Ind. Eng. Chem. Res. 1993,32, 911-916
911
Permeation and Separation of Zinc and Copper by Supported Liquid Membranes Using Bis(2-ethylhexy1)phosphoric Acid as a Mobile Carrier Ruey-Shin Juang Department of Chemical Engineering, Yuan-Ze Institute of Technology, Nei-Li, Taoyuan, 32026, Taiwan, ROC
The permeation of zinc and copper ions from sulfate solutions through supported liquid membranes containing bis(2-ethylhexy1)phosphoric acid (D2EHPA) dissolved in kerosene as a mobile carrier was investigated at 25 O C . A permeation model was presented taking into account the aqueous film diffusion of metal ions toward and out of the membrane and the membrane diffusion of D2EHPA and ita metal complexes. It was found that the calculated permeation rates were in good agreement with the measured results. Higher selectivity in the separation of zinc and copper was obtained when the diffusional resistance in the membrane phase was dominant. Introduction Liquid membrane process is a novel technique for the selective separation and concentration of the solute of interest from dilute solutions, since it combines the processes of extraction, stripping, and regeneration into a single stage (Loiacono et al., 1986; Noble et al., 1989). Supported liquid membranes (SLM), which use a porous polymer membrane impregnated with complexing carriers to separate the feed and strip phases, represent one of the feasible types of liquid membranes. They have been applied mostly to study the transport of individual species (Duffey et al., 1978; Fernandez et al., 1986,1987; Huang and Juang, 1987, 1988; Imato et al., 1981; Plucinski and Nitsch, 1988) and the separation of species from multicomponent mixtures (Huang and Tsai, 1991; Loiacono et al., 1986; Matsuyama et al., 1987). The enrichment or separation of Zn(I1) and Cu(I1)from sulfate solutions by solvent extraction is of major interest in the hydrometallurgicalprocess (Beneitezand Calderon, 1990; Forrest and Hughes, 1978; Hughes, 1975). Although an oxime type reagent, for example, LIX or SME 529, could be used to remove small amounts of Cu(I1) from an aqueous feed containing large amounts of Zn(II), a sharp separation of Zn(I1)from Cu(I1) can be achieved,according to the solution pH, by use of bis(2-ethylhexy1)phosphoric acid (abbreviated as D2EHPA or simply HR) (Cox and Flett, 1987;Forrest and Hughes, 1978;Hughes, 1975).The purpose of this paper is to experimentally study the permeation in the separation of Zn(I1) and Cu(I1) through SLM containing D2EHPA and to compare it with those calculated by a competitive permeation model using the transport and extraction equilibrium data. As has been demonstrated in several previous studies of solvent extraction (Cianettiand Danesi, 1983;Dreisinger and Cooper, 1989; Miyake et al., 19901, the rate of extraction of Zn(I1) or Cu(I1) by D2EHPA is not exactly controlled by the complex formation reaction at the aqueous-organic interface, but is controlled mainly by diffusion of metal ions in the aqueous stagnant layer or by diffusion of the complex in the organic layer. Using an SLM impregnated with DPEHPA, the transport of single Zn(I1) or Cu(I1) ions has been frequently investigated (Fernandez et al., 1986, 1987; Huang and Juang, 1987,1988);however, fewer works have been carried out in the separation of Zn(I1) and Cu(I1). On the other hand, Matsuyama et al. (1987) have modeled the permeation rate of Co(I1) and Ni(I1) using 2-ethylhexylphosphonic acid mono-2-ethylhexyl ester as a mobile carrier by neglecting the resistances of some steps such as
interfacial reactions, diffusion of hydrogen ions in both liquid phases, and diffusion of metal ions in the strip phase. It seems inapplicable to this system for ignoring the diffusional resistance of metal ions in the strip phase, as contrasted with the reaction mechanism deduced in solvent extraction and as also seen in the work of Danesi and Reichley-Yinger (1986). Accordingly, much work must be done to clarify the permeation mechanism over a wider range of operation conditions. Modeling of the Permeation Rate ExtractionEquilibrium. The reaction stoichiometry of Zn(I1) and Cu(I1) extractions with D2EHPA has been extensively studied (Grimm and Kolarik, 1974;Huang and Juang, 1986; Komasawa et al., 1981). The extraction equilibrium of divalent metals at low organic loading can be generally represented by the following reaction: M2++ [(2 + n)/2l(HR),
