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A diffusion model for reversible comsumption in emulsion liquid membranes
Journal of Membrane Elsevier
Science
21 (1984)
Science,
Publishers
B.V.,
55-71
Amsterdam
55
-
Printed
in The Netherlands
A DIFFUSION MODEL FOR REVERSIBLE COMSUMPTION IN EMULSION LIQUID MEMBRANES*
A.L.
BUNGE
Colorado
School
and RICHARD
National (Received
of Mines, Golden,
CO 80401
(U.S.A.)
D. NOBLE
Bureau of Standards, October
28,1983;
Boulder, accepted
CO 80303 m revised
(U.S.A.) form May 17, 1984)
Summary This work extends previous diffusion models for emulsion globules in which a solute reacts with an internal reagent This model allows for reversible consumption of the solute by the internal reagent. Local concentration of the internal reagent is nonzero and satisfies reaction and phase equilibria within the reacted zone. Predicted solute absorption rates are lower for the reversible consumption model than for irreversible models.
Introduction Emulsion liquid membranes were invented by Li [l] , who first applied this technique to the separation of hydrocarbons [2, 31. Since then, emulsion liquid membranes have been applied to other separations. These include recovery and purification of metal ions [4--81, removal of phenol from waste water [9-111, and biological applications such as an artificial kidney [ 12--14 J . Review articles discuss this separation and its uses in more detail [15,16]. Numerous researchers have mathematically modeled these membrane separation processes. Previously, model development has followed one of two approaches. The mathematically simpler representation characterizes the emulsion globule as a double shell around a single internal phase droplet. The spherical shell viewpoint is formally equivalent to assuming an internally well-mixed globule . In an early paper, Calm and Li [9] analyzed experiments involving phenol removal from wastewater by asserting that the rate of phenol transfer is directly proportional to the concentration difference of phenol across the membrane shell. Assuming that a reagent within the internal droplet instantaneously and irreversibly consumes phenol, they determined that the effec*Paper presented Chemical Process
0376-7388/84/$03.00
at the Symposium entitled “Membrane Separation Applications,” Denver, Colorado, August 28-31,
o 1984
Elsevier
Science
Publishers
B.V.
Technology 1983.
for
56
tive permeability varied with time. Boyadzhiev et al. [ 171 follow this same analysis. However, their treatment does not account for internal reagent consumption. Kremesec [18] and Kremesec and Slattery [19] use planar geometry and sum mass transfer resistances through the continuous, membrane, and internal phases. In their method geometric and internal circulation effects are lumped into the overall mass transfer coefficient. Gladek et al. [ 201 examine the unsteady-state solute absorption when the solute partition coefficient varies with concentration. Several researchers choose the spherical shell approach to describe carrierfacilitated transfer for a variety of situations. Matulevicius and Li [ 211 consider theoretically the time-dependent permeability by solving the unsteadystate mass transfer equations when solute diffusion through the membrane layer controls. In a model with limitations similar to that of Matulevicius and Li, Hochhauser and Cussler [4] consider chromium concentration by coupled transport in the membrane phase. Way and Noble [22] model copper ion uptake in a continuous flow separation scheme which include residence time and particle size distributions for the double emulsion drops. Noble [23] calculated the steady, facilitated flux for either diffusion- or reaction-limited cases when the carrier molecule reacts reversibly. Folkner and Noble [ 241 examine the initial transient response for this system. While mathematically more complex, a second approach more correctly accounts for mass transfer contributions accompanying solute diffusion and reaction in an emulsion globule. This approach assumes that solute removed from the bulk phase diffuses through the globule to a reaction front, where it is removed instantaneously and irreversibly by reaction with an internal reagent. In turn, the reaction front progresses towards the center as internal reagent is consumed. Kopp et al. [25] adopted this approach by examining the analogous planar problem with constant bulk solute concentration. Three papers, Ho et al, [ 261, Kim et al. [ 271, and Stroeve and Varanasi [ 281, formulate advancing front theories which include both spherical geometry and depletion of solute in the bulk phase. All three treatments assume homogeneous distribution of noncirculating internal droplets within the globule, although Kim et al. assume a thin outer liquid membrane layer which contains no internal droplets. For globule diffusion controlling solute transfer, Ho and coworkers use a perturbation method to solve the resulting system of nonlinear equations. From the zero-order or pseudo-steady-state solution, they calculate phenol removal from the continuous phase as a function of time. Using a Sauter mean diameter for the average emulsion drop size and no adjustable parameters, they predict somewhat higher removal rates than observed experimentally. Stroeve and Varanasi also use a zero-order advancing front model but include a mass transfer resistance in the continuous phase. They show that their model reduces to that of Ho et al. when the mass transfer resistance becomes negligible.
Kim et al. include diffusion through a thin membrane film without droplets as an additional resistance. As expected, their model successfully predicts the experimental results given, since the thickness of the membrane film essentially constitutes an adjustable parameter. The advancing front approach overpredicts the removal rate of solute since this approach requires the reagent concentration in the reacted zone between the globule surface and the reaction front to be zero. When the internal reagent is a base such as sodium hydroxide, this requirement is not physically correct. Furthermore, advancing front models assume that the local solute concentration does not affect the amount of reagent to react and that the reagent permanently traps reacted solute. Thus, advancing front models incorrectly predict that even low solute concentrations can force reagent at the reaction front to react completely. These limitations arise because one of the principal assumptions in advancing front theories is reaction irreversibility. We present a model which incorporates reaction equilibrium into a description reflecting the controlling membrane transport processes. This theory predicts nonzero reagent concentration, and interdependent solute, reagent, and product concentrations. An additional feature is that the differential equation locating the reaction front becomes unnecessary since satisfying reaction equilibrium turns the reaction on and off. Without any adjustable constants this reversible reaction model predicts batch extraction data from measurable physical parameters: diffusion coefficients, solute partition coefficients, and average globule size. While the model presented by Teramoto and co-workers [29] does include reaction reversibility, specification of two additional system variables, both external and internal mass transfer coefficients, are required, Theory Figure 1 identifies the pertinent variables for a typical globule of emulsion. The membrane phase separates the encapsulated internal droplets from the Emulsion Globule Bulk Phase
Membrane Internal Droplet
Fig. 1. Schematic diagram of emulsion globule.
Phase Phase
continuous, bulk phase. We consider a solute A which diffuses through the globule reacting with reagent B to produce product P. Solute A distributes through all three phases while reagent B and product P are insoluble in the membrane phase. An equilibrium constant, K, characterizes the reversible reaction: A+B$P
(I)
Several additional assumptions are made in this development. (1) The membrane and bulk phases as well as membrane and internal phases are totally immiscible. (2) Local phase equilibrium holds between membrane and droplet phases. (3) No internal circulation occurs within the globule. (4) Globule size variations can be lumped into a single effective mean diameter. (5) The system is well-agitated, eliminating external bulk phase mass transfer resistence. (6) Concentration within the internal droplets is independent of position. (7) Diffusion within the membrane is slow relative to the rate of chemical reaction. Consequently, local reaction equilibrium applies throughout the globule. (8) Membrane breakage is neglected. A few additional comments are warranted. Assumption (1) removes the complication of globule diameter changes resulting from osmotic swelling. In a typical system, the diffusion coefficients for the internal phase are larger than those in the membrane phase, Coupling this fact with the significantly smaller average diameter of encapsulated internal droplets compared to the globule, we conclude that assumptions (2) and (6) are reasonable. The presence of surfactant minimizes surface shear induced circulation [30, 311. Because B and P are insoluble in the membrane phase, reaction equilibrium applies only in the internal droplets. Neglecting activity effects, we assume mass action correctly relates the internal phase concentrations of A, B and P: CPi
K=
CAi CBi
(2)
Reagent conservation and reaction stoichiometry insist that
cgi=CBi + cpi
(3)
where C~i denotes the original concentration of B in the internal droplets. Combining eqns. (2) and (3) yields Cpi =
KCAi Cgi
1 + KCAi
(4)
which specifies the product concentration based upon the solute and initial reagent concentrations.
