Ind. Eng. Chem. Res. 1992,31,1362-1366
1362
Effect of a Variable Solute Distribution Coefficient on Mass Separation in Hollow Fibers Ane M. Urtiaga and Angel Irabien Department of Chemical Engineering, Facultad de Ciencias, Universidad del Pais Vasco, Apartado 644, 48080 Bilbao, Spain
Pieter Stroeve* Department of Chemical Engineering, University of California at Davis, Davis, California 95616
The effect of a variable solute distribution coefficient on mass transfer in a hollow fiber separator is analyzed. This problem arises when liquid membranes are utilized in porous hollow fibers for mass separation. An example is the removal of toxic phenolic solutes from water. A variable solute distribution coefficient in the liquid membrane can have a significant effect on the separation performance of a hollow fiber module if the wall Sherwood number is less than 10. A distribution coefficient which increases with solute concentration gives increased mass transfer, and hence better separation, in comparison to a constant distribution coefficient. Vice versa, a distribution coefficient which decreases with solute concentration gives reduced separation in a hollow fiber module, particularly in the inlet region of the module.
Introduction Membrane-based qoass separation processes are of considerable interest in the areas of dialysis, metal extraction, non dispersive solvent extraction, gas separation, and artificial oxygenation (Lonsdale, 1982;Cussler, 1986,Belfort, 1987). Recently there has been increasing interest in the use of supported liquid membranes as selective separation barriers. In this technique an organic liquid, immobilized in the pores of a microporous support, can be used to transfer a solute between two aqueous solutions (Noble and Way, 1987). Mass transfer is accomplished by diffusion of the organic solute through the organic solvent in the membrane. Mass-transfer rates attainable in membrane separation devices are limited by solute transport through the membrane. The wall Sherwood number characterizes solute transport through the membrane, and it is defined as the ratio of the mass-transfer resistance in the fluid to that in the membrane. In many practical cases, membranes used in mass separators may have a relatively low wall Sherwood number, which leads to a low mass-transfer rate of the solute. For liquid membranes, a relatively high Sherwood number can be obtained by using an appropriate liquid with a high distribution coefficient for the solute. Supported liquid membranes are an alternative separation method for the removal of pollutanta from industrial waste streams. For removal of toxic phenolic solutes from water, both polar and nonpolar solvents can be used as liquid membranes. Several authors have reported distribution coefficients for phenol between water and organic solvents (Kiezyck and MacKay, 1973; Abrams and Prausnitz, 1975;Won and Prausnitz, 1975). For many of the solvents, the distribution coefficient for phenol was dependent on the phenol concentration. The dependency of the distribution coefficient on the solute concentration is a factor that must be considered when predicting the performance of supported, liquid membrane, mass separators. Mathematical models for mass transfer in hollow fiber mass separation devices have made used of two different approaches. A solution for a fluid in plug flow has been given by Noble (1984)and Danesi (1984)for reactive separations. However, the use
* Author to whom correspondence should be addressed.
