Ind. Eng. Chem. Res. 1992,31, 1745-1753
1745
Supported Liquid Membranes for the Separation-Concentration of Phenol. 2. Mass-Transfer Evaluation According to Fundamental Equations Ane M. Urtiaga, M. Inmaculada Ortiz, and Ernest0 Salazar Departamento de Zngenieria Quimica, Facultad de Ciencias, Universidad del Pais Vasco, Apartado 644, 48080 Bilbao, Spain
J . Angel Irabien* Departamento de Quimica, Escuela Politecnica Superior de Zngenieria, Universidad de Cantabria, Calle Sevilla, 6, 39001 Santander, Spain
The use of hollow fiber supported liquid membranes for the phenol separation from aqueous solutions and its simultaneous concentration into caustic stripping solutions has been studied. Kerosene and mixtures of kerosene and methyl isobutyl ketone were used as organic solvents. The mass-transfer rate of the phenol separation process is analyzed, making use of the steady-state mass conservation equation for fully developed laminar flow. The effect of the variation of the phenol distribution coefficient with the phenol concentration on the mass-transfer rate has been investigated. The results show that hollow fiber supported liquid membrane phenol separation is substantially influenced by the distribution coefficient. The mass-transfer parameters of three different support fibers have been evaluated from the experimental results. The kinetic model and parameters allow the mass-transfer simulation of the separation-concentration process.
Introduction The analysis of mass transfer in hollow fiber supported liquid membrane separation devices has been carried out, making use of two different approaches. First, many investigators have assumed plug flow for the fluid circulating through the hollow fiber lumen. This method leads to a description of the mass-transfer process using an overall permeability coefficient where the effects of the hydrodynamic conditions have been separated from the system properties by making use of the film theory. Noble (1984) solved the steady-state macroscopic mass balance through a reactive tube for the case of facilitated transport. The model can also be used for the simple diffusion case, showing that axial solute concentration gradients are important and should not be neglected. The option of using the local concentration difference at a particular position of the module gives rise to the often commonly d e d "local permeability coefficient" in contrast to an "average permeability coefficient" (Cussler, 1984). The last one is based on the difference of the average concentrations in the feed and the stripping phases between the inlet and the outiet of the hollow fiber supported liquid membrane module. Numerous authors (Danesi, 1984;Danesi and Reichley-Yinger, 1986;Loiacono et al., 1986) adopted the average permeability coefficient for the modeling of the experimental separation of metallic ions using supported liquid membranes when either the carrier facilitated or the counter transport is applied. However, a more accurate approach to the modeling of hollow fiber supported liquid membrane separations consists of considering the continuity mass conservation equation and the associated boundary conditions for the solute in the inner fluid. When linear boundary conditions are applied at the fiber wall, analytical solutions are available in the literature. Often transfer problems in maas or heat are analogous if the fluids are dilute and natural convection does not take place. The problem is similar to the classical 'Graetz problem" (Graetz, 1885),which has been studied by a number of investigators (Leveque, 1928; Brown, 1960) with the boundary condition of constant concentration at the fiber wall. Siege1 et al. (1957)and
Cess and Shaffer (1960)published the solution for heat transfer when a constant flux is applied at the wall. The problem of a mass flux through the wall proportional to the wall concentration has been analyzed by numerous workers (Sideman et al., 1964/65;Davis and Parkinson, 1970). Chapman et al. (1978)studied the problem for the initial region of the cylindrical and parallel-plate geometries. Numerical methods with the channel discretized in both axial and transverse directions have been used by Kim and Stroeve (1988,1989a,b,1990) to study the problem of nonlinear boundary conditions for reactive hollow fibers, solving the casea of facilitated transport, countertransport, facilitated ion-pair transport, and cotransport. An analytical approach to the problem of nonlinear boundary conditions was given by Rudisill and Levan (1990),who applied this method to the facilitated transport in hollow cylindrical fibers and between parallel plates. Mass-transfer rates attainable in hollow fiber supported liquid membrane modules