Ind. Eng. Chem. Process Des. Dev. 1984, 23, 79-87
Quitzsch, K.; Koehler, S.; Taubert, K.; Gelseler, G. J. frakt. Chem. 1969, 311, 429. Ragaini. V.; Santi, R. Carra, S. Lincei-Rend. S c . Fis. Mat. e Nat. 1968, 4 5 , 540. Ramalho, R. S.; Ruel, M. Can. J. Chem. Eng. 1988, 46, 456. Reid, R. C.; Prausnitz, J. M.; Sherwocd, T. K. “The Properties of Gases and Liquids”, 3rd ed.; McGraw-Hill: New York, 1977. Renon, H.; Prausnitz, J. M. Chem. Eng. Sci. 19678, 22, 299. Renon, H.; Prausnitz, J. M. Chem. Eng. Sci. I987b, 22, 1891. Rose, A.; Papahronls, B. T.; Williams, E. T. Chem. Eng. Data Ser. 1958, 3 , 216. Rose, A.; Suplna, W. K. J. Chem. Eng. Data 1961, 6 , 173. Schmidt, G. C. Z . fhys. Chem. 1926, 121, 221. Slobodyanik, I.P.; Babuskhina, E. M. Zh. frikl. Khim. 1966, 39, 1555. Sun, H. H.; Christensen, J. J.; Izatt, R. M.; Hanks, R. W. J. Chem. Thermodyn. 1980, 12,95. Tamir, A.; Wisniak. J. J. Chem. Eng. Data 1975, 2 0 , 391.
79
Tan, R. L.; Hanks, R. W.; Christensen, J. J. Thermochim. Acta 1977, 21, 157. Tan, R. L.; Hanks, R. W.; Christensen, J. J. Thermochim. Acta 1978, 23, 29. Taylor, E. L.; Bertrand, G. L. J. Solutbn Chem. 1974, 3 , 479. Timmermens, J.; “Physico-Chemical Constants of Pure Organic Compounds”; Elsevier: Amsterdam, 1950; Vol. 1. Timmermans, J. “Physlco-Chemlcal Constants of Pure Organic Compounds”; Elsevier: Amsterdam, 1965; Vol. 2. Udovenko, V. V.; Frid, T. S. B. Zh. F k . Khim. 1948, 22, 1135. Verhoeye, L.; DeSchepper, H. J. Appl. Chem. Biotechnol. 1973, 23,607. Wilson, G. M. 65th National Meeting American Institute of Chemical Engineers, Cleveland, OH, May 4-7, 1969, Paper 15c.
Received for review September 7 , 1982 Revised manuscript received March 7 , 1983 Accepted March 25, 1983
Permeation of Pure Gases under Pressure through Asymmetric Porous Membranes. Membrane Characterization and Prediction of Performance Ramamurtl Rangarajan, M. A. Mazld, Takeshl Matsuura, and S. Sourlrajan’ Division of Chemistry, National Research Council of Canada, Ottawa, Canada, K 1A OR9
Permeation rates of hydrogen, helium, methane, nitrogen, ethylene, and argon through asymmetric porous cellulose acetate, hydrolyzed cellulose acetate propionate, and polysulfone membranes have been measured at room temperature. The experimental data have been analyzed by a transport equation which incorporates terms for a pore size distribution on the membrane surface. The mechanism of transport involves simultaneous Knudsen, slip, viscous, and surface flow through the porous structure. The analysis is useful for membrane characterization and predictlon of performance with respect to a reference gas.
Introduction Sorption as well as unsteady- and steady-state permeation of different gases in a variety of membranes have been studied in great detail in view of the potential applications in gas separations and also to elucidate the nature of transport prevailing in such systems. As this communication is concerned only with the mechanisms of transport as it relates to the characterization of a given membrane and the prediction of membrane performance, it is deemed adequate to cite only the major accomplishments in the understanding of the nature of such transport as reported in some of the recent publications. An in-depth understanding of the different approaches to the subject may be pursued through the references cited therein. A historic perspective to transport of gases through synthetic polymer membranes is given by Stannett (1978). Reports of pertinent literature by Barrer (1967), Hopfenberg (1974), Hwang and Kammermeyer (1975), Stern (1976), Stannett et al. (1979), and Stern and Frisch (1981), based on their extensive researches, provide the necessary background material for the understanding of vapor and gas transport through membranes. From the mechanistic point of view of gas and vapor transport through “nonporous” membranes, the free volume theory and the dual sorption theory have been highly successful in the analysis of sorption, unsteady-, and steady-state permeation in such membrane systems. The concept of free volume as developed by Cohen and Turnbull (1959) for the case of self-diffusion in a liquid of hard spheres suggests that the permeant diffuses by a cooperative movement of the permeant and the polymer segments, from one “hole” to the other within the polymer, the creation of a 0196-4305/84/ 1123-0079$0 1.50/0
“hole” itself caused by fluctuations of local densiy. Based on the concept of redistribution of free volume to represent the thermodynamic diffusion coefficient (Fujita et al., 1960),and standard reference state for free volume (Frisch, 1970), Stern and Fang (1972) interpreted their permeability data for nonporous membranes, and Fang, Stern and Frisch (1975) extended the theories to the case of permeation of gas and liquid mixtures. The dual sorption model invokes the existence of two thermodynamically distinct populations of the penetrant gas, namely, molecules dissolved in the polymer by an ordinary dissolution mechanism (obeying Henry’s law) and molecules residing in a limited number of preexisting microcavities in the polymer matrix (obeying Langmuir type of isotherm) with rapid exchange between these two populations. The development and the illustration of the applicability of the dual sorption model for permeation through glassy polymers are the results of extensive investigations by Paul, Koros, and their associates, Vieth and his associates, and Stern and co-workers. The work of Koros et al. (1977) on sorption and transport of gases in polycarbonate, Chan et al. (1978) on hydrocarbon gas sorption and transport in ethyl cellulose, and on CO,, CHI, A, and N2 transport through polysulfone by Erb and Paul (1981) and the relaxation of immobilization assumption by Petropoulos (1970) resulted in a coherent approach to the understanding of the transport behavior of gases through glassy polymer membranes. Two exhaustive reviews by Vieth et al. (1976) and Paul (1979) furnish detailed information pertaining to different aspects of dual-sorption model. In addition to the free volume theory and dual sorption models, a few molecular models have also been proposed. Published 1983 by the American Chemical Society
80
Ind. Eng. Chem. Process Des. Dev., Vol. 23, No. 1, 1984
