908
Znd. Eng. Chem. Res. 1992,31,908-920
Nakayama, A.; Koyama, H. Integral treatment of buoyancy-induced flows in a porous medium adjacent to horizontal surfaces with variable wall temperature. Znt. J. Heat Fluid Flow 198713, 21, 297-300. Nakayama, A.; Koyama, H.; Kuwahara, F. Two-Phase Boundary Layer Treatment for Subcooled Free-Convection Film Boiling Around a Body of Arbitrary Shape in a Porous Medium. J. Heat Transfer 1987,109,997-1002. OSullivan, D. Novel Separation Technology May Supplant Distillation Towers (Imperial Chemical Industries). Chem. Eng. News 1983,61 (Mar 7), 26-28. Redman, J. Pervaporation heading for new horizons. Chem. Eng. 1990 (Feb), 4649. Reid, R. C.; Prauenitz, J. M.; Sherwood, T. K. The Properties of Case8 and Liquids, 3rd ed.; McGraw-Hill: New York, 1977. Seader, J. D. Continuous Distillation Apparatus. US. Patent 4 234 291,1980. Seok,D. R.; Hwang, S. Zero-Gravity Distillation Utilizing the Heat
Pipe Principle (Micro-Distillation). AZChE J. 1985, 31, 2059-2065. Smith, J. M. Modelo de poroa en desorden, Ingenieria de la Cinetica Quimica, 1st ed. in Spanish; Compaiiia Editorial Continental: Mexico, D.F., 1986 Chapter 11. Treybal, R. E. Mass-Transfer Data for Simple Situations. MassTransfer Operations, 3rd ed.; McGraw-Hik Tokyo, 1981; Chapter 3. Waller, K. V.; Finnerman, D. H.; Sandelin, P. M.; Hiiggblom, K. E.; Gustaffsson, S. E. An Experimental Comparison of Four Control Structures for Two-Point Control of Distillation. Znd. Eng. Chem. Res. 1988,27, 624-630. Whitaker, S. Flow in Porous Media 1. A Theoretical Derivation of Darcy's Law. Transp. Porous Media 1986, I , 3-25. Received for review March 26, 1991 Revised manuscript received November 26, 1991 Accepted December 4, 1991
Simulation of Mass Transfer in a Batch Agitated Liquid-Liquid Dispersion A. H.P.Skelland* and Jeffrey S . Kanelt School of Chemical Engineering, Georgia Institute of Technology, Atlanta, Georgia 30332
A simulation model is developed to compute the transient Sauter-mean drop diameter and the fractional mass transfer in batch agitated liquid-liquid dispersions when the continuous phase, dispersed phase, or both phases offer significant resistance to mass transfer. This model accounts for mass transfer during drop breakup, drop-drop coalescence or rebounding, drop oscillations, and free motion of the drops throughout the vessel. Using 16 different formulations, 8 of which were developed here, the transient fractional mass transfer was predicted with an average deviation of f20% for all four directions of diffusion and dispersion, using no adjustable parameters. However, to enhance the simulation, one constant in the breakage frequency was then varied to improve predictions to f15% for 10 simulations, including 4 with surfactants present. Furthermore, the transient Sauter-mean drop diameter was predicted with an average deviation of &8% for these simulations. The fraction of the total mass transferred due to drop-drop interactions was found to be indeterminate, but small, in this work. However, drop breakup and subsequent damped oscillations accounted for about 5 % of the total transfer in the simulation for dispersed-phasecontrolled systems and insignificantly for continuous-phase-controlled systems.
Introduction To design an agitated liquid-liquid contactor, appropriate models for the system must be constructed that accurately describe the phenomena occurring within it. Physical factors that influence the extraction rate are droplet phenomena such as breakage, coalescence, rebounding, oscillations, and free motion throughout the vessel; microscopic interfacial transport processes; and macroscopic hydrodynamics in the vessel. Earlier investigators developed models to predict the drop size distribution for agitated liquid-liquid dispersions in the absence of mass transfer using either drop population balances [Valentas et d. (1966), Valentas and Amundson (1966), Ramkrishna (19741, and Bajpai et al. (1976)] or simulation techniques [Zeitlin and Tavlarides (1972), Coulaloglou and Tavlarides (1977),Molag et al. (1980),and Hsia and Tavlarides (1983)l. When solute transfer was added to drop population balance models, a trivariant drop distribution was required to account for a drop's size, ita Present address: Eastman Chemical Company, Kingsport, TN.
age (time that the droplet has existed in the vessel), and its solute concentration. Both drop population balances and simulation techniques have been used to predict solute-transfer rates in these dispersions; the former are represented by Bayens and Lawrence (1969) and Shah and Ramkrishna (1973) and the latter by Curl (1963), Bapat et al. (19831, Bapat and Tavlarides (1985), and Jeon and Lee (1986). Models for mass transfer in agitated liquid-liquid dispersions are summarized in Table I. These previous models only computed solute transfer during the free movement of droplets through the vessel and assumed that coalescence and breakup occur instantaneously, tacitly neglecting transfer during these processes. However, during drop splitting, Bozorgzadeh (1980) showed a 100-200 "a enhancement in the mass-transfer coefficient (accounting for increased surface area) over oscillating drop predictions, and Bately and Thornton (1989) concluded that for drop-drop coalescence "it might be expected that the enhancement over and above stagnant diffusion rates would be appreciable, and the effect of inter-droplet coalescence should not be dismissed without experimental confirmation". Thus, the present model accounted for
0888-5885/92/2631-0908$03.00/00 1992 American Chemical Society
Ind. Eng. Chem. Res., Vol. 31, No. 3, 1992 909 Table I. Previous Drop Population Balance (DPB) Models and Simulations for Mass Transfer in Agitated Liquid-Liquid Dispersions DPB or batch or binary equal-size compared controlling workers simuln conta breakupb breakupb coalescb to expt coeff simuln C Yes yes Yes no kd Curl (1963) B yes Bayens and Laurence (1969) DPB no Yes no kd C yes yes no no kd Shah and Ramkrishna (1973) DPB simuln C Yes Bapat et al. (1983) no yes Yes kc C Yes Bapat and Tavlarides (1985) simuln no yes Yes kc C Yes no Yes no kd Jeon and Lee (1986) simuln a B, batch; C, continuous. *Transfer during drop breakup and coalescence was always assumed to be zero, despite observations to the contrary in some cases by Davies (1972, p 241).
drop volume
V
2v
2v
i n l t l a l drop size distrlbution:
0
0
0
2nv
."
