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Diffusion of Ideal in and Porous Solids D. S. SCOTT and F. A. L. DULLIEN University of British Columbia, Vancouver, British Columbia, Canada

The steady state diffusion of gases through capillaries or through the pores of a solid when fusion, bulk flow of the gases is free to the total in the system is constant is considered. It is shown that the ratio of the molar occur, but in the present situation diffusion rate of the lighter to that of the heavier gas must be equal to the square root such bulk flow is effectively prohibited of the ratio of the molecular of the heavier to the lighter. From simple momentum transfer considerations a diffusion equation is derived to describe by the mechanics of flow in the fine the diffusion rate as the nature of the process changes from ordinary mutual diffusion to pores* Therefore the rates Knudsen diffusion. This equation is shown to give good agreement with experimental measure- Of diffusion Of the molecular ments of diffusion rates in porous solids. A structural parameter of the porous solid, the diffusion species may be different, and the ratio, is calculated from the experimental results and compared with the experimental value usual concept Of equimolal counter of the same ratio found from electrical resistance ratios and from flow measurements. diffusion for a constant pressure sys- tem will be true here ody in spedial The process of gas diffusion through exists, or by a two dimensional surface circumstances. In general the Fick‘s capillaries, or through the fine chan- flow if one of the mokcular species is first law form of the diffusion equation nels of porous solids, is of considerable adsorbed to a considerable degree. often used is not correct with respect importance in operations concerned If the radius of the channel is small to a system of fixed coordinates for with chemical reactions in beds of por- relative to the , trans- ordinary diffusion in a small channel ous particles, in the drying of solids, port occurs by Knudsen flow or mo- when the total pressure is constant at etc. In the usual case the porous par- lecular streaming in the presence of both ends of the diffusion path. It is ticle is surrounded by a flowing gas either a total pressure or partial pres- necessary to how what the relative stream on all or much of its surface. sure gradient. In the present case the rates of transport of each gas will be If the pressure drop due to flow around transport mechanism will be assumed if appropriate diffusion equations are a single porous particle is counted as to be either Knudsen or ordinary dif- to be applied. In the following section, negligible, a constant total pressure fusion, depending on the relationship a proof will be offered that these rela- must exist on the exposed faces of the between radius and mean free path. tive rates are equal to the square root particle. Transport of gases into or out It is apparent that a region will exist Qf the inverse of the molecular weight of the interior void spaces of the solid in which the dif€usive process is inter- ratio of the two gases, when the total will then occur by the process of dif- mediate in nature between these two pressure is taken as constant through- fusion. extremes. Qualitatively this region ap- out the diffusing system. This relation- Some aspects of the particular case pears to extend from values of about ship has already been demonstrated of diffusion of a binary gas mixture in 0.1 to 10 T/A. For most gases at ordi- experimentally, (3, 4), but a complete small channels when the total pressure nary and these explanation has not previously been is maintained constant will be con- values correspond to pore radii from given. sidered here. These considerations form 0.005 to 0.5 p, a range of pore sizes a necessary preliminary to a more gen- encountered in a great many com- INDIVIDUAL RATES OF DIFFUSION eral analysis of gas transport in porous mercial solid catalysts. Thus this inter- IN CONSTANT PRESSURE SYSTEMS solids. mediate or transitional diffusive be- Suppose a gas mixture is diffusing In the following discussion it is as- havior is of considerable practical through a fine , and the pres- sumed that counter diffusion is occur- interest. sure and composition are maintained ring along a path of small diameter In fine pores, of the order of magni- at constant steady state values at each relative to its length, for example tude given above, Wheeler (I, 2) and end of the capillary. through a capillary