664
Ind. Eng. Chem. Process Des. Dev. 1986, 2 5 , 664-673
falling within the range which has been considered to be the most effective for the interfacial-transfer process (Calderbank and Moo-Young, 1961). Nomenclature A = amplitude of reciprocation, cm a = specific interfacial area, cm-' C, = orifice coefficient, dimensionless Cd = orifice coefficient when (VL - wA cos w t ) > 0, dimen-
sionless
C,, = orifice coefficient when (V, - wA cos w t ) < 0, dimen-
sionless = Sauter mean bubble diameter, cm F = frequency of reciprocation, Hz n = number of plates, dimensionless A?' = pressure difference between two stations along a column, mmHzO A?',, = two-phase pressuredrop when VL = 0 and wA = 0, mmHzO APL = instantaneouspressure drop for single liquid phase flow with agitation, mmH20 hph = pressure drop for single liquid phase flow through the perforated plates without reciprocating motion, mmHzO e,h, p L, , = instantaneous and time-averaged pressure drop for single liquid phase flow through perforated plates with upward motion, mmH20 M u , h p ~ =d instantaneous and time-averaged pressure drop for single liquid phase flow through perforated plates with -downward motion, mmH20 hpLg = time-averated two-phase pressure drop across reciprocating plates, mmHzO hpLBo = two-phase pressure drop across stationary plates, mmHzO t = time, s V = superficial gas velocity, cm/s v", = superficial liquid velocity, cm/s VLg = superficial velocity of two-phase mixture, VL/cL, cm/s 2 = distance, mm d32
Greek Letters tg = gas holdup with plate reciprocation, fractional ego = gas holdup without plate reciprocation tL = liquid holdup w = angular frequency = 27F, s-l p, = gas-phase density, kg/m3 pL = liquid-phase density, kg/m3 = fractional open area of plates w 2 = variance of bubble-size distribution, cm un2 = variance of bubble-size distribution, cm P L ~= density of two-phase mixtures, cLpL, kg/m3
Literature Cited Akita, K.; Yoshida, F. Ind. Eng. Chem. Process Des. Dev. 1973, 12, 76. AI Taweel, A. M.; Landau, J.; Picot, J. J. C. AIChE Symp. Ser. 1979, 75(No. 190), 217. Baird, M. H. I.; Grastang, S . H. Chem. Eng. Sci. 1972, 27,627. Brodkey, R. S."The phenomena of Fluid Motions"; Addison-Wesley: Reading MA 1967; p 215. Galderbank, P. H.; Moo-Young, M. B. Chem. Eng. Sci. 1961, 16, 39. Calderbank. P. H. I n "Mixing": Uhl, V. W., Gray, J. B.. Ed.; Academic Press: New York. 1964 Fair, J . R.;-Limbright, A. J.; Anderson, J. W. Ind. Eng. Chem. Process Des, D e v . 1962. 1 . 33. Gomaa, H. G.'Ph:D. Dissertation, University of New Brunswlck, New Brunswick, Canada, 1977. Hughmark, G. A. Chem. Eng. Prog. 1965, 61, 97. Joshi, J. B.; Sharma, M. M. Trans. IChE 1979, 57,244. Karr, A. E.; Lo, T. C. Chem. Eng. Prog. 1976, 72,68. Karr, A. E.; Gebett, W.; Wang, M. Can. J . Chem. Eng. 1980, 58, 249. Kim, S.D.; Baird, M. H. 1. Can. J . Chem. Eng. 1976, 5 4 , 81. Nishikawa, M.; Yonezawa, Y.; Toyada, H.; Nagata. S.J. Chem. Eng. Jpn. 1978, 11, 73. Noh, S.H.; Balrd, M. H. I . AIChE J. 1984, 30, 120. Pew, R. H., Ed. "Chemical Engineers Handbook", 5th ed.;McGraw-Hill: New York, 1973; pp 534-537. Relth, T.; Renken, S.;Israel, B. A. Chem. Eng. Sci. 1968, 23,613. Shulman, H. L.; Uirich, C. F.; Wells, N. AIChE J . 1955, 1 , 247. Top, K.; Miyanami. K., Yano, T. J . Chem. Eng. Jpn. 1974, 7, 123. Top, K.; Mlyanami. K.; Yano, T. J . Eng. Jpn. 1975, 8 , 122. Towell, G. D.; Strand, C. P.; Ackerman, G. H. AIChE/I. Chem. Eng. Symp. Ser. 1975, No. IO, 1.
Receiued for review February 19, 1985 Revised manuscript received October 10, 1985 Accepted November 15, 1985
Mass Transfer and Drop Sizes in Pulsed-Plate Extraction Columns Llan S. lung and Richard H. Luecke' Depariment of Chemical Engineering, University of Missouri-Columbia,
Columbia. Missouri 652 1 1
Stage efflclency In pulsed-plate extraction columns is modeled by computing the mass transfer as a function of the drop slze distribution. Two mess-transfer mechanisms of the drops are considered. For small drops, a rigid model is assumed; a twbubnt internal circulation model is used for large drops. The mass transfer is calculated for each drop size and integrated over the full drop size range. Using a correlation developed for the drop size transition, a forward mixing model predicts the observed number of stages with an average error of 19.2% for 299 sets of mass-transfer data from the literature.
Liquid-liquid extraction has been recognized as a powerful separation method for many years. Its past application on an industrial scale has been limited, however; it has been considered only when separation by other methods such as distillation, evaporation, or crystallization are unsuitable. As a result of escalating energy costs,there is noww a more favorable climate for liquid-liquid extraction. It can often be an economic alternative to other separation processes. Mechanical agitation and pulsation of the contents of an extraction column increase the intimate contact of two
phases by increasing the interfacial area. The rate of transport of the solute is also accelerated by increasing the flow turbulence. Many devices have been designed and/or invented to add outside energy to the contents of the column. The pulsed-plate extraction column is one of these devices. The pulsed-plate extraction column was seldom used in the chemical industry until its application to solvent extraction separation for radioactive chemicals in the late 1940s. Since then, this type of column has been widely used in the nuclear fuel reprocessing industry. Some ap0 1986 American Chemical Society
Ind. Eng. Chem. Process Des. Dev., Vol. 25, No. 3, 1986 685
plications have also been found in petrochemical and metallurgical industries (Schweitzer, 1979). Because of its relatively high efficiency, increased future use seems likely. In most previous models for extraction columns, mass transfer has been calculated only for a single representative drop size. In real extraction columns, the drop sizes are not uniform and thus more than one type of mass-transfer mechanism may occur. Very small drops are essentially stagnant with no internal circulation, drops of intermediate size have laminar toroidal internal circulation, and very large drops may be expected to haave turbulent internal circulation and/or oscillations and deformations. Korchinsky and Cruz-Pinto (1979) and Cruz-Pinto and Korchinsky (1980) showed that the actual column efficiencies in the RDC were between those predicted by the rigid drops and turbulent internal circulation drops but closer to those predicted by the turbulent internal circulation drops. Khemangkorn et al. (1978) and Miyauchi et al. (1967) have shown, however, that the drops in the emulsion region of the pulsed-plate columns are smaller than in an RCD, and thus many will exhibit ridid masstransfer behavior. We propose a model that takes into account the effects of drop size distribution. Only two mass-transfer mechanisms, the rigid and turbulent circulation, are included. Variations of the mass-transfer coefficient and interface area are computed, along with different drop velocities through the column which result from the different drop sizes.
