J. Chem. SOC., Faraday Trans. I , 1986, 82, 2459-2471 Isothermal and Non-isothermal Molecular Gas Transport in Model Non-homogeneous Porous Adsorbents John H. Petropoulos Physical Chemistry Laboratory, Democritos Nuclear Research Centre, Aghia Paraskevi, Athens, Greece The evaluation of isothermal and non-isothermal transport parameters for dilute gases in axially or radially non-homogeneous porous adsorbing diaphragms has been considered on the basis of the molecular transport approach of Nicholson and Petropoulos. The behaviour of model porous adsorbents with realistic axial or radial variation in porosity have then been considered and the salient features of the deviations from homogeneous medium behaviour have been investigated and illustrated with suitable examples. Analogies and differences between the effects of non-uniform porosity and of non-ideal pore structure are discussed.Particular attention has been paid to the effect of axially non-uniform porosity on the evaluation of heats of transport by means of integral-type experiments. The results obtained are of particular significance for the interpretation of experimental results. A common approach to isothermal and non-isothermal molecular gas flow in finely porous adsorbents is based on analysis of the relevant phenomena in idealised single pores.l-' The deviation of the observed transport parameters from the ideal values is then attributed to the complicated pore microstructure of real porous media, such as pore tortuosity, variation in pore size and shape etc., which is embodied in a suitable ' structure factor'.In a recent paper,s we pointed out that such interpretations are bound to be incomplete in the vast majority of cases because they ignore the macroscopic non- homogeneity of the porous solid test samples caused by the method of their preparation (compaction or other pelletization p r o c e ~ s ) . ~ - ~ ~ The fact that macroscopic non- homogeneity can contribute materially to the aforesaid deviations of real isothermal flow parameters from the ideal values was then demonstrated by means of suitable model calculationss and experimental data. l3 More particularly, ref. (8) was chiefly concerned with the effect of axially or radially non-uniform porosity on isothermal non-adsorbable and adsorbable gas molecular flow.For the latter case, the conventional surface flow treatment1-3. 6* was used. In the present paper, a recently developed, more refined and rigorous approach1*? l5 is employed for this purpose. The same methods are then applied to the study of the effect of axially and radially non-uniform porosity on non-isothermal molecular gas flow. Theory We consider steady-state permeation in the axial direction, X , through a porous diaphragm in the form of either a slab of dimensions Ix = 1, lq, I, or a cylinder of length 1 and radius 1,. At any position 0 < X < I in the porous medium, we have: (a) Isothermal flux J = - A , PdC,ldX where C,(X = 0) = c,"; C,(X = I ) = cgp 24592460 (b) Non-isothermal flux Gas Transport in Non-homogeneous Media where Cg(X = 0) = cgo; C,(X = I ) = c,, T(x = 0) = 6; T ( X = I ) = q.In eqn (1) and (2), C, denotes gas concentration in the gas phase and T the temperature (the corresponding boundary values Cgo, C,, and To, & being constant in any one experiment); qc = q,/RT, where qc is the (differential) heat of transport and R is the gas constant; P is the isothermal permeability of the gas; and A , = I, I, (slab) or A , = 711; (cylinder). We restrict ourselves to the Knudsen flow regime (assuming perfectly diffuse reflection of the gas molecules at the pore walls) and the Henry law region of adsorption. Under these conditions, P is independent of C, and, for a homogeneous medium, may be determined directly from an experimental measurement over a suitable concentration interval ACg = Cgo-Cgl by straightforward integration of eqn (1).In the case of a non-homogeneous medium, this operation will yield an effective isothermal permeability coefficient, the value of which will depend on the spatial variation of P in the manner indicated by Nicholson and Petropoulos.8 A convenient way of determining 4, is to impose a temperature difference across the diaphragm AT = To- q and then allow a dynamic equilibrium (JT = 0) to be established through the build-up of a