59
According to assumption (2), the membrane concentration, C’A~, at a given radial position inside the globule dictates the internal phase concentration of A for a droplet at the same position. That is, CAi
=
Corn
(5)
/Km
for a solute partition coefficient, Kh, between the internal and membrane phases. Hence, radial variation of CAm induces a corresponding position dependence on the internal phase concentrations of A, B and P. The equations describing the solute concentration in the membrane portion of the globule, CAM, and in the continuous, bulk phase, CAb, are: Globule: -
(‘3) (R > r-2 0)
(7)
CAm = Kbrn CAb
(t 2 0)
(8)
acAm=.
(for all t)
(9)
t=o
cAm
r=R r=O
= 0
i3r
Bulk phase:
dCAb_ -3Deff dt
(1 -
fb)fm
~CA,
ar
R fb
(10) r=R
where R is the globule radius, Kbm is the solute partition coefficient between bulk and membrane phases, fm is the volume fraction of the globule occupied by the membrane phase, and fb is the bulk phase fraction of the total volume. The mean effective diffusivity, Defr, based on the membrane phase driving force, includes diffusion of both the reacted and unreacted solute through the internal phase. Estimation of fieff from the membrane and internal phase diffusion coefficients is discussed m the Appendix. The time derivatives on the right-hand side of eqn. (6) account for changes in the membrane concentration of A by transfer into the internal droplet phase. Because the transferred solute appears in the internal phase in its reacted and unreacted forms, changes in both CAi and Cpi must be considered. If local phase and reaction equilibria are established between the internal phase droplets and the membrane phase, then the time derivatives of the internal phase concentrations of A and B can be related to the membrane concentration of A: KC’gi (1
+KQdKi,d2
(12)
60
Substituting this expression into eqn. (6) yields a differential equation in terms of the membrane phase solute concentration only. Using the following definitions, we cast these equations mto dimensionless form. =-.
rl
r
De&
CAb
4m
d’b =-;
CAm
= cib
%b 01
(13)
r=R2
R'
=fm(y)
(14)
Kbm
(15)
Kbm
02 =(l--ff,)
(yyKg
(16)
03
=
KC~i
(17)
04
=
KKbm C&,/Ki,
(16)
In dimensionless form, eqns. (6)-(11)
become: 1
1 + (cz/a1) [I + 037u + 0491n)21i 7
=o
bn
=o
17’1
@m=@b
q=o
-=
wrn a77
(1 > (r >
0
(all 7) (23) (24)
The physical significance of the dimensionless groups defined in eqns. (15)(18) is worth considering, Consisting of volume fractions and partition coefficients, ul measures the membrane capacity for solute relative to the bulk phase capacity, while o2 represents the internal phase capacity for unreacted solute relative to the bulk phase capacity. The original internal reagent and bulk solute concentrations are specified, respectively, as dimensionless groups u3 and u4. We note that the magnitude of physically possible values for u1 and u2 is restrictive. The membrane fraction of the globule, fm, can probably be no smaller than 0.5, and when fm equals one there are no internal droplets. Generally, the bulk phase fraction of the total volume is large since good
61
mixing and thorough contact with the globules require a relatively dilute emulsion. For typical partition coefficients, Kbm and Ki,, of one or less, u1 and (Towill both be less than about 0.1, A uZ value of zero corresponds to conventional liquid/liquid extraction with no internal droplet phase. Equation (19) is nonlinear and does not have an analytical solution. Equations (19) and (23) are solved simultaneously for conditions (20)-(22) and (24) using a computer package called PDECOL [ 321. Because the solute removed from the bulk phase, 1 - 4 b , must appear either in the membrane phase as unreacted A or in the internal phase as A and P, we assure accuracy of the simulation by requiring the globule and bulk phase material balances to agree within one percent. Knowing the membrane concentration profile, all internal phase concentrations can be calculated: @Ai=#m
CAi
=
Kim
(25)
Czb Kbm
(26) @Pi -0
CPi
04 @Ai
= l-~Bi=
1 + 04 @Ai
hi
(27)
These expressions reflect features characteristic of chemical reversibility. For example, the internal reagent can never be completely converted to product, although this limit can be approached when the solute concentration, indicated by U&Ai, is large. Furthermore, decreases in the membrane concentration, resulting from solute depletion in the bulk phase, force reaction (1) in reverse. Consequently, some product generated earlier reverts back to unreacted solute and reagent. The long time solution describing solute absorption and reaction consists of concentrations satisfying both phase and chemical equilibria. Overall material balances yield a quadratic equation for #b* from which the final equilibrium values are found: 4;
=#:,
Z&G
-b
+dK
(28)
where
(29)
(’ +
Ul
b=
+
u2
+
2
u203
-
04)c
(30)
The internal product and reagent concentrations are computed using eqns.
62
(26) and (27). If the original mass ratio of the internal reagent to bulk solute, u2u3/u4, is small, then eqns. (28)--(30) simplify to: (31) which considers only absorption because almost no reaction occurs. The large reagent to solute ratio limit is derived from the original quadratic equation for 4; (@g” and u4 are small):
4;s=
1 1 + 01 + 02 (1 + 03)
(32)
This reflects that the reagent concentration is large enough to force the for. ward reaction even at low solute concentrations. Results
The predictions of the reversible reaction models are shown in Figs, 2-7. To illustrate the difference between this model and the advancing front model, we include predictions from the advancing front model of Ho et al. [ 261 under identical conditions. Throughout these figures, results from our reversible reaction model are represented by solid lines, while those based on Ho’s advancing front theory are designated with dashes. Advancing front model calculations depend on two dimensionless parameters, E and E, which in our nomenclature are defined as ,tJ=-
302
03
(33)
04
(34) Physically, E is three times the original mass ratio of internal reagent to bulk solute. If E is greater than 3, then there is sufficient reagent to completely remove the solute; for E smaller than 3, there is insufficient reagent. The second dimensionless group, c, measures the globule capacity for unreacted solute relative to the reaction capacity provided by reagent B. Figure 2 compares calculations from our reversible reaction model with experimental data and computations presented by Ho et al. [26]. Table 1 summarizes the operational parameters of their experiments for phenol extraction using sodium hydroxide as the internal reagent. Computations for both models are made from measurable physical and system parameters. Since chemical reversibility slows solute uptake, the reversible equilibrium model more closely predicts experimental results than does the advancing front theory, which consistently overpredicts the rate of solute absorption. This discrepency between-laboratory and computed results grows larger for longer times.
63
Because operational values of u1 and u2 are limited, in all subsequent figures we use the same (I~and u2 arismg in the experiments summarized in Table 1. Differences between the reversible reaction model outlined here and the
0.2 0
Data
at
600
0 0
Fig. 2. Comparison
TABLE
rpm
(R = 0.3
mm)
I
I
2
4
of model
6
c:lculations
and experimental
results.
1
Experimental
conditions
and physical
Gb
= 8.19
‘OBi
= 0.375 eq-dmm3
fm
= 0.637
fb
= 0.938
KinI
= Kbm = 0 52
+D,
= 0.65
X
lO-‘O mz-set-’
+Di
= 9.98
X
lo-”
T
= 296 K
x
parameters
(from
Ho
et
al. [ 26 ] )
10m3 mol-dm-’
m2-sec.’
