- Email: [email protected]
Mechanistic models of mass transfer across a liquid membrane
Journal of Membrane Science, 20 (1984) l-24 Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands
1
Review MECHANISTIC MEMBRANE
MODELS OF MASS TRANSFER
&ROSS
A LIQUID
CHIH CHIEH CHAN and CHAU JEN LEE* Department
of Chemical Engineering,
National Tsing Hua University, Hsinchu (Taiwan)
(Received October 24, 1983; accepted in revised form January 27, 1984)
The five mechanistic models dealing with solute removal rate in liquid membranes, as they appear in the literature, are critically reviewed and discussed. The validity of these models is evaluated and compared against the experimental data of Ho et al. [ 231 and Kim et al. [ 281. It was interesting to find that Models 1, 2, 3 and 5, with the adjustable parameter of membrane film thickness, fitted the data of Kim et al. reasonably well; however, Models 4 and 5 with a proper choice of effective diffusivity fitted better the data of Ho et al. The results of this paper will direct our further studies on continuous separation processes utilizing liquid membranes.
1. Introduction Artificial membranes have been a topic of great interest in recent years because of their potential applications, ranging from desalination of water to timed release of drugs. One of the most intriguing classes of artificial membranes is the liquid membrane. Liquid membrane technology is taken to mean those processes in which a simultaneous extraction/stripping process takes place across a selective liquid separating phase. In general, there are two main types of liquid membrane systems that have been considered for practical applications - the liquid surfactant membranes or emulsion-type liquid membranes [2-8,11,18,19,21-23,29,31-36,40,41,43,46,49-541 and the supported liquid membranes [ 10,12,13,15-17,25-27,30,37,42,45]. Liquid surfactant membranes are made by forming an emulsion of two immiscible phases and then dispersing the emulsion in a third phase (the continuous phase). Usually, phases separated by a membrane are completely miscible. The emulsion is stabilized by surfactants. Supported liquid membranes are formed by absorbing a suitable solution into a microporous sheet. Here the liquid layers are stabilized by capillary or surface forces. The liquid membrane process has found applications in hydrometallurgy [6-19,29-34,43-46,49,50,54], hydrocarbon separation [ 1,11,21,36,48,51, 521, waste water treatment [ 9,18,22-24,29,35,40,41,53], and biomedical engineering [ 2-5,20,38,39], etc. The mathematical models of these systems *To whom correspondence 0376-7388/84/$07.20
should be addressed.
0 1984
Elsevier Science Publishers B.V.
2
are important in the interpretation of laboratory data and in the design and scale-up of such systems. A number of mathematical models which describe the real behavior of supported liquid membrane separation processes and also of liquid surfactant membrane separation processes have been proposed [ 10,12-17,25-28,37,42,45]. In this paper, the various modeis which are commonly used to describe the first type of facilitated mass transfer phenomena [ 231 in an emulsion drop are reviewed and discussed. Furthermore, the parameters and their physical significance of each model are clearly identified and compared. Finally, the type of experimental data from separation processes in batch-type operation which are required for model evaluation are identified and collected from the published literature. This enables the comparison in applicability and usefulness of each model in a specific separation process utilizing liquid membrane technology. 2. Mechanism of Type 1 facilitated transport of solute through an emulsion drop There are two types of facilitation method to improve the mass transfer rate of a solute by using a liquid surfactant membrane separation system called Type 1 and Type 2 facilitation, respectively [ 231. In the first type of facilitation, the concentration gradient of the membrane-soluble permeate is maximized by irreversibly reacting the solute with the reagent into an impermeable form in the receiving phase, and thereby maintaining the permeate concentration effectively zero in this phase. In Type 2 facilitation, a transport facilitator or “carrier” incorporated in the membrane phase carries the diffusing species across the membrane to the receiving phase. This is commonly known as “carrier-mediated” transport. Since there have been a number of investigations of the second type of facilitated transport, we do not deal with it in this paper. Consider an emulsion drop (say water-in-oil emulsion), as shown in Fig. 1, consisting of an aqueous solution of reagent B in internal fine droplets suspended in the external aqueous phase of permeable solute A. Solute A has an appreciable membrane solubility and readily permeates from the outside aqueous phase through the oil membrane into the encapsulated aqueous re-
External
phase
Membrane
Fig. 1. Schematic
phase
diagram bf emulsion liquid membrane system.
3
agent phase and it is trapped in the impermeable form, P, by an instantaneous and irreversible chemical reaction, aA+bB
ks
-P
(1)
In order to express the global transport rate in terms of the external bulk properties, expressions must be formulated for each of the transfer steps in the overall process. In brief, the sequence of steps for transporting solute A through the membrane into the encapsulated phase is as follows. (1) Transport of solute A from the bulk fluid phase to the fluid-membrane interface (external surface of the emulsion drop). (2) Solute A dissolves from the external phase into the emulsion membrane phase:
(3) Diffusion of the dissolved solute A across the membrane to the internal interface (membrane phase-internal phase interface). (4) The dissolved solute A is extracted from the membrane phase into the internal phase
(5) The extracted solute A reacts with the encapsulated reagent B to the impermeable form P, eqn. (1). (6) The impermeable product, P, cannot dissolve in the membrane phase and is trapped in the encapsulated droplet phase. At steady state, the rates of all the individual steps will be the same. This equality can be used to develop a global transport rate equation in terms of the concentrations of the external bulk fluid phase. The derivation of such equations will be considered in detail in the subsequent sections for different emulsion drop models. 3. Global transport
rate in various emulsion drop models
For modeling Type 1 facilitated transport within an emulsion drop, five types of models have been proposed: (1) Uniform flat-sheet model. (2) Hollow sphere model. (3) Hollow sphere-advancing front model. (4) Immobilized globule-advancing front model. (5) Immobilized hollow spherical globule-advancing front model. In the following sections, we first examine the various above conceptual models and compare these models in the light of the rate-controlling factors of mass transfer phenomena. Then, we attempt to show how a suitable model may be selected for a batch-type liquid membrane separation system. To
simplify the mathematical treatment, the following assumptions are made: (1) Isothermal process. (2) Constant physical and transport properties. (3) Uniform internal droplet and emulsion drop size. (4) No drop break-up and coalescence occurs. (5) Excess reagent concentration in the internal phase. (6) The chemical reaction is irreversible and can be represented by eqn. (1). Since the concentration of internal reagent B is so much greater than A at the reaction interface, it may be combined with the rate constant and the reaction rate is assumed to be linearized with respect to the concentrations of A and B, i.e., = kc/, (4) -rs = k&&A (7) The transport process is at pseudo-steady state. This approximation is reasonable for most liquid membrane cases of practical interest, since the parameter E = a(vi
+ Vm)Go
(5)
KG0
is less than 1, as pointed out by Ho et al. [ 231. 3.1 Uniform flat-sheet model (Model 1) This model assumes that all of the fine droplets within the emulsion are coalesced into a single large droplet, and the mass transfer process consists of diffusion across a stagnant membrane of thickness 6. Furthermore, the membrane thickness is assumed to be negligible compared to the drop radius, so that it may be considered that the mass transfer surface area is constant, i.e., uniform flat membrane thickness. The reagent is well mixed in the coalesced internal droplet. The solute diffuses into the interior interface where it is removed according to eqn. (1). The assumed physical situation for this model of transport system is depicted in Fig. 2a. At pseudo-steady state, the global transfer rate of solute A per unit external surface area, rA, can be expressed as 1 hA -_-= rA- 47rRZ dt
1 R-Ri -+++kL Q,
1 %k 3
-’
C, = K,C,
(6)
where Cme = QeCem
(7)
and the overall mass transfer resistance is 1 1 -_=-+ KI kL
R-Ri drn
+L hk
(8)
3.2 Hollow sphere model (Model 2) This model is similar to the uniform flatsheet model except that a hollow
5
spherical membrane film is used instead of the uniform flat membrane film. All assumptions used before also apply in this model. Referring to Fig. 2b, the global mass transfer rate of solute A per unit external surface area becomes, rA=
[$
+
(+&)
(t)
(R-&I+
(2)
(z) = K,Ce
‘]
-lCe (9)
and the overall mass transfer resistance is -
1
= &
+ (A)
(g)
(R-&I+
(&)
(c,’
(10)
K2
x=Ri
x=R
r=O
RiR
(Cl
Fig. 2. Schematic diagrams of emulsion drop models. a: uniform flat-she& model; b: hollow sphere model; c: hollow sphermdvancing front model; d: immobilized globule-advancing front model; e: immobilized boilow spherical globule-advancing front model.
6
3.3 Hollow sphere-advancing front model (Model 3) This is basically the same as the hollow sphere model, except that all of the fine droplets within the emulsion are coalesced into a single large droplet, and this droplet is encapsulated in a stagnant hollow spherical membrane film. However, the internal reagent phase in this model is assumed to be immobilized and the solute reacts irreversibly with the reagent at a reaction surface which advances into the droplet center while the reagent is being consumed. Referring to Fig. 2c and combining with the distribution relationship of eqns. (7) and (11) C&i = LYiCi,
(11)
the global transfer rate of solute A per unit external surface area, rA, becomes
rA=[i$ +(&I ($1
tR-Ri)
+
(2) (&)
+ (2)
(i)
(Rs)(Ri-rc)
(CT]
-lc,
=&Ce
(12)
and the overall resistance is 1 = K3
d +(&) ($,tRwRi)+ (&) (&)(Ri-rc) e!) (El2 +
(13)
In eqns. (12) and (13), r, must be expressed as function of time for these equations to become useful design equations for such separation processes. According to the spherical geometry of the emulsion drop, the rate of reaction of B may be obtained as (14) Combining eqns. (12)-( -
14) gives rC = f(t), i.e.,
p=(&) (E)‘[k +(A) (+)(R-&I+ (15)
Equation (15) can be integrated to give rc as a function of C, and time. Substituting this expression for rC in eqn. (12), it gives the expression for the global transport rate in terms of C, and t.
