Multiphase Flow Handbook

multiphase flow

handbook

edited by

Clayton T. Crowe

A CRC title, part of the Taylor & Francis imprint, a member of the Taylor & Francis Group, the academic division of T&F Informa plc.

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Library of Congress Cataloging-in-Publication Data Multiphase flow handbook / edited by Clayton T. Crowe.

p. cm.

Includes bibliographical references and index. ISBN 0-8493-1280-9

1. Multiphase flow--Handbooks, manuals, etc. I. Crowe, C. T. (Clayton T.) TA357.5.M84.M85 2005

620.1’064--dc22 2005048623 Visit the Taylor & Francis Web site at

http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com

Multiphase flow is the simultaneous flow of a mixture with two or more phases. They are encountered in many of our day to day activities. Numerous industrial and energy conversion processes rely on the flow of multiphase mixtures.

In 1982, Hemisphere Publishing produced Handbook of Multiphase Systems with G. Hetsroni as the editor in chief. This handbook provided information important to the design engineers, researchers, and students involved with multiphase flows. Since then, there have been numerous advances in analysis, modeling, and experimental methods. In particular, there has been such a surge of activity in numerical modeling that computational fluid mechanics is now regarded as a viable tool. The purpose of this hand-book is to review the fundamental principles and provide the reader with current information useful in research, industrial design, and education.

I am indebted to the many contributors who spent considerable time and effort in this undertaking. Obviously, this Handbook would not have been possible without their interest in the project. In several instances, the crunch of time put stress on the contributors (and the editor-in-chief, in particular), but, finally, the work was accomplished on time. I particularly want to thank Professor Dyakowski for helping select the contributors for Chapter 14. I also appreciate the efforts of Helena Redshaw, Preethi Cholmondeley and Cindy Carelli for responding to questions and monitoring the schedule. I also express my gratitude to the Production team of Macmillan India Limited for responding to some difficult issues. I am also indebted to the support of my family, Chad and Brenda, Kevin, Zoe, and Ezra, and my dear friend Jeannette.

Clayton T. Crowe is professor emeritus of mechanical and materials engineering at Washington State University. He has had over 30 years of experience in fluid mechanics and multiphase flows, and has been involved in a wide diversity of two-phase flow problems from geothermal power to food and materials process-ing. He developed the particle-source-in-cell (PSI-Cell) method for the numeri-cal simulation of multiphase flow that has been used extensively in industry and in most commercial software simulations. He has over a hundred publications in the literature and has coauthored an undergraduate textbook, Engineering Fluid Mechanics, which is cur-rently in its eighth edition. He has also been the lead author of a book on multiphase flows with droplets and particles published by CRC Press. He is an ASME Fellow and has received numerous awards for his work including the prestigious ASME Fluids Engineering Award. In May 2001, he received the Senior International Prize for Multiphase Flows.

Alain Berlemont, Director of Research at National Center for Scientific Research (CNSR), UMR 6614-CORIA, University of Rouen, France.

Steven Ceccio, Associate Vice President for Research, Professor, Department of Mechanical Engineering and Applied Mechanics, University of Michigan, Ann Arbor, MI. Former member of Defense Studies Group, Institute for Defense Analysis.

Yi Cheng, Associate Professor, Department of Chemical Engineering, Tsinghua University, Beijing, China. Formerly Research Associate at Powder technology Research Center, Department of Chemical and Biochemical Engineering, University of Western Ontario, London, Ontario, Canada.

Jacob Chung, Andrew H. Hines Jr./Progress Energy Eminent Scholar Chair, Department of Mechanical and Aerospace Engineering, University of Florida, Gainesville, FL. ASME Fellow. Associate editor of the ASME Journal of Heat Transfer.

Candis Claiborn, Professor, Department of Civil and Environmental Engineering and Associate Dean for Research and Graduate Programs in College of Engineering, Washington State University, Pullman, WA. James Cullivan, Institute of Particle Science and Engineering, School of Process, Environmental and materials Engineering, University of Leeds, North Yorkshire, UK. Previously with Camborne School of Mines, University of Exeter.

Shrikant Dhodapkar, Research Leader, Solids Processing Laboratory, The Dow Chemical Company, Freeport, TX. Vice-Chair Particle Technology Forum (AIChE), recipient of numerous technology awards at Dow.

Mingzhe Dong, Associate Professor, Petroleum Systems Engineering, University of Regina, Regina, Saskatchewan, Canada.

Yannis Drossinos, Senior Research Scientist, European Commission, Joint Research Centre, Ispra, Italy. Currently Visiting Professor in the School of Mechanical and Systems Engineering at University of Newcastle upon Tyne, United Kingdom.

Francis Dullien, Distinguished Professor Emeritus, Department of Chemical Engineering, University of Waterloo, Waterloo, Ontario, Canada. Fellow of the Royal Society of Canada and Fellow of the Chemical Institute of Canada. Member of Advisory Editorial Advisory Board for Transport in Porous Media and Journal of Porous Media.

Manchester, Manchester, UK. Also Manager of Applied Process Tomography Unit.

John Eaton, Professor and Vice Chairman, Department of Mechanical Engineering, Stanford University, Stanford, CA. ASME Fellow. Tau Beta Pi and Perin Awards for undergraduate teaching.

Said Elghobashi, Professor, Department of Mechanical and Aerospace Engineering, University of California, Irvine, CA. Fellow of the American Physical Society and American Society of Mechanical Engineers.

Thomas Frank, Head of Funded CFX Development Group, ANSYS Germany GmbH, Otterfing, Germany. Formerly Head of the Research Group on Numerical Methods for Multiphase Flows, Chemnitz University of Technology, Chemnitz, Germany.

Udo Fritsching, Director of research group for Multiphase Flow, Heat and Mass Transfer, Foundation Institute for Material Science, University of Bremen, Bremen, Germany.

Kamiel Gabriel, Associate Provost for Research and Graduate Programs, University of Ontario Institute of Technology, Oshawa, Ontario, Canada. Formerly established the Microgravity Research Group at the University of Saskatchewan.

John Grace, Professor of Chemical Engineering, University of British Columbia, Vancouver, British Columbia, Canada. Formerly Department Head and Dean of Graduate Studies. Currently holds Canada Research Chair and is a Fellow of the Canadian Academy of Engineering and of the Royal Society of Canada. Past winner of the R.S. Jane Award of the Canadian Society for Chemical Engineering.

Erling Hammer, Professor, Department of Physics, University of Bergen, Bergen, Norway. Formerly with the Chr. Michelsen Institute. Recipient of the STIMAREC Prize for malfunction analysis of pacemaker systems.

Yassin Hassan, Professor, Department of Nuclear Engineering and the Department of Mechanical Engineering, Texas A&M University, College Station, TX. Fellow of ASME and ANS. Recipient of the ASME George Westinghouse Gold Medal and the ANS Arthur Holly Compton Award.