corresponding concentration difference AC, = Cgo - Cgl. Since 4, is generally a function of T, integration of eqn (2) for a homogeneous medium yields an integral heat of transport P = Jl/A, ACg (3) which reduces to 4, only when AT--+ 0 (differential-type experiment). In practice 4, is often determined from a series of integral-type experiments in which and C,, are kept constant and T, is varied (cf., e.g. Ash et al.3).Then If the above procedures are applied to a non-homogeneous medium, the resulting effective differential heats of transport, ;,( T,), may reasonably be expected to depend on the spatial variation of P, qc and dT/dX (in the manner indicated below). The spatial variations of P etc. are, in turn, determined by the macroscopic structural non-homogeneity of the porous medium, or, more specifically, by the variation of the local porosity ( E ) and pore As already mentioned, the effect of pore structure is embodied in the appropriate structure factor K , which is defined as the property of interest of the porous solid relative to that of an idealised single pore of hydraulic radius r h equal to that of the porous medium.A cylindrical reference pore is mostly chosen, but other choices are, of course, possible.lg 6 y l6 The local isothermal permeability of a non-absorbed gas (subscript g) is given by8 l1 Pg = I C ~ BErh = I C ~ Bc2/A0( 1 - E ) where A , is the specific surface area exposed to gas within the porous medium per unit volume of the solid material, which may be taken as essentially constant;ll B involvesJ . H . Petropoulos 246 1 the mean molecular gas speed multiplied by a numerical factor depending on the chosen cross-sectional shape and length to radius ratio of the reference pore1$ l6 and E and K~ are local values. The overall isothermal permeability of the non-homogeneous medium is similarly given by Fg = k, BE2/Ao( 1 - 21) where 21 and t, are the experimental overall porosity and gas-phase diffusion structure factors, respectively.In the context of the theory of isothermal absorbable gas flow of Nicholson and P e t r o p o u l o ~ , ~ ~ ~ l5 the conventional distinction between ‘ gas-phase’ and ‘ surface ’ perme- abilities used in our previous paperE is not very meaningful. Accordingly, the local permeability coefficient is formulated here as (6) P = Pg 4 = P, ~4 de = I C ~ I C ~ BE^^^ / A o ( 1 - E ) (7) where eqn ( 5 ) has been used; de is calculated for the appropriate reference pore according to Nicholson and Petropou1os;l4~ l5 and I C ~ is the relevant structure factor, which is a function16 of pore geometry and of Do = Uo/RT; U,, measures the adsorbability of the gas (it represents the depth of the adsorption potential well at a single solid surface for the given gas).The overall observed value of 4 which characterizes a non-homogeneous medium is then given by (8) = PAo( 1 - qpg BE2. Non-isothermal flow of a non-adsorbed gas is characterized by qc = qcg = f, inde- pendently of porosity and pore structure.17 In the presence of adsorption, however, qc contains an additional pore-structure dependent term.18 Hence, the local heat of transport of interest here may be written qc = + + I C ~ ( ~ , , -f), where qce is calculated for the appropriate reference pore according to Nicholson and P e t r o p o u l o ~ ~ ~ and I C ~ is the corresponding structure factor (which, like ?c4, is also a function of Do).It is worth noting in passing that the gas-phase structure factor may be written as = K , R,, where IC, is an ‘orientation’ or ‘ anisotropy ’ factor (equal to for an isotropic medium) and R, contains contributions from all other pore structural features6 The ideal value of zg, rcCI and K~ is unity. Furthermore, it is obvious that I C ~ = 1 for a non-adsorbed gas. The evaluation and properties of kg for radially and axially non-homogeneous model porous media have been described in our previous paper.E Expressions for the calculation of 4 and tc in such media are given below. We note at the outset that, since qcg is independent of E and of pore structure, we must have GCg = qcg = t. Radially Non-homogeneous Medium Here, eqn (1) and (2) are modified tos J = - A c J o l P s d w dx JT = - A c ~ l P C , ( ~ + q c - - - & ) d w d In T 0 where w = y (slab) or w = y 2 (cylinder).