MR,(HR),
+ 2H+
(1)
Also, the extraction equilibrium constant K,, is given by
where n = 1 for Zn(I1) and n = 2 for Cu(I1). The value of K,, by eq 2 used in this work is found to be 0.15 ( m ~ l / m ~ )and ' / ~ 1.20 X 10-4 at 25 OC, respectively, in the extraction of Zn(I1) and Cu(I1)from 5 X lo2mol/m3 (Na,H)SO1 aqueous media with kerosene solutions of D2EHPA (Huang and Juang, 1986). It is worth noting that, based on the equilibrium studies of metal extraction, the reaction stoichiometry expressed by eq 1is applicable only over a range of loading ratio of DPEHPA, defined as [MR,lJ[(HR),lo, less than 0.12 for Cu(I1) and 0.16 for Zn(I1) (Huang and Juang, 1986; Komasawa et al., 1981). Here, [(HR),10 denotes - the initial concentration of dimeric DBEHPA and MR, represents the metal-D2EHPA complex MR,(HR), hereinafter in this work for simplicity. With an increase in loading ratio, the monomeric species MR, tends to be broken up and some aggregated species appear (Komasawa et al., 1981). Permeation Model in SLM. The transport and concentration profiles of each species through SLM using D2EHPA as a mobile carrier is illustrated in Figure 1.In the present permeation model, the followingtwo steps are taken into account, i.e., (1)diffusion of metal ions (Zn(I1) and Cu(I1)) across the aqueous stagnant layers of feed
0888-588519312632-0911$04.00/0 0 1993 American Chemical Society
912 Ind. Eng. Chem. Res., Vol. 32, No. 5, 1993 feed solution
bulk
,
liquid
strip
membrane
SOlUtiOn
film ,
film
L
bulk
Figure 2. Permeation cell used in the permeation experiments: (1) supported liquid membrane; (2) Teflon spacer; (3) stirrer. (Dimensions given in millimeters.)
s,i x=L
f,i x=o
Figure 1. Concentration profile of each species through SLM containing D2EHPA as a mobile carrier.
and strip phases and (2) diffusion of D2EHPA (monomer and dimer) and the complexes in the membrane phase. The permeation rates of these steps are as follows: 1,2. Diffusion of metal ions in the feed phase toward the membrane: J1 = kz,,,f([Zn2+lf
- [Zn2+lf,i)
(3)
J2 = kc,,f([C~~+If[Cu2+If,i)
(4)
3-6. Diffusion of the complex and carrier through the liquid membrane:
equation can thus be obtained.
In eq 12 it was implicitly assumed that the dimerization reaction of DBEHPA (eq 13) was in equilibrium (Huang and Juang, 1986; Komasawa et al., 1981).
- -
2HR e IHR\-
- -
K- = T(HR\-1/rHRl2
(I.?\
The dimerization constant K2 is taken to be 12.0 ms/mol (Juang and Su, 1992). Since the counterdiffusion occurs in the membrane, the continuity of the total flux of DBEHPA, originated from the species ZnR,(HR), CuR,(HR),, (HR),, and is expressed by
m,
+ 4 J d = % 7 5 + J6
(14) At steady state, the following equalities hold for Zn(I1) and Cu(I1) systems, respectively. 3J3
J1= J3 = J7 and J2 = J4 = Ja
7,8. Diffusion of metal ions in the strip phase outside the membrane:
J8
kcu,s([Cu2+la,i - [Cu2+1,)
(10)
In the description of the permeation model, the following assumptions were made: (1)the resistance of interfacial reactions between DBEHPA and metals was ignored, (2) the resistance of diffusion of hydrogen ions in both stagnant layers was also neglected, and (3) the diffusion processes could be described by Fick’s diffusion equations. It should be noted that the first assumption has been justified for the permeation of Zn(I1) through SLM containing DBEHPA (Fernandez et al., 1986,1987). The solubility of D2EHPA in acidic aqueous solutions is considered to be negligibly small (Huang and Juang, 1986; Komasawa et al., 1981), so the total amounts of DBEHPA in the membrane phase is kept constant:
-
E(HR),I0 = (l/L)SU(HR),l
+ (1/2)[mI +
