Journal of Membrane Science, 20 (1984) 231-248 Elaeviar Science Publishers B.V., Amsterdam - Printed in The Netherlands
A SIMPLIFIED MODEL FOR THE COUPLED TRANSPORT OF METAL IONS THROUGH HOLLOW-FIBER SUPPORTED LIQUID MEMBRANES
PIER R. DANESI Chemistry Division, Argonne IL 60439 (U.S.A.) (Received
9700 South Cass Avenue, Argonne,
27, 1983; accepted ifi revised form April 5,1984)
Summary This paper describes a simplified model for the carrier-facilitated transport of metal ions through hollow-fiber supported liquid membranes, HFSLM. The model leads to approximate and simple equations describing the concentration variations expected when an aqueous feed solution is flowing through the lumens of a HFSLM module. The equations incorporate simple and independently measurable parameters and apply to two situations: (a) a once-through mode, i.e., the feed solution passes only once through the module, and (b) a recycling mode, i.e., the feed solution is continuously recirculated through the module. The equations have been tested by measuring the transport of Cu*+ ions through microporous polypropylene hollow fibers containing a 0.3 F solution of bis(2-ethylhexyl)phosphoric acid in n-dodecane. HFSLM modules containing a variable number of fibers, fibers of different lengths and operated at different linear flow velocities have been used.
SLM, has been described in a number of papers [l-11]. In this type of transport, metal ions can be transported across the membrane against their concentration gradient. The “uphill” transport okurs at the expense of membranes,
of the membrane.
different concentration and/or composition. in a water-immiscible, low~dielectz-ic constant carrier adsorbed on a microporous polymeric
between by using
the two aqueous
The SLM consists of a solution, organic diluent, cf a metal film. The metal carrier is in
cases a metal extraction reagent,. The aqueous solution, initially containing all the metal ions which can permeate the SLM, is generally referred to as feed solution. The aqueous solution present on the opposite side of the membrane, which is initially free from the permeable metal ions, is generally referred to as strip solution. When the membrane carrier is an acidic extractan& the driving force for the facilitated transport is provided by the different acidities of the feed and strip solutions. most
o 1984 Elsevier Science Publishers B.V.
If the microporous polymeric support for the liquid membrane is shaped as a tiny hollow tube, we are dealing with a hollow-fiber supported liquid membrane, HFSLM. A schematic representation of the cross- and axial sections of a HFSLM is shown in Fig. 1. Single Hollow Fiber
203 + d,)
Carrier and Organic Dlluent
Hollow Fiber Lumen
Flber Wall (porou8) Feed Solution
Fig. 1. Schematic
of a hollow-fiber supported liquid membrane, HFSLM.
R isthe radius and d, is the wall thickness of the fiber.
HFSLMs represent a very attractive solution to the need of operating membrane modules with very high throughputs. With HFSLM modules membrane packaging densities as high as 1000 m*/m” can be reached. This value compares to about 500 m2/m3 for plate and frame and to about 50 m2/m3 for tubular membrane modules. Moreover, HFSLM modules should be characterized by low investment and operating costs because of the reduced hardware and the favorable hydrodynamics which minimizes aqueous concentration polarization effects and membrane fouling. Models describing facilitated transport of metal ions through low dielectric constant, flat-sheet SLMs have been previously reported [2,5,6-U]. On the other hand, the modelling of HFSLMs has been mainly restricted to the permeation of gases [ 5,X2,13]. In the present paper a simplified model for the carrier-facilitated transport
of metal ions through HFSLMs is described. The model extends to a cylindrical geometry our previously developed model for a flat-sheet SLM 16-111. The model describes in terms of simple and independently measurable quantities the concentration variations of the metal ions expected when the aqueous feed solution is flowing in a laminar regime through the lumens of HFSLMs, and the aqueous strip solution circulates on the shell side of the fibers. The derived equations apply to the two cases (a): a recycling mode, i.e., the feed solution is continuously recirculated through the HFSLM module, and (b) a once-through mode, i.e., the feed solution passes only once through the HFSLM module. The approximate equations for a once-through mode are experimentally tested in the case of feeds with low