Table Of ContentShih-1 Pai
Two-Phase Flows
VIEWEG TRACTS IN PURE
AND APPLIED PHYSICS
Volume 3
Advisory Board
Prof. Dr. H. Stumpf
University of Tlibingen, Germany
Prof. Dr. K. Wildermuth
University of Tlibingcn, Germany
Shih-I Pai
Two-Phase Flows
edited by K. Oswatitsch
With 82 figures
Springer Fachmedien Wiesbaden GmbH
Prof. Dr. Shih-I Pai
Institute for Physical Science and Technology
University of Maryland
College Park, Maryland, U.S.A.
CIP-Kurztitelaufnahme der Deutschen Bibliothek
Pai, Shih·j
Two-phase flows led. by K. Oswatitsch. - 1. Aufl. -
Braunschweig: Vieweg, 1977. (Vieweg tracts in pure
and applied physics; VoI. 3)
1977
AII rights reserved
Copyright © 1977 by Springer Fachmedien Wiesbaden
Originally published by Friedr. Vieweg & Sohn Verlagsgesellschaft mbH, Braunschweig in 1977
Softcover reprint of hardcover I st edition 1977
No part of this publication may be reproduced, stored in a retrieva1 system or transmitted
mechanically, by photocopies, recordings or other means, without prior permission of the
Copyright holder.
Set by Vieweg, Braunschweig
Printed by E. Hunold, Braunschweig
Bookbinder: W. Langeliiddecke, Braunschweig
ISBN 978-3-528-08340-3 ISBN 978-3-322-86348-5 (eBook)
DOI 10.1007/978-3-322-86348-5
To
Professor Shao-Ying Wang
VI
Preface
Matter may be divided into four phases or states: solid, liquid, gas and plasma (ionized
gas). The termfluid has been used as a general name for the last three states: liquid, gas,
and plasma because they may be deformed without applying any force, provided that the
change of shape occurs very slowly. Furthermore, when a large number of small solid
particles flow in a fluid, if the velocity of the fluid is sufficiently high, the behavior of such
solids is similar to ordinary fluid. We may consider these solid particles as pseudofluid.
Under proper conditions, we may treat fluid flow problems for solid (pseudofluid),
liquid, gas and/or plasma.
In ordinary fluid mechanics, we treat the flow problems of a fluid which is in one state
only, i.e., liquid, gas, or plasma, and the solids in this fluid medium are assumed to be
rigid bodies which may be considered as given boundary conditions of the flow problems.
Hence we study only one phase of the medium. Such a simple analysis of fluid mechanics
becomes insufficient in many engineering problems as well as many flow problems in
nature in which we have to consider simultaneously two or more states of matter in order
to solve the whole problem. Such a system is called Multi-phase flow. The most common
types of multi-phase flows consist of two phases only, i.e., any combinations of two of
the four states of matter. In this book, the author presents a systematic study of various
two-phase flows from the modern fluid dynamics point of view in which both macroscopic
and microscopic treatments are discussed. The analysis of these two-phase flow problems
may be generalized into multi-phase cases (see Chapter X).
There are many engineering and practical applications of two-phase flow problems
which we shall discuss in this book. For instance, pollution problems consist of many
two-phase flows. Hence two-phase flows have been extensively studied by many inves
tigators, particularly aerospace, chemical, civil, and mechanical engineers as well as
geophysicists and astrophysicists. However, because of the complexity of these issues,
most of the studies are empirical in nature and consider only specific problems. It seems to
the author that no textbook has been written to cover the entire field of two-phase flows.
In this book, tlie author intends to apply modern analytical techniques of fluid
mechanics to various two-phase flow problems so that the reader may have a better
understanding of the basic concepts of two-phase flows. It is felt that this would be
useful for further study and research in to this complicated subject. We shall not discuss
in detail the empirical facts of various two-phase flows which are enormous in quantity.
We shall point out only those essential facts to illustrate our analytical results. The reader
should refer to specific literature for these empirical facts, cited in the reference section.
In general, two-phase flows may be divided into two groups: The first group consists
of the study of the mixture of two phases of the four states: solid (pseudofluid), liquid,
gas, and plasma. These two phases may be mixed homogeneously or inhomogeneously.
Ordinary two-phase flows refer to problems in this first group only. The second group
consists of the study of the interaction between two phases of matter through their
interface. In each phase, the matter is a homogeneous medium, but we have to solve the
Preface VII
mechanics of these two phases simultaneously because they are coupled through their
interface. Many interesting problems belong to this second group of two-phase flows even
though such problems are not included in many two-phase flow books. In this book, both
groups of two-phase flows will be discussed so that we may have a general view of the
entire field of two-phase flows.
In Chapter I, we shall discuss qualitatively various two-phase flow problems so that the
reader may know the essential points of two-phase flows which are not usually included
in ordinary textbooks of fluid mechanics. In Chapter II, a brief review of the fundamental
properties of matter from a macroscopic point of view is given which is the foundation of
two-phase flow problems. We consider not only the essential properties of each phase of
a matter but also the behavior of phase transitions.
