FLOW THROUGH POROUS MEDIA SYMPOSIUM
DF ’I Fusion and Flow
of Gases in Porous
Solids Diffusion and flow fluxes
for gases through porous solids
are discussed
Gordon
52
R . Youngquist
INDUSTRIAL A N D ENGINEERING CHEMISTRY
he prediction and correlation of fluxes for diffusion and flow of gases through porous solids, such as catalysts and adsorbents, are of considerable importance, Reactions catalyzed by porous solids are limited in many cases by the rate of transport of reactant molecules through the catalyst pores to the active catalyst surface. Similarly, the rate of approach to adsorption equilibrium is frequently limited by transport through adsorbent pores. Additional examples are found in cases of noncatalytic reactions where one reactant is located in the porous solid and a second must be transported into the solid. Such reactions may include, for example, the combustion of carbon deposits involved with the regeneration of catalysts and the oxidation or reduction of ores. Mass transport in a porous solid is complex. Several mechanisms of transport, including molecular diffusion, Knudsen diffusion, viscous and slip flow, as well as surface migration, may contribute to the flux of gas through the pores of the solid. I n addition, the pore structure of solids is not well understood, so that the relationship between structure and fluxes is not easily delineated. As a consequence, most models for prediction of gaseous diffusion and flow in porous solids are based on reasonably well developed theories for diffusion and flow in capillaries. These capillary theories have been adapted for use with porous solids through the use of relatively simple structural models for the solid. Most such models contain a t least one adjustable parameter, which must be determined by experiment, and in this sense are not completely predictive. Given the structural complexity of porous solids, and the limited ability to determine the nature of the pore structure, it is quite possible that a completely predictive approach may never be developed. Nevertheless, the models formulated to date, when used with a minimum of experimental data, appear reasonably useful for extrapolation and interpolation of data, and give a t least a qualitative indication of the effects of pore structure. The developments leading to the formulation of various models for the description of mass fluxes in porous materials are reviewed. Experimental methods are discussed briefly, and comparison of the model predictions to experimental results is also described.
T
D I F F U S I O N A N D FLOW IN CAPILLARIES Consider a single cylindrical capillary of radius r and length L through which steady-state transport of components A and B of a binary gas mixture occurs. The gas mixture a t ends of the capillary are maintained at constant pressures PO and PL, and constant compositionsyAoandy,,, respectively, as shown in Figure 1.
T h e mean molecular velocity theory of gases as
is given by the kinetic
flA
and the fraction f of molecules, which undergo diffuse reflections a t the walls, is usually taken to be one. Thus, the Knudsen diffusion coefficient is
Diffusion
When the system pressure is uniform (PO= P,), the mass flux is diffusive in nature and may involve Knudsen diffusion, ordinary molecular diffusion, and surface migration. The latter will only oFcur when the diffusing gases are adsorbed in a mobile layer; here it will be considered negligible. If the radius of the capillary and the gas pressure is such that the mean free path is large compared to the diameter of the pores, the rate of transport of the molecules A and B is governed only by collisions with the capillary wall. This type of transport is usually referred to as Knudsen diffusion, after Knudsen who first investigated this type of behavior for capillaries (75). Qualitatively, Knudsen diffusion appears to dominate for values of r/X less than 0.1. Knudsen showed that the diffusive flux is given by
and is seen to be proportional to the capillary radius r, to T112and inversely proportional to the square root of molecular weight of the diffusing specie. Since the rate of transport is governed by collisions’ with the capillary wall, the gas A diffuses independent of the presence of other species, and Equation 1 applies regardless of the existence of a total pressure gradient. At steady state, Equation 1 may be integrated to give (3) Similarly, for component B,
At constant pressure, Equations 3 and 4 add to give
where
By use of Equation 2, the ratio of fluxes is given by NA/NB
=
-(z) M,
(5)
When the ratio r/X is greater than about 10, ordinary molecular diffusion predominates, and the flux of A may be described by
Figure 1 .
