Analogies between the transient momentum and energy equations of particles

Prog. Energy Combust. Sci. Vol. 22, pp. 147 162, 1996 Copyright © 1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0360-1285/96 $29.00

Pergamon PII: S0360-1285(96)00002-0

ANALOGIES BETWEEN THE TRANSIENT M O M E N T U M AND E N E R G Y EQUATIONS OF PARTICLES Efstathios E. Michaelides and Zhi-Gang Feng Department of Mechanical Engineering, Tulane University, New Orleans, LA 70118, U.S,A.

Al~traet--Analytical forms of the rectilinear equation of motion and energy equation of particles, droplets or bubbles have been developed for very low Reynolds and Peclet numbers. Some of the early work on the two equations is briefly explored and recent advances are presented in more detail. Particular emphasis is placed on the analogies and similarities between the momentum and energy equations, and the ways the similarities have been utilized in practice. The creeping flow assumption, on which most of the known analytical forms are based, is critically examined. The semiempirical and empirical versions of the momentum and energy equations, which are widely used in engineering practice, are also presented, as well as a numerical method to deal with the history terms. Implicit assumptions on the use of the empirical equations are exposed. An erroneous result pertaining to the droplet flows, and a paradox related to the typical history terms are examined, and their rectification is pointed out. Recent results on the motion and heat transfer at finite Reynolds and Peclet numbers are also exposed. In this case, the momentum and thermal wakes around the particle play an important role in the momentum and energy transport process. Copyright © 1996 Elsevier Science Ltd.

Keywords:particle, bubble, droplet, history terms, transient motion, transient heat transfer, creeping flow. CONTENTS 1. Historical background 2. The equations of motion and heat transfer at the limit of zero Re and Pe 3. The equations for a viscous and conducting sphere for zero Re and Pe 4. Empirical and semi-empirical equations 5. The case of a spheroid 6. The equations of motion and energy at finite Re and Pe 7. The equations of motion and energy as used in practice 8. The effect of the history terms Acknowledgements References

I. HISTORICAL BACKGROUND It is rather extraordinary that the first scientific works on the energy and m o m e n t u m equations of particles appeared a few decades before the development and general recognition of what we now call 'First Law of Thermodynamics' and ' N a v i e r - S t o k e s equations'. Fourier's celebrated book 1 on the analytical theory o f heat appeared in 1822, a couple of decades before the appearance of the pioneering work o f Joule on the first law of thermodynamics and many years before the work of Meyer was accepted in scientific circles. Poison 2 in 1831 (almost fifteen years before the formulation of the N a v i e r - S t o k e s equations) wrote the first paper on the transient equation of motion of a sphere in a potential flow field. He correctly predicted that the coefficient of what we now call 'the added mass term' is ½. Fourier's primary interest appears to have been in geophysics and especially in the global temperatures and the cooling o f the earth. In his book of 1822 and in a series o f 'memoirs' and articles, published by the

147 148 149 152 153 154 158 159 160 160

Academy of Sciences of Paris and the Annales de Chimie et de la Physique between 1816 and 1829, he defined conductivity, exposed lucidly the analytical theory of heat conduction, applied this theory to the problem of earth's cooling, presented solutions for the conduction equation of solids, and even commented on the physics of the radiation heat transfer. His treatment of the cooling earth is the first exposition of transient heat transfer. It is remarkable that Fourier subscribed to the caloric theory, which treats heat as a (weightless) fluid. In the numerous works by him we meet the 'movement' or 'communication' of heat, but not the 'transfer' of heat or the 'transfer of energy'. Regardless of this, he treated the subject of heat conduction with authority, precision and thoroughness, which left little r o o m for immediate extension or improvement on the theory. F o r this reason, there is very little in the literature on the heat transfer or the energy equation of particles until the mid- 1950s. Tait 3 in 1885 declared on Fourier's work that " . . . Its exquisitely original methods have been the source of inspiration of some o f the greatest mathematicians; 147

148

E.E. Michaelides and Z.-G. Feng

and the mere application of one of its simplest portions to the conduction of electricity has made the name of Ohm famous". Starting in the 1830's a number of scientists were studying the motion of the pendulum as a means of producing more accurate clocks. Poison 2 was the first to study the transient motion of a sphere in a nonviscous fluid. He was soon accompanied by Green 4 who studied the potential flow of an ellipse in an infinite fluid. Clebsch s completed these potential flow studies by examining the effect of rotation of the ellipse. Stokes 6'7 introduced the viscous effects in the studies of particulate flows and presented an expression for the hydrodynamic force on a sphere moving sinusoidally in a viscous fluid. He also showed that the steady-state drag coefficient on a sphere is: cn = 24~Re.* His results were obtained as a solution to what we now call 'the Stokes equations' of particle motion, which are essentially the Navier-Stokes equations without the non-linear inertia terms. In a strict sense, these equations are only valid at the limit of zero Reynolds number (based on the relative velocity of the sphere). The more modern term 'creeping flow' characterizes this type of flow around an object of any shape. This term implies that the object may move very slowly inside the viscous fluid. The transient equation of motion of a sphere moving with an arbitrary velocity V(t), at creeping flow conditions was first derived by Bousinesq. 8'9 Three years later Basset 1°'11 independently derived the same equation, under the same conditions. The pseudo-steady-state drag term, the added mass term, and the history integral term (which represents the effect of the diffusion of vorticity around the sphere) appear prominently in Boussinesq's and Basset's expressions of the hydrodynamic force exerted by the fluid. Given the chronological precedence of Boussinesq's work, it is ironic that the history term of the hydrodynamic force has since been called by many researchers the 'Basset force' or the 'Basset integral'. The first attempt to relax the creeping flow assumption was made by Whitehead, 12 shortly after the book by Basset H was published. He used Stoke's method and attempted to improve upon the latter's work by obtaining higher order approximations to the flow, when Re is small but finite. His method is equivalent to an expansion in terms of the Reynolds numbers. However, it does not yield valid results, because it fails to satisfy completely the boundary conditions far from the sphere, as pointed out by Oseen. 12 Oseen 13'14 is also credited, among other developments, with the improved prediction of the * The dimensionless group pVa/v was not denoted by Re and was not named 'the Reynolds number' until the twentieth century. For convenience and consistency we will u s e the modem notation and definitions, even though the original authors employed different symbols and names.

steady-state drag coefficient at finite Reynolds numbers [co = 24(1 + 3/8Re)/Re]. His student Faxrn 15 contributed to the equation of motion of a particle by including the non-uniform terms of the fluid velocity field. An account of the work, that was performed in the nineteenth and the first half of the twentieth century, on the hydrodynamic forces, the transient equation of motion of a particle and the transient heat transfer may be found in some of the older treatises, such as the ones by Lamb, 16 Dryden et al., 17 Oseen, 18 Villat, 19 Freeman's 2° translation of Fourier's book, Preston 21 and Carslaw and Jaeger. 22 Since the problems of the transient motion and heat transfer of particles, droplets and bubbles are important and intriguing from the scientific as well as the engineering point of view, several researchers contributed to them. In the following sections the more recent contributions will be examined, as they influenced the form and applications of the equations of energy and rectilinear motion. It must be pointed out that in a short survey, such as this one, it is impossible to present every single study on the subject. The works presented here have significantly contributed to the development of the equations of motion and energy from the scientific point of view, or have profoundly influenced the engineering applications.