In Chapters III to XI, various types of two-phase flows are discussed. Firstly, we
discuss the first group of two-phase flows in which the two phases are mixed homo
geneously. In Chapter III, we consider the mixture ofliquid and its own vapor. This is
a special case of a mixture of liquid and gas which has not been extensively studied from
the modern fluid mechanics point of view, even though considerable empirical materials
are available through the study of steam turbines, boilers, etc. This new approach is
important for very high speed and high temperature flows of liquid which may be called
supers peed hydrodynamics as shall be discussed in Chapter III. In Chapter IV, we consider
the mixture of liquid and ~as of different matters. This is the classical two-phase flow
which has been extensively investigated. In Chapter V, we discuses the mixture of a gas
and small solid particles which may be considered as a pseudo-fluid. Both macroscopic
and microscopic treatments will be presented.
Secondly, we deal with the inhomogeneous mixture of two-phase flows. In Chapter VI,
consideration is given to inhomogeneous mixture of liquid and gas, particularly the
atomization of a liquid in a gaseous medium or the motion of gas bubbles in a liquid
medium. In Chapter VII, we consider two extreme cases of the mixture of solid and fluid:
one is a small amount of solid particles in a large quantity of fluid which is the problem
of sedimentation, and the other is a small amount of fluid in a solid medium which is
essentially the flow through a porous medium.
Thirdly, we treat the second group of two-phase flows. In Chapter VIII, we consider
the ablation problems which are studies of interaction between gaseous and liquid layers
and which have been investigated recently in connection with hypersonic flow of space
flight. In Chapter IX, we study the interaction between solid and fluid flow which is
known as aeroelasticity or hydroelasticity according to whether the fluid is a gas or a
liquid respectively. It is very important in the design of high speed aircraft or hydrocraft,
and in biomechanics.
In Chapter X, we consider the multi-fluid theory of a mixture of several fluids or
pseudo-fluids with special treatment of plasma. In principle, the multi-fluid theory is
applicable to any multi-phase system.
We neglect the effects of electromagnetic fields in the study of two-phase flows from
Chapter III to IX. There are many interesting practical problems of two-phase flows in
which electromagnetic fields play important roles. In Chapter XI, we consider the two-
VIII Preface
phase flows with the influence of electromagnetic fields, which may by called electro
magneto-fluid dynamics. We consider both the first and second group of two-phase flows
with the effects of electromagnetic fields. These problems include such interesting subjects
as magnetogasdynamics, electrogasdynamics, ferrohydrodynamics, collection of aerosol in
strong electrostatic fields and others.
The author wishes to take this opportunity to thank his wife, Alice Yen-Lan Wang Pai,
for her constant interest and encouragement during the preparation of this manuscript.
College Park, Maryland, U.S.A. Shih-1 Pai
IX
Contents
Preface
Chapter I. Introduction: Classification of Two-Phase Flows
1. Introduction 1
2. Liquid-Gas Flows 2
3. Liquid-Solid Flows 7
4. Gas-Solid Flows 9
5. Liquid-Plasma Flows. Electrohydrodynamics 11
6. Plasma-Solid Flows. Electromagneto-Fluid Dynamics 11
7. Gas-Plasma Flows. Multifluid Theory of a Plasma 12
8. Methods of Treatment 12
9. References 13
Chapter II. Properties of Matter 15
1. General Description of Matter 15
2. Thermodynamic Relations 20
3. Gases and Vapors 25
4. Liquids and Wet Vapors 36
5. Plasma and Mixture of Gas and Vapor 41
6. Some Properties of Solids 48
7. Electromagnetic Properties of Matter 50
8. References 54
Chapter III. Superspeed Hydrodynamics. Mixture of Liquid and Its Own Vapor 56
1. Introduction 56
2. Fundamental Equations of Superspeed Hydrodynamics 58
3. Dryness Fraction 62
4. Thermodynamics of a Mixture of Liquid and Its Vapor 64
5. Gasdynamics of Condensing Vapors 67
6. One-Dimensional Flow of Liquid and Its Vapor 75
7. Boundary Layer Flow of Superspeed Hydrodynamics 80
8. Some Advanced Problems of Superspeed Hydrodynamics 83
9. References 85
Chapter IV. Two-Phase Flows of Gas and Liquid 87
1. Introduction 87
2. Empirical Treatment of the Flow of a Mixture of a Gas and a Liquid in Pipes 88
3. Fundamental Equations of a Homogeneous Mixture of Gas and Liquid.
Froth Flow 90
4. Sound Wave in a Froth Flow 97
5. Shock Wave of a Froth Flow 100
6. One-Dimensional Steady Froth Flow 102
7. Flow with the Effect of Condensation of Vapor 104
8. Flow with Effect of Evaporation of Liquid. Cavitation Flow 106
9. Stratified Flow 108
10. Two-Phase Flow with Non-Uniform Surface Tension 111
11. References 114
