Ind. Eng. Chem. Res. 1991,30, 1428-1433
1428
Effective Diffusion Coefficients and Tortuosity Factors for Commercial Catalysts Ramesh K. Sharma* and David L. Cresswellt Systems Engineering Group, ETH Zentrum, CH-8092Zurich, Switzerland Esmond J. Newsod Swiss Aluminum, Research and
Development, CH-8212 Neuhausen, Switzerland
The effective diffusion coefficients of 13 commercial catalysts and supports were measured under nonreacting conditions by a dynamic tracer-response method using a single pellet string reactor (SPSR). Helium, nitrogen, and n-butane were used as tracers. Axial dispersion in SPSR was measured independently by using identical nonporous glass particles and was found to be low. It was observed that the possibility of convective flow through the catalyst pores was small. An analysis of the diffusion results using simple pore models showed that the most reasonable values of tortuosity factors were obtained for a cylindrical pore model. The diffusivities predicted by a random pore model were nearly twice the measured values in many cases. The tortuosity factor was found to vary strongly with the particle porosity.
Introduction Major advances have been made in the fundamental design of catalytic reactors using sophisticated models (Froment and Hoffmann, 1986). The application of these models to commercial cases has been rather slow. Under these conditions, the reactor simulation is often influenced by the intraparticle diffusion limitations. Many of the commercial reactors, such as those for hydrocarbon oxidation, are operated near the stability limits so that an accurate account of intraparticle diffusion is essential for their simulation (Sharma et al., 1984, 1991). The intraparticle diffusion depends on the pore structure of the catalyst, which is characterized by a tortuosity factor, T . There is an uncertainty in how to evaluate T , especially when the distribution of pore volume is over a wide range of pore radii where several mechanisms, including bulk, Knudson, and surface diffusion, may contribute to the total flux. For an accurate estimation of the tortuosity factor, the effective diffusion coefficient may have to be measured experimentally. Both steady-state and dynamic methods have been developed to measure the effective diffusion coefficients in catalyst pores under inert conditions. The dynamic methods have the advantage that the contributions of micropores and dead-ended pores are taken into account (Satterfield, 1970; Cresswell and Orr, 1982). These methods employ either a single pellet diffusion cell (Dogu and Smith, 1975) or a packed bed containing a large number of particles (Davis and Scott, 1965; Van Deemter et al., 1956). Single pellet measurementa have the disadvantage that only mass transfer through the axial faces is accounted for and transfer through the side wall is excluded. The packed-bed methods are applicable to the pellets of arbitrary shape, and the operating conditions may also be varied in order to closely reproduce the flow pattern that exists under the reaction conditions. However, it is necessary to accurately account for the interparticle axial dispersion. Haynes and Sarma (1973) and Scott et al. (1974) developed a single pellet string reactor *Towhom correspondence should be addressed at the Department of Chemical Engineering, University of Saskatchewan, Saskatoon, Canada S7N OWO. 'Chemicals and Polymers Group, ICI, P.O. Box 8, The Heath, Runcorn, Cheshire WA7 4QD, England. $EnergyStorage Laboratory, Department F4B, Paul Scherrer Institute, CH-5303 Wurenlingen, Switzerland.
(SPSR) in which the contribution of the axial dispersion to the pulse broadening is small. A high voidage in SPSR permits the use of high carrier-gas velocities, with low pressure drop, which reduces the axial dispersion. A large number of studies have since been reported using SPSR for diffusion measurements (Cresswell and Orr, 1982; Baiker et al., 1982). In this work, the effective diffusivities of 13 commercial catalyst materials were measured under nonreacting conditions, using a single pellet string reactor. The measurements were made by injecting pulses of helium, nitrogen, and n-butane in a flow of nitrogen or helium, which were used as carrier gases. The axial dispersion in SPSR was measured independently in each case with use of identical nonporous glass particles. The data were analyzed in terms of simple pore models, and the tortuosity factors from different models were calculated and compared. The effect of particle porosity and the type of tracer on T was examined.
