THERMO-FLUID DYNAMICS OF TWO-PHASE FLOW THERMO-FLUID DYNAMICS OF TWO-PHASE FLOW Authored by MAMORU ISHII Purdue University TAKASHIHIBIKI Kyoto University ^ Springer Mamom Ishii School of Nuclear Engineering Purdue University 1290 Nuclear Engineering Building West Lafayette, IN 47906 U.S.A Takashi Hibiki Research Reactor Institute Kyoto University Noda, Kumatori, Sennan Osaka 590-0494 Japan Thermo-fluid Dynamics of Two-phase Flow Library of Congress Control Number: 20055934802 ISBN-10: 0-387-28321-8 ISBN-13: 9780387283210 ISBN-10: 0-387-29187-3 (e-book) ISBN-13: 9780387291871 (e-book) Printed on acid-free paper © 2006 Springer Science+Business Media, Inc All rights reserved This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, Inc., 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden The use in this publication of trade names, trademarks, service marks and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights Printed in the United States of America springer.com SPIN 11429425 Dedication This book is dedicated to our parents Table of Contents Dedication v Table of Contents vii Preface xiii Foreword xv Acknowledgments Part I xvii Fundamental of two-phase flow Introduction 1.1 Relevance of the problem 1.2 Characteristic of multiphase flow 1.3 Classification of two-phase flow 1.4 Outline of the book Local Instant Formulation 1.1 Single-phaseflowconservation equations 1.1.1 General balance equations 1.1.2 Conservation equation 1.1.3 Entropy inequality and principle of constitutive law 1.1.4 Constitutive equations 1.2 Interfacial balance and boundary conditions 1.2.1 Interfacial balance (Jump condition) 1 10 11 13 13 15 18 20 24 24 viii ThermO'Fluid Dynamics of Two-Phase Flow 12.2 Boundary conditions at interface 1.2.3 Simplified boundary condition 1.2.4 External boundary condition and contact angle 1.3 Application of local instant formulation to two-phase flow problems 1.3.1 Drag force acting on a spherical particle in a very slow stream 1.3.2 Kelvin-Helmholtz instability 1.3.3 Rayleigh-Taylor instability Part II 32 38 43 46 46 48 52 Two-phase field equations based on time average Various Methods of Averaging 1.1 Purpose of averaging 1.2 Classification of averaging 1.3 Various averaging in connection with two-phase flow analysis Basic Relations in Time Averaging 1.1 Time domain and definition of functions 1.2 Local time fraction - Local void fi-action 1.3 Time average and weighted mean values 1.4 Time average of derivatives 1.5 Concentrations and mixture properties 1.6 Velocity field 1.7 Fundamental identity Time Averaged Balance Equation 1.1 General balance equation 1.2 Two-fluid model field equations 1.3 Diffusion (mixture) model field equations 1.4 Singular case of Vj^=0 (quasi-stationary interface) 1.5 Macroscopic jump conditions 1.6 Summary of macroscopic field equations and jump conditions 1.7 Alternative form of turbulent heat flux Connection to Other Statistical Averages 1.1 Eulerian statistical average (ensemble average) 1.2 Boltzmann statistical average 55 55 58 61 67 68 72 73 78 82 86 89 93 93 98 103 108 110 113 114 119 119 120 Part III Three-dimensional model based on time average Kinematics of Averaged Fields 1.1 Convective coordinates and convective derivatives 129 129 ThermO'Fluid Dynamics of Two-Phase Flow 1.2 Streamline 132 1.3 Conservation of mass 133 1.4 Dilatation 140 Interfacial Transport 143 1.1 Interfacial mass transfer 143 1.2 Interfacial momentum transfer 145 1.3 Interfacial energy transfer 149 Two-fluid Model 155 1.1 Two-fluid model field equations 156 1.2 Two-fluid model constitutive laws 169 1.2.1 Entropy inequality 169 1.2.2 Equation of state 172 1.2.3 Determinism 177 1.2.4 Average molecular diffusion fluxes 179 1.2.5 Turbulent fluxes 181 1.2.6 Interfacial transfer constitutive laws 186 1.3 Two-fluid model formulation 198 1.4 Various special cases 205 10 Interfacial Area Transport 217 1.1 Three-dimensional interfacial area transport equation 218 1.1.1 Number transport equation 219 1.1.2 Volume transport equation 220 1.1.3 Interfacial area transport equation 222 1.2 One-group interfacial area transport equation 227 1.3 Two-group interfacial area transport equation 228 1.3.1 Two-group particle number transport equation 229 1.3.2 Two-group