation 2.3 Fluid dynamics 2.3.a Mathematical formalism 2.3.b Generic differential conservation laws 2.3.c Mass and momentum differential conservation equations 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 ConclusionsReferences 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 3.3 Resolving the energy dependence 3.4 One-dimensional micro-group pin cell calculations 3.4.a Introduction 3.4.b Transport correction 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 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 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 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 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 6.5 Conclusions References 7. Conclusions 7.1 Summary 7.2 Outlook References Index

- No. of pages: 368
- Language: English
- Edition: 1
- Published: November 18, 2019
- Imprint: Academic Press
- Paperback ISBN: 9780128150696
- eBook ISBN: 9780128150702

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### Christophe Demazière

*Modelling of Nuclear Reactor Multi-physics*on ScienceDirect