Equilibrium constant T*K = 1.1 x lo4 dm3-eq-’ Computed parameters E = 3.32 = 0.0418 b,ff = 2.0 X lo-‘> m2-sec.’ D,B (advancing front model) #;
U1 = 0.0221 cr2 = 0.0242 a, = 4125 = 1.65
x
lo-”
m2-set-’
= 0.0627
@;I,,
= 0.0
+Calculated using Wilke-Chang ++Ref. 1341.
correlation
[ 331.
o4 = 90
64
advancing front model of Ho and coworkers are demonstrated by concentration profiles such as those in Figs. 3 and 4. Figure 3a shows the solute concentration in the membrane, while Figs. 3b and 3c give internal phase concentrations for product and reagent. Figure 4 illustrates solute and product concentrations for a later time. 1.2 (0) r =4.352
1.:
(b)
I
I
1
I (b)
0.4 -
I.
Cc)
r
I
I
I
0.2
0.4
0.6
0.0
0
0.2
0.4
0.6
0.8
1 0
1.0
Fig. 3. Dimensionless concentration profiles for T = 2 176 and uJ/a4 = 50; a-solute concentration in membrane, b - internal phase concentration for product; c - internal phase concentration
for reagent.
Fig. 4. Dimensionless concentration profiles for T = 4.352 and a,/o, = 50; a centration in membrane; b - internal phase concentration for product.
sol~~te con-
65
The original mass ratio of reagent to bulk solute is held constant in both Figs. 3 and 4 as indicated by an unchanging E of 3.6. Concentration profiles computed from the reversible reaction model depend on the individual values of the original concentrations in addition to the ratio. The solid lines designate u3 varying from 100 to 10,000, with corresponding changes in u4. As u3 and u4 increase, our model more closely resembles Ho’s model. This can be viewed in two ways. Both u3 and u4 are increased by increasing the concentration of both the reagent and the solute. As the concentration of reactants increases, reaction in the forward direction is favored and the system appears to approach irreversibility. A second view is that proportionally increasing u3 and u4 suggests an increased equilibrium constant and consequent preference for the forward reaction. Except for large initial concentrations, reversibility allows a significant fraction of the internal reagent to remain unconverted. More significantly, at later times some of the reagent which originally reacted is restored by reaction (1) acting in reverse. The reversible reaction model predicts that the mass ratio of solute to reagent alone is an insufficient basis for describing solute removal from the bulk phase. Consequently, designing separation equipment from extrapolated laboratory results without considering reaction reversibility could lead to serious errors. The discrepancy between the reaction equilibrium and reaction front model grows with dimensionless time as exhibited in Fig. 4. As should be expected, the mass of solute and reagent becomes depleted at later times, thereby enhancing reaction (1) in the reverse direction. The product concentration at the globule surface is less in Fig. 4b than in Fig. 3b, because to maintain reaction equilibrium some product reacts, restoring reactant and solute. The influence of reversibility is most evident for long times when overall equilibrium is approached. The reversible reaction view correctly computes uniform, finite concentrations of A, B and P in all three phases. By contrast, when the original mass ratio of reagent to solute is greater than 1, advancing front models predict discontinuous profiles, with reacted and unreacted zones. Because these models assume reaction irreversibility in their development, long time solutions are not appropriate. The exception to this failure is when the mass ratio of reagent to solute is small. In this situation, all of the reagent is consumed and phase equilibrium rather than solute reaction controls. The parameters u3 and u4 provide a quantitative measure of the applicability of the reaction front model. When both u3 and u4 are large or when only short dimensionless times are considered, the computationally simpler algebraic description of Ho and coauthors is adequate. Figures 5a and 5b show model predictions for bulk solute uptake as a function of time. The advancing front model overpredicts solute absorption whether the original mass of the reagent exceeds the solute, E > 3, or the reverse case is true, E < 3. The bulk solute concentration at infinite time is indicated inside parenthesis. When E is larger than 3, all of the solute can
66
react. When E is less than 3, solute consumes all of the reagent and then establishes phase equilibrium. Under these circumstances, Ho’s model predicts a final equilibrium value of: 1 -E/3 &
\Ho =
1 +
eE/3
z
1 - 02
fJ3/@4
(35)
1 f u1 + u.2
When the mass ratio of reagent to solute becomes very small, eqn. (35) COincides with eqn. (31). As observed earlier in Figs. 3 and 4, our model approaches Ho’s results when u3 and o4 are proportionally increased while holding their ratio constant. 1.2
I
I
I
I
(0) < = 00384
E =3.6
o0
I 1
I 2
I 3
I 4
5
5
Fig. 5a. Bulk solute concentration 1.2
[
I E= 2.88
0’0
as function of T for u,/04 = 50. I
ai = 0.048
I
I
I
I
I
1
2
3
4
5
I-
Fig. 5b. Bulk solute concentration
as function of T for u,/o,
= 40.
Figures 6 and 7 demonstrate effects of u3 or u4 variations on solute uptake rates. Figure 6 considers variations in the reagent for fixed solute, 04, concentrations. Our model predicts higher solute concentrations, with the gap between models widening as u3 increases. For small u3, phase equilibrium rather than reaction equilibrium controls and the models nearly coincide. Unfortunately, this situation is unlikely in practical applications of this
67
process. For extremely large values of u3, the two models should again converge since this corresponds to very rapid decreases in bulk solute concentration. Figure 7 examines variations in solute concentration, (~4, for fixed (~3. Trends similar to Fig. 6 are observed. Results like those predicted in Figs. 6 and 7 were observed by Teramoto and coworkers [29] in experiments extract ing amines using HCl as the internal reagent.
4
0
8
50
(0.393)
55
(0.364)
30
(0)
-
_
I
I
12
16
20
T Fig.
6. Bulk solute concentration 12
-0
I
1
1
2
as function I
3
of T for varying
(Jo.
I
4
5
r
Fig. 7. Bulk solute concentration
as function
of T for varying
o3
Conclusions We present a new model for predicting and analyzing solute uptake in emulsion liquid membranes. Calculations are made using only measurable physical and system properties. The distinguishing feature compared to previous models is the inclusion of reversibility in the reaction of solute with reagent present in the internal droplets. The agreement between model predictions and experiments is excellent. Sample computations clearly illustrate that the original mass ratio of
68
reagent to solute is an insufficient criterion for specifying performance. Depending on the reaction equilibrium constant and the original reactant concentrations, reversibility effects can be large. Acknowledgements The authors have benefitted from discussions with Professor C.J. Radke and Mr. Frank Jahnke, both at the University of California. References 1 2 3 4 5 6
7 8
9 10 11 12 13
14
15 16 17 18 19
N.N. Li, Separating hydrocarbons with liquid membranes, U.S. Patent 3,410,794, 1968. N.N. Li, Permeation through liquid surfactant membranes, AIChE J., 17 (1971) 459. N.N. Li, Separation of hydrocarbons by liquid membrane permeation, Ind. Eng. Chem., Process Des. Dev., 10 (1971) 215. A.M. Hochhauser and E.L. Cussler, Concentrating chromium with liquid surfactant membranes, AIChE Symp. Ser., 71(152) (1975) 136. T.P. Martin and G.A. Davies, The extraction of copper from dilute aqueous solutions usmg a liquid membrane process, Hydrometallurgy, 2 (1976/1977) 315. K. Kondo, K. Kita, I. Koida, J. Irie and E. Nakashio, Extraction of copper with liquid surfactant membranes containing benzoylacetone, J. Chem. Eng. Jpn., 12 (1979) 203. W. Volkel, W. Halwachs and K. Schiigerl, Copper extraction by means of a liquid surfactant membrane process, J. Membrane Sci., 6 (1980) 19. J. Strzelbicki and W. Charewicz, The liquid surfactant membrane separation of copper. cobalt, and nickel from multicomponent aqueous solutions, Hydrometallurgy, 5 (1980) 243. R.P. Cahn and N.N. Li, Separation of phenol from waste water by liquid membrane technique, Sep. Sci., 9 (1974) 505. W. Halwachs, E. Flaschel and K. Schhgerl, Liquid membrane transport - A highly selective separation process for organic solutes, J. Membrane Sci., 6 (1980) 33. 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. S.W. May and N.N. Li, The immobilization of urease using liquid surfactant membranes, Biochem. Biophys. Res. Commun., 47 (1972) 1179. 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 Int., 7 (1975) 5-409. 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. Intern. Organs, 23 (1977) 673. W. Halwachs and R. Schhgerl, The liquid membrane technique - A promising extraction process, Int. Chem. Eng., 20 (1980) 519. J.D. Way, R.D. Noble, T.M. Flynn and E.D. Sloan, Liquid membrane transport: A survey, J. Membrane Sci., 12 (1982) 239. L. Boyadzhiev, T. Sapundzhiev and E. Bezensbek, Modeling of carrier-mediated extraction, Sep. Sci., 12 (1978) 541. V.J. Kremesec, Modeling of dispersed-emulsion separation systems, Sep. Purif. Methods, 10 (1981) 117. V.J. Kremesec and J.C. Slattery, Analysis of batch, dispersed emulsion separation systems, AIChE J., 28 (1982) 492.