Capillary schematic
The molecular diffusion coefficient D A B is, of course, independent of the size of the capillary. From kinetic VOL. 6 2
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theory, DAB is proportional to T3i2and inversely proportional to the total pressure. Integrating Equation 6 for the steady-state case and at constant total pressure gives (7) where
The transition region between Knudsen diffusion and ordinary molecular diffusion has received considerable attention in recent years, principally because many cases of diffusion in porous solids that are of practical interest lie in this region. Early attempts at describing the diffusion flux in the transition region were more or less intuitive-e.g., Wheeler (33). Bosanquet ( 3 ) )using a random walk argument, suggested that the flux be described by
where
Knudsen diffusion follows the inverse square root molecular weight relationship, Equation 5. This relationship has further been shown (12, 26) to apply to all open systems involving the diffusion of a binary gas at uniform pressure, regardless of the diffusion mechanism-Knudsen, molecular, or transition. Thus, at constant pressure
Combined Diffusion and Flow
In the presence of a pressure gradient (Po i PL) both diffusion and flow contribute to the net flux through the capillary. Evans et al. (8) have shown that the flux relationship given by Equation 10 for pure diffusion is also appropriate to combined diffusion and flow. Account must be taken of the pressurc dependence of the molecular diffusion coefficient, and of the fact that the flux ratio no longer obeys the inverse square root molecular weight relationship. To integrate Equation 10, the variation of pressure with x must be known. The total molar flux is given by the sum of contributions from diffusion and forced flow :
Ni =
Equation 9 is a special case derived from a more general flux relationship. More recently, Rothfeld (79) and Scott and Dullien (24))using a momentum balance, and Spiegler (27))using a friction factor analysis, have shown that the flux in the transition region is more properly described by
-DAB
dC.4 I_
+ (NA + NB~YA
,AT,
+ F = NA + NB + F = AT: + ICT;
(14)
where NA and N B are pure diffusion fluxes given by Equation 11, and F is the forced flow term. Combining Equations 14, 11, and 13 we obtain dC *
For component B,
or I n terms of the overall pressure drop,
When the fluxes are equal, a = 0 and the Bosanquet expression results. Integrating at constant pressure and temperature results in
Na =
D A Bp R TLa
When molecular diffusion dominates, D,, >> DAB and Equations 10 and 11 become identical to Equations 6 and 7, respectively. Similarly, when DAB >> DKA, the equations reduce to Equations 1 and 3. T o use Equation 12, some knowledge of the flux ratio N B / N A is required. At constant pressure the flux ratio for 54
INDUSTRIAL AND ENGINEERING C H E M I S T R Y
AUTHOR Gordon R. Youngquist is a n Associate Professor of Chemical Engineering at the Clarkson College of Technology, Potsdam, N . Y. 73676. This paper was #resented as part of the Symposium on Flow Through Porous Media, Washington D.C., J u n e 9-17 , 1969.
T h e nature of the forced flow term F has been the subject of some discussion. Clearly, where r/X > 1, Poiseuille flow is observed. I n the transition region, r / X = 1, slip flow results. Most treatments of this transition flow region use a modification of the slip flow term, which corrects for the nonzero velocity at the wall when slip occurs. Various methods of establishing the correction term have been proposed. Recently, Scott and Dullien (25) have used modern kinetic theory arguments which show that, for a pure gas, F is given by
-
-
0.4 I
Wakao et al. (30) suggest that for a binary gas,
AV
03
aim
SlNCLAlR RES.
0.2
4- -AUTHORS
so that Equation 18 becomes
0.I 0
Figure 2. Typical pore size distributions for compressed alumina powder [Cunningham and Geankoplis ( 5 ) ]
Equation 17 thus may be written
(1 - a ) N ; + N i =
C dP R T dx
where
Equations 15 and 21 must be integrated simultaneously over the length of the capillary to predict the fluxes for specified end conditions. This may be done by trial and error using numerical integration. However, it appears that little error is introduced if arithmetic averages for the compositions and pressure are used to evaluate the concentration and temperature dependent terms in C. Thus, the assumption of a linear pressure relationship for use with Equation 15 seems quite reasonable.
