Literature Cited Herrnsen, W., J . M., Prausnitz, J. Chem. Phys., 34, 108 (1961). Hildebrand, J. H., Proc. Nat. Acad. Sci. U.S., 64, 1331 (1969). Hildebrand, J. H., Prausnitz, J. M., Scott, R. L., "Regular and Related Solutions," p 40, Van Nostrand-Reinhold, New York, N. Y., 1971 Linford, R . G., Hildebrand, J . H., Trans. Faraday Soc., 65, 1470 (1969)
Miller, K . W., Hildebrand, J. H., J. Phys. Chem., 72, 2248 (1968) Walkley, J., Jenkins, W. I., Trans. Faraday Soc., 64, 19 (1968).
Received for reuieu J u l y 25,1973 Accepted J a n u a r y 14,1974
Diffusion in Bidisperse Porous Catalyst Pellets Noboru Hashimoto' and J. M. Smith* Department of Chemistry, University of California, Davis, Calif. 95616
Realistic treatment of diffusional effects in bidisperse porous catalysts requires separate accounting for macro- and micropore diffusion. Pulse-response data for pelletted microporous particles are used to illustrate that both macro- and micropore diffusivities can be obtained from such chromatographic techniques. The bidisperse-pore theory employed to analyze the data also yields the equilibrium and rate constants for adsorption. Results are obtained for the adsorption of n-butane on alumina (Boehmite) pellets at 1 atm pressure and 30-75°C. Macro- and micropore diffusivities at 30°C were 3.2 X and 4.0 X cm2/sec for a pellet with macro- and micro- mean pore radii of 1200 and 17 A, respectively. Adsorption equilibrium and first-order rate constants were about 870 cm3/g and 600 cm3/ (9) (sec) at 30°C. For successful analysis, the chromatographic data must be obtained at concentrations where the rate of adsorption is first order and the diffusion times are neither too short nor too long.
It has been demonstrated (Furusawa and Smith, 1973; Wakao and Smith, 1962) that erroneous diffusivities may result if diffusion data in a bidisperse catalyst pellet are analyzed as a monodisperse porous material. This has been recognized by many investigators, and Sargent and Whitford (1971) and Ruckenstein, et al. (1971), have used bidisperse theories to analyze dynamic diffusion and adsorption data. A method of analyzing response peaks of adsorbing gas injected as pulses into a fixed bed has been developed (Hashimoto and Smith, 1973) for bidisperse pellets. With data for different size pellets (composed of microporous particles) and particles, it is possible, in principle, to extract both intrapellet (macropore) and intraparticle (micropore) diffusivities. This was not possible for the data obtained by Hashimoto and Smith with pellets of 5A molecular sieve crystals. Presumably, the failure to obtain micropore diffusivities was because the transport rate in the 5A pores was too small to give an accurately measurable response peak. The slow desorption process from within the crystals caused a peak with a finite tail but one that was too long and of too small a height to analyze accurately. If this were true, both macro- and micro-diffusivities should be measurable for adsorbing pellets where the spread between macro- and micropore radii was less than in 5A zeolite pellets-for example, for alumina pellets. Accordingly, the objective of the work reported here was to see if these two diffusivities could be determined from chromatographic data for the adsorption of n-butane on pellets of microporous particles of alumina (Boehmite). Theory
the agglomeration constitute the macropores. When a pulse of adsorbable gas is introduced into the feed to a packed bed of the pellets (also assumed to be spherical), mass transport is supposed to occur by the following processes: (1) axial dispersion in the interpellet space of void fraction, C Y , according to a dispersion coefficient E; ( 2 ) gas-to-pellet surface transport with mass transfer coefficient, k , ; ( 3 ) intrapellet mass transfer according to a macropore diffusivity, D,; (4) intraparticle diffusion with micropore diffusivity, Di; ( 5 ) reversible adsorption at a site within the micropores with a first-order rate constant, k , and equilibrium constant, K. Adsorption in the macropores is neglected since the surface area is small with respect to the area of the micropores. When a square pulse of injection time t o is introduced to the bed, the differential equations describing the concentration of adsorbable gas, as a function of time and axial position in the bed, are linear. It has been shown (Hashimoto and Smith, 1973) that the first absolute moment, w ~ and , the second central moment, p~', of the response peak in the bed effluent are related to E, kf, D,, Di,k , and K by the equations
where
6,) =
l--cu a
+
[Ir
€1
(1
+
5 1
(3)
The pellet is assumed to consist of an agglomeration of spherical, microporous particles; the interparticle voids in
' On leave from Japan Gasoline Co., Yokohama, Japan. Ind. Eng. Chem., Fundam.,Vol. 13, No.2, 1974
115
6, = -
a
15
(5) The method of deriving these equations is similar to that for monodisperse porous materials presented in detail by Schneider and Smith (1968a). The two moments can be evaluated from the experimentally measured response peaks from the expressions