Assuming that the concentration profile of each species present in the membrane phase is linear, the following
(16) Provided that the mass-transfer coefficients and extraction equilibrium constants are known, it is possible to solve the system of eqs 3-10 with the additional conditions of eqs 12-15 using the trial and error method. In this work, assuming first that [M2+1,= 0 at least at the early , ~for simplicity, stage of permeation process and k ~= kM,f we could express [M’+]f,i as a function of both [(HR),lf,i and [(HR),],, from eq 16. The resulting expressionswere then substituted - into eqs 12 and 14 to calculate [(HR),],, and [(HR),IBiand hence the permeation rate from eqs 3 and 4. Ab the calculations were performed by IBM PC equipment ueing a powerful program EUREKA: THE SOLVER (product of Borland International Co.) for solving such a system of nonlinear equations. It is expected that the present permeation model is applicable and valid over a range of relatively low loading ratios of DBEHPA, as indicated above, since eqs 12, 14, and 15are basically established on the basis of the reaction stoichiometry expressed by eq 1. This point will be examined in the following section. Experimental Section Apparatus and Membrane. The apparatus used in this study was similar to the diaphragm cell as illustrated in Figure 2. It consisted of two chambers (feed and strip
Ind. Eng. Chem. Res., Vol. 32, No. 5, 1993 913 phases), each of about 1.20 X lo4 m3, separated by a membrane support of 9.40 X lo4 m2 area sandwiched between two 2-mm-thick Teflon spacers. Two chambers were synchronously stirred a t 600 rpm with a Cole-Parmer variable-speed agitator driven by a Master Servodyne system. The entire cell was immersed in a thermostat controlled a t 25 "C. The porous membrane support used was the hydrophobic Durapore HVHP disk filters, a product of Millipore Co., made of poly(viny1idene difluoride). It had a nominal mean thickness of 125pm, an average pore size of 0.45 pm, and a typical porosity of 75%. Reagents and Solutions. D2EHPA was the product of Merck & Co. and had a purity of approximately 98.5 % determined by potentiometric titration of an 80 vol % ethanol solution of the acid with 1 X lo2 mol/m3 NaOH in ethanol. It was further purified following the method reported elsewhere (Huang and Juang, 1986; Komasawa et al., 1981). The diluent kerosene, offered from Union Chemical Works Ltd.,Hsinchu, Taiwan, was washed twice with 20 vol % HzSO4 to remove aromatics and then with distilled water several times. All other inorganic chemicals were also supplied by Merck & Co., as analytical reagent grade. The membrane phase was prepared by diluting DPEHPA with kerosene and presaturated two times with the metal-free aqueous phase. The initial concentration of dimeric D2EHPA in the membrane phase ranged from 5 to 2.5 X 10, mol/m3. The pores in membrane support were fiied with D2EHPA carrier under vacuum as reported previously (Huang and Juang, 1987,1988;Huang and Tsai, 1991),and the resulting liquid membranes were immersed in the membrane phase before use. The feed phase was prepared by dissolving equimolar amounts of ZnSO4 and CuSO4 in distilled water, to which 5 X 102 mol/m3 (Na,H)S04 was added to maintain the ionic strength constant. The feed pH was changed by adjusting the fraction of the combinations of Na~S04and H2SO4. The initial concentration of Zn(I1) or Cu(I1) ions varied from 0.91 to 2.02 X 10, mol/m3 and the feed pH was in the range of 1.51-5.00. The strip phase also contained 5 x 102 m0l/m3 (Na,H)S04. All aqueous phases were presaturated with kerosene. Procedure. In the beginning of each run,the membrane impregnated with D2EHPA carrier was first clamped, and the apparatus was assembled as shown in Figure 2. Then the feed and strip phases were introduced into the chambers A and B, respectively, and the agitator was immediately started at a chosen stirring speed. It was found that the evaporation of kerosene solvent from membrane supports was negligible in this work