Unsteady Conjugate Mass Transfer between a Single Droplet and an

Ind. Eng. Chem. Res. 1996, 35, 2875-2888

2875

Unsteady Conjugate Mass Transfer between a Single Droplet and an Ambient Flow with External Chemical Reaction Leonid S. Kleinman† and X B Reed, Jr.* Department of Chemical Engineering, University of MissourisRolla, Rolla, Missouri 65401

Unsteady conjugate mass transfer between a spherical bubble, droplet, or particle and a surrounding fluid flow with chemical reaction in the continuous phase has been analyzed. An algorithm based on spectral methods has been developed and implemented for the simulation of the interphase mass transfer. The influence of Pe (convection) and Da (rate of external reaction), but in particular of D (diffusivity ratio) and H (distribution coefficient), which characterize the coupling features of the conjugate mass transfer, is studied in a wider range of parameters than earlier investigations. The asymptotic regime of mass transfer is investigated for long times, and an upper limit for the corresponding dimensionless mass transfer coefficient is established. A method for decoupling the conjugate problem is proposed that results in a modified version of the addition rule, which is a higher level of approximation than the original one. As measured against the results of our numerical simulations, the modified addition rule yields results in the asymptotic regime that are superior to those obtained using the customary addition rule. Introduction Interphase mass transfer between a spherical particle (solid, liquid, or gas) and a surrounding continuous phase (liquid or gas) is basic to many practical applications, not least in the chemical process industries. Historically, this class of problems was investigated as two entirely separate problems: “internal”, for mass transfer inside the particle, and “external”, for mass transfer in the continuous phase, with the surface concentration presumed constant for each of the problems. The solutions to these uncoupled problems are exact descriptions of situations in which each of the phases (dispersed and continuous) are single-component media, and in conjugate multicomponent systems, they are accurate descriptions of limiting cases when the resistance to interphase transfer is concentrated in one of the phases. Investigation of the internal problem in the absence of flow (Pe ) 0) is an elementary problem today (Newman, 1931). The solution of the opposite limiting case (Pe f ∞) is also well-known (Kronig and Brink, 1950). Johns and Beckmann’s (1966) paper may be considered a classic for this problem; it included numerical solutions and a thorough physical analysis of the problem with internal circulation modeled by the Hadamard-Rybczinski flow. Further research in this direction included complications connected with higher Reynolds numbers for the external flow, internal chemical reactions, and droplet oscillations. A reasonably complete bibliography is available in Clift et al. (1978) and Brounstein and Shegolev (1988). The external problem attracted the attention of investigators of both heat and mass transfer, and numerical solutions for both solid and fluid spheres have been published. Further developments included external heterogeneous and homogeneous chemical reaction, higher Reynolds numbers, and influence of surface contamination. Again, Clift et al. (1978) and Brounstein and Shegolev (1988) provide further background. * To whom correspondence should be addressed. Fax: (573) 314-4377. † Current address: Koch Engineering Co., Wilmington, MA 01887.

S0888-5885(95)00671-3 CCC: $12.00

Nevertheless, as Steiner (1986) points out, “After nearly a century of research, documented by hundreds of publications, mass transfer rates are still predicted with low accuracy, even in the case of single drops ascending through a quiscent continuous liquid”. For interphase mass transfer in multicomponent systems having comparable resistances in both the dispersed and continuous phases, the conjugate problem must be solved. So also must conjugate heat-transfer problems in two-phase flows in such applications as direct contact heat transfer. Numerical solution of a conjugate problem is a much more involved procedure than are the separate internal or external subproblems. Two additional parameters characterizing the conjugate features of the problem must be included, namely, the diffusivity ratio D and the distribution coefficient H. There are analytical results for the conjugate mass transfer problem available based on the assumption of a diffusion boundary layer and valid only for high Peclet numbers (Levich et al., 1965; Chao, 1969; Ruckenstein et al., 1971; Konopliv and Sparrow, 1972). For conjugate heat-transfer problems, the first numerical results for a range of Peclet numbers for which analytical solutions were not valid were published by Abramzon and Borde (1980). In addition to a set of numerical results, it included a detailed discussion of the physics of the problem, making it reminiscent of the paper by Johns and Beckmann (1966); in this sense, it may also be called a classical paper. Oliver and Chung (1986) solved the same heat-transfer problem for a much wider range of parameters, and they extended their investigation to include the effect of higher Reynolds numbers (Oliver and Chung, 1990). Nguyen et al. took into account simultaneous development of the velocity and temperature fields (1993a) and mixed convection (1993b). While including these and other physical phenomena in the mathematical model makes the problem richer, it unfortunately also makes the numerical solution much more CPU intensive, thereby precluding the investigation of a wide range of the parameters involved. Moreover, the more complicated physical effects can complicate the understanding of the basic features of dispersed-continuous phase mass transfer, which remain ill-understood at their most fundamental level. © 1996 American Chemical Society

2876 Ind. Eng. Chem. Res., Vol. 35, No. 9, 1996

Again citing Steiner (1986), “It is the fundamental description of the mass transfer behaviour of the drops which is most difficult to predict”. The objective of the present research is to investigate the fundamental features deriving from the conjugate character of the interphase transfer: connection of its parameters with the parameters of the individual mass transfer processes in each of the participating phases, and influence on them of the parameters characterizing the coupling of the process. In addition, we investigate the influence of a chemical reaction in the continuous phase on the mass transfer of solute from the droplet, which has practical implications for reactive extraction and for two-phase reactors. For numerical simulation of the interphase mass transfer process under consideration, we have developed a numerical algorithm based on Galerkin spectral methods (Kleinman and Reed, 1992, 1994). This algorithm enabled us to investigate a range of parameters wider than has been done by earlier investigators. We were, moreover, able to simulate limiting situations of mass transfer and to investigate realizations of those situations having different sets of characteristic parameters. Interphase mass transfer between a particle, drop, or bubble and the surrounding continuous phase is inherently unsteady, leading to vanishing values of local and average concentrations and their gradients as τ f ∞. The same variables, when normalized by the instantaneous average droplet concentration, approach timeindependent nontrivial distributions as τ f ∞. The final state, or asymptotic regime, is independent of the initial concentration distribution. A dimensionless, timeindependent overall mass transfer coefficient realized during the asymptotic regime is often called an asymptotic Sherwood number (or Nusselt number, in the case of heat transfer) and denoted by Shas. It may serve as a convenient measure for comparisons between the mass transfer rates for different sets of parameters (Pe, Da, D, H). Our computations have confirmed the intuitively obvious conjecture that its value has an upper limit of approximately 17.9, if defined on the basis of the dispersed-phase properties. Most investigators who have studied the conjugate mass/heat transfer problem (Abramzon and Borde, 1980; Abramzon and Elata, 1984; Oliver and Chung, 1986, 1990; Nguyen et al., 1993a,b) have examined the applicability of the well-known addition rule for the calculation of the overall asymptotic Sherwood number for the conjugate problem in terms of the partial Sherwood numbers for the separate internal and external problems. Such an approximation offers the possibility of using known approximations for the partial Sherwood numbers (which are independent of D and H) in the calculation of Shas. The addition rule (Abramzon and Borde, 1980; Abramzon and Elata, 1984; Oliver and Chung, 1986, 1990; Nguyen et al., 1993a,b) has not yet been seriously examined because of the limited range of results available. We have investigated this issue for nonconvective diffusion-reaction mass transfer (Kleinman and Reed, 1995) and found that in some parameter domains the relative errors in the addition rule are often as high as 30% and are occasionally even higher. Abramzon and Borde (1980) and Abramzon and Elata (1984) suggested approximate methods based on the “film model”, which imposes severe limitations on their methods. In any event, the lack of suitable numerical results precluded analysis of the range of validity of their methods.