7
3.4 Immobilized globule--advancing front model (Model 4) The schematic representation of a typical liquid surfactant membrane system, as shown in Fig. 1, indicates two key features which must be adequately described in any realistic model of transport and reaction in such a system. One is the emulsion heterogeneity resulting from the presence of internal droplets, the other is the non-uniform size distribution of the globules. The proposed model (Fig. 2d) attempts to approach this realistic behavior. In this model, the solute diffuses through the external boundary layer and the re acted globule to the reaction front where it is removed by an irreversible then ical reaction, eqn. (1). The reaction front advances towards the center as the reagent is consumed. To simplify the mathematical treatment, the following assumptions were made in addition to the previous assumptions (l)-( 7): (8) No internal circulation within the emulsion drop. (9) Local equilibrium exists between the internal phase and the membrane phase. Based on the above assumptions and the definition of solute concentration in the exhausted region of emulsion globules, as used in the paper by Ho et al. [ 231, i.e.: ViCi + VmCm = (vilai
r_
,-
Vi + Vm
+ vUl
\ Vi + Vm
c
1 m
(17)
Cl ,.=R = “‘,,
( I(
a=
2.E ai
(16)
Vi + CuiVm Vi + Vm 1
(IS)
then the global transport rate of solute A per unit external surface area in this model becomes (19) [$ +(2) @R-rd+($7)(5)’ J-1ce=K4ce
rA=
and the overall mass transfer resistance is 1 -
&
=
& +(&) (t)(R-r.)+($) (;)
(20)
Since r, is a time-dependent variable, we must express r, as a function of time. A material balance over the reaction front gives -
$[+;
!,+“,)c&,] =($)9=($)(4”R2)r_4
Substituting eqn. (19) into eqn. (21) we obtain after rearranging
(21)
8
-
%=(&)
(“+,“-)(:)*[$
+
(&)
(E)@-rJ
(22) and thus, eqns. (19) and (22) may be solved simultaneously to give the global transport rate of A in terms of C, and t. 3.5 Immo bilked hello w spherical globule-advancing front model (Model 5) Similar to the “immobilized globule-advancing front” model, this model assumes that the internal droplets are distributed heterogeneously in the emulsion globule. In addition, from a practical point of view, a peripheral membrane layer with an infinitesimal thickness is assumed to exist around the droplets. Therefore, the transfer of solute from the external phase towards the emulsion is in the following three steps: (1) diffusion through the external boundary layer, (2) diffusion through the peripheral membrane layer, and (3) diffusion through the interior bulk emulsion globule. Referring to the schematic diagram as shown in Fig. 2e, and combining all the transport steps and rearranging, the global transport rate of solute A per unit external surface area becomes
vi+v’ Vi& +
X
vz )+
(2)
(v;a++TE) e
i
(;,‘]-lCe=K&e
(23)
i
Therefore, the overall resistance is 1 -
=& +(--&--)(g)CR-&)+ (ak) (&) (Ri-rc)
KS
X
(
Combining eqn. (23) and the material balance equation over the reaction front gives
(24)
9
(25) Therefore, the global transport rate in terms of C, and t can be obtained by solving eqns. (23) and (25) simultaneously. Table 1 summarizes the global transport rate equations based on the, five emulsion drop models previously discussed. In order to evaluate the usefulness of these models, a separation process utilizing membranes in a batch stirredtank system was selected for consideration and comparison. 4. Liquid membrane process in batch stirred-tank system In a batch stirred-tank system, the material balance on solute A in the external phase may be given as
dCe -=
-v,
dt
(4nnR2)rA =
I.C. t = 0,
+ R
3(Vi
vm)
rA (27)
c, = c,
With the expressions for global transport rate, rA, as obtained from each model (Table l), we can solve eqn. (26) to give the “conversion” of solute A as a function of time. This information enables us to design a batch separation process for a specified system. The results based on; such analysis can be summarized as follows: (1) Uniform flat-sheet model:
” =exp[c,
i
(vi\:)R1t]
(28)
(2) Hollow sphere model:
c,
GO
=exp [-
i
( viievm)K2t]
(3) Hollow sphere-advancing
(29)
front model: (36)
t=
s
ln(A+Br3)12+
+Ep 3A
Y[tln 3RP
“,:f)y
10
I
.
. b
I
:
1
3
’ .
0
I
,
.
.
> . I
. .
.
I
.
! Y .
,
.
.
a
3
J
Y
.
I
1 /
a J
.
a , I I
II
r
P
-h
a
-0 I
11
where
(4) Immobilized globule-advancing
front model: (32)
t= g
In (A’ +B’r”)IE
+ -
D’
3B’P’
$ In
A’ + B’r3 (8’ + r13
(33)
where A’=p B’=
(2) $(v’;evm)
( vi :,“)_p
( “ievm)
12
G&f-_
-
h
R
a&
p'= 3 $ 1/ (5)Immobilized hollow spherical globule-advancing
front model
2
[I-
=1-(Z)
(vi;vm)(vi,“;.)(;)3 eo
t=
-$-
E”P”
+3A”
t [
m
e
In (A” +B”r)
(34)
($)‘I
1% + -I!?-
rc
3B”P”
[
i In
” + r)3 In @ +fi A” + B”y3
tan-’
(‘iii’:)
(“i’rn)
A” + B”r3
(P” + r)3
(%)]/I
ul Ri
+fitan-’
_II (>p”
i r,
(35)
where AJ~=R~ (:z;)
_ f
e gr=
$
(viievm)
D” = RZ (2)
(vJy+z
m
1
vi + vg vi /C$+ vf
G”=;
+ (-$-)
)
cf)W
The solution form in the last three models as shown in eqns. (31), (33), and (35) is similar to the zero order or pseudo-steady state solution in the papers by Ho et al. [23] and Terry et al. [53].
13
5. Evaluation of parameters - Diffusion coefficients Prior to the mechanistic models being tested against the experimental data which appeared in the published literature, the effective diffusivity, De (appearing in models 4 and 5) must be correlated with the individual diffusivities of both phases in the emulsion. Casamatta et al. [ 111 assumed that the effective radial mass transfer area is proportional to the volumetric fraction of membrane phase in the emulsion. Ir other words, the total mass transfer rate is given by the following equations: NA = (1 - @)NA~ + @NAP i.e., dc = -(l dr
-#)D,
c = (1- $)C,
+#Ci
-D,
dGn 7
-#Di
$
(36)
since (37)
Equation (36) can then be written as
$=-
_D,
$/ffi
[
D
1 - 4 + #/ffi )I
iz
Cu:
(33)
Therefore, the effective diffusivity of solute in the emulsion mixture may be given by D,
=
l-4 1 - G+
#/ai
i
Ho et al. [ 231 used the effective diffusivity, DL, based on a concentration driving force defined in terms of the membrane phase concentration, C, , to calculate the effective diffusivity, De, based on the average concentration, C, in the emulsion mixture, that is D,
dC dr
=D;
dC, dr
(46)
Therefore, we can obtain (41) where O!=
(%1
[(I-
1
@l&i +
$1
(42)
14
The effective diffusivity, I&, can be estimated from the Jefferson-WitzellSibbett equation [ 141, as suggested by Ho et al. [ 231: Dk =Dm
n (I+
1
*)DAD,
4(1 + !P)2 - II
+ 4(1 + 2P)2
4(1 + 2P)2
D,
+ ~PDA
1
(43)
where
Dilai
ln
DA =
(44)
1
--
1
Dp
P = 0.4039-1’3
- 0.5
(45)
The individual molecular diffusivity of a given solute in each phase may be estimated by the Wilke-Chang correlation [ 551, the Scheibel correlation [ 471, or the Reddy-Doraiswamy correlation [ 441. 6. Comparison of model predictions with experimental data Figures 3-8 give the comparison of the model predictions with the experimental results given by Ho et al. [ 231 and Kim et al. [ 281, respectively. The experimental conditions of both Ho and Kim are shown in Table 2. In both experimental systems, phenol was extracted by using NaOH as an internal reagent. Since it is an acid-base reaction, the reaction rate is always fast.
---.__
0.6 --\.
.L_ R
0.4 .
.
4.’
..
-._
01 0
972
1944
-.
--.
---__
2916
4860
3808 Time,
5032
---____ . 6804
10
set
Fig. 3. Comparisonof model predictionswith experimentaldataof Ho et al. [ 231; N = 400 rpm; R = 0.05 cm. Curve1 (Ri = 0.9410R); curve 1’ (Ri = 0.7133R); curve 2 (Ri =
0.9443R); curve 2’ (Ri t 0.7133132); curve 3 (Ri = 0.9443R); curve 4 (Ri = R); curve 4’ (Ri = R); curve 5 (Ri = 0.9600R); curve 6: computed shrinking rate based on curve 4’.
curve 3’ (Ri = 0.7133R); curve 5’ (Ri = 0.9910R);
15
Therefore, the chemical reaction term in the model transport equations is negligible. In the figures, the circles represent the experimental data. Curves 1 and 1’ are the results for the uniform flat-sheet model computed with eqn. (28); curves 2 and 2’ those for the hollow sphere model with eqn. (29); curves 3 and 3’ those for the hollow sphere-advancing front model with eqns. (30) and (31); curve 4 for the immobilized globule-advancing front model with eqns. (32), (33) and (39); curve 4’ for the immobilized globuleadvancing front model with eqns. (32), (33) and (41); curve 5 for the immobilized hollow spherical globule-advancing front model with eqns. (34), (35) and (39); and curve 5’ for the immobilized hollow spherical globuleadvancing front model with eqns. (34), (35) and (41). All these emulsion drop models, except model 4, i.e., curves 4 and 4’, have assumed a “fictitious” men brane film to exist in the emulsion. However, no theoretical correlation has been proposed to estimate this membrane film thickness successfully. In some approximations, the internal phase was assumed to be coalesced into an internal large droplet inside the emulsion. In other words, the membrane phase became a hollow spherical film surrounding the whole emulsion drop. Therefore, the membrane film thickness (used in models 1, 2, or 3), was estimated by the following equation, 6
=K!-Ri=R[l-tviybm)“‘]
(46)
Fig. 4. Comparison of model predictions with experimental data of Ho et al. [ 231; N = 600 rpm; R = 0.03 cm, Curve 1 (Ri= 0.8944R);curve 1'(Ri= 0.7133R); curve 2 (Ri= 0.9045R),curve 2’ (Ri= 0.7133R); curve 3 (Ri= 0.904512); curve 3’ (RiE 0.7133R); curve 4 (Ri= R);curve 4’ (Ri=R); curve 5 (Riz0.9360R);curve 5'(Ri=0.9850R); curve 6: computed shrinking rate based on curve 4’.
