Ind. Eng. Chem. Res. 1995,34, 3621-3631
3621
Interphase Mass Transfer from Bubbles, Drops, and Solid Spheres: Diffusional Transport Enhanced by External Chemical Reaction Leonid 5. Kleinmant and X B Reed, Jr.* Department of Chemical Engineering, University of Missouri-Rolla, Rolla, Missouri 65401
Interphase mass transfer has been analyzed for a reactant being transferred from a sphere into a quiescent, infinite continuous phase where it undergoes a chemical reaction. Ranges of the diffusivity ratio D,the distribution coefficient H,and the Damkohler number Da have been delineated, within which the overall resistance to mass transfer may be reasonably ascribed to the s u m of the individual resistances in each of the two phases. Outside these parameter ranges, the resistances for two decoupled mass transfer problems may not be added to give the overall resistance to the coupled mass transfer problem. An alternative procedure for decoupling the conjugate problem is proposed and an expression for the upper bound for the overall mass transfer rate established. Also, because chemical reactions are widely used to enhance interphase mass transfer, we present minimum rates of reaction Da for given physical propePties D and H that will produce maximum rates of mass transfer.
Introduction The physical process of diffision of a chemical species from a particle or an occlusion into a surrounding phase is basic to many phenomena in industrial applications and in nature, as is also true for that from a bubble or drop t o a surrounding fluid phase. Often another process occurs simultaneously with the physical process of diffision, either by design or naturally. The most common physical process that tends to enhance interphase mass transfer is that of convection by an external flow field, which also induces circulation within bubbles and drops, thereby modifying the overall transfer relative to that of a solid particle. The most common chemical process that enhances interphase mass transfer from a bubble, drop, or particle is of course a chemical reaction. There is an extensive literature on the influence of the physical mechanisms on interphase mass transfer (CliR et al., 1978;Brounstein and Shegolev, 1988). Most of the published research, however, has been restricted to cases in which the resistance to mass transfer occurs in one of the phases, while that in the other phase is negligible, Exceptions are the computationally intensive investigations of Chung and his co-workers (Oliver and Chung, 1986,1990; Nguyen et al., 19931, as well as of Abramzon and Borde (1980). If there is a chemical reaction present, then the problem is even more computationally intensive than that for conyedive diffision, for which only the role of the Peclet number Pe has been studied extensively (Abramzon and Borde, 1980; Oliver and Chung, 1986,1990), although a rather wide range of the diffisivity ratio D has been considered (Jungu and Mihail, 1986). To our knowledge the only comparable coupled convective diffision-reaction research on interphase mass transfer is our own (Kleinman and b e d , 1992, 1994). Moreover, as long as convection participates, it both precludes exact analytical solution and inhibits the computational research from exploiting the full range of all the parameters in the problem (Pe, D , H , Da), with H the distribution coefficient and D a the Damkbhler number.
* To whom correspondence
should be addressed. Current address: Koch Engineering Co., 850 Main Street, Wilmington, MA 01803. +
0888-588519512634-3621$09.00/0
The present research is devoted to interphase mass transfer between a quiescent sphere and an infinite continuous phase at rest, where a chemical reaction takes place. This problem has the basic features pertinent to the general conjugate mass transfer problem, yet it can be solved by Laplace transforms. Important in its own right, this problem can serve as a means for investigating the role of the parameters coupling mass transfer in both phases (Hand D).We will use the terms “coupled” and “conjugate” interchangeably in this paper, with the latter being used in the mechanical engineering literature to indicate coupling in two phases through the boundary condition at the common interface. It is shown that characteristics of the conjugate interphase mass transfer process are determined by the existence and values of the roots of a certain nonlinear algebraic equation. This provides an opportunity to connect physical features of the problem with the parameters of the mathematical solution, The existence of an explicit solution enables us to investigate the influence of parameters characterizing the inherently conjugate features of the problem in a much wider range than was done before. On this basis it is possible to clarify some of the issues involved in the customary engineering treatment of resistance to interphase mass transfer in terms of the sum of the individual resistances in each of the phases (e.g., Foust et a l . , 1980; Laddha and Degaleesan, 1976). The utility of an addition rule for resistances predicted on solutions to decoupled problems is obvious. It is the validity of such an addition rule that is suspect. For it is certainly true that not every coupled (conjugate) problem can be described in terms of a decoupled one without first solving the coupled problem. However, if the coupled problem has been solved, there is little need for decomposition into an addition rule. An addition rule for mass transfer resistances is thus an engineering approximation, and what is crucial is the range of validity of the approximation. For the present conjugate diffisionreaction problem we have figured out the ranges of the characteristic parameters where the application of this rule leads to unacceptable errors. We also have suggested the alternative procedure for the calculation of the overall conjugate mass transfer coefficient on the basis of mass transfer coefficients for uncoupled prob0 1995 American Chemical Society
3622 Ind. Eng. Chem. Res., Vol. 34,No. 10, 1995
lems. For the present problem it leads to results which are exact. The application of this procedure to the convective conjugate mass transfer will be considered in a forthcoming article (Kleinman and Reed, 1995b). Finally, it is well-known that a chemical reaction can be diffusion-limited. That aspect of the present class of coupled diffision-reaction problems has not yet been investigated. We rephrase the task by posing the following question: For interphase mass transfer enhanced by a chemical reaction, what level of reaction rate would produce effectively the same mass transfer rate as an infinitely fast reaction? We answer this question for quiescent phases with a chemical reaction in the continuous phase.
