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Optimal regimes of facilitated transport
Journal of Membrane Science, 15 (1983) 259-274 Elsevier Science Publishers B.V., Amsterdam -Printed
259
in The Netherlands
OPTIMAL REGIMES OF FACILITATED TRANSPORT L.L. KEMENA, National (U.S.A.)
R.D. NOBLE and N.J. KEMP
Bureau of Standards,
(Received September
Center for Chemical
Engineering,
773.1 Boulder,
CO 80303
23, 1982; accepted in revised form April 22, 1983)
Summary An optimization of facilitated transport in liquid membranes is accomplished to determine the maximum facilitation factor and corresponding dimensionless equilibrium constant for a given inverse Damkohler number, E, and a parameter, 01, which is directly proportional to the initial carrier concentration. The analysis covers the entire range of diffusion-limited to reaction-limited mass transport. A wide range of downstream permeate concentrations are also considered. The existence of the maximum is demonstrated. The optimal facilitation factor and optimal dimensionless equilibrium constant are plotted for a range 0.001 < E d 10.0 and 0.01 < (Y < 100.0 For this range the optimal dimensionless equilibrium constant lies between 1.5 and 10. The optimal facilitation factor increases with decreasing E and is strongly dependent on 01. As the downstream permeate concentration increases, the optimal facilitation factor and dimensionless equilibrium constant both decrease. The results can be used to select optimal operating conditions and/or carriers, or to compare actual to optimal results.
Introduction Facilitated transport is a separation process by which an active chemical carrier will selectively bind with a permeate, transport this across a film, and release the permeate at the other boundary. It has high potential for obtaining high selectivity while achieving high enough flux rates to be commercially feasible. There are several review articles which describe this process in detail [l-9] . The most common mechanism postulated is tA+B
s AB 4
(1)
where A is the component being transported across the membrane, B the active chemical carrier, and AB the active carrier complex. The arrow next to A indicates that it is the component being transported, There have been various models developed to describe the steady-state flux of a permeate, Olander [lo] described simultaneous mass transfer combined with an equilibrium chemical reaction. He showed the effect of reaction on
260
the total mass transfer rate. Four different cases were analyzed. Friedlander and Keller [ 111 used a linearized form of the reaction rate expression to describe the flux of permeate with reversible chemical reaction to the simple diffusional flux. Their assumptions were based on the reacting system being near equilibrium. Ward [ 121 derived analytical solutions for the steady-state transport of NO through ferrous chloride solution under two limiting conditions, reaction-limited (“frozen” condition) and diffusion-limited (reaction equilibrium). He found that this system was not operating at a limiting condition but somewhere in between, He used a numerical method to solve for the flux. Kreuzer and Hoofd [ 131 used the concept of a reaction boundary layer to explain the physical problems with assuming instantaneous reaction equilibrium at the membrane boundary. Their results showed that an equilibrium core existed through the main portion of the membrane with reaction boundary layers at each boundary. Goddard et al. [14] and Smith et al. [15] also demonstrated this result. Yung and Probstein [16] used a similarity transform method to develop a single equation which described the concentration profiles for all components. The equation describes steady-state conditions in flat plate geometry. The equation could be evaluated by numerical methods to obtain the facilitated flux over the entire range of operating conditions. The solution is iterative and could cause some convergence problems. They also present some simplifications under limiting conditions. Smith and Quinn [17] developed an analytical solution for the facilitation factor for flat-plate membranes under steady-state conditions. They obtained this result by assuming that there is a large excess of carrier. This allows them to linearize the reaction rate expression. They show that their result has the correct limiting behavior for both reaction-limited and diffusion-limited cases. Their model shows good agreement with previously published results. Their model becomes less accurate when reaction equilibrium is approached (diffusion limited). Hoofd and Kreuzer [ 181 developed an analytical solution to steady-state onedimensional facilitated transport. They approximate the permeate concentration as having two components. One component is based on the assumption of chemical reaction equilibrium and is a function of carrier concentration. The second term accounts for departures from chemical equilibrium and is a function of position only. By solving for these two components, one obtains an analytical solution. Their result is consistent with Smith and Quinn’s for this case. Noble [ 191 developed analytical expressions for shape factors under both limiting conditions of being either diffusion-limited or reaction-limited. These expressions allowed one to determine facilitated flux for cylindrical or spherical geometries if the flux for a flat-plate geometry is known. He found that a flat-plate geometry yielded the highest flux under identical operating conditions.