16
. . .._...-.
---...
17 TABLE 2 Comparison of experimental conditions
of Ho et al. [23] and those of Kim et al. [ 281
Conditions
Ho et al.
Kim et al.
Qpe of separation Temperature Emulsification Type of emulsion R
Batch 23°C 12000 rpm w/o/w 0.05 cm; 0.03 cm 0.363 15.0 Aqueous phenol solution Phenol 400 rpm; 600 rpm 0.00819 N 780 ml 0.52 large Aqueous NaOH solution NaOH 0.375 N 18.876 ml 0.52 0.5 Mm 9.98 X 10e6 cm’/sec Oil phase 3 wt.% ENJ3029 + 96 wt.% SlOON 1 wt.% Span 80 33.124 ml 0.65 X lo-* cm’/sec
Batch 20°C 1300 rpm w/o/w 0.0264 cm 0.5 3.0; 4.0 Aqueous phenol solution Phenol 200 rpm 0.0106 N; 0.0318 N -
Vil(Vm + Vi)
vel(vm + vi)
External phase Solute Stirring
kL
Internal phase Reagent cBo vi ori
3 Membrane phase Oil base Surfactant
0.13 0.02 cm/set Aqueous NaOH solution NaOH 0.0626 N; 0.25 N 0.13 2.6 pm 9.3588 x 1Om6 cm’/sec Oil phase Paraffin oil (C,,_,,) Span 80 (HLB = 4.3) 0.4527 x lo-*
cm’/sec
The results computed using this approximation are shown in the upper curves (i.e., curves l’, 2’ and 3’) in Figs. 3-8. Due to the fact that the membrane film thickness estimated by this method was always too high and not quite realistic, the results using this approximation could not predict the removal rate successfully. We have tried several methods to estimate the film thickness from the knowledge of the phase volume fraction, 4, the emulsion drop size and the inFig. 5. Comparison of model predictions with experimental data of Kim et al. [ 281; Vel(Vi + Vm) = 3; Gee = 0.0106 N; Cne = 0.25 N. Curve 1 (Ri = 0.9892R); curve 1’ (Ri = 0.7937R); curve 2 (Ri = 0.9893R); curve 2’ (Ri = 0.7937R); curve 3 (Ri = 0.9893R); curve 3’ (Ri = 0.7937R); curve 4 (Ri = R); curve 4’ (Ri = R); curve 5 (Ri = 0.9906R); curve 5’ (Ri = R); curve 6: computed shrinking rate based on curve 4’. Fig. 6. Comparison of model predictions with experimental data of,Kim et al. [28]; Vel(vi + Vrn)= 3~ ceO = 0.0318 N; Cne = 0.25 N. Curve 1 (Ri = 0.9849R); curve 1’ (Ri = 0.7937R); curve 2 (Ri = 0.9851R); curve 2’ (Ri = 0.7937R); curve 3 (Ri = 0.9851R); curve 3’ (Ri = 0.7937R); curve 4 (Ri = R); curve 4’ (Ri = R); curve 5 (Ri = 0.9906R); curve 5’ (Ri = R); curve 6: computed shrinking rate based on curve 4’.
18
. .._.....
---
19
ternal micro-droplet size. However, they all failed to predict satisfactorily the experimental results. It seems, while Kim et al. [ 281 proposed a method to estimate the film thickness required for model 5, that one easy way to find the membrane film thickness is to back calculate it from experimental data by curve fitting, though this method may not be practical for application to design equations. The lower curves (i.e., curves 1, 2, 3, 4, 4’, 5, and 5’) in Figs. 3-8 show the computed results based on various models. Values of Ri (and thus, of the membrane film thickness) used in curves 1, 2, 3, 5, and 5’ were calculated by the equations mentioned above together with curve-fitting of the experimental data by the least-squares method. The calculated Ri values were shown in parentheses in the figures. Since no Ri value was required, curves 4 and 4’ were calculated directly by the equations given. Comparing with the published experimental data, curve 4 (i.e., combination of eqns. 32, 33 and 39) gave the worst predictions (too fast removal rate). Curves 1, 2, 3 and 5 predict well the data of Kim (Figs. 5--8), but not those of Ho (Figs. 3 and 4). Curves 4’ and 5’, however, better predict the data of Ho than those of Kim. The physical significance of such trends cannot be clearly concluded until further experimental data are gathered and analyzed. For curves 2 and 3, it is found that the calculated values of Ri of both curves are the same and that the removal rate predicted by these two models is almost identical. This result can be explained by the fact that the diffusivity of a solute in the internal phase is always larger by an order of magnitude than that in the membrane phase. Since model 2 is simpler mathematically, it would be preferred to model 3 when used in the design equations. Model 4 has the advantage over the other models in that the resulting equations are easy to apply, enabling the prediction of removal rate directly without the trouble of estimation of the film thickness. This is a significant improvement over the other four models, though it is not physically plausible. However, the results of predictions using this model are not always as good as we would wish to see. Curves 6 in the fiies show the computed shrinking rates which are calculated based on curve 4’ conditions (i.e., eqns. 33 and 41). It is found that model 4 (curve 4’) could not give good predictions as the reaction front is less than, roughly speaking, 80% of the drop size, i.e., r,/R < 0.8. However, further studies are needed to verify the usefulness of the more preferred model 4. Fig. 7. Comparison of model predictions with experimental data of Kim et al. [ 281; curve 1’ Ve/( Vi f Vm) = 3; Ce, = 0.0106 N; CBo = 0.0625 N. Curve 1 (Ri = 0.9835R); (Ri = 0.7937R); curve 2 (Ri = 0.9838R); curve 2’ (Ri = 0.7937R); curve 3 (Ri = 0.9838R); CUNe 3’ (Ri = 0.7937R); cUNe 4 (Ri = R); curve 4’ (Ri = R); curve 5 (Ri = 0.9906R); curve 5’ (Ri = R); curve 6: computed shrinking rate based on curve 4’. Fig. 8. Comparison of model predictions with experimental data of Kim et al. [ 281; Ve/( Vi * Vm) = 4; Cm = 0.0106 N; C,, = 0.25 N. Curve 1 (Ri = 0.9887R); curve 1’ (Ri = 0.7937R); curve 2 (Ri = 0.9888R); cuNe 2’ (Ri = 0.7937R); curve 3 (Ri = 0.9888R); curve 3’ (Ri = 0.7937R); cUNe 4 (Ri = R); curve 4’ (Ri = R); curve 5 (Ri = 0.9906R); curve 5’ (Ri = R); curve 6: computed shrinking rate based on curve 4’.
20
7. Conclusions
The five emulsion drop models proposed for the prediction of solute removal rate in the external phase using a liquid membrane system have been critically analyzed and discussed. By applying these models to the batch liquid membrane separation process and then comparing the computed results with the experimental data given in the literature, the following conclusions can be obtained: (1) A pseudo-steady state flux approximation can be used satisfactorily in the emulsion drop models for the liquid surfactant membrane systems dis cussed. This approximation is reasonable for most liquid membrane cases of practical interest, since the parameter e = a(Vi + V,)C,/ViC& was usually less than 1, as pointed out by Ho et al. [ 231. (2) Membrane film thickness estimated by using the hollow sphere emulsion drop method did not predict the removal rate successfully in the mathematical models, (3) Model 2 is more suitable for design equations than model 3 since the former gives almost the same predictions as the latter and is mathematically simpler. (4) Since no curve fitting is required, model 4 (curve 4’ combination) is an improvement over the other four models. However, it does not give good predictions when the reaction front advances deep inside the drop. The C,/Ck ratio must be carefully handled when applying this model in design equations. There are some inconsistencies existing between the mechanistic models and the published experimental data. More experimental data are needed for clarifying these inconsistencies. Further studies on these models and an extension of them to continuous liquid membrane separation processes will be pursued in our laboratory. List of symbols A A A’ A”
4 Ai &l a B B B’ B’ b
Component A. Constant used in eqn. (31). Constant. Constant. Component A in external phase. Component A in internal phase. Component A in membrane phase. Stoichiometric coefficient. Component B. Constant used in eqn. (31). Constant. Constant. Stoichiometric coefficient.
21
C
c’
c= CA CB
GO CC! GO C em
G Cm
Cme Cmi
D D’ D” DA De 0; Di Dm
E E’ E” G G’ G” K1 K? K3 K4 KS k kL ks NA NAi NAlll
n
nA
P P
Solute concentration in exhausted region of emulsion globule (N). Solute concentration in region I = CL = Cm (N). Solute concentration in region II (N). Solute concentration at chemical reaction (N). Concentration of internal reagent B (N). Initial concentration of internal reagent B (N). Concentration of solute in external phase (N). Initial concentration of solute in, external phase (N). Solute concentration (on external side) at the interface of membrane and external phase (N). Solute concentration in internal phase (N). Solute concentration in membrane phase (N). Solute concentration (on membrane side) at the interface of membrane and external phase (N). Solute concentration (on membrane side) at the interface of membrane and internal phase (N). Constant. Constant. Constant. Constant, eqn. (44). Effective solute diffusivity in the emulsion mixture (cm2 /set). Effective solute diffusivity in the emulsion mixture, eqn. (43) (cm?/sec). Solute diffusivity in internal phase (cm2/sec). Solute diffusivity in membrane phase (cm’/sec). Constant. Constant. Constant. Constant. Constant. Constant. Overall mass transfer coefficient based on model 1, eqn. (8) (cm/set). Overall mass transfer coefficient based on model 2, eqn. (10) (cm/set). Overall mass transfer coefficient; based on model 3, eqn. (13) (cm/set). Overall mass transfer coefficient based on model 4, eqn. (20) (cm/set). Overall mass! transfer coefficient based on model 5, eqn. (24) (cm/set). Apparent rate constant (cm/set). External mass transfer coefficient (cm/set). Intrinsic’rate constant (cm4/sec-mol). Molar flux of component A in\emulsion mixture (mol/sec-cm2). Molar flux of component A in internal phase (mol/sec-cm2). Molar flux of component A in membrane phase (mol/sec-cm2). Number of emulsion drops in continuous phase. Moles of component A (mol). Reaction product, P. Constant defined bv ean. (45).