(6)-(8), it is easily shown (cf. Nicholson and PetropoulosE) that Bearing in mind that dC,/dx and dT/dx are independent of y and using eqn (3) and f l / fl2462 Gas Transport in Non-homogeneous Media The corresponding expression for the effective heat of transport is Gc = Jol qc dw/jol Pdw 1 jol Kg K+ K Q [&'/(I -&>I 4e(qce -i) dw = -+ (10) Jo1KgK4[E2/(1 -&)14& and the result of a differential-type experiment, or of a series of integral experiments using eqn (4), should be the same. Axially Non-homogeneous Medium For isothermal flow we have8 6 = i' Pi1 dx/jol where eqn (6)-(8) have been used. 0 In the case of non-isothermal flow, one should bear in mind that the thermoconductivity of the porous solid, G, is also a function of its macroscopic structure (cf.De Vries'O). The thermal flux at any position x is given by d T dx JH = - A , G(x) - = const. Integrating this equation first between limits (x, l), then between (0, l), and dividing, we obtain (12) TW- T = (T,- T ) [1 -~(X)/W)l where F(x) = G(x')-ldx'. iox Application of the method of determining qc described previously then yields wherein T(x) is given by eqn (12). Hence, we get which, for a differential-type experiment (T, -+ q), reduces to It is obvious that eqn (1 3) and (14) will, in general, yield different results. Furthermore, the result of eqn (14) is independent of the direction in which the temperature gradient is applied.Reversal of the direction of the temperature gradient is equivalent to conversion of the functions G(x), qc(x, T,) into their mirror images about the mid-section of the diaphragm. Using asterisks to denote the new functions, we have G*(x) = G( 1 - x), q:(x, T,) =qc(l -x, To), P ( 1 ) = F(l), whence it follows that G: = Gc (cf. Petropoulos et ~ 1 . ' ~ ) . In the case of eqn (13), however, the relation between @(x, T) and qc(x, T) is complicated; furthermore21 P ( x ) = F( 1) - F(x). Hence, in general, <:(To) # ic(q). -J. H. Petropoulos 2463 0.7 0.6 0.5 W IL, 0.4 0.3 0.2 1 I I I I 0 0.2 0.4 0.6 0.8 1 U Fig. 1. Types of spatial variation of porosity examined: (a) c(u) = E" = 0.5(A); (b) ~ ( u ) = ~ ( 0 ) (1 + k , u + k , u2) with k , > O(B, D, F) or k , < O(C, E) and k, = O(B, C , D, E) or k , = -k,(F); emax/emin = l(A), 1.6 (D, E) or 2.4 (B, C, F).The computed d, or 4, in subsequent figures are labelled with the letter denoting the particular ~ ( u ) function used, followed (except for A) by a numerical suffix identifying the different cases considered here, namely: (i) $ or 4, for radially variable porosity with u = y (suffix 1) or u = y2 (suffix 2); (ii) 4, or GP derived from differential-type experiments assuming G = constant, for axially (u s x ) variable porosity (suffix 3); (iii) 4, for axially variable porosity (u = x) derived from differential-type experiments assuming G = G(0)(1 - E ) (suffix 4); (iv) tP for axially variable porosity (u = x) derived from integral-type experiments with G = constant (suffix 5) or G = G(O)(l - E ) (suffix 6).Results and Discussion Computations based on the expressions derived in the previous section were carried out along the lines of our previous study.8 The integrals appearing in eqn (9), (lo), (1 l), (1 3) and (14) were evaluated by Gauss quadrature. It was found more convenient to present the heat of transport results in terms of -4, = 1 -4,. In contrast to our aforementioned earlier study, the evaluation of the local values q5e and -qpe = 1 -qce here is time- consuming. We have shown l5 that computer demands may be drastically reduced, without affecting the salient features of the computed flow behaviour, by modelling pores as two-dimensional slits with a ' triangular ' adsorption potential-energy well. This practice was adopted (and further justified) in our subsequent model studies of pore structure effecW9 and is continued here.More specifically, the local q$e and qpe were computed numerically on the basis of eqn (1 2), (1 3) and (1 5 ) of Petropoulos,ls putting R = &/Aozo(l - E ) , with a pore length to radius ratio equal to 20 and all other specifications unchanged. The axial (u = x) or radial (u = y or u = y,) variation of the porosity was represented, as in our previous studys by (1 5 ) where k , = constant and k, = 0 (lines B-E in fig. 1) or k, = - k , (line F of fig. 1). These functions were chosen in order to represent realistically the axial variation of porosity