Ind. Eng. Chem. Process Des. Dev.
1985,2 4 , 907-913
907
Separation of Hydrogen-Methane Gas Mixtures by Permeation under Pressure through Porous Cellulose Acetate Membraned M. A. Mazld, Ramamurtl Rangarajan, Takeshl Matsuura, and S. Sourlrajan" Division of Chemistty, National Research Council of Canada, Ottawa, Ontario K1A OR9
The separation of H2-CH, gas mixture was accomplished by permeation through dry porous cellulose acetate membranes of different surface porosities at operating pressures in the range 515 to 2310 kPa abs at room temperature (23-25 "C).The mole fraction of hydrogen in the feed gas mixture was in the range 0.484-0.883. The highest separation factor achieved in these studies was 17.5. An attempt was made to predict total gas permeation rate and separation factor on the basis of the transport equations developed in our earlier work. The results obtained are discussed.
Introduction Though there are a number of fundamental studies on the separation of gas mixtures by permeation under pressure through a variety of membranes (Michaels and Bixler, 1968; Stern, 1972) and virtually all the important engineering and technological concepts seem to have been well established (Hwang and Kammermeyer, 1975; Pan and Habgood, 1978a,b), membrane gas separations did not reach the commercial stage of utilization until recently when Monsanto announced and started marketing their PRISM systems (Henis and Tripodi, 1980a,b, 1981). In all the above developments, there is a common notion that the gas flow through membrane pores does not contribute to gas separations. Therefore, gas flow through membrane pores should be either stopped with nonporous materials such as silicone rubber (Henis and Tripodi, 1980a, 1981), or the membrane should be composed of a nonporous polymer material (Stern, 1976) in order to achieve efficient separations of gas mixtures. However, it has also been established that cellulose acetate membranes whose pore structures are preserved during the drying procedure can also be used for gas separations (Agrawal and Sourirajan, 1969, 1970; Schell and Houston, 1982). In fact, the separation factor of He/Kr mixture achieved by a cellulose acetate membrane is even higher than that achieved by a silicone rubber membrane, and the preference to the permeation of the above gases can also be completely reversed (Ohno et al., 1976). The above results clearly indicate that nonporous rubbery polymer is neither necessary nor preferable to porous membranes for achieving gas separations. This work is concerned only with gas transport through porous reverse osmosis membranes. In our previous work (Rangarajan et al., 1984) a model was developed for the permeation of single gases through porous membranes, in which gas permeation through a membrane pore was split into contributions from Knudsen, slip, viscous, and surface transport mechanisms. A basically similar approach was applied for single gas permeation through membranes with pore radii in the range of >10 nm by Kamide et al. (1982). It was found in our study that the magnitude of the contribution from each flow mechanism depends on the nature of the permeant gas, the membrane material and the pore size distribution, and the operating conditions such as the operating pressure. As a result, the shape of the curve correlating permeability Issued as N.R.C. Contribution No. 24343.
coefficient vs. operating pressure changes significantly as the above variables are changed. More importantly, we were able to compute the permeability coefficient of different pure gases from experimental data on such coefficients for a reference gas using the transport equations developed in our work. The natural consequence of the above approach is that the separation of a gas mixture also changes with the change in the membrane material and in the pore size distribution and we would be able to compute such changes on the basis of the pore flow model subject to restrictions required for such systems. The object of this paper is, therefore, to perform the calculations according to the pore flow model, to predict the separation factor of a mixed gas system with membranes of different pore size distribution, and to test the predicted values by experiments. It is particularly important to know the contribution of the surface flow to the separation factor as compared to that of the rest of gas flow mechanisms, since the gas-polymer interaction affects primarily the surface flow. The permeation of H2/CH4gas mixture was chosen in view of its relevance to the development of energy sources from coal, petroleum, and biomass conversion processes. Theoretical It was shown earlier (Rangarajan et al., 1984) that characterization of pore structure by nitrogen permeation data, using an expression for gas permeability coefficient involving a pore size distribution and four mechanisms of transport, viz., Knudser, slip, viscous, and surface flows, could conveniently lead to a framework for the prediction of permeation rates of different gases through a porous membrane. Here the same treatment is extended to deduce expressions for fluxes of component gases in a mixture of two gases, and relationships between separation factors and membrane characterization are derived. Basic Gas Transport Equations. A brief description of the underlying principles in the computation of fluxes of gas through a membrane is in order. First of all, the porous structure of a given membrane is represented by the distribution function
Through these systems of capillary pores, gases permeate by different flow mechanisms. The governing quantity which provides a guideline in determining which mechanism is operative in a given pore under the given experimental conditions is the ratio ( R l h ) . However, i t should 0 1985 American Chemical Society
908
Ind. Eng. Chem. Process Des. Dev., Vol. 24, No. 4, 1985
be remembered that only surface transport occurs in all the pores. In addition to the surface flow, one of the three mechanisms, viz., Knudsen, viscous, and slip, may also operate depending on the RIA ratio. Knudsen flow is predominant in the range of RIA between 0 and 0.05; slip flow is predominant in the range 0.05-50; and viscous flow predominates beyond 50. With this background information, using the relationships provided by the kinetic theory of gases, the following expression for the permeability coefficient, (AG),of a gas has been deduced.