0
9 start
read l n l t l a l drop slze dlstrlbutlon, physlcal
f i r s t t i m e step: number of tlme s t e m
a1 I drops are experiencing f r e e motion throughout the vessel
L
compute the mass transfer durlng free drop motlon
i
second t i m e step: drop Of VOlUme 2V breaks, and a l l other drops experlence free motion throughout the vessel
calculate the number
calculate the mass
I
o t o
.
drop breakup
0
4 coalescence
t h l r d t l m e step: t W O drops Of Volume V coalesce and all other drops experience f r e e motlon throughout the vessel
calculate the mass transferred during coalescence
'x' write output
fourth t i m e step: t w o drops of volume V rebound and a l l other drops experience free motlon throughout the vessel damped
oscillatton t l m e
Figure 1. Diagram for the method of simulating mass transfer in batch agitated liquid-liquid dispersions.
mass transfer during breakage, coalescence, and rebounding of drops. The possibilities of drop oscillations and circulations have also been excluded in the previous models. However, Scott and Byers (1989),modeling mass transfer in oscillating-circulating liquid drops, concluded "The results clearly indicate that a significant enhancement in mass transfer can be anticipated as a result of forced oscillation of droplets in a continuum." Therefore, oscillation of large droplets was incorporated in the present model. Furthermore, the currently available models assumed that the dominant resistance to transfer resides in either one phase or the other. This restriction was relaxed in the current simulation so that either the continuous phase, the dispersed phase, or both phases may contribute significant resistance to interfacial transfer. In the previous simulationsthe breakage and coalescence frequencies were developed in the absence of mass transfer. However, a
Figure 2. General flowsheet for simulating mass transfer in batch agitated liquid-liquid dispersions.
breakage frequency developed by Kanel (1990) that incorporated the effect of mass transfer was used in this simulation. Finally, the currently available simulations were limited to continuous flow systems in contrast to the simulation developed here. This paper is in some ways the analogue for the agitated vessel of the paper by Skelland and Conger (1973) on perforated-plate extraction columns. The latter collected theoretical and empirical relationships for all hydrodynamic and mass-transfer aspects of drops in liquid-liquid systems in a quantitative description of the mass transfer attained in such columns. Similarly, in this paper, all relevant hydrodynamic phenomena, including drop size and size distribution, breakage times, circulation times, and coalescence and rebounding periods, are formulated and combined with corresponding mass-transport quantities during drop breakup, coalescence, and rebounding, damped oscillations after each of these occurrences, and free movement of each drop between such events. The resulting prediction of the transient mass transfer and Sauter-mean drop diameter in agitated liquid-liquid systems is found to agree well with experimental data.
910 Ind. Eng. Chem. Res., Vol. 31, No. 3, 1992 Table 11. Expressions Used in the Simulation Breakage interfacial area ‘new” and ‘old”
AB = rd,2
breakage time
Kanel (1990) = 0.447
mass-transfer coeff for ‘newn surface
( )’”
Kanel (1990)
T:B
Kanel (1990) mass-transfer coeff for “old” surface breakage freq
extend expressions for free motion through vessel dd,) = kd,2
Skelland et al. (1990)
(T from Table VII) Hong and Lee (1985) interfacial area
An: = rd,2
circulation time
tc = 0.85(
Free Circulation
$!:
Holmes et al. (1964)
mass-transfer coeff
Grober (1925)
Glen (1965) interfacial area coalescence time
mass-transfer coeff for plane surface
spherical Ac = rd,2 planar Ac = rRz tc = 0.446~*
k*ccmm
Coalescence Chen et al. (1984)
-+
=
Kanel (1990)
2 327 - I ] + ..*) In (2)
for spherical surface
extend the expression for free motion
coalesce freq
f(dpIdp2) =
C3df3
(dpl + d,2)2(dp12/3 + dPZ2/3)1/2x 1+4
c3= 1.9 x 10-3 c,
= 2.0 x 108
Simulation Development The input requirements for the simulation include physical and transport properties (u, p,, Pd, CL,,& D,,&) and the solute distribution coefficient between the solvents;
Coulaloglou and Tavlarides (1977)
Bapat and Tavlarides (1985)
equipment geometry, including impeller type, speed, and diameter, vessel diameter, filled height, and baffles; and an initial drop size distribution that can be obtained by a photograph of the dispersion shortly (3-6 s) after injection of the dispersed phase into the agitated continuous phase.
Ind. Eng. Chem. Res., Vol. 31, No. 3,1992 911 Table 111. Additional Expressions Used in the Simulation purpose expression Rebounding interfacial area spherical AR = rdp2 planar AR = r R 2 rebounding time t R = 2tc
reference
Kanel (1990)
for spherical surface
extend the expression for free circulation Coulaloglou and Tavlarides (1977)
rebounding frequency
cs= 1.9 x 10-3 c, = 2.0 x 108
Bapat and Tavlarides (1985) Undamped Oscillations
interfacial area circulation time
Holmes et al. (1964)
mass-transfer coeff
Clift et al. (1978, pp 196-197) Schroeder and Kintner (1965)
Clift et al. (1978, pp 196-197)
pcomuo.u oscillation criterion
interfacial area oscillation time
Klee and Treybal (1956)
d, > 0.33,,,0.14~~0.43[ =]cm Am = rd; tos = eo.oo8~o.B4~l.ol
Damped Oscillations Kanel(1990)
tos = t, if ( t d d C > t , mass-transfer coeff
Kanel (1990)
Kanel (1990)
Isotropic turbulence throughout the vessel was assumed since for NbI > 100oO the flow field in an agitated liquid-liquid dispersion may be considered fully isotropic [Coulaloglou and Tavlarides (1976)]. The minimum NReI used in our experiments were 39 400 and 33 600 for the large and small impellers, respectively. Therefore, the simulation could appropriately be compared to our experimental data. The time step used in the simulation was taken as the circulation time of Holmes et al. (1964), which was defined as the average time required for a fluid element to move from the impeller region, around the vessel, and back to the impeller. The expression
jq
t, = (0.85 f 0.05)pdI )11
was developed for six-blade disk turbines in baffled vessels with Reynolds numbers above lo4. The circulation time was chosen as the time step in the simulation since Park and Blair (1975) observed that "Beyond distances from the impeller region of order of the impeller diameter, breakup was virtually nonexistent." Furthermore, for coalescence phenomena to occur, they noted that "the coalescence rate is directly proportional to the turbulence level; that is, the highest coalescence rates occur closest
912 Ind. Eng. Chem. Res., Vol. 31, No. 3, 1992 Table IV. Physical and Transport Properties of Water and Chlorobenzene at 25 O C density, viscosity, diffusivity of N s/mz TBAB, mz/s liquid kg/m3 0.00073 2.62 X 10-lo chlorobenzene 1083.0 6.24 x 10-lo 0.00087 water 997.1
8
0
9 00
05
10
15
25
20
30
35
40
b t o n X-100 concentration (g/l HZO)
Figure 3. Interfacial tension between water and chlorobenzene versus Triton X-100 concentration at 25 O C .
