J. Chem. SOC., Faraday Trans. I, 1982, 78, 3581-3593 Influence of Macroscopic Structure on the Gas- and Surface-phase Flow of Dilute Gases in Porous Media BY DAVID NICHOLSON* Department of Chemistry, Imperial College, London SW7 2AY AND JOHN H. PETROPOULOS Department of Chemistry, Democritos Nuclear Research Centre, Aghia Paraskevi, Athens, Greece Received 16th March, 1982 Calculations using model systems are presented which illustrate the effect of macroscopic heterogeneity of the porosity on the gaseous- and surface-phase flow of gas through porous media. It is argued that macroscopic heterogeneity is likely to be present in many systems studied in the laboratory and that its effect on flow properties could be significant. Analogies with previous work on pore structure are drawn.INTRODUCTION In previous paperP4 we considered the effect of pore structure on gaseous Knudsen flow in porous solids both with and without accompanying surface flow. For this purpose, structure factors K , and K , were defined and their behaviour studied in detail theoretically. If D, and D, are the gas-phase and surface diffusion coefficients of the porous medium and D; and Di are the corresponding coefficients for an idealized ‘ standard medium ’, then K , = D,/D; and K , = D,/Di. (1) The standard medium consists of a bundle of identical long uniform cylindrical capillaries oriented in the direction of flow of radius re = 2&/A, where E is the pore volume and A is the specific surface area of the pore walls per unit volume of the real porous medium.If the mean speed of the gas molecules is u and diffuse reflection of these molecules from the solid surfaces is assumed, then (2) In our studies2y4 the real porous medium was simulated by a network consisting of a regular square or cubic array of junctions interconnected by cylindrical capillaries, the radii of which were assigned at random from a distributionflr). D, and D,, and hence K~ and K , , were then calculated as a function off(r) (defined between limits rl, r2), the connectivity (i.e. the number, nT, of capillaries meeting at a typical junction) and the surface to gas-phase flux ratio in a capillary of radius rm = +(rl + rz), namely DZ = &ire = Br,. where k , is the Henry’s law adsorption coefficient. In practice, only K~ can be determined, because there is no expression for DZ comparable to eqn (2) for Di.Thus, model studies, like the aforementioned one 35873588 FLOW OF GASES IN POROUS MEDIA undertaken by US,^^^ are very useful for the purpose of providing guidance for (i) understanding the observed behaviour of K~ and (ii) making informed guesses about K , . The network model suggests2 that I C ~ and K , can be written as I C ~ = K E E ~ and K , = K:R, = K ~ Q (4) where K: = K: is the 'orientation' or 'anisotropy' factor, which has the ideal value of 1 / 3 in an isotropic All other pore structure effects (which in our network model are essentially the radius distribution, the connectivity and the randomness of radius assignment) are therefore included in E~ and E,, which have ideal values of unity, corresponding to a network of uniform radius.It is now becoming increasingly evident that the porous media studied in practice are often non-homogeneous on the macroscopic ~ c a l e . ~ - ~ This is not surprising because the most common objects of study are pellets formed by uniaxial compaction of powders in die^,^-^ which is known to lead to uneven local density of packing (i.e. non-uniform porosity).1° Other methods of pelletization also do not necessarily guarantee macroscopically homogeneous products. It is therefore necessary to examine the effect of macroscopically non-homogeneous structure on both gaseous and surface flow before a realistic theoretical analysis of the flow behaviour of porous media described in the literature can be made. Here we examine the effect of macroscopic heterogeneity, described by simple functions, on the behaviour of icg and K , .DEFINITION AND EVALUATION OF MACROSCOPIC STRUCTURE FACTORS Consider a porous medium in the form of a slab of dimensions I, = I, lg, It or a cylinder of length 1 and radius Zu. A gas concentration difference ACg is applied in the axial direction X , i.e. across the flat surfaces at X = 0 and X = I (the remaining surfaces being blocked). The resulting steady-state permeation flux at any X is J = UPdC,/dX ( 5 ) where dCg/dXis the local gas-phase concentration gradient, Pis the local permeability coefficient and U is the cross-sectional area normal to the direction of flow, given by either Z u l , or npY in the case of the slab or cylinder, respectively. The permeability coefficient is made up of gas-phase and surface components, namely (6) where A,, is the specific surface area per unit volume of the solid material making up the porous medium and e, K~ and K , are local value^.^ Experimentally an integral permeability coefficient is determined from the integrated P = Pg+P, = EDg+AksD, = ~ K ~ B E ~ / A , , ( ~ - E ) ~ A , ~ , ( ~ - E ) K , D ~ form of eqn ( 5 ) F = JI/UACg (7) where p is identical with P only when the porous medium is macroscopically homogeneous.In macroscopically non-homogeneous media, E and/or 7cg and K , , and hence P, vary locally and P is some average value, which can also be analysed into gas-phase and surface components, namely where b, and b , a r e integral diffusion coefficients and kg and Z, are analogous toD.NICHOLSON AND J. H. PETROPOULOS 3589 I C ~ and K , but also include the departure from ideal behaviour caused by the inhomogeneous macroscopic structure, E" is the observed overall porosity and A , 5 is considered not to vary appreciably with E, a usually reasonable appr~ximation.~ Pg is conventionally determined from the corresponding helium permeability, Pg, He, as p g = pg, He(MHe/Wi where A4 is the molecular weight of the gas. B g ( ~ g ) and 8, then follow from eqn (8). PROPERTIES OF 2,AND Es We now proceed to consider the properties of the macroscopic structure factors when P is a function of position in the axial or lateral directions, i.e. when P = P(X) or P = P(Y). CASE OF P = P ( x ) Integration of eqn (5) between X = 0 and X = I and comparison with eqn (7) shows that P = [ S,'dx/P(x)]-' where x = X / I .Similarly, Also E" = J:Edx. Hence, the expressions for the observed structure factors are (9) where P and Pg are given by eqn (6). It can be shown quite generally (see Appendix) that, if the macroscopic non- homogeneity of the porous medium is due principally to the local variation of E , whilst K, 1: constant, then R, < K , . If the local variation of icg is also appreciable, the corresponding result is f l tg -= J I C ~ ~ X . 0 The above results may be illustrated by numerical calculations using linear and parabolic E(X) functions E ( X ) = ~ , ( l +ax) E(X) = E,( 1 + ax - ax2) where E , = E (x = 0) and a = constant. These functions should be representative of porosity distributions resulting from one-ended or symmetrical two-ended powder compaction, respectively [cf- fig.96 of ref. (1 O)]. In each case I C ~ = K , = K , = constant was assumed and E , and a were chosen so as to have E" = 0.5 and E,,,/E, = 1.3, 1.6, 2.0 and 2.4 [where is the maximum value of E ( X ) in 0 < x < 11. The results3590 FLOW OF GASES IN POROUS MEDIA obtained under these conditions are practically the same for both functions and indicate a smooth decrease of R,/Ic, below unity as the degree of non-homogeneity, measured by E ~ ~ ~ / E ~ , increases (see fig. 1). The behaviour of CS was studied by analogous computations, using the same e(x) functions, as a function of the surface to gas-phase permeability ratio of the standard medium, namely A: k, DE (1 --E‘)”)22BE2 = 0.01,O.1, 1 and 10. The results obtained are given in fig. 2. Note that I~,/IIc, tends to increase with E ~ ~ ~ / E ~ substantially, especially at low surface to gas-phase permeability ratios. 1.4 1.2 c 1.0 Kg K O - 0.8 0.E 1 I I 0 1.5 2.0 2.5 EITlax/fO FIG. 1 .-Gas-phase structure factors for P = P( Y), (C/M,, > 1) and P = P(X), ( I ~ / M , < 1). Full l k s , linear variation of E . Broken lines, parabolic variation of E . The behaviour of R, and R, for the case when the macroscopic porosity varies along the direction of flow is thus very similar to that of IC, and IC, for the ‘ serial capillary model’,’* as would be expected intuitively. CASE OF P = P(Y) For a porous medium in the form of a slab with P varying along one of the lateral directions Y, eqn (5) is modified to and comparison with eqn (7) shows that = f P(y)dy 0D.NICHOLSON AND J. H. PETROPOULOS where y = Y/lu. Similarly, B ugc2 Pg(y)dy =2 -[--dy. A , 0 1-& Also E"= [:&dy. 3591 1 I I I 1.5 2 .o 2.5 ~m,x/EO FIG. 2.