while mounting and starting experiments. When steady state was reached (about 30 min), sample (1-2 cm3) was taken from the strip solution a t preset intervals and the original strip solution was added to maintain the volume unchanged. The feed pH was measured with a pH meter (Radiometer Model PHM82). The concentrations of Zn(I1) and Cu(I1) in the sample were determined with an atomic absorption spectrophotometer (Perkin-Elmer Model 5100 PC) at wavelengths of 213.9 and 324.8 nm, respectively, and corrections due to volume replacement were made. The permeation rate was thus obtained according to JM = (Va/A)(d[M2+la/dtj (16) Results and Discussion Evaluation of the Mass-Trwsfer Coefficients. The values of kznp and kcu,fwere determined from the per-
Table 1. Values of Parameters Used for the Calculation In This Work parameter Zn(I1) system Cu(II) system D2EHPA dimerization const KP 12.0 mg/moi equilibrium const Ke I 0.15 (m01/m3)1/2 1.20x 1 0 - 4 mass-transfer coeff 2.64 X 106 m/s 2.76 X 10-6m/s kM,f = kM,s kMR2,m ~ ( H R ) ~ , ~
kHR,m
3.16 X 1C6m/s
2.74 X 10-6m/s
3.74 X 106 m/s 5.20 X 10-6 m/s
meation data under conditions of low metal concentration, high feed pH, and high extractant concentration, in which diffusion of metal ions in the feed solution was ratecontrolling (de Haan et d.,1989; Matsuyama et al., 1987; Plucinski and Nitach, 1988). In this study, such values were determined when [Zn2+lf= [Cu2+1f= 1.82 mol/m3, [H+lf= 1 X IO-' mol/m3, and [(HR),],, = 5 X 10 mourn3 (not shown). The parameters obtained and used in this work are compiled in Table I. The diffusivities of Zn(I1) and Cu(I1) in the aqueous phase containing 5 X 10, mol/m3(Na,H)S04are estimated to be 6.42 X 10-lo and 6.53 X m2/s, respectively (Awakura et al., 1988); a reasonable thickness of the aqueous stagnant layer of about 3.0 X 1V m is hence obtained (Huang and Juang, 1987;Huang and Tsai, 1991). Furthermore, the value of kZnRs,m was calculated from the flux obtained in the experiment using the diffusion cell where Zn(I1) was transported from an organic phase loaded with trace Zn(I1) to the strip phase through the same membrane support as employed in SLM. The values ,, and k H R p were thus estimated from of ~ C ~ R ~ . , ,k(HR)z,m, kznRz,m by the relation between the diffusivity and molar volume (Miyake et al., 1990; Reid et al., 1987). In fact, the diffusivities of D2EHPA and its metal complexes in the bulk kerosene phase can also be estimated as 7.45 X m2/s for(HR),, 1.22 X m2/s for HR, 5.84 X W0 m2/s for ZnR,, and 4.76 X 10-lo m2/sfor CUR, at 25 "C by the Hayduk and Minhas correlations (Reid et al., 1987)on the basis of a viscosity of pure kerosene of 1.34 X Pas. In this estimation, the molar volume of DPEHPA monomer at boiling point is taken as 405 cm3/mol (Schotte, 1992), and those of Zn(I1)- and Cu(II)-DSEHPA complexes are assumed to be 3 and 4 times that of DPEHPA monomer, respectively (Miyake et al., 1990). Considering the actual thickness (112 pm), porosity (0.75),and tortuosityfactor (1.67) of themembranesupport used (Duffey et al., 1978; Huang and Juang, 1987; Imato et d.,1981),the values of kZnRn,m and k C a z , m of 1.41 X 10-6 and 1.15 X 1Pm/s, respectively,are thus estimated, which are still considered to be in broad agreement with the measured ones as listed in Table I. Such discrepancies, on the other hand, may result from the anomalously high diffusivitiesoccurring in small pores, as also seen by Duffey et al. (1978). They suggested that this type of effect is a consequence of surface diffusion or rapid transport along the wall. Comparison of the Calculated and Measured Permeation Rates. Figure 3 shows the effect of the concentration of metal ion in the feed solution, [M2+lf,on the permeation rates of Zn(I1) and Cu(II), J z n and Jcu. It should be reminded that equimolar amounts of Zn(I1) and Cu(I1) are present in the feed solution. The solid lines are calculated by the present permeation model. Close agreement between the experimental and calculated rates is obtained.