APPLICATION TO POROUS SOLIDS T h e pore structure of solids is exceedingly complex and for the most part unknown. Gross properties of the
solid are easily measured-e.g., the porosity-but the details of the structure defy direct observation. Indeed, if a packed bed is visualized as a large-scale representation of a porous solid, with the interstices between the packing as pores, one gets some appreciation of the complex structure of the solid. T h e size of pores is in most cases determined by either or both of two techniques : mercury porosimetry (78) for relatively large pores (usually r > 100 A) and low temperature nitrogen adsorption (2) for small pores ( < l o 0 A). Neither method provides the means for determining the shape or length of the pores, and in each case the pores are normally represented as interconnected cylindrical capillaries having some effective radius. While the pore sizes for some solids are more or less uniform, many solids have a fairly broad pore size distribution. T h e data of Figure 2 are typical. In treating diffusion and flow through porous solids, frequent use has been made of the capillary equations describing these phenomena and discussed earlier. Relatively simple structural models for the solids based on capillaries have been proposed. T o account for the shape and orientation of pores, and for the effective length of the pores, an adjustable constant (the “tortuosity”) is often included. Various attempts have been made to deduce theoretically the nature of the tortuosity, but with only limited success, so this constant must be determined experimentally. More detailed structural models have also been suggested, but often these are of such complexity that they become both difficult to validate and to use conveniently. VOL. 6 2
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Parallel Pore Models
Perhaps the simplest model for diffusion and flow in a porous solid pictures transport as occurring through a number of parallel capillaries all of which are the same size, As proposed by Wheeler (33), the model suggests that the flux be given by
where iVAS represents the superficial molar flux (mol/ unit area of solid X time), N A is the molar flux in each capillary of radius r, nar2 is the total pore area per unit solid area, n being the number of pores and r the radius of the pores, and L J L is the ratio of the actual diffusion path length to the straight line distance through the solid. The porosity of the solid, E , equals nm2Le/L,so Equation 23 becomes
The use of Equation 25 is likely to be satisfactory only for solids having a relatively narrow pore size distribution unless ordinary molecular diffusion dominates so that the diffusion flux becomes independent of pore size. The latter is unlikely, since this would be true only for large pores and/or very high pressures, neither of which is probable in most cases of interest. The parallel pore model may be adapted (74) to consider the variation in pore size as follows. A pore size distribution function f ( r ) is defined such that f ( r ) d r is the void fraction of dr. Using Equation pores with radii between r and r 25, the contribution to the flux froin pores in this size range is
+
Assuming constant T and integrating over all possible pore sizes, NAs
Wheeler takes the length ratio L,/L = fi.However, some account must also be taken of the shape and orientation of the pores. Petersen (16)) for example, by examining diffusion in a pore of varying cross section shows that the shape of the pore may markedly reduce the flux from that obtained in an equivalent cylindrical capillary. T h e form o f Equation 24 in many cases has been retained, and the flux expressed as
where T is the tortuosity. According to Wheeler's model, T = (L,/L)2. Obviously, T must include shape factors as well. I n most instances, I'is determined from experimental data because of the lack of detailed information about pore geometry. I n principle, I' could be detcrmined from a single flux measurement using Equation 25 and the diffusion flux hiA predicted by Equation 12 for a single capillary. T o use Equation 12, a single average pore size must be chosen in order to evaluate the Knudsen diffusion coefficient. Given the pore size distribution, the average pore size may be calculated from y
= --
v,
Lv'rdV
(26)
where V represents the pore volume as determined by mercury porosimetry or nitrogen adsorption. Alternatively, an estimate of the average pore size may be obtained from the void volume-surface area ratio :
Equation 14 assumes the pores to be equal cylinders of radius T . 56
I N D U S T R I A L A N D E N G I N E E R I N G CHEMISTRY
=
$ Lm
(23)
f(r)dr
N A ( ~ )
For pure diffusion a t constant temperature and pressure, N,,,, may be replaced by Equation 1 2 to give 1
The integration with respect to r may be performed numerically. If the pore size distribution function f ( r ) may be assumed to consist of one or more sharp peaks (6 functions), the integral may be evaluated directly. Indeed, for a single peak Equation 25 is obtained. Alternatively, the integral of Equation 30 may be evaluated incrementally as indicated by Satterfield and Cadle (27). F
NA
= 7
DAB
max
-
aYAa
+
DKd-
where A€(?) is the void fraction for pores of average size r. Rundom Pore Models
One of the few completely predictive models is that proposed by Wakao and Smith (2.9). These authors noted that many porous solids are pelleted materials prepared by compressing powder particles that are themsclves porous. Thus, the resulting pellet may have a bidisperse pore structure consisting of micropores (pores within the powder particles) and macropores (spaces between the powder particles). They represented the solid, as shown in Figure 3, and noted that diffusion through the solid could occur by three parallel paths :
thus implying that the tortuosity of Equation 25 equals Cunningham the reciprocal of the porosity, T = l / e . a n d Geankoplis (5) have extended the Wakao and Smith treatment to tridispersed pore size distributions. Other Models
I. MACROPORES IN SERIES 2 MICROPORES IN SERIES 3.MACROPORES AND MICROPORES IN SERIES
Figure 3. Random pore model for bidisperse solids [Wakao and Smith ( 2 9 ) ]
(1) through the macropores of average radius f , (2) through the micropores of average radius f i (3) through macropores and micropores in series Associated with the macropores is a void fraction e,, and with the micropores a void fraction e I . T o predict the contributions to the flux due to each path, a sample is pictured as being cut and rejoined a t random. T h e flux area for each mechanism a t the rejoined plane is taken as the probability of lining up macropores with macropores, micropores with micropores, and macropores with micropores. The flux per unit area of pellet is the sum of the three contributions. For diffusion at constant pressure, Wakao and Smith (29) find
Recently, Foster and Butt ( 9 ) have proposed a somewhat more detailed computational model for the prediction of diffusion fluxes in porous solids. The entire pore volume is represented by two approximately conical ducts, shown in Figure 4, which are themselves made up of straight cylindrical capillary segments. One of the ducts is centrally convergent and the other centrally divergent, each being the inverse of the other. T h e exact shape of these ducts is determined from the pore size distribution of the solid by assigning the pore volume corresponding to a given pore radius to capillary segments of that radius in the two ducts. Given the gas composition at opposite ends of a solid sample, the diffusional flux may be calculated by an iterative procedure. By using a n assumed value for the flux, the capillary equation (Equation 12) is rearranged to give the change in composition across each capillary segment. For the j t h segment,
Starting a t one end, Equation 34 is applied consecutively to each capillary segment. Mixing between the two ducts may be allowed a t various points, and the extent to which it is allowed provides an adjustable parameter for data correlation. The computations are carried through the total length of the ducts, and correctness of the assumed flux is determined by comparing the calculated and given end compositions.
