THE CO-CURRENT STRATIFIED FLOW OF TWO IMMISCIBLE LIQUIDS IN A RECTANGULAR CONDUIT
MASS CONSERVATION…… 1 1 INTRODUCTION 1 2 BASIC DEFINITIONS 1 3 NONSTRUCTURED AND STRUCTURED FIELDS 1 4 THE SLATTERY AND WHITAKER 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 MONODISPERSITY 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 AVERAGING ON THE EFFECTIVE VELOCITY DIFFERENCE 1 12 STRATIFIED STRUCTURE 1 13 FINAL REMARKS AND CONCLUSIONS REFERENCES CONSERVATION OF MOMENTUM…… 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 A LAMINAR CONTINUUM – PSEUDO TURBULENCE 2 6 VISCOUS AND REYNOLDS STRESSES 2 7 NONEQUAL BULK AND BOUNDARY LAYER PRESSURES 2 7 1 CONTINUOUS INTERFACE 2 7 2 DISPERSED INTERFACE 2 8 WORKING FORM FOR THE DISPERSED AND CONTINUOUS PHASE 2 9 GENERAL WORKING FORM FOR DISPERSED AND CONTINUOUS PHASES 2 10 SOME PRACTICAL SIMPLIFICATIONS 2 11 CONCLUSION 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 ( ) MAX 2,, , , I P T C ? ? = 3 2 3 PARTIAL DERIVATIVES IN THE EQUATION OF STATE ( ) MAX 2,, , , I T T P C ? = , WHERE , , S H E ? = 3 2 4 CHEMICAL POTENTIAL 3 2 5 PARTIAL DERIVATIVES IN THE EQUATION OF STATE ( ) MAX 2,, , , I P C ? ? ? = , WHERE , , S H E ? = 3 3 MIXTURE OF LIQUID AND MICROSCOPIC SOLID PARTICLES OF DIFFERENT CHEMICAL SUBSTANCES 3 3 1 PARTIAL DERIVATIVES IN THE EQUATION OF STATE ( ) MAX 2,, , , I P T C ? ? = 3 3 2 PARTIAL DERIVATIVES IN THE EQUATION OF STATE ( ) MAX 2,, , , I T T P C ? = WHERE , , H E S ? = 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 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 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 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 SOME SIMPLE APPLICATIONS OF 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 HEAT REMOVAL FROM A CLOSED VOLUME CONTAINING EQUILIBRIUM TWO-PHASE MIXTURE 6 7 CHEMICAL REACTION IN A GAS MIXTURE IN A CLOSED VOLUME 6 8 HYDROGEN COMBUSTION IN AN INERT ATMOSPHERE 6 8 1 SIMPLE INTRODUCTION TO COMBUSTION KINETICS 6 8 2 IGNITION TEMPERATURE AND IGNITION CONCENTRATION LIMITS 6 8 3 DETONABILITY CONCENTRATION LIMITS 6 8 4 THE HEAT RELEASE DUE TO COMBUSTION 6 8 5 EQUILIBRIUM DISSOCIATION 6 8 6 SOURCE TERMS OF THE ENERGY CONSERVATION OF THE GAS PHASE 6 8 7 TEMPERATURE AND PRESSURE CHANGES IN A CLOSED CONTROL VOLUME; ADIABATIC TEMPERATURE OF THE BURNED GASES 6 9 CONSTITUENTS OF SODIUM VAPOR 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 (L’ÉNERGIE 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 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 MULTIPHASE FLOW 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 DISCONTINUOUS 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 XXXVI TABLE OF CONTENTS 9 4 DETONATION WAVES IN WATER MIXED WITH DIFFERENT MOLTEN MATERIALS 9 4 1 UO2 WATER SYSTEM 9 4 2 EFFICIENCIES 9 4 3 THE MAXIMUM COOLANT ENTRAINMENT RATIO 9 5 CONCLUSIONS 9 6 PRACTICAL SIGNIFICANCE 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 10 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 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 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 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 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 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 VISUAL DEMONSTRATION OF THE METHOD 14 1 MELT-WATER INTERACTIONS 14 1 1 CASES 1 TO 4 14 1 2 CASES 5, 6 AND 7 14 1 3 CASES 8 TO 10 14 1 4 CASES 11 AND 12 14 1 5 CASE 13 14 1 6 CASE 14 14 2 PIPE NETWORKS 14 2 1 CASE 15 14 3 3D STEAM-WATER INTERACTIONS 14 3 1 CASE 16 14 4 THREE-DIMENSIONAL STEAM-WATER INTERACTION IN PRESENCE OF NON-CONDENSABLE GASES 14 4 1 CASE 17 14 5 THREE DIMENSIONAL STEAM PRODUCTION IN BOILING WATER REACTOR 14 5 1 CASE 18