lo
"kt
10 4
-I,
n
10 -I 10 -'
1
8
1
10
10 -'
lo-',
W
8
c,
10
-'
10
10
1
10
-"L lo-'
10
nl
lo-'
lo-'
lo-'
[H+I, Figure 3. Effect of m g o n concentration in the feed solution on the permeationrate: [(HR),!, = 50 mol/m3;[H+l, = 1 X 103 mol/m3; [H+]f= 1 (line 1 and data points) and 10 mol/m3(line 2). The solid lines are calculated by the present permeation model. The dashed line is obtained when diffusion of Zn(I1) ions in the feed solution is rate-controlling,
The calculation results when the permeation process is controlled by diffusion of Zn(11) ions in the feed solution are also shown as the dashed line in the figure. In Figure 3, the region of lower [M2+lfcorrespondsto this case, and a smaller difference between Jzn and Jc,,is apparently observed. It is also found that& increases with increasing [M2+lf and finally reaches a plateau, as commonly seen in the single permeation run (Fernandez et al., 1986,1987;Huang and Juang, 1987, 1988). At low [H+lf,JcU increases as [Cu2+]fincreases for lower [M2+lfbecause the concentration of free DBEHPA at the interface is in excess compared to [MZ+]f,due to a relatively low distribution ratio of Zn(I1). However, Jc,, is drastically lowered by the presence of Zn(I1) at higher [M2+]fbecause most of the extractant at the feed-membrane interface is exhausted by the complexation with Zn(I1) which has a much higher distribution ratio than Cu(I1) (Huang and Juang, 1986; Komasawa et al., 1981). Therefore, the plot of Jc,, versus [Cu2+1fhas a maximum. This is supported by the lack of such phenomena in the case of high [H+lf,in which the conversion of DBEHPA is always small (Le., low organic loading). The effect of the concentration of hydrogen ion in the feed solution, [H+lf,on Jzn and Jcuis shown in Figure 4. It is evident that both J z n and Jcudecrease with increasing [H+]f,as also seen in the single permeation run. In the case of high [H+]f,resistance of the diffusion of metal ions in the feed solution becomes less dominant. In this regard, the permeation process may be governed mainly by the membrane diffusion and by some extent of equilibrium distribution at the interface (Plucinski and Nitsch, 1988). Figure 5 shows the effect of the concentration of dimeric DBEHPA in the membrane solution, [(HR),],, on Jzn and Jc,,. Under the ranges studied, both Jznand JcUincrease as [(HR),], increases and finally approaches a plateau. Evidently, the permeation process is practically controlled by diffusion of metal ions in the feed solution over a wider range of [(HR),], at low [H+lfthan that at higher [H+lf. In fact, the role of membrane diffusion becomes more
1
I
1111,1111
10
10'
(moi/m3)
Figure 4. Effect of hydrogen ion concentration in the feed solution on the permeatione: [H+l, =I 1 X lo3mol/m3;[Zn2+lf= [Cu2+1r = 25.3 mol/m3, [O,], = 50 mol/m3(line l),and [Zn2+lf= [Cu2+lf = 1.82 mol/m3,[(HR),], = 10 mol/m3(line 2 and data points). The meaning of the solid and dashed lines is the same as that in Figure 3. 10 -
3
3
1
!-
lo
4
-I
10 - 6 k
e
-'
10 lo-/
b
b
lo
-'I' 1 Zn: o
cu:0 * ' - o10 l -IQ
10
[oelo
10'
10'
(mo1/m3)
Figure 5. Effect of initial D2EHPA concentrationin the membrane solution on the permeation rate: [Zn2+lf= [Cu2+1f= 25.3 mol/m3; [H+], = 1 X 103 mol/m3; [H+]f= 0.1 (line 1) and 1 mol/m3 (line 2 and data points). The meaning of the solid and dashed lines is the same as that in Figure 3.
important for low [(HR),], as mentioned above. Moreover, the lower the value of [(HR),],, the greater the extent of resistance of the membrane diffusion to the overall permeation resistance, hence the larger the difference between Jzn and Jc,,(de Haan et al., 1989). Figure 6 shows the effect of the concentration of hydrogen ion in the strip solution, [H+l,, on Jzn and Jcu. It is evident that, under the ranges studied, an increase in [H+l, from about 10 mol/m3 has no effect on both Jzn and Jcu.This indicates once more that for the described condition the permeation processis controlledby combined diffusion of metal ions in the feed solution and of metal complexes through the liquid membrane (de Haan et al., 1989; Plucinski and Nitsch, 1988). However, decreasing
Ind. Eng. Chem. Res., Vol. 32, No. 5, 1993 915 attractive by SLM using D2EHPA as a mobile carrier under suitable conditions. Conclusions
8
c,
Zn:o cu: e 10 - 7 b , 10 -'
, ,,,,#,1
, , , , ,,,,, , , ,,,,,,1
, , ,,,,,d , ,J
10
PI+],
10'
(mo1/m3)
Figure 6. Effect of hydrogen ion concentration in the strip solution on the permeation rate: [Znz+lf = [Cu*+]f= 25.3 mol/m3; [(HR),], = 25 mol/m3; [H+lf = 0.1 (line 1 and data points) and 1 mol/m3 (line 2). The meaning of the solid and dashed lines is the same as that in Figure 3.