E(U) = E(O)(l +k, U + k , U 2 )2464 Gas Transport in Non-homogeneous Media 1 0.5 1 - r: - 0 -0.5 I 1 I I I 0 2 4 6 vo = UO/(RT) Fig.2. Examples of computed 4 for radially (Bl, D1, F l ) or axially (B3, D3, F3) variable porosity, in comparison with the corresponding 4, for uniform porosity (A), in the medium to low go range. Labels on curves as defined in fig. 1. in porous adsorbents in the form of pellets produced by one-ended (k, = 0), or symmetrical two-ended (k, = -kl), compaction of a powdered l1 The fabrication c process can also proauce raaiai variation in porosity. ine case or u --= y is appropriate for a porous material made in the form of a sheet: E may increase or decrease from one edge of the sheet to the other (k, = 0), or from the middle of the sheet to the edges (k, = -kl). The porosity of pellets made by compaction of a powder in a cylindrical die (case of u = y 2 ) may also vary significantly from the axis to the peripher~.~.l2 In each case, the computed 4 and 6, are compared with the corresponding Je,Gpe values characteristic of a homogeneous porous medium with E = E, where Z. = Jol e(u)du. (16) Since the aim of the present investigation was to study the effect of non-uniform porosity, ideal pore structure was assumed throughout, thus eliminating I C ~ , I C ~ and ‘c& (where relevant) from the expressions for 4 and 6,. Under these conditions, the behaviour of the homogeneous medium with E = E is given by that of the reference pore, i.e. #e = be(Re), cce = #,,(Re), where Re = rh/zO =-E/A,(l -%)zo. Values of ~ ( 0 ) and kl(k,) were chosen yielding e = 0.5 and different degrees of non-homogeneity, corresponding to E , .J E , ~ ~ = 1.6 or 2.4 (cf. fig. 1). We also used A , z, = 0.5 (corresponding to Re = 2). In the case of axial non-homogeneity, computations were performed for either (i) a linear temperature gradient (G = constant) ; or (ii) a non-linear temperature gradient, assuming for the sake of simplicity that the effective local thermal conductivity of the porous medium is proportional to the local fractional volume of solid, i.e. G(E) = G(E = 0)( 1 - E ) .J . H. Petropoulos I 2465 -'t I I I I I I 5 6 7 8 9 1 0 g o = Uo/(R T ) Fig. 3. Examples of computed 6 as in fig. 2, but in the medium to high Uo region: A (0); Bl (O), Dl (A), F1 (V); €33 (W, D3 (A), F3 (V). Isothermal Flow Examples of the behaviour of $ are given in fig.2 and 3. The most ncticeable features in the lower uo range (fig. 2) are a tendency of the minimum in the 3 us. Uo plot to become - shallower in the case of radially variable porosity (lines B 1, D 1, F I), and to shift to lower Uo values in the case of axially variable porosity (lines B3, D3, F3), by comparison with 4, (line A). In the higher go region, (fig. 3), 3 for axially or radially variable porosity tends to lie above or below $, respectively. These data have been plotted in the form appropriate to the analytical expression found to apply to a good approximation, at sufficiently high Vo and low R, to single model pores of the type used here, namely18 4 = 1 + K exp (auo) = 1 +KO R-" exp (ago) 6 = 1 +I? exp (atTo) (17) where KO x 0.045, rn z 2.1 and a x 0.8.Introduction of eqn (17) into eqn (9) (with 7cg = constant, q = 1) yields (174 where Eqn (1 7a) also applies to the case of E = 2. with I? equal to Re % KO [Aozo( 1 -E)/E]m. Since m z 2 in eqn (18) and (18a) it follows [for proof, see Appendix in ref. (S)] that2466 Gas Transport in Non-homogeneous Media I I I 0 1 2 3 L Go = Uo/(RT) or Uo/(RTo) Fig. 4. Examples of computed 4 and 4, for radially (Bl, B2) or axially (B3, B4, B5) non-uniform porosity, in comparison with the corresponding ge and &,e for uniform porosity (A). Labels on curves as defined in fig. 1. Conditions applicable to B5 as specified in fig. 5. Hence, for values of Do for which eqn (17) is applicable to sufficient approximation in the range emin < E < the effect of radially non-uniform porosity is to shift the In (4- 1) us.Do plot downward without changing its slope. The relevant data of fig. 3 conform to this conclusion reasonably well. Comparable simple expressions cannot be obtained in the case of axially variable porosity. As shown in fig. 3, the respective In (4 - 1) us. Do plots exhibit a fairly extensive nearly linear region of slope appreciably les than a. (A tendency for the slope