to the impeller.” However, Sprow (1967) postulated that coalescence processes occur predominantly in the region away from the impeller. Nonetheless, it seemed appropriate to teat each drop at the end of every circulation time to determine whether it would undergo breakage, coalescence, or rebounding. The functions used to test whether or not these events would occur are in Tables I1 and 111. Mass-Transfer Calculations. Mass transfer during the processes of free motion through the vessel, drop breakup, coalescence, rebounding, damped oscillation, and undamped oscillations was calculated as outlined in Figure 1. The interfacialareas, exposure times, and mass-transfer coefficients used in the simulation are presented in Tables I1 and 111. It was assumed that the continuous phase was completely mixed at the end of each time step, but the concentration profiles in the dispersed phase were allowed to develop over the drop’s age, which could be several time steps in length. Only when a droplet underwent an event was its contents assumed to be completely mixed. When two drops coalesced, the concentration of solute in the resulting drop was taken as the average value. When breakage occurred, each daughter initially contained identical solute concentrations. Sauter-Mean Drop Diameter. The Sauter-mean drop diameter was computed at each time step as N’
N’
d32 = C n l d p r 3 / C n r d p l 2 t=l
r=l
(2)
where N’ is the total number of drops in the simulation and n, is the number of drops of diameter dpl. Program Details. A general flowsheet of the simulator appears in Figure 2, and the code, in FORTRAN 77, is deposited with the American Document Institute, ADI. A nomenclature section was included in the program which defined the variables and noted their dimensions when appropriate. The user was prompted for the number of time steps and the initial drop size distribution to be used. Geometrical and physical properties were set, followed by the computation of various dimensionless groups. Breakage, coalescence, and rebounding frequencies for each possible drop size were then computed along with k,. The large DO loop advanced through the time steps while a nested loop stepped through the drop sizes. Mass transfer during the free motion of droplets in the vessel, including undamped oscillations, was then computed for all drops. The resulting solute concentration in each drop was also calculated. The number of drops that undergo breakage, coalescence, and rebounding was then computed. If more drops were predicted to experience an event than existed in that size, preference was given in the order of breakage,
coalescence, and rebounding. Drops undergoing an event were randomly chosen from all the drops of that size by a random number generator. Here, the congruential method was used. Solute transfer during any event was then calculated only for drops predicted to undergo that event while the rest of the population was returned for the next time loop. For mass transfer during drop breakup with subsequent damped oscillations, the resulting daughters were assumed to be completely mixed with equal concentrations of solute. The age of each daughter was then set to zero. For transfer during coalescence or rebounding between two equal size droplets, the resulting drop age was set to zero while the contenb of the two drops or subsequent larger drop were assumed to be completely mixed. After each drop size was evaluated, the solute concentration in the continuous phase was updated and the next time increment was analyzed. The efficiency of the program was signifhntly enhanced by using pointers. An array DROP was used to keep track of the drop’s age, whether or not that array element was used, and a pointer denoted the next drop to be evaluated. Typical simulation times for 50 time increments ranged between 1 and 7 CPU min on a VAX 6210 computer. Effects of Interaction Terms. The effects of different expressions for the breakage and coalescence frequencies on the predicted transient Sauter-mean drop diameter data were ascertained so that the expressions for g and f which best fitted the experimental data and most accurately represented the physical situation could be used. The tested equations were Coulaloglou and Tavlarides’ (1977) breakage and coalescence frequencies, Howarth’s (1964) coalescence frequency, and the breakage frequency expression of Skelland et al. (1990) [see also Kanel (1990)l. All four possible combinations were examined. Upon inspecting runs with the same breakage expression and either coalescence frequency, both equations appeared to give very similar results, possibly due to the relatively low # values of 0.07 or 0.03. Thus, the coalescence frequency of Coulaloglou and Tavlarides (1977) was used [with the constants of Bapat and Tavlarides (1985)l in this study because of its successful application by Bapat and Tavlarides (1985) to dispersions in agitated vessels with solute transfer. The effect of the breakage frequency relationship upon the agreement between predicted transient Sautermean drop diameter and that from experiment is significant. Coulaloglou and Tavlarides’ (1977) expression consistently underpredicted the d32,while the expression of Skelland et al. (1990) predicted values close to those determined experimentally with solute transfer from the dispersed phase, but less faithfully when transfer was in the opposite direction. Therefore, since Skelland et al.’s (1990) relationship for g more accurately predicted the experimental data and is the only expression that incorporates the effect of solute transfer, it was chosen for the simulation. Effect of the k,Expression. Glen’s (1965) expression for k, during droplet free motion between events was selected as the best currently available for an agitated liquid-liquid dispersion, based on experimental evidence of Skelland and Moeti (1990). To confirm this choice of k,, the simulator was run using several of the available ex-
Ind. Eng. Chem. Res., Vol. 31, No. 3, 1992 913 M o n a l mass trans for run 108
0.0
10.0
20.0
30.0
40.0
50.0
time (sec)
Efactiohal mass trans for run 89
0.0
10.0
20.0
30.0
40.0
50.0
time (sec) Simulated and arp. d32 w t Run 89
0
s 0
’
Chloro(d) C->D k 3 s
T &?s!
tu
Chloro(d) D->C M s
- y.....