(AG)=
($)[ (&) 0.05A
R3 exp[
1
1i2
X
-X(
e)’] dR +
pressure in the boundary gas phase, mole fraction of the component 1 in the boundary gas phase, flux of the component 1, the quantity (G,) with respect to the component 1, and the viscosity of the gas mixture of components 1 and 2, respectively. Furthermore, the total pressure and the mole fraction of each component in the boundary gas phase were equated to those of the bulk feed phase. This is justified since the pressure drop takes place only across the membrane and since the diffusivity of gas molecule is high. It may be recalled that eq 2 has been derived assuming that the concentration of the adsorbed species is proportional to the gas pressure, which implies adsorption represented by a form similar to Henry’s law, and as the situation warrants, such a dependence may be replaced by another expression representing a suitable adsorption isotherm; also the flux of the gas is given by multiplying (AG) by the pressure differential, hp = (P, - P3). Applicable Flux Equations for the Gas Mixtures. Flux Equations of Component Gases. In developing flux equations for the component gases in a mixture of two gases, it is assumed that the sorption of the individual gases on the surface is independent. However, in the gas phase, the quantities, A, d, etc., are different from the correspondingquantities for the pure gases. Thus, we have the following applicable relationships
and AlZ
where -
P2
p=-
+ P3 2
(3)
=
RT 21~z~d12zP
Moreover, the pressure differentials are different for two different gases, viz. (10) U 1 = XlZPZ - x13p3 and
(4)
and A =
RT 2‘&d2NP
(5)
Following the previous paper (Rangarajan et al., 1984) we set two parameters A , and A z as A I = Nt/S6 (6) and
A2 =
RTPappk 20OOTC~62S
xZZpZ - x23p3 Also we have the following relationships hp = h p 1 + AP, = P, - P3 U
Z =
X12+ xz2= 1 Xi3 + X23 = 1 xlZpZ + x13p3
P, = Pz =
(7)
All symbols are defined a t the end of the paper. A note here is in order on the subscripts used in our nomenclature. Numerical subscripts inside the bracket have no particular meaning other than distinguishing the preceding symbol; for example, (Il), (Iz),(I3),(I4),and (I5)are integrals defined differently by eq 19 to 23, and (G,), ( G 2 ) ,and (G3)are different quantities defined by eq 24 to 26. With respect to other numerical subscripts, 1 and 2 refer to the indicated component of gas mixture (1 = hydrogen gas molecule and 2 = methane gas molecule in this paper), while 1, 2, and 3 refer to the bulk feed gas phase, the boundary gas phase on the high-pressure side of the membrane, and the product gas phase on the low-pressure side of the membrane. Thus, hpl, P p ,Xlz,J1,(Gl)l, and q12indicate the difference in the partial pressure of the component 1,
2 x22pZ
+ x23p3 2
(11)
(12)
(13) (14)
Ind. Eng. Chem. Process Des. Dev., Vol. 24, No. 4, 1985 909
where 1
R3 exp(-y2(
q)} dR;
(i = 1, 2) (19)
where Rref is the effective pore radius obtained by the regression analysis of the permeation data, viz., (AG) vs. P2,with respect to the reference gas. The quantity, Ai, called the radius correction factor, is listed in Table IV of the previous work (Rangarajan et al., 1984) on the basis of the nitrogen reference gas. We also have the following definitions for the mole fraction of gases 1 and 2 in the permeate
(i = 1, 2) (20)
Xi3
=
J1 ~
J1
+ J2 JZ
(33)
x23 =
and X13 + X23 = 1
(34)
Rewriting eq 32 as (35) ( J 1 + JZ)X13 - J 1 = 0 and substituting for J1and J2using eq 17 and eq 18, we obtain the following cubic equation in terms of X13 after appropriate simplifications effected by eq 10 to 16.