to increase at very high Do is evident and can be inferred analytically; but this is hardly likely to be relevant from the practical point of view, because of breakdown of the dilute gas assumption inherent in the basic treatment.)l59 l6 All the above effects of radially or axially non-uniform porosity closely resemble the pore-structure effects represented by the parallel (P) or serial (S) capillary models, respectively, investigated by Nicholson and Petropoulos.16 Also, we note that the deviation of d, from 4, revealed by the present calculations depends more strongly on the degree of non-homogeneity (cf.lines D1, D3 with lines B1, B3, respectively) than on the functional form of E(U) (cf. lines BI, B3 with lines F1, F3, respectively). This conforms to the conclusions drawn from our earlier study based on the conventional theory of adsorbable gas flow.* Heats of Transport Examples of the behaviour of the computed &,, in comparison with the corresponding tpe are given in fig. 4-6. Here again, the effect of radial non-uniformity in porosity corresponds to that of P-type pore structure; but axial variation in E can also produce effects which have no counterpart in S-type pore structure, as a result of (i) non-linear temperature gradients, or (ii) derivation of 4, from integral-type experiments.Effect (i) is illustrated by lines B3, B4 (fig. 4 and 5 ) and F3, F4 (fig. 6) for Gp derived from experiments of the differential type and by lines B5, B6 (fig. 5 ) for GP determinedJ . H. Petropoulos 2467 1 1 2 1 L TOIT, Fig. 5. Examples of computed 4, for, radially (Bl,B2) or axially (B3, B4, B5, B6) non- uniform porosity (in compariso_n with gpe for uniform porosity denoted by A) demonstrating the temperature dependence of gp, determined by differential (T, + q; lines Bl-B4) or integral- type (& = constant, &/& % 1-1.6; lines B5, B6) for gases of different adsorbability (U,,).Labels on curves as defined in fig. 1. U,,/R&: (a) 0.5, (b) 2.0, (c) 4.0. from experiments of the integral type. In all cases, the Gp us. O,, or TJT plot, corresponding to the linear temperature gradient, suffers an appreciable downward shift. Effect (ii) is the one more worthy of attention, because it introduces important new features. More specifically ip determined at T, via eqn (4) from an integral-type experiment is not a function of Uo/RT, alone; but depends on both Uo/RT, (or U,/RT) and ZJT, as illustrated in fig. 4 and 5 (lines B5, B6). The deviation from the corresponding &, derived from a differential-type experiment at T, (lines B3, B4, respectively), can be marked and increases with increasing TJT. It is accentuated as the degree of non-homogeneity becomes more marked (cf.lines D3, D5 us. B3, B5 or lines E3, E5 vs C3, C5 in fig. 6). Also, both the magnitude and the direction of the said deviation depend on the functional form of E ( X ) (cf. the line pairs B3, B5; C3, C5; F3, F5; F4, F6 in fig. 6). In particular, substantial positive (lines C3, C5) or negative (lines B3, B5) deviations are produced when the porosity increases in the direction of increasing or decreasing temperature, respectively. In the symmetrical function F, these opposing tendencies very nearly cancel out, leaving only a small net deviation (lines F3, F5 or F4, F6). Bearing in mind that reversal of the direction of the temperature gradient is2468 Gus Transport in Non-homogeneous Media o.8 t 1 1.2 1.b TOIT, Fig.6. Examples of computed &,, as in fig. 5, for axially non-uniform porosity, illustrating the effect of the functional form of E(X) and of the degree of non-homogeneity on the discrepancy between ip determined by differential (B3, C3, D3, E3, F3, F4) and integral-type (B5, C5, D5, E5, F5, F6) experiments, respectively. Labels on curves as defined in fig. 1 ; U,/RT = 4. equivalent to conversion of function B(D) into C(E) or vice versa, whilst F is unaffected (cf. theoretical section), we conclude that the difference between heats of transport determined from differential- and integral-type experiments can be expected to be comparatively large (small) and sensitive (insensitive) to the direction of the temperature gradient, in the case of non-homogeneous porous diaphragms formed by one-ended (two-ended) powder compaction.Relation between 5, and 4 The relation between &, and 6 is of considerable