metal concentrations by measuring the facilitated transport of Cu*’ ions. The HFSLM is a n-dodecane solution of the acidic carrier bis (2-ethylhexyl) phosphoric acid, HDEHP, adsorbed on microporous polypropylene hollow-fiber supports. Model When an aqueous feed solution containing metal ions is flowing through the lumen of an HFSLM, the concentration of the metal ions at the exit of the hollow fiber, Gout, will be related to the concentration at the entrance, C,, through a number of chemical and hydrodynamical parameters. In principle, the correlation between COutand Ci, could be obtained by solving the differential equation describing the mass transport in terms of convection and molecular diffusion, provided the chemical boundary conditions at the aqueous solution-membrane interfaces are known. In practice, such an approach cannot be followed because the physico-chemical hydrodynamics of the system are so complex to escape analytical solutions. Bessel functions have sometimes been used to solve equations describing the motion of a fluid in a cylinder in the presence of interfacial chemical transformations. Altematively, numerical solutions of the mass transfer equations can be worked out taking advantage of high speed computers. In both cases the simultaneous presence of a two-dimensional mass transport process occurring inside the fiber lumen, and a one-dimensional diffusion process occurring inside the fiber wall, coupled to chemical reactions occurring at the aqueous solutionsliquid membrane interfaces, requires the use of complicated mathematical algorithms. As a result, the direct physical meaning of the role that the various chemical, hydrodynamical and geometrical parameters play on C,, easily gets lost in the complexity of the mathematics used. Therefore a simple correlation between C,, and Gin, which can be used as a rule of thumb to predict to a first approximation the behavior of HFSLMs - in terms of the linear flow velocity of the feed solution in the fiber lumens, the radius and length of the fibers and the chemical composition of the system - can be useful. The following derived model leads to such a correlation. The model is developed using approximations and assumptions similar to those previously
used by us to derive the permeability coefficient equation of flat-sheet SLMs [6-111. In Refs. [ 7-111 the following equation was derived and experimentally tested for the membrane flux, J(cm-set-* -M), across a flat-sheet SLM: KdC
K&L, +A, The symbols in eqn. (1) represent: P = permeability coefficient (cm/set), C = aqueous feed concentration of the metal species permeating through the SLM at time t (M), Kd = organic/aqueous partition coefficient of the metal species at the aqueous feed solution-membrane interface, A, = c&/D, = aqueous diffusional parameter (set/cm), A0 = d, /Do = membrane diffusional parameter (set/cm), d, = thickness of the aqueous boundary layer (cm), D, = aqueous diffusion coefficient of the permeating metal ions (cm*/sec), d, = membrane thickness (cm), and D, = diffusion coefficient of the metal ion-carrier complex in the SLM (cm*/sec). Equation (1) was derived by assuming: (a) steady state, (b) linear concentration gradients, (c) organic/aqueous partition coefficients of the metal species at the membrane-aqueous feed solution interface which are much larger than those at the membrane-aqueous strip solution interface, (d) instantaneous interfacial chemical reactions between the metal species and the membrane carrier, (e) very low metal concentrations. By considering that J = -(dC/dt) V/A, eqn. (1) can be easily integrated to:
where Co = aqueous feed concentration of the metal species permeating the SLM at time zero, A = area of the flat-sheet SLM available for the mass transport process, and V = volume of the aqueous feed solution. When assumption (e) is removed, i.e., the aqueous feed metal concentration is so high that the carrier is completely loaded with the metal ion, eqns. (1) and (2) must be replaced with [8,11] : J
respectively. In eqns. (3) and (4) [E] stands for the carrier molar concentration in the liquid membrane and n for the number of carrier molecules bound
to each metal ion in the metal-carrier complex formed in the membrane phase. Equations (l)-(4) were derived by assuming that the diffusion coefficient of the free carrier is much higher than the diffusion coefficient of the metal-carrier complex. In this way the counter-diffusion of the free carrier is never rate-determining and it can be ignored in the steady-state membrane flux.