Previous Work Column models using a single drop size to calculate the mass-transfer interface area and mass-transfer coefficients may include stage, backflow, plug flow, and diffusion models. The work of Olney (1964) showed that uniform drop models do not describe the mixing of the dispersed phase with sufficient accuracy in those cases where the dispersed phase has a wide drop size distribution. Those cases are often encountered in mechanically agitated extraction columns with low holdup. A drop size distribution induces another form of longitudinal mixing due to drops of different sizes having different axial velocities and hence different residence times. This kind of mixing has been called forward mixing to distinguish it from backmixing (Rod, 1966). In a study of the asymmetric rotating disc extractor, Misek and Marek (1970) considered a stage model with forward mixing. The solute composition was assumed uniform for a given drop size fraction throughout a stage; actual drop concentrations would change as the drop flows through the stages. Korchinsky and Azimzadeh-Khatayloo (1976) improved this aspect by assuming complete mixing for the continuous phase only; concentrations in the drops were allowed to vary with the axial position. They did not, however, consider the coalescence of drops, and they neglected the backmixing or entrainment of the dispersed phase. This model predicted a column height up to 200% greater than the height predicted by a single mean drop size. Rod (1966) calculated the influence of forward mixing of the dispersed phase and backmixing of the continuous phase on the extraction efficiency. The height of a transfer unit with fully developed forward mixing was 1.66 times greater than that for the uniform drop model with plug flow. If backmixing of the continuous phase was included with forward mixing, the height of a transfer unit was 2.35 times larger than for the uniform drop plug flow model. Chartres and Korchinsky (1975) found that the rigid drop model gave considerably greater differences in the
concentration profiles between small and large drops than did the oscillating drop model. Nevertheless, the variation with drop size was also considerable for the oscillating drop model. Cruz-Pinto and Korchinsky (1980) experimentally confirmed that influence of the drop size distribution (forward mixing) on the column performance. Different dispersed-phase distributors, and not the operating conditions, were used to generate the different drop size distributions. The number of transfer units of the narrow drop size distributions was 1.28-1.56 times as large as those of the wide drop size distributions.
Design Approach The design approach developed here includes the effects of the drop size distribution. The Sauter mean diameter correlation is predicted from system parameters by using empirical correlations. A drop size distribution is calculated based on the mean diameter and is divided into 25 finite compartments. The drop velocity, the mass-transfer coefficient, and interfacial area are computed as a function of the median drop size of each compartment. Drops smaller than a certain transition drop size are assumed to be rigid drops; mass transfer in the larger drops is computed with the turbulent internal circulation drop model. Mass transfer is then integrated through the column by using different velocities, concentrations, and mass-transfer rates for each drop size. Backmixing in the continuous phase is computed by using a new correlation developed in this work (Appendix B). Source of Mass-Transfer Data The mass-transfer data were taken from Khemangkorn et al. (1978), Smoot and Babb (1962), Thornton (19571, Logsdail and Thornton (1957), and Sege and Woodfield (1954). Column diameters from these references varied from 5.0 to 30.5 cm and stage heights from 1.25 to 5.58 cm, hole diameters were 0.1580.318 cm, and the plate free area fractions were from 0.131 to 0.621. Water was the continuous phase for all the systems. The dispersed phases (and solutes) included carbon tetrachloride (iodine), MIBK (acetic acid), 1,1,2-trichloroethane (acetone), toluene (acetone), butyl acetate (acetone), and 12.5% tributyl phosphate in a kerosene-type hydrocarbon (UO,(NO,),). Pulse intensities ranged from 1.90 to 8.91 cm/s and superficial velocities of the continuous and dispersed phases from 0.0745 to 0.872 cm/s and from 0.0141-1.41 cm/s, respectively. The kinetic energy distribution e, ranged from 64 to 7243 erg/(s cm2) (Tung, 1984). There were 299 data sets including 51 with interfacial convective instability. Other data available in the literature were either incomplete or lacked the values of parameters needed in our correlations. Column Model Since the backmixing in the dispersed phase has been indicated to have a negligible effect on mass transfer (Chartres and Korchinsky, 1975) and since measurements of this variable are very limited, this factor was not considered in the model. The column model follows the forward mixing model developed by Rod (1966) and Chartres and Korchinsky (1975). The following assumpions were made to formulate the model equations: 1. Coalescence,redispersion, and breakup of drops are absent in the column. Individual drops maintain their identities through the column. It was necessary to modify this assumption with a correction factor at high holdup for the dispersed phase.
666
Ind. Eng. Chem. Process Des. Dev., Vol. 25, No. 3, 1986
2. The drop size distribution is uniform throughout the column. 3. Each drop size fraction is represented by a median drop size of that fraction. 4. There is uniform holdup through the column. 5. The continuous phase has a uniform radial concentration. 6. Backmixing occurs only in the continuous phase. Forward mixing in the dispersed phase occurs because of unequal axial velocities of various drop sizes. 7. Drops are spherical. 8. Phase flow rates are constant, and solute concentrations are low. 9. The physical properties of the liquid systems were taken as those of the inlet streams. Although some properties may change through the system, in situ data are generally not available. Concentrations changes were partially included operationally in the correlation; egg.,a factor is changed if mass transfer is present.