261
While the above models will allow one to determine the actual facilitated flux for a set of operating conditions, it would be desirable to know what the maximum facilitated flux and corresponding dimensionless equilibrium constant is under the same operating conditions. This could aid one in selecting carriers and membrane thickness. Also, it would be usable as a guide in comparing actual to optimum performance. Schultz et al. [l] discussed the fact that there is an optimum value of the dimensionless equilibrium constant, K, which gives maximum facilitation. They limited their analysis to the reaction equilibrium regime. For a simple reversible reaction, they found that K is equal to the negative square root of the product of the permeate concentration on each side of the membrane. For large K, facilitation is very sensitive to downstream permeate concentration and is easily stopped by back pressure. The objective of this study is to determine the optimal facilitation factor and corresponding dimensionless equilibrium constant for a given set of operating conditions. The analysis is not limited to the reaction equilibrium regime but covers the entire range between diffusion- and reaction-limited mass transport. The existence of the maximum will be demonstrated. The maximum facilitation factor and optimal equilibrium constant are plotted as a function of an inverse Damkohler number, E, and a parameter, 1y,which is directly proportional to the initial carrier concentration. This analysis is done for a range of downstream permeate concentrations. Mathematical formulation Optimization of facilitated transport is a two-step process. First, the governing differential equations need to be written, with appropriate assumptions described. Secondly, this system of equations is optimized to provide the operating point which corresponds to the maximum facilitation factor. As a first step, one can define the reaction scheme of interest. Equation (1) was chosen due to its wide interest in the literature [ 12-171. A steady-state differential mass balance can be written for each component. A flat-plate geometry with one-dimensional transport is assumed. O=D,
d2CA - klCA cri + &CAB d&X2
o=D,
d2C, dx2
O=D,B-
d2CAB
dx2
- k&A
CB
+ k,CA CB
+ bC/iB
-&CAB
Here we have assumed that each diffusion coefficient is a constant. The solute is often much smaller than the carrier and so the carrier and the solute--carrier
262
complex are roughly equivalent in size. This allows us to assume that D, = DAB. Based on this assumption, we can state c,
= Ca + CAB
where CT is the amount of carrier initially present in the system. The boundary conditions are: @x=0
cA = cA,
@x=L
CA = CAL
aCB
-z-z
aCAB
ax
a~ acB -z-=0 ax
0
acAB
(7)
ax
The boundary conditions on B and AB represent the fact that the carrier and solute-carrier complex are non-volatile and constrained to stay within the system. The boundary conditions on A indicate that there is a constant source of A at one boundary, and that A is removed from the opposite boundary so that the concentration at that point is a constant. The total flux of A across the membrane is:
= -DA
NAP
-dCA dx
-DAB
dc,, dx
To aid in characterizing the facilitation effect in this system, the above equations are solved in a dimensionless form.
o=!s$+_ aK E
1
(
,=~+!L 2
E
O= F+'
jf CAB*
--A*CB*
;
-cA*CB*
CAB*
(9) (10)
)
(
C**CB* -$
(11)
CAB*
f
@x=1
CA*=-
%
%I 1 = cB* + CAB*
NAT
acg*
-=_=
ax
acABe
ax
0
(13) (14)
(15) NA, In equation (15), F is the facilitation factor as used by Smith and Quinn [17], and NAOis the diffusion flux of A in the absence of any facilitation (Cr = 0).
F=-
263
The dimensionless variables are
D AB f =-
(16)
K=2 k,CA k,
(17)
k2L2
(18) =_
x c
X
(19)
L CA
A*
=%
cg* =-CB CT
CAB*
(21)
C AB =-
(22) CT
E is the inverse of a Damkohler number and gives a measure of the relationship between diffusion and the reverse reaction rate. A small value of E indicates a rapid reaction step and the system is diffusion-limited. A large value of E indicates rapid diffusion and the system is reaction-limited. This is the same e as used by Yung and Probstein [ I.61 . K is a dimensionless equilibrium constant; LYis directly proportional to the amount of carrier initially present. Results The objective of this study was to determine the value of K which gave the maximum value of the facilitation factor F for a given e and 01.This would a.Jlowone to generate a map of optimal operating conditions. For this system, the facilitation factor becomes
(23)
To evaluate F, a computer package DOZGAF (NAG library) was used. Equations (9) and (11) were solved using eqns. (12)-(14). This package allowed direct evaluation of eqn. (23) for a given K, E, and e. The accuracy of the solution procedure was tested by comparing results with solutions of Yung
264
and Probstein [ 161 and Folkner and Noble [ 203 . For each case, the agreement was excellent. Initially, E and a were fixed and K was varied. This was used to determine a good initial guess for the optimum value of K. Typical results from this calculation are shown in Fig. 1. This clearly shows that there is a maximum value of F asK is varied. 24
, l,,,,,, / /,,,/,,,, ,,,,,,, ,,,., a=50
F
8’.l K Fig. 1. Facilitation
factor vs. dimensionless equilibrium constant.
To determine the actual optimum value of K, a computer program ZXMIN (IMSL library) was used. It does require a good initial guess for K to converge in a reasonable amount of computer time. The program uses a quasiNewton method for evaluating the optimum. The results for this computation for CA* Ix=1 = 0 are shown in Figs. 2-4. Figures 2 and 3 show optimum values of K for a given .Eand 01,while Fig. 4 shows the corresponding optimum value of F. The minimum in the curves in Figs. 2 and 3 corresponds to the inflection point of the curves for F in Fig. 4. As E decreases, the facilitated flux increases faster than the diffusion flux, so the rate of facilitation factor increase becomes larger. As this occurs, the value of K required to obtain the optimum F decreases. This corresponds to the binding constant k1 decreasing. As E decreases further, the inflection point in the curves for F is passed through, and the rate of increase in F slows. The rate of increase in the facilitated flux
265 1c
9
a
7
Kapt 6
5
4
13
Fig. 2. Optimum equilibrium
)’
constant vs. inverse Damkohler number.