PORE RADIUS, A '
where C is the concentration of adsorbable gas in the peak in the effluent leaving the bed. Equations 1-5, along with values of p1 and pz' determined for runs with pellets of various radii, R, and particles of various radii, ro, are sufficient to evaluate the two diffusivities, k, and K. The data obtained for n-butane pulses passed through beds of alumina pellets were analyzed with these equations, and the results are given later. Experimental Section The pellets were prepared from two sizes of microporous particles of spray-dried Boehmite. The small particles had an average diameter of 0.048 mm and consisted of particles within the sieve sizes 270-325 mesh; the larger had an average diameter of 0.114 mm (sieve size range 115-150 mesh). In analyzing the moments it was desirable to prepare pellets of the same apparent density, pp. This was approached by pelletizing the same weight of particles in a cylindrical mould, 2.7 cm i.d. and 2.55 cm long. Pellets of various sizes were prepared by crushing and sieving these large pellets. The properties of the pellets are given in Table I. The true density was determined in a helium pycnometer, and the apparent density of the crushed pellets was measured by mercury displacement at atmospheric pressure. The macropore volume distribution was obtained by mercury penetration under pressure in a porosimeter. The microTable I. Properties of Crushed Pellets Particle radius 0.024 0.057
Method True density, pt, g/cm3 Apparent density, pp, g/cm3 Total pore volume, Vt, cm3/g Total porosity, E t Macropore volume, V,, cm3,lg Macropore porosity, ea Macroport mean radius, A Micropore volume, Vi,
Vt
= l/pp
et =
1
ei
116
l/pt
- pp/pt
Ea
a.
= =
Vi =
$
ei =
ai -
svi
-
= i:
0.362 0.358
Vf
a,dV, 1300
1100
0,625 0.635
Ei/pp
1
1.171 1.178 0.777 0.776 0.551 0 :547
PPVB
ea
Micropore mean radius, A
-
mm
2.98 2.94 0,664 0.659
Hg porosimeter
cm3/g
Micropore porosity,
mm
He pycnometer Hg porosimeter
pp/pt
Lvi
-
0,415 0.418
ai dVi
Ind. Eng. Chem., Fundarn., Vol. 13, No. 2 , 1 9 7 4
17
Figure 1. Pore-size distributionsof alumina pellets
pore volume distribution was measured by low-temperature nitrogen adsorption using a Perkin-Elmer Sorptometer. These distributions, along with apparent densities for the pellets prepared from the two particle sizes, are shown in Figure 1. The macropore distributions and apparent densities are seen to be about the same for the two particle sizes. The various porosities and mean radii were obtained by the methods indicated in the second column of Table I. The various properties are based upon the weight obtained after heating at 280°C for 12 hr with helium gas flow to eliminate adsorbed water. The division between macro- and micropore radii was taken at the minimum of the distribution curve, or a t about 120 b, (Figure 1). The beds were prepared by packing the pellets in nominal 3/s-in. 0.d. (0.775 cm i d . ) copper tubing in the combinations of particle and pellet sizes shown in Table 11. Packed bed lengths and bed porosities, C Y , are also given. Preliminary measurements of adsorption capacities indicated that the isotherms were linear up to concentrations of at least 0.75 mole % butane in helium. Hence, pulses containing 0.5 mol % n-butane were introduced in a stream of pure helium at the bed entrance. This concentration ensured that the adsorption rate was first order. The response peaks were measured using a flame-ionization detector, and the signal was recorded. The flow diagram of the apparatus was essentially identical with that described by Schneider and Smith (1968a). Before introducing the pulses of n-butane in helium, the bed was pretreated by heating at 280°C for 12 hr with a helium flow of 10 to 20 cm3/min. After many pulses, some deactivation of the bed for further adsorption was observed, probably due to the accumulation of small amounts of irreversible adsorption. In such cases, the beds were regenerated by heating with helium flow for 1 hr at 280°C. The measured moments include the effects of retention time and dispersion in the dead volumes (between the point of pulse injection and bed entrance and between the bed outlet and detector), while eq 1 and 2 do not include such effects. In the construction of the apparatus these volumes were made as small as possible. Also corrections were evaluated by making blank runs in which the column was replaced by a 3-cm length of 0.16-cm i.d. tubing; that is, moments were measured by introducing n-butane pulses and measuring the response peak. In order to separate the axial dispersion contribution from the second moments and to obtain more reliable adsorption equilibrium constants, runs were made for each
Table 11. Characteristics of Packed Columns
Pellet size
a
Mesh size
Av radius, R , mm
Mesh size
14-16 14-16 20-24 28-32
0.537 0.537 0.384 0.272
115-150 270-325 270-325 270-325
L
=
Temp, OC
Particle size
packed bed length, cm.
