Symposium on Distillation,” Brighton, England, May 1960. Calderbank, P. H., Moo-Young, M. B., Chem. Eng. Sci., 16, 39 (1961). Frankel, N. A., Acrivos, A., Phys. Fluids, 11 (9), 1913 (1968). Friedlander, S. K., AIChE J., 3 (l),43 (1957). Friedlander, S. K., ibid., 7 (2), 347 (1961). Frossling, N., Gerlands Beitr. Geophys., 52, 170 (1938). Garner, F. H., Suckling, R. D., AIChE J., 4, 114 (1958). Garner, F. H., Keey, R. B., Chem. Eng. Sci., 9, 119 (1958). Harriott, P., AIChE J., 8, 93 (1962). Hixon, A. W., Baum, S. J., Ind. Eng. Chem., 33, 478 (1941). Humphrey, D. W., Van Ness, H. C., AIChE J . , 6, 289 (1960). Keey, R. B., Glen, J. B., Can. J . Chem. Eng., 42, 227 (1964). Kneule, F., Chem. Eng. Tech., 28, 221 (1956). Kramers, H., Physica (Utrecht), 12, 61 (1946). Langmuir, I., Phys. Reu., 12, 368 (1918). Levich, V. G., “Physicochemical Hydrodynamics,” Trans-
lation by Scripta Technica, Ind., Prentice-Hall, Englewood Cliffs, N. J., 1962. Marangozis, J., Johnson, A. I., Can. J . Chem. Eng., 40, 231 (1962). Middleman, S., AIChE J . , 11, 751 (1965). Miller, D. N., Ind. Eng. Chem., 56 ( l o ) , 18 (1964). Miller, D. N., Chem. Eng. Sci., 22, 1617 (1967). Nienow, A. W., Can. J . Chem. Eng., 47, 248 (1969). Shinnar, R., Church, J. M., Ind. Eng. Chem., 12 (3), 253 (1960). Steinberger, R. L., Treybal, R. E., AIChE J . , 6 (2), 227 (1960). Sykes, P., Gomezplata, Can. J . Chem. Eng., 45, 181 (1967). Ward, D. M., Trass, O., Johnson, A. I., ibid., 40, 164 (1962). Weisman, J., Efferding, L. E., AIChE J., 6 (3), 419 (1960). Wieselberger, C., Phyi. Z., 23, 219 (1922). Yuge, T., Sci. Per. Res. Inst., Tohoku Univ., Ser. B., 6 , 143 (1956). Zwietering, T. N., Chem. Eng. Sci., 8, 224 (1958). RECEIVED for review March 5, 1970 ACCEPTED October 6, 1970
Membrane Separation of Gases Using Steady Cyclic Operation Donald R. Paul Department of Chemical Engineering, University of Texas, Austin, Tex. 78712
Membrane permeation for separation of gas mixtures is usually considered only in terms of steady-state operation. The steady-state separation factor for two species is the product of the diffusion coefficient and solubility coefficient ratios-i.e., cyss =
>
1 and SI/
Si< 1.
