Perhaps a more instructive comparison would be t o compare sine and square potentials of equal displacement. In relation to the present study, this would mean t h a t the sine velocity is everywhere (except in the length expression, eq 9) multiplied by x/2. If this is done, i t can be easily shown that the square and sine characteristic lines begin and terminate a t the same points in the region 0 < ( < 1. However, a t the points of intersection along { = 1 (which determines the rate of enrichment) the sine and square velocities produce characteristic lines which intersect in general, at different points. If eq 9 still holds, the ratio of the sine wave interaction (27r + a d ) t o the square wave intersection (27r ac) along ( = 1,and 27r < % < 37r, can be shown t o be
+
and for the specific value of b = 0.185 used earlier, this relation shows that nc'/nc = 0.6225/0.687 = 0.906
(24)
Hence for this case, the sinusoidal velocity of displacement equal to the square velocity will always produce the slower enrichment. In general, discussions relating to the best selection of velocity wave form will obviously depend on the value of the parameter b, even under conditions of equal displacement; for the specific case cited in eq 24, the rate of enrichment will be slightly, but distinctly, different. Acknowledgments T h e author gratefully acknowledges the constructive comments of Professor Philip C. Wankat, who reviewed the manuscript.
L = totalbedlength M ( 7') = temperature-dependent equilibrium constant m = p,(l - c)M(T)/pp rno = steadyvalueof rn n = number ofcycles uo = Vo/(l + mol u = interstitial velocity Vo = velocity amplitude x = solids concentration y = fluid concentration z = axial position in bed Greek Letters a , = separation factor, ( Y T ) n / ( y B ) n (Y = fraction defined by eq 15 @ = fraction defined by eq 12 c = bed void fraction p = phasedensity w = cycle frequency % = dimensionless time, wt ( = dimensionless distance, z/L Subscripts B = bottom (lean) reservoir n = cyclenumber f = fluid phase s = solid phase T = top (rich) reservoir L i t e r a t u r e Cited Aris, R., Ind. Eng. Chem., fundam., 8 , 603 (1969). Chen, H.T.. Manganaro, J. A,. AIChEJ., 20, 1020 (1974). Hildebrand. F. B., "Advanced Calculus for Applications", pp 379-388, Prentice-Hall, Englewood Cliffs, N.J.,1965. Pigford, R. L., Baker, B., Blum, E. E., Ind. Eng. Chem., fundam., 8, 144 (1969). Rice, R. G.,Ind. Eng. Chem., Fundam., 12,406 (1973).
Department of Chemical Engineering University of Queensland Brisbane, Australia 4067
Nomenclature a = change in equilibrium constant b = a / ( l ma)
Richard G . Rice
Received for reuiew December 2,1974 Accepted June 5,1975
+
Mass Transfer from Spherical Drops at High Reynolds Numbers
The mass transfer resistance external to a spherical drop at high Reynolds and Peclet numbers is obtained using the thin concentration boundary layer assumption and the interfacial velocity of Harper and Moore (1968). Earlier treatments of the same problem are in error because an incorrect expression was used for the interfacial velocity. The available experimental data agree well with the present analysis.
-
The resistance to mass transfer external to a spherical drop for Pe 00 is given by
-
where ui is the velocity in the 8 direction a t the interface. the flow field external to the drop is given by For Re potential flow with ui 3 . - = - sin 8
u 2 substitution yields the well-known Boussinesq equation
(3)
A number of authors (Lochiel and Calderbank, 1964; Winnikow, 1967; Cheh and Tobias, 1968) attempted to extend the analysis to finite Reynolds numbers using the velocity boundary layer approach of Moore (1963). In all cases the velocities they used were in error. The error, which consisted of neglecting the circulation of vorticity inside the drop, is described by Harper and Moore (1968) and Harper (1972). The correct interfacial velocity is given by Ind. Eng. Chem., Fundam., Vol. 14, No. 4. 1975
365
1.2
1.0
I
1
EQ.3
I
0.5
/
previous
C U"
~
25
50
75
100
125
150
Re
0.6
'
1000
100
10
Figure 2. Comparison of analysis with the mass transfer data of Griffith (1960) for ethyl acetate drops in water, and with the results of previous analysis (Lochiel and Calderbank, 1964; Winnikow, 1967).
I
I
Re
Figure
, Predicted
effect of K and y on mass transfer.
g(z) exp(-4z2/X) dz
Jrn
(4)
where g ( z ) is a function given by Harper and Moore (1968). Since the earlier authors used only the first two terms on the right-hand side of eq 4, their analyses gave the correct result only for gas bubbles ( K = 0) (5) All of the previous results are incorrect for nonzero values O f K. In the present work eq 4 was substituted into eq 1 and the integration was performed. The integration of the first two terms resulting from eq 4 can be completed analytically (Winnikow, 1967) while the integration of the last term was completed numerically using g(z) from Harper and Moore (1968) and Simpson's rule. The results can be put into the following form
where R is a function of K and y. Values of B were computed for y d 4 and K d 2. At higher values of K the boundary layer approximations on which eq 1 is based fail to apply (Lochiel and Calderbank, 1964). Figure 1 presents values of Sh computed from eq 6 for several values of K and y.The computations show that y has very little effect on Sh/Pel/' for y > 0.25. For values of y greater than this, Sh/Pel/' is essentially a function of K only. This finding permits the correlation of the numerical results in the following approximate form for K d 2 , O < y d 4 Sh = -
1
(2.89
+
(7)
with an error of less than 5% in Sh/Pel/'. Figure 2 presents a comparison between the experimental data of Griffith (1960) for ethyl acetate drops in water
366 Ind. Eng. Chem., Fundam., Vol. 14, No. 4, 1975
( K = 0.41, y = 0.90) and the values computed from the exact equation, eq 6, the approximate equation, eq 7, and the relationship derived by the earlier authors (Lochiel and Calderbank, 1964; Winnikow, 1967). The earlier relationship is in error, yielding values about 10-15% too high. The data are in good agreement with the results of this work, eq 6 or 7, especially for Re > 70 where the boundary layer approximations are expected to be valid.
Nomenclature B = functionof~andy d = dropdiameter D = diffusivity of solute in external phase g(z) = function given by Harper and Moore (1968) k = mass transfer coefficient Re = Reynolds number based on the diameter and the properties of the external phase (= d U p / / . ~ ) Pe = Peclet number based on the diameter and the properties of the external phase (= d U / D ) S h = Sherwood number based on the diameter and the properties of the external phase ( = k d / D ) U = free stream velocity or terminal velocity ui = $-direction velocity in the interface X = 2 - 3 COS B cos3 B z = integration variable K = viscosity ratio, p d / / . ~ y = density ratio, p d / p p = density 0 = angle measured from front stagnation point p = viscosity
+
Subscript d = droplet (internal) phase L i t e r a t u r e Cited Cheh, H. Y., Tobias, C. W., Ind. Eng. Chem. Fundam., 7 , 48 (1968). Griffith, R. M., Chem. Eng. Scb, 12, 198 (1960). Harper, J. F., Adv. Appl. Mech., 12, 59 (1972). Harper, J. F., Moore, D. W., J. Fluid Mech., 32, 367 (1968). Lochiel, A. C., Calderbank. P. H., Chem. Eng. Sci., 19, 471 (1964). Moore, D. W., J. Nuid Mech., 16, 161 (1963). Winnikow. S.. Chem. Eng. SCL, 22, 477 (1967).
Department of Chemical Engineering McCill Uniuersity Montreal, Quebec, Canada
Martin E. Weber
Received for review March 24, 1975 Accepted June 30,1975