Christos Housiadas, Senior Research Scientist, Institute of Nuclear Technology and Radiation Protection, “Demokritos” National Centre for Scientific Research, Athens, Greece. Formerly with the European Commission’s Joint Research Center, Ispra, Italy and with the French Commissariat à l’Energie Atomique. Shenggen Hu, Principle Research Engineer/Project Manager, Division of Energy Technology, Commonwealth Scientific and Industrial Research Organisation (CISRO), Brisbane, Australia.

Karl Jacob, Scientist/Technical Leader, Solids Processing Laboratory, The Dow Chemical Company, Midland, MI.

Geir Anton Johansen, Professor, Department of Physics and Technololgy, University of Bergen, Bergen, Norway. Also Scientific Advisor at Christian Michelsen Research AS.

Director, Institute of Nature and Environmental Technology, Kanazawa University, Kanazawa, Japan. Satish Kandlikar, Gleason Professor of Mechanical Engineering at Rochester Institute of Technology, Rochester, NY. ASME Fellow, Associate editor for Heat Transfer Engineering, the ASME Journal of Heat Transfer and the Journal of Nanofluids and Microfluids.

Bo Leckner, Professor, Department of Energy and Environment, Chalmers University of Technology, Göteborg, Sweden. Previously acting Professor of Energy Conversion.

Eric Loth, Professor, Aerospace Engineering, University of Illinois at Urbana-Champaign, Urbana, IL. Professor of Aerospace Engineering, Willet Faculty Scholar, Associate Fellow of American Institute of Aeronautics and Astronautics.

Robert Lyczkowski, Chemical Engineer, Biotechnology and Biodefense Applications Group, Energy Systems Division, Argonne National Laboratory. AIChE Fellow. Editorial advisory board for Advances in Transport Processes.

Hisao Makino, Senior Research Scientist, Energy Engineering Research Laboratory, Staff (Director) Central Research Institute of Electric Power Industry (CRIEPI), Kanagawa, Japan. Formerly Director of CS Promotion Division of CRIEPI, Currently Director of the Combustion Society of Japan and the Society of Particle Technology in Japan. Also Editor of the Journal of the Society of Particle Technology, Japan.

Farzad Mashayek, Professor and Director of Graduate Studies, Department of Mechanical and Industrial Engineering, University of Illinois at Chicago, Chicago, IL. Received CAREER Award from U.S. National Science Foundation, Young Investigator Award from U.S. Office Of Naval Research and is Associate Fellow of AIAA.

Yoichiro Matsumoto, Vice Dean, School of Engineering, University of Tokyo, Tokyo, Japan. JSME Computational Mechanics Award, JSME Fluids Engineering Frontier Award, former editor of JSME International Journal Series B, former associate editor of ASME Journal of Fluids Engineering. JSME Fellow.

Efstathios (E.) Michaelides, Leo S. Weil Professor and Associate Dean for Graduate Studies and Director, Southcentral Regional Center of the National Institute for Global Environmental Change, School of Engineering, Tulane University, New Orleans, LA. Chair of ASME Fluids Engineering Division, ASME Fellow and ASME Freeman Scholar Award.

Paul Mort, Principal Engineer, Procter & Gamble Company, Cincinnati, OH. Chair of Particle Formation area in International Fine Particle Research Institute and the Particle Production area within the Particle Technology Forum.

René Oliemans, Director Multiphase Flow B.V., Part-time Professor Multiphase Flow, Delft University of Technology, Delft, Netherlands. Previously Department Manager for Engineering Physics and Mathematics, Shell Research, Amsterdam.

ratory, Richland, WA. Graduate Adjunct Faculty, Civil and Environmental Engineering Department, Washington State University.

Ghanem Oweis, Post-Doctoral Research fellow, Mechanical Engineering Department, University of Michigan, Ann Arbor, MI.

Bert Pots, Senior Staff Corrosion Engineer, Shell Global Solutions, Houston, TX. Formerly with Royal Dutch/Shell Laboratories in Amsterdam and Shell International Gas in London, UK.

Michael Reeks, Professor in Multiphase Flow, School of Mechanical and Systems Engineering, University of Newcastle upon Tyne, United Kingdom. Formerly Marie Currie Senior Research Fellow at the European Research Center at Ispra, Italy. Formerly associate editor of the Journal of Fluid Mechanics Research and ASME Journal of Fluids Engineering.

Ted Roberts, Senior Lecturer, School of Chemical Engineering and Analytic Science, University of Manchester, United Kingdom. Current chairman of the Society of Chemical Industry Electrochemical Technology Group.

Ilia Roisman, Chair of Fluid Mechanics and Aerodynamics, Technical University of Darmstadt, Darmstadt, Germany. Formerly Senior Lecturer at the Technion-IIT, Israel.

Akimi Serizawa, Professor, Division of Nuclear Engineering, Kyoto University, Kyoto, Japan. Heat Transfer Society of Japan Academic Award, Societe de Thermique International Award, Fellow of JSME and Atomic Society of Japan.

Olivier Simonin, Professor and Director of the Institute of Mechanics of Fluids of Toulouse, Toulouse, France.

Gabriel Tardos, Professor, Department of Chemical Engineering, The City College of the City University of New York, New York, NY., Formerly department chair.

Cameron Tropea, Chair of Fluid Mechanics and Aerodynamics, Technical University of Darmstadt, Darmstadt, Germany. Editor-in-chief of the journal Experiments in Fluids.

Yutaka Tsuji, Professor, Department of Mechanical Engineering and Graduate School of Engineering, Osaka University, Osaka, Japan. JSME Fluids Engineering Award, Fluid Science Research Award, AIChE Thomas Baron Award, JSME Medal, Jotaki Award from the Society of Powder Technology, JSME Fellow. Timothy Troutt, Professor, School of Mechanical and Materials Engineering, Washington State University, Pullman, WA.

Gretar Tryggvason, Professor and Head of the Mechanical Engineering Department, Worcester Polytechnic Institute, Worcester, MA. Currently editor-in-chief of the Journal of Computational Physics, associate editor of the International Journal of Multiphase Flows and Fellow of the American Physical Society.

Richard Williams, Anglo American plc Chair in Mineral and Process Engineering at the University of Leeds, Leeds, UK. Awarded the Ambrose Medal & Premium, the Isambard Kingdom Brunel Award, Beilby Prize and Gold Medal, Noel Webster Award and the Silver Medal of The Royal Academy of Engineering. Fellow of the Royal Academy of Engineering.

Hideto Yoshida, Professor, Department of Chemical Engineering and Graduate School of Engineering, Hiroshima University, Hiroshima, Japan. Editor of Chemical Engineering Journal in Japan.

Jesse Zhu, Director, Powder Technology Research Center, and Professor in the Department of Chemical and Biochemical Engineering, University of Western Ontario, London, Ontario, Canada. Canada Research Chair of Powder Technology Applications, Senior Ontario Research Chair for Ultrafine Powder Technology, Fellow of Canadian Society of Chemical Engineering.