Pulsed-Membrane Operation
Description. T o be effective, the principle illustrated above must be adapted t o continuous operation. This can be done by increasing and decreasing the upstream pressure periodically in time. One approach would be to vary the pressure sinusoidally with a certain frequency; however. this would be physically difficult to do. Another approach is to pulse the upstream pressure in a square wave form as illustrated in Figure 2 . This is mechanically easier to accomplish and results in a more effective utilization of this principle than the sinusoidal wave form. The effectiveness of this pulsed scheme of operation will depend on the relative time scale of the pressure pulses and the relaxation time-Le., T and A-and the natural time scale of the membrane-gas system-Le., o1 and 8_.. If D and S are constant for all species, the system behaves in a linear fashion and no enhancement in selectivity will be realized if permeate is collected continuously. T o reap the benefits of this method of operation, the permeate must be collected alternately in a t least two receiving vessels. Figure 3 illustrates schematically how this might be done. The upstream side of the membrane has a volume, V H ,and is well-mixed t o prevent concentration polarization. This side is connected to a reservoir on the left containing feed a t a pressure, pH, and composition, x, and to a reservoir on the right containing rejected feed . pulses are generated by simula t a pressure, p ~ Pressure taneously opening valve HA and closing HB. This valve arrangement is maintained for a time, A T . Then, a t just the proper time, valve HA is closed simultaneously with the opening of HB. This arrangement is maintained for a time. (1 - A ) T . Periodic repetition of this sequence will produce the form shown in Figure 2. Depending on the rapidity of opening and closing and the flow resistances, the wave form may be slightly distorted; however, we will deal here with the idealized form of Figure 2 . The downstream side of the membrane has a volume, V i , upstream to either valves L A or L B , and this side . side should is maintained a t a constant pressure, p ~ This be well-mixed, but it is less important here than above provided P H >> p~ as we will consider. The composition of the permeate in the downstream compartment will oscillate in a complex fashion with a period equal t o T . If the permeate were collected for
one period and mixed, its composition, 3 , would be given by the steady-state Equations 2 and 3, but the total quantity would be only A times the steady-state quantity. T o get an enhanced separation coefficient, the permeate must be collected into cuts. This can be done by utilizing the two receiving vessels shown. At a certain time, valve LA should be opened and L B simultaneously closed. We will let the valves remain in this position for a time equal to AT, and then reverse their positions, LA closed, LB open, for a time equal to (1 - A ) T . The lower valves need not be in phase with the upper ones; however, in the following analysis we will consider them exactly in phase so that permeate is collected on the left when the upstream pressure is p H , and on the right when it is pL. Thus, for a binary system with D1 > D2, Y,H will be greater than y and yli will be less than y where y is the steady-state value. The total amount of gas 1 collected on the left in one cycle is Q H I while the amount collected on the right is QLI. Due to the linearity mentioned earlier
+ Q L I ) / ( Q H ~ + QLZ)
= ass (5) Analysis. The efficiency of operation of the system in Figure 3 depends on several factors which will be simplified for the purposes of this analysis. The case where the fraction of the feed which permeates the membrane is very small will be considered. This means that the upstream composition is always the same. Further, the volume, VL, is quite small so there is virtually no mixing of the cuts when the downstream valves are reversed. Departure from these ideal conditions will be evident in any real situation and will result in some diminution of efficiency. A more complete analysis for a particular situation, however, could account adequately for these factors. The analogous heat-transfer problem to this pulseddiffusion problem has been solved by Carslaw and Jaeger (1959). Their solution for the temperature profile can be readily converted t o the required concentration profile by an appropriate change of nomenclature. The information needed here, QHt and QLi, can be obtained from this profile by appropriate differentiations and integrations. The results of these manipulations are (Qm
QH~
= ~ S ~ ( L-PYH J I L ) [ %A X , - F ( A , X , ) ]
(6)
and Q L ~=
~ S , ( X ~- Py ,Hp r ) F ( A , X , )
(7)
where X , = T/O, (0, = time lag for gas i = I’/6 D , ) . The function F ( A , X , )is defined by the series ,Feed
Reservoir
I
Rejected /Feed
Membrane
1
time
Figure 2. Upstream pressure pulse shape
Figure 3. Schematic illustration of apparatus for pulsedmembrane operation Ind. Eng. Chem. Process Des. Develop., Vol. 10, No. 3, 1971
377
Table II. Limiting Behavior T A
n
=
n'
1
asX+O
1 + ,(-T'~'/G)X, - e ( - ~ ' ~ ' ' / 6 ) A-Xe
(-~'n'/6)(1 - A ) X
1 - e-(a'~'/6)X
QH
Q?$
QL
'Q,\
o s X i m
A? A(l - A)
-+
4
A
X-'
(8) The results of Equations 6 and 7 can be expressed more conveniently and informatively as fractions of the steadystate output of i given by Equation 1. If we insist that X ~ >> H y ( p ~then , these results are
[ Q H I Q ~ ~=] A~ - 6 F ( A , X , ) / X ,
(9)
and [ Q ~ / Q s s l ~=
6 F(A,X,)/X,
(10)
As pointed out earlier [(QH
+ Q L ) I Q ~ ~ I=! A
(11)
for all conditions. The ratios given by Equations 9 and 10 are shown graphically in Figures 4 and 5 where the subscript, i, has been dropped. Table I1 summarizes some limiting behavior of these relationships. Figures 4 and 5 allow us to see how much the productivity of a membrane for a given species is reduced in pulsed operation compared to steady-state operation. One of the most interesting features is the drastic minimum in QH/Qss a t small values of A . Because of this minimum, improved separation factors can be realized. Note that it does not appear a t larger values of A because of an insufficient "relaxation" time for the membrane. To express the selectivity of the membrane operated in this mode, separation factors for each of the two receiving vessels, H and L , are used and designated as O H and C Y L . These can be expressed quite conveniently relative to the steady-state separation factor, ass, by the following definitions CYHICYSS
= =
( Q H / Q ~ ~ I) /
( Q H I Q S J ~
C Y L / C Y ~ ~( Q ~ / Q s s ) i / ( Q ~ / Q s s ) 2
(12) (13)
These definitions are convenient in that relative improvement in selectivity over steady-state operation can be rapidly assessed from Figures 4 and 5 by considering the ratio D1/Drwithout regard for the Sl/S2ratio.