(i = 1, 2)
where
and
(37)
As explained in our previous work, the first three terms in eq 17 and eq 18 (the terms with (Il),(I2), and (I3)) correspond to Knudsen, slip, and viscous flow, respectively, while the last term (including (I4)/ (I5))corresponds to surface flow. The following relationship is used in eq 25
Further q12 in eq 26 is given by Wilke's equation (Reid and Sherwood, 1958) as 11
112
= 1 + (X2/X1)812
72 + 1 + (Xl/X2)821
(28)
+ (G2)i(Z2)i + (GJi(I3)i);
(i = 1, 2) (38) Quantities (AJ1 and (A& are the surface transport coefficients of component 1 and component 2, respectively, and are given by the following equations. (A2)i = ( A J r e d i (39) ai
=
(Al)I(Gl)i(Il)i
The parameters $1 and c&, called relative surface transport coefficients, are listed on the basis of reference nitrogen gas in our earlier work (Rangarajan et al., 1984). Now it can be seen that Xi3 is obtained by solving the cubic equation, given by eq 36. The separation factor, S12, defined by (x13/xZ3)
slz= (Xll/XZl)
A brief note here is in order on the average pore radius Ri. As stated in our earlier work (Rangarajan et al., 1984) the effective mean pore radius R is distinct from the geometrical mean pore radius R*. The former mean pore radius can be expected to vary from one gas to another, depending upon the combined effect of gas-membrane interaction and the mobility of the adsorbed gas molecule. According to the paper referred above, for any gas i, Ri is computed from the equation (31) Ri = Rref+ Ai
-
-
(x13/xZ3)
(X12/X22)
(41)
can be calculated from Xi3 and X23 (= 1 - X13) and the feed composition XI1 and XZl at any operating pressure once (AJIef,(= AJ, (A2)ref, Rref,and u are determined from the permeation data of the reference gas using the values of Ai and 4i for component 1 and component 2. The permeation rate [PR] can be given by [PR] = J1 + J 2 (42) According to the equations developed above, the prediction of S12and [PR] can be performed following the steps indicated below: (1)The given membrane is first characterized in terms of (Allref, Rref, and u by regression analysis of the permeation data, viz. (AG) vs. P2,
910
Ind. Eng. Chem. Process
Des. Dev., Vol. 24, No. 4,
1985
Table I. Membrane Characterization by Using Helium Permeation Data (R)He
membranes CA-33 CA-34 CA-11 CA-12 CA-32 CA-22 CA-23 CA-14 CA-42 CA-21 CA-44 CA-41
x
lolo, m
5.0 4.8 6.5 5.8 3.2 5.7 6.3 4.1 6.0 7.0 5.5 5.2
u X
(Ailhie, kmol/m3 s lolo, m (A&. m-3 PaZ 4.20 X 1019 1.07 X lo-' 1.0 1.7 1.04 X 2.23 X 0.6 2.76 X 10" 6.41 X 0.6 6.15 x 1017 1.68 x 10-9 4.24 X 10" 1.15 X lo-' 2.3 2.5 3.18 X lozo 6.60 X 2.0 4.39 x 1019 2.23 x 10-8 1.2 3.66 x 1019 4.12 x 10-9 1.6 2.46 x 1019 1.12 x 10-8 7.02 X lov9 0.8 5.76 X 10" 2.0 4.30 X 10'' 5.14 X 0.5 3.62 X lo1@ 6.86 X
of the reference gas. (2) From (A&, ( A 2 ) ,and (A2& are calculated using eq 39. Further, from Rref,R , and R2 are calculated using eq 31. (3) From the operating conditions for which Slzand [PR] are to be predicted, X12 is calculated using eq 9 and all the necessary integrals (Il)i through are evaluated using eq 19 to 23. Also quantities (GJi through (G3)i are evaluated using eq 24 to 27. (4) With numerical values generated above, eq 36 is solved in terms of X13. (5) With the values of X13, X Z 3 ,Xll, and Xzl so obtained, Slz is calculated using eq 41. (6) Using the quantities J1and Jz evaluated by eq 17 and eq 18, [PR] is calculated by eq 42. Experimental Section The method of preparation of the membranes used in this study was the same as that described in our earlier work (Rangarajan et al., 1984). The membrane material was, however, limited to cellulose acetate (Eastman E398-3). Cellulose acetate batch 316( 10/30) membranes were first produced according to the