Physicochemical Setting, Theoretical Formulation, and General Solution Always sound, dimensionless formulations are especially apt for two-phase systems that may contain solids, liquids, and gases in any of the several pairings, for a broad range of physical properties and property ratios may be encountered. For the present problem, these are comprised of the diffusivity ratio D = D(1)/D(2) and the Henry’s “law”constant H reflecting the distribution of the transferring species between the two phases at equilibrium. In addition, there is the second Damkohler group, Da = kR2/D(2),that characterizes the rate of chemical reaction. In the absence of convection, the purely diffusional fluxes are in the radial direction, and the fluxes and concentrations depend only upon radial position and time. The mathematical formulation t o follow suggests a first order irreversible chemical reaction, but two points bear emphasizing. An elementary first order reaction for a transferring species would hardly be likely to occur in the continuous phase alone. An elementary bi- or trimolecular reaction in which the second or the second and third reactant species were present in excess (as would be the case for a reacting solvent) would produce an effectively first order reaction as long as the transferring species participates as a first order reactant; this also occurs for a catalytic and for an empirical rate expression in which the transferring species is an effectively first order reactant and the other reacting species are present in excess. In short, the range of reaction schemes to which the formulation and its results are applicable is much broader than that which might be inferred from the first order kinetics embodied in (2). With the concentration in the sphere measured in terms of its initial constant concentration Eo and with that in the continuous phase measured in terms of EdH t o emphasize the overall mass transfer driving force under the usual assumption that no reactant reaches infinity ( E , = 01, and with time measured in units of R2/D(2),the mathematical description of the physicochemical problem may be cast as the following coupled initial-boundary-value problem:
(3)
The Laplace transform method is natural for (1)-(6). With a standard transformation for spherical coordinates (c = F/r) and an obvious reference to Cooper’s (1977) notation of
P ( t , r )= r(c“’(t,r>- I),
P’(t,r)= rcCC(2)(t,r) (7)
one obtains coupled ordinary differential equations for the Laplace transforms
of the solutions to the coupled transformed boundary value problems (1)-(6):
F1)(s,r) =-
(1
+ &TEi>s i n h ( r m )
(9)
sqxs)
in which the initialpalues (6) have been incorporated. Here, the function #(SI is given by &s) = (1- HD
+1-
sinh HD
m+ cosh d%
(11)
Inversion of (9) and (10) by contour integration is more complicated than for the purely diffusive case of interphase mass transfer (Cooper, 1977)because introduction of a reaction leads to the appearance of singular points other than zero. There is a branch point at s = -Da, a pole at s = so = 0, and a set of poles Sk, k = 1,2, ..., K - 1 lying between SO and -Da. If there exists a pole at the point -Da, then we set SK = -Da, otherwise SK = 0. Consequently, the inversion formulas are (12) in which the integrand is analytical in the half-plane Re(s) 2 y . Introducing in the usual way a branch cut from --m t o the branch point -Da, with the upper and lower edges of the cut being called AB and CD, one obtains K
Res [e”’%’)(s,r),s = sk1-
F”’(z,r) = k=O
Ind. Eng. Chem. Res., Vol. 34, No. 10, 1995 3623 Introducing e = u& and writing s = eei6, we have along AB, where 6 = n, I u < 00, & = iu and = iAl(u), in which
a,
A,(U)
= J u 2 -~DU
(14)
Along CD we have I? = -n, = - iu&, -iAl(u). The solutions (13) thus reduce to
6HD
rG
n
=
e-u2DT