A brief description of the molecular models is presented by Stern and Frisch (1981). To sum up, the current status of understanding of vapor and gas transport through nonporous membranes can be best described by quoting Stern and Frisch (1981). “Mechanistic interpretations of gas transport in rubbery polymers are now available in the form of molecular and free volume models, which permit, in principle, the prediction of diffusion and permeability coefficients if specified parameters are known...The development of similar models for glassy polymers will have to wait until considerably more insight is obtained on the nature of the glassy state.” The solution, free volume, and dual sorption theories indicated above have been successfully used in analyzing data on gas permeation through nonporous membranes (Stern and Frisch, 1981). However, these theories are not amenable (Stern et al., 1974) to meet the needs of the situations arising in studies on gas permeation through asymmetric porous reverse osmosis membranes. This paper is concerned with the latter studies which are particularly relevant at this time in view of the increasing importance of the applications of reverse osmosis membranes for industrial gas separations (Schell, 1982). Several investigations on permeation of gases through porous media are described in the literature (Agrawal and Sourirajan, 1970; Barrer, 1965; Bartholemew and Flood, 1965; Frisch, 1956; Gantzel and Merten, 1970; Gilliland et al., 1958; Higashi et al., 1963; Kakuta et al., 1978 and 1980; Kammermeyer, 1959; Kammermayer and Rutz, 1959; Nohmi et al., 1977; Stahl, 1971; Tomlinson and Flood, 1948; Weaver and Metzner, 1966; Weber, 1954; Wicke and Vollmer, 1952). From these investigations, it is clear that any model to analyze the transport data on permeation of gases through porous membranes must take into account contributions of (1)Knudsen, (2) slip, (3) viscous, and (4) surface transport. It is shown in this paper how these different contributions can be mathematically represented for a porous membrane with a given pore size distribution for the purpose of analysis of gas permeation data, and be able to characterize a membrane and also predict its performance for the permeation of any gas from the knowledge of its performance for the permeation of a reference gas. Theoretical For the purpose of analysis of gas permeation data the following assumptions are made. (1) The surface layer of an asymmetric porous membrane is considered to be composed of a bundle of capillary tubes with a normal distribution of pore size represented by the function Nt R-R N(R)= ~
(2?r)1/*u
The distribution is assumed to be constant irrespective of the permeating gas, but the mean radius can be expected to be affected by the extent of adsorption and the mobility of the adsorbed layer. Unless the pore size is very small, the effect of pore blocking is negligible. Thus u is assumed to be constant for a given_membrane, irrespective of the gas permeating, whereas R is assumed to be related to the factors controlling gas-membrane interactions and mobility of the adsorbed molecules. (2) The establishment of a pressure gradient across the thickness of the membrane results in the movement of the gas molecules through the pores in the membrane, with the simultaneous establishment of an adsorption equilibria at all possible adsorption sites at a rate faster than the overall rate of transport of gas molecules across the mem-
brane, thus permitting the neglect of the adsorption-desorption kinetics on the overall transport. This adsorption results in the existence of two different populations of gas molecules, one (the sorbed species) under the influence of the nature and magnitude of attractive and/or repulsive forces that the membrane material exerts on the gas molecules, and the other (the gas-phase species), by virtue of being remotely interacted upon by the membrane material, is free to move conforming to the conventional gas transport mechanisms as would exist in the bulk of the gas phase. (3) Once the allocation of gas molecules between the gas phase and the sorbed phase is accepted, the rapid exchanges between the two states for any gas molecules are ignored throughout during the transit of the molecules from one side of the membrane to the other, thus permitting the use of a linear pressure gradient to be applicable for any position coordinate in the membrane. This is a simplifying assumption without which it would be necessary to integrate the flux at every position coordinate within the membrane, requiring several unknown parameters to be included in the model. (4) The molecules adsorbed on the membrane surface and the walls of the membrane pores are assumed to be mobile, and thus contribute to the total flux. With these assumptions, expressions are deduced to relate the gas permeability coefficient, with the porous structure, gas-membrane interactions, and the physicochemical properties of the gas as follows. Transport of Molecules in the Gas Phase. Transport of gas through capillary tubes has been extensively studied and relationships have been developed based on the kinetic theory of gases. Three types of mechanisms have been proposed for gas transport through capillaries, namely, (1) Knudsen flow (2) slip flow, and (3) viscous flow. In the case of single capillaries, it has been found that, depending on the relative magnitude of pore radius, R, and mean free path, A, of the gas, gas molecules pass through the capillary by any one of the mechanisms stated above. Liepmann (1961) suggested that when (RIA) < 0.05, Knudsen type flow is predominant. According to Stahl(1971), slip flow occurs in the range of (RIA) = 1.5 to 50 and viscous flow occurs when (R/X) > 50. In the case of a membrane with a distribution of pore size, all the three mechanisms can occur simultaneously, but to different extents, depending on the operating conditions of pressure and temperature and the gas under study. Hence the total quantity of gas permeating can be estimated by considering each pore and applying the relevant flow equation for gas transfer through it, and integrating the flow over the entire system of capillaries or pores. This is done as follows. For a given gas, at the given pressure and temperature, flowing through a membrane with a pore size distribution given by eq 1, first the mean free path, A, is calculated from the expression (Metz, 1976)
where P = (Pz+ PJ/2 is the mean pressure across the membrane. Now the contribution of each mechanism is considered as follows. Through all pores having radii from 0 to O.O5A, the gas molecules move by Knudsen mechanism. All pores having radii between 50X and maximum radius, R, lend themselves to the operation of viscous flow. All pores in the intermediate range 0.05 to 50X are considered to contribute to the total gas phase flow by a mechanism operative in the transition region, the so called “slip flow”.