2. THE EQUATIONS OF MOTION AND HEAT TRANSFER AT THE LIMIT OF ZERO Re AND Pe

BoussinesqS,9 and Basset 10,11 independently derived an expression for the transient hydrodynamic force (the prevalent name at the time was 'fluid resistance') exerted by an infinite, quiescent fluid on a rigid scheme, initially at rest. Since in their derivations they neglected the inertia terms, their expression is valid at the time of Re --o 0 (creeping flow conditions). dV

mv d V + 6oL2 /-~-p~ [t ~--r~--r dr

F 1 = 67rol~f V + -~- dt

J0 x/t - ~-

(1) where a is the radius of the sphere, p is the density, # the dynamic viscosity, m is the mass and t denotes the time. V(t) is the arbitrary velocity of the sphere and the implicit assumption in the derivation is V(0) = 0. The subscripts f and p pertain to the fluid and the sphere respectively. The first part of the right-hand side of Eq. (1) is the pseudo-steadystate drag, which in this case (creeping flow) is equal to the Stokes drag. The second term is now called the added mass. It represents the contribution of the mass of fluid, which is accelerated by the motion of particle. The third term originates from the diffusion of velocity gradients in the infinite fluid and is called by some the 'Basset term' and by others the 'history integral term'. The three terms of Eq. (1) appear as three independent components of the hydrodynamic

Transient momentum and energy equations of particles

149

force. For this reason, they were treated as indepen- essentially followed the method of Maxey and dent forces by several researchers who eventually Riley2s and adapted it to the subject of the transfer developed the semiempirical equations for the hydro- of a scalar (energy). The creeping flow hypothesis in the case of the transient heat transfer yields Pe --* O. dynamic resistance. Tchen 23 attempted to extend Eq. (1) to the case of a In accordance to the rigidity of the sphere in the fluid in motion and a particle with a finite initial equation of motion, the heat transfer analogy velocity V(0). He was successful in the first part. His assumed a sphere of infinite conductivity (Bi = 0). equation is valid for the motion of a particle, starting Their investigation exhibited the precise analogies from rest, when it moves in a fluid of uniform velocity between the momentum and the energy equation of a u(t). However, Tchen's equation is not valid in the sphere. They separated the imposed external temcase of a finite initial velocity of the particle. This fact perature field into an undisturbed field T°(x, t) and has led to an erroneous expression, which was used by the disturbance due to the presence of the sphere many others, in the area of the turbulent diffusion of Tl(x,t), and derived the following differential particles. 24-26 The reason for the inadequacy of equation for the temperature change of the sphere: Tchen's expression is that, when the particle does not start from rest, the history integral must be mscs d---t =- mfcf ~=o complemented by the quantity V(O)/t 1/2, which accounts for the contribution of the initial condition on the inverse-Laplace transform of a convolution - 47rc~kf [Ts(t ) - #(~', t)b= 0 integral. In this case, V(0) is the relative velocity of the sphere at the inception of motion. This omission was pointed out in the study of turbulent dispersion -- 610/2~2#(Z'0Zi0Zit) coefficients by Reeks and McKee. 27 The unabridged derivation of the equation of d motion of a rigid sphere by Maxey and Riley28 and its application to the case of turbulent dispersion by -6 ~ dr. Maxey, 29'3° clarified most of the issues and problems on the intrinsic assumptions of the equation of (2) motion of a sphere, extended the equation to a fluid of non uniform velocity field u(x, t) and settled the Of the terms which appear in the energy equation, question of the missing term in Tchen's expression. 2s the first one represents the rate of change of the Among the more recent studies, one should point out temperature of the sphere. The second term is the •the one by Feuillebois and Lasek, 31 who pointed out contribution of the undisturbed outer field and is the the existence of a history torque term in shear flow. rate of heat, that would have entered the control The analogue of the creeping flow conditions in the volume occupied by the sphere in the absence of it. heat transfer is the unsteady (transient) conduction. This term is analogous to the added mass term of the Until recently, research on this subject was restricted equation of motion. The third term accounts for the to specific thermal processes with practical applica- conduction from the sphere to the fluid, due to the tions, such as a step temperature change, sinusoidal temperature difference and to the curvature of the temperature variation, etc. For this reason, a general temperature field. This is the pseudo-steady-state expression for an arbitrary temperature field in the conduction term and is analogous to the similar drag surrounding fluid or in the interior of the sphere was term of the equation of motion. The fourth term is a not examined until very recently. In the early treatise history integral, which is the result of the diffusion of by Carslaw and Jaeger, 22 Duhamel's theorem was the temperature gradients inside the temperature field. used to derive a solution for the case where the It depends on the temporal as well as the spacial process is defined as a step temperature change. The variation of the temperature field and is analogous to appearance of the error function in the derived the history term of the equation of motion. For solution alludes to the presence of a history integral certain specific thermal processes, this term may be term, which may appear in a more general process. analytically integrated. Among the related studies with specific transient thermal processes, Cooper 32 and then Brunn 33 examined two related problems to the heat conduc3. THE EQUATIONS FOR A VISCOUS AND CONDUCTING SPHERE FOR ZERO Re AND Pe tion from a sphere to an infinite medium, when the initial temperatures of the sphere and the medium are different. In his well-known treatise on conduction, Equation (1) is derived as a solution to the NavierOzisik34 presents most of the known results and Stokes equations of the surrounding fluid, with the boundary condition that the surface velocity is equal methods for specific unsteady conduction processes. Michaelides and Feng 3s conducted a study, which to the uniform velocity of the sphere, V(t). Hence, it is analogous to the ones by Boussinesq and Basset, for applies to a solid sphere only, with no slip at the the transient heat transfer from a sphere. They interface. However, early researchers have used it

e=0] -- 47rc~2kf

150

E.E. Michaelides and Z.-G. Feng

extensively in studies pertaining to the transient motion of bubbles and droplets, where internal motion exists. This practice is recommended by Hughes and Gilliland, 36 who give an account of some of the early theoretical works and applications of droplet motion, and advocate its use in problems pertaining to droplets. Many of the researchers in the 1960s and 1970s on the heat transfer and evaporation of droplets also use the rigid particle equation of motion. 37-41 A review on droplets carried by a gaseous medium 42 is typical of this practice: it essentially treats the droplets as small rigid particles, which may evaporate. The case of a viscous sphere (bubble or droplet) differs from that of a solid sphere, because there is internal motion. This fact alters the boundary conditions at the surface of the sphere in the following ways: a) there is an internal velocity field, which makes the velocity at the surface of the sphere different than that of the center; b) the velocity of the outside fluid is equal to the local velocity on the surface of the viscous sphere, if no slip exists between the two phases; and c) the shear stresses of the two fluids at the interface of the sphere balance. These differences in the boundary conditions lead to important differences in the solution of the NavierStokes equations, from which (1) emanates. Similarly, the case of a sphere with finite conductivity (finite Biot number), includes internal temperature gradients, which are not accounted for in the derivation of Eq. (2). Hence, the boundary conditions of the energy equation at the interface are: a) the local temperatures of the fluid and the sphere are equal at the interface, and b) the local rates of heat transfer at the interface are balanced. The transient momentum and energy transfer from a viscous or a conducting sphere at the limit of creeping flow (Re ~ 0 and Pe ~ 0 respectively) are diffusion problems, which are modelled by parabolic differential equations. The momentum diffusivity (kinetic viscosity) and the thermal diffusivity of the problems are of course different in the interior and the exterior of the sphere. This implies that the processes are characterized by two timescales each, one for the interior and one for the exterior domain. The timescales for the two processes are: for the momentum transfer: c~2

~-f = - -

vs

Oz2

and

7-p = - up

(3)