1

Basic Concepts and Definitions . . . 1-1

2

Gas–Liquid Transport in Ducts . . . 2-1

3

Boiling and Condensation. . . 3-1

4

Fluid–Solid Transport in Ducts . . . 4-1

5

Fluidized Beds. . . 5-1

6

Aerosol Flows . . . 6-1

7

Particle Separation Systems. . . 7-1

8

Spray Systems . . . 8-1

9

Dry Powder Flows . . . 9-1

10

Porous Media Flows . . . 10-1

11

Microscale and Microgravity Flows . . . 11-1

G.F. Oweis, S.L. Ceccio, Y. Matsumoto, C. Tropea, I.V. Roisman, Y. Tsuji, R. Lyczkowski, T.R. Troutt, J. K. Eaton, and F. Mashayek

13

Modeling . . . 13-1

O. Simonin, Th. Frank, Y. Onishi, and B. van Wachem

14

Advanced Experimental Techniques . . . 14-1

1

Basic Concepts and

Definitions

1.1 General Features of Multiphase Flows ...1-1 Dispersed Phase and Separated Flows ● _Gas–Liquid Flows●_{Gas–Solid Flows}●_{Liquid–Solid Flows} ●_Three-Phase

Flows●_{Scope of the Handbook}

1.2 Fundamental Definitions...1-3 Volume Fraction and Densities●_{Superficial and Phase Velocities} ● _{Quality, Concentration, and Loading} ●_{Response Times} ● Stokes Number●_{Disperse Versus Dense Flows}● Phase Coupling 1.3 Size Distribution ...1-10

Size Distributions● Statistical Parameters ● Frequently Used

Size Distributions

1.4 Interactions of Fluids with Particles, Drops, and Bubbles 1-17 Introduction●_{Mass Transfer}●_{Momentum Transfer}●_Heat Transfer●_{Multiple Particle/Droplet/Bubble Effects}

1.1 General Features of Multiphase Flows

Clayton T. Crowe

A phase refers to the solid, liquid, or vapor state of matter. A multiphase flow is the flow of a mixture of phases such as gases (bubbles) in a liquid, or liquid (droplets) in gases, and so on. The purpose of this Handbook is to provide back ground and state-of-the-art information on multiphase flows, including industrial applications, analytic and numerical analyses, and experimental techniques.

1.1.1

Dispersed Phase and Separated Flows

Dispersed phase flows are flows in which one phase consists of discrete elements, such as droplets in a gas or bubbles in a liquid. The discrete elements are not connected. In a separated flow, the two phases are separated by a line of contact. An annular flow is a separated flow in which there is a liquid layer on the pipe wall and a gaseous core. In other words, in a separated flow one can pass from one point to another in the same phase while remaining in the same medium.

1.1.2

Gas–Liquid Flows

Gas–liquid flows occur in many applications. The motion of bubbles in a liquid as well as droplets in a conveying gas stream are examples of gas–liquid flows. Bubble columns are commonly used in several process industries. Atomization to generate small droplets for combustion is important in power

1-1 Clayton T. Crowe

Washington State University Efstathios E. Michaelides Tulane University

generation systems. Also droplet formation and impaction are important in spray forming for materials processing. Steam-water flows in pipes and heat exchangers are very common in power systems such as fossil fuel plants and nuclear reactors. Gas–liquid flows in pipes can assume several different configura-tions ranging from bubbly flow to annular flow, in which there is a liquid layer on the wall and a droplet-laden gaseous core flow.

1.1.3 Gas–Solid Flows

Gas–solid flows are usually considered to be a gas with suspended solid particles. This category of flow includes pneumatic transport as well as fluidized beds. Pollution control devices, such as cyclone separa-tors and electrostatic precipitasepara-tors, are based on the principles of gas–solid flows. The combustion of coal in fossil–fuel power systems depends on the dispersion and burning of coal particles. The micron-size particles in solid-propellant rocket exhausts affect the performance of the rocket. Another example of a gas–solid flow is the motion of particles down a chute or inclined plane. These are known as granular flows where particle–particle and particle–wall interactions are much more important than the forces due to the interstitial gas. If the particles become motionless, the problem reduces to flow through a porous medium in which the viscous force on the particle surfaces is the primary mechanism affecting the gas flow; an example is a pebble-bed heat exchanger. In this case, even though one phase is not “flowing,” it will be included as a type of multiphase flow.

1.1.4 Liquid–Solid Flows

Liquid–solid flows consist of flows in which solid particles are carried by the liquid, and are referred to as slurry flows. Slurry flows cover a wide spectrum of applications that range from the transport of coals and ores to the flow of mud. These flows can also be classified as dispersed phase flows and are the focus of considerable interest in engineering research. The flow of liquid through a solid is another example of porous media flow.

1.1.5 Three-Phase Flows

Three-phase flows are also encountered in engineering problems. For example, bubbles in a slurry flow give rise to three phases flowing together. Not much work has been reported in the literature on three-phase flows. For many years, the design of multiphase systems has been based primarily on empiricism. However, more sophisticated measurement techniques have led to improved process control and the quantification of funda-mental parameters. The increase in computational capability has enabled the development of numerical models that can be used to complement engineering system design. The improvement of numerical models is a rapidly growing field of technology, which have far-reaching benefits in upgrading the operation and effi-ciency of current processes, and in supporting the development of new and innovative technologies.

1.1.6 Scope of the Handbook

This Handbook is designed to provide a background for engineers and scientists new to the field and to serve as a source of information on current technology for those familiar with the field. Chapter 1 in tro-duces fundamental definitions including size distributions. It also includes the fundamentals of particle– fluid–bubble interaction, which is the kernel of multiphase flows. Chapter 2 addresses gas–liquid flows and provides information on methods of calculating system parameters. Chapter 3 deals with boiling and condensation in multiphase systems. Chapter 4 covers pneumatic transport and slurry transport, which are very important to multiphase systems. Chapter 5 addresses fluidized beds, covering the basic concepts, heat and mass transfer, and applications. The flow of aerosol particles is presented in Chapter 6 with application to respiratory systems. The removal of dust and droplets from gas streams is covered in

Chapter 7. The dynamics and features of sprays are covered in Chapter 8, with applications to energy conversion, food processing, and metallurgical processes. Granular and porous media flows are

introduced in Chapters 9 and 10. The flow of multiphase systems in microgravity and microscale dimen-sions are the subjects of Chapter 11. Basic multiphase interactions are addressed in Chapter 12, including cavitation, bubble physics, droplet breakup, particle–particle and particle–wall interactions, erosion, tur-bulent dispersion, turbulence modulation, and combustion. Chapter 13 introduces numerical modeling, including direct simulation, Lagrangian and two-fluid modeling, the PDF approach and applications to cyclone separators, slurry flow, and fluidized beds. Chapter 14, the final chapter in the Handbook, addresses the current state of the art in instrumentation for measuring multiphase flows.