are limited by solute transport through the liquid membrane. The membrane resistance to mass transfer is expressed in terms of the permeability of the membrane P,. In many practical cases, supported liquid membranes may offer a relatively low P,, which leads to a low mass-transfer rate of the solute. Different solute-transportmechanisms through the liquid membrane can be encountered from the pure Fickian diffusion of the solute to the counterdiffusion,where a mobile carrier which couples the flow of two or more species is used. In the experimental system under consideration the separation of phenol with supported liquid membranes is performed. Mass transfer is accomplished by diffusion of the phenol through the organic solvent immobilized on the porous wall of the hollow fiber membrane. In the previous works Wrtiaga et al., 1990,Urtiaga et aL,1992a),the hollow fiber supported liquid membrane technique was found to be a feasible process for the simultaneous separation and concentration of phenol. Kerosene was immobilized on polypropylene hollow fiber conducted to systems whose stability was experimentally checked for long operation times (up to 2000 h). The use of stripping solution con-
0888-5885/92/ 2631-1745$03.O0/0 @ 1992 American Chemical Society
1746 Ind. Eng. Chem. Res., Vol. 31, No. 7, 1992
, s t r i p p i n g solution
Table I. Dimensions, Pore Diameter, Tortuosity, and Porosity of the Supporting Hollow Fibers dfibmmm fiber supplier inner outer Accurel 2.6/1.8 ENKA, A.G. 1.8 2.6 Accurel 1.0/0.6 ENKA, A.G. 0.6 1.0 Celgard X-20 CELANESE 0.4 0.45
dPn, pm
Phenol d i f f u s e s through the organic l i q u l d membrane t o give:
e,
f
%
14.1 1.84 70 0.5 2.19 63 0.115 4 40
PhOH+NaOH#
PhO-+Na++ H 2 0
"According to MacKie and Meares (1955).
taining sodium hydroxide provided a significant enhancement on phenol separation rate compared to the use of water as stripping agent. The flux of the solute through the supported liquid membrane can be increased by using an appropriate solvent with a high distribution coefficient for the solute. For removal of phenol 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). In these works it is shown that polar organic solvents produce much higher distribution Coefficients than nonpolar hydrocarbons. Moreover, particular attention must be given to the variation of the distribution coefficient with phenol concentration. For most of the solvents tested the distribution coefficient showed a dependency on the phenol concentration in the aqueous phase. From the study (Urtiaga et al., 1992a) of the influence of the membrane composition, it was concluded that the value of the distribution coefficient shows an important effect on the mass-transfer rate. The use of kerosene as liquid membrane gave the major resistance of mass transfer to the immobilized liquid membrane, and the system showed no influence on the linear velocity of the fluids in the flow range under consideration. The aim of the present work is the development of the mass-transfer model and the evaluation of the design parameters for the optimum performance of hollow fiber supported liquid membrane separators for the simultaneous separation/concentration of phenol. For that reason, the solutions to the generalized steady-state mass balance and the associated boundary conditions are obtained and analyzed when either linear or nonlinear boundary conditions are applied. The type of boundary condition for this problem is determined by the dependency of the distribution coefficient on the solute concentration. When a nonpolar hydrocarbon with a constant distribution coefficient is used,linear boundary conditions are applied. The use of polar organic solvents with distribution coefficients dependent on the solute concentration in the aqueous phase leads to a nonlinear boundary condition applied at the fiber wall.
Experimental Section The experimental setup, procedure, reagents, and analytical techniques employed in this work have been described elsewhere (Urtiaga et al., 1992a). Three different types of fibers were used as support material. The characteristic dimensions, pore size, porosity, and tortuosity of the fibers are shown in Table I. The components of the organic phase supported on the porous wall of the fibers are kerosene and methyl isobutyl ketone (MIBK). Phenol separation experiments were run in a continuous operation mode using modulw containing one hollow fiber. The system was allowed to reach a steady state. Samples were collected at the module outlet, and phenol and MIBK concentrations were measured by liquid chromatography.
2ri
/
1
2 1 0
\
1
Microporous W a l l
Impermeable Wall
I
Liquid Membrane
Figure 1. Diagram of a hollow fiber supported liquid membrane.
