Fundamentals of Multiphase Flows Christopher E. Brennen California Institute of Technology Pasadena, California Cambridge University Press 2005 ISBN 0521 848040 1 Preface The subject of multiphase flows encompasses a vast field, a host of different technological contexts, a wide spectrum of different scales, a broad range of engineering disciplines and a multitude of different analytical approaches. Not surprisingly, the number of books dealing with the subject is volumi- nous. For the student or researcher in the field of multiphase flow this broad spectrum presents a problem for the experimental or analytical methodolo- gies that might be appropriate for his/her interests can be widely scattered and difficult to find. The aim of the present text is to try to bring much of this fundamental understanding together into one book and to present a unifying approach to the fundamental ideas of multiphase flows. Conse- quently the book summarizes those fundamental concepts with relevance to a broad spectrum of multiphase flows. It does not pretend to present a com- prehensive review of the details of any one multiphase flow or technological context though reference to books providing such reviews is included where appropriate. This book is targeted at graduate students and researchers at the cutting edge of investigations into the fundamental nature of multiphase flows; it is intended as a reference book for the basic methods used in the treatment of multiphase flows. I am deeply grateful to all my many friends and fellow researchers in the field of multiphase flows whose ideas fill these pages. I am particularly in- debted to my close colleagues, Allan Acosta, Ted Wu, Rolf Sabersky, Melany Hunt, Tim Colonius and the late Milton Plesset, all of whom made my pro- fessional life a real pleasure. This book grew out of many years of teaching and research at the California Institute of Technology. It was my privilege to have worked on multiphase flow problems with a group of marvelously tal- ented students including Hojin Ahn, Robert Bernier, Abhijit Bhattacharyya, David Braisted, Charles Campbell, Steven Ceccio, Luca d’Agostino, Fab- rizio d’Auria, Mark Duttweiler, Ronald Franz, Douglas Hart, Steve Hostler, 2 Gustavo Joseph, Joseph Katz, Yan Kuhn de Chizelle, Sanjay Kumar, Harri Kytomaa, Zhenhuan Liu, Beth McKenney, Sheung-Lip Ng, Tanh Nguyen, Kiam Oey, James Pearce, Garrett Reisman, Y C. Wang, Carl Wassgren, Roberto Zenit Camacho and Steve Hostler. To them I owe a special debt. Also, to Cecilia Lin who devoted many selfless hours to the preparation of the illustrations. A substantial fraction of the introductory material in this book is taken from my earlier book entitled “Cavitation and Bubble Dynamics” by Christopher Earls Brennen, c 1995 by Oxford University Press, Inc. It is reproduced here by permission of Oxford University Press, Inc. This book is dedicated with great affection and respect to my mother, Muriel M. Brennen, whose love and encouragement have inspired me throughout my life. Christopher Earls Brennen California Institute of Technology December 2003. 3 Contents Preface page 2 Contents 10 Nomenclature 11 1INTRODUCTIONTOMULTIPHASEFLOW 19 1.1 INTRODUCTION 19 1.1.1 Scope 19 1.1.2 Multiphase flow models 20 1.1.3 Multiphase flow notation 22 1.1.4 Size distribution functions 25 1.2 EQUATIONS OF MOTION 27 1.2.1 Averaging 27 1.2.2 Conservation of mass 28 1.2.3 Number continuity equation 30 1.2.4 Fick’s law 31 1.2.5 Equation of motion 31 1.2.6 Disperse phase momentum equation 35 1.2.7 Comments on disperse phase interaction 36 1.2.8 Equations for conservation of energy 37 1.2.9 Heat transfer between separated phases 41 1.3 INTERACTION WITH TURBULENCE 42 1.3.1 Particles and turbulence 42 1.3.2 Effect on turbulence stability 46 1.4 COMMENTS ON THE EQUATIONS OF MOTION 47 1.4.1 Averaging 47 1.4.2 Averaging contributions to the mean motion 48 1.4.3 Averaging in pipe flows 50 1.4.4 