which goes to zero with increasing ‘t, as expected. The rate of mass transfer can be described by the overall mass transfer coefficients Ki,i = 1, 2, through the relationships
k=O
u(r)= Ki(T)
Fdr(r)
All singular points Sk, K = 1,2, ...,K can be proven to be real and lie in the domain [-Da, 0). We denote them by yk = -Sk, SO that 0 < yk I Du, and the yk become the nonzero roots of &,J) = (1- HD
+ e) sin m+ H
Al(u) A 2 2 ( ~ ) du (22) u3G(u)
m = o (16)
= Kz(‘t>F d , ( t ) / H
(23)
where is the instantaneous rate of mass transfer through unit area of the sphere surface, which is also equal to
Fdr(r)
is the instantaneous driving force,
D COS ~
U on substituting Res[esTfi1)(s,r), s = 01 = -r and Res[eSTg2)(s,r), s = 01 = 0 and calculating the expressions for the residuals at the other singular points in (15) in the usual way, we arrive a t
Fdr(‘t)
= $‘)(‘t) - HE,
$‘)(‘t)
(25)
Corresponding dimensionless values of the overall mass transfer coefficientsKi are called overall Sherwood numbers and defined as
Restricting ourselves to the internal overall Sherwood number and for brevity dropping both indices, i.e., making the identification and i
we arrive after transformation to dimensionless variables a t
K
where
In (17) and (181, A,(u) = u cos u - sin u, A,(u) = HDA,(u)
+ sin u (19)
+
A,(u,r) = A,(u) sin u cos[(r - l)A,(u)l A&) s i n k - l)Al(u)l (20)
is the internal overall Sherwood number based on the initial driving force Fdr,o
= Eo - HE-
E
Eo
(30)
rather than on the instantaneous one (25). m e r substitution of (17) into (29), we obtain
and G(u)= A12(u)sin2 u
+ A,,(u)
(21)
Quantities of Interest Of more directly practical utility than (171, (18) is the evolution of the average sphere concentration,
(31) From (1)it follows that
3624 Ind. Eng. Chem. Res., Vol. 34, No. 10, 1995
(32) and upon introducing definition (28), that (33)
values of z the instantaneous Sherwood number becomes a nonzero constant given by (38),a regime called asymptotic; if there are no nonzero roots of (161, then an expression for Sh(z-=) in the asymptotic regime is given by (39). If there is no reaction, then (39) clearly shows that Sh(z---) = 0. When DaID 1,independently of the value of Du; (ii) Du 5* 1/@, which can be viewed as analogous to the high Peclet condition obtained by Abramzon and Borde (1980). Provided one of these conditions is satisfied, the addition rule in both forms (51) and (61)will give results close to one another. Dispersed Phase Resistance Dominant. The opposite limiting situation, that of the resistance inside the sphere being limiting, can be realized in several ways. For given H and Da, the ratio D may be so small that its further decrease does not influence the value of the asymptotic Sherwood number Sh,. The same thing occurs for given H and D if the nondimensional reaction rate Du is larger than some limiting value Dalim. This situation is presented in Figure 2, where for H = 0.1, D = 1.0, D a l i m = 100 is evident. For this case, condition (54) is violated and the minimum positive root of (16) is close to its maximum value of 2n22/3, which coincides with the asymptotic Sherwood number
4 0.00
0.05
0.10
z
0.20
0.15
Figure 2. Temporal evolution of ShD a t different values of Da.
for the internal problem. Thus, in (51) and (611,
Sh,,,, = 2213
(66)
should be substituted. For D and Da held constant, an increase in H leads to the uninteresting limiting case of vanishing mass transfer, whereas a decrease in H results in different limiting cases, depending upon the ratio In one instance, when < n,(16) does not have positive roots as H 0, and the resistance will lie entirely in the continuous phase. In the other, if I n, (16) has at least one positive root, and as H 0 the resistance shifts into the dispersed phase. This is illustrated in the Figure 3, where for D = 1.0 and Da = 100, the values of overall Sherwood numbers corresponding to H = 0.1 and lower are not changing and are equal to 2n2/3.