void fraction transport equation 230 1.3.3 Two-group interfacial area transport equation 234 1.3.4 Constitutive relations 240 11 Constitutive Modeling of Interfacial Area Transport 243 1.1 Modified two-fluid model for the two-group interfacial area transport equation 245 1.1.1 Conventional two-fluid model 245 1.1.2 Two-group void fraction and interfacial area transport equations 246 1.1.3 Modified two-fluid model 248 1.1.4 Modeling of two gas velocity fields 253 1.2 Modeling of source and sink terms in one-group interfacial area transport equation 257 1.2.1 Source and sink terms modeled by Wu et al (1998) 259 1.2.2 Source and sink terms modeled by Hibiki and Ishii (2000a) 267 ix Thermo-Fluid Dynamics of Two-Phase Flow 1.2.3 Source and sink terms modeled by Hibiki et al (2001b) 1.3 Modeling of source and sink terms in two-group interfacial Area Transport Equation 1.3.1 Source and sink terms modeled by Hibiki and Ishii (2000b) 1.3.2 Source and sink terms modeled by Fu and Ishii (2002a) 1.3.3 Source and sink terms modeled by Sun et al (2004a) 12 Hydrodynamic Constitutive Relations for Interfacial Transfer 1.1 Transient forces in multiparticle system 1.2 Drag force in multiparticle system 1.2.1 Single-particle drag coefficient 1.2.2 Drag coefficient for dispersed two-phase flow 1.3 Other forces 1.3.1 Lift Force 1.3.2 Wall-lift (wall-lubrication) force 1.3.3 Turbulent dispersion force 1.4 Turbulence in multiparticle system 13 Drift-flux Model 1.1 Drift-flux model field equations 1.2 Drift-flux (or mixture) model constitutive laws 1.3 Drift-flux (or mixture) model formulation 1.3.1 Drift-flux model 1.3.2 Scaling parameters 1.3.3 Homogeneous flow model 1.3.4 Density propagation model Part IV 275 276 277 281 290 301 303 308 309 315 329 331 335 336 336 345 346 355 372 372 373 376 378 One-dimensional model based on time average 14 One-dimensional Drift-flux Model 381 1.1 Area average of three-dimensional drift-flux model 82 1.2 One-dimensional drift velocity 387 1.2.1 Dispersed two-phase flow 387 1.2.2 Annular two-phase Flow 398 1.2.3 Annular mist Flow 403 1.3 Covarianceof convectiveflux 406 1.4 One-dimensional drift-flux correlations for various flow conditions 411 1.4.1 Constitutive equations for upward bubbly flow 412 1.4.2 Constitutive equations for upward adiabatic annulus and internally heated annulus 412 Thermo-Fluid Dynamics of Two-Phase Flow 1.4.3 Constitutive equations for downward two-phase flow 1.4.4 Constitutive equations for bubbling or boiling pool systems 1.4.5 Constitutive equations for large diameter pipe systems 1.4.6 Constitutive equations at reduced gravity conditions 15 One-dimensional Two-fluid Model 1.1 Area average of three-dimensional two-fluid model 1.2 Special consideration for one-dimensional constitutive relations 1.2.1 Covariance effect in field equations 1.2.2 Effect of phase distribution on constitutive relations 1.2.3 Interfacial shear term xi 413 413 414 415 419 420 423 423 426 428 References 431 Nomenclature 441 Index 457 Preface This book is intended to be an introduction to the theory of thermo-fluid dynamics of two-phase flow for graduate students, scientists and practicing engineers seriously involved in the subject It can be used as a text book at the graduate level courses focused on the two-phase flow in Nuclear Engineering, Mechanical Engineering and Chemical Engineering, as well as a basic reference book for two-phase flow formulations for researchers and engineers involved in solving multiphase flow problems in various technological fields The principles of single-phase flow fluid dynamics and heat transfer are relatively well understood, however two-phase flow thermo-fluid dynamics is an order of magnitude more complicated subject than that of the singlephase flow due to the existence of moving and deformable interface and its interactions with the two phases However, in view of the practical importance of two-phase flow in various modem engineering technologies related to nuclear energy, chemical engineering processes and advanced heat transfer systems, significant efforts have been made in recent years to develop accurate general two-phase formulations, mechanistic models for interfacial transfer and interfacial structures, and computational