b
(Y
Av radius, T O , mm 0.057 0.024 0.024 0.024 =
30
50
La
L
ffb
0.371 0.361 0.395 0.397
11.3 11.1 11.2 11.5
75
L
cy
0.384 0.362 0.395 0.397
11.4 10.4 10.8 10.8
a!
0.372 0,372 0.395 0.397
11.0 11.5 10.8 10.8
bed void fraction.
c
g* =-. $
103 8 6 1
az
EL
L Z El-
2
8 % C Iv < O
102 2.8
3.0
29
3.1
3.2
3.3
3.1
1O3/T,
Figure 3. Adsorption equilibrium constants for n-butane on alumina (Boehmite)
perature should be on the same line, since K is dependent only on temperature, and p p is essentially constant. Figure 2, which includes all the first moment data, shows that these requirements are met reasonably well. The correction for dead volumes, ( p l ) d , was never more than 1% of ( P I ) exp.
0
0.e
0.01
0.03
0.01
0.e
The slopes ,of the lines in Figure 2 determine the adsorption equilibrium constant, and these K values are plotted us. 1 / T in Figure 3. The heat of adsorption determined from the van't Hoff equation and the slope of the line in Figure 3 was A H = -9.2 kcal/mole. Other values of A H for n-butane on alumina were not found in the literature. However, Schneider and Smith (1968b), reported -7.8 kcal/mole for n-butane adsorption on silica gel. Both results are isosteric heats of adsorption corresponding to essentially zero surface coverage. This is because the concentration of n-butane falls from 0.5 mole % to a very small value in a very short initial section of the bed. A curve of concentration us. bed length, for another system, verifying this steep curvature, has been presented by Schneider and Smith (1968b).
0.M
Z i r . mfl.
Figure 2. First-moment results
bed at many flow rates ( i e . , bed velocities) over the range 30 to 180 cm3/min (at 25°C and 1 atm). All the data were obtained a t 1 atm pressure. First Moment Results
The experimentally determined first moment is related to eq 1and 3 by the expression
Second Moment Results
Axial Dispersion. Before the diffusivities and rate constant k can be obtained from eq 2, it is necessary to extract the axial dispersion effect on p2'. This effect depends upon the geometry of the packed bed, @, and DAB. For the measurements reported here @/DAB varied from 0.2 to 1.6 mm-I. Within this range Suzuki and Smith (1972) were able to correlate axial dispersion in similar beds of small particles by the expression
where (p1)d is the first moment of the dead volumes, measured in the blank runs. To express eq 8 in a simpler way, define the first moment of a n inert (nonadsorbable) gas by taking K = 0 in eq 1 and 3 . Then
E
DAB = 7 + 1 ( $ $
Combining eq 9 and 8 to eliminate to gives
where the first term on the right represents the molecular contribution to E and the second represents the velocitydependent, turbulent contribution. The diffusibility 9 and scale-of-dispersion 1 are characteristic of the tube and particle diameters and shapes. Equation 2 can be expressed in terms of 9 and 1 by eliminating E with eq 11
a Since (pl)i can be calculated for any velocity from the bed and pellet properties given in Tables I and 11, and ( ~ 1 and (fi1)d were measured, the left side of eq 10 is known. When this is plotted us. Z / U a straight line should result. The data for all particle and pellet sizes for the same tem-
)
~
~
~
Ind. Eng. Chem., Fundam., Vol. 13, No. 2,1974
117
5O'C
1S'C
10-1
1
I
I
1
3O'C I
2.9
3.0
3.1
3.1
3.3
350
3w
-.. f
250
, N , a m ,
"Lo I
's( 150
1w
J
I
2.8 50
3.4
103/T, *KV1
Figure 5. Diffusivities and adsorption rate constants 0 0
2
1
1
3
Id/,, min./cm. Figure 4. Second moments for n-butane at 30°C