Nonetheless, an idea of Abramzon and Elata (1984) for calculation of the asymptotic Nusselt number for heat transfer between a solid sphere and a surrounding flow was applied to the conjugate nonconvective diffusion-reaction problem (Kleinman and Reed, 1995), where it yielded exact results. In the present paper we develop this idea further. We derive an equation for Shas that is exact. Its solution requires an effort comparable to that of the general conjugate problem. We show that this equation can be transformed into the usual addition rule by making a set of simplifying assumptions. Dropping one of the assumptions enables us to obtain a modified addition rule which preserves the main advantages of the usual version and yet yields dramatic improvements in the results. Moreover, in our framework for decoupling the coupled (conjugate) problem, further modifications can be made in other ranges of parameters or for other applications. This method of decoupling the conjugate mass/heat transfer problem is not limited to a specific form of the velocity field and can be applied to other classes of problems, including non-Newtonian flows or Newtonian ones having high Reynolds numbers. Mathematical Formulation and Numerical Algorithm We consider a reactant species undergoing interphase mass transfer from a circulating spherical drop to a continuous phase in which a chemical reaction takes place and introduce the dimensionless solute concentration in the dispersed and continuous phases, i ) 1, 2:

c(1) ) c˜ (1)/c˜ 0,

c(2) ) Hc˜ (2)/c˜ 0

(1)

The forced convective diffusion equations governing mass transfer of the solute from the droplet into the continuous phase can be written in the following dimensionless form:

∂c(i) Pe (i) (i) D(i) 2 (i) + ‚v ‚∇c ) (2)‚∇ c - Da(i)‚c(i), ∂τ 2 D

i ) 1, 2 (2)

in which Da(1) ) 0. The concentrations c(i)(τ,ϑ,r), i ) 1, 2, are subject to initial and boundary conditions at the droplet interface of

τ ) 0: r ) 1:

c(1) ) 1, c(2) ) 0

(3)

c(i) ) c(2) ∂c(1) ∂c(2) ) H‚D ∂r ∂r

(4)

{

and at the limits of the overall domain of

r ) 0:

c(1) < ∞

(5)

r f ∞:

c(2) f 0

(6)

as well as to symmetry conditions of

|

∂c(i) ∂ϑ

ϑ)0,π

) 0,

i ) 1, 2

(7)

Our numerical algorithm was first described in Kleinman and Reed (1992) and later in 1994. In the appendix

Ind. Eng. Chem. Res., Vol. 35, No. 9, 1996 2877

we present a shortened but reasonably detailed description because Kleinman and Reed (1994) is not widely available. Quantities of Primary Interest The evolution of the concentration field inside and outside the drop reflects the detailed physics and chemistry of the interphase mass transfer for this conjugate problem, but the local instantaneous solute concentrations contain more detail than can be profitably used. A reduced set of information is desirable to simply and practically characterize the mass transfer process. The average droplet concentration

jc(1) )

∫01∫-11r2c(1)(τ,λ,r) dλ dr

3 2

(8)

is a more convenient measure than the full concentration field. Its evolution directly characterizes the overall mass transfer process. Usually, the mass transfer coefficients K are used to quantify the rate of mass transfer. By definition, the mass transfer coefficient K is a ratio of the mass flux to the driving force that causes the flux. The dimensionless overall mass transfer coefficients based on the instantaneous driving force ch˜ (1)(τ) - Hc˜ ∞ ≡ ch˜ (1)(τ) are called instantaneous Sherwood numbers:

Sh(i)(τ) )

K(i)(τ)‚2R , i ) 1, 2 D(i)

(9)

The Sherwood number for the internal domain is our focus, and for simplicity we omit the subscript “(1)” and denote it by Sh(τ). The corresponding Sherwood number based on the initial driving force c˜ 0 - Hc˜ ∞ ≡ c˜ 0 is identified as Sh0(τ):

∫-11

Sh0(τ) ) -

|

∂c(1)(τ,r) ∂r

r)1

dλ

(10)

It is easy to prove the following useful relationships:

Sh(τ) )

Sh0(τ) jc(1)(τ)

3 ∂ ln cj(1) ) - D‚Sh(τ) ∂τ 2

(11)

(12)

Asymptotic Regime for Mass Transfer The asymptotic regime plays an important part in the overall process of unsteady conjugate mass transfer for droplets. It was first discovered by Johns and Beckmann (1966) for mass transfer into a droplet by convective diffusion with negligible resistance in the continuous phase (the internal problem). The concentration profile and rate of mass transfer normalized by the instantaneous concentration driving force behave qualitively differently from the same quantities normalized by the initial (maximum) driving force. In the latter, the asymptotic limit is zero. In the former, the normalized concentration has a nonzero, nonuniform asymptotic distribution and the corresponding mass transfer rate a nonzero asymptotic value. This means, in particular, a nonzero limit designated as Shas for the average instantaneous Sherwood number Sh(τ) as τ f ∞. Johns and Beckmann (1966) also made the connec-