methods to solve these predictive models A strong emphasis has been put on the rational approach to the derivation of the two-phase flow formulations which represent the fundamental physical principles such as the conservations laws and constitutive modeling for various transfer mechanisms both in bulk fluids and at interface Several models such as the local instant formulation based on the single-phase flow model with explicit treatment of interface and the macroscopic continuum formulations based on various averaging methods are presented and 447 Np^ N^ N^ N^ n n n^ rig Prandtl number number of bubbles inside effective volume viscosity number density ratio /^ P^ P^j p p^ particle number per unit mixture volume unit normal vector bubble number density number of eddies of wave number per volume of twophase mixture outward unit normal vector for phase k production of shear-induced turbulence probability for a bubble to move toward neighboring bubble partial pressure tensor interfacial wetted perimeter wall wetted perimeter pressure critical pressure P k ^ ^ ^IPH p^ q partial, bulk mean and interfacial mean pressure mixture pressure heat flux q^ diffusion (drift) heat flux qj^, q^ q, ql Qj^ mean conduction and turbulent heat fluxes mixture conduction and turbulent heat fluxes local instant body heating qj^ average heat transfer pert interfacial area (energy gain) q^ R R R R^ mean conduction heat flux ideal gas constant radius of a pipe radius of curvature variable defined by RvJv^ R^ R mean radius of fluid particles particle number source and sink rate n^ P^^ PQ fluid 448 R^ Re (i?e)^ tube radius Reynolds number particle Reynolds number r r^ SQ , S( Sj radial coordinate non-dimensional radius surface available to collision particle source and sink rates per unit mixture volume Spf^ due toy-th particle interactions such as disintegration or coalescence particle source and sink rates per unit mixture volume s s^ 5^ , % s^ T due to phase change entropy surface entropy per area weighted mean entropy at bulk phase and at interfaces mixture entropy temperature T^, T^ instant and mean interface temperature Tj^, T^ T^ mean temperature at bulk phase and at interface stress tensor t tfj t^ time time required for bubble coalescence time when they^-interface passes the point C (^^ K) hybrid tensor of interface, see Aris (1962) U UQ velocity of shock in mixture velocity of stream ^B > ^c u u^ u^ UQ, UQ u^ u^^, % u^ volume available to collision internal energy surface energy per area mean fluctuation velocity bubble velocity eddy velocity weighted mean internal energy at bulk phase and at interfaces mixture internal energy 449 u^^^ averaged relative velocity between leading bubble and u^ \cnt bubble in wake region root-mean-square approaching velocity of two bubbles critical fluctuation velocity V V V^ volume time derivative of volume V critical bubble volume V^ non-dimensional drift velocity V^ Vj^ interfacial region drift velocity Vj^ diffusion velocity V^ fixed mass volume Vs ^atio of V^^^, to V^^^^ V^ Fj^ effective wake volume peak bubble volume in group V v^^ velocity liquid velocity fluctuation independent of bubble 'y^' agitation liquid velocity fluctuation dependent on bubble v^ agitation fi-iction velocity v^ average center-of-volume velocity of dispersed phase v^ v^, v^ velocity of interface weighted mean velocity at bulk phase and at interfaces \y[ j /2 mean turbulent kinetic energy v^ v^^ mixture center of mass velocity average local particle velocity weighted by particle v^ \ number relative velocity difference between area averaged mean velocities of v^^ phases relative velocity of a single particle in an infinite medium 450 V velocity of interfacial particles w;i work due to fluctuations in drag forces We Weber number critical Weber number X y convective coordinates spatial coordinates spatial coordinate spatial coordinate y' variable defined by yvJv^ Z spatial coordinate X X Greek OL drop void fraction in slug bubble section ratio of liquid-film cross-sectional area to total crosssectional area average overall void fi-action ratio of cross-sectional area of drops to cross-sectional a g,cnt area of core critical void fraction when center bubble caimot pass a 'g,Tnax