Feed aqueous solutions with an initial phenol concentration of 5 g/L and caustic stripping solutions with an initial concentration of 1mol/L were employed. The outer solution was continually recirculated in order to obtain a concentrated stripping solution. In the extraction equilibrium experiments, the aqueous and organic phases were put into contact in a rotary SBS stirrer (5-140 rpm). For each phase 10 cm3 was equilibrated. Both phases were allowed to settle and were separated; the concentrations of phenol and MIBK in the aqueous phase were analyzed.
Theory For the analysis of the experimental system of interest, it is sufficient to solve for the concentration profile in a single fiber in order to predict the performance of the supported liquid membrane (SLM) module. A schematic diagram of the fiber is shown in Figure 1. The aqueous solution containing the phenol flows in fully developed, one-dimensional laminar flow inside the lumen of the hollow fiber. Phenol removal takes place by diffusion through the organic solvent immobilized on the porous walls of the fiber. A sodium hydroxide aqueous solution flows along the outer side of the fibers. The phenol concentration in the stripping side is taken to be zero at all axial positions of the module due to the instantaneous reaction with the caustic stripping solution. The fundamental analysis starta from the steady-state mass conservation equation and the associated boundary conditions:
B.C.l.
C = C,
z = 0, all r
(2)
B.C.2.
ac - = 0,
r = 0, all z
(3)
B.C.3.
ar
-DA
ac = P,,,sC, ar
r= ri, all z
(4)
Here C is the solute concentration in the inner fluid, DA is the diffusivity of the solute in the inner fluid, and P, is the permeability of the membrane for the solute. The shape factor s is given by Noble (1983). r, - ri s=(5) TO ri In "i
Ind. Eng. Chem. Res., Vol. 31, No. 7, 1992 1747 For the case where the system is diffusion-limited, s is based on the inside surface area and relates the flux in a cylindrical geometry to the flux in a planar geometry. The third boundary condition imposes the continuity of the solute flux across the membrane-fluid interface, according to Kim and Stroeve (1988). The left-hand side of the equation gives the solute flux arriving at the membrane wall from the bulk. The right-hand side of the equation accounts for the transport of solute through the supported liquid membrane due to pure Fickian diffusion. The parameter P , is a function of the distribution coefficient of the solute between the organic liquid membrane and the inner fluid, H: P , = k,H (6) where k , is the mass-transfer coefficient of the supported liquid membrane. Organic solvents with a constant distribution coefficient for the solute will generate linear boundary conditions. By introducing eq 6 into eq 4, the third boundary condition is reduced to
(7) Equation 7 is a linear expression, and analytical solutions for the system of differential eqs 1-3 and 7 are available. When linear boundary conditions are applied, the following dimensionless variables and parameters are defined: z* =
ZDA
(8)
Uri2
C c* = -
where Am are the eigenvalues, Ym(r*)are eigenfunctions, and A, are constants. The dimensionless mixing-cup concentration, CB*,is a more useful result than the concentration gradient; CB* is defined by xlC*(r*,z*)(l - r*2)r* dr* CB* =
From eq 18 the mixing-cup concentration is determined as
Details on the evaluation of eigenvalues, derivatives, and constants for a cylindrical geometry are given in Appendix I. However, the dependency of the distribution coefficient on the solute concentration gives rise to nonlinear boundary conditions. In this work, according to the experimental results, a linear dependency of the distribution coefficient on the solute concentration in the aqueous phase is considered. In the experimental range of phenol concentration, this dependency is given as
H = h , + hC (20) where h, is the value of the distribution coefficient for infinite dilute solutions and h is the slope of the distribution coefficient. The substitution of eqs 6 and 20 into equation 4 gives B.C.3. -DA
C O
r r* = ri k,Hsri Sh, = -
(10)
D A
The dimensionless axial distance z* down the hollow fiber can also be written as
which is the inverse of the Graetz number. The parameter Sh, is the wall Sherwood number. Sh, is defined as the ratio of the mass-transfer resistance in the inner fluid to that in the membrane. For Sh, = m the supported liquid membrane resistance to mass transfer is negligible, and for Sh, = 0 the membrane resistance is dominant. The introduction of the dimensionless variables and parameters into eqs 1-3 and 7 gives
(18)
ac = k,s(h, + hC)C, ar
r = pi, all z
(21)
which is a nonlinear expression. Equation 20 can be expressed in terms of dimensionless variables, so that H = h, + (1 + yC*) (22) where 7 = C,h/h,
(23)
The parameter y takes into account the dependency of the solute distribution coefficient on the solute concentration. A value of y = 0 will yield a constant distribution coefficient. Negative values of y denote decreasing distribution coefficients with increasing solute concentration. For positive values of y the solute distribution coefficient increases with increasing solute concentration. In this way, the boundary condition at the fiber wall can be made dimensionless, so that
ac* = Sh,,s(l + yC*)C*, r = 1, ar*
B.C.3. --
all Z* (24)
where B.C.l.
