INT. COMM. HEAT MASS TRANSFER Vol. 16, la0. 843-849, 1989 ©]Pergamon Press pie

0735-1933/89 $3.00 + .00 Printed in the United States

Mass Transfer Rates from a Sphere in Potential Flow: Moderate Peclet Numbers

Douglas Oliver and Kenneth DeWitt College of Engineering

The University of Toledo Toledo, Ohio 43606

(Communicated by J.P. Hartnett and W.J. Minkowycz)

ABSTRACT Rates of mass transfer have been numerically investigated for the case of a sphere in a potential flow. It is demonstrated that a single equation can predict the steady-state rate of mass transfer from a sphere in a potential flow for the total range of Peclet numbers.

I n t r o d u c t i o n There have been many investigations of the heat and/or mass transfer

from a sphere in a viscous fluid, (for example see Clift et al. [1]). Only a

few investigations, however, have considered transfer from a sphere in a

potential flow. The potential flow solution for the flow field about a sphere

is only valid for high Reynolds number flows, and then only in the up-

stream region since the flow separates aft of the sphere. With the limited

applicability of investigations into the heat or mass transfer from a sphere

in a potential flow field, few investigators have endeavored to obtain

solutions for such a case.

Recent interest in bubble and droplet behavior in a micro-gravity

environment has resulted in a new class of fluid flow problem which has

the same flow field as the idealized potential flow about a sphere. Young

i t al. [2] demonstrated that a droplet in a thermally stratified fluid will

slowly migrate due to surface tension effects. If no body forces (such as

gravity) are present, the flow field about the droplet will be the same as

843

844 D. Oliver and K. DeWitt Vol. 16, No. 6

the potential flow solution for flow about a sphere. Thompson et al. [3]

performed drop-tower experiments to confirm the predictions of Young et

al. in a micro-gravity environment. More recently, Balasubramaniam and

Chai [4] demonstrated that the potential flow solution was valid for a

droplet regardless of the Reynolds number, so long as the Marangoni

number or Peclet number (the product of the Reynolds number and the

Prandtl number) remained low.

With this recent interest in potential flow about a sphere, it is worth

further investigating the mass transfer rates that occur in this situation.

The literature contains estimates for the mass transfer rates for the two

limiting cases of low and high Peclet number flows. For low Peclet

numbers, Sano [5] obtained the following equation (the following relations

differ from the original texts due to differing definitions of the

characteristic lengths used):

1 ~ 0 7 a 48407 ~ 4 S h = 2 + P e - PeZ+3--~-~Pe 7 5 ~ o r e + .... P e < l (1)

For large Peclet numbers, Hirose [6] obtained:

Sh = 1.128(Pe) 1/2 + 0 . 8 2 7 + O(Pe) -1/2, Pe >> 1 (2)

In the above equations, Sh is the Sherwood number, kx D/DAB.

No closed form solution exists in the literature which will accurately

predict the mass transfer rates for moderate Peclet numbers. Numerical

solutions could be used to establish the upper and lower limits of

equations (1) and (2), respectively, and to predict mass transfer rates in

the moderate Peclet number range. However, numerical solutions which

are presented in the literature are often difficult and time consuming to

r ep roduce .

Brunn and Isemin [7] proposed a single equation which could be used

to accurately predict heat transfer rates from fluid spheres for low

Reynolds number flows. Their equation may be adapted to predict mass

transfer rates for the special (yet important) limiting case of steady mass

transfer from a sphere for which: (1) the mass flux at the droplet surface is

sufficiently small such that the droplet diameter is constant with time, and

Vol. 16, No. 6 MASS TRANSFER RATES FROM A SPHERE 845

(2) the concentration level in the droplet is constant with time. With these

modifications the equation of Brunn and Isemin may be adapted to predict

the Sherwood number for mass transfer from a fluid sphere in creeping

flow as follows:

