I
C. E. ROGERS, VIVIAN STANNETT, and MICHAEL SZWARC Chemistry Department, New York State College of Forestry a t Syracuse University, Syracuse 10, N. Y.
Permeability Valves Permeability of Gases and Vapors through Composite Membranes
Membranes can b e constructed so that rate of permeation depends on direction of flow and they may b e of value as permeation “valves”
IN
THE STATIONARY STATE the rate of permeation of a gas through a film is g:ven by the familiar equation (stationary flow of a gas or vapor through a membrane) :
R = P X -AP
L
I
r
where R denotes the rate of permeation, Ap is the difference of pressures on both sides of the film, and L its thickness. I n the most general case, P is a function of pressure, as well as of temperature, and in this case one must distinguish between differential 6, defined by the equation:
R = @ X -dP dx
and the integrated P, defined by Equation l . The integrated P is related to the differential 6 by relation 3
unaltered in its magnitude although the direction of the flow is reversed. It would be desirable, however, to find systems exhibiting different permeabilities for different flow directions, since such systems might be used as “valves” in some special applications. The purpose of this article is to describe such a system and to discuss the conditions required for its operation. I n order to understand the problem better, the permeability of a gas or vapor through a membrane composed of two or more parallel films should be discussed, and its permeability should be calculated in terms of permeabilities of the individual films composing the membrane. Let us start by considering a set of films the permeability constants of
f
t
which are pressure independent4.e. P = @-and then extending the discussion to systems composed of films, the permeability constants of which are pressure dependent. Permeation through n Layers of Different Films. Consider n layers of films (Figure l), and denote the thickness of the i’th film by xi, and its permeability constant (assumed to be pressure independent), by P,. The concentrations in various layers are represented by a discontinuous variable, the discontinuity might occur at each interphase surface. T o avoid any discontinuities, denote by pi the pressure of the gas (or the vapor) which would be in equilibrium with the lower layer of z’th film or the upper layer of the i - i’th film by bo, the initial
t
t
4
3 2
I These equations indicate that the rate of permeation of a gas through a film is independent of the directioni.e., if the gradient of pressure is reversed, the rate of permeation remains
Figure 1,
Layers of film denoting thickness and permeability constants VOL. 49, NO. 11
NOVEMBER 1957
1933
pi
0
P
0
P
PI
Pressure dependence of gas transmission through a membrane
Figure 2.
pressure, and by p,&, the final pressure (the latter follows from the definition of p j ) , Since the authors are concerned with a stationary flow, the rate of permeation R through each layer must be constant through the whole film-Le. :
where Apnistands for p 1 - p i + 1. If P denotes the permeability constant of the membrane, then P is defined by the equation :
n
where
&
=
z a p , = PO 1
and
L =
Xi>
1
one derives the following relation between P and P,’s: n
R = P X zAP
Q
TP
This means that the Permeability constant of the composite membrane is given by the harmonic average of properly weighted permeability constants of the individual layers (x,/L being the respective weight). The result deduced above means also that the rate of permeation through a composite membrane is independent of the order in which the individual layers are assembled. Any permutation of these layers does not change either R or the gradient of pressure across any individual layer. In a special case of a membrane composed of two layers, the permeation through a membrane containing film 1 on top and film 2 on the bottom is the same as the permeation through a membrane in which film 2 is on top and film 1 on the bottom Permeation through a Composite Membrane. When P is pressure dependent the result deduced in the preceding section folloivs from the assumption that the permeabiLity constants of individual layers are independent of prrssure-Le., that the system of Equations 4 is linear. However, if the permeability constants are pressure dependent, Equations 4 cease to be linear, and then the permeability of the composite membrane is affected bv the order in which the layers are assembled. This discussion is restricted to a membrane composed of two films, permeability constants of which are pressure dependrnt. This dependence, shown by Figure 2, represents the flow of vapor through a film as a function of pressure. If the permeability constant is pressure independent the curve, shown in Figure 2, would be reduced to a straight line passing through the origin of the coordinates. The deviation of the curve from the straight line indicates an increase in the permeability constant with increasing pressure of the vapor. Having the curve (Figure 2), representing the flow, Q,. of a vapor maintained at pressure p on one side of the film and ai: zero pressure on the other, the flow Q,,,, can be calculated for a vapor maintained at pressure p on one side and at pressure p l on the other. The graphical method of calculation is shown clearly on Figure 2, and its justification is obvious since
Q,
=
;x fP
dP
04
II
I, is the thickness of the film, and
QP
P
O
Figure 3.