C* = 1, z* = 0, allr*
B.C.2.
ac* = 0, &*
B.C.3.
--"* ar*
(14)
Sh,, =
r* = 0, allz*
- Sh,C*,
(15)
r* = 1, all z*
Following Sideman et al. (1964/65), by separation of variables the solution to eqs 13-16 is PI
C* = C AmY,(r*) exp m=l
( ) --z* :A
(17)
kmhosri ~
DA In this work the problem of nonlinear boundary conditions has been solved by applying the method given by Rudisill and Levan (1990). The concentration at the fiber wall and the mixing-cup concentration were calculated following the procedure of Appendix 11. The method is a largely analytical approach to solve the system of differential eqs 1-3 and 21. However, numerical calculations are necessary in order to predict accurately the solute separation. The application of this technique allowed us to simulate the
1748 Ind. Eng. Chem. Res., Vol. 31, No. 7, 1992
Table 11. Phenol and MIBK Equilibrium Distribution between Kerosene and an Aqueous Phase Containing 1 % MIBK by Volume phenol concn in aq phase, g/L 5.084 3.963 3.322 2.123 1.139 0.604 0.067 0.029
MIBK content of org phase, % by vol 35 27.5 25 21.25 20.75 19 17.5 17.5
dietrib coeff 13.75 11.62 10.29 8.41 7.78 7.28 6.46 6.00
effect of a variable distribution coefficient on the solute separation rate. Calculations were performed for Sh, , numbers of 0.1, 1, and 10 where Sh,, = 0.1 represents a major resistance to mass transfer in the fiber wall and for Sh,, = 10 a negligible wall resistance is approached. For each Sh, ,a range of y values of -0.9 < y < was studied (Urtiaga et al., 1992b). QD
Results and Interpretation Equilibrium Study. Two types of organic solvents were used as liquid membranes: (i) kerosene, a commercially available mixture of hydrocarbons, and (ii) mixtures of kerosene and methyl isobutyl ketone (MIBK). The latter is a polar solvent. As a result of the study of the distribution of phenol between kerosene and water, a constant value of the distribution coefficient was obtained. The average value of H in the experimental range of equilibrium concentrations of phenol in the aqueous phase is H = 0.22. In the study of the distribution of phenol between mixtures of keroseneMIBK and water, the distribution of the phenol and the MIBK between the aqueous and the organic phase must be considered. Furthermore, the effect of the phenol concenhtion on the value of the distribution coefficient is known (Won and Prausnitz, 1975). In this way, the reduction of the phenol concentration in the inner fluid due to the phenol transport through the liquid membrane will produce a continuous change in the distribution coefficient along the fiber length. If we center on the experimental conditions, the composition of a liquid membrane formed by a mixture of kerosene and MIBK was determined by the MIBK content of the aqueous phases. This extractant was added to the feed and stripping solutions in order to avoid MIBK losses from the membrane phase due to the MIBK solubilization in water. In a continuous operation mode it was experimentally found that the MIBK content in the inner fluid was held constant along the hollow fiber module. For this reason, the equilibrium study was performed at constant MIBK composition of the aqueous phase. The equilibrium experiments were carried out using organic phases with compositions ranging from 10% MIBK-90% kerosene to 35% MIBK-65% kerosene; the initial concentration of the aqueous phase was in the range 0.1 < CphoH < 75 g/L. The distribution coefficients for the samples containing 1% MIBK (8 g/L) and a phenol concentration in the range 0.025 CphoH < 5 g/L were interpolated. The results are shown in Table 11. The linear regreasion of the results in Table 11yields the following expression for the effect of the phenol concentration on the distribution coefficient: H = 6.082 + 1.415CphOH (26)
9 = 0.9777 for aqueous phases containing 1% MIBK by volume.