-
-
3628 Ind. Eng. Chem. Res., Vol. 34,No. 10,1995 lead to almost complete mass transfer by the time the asymptotic regime is reached. With increasing Damkohler number, the resistance shifts into the dispersed phase, and the results obtained on the basis of the quasi-steady approximation become much better. For Da L 500, expression (51) is applicable in the most of the ( D m domain under consideration. Diffusion Limitation for the External Reaction. The existence of a limiting value for the Damkohler number, Dali,, beyond which mass transfer is not further enhanced, has practical implications. To determine this lowest value of the dimensionlessreaction rate constant yielding the (effectively) highest possible mass transfer rate a(DSh,,),,, where a is some number smaller than but close to 1,y = m z D may be substituted in (16). After rearrangement we arrive at
9-
8-
76 -
-
Sh
5-
Da=100 D=1.0
:' \ \ .'
2
0.00
.._
- - -. 0.02
---
- H= - ._ 5.000
0.06
0.04
0.08
0.10
Dalim= Dw2
+ [HD(1- w_) - 11 tan w
(67)
W = &
(68)
where
7 Figure 3. Temporal evolution of ShD at different values of H.
As the rate constant made dimensionless with the diffusivity in the phase other than the one where the reaction takes place, the ratio DalD has an important physicochemical meaning. It characterizes the relative roles of the reaction in the continuous phase versus the mass transfer from the dispersed phase that feeds the reaction. Ranges of Validity of Addition Rule. We are now able t o compare the validity of the two versions of the addition rule (51) and (61) over the entire range of relevant parameters. As was shown above, the set of parameters that leads t o the absence of positive roots of (16) always results in negligible resistance inside the droplet for large values of time z, and the addition rule in the form of (511, based on the quasi-steady approximation, is not valid. In Figure 4 we present the relative errors in the overall asymptotic Sherwood numbers, computed with both versions of addition rule, compared with the exact values in the range IH,D 5 lo3, 0.1 IDa 5 500. Version (51) of the addition rule, based on the quasisteady approximation, gives poor results for small values of the Damkohler number (Da I 1). This was anticipated because it is the very case when (16) has no positive roots. For Da 2 10 both versions (51)and (61)of the addition rule have sizable regions of applicability, large portions of which overlap. These correspond roughly to domains where D and H are either both large or both small. For large values of HD, this is an obvious consequence of the negligible resistance in the dispersed phase. For small values of HD, it is a consequence of negligible resistance in the continuous phase. As the addition rule is to an extent an interpolation formula, it yields good results at the ends of the interpolation domain. By the same token, the worst results are obtained when the values of the resistances in both phases are comparable. This is the case in the parametric region having small values of H and large values of D , which corresponds t o cases when the high diffisivity of the solute in the dispersed phase and its high solubility in the continuous phase, together with chemical reaction,
For an absorbed vapor in a drop that subsequently desorbs into a gas phase where a reaction occurs, D would be small. The curves of constant values of Dalim appropriate to this case are plotted in Figure 5. The value of a was taken t o be 0.97. For example, for D = 0.06 and H = 0.4, Dalim w 1.
Alternative Approach to Decoupling the Conjugate Mass Transfer Problem
As was mentioned earlier, the biggest advantage of the addition rule is that it offers a means of formally decoupling the problem, i.e., of obtaining the value of the overall asymptotic Sherwood number for the conjugate problem by solving separate internal and external problems individually, so that two additional parameters D and H are involved only in the final, algebraic step in the solution of the problem. The limited range of applicability of this rule serves as an incentive to develop more precise methods which would nonetheless preserve the advantages of the addition rule. The germ of the method we develop in this section appeared already in the work of Abramzon and Elata (19841, who considered forced convective heat transfer from a high conductivity solid sphere to a fluid in creeping flow. Until the present research, however, neither in their work nor in that of other authors subsequently has their idea been further elaborated. For quasi-steady behavior, we are thus motivated to substitute the normalized concentrations
-
that are constant as z = in (1) and (2) with an apparent homogeneous chemical reaction being posited in (1)that enables (quasi-) steady concentrations and mass transfer t o be maintained:
Ind. Eng. Chem. Res., Vol. 34, No. 10,1995 3629
a%
I: n E
0"W
d'm
e
.I
1
.em Y
3
3630 Ind. Eng. Chem. Res., Vol. 34,No. 10, 1995 The solution of problem (73) leads to
D C
B A 9
0.02
, , ,
,
,
0.04
I8 14
8
IO
7 6
8 6
5
4
4
2
3
1 0.5 0.2
:::L3, ,3xq ,
30 26 22
2 1
while that of (74) leads (cf. (52)) to
We thus arrive finally a t the equation
, , ,
D
0.06
0.08
-HD
0.10
(78)
Figure 5. Values of Daiimaccording to (67).
