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

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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
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