Sept., 1956
GASPERMEATION THROUGH MEMBRANES
1177
GAS PERMEATION THROUGH MEMBRANES DUE TO SIMULTANEOUS DIFFUSION AND CONVECTION BY H. L. FRISCH Departmat of Chemistry, University of Southern Calaj’ornia, Los Angeles 7 , California Heckivsd February I,$,1966
The theory of gas permeation by Pimultaneous diffusion and convection through membranes possessing pores or channels is developed particular y in reference to gas transmission measurements through polymer films. Considerable simplification is obtained when the gas dissolved in the membrane matrix is in e uilibrium across an interface with the gas in the channels (solution equilibrium). This state is generally attained rapidly. Reasurements of the rate of gas transmission, particularly the time lag, under various applied pressure gradients can be shown to yield, besides the value of the diffusion constant, the gas solubility, the specific convection velocity of the gas as well as other parameters, characteristic of the permeation mechanism, information concerning the membrane porosity (and thus something of the structure). The effect of the absence of solution equilibrium is investigated and the magnitude of such an effect is estimated. The temperature dependence of the convective contribution to the permeability coefficient is derived.
and k’ for Darcy flow is generally taken as an empirically determined ~ o n s t a n t . ~These relations have been used to interpret gas permeation through papers, glassine, certain cellulosic membranes, etc.5 as well as other types of membranes.’ Further modifications which have t o be introduced as a result of gas adsorption, capillarity, the effect of the channel orifice, etc., are discussed in the literature.‘Sa The simultaneous transmission of gases through membranes by both diffusion and convection has in the past received little (if any) attention. The present communication is an attempt a t some elementary considerations bearing on the theory of this type of transport. We will find that measurements of the rate of gas transmission through such “porous” membranes can be used to study structural characteristics I f the membrane as well as details of the mechan sm of gas flow. 11. Gas Permeation due to Both Diffusion and Convection.-Consider a membrane possessing small channels whose radius does not fluctuate too widely, so that the gas pressure in each channel does not differ markedly from the average gas pressure p(z, t ) a t a thickness 2 of the membrane. We assume that flow in the perpendicular y and z directions can be neglected (ie., we are dealing with a relatively thin membrane). Let c(x, t) be the concentration of dissolved, diffusing gas and Cg(z, t ) the average total gas concentration a t x in all channels where c, (2,t ) = P ( Z 1 t)/RTs (3 1 where T is the average channel radius, p is the gas pressure, bp/bx the pressure gradient across the with R the gas constant, T the absolute temperature membrane and p the gas density. For larger chan- and g a constant factor which is inversely propor. nels (roughly 0.1 p or larger in diameter for or- tional to the total channel volume and whose dinary diatomic gases a t N.T.P.) or higher pres- magnitude depends on the concentration units sures the flow changes to viscous, compressible employed. We neglect small deviations from ideal capillary flow satisfying Poiseuille’s Law for fairly behavior of the gas. Gas present in the channels straight channels or Darcy’s Law for very tortuous may dissolve in the polymer matrix adjoining a t a channels. The velocity b satisfies rate given by Ki(s)p(zlt). Similarly gas dissolved in polymer may evaporate at the channel surface with a rate given by K i ( e ) c ( z , t ) . The K i ’ s may be functions of both c and Cg, although when the with vg the gas viscosity (in the gaseous phase). concentrations are small we will neglect this where k‘ for Poiseuille flow is given by k’ = r 2 / 8 variation in the partition constants. The internal
I. Introduction In the absence of stress or thermal gradients within a membrane, the transmission of gases or vapors through membranes (polymer or otherwise) takes place as a result of diffusion and/or convection. I n polymer membranes diffusion controlled permeation is apparently the principal mechanism. The gas dissolves at the high pressure side of the membrane, passes through by an activated diffusion process and evaporates at the opposite surface. Above the second-order transition temperature of the polymer composing the membrane the diffusion is Fickian with a concentration dependent diffusion coefficient.2 Membranes containing small channels, cracks or flaws in the long-range structure of the membrane permit the convection of the gas or vapor through such channels. In principle, many types of convective mechanisms are possible.‘ Structural and experimental considerations present some limitations on the variety; for example, the small pressure gradients across the membranes used in studying them prevent the establishment of turbulent flow. When the gas pressure applied to the membrane is small and the channel-radius is small in comparison with the free path of the gas, the gas flow is of the Knudsen type with a convection velocity VI, satisfying3
(1) R. M. Barrer, “Diffusion I n and Through Solids,” Cambridge, 1951, P. 382 ff.. etc. (2) R. J. Kokes and F. A. Long, J . Am. Cham. Soc., 7 6 , 6142 (1953). (3) C. Zwikker, “Physical Properties of Solid Materials,” Interscience Publislicrs, Inc., New York. N. Y., 1954.
(4) 8. Tsakonas and R. Skaluk, “Laminar Flow through Granular Media,” Technical Report No. 2, CU 2-53-ONR-266(10)-CE, Columbia Univ., New York, N. Y.,1953. (5) J. P. Casey, “Pulp and Paper,” Vol. 11, Interscience Publishers Inc., New York, N. Y., 1952, p. 846 ff.
H. L. FRISCH
1178
solubility coefficient, Si = Ki(s'/Ki(e), will equal the external solubility coefficient S, unless the average channel radius r is so small that a correction due to activity effects has to be applied. Applying the principle of conservation of mass to a small volume element (x,x dx) of unit area of the membrane we see that the change of c(z, t ) with time is due to three effects: (a) diffusion, (b) loss due to evaporation, and (c) gain due to solution of gas in the channels, i.e.