throughfi-eespace among neighboring bubbles maximum void fraction a Oi„ a (3 r 7k A A At At time (void)firactionof phase k ratio of mixing length and width of wake variable to take account of overlap of excluded volume thermal expansivity based on averaged properties constant mass generation for phase k constant ratio of specific heats interfacial entropy generation per area entropy generation for phase k inter-group mass transfer rates fi'om group to group time interval of averaging time interval to drive daughter bubble apart with 451 At^ At„At^ At^ 6' ^crit init hk Sfx e 2s{or2e^) e'' e',e" Vph Vo characteristic length of D^ time interval for one collision time intervals associated with phase k and interfaces average time interval for a bubble in wake region to catch up with preceding bubble thickness of interface film thickness collective parameter critical film thickness where rapture occurs initial fihn thickness pressure deviation from saturation pressure volume element in /i space energy dissipation rate per unit mass time associated with they^-interface dissipation of shear-induced turbulence eddy diffusivity rate of volume generated by nucleation source per unit mixture volume amplitude ^» contact angle angle in cylindrical coordinates wake angle i^fi- variable defined by — expl—Cf^/ e ^Sk ' ^Tk A isentropic and isothermal compressibilities of phase k interfacial thermal energy transfer term in the averaged \ /^ equation wavelength constant breakup efficiency coalescence efficiency critical wavelength bulk viscosity viscosity K^ f^I mean molecular and turbulent viscosities A A K >^c \ D' j 452 T* /Jij^ fi^ u jy^ ^ mixing length coefficient mixture viscosity kinematic viscosity turbulent kinematic viscosity particle (phase) velocity in Boltzmann statistical ^ average ratio of V^^ to V^ ^ variable defined by (l - 02S94D*J ^ variable defined by P^IP^f p density p^ surface mass per area "p^, ~p^ partial and mean densities PI modified density defined by /^^coth (/c/i^) p^ a W W^ ^^^ mixture density surface tension viscous stress tensor diffusion (or drift) stress tensor bubble-induced turbulent stress tensor ^"^^ shear-induced turbulent stress tensor ^ , W^ mixture viscous and turbulent stress tensors ^ ,^^ Sl^^ average viscous and turbulent stress tensor average viscous stress ^ , ^ r^ r r^ ' ^ t k ' '^nk T^f interfacial shear stress contact time for two bubbles interfacial shear stress reference time constant tangential and normal stresses at interface wall shear ^ ^J ^^ velocity potential turbulent work effect in enthalpy energy equation interfacial mechanical energy exchange effect in the mixture thermal energy equation viscous dissipation ^^ f 453 m m 0a 0.0 X mixture viscous dissipation surface tension effect in the mixture thermal energy equation source term interfacial source per area source and sink rate for interfacial area concentration velocity potential coefficient accounting for contribution from inter- ^ group transfer property of extensive characteristics shape factor ^.A A mass weighted mean values for mixture and phase k property per interfacial area Q potential function ^ Subscripts and Superscripts a c d f i k ki surface (property per area) continuous phase dispersed phase liquid phase vapor phase interface 7*-interface each phase : (^=1 & 2), ( ^ c & d), (A=f & g) A:^-phase at interfaces [mixture (in macroscopic formulation) 7Y} III n RC sat [fixed mass (in local instant formulation) normal to interface reference random collision cylindrical coordinate saturation surface (surface property per mass) Q o solid phase 454 SI SO TI WE t w x,y ,z +, - surface instability shearing off turbulent impact wake entrainment tangential to interface wall rectangular coordinate + and - side of shock in macroscopic field 1,2 phase and phase Symbols and Operators A A A A- B AB A:B V• V V^ • {AY Dt D_ Dt A Dt tensor vector scalar dot product dyadic product of two vectors (=tensor) double dot product of two tensors (=scalar) divergence operator gradient operator surface divergence operator (Aris, 1962) transposed tensor _ ~ _ ~ _ Dt dt F =w JF F * F dt + '"k' d + Vm V dt d + c, dt ^ ~ dt + v^- surface convective derivative with v^ (Aris, 1962) time average weighted mean value A:^-phase weighted mean value phase average 455 V^^ ^^-phase mass weighted mean value ip mixture mass weighted mean value Fl fluctuating component with respect to mean value F^ fluctuation component with respect to surface mean value F(^.