P)
v
5-.___
C In
0 -
I
0
0
C
O
D
0
Q
20.0
30.0
8 0
0.0
10.0
40.0
50.0
time (see) Figue 4. Simulation predictions compared to experimental data for runs 108and 89 when Skelland et al.’s (1990) breakage frequency is used in conjunction with Bapat and Tavlarides’ (1985) coalescence frequency. Both runs exhibit dispersed-phase-controlledmass transfer. (No added surfactant.)
pmaions [Friedlander-Stokes in Calderbank (1967),Levins and Glastonbury (19721,Lamont and Scott (1970),Glen (1965) (turbulent flow model), Calderbank’s (1967) boundary layer model, Calderbank’s (1967)isotropic turbulence model, and Boyadzhiev and Elenkov (1966)I.The predicted fractional mass transfer, FMT, versus time curves were compared with experimental data for run 205 [ t e t r a b u t y h o n i u m bromide (TBAB)diffusing from the continuous chlorobenzene phase in the absence of surfactants]. The average percent deviation was calculated
dispersed phase were used. The average percent deviations for the Sauter-mean drop diameter versus time curves were -0.62% and -2.74% for the initial distributions at 3 and 6 a, respectively, for run 205 with the corrected constant (seelater) in the breakage frequency. The use of the latter distribution (smaller initial d32)forced the simulator to slightly underpredict the d32 versus time curve.
as
System Studied. Tetrabutylammonium bromide, TBAB, was chosen for transfer as solute between chlorobenzene and water, because the electrical conductivity of both phases increased measurably with increasing concentration of TBAB. The distribution coefficient, m,was 46.96 in favor of the aqueous phase (mass/volume units). The chlorobenzene was “ p d i e d ” grade, supplied by the Fisher Scientific Company, and the TBAB was provided by the Sigma Chemical Company. The water was deionized by passage through packed beds of mixed anionic and cationic ion-exchange resins from the Rohm and Haas Company. Triton X-100(C8H1,C6H4(OCH2CH2)gloOH), an octylphenoxypolyethoxyethanol,supphed by the Sigma Chemical Company, was chosen as the surfactant because ita
average 90deviation = M FMT,
.(
is1
- FMTeqi
FMTdq
)(?)
(3)
where M is the total number of data points. The expression offered by Glen (1965)yielded the minimum average percent deviation of 6.3%, justifying ita use in the simulation. Effect of Initial Drop Size Distribution. The effects of the initial drop size distribution on the predicted transient ds2 and fractional mass transfer curves were examined by comparing the simulation results with data when the distributions at 3 and 6 s after injection of the
Experimental Work
914 Ind. Eng. Chem. Res., Vol. 31, No. 3,1992
Ffactional mass trans.for run 205
Ftadional mass trans for run 165
C
C
Chloro (c) D->C, C3 sec 00
10.0
20.0
30.0
50.0
40.0
,
U
0.0
time (sec)
10.0
90.0
30.0
40.0
50.0
time (sec)
Simulated and exp. d32 vs t Run 205
Simulated and exp. d32 vs t Run 165
Chloro(c) D->C k 3 s
~
,00
100
200
300
400
500
* L
0
0.0
time (sec)
10.0
20.0
30.0
400
500
time (sec)
Figure 5. Simulation predictions compared to experimental data for runs 205 and 165 when SkeUand et d.'s (1990)breakage frequency is used in conjunction with Bapat and Tavlarides' (1985)coalescence frequency. Both runs exhibit continuous-phase-controlled mam transfer. (No added surfactant.) Table V. Interfacial Tension between Chlorobenzene and Deionized Water Corresponding to a Given Concentration of Triton X-100 at 25 "C concn of SAA in water, corresp u, solution no. g of SAA/ L of water dyn/cm 1 0.000 33.5 2 0.038 27.1 21.0 3 0.100 4 2.000 16.0 Table VI. Summary of Average Percent Deviations between Simulated and Experimental Transient Fractional Mass Transfer and Sauter-Mean Drop Diameter Curves Using Skelland et a1.b (1990) g and Bapat and Tavlarides' (1986) I (No Surfactant Added) av ?& deviation fractional run Sauter-mean mass direct. of no. drop diam transfer cont phase transfer 108 +26.8 +12.3 water C-.D water D.sC 89 +14.6 -19.4 C D -1.2 chlorobenzene 205 +23.5 chlorobenzene D4C 165 +3.8 +18.0 =$
nonionic nature eliminated electrical effects at the interface. Furthermore, Moeti (1984) claimed that Triton X-100produced clear dispersions in agitated liquid-liquid systems as opposed to anionic or cationic surfactants.
Table VII. Summary of the New Constants T for the Breakage Frequency of Skelland et al. (1990)
direction of transfer D-C D-C D-C C-D D-C D-C D*C C 4 D C-D C-D
cont phase water water water water chlorobenzene chlorobenzene chlorobenzene chlorobenzene chlorobenzene chlorobenzene
[SAAI,g/ (L of water) 0.000 0.038 2.000 0.000 0.000 0.038 2.000 0.000 0.038 2.000
run no.a 92 216 42 110 186 236 139 193 197 61
T 0.12 0.18 0.16 0.14 0.16 0.14 0.14 0.18 0.18 0.16
aComplete details are in Kanel (1990).
Physical and Transport Properties. AU physical and transport properties were measured at 25 O C . Viscosity was determined by using a Cannon-Fenske routine viscometer, and the density was measured via a Troemner specific gravity chain balance Model 5-101. Interfacial tension was determined by the du Nouy ring method, using a Fisher Surface Tensiomat Model 20. The resulting in-
Ind. Eng. Chem. Res., Vol. 31, No. 3, 1992 916 Table VIII. Summary of Simulation Predictions Compared to Samples of direct. of block cont phase diffusion 4 [SAAI no. D-C 0.03 1 5 water 0.07 1 5 water D-C D-C 0.03 3 2 water 6 C-D 0.03 1 water water C-D 0.07 1 6 0.03 1 9 chlorobenzene D-C D-rC 0.07 1 9 chlorobenzene D-C 0.03 4 7 chlorobenzene 0.07 4 7 chlorobenzene D4C C-D 0.03 1 10 chlorobenzene C-D 0.07 1 10 chlorobenzene 0.03 4 3 chlorobenzene C-D
Experimental Data" run av % deviation no.& for dg2 93 -1.73 103 +4.03 42 +4.04 112 -6.48 +11.32 107 189 +6.50 207 -6.72 141 -6.23 136 -3.96 190 +7.90 -2.30 202 -6.10 145
av % deviation FMT -6.19 -2.83 +6.53 +11.69 +23.57 +27.09 +8.15 +15.11 +13.97 -6.79 +5.12 +8.19
OThe impeller diameter was 0.1015 m and the impeller speed was 240 rpm, except for run 42 where N = 220 rpm. Surfactant concentrations are represented as 1 = no surfactant, 2 = 0.038 g of SAAf L of water, 3 = 0.100 g of SAAfL of water, and 4 = 2.000 g SAAfL of water. bComplete details are in Kanel (1990).
hctional mass trans.for run 103
0.0
10.0
20.0
40.0
30.0
hetionel mass trans. for run 112
.