Experimental Section Experimental Apparatus and Run Procedure. The experimental setup and run procedures have been described previously by Cresswell and Orr (1982) and are presented here only briefly. The setup consisted of a 2-m-long perspex column packed with the particles for which the diffusion coefficient was to be measured. It was necessary to keep the ratio of column/particle diameters in the range 1.1-1.4 (Scott et al., 1974) in order to eliminate the possibility of fluid bypassing along the wall. Tracer pulses injected at the top of the bed were detected upon entering and leaving the bed by means of Taylor-Servomex MK-158 microkatharometers as detectors, having a volume of 2 X lo-' m3 and a time constant of 40 ms. Diffusion measurementa were made at ambient temperature and at 106-131 kPa for three gas systems: pulses of nitrogen and n-butane in helium and pulses of helium in nitrogen. Measurement of the Diffusion Coefficient. The diffusion coefficients were calculated from the response measurements using the method of moments (Cresswell and Orr, 1982). It involved the solution of inter- and intraparticle material balance equations for the tracer in the Laplace domain to obtain the transfer function, E. Various assumptions in this method were presented earlier (Cresswell and Orr, 1982). The operating conditions were chosen such that the external mass-transfer resistance was minimized and the chromatogram tailing was s m d . Axial
0888-6885/91/2630-1428$02.50/00 1991 American Chemical Society
Ind. Eng. Chem. Res., Vol. 30, No.7, 1991 1429 Table I. Physical Properties of Catalyst Materials" codeb Q17/3 (f) Q17/3 (c) G 40 G 40 (calcined) K550 435 AM M 8 (0 M8 (u) ALC 428 ALC 426 ALC 441 ALC 483 ALC 182 S 829 141 AM (f) 141 AM (u)
shape sphere sphere sphere
characteristic length/" 3.3 3.4 4.8
sphere cylinder sphere sphere ring ring ring ring sphere ring ring ring
4.6 3.4 4.8 4.8 8.7 8.4 8.8 9.1 7.5 8.3 5.6 5.4
surface area/(mz/g) 96.5 332 117 171 9.4 0.8 3.1 2.5 2.2 0.9 0.6 0.2 2.6
particle density/ kg/m3) 1.32 1.16 1.39 1.27 1.42 1.54 2.15 2.21 1.66 1.77 1.81 1.35 1.22 2.19 2.01 2.15
av pore radius/(lO' Pm) 50
radius/pm
porosity micro macro 0.33
micro 39
macro
30
28 91 54 635 1511
0.65 0.68 0.49 0.24 0.09
0.17 0.22 0.04
50 400 1900
2098 2124 2310 8066 15846 17351 497
0.11 0.1 0.21 0.07 0.17 0.11 0.05
0.43 0.37 0.25 0.42 0.41 0.27 0.08
800
800 1500 3000 1100 8200 2000
900
2000 19OOO 2800 2800 12000 9500 57500 18OOO
loo00
OAverage pore radius from pore volume and surface area, b f denotes fresh catalyst, c, coked, and u, used.
dispersion was measured by using identical nonporous particles; i.e., the particle shapes and sizes were reproduced. The dispersion and diffusion coefficients are based on the unit cross section of bed and particle, respectively, perpendicular to the direction of diffusion. The diffusion coefficient values may be affected by channeling, especially at low flow rates of the carrier gas and with finer particles (Edwards and Richardson, 1968). Significant channeling results in an apparent dependence of diffusion coefficient on the carrier gas flow rate. The measurements were, therefore, made at different flow rates of the carrier gas. Physical Properties of the Catalyst Materials. The physical properties of the catalyst materials used in this study are given in Table I. All the materials are commercial catalysts or supports. Q17/3 is a Si02/A1203 catalyst for the isomerization of xylenes. 435 AM is an unsupported V206-P206 catalyst for the oxidation of butane while M8 and 141 AM are V206-Mo03 catalysts supported on y-alumina for benzene oxidation. S829 is a carborundum catalyst for ethylene oxidation, and ALC 182, ALC 426, ALC 428, ALC 441, and ALC 483 are CYalumina supports for ethylene oxidation catalyst. G40 and K550 are y-alumina water adsorbents. The surface area of the pellets varies between 0.5 and 332 m2/g (Table I), while the porosity is in the range 0.13-0.66. The pore size distribution indicated that both Q17/3 and G40 are essentially monodisperse materials with the majority of pores less than 0.004 pm, while M8, ALC 441, ALC 483, ALC 182, S829, and 141 AM have a wide pore size distribution with pores larger than 0.1 pm. The size and volume of the micro- and macropores in these materials were found at the pointa of maximum and minimum slope, respectively, on the pore distribution curve. Differential pore size distribution curves were also used. In some cases, the pore size distribution into macroand micropores was less marked, which probably led to high values of micropore radii. An average pore radius, r was calculated from the pore volume and surface area. &e average pore radius, shown in Table I, is closer to the micropore radius for K550,435 AM, and ALC 441 and closer to the macropore radius for Q17/3, G40, ALC 428, ALC 426, ALC 483, and ALC 182. For M8 and 141 AM, the average pore radius is much smaller than the micropore radius.