22
R Ri R, r rA rc rs
t Vf? vi VlXl
Radius of an emulsion drop (cm). Radius of inner core in an emulsion drop (cm). Radius of internal droplet (cm). Radius (cm). Global mass transfer rate of solute A (mol/sec-cm?). Reaction front position (cm). Reaction rate (mol/sec-cm2). Time (set). Total volume of external phase (cm3). Total volume of internal phase (cm’). Total volume of membrane phase (cm”). Total volume of membrane phase in region II (cm3). Position (cm).
Greek letters Distribution coefficient of solute between the external and exhausted a emulsion phase, eqns. (18) and (42). Distribution coefficient of solute between the external and membrane cu, phase, eqn. (7). (Y. Distribution coefficient of solute between the internal and membrane 1 phase, eqn. (11). Constant. P Constant. P’ Constant. 0” Parameter, eqn. (5). E Volume fraction of the internal phase in the emulsion drop; I$ = @ Vi/(Vi + V,) for models l-4; 4 = Vi/(Vi + VE) for model 5. References P. Alessi, I. Kikic and O.V. Mirella, Liquid membrane permeation for the separation of C!, hydrocarbons, Chem. Eng. J., 19 (1980) 221. W.J. Asher, K.C. Bovee, J.W. Frankenfeld, R.W. Hamilton, L.W. Henderson, P.G. Holtzapple and N.N. Li, Liquid membrane system directed toward chronic uremia, Kidney J., 7 (1975) 5409. W.J. Asher, T.C. Vogler and K.C. Bovee, In vivo performance of liquid membrane capsules, Trans. Amer. Sot. Artif. Int. Organs, 22 (1976) 605. W.J. Asher, T.C. Vogler, K.C. Bovee, P.G. Holtzapple and R.W. Hamilton, Liquid membrane capsules for chronic uremia, Trans. Amer. Sot. Artif. Int. Organs, 23 (1977) 673. W.J. Asher, KC. Bovee, T.C. Vogler, R.W. Hamilton and P.G. Holtzapple, Secretion moderated release of urease from liquid membrane capsules, Trans. Amer. Sot. Artif. Int. Organs, 26 (1980) 120. M.P. Biehl, R.M. Izatt, J.D. Lamb and J.J. Christensen, Use of a macrocyclic crown ether in an emulsion membrane to effect rapid separation of Pb++ from cation mixtures, Sep. Sci. Technol., 17(2) (1982) 289. L. Boyadzhiev and G. Kyuchoukov, Further development of carrier-mediated extraction, J. Membrane Sci., 6 (1980) 107.
23 8 9 10 11
12
13 14 15 16
17 18 19 20 21 22 23 24 25
26
27 28 29 30 31
32
R.P. Cahn, J.W. Frankenfeld, N.N. Li, D. Naden and K.N. Subramanian, Extraction of Cu* by liquid membranes, Rec. Dev. Sep. Sci., 4 (1981) 51. R.P. Cahn and N.N. Li, Separation of organic compounds by liquid membrane process, J. Membrane Sci., 1 (1976) 129. F. Carraciolo, D.F. Evans and E.L. Cussler, Membranes with common ion pumping, AIChE J., 21(l) (1975) 160. G. Casamatta, C. Chavarie and H. Angelino, Hydrocarbon separation through a liquid water membrane - Modeling of permeation in an emulsion drop, AIChE J., 24( 6) (1978) 945. 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. E.M. Choy, D.F. Evans and E.L. Cussler, A selective membrane for transporting Na+ against its concentration gradient, J. Amer. Chem. Sot., 96(22) (1974) 7085. J. Crank, The Mathematics of Diffusion, 2nd edn., Ciarendon Press, Oxford, 1975, Chap. 12. E.L. Cussler, Membranes which pump, AIChE J., 17(6) (1971) 1300. P.R. Danesi, E.P. Horwitz and G.F. Vandegrift, Mass transfer rate through liquid membranes - Interfacial chemical reactions and diffusion as simultaneous permeability controlling factors, Sep. Sci. Technol., 16(Z) (1981) 201. P.R. Danesi, C. Cianetti and V. Violante, Induction time in metal permeation processes through supported liquid membranes, J. Membrane Sci., 14 (1983) 175. J. W. Frankenfeld and N.N. Li, Wastewater treatment by liquid ion exchanger in liquid membrane systems, Rec. Dev. Sep. Sci., 3 (1977) 285. J.W. Frankenfeld, R.P. Cahn and N.N. Li, Extraction of Cu++ by liquid membranes, Sep. Sci. Technol., 16(4) (1981) 385. J.W. Frankenfeld, W.J. Asher and N.N. Li, Biochemical and biomedical separations using liquid membranes, Rec. Dev. Sep. Sci., 4 (1978) 39. L. Gladek, J. Stelmaszek and J. Szust, Modeling of the mass transfer through a liquid membranes, Rec. Dev. Sep. Sci., 4 (1981) 29. W. Halwachs, E. Flaschel and K. Schiigerl, Liquid membrane transport - A highly selective separation process for organic solutes, J. Membrane Sci., 6 (1980) 33. W.S. Ho, T.A. Hatton, E.N. Lightfoot and N.N. Li, Batch extraction with liquid surfactant membranes - A diffusion controlled model, AIChE J., 28(4) (1982) 662. A. Hochhauser and E.L. Cussler, Concentrating Cr+6 with liquid surfactant membranes, AIChE Symp. Ser., 71 (1975) 136. H.B. Hopfenberg, M.Z. Ward, R.D. Rierson and W.J. Koros, The effect of component partitioning on potassium picrate transport across multicomponent supported liquid membranes, J. Membrane Sci., 8 (1981) 91. M. Igawa, M. Seno, H. Takashi and T. Yamabe, Selective concentration of metal ions through a hydrophobic membrane with a chelating agent, J. Membrane Sci., 2 (1977) 263. T. Imato, H. Ogawa, S. Morooka and Y. Kato, Transport of Cu* through supported liquid membrane containing LIX-65N, J. Chem. Eng. Jpn., 14(4) (1981) 289. K.S. Kim, S.J. Chol and S.K. Ihm, Simulation of phenol removal from wastewater by liquid membrane emulsion, Ind. Eng. Chem., Fundam., 22(Z) (1983) 167. T. Kitagawa, Y. Nishikawa, J.W. Frankenfeld and N.N. Li, Wastewater treatment by liquid membrane process, Environ. Sci. Technol., ll(6) (1977) 602; J. Klein, R.W. Baker and E.L. Cussler, Optical isomer separation by diffusion, Ind. Eng. Chem., Fundam., 10(l) (1971) 183. K. Kondo, K. Kita, I. Koida, J. Irie and F. Nakashio, Extraction of Cu++ with liquid surfactant membranes containing benzoylactone, J. Chem. Eng. Jpn., 12( 3) (1979) 203. K. Kondo, K. Kita and F. Nakashio, Extraction kinetics of Cu’+ with liquid surfactant membranes in a stirred transfer cell, J. Chem. Eng. Jpn., 14(l) (1981) 20.
24 33
34 35 36 37
38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55
V.J. Kremesec, Modeling of dispersed - emulsion separation systems, Sep. Purif. Methods, lO(2) (1981) 117. K.H. Lee, D.F. Evans and E.L. Cussler, Selective copper recovery with two types of liquid membranes, AIChE J., 24(5) (1978) 860. N.N. Li and AL. Shrier, Liquid membrane water treating, Rec. Dev. Sep. Sci., l(l972) 163. N.N. Li, Separation of hydrocarbons by liquid membrane permeation, Ind. Eng. Chem., Process Des. Dev., lO(2) (1971) 215. S.L. Matson, C.S. Herrick and W.J. Ward, Progress on the selective removal of H,S from gasified coal using an immobilized liquid membrane, Ind. Eng. Chem., Process Des. Dev., 16(3) (1977) 370. T.H. Maugh, Liquid membranes - New techniques for separation, purification, Science, 193 (1976) 134. S. W. May and N.N. Li, The immobilization of urease using liquid surfactant membranes, Biochem. Biophys. Res. Commun., 47 (1972) 1179. R.R. Mohan and N.N. Li, Reduction and separation of nitrate and nitrite by liquid membrane - Encapsulated enzymes, Biotechnol. Bioeng., 16 (1974) 513. R.R. Mohan, Nitrate and nitrite reduction by liquid membrane - Encapsulated whole cells, Biotechnol. Bioeng., 17 (1975) 1137. W.E. Morf, P. Wuhrmann and W. Simon, Transport properties of neutral carrier ion selective membranes, Anal. Chem., 48(7) (1976) 1031. F. Nakashio and K. Kondo, Extraction of Cu* with liquid surfactant membranes, Sep. Sci. Technol., 15(4) (1980) 1171. K.A. Reddy and K.K. Doraiswamy, Ind. Eng. Chem., Fundam., 6 (1971) 424. C.F. Reusch and E.L. Cussler, Selective membrane transport, AIChE J., 19(4) (1973) 736. S. Schlosser and E. Kossaczky, Comparison of pertraction through liquid membranes and double liquid-liquid extraction, J. Membrane Sci., 6 (1980) 83. E.G. Scheibel, Ind. Eng. Chem., 46 (1954) 2007. N.D. Shah and T.C. Owens, Separation of benzene and hexane with liquid membrane technique, Ind. Eng. Chem., Process Des. Dev., 11 (1972) 58. J. Strzelbicki and W. Charewicz, Separation of Cu* by liquid surfactant membranes, J. Inorg. Nucl. Chem., 40 (1978) 1415. J. Strzelbicki, Separation of Co ++ by liquid surfactant membranes, Sep. Sci. Technol., 13(2) (1978) 141. J. Stelmaszek, Mathematical models of liquid separation through a liquid membrane, Rec. Dev. Sep. Sci., 4 (1981) 11. J. Stelmaszek and B. Borkowska, Separation of mixtures using liquid membranes, Int. Chem. Eng., 17(3) (1977) 560. R.E. Terry, N.N. Li and W.S. Ho, Extraction of phenolic compounds and organic acids by liquid membranes, J. Membrane Sci., 10 (1982) 305. W. Volkel, W. Halwachs and K. Schiigerl, C!u* extraction by means of a liquid surfactant membrane process, J. Membrane Sci., 6 (1980) 19. C.R. Wilke and P. Chang, Correlation of diffusion coefficients in dilute solution, AIChE J., 1 (1955) 264.