or the pores of a others have shown that appreciable If T >> A, then ordinary diffusion solid. At the ends of the diffusion path total pressure differences are required occurs, and from the simple momen- the gas has a uniform composition, if Poiseuille flow is to be of the same tum transfer theory of Maxwell that is it is well mixed, and both ends importance as a transport mechanism of the path are at the game constant as is diffusive %ow. Therefore, for the total pressure. usual case of a porous particle bathed in a , the pore channel would NATURE OF DIFFUSION IN have to be of the order of 10-p radius The above equation which is not SMALL CHANNELS or greater before any appreciable frac- confined to the constant total pressure Transport of gas through a small tion of the total gas transport could be case cannot be used without some channel may occur in a variety of attributed to forced flow caused by knowledge of the value of NJN,. AI- ways. If the equivalent radius of the small differences in total pressure. In though constant total pressure is usu- channel is large relative to the mean the particular case of a porous solid ally taken to imply that equimolar free path of the molecules, then trans- immersed in a steady state flowing gas counter diffusion exists, the conditions port may take place because of the total pressure differences would described earlier allow a constant total Poiseuille or forced flow if a total usually be small over the length of one pressure to be maintained without the exists, by ordinary particle. necessity for equimolar counter diffu- diffusion if a gradient The situation described above has sion also occurring. some important consequences. In the Tf binary diffusion of gases at con- F. A. L. Dullien is at the University of Okla- homa, Stillwater, Oklahoma. usual physical concept of mutual dif- stant total pressure is considered in Vol. 8, No. 1 A.1.Ch.E. Journal Page 7 73 any individual capillary, then an over- all momentum balance written be- tween the two ends of the capillary shows that, in general, a net shearing stress exists at the wall of the channel. This shearing stress is very small and will not exceed a value of more than a few dynes per squared centimeter in ordinary situations. It is apparent that as the capillary radius decreases, such small shearin must be of de- creasing sign' cance in determining the total transportB rate. Therefore in capil- laries of sufficiently small diameter the net axial momentum transported to the wall per unit per unit time can be set equal to zero in a good approximation. Calculations show that this approximation would be valid, for Fig. 1. Apparatus for measuring diffusion rates in porous solids at example, for all but the coarsest of varying total pressures. porous solids normally encountered in practice. Obviously the above argument can tion that the relative rates of diffusion Therefore if the limitation is im- be extended to any surface element, of two gases are given also by Equa- posed that, effectively, the net axial and Equation (4) applied to the en- tion (4). momentum transported to the wall per tire surface. It can be said then that From these arguments gas-diffusion unit area per unit time is zero, and if a layer in the form of a thin film exists processes at constant total pressure diffuse molecular reflection is assumed, adjacent to the wall such that all can be summarized as follows. In large then these conditions can be written molecules colliding in this layer travel channels equimolal counter diffusion for an arbitrarily small element of wall either back into the interior of the must occur, and as channel radius de- surface as - - capillary or strike the wall. If a second creases, the ratio of diffusion rates of similar thin layer is considered adja- the two species changes and finally cent and parallel to the wall layer, approaches the ratio ( MA/MB)1/2 well then it is also required that there be before the point at which the radius is where U, and U, are the average vel- no net shearing stress between these of the order of a mean free path ocities of gases A and B. Inasmuch as two layers; that is there must be no length. With further decrease in chan- net momentum transfer through any nel radius the ratio of diffusion rates surface element. If the average differ- remains at this value as the transport ences in the axial diffusive transport process changes in nature from ordi- it follows from Equations (2) and (3) velocities of species A and B passing nary to Knudsen