L
: CONVERGING DUCT
where
1 DIVERGING DUCT
If either the macropores in the micropores do not pore size distribution exist--i.e., ea = 0 or et = 0-the is monodispersed and Equation 32 reduces to
Figure 4. Converging-diverging pore model [Foster and Butt ( 9 )1 VOL. 6 2
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DTkTY-
FLOWMETER
GAS A
L-
+THERMOS TAT
--.I
1 -”
GAS
B
-
U Figure 5.
THERMAL CONDUCTIVITY CELL
W’icke-Kallenbach dizusion apparatus
EXPERl M E N T A L M E T H O D S Perhaps the most common method presently in use for the study of diffusion and flow in porous solids is that first developed by Wicke and Kallenbach (34) and later modified by M’eisz (31). I n this method, a sample of the test solid is placed in a diffusion cell between gas streams of different composition. For pure diffusion measurements the pressure and temperature are maintained uniform. The gases on either side of the solid are normally pure, although this is not a necessary restriction. A typical apparatus is shown schematically in Figure 5 . Care must be taken in the design of the diffusion cell to ensure that boundary layer effects are not present at the cell ends. This is normally done by using a high gas velocity and contacting the plug ends uniformly with the gas stream. Concentrations of the effluent streams often have been determined by either thermal conductivity measurement or gas chromatography. From the flow rates and compositions of the feed and exit streams, the fluxes may be readily determined. Brief descriptions of apparatus of this type are given in several papers (5, 70,27,37). Although very useful and perhaps the best of available techniques, the Wicke-Kallenbach technique has its disadvantages. Single samples must be used and these must be cylindrical or shaped into cylinders. Since the axial flux is measured, no indication of the effects of the possible anisotropy of the solid or of dead end pores is possible. If the solid has a wide distribution of the pore sizes, most of the measured flux is likely to be due to the large pores. Since small pores contribute greatly to the active surface involved with catalysis or adsorption, fluxes measured in this way may 58
I N D U S T R I A L A N D E N G I N E E R I N G CHEMISTRY
not reveal the importance of the small pores in these applications. Moreover, the temperatures and pressures conveniently used with this type of apparatus are somewhat limited, so measurements at reaction conditions cannot easily be made. An unsteady-state method, which overcomes some of these objections, has been discussed by Davis and Scott (6). Based on an elementary theory of gas chromatography of van Deemter et al. (28), the method allows the determination of an effective diffusion coefficient for particular gas-solid systems. A carrier gas is passed continuously through a packed bed of the particles of interest, and the response of the system to an injected pulse of test gas is measured, much as in gas chromatography, although the gases used here must be nonadsorbing. By measuring the H E T P (height equivalent to a theoretical plate) for the bed as a function of gas velocity, the effective diffusion coefficient may be determined. Davis and Scott give the details. The method appears adaptable to a wide range of operating conditions, gives results reflecting the behavior of a large number of particles, and since it is a transient technique should reflect the presence of small and dead-end pores. However, the theory on which the method is based is only approximate, and some difficulties associated with wall effects and external mass transfer effects arise. Improvements to the chromatography theory as well as experimental results have been given by Schneider and Smith (23). Their results indicate that the chromatographic technique should develop into a very useful experimental tool. An experimental approach that is particularly useful for cases where Knudsen diffusion dominates is the “timelag” method developed by Barrer (7). I n this method, the transient flow of‘ pure gas through a porous plug is observed. The pressure on the upstream side of the plug is increased suddenly, and the downstream pressure maintained constant. The total quantity Q of gas that has passed through the plug is measured as a function of tinie. The transient condition that exists in the medium is described by (35) where D e is the effective diffusivity for Knudseri flow in the medium. Subject to the conditions of the experiment, Q approaches the asymptote for large t
Thus D e may be determined conveniently from the intercept of the asymptote with taxis: (37)
Z is referred to as the “time lag” and corresponds essentially to the time required to reach steady-state flow. Goodknight and Fatt ( 7 7 ) have shown that the magnitude of the time lag depends on the dead-end pore volume
so that values of De determined from the time lag are likely to be somewhat different from those obtained by steady-state methods.