1
Review MECHANISTIC MEMBRANE
MODELS OF MASS TRANSFER
&ROSS
A LIQUID
CHIH CHIEH CHAN and CHAU JEN LEE* Department
of Chemical Engineering,
National Tsing Hua University, Hsinchu (Taiwan)
(Received October 24, 1983; accepted in revised form January 27, 1984)
The five mechanistic models dealing with solute removal rate in liquid membranes, as they appear in the literature, are critically reviewed and discussed. The validity of these models is evaluated and compared against the experimental data of Ho et al. [ 231 and Kim et al. [ 281. It was interesting to find that Models 1, 2, 3 and 5, with the adjustable parameter of membrane film thickness, fitted the data of Kim et al. reasonably well; however, Models 4 and 5 with a proper choice of effective diffusivity fitted better the data of Ho et al. The results of this paper will direct our further studies on continuous separation processes utilizing liquid membranes.
1. Introduction Artificial membranes have been a topic of great interest in recent years because of their potential applications, ranging from desalination of water to timed release of drugs. One of the most intriguing classes of artificial membranes is the liquid membrane. Liquid membrane technology is taken to mean those processes in which a simultaneous extraction/stripping process takes place across a selective liquid separating phase. In general, there are two main types of liquid membrane systems that have been considered for practical applications - the liquid surfactant membranes or emulsion-type liquid membranes [2-8,11,18,19,21-23,29,31-36,40,41,43,46,49-541 and the supported liquid membranes [ 10,12,13,15-17,25-27,30,37,42,45]. Liquid surfactant membranes are made by forming an emulsion of two immiscible phases and then dispersing the emulsion in a third phase (the continuous phase). Usually, phases separated by a membrane are completely miscible. The emulsion is stabilized by surfactants. Supported liquid membranes are formed by absorbing a suitable solution into a microporous sheet. Here the liquid layers are stabilized by capillary or surface forces. The liquid membrane process has found applications in hydrometallurgy [6-19,29-34,43-46,49,50,54], hydrocarbon separation [ 1,11,21,36,48,51, 521, waste water treatment [ 9,18,22-24,29,35,40,41,53], and biomedical engineering [ 2-5,20,38,39], etc. The mathematical models of these systems *To whom correspondence 0376-7388/84/$07.20
should be addressed.
0 1984
Elsevier Science Publishers B.V.
2
are important in the interpretation of laboratory data and in the design and scale-up of such systems. A number of mathematical models which describe the real behavior of supported liquid membrane separation processes and also of liquid surfactant membrane separation processes have been proposed [ 10,12-17,25-28,37,42,45]. In this paper, the various modeis which are commonly used to describe the first type of facilitated mass transfer phenomena [ 231 in an emulsion drop are reviewed and discussed. Furthermore, the parameters and their physical significance of each model are clearly identified and compared. Finally, the type of experimental data from separation processes in batch-type operation which are required for model evaluation are identified and collected from the published literature. This enables the comparison in applicability and usefulness of each model in a specific separation process utilizing liquid membrane technology. 2. Mechanism of Type 1 facilitated transport of solute through an emulsion drop There are two types of facilitation method to improve the mass transfer rate of a solute by using a liquid surfactant membrane separation system called Type 1 and Type 2 facilitation, respectively [ 231. In the first type of facilitation, the concentration gradient of the membrane-soluble permeate is maximized by irreversibly reacting the solute with the reagent into an impermeable form in the receiving phase, and thereby maintaining the permeate concentration effectively zero in this phase. In Type 2 facilitation, a transport facilitator or “carrier” incorporated in the membrane phase carries the diffusing species across the membrane to the receiving phase. This is commonly known as “carrier-mediated” transport. Since there have been a number of investigations of the second type of facilitated transport, we do not deal with it in this paper. Consider an emulsion drop (say water-in-oil emulsion), as shown in Fig. 1, consisting of an aqueous solution of reagent B in internal fine droplets suspended in the external aqueous phase of permeable solute A. Solute A has an appreciable membrane solubility and readily permeates from the outside aqueous phase through the oil membrane into the encapsulated aqueous re-
External
phase
Membrane
Fig. 1. Schematic
phase
diagram bf emulsion liquid membrane system.
3
agent phase and it is trapped in the impermeable form, P, by an instantaneous and irreversible chemical reaction, aA+bB
ks
-P
(1)
In order to express the global transport rate in terms of the external bulk properties, expressions must be formulated for each of the transfer steps in the overall process. In brief, the sequence of steps for transporting solute A through the membrane into the encapsulated phase is as follows. (1) Transport of solute A from the bulk fluid phase to the fluid-membrane interface (external surface of the emulsion drop). (2) Solute A dissolves from the external phase into the emulsion membrane phase:
(3) Diffusion of the dissolved solute A across the membrane to the internal interface (membrane phase-internal phase interface). (4) The dissolved solute A is extracted from the membrane phase into the internal phase
(5) The extracted solute A reacts with the encapsulated reagent B to the impermeable form P, eqn. (1). (6) The impermeable product, P, cannot dissolve in the membrane phase and is trapped in the encapsulated droplet phase. At steady state, the rates of all the individual steps will be the same. This equality can be used to develop a global transport rate equation in terms of the concentrations of the external bulk fluid phase. The derivation of such equations will be considered in detail in the subsequent sections for different emulsion drop models. 3. Global transport
rate in various emulsion drop models
For modeling Type 1 facilitated transport within an emulsion drop, five types of models have been proposed: (1) Uniform flat-sheet model. (2) Hollow sphere model. (3) Hollow sphere-advancing front model. (4) Immobilized globule-advancing front model. (5) Immobilized hollow spherical globule-advancing front model. In the following sections, we first examine the various above conceptual models and compare these models in the light of the rate-controlling factors of mass transfer phenomena. Then, we attempt to show how a suitable model may be selected for a batch-type liquid membrane separation system. To
simplify the mathematical treatment, the following assumptions are made: (1) Isothermal process. (2) Constant physical and transport properties. (3) Uniform internal droplet and emulsion drop size. (4) No drop break-up and coalescence occurs. (5) Excess reagent concentration in the internal phase. (6) The chemical reaction is irreversible and can be represented by eqn. (1). Since the concentration of internal reagent B is so much greater than A at the reaction interface, it may be combined with the rate constant and the reaction rate is assumed to be linearized with respect to the concentrations of A and B, i.e., = kc/, (4) -rs = k&&A (7) The transport process is at pseudo-steady state. This approximation is reasonable for most liquid membrane cases of practical interest, since the parameter E = a(vi
+ Vm)Go
(5)
KG0
is less than 1, as pointed out by Ho et al. [ 231. 3.1 Uniform flat-sheet model (Model 1) This model assumes that all of the fine droplets within the emulsion are coalesced into a single large droplet, and the mass transfer process consists of diffusion across a stagnant membrane of thickness 6. Furthermore, the membrane thickness is assumed to be negligible compared to the drop radius, so that it may be considered that the mass transfer surface area is constant, i.e., uniform flat membrane thickness. The reagent is well mixed in the coalesced internal droplet. The solute diffuses into the interior interface where it is removed according to eqn. (1). The assumed physical situation for this model of transport system is depicted in Fig. 2a. At pseudo-steady state, the global transfer rate of solute A per unit external surface area, rA, can be expressed as 1 hA -_-= rA- 47rRZ dt
1 R-Ri -+++kL Q,
1 %k 3
-’
C, = K,C,
(6)
where Cme = QeCem
(7)
and the overall mass transfer resistance is 1 1 -_=-+ KI kL
R-Ri drn
+L hk
(8)
3.2 Hollow sphere model (Model 2) This model is similar to the uniform flatsheet model except that a hollow
5
spherical membrane film is used instead of the uniform flat membrane film. All assumptions used before also apply in this model. Referring to Fig. 2b, the global mass transfer rate of solute A per unit external surface area becomes, rA=
[$
+
(+&)
(t)
(R-&I+
(2)
(z) = K,Ce
‘]
-lCe (9)
and the overall mass transfer resistance is -
1
= &
+ (A)
(g)
(R-&I+
(&)
(c,’
(10)
K2
x=Ri
x=R
r=O
RiR
(Cl
Fig. 2. Schematic diagrams of emulsion drop models. a: uniform flat-she& model; b: hollow sphere model; c: hollow sphermdvancing front model; d: immobilized globule-advancing front model; e: immobilized boilow spherical globule-advancing front model.
6
3.3 Hollow sphere-advancing front model (Model 3) This is basically the same as the hollow sphere model, except that all of the fine droplets within the emulsion are coalesced into a single large droplet, and this droplet is encapsulated in a stagnant hollow spherical membrane film. However, the internal reagent phase in this model is assumed to be immobilized and the solute reacts irreversibly with the reagent at a reaction surface which advances into the droplet center while the reagent is being consumed. Referring to Fig. 2c and combining with the distribution relationship of eqns. (7) and (11) C&i = LYiCi,
(11)
the global transfer rate of solute A per unit external surface area, rA, becomes
rA=[i$ +(&I ($1
tR-Ri)
+
(2) (&)
+ (2)
(i)
(Rs)(Ri-rc)
(CT]
-lc,
=&Ce
(12)
and the overall resistance is 1 = K3
d +(&) ($,tRwRi)+ (&) (&)(Ri-rc) e!) (El2 +
(13)
In eqns. (12) and (13), r, must be expressed as function of time for these equations to become useful design equations for such separation processes. According to the spherical geometry of the emulsion drop, the rate of reaction of B may be obtained as (14) Combining eqns. (12)-( -
14) gives rC = f(t), i.e.,
p=(&) (E)‘[k +(A) (+)(R-&I+ (15)
Equation (15) can be integrated to give rc as a function of C, and time. Substituting this expression for rC in eqn. (12), it gives the expression for the global transport rate in terms of C, and t.