Hy drodyamics Experimental data for the velocity of drops relative to the column are scarce. The work of Korchinsky and Azimzadeh-Khatayloo (1976) was followed here to develop these values. The velocity of the drops is related to the velocity of the continuous phase, holdup, and terminal velocity of the drop by
If there are no drop interactions and if all drops are at the terminal velocity, the constant c would be unity. This equation could be made exact if there were a different c, for each drop size. Multiplying both sides of eq 1by the drop volume fraction and summing gives
By mass balance on drops, we get (3)
Drop Size Distribution There have been many experimental studies on the drop size distribution such as Sprow (1967), Chen and Middleman (1967), Olney (1964), and Khemangkorn et al. (1977), for example, but only a few of them were conducted in pulsed-plate extraction columns. Khemangkorn et al. (1977) studies the drop size distribution in a pulsed-plate column. They found that drop sizes varied over a wide range a t low pulsation intensity, and the range narrowed at high pulsation intensity. This experimental study was conducted under very low holdup conditions so that drop interactions were minimized. The Weibull distribution was the most appropriate for this experimental data. It is characterized by the probability density function with two parameters, { and w , as fn(dil{,w) = {wdir-'-' exp(-wdi3
(9)
The parameters of the Weibull distribution may be defined by the values of the Sauter mean drop diameter and the variance of the distribution. The standard deviation is related to the Sauter mean diameter by
Thus, the parameters w and t are defined by eq 9 and 10, if values of d32and the ratio of u to dS2are available. The ratio of u to d32for the Weibull distribution has a maximum value of approximately 0.36. Unfortunately, there is little information about the dependence of the parameters of the distribution on operating conditions, column geometries, and properties of liquid systems. From a few known drop distribution data, the ratio of u to d,, was observed to be near 0.30. Hence, the value of this ratio was fixed as 0.30 for design purposes. From this, all distributions have the same general shape but the dispersion range is proportional to the diameter (and inversely proportional to the pulsation intensity). The Sauter mean drop diameter was calculated from an empirical correlation developed by Boyadzhiev and Spassov (1982) using the Kolmogorov-Obukhov theory of the local structure of turbulent pulsation.
Thus, an average constant c in eq 1 can be calculated.
-vd4+ -
vc
1-4
(4)
Fluid properties in this correlation are bulk values. They reported that this equation predicts the Sauter mean drop diameter to within 4~20%accuracy.
The terminal velocity of the drop is determined with the correlations of Klee and Treybal (1956) Vti = 38.3pc*.45Ap0.68CL C*.11d.o.70 L (5)
Mass-Transfer Models of Drops The overall mass-transfer coefficient of the drops in the ith size compartment was obtained by using the two-film model, i.e., the addition of two individual resistances of the dispersed and continuous phases. m l -l- - -
c=
n
if di < d, and = 17.6pc-0.56~p0.28CL 0.10 0.18 C
Y
(6)
if di < dc where dc = 0.33pc-Q.14&,-0.43
0.38 0 2 4
(7) The terminal velocity is corrected for the effect of the column diameter (Uno and Kintner, 1956; Happel and Byme, 1954). CLC
Y
'
+-
&di
lZci
kdi
The rigid and turbulent circulation drop models were used for the film mass-transfer coefficients. For drop sizes smaller than the transition size, the rigid drop model was used. The mass-transfer coefficient inside rigid drops was taken as (Newman, 1931; Treybal, 1963)
kdi = -2T2 Dfd 3 di The mass-transfer coefficient of the continuous phase with rigid drops was given by Linton and Sutherland (1960) as
Ind. Eng. Chem. Process Des. Dev., Vol. 25, No. 3, 1986
kei = 0.582(0f,/di)(Re)1/2(S~)1/3(14) For drops sizes larger than the transition size, a turbulent mass-transfer coefficient in the drop phase was derived by Handlos and Baron (1957).
For the continuous phase with turbulent (oscillating) drops, Gamer and Tayeban (1960) gave the mass-transfer coefficient as
+
kCi= [50 0.0085ReS~~.~](D~,/di) (16)
Mass-Transfer Equations The mass balances for a representative drop size of the ith fraction of the dispersed phase is
and for the continuous phase is
With the introduction of a kinetic drop size fraction, reduced length, and dimensionless concentrations based on recovery fractions from equilibrium, these equations become, for the ith fraction of the dispersed phase
and for the continuous phase d2rc + Pe,-dr, dv2
dq
=
By introducing a dimensionless plug flow concentration of the continuous phase,
.
A-
the mass-transfer equation of the continuous phase can be broken in two equations, i.e., the actual extraction fraction gradient of the continuous phase
dr,/d.rl = Pe,(R, - r,)
(22)
and the plug flow extraction fraction gradient of the continuous phase
The boundary conditions are r& = rdm rc = rcout or dr,/dq = 0 a t q = 0
(24)
and
With the initial boundary conditions, these first-order ordinary differential equations can be integrated until the final bouundary conditions are satisfied and predict the
667
number of stages required for the given extent of extraction.
Calculation Procedure for the Transition Drop Size 1. Calculate an average drop diameter by using a correlation developed by Boyadzhiev and Spassov (1982), eq 11. 2. Drops form a Weibull distribution for which the ratio ~ / d 3 2is taken as 0.30. The values of cr and d32 allow complete characterization of the distribution. 3. A specific protocol was chosen for the finite representation of the drop size distribution. (a) The number of drop size compartments is chosen as 25 for all the calculations. Nearly identical results of the computer solution were obtained as the number of compartments was varied between 20 and 60 by Chartres and Korchinsky (1975). (b) The compartment width is specific to allow 12.5 compartments between 0 and da2. (c) Each drop size compartment is the same width and is represented in the transport equations by the median drop size of that compartment. (d) Terminal velocities are computed for each drop size fraction by using eq 5-8. For a few cases, the terminal velocity of the drops in the smallest drop size fraction was negative. When that occurred, the volume fraction of that compartment was added to that of the next larger compartment. This procedure was repeated if necessary. A total of 25 compartments were retained by increasing the upper size limit. (e) Since the distribution has no finite largest size, the largest drop size compartment would have no median. The representative drop size for that compartment is taken as one compartment width larger than the second largest drop size. (f) Each entire drop size compartment is assumed to act the same as the representative drop size. All drops in the compartment have the same velocity, concentration, mass-transfer rates, etc. (g) The holdup of the dispersed phase is calculated from the new correlation developed for this work (Appendix A). (h) The constant c in eq 1or 4 is determined. For a theoretical mass balance and with an absence of drop interactions, c would be unity. In practice, c varies. In this work, the calculated values of c were between 0.7 and 1.5 for most cases. For some experiments, c was found to be as low as 0.38 and for others larger than 4. The variation in c is the result of such factors as the deviations in predicted holdup, the Sauter mean drop diameter, the shape of the drop size distribution assumed, and many others. (i) The range of transition drop sizes is represented by a single drop size, d,. 6)The overall mass-transfer coefficient was determined for each drop compartment by using eq 12-16. 4. The longitudinal mixing coefficient for use in eq 21 and 22 was calculated for the continuous phase by the new correlation developed for this work (Appendix B). Axial mixing is a function of operating conditions and column geometry. 5. The 27 simultaneous firsborder differential equations (17), (22), and (23) were integrated. The concentration profile through the column for each drop size compartment was obtained from this computation. The IMSL subroutine, DGEAR, was used which is an implicit multistep type for stiff equations (Hindmarsh, 1974; Gear, 1971). It was found that frequently the system of equations was "stiff". 6. With all the initial boundary conditions known, the marching solution was halted when the desired outlet concentration was reached and the number of stages determined. Determination of the transition drop size was accomplished by locating the two adjacent transition drop size
668
Ind. Eng. Chem. Process Des. Dev., Vol. 25, No. 3, 1986
compartments that gave too many and too few stages. The transition size to be used in a correlation was taken as the linear interpolation between these two adjacent compartments. This search was conducted interactively (manually); experience with the calculations allowed rapid determination of these sizes.