1.2 Fundamental Definitions

Clayton T. Crowe

This section introduces a few definitions that are fundamental to multiphase flows. For convenience, the term discrete or dispersed phase will be used for the particles, droplets, or bubbles, while carrier or con-tinuous phase will be used for the carrier fluid. The dispersed phase is not materially connected.

1.2.1

Volume Fraction and Densities

The volume fraction of the dispersed phase is defined as

αd
lim_δ

V→Vo (1.1)

where δV_dis the volume of the dispersed phase in volume δV. The volume δVo_{is the limiting volume}

that ensures a stationary average (Crowe et al., 1998). Unlike a continuum, the volume fraction cannot be defined at a point. Equivalently, the volume fraction of the continuous phase is

αc
lim_δ

V→Vo (1.2)

where δV_cis the volume of the continuous phase in the volume. This volume fraction is sometimes referred to as the void fraction and in the chemical engineering literature, the volume fraction of the dispersed phase is often referred to as holdup. By definition, the sum of the volume fractions must be unity, i.e.,

αdαc
1 (1.3)

The bulk density (or apparent density) of the dispersed phase is the mass of the dispersed phase per unit volume of mixture or, in terms of a limit, is defined as

ρ

苶d
lim_δV→Vo (1.4)

where δM_dis the mass of the dispersed phase. The bulk density is related to the material density ρ_dby

ρ

苶d
αdρd (1.5)

The sum of the bulk densities for the dispersed and continuous phases is the mixture density

ρ

苶dρ苶c
ρm (1.6)

1.2.2

Superficial and Phase Velocities

For multiphase flow in a pipe, the superficial velocity of each phase is the mass flow rate M. of that phase divided by the pipe area A and material density. The superficial velocity for the dispersed phase is

Ud
(1.7) M.d _ρ dA δMd _δV δVc _δV δVd _δV

In other words, it is the velocity of the phase if the phase occupied the whole pipe area. The phase velo-city u is the actual velovelo-city of the phase. The superficial velovelo-city and the phase velovelo-city are related by the volume fraction

Ud
αdud (1.8)

The same relations hold for the carrier phase.

1.2.3

Quality, Concentration, and Loading

Another parameter important to the definition of dispersed-phase flows is the dispersed-phase mass concentration,

C
(1.9)

which is the ratio of the mass of the dispersed phase to that of the continuous phase in a mixture. This parameter will sometimes be referred to as the particle or droplet mass ratio. Sometimes dispersed-phase volume fraction is designated as concentration.

The quality of a liquid–vapor mixture where the liquid in the dispersed phase is

x
(1.10)

Another term in common use in multiphase flows is loading, which is the ratio of mass flux of the dispersed phase to that of the continuous phase:

z
(1.11)

However, loading has also been used to denote concentration. The terminology is not uniform, as each author appears to have his/her own choice of symbols for multiphase flow parameters.

1.2.4

Response Times

The response time of a particle or droplet to changes in flow velocity or temperature is important in establishing nondimensional parameters to characterize the flow. The momentum response time relates to the time required for a particle or droplet to respond to a change in velocity. The equation of motion for a spherical particle in a gas is given by

m
CD ρc(u v)u  v (1.12)

where v is the particle velocity and u the gas velocity. Defining the dispersed-phase Reynolds number as

Re_r
(1.13)

and dividing through by the particle mass gives

(u v) (1.14)

where µcis the viscosity of the gas (continuous phase). For the limits of low Reynolds numbers (Stokes

flow), the factor CDRer/24 approaches unity. The other factor has dimensions of reciprocal time and

defines the momentum (velocity) response time

τV
(1.15) ρdd2 _18µ CDRer ₂₄ 18µ _ρ dd2 dv _dt ρcdu  v _µ c πd2 ₄ 1 ₂ dv _dt m.d _m. c ρ_苶d _ρ 苶m ρ 苶d _ρ 苶c

So the equation can be rewritten as

(u v) (1.16)

The solution to this equation for constant u and an initial particle velocity of zero is

v
u[1  exp(t/τ_V)] (1.17)

Thus, the momentum response time is the time required for a particle to be released from rest to achieve 63% (e-1)/e of the freestream velocity. This characteristic time has been used extensively in the multi-phase flow literature. Often τpis used to denote velocity response time.

The thermal response time relates to the responsiveness of a particle or droplet to changes in temper-ature in the carrier fluid. The equation for particle tempertemper-ature, assuming that the tempertemper-ature is uni-form throughout the particle, and radiative effects are negligible, is

mcd
Nuπkcd(Tc Td) (1.18)

where Nu is the Nusselt number, cdthe specific heat of the particle material and kcthe thermal

conduc-tivity of the continuous phase. Dividing through by the particle mass and specific heat gives

(Tc Td) (1.19)

For low Reynolds numbers, the ratio Nu/2 approaches unity; the other factor is the thermal response time defined as

τT
(1.20)

Thus the thermal equation for the particle becomes

(Tc Td) (1.21)

which is the time required for a particle to achieve 63% of a step change in the temperature of the carrier phase.

The momentum and thermal response times are related through the properties of the fluid and the particles, i.e.,

(1.22)

where Pr is the Prandtl number. For gases, the Prandtl number is of the order of unity; thus response times are of the same order of magnitude. For a liquid, the Prandtl number can be of the order of 102_{, which}

means that velocity equilibrium is achieved much more rapidly than thermal equilibrium in a liquid. Even though the above relations for the ratio of response times have been derived for low Reynolds num-ber (Stokes flow), the ratio changes little for higher Reynolds numnum-bers.

1.2.5

Stokes Number

The Stokes number is a very important parameter in fluid-particle flows. The Stokes number related to the particle velocity is defined as

Stk
τ_τV (1.23) F 1 _Pr c_c _c d 2 ₃ τV _τ T 1 _τ T dTd _dt ρdcdd2 _12k c 12kc _ρ pcdD2 Nu ₂ dTd _dt dTd _dt 1 _τ V dv _dt

where τFis the characteristic time of the flow field. If Stk  1, the response time of the particles is much

less than the characteristic time associated with the flow field. In this case the particles will have ample time to respond to changes in flow velocity and, the particle and fluid velocities will be nearly equal (velocity equilibrium). On the other hand, if Stk  1, then the particle will have essentially no time to respond to the fluid velocity changes and the particle velocity will be little affected by fluid velocity change. The sym-bol St is often used to denote the Stokes number of the particle but symsym-bol Stk has been used here to avoid confusion with the Strouhal number. Also Stk is used more frequently in the aerosol literature.

An approximate relationship for the particle/fluid velocity ratio as a function of the Stokes number can be obtained from the “constant lag” solution. The velocity ratio is expressed as φ
v/u and is assumed to vary slowly with time. By substituting this value into the particle motion equation, Eq. (1.16), one has

φ
(1φ) (1.24)

The carrier-phase acceleration can be approximated by

~ (1.25)

which, when substituted into Eq. (1.24) yields

φStk ~ (1φ) (1.26)

Finally, solving for φ, gives

φ
~ (1.27)

One notes that, as StV→0, the particle velocity approaches the carrier-phase velocity and as Stk →, the

particle velocity approaches zero, i.e., the particle velocity is unaffected by the fluid. The same idea extends to Stokes number based on thermal response time.

