Starting with a summary of the fundamental principles and equations governing the behavior of bubbles, drops, and solid particles in Newtonian fluids, the text proceeds to a survey of the parameters used to characterize the shape of rigid particles, and of the factors that determine the shape of bubbles and drops. Succeeding chapters examine the behavior of solid and fluid particles under steady incompressible flow in an extended external phase. The text concludes with an exploration of effects that complicate the relatively simple case of a particle moving steadily through an unbounded fluid.
Starting with a summary of the fundamental principles and equations governing the behavior of bubbles, drops, and solid particles in Newtonian fluids, the text proceeds to a survey of the parameters used to characterize the shape of rigid particles, and of the factors that determine the shape of bubbles and drops. Succeeding chapters examine the behavior of solid and fluid particles under steady incompressible flow in an extended external phase. The text concludes with an exploration of effects that complicate the relatively simple case of a particle moving steadily through an unbounded fluid.
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Starting with a summary of the fundamental principles and equations governing the behavior of bubbles, drops, and solid particles in Newtonian fluids, the text proceeds to a survey of the parameters used to characterize the shape of rigid particles, and of the factors that determine the shape of bubbles and drops. Succeeding chapters examine the behavior of solid and fluid particles under steady incompressible flow in an extended external phase. The text concludes with an exploration of effects that complicate the relatively simple case of a particle moving steadily through an unbounded fluid.
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| ISBN-13: | 9780486317748 |
|---|---|
| Publisher: | Dover Publications |
| Publication date: | 03/25/2013 |
| Series: | Dover Civil and Mechanical Engineering |
| Sold by: | Barnes & Noble |
| Format: | eBook |
| Pages: | 400 |
| File size: | 26 MB |
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Bubbles, Drops, and Particles
By ROLAND CLIFT, JOHN GRACE, MARTIN E. WEBER
Dover Publications, Inc.
Copyright © 1978 R. Clift, J. R. Grace, and M. E. WeberAll rights reserved.
ISBN: 978-0-486-31774-8
CHAPTER 1
Basic Principles
I. INTRODUCTION AND TERMINOLOGY
Bubbles, drops, and particles are ubiquitous. They are of fundamental importance in many natural physical processes and in a host of industrial and man-related activities. Rainfall, air pollution, boiling, flotation, fermentation, liquid-liquid extraction, and spray drying are only a few of the phenomena and operations in which particles play a primary role. Meteorologists and geophysicists study the behavior of raindrops and hailstones, and of solid particles transported by rivers. Applied mathematicians and applied physicists have long been concerned with fundamental aspects of fluid-particle interactions. Chemical and metallurgical engineers rely on bubbles and drops for such operations as distillation, absorption, flotation, and spray drying, while using solid particles as catalysts or chemical reactants. Mechanical engineers have studied droplet behavior in connection with combustion operations, and bubbles in electromachining and boiling. In all these phenomena and processes, there is relative motion between bubbles, drops, or particles on the one hand, and surrounding fluid on the other. In many cases, transfer of mass and/or heat is also of importance. Interactions between particles and fluids form the subject of this book.
Before turning to the principles involved, the reader should be aware of certain terminology which is basic to understanding the material presented in later chapters. Science is full of words which have very different connotations in the jargon of different disciplines. The present book is about particles and the term particle needs to be defined carefully within our context, to distinguish it from the way in which the nuclear physicist, for example, might use the word. For our purposes a "particle" is a self-contained body with maximum dimension between about 0.5 µm and 10 cm, separated from the surrounding medium by a recognizable interface. The material forming the particle will be termed the "dispersed phase." We refer to particles whose dispersed phases are composed of solid matter as "solid particles." If the dispersed phase is in the liquid state, the particle is called a "drop." The term "droplet" is often used to refer to small drops. The dispersed phase liquid is taken to be Newtonian. If the dispersed phase is a gas, the particle is referred to as a bubble. Together, drops and bubbles comprise "fluid particles." Following common usage, we use "continuous phase" to refer to the medium surrounding the particles. In this book we consider only cases in which the continuous phase is a Newtonian fluid (liquid or gas). In subsequent chapters we distinguish properties of the dispersed (or particle) phase by a subscript ρ from properties of the continuous phase which are unsubscripted. Occasionally the dispersed and continuous phases are referred to as the "inner" and "outer" phases, respectively.
Another distinction we use throughout the book is between rigid, non-circulating, and circulating particles. "Rigid particles," comprising most solid particles, can withstand large normal and shearing stresses without appreciable deformation or flow. "Noncirculating fluid particles" are those in which there is no internal motion relative to a coordinate system fixed to the particle. "Circulating particles" contain fluid which has motion of its own relative to any fixed coordinate system. We consider only cases in which the dispersed phase is continuous. Hence the scale of the particle must be large compared to the scale of molecular processes in the dispersed phase.