E X P E R I M E N T A L RESULTS A N D COM PAR I SON W I T H TH EORY Although a wealth of experimental data for diffusion and flow of gases in porous solids do not exist, there is sufficient data available for a t least a preliminary evaluation of the models that have been proposed to describe these phenomena. Most diflusion and flow measurements have been made with such gases as helium, nitrogen, hydrogen, and argon. These gases have been chosen partly because of their ready availability, partly because of the high molecular weight ratios between pairs such as helium-nitrogen, and also because surface diffusion effects are often negligible. Experiments have been carried out at pressures as low as a few m m Hg, and less, and as high as 65 atm. The effect of temperature has not been as widely investigated, with most measurements made around room temperature. A reasonable spectra of solids have been tested, ranging from commercial catalysts to materials such as alumina, which have been pelleted especially so as to control porosity and macropore size. Diffusion
Diffusion measurements in the Knudsen region have been reported for some 59 different samples of porous catalysts by Weisz and Schwartz (32). The WickeKallenbach technique was employed, using apparatus described by Weisz (37)) and hydrogen and nitrogen at atmospheric pressure as the test gases. The solids were various silica-alumina, silica-magnesia, and chromiaalumina catalysts with pore sizes usually well below 200 A. Because the mean free path at atmospheric pressure is of the order of 1000 A, Knudsen diffusion was assumed to dominate. Effective diffusivities, defined as
were reported for hydrogen, but no data are given for nitrogen. The data were compared initially to values calculated using the method proposed by Wheeler (33). Knudsen diffusivities were evaluated using an average pore radius defined as
The experimental effective diffusivities were generally found to be somewhat lower than those calculated. From the structure of gel-type solids, Weisz and Schwartz argue that the pore structure is cellular rather than tubular as suggested by Wheeler. They subsequently take the effective diffusivity to be
(39) The porosity appears as 2 because of an area correc-
0
I
I
I
I
I I IIIII
I
I
I I Ill1
IO De (calculated), cm2/sec
100
Figure 6. Comparison of experimental to calculated di$usivities [ Weisz and Schzuartz ( 3 2 ) ]
tion, and also from a random walk argument for progappears as a correction to ress from cell to cell. the diffusion length. D K is calculated using F = 3 V,/S, which represents the pores as spherical cells. With this model, the agreement between experimental and calculated results is excellent, as indicated by Figure 6. The few data points that deviate widely have bimodal structures or high densities. For the bimodal solids, the deviations are possibly caused by the broad pore size distribution, with high experimental values for D E arising from large fluxes through large pores, which are not properly accounted for in the calculated data. The high density solids give low values for De, perhaps because of dead-end pores which contribute to the surface area and hence 9, but not to the flux. Thus it appears that the simple model given by Equation 39 will give good predictions for D Eif the pores are small with a narrow size distribution and if there are few deadend pores. Similar measurements have been made by Johnson and Stewart (74) for pelleted alumina catalysts having somewhat broader pore size distributions. The distributions were either bimodal or trimodal with pore sizes from 20 to 4500 8. Some 26 samples having different macropore and micropore volumes were tested using nitrogen and hydrogen at atmospheric pressure and room temperature. Because of the pore size range, the data obtained were in the transition region. Effective diffusivities for Hz were calculated from Equation 38 and compared with values predicted by using Equation 30 and a tortuosity I' = 3. No diffusivity data are given for Nz and the observed flux ratio is not mentioned. Figure 7 shows the result. I n most cases the experimental diffusivities agree with predicted
4:
VOL. 6 2
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59
I Ill
0.I
1 I IIIII
I 1
E
values within about *20%, although some of the data show considerably greater variation. Satterfield and Cadle (27) made diffusion measurements with hydrogen and nitrogen for seventeen commercial pelleted catalysts and catalyst supports. Measurements were made at atmospheric pressure and 20" to 30°C. With the exception of two methanol synthesis catalysts, the experimental flux ratio for H*/Nz was -3.75 f 0.11, which agrees very well with the theoretical value of -3.73. The flux ratios for the methanol synthesis catalysts were -3.46 and -3.53, and these low values were attributed to surface diffusion of nitrogen. The catalysts generally had wide pore size distributions, so Equation 31 was used to calculate tortuosities. The tortuosities were between 2.8 and 7.3 except for two materials, which were calcined at very high