50.0
0.0
time (sec)
Chloro.(d) C->D, tF3 se 10.0
20.0
30.0
40.0
50.0
time (sec)
Simulated and exp. d32 vs t Run 103
Pd
Simulated and exp. d32 vs t Run 112 0
Y d
zR d
Chloro(d) D-X, t = 6 ~
Chlom(d) C->D, fis3~
us!
a
8
u~.~.o.,.,,
Y) 0
0' .......
..............
d
0 0
d
-
I
.
"'-""..p .......
1
0
.......P...:,.
a.............JL .............Q............r O ...............
I
* L 0
0.0
10.0
20.0
30.0
40.0
50.0
time bet)
Figure 6. Simulation results versus experimental data for transfer of TBAB across the water-chlorobenzene interface for dispersed-phasecontrolled systems in the absence of surfactants. (T = 0.12 for run 103 and 0.14for run 112.)
terfacial tension versus surfactant concentration curve is presented in Figure 3. The effect of TBAB concentration on interfacial tension in the presence of Triton X-100was found to be negligible in the concentration range used here. The diffusivity of TBAB in various solvents was measured with diaphragm diffusion cells. The diaphragms were Ace Glass E-Type sintered glass filters with pore diameters between 4 and 8 pm. The principles and techniques of this
method have been described by Wilke and Chang (1955), Holmes (19651, Holmes et al. (1963), and Bidstrup and Geankoplis (1963). A summary of the physical and transport properties of water and chlorobenzene is presented in Table IV. Deecription of the Apparatus. The equipment doeely resembled that used by Skelland and Lee (1981) and Skelland and Moeti (1990), with some minor differences
916 Ind. Eng. Chem. Res., Vol. 31, No. 3, 1992
Ractional mass trans.for run 207
0.0
20.0
10.0
30.0
50.0
40.0
hctional mass trans.for run 190
0.0
10.0
Simulated and exp. d32 vs t Run 207
20.0
30.0
40.0
50.0
time (sec)
time (sec)
Simulated and exp. d32 vs t Run 190
C h l o r ~ ( c )D->C, t-6~
- "'p ..-.e....0.... -.......
............... Q--,.- .....Q ................
Q.
I
0.0
10.0
20.0
30.0
40.0
60.0
time (sec)
0.0
10.0
20.0
30.0
40.0
50.0
time (sec)
Figure 7. Simulation results versus experimental data for transfer of TBAB across the water-chlorobenzene interface for continuousphase-controlled systems in the absence of surfactants. (T = 0.18 for run 190 and 0.16 for run 207.)
as noted below. Briefly, a Model ELB experimental agitator kit made by Chemineer Kenics was used. The impellers were stainless steel six-flat-blade turbines, with diameters (dI) of 0.063 and 0.102 m. The cylindrical, flat-bottomed glass vessel was 0.213 m in diameter (5") and was f i e d to a height equal to ita diameter. The impeller was centrally located, and four radial baffles, of width equal to 0.089611, were employed. The agitated vessel was immersed in a plane glass aquarium filled with water maintained at 25 "C, and photographs were taken during each run. A Nikon F3 camera with a Micro-Nikkor 55-mm f / 2 . 8 lens was used, together with Kodak Technical pan film, for black and white prints. One Hedler Halogen-U lamp was placed beside the vessel. The solute concentration in the continuous phase was measured by an electrical conductivity cell constructed from two 7-mm-diameter glass capillary tubes 21 cm in length, both parallel and side by side with 15 mm of 20gauge platinum wire exposed at one end. A glass bead was placed at the tip of the platinum wires to prevent them from moving in the turbulent flow field. The platinum wires were spot welded to a nickel-copper wire which ran the length of the capillary tube and the cell constant was determined to be 609.1 l/cm. The cell was connected to either a Yellow Springs Instrument Conductivity Meter
Model 32 or a Hewlett-Packard 4329A High Resistance Meter. The output was then recorded by a Fisher Recordall Series 6ooo strip chart recorder. Experimental Procedure. For all runs the two solvents (chlorobenzene and water) were presaturated with each other and the solute and surfactant concentrations established (Triton X-100 in the aqueous phase, TBAB in the appropriate phase). The camera was located with the focal plane about 0.7 cm inside the veasel [and the axial position chosen to give an "average" drop size based on the work of Weinstein and Treybal (1973)], which initially contained only the continuous phase. The latter was agitated long enough to establish steady flow patterns, the strip chart recorder, the Hedler light, and the electrical conductance meter were all turned on, and the disperse phase was then rapidly poured from two beakers into opposite sides of the vessel at a radial position equidistant between the impeller shaft and the vessel wall. Photographs of the diapersion were taken at 3-5 intervals for the first 15 s and then at 10-8 increments for the next 40 8. About 500 drops were sized from each photograph with a Carl-Zeiss Particle Size Analyzer. The vessel, impellers, baffles, and conductance probe were throughly cleaned with hot soapy water, followed by rinses with copious amounts of tap water and further rinsing with deionized water. The equipment was then air
:v Ind. Eng. Chem. Res., Vol. 31, No. 3, 1992 917
hctional mass trans. for run 145
hctional mass trans. for run 42
Fa & &!
32 -
...'
-
t!q
e o
a
.VI 0
EX-
22
!
l!Q
:;o
-
nro
I
...........................""0
4 0
:
:o
:
~~
0.0
10.0
20.0
30.0
40.0
50.0
time (sec)
0.0
10.0
20.0
30.0
40.0
50.0
time (sec)
Simulated and exp. d32 vs t Run 145
Simulated and exp. d32 vs t R u n 42
0
0.0
10.0
20.0
30.0
40.0
50.0
time (sec)
0.0
10.0
20.0
30.0
time 6ec)
40.0
50.0
Figure 8. Simulation results versus experimental data for run 145 with transfer of TBAB to water drops with 2.0 g of surfactant/L of water (T = 0.16), and run 42,where transfer of TBAB was from chlorobenzene drops with 0.1 g of surfactant/L of water (T = 0.16).
dried before the next run. Experimental Design. In the factorial experimental program, four surfactant concentrations were chosen to give the maximum interfacial tension (no surfactant), a minimal interfacial tension (2.0 g of SAA/L of H,O), and two equally spaced intermediate Q values (0.038 and 0.10 g of SAA/L of HzO). The desired surfactant concentrations and resultant interfacial tensions are given in Table V. Two impeller diameters were selected, and four impeller speeds were used. They were chosen to be below the speed at which air would be drawn into the continuous phase when it was being agitated alone, and above the minimum impeller speed for complete dispersion as calculated from Skelland and Ramsay (1987). Two volume fractions of the dispersed phase were examined, namely, 3 and 7 vol 9%. The latter was the maximum value that still allowed clear photographs of the dispersion to be obtained 80 that the interfacial area for mass transfer could be ascertained. In addition these 4 values were low enough that the disperse phase contributed negligibly to the measured electrical conductivity of the heterogeneous mixture, according to Maxwell's equation (1881).