Fig. 2. Axial section of a microporous HFSLM. z = axial coordinate, r = radial coordinate, Ci, = input concentration (2 = 0), C,, = output concentration (z = L), L = fiber length (cm), R = fiber radius (cm), da = aqueous annular boundary layer thickness (cm), d, = hollow-fiber wall thickness.
Making the same assumptions from (a) to (e) when the feed solution is axially flowing in the hollow-fiber lumen in a laminar regime and further assuming that: (f) at a fixed value of the axial coordinate, z, the metal concentration in the fiber lumen is the same except for an annular zone of thickness d,, a correlation between the aqueous diffusional flux, J, , and the membrane diffusional flux, J,,, is easily derived for the cylindrical geometry of the hollow fiber. Figure 2 shows the axial section of a HFSLM. The dashed line indicates that the annular aqueous boundary layer of constant thickness d, extends over the entire length of the fiber. Figure 3 illustrates assumptions (b), (c) and (f), and the cross-section of the HFSLM is shown in Fig. 4. At the steady state, the relationship between the fluxes at r = R - d,, J,, and at r = R, Jo, is
JCl When the equations describing the aqueous and membrane fluxes
Do ciJO = 47
Fig. 3. Concentration profiles at z along the radial coordinate r. Ci = membrane-side interfacial concentration at the aqueous feed solution-membrane interface (r = R), Ci = aqueous side interfacial concentration at the aqueous feed solution-membrane interface (r = R), C, = bulk concentration at z (0 G r f R). R, d, and d, have the same meaning as in Fig. 2.
Fig. 4. Cross-section of the HFSLM. Ja = flux at r = R -da, Jo = flux at r = R. C,, Ci, G, R, d, and d, have the same meaning as in Fig 3.
and the interfacial equilibrium relationship Kd = s 1
are simultaneously solved, it follows that J, =
4)l c, & [R/W - a1 WL + &~/al & BI(R
where C,, Ci and Ci are the bulk and interfacial (aqueous side and membrane side) concentrations of the metal species at z; 7 is a tortuosity factor which takes into account the tortuous shape of the pores connecting the inside and the outside of the hollow fiber. The total loss of moles of metal species per unit time, m, over the entire fiber length is obtained from: L m=
J, 2n (R -d,)
To integrate eqn. (lo), the function C, (z) has to be known. If small z values and/or very low fluxes and metal concentration are considered, the first order exponential decay of the metal ion concentration with z can be approximated to a linear function, i.e. tin
c, = ci,-
- Gout L
In this region of low product recovery, the integration of eqn. (10) yields:
(14) e is the porosity of the SLM, i.e., the fraction of geometrical area available for mass transport. A total mass balance over the entire fiber also yields: 111= Q (C, - C,,)
where Q = flow rate through a single fiber (cm3/sec), and 0 = average linear flow velocity (cm/set).
By equalizing the right-hand-side terms of eqns. (12) and (15), we finally obtain:
@ = P”Le and p*
KdA; + A:
When the HFSLM module contains N equal fibers and Qr (cm3/sec) is the total flow rate through the module a more convenient expression for 4 is
QT ’ = P”LmNR
P* is a “modified” permeability coefficient which equals the permeability coefficient measured for a flat-sheet membrane (eqn. (1)) when d, -4 R and membranes having the same tortuosity factor, porosity, thickness and aqueous boundary layers of equal thickness are compared. Equation (16) only holds for @ > 1. In the region where @ < 1, the firstorder exponential decay of C, with z cannot be approximated to a linear function and eqn. (16) is no longer valid. In this region there is a high product recovery in a single pass through the hollow fiber and C,, asymptotically approaches zero as long as L increases. Equations (16) and (17) show that, for a given chemical system, the amount of metal removed decreases by increasing the fiber radius and the linear flow velocity. On the other hand, high permeability coefficients and long fibers increase the amount of metal removed per pass. In many practical cases the wall thickness of the hollow fibers and the tortuosity factor lead to a situation where AZ > Kd AZ. Therefore, P* - K,/Az and eqn. (17) can be used to predict the performance of an HFSLM module, once the chemistry of the system and the physical properties of the fibers are known, and providing the assumptions made on the linear concentration variations still hold. Although practical limitations on the values of fl, L and R are imposed by the maximum pressure that the HFSLM can withstand, some de-see of flexibility exists in adjusting th chemistry (&), the hydrodynamics (U), and the geometry (L, R) of t ‘e HFSLM module, to achieve the same degree of metal 1 species removal with idifferent combinations of the parameters controlling Q. When the HFSLM is operated in a recycling mode, i.e., the feed solution is continuously recirculated through the module, an equation describing the
variation with time of the metal concentration in terms of @Ican also be derived. The situation occurring in this case is illustrated in Fig. 5. If the total volume of the feed solution is V, the following mass balance equations can be written: -v-z dCin
for the stirred tank, and P* 21rRLe
Q CC, - Gut)
for the HFSLM module. Hollow-fiber
Stlrred Tank c = c,,
Fig. 5. Schematic representation
of an HFSLM module operating in a recycling mode.