diffusion. that through this element of the surface GA uA~A MB from the wall side and from the di- ----=-(%) (4) RATE OF DIFFUSION IN THE GB uB~B rection of the capillary interior are TRANSITION ZONE du, and duB, then the above condition where GA and GB refer to the rate of is met by writing diffusive transport of those molecules If the conditions of diffusion are such that r then the diffusion striking the wall. This equation has - - - 1, process will have a mixed character. also been given by Hoogschagen (3). mn Vn du, = - mB nB 2111 dun (5) Of these molecules which strike the As yet the only expressions which This equation can be rearranged to have been presented for describing wall some have obviously made their give last intermolecular collisions very close the rate of diffusion in this region are to the wall, while others will have either semiempiripal in character, such come from a greater distance. How- as those of Wheeler (1, 2), or are ever the of any significant complex and of limited applicability, such as that of Pollard and Present number of these molecules coming Therefore GA and GB must have the from a distance that is very much same ratio as that in the layer adjacent (5),for self-diffusion. greater than the mean free path is to the wall. Also it is apparent that Apparently,' elementary momentum negligible. Therefore all the molecules Equation (6) will apply to a surface transport theory in gases has not yet striking the wall can be thought of as element on the inner side of the thin been applied to this transition region. coming from a volume element that is cylindrical layer. When one extends In this region the frequency of wall thin in comparison with the radius of the above argument, Equation (4) collisions relative to intermolecular col- the capillary. can be generalized for all the gas in lisions must increase. Therefore the Not all molecules in such a volume the entire cross section of the capil- total pressure drop for gas A in a binary mixture of and can be element strike the wall. However it is lary. It is not required that Gn and GB A B reasonable to suppose that the - have constant values in the direction considered as being due to the sum of cuIes which leave the volume element normal to flow, but only that the ratio two momentum transfer processes, the in other directions after collision have of the two transport rates must always rate of momentum transfer to the wall, the same average diffusive rate as be the same. and the rate of transfer of momentum those which strike the wall. Therefore If the capillary radius is such that arising from unlike molecular colli- Equation (4) should apply to all the r << A, then Knudsen diffusion will sions. molecules that have already collided prevail, and it is evident from the Therefore in a section of capillary in the volume element. well-known Knudsen diffusion equa- tube of length L one can write

Page 114 A.1.Ch.E. Journal March, 1962 If the quantity Xa = 1 -ayAa + DAB/DKAis introduced, together with a similar definition for X,, then (13) can be written as DAB u, = - (14) XZY where XLz is the logarithmic mean average dehed by Xa - X,, = -Xi = In XJX,

ff (Y4- pz) 1- a pz+ DAB/DKA h( 1 - (I! YAi + DA~JD~A1 (15) In employing the quantity 01 = 1 + (NB)/(NIL),it should be remembered that Nn/NA is positive for concurrent diffusion and negative for counter dif- fusion. The apparent dependence of this mean diffusion coefficient on the composition arises because of the dif- PRESSURE, ATMS ferent functional form of the two Fig. 2. Diffusion of or hydrogen through porous solids in the transition diffusion equations applying in this region: open circle, half open circle, half open circle dark on right side experi- case. mental, curve Equation (9). In the special case of self-diffusion, or if two counter diffusing gases have The above equation must reduce to equal molecular , a = 0. the expression for ordinary diffusion in Equation (8) then integrates to one limit (r>> h) and to the expres- D sion for Knudsen diffusion in the other limit (7 h) For ordinary diffusion << . 