Results and Diecussion Convective Flow through the Catalyst Pores. In order to investigate the extent of convective flow through
the catalyst pores, pressure-drop measurements across SPSR were made. It was found that the highest pressure drop across the column was about 2% of the inlet pressure and the possibility of any convective flow through the pores was small. For such low pressure drops, the permeability constant has to be unusually large for convection through the pores to be significant. Another indication of the absence of convection is the independence of the effective diffusivity values with the superficial velocity of the carrier gas (Rodriques and Ferreira, 1988), as mentioned later. Satterfield and Cadle (1968b) have reported that, a t atmospheric pressure, the forced-flow convection and surface diffusion may be of minor significance due to small pressure gradients across the particle. The pressure drop with ringshaped particles was found to be higher than that predicted by the Ergun equation (Ergun, 1952), probably due to more oriented packing. In order to account for the difference, a shape factor of 0.4 was necessary. The equivalent diameter of the particle was calculated as 6/S, For a single-pellet string column, Scott et al. (1974) have defined S, as the total surface of pellets and wall divided by the total volume of pellets. Axial Dispersion in SPSR. An estimate of axial dispersion in SPSR was necessary in order to account for its contribution to the pulse broadening. Scott et al. (1974) and Cresswell and Orr (1982) have presented some axial dispersion data for the single pellet string column containing spherical particles. The data were correlated in terms of Peclet number, Reynolds number, and Schmidt number using the relation Npe = 1/[y/N&sC
+ A / ( 1 + S/NRdV,)l
(1)
where y, A, and 6 are constants related to various dispersive mechanisms. In order to extend the results, axial dispersion measurements were made for both spherical and ring-shaped particles of different sizes with use of various tracer/carrier gas combinations. In order to minimize channeling, extremely low Reynolds numbers were avoided (Edwards and Richardson, 1968). The results, presented in Figures 1and 2, were correlated by using the above relation with a modified Peclet number, defined as Np:
uGdp/&/(l-
a)
(2)
where d is the diameter of a sphere of equal volume. The bed voiJage, a,in these experiments varied between 0.56 and 0.76 and the tube/particle diameter ratio was within 1.1-1.4.
1430 Ind. Eng. Chem. Res., Vol. 30,No. 7,1991 T m e r : nitrogen, butane Carrier : helium
Tracer : helium Carrier : nitrogen
Legend
Legend
-. Ql7/3fmh 0 QV/3 c d t d
MBlnah
4.5 m m dass rings, N,
tr..?."~!??.rinQ%.h ....... 9.2 mm gass rings, t$ 0
4.7 m m &s?pbres.
C&
i
3 % i 10:
........
I
&GAL-
* K U !....... 0
m
,--.--.;I"lo
IO0
u-ts 828
Reynolds numhr. Re
Figure 1. Axial dispersion in SPSR with nitrogen and butane as tracers.
01
Suprrfklol v.losity X 10'. mja
Figure 3. Effective diffusion coefficient for helium.
Tracer : nitrogen Carrier : helium
Legend OW3
100.
,,,KC,,!?!,, ,, , ,
l a
c-40........ 0
AE426
A
KW?L_
s # M 4 p " & 3 P ' M . . u . . . x ~ G4ocdclned
M S. . . . . . . . . .
0-
schn#tnmkr -=022 .