1 -(r2 d 7 d(d2' ) - (Da r2 dr
- iDShas)(c)(2)= 0
(71)
with the conditions a t the interface being
(72b) If we consider two uncoupled problems
and
(c)(2)(r=1) = 1 (74) assuming that for each of these problems we are able to derive an expression for the concentration gradient a t the boundary r = 1as a function of the corresponding parameter (Da)"),i = 1, 2, then using (70), (71), (731, (74), and the second condition from (721, we arrive finally a t a nonlinear equation for the unknown Sh,,:
In the absence of convection the values of the concentration gradients at the interface can be obtained analytically, but otherwise appropriate empirical relationships might be developed. In (73) there is a source term present which has the sense of a generalized production by chemical reaction.
which coincides with (16), taking (57)into consideration. Consequently, (78) should yield the exact values of Sh,. Of course, the real value of this approach would be most important when the coupled (conjugate) and the decoupled problems do not have analytical solutions, yet when approximate expressions for the concentration gradients at the interface can be somehow obtained as functions of the main dimensionless parameters (say Pe and Da). This issue is taken up elsewhere (Kleinman and Reed, manuscript in preparation).
Conclusions The problem of mass transfer of a reactant from a quiescent droplet into a quiescent ambient fluid where it undergoes chemical reaction has been solved analytically using Laplace transforms. We find that the upper bound for the asymptotic value of the instantaneous overall Shenvood number based on the internal concentration gradient and internal diffisivity D(l)is given by the upper bound for the corresponding Sherwood number for the internal problem. This leads in the absence of the circulation inside the sphere to an upper limit for the mass transfer rate of (2n2/3)D. As a consequence,there exists an upper limit for the enhancement of mass transfer by chemical reaction, and plots for the minimum values of the reaction rate Da leading to maximum values of the mass transfer rate were obtained. T w o versions of the addition rule for the calculation of the asymptotic value of the Sherwood number for the coupled (conjugate) problem based on Sherwood numbers for the uncoupled internal and external problems were compared. The version based on the quasi-steady approximation proved to be invalid for small values of Da. For large values of Da both versions yield results close t o the exact ones (except for small H and large D ) . For intermediate values of Da, the addition rule yields reliable results in cases when the resistance to mass transfer is concentrated in one of the phases. We have suggested a method for the specific problem under consideration that provides a means of obtaining an exact value for the overall Shenvood number based on values of the Sherwood numbers for decoupled problems. Application of this method t o more complicated interphase mass transfer
Ind. Eng. Chem. Res., Vol. 34, No. 10, 1995 3631 problems having convection, as well as reaction, will be considered in a subsequent article (Kleinman and Reed, 1995b).
Nomenclature A, = functions defined by (141, (19), and (20), i = 1, ..., 4 EO = dimensional value of the concentration in the droplet att=0 E , = dimensional value of the concentration at infinity (=O) c(’) = dimensionless concentration of species in phase i E(’) = Laplace transform of c ( ~ ) D(’)= molecular diffisivity of species in the ith phase D = molecular diffisivity ratio, D(1)/D(2) Da = second Damkohler number in the continuous phase, kR2/D(2) Dallm= limiting value for the Damkohler number, beyond which mass transfer is not further enhanced G = function defined by (21) H = distribution coefficient Im = imaginary part of the expression K = overall mass transfer coefficient defined in (23) k , = partial mass transfer coefficient in the ith phase k = chemical reaction rate constant in the continuousphase r = dimensionless radial coordinate R = droplet radius Re = real part of the expression Res = residue of the expression Sho = overall Sherwood number based on the initial driving force Sh = overall Sherwood number based on the instantaneous driving force Sh, = instantaneous Sherwood number in the ith phase defined by (48) t = dimensional time y, = positive roots of (161,j = 1, 2, ... Greek Symbols a = constant 4 = function defined by (16) t = dimensionless time Subscripts