1.2.6

Disperse Versus Dense Flows

A disperse flow, is one in which the particle motion is controlled by the fluid forces (drag and lift). A dense flow, on the other hand, is one in which the particle motion is controlled by collisions. A qualitative esti-mate of the disperse or dense nature of the flow can be made by comparing the ratio of momentum response time of a particle to the time between collisions. Thus, the flow can be considered disperse if

 1 (1.28)

where τCis the average time between particle–particle collisions, because the particles have sufficient time

to respond to the local fluid dynamic forces before the next collision. On the other hand, if

 1 (1.29)

then the particle has no time to respond to the fluid dynamic forces before the next collision and the flow is dense. The time between collisions is generally estimated by using relationships from kinetic theory.

There is a further classification of dense flows: collision- and contact-dominated. In collision-dominated flow the collisions between the particles control the features of the flow, such as in a fluidized bed. In a con-tact dominated flow, the particle motion is controlled by continuous concon-tact such as in a granular flow.

1.2.7

Phase Coupling

An important concept in the analysis of multiphase flows is coupling. If the flow of one phase affects the other while there is no reverse effect, the flow is said to be one-way-coupled. If there is a mutual effect between the flows of both phases, then the flow is two-way-coupled.

τV _τ C τV _τ C 1 _{1 Stk} ud _u c u _τ F du _dt u _τ V du _dt

A schematic diagram of coupling is shown in Figure 1.1. The carrier phase is described by the density, tem-perature, pressure, and velocity field. However, it may be important to include the concentrations of the gaseous species in a gas phase. The particle or droplet phase is described by concentration, size, temperature, and velocity field. Coupling can take place through mass, momentum, and energy transfer between phases. Mass coupling is the addition of mass through evaporation or the removal of mass from the carrier stream by condensation. Momentum coupling is the result of an interaction force, such as a drag force, between the dis-persed and continuous phase. Momentum coupling can also occur with momentum addition or depletion due to mass transfer. Energy coupling occurs through heat transfer between phases. Thermal and kinetic energy can also be transferred between phases owing to mass transfer. Also, the presence of particles and droplets can also affect the carrier phase turbulence. Obviously, analyses based on one-way coupling are straightforward. One must estimate through experience or parameter magnitude if two-way coupling is important.

1.2.7.1 Mass Coupling

Suppose there are n droplets per unit volume in a box with side L as shown in Figure 1.2. Each droplet is evaporating at a rate m.. Thus, the mass generated by the dispersed phase per unit time due to evapora-tion is

M.d
nL3m

.

(1.30) The mass flux of the continuous phase through this volume is

M.c~ρ苶cuL2 (1.31) Pressure Temperature Velocity Species concentration Temperature Velocity Loading Particle size Mass Momentum Energy

Carrier fluid Dispersed phase

FIGURE 1.1 Schematic diagram of coupling effects.

L U

A mass coupling parameter is defined as

Πmass
(1.32)

IfΠ mass 1, then the effect of mass addition to the continuous phase would be insignificant and mass coupling could be treated as one-way coupling. This ratio can be expressed as

~ (1.33)

The ratio m./m scales as the reciprocal of a characteristic evaporation, burning, or condensation time,τm;

Thus, the ratio in Eq. (1.32) can be rewritten as

Πmass~ C (1.34)

By taking L as some characteristic dimension of the system, this ratio can be used to assess the impor-tance of mass coupling on the continuous-phase flow. Note that the ratio τ_mu/L can be regarded as the ratio of a time associated with mass transfer to a time characteristic of the flow. Thus, the ratio can be thought of as the Stokes number associated with mass transfer

Stkmass
(1.35)

hence, the mass transfer parameter can be expressed as

Πmass
(1.36)

Thus, if the droplet phase mass concentration and the mass exchange rate of the dispersed phase are low (large τmand Stkmass), mass coupling can probably be neglected.

1.2.7.2 Momentum Coupling

The importance of momentum coupling can be assessed by comparing the drag force due to the dis-persed phase with the momentum flux of the continuous phase. The momentum coupling parameter can be defined as

Πmom
(1.37)

where Ddis the drag force due to the disperse phase in the volume and Momcis the momentum flux

through the volume. The drag associated with disperse phase in a volume with side L is

Dd
nL33π µd(uv) (1.38)

based on Stokes drag. The momentum flux of the continuous phase is given by

Mom_c
ρ_苶cu2_L2 _(1.39)

The momentum coupling parameter can be expressed as

Πmom

冢

冣

where m is the mass of an individual element of the dispersed phase. The product nm is the bulk phase den-sity of the dispersed phase and the velocity ratio can be related to the Stokes number through Eq. (1.27).

Πmom
(1.41) C _1Stk mom v _u nmL _ρ∼ cuτV Dd _Mom c C _Stk mass τmu _L L _uτ m Lm. _um ρ 苶d _ρ 苶c M.d  M.c M.d  M.c

which shows the correct limit as the Stokes number approaches zero. Momentum coupling effects become less important for small concentrations (loadings) and large Stokes numbers.

1.2.7.3 Energy Coupling

Energy coupling follows the same model as used for mass and momentum coupling. The significance of energy coupling can be assessed by comparing the heat transfer to (or from) the dispersed phase and the energy flux of the continuous phase. The energy coupling parameter is defined as

Πener
(1.42)

Obviously, if Πener  1, then energy coupling is unimportant.

The heat transfer associated with the dispersed-phase elements in volume L3 _is

H._d
nL3π _Nuk

cd(Td  Tc) (1.43)

where Nu is the Nusselt number and kc is the thermal conductivity of the continuous phase. The energy flux of the continuous phase is

E.c
ρ苶cucpTc L2 (1.44)

where cp is the specific heat of the continuous phase. The thermal coupling parameter can be expressed as

Πener ~

冢

冣

where StT is the Stokes number based on the thermal response time (StkT
uτT /L). As in the case of

momentum coupling parameter, there is an indeterminacy as the Stokes number approaches zero. Using the same arguments as before, the energy coupling parameter can be expressed as

Πener
(1.46)

For gaseous flows, the momentum and thermal response times are of the same order; hence,

Πmom ~ Πener

and justification of one-way coupling for momentum usually justifies one-way coupling for energy transfer. If most of the energy transfer in a system is associated with latent heat in the dispersed phase, there is another form of the energy coupling parameter that may be appropriate:

ΠL ~ C (1.47)

which is the mass coupling parameter with an additional factor, hL /cp Tc . This factor can be large,

caus-ing energy transfer due to phase change to be important even though mass transfer itself is unimportant. In many spray problems, the energy coupling due to phase change may be the only two-way coupling that must be included in developing an analysis for a gas-droplet flow.