procedure described by Pageau and Sourirajan (1972). The membranes were then shrunk at different temperatures (75-85 "C) to obtain different average pore size and pore size distributions and subsequently dried before carrying out the permeation and the separation studies. The drying procedure used involves successive equilibration of the wet membrane in solventwater mixtures of progressively higher solvent content and the final removal of the solvent either by drying in the air or by freeze drying (Rangarajan et al., 1984). The method of freeze drying has been described (Agrawal and Sourirajan, 1970). The membranes so produced were characterized by the parameters Rref, u, (A,),,, and (AJref computed on the basis of the gas permeation data using transport equations developed in our earlier work (Rangarajan et al., 1984). All the above parameters are listed in Table I with respect to all the membranes involved in this study. The reason for the choice of He as reference gas for membrane characterization instead of N2 gas, which was used in the previous study, is that there is less interaction with the membrane material and also higher reliability of the He permeation data due to its higher permeability. Permeation rates of pure He gas under the steady-state condition were measured as a function of operating pressure in the range 515 to 2310 kPa at 26 f 1 "C. Experimental data for the permeation and the separation of H,-CH, mixtures were obtained with different feed compositions (H, mole fraction ranging from 0.484 to 0.887) at different operating pressures (ranging from 515 to 2310 kPa a b 4 a t 26 f 1 "C. The test cell was thermostated in order to maintain a constant temperature. The details of the experimental procedure have already been reported (Rangarajan et al., 1984). Some special
features of the experimental procedure involved in this study have to be noted, however. Air was removed from the cell as well as from the permeate line prior to any experiments by purging with the appropriate gas or the gas mixture and also using a vacuum pump on the downstream side of the membrane. The upstream pressure was controlled by a back-pressure regulator while the permeate pressure was maintained at the atmospheric pressure. The permeate flow rate was measured with a rotameter or a flowmeter. In some cases, when the gas permeation rate was extremely low, it was measured by the rate a t which the permeate displaced mercury in a calibrated permeate line. The composition of the gas mixture was analyzed by gas chromatography with a Tracor MT160/220 Model equipped with a Porapak Q column. The permeation rates were reproducible within f2% in all cases and the accuracy involved in the determination of the gas composition is fl%. All pure gases as well as the mixtures of known compositions were obtained from the Union Carbide Gas Products Ltd., designated as Linde Speciality Gases of more than 99.9% purity. Results and Discussion Membrane Characterization and Separation Factor. All the membranes are characterized in Table I. It is clear in the table that there are over 2-fold variations in and &fold variations in g. Furthermore, there is 750-fold variation in the values of (A1)He reflecting that in the effective number of pores. There is also about 60-fold variation in the values of (A2)He, which reflects the change in the magnitude of sorption contribution to the gas permeation. Thus, the membranes tested have a wide range of porosity and also a wide range in the surface flow contribution, which should naturally result in a wide variation in separation factors. In fact, separation factors in the range 1to 17.5 were obtained with the membranes under study (see Table 11), depending on the