Comparison of the Simulation with Experimental Data The simulated transient Sauter-mean drop diameter curves and the fractional mass transfer curves were com-
pared with experimental data as shown in Figures 4 and 5 for dispersed-phase- and continuous-phase-controlled systems, respectively. Note that Skelland et al.'s (1990) breakage frequency and Coulaloglou and Tavlarides' (1977) coalescence frequency expressions [with the constants of Bapat and Tavlarides (1985)] were used with the other terms in Tables 11and 111. The average percent deviations for these runs are summarized in Table VI. The average percent deviations for the Sauter-mean drop diameter in the runs where solute transferred to the dispersed phase are notably higher than with transfer in the opposite direction. This may result from Skelland et al.'~(1990) breakage frequency being developed for solute diffusion from the drops. Noting the complexity of the modeled system and the number of expressions (16) involved in computing these curves, these predictions are felt to be quite good when one remembers the absence of any adjustable parameters. Furthermore, this is the first application of any simulation to all four directionsof diffusion and dispersion in batch agitated liquid-liquid systems. The constant in the breakage frequency presented by Skelland et al. (1990) was developed for transfer from the dispersed organic phase in the absence of surfactants. Therefore new values were determined for the constant T in their expression for transfer here in both directions with and without surfactant, respectively. They are given in Table VII. These values were determined by iteratively
918 Ind. Eng. Chem. Res., Vol. 31, No. 3, 1992
all of these runs are seen to be quite low. As would be expected, generally when the Sauter-mean drop diameter was overpredicted (+), the fractional mass transfer was underpredicted. The simulation overpredicted the fractional mass transfer in the presence of surfactants even when the diminished interfacial tension was used for a. Only two runs had average percent deviations above 15% for the transient fractional mass transfer data, and only one transient d32 point was above 8% average deviation. Therefore, with these new T values, the simulation can accurately predict mass transfer in agitated liquidliquid dispersions in all four directions of dispersion and diffusion, with or without surfactants present.
Ihctional mass trans.for run 141 ......
:
0.0
10.0
o
20.0
30.0
40.0
50.0
time Qec) Simulated and exp. d32 vs t R u n 141 0
0.0
10.0
20.0
30.0
40.0
50.0
time (sec) Figure 9. Simulation resulta versus experimental data for transfer of TBAB from water drops with 2.0 g of surfactant/L of water. The constant in the breakage frequency was 0.14. Table IX. Average Percent of the Total Mass Transfer Due to Drop Breakup and Subsequent Damped Oscillations for All Four Directions of Dispersion and Diffusion av % of total transfer due to breakage and damped oscilln cont phase direct. of transfer 6 = 0.03 6 = 0.07 6.2 4.3 water D-C C-D 2.0 7.0 water D-C 0.45 0.45 chlorobenzene C d D 0.65 0.13 chlorobenzene
running the simulation to minimize the s u m of squares of relative deviations, D,, defined as (4)
between the predicted and experimental Sauter-mean drop diameters. All experimental runs used to determine T were at 4 = 0.03 to minimize the possible effects of coalescence on k. These newly determined constants were then used to simulate other experimental data with the results summarized in Table VI11 and in Figures 6-9. The average percent deviations between simulation and experiment for
Magnitude of the Breakage Effect An estimate of the relative magnitude of mass transfer during the individual events of drop breakup, free motion throughout the vessel, and drop-drop interactions will now be developed. The simulation computed the mass transfer from each of these events for every time step. Therefore, the average percent of transfer resulting from each process was determined by averaging the percent transferred at the times 5,10,15, and 20 s after tl. Unfortunately, very few coalescence or rebounding events were predicted in any simulation. This probably results from the low 4 values used (4 = 0.03 and 4 = 0.07) and from the requirement in the simulation that only equal-size drops interact, However, since there was good agreement between the simulation and experimental values, the interaction effects must have contributed only slightly to the total mass transfer for these experimental conditions. The results are summarized in Table IX. Note that mass transfer during drop breakup and resulting damped oscillations accounted for between 2% and 7 % of the total transfer when water was the continuous phase. However, breakage contributed only between 0.13% and 0.65% when water was dispersed. These observations may be explained as follows. When water is continuous, the mass transfer is dispersed phase controlled and a concentration gradient builds up within the drops. Since breakage is a relatively violent process, these concentration gradients are destroyed during breakup, which results in greater extraction. However, when water is the dispersed phase, internal mixing of the drop resulting from breakage is not as important since the continuous phase now offers the dominant resistance to transfer. It may also be noted from Table IX that the percentage of transfer due to breakage and resulting damped oscillations appears to be independent of the direction of mass transfer, as well as of C$in the range studied. Design Applications The simulation detailed here enables batch units to be sized on the basis of the residence time needed to ensure some specified fractional extraction in a wide variety of cases including solute transferring either to or from drops; the aqueous phase being either continuous or dispersed; and in either surface actively contaminated or uncontaminated systems. Continuous units may also be characterized regarding mean residence times needed to ensure some specified fractional extraction. Drop size information as the run proceeds is available from the simulation if necessary. Conclusions A simulation model was developed to compute the transient Sauter-mean drop diameter as well as the frac-