. / d
$ 1 0.14
K
Transition Drop Size and Kolmogorov’s Length For droplets in a typical pulsed-plate column, there is a range of drop sizes for which the mass-transfer mechanism gradually changes. For design purposes, a single representative drop size is proposed called the transition drop size, dtr. In the following, a relationship is derived for the transition drop size determined by the procedure described above, as a function of the operating conditions, column geometry, and the properties of the liquid systems. The calculation of the transition drop size is based on the rate of the dissipation of the mechanical energy over the column of the flow system for incompressible Newtonian flow (Bird et al., 1960).
0.16
0.12
4 f I-
0.10
0.08
W
O.O2C’,
I
I
,
I
I
0.05
I
I
I
I
I
,
0.10
CORRELATED
TRANSITION DROP S I Z E ,
I
1
I
*
0.15
cm.
Figure 1. Prediction of transition drop size using eq 36.
The quantities on the right-hand side can be made dimensionless by introducing a characteristic velocity uo and a characteristic length Lo,and the equation can be changed to
=
--I*.’ dVo = S(27) Lo Re LO uO3 1
uo3
where e, is the rate of energy dissipation per unit mass, and S is a dimensionlessfunction of the Reynolds number and geometrical ratios. The uo can be considered as the eddy velocity, and Lo can be considered as the eddy length in a turbulent flow field. At certain eddy length Lo,the Reynolds number is found to be approximately unity.
(LOE~)’/~LOP~
R e = -u&Opc PC
s1/3pc
-1
(28)
Thus, a characteristic length, Lk, known as the Kolmogorov’s length is defined as (Sprow, 1967)
Turbulent flow past a drop will cause deformation of the drop at a Reynolds number of approximately 10 and will not cause deformation of the drop at a Reynolds number less than unity (Levich, 1962). Thus, we may postulate that the transition Reynolds number at which mass transfer switches from one mechanism to another is some value above unity and of the order of 10. Retr = UotrdtrPc/Pc (30) The energy dissipation rate per unit mass, E,, is a constant for any size of eddy in a homogeneous isotropic turbulent field. Thus, t, for the eddy length of d, can be expressed as
= Sl(uotr3/dtr) (31) where SI is a dimensionless function of the Reynolds number and various geometrical ratios. Substituting the uotI in the expression for Retr and rearranging gives Em
1
J
Thus,a form of the correlation between the transition drop size and the Kolmogorov’s length can be expressed as d t z / L k = s2 (33) S2 is a dimensionless function of operating conditions, geometry, and system properties. The energy dissipation rate per unit mass in the pulsed-plate columns is needed in eq 29 to evaluate the Kolmogorov’s length. If, in eq 31, the eddy velocity and length are replaced by the pulse intensity and height of the stage, respectively, the energy dissipation rate will be approximately
Pi3/h, (34) The complete expression can be derived as (Miyauchi and Oya, 1965) e,,
N
z,, = 9.5Pi3/Bhh,
(35)
Correlation of Transition Drop Diameter with Kolmogorov’s Length The exact observed number of stages could be calculated for 188 sets of mass-transfer data out of 248 seta of data by using the column model and appropriate transition diameters. The ratios of these diameters to Kolmogorov’s length, d,,/Lk, vary from 6 to 35 for those data. Levich (1962) indicated that the drop deformation started at the ratio of approximately 10. These data were examined to find an empirical relationship between the transition drop size, d,, and the Kolmogorov’s length, Lk. The relation was found as
(36) where the first group of variables on the right-hand size of eq 36 is a modified Froude number. The second is the ratio of the Reynolds number to the Weber number which we call the Luecke-Tung number. It is a measure of interfacial tension forces to viscous forces due to pulsation. The correlation result shown on Figure 1has an average deviation of 9.1% for the d,. In this figure, the “experimental transition drop size” is that obtained from the experimental data with the calculation procedure described earlier; the “correlated transition size” is from the correlation equation (36).
Ind. Eng. Chem. Process Des. Dev., Vol. 25, No. 3, 1986 669 Table I. Data with Extremely Large and Small Values of c Calculated column geometry
operating cond.
values of
fluid system toluene-acetone-water
4
h,
dh
4
Pi
vc
vd
C
7.32
5.08
0.318
0.621
3.81
0.190 0.192 0.193 0.197 0.422
CC4-iodine-water
5.0
5.0
0.2
0.188
6.0
0.566
0.392 0.550 0.730 0.897 0.178 0.028
3.84 3.73 3.64 3.58 4.14 0.43 0.42 0.40 0.39 0.38 0.39 0.38
0.042
6.0 7.5 7.5 7.5 8.0 8.0
9
0.014 0.028 0.042 0.014 0.028
I.0Or. I
c.
W (I
I
*
a
3
. 2
.
V
4 0 '1. 20
n'
W
e/.
(I
3
I-U
d W 0
K
K
it z
* .
-1.00
3
ki
-40%
*'
t
1
-20%
-
n
w
-0.8 I
0.2
1
0.4 MOLDUP OF D I S P E R S E 0 PHASE
I
I
0.6
Figure 2. Plot of the ratio of the predicted stage number to the actual stage number as a function of holdup.
The correlation may be used to calculate the transition diameters separating the type of mass transfer for each set of data. The model was used to compute the number of stages for those 188 sets of data. The average error of prediction of the stage number is 17.9%. While the correlated results might be improved if three mass-transfer models were used instead of two, the data were not sufficiently precise to warrant this addition.
Correction of Mass-Transfer Data at Higher Holdup Drop interactions increase as the holdup increases, and under these conditions, many assumptions of the column model are no longer valid. Each drop does not maintain its identity through the column, and the overall masstransfer interface area may be overestimated. As shown in Figure 2, the column model predicts higher than observed extraction efficiencies at holdup values above 0.15. The phenomenon of drop interactions has not been completely studied, and in any case, such relations are exceedingly complex. An empirical correlation was made for the systematic distortion of the predicted results at higher holdups. The holdup distortion correction is given as A In (Pn/R,) = 0.174 - 2.124 (37) (38) In (Pn)ch = ln (Pn)pr - A In (Pn/RJ After the correction, the average error of the predicted stage number is 19.9% for all 248 sets of data. The results, shown in Figure 3, corrected the predictions of higher holdup but at the expense of causing slightly higher scatter a t lower holdup.
2
0.0
I
0.2
I
I
0.4
0.6
HOLDUP OF TME DISPERSED PHASE
Figure 3. Plot of corrected prediction for both interfacial turbulence and holdup.