is not as great as the diffusion flux. This requires a larger value of K to reach this point. The binding constant k 1 then increases to accomplish this. Figures 2,3, and 4 can be used in different ways. If a carrier and a permeate are selected, one can fix OLand K. The figures can then be used to obtain the value of e which will provide the maximum facilitation. This corresponds to selecting the membrane thickness. If the membrane thickness is fixed for a particular permeate separation, the figures can be used to determine the optimal properties of a candidate carrier. These properties are included in ~1,E, and K, so there can be a multitude of choices. For a particular carrier, permeate, and membrane, the figures can be used to compare experimental results to the optimal operating conditions. This allows one to determine if the operation can be improved. Some examples at this point will help to clarify these points. Smith and Quinn [ 211 provide some data for CO facilitated transport using CuCl solutions. From their data for pure CO, K = 1.49,and (11ranges from 11.3 to 113. Assuming a value of QI= 50, the closest optimum K value is 5.5 at E = 0.02. This corresponds to a facilitation factor of 15.5 and a path length of 0.102 mm. Thus, a path length of 0.102 mm should provide the optimal facilitation
266
K
opt
c Fig. 3. Optimum equilibrium constant vs. inverse Damkohler number.
but it will fall short of the theoretical maximum of 15.5. Kutchai et al. [22] studied O2 transport in hemoglobin solutions. Using their data for the section starting on their p. 51, K = 4.55 and QI= 0.98. Using Fig. 3, this corresponds to optimal E values of 0.04 and 0.11. The value E = 0.04 provides a higher facilitation factor, as seen in Fig. 4. Calculating the path length under these conditions, one obtains 1.36 pm. They report that the actual value is between 1 and 2 pm, so this corresponds to an optimal operating condition. Ward [ 12 3 provides data on NO separation using ferrous chloride solutions. His values correspond to K = 2.08,01 = 2.54, and E = 0.02. Using Figs. 2 and 3 and interpolating, the optimal K value is K = 5.14. The actual facilitation factor that Ward measured is 2.29. The optimal facilitation factor in Fig. 4 for these conditions is 2.15. Considering the uncertainty in his data, this demonstrates that he was operating very close to optimal conditions. Murray and Wyman [ 231 provide some data on facilitated transport of CO in hemoglobin and myoglobin solutions. For hemoglobin, their data correspond to K = 3900,E = 0.065, and LY= 5.34 X 10-3. From Fig. 4 it is apparent that the K value is orders of magnitude too large. This directly supports their conclusion that there is no noticable facilitation of CO under these conditions
267
F
opt
E
Fig. 4. Optimum
facilitation
factor
vs. inverse
Damkohler
number.
because it is too strongly bound to the carrier (hemoglobin). Hoofd and Kreuzer [24] show facilitation of CO for very low CO concentrations. This would correspond to a low K value and a high 01value. Under these conditions, facilitation would be possible. Figures 5 through 13 show the effect of a non-zero concentration on the downstream side of the membrane. The value of the permeate concentration on the downstream side of the membrane, CA* Ix= 1, varies from 10% to 30% of the inlet permeate concentration. This provides a wide range in downstream permeate concentration and allows the reader to evaluate Kept and F opt for a broad range in operating conditions. Figures 7,10, and 13 show that the maximum facilitation factor decreases as the downstream permeate concentration increases for a given (Yand E. This decrease is due to the fact that the increased downstream permeate concentration inhibits the reverse reaction in eqn. (1). This retards the facilitation effect. For CA* Ix=1 f 0, the shape of the Kept vs. E curves is somewhat different from Figs. 2 and 3. As E decreases, the value of Kept does decrease to a
268
5 Ko,,t 4
‘0.01 2
8-
7-
6-
5-
4-
1
269
Fig. 7. Optimum facilitation factor vs. inverse Damkohler number.
minimum as in Figs. 2 and 3. The curves then flatten out and do not rise again as one approaches the reaction equilibrium regime. This is due to the fact that the reverse reaction has been slowed by a non-zero permeate concentration on the downstream side of the membrane. Thus, the permeate concentration and not the reaction rate constants control the facilitation. Also, as CA* IX= 1 increases, KOpt decreases for a given LYand E, Schultz et al. [l] derived the equation that
K opt
= [CC,*
Ix=01 CC,* Ix=l)l
-1’2
(24)
as E becomes very small. By examining the Kept vs.E curves, one can determine that the figures contained in this study have the proper limiting value. This agreement provides a check on the results. At this point, it is appropriate to comment on the limitations of the facilitation factor. The facilitation factor can be related directly to selectivity. The flux of the permeate is enhanced relative to the diffusion flux of other components in the mixture to be separated. Therefore, a higher facilitation factor corresponds to better selectivity. The facilitation factor does not Fig. 5. Optimum equilibrium constant vs. inverse Damkohler number.
Fig. 6. Optimum equilibrium constant vs. inverse Damkohler number.