1 where the first term on the right-hand at constant pressure in the steady state (16) 1 1 side describes the rate of momentum the integrated Maxwell equation is transfer to the wall, and the second (D,fD,,) term is the Maxwell expression for the Equating this with Equation (12) one rate of momentum transfer between Similarly, for Knudsen diffusion, Equa- sees that unlike colliding molecules. tion ( 11) applies: The expression defining a Knudsen diffusion coefficient for a circular tube can be introduced into the above an expression which has been pre- sented for the transition zone by equation; that is DKA = 2/3 r)d. In For ordinary diffusion DKA >> DAB, addition, since the total pressure is and the ratio DAB/D,A is very small. Bosanquet (6). It was shown by Pol- lard and Present (5) that Equation constant, PA = 1.0 = YA It is apparent that Equations (9) and + pB/P + yn, (17) gives almost the same values as where yA and yD are the respective (10) then become identical. On the mole fractions of A and B. other hand for Knudsen diffusion DgA their more complex expression for When one introduces these relation- <> 1- ayA, as would be expected, iently based on unit geometric cross- then Equations (9) and (11) become sectional area of the solid, and the diffusion path length is taken to be where GA = uAnn, Gn = uBns, and (I! identical. the actual length of the solid. The = 1 + GB/GA. If it is desired to define a single This equation can be readily inte- mean or average diffusion coefficient diffusion coefficients then become ef- grated for the steady state to give an DN for the transition zone by means fective values, which differ from the expression for NA, the total rate of of the usual simple diffusion equation, true values by a geometric factor transport of A, in moles per unit time that is which describes the increased path per unit area: length due to of the pores, and the reduction in cross-sectional N ---- P DAB area because of the limited porosity then Equations (9) and (11) can be of the solid. '-RTL a equated to give (1 - ap,) + DdD, It might be expected that Equation } (9) DAB (9) would not apply precisely to a In{ 1- apI) + DAB/DKA D, = In porous solid because the pores are where y~,and YA. refer to the two ends never of a completely uniform size, of the diffusion path, and cy is now nor are they, in general, circular in 1 + (NB)/(NA)* shape. These factors would influence

Vol. 8, No. 1 Page 115 TABLE1 DifT. Effective Equivalent pore ratio, Test gas. Poros- diff. coeffs..* True Diff. Elec. res. True radius microns Equation Sample A ' ity, % DABP DKA DABP ratio ratio DEA 1 2 122) (See footnotes below.) Selas 015 microporous porcelain Ox-A 61.4 0.072 1.31 0.200 2.78 2.90 3.64 1.17 1.24 2.73 (Selas Corp.) Selas 03-2 Hz-Nz 28.6 0.085 . 0.54 0.763 8.98 8.00 4.85 0.60 0.47 8.78 Celite catalyst support (diato- H2-Ne 56.1 0.0945 0.555 0.763 8.08 8.87 4.49 0.48 0.39 7.70 macwus earth, Johns-Manville co. ) Kaolin-unglazed porcelain (Coors &-N, 34.7 0.053 0.117 0.763 14.4 19.3 1.69 0.16 0.144 13.4 Porcelain Co. ) * All diffusion coelcients at 20°C. in sq. cm./sec. 1. By penetration. 2. Calculated from true value of DXA. Diffusion ratio is ratio of true to effective value of DAB. the coefficient DE, primarily because the value of the ordinary diffusion cury penetration measurement. The of its specific dependence on the pore coefficient was obtained, and the agreement again is good. shape and . However if the value of the Knudsen coefficient was The statement has also been made solid had a fairly uniform pore struc- taken from the low-pressure values. (9) that electrical resistivity ratio in ture, reasonable average values might Inasmuch as the geometric length and a porous solid should be analogous to be obtained for these coefficients. area of the porous solid were used in the ratio of the true ordinary diffusion Obviously, the best values might be these calculations, the value of the coefficient to the effective coefficient. expected to result from the use of ordinary diffusion coefficient so found The resistance ratios were measured experimental diffusion data, rather than was an effective value. The true value for these solids, and the values of the from other less direct measurements of this coefficient was obtained from two ratios are compared in Table 1. (for example premeability tests). the literature. If it is assumed that the Agreement is satisfactory in view of The rates of counter diffusion of geometric parameter given by the the diaculty of measuring the electri- hydrogen and nitrogen, or of oxygen ratio of the true value to the effective cal resistance for fine pored solids. and argon, were measured through a value of the ordinary diffusion coeffi- In another paper (10) the authors number of porous solids. A constant cient also applies to Knudsen diffusion, have shown that it should be possible pressure steady state flow technique then the true and effective Knudsen to calculate an effective Knudsen