average pore size and pore size distribution. The results confirm that significant separation factors are obtainable by gas permeation under pressure through porous membranes. Agreement of Predicted and Experimental Values. The prediction of the total gas permeation rate, [PR], and the separation factor, SL2, was attempted with respect to H2/CH4gas mixtures. The prediction procedure requires a knowledge of the radius correction factor, Ai, and the relative surface transport factor, cpi, for each component gas. The above parameters generated and listed in our previous work on the basis of nitrogen reference gas were transformed to the parameters based on helium reference gas by the equations and where the subscript outside of the bracket indicates the reference gas. According to the above equations (AHz)He, (&H,)He, (6H2)Her and ($cH,)H~ are 0, -0.5 x lo-'' m, 1.8,and 1.1,respectively. The totalpermeation rate calculated with the above numerical values are correlated with the experimental values in Figure 1. The data in the figure include those obtained for three different feed gas compositions, three different operating pressures, and membranes of different pore size distributions. The agreement of the calculated values with the experimental ones is either fair or unsatisfactory. At low values of [PR], the experimental value is far lower than the calculated value. The prediction of the permeate gas composition was tested by comparing the predicted and experimental values
Ind. Eng. Chem. Process Des. Dev., Vol. 24, No. 4, 1985 911 Table 11. Comparison of Experimental and Calculated Values of X I S Xz3, , and x13
run no. 1
2 3 4 5 6 7 8 9 10 11 12 13 14
film no. CA-33 CA-34 CA-11 CA-12 CA-32
15 16 17 18 19 20 21 22 23
CA-34
24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44
CA-22
a
CA-12 CA-32
CA-23 CA-14 CA-42 CA-21 CA-44 CA-41
S 1 2 ' SlZ
x23
operating press., kPa abs exptl calcd exptl = 0.883 Hvdroeen Mole Fraction in Feed Gas Mixture - 0.033 515 0.967 0.966 1410 0.967 0.974 0.033 2310 0.962 0.981 0.038 0.034 515 0.966 0.965 1410 0.969 0.974 0.031 2310 0.971 0.979 0.029 515 0.982 0.977 0.018 1410 0.977 0.986 0.023 515 0.984 0.980 0.016 0.986 0.014 1410 0.986 2310 0.986 0.988 0.014 0.014 515 0.986 0.984 1410 0.991 0.988 0.009 2310 0.9925 0.989 0.0075 Hydrogen Mole Fraction in Feed Gas Mixture = 0.781 515 0.949 0.922 0.051 1410 0.950 0.935 0.050 2310 0.951 0.942 0.049 515 0.972 0.941 0.028 1410 0.967 0.950 0.033 2310 0.974 0.954 0.036 515 0.963 0.037 1410 0.973 0.027 2310 0.978 0.022 Hvdroeen Mole Fraction in Feed Gas Mixtures = 0.484 0.484 0.709 0.516 515 1410 0.484 0.706 0.516 2310 0.484 0.695 0.516 0.470 515 0.530 0.698 1410 0.508 0.678 0.492 0.484 2310 0.664 0.516 515 0.600 0.726 0.400 1410 0.600 0.723 0.400 2310 0.600 0.700 0.400 0.700 0.470 515 0.530 1410 0.549 0.679 0.451 2310 0.530 0.666 0.470 515 0.508 0.680 0.492 1410 0.484 0.651 0.516 2310 0.484 0.639 0.516 0.549 0.673 515 0.451 1410 0.643 0.530 0.470 2310 0.508 0.631 0.482 0.724 0.654 0.276 515 1410 0.615 0.633 0.385 2310 0.549 0.622 0.451
calcd
exptl
calcd
0.034 0.026 0.019 0.035 0.026 0.021 0.023 0.016 0.020 0.014 0.012 0.016 0.012 0.011
3.9 3.9 3.5 3.8 4.2 4.4 7.3 5.7 8.2 9.1 9.3 9.4 14.5 17.5
3.8 4.9 6.8 3.e 4.9 6.1 5.6 9.4 6.5 9.3 11.1 8.0 10.9 11.8
0.077 0.065 0.058 0.059 0.050 0.046
5.2 5.3 5.4 9.6 8.2 7.5 7.3 10.0 12.7
3.3 4.0 4.6 4.5 5.3 5.9
0.291 0.294 0.305 0.302 0.322 0.336 0.274 0.277 0.300 0.300 0.321 0.334 0.320 0.349 0.361 0.327 0.357 0.369 0.346 0.367 0.378
1.0 1.0 1.0 1.2 1.1 1.0 1.6 1.6 1.6 1.2 1.3 1.2 1.1 1.0 1.0 1.3 1.2 1.1 2.8 1.7 1.3
2.6 2.6 2.4 2.5 2.3 2.1 2.8 2.8 2.5 2.5 2.3 2.1 2.3 2.0 1.9 2.2 1.9 1.8 2.0 1.8 1.8
All separation factors correspond essentially to zero product recovery.