tion between this value and the minimum eigenvalue of the spatial operator in the forced convective diffusion equation when the latter is solved by the method of separation of variables. They refer to the eigenvalues for the internal problem obtained analytically for Pe ) 0 (Newman, 1931) and for Pe f ∞ (Kronig and Brink, 1950), which give respectively 2π2/3 and 17.9 for the asymptotic Sherwood numbers. In Kleinman and Reed (1995), devoted to the investigation of nonconvective conjugate mass transfer with an external chemical reaction, we showed an obviously similar connection between the asymptotic Sherwood number and the leftmost singular point (which was proven to be real) of the analytical solution for the Laplace-transformed concentration. It was shown that for the conjugate problem (which depends on D, H, and Da(2)) this point has an upper bound equal to the value of the corresponding singular point for the internal problem (π2) multiplied by the diffusivity ratio D. This led to the existence of an upper limit equal to 2π2/3 for the asymptotic average Sherwood number for the conjugate nonconvective problem, which coincides with the corresponding value for the internal problem. We have no rigorous proof of the analogous statement for the convective conjugate mass transfer problem under consideration, but all the numerical results presented in the following section are compatible with this intuitively inviting conjecture, which can be formulated as follows: An asymptotic average Sherwood number for a conjugate convective diffusion problem with external chemical reaction has an upper limit equal to the asymptotic Sherwood number for the independent internal problem having the same internal Peclet number as the conjugate problem. As long as the asymptotic Sherwood number for the internal problem has an upper limit as Pe f ∞ (Kronig and Brink, 1950), it is also the upper limit for the Sherwood number for the conjugate problem. The latter can therefore never exceed the value of approximately 17.9. Because the actual rate of mass transfer according to (12) is determined by the product of 3/2DSh, in its turn it cannot exceed the approximate value of 26.8D. Mathematically, the existence of an asymptotic regime is easily explained on the basis of what would be the solution of system (2) by the method of separation of variables (Johns and Beckmann, 1966). The concentration field can be expressed as a series of products of spatially dependent functions by exponential functions of the product of time by the respective eigenvalues of the spatial operators. For sufficiently large time, all terms in this series become negligible except the first one, which corresponds to the minimum eigenvalue. Such parameters as the average droplet concentration and concentration gradient obey the same exponential decrease in time. Thus, the instantaneous Sherwood number, being the ratio of the average of the gradient of the internal concentration over the droplet surface to the average concentration of the droplet, becomes time-independent, that is, attains its asymptotic value. The importance of the asymptotic regime lies in the independence of its parameters from the initial conditions, thereby enabling the use of Shas (more precisely, DShas) as a convenient measure of the mass transfer rate for comparison of different sets of parameters (Pe, Da(2), D, H). Computational Results and Discussion One of the objectives of this research was to develop an understanding of the influence of external convec-


Figure 1. Evolution of the local Sherwood number Shloc at (a) Da ) 0 (b) Da ) 100 for Pe ) 500 and D ) 0.1.

tion, internal circulation (a consequence of external convection), and external chemical reaction on the rate of conjugate mass transfer over a wider range of parameters than those of earlier investigators of conjugate mass transfer with or without chemical reaction. Because we wanted to investigate the influence of parameters other than Reynolds number on interphase mass transfer, we used the Hadamard-Rybczinsky solution for the convecting velocities. Whether these velocity fields are valid for this solution beyond Re ≈ 1 is a moot point because any prescribed arbitrary velocity field can be implemented in our numerical algorithm. Here we present a sampling of our results for computations that cover the following ranges of parameters: 0 e Pe, Da(2) e 1000, 0.05 e D e 10, 0.25 e H e 4, 0.1 e µ e 10, with one case of µ ) 107.

In Figure 1a,b the evolution of the local instantaneous Sherwood number Shloc(τ,ϑ) for Pe ) 500, H ) µ ) 1, D ) 0.1 , and Da ) 0 and 100 is presented. The behavior of Shloc(ϑ) for cases such as these, when the resistance is shifting with time into the droplet, and in the asymptotic regime, when it is almost totally concentrated there, is very different from that for the cases presented in (Kleinman and Reed, 1994). There, a significant part of the resistance to mass transfer is associated with the continuous phase. The existence of a strong external wake (isocontours demonstrating this effect have been omitted for brevity) leads to a pronounced fore-aft asymmetry in Shloc, with its value at the forward stagnation point being much higher than that at the rear one. The angular distribution of Shloc in the presence of an external chemical reaction is more


Figure 2. Evolution of the average Sherwood number Sh at different values of Pe for D ) 0.1 and µ ) 1.0 (a) and 107 (b).

symmetric, with the maximum occurring at ϑ ) π/2, where the internal diffusional boundary layer has the smallest thickness. This again contrasts with the results presented in Kleinman and Reed (1994) for cases with lower internal Peclet number, when the thinnest internal boundary layer was situated at the forward stagnation point. Figure 2 reflects the influence of the external Peclet number Pe on the instantaneous Sherwood number Sh(τ). The value of the diffusivity ratio D was chosen to be 0.1 in order to illustrate the limiting situation when the entire resistance to mass transfer is concentrated in the dispersed phase. The oscillations in Sh(τ), first discovered and discussed by Johns and Beckmann (1966) for the internal problem, reflect the effect of

circulation in the drop. The amplitude of these oscillations is higher, the period is shorter, and the asymptotic regime is more quickly established, the higher the Peclet number is. The realization of the limiting situation discussed in the previous section is confirmed: values of asymptotic Sherwood number for all Pe g 200 in Figure 2a where µ ) 1 are very close to one another and tend to the approximate limiting value of 17.9 for infinite internal Pe. The very high viscosity ratio in Figure 2b (µ ) 107) suppresses internal circulation, and the values of Shas tend to those for a solid sphere, that is, to about 6.58. From Figure 3a, in which the evolution of the instantaneous Sherwood number for the convective diffusion nonreaction case is presented at constant external


Figure 3. (a) Evolution of the average Sherwood number Sh at different values of D for Da ) 0 and Pe ) 1000. (b) Evolution of the product DSh at different values of D for Da ) 0 and Pe ) 300.

Peclet number Pe ) 1000 for different values of the diffusivity ratio D, one infers that the frequency of the oscillations in Sh(τ) is virtually independent of the internal diffusivity, confirming the explanation that these oscillations are due only to the internal circulation and not to diffusion (Johns and Beckmann, 1966). A decrease in D shifts the resistance into the drop, and the value of Shas again converges to an upper limit of about 17.9. For the smallest values of D, the Shas have not yet attained their asymptotic values in the time shown, but our computational results show that, for D ) 0.25, Shas ≈ 17.38, and for D ) 0.1, Shas ≈ 17.52. The actual rate of mass transfer, as follows from (12) and as was discussed in detail in Kleinman and Reed (1995), is determined by the product DSh(τ) and not by

the value of Sh(τ) alone. That is why the product DSh(τ) decreases, even though the values of Sh(τ) (and Shas) increase as D decreases, and it is evident that a lower internal diffusivity yields a lower mass transfer rate, all other parameters remaining unchanged. An increase in the diffusivity ratio D shifts the resistance into the continuous phase, leading to the actual rate of mass transfer that is proportional to DSh(τ) (according to (12)) becoming independent of D. This limiting situation is illustrated in Figure 3b for Pe ) 300. An increase in the distribution coefficient H (Figure 4a), which means a decrease in the solubility of the reacting species in the continuous phase, obviously leads to a decrease in the mass transfer rate (indeed, to zero