1.2.7.4 Three- and Four-Way Coupling

Further definitions of coupling include three-way coupling in which wakes and other disturbances in the carrier phase affect the motion of the discrete phase (particles and droplets). Four-way coupling addresses the situation where, in addition to discrete-phase–carrier-phase interaction, particle–particle collisions also affect the multiphase motion. This is the situation in dense flows. Four-way coupling effects become important when particle volume fraction exceeds 103. A schematic diagram of coupling is shown in

Figure 1.3. Strictly speaking, four-way coupling does not apply to granular flows because the effect of the interstitial fluid is negligible.

hL _c pTc 1 _Stk mass C _Stk T1 Tc _T d C _Stk T Q.d _E. c

However, the definition of three-way coupling could be included in the general category of two-way coupling. If this were the case, three-way coupling would be the case where particle–particle interaction is important.

1.3 Size Distribution

Clayton T. Crowe

Droplets or particle sizes are important parameters that govern the flow of a dispersed two-phase mix-ture. Hence, it is important to have a basic knowledge of the statistical parameters relating to particle size distributions. For spherical particles or droplets, a measure of the size is the diameter; for nonspherical particles, an equivalent diameter must be selected to quantify the size.

The most general definition of the spread of the particle size distribution is monodisperse or poly-disperse. A monodisperse distribution, is one in which the particles are close to a single size, whereas polydisperse distribution suggests a wide range of particle sizes. In monodisperse distribution, the stan-dard deviation is less than 10% of the mean particle diameter.

1.3.1 Size Distributions

Particle or droplet size distributions can be classified as discrete or continuous. The continuous size dis-tribution derives from the discrete disdis-tribution as the sampling interval approaches zero.

1.3.1.1 Discrete Size Distribution

Assume that the sizes of many particles in the sample have been measured by a technique, such as photo-graphy. One would choose size intervals, ∆d that would be large enough to contain many particles, yet small enough to obtain sufficient detail. The representative size for the interval could be the diameter cor-responding to the midpoint of the interval. The number of particles in each size interval is counted, recorded, and divided by the total number of particles in the sample. The results are plotted in the form of a histogram (bar chart) shown in Figure 1.4. This is identified as the discrete number frequency dis-tribution for the particle size.∗ The ordinate corresponding to each size interval is known as the number frequency ˜fn (di ). The sum of the number frequency over all the size categories is unity, i.e.,

冱

coupling Two-waycoupling

Particle Particle

Fluid

FIGURE 1.3 Schematic diagram of coupling.

∗

where N is the total number of intervals. When the distribution satisfies Eq. (1.48), it is a normalized distribution.

The number-average particle diameter of the distribution is obtained from d – n

冱

and the number variance is defined as†

σ2 n

冱

The variance is a measure of the spread of the distribution; a large variance implies a wide distribution of sizes. An alternative expression for the number variance is

σ2 n

冱

which is sometimes more convenient for calculations. The standard deviation is the square root of the variance.

σn
兹σ苶n2苶 (1.52)

Another approach to describe size distribution is to use the particle or droplet mass (or volume) in lieu of the number as the dependent variable. Thus, the mass of each particle would be obtained, or inferred, from measurement and the fraction of mass associated with each size interval would be used to construct the distribution. This is known as the discrete mass frequency distribution and identified as ~fm(di ). With

this distribution, one can calculate the mass-average particle diameter and mass variance as d–m

冱

冱

冱

FIGURE 1.4 Discrete number frequency distribution of particle diameter.

∗

The sum should be multiplied by the factor N/(N−1) to account for one degree of freedom removed by using the average diameter. However for N  1 we have N/(N1) ~ 1.

A very large number of particles or droplets have to be counted to achieve a reasonably smooth frequency distribution function. This is feasible with modern experimental techniques.

Another commonly used method to quantify particle size is the cumulative distribution, which is the sum of the frequency distribution. The cumulative number distribution associated with size dk is

F~n(dk)

冱

i
1

~

fn(di) (1.55)

The cumulative number distribution corresponding to the number frequency distribution shown in

Figure 1.4 is illustrated in Figure 1.5. The value of F~is the fraction of particles with sizes less than d_k. The value of F~n for the largest particle is equal to unity for normalized distributions. Both cumulative

num-ber and mass distributions can be generated from the corresponding frequency distributions. 1.3.1.2 Continuous Size Distributions

If the size intervals were made progressively smaller, then in the limit as ∆d approaches zero the contin-uous frequency function obtained, is

f_n(d)
lim

∆d→0 (1.56)

The number fraction of particles with diameters between d and d  d(d) is given by the differential quan-tity f_n (d) d(d). The variation of the frequency distribution with particle size is a continuous function as shown in Figure 1.6. Similarly, the differential quantity f_m (d) d(d) is the fraction of particle mass associ-ated with sizes between d and d d(d)

If the distribution is normalized, then the area under the frequency distribution curve is unity:

冕

0

f(d) d(d)
1 (1.57)

where dmaxis the maximum particle size.

The continuous cumulative distribution is obtained from the integral of the continuous frequency distribution Fn(d)

冕

and illustrated as the S-shaped curve in Figure 1.7. By definition, the cumulative distribution approaches unity as the particle size approaches the maximum size.

The equivalent continuous frequency and cumulative distributions for mass (or volume) fraction are obvious.

In general, size data are not available as a continuous distribution. It is common practice to consider the data obtained for a discrete distribution as values on a curve for a continuous distribution and to proceed accordingly to evaluate the appropriate statistical parameters.

1.3.2

Statistical Parameters

There are several parameters used to quantify distribution functions. Those used most commonly for dispersed phase flows are presented below.

1.3.2.1 Mode

The mode corresponds to the point where the frequency function is a maximum. The modes of a number and mass frequency distributions for a given sample will not be the same. A distribution that has two local

0.3

0.2

0.1

Particle diameter, d

Continuous frequency distribution,

fn

(

d

)

FIGURE 1.6 Continuous number distribution of particle diameter.

1.0 0.8 0.6 0.4 0.2 dnM Particle diameter, d

Continuous cumulative distribution,

fn

(

d

)

maxima is referred to as a “bimodal” distribution. The corresponding parameter for a discrete size distribu-tion is the most frequent diameter, i.e., the size that corresponds to the highest value for the frequency curve.

1.3.2.2 Mean

The mean of a continuous distribution is analogous to the average of a discrete distribution. The mean is calculated from the frequency distribution by evaluating the integral:

µ

冕

df(d) d(d) (1.59)

However, there is a number mean µ_nor mass mean µmsize depending on the frequency function used.