Correction for Interfacial Convective Instability According to the Marangoni principle, when the solute is transferred out of phase with lower solute diffusivity, or out of phase with higher viscosity, interfacial convective instability is usually promoted (Sternling and Scriven, 1959; Hanson, 1971). When using the correlation of d,/Lk obtained from the interfacial stable system for the interfacial convective unstable systems, the predictions indicated more stages than were actually required. Thus, an empirical correction was introduced for the case of interfacial convective unstable systems, i.e., systems where the ratio of higher diffusivity to lower diffusivity exceeds 1.30. The best average correction was found as (Pn)chc = (pn)ch/1*87 (39) The predictions that include corrections for both instability and holdup have an average error of 19.2% in the predicted stage number for 299 sets of mass-transfer data. Discussion of Results Although we have used improved correlations for holdup and axial mixing, the remaining inaccuracies contribute some error to the stage predictions. The holdup varies along the length of the column (Bell and Babb, 1969) and even radially over the cross section of the larger columns (Rouyer et al., 1974). We have used uniform overall holdup in the formulation of the model. Improvements in correlations for other parameters may also decrease prediction scatter. For example, the simple Handlos-Baron model for mass transfer (eq 15) was reported by Brunson and Wellek (1970) to be less accurate than the model of Skelland and Wellek (1964). This
670
Ind. Eng. Chem. Process Des. Dev., Vol. 25, No. 3, 1986
''OOOf
SYMBOLS FOR M A S S T R A N S F E R 0
: d -
e
C - J
A
NO
DIRECTION
/
C
d
MASS T R A N S F E R
%0 0.100 -
1
-1
0 z 0
w
a >
Table 11. Range of Variables for Data Included in This Correlation column diameter, cm 5-30 plate spacing, cm 3.75-10 plate hole diameter, cm 0.158-0.5 plate free area fraction 0.082-0.23 pulse intensity superficial velocity, cm/s continuous phase 0.0579-1.01 dispersed phase 0.051-0.605 e,, erg/(s cm2) 62.8-3625 organic (dispersed) phase MIBK, hexane, 1,1,2-trichloroethane, benzene, kerosene, carbon tetrachloride, trichloroethylene, and toluene
'C
lo
0.001
0.001
0.010 CALCULATED
0.100
1.000
t
HOLDUP
Figure 4. Prediction of holdup data using the correlation equations (A-3) and (A-4).
coefficient is seldom the limiting transfer mechanism, however. Our model uses a single transition drop size that separates the two mass-transfer mechanisms of drops. With the wide drop size distribution in a pulsed-plate column, it is expected that all three types of drop models, i.e., rigid, laminar internal circulation, and turbulent internal circulation or oscillating, must be present. More than two drop mass-transfer mechanisms could be modeled, but available mass-transfer data did not appear to be precise enough to support physically meaningful results. The laminar internal circulation drop model was omitted in this study. The fixed ratio, 0.30 of a/d32used in the calculations gives a wide drop size distribution at low pulsation conditions and narrow drop size distribution at high pulsation conditions. Nonetheless, the ratio a/d32is probably not constant over the range of operating conditions. Drop size distributions have significant influence on the predicted column efficiency (Chartres and Korchinsky, 1975; Korchinsky and Cruz-Pinto, 1979). Lack of detailed experimental information about the parameters of distribution, however, prevents a meaningful correlation of this ratio.
Conclusion This paper presents a correlation for a "transition drop size". This is the drop size at which the mass-transfer mechanism of a drop shifts because of a shift in the character of circulation and turbulence in the drop. For smaller drops, mass transfer is assumed to be diffusional, with the drop being a rigid sphere with little internal macrocirculation. For larger drops, mass transfer is calculated on the basis of turbulent internal circulation. The other calculations and design correlations discussed, while mathematically complicated, are merely extensions of simple and well-known concepts. We found that to achieve a reasonable level of accuracy, new and improved empirical correlations were needed for drop holdup and for axial mixing (see Appendex sections A and B). These correlations are not central to the development, however, and if better methods were found to predict these quantities, our model would be affected only by changes in the numerical values of the parameters in the prediction equation (36). Ideally, the mass-transfer mechanism (and the transition diameters) would be found by some clever direct measurement. This has not been done, and therefore we have
t
u
w
MEASURING METHOD A : DYNAMIC o : STEADY STATE : UNKNOWN
'i
6 -
O
w
>
-
K W
: 0
4 -
0
2 CALCULATED
4
E, , c m
6 ' 1 s
Figure 5. Prediction of longitudinal mixing coefficient of the continuous phase using eq B-3.
inferred the transition from the overall mass-transfer data as the residual effect after accounting for all known effects. Of course, considering the nature of the process, scatter could be expected. Our model in its present form represents a practical design tool. The prediction that was achieved is based only on accessible measurements of the system and liquid properties. We think our approach represents a new and important improvement in model accuracy.