Figure 4. Evolution of the average Sherwood number Sh at different values of (a) H and (b) µ for Da ) 0 and Pe ) 1000.

as H f ∞). Variation in H does not influence the frequency of the initial oscillations of Sh(τ), and their amplitudes decay with time. Decreasing H (Kleinman and Reed, 1995) leads to one of the possible limiting situations that depends on the values of the other parameters. In Figure 4a, the resistance has shifted into the dispersed phase as H f ∞, although for the range of H presented, this limiting situation has not yet been attained. The influence of internal convection on the mass transfer rate is illustrated in Figure 4b, where the evolution of the instantaneous Sherwood number is shown for different values of the viscosity ratio µ at Pe ) 1000. Qualitively, the influence of the variation in µ on the oscillations in Sh(τ) is the same as that of the

variation in Pe with all other parameters unchanged (Figure 2a). There is nothing unexpected in this similarity because in both kinds of cases we vary the internal Peclet number defined as proportional to Pe/(µ + 1). For higher µ, the correspondingly lower internal Peclet number leads to a thicker internal concentration boundary layer and consequently to a lower Sh(τ). Neither an increase nor a decrease in µ leads to a shift of the entire resistance into one of the phases, that is, to one of the limiting situations. Figure 5 illustrates the effect of the rate of external reaction on the average instantaneous Sherwood number for Pe ) 500, D ) 0.1, H ) µ ) 1. It corresponds to the case when the mass transfer resistance shifts with time into the dispersed phase. After it has become


Figure 5. Evolution of the average Sherwood number Sh at different values of Da for Pe ) 500 and D ) 0.1.

almost entirely concentrated there, the influence of Da(2) becomes negligible; for τ g 0.1 in Figure 5, the values of Sh for different Da(2) are very close to one another. Moreover, as in the previous cases for such a limiting situation, the value of the asymptotic Sherwood number tends to about 17.9, corresponding to an internal problem with Peclet number ∼ Pe/D ) 104. Asymptotic Decoupling of the Conjugate Mass Transfer Problem The most common procedure for calculating the overall mass transfer coefficient is the application of the addition rule: the representation of the overall resistance as a sum of partial resistances for two separate (specifically, uncoupled) internal and external problems. Abramzon and Borde (1980) first introduced the addition rule for heat transfer between a drop and a continuous phase in motion. In the present variables for mass transfer problems, we have reexpressed it as (Kleinman and Reed, 1995)

1 DH 1 ) + Shas Sh(1) Sh(2) as

(13)

st

(1) Here Shas is an asymptotic Sherwood number for the internal problem having a uniform, time-independent external concentration and an internal Peclet number the same as that in the droplet in the original conjugate (2) problem. Shst for the external problem corresponds to a steady-state concentration distribution in the continuous phase for uniform and constant droplet concentration. (1) (2) Because expressions for Shas and Shst were obtained for decoupled internal and external problems, the addition rule for mass transfer in the dispersed and continuous phases being coupled is at best an approximation to reality. Unfortunately, the range of its applicability has not been subjected to a detailed investigation. Although the computations performed by Abramzon and Borde (1980), Oliver and Chung (1986, 1990), and Jungu and Mihail (1987) have shown that it is fairly accurate in certain ranges of parameters,

application of this rule can lead to errors of 30% and larger in other ranges. The limitations imposed on this research by the high consumption of CPU time in numerical simulations and by the limited ranges of parameters for which convergence of the numerical algorithms could be obtained have precluded a systematic investigation. Our attempt (Kleinman and Reed, 1995), though restricted to the nonconvection case in order to obtain an analytical solution, sheds light on this issue, but the existence of an error inherent in the addition rule as an approximation formula is still not well understood. Inaccuracy in the addition rule has been ascribed to the poor representation of its individual (1) (2) and Shst ), which is, of course, a components (Shas separate source of error. Efforts have been made to improve the accuracy of the addition rule by the selec(1) and tion of more appropriate approximations for Shas (2) Shst (Laddha and Degaleesan, 1976), but this is not the only way to improve the addition rule. The method described in this section for improving the addition rule was mentioned in Kleinman and Reed (1995) as a development of an idea of Abramzon and Elata (1984). It offers a way to improve the accuracy of results for the asymptotic overall mass transfer coefficient obtained by decoupling the conjugate problem into internal and external problems, where the “uncoupled” ones are more closely linked to one another than is customary in the original addition rule (13). The method is not proposed as a panacea, not least because (1) (2) and Shst must still be computed numerically. Shas Although these computations are much less intensive than the original conjugate version, more remains to be done before the modified addition rule can be recommended as both simple and practical. We introduce concentrations cˆ (i)(τ,λ,r), i ) 1, 2, in both phases that are normalized by the surface concentration cs(τ,λ),

c(i)(τ,λ,r) ) cˆ (i)(τ,λ,r) cs(τ,λ)

(14)

In the asymptotic regime, ∂cˆ (i)/∂τ ) 0, i ) 1, 2, and ∂ log cs/∂τ ) ∂ log cj(1)/∂τ. Consequently, it follows from


(2) that the cˆ (i), i ) 1, 2, satisfy the steady equations and boundary conditions

Pe ˆ(i) (i) (i) v ‚∇cˆ ) ∇2cˆ (i) + Da ˆ(i)cˆ (i), 2

|

∂cˆ (1) ∂r

)

r)1

|

∂cˆ (2) ∂r

r)1

ˆ(i)(λ,r) ) Da

(

Upon substituting (24) into (23) and then eliminating (1) Sh ˆst between (22) and (23), we arrive at

(17)

1 ) Shas

)

3D Sh ˆ - Da(i) + G ˆ (i)(λ,r) (18) 2 D(i) as

ˆ(i),λ,r) ) G ˆ (i)(Pe (i) ∂c ∂c ˆ(i) υλ 1 Pe x1 - λ2 s + 12 ∂ (1 - λ2) s cs 2 r ∂λ r ∂λ ∂λ

[

{

ˆ(1) ≡ Pe/D, Pe

]}

ˆ Pe(2) ) Pe

(19) (20)