This is also called the arithmetic mean, diameter dAM. Other mean diameters include the surface mean

dSM

冤

冕

冥

dVM

冤

冕

冥

1.3.2.3 Geometric and Harmonic Mean Diameters

The geometric mean diameter is defined as ln dGM

冕

dmax

0

f(d)(ln d)d(d) (1.62)

The geometric mean corresponds to the mode of the log-normal distribution. The harmonic mean is obtained from dHM

冦

冕

冧

1.3.2.4 Sauter Mean Diameter

The Sauter mean diameter (SMD) is encountered frequently in the spray and atomization literature. It is defined as

d32
(1.64)

The SMD can be regarded as the ratio of the particle volume to surface area in a distribution that may have physical significance in some applications.

1.3.2.5 Variance

The variance of the distribution is calculated from

σ2

冕

0

(dµ)2_{f(d) d(d) (1.65)}

or by the equivalent expression

σ2

冕

0

d2_{f(d) d(d)}µ2 _(1.66)

Once again, the variance can be based on the number or mass distribution for particle size. As mentioned, previously, the variance is a measure of the spread of the distribution. The standard deviation is the square root of the variance and has dimensions of diameter.

冕

冕

A distribution can be classified as monodisperse if  0.1. 1.3.2.6 Median

The median diameter, dM, corresponds to the diameter for which the cumulative distribution is 0.5. The

number median diameter, dnM, is shown in Figure 1.7. The corresponding mass median diameter, dmM, is

determined from the cumulative mass distribution function.

1.3.3

Frequently Used Size Distributions

There are several size distribution functions frequently used to correlate particle or droplet size measure-ments. The characteristics of these distributions are discussed below.

1.3.3.1 Log-Normal Distribution

The log-normal number frequency distribution (Marshall, 1954) can be expressed as

fn(d)
exp

冤

冢

冣

2

冥

where dnMis the number median diameter and σois the geometric standard deviation. The

correspon-ding log-normal mass frequency distribution is

fm(d)
exp

冤

冢

冣

2

冥

where dmMis the mass median diameter. The geometric standard deviation can be found by plotting the

cumulative distribution on log-probability coordinates, which yields a straight line with a positive slope. The value for σ_ocan be found from

σo
ln (1.69)

where d84%is the diameter corresponding to the 84th percentile on the log-probability plot and dMis the

median diameter (value at the 50% point). The value for σ_ois the same for both the number and mass distributions.

A very useful relationship for manipulation of the log-normal distribution is dk Mexp(σo2k2/2)

冕

冤

冢

冣

冥

冕

which can be used to evaluate the various statistical parameters.

1.3.3.2 Upper-Limit Log-Normal Distribution

An off-shoot of the log-normal distribution is the upper-limit log-normal distribution (Mugele and Evans, 1951), which is designed to set a maximum particle diameter as the upper limit of the distribu-tion. The distribution is represented by

fn(d)
exp

冤

冢

冣

冥

where d_maxis the maximum diameter in the distribution, a is a constant, and σ_ois the geometric standard deviation. As the particle diameter varies from 0 to dmax,the argument of the exponent varies from 

to . The value of a is related to the median diameter by

a
 1

The standard deviation can be obtained from the slope of the log-probability plot of the data This distribution has been used to fit data from spray nozzles.

1.3.3.3 Square-Root Normal Distribution

The square-root normal distribution is given as (Tate and Marshall, 1953)

f_n(d)
exp

冤

冥

where σis the variance of the 兹d苶distribution. This distribution appears to fit data from vane-type atom-izers. Care must be taken when using the distribution because it does not have the tails of a normal distribution, so relationships for normal distributions may not be applicable here.

1.3.3.4 Rosin–Rammler Distribution

The Rosin–Rammler distribution (Mugele and Evans, 1951) is frequently used for representing droplet size distributions in sprays. It is expressed in terms of the cumulative mass distribution in the form

Fm(d)
1  exp

冤

冢 冣

冥

where δand n are two empirical constants. Note that Fm(0)
0 and Fm()
1.

The empirical constants can be determined by plotting the cumulative distribution on log–log coordi-nates. By taking the logarithm of the above equation twice gives

ln[ln(1  Fm(d))]
n ln d  n lnδ (1.75)

Thus, the slope, n, of the line is obtained by plotting ln[1Fm(D)] versus the diameter on log–log

coor-dinates. The parameter δcan be obtained by using n and the mass median diameter as

δ
(1.76)

The mass frequency distribution is obtained by taking the derivative of the cumulative distribution: fm(d)

e(D/δ)

n

冢 冣

(1.77) A very useful function for evaluating the statistical parameters for the Rosin–Rammler distribution is the gamma function, which is defined as

Γ(t)

冕

eλλtl_{dλ (1.78)}

It can be shown that

冕

冢 冣

冤

冢 冣

冥

冢 冣

冢

冣

冢

冣

冕

can be used to find statistical parameters from the Rosin–Rammler distribution.

α _n α _n d _δ d _δ d _δ d _δ n _δ dFm _dd dmM _0.6931/n d _δ (兹d苶兹d苶_苶)2 _2σ 1  2兹2苶πσ苶d苶 dmax _d M

1.3.3.5 Nukiyama–Tanasawa Distribution

The Nukiyama–Tanasawa distribution represents a more general distribution for which the Rosin–Rammler distribution is a special case. The Nukiyama distribution is

fn(d)
Bd2exp(Cdq) (1.81)

where B, C, and q are parameters chosen to fit the distribution. In a spray, q can vary from 1/6 to 2 with higher values corresponding to a narrower distribution.

1.3.3.6 Log-Hyperbolic Distribution

In some studies regarding sprays, it has been shown that neither the log-normal nor the Rosin–Rammler distributions fit the data sufficiently well. Two droplet size distributions fitted with a Rosin–Rammler distribution may have the same parameters but the actual distribution may be quite different.

One of the shortcomings of both the log-normal and Rosin–Rammler distributions is the representa-tion of the tails of the distriburepresenta-tion (values of the frequency funcrepresenta-tion for large and small values of the inde-pendent variable). Another distribution which is a better fit for the tails of the distributions is the log-hyperbolic distribution proposed by Barndorff-Nielsen (1977). The form of the frequency function is fn(d)
A exp

冤

冥

where α,β,δ, and µare fitting parameters and x is the random variable which, in this case, is the logarithm of the particle diameter. The coefficient A is a normalization factor that is related to the parameters by

A

where K1is the third-order Bessel function of the third kind. The logarithm of the frequency function is

lnfn(d)
lnA α兹δ苶2苶苶(苶d苶苶µ)苶2β(dµ) (1.83)

which is the equation of a hyperbola.For (dµ)/δ 0 the slope of the asymptote is δβ, while for (dµ)/δ 0 the slope of the asymptote is αβ. Thus, the logarithm of the frequency data can be plotted versus ln D and the slopes of the asymptotes can be measured to find αand β. The parameter µ is the mode of the distribution. The value for δmust be obtained from some fitting procedure.