7
3.4 Immobilized globule--advancing front model (Model 4) The schematic representation of a typical liquid surfactant membrane system, as shown in Fig. 1, indicates two key features which must be adequately described in any realistic model of transport and reaction in such a system. One is the emulsion heterogeneity resulting from the presence of internal droplets, the other is the non-uniform size distribution of the globules. The proposed model (Fig. 2d) attempts to approach this realistic behavior. In this model, the solute diffuses through the external boundary layer and the re acted globule to the reaction front where it is removed by an irreversible then ical reaction, eqn. (1). The reaction front advances towards the center as the reagent is consumed. To simplify the mathematical treatment, the following assumptions were made in addition to the previous assumptions (l)-( 7): (8) No internal circulation within the emulsion drop. (9) Local equilibrium exists between the internal phase and the membrane phase. Based on the above assumptions and the definition of solute concentration in the exhausted region of emulsion globules, as used in the paper by Ho et al. [ 231, i.e.: ViCi + VmCm = (vilai
r_
,-
Vi + Vm
+ vUl
\ Vi + Vm
c
1 m
(17)
Cl ,.=R = “‘,,
( I(
a=
2.E ai
(16)
Vi + CuiVm Vi + Vm 1
(IS)
then the global transport rate of solute A per unit external surface area in this model becomes (19) [$ +(2) @R-rd+($7)(5)’ J-1ce=K4ce
rA=
and the overall mass transfer resistance is 1 -
&
=
& +(&) (t)(R-r.)+($) (;)
(20)
Since r, is a time-dependent variable, we must express r, as a function of time. A material balance over the reaction front gives -
$[+;
!,+“,)c&,] =($)9=($)(4”R2)r_4
Substituting eqn. (19) into eqn. (21) we obtain after rearranging
(21)
8
-
%=(&)
(“+,“-)(:)*[$
+
(&)
(E)@-rJ
(22) and thus, eqns. (19) and (22) may be solved simultaneously to give the global transport rate of A in terms of C, and t. 3.5 Immo bilked hello w spherical globule-advancing front model (Model 5) Similar to the “immobilized globule-advancing front” model, this model assumes that the internal droplets are distributed heterogeneously in the emulsion globule. In addition, from a practical point of view, a peripheral membrane layer with an infinitesimal thickness is assumed to exist around the droplets. Therefore, the transfer of solute from the external phase towards the emulsion is in the following three steps: (1) diffusion through the external boundary layer, (2) diffusion through the peripheral membrane layer, and (3) diffusion through the interior bulk emulsion globule. Referring to the schematic diagram as shown in Fig. 2e, and combining all the transport steps and rearranging, the global transport rate of solute A per unit external surface area becomes
vi+v’ Vi& +
X
vz )+
(2)
(v;a++TE) e
i
(;,‘]-lCe=K&e
(23)
i
Therefore, the overall resistance is 1 -
=& +(--&--)(g)CR-&)+ (ak) (&) (Ri-rc)
KS
X
(
Combining eqn. (23) and the material balance equation over the reaction front gives
(24)
9
(25) Therefore, the global transport rate in terms of C, and t can be obtained by solving eqns. (23) and (25) simultaneously. Table 1 summarizes the global transport rate equations based on the, five emulsion drop models previously discussed. In order to evaluate the usefulness of these models, a separation process utilizing membranes in a batch stirredtank system was selected for consideration and comparison. 4. Liquid membrane process in batch stirred-tank system In a batch stirred-tank system, the material balance on solute A in the external phase may be given as
dCe -=
-v,
dt
(4nnR2)rA =
I.C. t = 0,
+ R
3(Vi
vm)
rA (27)
c, = c,
With the expressions for global transport rate, rA, as obtained from each model (Table l), we can solve eqn. (26) to give the “conversion” of solute A as a function of time. This information enables us to design a batch separation process for a specified system. The results based on; such analysis can be summarized as follows: (1) Uniform flat-sheet model:
” =exp[c,
i
(vi\:)R1t]
(28)
(2) Hollow sphere model:
c,
GO
=exp [-
i
( viievm)K2t]
(3) Hollow sphere-advancing
(29)
front model: (36)
t=
s
ln(A+Br3)12+
+Ep 3A
Y[tln 3RP
“,:f)y
10
I
.
. b
I
:
1
3
’ .
0
I
,
.
.
> . I
. .
.
I
.
! Y .
,
.
.
a
3
J
Y
.
I
1 /
a J
.
a , I I
II
r
P
-h
a
-0 I
11
where
(4) Immobilized globule-advancing
front model: (32)
t= g
In (A’ +B’r”)IE
+ -
D’
3B’P’
$ In
A’ + B’r3 (8’ + r13
(33)
where A’=p B’=
(2) $(v’;evm)
( vi :,“)_p
( “ievm)
12
G&f-_
-
h
R
a&
p'= 3 $ 1/ (5)Immobilized hollow spherical globule-advancing
front model
2
[I-
=1-(Z)
(vi;vm)(vi,“;.)(;)3 eo
t=
-$-
E”P”
+3A”
t [
m
e
In (A” +B”r)
(34)
($)‘I
1% + -I!?-
rc
3B”P”
[
i In
” + r)3 In @ +fi A” + B”y3
tan-’
(‘iii’:)
(“i’rn)
A” + B”r3
(P” + r)3
(%)]/I
ul Ri
+fitan-’
_II (>p”
i r,
(35)
where AJ~=R~ (:z;)
_ f
e gr=
$
(viievm)
D” = RZ (2)
(vJy+z
m
1
vi + vg vi /C$+ vf
G”=;
+ (-$-)
)
cf)W
The solution form in the last three models as shown in eqns. (31), (33), and (35) is similar to the zero order or pseudo-steady state solution in the papers by Ho et al. [23] and Terry et al. [53].
13
5. Evaluation of parameters - Diffusion coefficients Prior to the mechanistic models being tested against the experimental data which appeared in the published literature, the effective diffusivity, De (appearing in models 4 and 5) must be correlated with the individual diffusivities of both phases in the emulsion. Casamatta et al. [ 111 assumed that the effective radial mass transfer area is proportional to the volumetric fraction of membrane phase in the emulsion. Ir other words, the total mass transfer rate is given by the following equations: NA = (1 - @)NA~ + @NAP i.e., dc = -(l dr
-#)D,
c = (1- $)C,
+#Ci
-D,
dGn 7
-#Di
$
(36)
since (37)
Equation (36) can then be written as
$=-
_D,
$/ffi
[
D
1 - 4 + #/ffi )I
iz
Cu:
(33)
Therefore, the effective diffusivity of solute in the emulsion mixture may be given by D,
=
l-4 1 - G+
#/ai
i
Ho et al. [ 231 used the effective diffusivity, DL, based on a concentration driving force defined in terms of the membrane phase concentration, C, , to calculate the effective diffusivity, De, based on the average concentration, C, in the emulsion mixture, that is D,
dC dr
=D;
dC, dr
(46)
Therefore, we can obtain (41) where O!=
(%1
[(I-
1
@l&i +
$1
(42)
14
The effective diffusivity, I&, can be estimated from the Jefferson-WitzellSibbett equation [ 141, as suggested by Ho et al. [ 231: Dk =Dm
n (I+
1
*)DAD,
4(1 + !P)2 - II
+ 4(1 + 2P)2
4(1 + 2P)2
D,
+ ~PDA
1
(43)
where
Dilai
ln
DA =
(44)
1
--
1
Dp
P = 0.4039-1’3
- 0.5
(45)
The individual molecular diffusivity of a given solute in each phase may be estimated by the Wilke-Chang correlation [ 551, the Scheibel correlation [ 471, or the Reddy-Doraiswamy correlation [ 441. 6. Comparison of model predictions with experimental data Figures 3-8 give the comparison of the model predictions with the experimental results given by Ho et al. [ 231 and Kim et al. [ 281, respectively. The experimental conditions of both Ho and Kim are shown in Table 2. In both experimental systems, phenol was extracted by using NaOH as an internal reagent. Since it is an acid-base reaction, the reaction rate is always fast.
---.__
0.6 --\.
.L_ R
0.4 .
.
4.’
..
-._
01 0
972
1944
-.
--.
---__
2916
4860
3808 Time,
5032
---____ . 6804
10
set
Fig. 3. Comparisonof model predictionswith experimentaldataof Ho et al. [ 231; N = 400 rpm; R = 0.05 cm. Curve1 (Ri = 0.9410R); curve 1’ (Ri = 0.7133R); curve 2 (Ri =
0.9443R); curve 2’ (Ri t 0.7133132); curve 3 (Ri = 0.9443R); curve 4 (Ri = R); curve 4’ (Ri = R); curve 5 (Ri = 0.9600R); curve 6: computed shrinking rate based on curve 4’.
curve 3’ (Ri = 0.7133R); curve 5’ (Ri = 0.9910R);
15
Therefore, the chemical reaction term in the model transport equations is negligible. In the figures, the circles represent the experimental data. Curves 1 and 1’ are the results for the uniform flat-sheet model computed with eqn. (28); curves 2 and 2’ those for the hollow sphere model with eqn. (29); curves 3 and 3’ those for the hollow sphere-advancing front model with eqns. (30) and (31); curve 4 for the immobilized globule-advancing front model with eqns. (32), (33) and (39); curve 4’ for the immobilized globuleadvancing front model with eqns. (32), (33) and (41); curve 5 for the immobilized hollow spherical globule-advancing front model with eqns. (34), (35) and (39); and curve 5’ for the immobilized hollow spherical globuleadvancing front model with eqns. (34), (35) and (41). All these emulsion drop models, except model 4, i.e., curves 4 and 4’, have assumed a “fictitious” men brane film to exist in the emulsion. However, no theoretical correlation has been proposed to estimate this membrane film thickness successfully. In some approximations, the internal phase was assumed to be coalesced into an internal large droplet inside the emulsion. In other words, the membrane phase became a hollow spherical film surrounding the whole emulsion drop. Therefore, the membrane film thickness (used in models 1, 2, or 3), was estimated by the following equation, 6
=K!-Ri=R[l-tviybm)“‘]
(46)
Fig. 4. Comparison of model predictions with experimental data of Ho et al. [ 231; N = 600 rpm; R = 0.03 cm, Curve 1 (Ri= 0.8944R);curve 1'(Ri= 0.7133R); curve 2 (Ri= 0.9045R),curve 2’ (Ri= 0.7133R); curve 3 (Ri= 0.904512); curve 3’ (RiE 0.7133R); curve 4 (Ri= R);curve 4’ (Ri=R); curve 5 (Riz0.9360R);curve 5'(Ri=0.9850R); curve 6: computed shrinking rate based on curve 4’.