Table 111. Phenol Concentration at the Module Outlet as a Function of the Average Velocity of the Feed Solution When Uaing Kerosene aa Organic Liquid Membrane u, cm/s c, B/L Z* CB* (a) Fiber: Accurel 1.0/0.6 mm, L = 48 cm 1.150 0.851 0.649 0.356 0.169 0.097 0.091
4.180 3.893 3.568 2.811 1.549 0.614 0.527
0.463 0.626 0.812 1.495 3.158 5.489 5.863
0.837 0.779 0.714 0.562 0.310 0.123 0.105
(b) Fiber: Accurel 2.6/1.8mm, L = 48 cm 0.120 0.096 0.042 0.019
3.581 3.201 1.901 0.636
0.498 0.621 1.418 3.101
(c) Fiber: 1.621 1.604 0.887 0.568 0.524
Celgard X-20,L = 67.5 cm 2.343 1.039 2.892 1.050 1.900 1.803 0.994 2.894 0.977 3.215
0.716 0.649 0.382 0.127 0.569 0.578 0.361 0.199 0.195
Analysis of Phenol Separation The analysis of the separation of phenol with hollow fiber supported liquid membrane begins by considering the type of boundary condition at the fiber walls in the experimental system under consideration: (a) The use of kerosene as organic solvent leads to a constant distribution coefficient along the fiber length. In this case,the boundary conditions associated with eq 1are linear. The phenol mixing-cup concentration at the module outlet is obtained from eq 19. (b) The use of mixtures of keroseneMIBK as organic solvents leads to nonlinear boundary conditions due to the dependency of the phenol distribution coefficient on phenol concentration in the aqueous feed stream (eq 26). The mixing-cup concentration is obtained according to the method in Appendix 11. Influence of the Support Fiber. Table I11 lists experimental results for the variation of the mixing-cup concentration at the module outlet with the average linear velocity of the inner fluid and with the fiber length. Using keroeene as organic solvent immobilized on three different support fibers, all the experiments were performed starting from feed solutions with an initial phenol concentration of 5 g/L. The experimental values can be expreaaed in the form of dimensionless mixing-cup concentration CB*,as a function of the dimensionless axial distance z+. Since the use of kerosene leads to a linear boundary condition at the fiber wall, the interpretation of these experimental results is carried out according to eq 19. Figure 2 shows the simulated Shg-z* coursea for the wall Sherwood numbers of 10,5, 1,0.395,0.3025,0.21, and 0.1. The points correspond to the experimental results in Table 111. The overall Sherwood number is given as (Kooijman, 1973) Sh, = -In (C,*)/Z* (27) From Figure 2 it is observed that each set of experimental results shows a trend that can be associated with a certain value of the wall Sherwood number. In order to obtain the mass-transfer parameter of each of the support fibers impregnated in kerosene, a comparison between the experimental results and simulated values was established. CB* values were sequentially calculated for different wall Sherwood numbers. Comparison between experimental and simulated results was made on the basis of the minimum weighted standard QD,
Ind. Eng. Chem. Res., Vol. 31, No. 7, 1992 1749
9 I Oo2
loo
1
'
.1 .001
. . . .o 1 ...''I
'
.
. . . . ' . I
.1
.
' ' . . . . . I
1
'
. . . a d1 0
Z*
Figure 2. Effect of wall Sherwood number (Sh,) on overall Sherwood number vs dimensionless axial distance for a constant distribution coefficient. Experimental results correspond to kerosene impregnated in ( 0 )Accurel 2.6/1.8,(X) Celgard X-20,and (m) Accure1 1.0/0.6.
Shw
Figure 4. Weighted standard deviation as a function of wall Sherwood number for simulation of experimental results of phenol separation with kerosene impregnated on an Accurel 2.6/1.8 hollow fiber. I 0.14
-
0.09
-
1 D
0.04
'
0.30
0.28
ShW Figure 3. Weighted standard deviation as a function of wall Sherwood number for simulation of experimental results of phenol separation with kerosene impregnated on an Accurel 1.0/0.6 hollow fiber.
deviation, uw The weighted standard deviation is defined as
An effective diffusion coefficient for the solute in the immobilized organic liquid membrane can be defined as follows:
(30)
For phenol transport using kerosene as organic liquid
0.34
ShW
Figure 5. Weighted standard deviation as a function of wall Sherwood number for simulation of experimental results of phenol separation with kerosene impregnated on a Celgard X-20hollow fiber.