Acknowledgment We thank Dr. D. H. Logsdail for kindly providing us with some unpublished data. Nomenclature A = pulse amplitude, cm B, = defined as F:/[(1 - FJ(1 - F:)], dimensionless c = constant, defined as eq 1, dimensionless C, = concentrations of the continuous phase, g/cm3 C,, = inlet concentration of the continuous phase, g/cm3 CCh*= concentration of the continuous phase in equilibrium with the inlet concentration of the dispersed phase, g/cm3 Cd = concentration of the dispersed phase, g/cm3 c d , = concentration of drops in the ith compartment, g/cm3 Cdin = inlet concentration of the dispersed phase, g/cm3 CdinT = concentration of the dispersed phase in equilibrium wlth the inlet concentration of the continuous phase, g/cm3 d, = critical diameter of a drop, at the transition from one velocity region to another, cm dh = plate hole diameter, cm d, = representative drop diameter in the ith compartment, cm d,, = transition drop size, cm
Ind. Eng. Chem. Process Des. Dev., Vol. 25, No. 3, 1986 671
d32 = Sauter mean drop diameter, cm Df, = diffusivity of solute in the continuous phase, cm2/s Dfd = diffusivity of solute in the dispersed phase, cm2/s D, = diameter of column, cm E, = longitudinal mixing coefficient of the continuous phase, cmz/s f = pulse frequency, s-l f i = volume fraction of the ith drop compartment, dimensionless fni = number fraction of the ith drop compartment, dimensionless ft = transition frequency from mixer-settler to emulsion, s-' Fa = plate free area fraction, dimensionless g = acceleration due to gravity, cm/s2 gi = kinetic drop volume fraction = fivdi/v d , dimensionless h, = height of column, cm h, = plate spacing or height of stage, cm k, = mass-transfer coefficient in the continuous phase, cm/s kd = mass-transfer coefficient in the dispersed phase, cm/s Kd -= overall mass-transfer coefficient based on the dispersed phase, cm/s Kdi = overall mass-transfer coefficient based on the ith drop compartment, cm/s Lk = Kolmogorov's length, cm Lo = characteristic length in the turbulent field, cm m = distribution coefficient, Cd*/Cc, dimensionless n = total number of stages, dimensionless Pe, = Peclet number for the continuous phase based on the stage height, V,h,/E,, dimensionless Pi = product of frequency and amplitude, cm/s P, = predicted stage number, dimensionless r, = fraction of solute recovered in the continuous phase, [(C, - C,) / (Ccin- Ccin*)],dimensionless rd = fraction of solute recovered in the dispersed phase, [(Cdin - Cd)/ ( Cdin - Cdin*)], dimensionless R, = fraction of solute recovered in the continuous phase based on the plug flow, dimensionless Re = Reynolds number, dimensionless Retr = transition Reynolds number, dimensionless R, = observed stage number, dimensionless S, S1, Sz = functions, dimensionless Sc = Schmidt number, dimensionless uo = characteristic velocity in a turbulent flow field, cm/s uOtr= characteristic velocity for a transition size, cm/s V = volume, cm3 VO = defined as V/L,3, dimensionless V, = superficial velocity of the continuous phase, cm/s v d = superficial velocity of the dispersed phase, cm/s v d i = drop velocity, cm/s V, = slip velocity, cm/s Vti = drop terminal velocity, cm/s Greek Letters r = gamma function, dimensionless y = interfacial tension, dyn/cm c = rate of energy dissipation over the volume of the entire flow system, erg/s cm = rate of energy dissipation per unit mass, erg/(s g) { = parameter of Weibull distribution, dimensionless 9 = reduced height based on stage height, dimensionless p, = viscosity of the continuous phase, g/(cm s) wd = viscosity of the dispersed phase, g/(cm s) Ap = density difference, g/cm3 p, = density of the continuous phase, g/cm3 u2 = variance of drop size distribution, cm2 qv= dissipation function, qvo = defined as \k,(Lo/uo)2,dimensionless C#I = holdup fraction of the dispersed phase, dimensionless w = parameter of Weibull distribution, dimensionless Superscripts = dimensionless quantity * = equilibrium - = average
Subscripts c = continuous phase ch = correction for holdup chc = correction for holdup and convection d = dispersed phase i = ith compartment in = in stream o = overall characteristic out = out stream pr = predicted tr = transition m = free stream Appendix A. Holdup of the Dispersed Phase in Pulsed-Plate Extraction Columns. The dispersed-phase holdup is a major factor affecting the mass-transfer area in countercurrent extraction columns. An accurate model of the pulsed-plate extraction column must be based on good correlations of holdup with column parameters and operating conditions. For the pulsed-plate extraction column, there are a number of existing correlations for the holdup of the dispersed phase, but none is satisfactory for general column geometry, or with general liquid properties. In this work, we present improved correlations for holdup of the dispersed phase for the pulsed-plate extraction column. While the correlations are developed in terms of the liquid properties of the dispersed phase, the continuous phase is water in all cases. Very little data are available for the nonaqueous continuous phase. Operating Characteristics of the Columns. Two distinct types of phase dispersion may occur during normal operation of a pulsed-plate extraction column: the mixer-settler and emulsion regions. The mixer-settler type which occurs at low pulsation intensity is characterized by the separation of the dispersed and continuous phases into discrete and clear layers during the quiescent portions of a pulse cycle. At higher pulsation intensity, the dispersion gradually becomes the emulsion type which is characterized by small drops, with fairly uniform dispersion of the dispersed phase, and with little change in phase dispersion in the course of the pulse cycle. For the pulsed-plate extraction column, the region of industrial interest is the emulsion region with intensive pulsation. Data chosen for analysis in this work were from intensive pulsation operation only. At extremely intensive pulsation, an unstable dispersion will result (Sege and Woodfield, 1954) which consists of the mixing of small and large drops, formation of irregular-shaped large globules of the dispersed phase, and periodic phase reversals in a short section of the column. This conditions is seldom observed within usual operating ranges and does not occur in the data base used in this work. In order to predict the holdup in intensive pulsation conditions, the general data selection criteria used here to define when these conditions exist follow Boyadzhiev and Spassov (1982). A pulsation parameter em was defined as the lowest limit of the maximal local kinetic energy dissipated into the liquid per unit plate area: Intensive pulsation (emulsion region) is assumed when the parameter e, is greater than 60 erg/(s cm2). Results of other investigators tend to support this decision (see: Tung, 1984). Data Sources. The holdup data were selected from nine different groups of investigatrors: Sehmel and Babb (1963), Bell and Babb (1969), Khemangkorn et al. (1977, 19781, Arthayukti et al. (19761, Cohen and Beyer (1953),
672
Ind. Eng. Chem. Process Des. Dev., Vol. 25, No. 3, 1986