Discussion
Once the membrane material and the operating temperature are established, values for D I and 0 2 become fixed material properties for a given gas pair. The values of XI and X p are not fixed until the period, T , and the membrane thickness, I , are specified; however, their ratio is fixed since X1/X2= D 1 / D 2 .For optimum performance, X 2 (2 has lower D ) should be set a t a value near the minimum of the QH/Q19curve through a judicious choice of I and T . T o illustrate the calculation procedure, suppose a membrane was selected that has a ratio of D I / D , = 25, and the authors chose to set X p = 3.0 and A = 0.1. From Figures 4 and 5 , (QH/Qss)p = 1.28 x lo-' and (Qr/Qss):! = 9.9 x 10 ' are obtained. Now Xi = (25)(3) = 7 5 , so again from these graphs we obtain (Q~/Qss)i = 8.6 x and (QL/Qs3)l= 1.34 x lo-'. From Equations 12 and 13, aH/LYsi = 67.2 and a L / a S=s 0.135 are obtained. ~ ~ Table I11 summarizes numerous calculations of o ~ H / ( Yfor a range of parameters. All of these calculations are for X 1 a t the minimum of the (QH/Qss) curve for that A value. Spectacular enhancements in selectivity occur a t small values of A At values as high as 0.8, virtually no improvement in CYH is seen, and, of course, A = 1 represents steady state. For a given A , the enhancement increases as D 1 / D 2becomes larger but a finite limit is reached. Similar calculations show that a t constant DI/ D2, ( a L / a S 9increases ) to one as A approaches one. I t decreases t o even smaller values without reaching a limiting value as D l / D2 increases a t constant A . Once the selectivity characteristics are fixed as shown above, 1 and T must be selected to reach a suitable compromise between productivity, membrane fabrication capability, and operating ease. Because there are many variables, it would be wise to optimize every specific separation since it is not possible to state the needs for every application in general terms.
E Id'
,.I6
10-1
I00
IO"
10'2
X.T/B
Figure 4. Permeate collected during high-pressure portion of cycle relative to steady-state operation
378
Ind. Eng. Chem. Process Des. Develop., Vol. 10, No. 3, 1971
Figure 5. Permeate collected during low-pressure portion of cycle relative to steady-state operation. Curves for A and 1 - A are identical
~-
~
Acknowledgment
Table 111. Illustration of Improvements in Separation Factors oIH/cYSl
Di/Dz
A
X, 2 5 10 33
The author is grateful to Anita J. Beasley for doing many of the computations.