Sh = 1 + [1 + (1.127,~/(1 - Fo)Pe) for 2 / 3 _ F o < 1 (3)

where Fo is given by: Fo = (2 + 3X)/(3 + 3X), X = IXs/l~** (4)

and cx is given by: tx = 2.058(1 - Fo )0.395 (5)

(The original equation in Brunn and Isemin is modified such that the

sphere diameter is used for the characteristic length for the dimensionless

parameters Sh and Pe). Equation (3) was verified by Brunn and Isemin for

the special cases of a gas bubble (X = 0, Fo = 2/3), for the limiting case as X approaches ** (a spherical particle with Fo = 1), and for the special case of a

liquid droplet with Fo = 0.781.

The intent of the present work is to investigate the validity of

extrapolating Equation (3) for use in predicting mass transfer from a

sphere in a potential flow field. If Equation (3) were applicable to

potential flow, the corresponding value for 'Fo' would be Fo = 0. Thus an

extrapolation of Equation (3) for a sphere in a potential flow field is:

0.4859

Sh = 1 +[1 + 1.27 9(Pc) 1"029] (6)

Equation (6) agrees well with Equation (1) for low Peclet numbers and with Equation (2) for high Peclet numbers. Thus, the predictive equation of Brunn and Isemin may be extrapolated to potential flow about a sphere for the limiting cases of low and high Peclet numbers. However it remains to be demonstrated that Equation (6) is also valid for moderate Peclet numbers.

Numerical Confirmation of Eouat ion (6~ for Moderate Peclet Numbers

A numerical procedure is required to confirm the validity of

Equation (6) for moderate Peclet numbers. This section includes such a

numerical procedure which was used to independently predict the

846 D. Oliver and K. DeWitt Vol. 16, No. 6

Sherwood number as a function of the Peclet number for potential flow about a sphere.

The dimensionless species equation in spherical coordinates is given by:

Pet" Oc v ()c'~ ()2C 2 3C 2 ~.U ~ " + "T"3"-O) = "3"Q" + "F"~ " + -

co t0 3c 1 ~2c r 2 O0 + r 2 302 (7)

The boundary conditions imposed on Equation (5) are:

c(r=l,O) = 1, (8)

l im c(r,0) = 0 r---~ (9)

Oc] ~-ff = 0 , (due to s y m m e t r y )

0 = 0,~ (10)

Equation (7) is solved for moderate Peclet numbers using a series truncation method similar to that used by Dennis, Walker, and Hudson [8], who investigated heat transfer rates about particles at moderate Reynolds and Peclet numbers. The dimensionless concentration is approximated by a series of Legendre polynomials and corresponding unknown radial funct ion:

Letc(r l ,0 ) = 2 f n ( r l ) P n _ l ( z ) , w h e r e z = c o s 0 , a n d r l = l / r n =1

This series is then truncated and substituted into Equation (7), resulting in the following truncated version of the Equation (7):

(n - 2)(n - 1) n(n + 1 ) ) -rl4f"n + n(n - 1)l]2fn + rlG(rl) fn-1 2n --3" - fn+l 2n + 1 '

_TlaF(T1)(f'n n - 1 n )__ 0 -12~--3 +fn+12n +1 ( I I )

w h e r e G(TI) = - - ~ ( 1 +0.5T13), and F(TI)=-~-(1 -)13 )

Vol. 16, No. 6 MASS TRANSFER RATES FROM A SPHERE 847

The boundary conditions imposed on Equation (11) are as follows:

at the surface of the sphere, (rl= 1): fn (1) = ~nl at the free stream conditions, (rl= 0): fn (0) = 0.