69
L. Gladek, J. Stelmaszek and J. Szust, Modeling of mass transport with a very fast reaction through liquid membranes, J. Membrane Sci., 12 (1982) 153. 21 E.S. Matulevicius and N.N. Li, Facilitated transport through liquid membrane, Sep. Purif. Methods, 4 (1975) 73. 22 J.D. Way and R.D. Noble, A macroscopic model of a continuous emulsion liquid membrane extraction system, in: A. Petho and R.D. Noble (Eds.), Residence Tie Distribution Theory in Chemical Engineering, Verlog. Chemie, Deerfield Beach, FL, 1982, p. 247. 23 R.D. Noble, Shape factors in facilitated transport through membranes, Ind. Eng. Chem. Fundam., 22 (1983) 139. 24 CA. Folkner and R.D. Noble, Transient response of facilitated transport membranes, J. Membrane Sci., 12 (1983) 289. 25 A.G. Kopp, R.J. Marr and F.E. Moser, A new concept for mass transfer in liquid surfactant membranes without carriers and with carriers that pump, Inst. Chem. Eng., Symp. Ser., 54 (1978) 279. 26 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 (1982) 662. 27 K. Kim, S. Choi and S. Ihm, Simulation of phenol removal from wastewater by liquid membrane emulsion, Ind. Eng. Chem. Fundam., 22 (1983) 167. 28 P. Stroeve and P.P. Varanasi, Extraction with double emulsions in a bath reactor: Effect of continuous phase resistance, AIChE J., submitted for publication. 29 M. Teramoto, H. Takihana, M. Shibutani, T. Yuasa, Y. Miyake and H. Teranishi, Extraction of amine by W/O/W emulsion system, J. Chem. Eng. Jpn., 14 (1981) 122. 30 F.D. Rumscheidt and S.G. Mason, Particle motions in sheared suspensions: XI. Internal circulation in fluid droplets, J. Colloid Sci., 16 (1961) 210. 31 V.G. Levich, Physicochemical Hydrodynamics, Prentice-Hall, Englewood Cliffs, NJ, 1962. 32 N.K. Madsen and R.F. Sincover, PDECOL, general collocation software for partial differential equations [ D3 1, ACM Trans. Math. Software, 5 (1979) 326. 33 R.C. Reid, J.M. Prausnitz and T.K. Sherwood, The Properties of Gases and Liquids, McGraw-Hill, New York, NY, 1977, Chap. 11. 34 R.T. Morrison and R.N. Boyd, Organic Chemistry, 2nd edn., Allyn and Bacon, Boston, MA, 1966, p. 790. 35 J. Crank, The Mathematics of Diffusion, 2nd edn., Clarendon Press, Oxford, 1975, Chap. 12. 36 W.J. Ward, Analytical and experimental studies of facilitated transport AIChE J., 16 (1970) 405. 20
Appendix Although the mean effective diffusivity, Deff, is based on the membrane phase driving force, it must include contributions from diffusion of both the reacted and unreacted solute through the internal phase. Diffusion through the membrane phase can be consequential when the internal phase molecular diffusion coefficient is significantly larger than the membrane.phase, or when solute reaction induces large solute concentrations in the internal phase relative to those in the membrane phase. Following Ho et al. [26], we calculate the effective globule diffusivity based on a local membrane phase concentration driving force, D,ff, using the Jefferson-Witzell-Sibbit equation for diffusion through a composite medium [ 26,351: D eff = Dm
1 -
7l
4 (1 + 2p)2
1
+
DADrn
71
4 (1 + 2P)
Dm
+~PDA
(AlI
70
DA = P
Dir IKim
2 (Dir /Kim 1 Drn
(Dir /Kim 11
N DiT IKti
Dm
= 0.403 (1 - fm)-l’3 -
- Dm
)
ln ( Ksm)-l]
0.5
(A21 (A31
In eqn. (A2), D, is the molecular diffusion of the transferring solute in the membrane phase. The internal phase molecular diffusion coefficients for both the reacted and unreacted solute are assumed to be the same value, D,. A reversible reaction in the internal phase permits reaction of the solute on one side of an internal droplet, diffusion of the reacted species to the other side of the droplet, followed by the reverse reaction and solute transfer back into the membrane phase. Reaction and diffusion in the internal phase can enhance the apparent diffusivity based on the membrane phase driving force. This enhancement of D,ff is incorporated into eqn. (A2) through the parameter y which depends on the reaction equilibrium, the initial concentration of reactant B, the distribution coefficient between the membrane and internal phases, and the local membrane concentration of the solute. To find an expression for 7, we follow the analysis of Ward [ 361 in his study of facilitated transport. Consistent with the globule model, we assume chemical equilibrium is always satisfied in the internal phase. (53 r=l+
(A4)
(1 +
~4hl)”
The development of eqn. (A4) assumes that the internal droplets are small enough that the membrane concentration is essentially the same on opposite sides of the droplet. In general, the effective diffusion coefficient will depend on the local membrane concentration, om . The dependence of D,ff on r#~~is illustrated
no reactmn
1.01 0
Fig. Al oJo‘$.
I
I
0.2
0.4
I 06
I
0.8
1.0
Effective diffusivity as a function of dimensionless membrane concentration
for fixed
71
in Fig. Al. The membrane fraction of the globule volume is 0.637 in these calculations. Because the sensitivity of D,ff to changes in om are not large, and because tremendous mathematical complexity arises by allowing D,ff to changes in om are not large, and because tremendous mathematical complexity arises by allowing D,ff to vary with q&, we use Deff at its mean value over the range of q& varying from 0 to 1. The mean effective diffusivity for analyzing the results of Ho et al. [26] in Fig. 2 was found to be 2.0 X lo-” m’/sec for u3 and o4 of 4125 and 90, respectively. This compares to 1.65-X 10-l’ m’/sec if no reaction takes place.
1.6 10-l
I
I 10'
1
I 103
I
I 105
I
I 10'
c!
Fig. A2. Mean effective diffusivity as a function of dimensionless reagent concentration for fixed o,/o,.
Figure A2 is a plot of the mean Deff as a function of dimensionless reagent concentration, u3, while the mass ratio of reagent to solute remains constant. Again, fm is 0.637,As the reaction approaches irreversibility, as indicated by large values of u3, D,ff approaches the no reaction case. As u3 becomes small, enhancement of Beff disappears because the amount of solute reacting was decreased. In the advancing front model, diffusion occurs solely through the fully reacted shell. Accordingly, the effective diffusivity should not include the effect of reaction.