of quantities such as hydrogen mole fraction, X13, the methane mole fraction, X23, in the permeate gas, and the separation factor S12.As the results show (Table 11), the agreement of predicted and experimental values is generally good for the set of runs 1 to 14, and not equally satisfactory for other runs. With respect to the membrane CA-32, one notes a considerable difference between the predicted and the experimental values of S12,though the agreement in mole fractions looks far more reasonable. The discrepancy in the value of S12is partly due to the sensitivity of the latter quantity to X23. Since, in the calculation of SI2,the division by X23 is involved (see eq 41) and since the latter mole fraction is very small, a small error in the evaluation of X23 causes a large error in the resulting S12. Effect of Pore Flow and Surface Flow Contributions on S12.Our analytical approach reveals the respective contribution of the pore flow, including Knudsen, slip, and viscous flow, and of the surface flow to the separation factor, SI2,of H2/CH4mixtures. Looking into eq 17,18,37, and 38, it is obvious that X13 in the absence of the surface flow can be calculated from eq 36 by setting
PI = P2 = 0. In order to do this calculation, each term involved in the left side of eq 36 is multiplied by P32((A2)lP1 + (A2)&. By setting P1 = P2 = 0 the first term disappears and eq 36 is reduced to a quadratic equation in terms of X I 3 . On the other hand, X13 in the absence of the pore flow can be calculated by setting al = cy2 = 0. The Xi3 values so obtained further allow the calculation of the Slz value in the absence of the surface flow and that in the absence of the pore flow. The former S12is designated as (S12)p since the gas separation is entirely due to the pore flow, and the latter S12is designated as (S,,),, indicating that the separation is totally due to the surface flow. Such calculations were conducted with respect to the experiment 14, in which membrane CA-32 was used a t the operating pressure of 2310 kPa abs and a t the feed H2 mole fraction and (S12),, of 88.3%. The result was 3.5 and 13.5 for (S12)p respectively. Note that (S12), > (S1& The values of (Sl2Ip and (S12), were found to be essentially the same for other membranes and operating pressures if the hydrogen mole fraction in feed gas mixture is the same. For example, (S12)p ranged from 3.4 to 3.7, while (S12), ranged from 13.5 to 14.0 throughout runs 1 to 14. Equations 17 and 18
912
Ind. Eng. Chem. Process Des. Dev., Vol. 24, No. 4, 1985
e
I
CA-23 e
-
-
CA-47
e
CA-41 :A-21 e * CA-II * C A - 4 4 1
2
CA-I2
l
eCA-33
CA-14 e
CA-34 e
v
CA.32 ~
/
0-9
Figure 2. Plots of
J 100
[PR],a,cd
a IO',
IO00
kmOl/m2.s
Figure 1. Comparison of experimental and calculated permeation rate of H2/CH4 gas mixture. Hydrogen mole fraction in feed gas mixture, 0.484-0.883; operating pressure, 515-2310 kPa abs; membranes, cellulose acetate (CA-398) material, porosities given in Table I; operating temperature, room temperature.
further indicate that the relative contribution of the surface flow as compared to the pore flow increases as the operating pressure P2 increases, while P3 is maintained a t atmospheric pressure. As a consequence, Slzis expected to increase gradually with the increase in the operating pressure within the calculated limits of 3.4 ((Slz)p) and 14.0 ((SlZ)J, Predicted Slzvalues for runs 1-14 follow the above tendency, which is of course natural. The experimental values also follow the predicted tendency in general. However, there are some notable discrepancies between the experimental and predicted values. First, with respect to CA-11 and CA-33 membranes the increase in the operating pressure decreases Slzvalues slightly. Secondly, with respect to CA-32 membrane some of the experimental Slzvalues surpassed the calculated limit. Considering the sensitivity of Slzto minor errors in the values of Xi3 (and X,,), as described earlier, the above discrepancy may be attributed largely to experimental error. Another possibility for the smaller calculated limit of (Slz)s than the experimental value is the underestimation of the adsorption of hydrogen gas relative to methane gas in the theoretical calculation. This induces the underevaluation in the ratio of ( A J 1 / ( A & (also the underevaluation of the ratio @l/@z) and brings down the ratio J1/Jzwhen there is only surface flow (see eq 17 and eq 18). Thus, the quantity (Slz)sis lowered as a result of calculation by eq 32, 33, and 41. It is expected therefore a ratio of 41/&higher than that which is currently used (1.8/1.1= 1.64) will bring the calculated (Slz)s value closer to or identical with the experimental value. In fact d1/& value of 2.17 was necessary to bring (Slz)s value up to 17.5. The change in membrane porosity also affects the relative contribution of the surface flow in comparison to the pore flow (and consequently S12).For example, the decreasing order in (A1)He values of membranes is CA-33 > CA-34 > CA-11 > CA-12 >CA-32 Since value represents the magnitude of the pore flow, we may expect an increase in the relative contribution of surface flow and also an increase in Slzas (A1)He decreases. Both calculated and experimental results satisfy this expectation. In the other extreme of the lowest hydrogen mole fraction in the feed mixture (48.4%) the tendency is reand (SI,),,were versed. The two calculated limits, (S12)p computed to be 3.9 and 1.5, respectively. Note that (S12)p > (S12), in this case. Again, these values are practically
