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A few words about the third extended edition
A few words about the second extended edition
Summary
Table of contents
Nomenclature
Introduction
References
1 Mass conservation
1.1 Introduction
1.2 Basic definitions
1.3 Non-structured and structured fields
1.4 Slattery and Whitaker's local spatial averaging theorem
1.5 General transport equation (Leibnitz rule)
1.6 Local volume-averaged mass conservation equation
1.7 Time average
1.8 Local volume-averaged component conservation equations
1.9 Local volume- and time-averaged conservation equations
1.10 Conservation equations for the number density of particles
1.11 Implication of the assumption of mono-dispersity in a cell
1.11.1 Particle size spectrum and averaging
1.11.2 Cutting of the lower part of the spectrum due to mass transfer
1.11.3 The effect of the averaging on the effective velocity difference
1.12 Stratified structure
1.13 Final remarks and conclusions
References
2 Momentums conservation
2.1 Introduction
2.2 Local volume-averaged momentum equations
2.2.1 Single-phase momentum equations
2.2.2 Interface force balance (momentum jump condition)
2.2.3 Local volume averaging of the single-phase momentum equation
2.3 Rearrangement of the surface integrals
2.4 Local volume average and time average
2.5 Dispersed phase in laminar continuum - pseudo turbulence
2.6 Viscous and Reynolds stresses
2.7 Non-equal bulk and boundary layer pressures
2.7.1 Continuous interface
2.7.1.1 3D flows
2.7.1.2 Stratified flow in horizontal or inclined rectangular channels
2.7.1.3 Stratified flow in horizontal or inclined pipes
2.7.2 Dispersed interface
2.7.2.1 General
2.7.2.2 Virtual mass force
2.7.2.3 Form drag and stagnation pressure force
2.7.2.4 Lift force
2.7.2.5 Interfacial structure forces
2.7.2.6 Force in the wall boundary layer
2.7.2.7 Force causing turbulent diffusion
2.7.2.8 Force causing rejection of droplet deposition at the wall
2.8 Working form for dispersed and continuous phase
2.9 General working form for dispersed and continuous phases
2.10 Some practical simplifications
2.11 Conclusion
Appendix 2.1
Appendix 2.2
Appendix 2.3
References
3 Derivatives for the equations of state
3.1 Introduction
3.2 Multi-component mixtures of miscible and non-miscible components
3.2.1 Computation of partial pressures for known mass concentrations, system pressure and temperature
3.2.2 Partial derivatives of the equation of state
3.2.3 Partial derivatives in the equation of state
3.2.4 Chemical potential
3.2.4.1 Gibbs function
3.2.4.2 Definition of the chemical equilibrium
3.2.4.3 Partial pressures of perfect fluid compounds in chemical equilibrium
3.2.4.4 Phase equilibrium
3.2.4.5 Equilibrium of gas solution in water
3.2.5 Partial derivatives in the equation of state
3.3 Mixture of liquid and microscopic solid particles of different chemical substances
3.3.1 Partial derivatives in the equation of
3.3.2 Partial derivatives in the equation of state
3.4 Single-component equilibrium fluid
3.4.1 Superheated vapor
3.4.2 Reconstruction of equation of state by using a limited amount of data available
3.4.2.1 Constant thermal expansion coefficient and isothermal compressibility coefficient
3.4.2.2 Properties known only at given a pressure as temperature functions
3.4.2.3 Constant density approximations
3.4.2.4 Perfect gas approximations
3.4.3 Vapor-liquid mixture in thermodynamic equilibrium
3.4.4 Liquid-solid mixture in thermodynamic equilibrium
3.4.5 Solid phase
3.5 Extension state of liquids
Appendix 3.1 Application of the theory to steam-air mixtures
Appendix 3.2 Useful references for computing properties of single constituents
Appendix 3.3 Useful definitions and relations between thermodynamic quantities
References
4 On the variety of notations of the energy conservation for single-phase flow
4.1 Introduction
4.2 Mass and momentum conservation,energy conservation
4.3 Simple notation of the energy conservation equation