Ind. Eng. Chem. Process Des. Dev., Vol. 23, No. 1, 1984
However, the literature is uncertain about the range of the transition flow, and suggestion that all the three mechanisms could be assumed to be operative in the transition range was made by Weber (1954). In view of the fact that such a consideration would be tantamount to the inclusion of both Knudsen and viscous effects twice, it is arbitrarily assumed that only slip mechanism is operative in the (RIA) range 0.05 to 50, as there is considerable evidence to support the upper limit for (RIA) of 0.05 for Knudsen (Liepmann, 1961) and lower limit for (RIA)of 50 for viscous flow (Present, 1958). For a single capillary, the following equations have been derived (Present, 1958), for all the three mechanisms of flow. Knudsen flow, q K
viscous flow, qv
aR4F'AP qv =
=
of different mechanisms can vary over a considerable range as warranted by eq 8. Transport of Gas Molecules under the Influence of Gas-Polymer Interactions: Surface Flow. We have considered so far the flow of gas uninfluenced by the forces of interaction between the gas and the membrane material. The possibility of the gas "dissolving" in the polymer and diffusing through the body of the polymer is valid only for the limiting case of transport through nonporous membranes. For porous membranes exhibiting relatively high permeability, it is reasonable to assume a negligible contribution from such effects (Kakuta et al., 1978). Consequently, one needs to consider the contribution due to surface flow only, namely, the component of the total flow resulting from the mobility of the layer of the gas adsorbed on pore walls of the capillaries comprising the membrane. Gilliland et al. (1958) derived an expression for surface flow based on two-dimensional force balance on an adsorbed film as given by
(4)
-ZJiiT
slip flow, qsl Qsl
81
aR3AP
(5)
where e, the mean speed, is given by (Metz, 1976) E = (8RT/aM)lJ2
(6)
The total transport of gas, Qg, in the gas phase is given by the sum off&, qd, and qv, by suitably integrating over the pore size ranges applicable for each mechanism of transport
where A , is the plug (or pore) cross-sectional area, S, is the specific surface area of the solid over which the adsorbed molecules are mobile, CR is the coefficient of resistance, 7 is the tortuosity factor, L, is the pore length, pap, is the apparent density of the membrane, and x is the amount adsorbed per unit weight of the membrane. Equation 13 can be recast to fit the terminology of the present approach to include pore size distribution with the following expressions, modifying suitably the units wherever necessary. surface flow: Q, = N ,
(14)
8, = QK + Qsi + Qv R=0.05X
R=50X
R=O
R=0.05X
R=R,
C N(R)qK + C N(R)qsi + C N(R)qv (7)
=
R=50X
As a pore size distribution given by eq 1is assumed, the summation in eq 7 can be replaced by integration to give the following expression
Q, =
Nt -[G1I1 6
+ G2I2 + G3131AF'
(
32a
I1
=1
0.05X
G2 =
a TP E; G3 = 8qRT
R3 exp{-y2(
where 6 = TL,. In view of eq 14 to 16, eq 13 becomes
(8)
where
G1 = - ) I 2 ; 9MRT
S, = I , = 2a6L:rN(R)R dR
(9)
v)2) dR
(10)
and
It is to be noted that the upper limit of each integration is altered by the relative values of R,, (arbitrarily set at 100 A) and the upper limit. The following cases arise. (1) If R,, < 0.05X only Knudsen mechanism prevails. ( 2 ) If 50X > R, viscous mechanism is absent and the upper (3) If R , > 50X, all mechanisms are limit of slip is R,. operative simultaneously. Depending on the pore size distribution and the operating conditions, the contributions
Thus the evaluation of Q,, using eq 17, requires the knowledge of (1)the porous structure which is related to pa , 7 , 6 , R, and c, (2) the type of adsorption equilibrium, reyated to x , and ( 3 ) the mobility of the adsorbed phase which is related to CR. At this stage it is sufficient to state that by choosing a reference gas, transport analysis could be usefully carried out, by a procedure to be discussed later, without the the detailed information on pap,, 7,and 6. However, it is necessary to consider applicable cases of adsorption isotherms for incorporating an analytical expression for x in the transport equation. Applicable Cases of Adsorption Isotherms. Thomas and Thomas (1967) have tabulated different adsorption isotherms of which the following are useful in expressing x as a function of pressure, and eq 17 takes the following forms. case (i) Henry's form: x = k H P
82
Ind. Eng. Chem. Process Des. Dev., Vol. 23,No. 1, 1984
where P = (P, + P3)/2,AP = P2- P3, and