0 = initial moment 1 = sphere 2 = continuous phase = free stream conditions as = asymptotic regime st = steady regime s = surface of the particle int = “internal” domain (sphere interior) ext = “external” domain (sphere exterior)
Superscripts (1) = sphere (2) = continuous phase (0) =
Bird, R. B.; Stewart, W. E.; Lightfoot, E. N. Transport Phenomena; Wiley: New York, 1960. Bleistein, N.; Handelsman, R. A. Asymptotic Expansion of Zntegrals; Dover Publications: New York, 1986. Brounstein, B. I.; Shegolev, V. V. Hydrodynamics, Mass and Heat Transfer in Column Devices (in Russian); Khimiya: Leningrad, 1988. CliR, R.; Grace, J. R.; Weber, M. E. Bubbles, Drops and Particles; Academic Press: London, 1978. Cooper, F. Heat Transfer from a Sphere to an Infinite Medium. Znt. J . Heat Mass Transfer 1977,20,991-993. Crank, J. The Mathematics of Diffusion, 2nd ed.; Clarendon Press: Oxford, England, 1955. Foust, A. S.; Wenzel, L. A.; Clump, C. W.; Maus, L.; Andersen, L. B. Principles of Unit Operations, 2nd ed.; Wiley: New York, 1980. Johns, L. E.; Beckmann, R. B. Mechanism of Dispersed-Phase Mass Transfer in Viscous, Single-Drop Extraction Systems. AIChE J . 1966,12,10-16. Jungu, Gh.; Mihail, R. The Effect of Diffisivities Ratio on Conjugate Mass Transfer From a Droplet. Znt. J . Heat Mass Transfer 1987,30,1223-1226. Kleinman, L. S.;Reed, X B, Jr. Methods to Solve Forced Convection Interphase Mass Transfer Problems for Droplets in General Stokes Flows. Paper presented at the Thirteenth Symposium on Turbulence, Rolla, MO, 1992. Kleinman, L. S.; Reed, X B, Jr. Single-Drop Reactive Extraction/ Extractive Reaction With Forced Convective Diffusion and Interphase Mass Transfer. NASA Conference Publication 10161; Proceedings of Sixth Annual Thermal and Fluids Analysis Workshop; NASA-Lewis Research Center, Cleveland, OH, 1995a; pp 291-315. Kleinman, L. S.; Reed, X B, Jr. Conjugate Mass Transfer From a Circulating Single Droplet to a Surrounding Flowing Fluid Enhanced by a Chemical Reaction. I n d . Eng. Chem.Res. 199513, submitted. Kronig, R.; Brink, J. C. On the Theory of Extraction From Falling Droplets. Appl. Sci. Res. 1950,A2, 142-155. Laddha, G. S.; Degaleesan, T. E. Transport Phenomena in Liquid Extraction; Tata McGraw-Hill Publishing Co. Ltd.: New Delhi, 1976. Levenspiel, 0. Chemical Reaction Engineering; John Wiley & Sons: New York, 1972. Newman, A. B. The Drying of Porous Solids: Diffusion and Surface Emission Equations. Trans. Am. Znst. Chem. Eng. 1931,27, 203-211. Nguyen, H. D.; Paik, S.; Chung, J. N. Unsteady Conjugate Heat Transfer Associated with a Translating Droplet: A Direct Numerical Simulation. Numer. Heat Transfer,Part A 1993,24, 161-180. Oliver, D. L. R.; Chung, J. N. Conjugate Unsteady Heat Transfer of a Translating Droplet at Low Reynolds Numbers. Znt. J. Heat Mass Transfer 1986,29,879-887. Oliver, D. L. R.; Chung, J. N. Unsteady Conjugate Heat Transfer From a Translating Fluid Sphere at Moderate Reynolds Number. Znt. J . Heat Mass Transfer 1990,33,401-408. Ruckenstein, E.; Dang, V.-D.; Gill, W. N. Mass Transfer With Chemical Reaction From Spherical One or Two Component Bubbles or Drops. Chem. Eng. Sci. 1971,26,647-668.
overall parameter
Received for review January 31, 1995 Revised manuscript received July 25, 1995 Accepted August 1, 1995@
’ = derivative of the expression Literature Cited Abramzon, B.; Borde, I. Conjugate Unsteady Heat Transfer From a Droplet in Creeping Flow. AZChE J . 1980,26,536-544. Abramzon, B.;Elata, C. Unsteady Heat Transfer From a Single Sphere in Stokes Flow. Znt. J . Heat Mass Transfer 1984,27, 687-695.
IE950094Z
Abstract published in Advance ACS Abstracts, September 1, 1995. @