4.4 The entropy
4.5 Equation of state
4.6 Variety of notation of the energy conservation principle
4.6.1 Temperature
4.6.2 Specific enthalpy
4.7 Summary of different notations
4.8 The equivalence of the canonical forms
4.9 Equivalence of the analytical solutions
4.10 Equivalence of the numerical solutions?
4.10.1 Explicit first order method of characteristics
4.10.2 The perfect gas shock tube: benchmark for numerical methods
4.11 Interpenetrating fluids
4.12 Summary of different notations for interpenetrating fluids
Appendix 4.1 Analytical solution of the shock tube problem
Appendix 4.2 Achievable accuracy of the donor-cell method for single-phase flows
References
5 First and second laws of the thermodynamics
5.1 Introduction
5.2 Instantaneous local volume average energy equations
5.3 Dalton and Fick's laws, center of mass mixture velocity, caloric mixture properties
5.4 Enthalpy equation
5.5 Internal energy equation
5.6 Entropy equation
5.7 Local volume- and time-averaged entropy equation
5.8 Local volume- and time-averaged internal energy equation
5.9 Local volume- and time-averaged specific enthalpy equation
5.10 Non-conservative and semi-conservative forms of the entropy equation
5.11 Comments on the source terms in the mixture entropy equation
5.12 Viscous dissipation
5.13 Temperature equation
5.14 Second law of the thermodynamics
5.15 Mixture volume conservation equation
5.16 Linearized form of the source term for the temperature equation
5.17 Interface conditions
5.18 Lumped parameter volumes
5.19 Steady state
5.20 Final remarks
References
6 Some simple applications of the mass and energy conservation
6.1 Infinite heat exchange without interfacial mass transfer
6.2 Discharge of gas from a volume
6.3 Injection of inert gas in a closed volume initially filled with inert gas
6.4 Heat input in a gas in a closed volume
6.5 Steam injection in a steam-air mixture
6.6 Chemical reaction in a gas mixture in a closed volume
6.7 Hydrogen combustion in an inert atmosphere
6.7.1 Simple introduction to combustion kinetics
6.7.2 Ignition temperature and ignition concentration limits
6.7.3 Detonability concentration limits
6.7.4 The heat release due to combustion
6.7.5 Equilibrium dissociation
6.7.6 Source terms of the energy conservation of the gas phase
6.7.7 Temperature and pressure changes in a closed control volume; adiabatic temperature of the burned gases
References
7 Exergy of multi-phase mUlti-component systems
7.1 Introduction
7.2 The pseudo-exergy equation for single-fluid systems
7.3 The fundamental exergy equation
7.3.1 The exergy definition in accordance with Reynolds and Perkins
7.3.2 The exergy definition in accordance with Gouy(I'energie utilisable, 1889)
7.3.3 The exergy definition appropriate for estimation of the volume change work
7.3.4 The exergy definition appropriate for estimation of the technical work
7.4 Some interesting consequences of the fundamental exergy equation
7.5 Judging the efficiency of a heat pump as an example of application of the exergy
7.6 Three-fluid mUlti-component systems
7.7 Practical relevance
References
8 One-dimensional three-fluid flows
8.1 Summary of the local volume- and time-averaged conservation equations
8.2 Treatment of the field pressure gradient forces
8.2.1 Dispersed flows
8.2.2 Stratified flow
8.3 Pipe deformation due to temporal pressure change in the flow
8.4 Some simple cases
8.5 Slip model - transient flow
8.6 Slip model - steady state. Critical mass flow rate
8.7 Forces acting on the pipes due to the flow - theoretical basics
8.8 Relief valves
8.8.1 Introduction
8.8.2 Valve characteristics, model formulation
8.8.3 Analytical solution
8.8.4 Fitting the piecewise solution on two known position - time points
8.8.5 Fitting the piecewise solution on known velocity and position for a given time
8.8.6 Idealized valve characteristics
8.8.7 Recommendations for the application of the model in system computer codes