16
. . .._...-.
---...
17 TABLE 2 Comparison of experimental conditions
of Ho et al. [23] and those of Kim et al. [ 281
Conditions
Ho et al.
Kim et al.
Qpe of separation Temperature Emulsification Type of emulsion R
Batch 23°C 12000 rpm w/o/w 0.05 cm; 0.03 cm 0.363 15.0 Aqueous phenol solution Phenol 400 rpm; 600 rpm 0.00819 N 780 ml 0.52 large Aqueous NaOH solution NaOH 0.375 N 18.876 ml 0.52 0.5 Mm 9.98 X 10e6 cm’/sec Oil phase 3 wt.% ENJ3029 + 96 wt.% SlOON 1 wt.% Span 80 33.124 ml 0.65 X lo-* cm’/sec
Batch 20°C 1300 rpm w/o/w 0.0264 cm 0.5 3.0; 4.0 Aqueous phenol solution Phenol 200 rpm 0.0106 N; 0.0318 N -
Vil(Vm + Vi)
vel(vm + vi)
External phase Solute Stirring
kL
Internal phase Reagent cBo vi ori
3 Membrane phase Oil base Surfactant
0.13 0.02 cm/set Aqueous NaOH solution NaOH 0.0626 N; 0.25 N 0.13 2.6 pm 9.3588 x 1Om6 cm’/sec Oil phase Paraffin oil (C,,_,,) Span 80 (HLB = 4.3) 0.4527 x lo-*
cm’/sec
The results computed using this approximation are shown in the upper curves (i.e., curves l’, 2’ and 3’) in Figs. 3-8. Due to the fact that the membrane film thickness estimated by this method was always too high and not quite realistic, the results using this approximation could not predict the removal rate successfully. We have tried several methods to estimate the film thickness from the knowledge of the phase volume fraction, 4, the emulsion drop size and the inFig. 5. Comparison of model predictions with experimental data of Kim et al. [ 281; Vel(Vi + Vm) = 3; Gee = 0.0106 N; Cne = 0.25 N. Curve 1 (Ri = 0.9892R); curve 1’ (Ri = 0.7937R); curve 2 (Ri = 0.9893R); curve 2’ (Ri = 0.7937R); curve 3 (Ri = 0.9893R); curve 3’ (Ri = 0.7937R); curve 4 (Ri = R); curve 4’ (Ri = R); curve 5 (Ri = 0.9906R); curve 5’ (Ri = R); curve 6: computed shrinking rate based on curve 4’. Fig. 6. Comparison of model predictions with experimental data of,Kim et al. [28]; Vel(vi + Vrn)= 3~ ceO = 0.0318 N; Cne = 0.25 N. Curve 1 (Ri = 0.9849R); curve 1’ (Ri = 0.7937R); curve 2 (Ri = 0.9851R); curve 2’ (Ri = 0.7937R); curve 3 (Ri = 0.9851R); curve 3’ (Ri = 0.7937R); curve 4 (Ri = R); curve 4’ (Ri = R); curve 5 (Ri = 0.9906R); curve 5’ (Ri = R); curve 6: computed shrinking rate based on curve 4’.
18
. .._.....
---
19
ternal micro-droplet size. However, they all failed to predict satisfactorily the experimental results. It seems, while Kim et al. [ 281 proposed a method to estimate the film thickness required for model 5, that one easy way to find the membrane film thickness is to back calculate it from experimental data by curve fitting, though this method may not be practical for application to design equations. The lower curves (i.e., curves 1, 2, 3, 4, 4’, 5, and 5’) in Figs. 3-8 show the computed results based on various models. Values of Ri (and thus, of the membrane film thickness) used in curves 1, 2, 3, 5, and 5’ were calculated by the equations mentioned above together with curve-fitting of the experimental data by the least-squares method. The calculated Ri values were shown in parentheses in the figures. Since no Ri value was required, curves 4 and 4’ were calculated directly by the equations given. Comparing with the published experimental data, curve 4 (i.e., combination of eqns. 32, 33 and 39) gave the worst predictions (too fast removal rate). Curves 1, 2, 3 and 5 predict well the data of Kim (Figs. 5--8), but not those of Ho (Figs. 3 and 4). Curves 4’ and 5’, however, better predict the data of Ho than those of Kim. The physical significance of such trends cannot be clearly concluded until further experimental data are gathered and analyzed. For curves 2 and 3, it is found that the calculated values of Ri of both curves are the same and that the removal rate predicted by these two models is almost identical. This result can be explained by the fact that the diffusivity of a solute in the internal phase is always larger by an order of magnitude than that in the membrane phase. Since model 2 is simpler mathematically, it would be preferred to model 3 when used in the design equations. Model 4 has the advantage over the other models in that the resulting equations are easy to apply, enabling the prediction of removal rate directly without the trouble of estimation of the film thickness. This is a significant improvement over the other four models, though it is not physically plausible. However, the results of predictions using this model are not always as good as we would wish to see. Curves 6 in the fiies show the computed shrinking rates which are calculated based on curve 4’ conditions (i.e., eqns. 33 and 41). It is found that model 4 (curve 4’) could not give good predictions as the reaction front is less than, roughly speaking, 80% of the drop size, i.e., r,/R < 0.8. However, further studies are needed to verify the usefulness of the more preferred model 4. Fig. 7. Comparison of model predictions with experimental data of Kim et al. [ 281; curve 1’ Ve/( Vi f Vm) = 3; Ce, = 0.0106 N; CBo = 0.0625 N. Curve 1 (Ri = 0.9835R); (Ri = 0.7937R); curve 2 (Ri = 0.9838R); curve 2’ (Ri = 0.7937R); curve 3 (Ri = 0.9838R); CUNe 3’ (Ri = 0.7937R); cUNe 4 (Ri = R); curve 4’ (Ri = R); curve 5 (Ri = 0.9906R); curve 5’ (Ri = R); curve 6: computed shrinking rate based on curve 4’. Fig. 8. Comparison of model predictions with experimental data of Kim et al. [ 281; Ve/( Vi * Vm) = 4; Cm = 0.0106 N; C,, = 0.25 N. Curve 1 (Ri = 0.9887R); curve 1’ (Ri = 0.7937R); curve 2 (Ri = 0.9888R); cuNe 2’ (Ri = 0.7937R); curve 3 (Ri = 0.9888R); curve 3’ (Ri = 0.7937R); cUNe 4 (Ri = R); curve 4’ (Ri = R); curve 5 (Ri = 0.9906R); curve 5’ (Ri = R); curve 6: computed shrinking rate based on curve 4’.
20
7. Conclusions
The five emulsion drop models proposed for the prediction of solute removal rate in the external phase using a liquid membrane system have been critically analyzed and discussed. By applying these models to the batch liquid membrane separation process and then comparing the computed results with the experimental data given in the literature, the following conclusions can be obtained: (1) A pseudo-steady state flux approximation can be used satisfactorily in the emulsion drop models for the liquid surfactant membrane systems dis cussed. This approximation is reasonable for most liquid membrane cases of practical interest, since the parameter e = a(Vi + V,)C,/ViC& was usually less than 1, as pointed out by Ho et al. [ 231. (2) Membrane film thickness estimated by using the hollow sphere emulsion drop method did not predict the removal rate successfully in the mathematical models, (3) Model 2 is more suitable for design equations than model 3 since the former gives almost the same predictions as the latter and is mathematically simpler. (4) Since no curve fitting is required, model 4 (curve 4’ combination) is an improvement over the other four models. However, it does not give good predictions when the reaction front advances deep inside the drop. The C,/Ck ratio must be carefully handled when applying this model in design equations. There are some inconsistencies existing between the mechanistic models and the published experimental data. More experimental data are needed for clarifying these inconsistencies. Further studies on these models and an extension of them to continuous liquid membrane separation processes will be pursued in our laboratory. List of symbols A A A’ A”
4 Ai &l a B B B’ B’ b
Component A. Constant used in eqn. (31). Constant. Constant. Component A in external phase. Component A in internal phase. Component A in membrane phase. Stoichiometric coefficient. Component B. Constant used in eqn. (31). Constant. Constant. Stoichiometric coefficient.
21
C
c’
c= CA CB
GO CC! GO C em
G Cm
Cme Cmi
D D’ D” DA De 0; Di Dm
E E’ E” G G’ G” K1 K? K3 K4 KS k kL ks NA NAi NAlll
n
nA
P P
Solute concentration in exhausted region of emulsion globule (N). Solute concentration in region I = CL = Cm (N). Solute concentration in region II (N). Solute concentration at chemical reaction (N). Concentration of internal reagent B (N). Initial concentration of internal reagent B (N). Concentration of solute in external phase (N). Initial concentration of solute in, external phase (N). Solute concentration (on external side) at the interface of membrane and external phase (N). Solute concentration in internal phase (N). Solute concentration in membrane phase (N). Solute concentration (on membrane side) at the interface of membrane and external phase (N). Solute concentration (on membrane side) at the interface of membrane and internal phase (N). Constant. Constant. Constant. Constant, eqn. (44). Effective solute diffusivity in the emulsion mixture (cm2 /set). Effective solute diffusivity in the emulsion mixture, eqn. (43) (cm?/sec). Solute diffusivity in internal phase (cm2/sec). Solute diffusivity in membrane phase (cm’/sec). Constant. Constant. Constant. Constant. Constant. Constant. Overall mass transfer coefficient based on model 1, eqn. (8) (cm/set). Overall mass transfer coefficient based on model 2, eqn. (10) (cm/set). Overall mass transfer coefficient; based on model 3, eqn. (13) (cm/set). Overall mass transfer coefficient based on model 4, eqn. (20) (cm/set). Overall mass! transfer coefficient based on model 5, eqn. (24) (cm/set). Apparent rate constant (cm/set). External mass transfer coefficient (cm/set). Intrinsic’rate constant (cm4/sec-mol). Molar flux of component A in\emulsion mixture (mol/sec-cm2). Molar flux of component A in internal phase (mol/sec-cm2). Molar flux of component A in membrane phase (mol/sec-cm2). Number of emulsion drops in continuous phase. Moles of component A (mol). Reaction product, P. Constant defined bv ean. (45).