270 B-
7-
6-
sKept 4-
3-
z-
:a
I 1c
271
cJ,,,=o.2
20 50 16
I“\\
Fig. 10. Optimum facilitation factor vs. inverse Damkohler number.
directly correspond to absolute permeate flux or capacity. For example, Fig. 4 demonstrates that increasing the membrane thickness (i.e., decreasing E) will continuously increase F. But the simple permeate diffusion flux (the denominator in F) decreases as the membrane length increases. So increasing F does not directly correspond to an increase in capacity since some of the increase in F is due to a decrease in the denominator and not to any increased capacity. So the optimal facilitation factor will give the reader a measure of the best selectivity obtainable for a given set of operating conditions, This has to be weighed along with the permeate flux obtained when designing such a system. Conclusions An optimization of the steady-state facilitated transport of a permeate through a flat-plate membrane has been performed, The transport is assumed to be onedimensional. The optimization was performed to obtain the maximum facilitation factor F and corresponding optimal dimensionless equiFig. 8. Optimum equilibrium constant vs. inverse Damkohler number. Fig. 9. Optimum equilibrium constant vs. inverse Damkohler number.
272
“r 7
6I
Kept
CJ,,,
= 0.3
273
50 16
\
f l
Fig. 13. Optimum
facilitation
factor
vs. inverse Damkohler
number,
librium constant K for a given set of values of E (inverse Damkohler number) and CI(directly proportional to initial carrier concentration). The downstream permeate concentration was also varied. The results are valid over the entire range between diffusion-limited and reaction-limited mass transport. The results allow one to obtain desirable properties for facilitated transport systems and compare actual to optimal operating conditions. References J.S. Schultz, J.D. Goddard and S.R. Suchdeo, Facilitated diffusion via carrier-mediated diffusion in membranes, part I. Mechanistic aspects, experimental systems, and characteristic regimes, AIChE J., 20(3) (1974) 417. J.D. Goddard, J.S. Schultz and S.R. Suchdeo, Facilitated transport via carrier-mediated diffusion in membranes, part II. Mathematical aspects and analyses, AIChE J., 20(4) (1974) 625. D.R. Smith, R.J. Lander and J.A. Quinn, Carrier-mediated transport in synthetic membranes, Recent Dev. Sep. Sci., 3 (1977) 225.
Fig. 11. Optimum
equilibrium
constant
vs. inverse
Damkohler
number.
Fig. 12. Optimum
equilibrium
constant
vs. inverse
Damkohler
number.
274 4
J .D. Goddard,
Further
applications
of carrier-mediated
transport
theory
A survey,
8
Chem. Eng. Sci., 32 (1977) 795. S.G. Kimura, S.L. Matson and W.J. Ward, III, Industrial applications of facilitated transport, Recent Dev. Sep. Sci., 5 (1979) 11. W. Halwachs and K. Schugerl, The liquid membrane technique - A promising extraction process, Int, Chem. Eng., 20(4) (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. F. Kreuzer and L. Hoofd, Facilitated diffusion of 0, and CO,, Handbook of Physioi-
9
ogy , in press. J.H. Meldon, P. Stroeve
5 6 7
10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
and C.K. Gregoire,
Carbon
dioxide
tions: A review, Chem. Eng. Comm., 16 (1982) 263. D.R. Olander, Simultaneous mass transfer and equilibrium
transport chemical
in aqueous reaction,
solu-
AIChE
J., 6(2) (1960) 233. S.K. Friedlander and K.H. Keller, Mass transfer in reacting systems near equilibrium, Chem. Eng. Sci., 20 (1965) 121. W.J. Ward, III, Analytical and experimental studies of facilitated transport, AIChE J., 16(3) (1970) 405. F. Kreuzer and L. Hoofd, Facilitated diffusion of oxygen in the presence of hemoglobin, Respir. Phys., 8 (1970) 280. J.D. Goddard, J.S. Schultz and R.J. Bassett, On membrane diffusion with near-equilibrium reaction, Chem. Eng. Sci., 25 (1970) 665. K.A. Smith, J.H. Meldon and C.K. Colton, An analysis of carrier-facilitated transport, AIChE J., 19(l) (1973) 102. D. Yung and R.L. Probstein, Similarity considerations in facilitated transport, J. Phys. Chem., 77(18) (1973) 2201. D.R. Smith and J.A. Quinn, The prediction of facilitation factors for reaction on augmented membrane transport, AIChE J., 25(l) (1979) 197. L. Hoofd and F. Kreuzer, A new mathematical approach for solving carrier-facilitated steady-state diffusion problems, J. Math. Biol., 8 (1979) 1. R.D. Noble, Shape factors in facilitated transport, Ind. Eng. Chem. Fundam., 22(l) (1983) 139. C.A. Folkner and R.D. Noble, Transient response of facilitated transport membranes, J. Membrane Sci., 12 (1983) 289. D.R. Smith and J.A. Quinn, The facilitated transport of carbon monoxide through cuprous chloride solutions, AIChE J., 26(l) (1980) 112. H. Kutchai, J.A. Jacquez and F.J. Mather, Nonequilibrium facilitated oxygen transport in hemoglobin solution, Biophys. J., 10 (1970) 38. J.D. Murray and J. Wyman, Facilitated diffusion: The case of carbon monoxide, J. Biol. Chem., 246(19) (1971) 5903. L. Hoofd and F. Kreuzer, Calculation of the facilitation of 0, or CO transport by Hb or Mb by means of a new method for solving the carrier-diffusion problem, Oxygen Transport to Tissue - III, Plenum Press, New York, 1978, p. 163.