8.8.8 Some illustrations of the valve performance model
8.8.9 Nomenclature for Section 8.8
8.9 Pump model
8.9.1 Variables defining the pump behavior
8.9.2 Theoretical basics
8.9.3 Suter diagram
8.9.4 Computational procedure
8.9.5 Centrifugal pump drive model
8.9.6 Extension of the theory to multi-phase flow
Appendix 1: Chronological references to the subject critical two-phase flow
References
9 Detonation waves caused by chemical reactions or by melt-coolant interactions
9.1 Introduction
9.2 Single-phase theory
9.2.1 Continuum sound waves (Laplace)
9.2.2 Discontinuum shock waves (Rankine-Hugoniot)
9.2.3 The Landau and Liftshitz analytical solution for detonation in perfect gases
9.2.4 Numerical solution for detonation in closed pipes
9.3 Multi-phase flow
9.3.1 Continuum sound waves
9.3.2 Discontinuum shock waves
9.3.3 Melt-coolant interaction detonations
9.3.4 Similarity to and differences from the Yuen and Theofanous formalism
9.3.5 Numerical solution method
9.4 Detonation waves in water mixed with different molten materials
9.4.1 U02 water system
9.4.2 Efficiencies
9.4.3 The maximum coolant entrainment ratio
9.5 Conclusions
9.6 Practical significance
References
10 Conservation equations in general curvilinear coordinate systems
10.1 Introduction
10.2 Field mass conservation equations
10.3 Mass conservation equations for components inside the field - conservative form
10.4 Field mass conservation equations for components inside the field - non-conservative form
10.5. Particles number conservation equations for each velocity field
1 0.6 Field entropy conservation equations -conservative form
10.7 Field entropy conservation equations - non-conservative form
10.8 Irreversible power dissipation caused by the viscous forces
10.9 The non-conservative entropy equation in terms of temperature and pressure
10.10 The volume conservation equation
10.11 The momentum equations
10.12 The flux concept, conservative and semi-conservative forms
10.12.1 Mass conservation equation
10.12.2 Entropy equation
10.12.3 Temperature equation
10.12.4 Momentum conservation in the x-direction
10.12.5 Momentum conservation in the y-direction
10.12.6 Momentum conservation in the z-direction
10.13 Concluding remarks
References
11 Type of the system of PDEs
11.1 Eigenvalues, eigenvectors, canonical form
11.2 Physical interpretation
11.2.1 Eigenvalues and propagation velocity of perturbations
11.2.2 Eigenvalues and propagation velocity of harmonic oscillations
11.2.3 Eigenvalues and critical flow
References
12 Numerical solution methods for multi-phase flow problems
12.1 Introduction
12.2 Formulation of the mathematical problem
12.3 Space discretization and location of the discrete variables
12.4 Discretization of the mass conservation equations
12.5 First order donor-cell finite difference approximations
12.6 Discretization of the concentration equations
12.7 Discretization of the entropy equation
12.8 Discretization of the temperature equation
12.9. Physical significance of the necessary convergence condition
12.10. Implicit discretization of momentum equations
12.11 Pressure equations for IVA2 and IVA3 computer codes
12.12 A Newton-type iteration method for multi-phase flows
12.13 Integration procedure: implicit method
12.14 Time step and accuracy control
12.15 High order discretization schemes for convection-diffusion terms
12.15.1 Space exponential scheme
12.15.2 High order upwinding
12.15.3 Constrained interpolation profile (CIP) method
12.15.3.1 Exactly conservative scheme for transport equations in non-conservative form
12.15.3.2 Computing the weighted averages in the new time plane
12.15.3.3 Choice of the gradients
12.15.3.4 Phase discontinuity treated with CIP
12.16 Problem solution examples to the basics of the CIP method
12.16.1 Discretization concept
12.16.2 Second order constrained interpolation profiles
12.16.3 Third order constrained interpolation profiles
12.16.4 Fourth order constrained interpolation profiles