Equations 15 with boundary conditions (16) can be considered as two separate steady-state convective diffusion-reaction problems for i ) 1, 0 e r e 1 and for i ) 2, 1 e r < ∞. The rate “constants” of the reactions are, however, spatially dependent and include the unknown function cs, a part of the solution of the conjugate problem. Together with (17), this provides the interconnection between the two uncoupled problems for i ) 1 and i ) 2 in (15). Introducing Sherwood numbers

Sh(i) ˆ st ) -

∫-1∂c∂rˆ |r)1 dλ, 1

(i)

i ) 1, 2

(21)

for these steady problems for the dispersed and the continuous phases and averaging condition (17) over the droplet surface gives

ˆ(1) ˆ(2) HDSh st ) Sh st

(22)

Because of the artificial generative reaction term in (15) for i ) 1, the corresponding asymptotic Sherwood number is defined as (1) Sh âs

≡

ˆ Sh(1) st chˆ (1) - 1

(24)

(16)

upon taking (12) into account. In (15) we have introduced the following definitions: (1)

chˆ (1) ) Sh ˆ(1) st /Shas

i ) 1, 2 (15)

cˆ (i)(τ,λ,r)1) ) 1 HD

Averaging (15) for i ) 1 over the drop volume while taking into account Da(1) ) 0 and the vanishing of the averaged G ˆ (1) yields

(23)

This definition is justified by the observation that, in the absence of a reaction term in (15) for i ) 1, a steady (1) solution is a trivial one with chˆ (1) ≡ 1 and Sh ˆst ≡ 0. The solution of the time-dependent version of (15), nonethe(1) less, results in a nonzero value of ˆ Shas in (23): being equal to 2π2/3, as obtained analytically by Newman (1931) for the nonconvection case ˆ Pe(1) ) 0; being equal to ≈17.9, as obtained by Kronig and Brink (1950) for infinite Pe ˆ(1); and being equal to different Pe ˆ(1)dependent values which can be determined numerically, as was done by Johns and Beckmann (1966) for finite nonzero Pe ˆ(1).

1

+ 3 Pe ˆ )Pe/D,Da ˆ(1)) Shas+G ˆ (1)(λ,r) 2 HD (25) 3 (2) (2) (2) (2) DSh Sh ˆ(2) Pe ˆ )Pe,Da ˆ ) -Da +G ˆ (λ,r) st as 2

(

(1) Sh âs

)

(1)

(

(

)

)

Equation 25 has the form of addition rule (13) and is exact for the asymptotic regime because no other assumptions have been made to this point. In general, solutions of problems (15), i ) 1, 2, could be obtained by assuming an approximate expression for ∂cs(λ)/∂λ in the asymptotic regime. If we neglect both the constants Shas and the functions G ˆ (i)(λ,r), i ) 1, 2, on the right side of (25), then we recover the usual addition rule, (eq 13). In that case, the internal and (1) (2) external problems that yield Sh âs and Sh ˆst become totally independent, the customary way to calculate the overall mass transfer coefficient. We propose the next, logical step of approximation within this new framework and neglect the functions G ˆ (i)(λ,r), i ) 1, 2, in (25) while retaining the unknown constant Shas. This is equivalent to neglecting the angular variation of the surface concentration cs. Then (25) becomes

1 ) Shas

1 + 3 )Pe/D,Da ˆ(1)) Shas 2 HD 3 Sh ˆ(2) ˆ(2))Pe,Da ˆ(2)) DShas-D(2) st Pe 2

ˆ (Pe

(1) Sh âs

)

(1)

(

(

))

(26)

which differs from the usual addition rule (13) in that both terms on the right are functions of the unknown asymptotic Sherwood number for the conjugate problem. This version of the addition rule, which we term the modified addition rule, is a nonlinear equation for Shas. To solve it, we need to be able to compute an asymptotic Sherwood number for the internal problem with homo(1) geneous generative reaction, Sh âs , and a steady-state (2) Sherwood number Sh ˆst for the external problem with either generative or consumptive homogeneous reaction (depending on the value of Da(2)). No value of Shas can exceed an asymptotic Sherwood number for the internal problem having the same effective internal Peclet number ˆ Pe(1) ) Pe/D. Consequently, the rate constants for generative reaction in the internal and external problems (with Da(2) ) 0) should span the range from Da ˆ(i) ) 0 to Da ˆ(i) ) (i) (1) (3/2)(D(1)/D(i))Sh âs (Pe ˆ ), for i ) 1, 2, respectively. Moreover, for the internal problem this restriction is satisfied naturally: the numerical solution of this steady problem by the stabilization method (i.e., as a timedependent approach to the steady state) showed that when rates of generative reaction exceed the value mentioned above, this problem has no steady solution. It is not difficult to figure out when the customary version of the addition rule (13) yields results close to


or

Table 1. Values of Constant Parameters in the Approximation Formula (32) for µ ) 1

(1) Shas ≈ Sh âs (Pe ˆ(1))Pe/D,Da ˆ(1))0)

Pe

R β (2) Shst

10

50

100

300

500

1.273 1.344 3.448

1.223 0.063 5.458

1.213 0.031 6.906

1.042 0.161 10.430

1.017 0.010 12.816

those obtained with the modified version (25). Numerical solution of the internal problem (15), i ) 1, (1) demonstrates that the value of Sh âs (Pe ˆ(1),Da ˆ(1)) start(1) ing at Da ˆ ) 0 quickly reaches its maximum near Da ˆ(1) ≈ 1 and then gradually decreases to the end of the interval

3 (1) (1) (1) Da ˆmax ) Sh ˆ (Pe ˆ ,Da ˆ(1))0) 2 as

(27)

where it attains the same value as at ˆ Da(1) ) 0. Thus, the first term in the modified addition rule (26) is close to that of the original version (13) when either

Shas, DShas , 1

(28)

(29)

If inequalities (28) are satisfied, the modified addition rule (25) reduces to the customary version (13). It is obviously realized when either of the following inequalities is satisfied:

H.1

(30)

Pe, Da(2) , 1

(31)

Inequality (30) occurs at a very low solubility of the reactant in the continuous phase. If inequalities (31) are valid, neither version of the addition rule offers a correct description because neither (13) nor (26) can have vanishing Shas, which is known to result from the solution for this case (Cooper, 1977; Kleinman and Reed, 1995). Inequality (29) usually holds when most of the resistance is concentrated in the dispersed phase. As may be seen from the results presented below, when a significant part of the resistance is concentrated in the continuous phase, the usual version of the addition rule (13) yields much larger errors than our modified version (26).