0.1 3 .O 6.3 31.5 52.7 78.8
0.2 2.25 2.82 7.95 10.9 14.1
0.3 2.0 1.98 4.17 5.25 6.25
0.5 1.7 1.28 1.98 2.26 2.57
The permeate collected in the receiving vessel on the left in Figure 3 will always be more enriched in the component with the higher diffusion coefficient while the one on the right will coutain permeate more enriched in the component with the lower diffusion coefficient. I n some cases, both permeates may be viewed as valuable products while in others only one may be retained and the other discarded or recycled. This method of operation and the analysis given here are not restricted to binary gas systems. Figures 4 and 5 may be applied to as many components as desired since they are applicable to any gas with constant D and S. The analysis given here was restricted to the two cuts of permeate being taken in a time sequence exactly in phase with the upstream pressure variations. Further improvements can probably be realized by imposing a certain phase shift. Also there may be some advantages to using collection times not equal to A T and (1 - A ) T . In fact, there is no reason why more than two cuts cannot be taken. Each of these factors can be investigated by going back to the concentration profile and performing the proper manipulations on it. These modifications will not be considered further as it would be more appropriate to do this for a particular system of interest. The objective here is to convey the basic idea. A similar analysis can be developed for hollow fiber geometry which is more favorable for practical separation processes; however, to do so it is necessary to start with solutions to Fick’s second law in cylindrical geometry. The recovery of helium from natural gas (predominately methane) was used here merely as an example of the kinds of separations that can advantageously employ pulsed-membrane operation. There are certainly others. The He-CH, case is a very important one that has received considerable attention as evidenced by the work of Stern et al. (1965) and numerous patents-e.g., Kohman and McAfee (1962), Jolley (1965), Jones (1964), and Niedzielski and Putnam (1967) to mention a few. One of the major problems was to secure adequate selectivity for the reasons pointed out earlier. Selectivity can be obtained, but usually it requires going to membrane materials that have very low permeabilities. Pulsed-membrane operation can give adequate high selectivity, but this, too, comes with a loss in productivity: however, this loss need not be as large as would be experienced in going to glass membranes for example. Further, this method offers a great deal of flexibility that may permit one unit to be adapted to a variety of separations by altering operating variables without changing the membrane material or construction variables.
Nomenclature
A C D F 1
= = = = =
fraction of period that the pressure is high gas concentration in polymer, moles cm’ diffusion coefficient, cm-/sec a defined function membrane thickness, cm P = gas pressure, atm P = permeability coefficient, moles/ cm-sec atm Q = permeate, moles s = gas solubility coefficient, moles/ cm’ atm T = period of pressure cycle, sec cell volume, cm’
v = x =
TO x = mole fraction in feed Y = mole fraction in permeate
Greek Letters 01
= separation factor
6’ =
time lag, sec
Subscripts
1,2 = gas species, 1 and 2 H = upstream conditions and factors associated with the high-pressure part of the pressure cycle L = downstream conditions and factors associated with the low-pressure part of the pressure cycle ss = steady state literature Cited
Carslaw, H . S., Jaeger, J. C., “Conduction of Heat in Solids,” 2nd ed., Oxford Clarendon Press, Oxford, England, 1959, pp 107-9. Crank, J., Park, G. S., “Diffusion in Polymers,” J. Crank and G. S. Park, Eds., Academic Press, New York, N. Y., 1968, Chap. 1. Jolley, J. E. (to E. I. d u P o n t deNemours & Co., Inc.), U. S. Patent 3,172,741 (March 9, 1965). Jones, A. L. (to Standard Oil Co.), U. S. Patent 3,135,591 (June 2, 1964). Kammermeyer, K., in “Progress in Separation and Purification” Vol. I , E . S. Perry, Ed., Interscience, New York, N. Y., 1968, pp 335-72. Kohman, G. T., McAfee, K. B. (to Bell Telephone Laboratories, Inc.), U. S. Patent 3,019,853 (February 6, 1962). Li, N. N., Long, R. B., Henley, E . J., Ind. Eng. Chem., 57 (3), 18-29 (1965). Michaels, A. S., Bixler, H. J., in “Progress in Separation and Purification,” Val. I , E. S. Perry. Ed., Interscience, New York, N.Y., 1968, pp 143-86. Niedzielski, E. L., Putnam, R. E . (to E. I. du Pont de Nemours & Co., Inc.), U.S. Patent 3,307,330 (March 7, 1967). Rickles, R. X., Ind. Eng. Chem., 58 (6), 19-35 (1966). Schrodt, V. N., ibzd., 59 (6), 58 (1967). Simmons, P. J., Spinner, I . H., A I C h E J . , 15, 489 (1969). Stannett, V., in “Diffusion in Polymers,” J. Crank and G. S. Park, Eds., Academic Press, S e w York, N Y . , 1968, Chap. 2. Stern, S. A., Sinclair, T. F., Gareis, P. J., Vahldieck, N. P., Mohr, P. H., I n d . Eng. C h e m , 57 ( 2 ) , 49-60 (1965). RECEIVED for review June 2 2 , 1970 ACCEPTED January 15,1971 Paper presented at Symposium on Membrane Separations, 68th meeting, AIChE. Houston. Tex., March 1971. Ind. Eng. Chem. Process Des. Develop., Vol. 10, No. 3, 1971
379