of small hollow fibers leads to fluids flowing in the laminar flow regime with a parabolic velocity profiie and Fickian diffusion in the direction perpendicular to flow. When the wall Sherwood number is constant, analytical solutions are available in the literature (Sideman et al., 1964/65;Davis and Parkinson, 1970). The use of'reactive membranes leads to nonlinear boundary conditions, and these problems have been solved numerically (Kim and Stroeve, 1988, 1989). An analytical approach to the problem of nonlinear boundary conditions has recently been given (Rudisill and Levan, 1990),but in order to use the method to predict the solute s'eparation, numerical calculations need to be made. In this paper we treat the problem of the variation of the distribution coefficient with the solute concentration and its effect on the mass-transfer rate in hollow fiber supported liquid membrane modules. The fluid to be treated flows in fully developed, one-dimensional laminar flow inside the lumen of the hollow fiber, and solute removal takes place by diffusion within the membrane wall. A second fluid, the stripping solution, circulates along the outside of the membrane and removes the solute. We apply the method given by Rudisill and Levan (1990)to solve the problem of the nonlinear boundary condition at the fiber surface. The effects of the physical parameters of the solute-solvent system on the mass-transfer rate are investigated. The results of this study can be used to provide design criteria for supported liquid membrane modules for the removal of solutes, such as phenol, from aqueous solutions. Theory
The supported liquid membrane module contains a large number of hollow fibers. It is sufficient to analyze the problem for one hollow fiber in order to predict the performance of the mass separator. A schematic diagram of the fiber is shown in Figure 1. A Newtonian fluid with the solute to be extracted flows within the fiber in fully developed, one-dimensionallaminar flow. The extractive membrane begins at z* = 0. A t this point the concentration of the solute is uniform and equal to C*+ The solute permeates through the liquid membrane by diffusion to react with a second fluid (the stripping solution) in the outer side of the fiber. The chemical reaction in the outer part of the fiber is instantaneous and makes the
0888-5885/92/2631-1362$03.00/00 1992 American Chemical Society
Ind. Eng. Chem. Res., Vol. 31, No. 5, 1992 1363 strlpping solution
.
Following Rudisill and Levan (19901, the solution to eqs 3-5 with eq 6 replaced by
t
c=o
C = C,(z), r = 1, all z
(8)
is
Here Cl(r,z-z? is the solution to eqs 3-5 with eq 6 replaced by
I
\
Supported Liquld Membrane Variable Distribution Coefficient
Impermeable Membrane
C=O,
Figure. 1. Diagram of the hollow fiber. The liquid membrane is supported in the porous walls of the hollow fiber.
concentration of the solute at the stripping phase side equal to zero. The equilibrium distribution coefficient H is defined as the equilibrium distribution ratio of the solute concentration in the liquid membrane to the concentration in the fluid side. In this work, as a first approximation, we assume a linear dependency of the distribution coefficient on the solute concentration in the aqueous phase (Urtiaga et al., 1990). The linear dependency of the distribution coefficient in dimensional form is given as H = ho h*C* (1)
r = l , allz
(10)
which is the Graetz problem for mass transfer with an infinite wall Sherwood number (Brown, 1960; Sideman, 1964/65). For the Graetz problem the solution to eq 3 can be obtained by separation of variables and is
where Y,(r) is the eigenfunction, A, is the eigenvalue, and A, is the associated constant. The values of A, are determined from
+
where ho is the value of the distribution coefficient for infinite dilute solutions, C* is the concentration, and h* is the slope of the distribution coefficient. Equation 1 can be made dimensionless so that
Substitution of eq 9 into eq 6 gives &'=
Sh,(l where y = C*ih*/hois the dimensionless slope. The parameter y takes into account the dependency of the solute distribution coefficient on the solute concentration. For a constant distribution coefficient y = 0. At steady state the mass conservation equation for the solute and the associated boundary conditions can be expressed in terms of dimensionless variables. Axial diffusion is neglected compared to axial convention. The equations are given by 2'1-?)aZ=;F[r(T)] ac 1 a B.C.l:
C=l
B.C.2:
ac -- 0 _ ar
B.C.3: -- = Sh,(l ar
z=O,
ac allr
r = O , allz
+ 7C)C
r = 1, all z
(3)
+ yC,)C,
(13)
where eq 13 is the Volterra integral. The constants Bnare expressed in terms of A, as follows:
The solution of eq 13 gives the concentration at the wall, C,(z). Equation 13 can be solved using a backward difference approximation for dC,(z?/dz! The derivative over any segment length Az is a constant and can be moved outaide the integral. The integral of the summation can be evaluated analytically. Equation 13 becomes
(4)
(5) (6)
where
k,SRho
Sh, = (7) D The third boundary condition imposes the continuity of solute flux across the membrane-fluid interface. The left-hand side of the equation givm the solute flux arriving at the wall from the lumen of the hollow fiber. The right-hand side of the equation accounts for the transport of the solute across the cylindrical wall by Fickian diffusion.
