Dr. William T. Sha's longstanding technical achievements and outstanding contributions in the nuclear reactor field are well known both in the United States and abroad. As the director of the Argonne National Laboratory (ANL), I had the privilege of working with Dr. Sha for more than a decade during which he markedly enhanced the reputation of ANL's international reactor programs as the director of the Analytical Thermal Hydraulic Research Program and Multiphase Flow Research Institute. Over many years, his rare combination of analytical rigor and creative insight allowed him to earn international recognition as a leader in the field of thermal hydraulics in both theoretical formulation and reactor design and safety analysis.

Figures and Table page xv

Foreword by Alan Schriesheim xix

Foreword by Wm. Howard Arnold xxi

Foreword by Charles Keiber xxiii

Nomenclature xxvii

Preface xxxv

Acknowledgments xliii

1 Introduction 1

      1.1 Background information about multiphase flow 2

      1.2 Significance of phase configurations in multiphase flow 6

      1.3 Need for universally accepted formulation for multiphase flow conservation equations 8

2 Averaging relations 12

      2.1 Preliminaries 13

      2.2 Local volume average and intrinsic volume average 14

      2.3 Local area average and intrinsic area average 15

      2.4 Local volume averaging theorems and their length-scale restrictions 17

      2.5 Conservative criterion of minimum size of characteristic length of local averaging volume 21

3 Phasic conservation equations and interfacial balance equations 23

      3.1Phasic conservation equations 23

      3.2 Interfacial balance equations 25

4 Local volume-averaged conservation equations and interfacial balance equations 27

      4.1 Local volume-averaged mass conservation equation of a phase and its interfacial balance equation 27

      4.2 Local volume-averaged linear momentum equation and its interfacial balance equation 29

      4.3 Local volume-averaged total energy equation and its interfacial balance equation 33

      4.4 Local volume-averaged internal energy equation and its interfacial balance equation 36

      4.5 Local volume-averaged enthalpy equation and its interfacial balance equation 38

      4.6 Summary of local volume-averaged conservation equations 41

      4.6.1Local volume-averaged mass conservation equation 41

      4.6.2 Local volume-averaged linear momentum conservation equation 42

      4.6.3 Local volume-averaged energy conservation equations 43

      4.6.3.1 In terms of total energy E_(k), E_(κ) = u_(κ) + ½U_(κ)·U_(κ) 43

      4.6.3.2 In terms of internal energy u_(κ) 44

      4.6.3.3 In terms of enthalpy h_(κ) 45

      4.7 Summary of local volume-averaged interfacial balance equations 45

      4.7.1 Local volume-averaged interfacial mass balance equation 45

      4.7.2 Local volume-averaged interfacial linear momentum balance equation 46

      4.7.3 Local volume-averaged interfacial energy balance equation 46

      4.7.3.1 Total energy balance (capillary energy ignored) 47

      4.7.3.2 Internal energy balance (dissipation and reversible work ignored) 47

      4.7.3.3 Enthalpy balance (capillary energy ignored) 47

5 Time averaging of local volume-averaged conservation equations or time-volume-averaged conservation equations and interfacial balance equations 48

      5.1 Basic postulates 48

      5.2 Useful observation without assuming υ'κ=0 53

      5.3 Time-volume-averaged mass conservation equation 54

      5.4 Time-volume-averaged interfacial mass balance equation 59

      5.5 Time-volume-averaged linear momentum conservation equation 60

      5.6 Time-volume-averaged interfacial linear momentum balance equation 73

      5.7 Time-volume-averaged total energy conservation equation 75

      5.8 Time-volume-averaged interfacial total energy balance equation (capillary energy ignored) 88

      5.9 Time-volume-averaged internal energy conservation equation 90

      5.10 Time-volume-averaged interfacial internal energy balance equation 100

      5.11 Time-volume-averaged enthalpy conservation equation 101

      5.12 Time-volume-averaged interfacial enthalpy balance equation (capillary energy ignored) 109

      5.13 Summary of time-volume-averaged conservation equations 110

      5.13.1 Time-volume-averaged conservation of mass equation 110

      5.13.2 Time-volume-averaged linear momentum conservation equation 111

      5.13.3 Time-volume-averaged total energy conservation equation 112

      5.13.4 Time-volume-averaged internal energy conservation equation 113

      5.13.5 Time-volume-averaged enthalpy conservation equation 113

      5.14 Summary of time-volume-averaged interfacial balance equations 114

      5.14.1 Time-volume-averaged interfacial mass balance equation 114

      5.14.2 Time-volume-averaged interfacial linear momentum balance equation 115

      5.14.3 Time-volume-averaged interfacial total energy balance equation 115

      5.14.4 Time-volume-averaged interfacial internal energy balance equation 115

      5.14.5 Time-volume-averaged interfacial enthalpy balance equation 116

6 Time averaging in relation to local volume averaging and time-volume averaging versus volume-time averaging 117

      6.1 Time averaging in relation to local volume averaging 117

      6.2 Time-volume averaging versus volume-time averaging 121

7 Novel porous media formulation for single phase and single phase with multicomponent applications 125

      7.1 COMMIX code capable of computing detailed microflow fields with fine computational mesh and high-order differencing scheme 128

      7.1.1 Case (1): Von Karmann vortex shedding analysis 128

      7.1.2 Case (2): Shear-driven cavity flow analysis 134

      7.1.3 Some observations about higher-order differencing schemes 138

      7.2 COMMIX code capable of capturing essential both macroflow field and macrotcmperature distribution with a coarse computational mesh 140

      7.2.1 Case (6): Natural convection phenomena in a prototypical pressurized water reactor during a postulated degraded core accident 140

      7.2.1.1 Heat transfer 144

      7.2.1.2 Natural convection patterns 145

      7.2.1.3 Temperature distribution 145

      7.2.2 Case (7): Analysis of large-scale tests for AP-600 passive containment cooling system 149

      7.2.2.1Circumferential temperature distribution 152

      7.2.2.2 Condensation and evaporation rate 154

      7.2.2.3 Air partial pressure and containment pressure 155

      7.2.2.4 Condensation and evaporating film thickness 156

      7.2.2.5 Temperature distributions at various locations 156

      7.3 Condusion 158

8 Discut^ion and concluding remarks 160

      8.Time averaging of local volume-averaged phasic conservation equations 161

      8.1.1 Length-scale restriction for the local volume average 162

      8.1.2 Time scale restriction in the time averaging 163

      8.1.3 Time-volume-averaged conservation equations are in differential-integral form 165

      8.1.4 Unique features of time-volume-averaged conservation equations 166

      8.2 Novel porous media formulation 168

      8.2.1 Single-phase implementation 170

      8.2.2 Multiphase flow 171

      8.3 Future research 172

      8.4 Summary 177

APPENDIX A: Staggered-grid computational system 179

APPENDIX B: Physical interpretation of ▽α_(κ)=-υ~(-1)∫∧_(κ)n_(κ)dA with γ_(υ) 184

APPENDIX C: Evaluation of ~(τ)〈~(3i)〈τ_(κ)〉〉for non-Newtonian fluids with γ_(υ)=1 188

APPENDIX D: Evaluation of~(τ)〈~(2i)（J_(qk)〉〉for isotropic conduction with variable conductivity and with γ_(υ)= 1 191

APPENDIX E: Further justifications for assuming α'_(κ),A'_(κ), and υ'_(κ) are negligible 192

References 193

Index 201

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