DIFFUSION AND REACTION IN POROUS CATALYSTS N O R I A K I WAKAO C;lirersitj o/
AND J .
M . SMITH
Culi/ornin. n a r i s . CulJ
Using a previously developed concept of diffusion in bidisperse porous catalyst pellets, an expression for the effective diffusivity i s derived for diffusion under reaction conditions. This diffusivity i s a function of the effectiveness factor, E,, of the microporous particles composing the catalyst pellet and reduces to the normal diffusivity when Ei becomes unity. The diffusion results are applied to the problem of evaluating the effectiveness factor, E,, of the entire pellet. Charts are given for predicting E, for a first-order, isothermal reaction, in terms of a micropore diffusion parameter, a macropore diffusion parameter, and a reaction rate parcmeter. To evaluate E, requires a knowledge of the macro- and micropore size distributions, the micro effectiveness factor, E t , and either the average reaction rate per pellet or the reaction rate constant, k,. The method i s based upon a simple model of the pore structure. The model contains some unrealistic simplifications with respect to the actual mass transfer processes in porous materials. Hence the procedure should not be regarded as a general one applicable to all types of porous catalysts.
m m the effectiveness factor \vas introduced by Thiele S:13)there has been copciderable ipterest-for example il. -7. 1 5 ) P i n evaluating the averaqe reaction rate in a porous catalyst. This reqilirrs a n effective diffusivity for the reactant fluid in the porei of the solid material. Accordinqly, attention has been focused recently on measurement and prediction of diff(ision rates (1-6.9. 7 7 . 72. 77). These studies have al\va>-s been considered from the standpoint of difl‘usion in the abpence of chemical reaction. Experimental nieasuremer ts have been geperally made for diffusion thr0ug.h catalyqt pellets. which in itself is not equivalent. in most instance-. to diffurion under reaction conditions. I n the present Lvork the first objective is the effect of chemical reaction on the difTusion rates. Using the results E O obtained. expressions are then developed for predicting the effectiveness factor for a n isothermal. first-order reaction in a spherical catalyst pellet of known pore size diqtribution. In this \vork the catalVtic material is supposed to be a pellet Lvhich coptains both macro- and micropores. This type of material is ohtainrd Lvhen the catalyst is prepared by pelleting microporous po\vder. T h e effectivenecs factor development i r rirnilar to that of Mingle and Smith (7). in that catalysts Lvith bidicperse pore systems are treated. Hokvever. since the earlier paper a more accurate expression has been derived f,?. (?. 7-71 for diffu5ion in pores where both Knudsen and bulk diffusion are significant. This result has been used by LVakao and Smith ( 7 I ) . along \vith a new concept of pore geometry. to develop a method of predicting diffusion rates as a function of pore 4ze distribution. T h e method gave results in good agreement with ruperimental diffusion d a t a for alumina pellets covering a wide denrity range. T h e baiic conclusion that the diffu.ion rare is proportional to the square of the porosity (Equation 3 ) \vas also drveloped independently by \l’eisz and Schwartz ( 7 6 ) . This permits one to determine the effect of reaction upon diffusion-the first objective of this paper.