259
in The Netherlands
OPTIMAL REGIMES OF FACILITATED TRANSPORT L.L. KEMENA, National (U.S.A.)
R.D. NOBLE and N.J. KEMP
Bureau of Standards,
(Received September
Center for Chemical
Engineering,
773.1 Boulder,
CO 80303
23, 1982; accepted in revised form April 22, 1983)
Summary An optimization of facilitated transport in liquid membranes is accomplished to determine the maximum facilitation factor and corresponding dimensionless equilibrium constant for a given inverse Damkohler number, E, and a parameter, 01, which is directly proportional to the initial carrier concentration. The analysis covers the entire range of diffusion-limited to reaction-limited mass transport. A wide range of downstream permeate concentrations are also considered. The existence of the maximum is demonstrated. The optimal facilitation factor and optimal dimensionless equilibrium constant are plotted for a range 0.001 < E d 10.0 and 0.01 < (Y < 100.0 For this range the optimal dimensionless equilibrium constant lies between 1.5 and 10. The optimal facilitation factor increases with decreasing E and is strongly dependent on 01. As the downstream permeate concentration increases, the optimal facilitation factor and dimensionless equilibrium constant both decrease. The results can be used to select optimal operating conditions and/or carriers, or to compare actual to optimal results.
Introduction Facilitated transport is a separation process by which an active chemical carrier will selectively bind with a permeate, transport this across a film, and release the permeate at the other boundary. It has high potential for obtaining high selectivity while achieving high enough flux rates to be commercially feasible. There are several review articles which describe this process in detail [l-9] . The most common mechanism postulated is tA+B
s AB 4
(1)
where A is the component being transported across the membrane, B the active chemical carrier, and AB the active carrier complex. The arrow next to A indicates that it is the component being transported, There have been various models developed to describe the steady-state flux of a permeate, Olander [lo] described simultaneous mass transfer combined with an equilibrium chemical reaction. He showed the effect of reaction on
260
the total mass transfer rate. Four different cases were analyzed. Friedlander and Keller [ 111 used a linearized form of the reaction rate expression to describe the flux of permeate with reversible chemical reaction to the simple diffusional flux. Their assumptions were based on the reacting system being near equilibrium. Ward [ 121 derived analytical solutions for the steady-state transport of NO through ferrous chloride solution under two limiting conditions, reaction-limited (“frozen” condition) and diffusion-limited (reaction equilibrium). He found that this system was not operating at a limiting condition but somewhere in between, He used a numerical method to solve for the flux. Kreuzer and Hoofd [ 131 used the concept of a reaction boundary layer to explain the physical problems with assuming instantaneous reaction equilibrium at the membrane boundary. Their results showed that an equilibrium core existed through the main portion of the membrane with reaction boundary layers at each boundary. Goddard et al. [14] and Smith et al. [15] also demonstrated this result. Yung and Probstein [16] used a similarity transform method to develop a single equation which described the concentration profiles for all components. The equation describes steady-state conditions in flat plate geometry. The equation could be evaluated by numerical methods to obtain the facilitated flux over the entire range of operating conditions. The solution is iterative and could cause some convergence problems. They also present some simplifications under limiting conditions. Smith and Quinn [17] developed an analytical solution for the facilitation factor for flat-plate membranes under steady-state conditions. They obtained this result by assuming that there is a large excess of carrier. This allows them to linearize the reaction rate expression. They show that their result has the correct limiting behavior for both reaction-limited and diffusion-limited cases. Their model shows good agreement with previously published results. Their model becomes less accurate when reaction equilibrium is approached (diffusion limited). Hoofd and Kreuzer [ 181 developed an analytical solution to steady-state onedimensional facilitated transport. They approximate the permeate concentration as having two components. One component is based on the assumption of chemical reaction equilibrium and is a function of carrier concentration. The second term accounts for departures from chemical equilibrium and is a function of position only. By solving for these two components, one obtains an analytical solution. Their result is consistent with Smith and Quinn’s for this case. Noble [ 191 developed analytical expressions for shape factors under both limiting conditions of being either diffusion-limited or reaction-limited. These expressions allowed one to determine facilitated flux for cylindrical or spherical geometries if the flux for a flat-plate geometry is known. He found that a flat-plate geometry yielded the highest flux under identical operating conditions.