1 934
Rate of flow through a composite membrane
INDUSTRIAL A N D ENGINEERING CHEMISTRY
Q
the differential permeability constant. Having obtained curve 2 directly from experimental data, curve 2a can be constructed, representing QpI.p as a function of p l for a constant p . A membrane composed of two films, I and 11, is now considered. Let curve
PERMEABILITY V A L V E S Experimental Procedure Table I. Transmission of Water Vapor at 25' C., Single Films Transmission,
Pressure,
Mm. Hg
Film mplon 6
Qp
23.2 22.5 20.0 18.0 15.1 9.2 5.6
Ethocell 610
X 106
91.7 76.6 38.8 19.8 10.0 3.8 2.1
22.5 20.7 18.1 14.0 9.8 4.5
171 f 2 151 + 2 125 f 2 93.6 zk 2 . 5 61.3 29.4
QF
I
I1 (Figure 3) represent the curve for the second film as a function of p , while curve I represents Q',,, for the first film as a function of p l at constant p . The crossing point of these two curves determines the rate of flow through a membrane composed of both films kept at zero pressure above the second film and at pressure p below the first film (Figure 1). (This construction follows from the fact that in stationary flow QL1 = Q',,,.) If the gradient is reversed, the flow through the membrane would be given by the crossing point of curves Q i and Q&,. From these constructions it follows that a membrane composed of two films, permeability constant of one increasing rapidly with pressure while the permeability constant of the second only slightly (or not at all), will be more permeable if the first film is exposed to the high pressure than if the gradient of pressure is reversed. This conclusion has been verified by using a membrane composed of nylon and Ethocell films. Experimental determinations show that there is reasonable agreement between the calculated and observed rates of permeation.
Table II.
Transmission of Water Vapor
at 25" C. through Two Films in Series
1801-
170b
The experimental method used to measure the rate of gas transmission is essentially similar to that used by Barrer (7, 2 ) . The permeability cell, the equipment, and the method for measuring water vapor permeability have been described in detail in previous articles (3, 4). The transmissions are given in units of cubic centimeters of gas at standard temperature and pressure per second per square centimeter of area. A constant pressure of water vapor was maintained over the film by using a supply of thoroughly degassed liquid water thermostated at the appropriate temperature. The vapor pressure was measured with a Zimmerli gage. The polymer films used were Dow Ethocell (plasticized ethvlcellulose, thickness of 0.254 mm.) and Du Pont Nvlon 6 (polyamide, thickness of 0.1 14 mm.). For each determination the system was allowed to reach a steady-state transmission by continual evacuation of the low pressure side, keeping a constant vapor pressure on the high pressure side. At appropriate intervals, the low pressure side was disconnected from the vacuum line and the increase of pressure determined as a function of time. Re-evacuation and subsequent measure of transmission were repeated until successive measurements yielded constant rates of transmission. With the two films in series it was necessary to pump out the membrane for a long period, in order to evacuate thoroughly the space between the films. Precision of the data was ensured by making several measurements a t each vapor pressure and the result expressed as the mean =t the standard error of the mean. The transmissions of each single film were determined both before and after the measurements on the composite membrane in order to check whether irreversible changes in the individual films, due to crystallization or degradation, did take place. No significant change in the transmission was found in the course of these measurements.