Table IV. Optimum Wall Sherwood Number, Membrane Mass-Tranrport Coefficients, and Effective Diffurivity for Phenol Diffusion in Three Support Hollow fibers Impregnated with Kerosene fiber Accurel 2.6/1.8 Accurel 1.0/0.6 Celgard X-20
Figures 3-5 show the evolution of uw as a function of the theoretical wall Sherwood number for each set of experimental results listed in Table 111. The value of the wall Sherwood number that supplies the lowest uw expresses the mass-transfer resistance of each of the support fibers impregnated with kerosene. From the definition of the parameter Sh, (eq ll), the membrane mass-transfer coefficient is obtained as
0.32
Sh, 0.395 0.210 0.303
k,, cm/s
D.#, cm2/s
1.666 X lo4 2.461 X lo-' 6.540 X lo-'
1.75 X loa 1.71 X 1.66 X 10"
Table V. Experimental and Simulated Values of Ce* Obtained with Liquid Membranes of KeroseneMIBK in an Accurel 1.0/0.6 Fiber CB*.ilU
z* 0.2122
cB*exptl
0.4835
Sh, = 5.5 0.4925
Sh, = 6 Sh, = 6.05 Sh, = 6.5 0.4842 0.4835 0.4771
membrane, Table IV shows experimental values of k, and obtained according to the above procedure (eqs 29 and 30). The valuea of for the three different porous hollow fibers are nearly coincident, showing the possibility of correlating the mass-transfer parameter of the supported liquid membrane, k,, to the properties of the porous support (wall thickness, porosity, and tortuosity of the pores) and the effective diffusion coefficient of phenol in kerosene. D& values in Table IV are in the range of solute diffusion coefficients in liquids. Separation of Phenol from Dilute Solutions. The analysis of phenol separation from dilute solutions was performed starting with aqueous solutions with an initial
1750 Ind. Eng. Chem. Res., Vol. 31, No. 7, 1992
phenol concentration of 0.06 g/L. The experimental results are shown in Table v. The liquid membrane is formed by a kerosene-MIBK mixture in equilibrium with aqueous phases containing 1% MIBK by volume. In the analysis of the experimental results, the following aspects were considered: (i) The value of the distribution coefficient remains constant for phenol concentrations below 0.1 g/L. (ii) The value of the parameter y = (h/h,)C, = (1.42/ 6.08)C0 = 0.014 is close to 0. Thus, linear boundary conditions can be assumed and the analysis of the results is carried out according to eq 19. From the simulated results in Table V, it was determined that the parameter Sh, = 6.05 predicts the experimental results up to the fourth significant figure. From the definition of the wall Sherwood number (eq l l ) , the maas-transfer parameter of the supported liquid membrane is obtained: k, = 2.475 X cm/s corresponding to an Accurel1.0/0.6 fiber impregnated with a 82.5% kerosene-17.5% MIBK mixture. The latter experimental value of k, is very close to the k, value obtained for the Accurel1.0/0.6 mm fiber impregnated with kerosene. This similarity indicates that k, is a parameter basically dependent on the supporting hollow fiber. Influence of the Inner Fluid Flow Rate. The influence of the flow rate of the inner and the outer streams on the solute separation rate has often been studied in order to predict the performance of hollow fiber mass separation devices. Most authors have made use of the film theory; in this way, the effect of the flow rate is included in the interfacial mass-transfer coefficients, corresponding to the aqueous interfaces created close to the fiber wall. This approach considers plug flow and linear concentration gradients, where the changes in concentration are limited to the thin boundary layers. Mass-transfer coefficients are usually reported as correlations of dimensionless numbers. Existing