Science
21 (1984)
Science,
Publishers
B.V.,
55-71
Amsterdam
55
-
Printed
in The Netherlands
A DIFFUSION MODEL FOR REVERSIBLE COMSUMPTION IN EMULSION LIQUID MEMBRANES*
A.L.
BUNGE
Colorado
School
and RICHARD
National (Received
of Mines, Golden,
CO 80401
(U.S.A.)
D. NOBLE
Bureau of Standards, October
28,1983;
Boulder, accepted
CO 80303 m revised
(U.S.A.) form May 17, 1984)
Summary This work extends previous diffusion models for emulsion globules in which a solute reacts with an internal reagent This model allows for reversible consumption of the solute by the internal reagent. Local concentration of the internal reagent is nonzero and satisfies reaction and phase equilibria within the reacted zone. Predicted solute absorption rates are lower for the reversible consumption model than for irreversible models.
Introduction Emulsion liquid membranes were invented by Li [l] , who first applied this technique to the separation of hydrocarbons [2, 31. Since then, emulsion liquid membranes have been applied to other separations. These include recovery and purification of metal ions [4--81, removal of phenol from waste water [9-111, and biological applications such as an artificial kidney [ 12--14 J . Review articles discuss this separation and its uses in more detail [15,16]. Numerous researchers have mathematically modeled these membrane separation processes. Previously, model development has followed one of two approaches. The mathematically simpler representation characterizes the emulsion globule as a double shell around a single internal phase droplet. The spherical shell viewpoint is formally equivalent to assuming an internally well-mixed globule . In an early paper, Calm and Li [9] analyzed experiments involving phenol removal from wastewater by asserting that the rate of phenol transfer is directly proportional to the concentration difference of phenol across the membrane shell. Assuming that a reagent within the internal droplet instantaneously and irreversibly consumes phenol, they determined that the effec*Paper presented Chemical Process
0376-7388/84/$03.00
at the Symposium entitled “Membrane Separation Applications,” Denver, Colorado, August 28-31,
o 1984
Elsevier
Science
Publishers
B.V.
Technology 1983.
for
56
tive permeability varied with time. Boyadzhiev et al. [ 171 follow this same analysis. However, their treatment does not account for internal reagent consumption. Kremesec [18] and Kremesec and Slattery [19] use planar geometry and sum mass transfer resistances through the continuous, membrane, and internal phases. In their method geometric and internal circulation effects are lumped into the overall mass transfer coefficient. Gladek et al. [ 201 examine the unsteady-state solute absorption when the solute partition coefficient varies with concentration. Several researchers choose the spherical shell approach to describe carrierfacilitated transfer for a variety of situations. Matulevicius and Li [ 211 consider theoretically the time-dependent permeability by solving the unsteadystate mass transfer equations when solute diffusion through the membrane layer controls. In a model with limitations similar to that of Matulevicius and Li, Hochhauser and Cussler [4] consider chromium concentration by coupled transport in the membrane phase. Way and Noble [22] model copper ion uptake in a continuous flow separation scheme which include residence time and particle size distributions for the double emulsion drops. Noble [23] calculated the steady, facilitated flux for either diffusion- or reaction-limited cases when the carrier molecule reacts reversibly. Folkner and Noble [ 241 examine the initial transient response for this system. While mathematically more complex, a second approach more correctly accounts for mass transfer contributions accompanying solute diffusion and reaction in an emulsion globule. This approach assumes that solute removed from the bulk phase diffuses through the globule to a reaction front, where it is removed instantaneously and irreversibly by reaction with an internal reagent. In turn, the reaction front progresses towards the center as internal reagent is consumed. Kopp et al. [25] adopted this approach by examining the analogous planar problem with constant bulk solute concentration. Three papers, Ho et al, [ 261, Kim et al. [ 271, and Stroeve and Varanasi [ 281, formulate advancing front theories which include both spherical geometry and depletion of solute in the bulk phase. All three treatments assume homogeneous distribution of noncirculating internal droplets within the globule, although Kim et al. assume a thin outer liquid membrane layer which contains no internal droplets. For globule diffusion controlling solute transfer, Ho and coworkers use a perturbation method to solve the resulting system of nonlinear equations. From the zero-order or pseudo-steady-state solution, they calculate phenol removal from the continuous phase as a function of time. Using a Sauter mean diameter for the average emulsion drop size and no adjustable parameters, they predict somewhat higher removal rates than observed experimentally. Stroeve and Varanasi also use a zero-order advancing front model but include a mass transfer resistance in the continuous phase. They show that their model reduces to that of Ho et al. when the mass transfer resistance becomes negligible.
Kim et al. include diffusion through a thin membrane film without droplets as an additional resistance. As expected, their model successfully predicts the experimental results given, since the thickness of the membrane film essentially constitutes an adjustable parameter. The advancing front approach overpredicts the removal rate of solute since this approach requires the reagent concentration in the reacted zone between the globule surface and the reaction front to be zero. When the internal reagent is a base such as sodium hydroxide, this requirement is not physically correct. Furthermore, advancing front models assume that the local solute concentration does not affect the amount of reagent to react and that the reagent permanently traps reacted solute. Thus, advancing front models incorrectly predict that even low solute concentrations can force reagent at the reaction front to react completely. These limitations arise because one of the principal assumptions in advancing front theories is reaction irreversibility. We present a model which incorporates reaction equilibrium into a description reflecting the controlling membrane transport processes. This theory predicts nonzero reagent concentration, and interdependent solute, reagent, and product concentrations. An additional feature is that the differential equation locating the reaction front becomes unnecessary since satisfying reaction equilibrium turns the reaction on and off. Without any adjustable constants this reversible reaction model predicts batch extraction data from measurable physical parameters: diffusion coefficients, solute partition coefficients, and average globule size. While the model presented by Teramoto and co-workers [29] does include reaction reversibility, specification of two additional system variables, both external and internal mass transfer coefficients, are required, Theory Figure 1 identifies the pertinent variables for a typical globule of emulsion. The membrane phase separates the encapsulated internal droplets from the Emulsion Globule Bulk Phase
Membrane Internal Droplet
Fig. 1. Schematic diagram of emulsion globule.
Phase Phase
continuous, bulk phase. We consider a solute A which diffuses through the globule reacting with reagent B to produce product P. Solute A distributes through all three phases while reagent B and product P are insoluble in the membrane phase. An equilibrium constant, K, characterizes the reversible reaction: A+B$P
(I)
Several additional assumptions are made in this development. (1) The membrane and bulk phases as well as membrane and internal phases are totally immiscible. (2) Local phase equilibrium holds between membrane and droplet phases. (3) No internal circulation occurs within the globule. (4) Globule size variations can be lumped into a single effective mean diameter. (5) The system is well-agitated, eliminating external bulk phase mass transfer resistence. (6) Concentration within the internal droplets is independent of position. (7) Diffusion within the membrane is slow relative to the rate of chemical reaction. Consequently, local reaction equilibrium applies throughout the globule. (8) Membrane breakage is neglected. A few additional comments are warranted. Assumption (1) removes the complication of globule diameter changes resulting from osmotic swelling. In a typical system, the diffusion coefficients for the internal phase are larger than those in the membrane phase, Coupling this fact with the significantly smaller average diameter of encapsulated internal droplets compared to the globule, we conclude that assumptions (2) and (6) are reasonable. The presence of surfactant minimizes surface shear induced circulation [30, 311. Because B and P are insoluble in the membrane phase, reaction equilibrium applies only in the internal droplets. Neglecting activity effects, we assume mass action correctly relates the internal phase concentrations of A, B and P: CPi
K=
CAi CBi
(2)
Reagent conservation and reaction stoichiometry insist that
cgi=CBi + cpi
(3)
where C~i denotes the original concentration of B in the internal droplets. Combining eqns. (2) and (3) yields Cpi =
KCAi Cgi
1 + KCAi
(4)
which specifies the product concentration based upon the solute and initial reagent concentrations.