Starting at z = 0 where C, = 1, C, is calculated at each successive point. Once C,(z) has been determined, the concentration at any point in the channel can be obtained from eq 9. The dimensionless mixing-cup concentration is a more useful result than the local concentration field, and it can be obtained from J12C(r,z)(1
c,
- P)r dr (16)
=
x 1 2 ( 1 - r2)rdr
where C(r,z) is given by eq 9.
1364 Ind. Eng. Chem. Res., Vol. 31, No. 5, 1992 Table I. Comparison of the Numerical Results (N) to the Analytical Results (A) for the Case of a Constant Distribution Coefficient ( y = 0) and the Numerical Results for a Variable Distribution Coefficient dimensionless mixing-cup concentration Sh, = 1 Sh, = 0.1 constant H (y = 0) constant H (y = 0) A N Y = 0.1 r = l 2 A N y = 0.1 y=l 0.980048 0.982961 0.982 283 0.997739 0.995759 0.971118 0.01 0.998034 0.997937 0.859 620 0.978829 0.963544 0.860585 0.851 779 0.811 091 0.980597 0.1 0.980814 0.749 808 0.748813 0.738177 0.961899 0.958534 0.933780 0.682304 0.2 0.962185 0.486580 0.500057 0.499920 0.908104 0.900 555 0.840861 0.421381 0.5 0.908536 0.812 170 0.716369 0.255004 0.254408 0.245085 0.825084 1.0 0.825714 0.198930 0.660485 0.532879 0.066 316 0.066 080 0.063OOO 0.681121 2.0 0.682032 0.048622 '
The integrals in eq 16 can be evaluated using the properties of the Graetz functions. The mixing-cup concentration at zk, the axial point of interest, is obtained from
Results and Discussion A value of 0.004 or less was used for Az in the calculations for z greater than 0.1. At the inlet region where z 0, it is necessary to set the value of the step size at 2 X lo4 because very steep changes in the concentration at the wall are observed. As the axial distance increases, step sizes larger than 2 X can be used. The concentration at the wall is calculated using eq 15. Starting at z = 0 where C = 1,the only unknown at each successive step is C,, which is solved for. Finally, eq 17 is used to calculate the mixing-cup concentration at each dimensionless axial distance. Evaluation of the concentration at the wall through eq 15 requires the use of the eigenvalues An and coefficients B, obtained from the Graetz problem. These values were calculated using the procedure given by Sideman et al. (1964/65). Up to 20 eigenvalues were calculated, and these were coincident to within the tenth significant figure with the eigenvalues given by Brown (1960). In calculating the mixing-cup concentration we compared the technique of Rudisill and Levan (1990) for the case of a constant distribution coefficient (y = 0) to results from a known analyticalsolution. Sideman et al. (1964/65) have given analytical solutions for mass transfer in hollow fibersfor a constant distxibution Coefficient. Table I shows the dimensionless mixing-cup concentration for wall Sherwood numbers of 0.1 and 1for a constant distribution coefficient and variable distribution coefficients (y = 0.1 and y = 1). The analytical results from the method of Sideman et al. (1964/65) and the numerical calculations using the method of Rudisill and Levan (1990) are compared to six significant figures. When 20 eigenvalues are used, the two methods show agreement for the case of y = 0 to three significant figures only. The analyticalmethod is exact, and the differences beyond the third significant figure are due to the fact that the Rudisill and Levan technique employs the wall concentration to obtain the solution for the mixing-cup concentration. The wall concentration is a steep function of z, and the numerical errors in determining this steep function are responsible for the differences observed in Table I. Decreasing the step size Az below 2 X does not increase the accuracy. Thus, the results for a variable distribution coefficient are ac-
-