correlations for hollow fibers relate the Sherwood number, involving the masstransfer coefficient, Shi = kidi/DA, with the Reynolds number, characteristic of the forced convection. The more common correlations have been obtained for heat transfer and are applied to analogous mass-transfer problems. Sengupta et al. (1988) used the general expression given by Skelland (1974) to evaluate boundary layer coefficients for flow inside hollow fibers. Yang and Cussler (1986) obtained empirical correlations of inner and outer boundary coefficients for gas separations in hollow fiber contactore. Further studies in liquid-liquid hollow fiber contadors (Dahuron and Cwler, 1988)are consistent with the previous work. However, the small inner diameter of the hollow fibers employed in SLM modules gives rise to low Reynolds numbers, so that the fluid flowing through the lumen of the hollow fiber is in the laminar flow regime. In these conditions, parabolic velocity profiles must be considered. Also, the concentration gradients are extended over the whole of the hollow fiber cross section. When the fundamental equations are used, the flow rate is included in the dimensionless axial distance. Therefore, increasing values of the fluid flow rate will lead to decreasing values of z*, when the length and inner radius of the fiber and the diffusivity of the solute are held constant. Table VI shows the experimental values of phenol concentration at the module outlet as a function of the average velocity of the inner fluid. In the experiments, an organic mixture of kerosene and MIBK in equilibrium with aqueous phases containing 1% MIBK was used as liquid
Table VI. Influence of the Inner Fluid Flow Rate on Phenol Separation When Using Kerosene-MIBK Mixtures in Equilibrium with Aqueous Phases Containing 1 % MIBK in an Accurel 1.010.6 Fiber u, cm/s L,cm C, d L z* CR*..~+~ CR*.;, 0.615 1.130 1.231 1.257 1.768 1.796 1.834 2.050 2.172 2.541 3.144 3.275 3.556 3.583 5.665 6.693 8.270 8.842 11.600 12.326 14.701 18.990 33.992 34.945
45 45 48 45 45 45 45 45 45 45 45 48 45 48 45 48 45 46 45 45 45 48 45 45
0.409 1.153 1.071 1.175 1.832 1.787 2.076 2.032 2.138 2.349 2.500 2.581 2.682 2.679 3.397 3.304 3.681 3.679 3.909 3.928 4.103 4.166 4.518 4.535
0.811 0.442 0.432 0.397 0.282 0.278 0.272 0.243 0.230 0.196 0.159 0.163 0.140 0.149 0.0881 0.0795 0.0603 0.0577 0.0434 0.0405 0.0339 0.0280 0.0147 0.0143
0.082 0.231 0.214 0.235 0.367 0.357 0.415 0.406 0.428 0.470 0.450 0.516 0.536 0.536 0.679 0.661 0.736 0.736 0.782 0.786 0.821 0.833 0.904 0.907
0.085 0.248 0.254 0.281 0.394 0.400 0.406 0.445 0.459 0.504 0.564 0.558 0.596 0.582 0.702 0.722 0.771 0.777 0.818 0.826 0.847 0.868 0.918 0.920
membrane. The support fiber is an Accurel1.0/0.6 hollow fiber. The initial phenol concentration of the feed solution is 5 g/L. For this system, the parameters Sh,, and y are obtained as follows: h,k,ris Sh,, = n - 5.85
--
y
h 1.42 = -C, = -Co h, 6.08
= 1.16
where k, = 2.475 X lo-* cm/s has been obtained in the previous section. As shown in our previous work (Urtiaga et al., 1992b), a variable distribution coefficient can have a significant effect on the mass-transfer rate if the wall Sherwood number is less than 10. Thus, in order to predict the performance of hollow fiber SLM modules, the variation of the distribution coefficient must be included. The dimensionless mixing-cup concentration, CB*,was simulated at each of the experimental dimensionless axial distances for Sh,, = 5.85 and y = 1.16. These values are shown in Table VI. The simulation with the latter values of the Sh,, and y parameters generates an error of 8% in the prediction of the experimental results.
