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).
INDUSTRIAL AND ENGtNEERING CHEMISTRY
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 .