dif- was used, similar to that employed by diffusion coefficients are also known; fusion coefficient from flow tests in Weisz (7) but with larger cylindrical that is which the specific flow rate is obtained samples and somewhat higher flow as a function of the mean total pres- rates. The apparatus is shown sche- sure. Extrapolation of the straight-line matically in Figure 1. The major plot obtained to zero pressure can be modification in the apparatus consisted The diffusion coefficients obtained taken to give the value of the slip flow of combining the outlet streams from as described above were used to cal- coefficient. On the basis of the flow the two halves of the diffusion cell culate the diffusion rate at intermedi- equation presented by the authors it and removing the combined flow ate pressures. The results are shown in was shown that the ratio of the slope through a manostat and a vacuum Figure 2 for three different types of A to the zero pressure intercept B is system. Analysis of the outlet streams porous solid and for both gas pairs. given by - was accomplished by withdrawing a The solid line represents the calculated A 3r small sample continuously from either values in accordance with Equation -=- (19) stream through a separate pump (not (9), and the points give experimen- B 4.,-&; shown). The sample stream was usu- tally measured values. The properties It was further shown that the limiting ally only 10 to 20% of the total flow. of the three soIids are given in Table Knudsen molecular flow CAP is given With this arangement it was possible 1. The Selas and porcelain (kaolin) by the equation to operate the diffusion cell at any samples had a narrow range of pore' absolute pressure from 1.0 down to sizes, while the celite had a fairly uni- 4BAp about 0.01 atm. Analysis was by means form distribution of pore diameters CAP = -= DE. A, -Ap (20) of a modified thermal-conductivity over a much wider range. x kTL, cell described elsewhere (8). An inspection of the results shows The right-hand side of the above In all tests pure gases were used on that Equation (9) satisfactorily de- equation serves as a definition of D,,, each face of the solid, and outlet con- scribes the rate of diffusion through the effective Knudsen diffusion coeffi- centrations were low. Therefore the these solids. Curves plotted in the cient. Rearranging the above equation total partiaI pressure difference for a fashion shown in Figure 2 should be- one gets diffusing gas was always in the range come asymptotic to a 45-deg. line at of 95 to 99% of the absolute pressure. the low-pressure end and approach a Results were expressed as the rate of constant vaIue in the high-pressure diffusion of hydrogen, or of oxygen, range. The approximate location where depending on the gas pair used. r = k is also shown on each plot. Combining this equation with DxA = Experimentally measured values were It should be possible to calculate an 2/3 pu, and with Equation (19), one substituted in Equation (9) from runs equivalent pore radius for diffusion obtains the result for the diffusion at the highest and lowest pressures from the values of the Knudsen dif- ratio based on Knudsen flow: and the resulting equations solved fusion coefficient obtained. This value Dg.4 1% A TA* (22) simultaneously for values of DKa and is also given in Table 1 and is com------DARP. From the high-pressure results pared with the value obtained by mer- Dg. 9 B' mL, Page 116 A.1.Ch.E. Journal March, 1962 Table 1 includes the values of the dif- ACKNOWLEDGMENT ferred to a fixed set of coor- fusion ratio calculated in this way The financial and technical assistance of - dinate s from flow rate vs. mean total pres- the Petroleum Research Fund of the 0 = average molecular speed sure plots. The agreement is well American Chemical Society, the Research = mole &action Corporation, and of Imperial Oil Ltd. for Y within the probable experimental error. h various aspects of the project of which this = mean free path = gas DISCUSSION is a part are gratefully acknowledged 1) by the authors. AP = pressure differential causing The form of Equations (9) and (13) flow indicate a rather slow convergence to NOTATION Subscripts the asymptotic values at high and low A,,L, = geometric area and length pressures. Therefore a mixed mode of of porous sample A,B = gas A and gas B, respec- dBtlsion does occur to an appreciable DAB = binary mutual diffusion co- tively degree over a wide range of values efficient for gases A and B, of the ratio r/A. On the basis of