temperatures, that showed abnormally large values (11.1 and 79). These tortuosities agree well with those observed by Johnson and Stewart for similar solids. For comparison, Satterfield and Cadle also predicted the fluxes using the random pore model of Wakao and Smith (29). The agreement between predicted and experimental values was generally not very good, with deviations ranging from 8 to 160%. I n a second study, Satterfield and Cadle (20) made pure diffusion measurements for helium and nitrogen on five commercial catalysts over the pressure range 1 to 65 atm and at ambient temperature. Combined diffusion and flow measurements were also made for a single catalyst under similar conditions. The results of the diffusion studies indicated that surface diffusion of 60
INDUSTRIAL AND ENGINEERING CHEMISTRY
nitrogen was important for four of the five catalysts tested. The observed dependence of the helium flux on pressure was as predicted by Equation 31. At low pressures, the flux increases rapidly and at high pressures, when bulk diffusion dominates, the flux becomes essentially independent of the pressure. With the exception of one catalyst, the nitrogen flux continued to increase with pressures, thus indicating the presence of surface diffusion for Nz. Correspondingly, the flux ratios -"He/NN2 decreased markedly as a function of pressure to values much below the theoretical value of 2.65. For the one catalyst where no surface diffusion was evident, the flux ratio was very close to the theoretical at all pressure levels. Tortuosities were calculated from the diffusion data using Equation 31 and as with the previous Hz-N2 experiments all fall in the range 2.87.5. The tortuosities were largely independent of pressure, so that in principle a single diffusion measurement would suffice for extrapolation. The random pore model of Wakao and Smith (29) was found inadequate for extrapolation purposes, giving poor agreement with experiment and predicting trends with pressure incorrectly. I n contrast to the generally poor agreement between experimental and predicted results from the random pore model obtained by Satterfield and Cadle, several studies that show excellent agreement have been reported. Wakao and Smith (29) did nitrogen-helium counterdiffusion experiments in the pressure range 1-1 2 atmospheres using Boehmite pellets of various densities and
01
1 /@q 00
I
m O4 \
n"
III
IIII
0.01
I
I I111111
I
I l l
0.1 De/Dne (c a Iculat ed)
Figure 8. Comparison of experimental results to random pore mqde! [ Wakao and Smith ( 2 9 )]
I
I I I11111
I
I I I11111
I I11111
I
I I 1 Illll
c
( - 1
t-
9%/
!-
t q /
e 0.01
P
-
I I11111
I
I 1111111
I
I I I I Ill,
1.0 De (calculated), cm2/scc 0.1
Figure 9. Comparison of experimental results to random pore model [Cunningham and Geankoplis ( 5 ) ]
Figure 70. Comparison of experimental results to random pore model [Henry et al. (72)]
porosities. Henry et al. (72) did similar experiments for a much lower pressure range (0.5 to 600 mmHg) using alumina and Vycor, as did Cunningham and Geankoplis (5) using alumina. I n all three studies, good agreement between experimental and predicted results was obtained as is indicated by Figures 8, 9, and 10. T h e flux ratios also agreed well with theoretical values. The test pellets were made from powder in a pellet press, so that the structure of the pellets closely approximates that used as the basis for the random pore model. T h e powder particles were themselves microporous (radii generally less than 100 &. The size of the macropores, which consist largely of spaces between the particles, could be controlled over rather wide limits by the pressure applied in making the pellets. As is intuitively expected, the random pore model predicts that where the number and size of macropores is largei.e., low pellet density and large macropore radiusmost of the diffusion occurs through the macropores with relatively small contributions from micropores and macropores in series with micropores. Cunningham and Geankoplis also used a random pore model for tridispersed pore size distributions to predict fluxes for two samples that had quite broad distributions. Agreement between experimental and predicted values was only slightly better using the tridispersed model rather than the bidispersed model. Foster et al. (70) measured diffusion fluxes for the argon-helium system at pressures from 1 to 14.6 atm and temperatures from 0" to 69°C. Pelleted test samples