-Surface flow structure factors for P = P( Y). Full lines and broken lines are for linear and parabolic porosity functions, respectively, as in fig. 1. Hence the observed structure factors are given by u g = - - ~ f i g - FgAo(l-&) = - f L d y 1-E" K c 2 DE 2BE2 2 0 1 - & Again, in the case of tg it can be proved generally (see Appendix) that, if Eg N constant and the local variation of E is primarily responsible for the micro- scopic non-homogeneity of the porous medium, & > K ~ . If there is also appreciable local variation of icg, the corresponding result is dy -l 'g ' lo a1 * For a cylindrical medium, eqn (1 3) must be rewritten as J = -J:'2nYdypdCg = U B 2yPdy.dX3592 FLOW OF GASES I N POROUS MEDIA Introducing o = y 2 and comparing with eqn (7) we find which is exactly analogous to eqn (14). The expressions for p,, E, Rg and Rs are similarly analogous to eqn (15)-(18) and the result of eqn (19) is again obtained in the form The above results are illustrated by numerical examples, given in fig. 1, in which the same functions as before, namely e(w) = E,( 1 + am) e(o) = &,( 1 + ao - ad) were employed with the same values of ern ax/^, and E and with o = y or o = y 2 representing a slab or cylinder, respectively. As expected, f g / ~ , increases smoothly above unity with E,,,/E,.The behaviour of Rs is quite simple in this case, as shown by eqn (IS), and should not normally deviate materially from the mean value of K,. Thus, as might be expected, the behaviour of R, and Rs in the case of POI) closely resembles that of K~ and K , for the ‘parallel capillary model’.1y3 CONCLUSION There now exists convincing evidence that many porous media studied in practice are macroscopically non-uniform and, as indicated in the Introduction, some general trends in the pattern of the non-uniformity can be given, although precise quantitative description may be much more difficult to achieve. In this work we have shown, by choosing physically reasonable models for the spatial variation in porosity, how measured gas and surface diffusion coefficients would be affected by macroscopic non-uniformity .Similarly macroscopic heterogeneities may also occur in polymer films, usually cast in the form of a rectangular slab, and although the equations have been derived here in terms of the properties of a porous medium, the trends indicated would also apply in such cases. Note that, according to the curves shown in fig. 1, the effects of radial and axial variations in porosity act in opposite directions, and it is conceivable that in some systems both types of macroscopic heterogeneity could coexist and only very small modifications to the structure factor E~ would be anticipated. Although interpretation of actual data is not feasible until precise information about this property is available, it clearly emerges from this work that macroscopic heterogeneity is likely to be important in practice. APPENDIX The following results are obtained by repeated application of the Schwarz inequality Ibfi(~)~ a dx f f 2 ( x ) . a dx. . . > ( ff,(x)f2(x) a . . . d x y in conjunction with (a) eqn (9) and (10) or (b) eqn (1 6) and (1 7).D. NICHOLSON AND J. H. PETROPOULOS 3593 CASE OF P = P(x) D. Nicholson and J. H. Petropoulos, J . Phys. D, 1968, 1, 1379. D. Nicholson and J. H. Petropoulos, J. Phys. D, 1971, 4, 181. D. Nicholson and J. H. Petropoulos, J. Phys. D, 1973, 6, 1737. D. Nicholson and J. H. Petropoulos, J. Phys. D, 1975, 8, 1430. M. F. L. Johnson and W. E. Stewart, J . Catal., 1965, 4, 248. C. N. Satterfield and S. K. Saraf, Znd. Eng. Chem., Fundam., 1965, 4, 451. ’ J. H. Petropoulos and P. P. Roussis, J . Chem. Phys., 1968,48,4619. * P. P. Roussis and J. H. Petropoulos, J. Chem. Soc., Faraday Trans. 2, 1977, 73, 1025. K. Tsimillis and J. H. Petropoulos, J. Phys. Chem., 1977, 81, 2185. lo e.g. C. G . Goetzel, Treatise on Powder Metallurgy (Interscience, New York, 1949), vol. 1, chap. 8 and 9. (PAPER 2/454)