curate to only three significant figures. Fortunately the accuracy is sufficient for practical applications of determining the mixing-cup concentration versus dimensionless length. Variable distribution coefficients are encountered in phenol removal form water in hollow fiber mass separators with supported liquid membranes (Urtiaga et al., 1990; Urtiaga, 1991). For example, the distribution coefficient of phenol between kerosene and water at 25 "C is only a weak function of phenol concentration with y = 0.014 (Urtiaga, 1991). However, the dissolution of only 1% by volume of methyl isobutyl ketone into kerosene causes both a significant increase of the phenol distribution coefficient, between the kerosene/methyl isobutyl ketone mixture and water, and an increase of y. In this case y = 1.16 (Urtiaga, 1991). The effect of the change in the distribution coefficient on the mixing-cup concentration with axial distance is shown in parts a, b, and c of Figure 2 for wall Sherwood numbers of 0.1, 1, and 10, respectively. The dashed lines are for negative values of the slope y and denote a decreasing distribution coefficient with increasing solute concentration. For this case the lowest value of the distribution coefficient is found at the inlet of the membrane module, where the solute concentration is at its maximum value. The solid lines are for positive values of y, where the distribution coefficient increases with increasing solute concentration. In this case the maximum value of the distribution coefficient occurs at the inlet region. The variation of the dimensionless mixing-cup concentration with z for Sh, = 0.1 as a function of y is given in Figure 2a. For the constant distribution coefficient (y = 0) the change in the dimensionless mixing-cup concentration is reduced to 68% of the initial concentration for an axial distance of z = 2. When y = 0.1, a small improvement in mass separation is observed. With increasing y the separation of solute from the flowing fluid in the lumen of the hollow fiber is more effective. For y = 1only 53% of the initial solute concentration remains at z = 2. When y = 100 the limiting case of Sh, = for a constant distribution coefficient is approached. The effect of negative slopes of the distribution coefficient on mass separation is opposite to that of positive slopes. The loss of solute in the inner fluid reduces the distribution coefficient at axial distances from the inlet point, and separation becomes less effective. Figure 2b shows the results for Sh, = 1. In this case the effect of the slope of the distribution coefficient on mass separation is reduced compared to Sh, = 1. For y = 0.1 the variation of the dimensionless mixing-cup concentration with the dimensionless axial distance is very close to the case of a constant distribution coefficient. However, when y = 1one can observe a significant improvement in the solute separation compared to 7 = 0. For certain values of y an unusual behavior of the C, versus z curve is observed. When the slope of the distribution coefficient
Ind. Eng. Chem. Res., Vol. 31, No. 5, 1992 1366 1 .o
E
0
2 0.8
P d
0.6
W
u
z
0
0.4 n 3 0 I
p- 0.2
-.
x
I
0.0 0:0
0.5
1.o
1.5
2.0
AXIAL DISTANCE, z (a) Shw=O.l 1.o
E
u
Conclusions A variable distribution coefficient can have a significant effect on the mass transfer in hollow fiber mass separation devices if the wall Sherwood number is less than 10 and y is greater than 0.1 From experimental data on distribution coefficients for phenol between water and organic solvents (Kiezyck and MacKay, 1973; Abrams and Prausnitz, 1975; Won and Prausnitz, 1975; Urtiaga et al. 1990, Urtiaga, 19911, wall Sherwood numbers of order 1 and dimensionless slopes of the distribution coefficient, y, of order 1are typical. In order to predict the performance of a hollow fiber separation device or to design mass separation devices, the variation of the distribution coefficient must be included. The technique of Rudisill and Levan (1990) can give predictions accurate to three significant figures in the mixing-cup concentration.