Niebuhr and Vogelpohl(1980), Reissinger et al. (1981), and Biery (1961). Each of these investigators conducted experiments at a single column diameter. The column diameter for different workers varied from 2.54 to 7.6 cm. The plate spacing was from 5.0 to 10 cm. The diameter of the plate hole was from 0.102 to 0.318 cm. The plate free area fraction was from 0.09 to 0.23. Each of investigators conducted their experiments for only a single liquid system with the exceptions of Sehmel and Babb (1963) and Bell and Babb (1969), who used three and two liquid system, respectively. The criterion, t, > 60 eg/(s cm2), was used to determine that the system was within intensive pulsation conditions. Water was the continuous phase for all the systems. The dispersed solvents included hexane, benzene, MIBK, carbon tetrachloride, isoamyl alcohol, toluene, and 50% TBP in kerosene. The pulse intensities, i.e., the products of the pulse amplitude and frequency, ranged from 1.224 to 8.7 cm/s; the superficial velocities of the continuous and dispersed phases ranged from 0.098 to 0.566 and from 0.0142 and 0.576 cm/s, respectively. The t, range was from61 to 8231 erg/(s cm2). Only the data of Khemangkorn et al. (1977,1978), Cohen and Beyer (19531, and Biery (1961) involved mass transfer during the measurement of the holdup data. The total number of data points is 240, including 65 data with mass transfer. Evaluation of Existing Correlations. The holdup correlations of Thornton (1957), Miyauchi and Oya (1965), and Burkhart (1979) and Mia0 (1978) were tested with the holdup data base described above. With the reported coefficients, there was a great amount of scatter in these three correlations. The average errors of prediction on the data set were 42.4, 140, and 127%,respectively. The holdup correlation of Burkhart (1979) and Mia0 (1978) was moderately efficient for those holdup data with higher values but predicted poorly a t lower holdups. Similar results were obtained with the correlation of Miyauchi and Oya (1965). None of the existing correlations were really satisfactory for the holdup data set used in this work. A Revised Correlation. Changes were made to the basic form of the correlation of Miyauchi and Oya (1965). The group variable, 4, which includes parameters for pulse intensity and column characteristics, as well as for liquid properties, was separated into two groups. One contained only the column and pulse characteristics; the other contained only the liquid properties. In addition, the exponent on the holdup term in the correlation was made a free parameter. The generalized least-squares algorithm of Britt and Luecke (1973) was used to determine the coefficients. This algorithm is an extension of ordinary regression and finds the maximum likelihood estimate for parameters when independent variables, as well as the dependent variables, are subject to error. Correlation of the 240 data points yielded the equation
The effect of V, on the holdup was not significant at 0.05 probability level. The average error of prediction on the data set is 32.4%. Mass-Transfer Effect on the Holdup. Equation A-2 tends to predict the holdup too high when mass transfer occurs. Thornton (1957) had indicated that the values of the characteristic velocity for the mass transfer from the dispersed phase to the continuous phase were higher (and the holdup lower) than those without mass transfer. The
work of Logsdail and Thornton (1957) showed that the holdup with mass transfer from the dispersed to the continuous phase was lower than it was without mass transfer. Hanson (1971) suggested that the Marringoni effect induces coalescence of drops in the mass-transfer direction of the dispersed phase to the continuous phase. The coalescence produces large drops which have shorter residence times, and the holdup is decreased. It is likely that the holdup should be a continuous decreasing function of mass-transfer rate and would be different for different mass-transfer directions. Insufficient data are available to determine the form of such a function or the magnitude of the difference. On the other hand, the experimental observations of Khemangkorn et al. (1977) showed that the holdup was larger when the mass transfer was from the dispersed phase to the continuous phase than in the opposite direction. In these low holdup conditions, the factor determining the drop size and thus the holdup is not coalescence but solute concentration. The solute concentrations lowers the interfacial tension; thus, smaller drops are produced with mass transfer than without. Since there is little coalescence at these conditions, there is, thus, increased holdup for this data set with mass transfer in both directions. To account for the effect of mass transfer in both directions on the holdup, a single factor of 1.579 was incorporated for the decrease in holdup that occurs when mass transfer occurs in either direction (Figure 4). 4 = 0.9785vd1~0843[ 1.8986 0.3557 (A-3)
“-1
(”)
(B,h,)1/3 with mass transfer, and
-1
4 = l.545Vd1.OM3[ Pi (B,h,) 1/3
YAP
(”)
1.8986
YAP
0.3557
(-4-4)
without mass transfer. For this correlation, the average error of prediction is 28.9%. The F test supported the holdup factor at the 0.05 level of significance. This correlation showed that the two subgroups formed from the variable of Miyauchi and Oya (1965) presented quite different effects on holdup. The effect of superficial velocity of the continuous phase on the holdup was found to be statistically insignificant. The column diameter was not incorporated into the correlation in our work, since the column diameters in our data varied only from 2.54 to 7.6 cm. Additional data are needed for columns of large diameter (Garg and Pratt, 1981). B. Longitudinal Mixing in Pulsed-Plate Extraction Columns. Longitudinal mixing impairs the performance of the extraction columns. It decreases the mass-transfer driving force by formation of concentration jumps at stream inlets and by longitudinal bulk transport of solute. This decrease in mass transport driving force due to the longitudinal mixing is reported to account for 60-75% of the effective height of production size extraction columns (Hanson, 1971). There have been four previous major investigations that resulted in correlations of the longitudinal mixing coefficient of the continuousphase in the pulsed-plate extraction columns. They were Mar and Babb (1959) and Sehmel and Babb (1964), Miyauchi and Vermeulen (1963) and Miyauchi and Oya (19651, Kagan et al. (1965, 1973), and Mia0 (1978). The first two of these correlations were for a relatively narrow range of conditions. In fact, Sehmel and Babb (1964) used equal superficial velocities in both phases.
Ind. Eng. Chem. Process Des. Dev., Vol. 25, No. 3, 1986 673