261
While the above models will allow one to determine the actual facilitated flux for a set of operating conditions, it would be desirable to know what the maximum facilitated flux and corresponding dimensionless equilibrium constant is under the same operating conditions. This could aid one in selecting carriers and membrane thickness. Also, it would be usable as a guide in comparing actual to optimum performance. Schultz et al. [l] discussed the fact that there is an optimum value of the dimensionless equilibrium constant, K, which gives maximum facilitation. They limited their analysis to the reaction equilibrium regime. For a simple reversible reaction, they found that K is equal to the negative square root of the product of the permeate concentration on each side of the membrane. For large K, facilitation is very sensitive to downstream permeate concentration and is easily stopped by back pressure. The objective of this study is to determine the optimal facilitation factor and corresponding dimensionless equilibrium constant for a given set of operating conditions. The analysis is not limited to the reaction equilibrium regime but covers the entire range between diffusion- and reaction-limited mass transport. The existence of the maximum will be demonstrated. The maximum facilitation factor and optimal equilibrium constant are plotted as a function of an inverse Damkohler number, E, and a parameter, 1y,which is directly proportional to the initial carrier concentration. This analysis is done for a range of downstream permeate concentrations. Mathematical formulation Optimization of facilitated transport is a two-step process. First, the governing differential equations need to be written, with appropriate assumptions described. Secondly, this system of equations is optimized to provide the operating point which corresponds to the maximum facilitation factor. As a first step, one can define the reaction scheme of interest. Equation (1) was chosen due to its wide interest in the literature [ 12-171. A steady-state differential mass balance can be written for each component. A flat-plate geometry with one-dimensional transport is assumed. O=D,
d2CA - klCA cri + &CAB d&X2
o=D,
d2C, dx2
O=D,B-
d2CAB
dx2
- k&A
CB
+ k,CA CB
+ bC/iB
-&CAB
Here we have assumed that each diffusion coefficient is a constant. The solute is often much smaller than the carrier and so the carrier and the solute--carrier
262
complex are roughly equivalent in size. This allows us to assume that D, = DAB. Based on this assumption, we can state c,
= Ca + CAB
where CT is the amount of carrier initially present in the system. The boundary conditions are: @x=0
cA = cA,
@x=L
CA = CAL
aCB
-z-z
aCAB
ax
a~ acB -z-=0 ax
0
acAB
(7)
ax
The boundary conditions on B and AB represent the fact that the carrier and solute-carrier complex are non-volatile and constrained to stay within the system. The boundary conditions on A indicate that there is a constant source of A at one boundary, and that A is removed from the opposite boundary so that the concentration at that point is a constant. The total flux of A across the membrane is:
= -DA
NAP
-dCA dx
-DAB
dc,, dx
To aid in characterizing the facilitation effect in this system, the above equations are solved in a dimensionless form.
o=!s$+_ aK E
1
(
,=~+!L 2
E
O= F+'
jf CAB*
--A*CB*
;
-cA*CB*
CAB*
(9) (10)
)
(
C**CB* -$
(11)
CAB*
f
@x=1
CA*=-
%
%I 1 = cB* + CAB*
NAT
acg*
-=_=
ax
acABe
ax
0
(13) (14)
(15) NA, In equation (15), F is the facilitation factor as used by Smith and Quinn [17], and NAOis the diffusion flux of A in the absence of any facilitation (Cr = 0).
F=-
263
The dimensionless variables are
D AB f =-
(16)
K=2 k,CA k,
(17)
k2L2
(18) =_
x c
X
(19)
L CA
A*
=%
cg* =-CB CT
CAB*
(21)
C AB =-
(22) CT
E is the inverse of a Damkohler number and gives a measure of the relationship between diffusion and the reverse reaction rate. A small value of E indicates a rapid reaction step and the system is diffusion-limited. A large value of E indicates rapid diffusion and the system is reaction-limited. This is the same e as used by Yung and Probstein [ I.61 . K is a dimensionless equilibrium constant; LYis directly proportional to the amount of carrier initially present. Results The objective of this study was to determine the value of K which gave the maximum value of the facilitation factor F for a given e and 01.This would a.Jlowone to generate a map of optimal operating conditions. For this system, the facilitation factor becomes
(23)
To evaluate F, a computer package DOZGAF (NAG library) was used. Equations (9) and (11) were solved using eqns. (12)-(14). This package allowed direct evaluation of eqn. (23) for a given K, E, and e. The accuracy of the solution procedure was tested by comparing results with solutions of Yung
264
and Probstein [ 161 and Folkner and Noble [ 203 . For each case, the agreement was excellent. Initially, E and a were fixed and K was varied. This was used to determine a good initial guess for the optimum value of K. Typical results from this calculation are shown in Fig. 1. This clearly shows that there is a maximum value of F asK is varied. 24
, l,,,,,, / /,,,/,,,, ,,,,,,, ,,,., a=50
F
8’.l K Fig. 1. Facilitation
factor vs. dimensionless equilibrium constant.
To determine the actual optimum value of K, a computer program ZXMIN (IMSL library) was used. It does require a good initial guess for K to converge in a reasonable amount of computer time. The program uses a quasiNewton method for evaluating the optimum. The results for this computation for CA* Ix=1 = 0 are shown in Figs. 2-4. Figures 2 and 3 show optimum values of K for a given .Eand 01,while Fig. 4 shows the corresponding optimum value of F. The minimum in the curves in Figs. 2 and 3 corresponds to the inflection point of the curves for F in Fig. 4. As E decreases, the facilitated flux increases faster than the diffusion flux, so the rate of facilitation factor increase becomes larger. As this occurs, the value of K required to obtain the optimum F decreases. This corresponds to the binding constant k1 decreasing. As E decreases further, the inflection point in the curves for F is passed through, and the rate of increase in F slows. The rate of increase in the facilitated flux
265 1c
9
a
7
Kapt 6
5
4
13
Fig. 2. Optimum equilibrium
)’
constant vs. inverse Damkohler number.