, ppcpa: Tp= kp

(4)

and for the energy transfer: T)=

psCsa~ ks

and

where a is the radius of the sphere, u is the kinematic viscosity, c the specific heat capacity and k the thermal conductivity. Sy e t al. 43 conducted a detailed study on the transient creeping flow of a sphere at the two limits of

infinite internal viscosity (solid particle) and zero internal viscosity (inviscid bubble). Their result for a solid sphere is equivalent to that of Boussinesq and Basset, but the method they used is slightly different from the one of the last authors. One year later, Sy and Lightfoot44 extended the previous results to spheres of finite viscosity. Konoplin, 45 in the light of the last two studies, hastened to proclaim an analogous expression for the heat transfer. However, it is rather unfortunate that in their derivation, Sy and Lightfoot44 combined a dimensional boundary condition with dimensionless governing equations. This has led to an inaccurate expression for the hydrodynamic force on the sphere, when the viscosity ratio of the sphere to the fluid is finite. The correct formulation of this problem and its solution was obtained by Chisnell, 46 who also pointed out that the process of momentum transfer has two timescales. Chisnell derived an accurate (albeit complicated) expression, in the Laplace domain, for the transient velocity of a viscous sphere in a gravitational field at Re ~ O. This expression cannot be analytically transformed in the time domain. He also obtained the asymptotic behavior of the solution at short and long times and asserted that the viscous drop has to remain spherical in shape as long as the motion may be described by the Stokes equations. It appears that Chisnell's work was not immediately noticed, because several studies were conducted afterwards, without reference to this work. A recently published and otherwise meritorious text, by Kim and Karilla, 47 has used the misleading formulations and derived erroneous results similar to the ones described in the last paragraph, while the monograph by Lea148 avoids any mention to the unsteady processes. A short paper by Yang and Lea149 on a new memory term also suffers from the same drawback, although the basic premisses of the paper, that there are new memory (history) terms and that the equation of motion of viscous sphere is a great deal more complex than (1), are correct. The question of the transient motion of a viscous sphere was revisited more recently by several other researchers, 5°-s2 who extended ChisneU's work and presented the appropriate formulation and solution to the problem of the accelerating viscous sphere under creeping flow conditions. In the last of these studies, Michaelides and Feng 52 examined the possibility of interfacial slip between the external fluid and the sphere. Slip was observed in some materials processes involving the formation of nanoclusters, 53 as well as the aerosol motion in the upper atmosphere. 29 The two timescales in the equation of motion render the problem of the transient momentum transfer from a viscous sphere very complex. Thus, the derivation of a general analytical solution of the problem, in the time domain, is an impossible task. Analytical solutions to this problem may only be derived in the Laplace domain. 46 '50- 52 From these, asymptotic solutions were obtained in the time

Transient momentum and energy equations of particles domain, for short- and long-times since the inception of motion. In the most complex case, which includes the presence of slip at the interface, the expression for the hydrodynamic force in the Laplace domain is as follows: 52

~, =-67m#f[Ti(s)-~(f(t),s)]{ ~+ Af + l _

()~f + I)2([Ap3 - Ap2 tanh(Ap) - 2f(Av)]~a + f ( A p ) ) "~ [1 + a(A t- + 3)][Ap3 - Ave tanh(Ap) - 2f(Ap)]e~ + (Af + 3ff'(Ap) J '

(5) where the overbar denotes the Laplace transform of a function, n is the ratio of the dynamic viscosities (~ = #p/#f) and cr is a dimensionless parameter related to the slip coefficient (a = #f//3a). The slip coefficient, /5 (which is the same as the one contemplated by Basset, n and used in the case of steady-state motion by Happel and Brenner54), is equal to the shear stress on the surface of the particle divided by the slip velocity at the surface of the sphere (/3 ~-O'rO/6Wo).The parameters Af and Ap may be considered as two dimensionless variables in the Laplace domain, which are related to the two timescales of the problem: Af = ~

and

Ap = sv/3~p,

(6)

where s is the Laplace transformation variable. The function f of Eq. (5) is defined as: f ( ( ) = (~2 + 3) tanh(() - 3~.

(7)

Equations (5) through (7) represent the hydrodynamic force in the most general case of the rectilinear motion of a spherical particle, bubble or droplet, to be encountered under the creeping-flow conditions. It is no surprise that the equations are complex and that a general analytical solution in the time-domain is unfeasible. All cases of practical interest may be modelled as special cases of this set of equations. For example, when the interfacial slip is zero (cr = 0), the resulting expression is the same as the one derived by Galindo and Gerbeth, 5° or Lovalenti and Brady. 51 In the limit of infinite viscosity for the sphere, Eq. (5) results in the expression derived for the rigid sphere. When the internal viscosity is zero, the equation of motion for an inviscid bubble is obtained. Remarkably, the cases of infinite slip coefficient (/3 = oc, or = 0) and of an inviscid sphere (#p = 0) are equivalent and result in the same equation of motion. The Laplace transform of the hydrodynamic force for the latter case was obtained by Morrison and Stewart 55 in an implicit form, while Eq. (5) yields this force in an explicit form. s2 In the limiting case of an inviscid sphere, the pseudo-steady-state drag is confirmed to be two thirds of that for a solid sphere (r'n'#ozV).52'54'55 This value agrees with the classical result for an inviscid sphere, which was derived early in the century by Hadamard 56 and RybczynskiY It must be pointed out that, Eq. (5) is very complex even at zero slit. It is obvious that a general analytical

151

expression for the total hydrodynamic force exerted on a viscous sphere may only be obtained in the Laplace domain and not in the time domain. It is only in special cases, where the motion of the sphere is well-defined, that such an expression may be obtained in the time domain. All the terms associated with the Basset/Boussinesq expression for the hydrodynamic force may also be identified in Eq. (5). Thus, the first term in the curly brackets (A~/9) would yield the added mass term of the expression for the solid sphere; the second term (Af) would yield the history integral term; and the third term (1) would yield the pseudo-steady-state drag. The last term in the curly brackets is a residual term, which demonstrates the complexity of the flow field and of the vorticity created by the viscous sphere. A term similar to this was called 'a new memory term'. 49'50 The energy analogue of the case of a viscous sphere is a sphere with finite conductivity. The Biot number in this case is also finite. As in the case of the momentum transfer, an analytical expression for the rate of heat transfer may be obtained in the Laplace domain. Such analytical expressions for the rate of heat transfer, in the time domain, may only be obtained at the asymptotic limits of long and short times from the inception of the heat transfer process. Feng and his co-workers 5s-6° have tackled the problem of the transient energy equation in an analogous way others solved the problems of the transient equation of motion. They derived the corresponding expressions for the heat transfer, with an arbitrary initial temperature field Tp(xi, 0), which in the Laplace domain is as follows:

-

(~p. ~ as = 47rakf [t~f(A~)+ (1 + A)a) sinh(A~a)] × [(1 + ~).~)~(~ = o) + (½+ ~ ~ , ~ ) ~ 2 v ~ ( ~ [(1 +,k~a

a2

-

= o)]

] (V2Tp)(0, 0) J

~'~kp(1 + ~)~) -+ s[mV(A~)+ (1 + A~a) sinh(A;a)] sinh(Apr)

× fvpVTp(-~,O)'V[~]

dv.