The result can then be used to calculate effectiveness factors without making assumptions regarding. the controlling type of diffusion in the micro- avd macropores. ar \vas necessary in the earlier paper (7). For example. diffuriori in the macropores (the space betlveen powder particle?) may be predominantly by a bulk mechanism in lo\v-den ----)Iz
B+1
(B-+x
-
1 (32)
This analytical result was compared to the results obtained by extrapolating the numerical solution for finite values of a to n = 0. T h e agreement was good in a11 cases, lending confidence to the numerical rvork. Figure 2 shows the expected decrease in E, with an increase in reaction rate--i.e., reaction rate parameter X. \Vhen the average pore radii increase. D,, and D,i also increase. 'This results in higher values of u and B . Figure 2 shows the magnitude of the increase in E, due to this reduction in diffusional resistances. The method of using Figure 2 ro determine the pellet effectiveness factor may be briefly summarized. First it is assumed that the reaction rate is first-order-that is. Equation 15 is applicable. The effectiveness factor for particle E imust be krio\vn. I t can be determined from rate measurements on the particles as described by Rao. Lt-akao. and Smith (8)for the orthohydrogen conversion. If the poxvder particles are small (less than a few hundred microns). E , is likel>-to be nearly 1 .O. From pore size measurements rhe quantities e,. e l , and a,, and can be obtained, as described by Wakao and Smith ( I 1 ) . 'The value of a w n be ascertained from the stoichiometry of the reaction. Then parameters B and a are established from Equations 26 and 2'. If rhe reaction rate constant. kw. is known, X is determined from Equation 28 and E, read from Figure 2. If instead the reaction rate: i.. for the pellet has been established, a trial and error procedure is necessar>-. A value of k , or X is assumed. and & is obtained from Figure 2. Then the value of the product (k,E,) is checked by comparison with Equation 15. Evaluation of the method Jvith data for the ortho-parahydrogen reaction is considered in the following paper (8). Acknowledgment
The authors express their appreciation to R. De Vogelaere, Department of Mathematics. University of California. Berkeley, for devising a numerical method for the solution of Equation 13. Yu Chang carried out the necessary computer rvork and his assistance is gratefully acknowledged. ID
.,?-
loo
Nomenclature R
=
macropore diffusion parameter defined by Equation
0, fii
I)
= = = =
Dh
=
mean radius for diffusion in macropores? cm. mean radius for diffusion in micropores. cm. micropore diffusion parameter defined by Equation 26 effective diffusivit?. in catalyst pellet under reaction conditions. s q . cm. sec. binary bulk diffusivity. sq. cm., sec.
27
Figure 2.
126
I&EC
Pellet effectiveness factor
FUNDAMENTALS
B
D,D
effective diffusivity for microporous particle. defined by Equation 18: sq. cm.,’sec. fi,, = mean Knudsen diffusivity of gas A in macropores: sq. cm.,’sec. Dk, = mean Knudsen diffusivity of gas A in micropores, sq. cm. Isec. E, = effectiveness factor of catalyst pellet E, = effectiveness factor defined for microporous powder particles hI,, h, = Thiele modulus defined by Equations 16 and 17, respectively = reaction rate constant defined per unit time per unit mass of catalI,st, g. mole/’g. sec. = half thickness of slab-shaped pellet. cm. = diffusion rate per unit time per unit area of pellet, g. mole,./sq. c m ~sec. = total pressure, atm. = average reaction rate per unit time per unit mass of catalyst pellet, g. mole/g. sec. = reaction rate (rate of conversion of reactant) per unit time per unit mass of powder particle, g. mole/ g. sec. = gas constant, cc. atm./’g. mole O K . = X “ 2 X/X, = temperature, K . = mean molecular velocity, cm./sec. = distance variable, cm. = mean radius of powder particles, cm. = mean radius of spherical pellet, cm. = reactant gas mole fraction in macropores of pellet = equilibrium mole fraction = reactant gas mole fraction at external surface of pellet =
aiy = 1
=
=
- Je)/’!Jo-
+
et
= rnicrovoid fraction in peliet
elp
=
e,
= = = =
X PB
pp
microvoid fraction in particle solid fraction in pellet reaction parameter defined by Equation 28 density of pellet, g I’cc. densit! of particle, g.;cc.