Modeling with the combined phase equations 50 1.4.5 Mass, force and energy interaction terms 51 4 2 SINGLE PARTICLE MOTION 52 2.1 INTRODUCTION 52 2.2 FLOWS AROUND A SPHERE 53 2.2.1 At high Reynolds number 53 2.2.2 At low Reynolds number 56 2.2.3 Molecular effects 61 2.3 UNSTEADY EFFECTS 62 2.3.1 Unsteady particle motions 62 2.3.2 Effect of concentration on added mass 65 2.3.3 Unsteady potential flow 65 2.3.4 Unsteady Stokes flow 69 2.4 PARTICLE EQUATION OF MOTION 73 2.4.1 Equations of motion 73 2.4.2 Magnitude of relative motion 78 2.4.3 Effect of concentration on particle equation of motion 80 2.4.4 Effect of concentration on particle drag 81 3 BUBBLE OR D ROPLET TRANSLATION 86 3.1 INTRODUCTION 86 3.2 DEFORMATION DUE TO TRANSLATION 86 3.2.1 Dimensional analysis 86 3.2.2 Bubble shapes and terminal velocities 88 3.3 MARANGONI EFFECTS 91 3.4 BJERKNES FORCES 95 3.5 GROWING BUBBLES 97 4 BUBBLE GROWTH AND COLLAPSE 100 4.1 INTRODUCTION 100 4.2 BUBBLE GROWTH AND COLLAPSE 100 4.2.1 Rayleigh-Plesset equation 100 4.2.2 Bubble contents 103 4.2.3 In the absence of thermal effects; bubble growth 106 4.2.4 In the absence of thermal effects; bubble collapse 109 4.2.5 Stability of vapor/gas bubbles 110 4.3 THERMAL EFFECTS 113 4.3.1 Thermal effects on growth 113 4.3.2 Thermally controlled growth 115 4.3.3 Cavitation and boiling 118 4.3.4 Bubble growth by mass diffusion 118 4.4 OSCILLATING BUBBLES 120 4.4.1 Bubble natural frequencies 120 5 4.4.2 Nonlinear effects 124 4.4.3 Rectified mass diffusion 126 5 CAVITATION 128 5.1 INTRODUCTION 128 5.2 KEY FEATURES OF BUBBLE CAVITATION 128 5.2.1 Cavitation inception 128 5.2.2 Cavitation bubble collapse 131 5.2.3 Shape distortion during bubble collapse 133 5.2.4 Cavitation damage 136 5.3 CAVITATION BUBBLES 139 5.3.1 Observations of cavitating bubbles 139 5.3.2 Cavitation noise 142 5.3.3 Cavitation luminescence 149 6 BOILING AND CONDENSAT ION 150 6.1 INTRODUCTION 150 6.2 HORIZONTAL SURFACES 151 6.2.1 Pool boiling 151 6.2.2 Nucleate boiling 153 6.2.3 Film boiling 154 6.2.4 Leidenfrost effect 155 6.3 VERTICAL SURFACES 157 6.3.1 Film boiling 158 6.4 CONDENSATION 160 6.4.1 Film condensation 160 7 FLOW PATTERNS 163 7.1 INTRODUCTION 163 7.2 TOPOLOGIES OF MULTIPHASE FLOW 163 7.2.1 Multiphase flow patterns 163 7.2.2 Examples of flow regime maps 165 7.2.3 Slurry flow regimes 168 7.2.4 Vertical pipe flow 169 7.2.5 Flow pattern classifications 173 7.3 LIMITS OF DISPERSE FLOW REGIMES 174 7.3.1 Disperse phase separation and dispersion 174 7.3.2 Example: horizontal pipe flow 176 7.3.3 Particle size and particle fission 178 7.3.4 Examples of flow-determined bubble size 179 7.3.5 Bubbly or mist flow limits 181 7.3.6 Other bubbly flow limits 182 6 7.3.7 Other particle size effects 183 7.4 INHOMOGENEITY INSTABILITY 184 7.4.1 Stability of disperse mixtures 184 7.4.2 Inhomogeneity instability in vertical flows 187 7.5 LIMITS ON SEPARATED FLOW 191 7.5.1 Kelvin-Helmoltz instability 192 7.5.2 Stratified flow instability 194 7.5.3 Annular flow instability 194 8 INTERNAL FLOW ENERGY CONVERSION 196 8.1 INTRODUCTION 196 8.2 FRICTIONAL LOSS IN DISPERSE FLOW 196 8.2.1 Horizontal Flow 196 8.2.2 Homogeneous flow friction 199 8.2.3 Heterogeneous flow friction 201 8.2.4 Vertical flow 203 8.3 FRICTIONAL LOSS IN SEPARATED FLOW 205 8.3.1 Two component flow 205 8.3.2 Flow with phase change 211 8.4 ENERGY CONVERSION IN PUMPS AND TURBINES 215 8.4.1 Multiphase flows in pumps 215 9 HOMOGENEOUS FLOWS 220 9.1 INTRODUCTION 220 9.2 EQUATIONS OF HOMOGENEOUS FLOW 220 9.3 SONIC SPEED 221 9.3.1 Basic analysis 221 9.3.2 Sonic speeds at higher frequencies 225 9.3.3 Sonic speed with change of phase 227 9.4 BAROTROPIC RELATIONS 231 9.5 NOZZLE FLOWS 233 9.5.1 One dimensional analysis 233 9.5.2 Vapor/liquid nozzle flow 238 9.5.3 Condensation shocks 242 10 FLOWS WITH BUBBLE DYNAMICS 246 10.1 INTRODUCTION 246 10.2 BASIC EQUATIONS 247 10.3 ACOUSTICS OF BUBBLY MIXTURES 248 10.3.1 Analysis 248 10.3.2 Comparison