from powders of iron and nickel oxide were supported on kieselguhr. The average of 102 observations of the was 3.04 f 0.13 compared to the flux ratio theoretical value of 3.16. Several low values were obtained, especially a t high pressures, indicating that surface diffusion of argon may have occurred. The dependence of the measured fluxes with pressure agrees with model predictions. At low pressures the flux increases rapidly with pressure and a t high pressures the flux becomes independent of the pressure. T h e effect of temperature on the fluxes is also that expected from theoretical predictions. For low pressures and/or small pores where Knudsen diffusion dominates, the flux is expected to be inversely proportional to T1jza t constant pressure. I n large pores and/or high pressures where molecular diffusion dominates, the flux is directly proportional to I n the intermediate region, fluxes will be more or less independent of temperature. These expected trends were confirmed by the experiments. At 1 atm the measured fluxes were essentially independent of temperature, while a t 14.6 atm a modest increase in the fluxes was observed over the range tested. T h e flux data correlated well using the convergent-divergent pore model devised by Foster and Butt (9) and the mixing efficiency as an adjustable parameter. T h e predictive Wakao and Smith (29) random pore model, in contrast, gave very poor agreement. I t was determined that the mixing efficiency could be correlated with the properties of the porous structures tested by a relationship of the form VOL. 6 2 NO. 8 A U G U S T 1 9 7 0
61
where L = particle thickness, d the average pore diameter, E~ the surface porosity taken as 2/3 the volume porosity, q y the mixing efficiency, and P = a constant determined from experiment. For five materials examined, P was constant within k l 4 % . The form of Equation 40 was justified by considering the solid as an Emmenthal structure.
for diffusion and flow. For pure gas flow, average permeabilities were calculated from
K AP ATAS = - RT L
using measured values of the flux. For a single capillary, the permeability for a pure gas is given by
Combined Diffusion and Flow
Very few experimental measurements for combined diffusion and flow have been reported. Such experiments were conducted by Satterfield and Cadle (20) for one of the catalysts noted above. They used three methods to correlate their data: a parallel pore model similar to that used for interpretation of diffusion measurements, a capillary tube model, and a dusty gas model. The parallel pore model is given by Equation 31. The experimental data were fit to the model using an iterative technique after correction of measured fluxes for surface diffusion. Various values of the tortuosity were assumed until values of end compositions determined by numerical integration matched the experimental values. The pressure was assumed to be a linear function of distance through the pellet. Sufficient cross passages between pores to assure uniform pressure and composition at any axial position were assumed to exist. Tortuosities determined in this way were remarkably constant, showing an 8% variation from an average value of 7.1 in excellent agreement with the tortuosity obtained from pore diffusion measurements, which was 6.9. The capillary tube model was not nearly as satisfactory for correlation of the data. Here, the flux is taken as the sum of the pure diffusion flux and the forced flow flux
AT;
= NT $.
F
The diffusion flux ATTwas evaluated using the pure diffusion equation at the mean pressure. The forced flow contribution was evaluated using the developments of Wakao et al. (30). If the same tortuosity factor was applied to both terms, wide variations of T resulted, including several negative values. This deficiency was overcome somewhat by using different values for the diffusion and flow terms, but these results are difficult to reconcile with the results from the parallel pore model above since the two should be equivalent. I n the dusty gas model, fluxes are described by
where CD is the dusty gas constant. Evans et al. (8) neglecting Knudsen and slip flow terms, suggest that C, should be linear with pressure. Satterfield and Cadle show that their data for C, agree well with this prediction over the entire pressure range used. Wakao et a l . (30) obtained data for pure gas flow and combined diffusion and flow for a binary gas system using pellets of Boehmite and Vycor. The data, for nitrogen and hydrogen, were used to test their random pore model 62
INDUSTRIAL A N D ENGINEERING CHEMISTRY
When we use the random pore model, the permeability for the porous solid is
K
=
s2K,
+ E A 11 -+ 3%) Kt
(44)
E,
For comparison with the experimental results, average values were calculated from this relationship using the arithmetic average pressure to evaluate the pressure dependent terms. As Wakao et al. note, little error is introduced by doing this. I n most cases the predicted permeabilities were somewhat higher (by a few yo to about 1OOyo)than experimental values, but showed the same trends with pressure and pressure drop. For combined diffusion and flow an effective flow parameter C , was introduced and evaluated from the experimental data :
(45) Values of C, were predicted from the random pore model using arithmetic average values of y and P to evaluate the macropore, micropore, and series contributions to the flux from Equation 43. These contributions were combined according to the random pore model by
Because of the differences in pore size and porosity, as well as pressure, C,varied by a factor of several thousand. Agreement between experimental and predicted values of C, was quite good and generally better than for the pure gas experiments. Conclusions