160-
150L
-
h x
140-
'
1 3 0 i
.
t 2 O F
e
N
110-
;;1 0 0 c 63
90-a 6 0 1
z -
1
roc 601-
x
5040-
f
20130-
0
2
4
6 8 10 12 14 16 18 20 WbTZR V A P O R P R E S S U R E imm.HgI
Figure 4. Transmission of vapor a + 250 c.; single films E
e
22
24
water
Permeation through Ethocell 61 0 Permeation through Nylon 6
The predicted and the observed values are listed in Table I1 and the extent of the agreement is shown graphically in Figure 5. The predicted "two-sidedness" of the barrier is found and the experimental results are in reasonable agreement with the theory. The small departures from theory at the high vapor pressures are probably due to experimental error, although all precautions were made to avoid such errors.
Transmission, Pressure, Mm. H g
Film
Q x 10'
Exptl.
80.5 61.0 35.0 29.0 9.1 1.7
610
Ethocell
610 Nylon6
1
Results
Calcd.
19.3 14.8 4.5
30.9 10.4 1.7
22.7 20.5 17.4 14.9 10.2 4.5
26.2 rt 2 . 0 18.1 1.2 12.6 7.9 4.0 1.1
+
35.0 21.5 11.8 8.0 4.0 1.7
The data of Table I and Figure 4 indicate that the transmission of water vapor through the single films are pressure dependent, probably due to the nonlinear increase with pressure in the solubility of water in the hydrophilic films, and to an increase in the diffusion constant due to the plasticizing effect of the sorbed water vapor. From these experimental values, the theoretical transmission through the two films in series have been calculated.
0
2
4
6 8 10 WATER VAPOR
12 14 16 18 2 0 PRESSURE ~ m m . H g )
22
24
Figure 5. Transmission of water vapor at 25' C.; films in series
i 0
Nylon 6, Ethocell 61 0 Ethocell 61 0, Nylon 6
VOL. 49, NO. 1 1
NOVEMBER 1957
1935
Membranes with Composition Continuously Variable Membranes of this type show a behavior similar to that exhibited by membranes composed from different layers. Particularly, the rate of flow through such a membrane might depend on the direction of the flux. Some simple examples of such a membrane are given. Let us consider a membrane for which the rate of flow is given by the equation Q = cu.ds/dx
and a is a constant throughout the whole membrane and p denotes the chemical potential. Let us assume further that the gas or the vapor flowing through the membrane obeys the ideal laws -Le., p = - R T log ;b where p is the pressure of the gas in equilibrium with the particular layer of the membrane. From the preceeding equation it follows thatp changes
linearly through the membrane, fallinge.g., from the highest value pa for x = 0 t o 0 f o r x = 1. Let us now assume that the solubility coefficients of this membrane varies also linearly ivith the depth x according to the equation
+
S = So k . x Hence, the concentration of the gas in the membrane varies with the depth x according to the equation
c
= (So
-t kx).po.(l
- x/lj
and the gradient of concentration is given by the equation dc/dx = po(k
- So/l - (2k/l).x}
Hence, if So/k < 1 the concentration of the gas goes through a minimum value for x = l / a . ( l - Salk). If we write formally that the rate of flow is given by the equation Q = -D.(dc/dx)
CORRESPONDENCE
then, the diffusion coefficient, D,in such a membrane varies from negative values to positive ones, being infinity for x = l/2(1 - So/k). In other words, at some stage the gas flows against the gradient of concentration, although not against the gradient of chemical potential. literature Cited (1) Barrer, R. M., “Diffusion In and Through Solids.“ Cambridge Univ. Press, Cambridge, Eng., 1951. ( 2 ) Barrer, R. M., Skirrow, G. J., J . Polymer Sei. 3, 549 (1948). (3) ETeilman, W.! Tarnmela, V., Meyer, J. A4., Stannett, V., Szwarc, M., ISD.ENG.CHEM.48, 821 (1956). ( 4 ) Meyer, J. A,, Rogers, C. E., Stannett, V., Szwarc, X i . , Tu@i 39, 737 (19561.