12.17 Pipe networks: some basic definitions
12.17.1 Pipes
12.17.2 Axis in the space
12.17.3 Diameters of pipe sections
12.17.4 Reductions
12.17.5 Elbows
12.17.6 Creating a library of pipes
12.17.7 Sub system network
12.17.8 Discretization of pipes
12.17.9 Knots
Appendix 12.1 Definitions applicable to discretization of the massconservation equations
Appendix 12.2 Discretization of the concentration equations
Appendix 12.3 Harmonic averaged diffusion coefficients
Appendix 12.4. Discretized radial momentum equation
Appendix 12.5 The Zi coefficients for Eq. (12.46)
Appendix 12.6 Discretization ofthe angular momentum equation
Appendix 12.7 Discretization of the axial momentum equation
Appendix 12.8 Analytical derivatives for the residual error of each equation with respect to the dependent variables
Appendix 12.9 Simple introduction to iterative methods for solution of algebraic systems
References
13 Numerical methods for multi-phase flow in curvilinear coordinate systems
13.1 Introduction
13.2 Nodes, grids, meshes, topology - some basic definitions
13.3 Formulation of the mathematical problem
13.4 Discretization of the mass conservation equations
13.4.1 Integration over a finite time step and finite control volume
13.4.2 The donor-cell concept
13.4.3 Two methods for computing the finite difference approximations of the contravariant vectors at the cell center
13.4.4 Discretization of the diffusion terms
13.4.4.1 General
13.4.4.2 Orthogonal coordinate systems
13.4.4.3 Off-diagonal diffusion terms in the general case
13.4.4.4 Final form of the finite volume concentration equation
13.5 Discretization of the entropy equation
13.6 Discretization of the temperature equation
13.7 Discretization of the particle number density equation
13.8 Discretization of the x momentum equation
13.9 Discretization of the y momentum equation
13.10 Discretization of the z momentum equation
13.11 Pressure-velocity coupling
13.12 Staggered x momentum equation
Appendix 13.1 Harmonic averaged diffusion coefficients
Appendix 13.2 Off-diagonal viscous diffusion terms of the x momentum
Appendix 13.3 Off-diagonal viscous diffusion terms of the y momentum equation
Appendix 13.4 Off-diagonal viscous diffusion terms of the z momentum equation
References
Appendix 1 Brief introduction to vector analysis
References
Appendix 2 Basics of the coordinate transformation theory
References
14 Visual demonstration of the method
14.1 Melt-water interactions
14.1.1 Cases 1 to 4
14.1.1.1 Case 1
14.1.1.2 Case 2
14.1.1.3 Case 3
14.1.1.4 Case 4
14.1.1.5 Model elements addressed in cases 1 to 4
14.1.1.6 Available file on CD
14.1.2 Cases 5, 6 and 7
14.1.2.1 Initial conditions
14.1.2.2 Driving instability for Case 5 and 6
14.1.2.3 Material relocation
14.1.2.4 Model elements addressed in cases 5,6 and 7
14.1.2.5 Available files on CD
14.1.3 Cases 8 to 10
14.1.3.1 Case 8
14.1.3.2 Case 9
14.1.3.3 Case 10
14.1.3.4 Model elements addressed in cases 8 to 10
14.1.3.5 Available files on CD
14.1.4 Cases 11 and 12
14.1.4.1 Case 11
14.1.4.2 Case 12
14.1.4.3 Model elements addressed in cases 11 and 12
14.1.4.4 Available files on CD
14.1.5 Case 13
14.1.5.1 Model elements addressed in case 13
14.1.5.2 Available files on CD
14.1.6 Case 14
14.1.6.1 Results of the computational analysis
14.1.6.2 Model elements addressed in case 14
14.1.6.3 Available files on CD
14.2 Pipe networks
14.2.1 Case 15
14.2.1.1 Model elements addressed in case 15
14.2.1.2 Available files on CD
14.3 3D steam-water interaction
14.3.1 Case 16
14.3.1.1 Model elements addressed in case 16
14.3.1.2 Available files on CD
14.4 Three dimensional steam-water interaction in presence of non-condensable gases
14.4.1 Case 17
14.4.1.1 Model elements addressed in case 17
14.4.1.2 Available files on CD
14.5 Three dimensional steam production in boiling water reactor
14.5.1 Case 18
14.5.1.1 Model elements addressed in case 18
14.5.1.2 Available files on CD
References
Index