Equalizing the left hand terms of eqns. (20) and (21) and using eqn. (16) to correlate C,,, and Ch, it follows that: = _Ap*_
where C$ is the value of Ci, at time zero, and A is the total internal area of the HFSLM module and is equal to 2nRLNe. Equation (22) has physical meaning only when 9 > 1. When $3 1, C,, will always be very close to Ci, at each pass of the feed stream through the module, and eqn. (22) becomes identical to eqn. (2). In this case, the HFSLM behaves as a flat-sheet SLM of area 2nRNL.c in contact with a feed solution of volume V. When the metal concentration in the aqueous feed is high and the membrane carrier is completely loaded with the permeating metal, eqns. (3) and
(4) hold. In this case, the flux J, does not vary along the z-axis and the result of integrating eqn. (10) is 277RLe
m = -!i% nA2
Here, it is assumed that (R -cl,) - R. By using the mass balance equation, eqn. (15), we now obtain
c out =
[Q 2 LE
n AZ OR
For a HFSLM module operated in a recycling mode, the mass balance equations lead to dCi, _v-=dt
Equations (24) and (26) have physical meaning only when [E] 2Le/nAzuR 4 Ci, and ([E]A/nAzV)t Q C&. In the following sections a test of the equations derived in the case of feeds with low metal concentrations is reported. Experimental The equations derived for low metal concentrations have been tested by using the system shown in Fig. 6. Reagents Copper (II) nitrate hydrate, acetic acid, potassium acetate, nitric acid and n-dodecane were analytical purity grade reagents. HDEHP, bis (2-ethylhexyl)phosphoric acid, was a 99% pure reagent (Eastman Kodak Company). The initial Cu(I1) concentration of the aqueous feed solution was 1.0 X 10e3 M in all the once-through experiments and it ranged between 1 X 10s4 M and 1 X lop3 M in the recycling experiments. All aqueous feed solutions were made 0.5 M in CH3COOH and 0.5 M in CH3COOK. The strip solution was 1 M HN03. The liquid membrane was a 0.3 F solution of HDEHP in n-dodecane. Mem banes The HFSLM modules were fabricated by inserting the required number of
in n-dodeca ,%, ,
Strip pH -0
Feed pH -4.6
Fig. 6. Schematic representation of the SLM system used in the experiments. reaction responsible for the facilitated transport is also shown.