The second moment results are illustrated by the 30°C data points in Figure 4, where the left side of eq 12 is plotted us. l O 3 / u . Correction factors for dispersion in the dead volumes were evaluated from the blank run measurements. However, these corrections were less than 3% of the observed pz' for the bed. Since the accuracy of the second moments is, a t best, no better than 3%, such corrections were not made. To determine the values of 6, + d,, and hence the diffusivities, a least-square procedure, employing eq 12, was used with all the data first to determine the best values of q and 1. Twelve sets of data, each consisting of measurements at many velocities, were used in the evaluation. These twelve sets corresponded to four pellets and three temperatures; 17 and I are independent of temperature, while (6, 6,) is temperature sensitive. This procedure 6,). gave 17 = 0.344 and 1 = 0.032 cm and values for (6, The solid curves in Figure 4 illustrate how eq 12, with these results of q and I, agree with the data. The values of 6, and 6, are the intercepts on these curves at lO3/u = 0. Axial dispersion coefficients appear to be very sensitive to packing geometry and vary from one study to another. For example, in beds of CuO-ZnO catalyst pellets Suzuki and Smith (1972) found I) = 0.44 and 1 = 0.083 cm. Micropore Diffusivities. Moment data were determined for the same size pellets ( R = 0.537 mm) made from two particle sizes (r01 = 0.057 mm and r02 = 0.024 mm). Assuming that K, k , and the macropore diffusivity are independent of particle size, eq 4 and 5 may be added and applied to each particle size to give
+
+
[F]rOl-[F] + + 61
6'1
6,
6,
=
n2
Since p p for the two particle sizes is essentially the same (Table I), an average value may be used in eq 13. All other properties for these two pellets are known, and 6i + 118
Ind. Eng. Chem., Fundam., Vol. 13, No. 2,1974
Table 111. Micropore Diffusivities and Adsorption Rate Constants
Temp, OC 75 50 30
Di, cm2/sec
5.85 X 4.70 X 4.05 X
lo-' lo-* lo-*
DKi,
cm2/sec 4.04 X 3.90 X 3.76 X
Tortuosity factor,
k, cm3/(g) (set)
Ti
2.89 3.48 3.87
1 0 . 6 X lo2 7 . 6 7 X lo2 6 . 5 6 X lo2
6, have been determined for each temperature. Hence, eq 13 may be used to calculate Di; the results are given in Table I11 and plotted us. 103/T in Figure 5. Micropore
tortuosity factors ~i were calculated from the equation
where DK, is the Knudsen diffusivity for n-butane in the micropores, evaluated at the mean pore radius of 17 A (Table I). For these small pores bulk diffusion due to intermolecular collisions was negligible. The resulting tortuosity factors should be independent of temperature, but actually they decrease about 25% from 30 to 75°C. This variation could be due to surface diffusion. The level of the tortuosity factors (3-4) is in agreement with surveys of such information for a variety of microporous catalysts (Satterfield, 1970). Hence, it seems more likely that the change in T I with temperature is a measure of the accuracy with which diffusivities can be evaluated from chromatographic data. Adsorption Rate Constants. The (6, 6,) results for the three different pellet sizes at a constant particle size (r02 = 0.024 mm) can be used to determine D, and k at each temperature. Since k and Di are constant as the pellet size changes (for constant ro), eq 4 and 5 show that (6, 6,) plotted us. Rz should be a straight line. The intercept at R2 = 0 on such a plot is equal to d i . This type of figure is shown in Figure 6, and from it 6, can be obtained. Rearranging eq 4 gives
+
+
cy
Since Di, K, and the properties of the pellets and bed are
Table IV. Macropore Diffusivities
+
OC
5/kfR l/Da, sec/cm2
D , X lo2, cm2/sec
DAB, cm2/sec
cm2/sec
D, cm2/sec
75 50 30
27.4 32.9 37.9
4.67 3.81 3.25
0.556 0.488 0.438
0.262 0.252 0.244
0.178 0.166 0.157
Temp,
DKW
7%
1.36 1.56 1.73
known, the 6i values can be used in eq 15 to obtain the adsorption rate constant a t 30-75°C. The results are given in the last column of Table III and are plotted according to the Arrhenius equation in the central line in Figure 5. The results correspond to an activation energy for this physical adsorption process of 2.3 kcal/mole. For n-butane adsorption at 50°C on silica gel, Schneider and Smith (1968a) reported k = 1500 cm3/(g sec) in comparison with 767 cm3/(g sec) given in Table 111. Macropore Diffusivities. The slopes of the lines in Figure 6 are equal to