^, F^ surface average F^ ( ),/5 mass flux weighted mean value at interfaces surface covariant derivative (Aris, 1962) [At\^ with (^T,S,1,2); sets of time intervals y^ summation on both phases k y^ summation on the interfaces passing in At at x Index Angular momentum (Conservation of in single-phase flow), 16 Area averaging, 63-65 Area concentration (Surface), 108-109,192-195 Averaged fields (Kinematics of), 129-141 Averaging (Area), 62-65 Averaging (Botlzmann), 58-61,120-128 Averaging (Ensemble cell), 66 Averaging (Eulerian), 8-61 Averaging (Lagrangian), 8-61 Averaging (Statistical), 58-61, 65-66,119-128 Averaging (Time), 64-65 Averaging (Various in connection with two-phase flow analysis), 61-66 Averaging (Various methods of), 55-66 Averaging (Volumetric), 62-63 Balance equation (Single-phase flow general), Balance equation (Surface), Balance equation (Time averaged), Basset force, Boltzmann averaging, Center of mass velocity, Change (Phase boundary condition), Chemical boundary condition, Chum-turbulent-flow regune, 13-15 30 93-117 256, 302-308 58-61,120-128 86 37-38 37-38 6-8, 228,281,290, 307-308, 315, 325-327 330, 361, 396,404-405,417-418 458 Classification of two-phase flows, Clausius-Clapeyron equation, Concentration, Concentration (Surface area), Conservation equation (Single-phase), Constitutive axioms, Constitutive laws (Drift-flux model), Constitutive laws (Two-fluid model), Constitutive laws or equations, Contact angle, Continuity equation (Single-phase), Convective coordinates, Convective derivatives, Coordinates (Convective), Covariance, Creeping flow, 40 82-86 108-109,192-195 13-24 18 355-372 169-197 12,18-24 43-46 15 129-132 129-132 129-132 406-411,423-426 46 D Density propagation equation, 138 Density propagation model, 378-379 Derivatives (Convective), 129-132 Derivatives (Time average of), 78-82 Diffusion flux, 90 Dilatation, 140 Discontinuities (Shock), 110-112 Dispersed flows, 5-7 Displacement velocity, 80 Distorted-fluid-particle regime, 323-325, 361 Distribution parameter, 257, 328, 387-395,403,408,412-418,425 Distribution parameter ( for a flux), 407,424 Distribution parameter ( for enthalpy flux), 426 Distribution parameter ( for A:-phase enthalpy), 423 Distribution parameter ( for ^-phase momentum), 422 Distribution parameter (Mixture-momentum ), 409 Drag force in multiparticle system, 308-329 Drag force acting on a spherical particle in a very slow stream, 46-48 Drag force (Interfacial), 189-191 Drift-flux model, 62, 345-379 Drift-flux model constitutive laws, 345-372 Drift-flux model field equations, 103-108, 346-354 Drift-flux model formulation, 372-379 Drift velocity, 88,136-137, 372 Energy (Conservation of in single-phase flow), Enthalpy equation (Single-phase flow), Entropy inequality (Interfacial), Entropy inequality (Single-phase flow), 16-17 18 34-35 18-20 459 Eulerian averaging, Eulerian statistical average, External boundary condition, Extra deformation tensor (Interfacial), 58-61 119-120 43 179 Field equations, Field equations (Diffusion model), Field equations (Two-fluid model), Field equations (Two-phase based on time average), Fields (Kinematics of averaged), Fluctuating component, Flux (Diffusion), Flux (Volumetric), Fundamental identity, 12 103-108, 346-354 98-103,156-169 55-128 129-141 78 90 87,135-136 89-92 Green's theorem, 14 Green's theorem (Surface), 28 H Heat flux (Interfacial), 191-192 Homogeneous flow model, 376-378 I Identity (Fundamental), Instant (Local formulation), Interface (Quasi-stationary), Interfacial area transport equation, Interfacial area transport equation (One-group Interfacial area transport equation (Two-group Interfacial boundary condition, Interfacial conditions, Interfacial drag force, Interfacial energy balance, Interfacial energy source, Interfacial energy transfer, Interfacial entropy inequality, Interfacial extra deformation tensor, Interfacial heat flux, Interfacial mass balance, Interfacial mass transfer, Interfacial momentum balance, Interfacial momentum