is not as great as the diffusion flux. This requires a larger value of K to reach this point. The binding constant k 1 then increases to accomplish this. Figures 2,3, and 4 can be used in different ways. If a carrier and a permeate are selected, one can fix OLand K. The figures can then be used to obtain the value of e which will provide the maximum facilitation. This corresponds to selecting the membrane thickness. If the membrane thickness is fixed for a particular permeate separation, the figures can be used to determine the optimal properties of a candidate carrier. These properties are included in ~1,E, and K, so there can be a multitude of choices. For a particular carrier, permeate, and membrane, the figures can be used to compare experimental results to the optimal operating conditions. This allows one to determine if the operation can be improved. Some examples at this point will help to clarify these points. Smith and Quinn [ 211 provide some data for CO facilitated transport using CuCl solutions. From their data for pure CO, K = 1.49,and (11ranges from 11.3 to 113. Assuming a value of QI= 50, the closest optimum K value is 5.5 at E = 0.02. This corresponds to a facilitation factor of 15.5 and a path length of 0.102 mm. Thus, a path length of 0.102 mm should provide the optimal facilitation
266
K
opt
c Fig. 3. Optimum equilibrium constant vs. inverse Damkohler number.
but it will fall short of the theoretical maximum of 15.5. Kutchai et al. [22] studied O2 transport in hemoglobin solutions. Using their data for the section starting on their p. 51, K = 4.55 and QI= 0.98. Using Fig. 3, this corresponds to optimal E values of 0.04 and 0.11. The value E = 0.04 provides a higher facilitation factor, as seen in Fig. 4. Calculating the path length under these conditions, one obtains 1.36 pm. They report that the actual value is between 1 and 2 pm, so this corresponds to an optimal operating condition. Ward [ 12 3 provides data on NO separation using ferrous chloride solutions. His values correspond to K = 2.08,01 = 2.54, and E = 0.02. Using Figs. 2 and 3 and interpolating, the optimal K value is K = 5.14. The actual facilitation factor that Ward measured is 2.29. The optimal facilitation factor in Fig. 4 for these conditions is 2.15. Considering the uncertainty in his data, this demonstrates that he was operating very close to optimal conditions. Murray and Wyman [ 231 provide some data on facilitated transport of CO in hemoglobin and myoglobin solutions. For hemoglobin, their data correspond to K = 3900,E = 0.065, and LY= 5.34 X 10-3. From Fig. 4 it is apparent that the K value is orders of magnitude too large. This directly supports their conclusion that there is no noticable facilitation of CO under these conditions
267
F
opt
E
Fig. 4. Optimum
facilitation
factor
vs. inverse
Damkohler
number.
because it is too strongly bound to the carrier (hemoglobin). Hoofd and Kreuzer [24] show facilitation of CO for very low CO concentrations. This would correspond to a low K value and a high 01value. Under these conditions, facilitation would be possible. Figures 5 through 13 show the effect of a non-zero concentration on the downstream side of the membrane. The value of the permeate concentration on the downstream side of the membrane, CA* Ix= 1, varies from 10% to 30% of the inlet permeate concentration. This provides a wide range in downstream permeate concentration and allows the reader to evaluate Kept and F opt for a broad range in operating conditions. Figures 7,10, and 13 show that the maximum facilitation factor decreases as the downstream permeate concentration increases for a given (Yand E. This decrease is due to the fact that the increased downstream permeate concentration inhibits the reverse reaction in eqn. (1). This retards the facilitation effect. For CA* Ix=1 f 0, the shape of the Kept vs. E curves is somewhat different from Figs. 2 and 3. As E decreases, the value of Kept does decrease to a
268
5 Ko,,t 4
‘0.01 2
8-
7-
6-
5-
4-
1
269
Fig. 7. Optimum facilitation factor vs. inverse Damkohler number.
minimum as in Figs. 2 and 3. The curves then flatten out and do not rise again as one approaches the reaction equilibrium regime. This is due to the fact that the reverse reaction has been slowed by a non-zero permeate concentration on the downstream side of the membrane. Thus, the permeate concentration and not the reaction rate constants control the facilitation. Also, as CA* IX= 1 increases, KOpt decreases for a given LYand E, Schultz et al. [l] derived the equation that
K opt
= [CC,*
Ix=01 CC,* Ix=l)l
-1’2
(24)
as E becomes very small. By examining the Kept vs.E curves, one can determine that the figures contained in this study have the proper limiting value. This agreement provides a check on the results. At this point, it is appropriate to comment on the limitations of the facilitation factor. The facilitation factor can be related directly to selectivity. The flux of the permeate is enhanced relative to the diffusion flux of other components in the mixture to be separated. Therefore, a higher facilitation factor corresponds to better selectivity. The facilitation factor does not Fig. 5. Optimum equilibrium constant vs. inverse Damkohler number.
Fig. 6. Optimum equilibrium constant vs. inverse Damkohler number.