(8)

The parameters A' are the corresponding analogous quantities to the parameters A of Eq. (6). They are defined in terms of the characteristic timescales of the energy transfer process and the Laplace variable s: A~=~f

and

Ap=V/~p,

(9)

The dimensionless function g of the two parameters ,x) and ,X~ is: =

~'A;a cosh(A~a) - ~' sinh(A;a) a'A~acosh(A~a) - ~' sinh(A~a) + (1 + A~a) sinh(A~a)

(10)

152

E.E. Michaelides and Z.-G. Feng

convection over objects of several shapes as well as droplets. Many other expressions may readily be found in monographs and textbooks, which are widely used by the engineering community.~-6s These expressions cover the steady-state drag and heat transfer coefficients for a variety of shapes and ranges of Re and Pe (or Pr as the case may be). Numerous researchers have used these empirical expressions to successfully model the interaction between the fluid and a particle from the early days until very recently.36'69-72 In a comparative study, 73 the effect of using different empirical steady-state drag coefficients on the velocity and the total distance 4. EMPIRICALANDSEMI-EMPIRICALEQUATIONS travelled by a particle was investigated. It was concluded that the final time-averaged results are The vast majority of the practical applications of comparable for all the drag coefficients. In the cases where the time-dependent terms of the the rectilinear m o m e n t u m and energy equations of particles, bubbles and droplets are at finite relative momentum equation cannot be neglected, researchers have extended the application of Eq. (1) to higher velocity between these object and the surrounding fluid. In this case, both the Reynolds and Pcclct Reynolds numbers by using empirical coefficients numbers arc finite. Equations (1) and (2) for the for each term of the equation. Thus, Odar and transient hydrodynamic force and heat transfer are, Hamilton, 74 used an expression for the pseudostrictly speaking, accurate at zero relative velocity steady-state drag coefficient, el, and introduced two (Re ~ 0 and Pe ~ 0). For this reason, experimental more empirical coefficients (AA and An) to account results,obtained at finiteReynolds or Peclet numbers, for the remaining two parts of the hydrodynamic did not agree with calculations that used Eq. (I) for force. The expression they proposed is valid for a rigid the hydrodynamic force. In order to compensate for sphere, starting from rest and has the following form: this discrepancy, there has bccn a strong tendency to F I = Cl(6na.f V) + AA ~ ~f--~) use empirical factors for some, or for allthree terms of Eq. (1).This practice has spawned several semiempirical equations that are commonly used in engineering applications. In the case of the equation of motion, + A n a2 ~pvrff~JtoJ-~T drY. (11) the semicmpirical expressions are based on the Basset/ Boussincsq expression and include coefficientsfor all three terms. Given that the energy equation in the Early experiments75'76 suggested that the coefficients form shown in (2) is very recent, there are not any of the three terms of the transient hydrodynamic force empirical factors related to the two transientterms. In are functions of the Reynolds number of the particle, the past several decades, however, many empirical Re, and the acceleration number, Ac: expressions for the pseudo-steady-state heat transfer 18pf r~ d_~ Re=2apflVl ' A C = R e ~ -~c (12) from a particle wcrc derived experimentally or #: analytically. The field of gas-particle and gas-droplet flows is where Uc is the characteristicvelocity of the flow. The abundant with empirical equations for the steady- explicit expressions for the empirical coefficients state drag and heat transfer coefficients. At very low suggested by Odar and Hamilton 74 are: values of the p f / p , ratio, dimensional arguments 0.066 A A = 1.05show that the history and the added mass term may be O.12 + Ac 2 neglected, provided that the variation of the external and (13) field is slow in comparison to the timescale of the 3.12 An = 2.88 + particle (this implies low Strouhal numbers). In this (1 + Ac) 3" case, relatively accurate results may be obtained by The practice of using the empirical coefficients in only calculating the pseudo-steady-state part of the equations. Hence, researchers derived from experi- (11) implicitly assumes that the three parts of the ments empirical expressions for the drag and the heat hydrodynamic force are independent of each other, at transfer coefficients, in terms of the Reynolds and finite Reynolds numbers. It seems that this hypothesis Prandtl numbers. The experimental study by Rowe61 was reinforced by the fact that the three terms of is typical of such studies and yields an empirical Eq. (1) were given separate names in the past. However, the hydrodynamic resistance of the fluid is correlation for the steady-state drag coefficient. The studies of Whitacker 62 and Eisenklam et aL 63 a unique entity, which only manifests itself in the form are typical of the empirical expressions for the heat of (I) in the very narrow case of a rigid sphere, transfer and yield the pertinent coefficients for forced starting from rest,and moving slowly in an otherwise

and ~' is the ratio of conductivities, kplkf. In this case, it is analogous to the ratio, ~ of the dynamic viscosities, which was met in the expression for the hydrodynamic force. It must be emphasized that, although explicit expressions (in the time or the Laplace domain) for only the transient hydrodynamic force or the transient heat transfer are given in this study, the derivation of the entire transient momentum or energy equation is a rather trivial matter, that can be easily accomplished.28'35

(m

(

Transient momentum and energy equations of particles quiescent fluid at the limit of zero Reynolds number. Any variation of these conditions alters significantly the form of the resulting equation of motion, as was seen in the previous section. For example, in the equation pertaining to the viscous droplets, (5), the departure from rigidity for the sphere, creates a totally different form for the transient hydrodynamic force and by extension for the equation of motion. It is apparent that the theoretical justification for the use of (11) in many practical applications is questionable. However, regardless of its precarious theoretical grounds, the general practice of separating the transient force into three independent components and using empirical values for Cl, A A and A n was crowned with success and became very popular in engineering calculations. The reason for this, is the close agreement of the calculations with experimentally observed quantities, as was reported by many researchers. The agreement is due to the fact that the semiempirical equation has a sound experimental basis, because it has been derived from experimental measurements of the total hydrodynamic force. Since it has always been used to merely calculate the total hydrodynamic force (and not any of its three parts) it is not surprising that the calculations were frequently validated by experiments. It appears that this fortuitous agreement is the only reason why the semiempirical expressions are still in use (especially in engineering calculations) and their results are trusted by many. In a series of calculations on the motion of particles using the semiempirical equation, Odar 75'76 first verified the agreement of the results with his new experimental data and subsequently extended the method of empirical coefficients to include the effect of the curvature of the particle's path. He introduced two more empirical coefficients in combination with A H and A n, respectively, to account for the curvature of the trajectory of a sphere and presented a diagram for their variation with the curvature. This method implicitly made the additional, and rather questionable assumption, that the curvature of the path independently affects the three components of the transient force. Because of this, and of the complexity of the resulting expression, these coefficients have not been used extensively by others. Hence, the form of the semiempirical equation of motion has endured with only the three terms shown in (l 1). In the last twenty years, the range and utilization of the semiempirical equation of motion was extended beyond the range of the original study, 77-s° and relatively accurate calculations have been performed by many researchers, sl-s3 It must be pointed out, that there have not been any studies on the development of semiempirical expressions, which include all three terms, in the case of the transient heat transfer Eq. (2). This is due to the fact that Eq. (2) was developed recently and the effect of the two transient terms on the total heat transfer rate has not been correlated with available data,

153

Among the several types of empirical expressions for the momentum and energy transfer one must include the ones by Clamen and Gauvin, s4 Gilbert and Angellino, s5 and Hubbard et al. 39 who determined experimentally and quantified the effects of turbulence on the drag and heat transfer coefficients, by using explicit empirical factors, similar to the ones used by Odar and Hamilton. More recently Sirignano and his co-workers s6-ss obtained empirical expressions for the heat transfer, mass diffusion and drag coefficients of a series of droplets, under various flow conditions. It is of interest to note that the results of the last two studies emanate from well-defined numerical studies, rather than physical experimentation. In another recent study, Bellan and Harstad s9 examined the effects of two different empirical drag equations on the convective evaporation of droplet clusters. 5. T H E CASE OF A SPHEROID

Lawrence and Weinbaum 9°'91 conducted an analytical study on the motion of a rigid spheroid of revolution with small eccentricity, under the creeping flow conditions (Re ~ 0). Their study follows the same methods employed by others (e.g. Maxey and Riley 2s) and used the method of decomposition for the outer velocity field into an undisturbed component and the disturbance created by the presence of the spheroid. They represented the effect of the eccentricity by a second order expansion. The resulting expression for the transient hydrodynamic force exerted on the spheroid, correct to 2, is as follows: [ Fi I =

-- 6 7 r # f a

( H i

1-

~~

37~2'_

+ -i-~)

dHi

+~

a

(

2c

81~2'~ l'

dr.

dr

.1 - T + qS~) J0 e t - ~-

a2 ( e 26e2~ dHi +~v 1 +5--i'~] dt

a

ft ~dHi

+ ~J0

1 (G(t - r) dT" ,

(14)

d

where Hi is the relative velocity vector, Hi = Vi - ui, and the function G of the history integral is:

G(t) = ~

lm[v-~-e

erfc ~ '

with ~b= 3(1 + ix/3).