In this book we consider as particles only those bodies which are biologically inert and which are not self-propelling. To give some specific examples, raindrops, hailstones, river-borne gravel, and pockets of gas formed by cavitation or electrolysis are all considered to be particles. However, insects and microorganisms are excluded by their life, weather balloons and neutrons by their size, homogeneous vortices by the lack of a clearly defined interface, and rockets and airplanes by their self-propelling nature and size. Our attention is concentrated on particles which are free to move through the continuous phase under the action of some body force such as gravity. Thus heat exchanger tubes, for example, are not considered—not only because of their size but also because they are fixed in position. Some elements of our definitions are of necessity arbitrary. For example, a golf ball satisfies our definition of a particle while a football does not. In most cases, there is little ambiguity, however, so long as these general guidelines regarding terminology are borne in mind.
Other terms which can be defined quantitatively are introduced in the following sections. Some other terms, such as "turbulence," "viscosity," and "diffusivity" are used without definition. For a full explanation of these terms, we refer the reader to standard texts in fluid mechanics, heat transfer, and mass transfer.
II. THEORETICAL BASIS
The fundamental physical laws governing motion of and transfer to particles immersed in fluids are Newton's second law, the principle of conservation of mass, and the first law of thermodynamics. Application of these laws to an infinitesimal element of material or to an infinitesimal control volume leads to the Navier–Stokes, continuity, and energy equations. Exact analytical solutions to these equations have been derived only under restricted conditions. More usually, it is necessary to solve the equations numerically or to resort to approximate techniques where certain terms are omitted or modified in favor of those which are known to be more important. In other cases, the governing equations can do no more than suggest relevant dimensionless groups with which to correlate experimental data. Boundary conditions must also be specified carefully to solve the equations and these conditions are discussed below together with the equations themselves.
A. Fluid Mechanics
1. The Navier–Stokes Equation
Application of Newton's second law of motion to an infinitesimal element of an incompressible Newtonian fluid of density ρ and constant viscosity µ, acted upon by gravity as the only body force, leads to the Navier–Stokes equation of motion:
ρ Du/Dt = ρg - [nabla]p + µ [nabla]2u. (1-1)
The term on the left-hand side, arising from the product of mass and acceleration, can be expanded using the expression for the substantial derivative operator
D/Dt = [partial derivative]/[partial derivative]t + u · [nabla]. (1-2)
where the first term, called the local derivative, represents changes at a fixed point in the fluid and the second term, the convective term, accounts for changes following the motion of the fluid. The ρg term above is the gravity force acting on unit volume of the fluid. The final two terms in Eq. (1-1) represent the surface force on the element of fluid. If the fluid were compressible, additional terms would appear and the definition of p would require careful attention. For discussions of these matters, see Schlichting (SS1), Bird et al. (BB1), or standard texts on fluid dynamics. Equation (1-1) is written in scalar form in the most common coordinate systems in many texts [e.g. (BB1)].
In the simplest incompressible flow problems under constant property conditions, the velocity and pressure fields (u and p) are the unknowns. In principle, Eq. (1-1) and the overall continuity equation, Eq. (1-9) below, are sufficient for solution of the problem with appropriate boundary conditions. In practice, solution is complicated by the nonlinearity of the Navier–Stokes equation, arising in the convective acceleration term u·[nabla]u. In dimensionless form, Eq. (1-1) may be rewritten as
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1-3)
where the primes denote dimensionless quantities or operators formed using dimensionless variables. Reference quantities L, U0 and p0 are used together with the fluid properties to form the dimensionless quantities as follows:
u' = u/U0 (1-4)
x1' = x1/L; y1' = y1/L; z1' = z1/L; t' = tU0/L (1-5)
pm' = (p - p0) (1-6)
Re = ρLU0/µ = LU0/v (1-7)
where hv is a coordinate directed vertically upwards. The Reynolds number, Re, is of enormous importance in fluid mechanics. From Eq. (1-3) it can be interpreted as an indication of the ratio of inertia to viscous forces. For convenience we have defined a dimensionless modified pressure, pm', which gives the pressure field due to the flow (i.e., discounting hydrostatic pressure variations). Batchelor (BB1) gives a good discussion of the modified pressure. It is useful in a wide range of problems where gravity effects can be isolated from the boundary conditions.