Ind. Eng. Chem. Res., Vol. 31, No. 3, 1992 919 tional mass transfer in batch agitated liquid-liquid dispersions. This was accomplished by accounting for mass transfer during drop breakup, drop-drop interactions resulting in coalescence or rebounding, drop oscillations, and free motion of the drops throughout the vessel. Using 16 different expressions, 8 of which were developed in this study, the transient fractional maw transfer was felt to be well predicted with a i 2 0 % average deviation for all four directions of diffusion and dispersion, when no adjustable parameters were used. However, to further enhance the simulation, one constant in the breakage frequency was varied to improve predictions to f15% for 10 simulations, including 4 with surfactants present. Furthermore, the transient Sauter-mean drop diameter was predicted with a i8% average deviation for these simulations. The fraction of the total mass transferred due to dropdrop interactions was found to be indeterminate, but small, for the system studied here. However, drop breakup and subsequent damped oscillations accounted for approximately 5% of the total transfer in the simulation for dispersed-phase-controlled systems and insignificantly for continuous-phase-controlled systems. Acknowledgment Some support was received from National Science Foundation Grant CPE-8203872/01. Robert M. Kanel is gratefully acknowledged for his programming assistance on the simulation. Nomenclature AB = new surface area created during drop breakup, m2 AFC,A? Ab Am = interfacial area during free motion of drops, during coalescence, during rebounding, during damped oscillation, m2 A. = interfacial area of undeformed drop, m2 erg cm, dimensionless B = constant = 1 x B, = constant in the Grober (1925) expression, dimensionless C1,C2, C3,... = constants, appropriate units D,, Dd = diffusivity of solute in the continuous and dispersed phases, m2/s D,= sum of squares of relative deviation, dimensionless dI = impeller diameter, m d , = particle diameter, m d3Zc,d3Ze = computed and experimental Sauter-mean drop diameter, m d32, d*32 = Sauter-mean drop diameter, equilibrium value, m e(d l,dp2)= rebound frequency between drops with diameter !C and dp2, l / s FM% = fractional mass transfer, dimensionless f = coalescence frequency, l / s f, = oscillation frequency, l / s f* = complex function of time, dimensionless g = gravitational acceleration, or breakage frequency, m/s2 or l / s H = fluid height in vessel, m h, hl, h2 = distance between drops, at coalescence or start of rebounding, at initial deformation of drops, m k = breakage frequency constant, l/(s m2) k, =continuous-phase mass-transfer coefficient, m/s kcB, kdB= average continuous- and dispersed-phase masstransfer coefficients for new surface during drop breakup, m/s k * b = individual continuowphase mass-transfer coefficient for the plane region of the drop during the coalescence process averaged over both axial and radial positions, m/s k*= individual continuowphase mass-transfer coefficient for the plane region of the drop during the rebounding process averaged over both axial and radial positions, m/s
--
kcD,-,A,k d , = continuous- and dispersed-phase capacity coefficients during damped drop oscillations,time-averaged values, - - m/s k$, kdA = mean continuous and dispersed phase capacity coefficients, m/s &E, k,ac = overall disperse-phase mass-transfer coefficient and individual continuous-phase mass-transfer coefficient during free motion of drops, m/s M = total number of data points in eqs 3 and 4 m = distribution coefficient, dimensionless N = impeller speed, l / s N' = total number of drops in the simulation N F ~=, impeller Froude number, p,d?W/ApHg, dimensionless Npe = Peclet number, V,h/D,, dimensionless N w = impeller Reynolds number, pd?N/u, dimensionless N k = Schmidt number, p/Dp, dimensionless Nwq = impeller Weber number, Wd?p,/u, dimensionless ni = number of drops of diameter dPi R = radius of the plane disk during coalescence or rebounding, m SAA = surface-active agent T = tank diameter, m t, tl, to= time, also t represents the time after tl used to obtain the initial drop size distribution, time at which startup disturbance has ceased, time when damped oscillations begin, s tg, tc, t R , tOS, t, = breakage time, coalescence time, rebound time, oecillation time, circulation time as defined by Holmes et al. (1964), s V = drop volume, m3 V , = net approach velocity of two drops during interaction, m/s Greek Symbols /? = constant = 0 . 8 d Y ; d,[=]cm, dimensionless y = 1/3Npe,dimensionless e
= power input per unit mass, m2/s3
z: 1+ 7 is the ratio of the maximum to minimum surface area over one cycle for an oscillating drop, dimensionless
t = (0.438 p / u ) (32.5 u/pddp)lI2,dimensionless q
= separation constant, dimensionless
A, = constant in the Grober (1925) expression, dimensionless pc, C(d = viscosity of the continuous and dispersed phase,
kg/ (ms)
4 = constant, dimensionless pc, Pd = density of
continuous, dispersed phase, kg/m3 AP = I P C - Pdl, kg/lq3 u = interfacial tension, N/m 7 = (d / 2 ) [ d d / 3 2 . 5 ~ ] ' / ~s, r* = &fined% Table 11, s T = constant in the breakage frequency expression of Skelland et al. (1990), dimensionless 9 = volume fraction of dispersed phase, dimensionless Literature Cited Bajpai, R. K.; Ramkrishna, D.; Prokop, A. A Coalescence Redispersion Model for Drop Size Distributions in an Agitated Vessel. Chem. Eng. Sci. 1976,31, 913-920. Bapat, P. M.; Tavlarides, L. L. Mass Transfer in a Liquid-Liquid CFSTR. AZChE J. 1985,31 (4), 659-666. Bapat, P.M.; Tavlarides, L. L.; Smith, G. W. Monte Carlo Simulation of Mass Transfer in Liquid-Liquid Dispersions. Chem. Eng. Sci. 1983, 38 (12),2003-2013. Bately, W.; Thornton, J. D. Partial Mass-Transfer Coefficients and Packing Performance in Liquid-Liquid Extraction. Znd. Eng. Chem. Res. 1989,28, 1096-1101. Bayens, C. A.; Laurence, R. L. A Model for Mass Transfer in a Coalescing Dispersion. Znd. Eng. Chem. Fundam. 1969, 8 (l), 71-77. Bidstrup, D. E.; Geankoplis, C. J. Aqueous Molecular Diffusivities of Carboxylic Acids. J. Chem. Eng. Data 1963, 8 (21,170-173.