t15.' [ 1 + 2
(1
+
%)IZ(&
+ &)
+
From these slopes (5/kfR l/Da) was calculated a t each temperature. This sum, given in Table IV, represents the combined effect of gas-to-pellet mass transfer and macropore diffusion. In order to determine Da, it is necessary to estimate k f and then subtract 5/kfR from the combined contribution. This introduces some uncertainties, because kf is not accurately known for beds of small particles a t low Reynolds numbers. The modified Reynolds numbers for the various beds were from 0.05 to 0.57. Another reason for the uncertainty is the relatively large values of 5/kfR in comparison with those of l/D,, as seen from the data in Table IV. After consideration of the various possibilities it was decided that the following correlation of Wakao, et al. (1958), gave the most appropriate values of k f for the conditions of our experiments 2Ruop DAB
Using this equation for kf, and the experimental results for the combined contribution of 5/kfR 110, given in Table IV, the macropore diffusivities shown in the third column of Table IV were obtained. These results are about two orders of magnitude larger than Di, which is about what would be expected in view of the differences between macro- and micropore sizes. However, the absolute value of D, is probably not very accurate because of the uncertainties already mentioned. Since the macropore sizes are such that diffusion there is in the transition region, a composite molecular diffusivity, D , must be calculated before tortuosity factors can be estimated. Hence, D was calculated from the equation
+
1 D
1
-
f-
1
DAB
E
using the average mean macropore radius (1200 A) given in Table I in order to determine the Knudsen diffusivity DKa. Then, the macropore tortuosity factors were obtained from the expression
Da
€a.
D
~
Ta
(18)
Values of D and T, are also given in Table IV. The range of T, is in general agreement with other effective diffusivities in the literature (Satterfield, 1970) for the relatively large pore sizes corresponding to macropores.
0
5
10
15
M
25
30
~2 x 104, !c,
Figure 6. Relation between (61+ 6,)/[(1
- a)/.]
and pellet size
Discussion and Conclusions
It has been demonstrated that chromatographic experiments provide data from which both macro- and micropore diffusivities can be evaluated for bidisperse porous catalysts. The advantage of the method over other procedures is that the data can be obtained rapidly over a range of temperatures with small quantities of the adsorbing gas and with simple equipment. There are restrictions in application of the method: (1) the gas concentration in the inlet pulse must be low enough that the reversible adsorption process is first order, and (2) the micropores must be large enough to give diffusivities sufficiently high to eliminate the possibility of a long tail on the response peak in the effluent from the bed. In the experiments reported here the diffusion plus adsorption time, which is of an order given by ppK(ro2/Dl), was less than 75 sec. This is well within the total time of 200 to 300 sec corresponding to the complete effluent peak. In the earlier work with 5A molecular sieves (Hashimoto and Smith, 1973), the micropore diffusion plus adsorption time was about 8000 sec, much too large for D , values to be determined. Thirdly, the accuracy of the diffusivities determined chromatographically is not such as to give values within a few per cent, due basically to the difficulties in accurately measuring second moments. Macropore diffusivities are also subject to uncertainty due to the inaccuracy of fluid-toparticle mass transfer coefficients. It is estimated that diffusivities within 25% of true values can be obtained from chromatographic data taken at proper operating conditions. This is within the needs of most applications in heterogeneous catalysis. Finally, the moment data for the beds with R = 0.537 mm and ro = 0.057 mm were not used in the previous evaluation of D, and k. Hence, an overall check on the validity of these quantities can be obtained by comparing the values of 6, + 6, established from the experimental moments (that is, the intercepts illustrated in Figure 4) with the values predicted from eq 4 and 5 using D,, and D,, and k determined from the other data, as given in TaInd. Eng. Chem., Fundam.,Vol. 13,No. 2,1974