Conclusions The separation of phenol from aqueous solutions and its simultaneous concentration into caustic stripping solutions (1 mol/L NaOH) using hollow fiber supported liquid membranes have been investigated. Extensive data have been obtained from the study of the influence of the organic solvent, the type of support fiber, and the flow rate of the fluids. The study indicates that the dependency of the phenol distribution coefficient with the phenol concentration must be considered when predicting the performance of a hollow fiber supported liquid membrane module. The reduction of the phenol concentration in the inner fluid due to the phenol transport through the liquid membrane will produce a continuous change in the distribution coefficient along the fiber length. As a result of the experimental study of the equilibrium, a linear expression was obtained
Ind. Eng. Chem. Res., Vol. 31, No. 7, 1992 1751 for the dependency of the distribution coefficient on the phenol concentration, at constant MIBK composition of the aqueous phases. The dependency of the distribution coefficient on the phenol concentration determines the type of boundary condition at the fiber wall associated with the steady-state mass conservation equation with fully developed, laminar flow. In the experimental system under consideration, the use of kerosene as organic solvent leads to constant wall Sherwood numbers along the fiber length. An analytical solution allows obtaining the mixing-cup concentration as a function of the wall Sherwood number and the axial distance. The low value of the distribution coefficient (H = 0.22) causes an important resistance to mass transfer to be due to the supported liquid membrane, and consequently low values of the Sh, for each of the supporting fibers impregnated with kerosene have been obtained. The use of mixturea of kerosene-MIBK as organic solvent leads to nonlinear boundary conditions due to the linear dependency of the phenol distribution coefficient on phenol concentration in the aqueous feed stream. The technique of Rudisill and Levan (1990), a more straightforward method than the numerical solutions given in previous works (Kim and Stroeve, 1989), was used to solve the system of differential equations. The dimensionless mixing-cup concentration is obtained as a function of the dimensionless axial distance, the wall Sherwood number, and the dimensionless slope of the distribution coefficient, y. The mass-transfer rate is substantially increased in relation to the use of kerosene. The analysis of the influence of the flow rate of the inner fluid on the mass-transfer rate allowed obtaining the mass-transfer parameter of each of the support fibers, which basically is a characteristic of the support material.
Sh,,: wall Sherwood number defined in eq 25 t,: thickness of the hollow fiber wall, cm u: average velocity of the inner fluid, cm/s Y: eigenfunction Z: function defined in eq A2 z: axial coordinate z*: dimensionless axial coordinate Az*: step size in the axial direction
Greek Letters y: dimensionlessslope for a variable distribution coefficient
porosity eigenvalue weighted standard deviation 7: tortuosity Su bscriptslSuperscripts j : index in the z* direction m: order of the eigenfunctions, eigenvalues, and constants n: order of the coefficients in eq A6 e:
X: :,u
Appendix I The solution to eqs 13-16 is obtained by separation of variables. The separation of variables gives rise to two independent differential equations: C*(z*,r*) = Z(z*) Y(r*)
(AI)
The eigenfunctions are solutions to equations
B.C.1
Acknowledgment Special acknowledgment is given to Professor Pieter Stroeve from the University of California for his valuable help in the development mathematical analyis. This work was financially supported by the Basque Government under Project No. 069.310-0014/89 and by the Ministerio de Educacidn y Ciencia of Spain under a FPI Research Grant.
B.C.2
Nomenclature A: numerical constant a: coefficient in eq A6 d diameter C: solute concentration in the inner fluid, g/L C,: initial solute concentration, g/L C*: dimensionless solute concentration in the inner fluid CB*:dimensionless mixing-cup concentration C,*: dimensionless solute concentration at the wall DA: diffusivity of the solute in the fluid phase, cm2/s E: numerical constant H equilibrium distribution coefficient of the solute between the organic and the aqueous phase h,: distribution coefficient for infinite solute dilution h: dimensional slope of the distribution coefficient, L/g k,: membrane mass-transfer coefficient, cm/s L: length of the hollow fiber, cm P,: membrane permeability coefficient, cm/s r: radial coordinate, cm r*: dimensionless radial coordinate ri, r,: inner and outer radii of the hollow fiber, respectively, cm s: shape factor, defined in eq 5 Sh,: overall Sherwood number Sh,: wall Sherwood number defined in eq 11
where
-.-
ar*
= Sh,Y
r* = 1
Equation A3 is a Sturm-Liouville type equation, and it can be solved by assuming an infinite series for Y(r*): m
Y = Canr*n
(A61
n=O
an=O
n