No a priori method for predicting diffusion and flow fluxes for gases through porous solids is yet available. However, on the basis of the relatively meager experimental data available, it appears that simple capillary models coupled with elementary knowledge of the pore size distribution and a few flux measurements may be used with reasonable success to extrapolate data. For solids with rather broad pore size distributions, parallel pore models have given good correlations of data obtained under a wide range of pressure conditions. Random pore models in some cases have been used with considerable success in predicting fluxes through pellets made from pressed microporous powders. Whether these methods may be used to predict the
fluxes appropriate to diffusion limited reaction or adsorption conditions remains largely unanswered. For solids with broad pore size distributions, the models predict that most of the flux occurs through the large pores. No effect of dead-end pores is in the models. T h e surface area to volume ratio is greatest for small pores. Thus, the small pores, as well as dead-end pores, which may be of great importance to reaction or adsorption, are largely ignored by the models. I n addition, most experimental flux measurements have been made using low-molecular weight and inert gases a t relatively low pressures and temperatures. Little is known of the applicability of the models for large molecules or a t more extreme conditions. A further point of considerable importance should be made. Depending on the method of manufacture, the structure of porous solids may exhibit considerable heterogeneity that can cause the local diffusivity to vary with position. As Satterfield et a2. and (4, 22) have shown this effect may be quite large with the local diffusivity varying by about a factor of three. This suggests that agreement between theory and experiment in many cases may be a t least partly fortuitous. These facts point up the need for better experimental testing. Ideally, diffusivity or flux measurements obtained from reaction or adsorption rate measurements should be compared with model predictions and/or measurements calculated by other methods. Results obtained by transient methods such as the time-lag or chromatographic methods, when compared with steadystate results, may also provide some indication of the relative importance of small or dead-end pores. Experiments with a wider variety of diffusion gases and operating conditions are also indicated.
Nomenclature defined by Equation 22, sq cm/sec concentration of A , mol/cc concentration of B, mol/cc dusty gas constant, sq cm/sec effective diffusion and flow parameter, sq cm/sec diffusion and flow parameter for macropores, sq cm/sec diffusion and flow parameter for micropores, sq cm/sec upstream gas concentration, mol/cc molecular self-diffusion coefficient, sq cm/sec molecular diffusion coefficient, sq cm/sec Knudsen diffusion coefficient for A , sq cm/sec Knudsen diffusion coefficient for B , sq crn/sec Knudsen diffusion coefficient based on macropore radius, sq cm/sec Knudsen diffusion coefficient based o n micropore radius, sq cm/sec effective diffusion coefficient defined by Equation 9, sq cm/sec effective diffusion coefficient for macropores defined bv Equation 32, sq cm/sec effective diffusion coefficient for micropores defined by Equation 32, sq cm/sec effective diffusivity defined by Equation 38, sq cm/sec average pore diameter, cm forced flow flux, mol/sq cm sec fraction of molecules diffusely reflected a t pore walls pore size distribution function average permeability, sq cm/sec macropore permeability, sq cm/sec micropore permeability, sq cm/sec
= = = = = = = = = = = = = = = = = = =
capillary length or pellet thickness, cm pore length, cm pore segment length, cm molecular weight of A molecular weight of B diffusion flux of A , mol/sq cm sec diffusion flux of B , mol/sq cm sec
NA f NB
combined diffusion and flow flux, mol/sq cm sec flux based on total area of solid, mol/sq cm sec number of pores per unit area of solids total pressure, atm total pressure a t x = 0, atm total pressure a t x = L , atm partial pressure of A a t x = L , atm partial pressure of A a t x = 0, atm partial pressure of B a t x = L , atm partial pressure of B at x = 0, atm total quantity of gas passed through cell in time-lag apparatus, mol = gas constant, cc atm/OK = pore or capillary radius, cm = average pore radius, cm = surface area of solid, cm2/g solid = pore volume, cc/g solid = total pore volume, cc/g solid = mean molecular speed of A , cm/sec = mean molecular speed of B , cm/sec = axial distance through capillary or pellet, cm = mole fraction A a t x = 0 = mole fraction A a t x = L = 1 NA/NB = constant defined by Equation 40 = mixing efficiency = porosity = void fraction of macropores = void fraction of micropores = void area fraction = viscosity = mean free path = tortuosity
-+
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AUGUST 1970
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