2 0.8 o_ I-
d
2 0.6 w
u
z
0
0.4
n 3
u I
p - 0.2
xI
0.0 1 .o
1 .o
gives rise to a higher distribution coefficient and therefore improved solute separation. This phenomenon also occura in Figure 2a but a t much larger z than shown. When the wall Sherwood number is large (Sh, = lo), the change in the dimensionless mixing-cup concentration with the dimensionless axial distance approaches the change for infinite wall Sherwood number, which is shown in Figure 2c. For a value of y = 1 the mixing-cup concentration behaves almost identically to that obtained for the negative value of y = -0.9. The case of a constant distribution coefficient (y = 0) falls in between the two lines (not shown). The use of higher values of y gives variations of the mixing-cup concentration with the axial distance closer to those obtained for Sh, = From eq 15 it is clear that the effect of a variable distribution coefficient is determined not only by y but also by the value of the concentration a t the wall C ,. The contribution of the Sh,yCW2 term becomes negligible relative to the SkC, term when C, approaches zero. Thus at sufficiently larger axial distances where C, is small,the effect of y on mass transfer becomes less significant.
,
1.5
AXIAL DISTANCE, z (b) Shw=l
1
Acknowledgment This work was supported by the Spanish Education Ministry under a FPI Research Grant to A.N.U., and a University of California Faculty Research Grant to P.S.
0.8
0 I-
d
2 0.6
W
0
z
0
0.4 n
3 0
I
- 0.2 xI 0.0 1:o
AXIAL DISTANCE, (c) Shw=lO
1.5
210
z
Figure 2. Effect of variable distribution coefficient on dimensionless mixing-cupconcentration vs dimensionless axial distance for (a) Sh, = 0.1, (b) Sh, = 1, and (c) Sh, = 10.
is -0.9, an inflection point can be observed for the mixing-cup concentration versus axial distance. The solute concentration is larger for small axial distances than for large axial distances, and consequently the distribution coefficient is lower, which leads to less efficient masstransfer separation a t small axial distances. For longer axial distances the concentration decrease along the fiber
Nomenclature A = numerical constant E = numerical constant C = dimensionless concentration, C = C*/C*i C, = dimensionless mixing-cup concentration C, = dimensionless solute concentration at the wall C* = concentration, mol/m3 C*i = initial solute concentration, mol/m3 D = diffusivity of solute in the fluid phase, mz/s k, = membrane permeability coefficient, m/s H = equilibrium distribution coefficient of solute concentration in the membrane to that in the fluid h* = dimensional slope of the distribution coefficient, m3/mol ho = distribution coefficient for infinite solute dilution R = inner radius of hollow fiber, m Ro = outer radius of the hollow fiber, m r = dimensionless radial coordinate r* = radial coordinate, m s = (R,- R)/[RIn ( R o / R ) ] hollow , fiber shape factor Sh, = wall Sherwood number u = average fluid velocity, m/s Y = eigenfunction z = dimensionless axial coordinate z* = axial coordinate, m b = step size in the axial direction
I n d . Eng. Chem. Res. 1992,31,1366-1372
1366
Kim, J. I.; Stroeve, P. Mass transfer in separation devices with reactive hollow fibers. Chem. Eng. Sci. 1988,41, 247-257. Kim, J. I.; Stroeve, P. Uphill transport in mass separation devices X = eigenvalue with reactive membranes: counter-transport. Chem. Eng. Sci. 1989,44, 1101-1111. Subscripts Lonsdale, H. K. The growth of membrane technology. J. Membr. j = index in the z direction Sci. 1982, IO, 81-181. n = order of the eigenvalues and constants Noble, R. D. Two-dimensional permeate transport with facilitated transport membranes. Sep. Sci. Technol. 1984,19,469-478. Noble, R. D.; Way, J. D. Application of liquid membrane technology. Literature Cited ACS Symp. Ser. 1987, No. 347, 110-122. Abrams, D. S.; Prausnitz, J. M. Distribution of phenolic solutes Rudisill, E. N.; Levan, M. D. Analytical approach to mass transfer between water and non-polar organic solvents. J. Chem. Therin laminar flow in reactive hollow fibers and membrane devices modyn. 