(15)

It is apparent that the departure from sphericity resulted in the expansion of the added mass, history and pseudo-steady-state components, in terms of the eccentricity, a result which is expected. An unexpected result in Eq. (15) is the presence of the last term, which

154

E.E. Michaelides and Z.-G. Feng

is an additional history term, and is entirely due to the eccentricity of the spheroid. This term is absent in the case of the sphere. It is apparent from its form and from the function G(t) that the magnitude of this term depends on the frequency of variation of Hi. The studies by Lawrence and Weinbaum were the first ones, which reveal a frequency-dependent term in the expression for the transient drag of a body. The significant modification of the other terms and the presence of the last history term, raises questions on the applicability of equations such as (1) or (2) for non-spherical particles. In particular, it challenges the practice of using the concept of an 'equivalent diameter' to account for the shape of irregular solid particles. Equation (14) shows that the effect of any 'equivalent diameter' should depend on the shape of the particle as well as the frequency of the applied velocity field. The work by Gavze, 92 who examined the tensors associated with the motion of a rigid particle of arbitrary shape and their symmetry properties, also casts doubts on the practice of using 'equivalent diameters' indiscriminately. The solution of the energy equation for a spheroid of revolution, with a second-order expansion for the eccentricity function, yields remarkably similar results: Feng 58 followed the method by Lawrence and Weinbaum, decomposed the applied temperature field and approximated the effect of the sphericity by a second-order expansion. He derived the analogous expression for the heat transfer from a spheroid:

a(t) = - 4or/(l + 2e - l e 2 ) ( T s - Too) d + (i + 4, _ _~e2) [' ~r_ (T~-- T°°)

Jo

third term is one, which represents the effect of the far field and is equivalent to the added mass term. The last term in both expressions is the new history term, which depends entirely on the eccentricity, does not appear in the case of the sphere, and is frequency dependent. The kernel of this term does not necessarily follow the t -~/2 decay of the typical history term.

6. THE EQUATIONS OF MOTION AND ENERGY AT FINITE Re and Pe

The creeping flow assumption (Re---,0 and Pe---, 0) for the hydrodynamic force and the heat transfer have resulted in the only exact expressions of the transient momentum and energy equations of a sphere, and through this, in several useful results, such as the Stokesian drag coefficient and the semiempirical relations. However, it leads to the paradoxical result that transverse motion is impossible 4s and fails to yield accurate analytical expressions at finite Reynolds 93 or Peclet numbers (although the semiempirical expressions compensate for this). In addition, the history term as it appears in (1) leads to the paradox 27 that the particle retains the memory of its initial velocity at long times. The first attempts to relax the creeping flow assumption by Whitehead, 12 Oseen 13'14'1s and Faxen 15 were reported in the introductory section. Oseen identified two regions for the problem of the transient flow around the sphere, at finite Reynolds numbers: the first is the inside region, where the advective term has a negligible effect on the momentum transport. Here the problem may be adequately described by the so-called 'Stokes equations': -X3P + #fV2a = 0,

+1(i

,,Z>roo, . , ,

It is apparent that Eqs. (14) and (16) are remarkably similar. Their only differences are in the numerical coefficients of the corrections for the eccentricity. The function G'(t) is analogous to G(t) of the force expression (14) and is defined as follows:

G(t) = Im [V/-~e°terfc(x/~t)]

and

# = 3ei(r/3).

(17) A glance at Eqs. (14) and (16) demonstrates the similarities between the equation of motion and the energy equation for a spheroid. The first term on the right-hand side of both is the usual pseudo-steadystate drag force or heat conduction term; the second term is a typical history term, which emanates from the diffusion of vorticity (in the case of momentum transfer) or the temperature gradients (in the case of the heat transfer) in the external fluid domain; the

V.a=0.

(18)

The second region is the outer region, where the advective term contributes significantly to the momentum transport and, hence, cannot be neglected. In this region, the flow is better described by the 'Oseen equations':

-~7e+lzfV2a=#fv.va,

v . ~ ' = o.

(19)

The difference between the Oseen equations and Stokes equations is the advective term. The approximate boundary between the two regions is a sphere of radius o~Re-1 from the center of the original sphere. This distance has recently been referred to as 'the Oseen distance'. Oseen ls did not compute the velocity field around the sphere, but obtained by an expansion technique an improvement to Stokes's expression for the steady-state drag: F 1 = 67rat~V(1 + 3Re~8). Apparently, the correction represents a considerable improvement over the original expression by Stokes. An experimental study by Maxworthy 93 verified that the Oseen correction, which is analytically derived, is more accurate than other empirical formulae (known at the time) up to Re = 0.45.

Transient momentum and energy equations of particles Proudman and Pearson 94 were the first to calculate the velocity field around a solid sphere and a cylinder at steady-state, to O(Re). They devised a singular perturbation analysis to solve the singular perturbation problem, which is associated with the two regions around the sphere. Thus, they expanded the streamfunction in suitable polynomials, locally valid in the inner and outer regions of the flow. This method enabled them to extend Oseen's result and to calculate the pseudo-steadystate drag coefficient to O(Re2). Their expression for the drag coefficient is as follows:

cn = ~24 e ( 1 +3Re+9Re21nRe).

(20)

Acrivos and Taylor 95 conducted a parallel study for the energy equation of a spherical particle. They used the same expansion method as Proudman and Pearson, 94 and derived the following pseudo-steady-state Nusselt number (which is analogous to the drag coefficient) at finite but small Peclet numbers:

Nuo =

(

Fl(t) = 67ro~#fV H(t) +½6(t) -~ ~ +3Re[(1

+ (21)

Brenner and COX96 and later COX97 used the same technique, and analytically extended the results of Proudman and Pearson 94 to the case of solid particles of arbitrary shape. Brenner 98 also extended the solution by Acrivos and Taylor 95 to the case of heat transfer from a particle of arbitrary shape. His expression is in terms of a Nusselt number, Nuo (which is defined for a purely conduction process), and of the drag coefficient of the particle:

Nuo

expansion to calculate the transient force on a solid sphere, whose velocity undergoes a step temperature change. They also calculated the velocity field around a solid sphere, on which a constant force is suddenly applied. However, their method was proven by Sano 1°3 to be incomplete and inadequate to yield correct results. Sano succeeded in completing the matching asymptotic method of Bentwich and Miloh, for small but finite Reynolds numbers. He derived the unsteady drag on a solid sphere, which undergoes a step change in its velocity [VH(t)] in a quiescent fluid, correct to O(Re2). Sano's solution for the transient hydrodynamic force acting on the solid sphere at long times It = O(Re-2)] is:

2(1 + ½Pe + ½Pe2 In Pe + 0.41465Pe 2

+ Ipe3 In Pe + O(pe3))

l+~Pe+~cnpe21nPe+O(pe2).