hollow fibers (Celanese Plastic, Charlotte, N.C.) into a glass hollow tube, 1.5 cm i.d,, having at the extremes two openings of the type present in cooling jackets. The openings allowed the circulation of the 1 M HNOJ strip solution on the shell side of the fibers. After the insertion of the fibers into the glass tube, an epoxy resin (5 min curing time) was used to seal the fibers in the tube. The seal extended into the tube for a few millimeters, providing complete isolation of the feed and strip aqueous solutions. The HFSLM modules were made to contain from 20 to 180 fibers. Their lengths varied from 5.9 to 20.5 cm. The hollow-fiber material was microporous polypropylent The dimensions of the fibers, determined by microscopy, were: i.d. = 0.0400 cm, o.d. = 0.0460 cm. The liquid membrane was adsorbed on the Celgard hollow fibers by flowing at a slow rate the organic solution through the fiber lumens and then rinsing the lumen and the shell side of the fibers with distilled water. Copper analysis The Cu(I1) concentrations of the feed solution at the entrance and at the exit of the HFSLM modules were measured by an Orion cupric electrode and a double-junction Ag/AgCl reference electrode. A Beckman 4500 pH meter and a Cole Parmer chart recorder were used to measure the potentiometric signal. The e.m.f. values of the Cu(I1) electrode were related to Cu*+ concentrations through a calibration curve obtained in the same potassium acetateacetic acid buffer. The pH of each solution was measured by glass electrode potentiometry. The analyses were performed at 25°C. Distribution ratio measurements The distribution ratios of Cu(II), Kd = [Cu(II)] organic/[C~(II)]aqueous,were
obtained by shaking equal volumes of the aqueous phase ([Cu”‘] = 1.0 X 10d3 M, [CH3COOH] = 0.5 M, [CH3COOK] = 0.5 M) and organic phase (HDEHP in n-dodecane) at 25°C. After equilibration the phases were separated and the equilibrium aqueous Cu(I1) concentration and pH were measured potentiometrically. The Kd of Cu(I1) was measured as a function of HDEHP concentration. The results of the distribution experiments are reported in Fig. 7 as log Kd + 2 log [H+] vs. log an points, where an represents the activity of the dimeric HDEHP in n-dodecane, taken from Ref. [ 141. The data points fall on a straight line of slope +2.0, in agreement with the extraction stoichiometry reported in Fig. 6. The equilibrium ratio of the extraction reaction evaluated from the data points of Fig. 7, K, = K,[H’]*/a?,, is equal to 5.2 X 10W6.The Kd values at the feed solution-SLM and strip solution-SLM interfaces are 30 and -lo-’ respectively. These values support the validity of assumption
Fig. 7. Log I& + 2 log [H’] vs. log aD plot for the extraction of Cu2+ by a 0.3 F HDEHP n-dodecane solution at 25°C. aD is the activity of dimeric HDEHP.
Membrane permeation measurements The 1 M HN03 strip solution was circulated on the shell side of the HFSLM module by a peristaltic pump operated at a constant flow rate. The Cu(I1) concentration variations of the aqueous feed solution were independent of the flow rate of the strip solution in the range 1 ml/min to 40 ml/min. This result was expected on the basis of the low & of Cu(I1) at the strip solutionmembrane interface. The feed solution was circulated through the hollow-fiber lumens at different flow rates by a Master Flux, Cole Parmer, peristaltic pump
or an FM1 piston pump, Fluid Metering, Inc. The flow rate was experimentally checked when the pump was connected to the HFSLM modules. The flow rate covered the linear flow velocity interval from 0.018 to 20 cm/set. When the HFSLM module was operated in a recycling mode (Fig. 5) the volume of the stirred tank was always much larger than the volume of the connecting tubes and of the hold-up volume of the module. In these experiments the cupric ion and reference electrodes were inserted into the stirred tank. Figure 8 shows the results of a typical recycling mode experiment. In all recycling experiments, straight lines were obtained when log [Cu”‘] was plotted vs. time. The membrane permeation experiments were conducted at 22°C.
Fig. 8. Example of a recycling experiment through an HFSLM module. [CU”]~ is the concentration of Cu(I1) in the stirred tank at time t. Module characteristics: L = 20.0 cm, R = 0.02 cm, N = 24 fibers, total membrane area = 60.3 cm’, aqueous feed volume = 20 cm3, and linear flow velocity 0 = 5 cm/set.
Results Recycling mode experiments The results of the recycling mode experiments are reported in Table 1. They have been used to evaluate the relatiorship between P*E and fl holding for the HFSLMs used in this investigation. K represents the slope of the straight lines obtained by plotting In (Ci,/ Cio,)vs. time multiplied for the volume of the feed solution, V, and divided by the membrane geometrical area, 2nRLN. Ci, and CR are the Cu(I1) concentrations in the feed tank at time t and at time zero, respectively. J? is a function of p through P* and $, i.e., R o/L (R @LP*e)
The relationship between P*e and 0 was obtained through eqn. (27) by calculating P*E for the corresponding values of the four parameters 0, I,, R and K, and then drawing the best smoothed curve through the data points. The smoothed curve describing the function P*e vs. 0 is shown in Fig. 9. As expected, P*e first increases with a and then becomes independent of it. The increase of P*E with u is caused by the decrease of the thickness of the aqueous boundary layer, d, , when the linear flow velocity in the fiber lumen increases. Eventually P*E becomes independent of 0 when the term KdAz becomes negligible with respect to AZ in eqn. (18).
log u Fig. 9. SLM permeability coefficient times membrane porosity, P*e (cm/set), flow velocity of the feed solution, i? (cm/set).