1975, 7,61-72. with nonlinear kinetics. Chem. Eng. Sci. 1990,45, 2991-2994. Belfort, G. Membrane separation technology: an overview. AdSideman, S.;Luss, D.; Peck, R. E. Heat transfer in laminar flow in vanced Biochemical Engineering; Bungay, H. R., Belfort, G., a.; circular and flat conduits with (constant) surface resistance. Wiley Interscience: New York, 1987. Appl. Sci. Res. 1964/65,14, 157-171. Brown, G. M. Heat or mass transfer in a fluid in laminar flow in a Urtiaga, A. M. Applicacidn de las membranaa liquidas soportadas a circular or flat conduit. AIChE J. 1960,6,179-183. la recuperacidn de fenol en m6duloa de fibras huecas. Ph.D. Cuasler, E. L. Diffwion; Cambridge University Press: London, 1986, Thesis, Universidad del Pais Vasco, Bilboa, Spain, 1991. Chapter 15. Urtiaga, A. M.; Ortiz, M. I.; Irabien, A. J. Phenol recovery with Danesi, P. R. A simplified model for the coupled transport of metal supported liquid membranes. I. Inst. Chem. Eng. Symp. Ser. ions through hollow fiber supported liquid membranes. J. 1990, No. 119,35-46. Membr. Sci. 1984,20, 231-248. Won, K. W.; Prausnitz, J. M. Distribution of phenolic solutes beDavis, H. R.; Parkinson, G. V. Mass transfer from small capillaries tween water and polar organic solvents. J. Chem. Thermodyn. with wall resistance in the laminary flow regime. Appl. Sci. Res. 1975, 7,661-670. 1970, 2, 20-30. Received for review July 22, 1991 Kiezyck, P. R.; MacKay, D. The screening and selection of solvents Revised manuscript received November 26, 1991 for the extraction of phenol from water. Can. J. Chem. Erg. 1973, 51, 741-745. Accepted February 11, 1992 Greek Letters y = dimensionless slope for a variable distribution coefficient
GENERAL RESEARCH Settling Velocity of a Sphere Falling between Two Concentric Cylinders Filled with a Viscous Fluid Guang-HongZheng, Robert L. Powell, and Pieter Stroeve* Department of Chemical Engineering, University of California, Davis, California 95616
The method of reflections is used to analyze experimental data for spheres falling between two concentric cylinders filled with a viscous fluid. The method of reflections is shown to predict the settling velocity of a sphere to within 5 % or less if the sphere is a distance more than three sphere radii away from either the inner or outer walls of concentric cylinders. Near the inner cylinder the method of reflections overpredicts the settling velocity of the sphere, while near the outer cylinder the method underpredicts the velocity. Introduction The problem of a sphere moving in a viscous fluid is of interest in connection with various problems such as the flow of slurries in pipes, the settling of a suspension in a container, and the mixing of an emulsion. M a n y studies have been made to determine the forces acting on a sphere in both bounded and unbounded fluids. In bounded fluids, the geometry of the container walls can play an important role in determining the interaction of a sphere with the wall. There have been considerable theoretical and experimental investigations of a sphere falling in the axial direction inside a single cylinder (Ladenburg, 1907; FGxen, 1922, 1923; Suge, 1931; Francis, 1933; Paine and Sherr, 1975; Happel and Byme, 1954; Haberman, 1956; Haberman and Sayre, 1958; Bohlin, 1960; Fidleris and Whitmore,
1961). Brenner and Happel (1958) used the method of reflections to calculate the frictional force in the axial direction (the k direction) experienced by a sphere falling parallel to the axis of a cylinder but situated in an arbitrary position with the cylinder. T h e y f o u n d
F=
- k ( 6 w ~UJ
(1)
1-f(:);
where Uo is the sphere settling velocity in an off-center position in the cylinder and f(b/Ro)is the eccentricity function which has been evaluated by Famularo (1962), Greenstein (1967), and Hirschfeld et al. (1984). The other symbols used in eq 1 are d e f i i e d in the Nomenclature
0888-5885/92/2631-1366$03.00/00 1992 American Chemical Society