'
."'"io
1 '
'
"""'
*o
'
' " " ' 4
8
ALC483
Dpo
k*lolanmbr,k
Figure 2. Axial dispersion in SPSR with helium a~ tracer.
For nitrogen and helium as tracers, the values of y, A, and 6 were estimated from (1)and (2)by nonlinear regression as y = 0.18 f 0.05 h = 0.33 f 0.05 b = 13.1 f 5.1 (3) The uncertainty in the estimated values of y, A, and 6 is low, as indicated by their narrow 95% confidence limits. The values of the constants are slightly lower than those reported by Scott et al. (1974)and Cresswell and Orr (1982) for spherical particles. For n-butane, the values for the modified Peclet number were lower than those for nitrogen or helium, probably due to its adsorption over the catalysts. Effective Diffusivity. The diffusivity values for helium and nitrogen for various materiala are given in Figures 3 and 4. Similar data were obtained for n-butane. For most of the runs, the diffusivities do not vary with the velocity, suggesting the absence of bypassing or any external mass-transfer effects during the measurements. According to Rodriques and Ferreira (1988),the independence of the effective diffusivity with the superficial velocity is also an indication of the absence of convective flow through the catalyst pores. In the case of ALC 428, the diffusivity of butane appeared to increase with the carrier-gas velocity. The diffusivities for Q17/3,G40,M8, and K550 are low since they have fine pores and a small porosity. The diffusivity of helium (Figure 3) is lowest for Q17/3 at 2.5
Figure 4. Effective diffusion coefficient for nitrogen. X m2/s, which increases to nearly 2 X lo4 mz/s for G40,K550, M8,and 435 AM and to 4 X 10-'-1.5 X mz/s for S829,ALC 428,ALC 441,ALC 483,and ALC 182. The diffusivities for nitrogen (Figure 4)are slightly lower
than those for helium. The values for n-butane were, however, much smaller as compared to those for helium and were below 1 X lo4 m2/s except for ALC 182,which had a value of 7 X lo4 m2/s. An enhanced tailing in the chromatograms for 435 AM and ALC 428 indicated a possibility of some adsorption of n-butane over these catalysts. Pore Models. The experimental data were correlated using different pore models. The effective diffusivity, D d , was calculated from the relation, Dd = D ( r , ) p / ~ .D(rp) essentially represents the bulk diffusivity, DD, when the diffusion is predominantly in the bulk region (ALC 182), and Knudson diffusivity, Du(r ), when the diffusion is in the Knudson region (Q17/3,K k 0 , G40). D m was calculated by the Chapman Enskog Formula (Satterfield, 1970). For most other catalysts, the pore size distribution was relatively wider and the diffusion waa in the transition region between bulk and Knudson diffusion. For such cases, various models, described in the Appendix, were used to calculate the effective diffusivity. Models 1 and 2 are the simplest in which the effective diffusivity is calculated from the bulk and Knudson diffusivities using the average pore radius, rp (Evans et al., 1961;Wheeler, 1951). The average pore radius was calculated from the surface area and pore volume. Model 3
Ind. Eng. Chem. Res.,Vol. 30,No.7, 1991 1431 Table 11. Tortuosity Factor from Different Models model 1 model 2
G40 B17 K550 G40 ( c ) ~ 435 AM ALC 428 ALC 426 ALC 441 ALC 483 ALC 182 M8 S829 141 AM a
1.0 4.4 1.3 1.4 6.4 5.9 3.8 4.0 3.7 2.5 8.2 6.2 4.8
1.3 3.6 1.7 1.8 7.8 5.5 10.1 2.6 4.2 2.7 9.1 7.9
1.0 1.0 1.0 11.7 17.6
2.8
1.0 4.4 1.3 1.5 4.2 7.6 4.9 5.1 4.0 5.0 10.2 5.6 6.1
2.0 3.7 1.7 2.6 6.8 9.9 18.3 3.3 5.0 8.2 17.1 9.9
model 3 3.5 2.4
5.1 2.0
3.2
0.5 1.4
0.8 1.2
0.6
8.2 6.2 4.0 4.6 3.7 2.5 6.9 6.3 11.8
7.0 6.3 11.2 3.3 4.1 5.1 9.4 8.1
18.2 20.5
2.3 2.6 1.4 1.5 1.5 1.1 1.4 1.7 0.8
1.7 2.5 3.7 1.0 1.6 2.0 1.7 2.0
4.7 7.9
1.0 1.0 1.0 13.4 22.5
3.2
model 4O
2.8
1.1
Ratio of calculated and measured diffusivities. c, calcined.