22
R Ri R, r rA rc rs
t Vf? vi VlXl
Radius of an emulsion drop (cm). Radius of inner core in an emulsion drop (cm). Radius of internal droplet (cm). Radius (cm). Global mass transfer rate of solute A (mol/sec-cm?). Reaction front position (cm). Reaction rate (mol/sec-cm2). Time (set). Total volume of external phase (cm3). Total volume of internal phase (cm’). Total volume of membrane phase (cm”). Total volume of membrane phase in region II (cm3). Position (cm).
Greek letters Distribution coefficient of solute between the external and exhausted a emulsion phase, eqns. (18) and (42). Distribution coefficient of solute between the external and membrane cu, phase, eqn. (7). (Y. Distribution coefficient of solute between the internal and membrane 1 phase, eqn. (11). Constant. P Constant. P’ Constant. 0” Parameter, eqn. (5). E Volume fraction of the internal phase in the emulsion drop; I$ = @ Vi/(Vi + V,) for models l-4; 4 = Vi/(Vi + VE) for model 5. References P. Alessi, I. Kikic and O.V. Mirella, Liquid membrane permeation for the separation of C!, hydrocarbons, Chem. Eng. J., 19 (1980) 221. W.J. Asher, K.C. Bovee, J.W. Frankenfeld, R.W. Hamilton, L.W. Henderson, P.G. Holtzapple and N.N. Li, Liquid membrane system directed toward chronic uremia, Kidney J., 7 (1975) 5409. W.J. Asher, T.C. Vogler and K.C. Bovee, In vivo performance of liquid membrane capsules, Trans. Amer. Sot. Artif. Int. Organs, 22 (1976) 605. W.J. Asher, T.C. Vogler, K.C. Bovee, P.G. Holtzapple and R.W. Hamilton, Liquid membrane capsules for chronic uremia, Trans. Amer. Sot. Artif. Int. Organs, 23 (1977) 673. W.J. Asher, KC. Bovee, T.C. Vogler, R.W. Hamilton and P.G. Holtzapple, Secretion moderated release of urease from liquid membrane capsules, Trans. Amer. Sot. Artif. Int. Organs, 26 (1980) 120. M.P. Biehl, R.M. Izatt, J.D. Lamb and J.J. Christensen, Use of a macrocyclic crown ether in an emulsion membrane to effect rapid separation of Pb++ from cation mixtures, Sep. Sci. Technol., 17(2) (1982) 289. L. Boyadzhiev and G. Kyuchoukov, Further development of carrier-mediated extraction, J. Membrane Sci., 6 (1980) 107.
23 8 9 10 11
12
13 14 15 16
17 18 19 20 21 22 23 24 25
26
27 28 29 30 31
32
R.P. Cahn, J.W. Frankenfeld, N.N. Li, D. Naden and K.N. Subramanian, Extraction of Cu* by liquid membranes, Rec. Dev. Sep. Sci., 4 (1981) 51. R.P. Cahn and N.N. Li, Separation of organic compounds by liquid membrane process, J. Membrane Sci., 1 (1976) 129. F. Carraciolo, D.F. Evans and E.L. Cussler, Membranes with common ion pumping, AIChE J., 21(l) (1975) 160. G. Casamatta, C. Chavarie and H. Angelino, Hydrocarbon separation through a liquid water membrane - Modeling of permeation in an emulsion drop, AIChE J., 24( 6) (1978) 945. 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. E.M. Choy, D.F. Evans and E.L. Cussler, A selective membrane for transporting Na+ against its concentration gradient, J. Amer. Chem. Sot., 96(22) (1974) 7085. J. Crank, The Mathematics of Diffusion, 2nd edn., Ciarendon Press, Oxford, 1975, Chap. 12. E.L. Cussler, Membranes which pump, AIChE J., 17(6) (1971) 1300. P.R. Danesi, E.P. Horwitz and G.F. Vandegrift, Mass transfer rate through liquid membranes - Interfacial chemical reactions and diffusion as simultaneous permeability controlling factors, Sep. Sci. Technol., 16(Z) (1981) 201. P.R. Danesi, C. Cianetti and V. Violante, Induction time in metal permeation processes through supported liquid membranes, J. Membrane Sci., 14 (1983) 175. J. W. Frankenfeld and N.N. Li, Wastewater treatment by liquid ion exchanger in liquid membrane systems, Rec. Dev. Sep. Sci., 3 (1977) 285. J.W. Frankenfeld, R.P. Cahn and N.N. Li, Extraction of Cu++ by liquid membranes, Sep. Sci. Technol., 16(4) (1981) 385. J.W. Frankenfeld, W.J. Asher and N.N. Li, Biochemical and biomedical separations using liquid membranes, Rec. Dev. Sep. Sci., 4 (1978) 39. L. Gladek, J. Stelmaszek and J. Szust, Modeling of the mass transfer through a liquid membranes, Rec. Dev. Sep. Sci., 4 (1981) 29. W. Halwachs, E. Flaschel and K. Schiigerl, Liquid membrane transport - A highly selective separation process for organic solutes, J. Membrane Sci., 6 (1980) 33. W.S. Ho, T.A. Hatton, E.N. Lightfoot and N.N. Li, Batch extraction with liquid surfactant membranes - A diffusion controlled model, AIChE J., 28(4) (1982) 662. A. Hochhauser and E.L. Cussler, Concentrating Cr+6 with liquid surfactant membranes, AIChE Symp. Ser., 71 (1975) 136. H.B. Hopfenberg, M.Z. Ward, R.D. Rierson and W.J. Koros, The effect of component partitioning on potassium picrate transport across multicomponent supported liquid membranes, J. Membrane Sci., 8 (1981) 91. M. Igawa, M. Seno, H. Takashi and T. Yamabe, Selective concentration of metal ions through a hydrophobic membrane with a chelating agent, J. Membrane Sci., 2 (1977) 263. T. Imato, H. Ogawa, S. Morooka and Y. Kato, Transport of Cu* through supported liquid membrane containing LIX-65N, J. Chem. Eng. Jpn., 14(4) (1981) 289. K.S. Kim, S.J. Chol and S.K. Ihm, Simulation of phenol removal from wastewater by liquid membrane emulsion, Ind. Eng. Chem., Fundam., 22(Z) (1983) 167. T. Kitagawa, Y. Nishikawa, J.W. Frankenfeld and N.N. Li, Wastewater treatment by liquid membrane process, Environ. Sci. Technol., ll(6) (1977) 602; J. Klein, R.W. Baker and E.L. Cussler, Optical isomer separation by diffusion, Ind. Eng. Chem., Fundam., 10(l) (1971) 183. K. Kondo, K. Kita, I. Koida, J. Irie and F. Nakashio, Extraction of Cu++ with liquid surfactant membranes containing benzoylactone, J. Chem. Eng. Jpn., 12( 3) (1979) 203. K. Kondo, K. Kita and F. Nakashio, Extraction kinetics of Cu’+ with liquid surfactant membranes in a stirred transfer cell, J. Chem. Eng. Jpn., 14(l) (1981) 20.
24 33
34 35 36 37
38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55
V.J. Kremesec, Modeling of dispersed - emulsion separation systems, Sep. Purif. Methods, lO(2) (1981) 117. K.H. Lee, D.F. Evans and E.L. Cussler, Selective copper recovery with two types of liquid membranes, AIChE J., 24(5) (1978) 860. N.N. Li and AL. Shrier, Liquid membrane water treating, Rec. Dev. Sep. Sci., l(l972) 163. N.N. Li, Separation of hydrocarbons by liquid membrane permeation, Ind. Eng. Chem., Process Des. Dev., lO(2) (1971) 215. S.L. Matson, C.S. Herrick and W.J. Ward, Progress on the selective removal of H,S from gasified coal using an immobilized liquid membrane, Ind. Eng. Chem., Process Des. Dev., 16(3) (1977) 370. T.H. Maugh, Liquid membranes - New techniques for separation, purification, Science, 193 (1976) 134. S. W. May and N.N. Li, The immobilization of urease using liquid surfactant membranes, Biochem. Biophys. Res. Commun., 47 (1972) 1179. R.R. Mohan and N.N. Li, Reduction and separation of nitrate and nitrite by liquid membrane - Encapsulated enzymes, Biotechnol. Bioeng., 16 (1974) 513. R.R. Mohan, Nitrate and nitrite reduction by liquid membrane - Encapsulated whole cells, Biotechnol. Bioeng., 17 (1975) 1137. W.E. Morf, P. Wuhrmann and W. Simon, Transport properties of neutral carrier ion selective membranes, Anal. Chem., 48(7) (1976) 1031. F. Nakashio and K. Kondo, Extraction of Cu* with liquid surfactant membranes, Sep. Sci. Technol., 15(4) (1980) 1171. K.A. Reddy and K.K. Doraiswamy, Ind. Eng. Chem., Fundam., 6 (1971) 424. C.F. Reusch and E.L. Cussler, Selective membrane transport, AIChE J., 19(4) (1973) 736. S. Schlosser and E. Kossaczky, Comparison of pertraction through liquid membranes and double liquid-liquid extraction, J. Membrane Sci., 6 (1980) 83. E.G. Scheibel, Ind. Eng. Chem., 46 (1954) 2007. N.D. Shah and T.C. Owens, Separation of benzene and hexane with liquid membrane technique, Ind. Eng. Chem., Process Des. Dev., 11 (1972) 58. J. Strzelbicki and W. Charewicz, Separation of Cu* by liquid surfactant membranes, J. Inorg. Nucl. Chem., 40 (1978) 1415. J. Strzelbicki, Separation of Co ++ by liquid surfactant membranes, Sep. Sci. Technol., 13(2) (1978) 141. J. Stelmaszek, Mathematical models of liquid separation through a liquid membrane, Rec. Dev. Sep. Sci., 4 (1981) 11. J. Stelmaszek and B. Borkowska, Separation of mixtures using liquid membranes, Int. Chem. Eng., 17(3) (1977) 560. R.E. Terry, N.N. Li and W.S. Ho, Extraction of phenolic compounds and organic acids by liquid membranes, J. Membrane Sci., 10 (1982) 305. W. Volkel, W. Halwachs and K. Schiigerl, C!u* extraction by means of a liquid surfactant membrane process, J. Membrane Sci., 6 (1980) 19. C.R. Wilke and P. Chang, Correlation of diffusion coefficients in dilute solution, AIChE J., 1 (1955) 264.