with experiments 250 10.4 SHOCK WAVES IN BUBBLY FLOWS 253 7 10.4.1 Normal shock wave analysis 253 10.4.2 Shock wave structure 256 10.4.3 Oblique shock waves 259 10.5 FINITE BUBBLE CLOUDS 259 10.5.1 Natural modes of a spherical cloud of bubbles 259 10.5.2 Response of a spherical bubble cloud 264 11 FLOWS WITH GAS DYNAMICS 267 11.1 INTRODUCTION 267 11.2 EQUATIONS FOR A DUSTY GAS 268 11.2.1 Basic equations 268 11.2.2 Homogeneous flow with gas dynamics 269 11.2.3 Velocity and temperature relaxation 271 11.3 NORMAL SHOCK WAVE 272 11.4 ACOUSTIC DAMPING 275 11.5 LINEAR PERTURBATION ANALYSES 279 11.5.1 Stability of laminar flow 279 11.5.2 Flow over a wavy wall 280 11.6 SMALL SLIP PERTURBATION 282 12 SPRAYS 285 12.1 INTRODUCTION 285 12.2 TYPES OF SPRAY FORMATION 285 12.3 OCEAN SPRAY 286 12.4 SPRAY FORMATION 288 12.4.1 Spray formation by bubbling 288 12.4.2 Spray formation by wind shear 289 12.4.3 Spray formation by initially laminar jets 292 12.4.4 Spray formation by turbulent jets 293 12.5 SINGLE DROPLET MECHANICS 299 12.5.1 Single droplet evaporation 299 12.5.2 Single droplet combustion 301 12.6 SPRAY COMBUSTION 305 13 GRANULAR FLOWS 308 13.1 INTRODUCTION 308 13.2 PARTICLE INTERACTION MODELS 309 13.2.1 Computer simulations 311 13.3 FLOW REGIMES 312 13.3.1 Dimensional Analysis 312 13.3.2 Flow regime rheologies 313 13.3.3 Flow regime boundaries 316 8 13.4 SLOW GRANULAR FLOW 317 13.4.1 Equations of motion 317 13.4.2 Mohr-Coulomb models 317 13.4.3 Hopper flows 318 13.5 RAPID GRANULAR FLOW 320 13.5.1 Introduction 320 13.5.2 Example of rapid flow equations 322 13.5.3 Boundary conditions 325 13.5.4 Computer simulations 326 13.6 EFFECT OF INTERSTITIAL FLUID 326 13.6.1 Introduction 326 13.6.2 Particle collisions 327 13.6.3 Classes of interstitial fluid effects 329 14 DRIFT FLUX MODELS 331 14.1 INTRODUCTION 331 14.2 DRIFT FLUX METHOD 332 14.3 EXAMPLES OF DRIFT FLUX ANALYSES 333 14.3.1 Vertical pipe flow 333 14.3.2 Fluidized bed 336 14.3.3 Pool boiling crisis 338 14.4 CORRECTIONS FOR PIPE FLOWS 343 15 SYSTEM INSTABILITIES 344 15.1 INTRODUCTION 344 15.2 SYSTEM STRUCTURE 344 15.3 QUASISTATIC STABILITY 347 15.4 QUASISTATIC INSTABILITY EXAMPLES 349 15.4.1 Turbomachine surge 349 15.4.2 Ledinegg instability 349 15.4.3 Geyser instability 350 15.5 CONCENTRATION WAVES 351 15.6 DYNAMIC MULTIPHASE FLOW INSTABILITIES 353 15.6.1 Dynamic instabilities 353 15.6.2 Cavitation surge in cavitating pumps 354 15.6.3 Chugging and condensation oscillations 356 15.7 TRANSFER FUNCTIONS 359 15.7.1 Unsteady internal flow methods 359 15.7.2 Transfer functions 360 15.7.3 Uniform homogeneous flow 362 16 KINEMATIC WAVES 365 16.1 INTRODUCTION 365 9 16.2 TWO-COMPONENT KINEMATIC WAVES 366 16.2.1 Basic analysis 366 16.2.2 Kinematic wave speed at flooding 368 16.2.3 Kinematic waves in steady flows 369 16.3 TWO-COMPONENT KINEMATIC SHOCKS 370 16.3.1 Kinematic shock relations 370 16.3.2 Kinematic shock stability 372 16.3.3 Compressibility and phase change effects 374 16.4 EXAMPLES OF KINEMATIC WAVE ANALYSES 375 16.4.1 Batch sedimentation 375 16.4.2 Dynamics of cavitating pumps 378 16.5 TWO-DIMENSIONAL SHOCKS 383 Bibliography 385 Index 407 10 [...]... similar to Q Mean value of Q or complex conjugate of Q Small perturbation in Q Complex amplitude of oscillating Q Time derivative of Q Second time derivative of Q Laplace transform of Q(t) Coordinate with origin at image point Small change in Q Real part of Q Imaginary part of Q 17 NOTES Notation The reader is referred to section 1.1.3 for a more complete description of the multiphase flow notation... x3 directions The mass flow of component N through one of the faces perpendicular to the i direction is given by ρN jN i and therefore the net outflow of mass of component N from the cube is given by the divergence of ρN jN i or ∂(ρN jN i ) ∂xi (1.20) The rate of increase of the mass of component N stored in the elemental volume is ∂(ρN αN )/∂t and hence conservation of mass of component