is a cylindrical pore model developed by Johnson and Stewart (1965). It describes the porous pellet as a solid penetrated by variously oriented cylindrical pores of different sizes. The pore size and the pore orientation distributions are considered independent of each other, In this model, DKA(rp) is evaluated in the entire range of the pore radii. It was found by Johnson and Stewart (1965) that the same model equations are obtained by assuming the pores to be straight and nonintersecting. Model 4 is a random pore model, proposed by Wakao and Smith (1962), and describes the pellet as a bidisperse pore structure consisting of micropores and macropores. The model distinguishes among three types of parallel diffusion paths: (a) through macropores of average radius ra, (b) through micropores of average radius ri, and (c) through micropores and macropores in series (Wang and Smith, 1983). Since the model is completely predictive, it does not have an adjustable parameter in order to match the calculated values with the experimental observations. The above four models were used in this study since they have a maximum of one unknown parameter. Among other models in literature are the working models of Feng and Stewart (1973)based on a cross-linked, cylindrical pore structure. These models do not require any pore orientation information. One of these models is similar to the cylindrical pore model of Johnson and Stewart (1965), while the others have more than one adjustable parameters. The dusty-gas model (Mason et al., 1967)also has one unknown parameter (when the permeability term is not considered), but there appears to be a lack of a clear relation between the unknown and the pore structure. Tortuosity Factors. The calculated values of the tortuosity factors for the above pore models are presented in Table 11. For most of the materials, the tortuosity factors are between 1 and 6 for helium, 1 and 7 for nitrogen, and 1 and 3 for butane. McGreavy and Siddiqui (1980)have also reported higher tortuosity factors for nitrogen as compared to those of helium. The values of 7 for 435 AM and ALC 428 with butane are high probably due to ita adsorption over these materials as indicated by the chromatograms. The exact reason for the consistently high values of T for ALC 426,5289,and MS with nitrogen and for M8 with helium is not clear, but there may be severe pore constrictions in these cases or the materials may have been calcined a t very high temperatures. Satterfield and Cadle (1968a)have reported the tortuosity factors for commercial catalysta in the range 3-7. A comparison of the tortuosity factors for helium from various models shows that the values from models 1 and 2 are similar for Q17/3,G40,and K550. Small differences in 7 values from models 1 and 2 are seen for ALC 428,ALC
426,ALC 441,M8,ALC 483, ALC 182,and 141 AM, where the Knudson diffusivity was comparable to the bulk diffusivity. Tortuosity factors from the cylindrical pore model (model 3) are higher than from models 1 and 2 for G40, K550,435AM, and 141 AM, whereas, for Q17/3 and M8, the 7 values from model 3 are lower. It should be noted that the diffusion contribution in model 3 is integrated over the entire range of pore size distribution as compared to an average pore radius used in models 1 and 2. The values of T from model 3 may be higher or lower than those from models 1 and 2, depending on whether the macro- and micropore radii are larger or smaller than the average pore radius. The diffusivities calculated from the random pore model (model 4)were larger than the experimentally measured values. The ratio (f)of the diffusivity calculated from this model to the measured value is between 1 and 2 in most cases, except for 435 AM and ALC 428 (Table 11). Baiker et al. (1982)also found the diffusivities predicted by the random pore model to be about 1.5 times higher than the measured values (i.e., f of 1.5)and attributed the difference to the assumptions concerning the probability of interconnections of different pore types implied in the model. For bidisperse catalysta,the relative contribution of the macro- and micropore systems to the total diffusivity is of particular significance. Table 111shows the percentage contribution of different pore types to the diffusivity as calculated from random pore model (model 4). Both G40 and Q17/3 have monodisperse pore structures, and the diffusivities are considered to be mainly from the micropores. In all the other cases,the contribution of macropore diffusion to the total diffusivity is over 40%. Some materials, such as ALC 428,ALC 426,ALC 483,5829,and ALC 182,have over 70% of macropore diffusivity for helium, nitrogen, and butane. The highest contribution of micropore diffusion is 43% for helium (K550)and 30% for nitrogen and butane (K550),while the lowest contribution is about 2% in all the cases. The series contribution is highest at 30% for helium (ALC 441)and 23-2670 for nitrogen and butane, while the lowest contribution is 54%.