(22)

The analogy between the momentum and heat transfer equations of a particle of arbitrary shape is underscored in equation (22) by the inclusion of the drag coefficient cD in the expression for the Nusselt number. Among the analytical studies on particles in specific external velocity fields, Batchelor 99 presented an asymptotic solution for the problem of an arbitrary particle freely suspended in a linear flow regime. Shortly thereafter, Acrivos 1°° extended the last solution to Peclet numbers of higher order. The transient motion of a sphere at finite Reynolds numbers was revisited by Ockendon. 1°1 However, in his study he only considered one timescale and, under this condition, he proved that an asymptotic expansion in Re becomes invalid at long times. It is apparent that the problem must be posed in terms of at least two timescales. In the case where the timescale of the particle is very long, Ockendon also proved that the transient drag is considerably different than the steady-state drag. Bentwich and Miloh t°2 used a matched asymptotic

155

,

+R-~)erf(1Retl/2)

2 l2 exp ~ ---~-) (Trt)l/-~Re( ~e2t) {Re2t'~

3(Trt~/2Re] +9Re21nRe) + O(Re2). (23) Time in the last equation is made dimensionless by dividing with the characteristic time of the fluid rf, and H(t) is the Heavyside function. It must be pointed out that the expression in the square brackets results from the contribution of the outer field and is a consequence of the retention of the advective terms in the Oseen equations. For this reason, it has been referred to as the 'Oseen contribution'. At long times, the last term of the 'Oseen contribution' cancels the (Trt)-I/2 term of what may be called the 'Boussinesq/ Basset contribution'. The remaining transient terms, at long times, decay asymptotically as follows

t-2erf(t 1/2)

or

~ t -2,

t-1/2 exp(-- 1 Re2t), and t-1 exp(- 1 Re2t). The fact that the transients at long times decay faster than the typical t -1/2 rate, yields a satisfactory answer to the problem posed by Reeks and McKee 27 on the non-fading memory of the initial velocity of the particle, which was discussed in the second section. At long times the vorticity around the sphere is advected by the moving fluid and not simply diffused. As a result, the velocity field develops faster and the history term at long times decays faster than t -1/2, which is the diffusion result given by the solution to the Stokes equations. With the rates of decay given above, the particle in the viscous fluid would not retain any memory of its initial velocity at long times. In the last four years there have been several studies

156

E.E. Michaelides and Z.-G. Feng

on the transient equation of motion at finite but small Reynolds numbers: Mei et al. 1°4conducted a numerical study of the motion of a rigid particle for 0 < Re < 50 when the free-stream velocity fluctuates with a small amplitude. Their numerical results indicate that, in the low-frequency limit, the hydrodynamic force decays faster than the t -1/2 rate, predicted by the Boussinesq/Basset theory (a t-2 decay is apparent in their numerical solutions). An analytical study by two of the same authors 1°5 followed, and the faster rate was verified for motions at very low frequencies (Sl<
61~faV (t) (zrReSl ) 1/2 _ 9 #fa (TrReSl ) l/2 tl/2

(t - s) 3/2

] V(t) - --~ [--f~erf(A)

-exp(-AE)] ] V(s) } ds

(24)

where A is the magnitude of the vector:

Re

t - s Ii ~(q) dq

" 4 = T ~

(t-s)

(25)

Hinch 1°7 contributed an appendix to the last manuscript, which is rather remarkable. He presented dimensional arguments and simple asymptotic methods to explain the general solutions obtained by Lovalenti and Brady. He reduced the effect of the Oseen contribution (that is, of the advection terms) to the action of sources and sinks associated with the presence of a wake behind the particle. According to Hinch, the momentum and mass deficits in the wake behind the spherical particle are 67r/zfo~V and 67rlzfa, respectively. Therefore, during the process of start-up from rest, the wake is formed and diffuses downstream. Hence, the particle creates a force, that is proportional to t -2, in addition to any other forces derived from the Stokes equations. During a stopping process, there is already a wake downstream, which was created at the start-up. This wake diffuses cylindrically in the outer fluid. With no wake upstream, the particle experiences an additional force, proportional to t -1. After a step increase of the particle velocity there are two wakes, the 'old' and

the 'new', with the same total mass deficit 61r#fa, the same width (I/f t) 1/2, but different false origins. Their combined effect on the particle reduces to a velocity disturbance, which is due to the exponentially small diffusion of vorticity. During this stage of the process, there is 'an interesting competition' between the decay from the spatial variation and the growth from the spreading with the age of the wake. Thus, according to Hinch, 1°7 for a small velocity increase, the extra force is proportional to t -5/3 exp(-b2t), while, for a large step increase, the extra force is proportional to t - 2 e x p [ - 4 b ( 1 - b)t] (b defines the fractional increase of the velocity). It is of interest to note that in all the asymptotic expressions, the transients due to the wakes decay faster than the classical rate of t -1/2. Hence, the effects of the memory of an initial velocity of the particle completely fade at long times. The asymptotic behavior of the unsteady drag for a rigid sphere at finite but small Reynolds numbers have also been the subject of a few very recent papers, by Mei 1°s who examined the case of an oscillating sphere; by Lawrence and Mei 1°9 for the motion induced by an impulse, and by Lovalenti and Brady ll° for an abrupt step change of the velocity of the sphere. One may conclude from all these recent studies, that analytical expressions for the rectilinear equation of motion of a sphere are very complex in form, and depend on the Reynolds and Strouhal numbers, as well as on the details of the process considered. There is a remarkable similarity in the energy and momentum differential equations of a particle: the analogue of the Stokes equation for the particle is the unsteady conduction equation 00 pc-~ = kVEO (26) while the analogue to the Oseen equation is the convection equation:

pc[~t+ V.VO] =kV2O.

(27)

Feng and Michaelides m used dimensional arguments and the singular perturbation method of Sano 1°3 to solve the problem of heat transfer from a spherical particle at finite but small Peclet numbers, when the temperature of the particle undergoes a step temperature change. They established that in the immediate vicinity of the particle, Eq. (26) models adequately the energy transfer. Far from the particle the advective effects must be taken into account. Hence, one has to use Eq. (27). The characteristic distance, which separates the regions of validity of the two equations is equal to ape -I, and may be called the 'equivalent Oseen distance'. They derived an expression for the uniformly valid temperature field around the sphere, and finally an expression for the Nusselt number at short times (t < Pe),

Nu = 2(l +---~t) + O(Pel+),

(28)

Transient momentum and energy equations of particles and at long times (t > =

Pe-2)

Nu = 2(1 +pe[lerf(Pe-~) +~ox,(

+ ~3P~e+ 2

,)] 2

lnPe)\ + O(Pe '+)

(29)

(r~ = #p/#f). As with Sano's expression for the hydrodynamic force, the transients of the last expression decay faster than the typical rate of the history term (t-U2), which in this case emanates from the diffusion of the temperature gradients. Therefore, the inclusion of the advection term in both the energy and momentum equations, accelerates the decay of the transients. Feng ss conducted a more general study on the heat transfer from a particle of arbitrary shape at low Peclet numbers, which is analogous to the one by Lovalenti and Brady. t°6 He used a singular perturbation method to derive the general solution for the heat transfer from a particle of arbitrary shape. The particle moves with time-dependent velocity uS(t), relative to an ambient fluid velocity u~(t), and the temperature field of the fluid is given by an arbitrary time-dependent function O °° (t). The total rate of heat transfer in this case is: Qr