2ooo7cR62s),where k is the constant of adsorption isotherm (and should be suitably subscripted with H, F, or L to represent the adsorption isotherms, namely, Henry, Freundlich or Langmuir type) and A , (= 2/n or kL depending on whether the Freundlich or Langmuir adsorption isotherm is used) can be obtained by a nonlinear regression analysis. In the absence of adsorption data, the best form of adsorption equilibria is obtainable only by trial and error by attempting all three possible cases of adsorption independently to assess which one gives the best fit of experimental data. In this communication, as a first approximation, the Henry's form is applied for the gas-membrane equilibrium, and the corresponding relationship for the regression analysis is
RTpapp
A,' =
kH2
20OOTC$2
case (ii) Freundlich's isotherm: x = kFP'/"
AG = AI(G111 + G212
14 + G3131 + A,-P
I,
where
case (iii) Langmuir's isotherm:
x =
kLP
~
1 + kLP
1 + A3P2 + log 1 + A3P3
1
(174 where
Equations 17a to 17c represent surface flux as influenced by different adsorption isotherms. Total Gas Flow as a Result of Contributions by All the Mechanisms. The total gas flow, Qt, is given by the sum of gas phase flow, Qg,and surface flow, Q,. For the case of Henry's isotherm, Q, involves only one constant, A;, and the Freundlich and Langmuir isotherms involve A i and A S . Thus, a general expression for the total gas flow through an asymmetric porous membrane, Qt, can be written as Qt
Nt
= $GIIl
=
Qg
+ Q,
+ Gz12+ G313)AP+ A , /Iz4 f ( P ,A3...)
(18)
Now, we can define a gas permeability coefficient, AG, as the amount of gas permeating per second per unit area per unit pressure difference. From eq 18, denoting the membrane area as S, we get Qt
AG=:=SAP
where A2 = A,'/S. Equation 19 gives an expression for AG, and it has a fundamental mechanistic basis to be useful. From a set of experimental pressure vs: AG data, the average pore size on the membrane surface, R, its standard deviation, u, and the constants A , (= N t / S 4 ) and A , (= R T p a p p k z /
(20)
Significance of Regression Constants and Quantities Pertaining to Membrane-Gas Interactions. The permeability coefficient, A,, as represented in eq 20, is shown to depend on A,, A,, R, and u at any given pressure, P or (P,). As A l equals Nt/S-6,it should remain constant for a given membrane, irrespective of the gas. A2, on the other hand, should depend on the adsorption equilibria as well as the mobility of the sorbed species. As A2 depends on papp, r , s, and 6 , it is also dependent on the membrane pore structure. By being dependent on kH and CR, A2 is related to the membrane-gas equilibria and mobility of the adsorbed phase, respectively. However, if different gases do not affect the pore structure, a quantity, CPI, can be defined such that it is free from the effects of membrane morphology. Thus CP, is difined as the ratio of A , for gas i to that of a reference gas (nitrogen gas is arbitrarily chosen as the reference) and is called the relative surface transport coefficient.
a,=-=(A,), ( - 4 2 ) ~ ~
(cR)NZ(
(CRA
(kH)l
j2
(21)
( ~ H ) N ~
Thus, the quantity, 9,can be assumed to be a constant and independent of membrane pore structure for a given gas in a given membrane material. It is necessary to make a distinction between the effective mean pore radius, R , obtainable from the regresskn and the real mean pore radius (in the geometric sense), R*. With such a distinction in mind, the surface transport can be imagined to take place in the annular space between R* and R . Thus, R can be expected to vary from one gas to the other, depending upon the combined effect of gasmembrane interaction and the mobility of adsorbed gas molecules. On the basis of the above concept, a quantity called the radius correction factor, Al, is defined as A, = R, - R N ~
(22)
and is assumed to be a constant for a given gas in a given membrane material, applicable as a pressure-average quantity for the pressure range in which it is determined. On the other hand, the standard deviation, u, with respect to the effective mean pore radius remains constant. From the experimental data, namely, AG vs. P2 for different gases including that of nitrogen, by regression analysis, (A,),,R, (and other regression constants) can be obtained to generate values of a, and A, for different gases in different membranes. Experimental Section Membranes, approximately 0.01 cm (ascast) thick, were prepared from cellulose acetate (CA), hydrolyzed cellulose
Ind. Eng. Chem. Process Des. Dev., Vol. 23, No. 1, 1984 83 Table I. Experimental Conditions Used in the Preparation of Dry Membranes for Use in Gas Permeation Studies membr no.
compn of casting soln, wt %
casting cond.