920 Ind. Eng. Chem. Res., Vol. 31, No. 3, 1992 Boyadzhiev, L.; Elenkov, D. On the Mechanism of Liquid-Liquid Mass Transfer in a Turbulent Flow Field. Chem. Eng. Sci. 1966, 21,955-959. Bozorgzadeh, F. Study of Mass Transfer During Droplet Splitting Using a New Optical Technique. PbD. Dissertation, University of Newcastle upon Tyne, 1980. Calderbank, P. H. Mass Transfer. In Mixing Theory and Practice; Uhl, V. W., Gray, J. B., EMS.; Academic Press: New York, 1967; VOl. 2. Chen, J. D.; Hahn, P. S.; Slattery, J. C. CoalescenceTime for a Small Drop or Bubble at a Fluid-Fluid Interface. AIChE J. 1984,30(4), 622-630. Clift, R.; Grace, J. R.; Weber, M. E. Bubbles, Drops, and Particles; Academic Press: New York, 1978. Coulaloglou, C. A.; Tavlarides, L. L. Drop Size Distributions and Coalescence Frequencies of Liquid-Liquid Dispersions in Flow Vessels. AZChE J. 1976,22(2),289-297. Coulaloglou, C. A,; Tavlarides, L. L. Description of Interaction Proceeees in Agitated Liquid-Liquid Dispersions. Chem. Eng. Sci. 1977,32,1289-1297. Curl, R. L. Dispersed Phase Mixing: 1. Theory and Effects in Simple Reactors. AIChE J. 1963,9(2),175-181. Davies, J. T. Turbulence Phenomena; Academic Press: New York, 1972. Glen, J. B. Mass Transfer in Disperse Systems. Ph.D. Dissertation, Department of Chemical Engineering, University of Canterbury, 1965. Grober, H. 2. Ver. Dtsch. Zng. 192669,705. Holmes, D. B.; Voncken, R. M.; Dekker, J. A. Fluid Flow in Turbinestirred, Baffled Tanks-I Circulation Time. Chem. Eng. Sci. 1964,19,201-208. Holmes,J. T. SimplifiedTechniques for Using the Diaphragm Type, Liquid Diffusion Cell. Rev. Sci. Znstrum. 1965,36 (6), 831-832. Holmes, J. T.; Wilke, C. R.; Olander, D. R. Convective Mass Transfer in a Diaphragm Diffusion Cell. J. Phys. Chem. 1963, 67, 1469-1472. Hong, P. 0.;Lee,J. M. Changes of the Average Drop Size During the Initial Period of Liquid-Liquid Dispersions in Agitated Vessels. Znd. Eng. Chem. Process Des. Dev. 1985,24,86&872. Howarth, W. J. Coalescence of Drops in a Turbulent Flow Field. Chem. Eng. Sci. 1964,19,33-38. Hsia, M. A,; Tavlarides, L. L. Simulation Analysis of Drop Breakage, Coalescence and Macromixing in Liquid-Liquid Stirred Tanks. Chem. Eng. J. 1983,26,189-199. Jeon, Y. M.; Lee, W. K. A Drop Population Balance Model for Mans Transfer in Liquid-Liquid Dispersion, 1. Simulation and its Results. Ind. Eng. Chem. Fundam. 1986,%, 293-300. Kanel, J. S. Effects of Some Interfacial Phenomena on Mass Transfer in Agitated Liquid-Liquid Dispersions. Ph.D. Dissertation, Georgia Institute of Technology, Atlanta, GA, 1990. Klee, A. I.; Treybal, R. E. Rate of Rise or Fall of Liquid Drops. AZChE J. 1956,2,444-447. Lamont, J. C.;Scott, D. S. An Eddy Cell Model of Mass Transfer into the Surface of a Turbulent Liquid. AZChE J. 1970,16 (4), 513-519. Levins, D. M.; Glastonbury, J. R. Particle-Liquid Hydrodynamics
and Mass Transfer in a Stirred Vessel. E.Mass Transfer. Tram. Znst. Chem. Eng. 1972,50,132. Maxwell, J. C. A Treatise on Electricity and Magnetism, 2nd ed.; Clarendon Press: Oxford, 1881; Vol. I. Moeti, L. T. Effecta of Surface Active Agents on Minimum Impeller Speeds for Liquid-Liquid Dispersion in Baffled Vessels. M.S. Thesis, Georgia Institute of Technology, Atlanta, GA, 1984. Molag, M.; Joosten, G. E. H.; Drinkenburg, A. H. Drop Breakup and Distribution in Stirred Immiscible Two-Phaee Systems. Znd. Eng. Chem. Fundam. 1980,19,275-281. Park, J. Y.; Blair, L. M. The Effect of Coalescence on Drop Size Distribution in an Agitated Liquid-Liquid Dispersion. Chem. Eng. Sci. 1975,30, 1057. Ramkrishna, D. Drop-Breakup in Agitated Liquid-Liquid Dispersions. Chem. Eng. Sci. 1974,29,987-992. Schroeder, R. R.; Kintner, R. C. Oscillations of Drops Falling in a Liquid Field. AZChE J. 1966,11 (l), 5-8. Scott, T. C.; Byers, C. H. A Model for Mass Transfer in Oscillating-Circulating Liquid Drops. Chem. Eng. Commun. 1989,77, 67-89. Shah, B. H.; Ramkrishna, D. A Population Balance Model for Mass Transfer in Lean Liquid-Liquid Dispersions. Chem. Eng. Sci. 1973,243,289-299. Skelland, A. H. P.; Conger, W. L. A Rate Approach to Design of Perforated-Plate Extraction Columns. Znd. Eng. Chem. Process Des. Deu. 1973,12 (4),448-454. Skelland, A. H. P.; Lee, J. M. Drop Size and Continuous-Phase Mass Transfer in Agitated Vessels. AIChE J. 1981, 27 (l),99-111. Skelland, A. H. P.; Moeti, L. T. Mechanism of Continuous-Phase Mass Transfer in Agitated Liquid-Liquid Systems. Znd. Eng. Chem. Res. 1990,29,2258-2267. Skelland, A. H. P.; Ramaay, G. G. Minimum Agitator Speeds for Complete Liquid-Liquid Dispersion. Znd. Eng. Chem. Res. 1987, 26,77-81. Skelland, A. H. P.; Xien, Hu; Kanel, J. S. Drop Breakage in Agitated Liquid-Liquid Systems Undergoing Maas Transfer. Unpublished work at the Georgia Institute of Technology; reported in Kanel (1990). Sprow, F. B. Drop Size Distributions in Strongly Coalescing Agitated Liquid-Liquid Systems. AIChE J. 1967,13 (5),995-998. Valentas, K. J.; Amundson, N. R. Breakage and Coalescence in Dispersed Phase Systems. Znd. Eng. Chem. Fundam. 1966,5(4), 533-542. Valentas, K. J.; Bilous, 0.; Amundson, N. R. Analysis of Breakage in Dispersed Phase Systems. Znd. Eng. Chem. Fundam. 1966,5 (2),271-279. Weinstein, B.; Treybal, R. E. Liquid-Liquid Contacting in Unbaffled, Agitated Vessels. AZChE J. 1973,19 (2),304-312. Wilke, C. R.; Chang, P. Correlation of Diffusion Coefficients in Dilute Solutions. AZChE J. 1956,1, 264. Zeitlin, M. A.; Tavlarides, L. L. Fluid-Fluid Interactions and Hydrodynamics in Agitated Dispersions: A Simulation Model. Can. J. Chem. Eng. 1972,50,207-215. Received for review August 27, 1991 Accepted September 5, 1991