Table 2. Values of the Asymptotic Sherwood Numbers and Corresponding Relative Errors Computed through the Customary (13) and Modified (26) Versions of the Addition Rule for µ ) 1 Pe

Da(2)

D

H

Shas(num)

Shas (26)

, % (24)

Shas (13)

, % (11)

10 10 10 10 10 10 10 10 50 50 50 50 50 50 50 100 100 100 100 100 100 100 100 100 100 300 300 300 300 300 300 300 300 500 500 500 500 500 500 500 500 500 500 500 500 500

1 10 100 1000 0 0 0 0 10 100 500 0 0 0 0 10 100 1000 10 100 0 0 0 0 0 10 100 1000 0 0 0 0 0 10 100 500 10 100 1000 10 100 0 0 0 0 0

1.0 1.0 1.0 1.0 0.1 0.5 1.0 1.0 1.0 1.0 1.0 0.1 1.0 1.0 1.0 1.0 1.0 1.0 10.0 10.0 0.1 1.0 10.0 1.0 1.0 1.0 1.0 1.0 0.1 1.0 10.0 1.0 1.0 1.0 1.0 1.0 0.1 0.1 0.1 10.0 10.0 0.1 1.0 10.0 1.0 1.0

1.00 1.00 1.00 1.00 1.00 1.00 1.00 2.00 1.00 1.00 1.00 1.00 1.00 0.25 4.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 0.25 4.00 1.00 1.00 1.00 1.00 1.00 1.00 0.25 4.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 0.25 4.00

1.56 3.61 5.47 6.25 7.12 2.14 1.11 0.86 4.00 6.43 7.38 12.89 2.68 4.59 1.07 4.61 7.79 9.98 0.66 1.61 14.48 3.56 0.47 6.31 1.38 6.39 9.49 13.44 15.90 5.76 0.79 10.55 2.15 7.53 10.02 13.03 16.43 16.90 17.52 1.09 1.74 16.34 7.08 0.99 12.54 2.65

1.49 3.65 5.46 6.25 7.14 1.97 1.02 0.82 4.08 6.17 7.22 12.97 2.63 4.31 1.09 4.66 7.23 9.35 0.65 1.65 14.74 3.59 0.45 6.27 1.40 6.39 9.38 12.94 15.85 5.75 0.77 10.54 2.15 7.52 10.25 12.97 16.43 17.02 17.59 1.08 1.82 16.25 7.06 0.98 12.49 2.65

-4.31 1.13 -0.18 -0.01 0.29 -7.75 -8.44 -4.33 2.05 -4.06 -2.19 0.61 -1.83 -6.04 1.44 1.15 -7.20 -6.33 -1.24 2.06 1.78 0.71 -3.96 -0.62 1.48 -0.01 -1.06 -3.73 -0.32 -0.04 -2.76 -0.12 0.07 -0.13 2.28 -0.44 -0.01 0.71 0.40 -0.86 4.68 -0.54 -0.22 -1.33 -0.37 0.00

2.67 3.73 5.11 6.04 8.43 3.44 2.27 1.37 4.35 6.01 7.00 13.04 3.30 5.99 1.18 5.27 7.47 9.53 0.87 1.69 14.05 4.26 0.63 7.95 1.49 7.05 9.58 12.93 15.25 6.33 0.91 11.61 2.24 7.86 10.11 12.61 15.92 16.67 17.41 1.24 1.90 15.69 7.33 1.11 12.82 2.70

71.75 3.34 -6.61 -3.36 18.48 61.22 104.70 60.06 8.71 -6.61 -5.20 1.20 22.84 30.52 9.79 14.53 -4.15 -4.49 31.69 4.50 -2.94 19.70 33.33 25.94 8.11 10.35 1.05 -3.80 -4.10 9.96 14.75 10.06 4.56 4.30 0.91 -3.23 -3.10 -1.33 -0.63 13.83 9.49 -3.97 3.53 11.62 2.26 2.05


Both internal and external steady boundary value problems (15) and (16) were solved numerically by the stabilization method with the same numerical algorithm as was used for the conjugate problem. Computational results for the Sherwood number for the steady external problem (15) (i ) 2), (2) Sh ˆst (Pe ˆ(2),Da ˆ(2)), were approximated by empirical analytical relationships separately for the consumptive and for the generative reactions. For the consumptive reaction (Da ˆ(2) < 0) (Dilman and Polyanin, 1988),

between a solid sphere of high conductivity and an ambient flow that the value of the corresponding factor (2) including Nust overestimates Nuas. The modified version of the addition rule (26), by construction, should yield values of Shas lower than (2) (2) Shst /(DH) because Sh ˆst , as may be seen from (26), decreases with Da ˆ(2). The results in Table 2 show that for such a limiting situation (e.g., for large D), (26) yields results which are very close to the exact (numerical) ones.

(2) Sh ˆst ≈2+

Conclusion

x

(2) [Sh ˆst (Pe ˆ(2),Da ˆ(2))0)

2

- 2] +

(2) [Sh ˆst (Pe ˆ(2))0,Da ˆ(2))

2

- 2]

(32)

was used, where (Crank, 1975)

ˆ(2) ˆ(2))0,Da ˆ(2)) ) 2(1 + x|Da ˆ(2)|) Sh st (Pe

(33)

The corresponding Sherwood number for the convective (2) diffusion problem without reaction, Sh ˆst (Pe ˆ(2),Da ˆ(2) ) 0), was determined numerically, although any of the well-known empirical expressions (Clift et al., 1978) could be used. For the generative reaction (Da ˆ(2) > 0), the approximation

Sh(2) ˆ ˆ(2),Da ˆ(2)) ) ˆ Sh(2) ˆ(2),Da ˆ(2))0) [1 - β(Da ˆ(2))R] st (Pe st (Pe (34) fits numerical data with a relative error less than 3% (2) except for values of Da ˆ(2) leading to ˆ Shst ≈ 0. Values of R and β in (34) depend on Pe and are presented in Table 1 for several values of Pe for µ ) 1. In order for (26) to be of greatest practical utility, an (1) accurate approximation for Sh âs should be found that is also analytically simple, for instance, along the lines (2) of (34) for Sh ˆst . A spline approximation of the corresponding numerical solution has been used here. In Table 2, values of the asymptotic Sherwood number obtained with the modified addition rule (26) and its customary version (13) are compared for values of the Peclet number of the external flow, Pe ) 10, 50, 100, 300, and 500, and for equal viscosities of the dispersed and continuous phases, µ ) 1. It is obvious that application of (26) yields dramatic improvements over (13) across the entire range of parameters (Pe, Da, D, H). The upper limit for the relative error in applying (26) is about 8%, whereas the upper limit with (13) is more than 100%. There are very few points where the errors in (26) are larger than in (13), but even in those cases they do not exceed practically acceptable values of 8%. Neglecting G ˆ (i) and Shas in Sh ˆ(i) could have led to errors of different signs and hence to their partial cancellation, which, in turn, could have left the original version, in which both terms were neglected, more accurate than the modified one, in which one term was neglected. The results presented in Table 2 confirm our earlier conclusion, viz., that when a dispersed phase is mass transfer limiting, both versions of the addition rule yield accurate results. In contrast, the customary version of the addition rule (13) fails to adequately describe situations in which continuous phase mass transfer is limiting. According to (13), the value of Shas in such (2) cases should be well approximated by Shst /(DH). Abramzon and Elata (1984) found for heat transfer