source, Interfacial momentum transfer, ), ), 89-91 11 -46 108-110 10,195,217-299 227-228,257-276 228-242,246-248,276-299 13,32-38 12 190-191 32 196-197 149-154 34-36 179 191-192 31 143-144,188-190 32 192-195 145-149 460 Interfacial shear term, Interfacial structure, Interfacial transfer condition, Interfacial transport, Internal energy equation (Single-phase flow), 428-430 95 143-145 18 J Jump conditions, Jump conditions (Macroscopic), 13,24-32 110-113 K A:-^ model, Kelvin-Hehnholtz instability, Kinematic shock wave, Kinematic wave, Kinematics of averaged fields, Lagrangian averaging, Leibnitz rule, Lift force, Local instant formulation, 341-343 48-52, 313 13 8-140 136-138 129-141 58-61 14 331-335 11 -46 M Mass weighted mean values, Material derivative (Transformation on), Mean values (Mass weighted), Mean values (Weighted), Mechanical energy equation (Single-phase flow), Mixed flows, Mixture properties, Mixture viscosity, Momentum equation (Single-phase), Momentum source (Interfacial), Motion (Equation of in single-phase flow), 75-76 17 75-76 73-77 17 4-6 82-86 303, 316-320, 367, 376, 396 15 192-195 17 N Newton's regime, Normal vector, Number transport equation, One-dimensional drift-flux model, One-dimensional two-fluid model, 315, 320-323 80 219-220,229-230 381-418 419-430 461 One-equation model, 339-341 P Phase average, Phase change boundary condition, Propagation (Density equation), Propagation (Density model), Propagation (Void equation), 75 37-38 138 378-379 136-138 R Rayleigh-Taylor instability, Reynolds transport theorem, 52-53 14 Scaling parameters, 375 Scaling parameters (Drift-flux model), 373-376 Scaling parameters (Two-fluid model), 205-210 Second law of thermodynamics (Single-phase systems), 19 Separated flows, 3-6 Shock (Kinematic — wave), 13 8-140 Shock discontinuities, 110-112 Similarity groups, 375-376 Single-phase flow conservation equations, 13-24 Slip (No condition), 36 Slug-flow reghne, 315, 327-330, 361 Source and sink terms in one-group interfacial area transport equation, 257-276 Source and sink terms in two-group interfacial area transport equation, 276-299 State (Equation of), 20-22 Stationary (Quasi interface), 108-110 Statistical averaging, 58-61, 65-66,119-128 Streamline, 132-133 Structure (Interfacial), Surface area concentration, 108-109,192-195 Thermal boundary condition, Three-dimensional model based on time-averaged, Time average, Time average (Three-dimensional model based on), Time average (Two-phase field equations based on), Time average of derivatives, Time averaged balance equation, Time averaging, Time fraction (Local), Transfer condition (Interfacial), 36 129-216, 345-379 73-77 129-216, 345-379 55-128 78-82 93-117 64-65 72-73 95 462 Transitional flow, Transport (Interfacial), Transport theorem, Transport theorem (Surface), Turbulence in multiparticle system, Turbulent dispersion force, Two-equation model Two-fluid model Two-fluid model (Modified ), Two-fluid model constitutive laws Two-fluid model field equations Two-fluid model formulation Two-group void fraction transport equation Velocity (Center of mass) Velocity (Diffusion), Velocity (Displacement), Velocity (Drift), Velocity field Virtual mass force, Viscous regime Void fraction (Local), Void propagation equation, Volume transport equation Volumetric averaging Volumetric flux, 5-7 143-154 14 28 336-343 336 341-343 62,155-216 245-257 169-197 98-103,156-169 198-205 230-234,246 87 80 136-137, 372 86-89 256, 302-308 310-311, 315-320, 322, 330, 360 72-73 136-138 220-222 62 87,135-136 W Wall lift force, Wave (Kinematic), Weighted mean values Zero-equation model 302, 335-336 136-138 73-77 337-339 ... very often, the safe control of a great number of important systems depend upon the availability of realistic and accurate mathematical models of two- phase flow 1.2 Characteristic of multiphase flow. .. in great detail, because for most two- phase flow, thermo- fluid dynamics are dominated by the interfacial structures and interfacial momentum transfer Some other necessary constitutive relations... Fundamental of two- phase flow Introduction 1.1 Relevance of the problem 1.2 Characteristic of multiphase flow 1.3 Classification of two- phase flow 1.4 Outline of the book Local Instant Formulation