RECEIVED for review October 23, 1956 ACCEPTED Fehruary 28, 1957 Investisation supported by the Quartermaster Corps and Technical Association of the Pulp and Paper Industry (TAPPI).
) Vries, D. A . de, Bull. inst. zntern. froid,
Annexe 1952-1. 115-31 (1952).
) Vries, D. A . de, .Mededel. Landbouw-
hogeschoo( Wagenzngen 52, 1 (1952); D.Sc. thesis, Leiden University, h-etherlands. D. A. DE VRIES
Analysis of Porous Thermal Insulating Mate-riaIs
.
SIR: An article on the heat transfer in porous thermal insulating materials was discussed by Topper (70). In the introduction he states that this is “a first step toward the development of means to predict the effective thermal conductivity of a porous material from the properties of its component materials.” He continues: ”Experimental data beyond those now available in the lirerature will be needed to assess the practical usefulness of the proposed relations.” The object of this letter is to draw attention to the fact that the theoretical side of the problem has frequently received attention in scientific literature and that a large number of experimental results has been pitblished (72). The earlier theoretical Tvork (3, 6. 9)dealt with the electrical conductivitv of granular materials. while later contributions were also concerned with the electrical (7, 2, 8. 7 7 , 72) and magnetical (7) permittivities and the thermal conductivity (4, 77, 72) of these media. All these problems-to which can be added the stationary diffusion of gases through porous media-are mathematicallv identical, as they are governed by Laplace’s equation with similar boundary conditions. A few years ago this author ( 7 7 , 72) discussed the thermal conductivity of granular materials, with special reference
1 936
to soils, in the light of the available theoretical and experimental data. A theoretical treatment of the subject and an approximate method for calculating the thermal conductivity of soils resulted from this work. The influence of radiation on the heat transfer, also mentioned by Topper (70), was not discussed as its effect is negligible in soils. A theoretical discussion of this subject has been given by van der Held
(5).
D. A.
DE
VRIES
literature Cited
Burger; H. C.: Physik Z . 20,73 (1915). Eucken, A . , Forsch. Gebiete Ingenieurw. B3, VDI-Forschungsheft No. 353 (lY3L). (1932). Held, E. F. R;I. van der, A$$. Sci. Research 3A, 237 (1952); 4A, 77 (1953). 11953). Maxwell, C., “Treatise on Electricity and Magnetism:” Oxford, 1873. ( 7 ) Ollendorff, F., Arch. Elektrotech. 25, 436 (1931). ( 8 ) Polder, D., Santen, J. H. van, Physica 12, 257 (1946). (9) Raleigh, W. R.,Phil. Mag. [5], 34,481 (1892). (10) Topper, L.,IND. ENG.CHEW47,1377 (1955).
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C.S.I.R.O., DIVISION OF PLANT IXDUSTRY P.O. Box 226, DE>XLIQLW. N.S.W. AUSTRALIA
SIR: In reference to D A. de Vries’ comments, this article was written for technologists dealing with the manufacture and use of insulating materials and apparently has been well received by that group. de Vries’ work contains much information but it is relatively inaccessible to most readers. Also the information cannot be easily utilized for any kind of engineering work. (de Vries says that in his experience agronomists are able to use it intelligently, although their mathematical background is usually weaker than that of engineers.) For my own interest. I made one calculation to compare mv Equation 4 with de Vries’ Table I1 (72). Using his symbols, when €1 E is 0.1, Equation 4 gives € / E , = 0.49, xvhile de Vries’ Table 11 gives E ~ ’ E =~ 0.47 (de Vries states that-differences between the formulas proposed by me and Topper’s formula increase with increasing difference of €1,’~~ from 0.1, in particular for E ~ / E>~ 0.1. Also, the small difference in the example given bv Topper is therefore not at all surprising.) LEONARD TOPPER ENGINEERING CENTER COLUMBIA ~JNIVERSITY Nmv YORK27, N. Y .