I
1
vs. (AJHe
unchanged throughout runs 24 to 44, where the hydrogen mole fraction in feed was maintained at 48.4%. Since the relative surface flow contribution increases as the operating pressure increases, as eq 17 and eq 18 indicate, it is expected S12to decrease within the calculated limits of 3.9 and 1.5 as the operating pressure increases. This tendency is shown by the predicted values which is again natural. Generally, experimental values also follow the tendecy of decreasing S12with increase in the operating pressure; S12 either decreased or remained constant as operating pressure increased except in the case of film CA-42. However, the lowest experimental value of S12is below 1.5, which value was calculated as the lower limit. The discrepancy between the calculated and the experimental values can, in this case, be ascribed to the overestimation of hydrogen adsorption (which is equivalent to the overestimation of C $ ~ / I $in~ the ) theoretical calculation of (Slz)s. For example, X13 and (Slz)s were calculated to be 0.484 and 1.0, respectively, using the &/& ratio of 0.995 for run 26. The higher $ 1 / ~ 2 ratio (2.17) a t the higher hydrogen mole fraction in the feed (0.883), and the lower C$l/rp2 ratio (0.995) at the lower hydrogen mole fraction in the feed (0.484) required for the better agreement of the calculated and experimental values implies that a competitive adsorption between hydrogen and methane molecules is operative, which exerts a further effect on the surface flow of each individual component. The results for experiments for the feed hydrogen mole fraction of 78.1% lie somewhere between the above two extremes, though they are closer to those for the hydrogen mole fraction of 88.3%, Thus, the general tendency of experimental values is well represented by the calculated values, though there are some cases where notable differences between experimental and predicted values are observed. Furthermore, the above approach can predict precisely that (1)the hydrogen mole fraction in the feed gas mixture should be high, (2) the operating pressure should be high, and (3) (A1)He value of the membrane should be low in order to achieve a high separation factor. Among the three factors mentioned above, the first one increases the surface flow of hydrogen gas, while the other two enhance the surface flow contribution of both hydrogen and methane. A note here is in order on the surface flow. It is a common practice to conceive the membrane gas permeation other than the pore flow as flow through the membrane polymer matrix. The surface flow is different from such a flow. It is a flow which takes place on the surface of the membrane pore. One piece of evidence which supports the above notion is the plot of (A2)He vs. (AJHewhich is shown in Figure 2. Though there are a few exceptional data, the parameter (A2)He generally increases with increase in the parameter (A1)He, indicating that surface flow increases with increase in the number of pores. If the last term of equations for the gas permeation (see eq 17 and eq 181,
Ind. Eng. Chem. Process Des. Dev., Vol. 24, No. 4, 1985 913
which we call the surface flow, was in fact the permeation through the polymer matrix, (A2)He would be independent from (A1)He. Conclusion By applying the transport equations of gases developed in our previous work on the basis of a pore flow model to the separation of hydrogen and methane gas mixture, all the tendencies found in the experimental data on the total permeation rate and the separation factor with the change in the variables such as the hydrogen mole fraction in the feed gas mixture, operating pressure, and the membrane porosity were reproduced, though the agreement between numerically predicted and experimental values was not always satisfactory. The discrepancy between the predicted and experimental values indicates the need for (i) considering the effect of the competitive adsorption between hydrogen and methane gases on surface flow, (ii) examining alternative expressions for the adsorption isotherm in the basic transport equation as indicated in our earlier paper, and (iii) analyzing the effect of presence of more than one pore size distribution on the membrane surface.
Acknowledgment This work has been supported by the Bioenergy R & D Program of the National Research Council of Canada. One of the authors (M.A.M.) thanks NSERC for the award of a Visiting Fellowship. Another author (R.R.) is grateful to the CSIR, New Delhi, India, for the grant of leave of absence from CSMCRI, Bhavnagar, and thanks the NRC Canada for a Visiting Research Officership.