evap time
gelation
shrinkage drying" temp, "C method _____
_______l___l_
CA-1 CA-2 CA-4 CA- 5 CA- 3 HCAP-1 HCAP-2 HCAP-3 HCAP-4 HCAP-5 PS-1 PS-2
ps-3} PS-4 PS- 5
17% CA; 69.2% acetone; 1.45% Mg(ClO,),; 12.35% H,O ( a t 10 "C)
Cellulose Acetate Membranes cast a t 30 "C 60 s at a relative 60 s humidity of 55% 60 s 60 s b 30 s
ice-cold water a t 0-2 "C for 1 h
Hydrolyzed Cellulose Acetate Propionate Membranes b 60 s ice-cold water b 60s a t 0-2"Cfor 1 h 17% HCAP; 69.2% acetone 3.4% Mg(ClO,),; 10.4% b 60 s b 60 s H,O (at 25 "C) b 60 s Polysulfone Membranes 12% Victrex (300-P) PS; 84% CH,Cl,; 21 4% polyvinyl pyrrolidone 18% Victrex (200-P) PS; 82% CH,Cl, 21 21 13% Victrex (300-P); 37.4% CH,Cl,; 46.1% CHCl,; 3.5% CH,OH 21 21 20% U d e l ( l 7 0 0 ) PS; 80% N-methyl pyrrolidone
82 unshrunk 85 75 82
FD FD FD EG
M
70 80 unshrunk 70 unshrunk
FD FD FD FD FD
15s
methanol at 1 9 "C
unshrunk
AD
24 h 15 s
no gelation methanol at 19 "C
unshrunk unshrunk
AD AD
15s 5s
methanol at 1 9 "C 25% acetone in water a t 20 "C
unshrunk unshrunk
AD AD
a FD, freeze drying; AD, dried in air; EG, solvent exchange with ethylene glycol and air drying after leaching ethylene glycol by methanol; M, solvent exchange with methanol and subsequently dired in air. Cast a t room temperature ( - 20-22 "C) and relative humidity of about 66%.
acetate propionate (HCAP), and polysulfone (PSI materials. CA(E398-3) and HCAP(CAP-063) were obtained from Eastman Kodak Co. In the case of polysulfone three types of polymers were used, namely, Victrex (300-P), Victrex (200-P),and Udel(1700) of which the first two were supplied by Imperial Chemical Industries and the last by Union Carbide Co. The conditions of preparation of all membranes are presented in Table I. All CA and HCAP membranes except CA-3 were prepared by a procedure described previously (Pageau and Sourirajan, 1972). For the preparation of CA-3 membrane, the solution composition used was the same as that suggested in the literature (Manjikian et al., 1965). To obtain different porosities, CA and HCAP membranes were shrunk at different temperatures, and in the case of PS, different porosities were obtained by varying the composition of the casting solution and conditions of membrane making. It was necessary to dry all the membranes free of solvents before carrying out permeation experiments. All PS membranes were dried in air. In the case of CA membranes, the membranes were either freeze-dried or water was first removed by solvent exchange and then the solvent was dried in air. CA-3 membrane was equilibrated in successive water-methanol mixtures and finally methanol was removed by drying in air. CA-5 membrane was similarly treated with successive ethylene glycol-water mixtures and ethylene glycol was leached by methanol before drying in air. The effect of different solvent exchange procedures is yet to be established. However, the mechanical stability of the membranes obtained by different solvent exchange procedure seems to be affected. All HCAP membranes were freeze-dried before permeation studies were initiated. The method of freeze drying and the details of the apparatus used for the permeation studies were already reported (Agrawal and Sourirajan, 1970). The effective area of the membranes used in the permeation studies were 10.2 X lo4 m2. The steady-state permeation rates at different operating pressures (recorded as psig, but converted to Pa abs) were measured at 25 f 0.5 "C with a bubble flow meter. All gases were supplied by Union Carbide Gas
Products Ltd. with a purity of more than 99.9%.
Results and Discussion The experimental data, basically collected as volume flow rate for different gases at different pressure gradients, are converted to permeability coefficient, AG (kmol/m2 s Pa). The sets of AG vs. P2 (or P ) for nitrogen gas are used in the regression analysis to obtain RNP, UN , ( A ~ ) Nand ~, (A2)N2 for the given membrane. The results of such a regression analysis with nitrogen permeation data for all the membranes used in this work are shown in Table 11, which provides a complete characterization of all the membranes used from the point of view of pore structure and surface transport of nitrogen. For the purpose of prediction of performance with any gas in a given membrane from nitrogen characterization experiments, it is necessary to obtain data of Ai and for a given combination of a membrane material and a gas. Therefore, Ai and aivalues were generated by similar regression analysis with data collected for different gases in different membranes. From the point of view of model