=

~s,

The first term of Eq. (30) represents that part of the dimensionless rate of heat transfer, which is due to the pseudo-steady conduction from the surface of the particle, and is analogous to the pseudo-steady-state drag of the equation of motion. The second term accounts for the regular perturbation, due to the unsteady conduction. The analogue to this term in the equation of motion is the contribution from the regular perturbation to the unsteady Stokes problem. The third term is the contribution of the undisturbed temperature field. Its analogue in the equation of motion is the added mass term. The fourth and fifth terms, which appear as volume integrals, represent the combined advection and conduction effects on the fluid in the outer region. The outer region in this case is further than ~Pe -1 from the center of the particle. In the case of a sphere, Eq. (30) is reduced to the following expression for the dimensionless heat transfer at long times: QT(t ) _

4roPeS1 dO °~ 3

dt

i 2Pev~ f' -

x/~

(Op - O ~ ) V O ~. ffdS

157

47r(Op- 0 ~)

t Op - 0 °° erf IX'[ ( O p - O~)l~'Sl (t - "r)l/l' t -,

x [2-~erfJA l- exp(-,.4'12)]dT +o(Pel+).

- J v o~(r3PeSl--~dV+PeSlVp BO°°

(34)

- Pe Jv Oa(r-) [ P e S l ~ + (ff - ffs)'~Oo]dV

+ o(?ei+). (30) where SI is the Strouhal number [Pe = ULpycf/ky and Sl=k/(ULppcp) with U and L being the characteristic velocity and length of the process]; O 4 is an auxiliary temperature field, which may be conveniently defined for the calculations; Sp and Vp are the surface and volume of the particle; Vf is the entire volume of the fluid and V~ = Vf + Vm The other temperature fields arise from the singular perturbation analysis, as in the case of the momentum equation. The governing equations for these temperature fields are: 58'nl ~7200 = V201

- PeSI-~

PeSI O00t°,

= [/7- iTs(t)] • VOo,

(31) (32)

and V20~ =

P e S lO0 ~ p - PeffS(t) • ~0~.

dr

(33)

where fl represents the angular integration over a spherical surface, n is the outward unit vector normal to that surface, and the vector A' is analogous to the vector A, defined in (25):

~,

I'

Pe (t - r~ 1/2 , ffs(~) d~ = -2- \PeSI: t- r

(35)

Equations (34) and (35) are the analogous expressions to (24) and (25). It is not difficult to discern the obvious analogies in the structure and the terms of the four expressions. It must be pointed out that expressions emanating from the Stokes equations or its equivalent transient conduction equation do not make any distinction as to whether the particle accelerates or decelerates, heats up or cools down. The resulting forces and rates of heat transfer depend solely on the absolute values of the relative velocities or temperatures. Thus, the processes described by them may be called 'invertible' with respect to particle motion and temperature change (certainly these processes are not 'reversible' in the thermodynamic sense). Apparently, the inclusion of the advective terms brings 'non-invertibilities' into the processes. Thus, a particle starting from rest experiences different forces than a stopping particle, even if the relative

158

E.E. Michaelides and Z.-G. Feng

velocities are the same. z°6'1°7 Similarly, a heating sphere exchanges with its surroundings different quantities of heat than a cooling sphere, even if the temperature differences are the same. 5s'111 The 'noninvertibilities' may be always associated with the presence and advection of the momentum and thermal wakes, which are created by the motion of the particle.

7. THE EQUATIONS OF MOTION AND ENERGY AS USED IN PRACTICE

It is obvious that the transient hydrodynamic force and heat transfer from a sphere have a very complex form, even at the limit of zero Reynolds numbers. The two equations assume an even more complex form at finite (but still very small for many practical applications) Reynolds and Peclet numbers. Moreover, at finite Re and Pe, the exact value of the force and the heat transfer rate are still unknown, because all pertinent expressions emanate from singular perturbation theory. This fact precludes the possibility of performing calculations with any known degree of accuracy by using the analytical expressions derived for 0 < Re < 1 and 0 < Pe < 1, as they were introduced in the last section. Rigorously derived analytical expressions for the transient hydrodynamic force and for the rate of heat transfer at Re > 1 and Pe > 1 (where a great deal of the engineering applications fall) are unknown to-date. From what we know so far, we can only deduce that the drag and heat transfer coefficients will depend heavily on the Reynolds or Peclet number, respectively, as well as the Strouhal number. At higher Re and Pe there are several experimental studies and a few more recent numerical studies for the drag and the heat transfer coefficient, such as the ones by Sirignano and his co-workers 86-ss and Abramson and Elata. 112 They include several charts on the dependence of co and Nu on Re and Pe. However, the numerical studies are not as general as the analytical ones and, are only applicable to the group of the well defined processes, for which the numerical simulations were performed. When performing engineering calculations, one needs to use expressions that are accurate enough to satisfy the precision of the overall problem, and yet simple enough to be used easily and, oftentimes, repetitively. Long, repetitive computations are often necessary in Lagrangian computations of the dispersion of particles in gaseous streams, as indicated by Stock in his recent review article. 113 At first glance, it appears that none of the forms of the energy equation or the equation of motion of the particle may be suitable for engineering applications. A moment's reflection, however, proves that relatively accurate calculations were made in the past, and equipment design was successfully accomplished, by using simplified versions for the equations of motion and

energy for particles, bubbles or droplets. In their vast majority, these expressions are simplified forms of the semiempirical equations and have an experimental basis to some degree. In a very recent ASME symposium on Gas-Particle Flows, edited by Stock et al., 72 forty seven papers were contributed, all of which used, in one way of another, expressions for the hydrodynamic force on the particle. The vast majority of the contributed manuscripts pertained to the transient motion of particles. Most of the authors used Stokes's drag law or a simple empirical expression (emanating from steady-state drag) to model the instantaneous hydrodynamic force. Several of the authors mentioned the existence of the transient terms (history and added mass). However, by using dimensional arguments or by referring to higher authority, all of them ultimately neglected the transient terms in their calculations. The same applies to studies of transient heat transfer with particles: in a recent monograph by Kaviani 114 one observes a multitude of transient heat transfer problems solved by using Fourier's law or empirical equations emanating from steady-state experiments. Apparently, in both cases of the momentum and the energy transfer, an implicit assumption is made by the researchers, that the particle motion or heat transfer may be described in a quasistatic way. Hence, the transient terms do not influence appreciably the quantities of interest, which are usually time-averaged characteristics of the flow field, such as the dispersion of particles, the average velocity, the total amount of heat transfer, the total evaporation, etc. Despite this simplification, the calculations in most manuscripts appear to lead to reasonable results and to satisfy the required precision. When the pseudo-steady-state force or heat transfer is the major contributor to the final (time-averaged) result, the other transient parts do not play an important role and, hence, may be conveniently neglected. While studying the effect of the history term on the average velocity and distance travelled by a particle in simulated transient flows, Vojir and Michaelides 83 found ample support for this practice in the cases of slowly varying fluid velocity fields. However, neglecting the transient terms presumes an a priori knowledge of the relative importance of all the terms in the transport equation. Our knowledge on the complete form of the transient energy equation of spheres is very recent. 35 For this reason, the practice of not explicitly accounting for the transient terms was never challenged. Given that in most cases, calculations on the time-average heat transfer are needed, 114'115 this practice is not objectionable in view of our similar experience with the equation of motion. 83 For example, Soo 116 suggests a conservation equation for the dispersed phase of dilute suspensions, which is essentially an extension of Fourier's law. More recently, in a comprehensive study of particles in a