Recycling data D (cm/set)
0.018 0.024 0.094 0.21 0.56 2.6 5.0 10 15 20
9.4 10.0 10.1 9.4 14.2 14.2 20.2 20.0 18.7 20.0
80 44 44 80 104 104 24 24 24 24
15 25 25 10 15 15 20 20 20 20
1.2 2.1 6.9 9.9 1.4 2.2 2.5 2.5 2.5 2.6
x 1o-s x 1o-s x lo-’ x 1o-5 x 1o-4 x 1o-4 x 1o-4 X 1O-4 X 1O-4 X 1O-4
D = linear flow velocity, L = length of fibers, N = number of fibers, V = total volume of the feed solution, z = (R~/L)/[(R~/LP*e)+ 11, with R = 0.02cm in all cases.
Once-through mode experiments Once the relationship between P*e and u is available, eqn. (16) can be tested by comparing the Goutvalues obtained from the once-through experiments with the results expected from the model. This comparison is shown in Fig. 10. Here, the experimental C,, values obtained with HFSLM modules of different length, containing a different number of fibers and operated at different linear flow velocities, are plotted as function of the variable y = (C#I - l)/(# + 1). The parameters characterizing each set of data are reported in the figure. The values of P*E, corresponding to each I!?,necessary to evaluate 6, were obtained from the curve of Fig. 9. Although the experimental points of Fig. 10 cluster along the straight line calculated by C,, = 10m3y, deviations are observed at the two extremes of the plot. This slight curvature exhibited by the data points indicates that in the case of feed solutions containing the metal species at low concentration the derived model gives a simple but only approximate description of the behavior of the HFSLM modules.
Fig. 10. Comparison between experimental and calculated Gout values for HFSLM modules operated in a once-through mode at increasing Iinear flow velocities. Ci, = 1 x lo-’ M (Cu”) in all experiments. The different symbols refer to different modules: 0: L = 14.9 cm, R = 0.02 cm, N = 104 fibers, 0.044 Q v(cm/sec) G 0.952; q : L = 20.5 cm, R = 0.02, N = 24 fibers, 0.30 S o(cm/sec) Q 1.67; A: L = 5.9 cm, R = 0.02 cm, N = 180 fibers, 0.053 4 u(cm/sec) Q 0.53. The straight line is calculated through the relationship Gout = 1O-3 y.