+
with D(c) the differential diffusion coefficient which is in turn related to the integral diffusion coefficient by Similarly the change of Cg(z,t) with time is due to: (a) convection of gas in the channels, (b) gain due to evaporation of dissolved gas and (c) loss due to solution of the gas in the polymer, i.e. at
=
4 ( v ( ~ , ) c,) - K,W dx
= K , ( e ) c(z, 1 )
(5)
or
c, ( z , t ) = +c(zJ) where
+
= Ki(e)/Ki(s)RTg = l / S R T s
This type of behavior is generally exhibited by membranes composed of synthetic polymers with only a few notable exceptions.6 For membranes satisfying eq. 5 the transport is found, on adding eq. 4, to satisfy a single modified Smoluchowski equation in the total gas concentration, n(x,t) = c(x,t) C,(x,t) = (1 4) c(z,t)
+
128
S(1
+ +)pB
(7)
An arrangement which frequently is used' is one for which the gas reservoir at x = 1 is maintained at PB = p z and the other membrane surface a t x = 0 is in contact with a vacuum, i.e., p , = 0. This leads to particularly simple formulas and affords no real loss in generality and will henceforth be used wherever eq. 7 applies. We shall consider first membranes for which solution equilibrium applies before discussing the complications which arise when eq. 5 fails. 111. The Steady-state Permeation of Gas through a Membrane in Solution Equilibrium.At steady state (an/& = 0) we find on integrating eq. 6 that the steady-state flux of gas, q, is given by
Two cases immediately arise depending on whether
v is given by eq. 1 or 2. In the case of Knudsen flow, ie., for very small channels, a direct integration of eq. 8 leads to
C , R T ~ + K,(e) c (4))
where v(C,) is the convection velocity given by eq. 1 or 2. Equation 4 together with boundary conditions a t the exterior surfaces of the membrane completely specify the transport process given the concentration of gas free in channels and dissolved in the membrane matrix a t some initial instant. If the process of solution of the gas in the membrane proceeds at a faster rate than either convection or diffusion of the gas, then one may assume that local solution equilibrium has been attained K , ( d C, (x,t)RTg
VOl. 60
which by virtue of eq. 7 becomes
+ kK/RTgI
p l = PZ
where the constant
k K
(9)
is given by
with cs the local velocity of sound and CP and C, the heat capacities at constant pressure and volume respectively. The permeability coefficient, as measured experimentally, is thus found' to be P
= pl/pz =
Ds
+ kK/RTg
(10)
which is independent of the pressure p2 and consists of the usual diffusion dependent term DS and a convective contribution kk/RTg. More important is the case when the convection velocity is given by eq. 2. We shall be concerned with this case in the remainder of this paper and shall make use of the results derived in this section. Integrating eq. 8 and from 0 to 1, we find17on using eq. 2 with y = k'/qg
+
+
+
4). The where = a> ( n l l 4) and 2, = v (n/l ratio of gas carried by diffusion to that carried by convection is I/+. Since solution equilibrium has been attained the boundary conditions are also greatly simplified. They depend of course on the experimental arrangement chosen to study the gas transmission. In particular, if the membrane surfaces are in contact with gas reservoirs maintained a t constant pressures, then the boundary conditions are of the Nernst type; the concentration of gas a t the boundary n = n B is a constant determined by the pressure of the adjoining gas reservoir p~ by the relation ( G ) A. E . Korvezee 6nd E. A . J. Mol, J . Polvmer Sct. 8 , 371 (1947).
or
+
P = ql/Pz = P, DS apz T = +=rS2/2
(12)
Thus hy plotting P versus p2 we expect to find a straight line whose ordinate intercept gives the diffusion dependent term DE while the slope gives the parameter T. While the steady state, which is attained after a sufficiently long wait, allows one to determine P , it does not yield any information concerning the desired values of D, S , +, g and y separately. To obtain these parameters it is necessary to make measurements in the transient portion of (7) M. Swaro, V . T. Stannet, H. L. Frisrh, R . Waaok and N. H. Alex, "The Evaluation of Gas Transmission Rates of Flexilde Parkaging Material," Final Report, Contract DA 44-109-qni-1445, (1954).
THROUGH MEMBRANES GASPERMEATION
Bept. , 1956
the gas transmission process and in particular it will be shown that it suffices to measure the time lag L of attainment of the stationary state in analogy vvith ordinary diffusion controlled permeation. ' IV. The Time Lag in Gas Permeation through a Membrane in Solution Equilibrium.-The exact solution of eq. 6 offers considerable difficulty in view of the non-linearity of the equation. Again two cases can be distinguished according to whether diffusion or convection of gas is the more important mode of transport. In polymer membranes and even glassine membranes of certain types, it is molecular diffusion which is predominant in view of the smallness and rarity of channels or cracks in the membrane. In view of this predominance eq. 6 may be linearized8 but before we carry this out we introduce dimensionless quantities throughout by the substitution, assuming % = D a constant
1179
Following Barrer' the time lag L in the permeation is found by plotting the flow of gas (per unit volume) through the membrane, which is directly experimentally measured, versus the time. This flow is given by
by virtue of eq. 8 and 1Ga. As proaches the straight line.
t-m,
Q(t) ap-
where QBis the steady-state flow (per unit volume) X
y = -; l>>a = __ 9%; 1 20
=
+
tD lY1
(13)
which carries eq. 6 into
and L is the time lag
with u
-+
0
u -* 1 u -+ 0
as y
+
0;
7 > 7
or
0;
> 0;
asy+l; as740
O