The above results indicate that the macropore diffusion is dominant in bidisperse pellets with a high macroporoeity. In contrast the micropore contribution is high when the microporosity is higher than the macroporosity and the micropores are much smaller than the macropores. The series contribution is high when the micro- and macroporosities are comparable. When the diffusivity contribution from the micropores was high, f values for nitrogen
1432 Ind. Eng. Chem. Res., Vol. 30, No. 7, 1991 Table 111. Percentage Contribution of Different Paths in Random Pore Model He in N2 N2in He code macro micro series macro micro 100 100 100 100 46.7 29.8 43.3 8.4 48.3 21.4 27.8 55.5 25.5 46.7 2.6 10.2 89.4 3.4 86.4 3.2 4.1 9.4 89.4 86.5 19.4 23.3 30.5 55.2 46.2 2.1 91.7 2.5 7.3 90.2 4.7 19.0 82.4 7.1 73.9 14.6 73.1 18.6 15.4 66.0 79.6 8.2 10.9 16.3 72.8 18.4 37.8 43.8
Parallel pore model 1513-
Legend
11-
Helium
9?d
ig
5-
2
3-
B
8
7
01
Nitrogen
0
03
0.5
07
,
09
,
I
Partlck porosity
Figure 5. Dependence of tortuosity factor on particle porosity.
were higher than for helium, since the diffusivity of nitrogen was lower. Thus, consideration of the diffusion in both micro- and macropores is essential in order to obtain the reasonable values of 7'. Dependence of T on Particle Porosity. The tortuosity factors were found to vary with the particle porosity, as shown in Figure 5. As p increases above 0.4, 7 decreases sharply and approaches unity as p 1. I t appears that the pore geometry of these catalysts varied significantly with the porosity. The particle porosity may be one of the several factors that determine T , resulting in a data spread in Figure 5. Also, for materials with very low porosities (j3 < 0.2), the tortuosity factors were high but were smaller than those indicated in Figure 5. Comparison with the Results in Literature. A comparison of these results against those in literature showed that the tortuosity factors for helium and nitrogen, calculated from the cylindrical pore model of Johnson and Stewart (1965), are in the range indicated by Satterfield (1970) for different materials. The values of 7 changed dramatically with the pellet porosity, as also observed by Kim and Smith (1974). The results suggest that 7 depends on the type of pore model in addition to the pore structure. The type of pore model to be used may depend on the pore structure of the catalyst. The properties of the gas mixture may also influence the value of 7 , especially when the tracer is adsorbing. In order to predict the diffusion to within a factor of 2, Figure 5 shows that a tortuosity factor of 4 may be adequate in most of the cases. I t is interesting to note that Satterfield and Cadle (1968a,b) and Brown et al. (1969) have also proposed a tortuosity factor of 4 for the cylindrical pore model. The most consistent values of T for helium were obtained from the cylindrical pore model of Johnson and Stewart
-
CIHto in He micro 100
series
macro
series
23.5 23.1 8.0 7.4 25.4 6.2 12.9 12.3 12.2
45.1 52.4 88.7
30.5 21.7 2.9
24.4 25.9 8.4
80.2
14.4
5.4
(1965), except for 435 AM and 141 AM. The model appeared to be more suitable for materials in which the macropore diffusion was dominant. It should be added that the tortuosity factors for M8 and 435 AM, calculated from the cylindrical pore model, were also used in the simulation of pilot plant reactors for the selective oxidation of benzene and butane under commercial conditions, and an excellent comparison between the simulated and the observed product distributions was obtained (Sharma et al., 1984,1991). The cylindrical pore model was also found to be suitable for materials with micropores smaller than 0.1 pm since the random pore model led to higher diffusivity values in such cases. When both the macro- and micropores contributed significantly, i.e., both the microand macropore radii were over 0.1 pm, the random pore model was probably more appropriate. This model showed that the main contribution to the transport was due to macropore diffusion in most cases. Thus the random pore model may be useful when the active surface area of the catalyst is in the micropores so that both macro- and micropores contribute significantly to the diffusion. The predictions of the cylindrical pore model of Johnson and Stewart (1965) may be better than those of the random pore model since the former utilizes an adjustable parameter that is determined experimentally.