270 B-
7-
6-
sKept 4-
3-
z-
:a
I 1c
271
cJ,,,=o.2
20 50 16
I“\\
Fig. 10. Optimum facilitation factor vs. inverse Damkohler number.
directly correspond to absolute permeate flux or capacity. For example, Fig. 4 demonstrates that increasing the membrane thickness (i.e., decreasing E) will continuously increase F. But the simple permeate diffusion flux (the denominator in F) decreases as the membrane length increases. So increasing F does not directly correspond to an increase in capacity since some of the increase in F is due to a decrease in the denominator and not to any increased capacity. So the optimal facilitation factor will give the reader a measure of the best selectivity obtainable for a given set of operating conditions, This has to be weighed along with the permeate flux obtained when designing such a system. Conclusions An optimization of the steady-state facilitated transport of a permeate through a flat-plate membrane has been performed, The transport is assumed to be onedimensional. The optimization was performed to obtain the maximum facilitation factor F and corresponding optimal dimensionless equiFig. 8. Optimum equilibrium constant vs. inverse Damkohler number. Fig. 9. Optimum equilibrium constant vs. inverse Damkohler number.
272
“r 7
6I
Kept
CJ,,,
= 0.3
273
50 16
\
f l
Fig. 13. Optimum
facilitation
factor
vs. inverse Damkohler
number,
librium constant K for a given set of values of E (inverse Damkohler number) and CI(directly proportional to initial carrier concentration). The downstream permeate concentration was also varied. The results are valid over the entire range between diffusion-limited and reaction-limited mass transport. The results allow one to obtain desirable properties for facilitated transport systems and compare actual to optimal operating conditions. References J.S. Schultz, J.D. Goddard and S.R. Suchdeo, Facilitated diffusion via carrier-mediated diffusion in membranes, part I. Mechanistic aspects, experimental systems, and characteristic regimes, AIChE J., 20(3) (1974) 417. J.D. Goddard, J.S. Schultz and S.R. Suchdeo, Facilitated transport via carrier-mediated diffusion in membranes, part II. Mathematical aspects and analyses, AIChE J., 20(4) (1974) 625. D.R. Smith, R.J. Lander and J.A. Quinn, Carrier-mediated transport in synthetic membranes, Recent Dev. Sep. Sci., 3 (1977) 225.
Fig. 11. Optimum
equilibrium
constant
vs. inverse
Damkohler
number.
Fig. 12. Optimum
equilibrium
constant
vs. inverse
Damkohler
number.
274 4
J .D. Goddard,
Further
applications
of carrier-mediated
transport
theory
A survey,
8
Chem. Eng. Sci., 32 (1977) 795. S.G. Kimura, S.L. Matson and W.J. Ward, III, Industrial applications of facilitated transport, Recent Dev. Sep. Sci., 5 (1979) 11. W. Halwachs and K. Schugerl, The liquid membrane technique - A promising extraction process, Int, Chem. Eng., 20(4) (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. F. Kreuzer and L. Hoofd, Facilitated diffusion of 0, and CO,, Handbook of Physioi-
9
ogy , in press. J.H. Meldon, P. Stroeve
5 6 7
10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
and C.K. Gregoire,
Carbon
dioxide
tions: A review, Chem. Eng. Comm., 16 (1982) 263. D.R. Olander, Simultaneous mass transfer and equilibrium
transport chemical
in aqueous reaction,
solu-
AIChE
J., 6(2) (1960) 233. S.K. Friedlander and K.H. Keller, Mass transfer in reacting systems near equilibrium, Chem. Eng. Sci., 20 (1965) 121. W.J. Ward, III, Analytical and experimental studies of facilitated transport, AIChE J., 16(3) (1970) 405. F. Kreuzer and L. Hoofd, Facilitated diffusion of oxygen in the presence of hemoglobin, Respir. Phys., 8 (1970) 280. J.D. Goddard, J.S. Schultz and R.J. Bassett, On membrane diffusion with near-equilibrium reaction, Chem. Eng. Sci., 25 (1970) 665. K.A. Smith, J.H. Meldon and C.K. Colton, An analysis of carrier-facilitated transport, AIChE J., 19(l) (1973) 102. D. Yung and R.L. Probstein, Similarity considerations in facilitated transport, J. Phys. Chem., 77(18) (1973) 2201. D.R. Smith and J.A. Quinn, The prediction of facilitation factors for reaction on augmented membrane transport, AIChE J., 25(l) (1979) 197. L. Hoofd and F. Kreuzer, A new mathematical approach for solving carrier-facilitated steady-state diffusion problems, J. Math. Biol., 8 (1979) 1. R.D. Noble, Shape factors in facilitated transport, Ind. Eng. Chem. Fundam., 22(l) (1983) 139. C.A. Folkner and R.D. Noble, Transient response of facilitated transport membranes, J. Membrane Sci., 12 (1983) 289. D.R. Smith and J.A. Quinn, The facilitated transport of carbon monoxide through cuprous chloride solutions, AIChE J., 26(l) (1980) 112. H. Kutchai, J.A. Jacquez and F.J. Mather, Nonequilibrium facilitated oxygen transport in hemoglobin solution, Biophys. J., 10 (1970) 38. J.D. Murray and J. Wyman, Facilitated diffusion: The case of carbon monoxide, J. Biol. Chem., 246(19) (1971) 5903. L. Hoofd and F. Kreuzer, Calculation of the facilitation of 0, or CO transport by Hb or Mb by means of a new method for solving the carrier-diffusion problem, Oxygen Transport to Tissue - III, Plenum Press, New York, 1978, p. 163.