development, A,, which is equal to (Nt/S6),can be regarded as a constant for a given membrane, assuming 6 is independent of the permeating gas. The constancy of Al for different gases for the same membrane was verified for some cases and is shown in Table 111. The maximum deviation in A, for different gases in a given membrane can be expected to be about 10% and in many cases was found to be less than 5%. With the precision of experimental determination of AG (about f2-4%), a consideration of Al as constant to within the specified limit is quite reasonable, and therefore, Az and for different gases were determined by regression with A , the same as that of corresponding nitrogen data. The applicable Ai and values for different gases in different membrane materials are presented in Table IV. Experimental Verification of Prediction of Performance from the Parameters of Characterization. From the above considerations it is obvious that a membrane is characterized by the four quantities (AJN2, RNz,and uN,. Here steps are given to illustrate how eq 20
84
Ind. Eng. Chem. Process Des. Dev., Vol. 23, No. 1, 1984
Table 11. Characterization of All Membranes Used in This Work b y Nitrogen Permeation Data
ax
_~-__-1_-____-____1_______11____l______________l_--
membrane
l o l O ,
m
u
x
lo1', m
A , , m-3
Cellulose Acetate 4.5 3.4 3.0 5.0 1.5
CA-1 CA- 2 CA-3 CA-4 CA-5
30.0 16.0 9.7 13.4 5.5
HCAP-1 HCAP-2 HCAP-3 HCAP-4 HCAP-5
Hydrolyzed Cellulose Acetate 0.45 0.43 13.0 1.20 19.8 4.0 15.2 0.40
PS-1 PS-2 PS-3 PS-4 PS-5
17.0 16.5 20.0 16.2 19.0
Table 111. Results of -~ membrane CA-1
HCAP-1 HCAP-4 HCAP-5
PS-1 PS-2 PS-3
13.0 15.0
Polysulfone 3.6 2.8 3.5 5.5 6.5
4.98 x 3.41 x 6.32 X 1.32 x 1.25 X
A , , kmol/(m3 s p a z )
1.13 x 1.13 X 3.48 X 4.36 x 2.13 x
1017 1019 10"
lozo
loz
Propionate 9.29 x lo1' 5.76 X 10" 5.18 X 10" 2.82 X 10" 1.23 x l o 1 '
3.50 3.23 1.48 1.52 5.49
1.38 x lozo 5.50 X 10" 2.73 x 1019 6.21 x 1019 1.18 X lozo
6.39 X 1.22 x 1.02 x 2.53 x 1.37 x
10-9 lo-'
lo-''
10-9 10-9
x x x x
10-9 10-9 10-9 10-9 x 10-'0 10.' 10-9
lo-'
10-9 10-7
Regression on AG vs. P, Data for Some Gases in Different Membranes Showing Constancy of A gas A , , m-' A z , kmol/(m3 s PaZ) E x l o l o ,m u x l o l o ,m ~
nitrogen hydrogen helium methane argon nitrogen argon nitrogen helium nitrogen hydrogen methane nitrogen helium nitrogen hydrogen nitrogen hydrogen methane
-
-
~
~
_
Cellulose Acetate 30.0 4.5 30.0 4.5 30.0 4.5 29.0 4.5 30.0 4.5
_
_
4.98 4.74 5.09 4.86 5.08
_
_
_ _ _ I I _ _ _ _ -
x 1017 x 1017 x 1017 x 1017
x
1017
Hydrolyzed Cellulose Acetate Propionate 0.45 9.29 X 10" 13.0 0.45 8.77 x lo1' 14.2 4.0 2.82 X 10" 19.8 4.0 2.74 X 10" 17.3 0.40 1.23 X 10" 15.2 12.5 0.40 1.40 X 10" 0.40 1.08 X 10" 14.6 17.0 12.0 16.5 14.5 20.0 17.4 18.5
Polysulfone 3.6 3.6 2.8 2.8 3.5 3.5 3.8
could be used to calculate AG for any gas in the membranes characterized by nitrogen data, at p y operating pressure. Step 1. For any gas i, calculate Riusing the relationship (22) 8, = RN2 + ai
Step 2. For the given membrane and the given gas i, calculate (A2)&using the relationship = @i*(A2)N2 (21) Step 3. Calculate A, 0.05A, and 5 0 X for the given mean pressure, P using eq 2 and the collision diameter values obtainable from Reid and Sherwood (1958). Step 4. Evaluate all the integrals, 11, 12,13,14,and I5 given by eq 10, 11, 12, 15, and 16. Step 5. From the physicochemical properties of the gas evaluate G1,G2, and G3 using eq 9. Step 6. Substitute the values of A I (= ( A 1 ) ~ JR, , a(= aN2),GI,G2,G3,and the integrals, I I to Is,along with (A2), and obtain the value of AG for the given gas at the given pressure, using eq 20. From the nitrogen characterization, the values of AG for different gases in different membranes at several operating pressures were calculated by the procedure described above. These prediction calculations were done for the
1.38 X 1.42 X 5.50 X 5.61 X 2.73 x 2.57 x 3.18 x
10"
lozo lo1' 10" 1019 1019 1019
1.13 x 3.09 x 1.93 x 1.93 x 9.07 x
10-9 10-9 10-9 10-9
3.50 x 2.70 x 1.52 x 5.26 x 5.49 x 6.79 x 1.25 x
10-9 10-9 10-9 10-9 10-1° 10-9 10-9
6.39 X 3.47 x 1.22 x 7.21 x 1.02 x 6.14 X 1.66 X
lo-' 10-7 10-9 10-9
1010"
lo-'
Table IV. Radius Correction Factors, A I , and Relative Surface Transport Coefficients, @ i , for Different Gases in Cellulose Acetate, Hydrolyzed Cellulose Acetate Propionate and Polysulfone Membranes -___ __-hydrolyzed cellulose cellulose acetate propionate polysulfone acetate - A
A
gas ____hydrogen helium methane ethylene argon
x 1010, m 0 0 -0.5 0 0
rf,
_
2.7 1.5 1.7 1.5 0.9
x 10'0, m -3.7 -3.2 -1.0 -1.4 1.3
A
x 1010, @
m
@ __.