literature Cited
(1) Beek, John, A.I.Ch.E. J . 7 , 337 (1961). (2) Carberry. J. J.. Ibid., 7 , 351 (1961). (3) Evans, R. B., LVatson, G. M., Mason, E. .4.. “Gaseous Diffusion in Porous Media at Uniform Pressure,” IMP-AEC-15, Inst. for Molecular Phvsics. L’niv. of Marvland, June 1. 1961. (4) Henry, J. P.. Chennakesanan. B.. Smith. J. M.. A.Z.Ch.E. J . 7. 10 (1961) (5) ’Ho&sch&en, J., znd. Eng. Chem. 47, 906 (1955). (6) Masamune, S.. Smith. J. M., A.I.Ch.E. J . 8, 217 (1962). , , ( 7 ) Mingle, J. O., Smith. J. M., Ibid., 7, 243 (1961). (8) Rao: M. R., Wakao. Noriaki, Smith, J. M., IND.ENC.CHEM. FUKDAYESTALS 3, 127 (1964). (9) Rothfeld, I,. B., LVatson, C . C., “Gaseous Countrr Diffusion in Catalyst Pellets.” 54th ’4nnual Meeting: A.I.Ch.E., hTew York, Dec. 3-7. 1961. (10) Scott, I).S., Can. J . Chem. Ene., to br published. (11) Scott, I). S.. Cox. K. E., J . Chim. Phys. 57, 1010 (1960). (12) Scott. L). S.. Dullirn, F. A. L.. A.I.Ch.E. J.. to be published. (13) Thiele, E. i V . , Znd. Ene. Chem. 31, 916 (1939). (14) kl’akao. Noriaki, Smith, J. M., Chem. Erie. Sci. 17, 825 (1962). (15) Whmler, Alborn. “Catalysis,” Vol. 11, Reinhold, New York, 1955.
(16) Weisz, P. B., Schwartz, A. B., J . Catalyris 1, 399 (1962) (17) Wicke, E., Kallenback, R., Kolloid Z. 97, 135 (1941). RECEIVED for review April 5, 1963 ACCEPTEDDecember 30, 1963
J,)
(1VR/lVA)
Project carried out with the financial assistance of the U. S. Army Research Office, Grant No. DA-ARO(D)-31-124-G191.
macrovoid fraction in pellet
DIFFUSION AND REACTION RATES IN T H E ORTHO-HYDROGEN CONVERSION M. RAJA RAO, N O R l A K l WAKAO, AND J . M . S M I T H C’ni~ersityof Californza, Davis. Calif.
Rate studies were carried out for the ortho-para-hydrogen conversion using single-pellet catalysts of N i O on A1203. Measurements were also made with the powder particles of catalyst used to prepare the pellets, From these data the effectiveness factor, Ea, was evaluated for pellets of three different densities and, hence, different macropore properties.
Pore size distributions, void volumes, and particle sizes were also meas-
ured. This information was sufficient to apply the theory in the preceding paper to calculate theoretical effectiveness factors. The agreement between the experimental and predicted results indicated that the theory was satisfactory for the specific, bidisperse catalysts used in this study. The effectiveness factor, Et, for the microporous particles in the pellets was found to b e unity. It appears that E a will be close to 1.0 except for very rapid reactions using pellets prepared from unusually large catalyst particles.
effect of pole diffusion on rates of solid catalytic reacSince then a considerable volume of literature has accumulated-- for example (3. 5. 7, 72, 74. 75). However, no experimental work has been reported for diffusion measurements under reaction conditions Particularly the relation between diffusivity and pore geometrv. with simultaneous reaction, has not been studied I n a preceding paper, a model developed for diffusion in bidisperse pore systems (70) is applied to the reaction case. T h e object of the present study is to compare experimental results with the foregoing theory. HE
T tions was first analyzed by Thiele (6).
For the experimental proqram, reaction rates were measured for slab-type catalyst pellets under conditions analogous to those used to develop the diffusion theory. In addition. rate data were measured for the catalyst particles used to prepare the pellets. For these studies the ortho-para-hydrogen reaction was employed wing a 25% NiO on A1?03 catalyst. This reaction was chosen for several reasons: The diffusion process is equimolal and cquntercurrent. ’This simplifies the diffusivitv equations. because (Y = 0. Temperature gradients. even with large pellets. are negligible because of the low heat of reaction. VOL. 3
NO. 2
MAY
1964
127