The Sherwood number is obtained through the relation:

Sh-- sinO O ",1=1 ° (12)

which is equivalent to:

S h = 2 f'l(rl= 1) (13)

Equation (11) is solved using finite-differences techniques. The

radial domain is divided into 'm' equally spaced nodes (for this work

m =40). This results in a system of (m-1)x no equations (where no is the

truncation limit). This system has a block-tridiagonal matrix associated

with it; thus the solution time is rapid for moderate values of no. The

above solution procedure results in rapid convergence for low Peclet

numbers. The rate of convergence of the series solution is enhanced if the

results are extrapolated using the extrapolation procedure suggested by Shanks [9]. That is, if Shno, Shno+l, and Shno+2 represent the Sherwood

numbers predicted by truncation limits of no, no+l, and no+2, respectively, then the extrapolated value for the Sherwood number is given by;

= / -Sh2 )/(Shn +Shno+2-2Sh ) Shext ShnoShn 0 +2 o +1 o no+l (14)

For this work, a value of no = 37 was used for all values of the Peclet

n u m b e r .

R e s u l t s a n d D i s c u s s i o n

The numerical results for the Sherwood number obtained from

Equation (14) are tabulated on Table 1. Comparison is made with the predictive equations of Sano [Equation (1)], Hirose [Equation (2)], and the

extrapolation of Brunn and Isemin [Equation (3), with Fo = 0].

848 D. Oliver and K. DeWitt Vol. 16, No. 6

TABLE 1. A Comparison of Predicted Sherwood Numbers .

P e c l e t n u m b e r

0.1 0.2 0.5 1.0 2.0 5.0

10.0 20.0 50.0

100.0

S a n o Eq. (1)

2.05 2.10 2.23 2.43

H i r o s e Eq. (2)

5.87 8.80

12.11

Brunn & Isemin Eq. (3), Fo -- 0

2.06 2.11 2.27 2.49 2.87 3.70 4.69 6.13 9.02

12.31

N u m e r i c a l P r e d i c t i o n

Eq. (14)

2.04 2.10 2.23 2.44 2.79 3.59 4.58 6.02 8.94

12.18

As may be seen in Table 1, the predictive equation for the Sherwood

number proposed by Brunn and Isemin [Equation (3)] may be extrapolated

to predict mass transfer rates from a sphere in a potential flow (with

Fo = 0) for the total range of Peclet numbers. Thus, a single equation may

be used to predict the rate of mass transfer from a sphere in a viscous

fluid and for potential flow for the special case of steady-state mass

transfer where the rate of mass transfer is low enough such that the flow

field is not significantly perturbed.

N o m e n c l a t u r e a sphere radius

D sphere diameter

DAB diffusion coefficient of mass

kx mass transfer coefficient

Pe Peclet number

r dimensionless radial coordinate

Sh Sherwood number, kxD/DAB U~ sphere velocity

~nl Kronecker delta function

~t dynamic viscosity

0 tangential coordinate

Vol. 16, No. 6 MASS TRANSFER RATES FROM A SPHERE 849

Subscripts

s sphere oo free stream

A c k n o w l e d g e m e n t The authors wish to acknowledge support of this work by the

National Aeronautics and Space Administration (NASA grant NAG 3-910)

R e f e r e n c e s

I. R. Clift, J. R. Grace and M. E. Weber, Bubbles. Drops. and Particles, Academic Press, New York (1978).

2. N.O. Young, J. S. Goldstein and M. J. Block, L of Fluid Mech. 6, pp. 350- 356 (1959).

3. R.L. Thompson, K. J. DeWitt and T. L. Labus, Chem. Eng. Commun, 5, pp. 299-314 (1980).

4. R. Balasubramaniam and A. Chai, J. of Colloid and Interracial Sci. 119, No. 2, pp. 531-538 (1987)

5. T. Sano, J. of Engineering Mathematics, _6, No. 2, pp. 217-223 (1972).

6. T. Hirose, Int. Chem. Eng. 18, No. 3, pp. 514-520 (1978)

7. P .O. Brunn and D. Isemin, Int. J. Heat Mass Transfer 27, pp. 2339- 2345 (1984).

8. S .C .R. Dennis, J. D. A. Walker and J. D. Hudson, J. Fluid Mech. 60, pp. 273-283 (1973)

9. D. Shanks, J. of Mathematics and Physics 34, pp. 1-42 (1955).