10.9 5.0 2.2 1.8 0.8
-2.6 -3.8 -1.5 -3.2 -0.3
6.2 5.5 2.0 1.9 1.7
membranes whose experimental data of AG vs. pressure were not used for the generation of A, and ai.The comparison between experimental and predicted values of AG at different pressures for hydrogen, helium, methane, ethylene, and argon in CA, HCAP, and PS membranes are shown in Figures 1 to 5. The agreement between the predicted and experimental AG values in many cases are well within f10% and in a few cases vary between f10 and
Ind. Eng. Chem. Process Des. Dev., Vol. 23, No. 1, 1984 PERMEATION OF METHANE
PERMEATION OF HYDROGEN A o m oA
85
experimental
e o experimental calculated
A
E
-calculated 12.0 I
A CA-4
t
p
p
I
8
B
0
> 2.0
t
0.6F
0.4 A
0.2
PS-2
/-HCAP-3 0.05A"
0
0.5
1.O
2.0
1.5
PRESSURE, P* x
I
'
lo-*,
'
'
Pa ab8
Figure 3. Plot of permeability coefficient, AG, against operating pressure, Pz, showing comparison between experimental and calculated values for methane through some membranes. Experimental data: (a) PS-5; (A)PS-1; (m) CA-3; (0)CA-4; (e) HCAP-2; (0) HCAP-1. PERMEATION OF ETHYLENE
PERMEATION OF HELIUM A O ne expehental calculated
I
'
Pa abs
Figure 1. Comparison between experimental and calculated permeability coefficient, AG,as a function of applied pressure, P2, for hydrogen through several membranes. Experimental data: ).( CA-3; (A)PS-1; (0)CA-4; (B) CA-5; (A) PS-2; (0)HCAP-1; ( 0 ) HCAP-3.
h
"
PRESSURE, Pz x
25
A m 0 m0
-
5
n a
e*Perl,rm"t.I Calculated
7.0
I
"F
CA4
W
8
0
0.5
1.0 PRESSURE, P2 x
1.5
lo-',
2.0
2.5
Pa abs
Figure 2. Comparison of experimental and calculated permeability coefficient, AG, as a function of applied pressure, Pz,for helium through different membranes. Experimental data: (m) CA-3; (A) PS-1; ( 0 )CA-4; (E) PS-3; (e) HCAP-3.
15%. Considering the assumptions involved in the transport model, the above agreement is deemed satisfactory and practically useful. It is interesting to note that the A G vs. Pz follow different trends. Some examples of the trends are (1)continuous increase (methane in PS-5, HCAP-1, and HCAP-2; argon in HCAP-5), (2) continuous decrease followed by a plateau (hydrogen in CA-3 and CA-4; helium in CA-3; methane in CA-3; ethylene in CA-5; and argon in CA-3), (3) continuous increase approaching a plateau (ethylene in CA-4 and HCAP-l), and (4) existence of minima (hydrogen in PS-2; helium in PS-3). In addition, there are shapes other than those grouped above (methane in PS-1; argon in CA-5). All these trends are faithfully predicted by the model. One
I
@
I
*
.CA-3
7
1.5
6 CA-4
I
'
r I
0.050L PRESSURE, P2 x
lo-',
Pa abs
Figure 4. Plot of permeability coefficient, AG, against operating pressure, Pz, showing comparison of experimental and calculated values for ethylene through different membranes. Experimental data: (A)PS-1; (m) CA-3; (0)CA-4; (at) CA-5; (0) HCAP-1.
of the data sets which show a constant deviation between the experimental and predicted values of AG is the case of helium permeation in PS-1. The agreement in this case is poorer than in other cases, although the predicted trend seems to be quite similar to the experimental trend (shown by dotted line). The difference may be attributable to the inevitable change in the membrane due to continuous use and probable compaction. Some membranes showed noticeable change in performance during the initial runs, to varying degrees, with AG values at corresponding pressures showing differences of up to 20% between,the initial and final runs. To
86
Ind. Eng. Chem. Process Des. Dev., Vol. 23, No. 1, 1984
3o c m @
I
o a
l -
B,oc
PERMEATION OF ARGON
-
experimental calculated
CALCULATED PERMEATION Meihane in PS-5
(0
-..----.--=
--
..e---
,-,-.
0 a
E
1
p
*
oL 1 i
........-.TOT*,
SURFACE
Helium in CA-4
I -
(-.-.SLIP
!5i'I
.............KNUDSEN ..... /.-
2.01
-.
_._,-.
I
w
HCAP-5 -
a I
0
0.5
,
l
1.0
/
/
,
15
/
2.0
,
25
PRESSURE, P2 x lo-', Pa abs
Figure 5. Experimental and calculated values of permeability coefficient, AG, vs. applied pressure, Pz, for argon through several membranes. Experimental data: (L) CA-3; (a)CA-5; ( 0 )PS-3; (0) HCAP-1; ( 0 )HCAP-5.
minimize the uncertainty in using reference nitrogen data for the prediction of performance for any other gas, frequently the membranes were monitored for changes in performance with nitrogen and it was ensured that the performance did not change by more than f 5 % before experiments with any other gas were conducted. In view of such precautions, the deviations between the predicted and experimental A G for helium in PS-1 is difficult to explain and needs further investigation. To sum up, the model is able to predict the permeability coefficient to within f10% over the pressure range studied, and is also able to simulate the experimental AG vs. Pz trends. The permeability coefficients of any gas in different membranes of same material varied over 2 to 3 orders of magnitude and for different gases in any one membrane varied by a factor of 3 to 5. In view of this fact, it is obvious that the model is quite satisfactory and useful. Notwithstanding the apparent success of the prediction technique, it is necessary at this stage to point out briefly further considerations that each assumption is to be given for the improvement of the model. The assumption that the real pore size distribution could be approximated by normal distribution is difficult to verify unless other independent methods of determination of pore size distribution are developed. Attempts could be made to represent the pore size distribution by other more realistic mathematical functions. Unless 6 is determined unequivocally, the assumption of constancy of AI cannot be verified without uncertainty. The mean pore radius, R , as determined by regression, is only a "pressure-averagen value and may be a discontinuous pressure-dependent quantity if multimolecular layers are formed on the membrane surface. Unless the transport data are augmented by sorption data, it is not possible to verify the validity of this assumption. In the development of the model, effects due to pore blocking are totally disregarded as well as the fact that smaller molecules like hydrogen and helium can pass through pores through which a larger reference gas cannot. This would impIy that they can literally "see" a pore size distribution different from the one determined by the regression analysis on the reference gas data. Any further improvement of the model should consider the above aspects in addition to the improvement of the precision of the experiment in order to appraise better the
,71
05
,
I
,
1.0 PRESSURE, p2 x
,
I
1.5
20
io-',
,
25
Pa abs
Figure 6. Calculated values of AG vs. P2and the contribution of different mechanisms to the total AG for methane in PS-5 and helium in CA-4 and PS-3 membranes.