1.4 Interactions of Fluids with Particles, Drops, and Bubbles

Efstathios E. Michaelides

1.4.1

Introduction

Because the sphere is the simplest of the three-dimensional shape, heat transfer and fluid dynamics problems pertaining to a sphere have been the subjects of the first applications of most theoretical methods on the solution of the governing equations, as well as of intellectual curiosity among the scientists. These problems belong to the class of the most fundamental problems in fluid dynamics, heat transfer, and mass transfer, and have attracted the attention of many mathematicians, physicists, and engineers. They are also subjects that have numerous practical applications including combustion and propulsion, chemical reactions and catalysis, mixing and separation, boiling and condensation, environmental sedimentation and resuspension as well as biological flows. The transport of momentum, heat, and mass are of primary interest in all of these processes, which are often expressed in terms of transport coefficients or in terms of dimensionless parame-ters of these coefficients. For example, the instantaneous loss or gain of mass of a sphere during any process may be expressed in terms of the mass transfer coefficient, hM or its dimensionless representation, the

兹α苶2苶_苶苶β苶2

 2αδK1(δ兹α苶2苶苶苶β苶2)

Sherwood number, Sh, and the material properties of the sphere and the carrier fluid:

m.

πhMd2ρf(Ys Y)
π(Sh) dᑞρf(Ys Y) (1.84)

Similarly, the instantaneous temperature change of the sphere, in the case of convective heating (when radiation is negligible) may be expressed in terms of the heat transfer coefficient, h, or its dimensionless representation, the Nusselt number:

mscps
Q.
πhd2(Ts T)
πNukf(Ts T) (1.85)

Finally, the equation of motion of a sphere may be expressed by Newton’s second law in terms of all the instantaneous forces that act on the sphere. These forces are the hydrodynamic drag force on the sphere, FH(which is composed of two parts, the drag and lift) and the gravity/buoyancy force FB. The former may

be written in terms of two dimensionless factors: the drag coefficient, CDand the lift coefficient, CL. The

equation of motion becomes:

ms
FDi FLi Fbi
πd2ρcCD(ui vi)冨ui vi冨

 πd2ρ

cCL兹(u苶m苶苶苶v苶m)(苶u苶m苶苶苶vm苶)苶  (ρsρc) d3gi (1.86)

Therefore, for the performance of any calculations on motion of the sphere one would need accurate information on the transport coefficients, i.e., drag coefficient, lift coefficient, heat transfer coefficient, and mass transfer coefficient. The numerical values of these coefficients emanate from analytical or numerical results and from experimental data.

This section intends to be a succinct presentation of the known analytical, experimental, and numerical results that pertain to the transport processes between a carrier fluid and bubbles, drops or particles, and are useful for the performance of engineering calculations. In particular, mass transfer, momentum transfer, and heat transfer from a particle, bubble, or drop with the carrier fluid will be presented. For a historical back-ground and a more extensive description of the development of these subjects one may consult a recent article by Michaelides (2003). Of the other recent studies, the main results on the momentum transfer from a sphere to a fluid as well as the energy and mass transfer at wide ranges of Reynolds and Peclet numbers may be found in several treatises and monographs, such as the ones by Leal (1992), Kim and Karila (1991), Crowe et al. (1998), Sirignano (1999), or the forthcoming one by Michaelides (2005). Also in specific review articles, such as the ones by Leal (1980) on the motion of fine particles at low Reynolds numbers, Brady and Bossis (1988) on the Stokesian formulation of suspensions, Feuillebois (1989) on the asymptotic methods applied to the equation of motion of spheres in viscous liquids, Sirignano (1993) on drops and sprays, Stock (1996) on particulate dispersion, Michaelides and Feng (1996) on the analogies between the heat flux and the motion of particles, Michaelides (1997) on the transient equation of motion of particles, Loth (2000) on the numer-ical methods for the treatment of the motion of bubbles, drops, and particles, Koch and Hill (2001) on the inertia effects of suspended particles, and Michaelides (2003) on the transient equations of motion and heat transfer. Some older monographs by Levich (1962), Clift et al. (1978), Happel and Brenner (1963), Govier and Aziz (1977), and Soo (1990) contain very useful theoretical and empirical results on the motion and heat/mass transfer processes from spheres as well as irregularly shaped particles.

1.4.2

Mass Transfer

1.4.2.1 Evaporation, Sublimation, and Condensation

Consider a sphere of radius a, inside a carrier gas of density ρf, in a spherical system of coordinates that

fol-low the center of the sphere. The carrier gas and the sphere have different chemical composition and their species are denoted by c and l, respectively. Mass is allowed to cross the boundary of the sphere due to one of the processes of sublimation, evaporation, or condensation. Because of this, the gaseous phase is

1 ₆ e_ijk(u_kv_k)Ω_j  兹Ω苶l苶Ωl苶 1 ₈ 1 ₈ dvi _dt dTs _dt dms _dt

composed of both the species l and c, while the sphere, which may be either solid or liquid, is composed solely of the species l. Figure 1.8 is a schematic diagram of such a process. The directions and numerical signs of the pressure and temperature gradients in this figure are such that they correspond to the case of condensation. The total pressure in the gaseous mixture is constant and equal to pT; the total pressure is

comprised of two parts: the partial pressure, or vapor pressure, of the species l, pl, and the partial pressure

of the carrier gaseous species c, pc. In all the transport processes that are pertinent to mass transfer to or

from the sphere, both partial pressures are functions of the radial direction, r:

pT
pc(r) pl(r) (1.87)

If the carrier phase is composed of a number of species, c1, c2, c3,…, as for example in the case of air, then the total pressure is the sum of all the partial pressures of these species:

pT

冱

i

pcl(r) pl(r) (1.88)

When the sphere is a liquid drop, the pressure in the interior of the drop would be higher than pTby the

amount contributed by the surface tension:

pint
pT (1.89)

In most practical applications the radius of the sphere, a, is large enough to satisfy the condition pT 2σ/a. Hence, the internal pressure, pint, is approximately equal to the total pressure in the gas phase,

pT. Thermodynamic equilibrium implies that the temperatures of the species l and c anywhere in the

gaseous mixture are equal, i.e., Tc(r)
Tl(r)
T(r), where r  a. The temperature inside the sphere may

be non-uniform and, hence, a function of the radial distance. This temperature will be denoted as Tint(r),

where r a.

At the surface of the sphere, there is a phase transition, liquid–vapor or solid–vapor that normally occurs within a very short layer of molecular dimensions. The phase of the matter in this film layer is not well defined. Hence, the material and thermodynamic properties within this molecular layer are not well defined. For this reason, they are depicted by the dashed lines in Figure 1.8. It is known that the specific properties of matter exhibit a jump discontinuity within this layer, e.g., from liquid enthalpy to vapor enthalpy. The intensive properties of matter, temperature, and pressure, exhibit a discontinuity of much

2σ _a pint Tint(r) pl(r) pc(r) p_T Tc(r) = Tl(r) = T(r) a 2
a r 0 r

FIGURE 1.8 Pressure and temperature fields developed in the vicinity of a drop carried by a fluid of different composition.