Transient momentum and energy equations of particles turbulent flow field Maciejewski and Moffat 117'118 correlated the average heat transfer rate (Stanton number) with an average turbulence number and the Prandtl number. Jacobs and Golafshani 119 used a steady-state model to obtain the heat transfer characteristics in a turbulent spray column. It appears from these studies, that the average heat transfer coefficients are sufficient for most engineering calculations. Even when calculations of transient variables are made, time-averaged transport coefficients have been used, with various degrees of accuracy. As in the case of the equation of motion, this practice assumes implicitly a quasistatic process. The empirical coefficients (mainly of experimental origin), which are most often used in these studies, facilitate the computations in quasistatic calculations. They also yield sufficiently accurate results, provided that the range of the coefficients used is not extended beyond the range of the original experiments. There are indications, however, that the steadystate empirical coefficients are not adequate for the calculation of transient heat transfer. The recent study by Gopinath and Mills 12° (who followed an analogous momentum-equation study by Riley 121) indicates that the 'streaming Reynolds number' is a very important variable in the determination of the Nusselt number in problems involving acoustic excitation. As defined in that study, the streaming Reynolds number is actually the ratio Re/SI. This is evidence that the Strouhal number of the process (which features prominently in Eqs. 24 through 35) is an important variable in the calculation of the rate of heat transfer. Gilbert and Angellino85 also used a similar type of 'Reynolds number', to account for the characteristic frequency of the process. There are also indications in previous studies that Strouhal-type variables are needed for a more accurate determination of even the time-averaged Nusselt number (e.g. the turbulence number39). We also think that the strong sensitivity of the mass fluxes of droplets on the evaporation models used by a recent study 122 is due to the fact that neither the Strouhal number nor the transient terms were adequately taken into account in the model equations used. A more definitive study on the effect of the Strouhal number and on the transient terms of the energy equation seems to be very timely.

8. T H E EFFECT OF THE HISTORY TERMS

In the case of particulate equations of motion and heat transfer, where the ratio of densities is very large (pJpf >> 1) dimensional arguments lead to the conclusion that the transient terms of Eqs. (1) and (2) may be neglected. The cases of motion of bubbles, droplets or particles in liquids (pf/pp is not very small) and cases in which the desired result is an instantaneous and not a time-averaged quantity, are markedly JPECS 22-2-D

159

different. Li and Michaelides 123 calculated that the history term accounts for about 25% of the instantaneous force by a vapor on a droplet. However, its effect on the time-averaged force was negligible. They also found that the history term contributes significantly to the time-average results for particles and bubbles in liquids. In addition, the added mass term becomes significant, especially in bubbly flows, where the ratio of fluid to sphere densities is of the order of 1000. A previous study by Wang et al. 124 used a dimensionless argument, to conclude that the history force is an order of magnitude higher than the pseudo-steady state drag, during the flow of particles at the entrance of small pores. However, the last study appears to exaggerate the effect of the history force, because it did not take into account the fact that Eq. (1) and its outcomes are only valid in the case of a uniform, unbounded flow. The significance of the transient terms in the equation of motion of particles is corroborated by several other studies. 68'82's3'125 Regarding the transient terms of the energy equation, Lain and Michaelides 126 performed preliminary calculations on the effect of the transient terms of Eq. (2) and found that the history term contributes substantially to the instantaneous rate of heat transfer. These results strongly advocate the inclusion of the transient terms in the equation of motion or the energy equation, in the case of fast-varying external fields. The inclusion of the added mass term in the equation of motion (1) or its analogous term in the energy Eq. (2) does not present any difficulties. The resulting equation is still explicit in the relative velocity or temperature and may be solved analytically or numerically, without any added difficulty. What complicates the solution procedure considerably is the inclusion of the history integral term. Then both the equation of motion and the energy equation become integrodifferential equations, which are implicit in the relative velocity or temperature of the sphere. As a consequence, analytical solutions may be derived in the simplest cases only. In this case, a numerical solution of the resulting equation is necessary and an iterative scheme must be adopted, because the equation is not explicit in the dependent variable (V or Tp). In addition, because the memory terms decay slowly, a great number of the past accelerations of the particle must be stored in memory (they must be 'remembered' in every step of the computations). This results in considerable increase of the computational time and the amount of memory required. Especially when Lagrangian simulations of an ensemble of spheres are made, the inclusion of the history term slows the computations significantly (by an order of magnitude). Oftentimes this increase of the CPU time renders certain computations prohibitively expensive. This is the main reason why the history term has been conveniently neglected in many cases when other terms of lesser magnitude were included in the computations.

160

E.E. Michaelides and Z.-G. Feng

In order to speed up the computations when the history term must be included, one may use an integrodifferential transformation for the equation of motion. 83'127 The same transformation may be easily extended to the energy equation. Thus, the first order implicit equations are transformed to second order equations, which are explicit in the relative velocity or temperature. For example, the semiempirical equation of motion, when transformed, appears in the following form: d2Vi ( dt 2 F7 2Cl = -70 -

9(2A~)dVi+72c~Vi -~

~- d2ui du i ) ~ - "/2(1 - ~)Cl -~-

Acknowledgements--The original work of the authors was partly supported by grants from the NSF, NASA, a SPACE, DOE (through the Tulane/Xavier CBR) and LESQF to Tulane University.

d2ui + 7 2 ( 1 - ~)A~/~29~ Ito (t-d)-.~ o s d a + T A ~ 2 ~ t

x ( 7 ( 1 - ~)u' (O) - 7 ( 1 - ~)Oi + Cl ~tt ) + "/2(1 -- ~)ClG i + "12Vio9~--~22H~(t)

9/~

9~ 2 -F~/AHV~Vio~ (t)

(36)

with the following initial conditions for the velocity and for the acceleration:

Vi(O) = Vio and dVi dt (0) = - ~/ClVio - "r(1 - ~)u'i(0) + 7(1 - ~)Gi - V i o T A H i ~ 6 ( t ) .

above, the deviations from sphericity introduce complex terms in the equations of motion and energy. The concept of an 'equivalent diameter' is (at least from a theoretical point of view) inappropriate, and in the cases of drastic deviations from sphericity (e.g. dendritic shapes), appears to be inaccurate. A practical solution to this question will emerge from experimental or numerical work on the unsteady forces and heat transfer rates in complex shaped particles. 128 By extensive and accurate physical or numerical experimentation, 'empirical' solutions to this problem will be found, at least for the shapes and processes of the highest practical interest.

(37)

Equation (36) yields the transformed equation of the Boussinesq/Basset expression when all empirical coefficients are equal to 1, u ' ( 0 ) = 0, Gi = 0 and Vio = 0.127 In the above two equations, the parameter is defined as the fluid to sphere density ratio, and 7, which is equal to 1/(1 + ½Aa~), includes the added mass coefficient. The time has been made dimensionless by dividing with the quantity r = 2ppa2/9#y) and the velocity by using the characteristic velocity U0. The dimensionless gravity vector is Gi = gir/Uo, where gi is the gravitational acceleration. Since Eq. (36) is explicit in the relative velocity, an iterative numerical procedure is not needed and the equation may be solved by an explicit time marching method. This transformation reduces significantly the required CPU time for the solution of the implicit equation. However, even the transformed explicit equation is computationally intensive, when compared to an equation, which does not include the history term. For this reason it is always convenient and advantageous to neglect the history term, whenever this practice can be thoroughly justified. A more difficult problem to overcome seems to be that of irregularly shaped particles. As mentioned

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