119
Table V. Comparison of Experimental and Predicted for R = 0.537 mm, ro = 0.057 mm Values of (Si
u = velocity in the voids of bed, cm/sec Va = macropore volume in crushed pellet, based on
+ 6*) from
weight of crushed pellet, cm3/g Vi = micropore volume in crushed pellet, based on weight of crushed pellet, cm3/g Vt = total pore volume in crushed pellet, based on weight of crushed pellet, cm3/g z = axial distance from inlet to bed, cm Greek Letters a = void fraction in the bed 60,6i,6, = moment contributions defined by eq 3-5 t a = macropore porosity, based on volume of crushed pellet t i = micropore porosity, based on volume of crushed pellet t t = total porosity, based on volume of crushed pellet 7 = diffusibility p1 = first absolute moment in the bed, sec (pi)exp = experimentally determined first moment for nbutane pulse, sec ( p ~ ) d= first moment of dead volumes, sec (p1)i = first absolute moment in the bed for a pulse of nonadsorbable (inert) gas, calculated from eq 9, sec p = gas viscosity, g/(cm)(sec) p2' = second central moment in the bed, sec2 p = gas density, g/cm3 pp = apparent density of crushed pellets, g/cm3 pt = density of solid phase of crushed pellet, g/cm3 r , = tortuosity factor in macropores T i = tortuosity factor in micropores
+
Temp, "C 75 50 30
(ai
+min
6a)exp,
2.42 18.9 169
(Si
eq 4 and 5, min 2.40 19.0 167
bles III and IV. These comparisons, shown in Table V, lend some confidence to the results. Acknowledgment
The American Cyanamid Co. kindly provided the Boehmite powder. The financial assistance of the Japan Gasoline Co. in the form of a postgraduate fellowship is gratefully acknowledged. Nomenclature
a = pore radius; a, refers to macropores and ai to micropores, A C = concentration of adsorbable component in the gas phase in the bed, mole/cm3 D = composite diffusivity, defined by eq 17, cm2/sec DAB = bulk molecular diffusivity in n-butane-helium gas mixture, cm2/sec D, = effective diffusivity in macropores, based upon void plus nonvoid area of crushed pellet, cm2/sec Di = effective diffusivity in micropores, based upon void plus nonvoid area of particles, cm2/sec DK, = Knudsen diffusivity in macropores, cm2/sec D K ~= Knudsen diffusivity in micropores, cm2/sec E = axial dispersion coefficient, based on cross-sectional area of column, cm2/sec A H = enthalpy change on adsorption, kcal/mole k = adsorption rate constant, cm3/(g)(sec) k f = gas-to-pellet mass transfer coefficient, cm/sec K = adsorption equilibrium constant, cm3/g L = length of packed bed, cm 1 = scale of dispersion, cm ro = radius of microporous (spherical) particle, cm R = radius of crushed pellet, cm t = time, sec to = injection time of pulse, sec uo = superficial velocity (ug = ua), cm/sec
120
Ind. Eng. Chem.,Fundam., Vol. 13, No. 2, 1974
Literature Cited Furusawa,T.,Smith, J. M.,A./.Ch.E. J. 19, 401 (1973). Hashimoto, N., Smith, J. M., Ind. Eng. Chern., Fundarn. 12, 353 (1973). Ruckenstein, E., Vaidyanathan, A. S.. Youngquist, G . R., Chern. Eng. Sci. 26, 1305 (1971). Sargent, R . W. H., Whitford, C. J., Advan. Chem. Ser. No. 102, 155 (1971). Satterfield, C. N., "Mass Transfer in Heterogeneous Catalysis," pp 5677, M.I.T. Press, Cambridge, Mass., 1970. Schneider, P., Smith, J. M.,A.CCh.€. J. 14, 762 (1968a). Schneider, P., Smith, J. M.,A./.Ch.E. J. 14, 886 (1968b). Suzuki, M., Smith, J. M., Chern. Eng. J. 3, 256 (1972). Wakao, N., Oshima, T., Yagi, S.,Chem. Eng. Jap. 22, 780 (1958) Wakao, N., Smith, J . M., Chem. Eng. Sci. 17, 825 (1962).
Received for review April 25, 1973 Accepted August 27, 1973