2. Overall Continuity Equation
Application of the principle of conservation of mass to a compressible fluid yields
[partial derivative]p/[partial derivative]t] + [nabla] · ρu = 0, (1-8)
which for an incompressible fluid reduces to
[nabla] · u = 0 (1-9)
In dimensionless form, Eq. (1-9) becomes simply
[nabla]' · u = 0. (1-10)
3. Velocity Boundary Conditions
In order to solve the Navier–Stokes equations for the dispersed and continuous phases, relationships are required between the velocities on either side of an interface between the two phases. The existence of an interface assures that the normal velocity in each phase is equal at the interface, i.e.,
un = un)p (everywhere on interface) (1-11)
where the subscript n refers to motion normal to the interface. For a particle of constant shape and size the normal velocity is zero relative to axes fixed to the particle. The condition on the tangential velocity at the interface is not as obvious as that on the normal velocity. There is now ample experimental evidence that the fluid velocity at the surface of a rigid or noncirculating particle is zero relative to the particle, provided that the fluid can be considered a continuum. This leads to the so-called "no-slip" condition, which for a fluid particle takes the form
ut = ut)p (everywhere on interface) (1-12)
where the subscript t refers to motion tangential to the surface.
Additional velocity boundary conditions are provided by the velocity field in the continuous phase remote from the particle and the existence of points, lines, and/or planes of symmetry. These conditions are set out in subsequent chapters for specific situations.
4. Stress Boundary Conditions
For solid particles a sufficient set of boundary conditions is provided by the no slip condition, the requirement of no flow across the particle surface, and the flow field remote from the particle. For fluid particles, additional boundary conditions are required since Eqs. (1-1) and (1-9) apply simultaneously to both phases. Two additional boundary conditions are provided by Newton's third law which requires that normal and shearing stresses be balanced at the interface separating the two fluids.
The interface between two fluids is in reality a thin layer, typically a few molecular dimensions thick. The thickness is not well defined since physical properties vary continuously from the values of one bulk phase to that of the other. In practice, however, the interface is generally treated as if it were infinitesimally thin, i.e., as if there were a sharp discontinuity between two bulk phases (LL1). Of special importance is the surface or interfacial tension, σ, which is best viewed as the surface free energy per unit area at constant temperature. Many workers have used other properties, such as surface viscosity (see Chapter 3) to describe the interface.
A complete treatment of interfacial boundary conditions in tensor notation is given by Scriven (SS2). If surface viscosities are ignored, the normal stress condition reduces to
pp + (τnn)p - p - τnn = σ[(1/R1) + (1/R1)
where R1 and R2 are the principal radii of curvature of the surface and the τnn are the deviatoric normal stresses (BB1, SS1). Under static conditions Eq. (1-13) reduces to the Laplace equation. The tangential stress condition corresponding to Eq. (1-13) is
τnt - (τnt)p = [nabla]s σ (1-14)
where the τnt refer to the shearing stresses and [nabla]s is the surface gradient (SS2). For a spherical fluid particle with both bulk phases Newtonian and an incompressible axisymmetric flow field, Eqs. (1-13) and (1-14) become
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1-15)
and
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1-16)
The final term in Eq. (1-16) is especially important for cases in which σ varies around the surface of a fluid particle due to concentration or temperature gradients (see Chapters 3, 5, and 7).
5. Stream Functions, Streamlines, and Vorticity
From the definition of a particle used in this book, it follows that the motion of the surrounding continuous phase is inherently three-dimensional. An important class of particle flows possesses axial symmetry. For axisymmetric flows of incompressible fluids, we define a stream function, ψ, called Stokes's stream function. The value of 2pψ at any point is the volumetric flow rate of fluid crossing any continuous surface whose outer boundary is a circle centered on the axis of symmetry and passing through the point in question. Clearly ψ = 0 on the axis of symmetry. Stream surfaces are surfaces of constant ψ and are parallel to the velocity vector, u, at every point. The intersection of a stream surface with a plane containing the axis of symmetry may be referred to as a streamline. The velocity components, ur and uθ, are related to ψ in spherical-polar coordinates by
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1-17)
The vorticity is defined as
ζ = [nabla] x u. (1-9)
It can be shown that ζ is twice the angular rotation of a fluid element. When ζ = 0 throughout a region of a fluid, the flow in that region is said to be irrota-tional. Flows which are initially irrotational remain irrotational if all the forces acting are conservative. Since gravity and pressure forces are conservative, vorticity generation in flow fields which are initially irrotational, such as around a particle accelerating in a stagnant fluid, arises from nonconservative viscous forces. For axisymmetric flows, vorticity can be treated as a scalar function. It is then often convenient to define surfaces of constant vorticity or lines of constant vorticity in a plane containing the axis of symmetry. Examples of streamlines and lines of constant vorticity are given in later chapters (for example, in Figs. 5.1 and 5.2).