59
According to assumption (2), the membrane concentration, C’A~, at a given radial position inside the globule dictates the internal phase concentration of A for a droplet at the same position. That is, CAi
=
Corn
(5)
/Km
for a solute partition coefficient, Kh, between the internal and membrane phases. Hence, radial variation of CAm induces a corresponding position dependence on the internal phase concentrations of A, B and P. The equations describing the solute concentration in the membrane portion of the globule, CAM, and in the continuous, bulk phase, CAb, are: Globule: -
(‘3) (R > r-2 0)
(7)
CAm = Kbrn CAb
(t 2 0)
(8)
acAm=.
(for all t)
(9)
t=o
cAm
r=R r=O
= 0
i3r
Bulk phase:
dCAb_ -3Deff dt
(1 -
fb)fm
~CA,
ar
R fb
(10) r=R
where R is the globule radius, Kbm is the solute partition coefficient between bulk and membrane phases, fm is the volume fraction of the globule occupied by the membrane phase, and fb is the bulk phase fraction of the total volume. The mean effective diffusivity, Defr, based on the membrane phase driving force, includes diffusion of both the reacted and unreacted solute through the internal phase. Estimation of fieff from the membrane and internal phase diffusion coefficients is discussed m the Appendix. The time derivatives on the right-hand side of eqn. (6) account for changes in the membrane concentration of A by transfer into the internal droplet phase. Because the transferred solute appears in the internal phase in its reacted and unreacted forms, changes in both CAi and Cpi must be considered. If local phase and reaction equilibria are established between the internal phase droplets and the membrane phase, then the time derivatives of the internal phase concentrations of A and B can be related to the membrane concentration of A: KC’gi (1
+KQdKi,d2
(12)
60
Substituting this expression into eqn. (6) yields a differential equation in terms of the membrane phase solute concentration only. Using the following definitions, we cast these equations mto dimensionless form. =-.
rl
r
De&
CAb
4m
d’b =-;
CAm
= cib
%b 01
(13)
r=R2
R'
=fm(y)
(14)
Kbm
(15)
Kbm
02 =(l--ff,)
(yyKg
(16)
03
=
KC~i
(17)
04
=
KKbm C&,/Ki,
(16)
In dimensionless form, eqns. (6)-(11)
become: 1
1 + (cz/a1) [I + 037u + 0491n)21i 7
=o
bn
=o
17’1
@m=@b
q=o
-=
wrn a77
(1 > (r >
0
(all 7) (23) (24)
The physical significance of the dimensionless groups defined in eqns. (15)(18) is worth considering, Consisting of volume fractions and partition coefficients, ul measures the membrane capacity for solute relative to the bulk phase capacity, while o2 represents the internal phase capacity for unreacted solute relative to the bulk phase capacity. The original internal reagent and bulk solute concentrations are specified, respectively, as dimensionless groups u3 and u4. We note that the magnitude of physically possible values for u1 and u2 is restrictive. The membrane fraction of the globule, fm, can probably be no smaller than 0.5, and when fm equals one there are no internal droplets. Generally, the bulk phase fraction of the total volume is large since good
61
mixing and thorough contact with the globules require a relatively dilute emulsion. For typical partition coefficients, Kbm and Ki,, of one or less, u1 and (Towill both be less than about 0.1, A uZ value of zero corresponds to conventional liquid/liquid extraction with no internal droplet phase. Equation (19) is nonlinear and does not have an analytical solution. Equations (19) and (23) are solved simultaneously for conditions (20)-(22) and (24) using a computer package called PDECOL [ 321. Because the solute removed from the bulk phase, 1 - 4 b , must appear either in the membrane phase as unreacted A or in the internal phase as A and P, we assure accuracy of the simulation by requiring the globule and bulk phase material balances to agree within one percent. Knowing the membrane concentration profile, all internal phase concentrations can be calculated: @Ai=#m
CAi
=
Kim
(25)
Czb Kbm
(26) @Pi -0
CPi
04 @Ai
= l-~Bi=
1 + 04 @Ai
hi
(27)
These expressions reflect features characteristic of chemical reversibility. For example, the internal reagent can never be completely converted to product, although this limit can be approached when the solute concentration, indicated by U&Ai, is large. Furthermore, decreases in the membrane concentration, resulting from solute depletion in the bulk phase, force reaction (1) in reverse. Consequently, some product generated earlier reverts back to unreacted solute and reagent. The long time solution describing solute absorption and reaction consists of concentrations satisfying both phase and chemical equilibria. Overall material balances yield a quadratic equation for #b* from which the final equilibrium values are found: 4;
=#:,
Z&G
-b
+dK
(28)
where
(29)
(’ +
Ul
b=
+
u2
+
2
u203
-
04)c
(30)
The internal product and reagent concentrations are computed using eqns.
62
(26) and (27). If the original mass ratio of the internal reagent to bulk solute, u2u3/u4, is small, then eqns. (28)--(30) simplify to: (31) which considers only absorption because almost no reaction occurs. The large reagent to solute ratio limit is derived from the original quadratic equation for 4; (@g” and u4 are small):
4;s=
1 1 + 01 + 02 (1 + 03)
(32)
This reflects that the reagent concentration is large enough to force the for. ward reaction even at low solute concentrations. Results
The predictions of the reversible reaction models are shown in Figs, 2-7. To illustrate the difference between this model and the advancing front model, we include predictions from the advancing front model of Ho et al. [ 261 under identical conditions. Throughout these figures, results from our reversible reaction model are represented by solid lines, while those based on Ho’s advancing front theory are designated with dashes. Advancing front model calculations depend on two dimensionless parameters, E and E, which in our nomenclature are defined as ,tJ=-
302
03
(33)
04
(34) Physically, E is three times the original mass ratio of internal reagent to bulk solute. If E is greater than 3, then there is sufficient reagent to completely remove the solute; for E smaller than 3, there is insufficient reagent. The second dimensionless group, c, measures the globule capacity for unreacted solute relative to the reaction capacity provided by reagent B. Figure 2 compares calculations from our reversible reaction model with experimental data and computations presented by Ho et al. [26]. Table 1 summarizes the operational parameters of their experiments for phenol extraction using sodium hydroxide as the internal reagent. Computations for both models are made from measurable physical and system parameters. Since chemical reversibility slows solute uptake, the reversible equilibrium model more closely predicts experimental results than does the advancing front theory, which consistently overpredicts the rate of solute absorption. This discrepency between-laboratory and computed results grows larger for longer times.
63
Because operational values of u1 and u2 are limited, in all subsequent figures we use the same (I~and u2 arismg in the experiments summarized in Table 1. Differences between the reversible reaction model outlined here and the
0.2 0
Data
at
600
0 0
Fig. 2. Comparison
TABLE
rpm
(R = 0.3
mm)
I
I
2
4
of model
6
c:lculations
and experimental
results.
1
Experimental
conditions
and physical
Gb
= 8.19
‘OBi
= 0.375 eq-dmm3
fm
= 0.637
fb
= 0.938
KinI
= Kbm = 0 52
+D,
= 0.65
X
lO-‘O mz-set-’
+Di
= 9.98
X
lo-”
T
= 296 K
x
parameters
(from
Ho
et
al. [ 26 ] )
10m3 mol-dm-’
m2-sec.’