Unsteady conjugate mass transfer from a fluid sphere to an ambient fluid in which flow and a chemical reaction occur has been investigated numerically. The range of parameters characterizing the coupling in the conjugate problem, namely, D and H, was much wider than in earlier studies of other investigators, thereby making possible the simulation of limiting situations of conjugate mass transfer for this class of problems, which contains Da, an additional parameter itself. The asymptotic instantaneous dimensionless mass transfer coefficient Shas has been shown to have an upper limit, independently of the values of the parameters of the problem (Pe, Da, D, H). A modified addition rule has been developed that gives better results than the usual addition rule for the approximation of overall mass transfer coefficients in terms of partial ones for the uncoupled individual internal and external problems, especially when a significant part of the resistance resides in the continuous phase. The procedure introduced to decouple the conjugate problem may be used to develop further improvements in the addition rule. Nomenclature c ) dimensionless concentration, eq 1 cˆ ) concentration normalized by the surface concentration (dimensionless), eq 12 D(i) ) molecular diffusivity of the reactant in the fluid in the ith domain D ) molecular diffusivity ratio, D(1)/D(2) Da(i) ) second Damko¨hler number in the ith domain, k(i)R2/ D(i) Da ) Da(2) - second Damko¨hler number in the continuous phase G ˆ ) functions defined by eq 19 H ) distribution coefficient (Henry’s law constant) K ) overall mass transfer coefficient k ) chemical reaction rate constant Pe ) Peclet number of the external flow, 2U∞R/D(2) r ) radial coordinate nondimensionalized with the droplet radius r∞ ) dimensionless radial coordinate of the artificial external boundary R ) droplet radius Sh(i) ) overall instantaneous Sherwood number, based on the ith phase, eq 9 Sh ) Sh(1) - overall instantaneous Sherwood number Sh0 ) overall Sherwood number based on the initial driving force, eq 10 t ) dimensional time U∞ ) speed of the uniform flow at infinity v ) velocity field nondimensionalized by U∞ Greek Symbols R, β ) constant parameters in eq 34 ϑ ) polar angle in spherical coordinate system λ ) cos ϑ

2886 Ind. Eng. Chem. Res., Vol. 35, No. 9, 1996 µ ) molecular viscosity ratio, µ(1)/µ(2) µ(i)

) molecular viscosity of the fluid in the ith domain

is easy to show that the spherical symmetry properties impose restrictions on the functions c(1) m (τ,r) in (A2):

τ ) dimensionless time, tR2/D(2)

c(1) m (τ,r)0) ) 0,

|

Superscripts

∂c(1) m ∂r

∼ ) dimensional value j ) average over the drop volume

r)0

) 0,

(1) c2k+1 (τ,r) ) odd function of r

(i) ) ith domain, i ) 1, 2 (1) ) drop

(A3)

m*1

(1) (τ,r) ) even function of r c2k

ˆ ) one of the uncoupled problems (15)

}

(A4)

k ) 0, 1, ... (A5) c(1) m (τ,r)

can be The major benefit gained is that expanded in a series containing solely Chebyshev polynomials of even order, which form a complete set on the domain 0 e r e 1 (Spalart, 1984):

(2) ) continuous phase Subscripts

N(1)

0 ) initial time

c(1) m (τ,r)

1 ) sphere

km

) δm,0R0(τ) + r

(1) φm,n (τ) T2n-2(r), ∑ n)1

m ) 0, 1, ..., M (A6)

2 ) continuous phase ∞ ) free-stream conditions

Here the Tp(r) are Chebyshev polynomials of the first kind of order p, R0 is the concentration at the origin of the drop, and

as ) asymptotic regime st ) steady regime

k2j ) 2,

s ) surface of the droplet

k1 ) 1,

Appendix: Numerical Algorithm Upon introducing the variable λ ) cos ϑ, which leads to automatic satisfaction of the periodic boundary conditions (7), we rewrite the forced convective diffusionreaction equations (2) as

x ( { ( ) (i)

∂c(i) Pe (i) ∂c(i) υλ + υ ∂τ 2 r ∂r r

1 - λ2

)

∂c(i) ) ∂λ

[

]}

(i) 1 ∂ D(i) 1 ∂ 2 ∂c(i) 2 ∂c r + (1 λ ) 2 ∂r ∂λ r2 ∂λ D(2) r ∂r D(i) i ) 1, 2 (A1) Da(i) (2)c(i) D

For spatial discretization of the governing equations, we use Galerkin spectral methods (Gottlieb and Orszag, 1977). The unknown local concentrations are expanded in series with respect to different types of orthogonal polynomials of angular and radial spatial variables. The Legendre polynomials are used in the angular direction: M

c(i)(τ,λ,r) )

m*0

∑ c(i)m (τ,r) Pm(λ), m)0

i ) 1, 2

(A2)

For expansion in the radial direction, we use Chebyshev polynomials, though in a different manner for the domains inside and outside the droplet. Our method of representation of the solution in the internal domain is somewhat different from that used, for instance, by Nguyen et al. (1993a), who implemented spectral methods for a similar problem. Unlike the above-mentioned authors, we do not map the internal domain 0 e r e 1 onto [-1, 1] but leave it as it is. This allows us to take advantage of certain symmetry features of the solution that seem not to have been capitalized upon earlier. It

j ) 0, 1, ...

k2j+1 ) 3,

j ) 1, 2, ...