the sq. cm./sec. LITERATURE CITED results for the porous solids this would D,, = Knudsen diffusion coefficient 1. Wheeler, A., in “Advances in ,” appear to be a somewhat greater for gas A, sq. cm./sec. Vol. 3, Academic Press, New York range than the hundredfold value DN = mean diffusion coefficient ( 1951). given earlier. The convergence of 2. -, in “Catalysis,” Vol. 2, P. H. G 5 molecular transport rate, Emmett, ed., Reinhold, New York Equations (9) or (13) is intermediate molecules/ ( sec. ) (sq. cm. ) between the simple form of Equation (1955). k = 3. Hoogschagen, J., Ind. Eng. Chem., 47, (17) and the more rapidly converging L,x, = length of diffusional path, 906 ( 1955), intuitive equation proposed by Wheeler cm. 4. Scott, D. S., and K. E. Cox, Can. J. (21, g‘wen as rn = mass of a molecule Chem. Eng., 3, 201 (1960). 5. W. G., and R. D. DN = DAB(1 - e-DdB’DKA) (23) M = molecular weight Pollard, Present, Phys. Rev., 73, 762 ( 1948 ) . A more rigorous test of Equation n = molecular , molecuIes/cc. 6. Bosanquet, C. H., quoted by ibid., p. (9) would require experiments of the 770. same kind as those performed on the N = molar transport rate, moles/ 7. Weisz, P. B., 2. Phys. Chem., 11, 1 porous solids but with true capillaries (sec.) (sq. cm.) (1957). of known dimensions. Such work is p = partial pressure 8. Scott, D. S., and A. Y. Han, Anal. currently under way. P = absolute pressure Chem., 33, 160 (1961). However it would appear that the T = radius 9. Klinkenberg, L. J., Bull. Geol. Suc. - Am., 62, 559 ( 1951). diffusion equation presented here gives T = equivalent pore radius 10. Scott, D. S., and F. A. L. Dullien, an adequate description of the transi- R = gas constant A.1.Ch.E. JournaZ, to bc published. tion zone of diffusion for binary gas T = absolute mixtures and can be applied with Zl = diffusional velocity in the Manuscript receiued February 17, 1961; reoi- sion receiued August 22, 1961; paper accepted good accuracy to porous solids. axial direction, cm./sec., re- August 31, 1961. , Thermodynamic Properties of Polar Gases in the Dilute Phase P. T. EUBANK and J. M. SMITH Northwestern University, Evonston, Illinois

A procedure is proposed, based in part on theory and in part on experimental data, for pre- dicting the effect of polarity on the thermodynamic properties of polar, organic gases in the dilute phase. This correlation was used to predict compressibility factors Z, and the change of enthalpy with pressure, in the vapor phase. By use of this correlation the computed compressibilities indicated an average absolute deviation of 1.0% from available experimental data, which includes reduced pressures UP to sent the forces which are present in 0.9 and temDeratures to 1.0. Similar comoarisons were mode for the effect of Dressure on some comDounds but not in others at enthalpy. the same lconditions of reduced pres- Reasonable progress has been made Watson (5) assumed that volume and sure and temperature. These forces are usually due to one in developing prediction methods for energy were the two parameters char- of the following causes: deviation from thermodynamic properties of nonpolar acterizing a gas. Thus the reduced vol- spherical molecular shape, or the de- gases, particularly pure components. ume or the reduced pressure V/V, gree of acentricity; high electrical However progress has been slight for P/P, was used with the reduced tem- forces such as manifested by the polar substances where electrical prop- perature to provide the two vari- T/T, dipole moment and the existence of erties may exert a significant effect. ables which determined the compres- hydrogen bonding; and quantum forces The objective of this paper is to pre- sibility factor: which are important hydrogen and sent a prediction method for classes in Z = W/RT (1) . of organic, polar compounds. Experimental cornpressibiliiy factor This latter effect will not be con- PREVIOUS CORRELATIONS data have shown that Z is not the sidered here, since it is important in The original method of Hougen and same function of the reduced pressure only the compounds mentioned. All and reduced temperature for every compounds are effected by the first P. T. Eubank is at Texas Agricultural and Mechanical University, College Station, Texas, compound. Thus more parameters are effect with the exception of the inert and J. M. Smith is at the University of Califor- nia. Davis, California. needed; these parameters must repre- gases, which are spherical. For this

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