It is often convenient to work in terms of a dimensionless stream function and vorticity defined, respectively, as
Ψ = ψ/U0L2 (1-19)
and
Z = ζL/U0 (1-20)
6. Inviscid Flow and Potential Flow Past a Sphere
In practice all real fluids have nonzero viscosity so that the concept of an inviscid fluid is an idealization. However, the development of hydrodynamics proceeded for centuries neglecting the effects of viscosity. Moreover, many features (but by no means all) of certain high Reynolds number flows can be treated in a satisfactory manner ignoring viscous effects.
For µ = 0 or Re -> ∞, Eq. (1-1) may be rewritten
which is the well-known Euler equation. Integration of Eq. (1-21) along either a streamline or parallel to ζ for steady incompressible flows leads to Bernoulli's equation, i.e.,
(p/ρg) + ([absolute value of u]2/2g) + hv = constant. (1-22)
From Kelvin's theorem, inviscid motions in a gravity (conservative) field which are initially irrotational remain so. We may, therefore, write
ζ = [nabla] x u = 0 (1-23)
Hence u may be written as the gradient of some scalar function, i.e.,
u = [nabla]Φ (1-24)
where Φ is conventionally termed a "velocity potential." From this designation, irrotational motions derive the name "potential flow." For incompressible potential flows it can be shown that Bernoulli's equation, Eq. (1- 22), applies throughout the flow field and that Φ satisfies Laplace's equation:
[nabla]2Φ = 0 (1-25)
If the flow is axisymmetric, ψ can be shown to obey the following equation in spherical polar coordinates (BB1):
r2 ([partial derivative]2/[partial derivative]r2ψ [partial derivative]θ2 - cot θ ([partial derivative]ψ/[partial derivative]θ) = 0 (1-26)
Since ψ by definition satisfies Eq. (1-9), potential flow solutions can be found by solving Eq. (1- 26) for ψ subject to the required boundary conditions. The pressure field can then be found using Eq. (1- 22).
Consider the case of a stationary sphere of radius a centered at the origin in a uniform stream of velocity --- -U. Equation (1-26) is second order and hence we require two boundary conditions. Remote from the sphere, the velocity must everywhere be -U , i.e.,
ψ = (-Ur2/2) sin2 θ as r -> ∞ (1-27)
No fluid crosses the sphere boundary. Hence the surface is a stream surface and since this boundary also cuts the axis of symmetry
ψ = 0 at r = a (1-28)
Equations (1-26) to (1-28) are satisfied by
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1-29)
Application of Eq. (1-17) gives
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1-30)
Since the pressure field depends only on the magnitude of the velocity (see Eq. (1-22)) and since the flow field has fore-and-aft symmetry, the modified pressure field forward from the equator of the sphere is the mirror image of that to the rear. This leads to d'Alembert's paradox: that the net force acting on the sphere is predicted to be zero. This paradox can only be resolved, and nonzero drag obtained, by accounting for the viscosity of the fluid. For inviscid flow, the surface velocity and pressure follow as
These results are useful reference conditions for real flows past spherical particles. For example, comparisons are made in Chapter 5 between potential flow and results for flow past a sphere at finite Re. Other potential flow solutions exist for closed bodies, but none has the same importance as that outlined here for the motion of solid and fluid particles.
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1-31)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1-32)
(Continues...)
Excerpted from Bubbles, Drops, and Particles by ROLAND CLIFT, JOHN GRACE, MARTIN E. WEBER. Copyright © 1978 R. Clift, J. R. Grace, and M. E. Weber. Excerpted by permission of Dover Publications, Inc..
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Table of Contents
This volume offers a critical review of the literature concerning the fluid dynamics, heat transfer, and mass transfer of single bubbles, drops, and particles. Upper-level undergraduates and graduate students, as well as professionals in the fields of engineering, physics, chemistry, geophysics, and applied mathematics, will find it a unified treatment of solid particles, liquid drops, and gas bubbles.Starting with a summary of the fundamental principles and equations governing the behavior of bubbles, drops, and solid particles in Newtonian fluids, the text proceeds to a survey of the parameters used to characterize the shape of rigid particles, and of the factors that determine the shape of bubbles and drops. Succeeding chapters examine the behavior of solid and fluid particles under steady incompressible flow in an extended external phase. The text concludes with an exploration of effects that complicate the relatively simple case of a particle moving steadily through an unbounded fluid.
Dover (2005) unabridged republication of the edition published by Academic Press, New York, 1978. xvi + 384pp. Appendices. Nomenclature. Index. 61/8 x 91/4. Paperbound.
ISBN 0-486-44580-1 $XX.XX in USA
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Vectors, Tensors and the Basic Equations of Fluid Mechanics, Rutherford Aris. 300pp. 53/8 x 81/2. 0-486-66110-5
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