N requires that... with examples from a broad range of applications and types of flow Parenthetically, it is valuable to reflect on the diverse and ubiquitous challenges of multiphase flow Virtually every processing technology must deal with multiphase flow, from cavitating pumps and turbines to electrophotographic processes to papermaking to the pellet form of almost all raw plastics The amount of granular material, coal, grain,... position of every interface But the computer power and speed 20 required to do this is far beyond present capability for most of the flows that are commonly experienced When one or both of the phases becomes turbulent (as often happens) the magnitude of the challenge becomes truly astronomical Therefore, simplifications are essential in realistic models of most multiphase flows In disperse flows two types of. .. Volume fraction Volume quality Ratio of specific heats of gas Shear rate Rate of dissipation of energy per unit volume Boundary layer thickness Damping coefficient Fractional mass Thermal boundary layer thickness Momentum thickness of the boundary layer Kronecker delta: δij = 1 for i = j; δij = 0 for i = j Fractional volume Coefficient of restitution Rate of dissipation of energy per unit mass Attenuation... that could, admittedly, be assumed, a priori Namely that the rate of transfer of mass to the component D in each particle, ID /nD , is equal to the Lagrangian rate of increase of mass, ρD vD , of each particle It is sometimes convenient in the study of bubbly flows to write the bubble number conservation equation in terms of a population, η, of bubbles per unit liquid volume rather than the number per unit... thermodynamic quantities Mass flux of component N in direction i Mass flux of component N Specific enthalpy Height Height Total head, pT /ρg Henry’s law constant Haberman-Morton number, normally gµ4 /ρS 3 Indices Square root of −1 Acoustic impulse Rate of transfer of mass per unit volume Total volumetric flux in direction i Volumetric flux of component N in direction i Volumetric flux of component N Polytropic constant... models In trajectory models, the motion of the disperse phase is assessed by following either the motion of the actual particles or the motion of larger, representative particles The details of the flow around each of the particles are subsumed into assumed drag, lift and moment forces acting on and altering the trajectory of those particles The thermal history of the particles can also be tracked if... other measures of relative motion that are frequently used The drift velocity of a component is defined as the velocity of that component in a frame of reference moving at a velocity equal to the total volumetric flux, ji , and is therefore given by, uN Ji , where uN Ji = uN i − ji (1.10) Even more frequent use will be made of the drift flux of a component which is defined as the volumetric flux of a component... necessary in order that each averaging volume (of volume 3 ) 27 contain representative samples of each of the components or phases In the sections that follow (sections 1.2.2 to 1.2.9), we proceed to develop the effective differential equations of motion for multiphase flow assuming that these conditions hold However, one of the more difficult hurdles in treating multiphase flows, is that the above two conditions . mother, Muriel M. Brennen, whose love and encouragement have inspired me throughout my life. Christopher Earls Brennen California Institute of Technology December 2003. 3 Contents Preface page 2 Contents. with relevance to a broad spectrum of multiphase flows. It does not pretend to present a com- prehensive review of the details of any one multiphase flow or technological context though reference to. texts, we exclude those circumstances in which the components are well mixed above the molecular level. Conse- quently, the flows considered here have some level of phase or component separation