Nomenclature (Al){= (AJref] = constant for a given membrane related to the
porous structure, m-3 (A2), (AJi = constant related to surface transport, (A,) for reference gas and gas i, respectively, kmol/m3 s (Pa), (AG), ( A G ) ~= gas permeability coefficient, (AG) for gas i, kmol/m2 s Pa CR, (C&f = coefficient of resistance for the transport of adsorbed molecules, cR for reference gas, kg/s m2 F, ~i = mean speed of gas molecules, F for gas i, m/s d , di, d12= collision diameter, d for gas i and mixture of gas 1 and gas 2, respectively, m ( G I ) ,( G J i = a physicochemical constant, (G,) for gas i (G,), (G,)i = a physicochemical constant, (G,) for gas i (G3),(G3)i = a physicochemical constant, (G3) for gas i (Il)i,(12)i, (13)i, (14)i, (I&= quantities defined by eq 19-23 Ji = flux of gas i, kmol/m2 s k, = constant representing gas adsorption equilibrium, k for reference gas M, Mi = molecular weight of gas, M for gas i, kg/kmol N = Avogadro number N(R) = number of pores having a radius R, Nt = total number of pores P = pressure, Pa Pz = pressure (absolute) on the high-pressure side of the membrane, Pa
P3 = pressure (absolute) on the low-pressure side of the membrane, Pa
AP,h p i = pressure differential across the membrane, AP for gas i, Pa P = mean pressure across the membrane, Pa [PR] = total permeation rate of the gas mixture, kmol/m2 s R =pore_radius, m R, (R),,f, Ri = mean pore radius, R for reference gas and gas i, respectively, m R, = radius of the largest pore, m R = gas constant, m3 Pa/K kmol S = membrane area Slz= separation factor for gas mixture given by eq 41 T = absolute temperature, K Xi, Xi,, Xi3 = mole fraction of gas i, Xi on the high-pressure side and on the permeate side, respectively Greek Letters = quantities given by eq 38 = quantities given by eq 37 Ai = radius correction factor, a constant for gas i for a given
ai pi
membrane material, m 6 = equivalent thickness of the membrane 7, 71, v12 = coefficient of viscosity, 7 for gas i and mixture of gas 1 and gas 2, respectively, Pa s 812,821 = quantities defined by eq 29 and eq 30, respectively A, X12 = mean free path of gas, h for mixture of gas 1 and gas 2, respectively, m
apparent density of the membrane, kg/m3 standard deviation for the pore size distribution, m 7 = tortuosity factor for the pores +i = characteristic parameter, called the relative surface transport coefficient (= (A& (AJref),related to gas-membrane interaction Registry No. C H I , 74-82-8;H2, 1333-74-0;cellulose acetate, 9004-35-7. Literature Cited papp = u =
Agrawal, J. P.; Sourirajan, S.J. Appl. Po/ym. Sci. 1989, 13, 1065. Agrawal, J. P.; Sourirajan, S. J. Appl. Po/ym. Sci. 1970, 14, 1303. Henis, J. M. S.;Tripodi, M. K. Sep. Sci. Techno/. 1980a, 15. 1059. Henis, J. M. S.;Tripodi, M. K. U S . Patent 4230465, Oct 28, 1980b. Henis, J. M. S.;Tripodi, M. K. J. Membr. Sci. 1981, 8 , 233. Hwang, S.-T.; Kammermeyer, K. "Membranes in Separations"; Wiley-Interscience: New York, 1975; Chapter X I I I . Kamide, K.; Manabe, S.;Nohmi, T.; Makino, H.; Narita. H.; Kawai, T. "Mechanism of Gas Permeation through Porous Polymeric Membrane", Cooper, A. R., Ed.; Plenum: New York, 1982; pp 35-74. Michaels, A. S.;Bixler, H. J. "Membrane Permeation: Theory and Practice", Perry, E. S.,Ed.; Interscience; New York, 1968; pp 143-186. Heki, H.; Miyauchi, T. Radiochem. RadioaOhno, M.; Morisue, T.; Ozaki, 0.; rial. Left. 1976, 27, 299. Sci. 1972, 16, 3185. Pageau, L.; Sourirajan, S. J. Appl. Po&" Pan, C.-Y.; Habgood, H. W. Can. J. Chem. Eng. 1978a, 56, 197. Pan, C.-Y.; Habgood, H. W. Can. J. Chem. Eng. 1978b. 56, 210. Rangarajan, R.; Mazid, M. A.; Matsuura, T.; Sourirajan, S. Ind. Eng. Chem. Process D e s . Dev. 1984, 23, 79. Rea, R. C.; Sherwood, T. K. "The Properties of Gases and Liquids"; McGrawHill: New York, 1958; pp 199-202. Schell, W. J.; Houston, C. D. Chem. Eng. Prog. Ocl. 1982, 33. Stern, S.A. "Industrbl Processing with Membranes", Lacey, R. E.; Loeb, S., Ed.; Wiley-Interscience: New York, 1972; Chapter 13. Stern, S.A. "The Separation of Gases by Selective Permeation", Meares, P., Ed.; Eisevier: New York. 1976; pp 293-326.
Received for review October 17, 1983 Revised manuscript received June 25, 1984 Accepted August 8, 1984