interest. Nicholson and Petropo~losl~ showed that for single pores - 1 dlnq5 4 =--- 2 d 1nT' - 4 = 1-- This means that -i&, 2 according as d4/dT 5 0. Insertion of eqn (19) into eqn (10) or (14) and differentiation of eqn (9) or (1 1) (assuming icg = constant, K~ = 1, K& = 1 in all cases) shows that the above relation is recovered, i.e. in the case of radial non-homogeneity, but not in the case of axial non-homogeneity;J . H . Petropoulos 2469 as expected from the corresponding results for P-type and S-type pore structure, respectively, reported in a recent paper.18 In the said paper it was further shown that the breakdown of eqn (19) in S-type model porous media gives rise to a region in no in which -pp < $ is associated with d4/dT < 0, in line with what is found in the extensive data reported by Ash et aL3 The same arguments can be used in the case of axially variable porosity for tp determined by differential-type experiments with linear temperature gradient.A relevant example is afforded by lines B3 in fig. 4, where -<, < h, d$/dT < 0 in the range Uo z 0.7-1. The present calculations further show that this condition may be realized over a considerably wider go range, under suitable circumstances (function B), if tp is determined by integral-type experiments (cf. line B5 for Uo/R& = 2 in fig. 4 and 5). This result is of particular importance, in view of the fact that the porous diaphragms of Ash et aL3 were constructed by a one-ended powder compaction procedure (cf.Savvakis and Petropoulos12 for a fuller discussion of this procedure) and the relevant heats of transport were determined by means of integral-type experiments. One may, therefore, justifiably point to axially non-uniform porosity as an additional cause (possibly even the principal one) for the -tP < 4, d4/dT < 0 correlations shown by the experimental data in question. Conclusion The first task of the present paper was to develop the formalism required for a reasonably rigorous evaluation of isothermal Knudsen gas permeabilities and heats of transport in axially or radially non-homogeneous porous adsorbents. For this purpose, our previous treatment of isothermal permeability for such porous media8 was combined with our fundamental treatment of isothermal Knudsen gas flow in model pores14* l5 and the requisite heat of transport formalism was similarly developed on the basis of our corresponding fundamental approach to non-isothermal Knudsen flow? Particular attention was paid to the complications introduced by axial non-homogeneity into the determination of heats of transport as a result of (i) non-linear temperature gradients and (ii) the use of integral-type (rather than differential-type) experiments.The most important conclusion in this respect is that, GP determined from integral-type experiments not only differs from that derived from experiments of the differential type, but generally also depends on the direction in which the temperature gradient is applied across the diaphragm.Our second task was to assess realistically the nature and magnitude of the effect of non-uniform porosity on 4 and &,, likely to be encountered in practice. This was done by means of appropriate model calculations. The results for 8 indicate that the dependence on Do remains qualitatively the same as in a homogeneous porous medium, although there may be considerable quantitative differences. The nature of the latter corresponds closely to what was found in our study of pore structure;l6? l8 radially or axially non-uniform porosity being the counterparts of P-type or S-type pore structure, respectively. The above conclusions apply also to &,, with the reservation that the additional effects (i) and (ii) mentioned above must be considered in the case of axial non-homogeneity.Effect (ii) is of particular practical importance, because it destroys the simple relation between the dependence of GP on gas adsorbability and on temperature. It is also noteworthy that the correlation -pp 8 t, dd/dT 5 0 deduced for single pores, remains valid in the case of P-type pore structure or radial non-homogeneity, but breaks down in the case of S-type pore structure or axial non-homogeneity. The realisation that the breakdown of this correlation in the latter case can be considerably exacerbated by effect (ii) is important for the