(A7) (A8)

By using the representation (A6), we automatically avoid the singularity at the origin of the sphere, and having the same number of terms in the series doubles the highest order of the polynomials used, in contrast to customary treatments in which these quantities are equal. The semiinfinite external domain, being truncated at some appropriately large radius r∞, is mapped onto the domain [-1, 1]. The choice of the value of r∞ is to an extent an arbitrary one, and the rule of thumb we have used is a conventional one: it should be the smallest value that upon being doubled would not appreciably alter the final results. From various types of boundary conditions traditionally imposed on the boundaries of truncation (Fornberg, 1980; Patera, 1984; Boyd, 1989), we have selected the “hard” one,

r ) r∞:

c(2) ) 0

(A9)

which has the physicochemical interpretation of substituting an instantaneous chemical reaction at r∞ for an asymptotic approach to zero of the solute concentration as r f ∞. Replacing the physical mechanism of spreading of a finite amount of solute into an infinite domain by the chemical mechanism at the boundary of a finite domain can only be justified by the absence of influence of the distance at which the artificial reaction takes place on the interphase mass transfer results. An algebraic mapping was used in (Kleinman and Reed, 1994) to map the domain 1 e r e r∞ onto the interval z ∈[-1, 1], which is necessary for the following expansion over the Chebyshev polynomials constituting a complete set in this domain. Here we use an exponential mapping (Grosch and Orszag, 1977; Canuto et al., 1988; Boyd, 1989)

z)1-

2 ln r ln r∞

(A10)

which yields the same results as the algebraic one with


which we earlier carried out computations (Kleinman and Reed, 1994). The functions c(2) m are now expanded in a series of basis functions that are linear combinations of regular Chebyshev polynomials (Kleinman and Reed, 1994), Zn(z), n ) 1, 2, ...,

(1) (2) and φm,N(2) , m ) 0, 1, ..., M, from Expressing φm,N(1) (A20) and (A21) and reordering the two vectors (A18) and (A19) into a single one,

(1) (1) (2) (2) O ≡ (R0, φ0,1 , ..., φ0,N(1)-1 , φ0,1 , ..., φ0,N(2)-1 , ..., (1) (1) (2) (2) φM,1 , ..., φM,N(1)-1 , φM,1 , ..., φM,N(2)-1 )T (A22)

N(2)

c(2) m (τ,z) )

(2) φm,n (τ) Zn(z), ∑ n)1

m ) 0, 1, ..., M

(A11)

in which each of the Zn(z), n ) 1, 2, ..., satisfies

Zn(z)-1) ) 0,

n ) 1, 2, ..., N(2)

A (A12)

thus ensuring automatic satisfaction of (A9). For the internal and external domains, respectively, two inner products are defined:

(f, g)(1) ≡

∫-11dλ∫01fg

(f, g)(2) ≡

∫-11dλ∫-11fg

dr

x1 - r2 dz

x1 - z2

(A13)

(A14)

Two corresponding sets of test functions are introduced:

P0(λ) T0(r),

Pm(λ)rkm T2n1-2(r),

m ) 1, ..., M,

n1 ) 1, 2, ..., N(1) (A15) Pm(λ) Zn2(z),

(2)

m ) 0, 1, ..., M, n2 ) 1, 2, ..., N (A16)

We thereby arrive at the two vector differential equations

A(i) )

we arrive at the system of 1+ (M + 1)(N(1)+ N(2) - 2) linear ordinary differential equations:

dO(i) (i,c) (i,d) (i,r) ) (-K(i) + K(i) - K(i) )‚O(i) + c B dB r B dτ (i) i ) 1, 2 (A17) K(i) r b ,

after implementation of the conventional Petrov-Galerkin procedure (e.g., Canuto et al., 1988). Here (1) (1) (1) (1) , ..., φ0,N(1) , ..., φM,1 , ..., φM,N(1) )T (A18) O(1)(τ) ) (R0, φ0,1

dO ) (B(c) + B(d) + B(r))‚O + b dτ

(A23)

The reordering of (A22) leads to a blocked structure for A(i), B(i,d), B(i,r), and B(i,c). The first three matrices are block diagonal with M + 1 nonzero square matrices on their main diagonal, and the first 1 + (N(1) + N(2) 2) elements in the first row and the first column are also nonzero. The matrices B(i,c) that result from discretization of the convection terms have a similar blocked structure, but unlike the other matrices in (A23), they are block-banded, rather than block diagonal. The number of nonzero block diagonals depends on the highest degree of λ involved in the expressions for the velocity fields v(i), i ) 1, 2, which we assume to be given. The Hadamard-Rybczinsky velocity field (Churchill, 1988) leads to the block-tridiagonal structure of B(i,c). System (A23) is discretized in time using the fully implicit first-order backward Euler method

(A - ∆τB)‚∆On+1 ) ∆τB‚On + ∆τ‚b

(A24)

where On is the vector φ at the nth time increment,

∆On+1 ) On+1 - φn

(A25)

The matrix on the left side of (A24) has the same structure as B(i,c). At every time step this system is solved by Gauss elimination preceded by LU decomposition (which was actually performed only two times per solution process as only two different values of the time step were used for each run). Implementation of spectral methods is known to result in ill-conditioned linear systems (Canuto et al., 1988; Boyd, 1989), and, consequently, the solution obtained through Gauss elimination was improved by iterative refinement (Golub and Van Loan, 1989). Literature Cited

and (2) (2) (2) (2) , ..., φ0,N(2) , ..., φM,1 , ..., φM,N(2) )T O(2)(τ) ) (φ0,1

(A19)

are vectors of the unknown time-dependent functions. A(i) B(i,c), B(i,d), and B(i,r) are constant matrices, and b(i) are constant vectors. Boundary conditions at the interface (4) are implemented through the Lanczos τ-method (Gottlieb and Orszag, 1977; Fornberg, 1980; Boyd, 1989), which leads to two sets of M + 1 linear algebraic equations

Q(1)‚O(1) ) Q(2)‚O(2)

(A20)

HD‚S(1)‚O(1) ) S(2)‚O(2)

(A21)

where Q(i) and S(i) are {(M + 1), (M + 1)(1 + N(i))} matrices, i ) 1, 2.

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Newman, A. B. The Drying of Porous Solids: Diffusion and Surface Emission Equations. Trans. Am. Inst. 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. Nguyen, H. D.; Paik, S.; Chung, J. N. Unsteady mixed convection heat transfer from a solid sphere: the conjugate problem. Int. J. Heat Mass Transfer 1993, 36, 4443-4453. Oliver, D. L. R.; Chung, J. N. Conjugate Unsteady Heat Transfer of a Translating Droplet at Low Reynolds Numbers. Int. 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. Int. J. Heat Mass Transfer 1990, 33, 401-408. Patera, A. T. A spectral element method for fluid dynamics: laminar flow in a channel expansion. J. Comput. Phys. 1984, 54, 468-488. 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. Spalart, P. R. A spectral method for external viscous flows. Contemp. Math. 1984, 28, 315-335. Steiner, L. Mass-transfer rates from single drops and drop swarms. Chem. Eng. Sci. 1986, 41, 1979-1986.

Received for review November 6, 1995 Revised manuscript received June 27, 1996 Accepted June 28, 1996X IE950671J

X Abstract published in Advance ACS Abstracts, August 15, 1996.

Unsteady Conjugate Mass Transfer between a Single Droplet and an

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