Some of the reported dependencies of variables are opposite to what would be expected and what has been found over larger ranges in other studies. The correlations of Miyauchi and Oya (19651, Mia0 (1978), and Kagan et al. (1973) were tested against the data selected in this work. I t was found that the results were very scattered for all these correlations. The average errors were 207, 128, and 152% for Miyauchi and Oya (1968, Mia0 (1978), and Kagan et al. (1973), respectively. Since part of the error is due to bias, the parameters in the latter two correlations were refitted to the data used in this work. The results were still highly scattered with average errors of 54.7 and 53.6% for Miyauchi and Oya (1965) and Kagan et al. (1973), respectively. Source Data. The longitudinal mixing data correlated in this work is only for two-phase countercurrent flow. In order to limit the correlation to intensive pulsation conditions, a criterion is used based on the energy input (Tung, 1984; Appendix A). ,E
=
-> 60 erg/(s 2F,2
cm2)
03-1)
The longitudinal mixing data (see Table 11)were from Smoot and Babb (1962), Sehmel and Babb (1964), Kagan et al. (1965, 1973), Rozen et al. (1970), Reissinger et al. (1981), and Niebuhr and Vogelpohl(1980). Each of these investigators except Kagan et al. (1973) conducted their experiments in a single diameter column. Development of a New Correlation. Dimensional analysis was used to develop a new correlation for this data. We postulated that the longitudinal mixing coefficient was a function of the following variables: D,, h,, dh,Fa, Pi,V,, Vd, &, Ap, and 4. The properties of the continuous phase were excluded from the dimensional analysis because the continuous phase was always water for all measurements. The plate thickness was omitted because it had showed a negligible effect in the correlation of Mar and Babb (1959). In addition, plate thicknesses were not reported in some data. Many combinations of groups were considered, and the final form of the relationship was
-E-, - 0.0478F;1.21g( h, VC
e)
0.687
(B-2)
The average error for this equation was 24.8%. An improvement in eq B-2 can be obtained by addition of another dimensionless group:
This equation has an average error of 21.4%. It is a statistically significant improvement over eq B-2 at the 5% confidence level. The effect of column diameter was found to be not statistically significant within this data base. Some investigators have reported that the pulsed-column diameter affects axial mixing (Miyauchi and Oya, 1965; Burkhart, 1979; Garg and Pratt, 1981); others have reported no or small effect (Kagan et al., 1973; Rouyer et al., 1974). Garg and Pratt (1981) cite an “urgent need for reliable experimental data on backmixing ratios for large columns”. It has been suggested that the data scatter results from the different methods of measurement of longitudinal mixing. Some measurement methods reportedly include Taylor diffusion while others do not (Sleicher, 1959). The
data points in Figure 5 are plotted with different symbols for the different methods. No systematic bias is evident except that the dynamic tracer injection method seems to scatter more widely than data from the steady-state tracer injection method. Thus, we may conclude that either Taylor diffusion was a small factor in these data or its presence affected all the measurements in a similar manner. Literature Cited Arthayukti, W.; Muratet, G.; Angeiino, H. Chem. Eng. Sci. 1978, 37, 1193. Bell, R. L.; Babb, A. L. Ind. Eng. Chem. Process D e s . Dev. 1989, 8, 2, 392. Biery, J. C. PhD Dissertation, Iowa State University, Ames, 1961. Bird, R. B.; Stewart. W. E.; Lightfoot, E. N. “Transport Phenomena”; Wiley: New York, 1960. Brm. H. I.; Luecke, R. H. Technomtrics 1973, 15 (2), 233. Brunson, R. J.; Wellek, R. M. Can. J. Chem. 1970, 48, 267. Boyadzhiev, L.; Spassov, M. Chem. Eng. Sci. 1982, 37 (2), 337. Burkhart, L. U. S. Department of Commerce, UCRL-15101, 1979. Chartres, R. H.; Korchinsky, W. J. Trans. Inst. Chem. Eng. 1975, 53, 247. Chen. H. T.; Middleman, S. AIChE J. 1987, 73, 989. Cohen, R. M.; Beyer, G. H. Chem. Eng. Prog. 1953, 49, 279. Cruz-Pinto, J. J. C.; Korchinsky. W. J. Chem. Eng. Sci. 1980, 35, 2213. Endoh, K.; Oyama, T. J. Sci. Res. Inst. 1958, 52. 131. Eguchl, W.; Nagata, S. Chem. Eng. Jpn. 1958, 22, 218. Garg, M. 0.; Pratt, H. R. C. Id. Eng. Chem. Process D e s . Dev. 1981, 20, 492-495. Garner, F. H.; Tayeban, M. Anal. Real Soc. €span. f i s . Quim 1980, 479. Gayler. R.; Roberts, N. W.; Pratt, H. R. C. Trans. Inst. Chem. Eng. 1953, 37,57. Gear, C. W. “Numerical Initial Value Problems In Ordlnary Differential Equations”; Prentice-Hall; Englewood Cliff, NJ, 197 1. Handlos, A. E.; Baron, T. AIChE J. 1957, 3 , 127. Hanson, C. “Recent Advances in Liquid-Liquid Extraction”; Pergamon Press: Oxford, 1971. Happel, J.; Byme, 8. J. Ind. €ng. Chem. 1954, 4 6 , 1181. Hlndmarsh, A. C. Lawrence Livermore Laboratory, Report UCID-30001, Revision 3. Dec 1974. Kagan, S. 2.; Aerov, M. E.; Lonlk, V.; Volkova, T. S. Int. Chem. Eng. 1985, 5, 656. Kaaan. S. 2.; Velsbein. B. A.; Trukhanov, V. G.; Muzychenko, L. A. Znt. Chem. Eng. 1973, 73, 217. Khemangkorn, V.; Molinier, J.; Anwlino, H. Chem. Eng. Sci. 1978, 33, 501. Khemangkorn, V.; Muratet, G.; Anaelino, H. Proc. Int. Solvent Exir. Conf. 1977,-21, 429. Klee, A. J.; Treybal, R. E. AIChE J. 1958, 7 , 444. Korchinsky, W. J.; Azimzadeh-Khatayloo, S. Chem. Eng. Sci. 1978, 3 1 , 871. Korchinsky, W. J.; Cruz-Pinto, J. J. C. Chem. Eng. Sci. 1979, 3 4 , 551. Levlch, V. G. “Physicochemical Hydrodynamlcs”; Prentice-Hall: Englewood Cliffs, NJ, 1962. Llnton, M.; Sutherland, K. L. Chem. Eng. Sci. 1980, 72, 214. Longsdall, D. H.; Thornton, J. D. Trans. Inst. Chem. Eng. 1957, 3 5 , 331. Mar, B. W.; Babb, A. L. Ind. Eng. Chem. 1959, 57, 1011. Mlao, Y. W. MS Thesis, Iowa State University, Ames, 1978. Misek. T.; Marek, J. B r . Chem. Eng. 1970, 75, 202. Miyauchl, T.; Oya, H. AIChE J. 1985. 7 7 , 395. Miyauchi, T.; Oya, H.; Kikuchi. T.; Hashizume, H.; Kagawa, K. Kagaku Kogaku 1987, 5, 108. Miyauchi, T.; Vermeulen. T. Ind. Eng. Chem. fundam. 1983, 2 , 304. Newman, A. B. Trans. AIChE1931, 27, 310. Niebuhr, D.; Vogelpohl, A. Chem.-hg.-Tech. 1980, 52, 61. Olney, R. B. AIChE J. 1984, IO, 627. Reissinger. K. H.;Schroter, J.; Backer, W. Chem.-Ing.-Tech. 1981, 53 (a), 607. Rod, V. Br. Chem. Eng. 1988, 1 7 , 463. Rouyer, H.; Lebouhellec, J.; Henry, E.; Michel, P. Proceedings of the International Solvent Extraction Conference, Lyon, 1974. Rozen, A. M.; Rubezhnyy. T.; Martynov, B. V. Sov. Chem. Ind., 1970, 2 , 66. Ryan, B. US Atomic Energy Commission Report HW-36056, 1955. SatO, C.; Sugihara, Taniyama, I. Chem. Eng. Jpn. 1983, 27, 583. Schweitzer, P. A. Handbook of Separation Techniques for Chemlcai Engineers“; McGraw-Hill: New York, 1979. Sehmel. G. A.; Babb, A. L. Ind. Eng. Chem. Process D e s . D e v . 1964, 3, 210. Sege, 0.; Woodfieid, F. W. Chem. Eng. Prog. 1954, 50, 396. Sleicher. C. A., Jr. AIChE J. 1959, 5 , 145. Skelland, A. H. P.; Wellek, R. M. AIChE J. 1984, 10, 491. Smoot, L. D.; Babb, A. L. Ind. Eng. Chem. fundam. 1982, 7 , 93. Sprow. F. B. Chem. Eng. Sci. 1987, 22, 435. Sternllng, C. V.; Scriven. L. E. AIChE J. 1959, 5 , 514. Thornton, J. D. Trans. Inst. Chem. Eng. 1957, 3 5 , 316. Treybal, R. E. “Liquid Extraction”, 2nd ed.; McGraw-Hill: New York, 1963. Tung, L. S. PhD Dissertation, University of Missouri, Columbia, 1984. Uno, S.; Kintner, R. C. AIChE J. 1958, 2 , 420.
-
1;;
Received for review August 22, 1984 Revised manuscript received November 5, 1985 Accepted November 14,1985