Equilibrium constant T*K = 1.1 x lo4 dm3-eq-’ Computed parameters E = 3.32 = 0.0418 b,ff = 2.0 X lo-‘> m2-sec.’ D,B (advancing front model) #;
U1 = 0.0221 cr2 = 0.0242 a, = 4125 = 1.65
x
lo-”
m2-set-’
= 0.0627
@;I,,
= 0.0
+Calculated using Wilke-Chang ++Ref. 1341.
correlation
[ 331.
o4 = 90
64
advancing front model of Ho and coworkers are demonstrated by concentration profiles such as those in Figs. 3 and 4. Figure 3a shows the solute concentration in the membrane, while Figs. 3b and 3c give internal phase concentrations for product and reagent. Figure 4 illustrates solute and product concentrations for a later time. 1.2 (0) r =4.352
1.:
(b)
I
I
1
I (b)
0.4 -
I.
Cc)
r
I
I
I
0.2
0.4
0.6
0.0
0
0.2
0.4
0.6
0.8
1 0
1.0
Fig. 3. Dimensionless concentration profiles for T = 2 176 and uJ/a4 = 50; a-solute concentration in membrane, b - internal phase concentration for product; c - internal phase concentration
for reagent.
Fig. 4. Dimensionless concentration profiles for T = 4.352 and a,/o, = 50; a centration in membrane; b - internal phase concentration for product.
sol~~te con-
65
The original mass ratio of reagent to bulk solute is held constant in both Figs. 3 and 4 as indicated by an unchanging E of 3.6. Concentration profiles computed from the reversible reaction model depend on the individual values of the original concentrations in addition to the ratio. The solid lines designate u3 varying from 100 to 10,000, with corresponding changes in u4. As u3 and u4 increase, our model more closely resembles Ho’s model. This can be viewed in two ways. Both u3 and u4 are increased by increasing the concentration of both the reagent and the solute. As the concentration of reactants increases, reaction in the forward direction is favored and the system appears to approach irreversibility. A second view is that proportionally increasing u3 and u4 suggests an increased equilibrium constant and consequent preference for the forward reaction. Except for large initial concentrations, reversibility allows a significant fraction of the internal reagent to remain unconverted. More significantly, at later times some of the reagent which originally reacted is restored by reaction (1) acting in reverse. The reversible reaction model predicts that the mass ratio of solute to reagent alone is an insufficient basis for describing solute removal from the bulk phase. Consequently, designing separation equipment from extrapolated laboratory results without considering reaction reversibility could lead to serious errors. The discrepancy between the reaction equilibrium and reaction front model grows with dimensionless time as exhibited in Fig. 4. As should be expected, the mass of solute and reagent becomes depleted at later times, thereby enhancing reaction (1) in the reverse direction. The product concentration at the globule surface is less in Fig. 4b than in Fig. 3b, because to maintain reaction equilibrium some product reacts, restoring reactant and solute. The influence of reversibility is most evident for long times when overall equilibrium is approached. The reversible reaction view correctly computes uniform, finite concentrations of A, B and P in all three phases. By contrast, when the original mass ratio of reagent to solute is greater than 1, advancing front models predict discontinuous profiles, with reacted and unreacted zones. Because these models assume reaction irreversibility in their development, long time solutions are not appropriate. The exception to this failure is when the mass ratio of reagent to solute is small. In this situation, all of the reagent is consumed and phase equilibrium rather than solute reaction controls. The parameters u3 and u4 provide a quantitative measure of the applicability of the reaction front model. When both u3 and u4 are large or when only short dimensionless times are considered, the computationally simpler algebraic description of Ho and coauthors is adequate. Figures 5a and 5b show model predictions for bulk solute uptake as a function of time. The advancing front model overpredicts solute absorption whether the original mass of the reagent exceeds the solute, E > 3, or the reverse case is true, E < 3. The bulk solute concentration at infinite time is indicated inside parenthesis. When E is larger than 3, all of the solute can
66
react. When E is less than 3, solute consumes all of the reagent and then establishes phase equilibrium. Under these circumstances, Ho’s model predicts a final equilibrium value of: 1 -E/3 &
\Ho =
1 +
eE/3
z
1 - 02
fJ3/@4
(35)
1 f u1 + u.2
When the mass ratio of reagent to solute becomes very small, eqn. (35) COincides with eqn. (31). As observed earlier in Figs. 3 and 4, our model approaches Ho’s results when u3 and o4 are proportionally increased while holding their ratio constant. 1.2
I
I
I
I
(0) < = 00384
E =3.6
o0
I 1
I 2
I 3
I 4
5
5
Fig. 5a. Bulk solute concentration 1.2
[
I E= 2.88
0’0
as function of T for u,/04 = 50. I
ai = 0.048
I
I
I
I
I
1
2
3
4
5
I-
Fig. 5b. Bulk solute concentration
as function of T for u,/o,
= 40.
Figures 6 and 7 demonstrate effects of u3 or u4 variations on solute uptake rates. Figure 6 considers variations in the reagent for fixed solute, 04, concentrations. Our model predicts higher solute concentrations, with the gap between models widening as u3 increases. For small u3, phase equilibrium rather than reaction equilibrium controls and the models nearly coincide. Unfortunately, this situation is unlikely in practical applications of this
67
process. For extremely large values of u3, the two models should again converge since this corresponds to very rapid decreases in bulk solute concentration. Figure 7 examines variations in solute concentration, (~4, for fixed (~3. Trends similar to Fig. 6 are observed. Results like those predicted in Figs. 6 and 7 were observed by Teramoto and coworkers [29] in experiments extract ing amines using HCl as the internal reagent.
4
0
8
50
(0.393)
55
(0.364)
30
(0)
-
_
I
I
12
16
20
T Fig.
6. Bulk solute concentration 12
-0
I
1
1
2
as function I
3
of T for varying
(Jo.
I
4
5
r
Fig. 7. Bulk solute concentration
as function
of T for varying
o3
Conclusions We present a new model for predicting and analyzing solute uptake in emulsion liquid membranes. Calculations are made using only measurable physical and system properties. The distinguishing feature compared to previous models is the inclusion of reversibility in the reaction of solute with reagent present in the internal droplets. The agreement between model predictions and experiments is excellent. Sample computations clearly illustrate that the original mass ratio of
68
reagent to solute is an insufficient criterion for specifying performance. Depending on the reaction equilibrium constant and the original reactant concentrations, reversibility effects can be large. Acknowledgements The authors have benefitted from discussions with Professor C.J. Radke and Mr. Frank Jahnke, both at the University of California. References 1 2 3 4 5 6
7 8
9 10 11 12 13
14
15 16 17 18 19
N.N. Li, Separating hydrocarbons with liquid membranes, U.S. Patent 3,410,794, 1968. N.N. Li, Permeation through liquid surfactant membranes, AIChE J., 17 (1971) 459. N.N. Li, Separation of hydrocarbons by liquid membrane permeation, Ind. Eng. Chem., Process Des. Dev., 10 (1971) 215. A.M. Hochhauser and E.L. Cussler, Concentrating chromium with liquid surfactant membranes, AIChE Symp. Ser., 71(152) (1975) 136. T.P. Martin and G.A. Davies, The extraction of copper from dilute aqueous solutions usmg a liquid membrane process, Hydrometallurgy, 2 (1976/1977) 315. K. Kondo, K. Kita, I. Koida, J. Irie and E. Nakashio, Extraction of copper with liquid surfactant membranes containing benzoylacetone, J. Chem. Eng. Jpn., 12 (1979) 203. W. Volkel, W. Halwachs and K. Schiigerl, Copper extraction by means of a liquid surfactant membrane process, J. Membrane Sci., 6 (1980) 19. J. Strzelbicki and W. Charewicz, The liquid surfactant membrane separation of copper. cobalt, and nickel from multicomponent aqueous solutions, Hydrometallurgy, 5 (1980) 243. R.P. Cahn and N.N. Li, Separation of phenol from waste water by liquid membrane technique, Sep. Sci., 9 (1974) 505. W. Halwachs, E. Flaschel and K. Schhgerl, Liquid membrane transport - A highly selective separation process for organic solutes, J. Membrane Sci., 6 (1980) 33. 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. S.W. May and N.N. Li, The immobilization of urease using liquid surfactant membranes, Biochem. Biophys. Res. Commun., 47 (1972) 1179. 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 Int., 7 (1975) 5-409. 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. Intern. Organs, 23 (1977) 673. W. Halwachs and R. Schhgerl, The liquid membrane technique - A promising extraction process, Int. Chem. Eng., 20 (1980) 519. J.D. Way, R.D. Noble, T.M. Flynn and E.D. Sloan, Liquid membrane transport: A survey, J. Membrane Sci., 12 (1982) 239. L. Boyadzhiev, T. Sapundzhiev and E. Bezensbek, Modeling of carrier-mediated extraction, Sep. Sci., 12 (1978) 541. V.J. Kremesec, Modeling of dispersed-emulsion separation systems, Sep. Purif. Methods, 10 (1981) 117. V.J. Kremesec and J.C. Slattery, Analysis of batch, dispersed emulsion separation systems, AIChE J., 28 (1982) 492.