Conclusions A simplified model which describes the permeation of metal species through hollow-fiber supported liquid membranes, HFSLM, was derived. The main assumption introduced into the model is that the radial and the axial concentration gradients are linear. The model leads to approximate equations, summarized in Table 2, which describe the performance of HFSLM modules
operated in a once-through mode and in a recycling mode. The equations are simple and describe the concentration variations in terms of a few, simple and often independently measurable parameters. They were derived for the two limiting cases of low and high metal concentrations of the feed solutions. Wben the metal concentration is low, the concentration of the unbound carrier essentially equals the total membrane carrier concentration. In this case the membrane flux varies with time and along the hollow-fiber axial coordinate. The equations derived in this case only hold in a region of low product recovery. When the metal concentration is high, the concentration of the unbound carrier is close to zero and the membrane flux is constant with time and along the hollow-fiber axial coordinate. An experimental verification of the equations derived for feed solutions with low metal concentrations was performed. The experimental results indicate that the derived equations represent an approximate but nevertheless useful description of the performance of HFSLMs. TABLE
equations describing the behavior of HFSLM modules
1. Once- through mode (a) Low metal concentrations
RC7 0 = P*Le
for @> 1
(b) High metal concentrations
- Loaded carrier (24)
(a) Low metal concentrations
(b) High metal concentration
ci, = c& -
g$ t 0
181 A t ‘e cio,
- Loaded carrier
247 TABLE 2 continued: List of symbols
used in table
concentration of metal species at the entrance of the HFSLM module (M) concentration of metal species at the entrance of the HFSLM module at time zero (M) CALut concentration of metal species at the exit of the HFSLM module (M) [El total concentration of carrier in the liquid membranes (&f) number of metal ions bound to each carrier molecule at loading : hollow-fiber internal radius (cm) L hollow-fiber length (cm) r7 linear flow velocity of feed solution (cm/set) membrane porosity ;* Kd /(Kd A,* + A*,), modified permeability coefficient (cm/set) Kc3 organic/aqueous partition coefficient of metal species at the aqueous feed-membrane interface aqueous diffusional parameter (cm/set) A*, membrane diffusional parameter (cm/set) AZ t time (set) A total internal area of the fibers (cm’) V volume of the aqueous feed solution (cm3)
Acknowledgments The author wishes to thank Dr. E. Philip Horwitz for his continuous support and Mr. Paul Rickert for performing the experimental tests. The helpful suggestions of the referees concerning the limitations involved in the use of a linear axial concentration gradient are also acknowledged. This work was performed under the auspices of the Office of Basic Energy Sciences, Division of Chemical Sciences, U.S. Department of Energy under contract number W-31-109-ENG-38. References E.L. Cussler and D.F. Evans, How to design liquid membrane separations, Sep. Purif. Meth., 3 (1974) 399. E.L. Cussler, Multicomponent Diffusion, Elsevier, Amsterdam, 1976, Chap. 8. H.K. Lonsdale, The growth of membrane technology, J. Membrane Sci., 10 (1982) 81. R. Marr and A. Kopp, Liquid membrane technology - A survey of phenomena, mechanisms and models, Int. Chem. Eng., 22 (1982) 44. J.D. Way, R.D. Noble, T.M. Flynn and E.D. Sloan, Liquid membrane transport: A survey, J. Membrane Sci., 12 (1982) 239. P.R. Danesi, E.P. Horwitz, G.F. Vandegrift and R. Chiarizia, Mass transfer rate through liquid membranes: Interfacial chemical reactions and diffusion as simultaneous permeability controlling factors, Sep. Sci. Technol., 16 (1981) 201. P.R. Danesi, E.P. Horwitz and P. Rickert, Transport of Eu3+ through a bis(2-ethylhexyl)phosphoric acid, n-dodecane solid supported liquid membrane, Sep. Sci. Technol., 17 (1982) 1183.
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R. Chiarizia, A. Castagnola, P.R. Danesi and E.P. Horwitz, Mass transfer rate through solid supported liquid membranes: Influence of carrier dimerization and feed metal concentration on membrane permeability, J. Membrane Sci., 14 (1983) 1. P.R. Danesi, R. Chiarizia and A. Castagnola, Transfer rate and separation of Cd(I1) and Zn(I1) species by a trilaurylammonium chloride-triethylbenzene supported liquid membrane, J. Membrane Sci., 16 (1983) 161. C. Cianetti and P.R. Danesi, Facilitated transport of HNO, through a supported liquid membrane containing a tertiary amine as carrier, Solv. Extr. Ion Exch., l( 3) (1983) 565. P.R. Danesi, E.P. Horwitz and P.G. Rickert, Rate and mechanism of facilitated Am(II1) transport through a supported liquid membrane containing a bifunctional organophosphorous mobile carrier, J. Phys. Chem., 87 (1983) 4708. R. Noble, Shape factors in facilitated transport through membranes, Ind. Eng. Chem. Fundam., 22 (1983) 139. R. Noble, Two dimensional facilitated transport in hollow fiber membranes, Sep. Sci. Technol., in press. P.R. Danesi and G. Vandegrift, Activity coefficient of bis(2-ethylhexyl) phosphoric acid in n-dodecane, Inorg. Nucl. Chem. Lett., 17 (1981) 109.