interpretation of this aspect of the data of Ash et aL3 On the other hand, there are other aspects of the aforesaid data (cf.the unusually low values of qp reported for a Graphon diaphragm) which cannot be properly explained at present. This is not surprising, in view of the fact that our model calculations are based2470 Gas Transport in Non-homogeneous Media on highly idealised pore and surface geometries15 and that pore structure and axial and radial non-homogeneity are, in practice, combined in complicated ways. l2 We conclude that the results of the present paper, in conjunction with those of our previous related papers, have yielded fundamental new insights into both isothermal and non-isothermal gas transport in porous adsorbents. It seems likely that further work along these lines will provide answers to some at least of the remaining questions. Thanks are due to Dr D.Nicholson for useful discussions and to Mr J. Petrou for help with the computations. Glossary cross-sectional area of porous diaphragm (m2) specific surface area defined in eqn (5) (m-l) parameter defined in eqn (5) (m s-l) gas concentration in the gas phase (mol mA3) upstream and downstream boundary values of C,, respectively function defined in eqn (1 2) thermal conductivity of porous solid (W m-l K-l) isothermal or non-isothermal flux of gas (mol s-l), respectively constants defined in eqn (1 5 ) thickness of porous diaphragm (m) width (slab) or radius (cylinder) of porous diaphragm (m) breadth of slab (m) gas permeability (m2 s-l) differential heat of transport (J mol-l) defined with reference to gas concen- tration or pressure, respectively integral heat of transport (J mol-l) hydraulic radius (m) dimensionless pore radius (in units of zo) gas constant (J mol-1 K-l) temperature (K) upstream and downstream boundary values of T, respectively variable defined in eqn (1 5) maximum depth of adsorption potential-energy well (J mol-l) variable defined in eqn (1 a), (2a) axial coordinate (m) radial coordinate normalised with respect to ly(O < y < 1) maximum width of triangular adsorption potential-energy well (m) porosity structure factors for Pg, 4 and qc, respectively normalised permeability of adsorbable gas XI, (0 < x < 1) value relating to reference pore or medium parameter pertaining to non-adsorbed gas Superscripts - N parameter normalised with respect to RT overall effective value characterising a non-homogeneous porous medium * parameter pertaining to reversed temperature gradientJ .H. Petropoulos 247 1 References 1 R. M. Barrer, Appl. Muter. Res., 1963, 2, 129. 2 C. N. Satterfield, Mass Transfer in Heterogeneous Catalysis (M.I.T. Press, Cambridge, Massachussetts, 1970). 3 R. Ash, R. M. Barrer, J. H. Clint, R. J. Dolphin and C. L. Murray, Philos. Trans. R. SOC. London, Ser. A, 1973, 275, 255. 4 D. Nicholson and K. S . W. Sing, Specialist Periodical Report, Colloid Science (Royal Society o f Chemistry, London, 1979), vol. 3, p. 1. 5 E. R. Gilliland, R. F. Baddour and H. F. Engel, AIChE J., 1962,8, 530. 6 D. Nicholson and J. H. Petropoulos, J . Phys. D, 1971, 4, 181. 7 D. Nicholson and J. H. Petropoulos, J . Phys. D, 1975, 8, 1430. 8 D. Nicholson and J. H. Petropoulos, J . Chem. Soc., Faraday Trans. I , 1982, 78, 3587. 9 C. G. Goetzel, Treatise on Powder Metallurgy (Interscience, New York, 1949), vol. 1, chap. 8 and 9. 10 C. N. Satterfield and S . K. Saraf, Ind. Eng. Chem. Fundam., 1965, 4, 451. 1 1 J. H. Petropoulos and P. P. Roussis, J . Chem. Phys., 1968, 48, 4619. 12 C. Savvakis and J. H. Petropoulos, J . Phys. Chem., 1982, 86, 5128. 13 C. Savvakis, K. Tsimillis and J. H. Petropoulos, J . Chem. Soc., Faraday Trans. 1, 1982, 78, 3121. 14 D. Nicholson and J. H. Petropoulos, J . Colloid Interface Sci., 1979, 71, 570. 15 D. Nicholson and J. H. Petropoulos, J . Colloid Interface Sci., 198 1, 83, 420. 16 D. Nicholson and J. H. Petropoulos, J. Colloid Interface Sci., 1985, 106, 538. 17 D. Nicholson and J. H. Petropoulos, J . Membr. Sci., 1981, 8, 129. 18 J. H. Petropoulos, to be published. 19 D. Nicholson and J. H . Petropoulos, J. Colloid Interface Sci., 1981, 83, 371. 20 D. A. De Vries, Bull. Inst. Int. Froid, Annexe 1952-1, 1952, 115. 21 J. H. Petropoulos, P. P. Roussis and J . Petrou, J . Colloid Interface Sci., 1977, 62, 114. Paper 5/ 165 1 ; Received 23rd September, 1985