effect each assumption has on the overall analysis. The quantities @fi and A, have yet to be systematically related to the gas-membrane interactions by an appropriate theory of molecular interaction, with relevant experiment data on adsorption of gases by the membrane material. From the foregoing considerations, it can be easily discerned that the transport analysis based on the proposed framework is a useful working model for not only characterizing gas permeation membranes but also for the prediction of permeation data for any gas from the experimental data for a reference gas, namely, nitrogen. The model has already been shown to be able to predict a variety of AG versus P2 trends. Figure 6 shows some typical trends for the cases of helium in CA-4 and PS-3 and methane in PS-5 to illustrate how the interplay of different mechanisms of transport results in different trends. All the curves are calculated from reference nitrogen data and the experimental data are not presented as they are presented in Figures 1and 2. It is to be noted that in none of the membranes reported in this paper is viscous flow present. Conclusions The pore model incorporating Knudsen, slip, viscous, and surface flow mechanisms leads to a useful working transport analysis for interpreting gas permeation data and forms the basis for a predictive procedure. It is able to simulate all observed AG vs. P2 trends. It is expected that once a correlation is developed between the parameters characterizing a given membrane and the casting variables, it will be possible to make membranes meeting any requirement. Acknowledgment The authors are grateful to the N.R.C. Bioenergy Project for supporting this work and Mr. T. A. Tweddle for supplying polysulfone membranes. One of the authors (R.R.) thanks N.R.C. Canada for the Visiting Research Officership and CSIR, New Delhi, India, for the grant of leave of absence from CSMCRI, Bhavnagar.
Ind. Eng. Chem. Process Des. Dev., Vol. 23,No. 1, 1984 87
Nomenclature Al = constant for a given membrane related to the porous structure, m-3 A2 = constant related to surface transport, kmol/(m3 s Pa2) A i = constant related to surface transport, kmol/(m s Pa2) A3 = quantity related to adsorption equilibria A, = pore (or plug) cross-sectional area, m2 AG = gas permeability coefficient, kmol/(m2 s Pa) CR = coefficient of resistance for the transport of adsorbed molecules, kg/(s m2) E = mean speed of the gas molecules, m/s d = collision diameter of gas molecules, m G1, G2, Gq = constants depending on the physicochemical properties of gases given by eq 9 I J J 3 , I J ~ = numerical values of integrals dependent on the porous structure given by eq 10, 11, 12, 15, and 16 k, kH, kF,kL = constants relating to adsorption equilibria, pertaining to Henry's, Freundlich's, and Langmuir's, isotherms, respectively = length of the pore, m = molecular weight of the gas N = Avogadro number N ( R ) = number of pores having a radius, R, m-l N , = amount of gas transported in unit time across the membrane by surface flow mechanism, kmol/s Nt = total number of pores, having a radii from zero to R,, n = constant of Freundlich's adsorption isotherm P = pressure, Pa P2 = pressure (absolute) on the high pressure side of the membrane, Pa. P, = pressure (absolute) on the low pressure side of the membrane, Pa = P2 - P3,pressure differential across the membrane, Pa P = (P2 + P3)/2, mean pressure across the membrane, Pa Qg,QK, Qd, Q,, Q,, Qt = quantity of gas transported in the gas phase, by Knudsen, slip, viscous, and surface flow mechanisms and total quantity of gas transported, respectively, kmol/s q K , qvI qsl = quantity of gas transported through a single capillary by Knudsen, viscous, and slip mechanisms, respectively, kmol/s R = pore radius, m R = mean pore radius, m R,, = pore radius of the largest pore, m R = gas constant S = membrane area, m2 S, = specific surface area of the membrane over which the adsorbed molecules are mobile, m2/kg T = temperature, K x = amount of gas adsorbed in a given amount of membrane meterial, kmol/kg Greek Letters Ai = a constant for a given gas in a given m_embranematerial, called the radius correction factor (= Ri - RNJ, m 6 = equivalent thickness of the membrane, m 7 = coefficient of viscosity of gases, Pa s X = mean free path of gases, m pap,, = apparent density of the membrane, kg/m3 CT = standard deviation for the pore size distribution, m T = tortuosity factor for the pores ai = characteristic parameter called the relative surface transport coefficient (= (A2)i/(A2)N2), related to gas-membrane interaction.
2
Registry No. Hydrogen, 1333-74-0; helium, 7440-59-7; methane, 74-82-8; nitrogen, 7727-37-9;ethylene, 74-85-1; argon, 7440-37-1;cellulose acetate, 9004-35-7;celluloseacetate propionate, 9004-39-1.
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5.;
-. - .
Received for review October 1, 1982 Accepted May 2, 1983
Issued as N.R.C. No. 22746.