Conclusions The possibility of convective flow through the catalyst pores was extremely small. The magnitude of axial dispersion was low as compared to the diffusion. The macropores contributed over 40% of the total flux for most of the materials with a bimodal pore size dhtribution. The most reasonable values of 7 were obtained from the cylindrical pore model of Johnson and Stewart (1965). The values predicted by the random pore model were nearly twice the measured values for many catalysts. The tortuosity factor varied strongly with the catalyst porosity. Acknowledgment R.K.S. is grateful to ETH Zurich for the financial support. We also acknowledge the cooperation of Alusuisse Italia and Chemicals and Polymers Group, IC1 England, in providing the commercial catalyst samples.
Nomenclature D(rJ = diffusion coefficient through pores of radius r ,m2/s DAB = bulk diffusion coefficient of the gas mixture A,& m2/s DrA= effective diffusion coefficient, m2/s Do = interparticle axial dispersion coefficient, m2/s DKA(rp) = Knudson diffusion coefficient in pores of radius rp m2/s (DKA= 9.7 x 10"rP Dw, Dmi = Knudson diffusion coefficient in macro- and micropores, m2/s
Ind. Eng. Chem. Res., Vol. 30, No. 7, 1991 1433 d = particles equivalent diameter, m E!= packed-bed transfer function f ( r ) = pore size density function MA,M B = molecular weight of species A and B N b = Reynolds number, p U G d p / p Nsc = Schmidt number, p / p D m Npc= Peclet number, Uodp/Do
average pore radius, pm 2UG===temperature, K superficial carrier-gas velocity, m/s zA = mole fraction of A in gas phase
Greek Symbols bed voidage @ = particle porosity p,, pi = macropore and micropore porosity of the pellet p = viscosity of carrier gas, N-m s p = density of carrier gas, kg/m 7 = tortuosity factor a=
I
Appendix Model 1. Model of Evans et al. (1961): ~/D,A
= [ 1 / ~ . u+
~
~
Model 2. Wheelers' model: DeA = @DAB [ 1 - e-(Du/Dm)] /7
(4) 1
(5)
Model 3. Cylindrical pore model of Johnson and Stewart
de^ = P / r l m0 [ ( l - ~ ~ A ) / D A+B~ / D K A I W dr ) (6) where a = 1 - (MA/MB)'/', f ( r ) dr is the fraction of void
volume in pores of radii between r and r + dr, and xA is the mole fraction of the diffusing component A in the mixture. Model 4. Random pore model: DeA
=
+ (1- @a)'Di + 4Pa(1 - @a)(l/Da + l/Di)-' (7)
with
+ DB/DKAJ @,)'/(1- ~ X + A DAB/DKAi)
Da = D m / ( 1 -
~ X A
Di = D & / ( 1 where pa, Pi, and Dun, DKAirepresent the void fractions and Knudson diffusion coefficients associated with the macro- and micropores, respectively, and a is the ratio of diffusion rates for species A and B. Registry No. SiOz, 7631-86-9; A1203,1344-28-1;VZOt 131462-1; P20L,1314-56-3; Moos, 1313-27-5; He, 7440-59-7; N2, 7727-37-9; CIHIO,106-97-8; carborundum, 409-21-2. Literature Cited Baiker, A.; New, M.; Richarz, W. Determination of Intraparticle Diffusion Coefficients in Catalyst Pellets-A Comparative Study of Measuring Methods. Chem. Eng. Sci. 1982,37,643-656. Brown, L. F.; Haynes, H. W.; Monogue, W. H. The Prediction of Diffusion Rates in Porous Materials at Different Pressures. J. Catal. 1969,14,220-225. Crewwell, D. L.; Om,N. H. Measurement of Binary Gaseous Diffusion Coefficients Within Porous Catalysts. In Residence Time
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