69
L. Gladek, J. Stelmaszek and J. Szust, Modeling of mass transport with a very fast reaction through liquid membranes, J. Membrane Sci., 12 (1982) 153. 21 E.S. Matulevicius and N.N. Li, Facilitated transport through liquid membrane, Sep. Purif. Methods, 4 (1975) 73. 22 J.D. Way and R.D. Noble, A macroscopic model of a continuous emulsion liquid membrane extraction system, in: A. Petho and R.D. Noble (Eds.), Residence Tie Distribution Theory in Chemical Engineering, Verlog. Chemie, Deerfield Beach, FL, 1982, p. 247. 23 R.D. Noble, Shape factors in facilitated transport through membranes, Ind. Eng. Chem. Fundam., 22 (1983) 139. 24 CA. Folkner and R.D. Noble, Transient response of facilitated transport membranes, J. Membrane Sci., 12 (1983) 289. 25 A.G. Kopp, R.J. Marr and F.E. Moser, A new concept for mass transfer in liquid surfactant membranes without carriers and with carriers that pump, Inst. Chem. Eng., Symp. Ser., 54 (1978) 279. 26 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 (1982) 662. 27 K. Kim, S. Choi and S. Ihm, Simulation of phenol removal from wastewater by liquid membrane emulsion, Ind. Eng. Chem. Fundam., 22 (1983) 167. 28 P. Stroeve and P.P. Varanasi, Extraction with double emulsions in a bath reactor: Effect of continuous phase resistance, AIChE J., submitted for publication. 29 M. Teramoto, H. Takihana, M. Shibutani, T. Yuasa, Y. Miyake and H. Teranishi, Extraction of amine by W/O/W emulsion system, J. Chem. Eng. Jpn., 14 (1981) 122. 30 F.D. Rumscheidt and S.G. Mason, Particle motions in sheared suspensions: XI. Internal circulation in fluid droplets, J. Colloid Sci., 16 (1961) 210. 31 V.G. Levich, Physicochemical Hydrodynamics, Prentice-Hall, Englewood Cliffs, NJ, 1962. 32 N.K. Madsen and R.F. Sincover, PDECOL, general collocation software for partial differential equations [ D3 1, ACM Trans. Math. Software, 5 (1979) 326. 33 R.C. Reid, J.M. Prausnitz and T.K. Sherwood, The Properties of Gases and Liquids, McGraw-Hill, New York, NY, 1977, Chap. 11. 34 R.T. Morrison and R.N. Boyd, Organic Chemistry, 2nd edn., Allyn and Bacon, Boston, MA, 1966, p. 790. 35 J. Crank, The Mathematics of Diffusion, 2nd edn., Clarendon Press, Oxford, 1975, Chap. 12. 36 W.J. Ward, Analytical and experimental studies of facilitated transport AIChE J., 16 (1970) 405. 20
Appendix Although the mean effective diffusivity, Deff, is based on the membrane phase driving force, it must include contributions from diffusion of both the reacted and unreacted solute through the internal phase. Diffusion through the membrane phase can be consequential when the internal phase molecular diffusion coefficient is significantly larger than the membrane.phase, or when solute reaction induces large solute concentrations in the internal phase relative to those in the membrane phase. Following Ho et al. [26], we calculate the effective globule diffusivity based on a local membrane phase concentration driving force, D,ff, using the Jefferson-Witzell-Sibbit equation for diffusion through a composite medium [ 26,351: D eff = Dm
1 -
7l
4 (1 + 2p)2
1
+
DADrn
71
4 (1 + 2P)
Dm
+~PDA
(AlI
70
DA = P
Dir IKim
2 (Dir /Kim 1 Drn
(Dir /Kim 11
N DiT IKti
Dm
= 0.403 (1 - fm)-l’3 -
- Dm
)
ln ( Ksm)-l]
0.5
(A21 (A31
In eqn. (A2), D, is the molecular diffusion of the transferring solute in the membrane phase. The internal phase molecular diffusion coefficients for both the reacted and unreacted solute are assumed to be the same value, D,. A reversible reaction in the internal phase permits reaction of the solute on one side of an internal droplet, diffusion of the reacted species to the other side of the droplet, followed by the reverse reaction and solute transfer back into the membrane phase. Reaction and diffusion in the internal phase can enhance the apparent diffusivity based on the membrane phase driving force. This enhancement of D,ff is incorporated into eqn. (A2) through the parameter y which depends on the reaction equilibrium, the initial concentration of reactant B, the distribution coefficient between the membrane and internal phases, and the local membrane concentration of the solute. To find an expression for 7, we follow the analysis of Ward [ 361 in his study of facilitated transport. Consistent with the globule model, we assume chemical equilibrium is always satisfied in the internal phase. (53 r=l+
(A4)
(1 +
~4hl)”
The development of eqn. (A4) assumes that the internal droplets are small enough that the membrane concentration is essentially the same on opposite sides of the droplet. In general, the effective diffusion coefficient will depend on the local membrane concentration, om . The dependence of D,ff on r#~~is illustrated
no reactmn
1.01 0
Fig. Al oJo‘$.
I
I
0.2
0.4
I 06
I
0.8
1.0
Effective diffusivity as a function of dimensionless membrane concentration
for fixed
71
in Fig. Al. The membrane fraction of the globule volume is 0.637 in these calculations. Because the sensitivity of D,ff to changes in om are not large, and because tremendous mathematical complexity arises by allowing D,ff to changes in om are not large, and because tremendous mathematical complexity arises by allowing D,ff to vary with q&, we use Deff at its mean value over the range of q& varying from 0 to 1. The mean effective diffusivity for analyzing the results of Ho et al. [26] in Fig. 2 was found to be 2.0 X lo-” m’/sec for u3 and o4 of 4125 and 90, respectively. This compares to 1.65-X 10-l’ m’/sec if no reaction takes place.
1.6 10-l
I
I 10'
1
I 103
I
I 105
I
I 10'
c!
Fig. A2. Mean effective diffusivity as a function of dimensionless reagent concentration for fixed o,/o,.
Figure A2 is a plot of the mean Deff as a function of dimensionless reagent concentration, u3, while the mass ratio of reagent to solute remains constant. Again, fm is 0.637,As the reaction approaches irreversibility, as indicated by large values of u3, D,ff approaches the no reaction case. As u3 becomes small, enhancement of Beff disappears because the amount of solute reacting was decreased. In the advancing front model, diffusion occurs solely through the fully reacted shell. Accordingly, the effective diffusivity should not include the effect of reaction.