[H+], from 10 mol/m3 to a lower value causes the permeation rate to decrease. This is connected with the decrease of the driving force for diffusion through the lipid membrane; i.e., the concentration of metal complexes at the membrane-strip interface, [MR,l,,i, increases. As shown in Figures 3-6, the somewhat larger deviations between the measured and calculated permeation rates are generally, although not apparently, observed under the conditions of high metal concentration, high feed pH, and low DBEHPA concentration. This corresponds to the cases with relatively high loading ratios of DBEHPA, as indicated by eq 1. Selectivity Factor. The selectivity factor 0 is commonly defined as follows (Huang and Tsai, 1991; Matsuyama et al., 1987):
P
The permeation rate and selectivity in the separation of Zn(1I) and Cu(I1) ions from 5 X lo2mol/m3 (Na,H)S04 solutions through SLM containing DBEHPA have been examined at 25 "C. The competitive permeation rate equations are derived considering the aqueous film diffusion of metal ions toward and out of the membrane and the membrane diffusion of DBEHPA and ita metal complexes. It is found that the calculated rates are in good agreement with the measured ones under the conditions studied. The application of such a model first requires knowledge of the transport parameters for the relevant geometry. Consequently,with known equilibrium data the modeling of the permeation would be possible in the case of relatively fast interfacial reaction. It is also found that a higher selectivityin the separation of Zn(I1) and Cu(I1) is obtained when the diffusional resistance in the membrane phase is dominant. Acknowledgment This work was supported by the ROC National Science Council under Grant No. NSC82-0402-E155-025, which is greatly appreciated. The author would also thank Mr. J. Y. Su for performing the experiments. Nomenclature
A = effective membrane area, m2 HR = monomeric form of D2EHPA (HR)2 = dimeric form of D2EHPA J = molar flux, mol/(m2.s) k = mass-transfer coefficient, m/s K2 = dimerization constant of D2EHPA, m3/m0l K,,= extraction equilibrium constant defined in eq 2 L = membrane thickness, m M = metal ion, Zn(I1) or Cu(I1) MR2 = simplified form of the metal-D2EHPA complex = (J~,/J~~)/([z~~+I~/[CU~+I~) (17)
If diffusion of metal ions in the feed solution is ratecontrolling, 6 can be represented by eq 18.
P OC kZn,f/kCu,f 1 (18) This case is encountered in the region of low [M2+lfin Figure 3, low [H+lf in Figure 4, and high [(HR),], in Figure 5. If resistance of the diffusion of metal ions in the feed solution is negligibly small, the permeation process is governed by the membrane diffusion, including the effects of equilibrium distribution and the diffusivities of the complexes (Matauyama et al., 1987). Since in this work the diffusivities of these complexes are not significantly different, eq 19 can be approximately derived from eqs 2 and 17. P a Kex,zn/(Kex,Cu [(HR)J 01'2)
(19)
The regions of high [H+lfin Figure 4 and low [(HR),], in Figure 5 more or less correspond to this case. Apparently, a higher selectivity is obtained when the diffusional resistance in the membrane phase is dominant, for example at high [H+lfin Figure 4, as demonstrated in the above discussion. In conclusion, the separation of Zn(11)from Cu(I1) in sulfate solutions seems promising and
MRz(HRh
n = stoichiometry defined in eq 1 t = time, s V = volume, m3 x = distance from the feed-membrane interface, m [ I = molar concentration of species in the brackets, m0Vm3
Greek Letter
p = selectivity factor defined in eq 17 Subscripts f, m, s = feed, membrane, and strip phases, respectively i = aqueous-membrane interface t = total value at equilibrium 0 = initial Superscript - = species in the organic or membrane phase
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Received for review August 3, 1992 Revised manuscript received December 7, 1992 Accepted January 8,1993
