I
J.
D. TINKLER
and A. B. METZNER
University of Delaware, Newark, Del.
Reaction Rates in
. .
Nonisothermal Catalysts Temperature gradients within a catalyst particle can be important factors in a chemical reaction. “Effectiveness factors” for nonisothermal catalyst particles will be useful in achieving optimum reactor design and operation
I N
THE
DESIGN
AND
OPERATION
Of
chemical reactors using porous catalyst pellets, rates of both mass transfer and chemical reaction must be considered. Because of resistance to mass transfer within the catalyst structure, diffusion of reactants and products into and out of the pellet is often the slow or rate-controlling step. As a result, the reaction rate may be much smaller than it would be if a diffusional resistance were not present, The effectiveness factor, first used by Thiele (74) in 1939, is defined as the ratio of actual conversion rate to the rate which would occur if reactant concentrations everywhere inside the pellet were the same as a t the surface. Equations for the effectiveness factor have been derived by Thiele and by Wheeler (77, 78). However, in both cases it was assumed that the catalyst particle is isothermal and, therefore, that the reaction rate constant does not vary with position. Prater (10) has pointed out that a significant temperature difference may exist between the interior and the surface of the pellet because of the heat of reaction. For instance, in dehydrogenating cyclohexane on a platinumalumina reforming catalyst, Prater estimated that the center temperature may be 53’ C. lower than that of the surrounding fluid. Obviously, the assumption of no temperature gradients within the pellet is not valid here. Thus, a theory which accounts for heat as well as mass transfer is needed. The nonisothermal effectiveness factor is defined as the ratio of conversion rate to the rate which would occur if both reactant concentrations and temperature were the same everywhere inside the pellet as a t the surface. Schilson ( 7 7 ) calculated the effectiveness factor, as well as intraparticle temperature and concentration profiles, by solving an integral equation by the method of successive approximations. This approach clearly illustrates the magnitude of the effects involved, but
as only a few sample calculations were made, no general design criteria were developed. Pigford and Tinkler (9) considered the case in which the temperature difference across the pellet is small but not negligible. By means of a perturbation technique, they developed a simple extension of Thiele’s result to account for heat generation in the case of a first-order, irreversible reaction inside a spherical pellet. I n the work reported here the nonlinear differential equations expressing conservation of mass and energy for reactions in nonisothermal particles were considered. Solutions were obtained for first- and second-order irreversible reactions taking place in spherical catalyst pellets. Calculations were also made for a first-order reaction taking place in a “flat-plate,” to obtain some insight into the effect of particle geometry. The reader may compare the present results with those obtained independently and approximately simultaneously by Carberry (6) and Weisz (76)*
Working Equations Derivations of the differential equations are based upon the following assumptions : 0 Catalyst particles have a porous,
homogeneous structure in which catalytic activity per unit area is independent of position in the particle. 0 Surface concentrations of reactants and surface temperature are known. If external resistances to mass and heat transfer exist, the surface conditions may be calculated by means of transfer coefficient correlations presented in a variety of standard references and recent reports (3,5,8,73). 0 Mass transfer within the pellet occurs by diffusion only and may be expressed by means of a constant diffusion coefficient. I n many commercial silica-alumina catalysts, Knudsen diffusion prevails up to gas pres-
sures of about 10 atm. (77) ; as similar conditions of pore size prevail in most good catalysts this mechanism is one of some generality, and it is for this case that the assumed temperature and concentration independence of the diffLisivity is either a good approxirnation or a well established fact, respectively. Transport by thermal diffusion and by surface migration of physically adsorbed layers is ignored in this work. The latter may not be small under low temperature conditions, but as neither its importance nor mechanism is well defined at the present time no alternative assumption is possible. .Heat transfer within the particle occurs by conduction through the solid portion of the pellet only. Simple calculations show that the additional enthalpy transport of the gases is negligible unless heat capacities of reaction products differ enormously from those of the reactants. This would require that the heat of reaction be a much stronger function of temperature than for any catalytic reaction known a t the present time. .Constant values of the thermal conductivity and heat of reaction are used. These approximations are expected to be excellent under the conditions encountered. .The temperature dependence of the reaction rate constant is expressed by the Arrhenius equation. .Specific reaction rate and stoichiometric expressions must be chosen. The problems analyzed herein are for the first- and second-order reactions whose stoichiometry and mechanism are both given by:
Since the common application of these equations is in catalysis of gaseous reactants, concentration terms are expressed in terms of partial pressures for convenience of application. However, no restriction to gaseous reactants or products is either intended or necessary, and the results may be directly applied to liquids by making self-evident and consistent changes in the concentraVOL. 53, NO.
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663
tioiis, reaction rate constants. and diffusion coefficients. The assumption of a constant diffusjvity is not a particularly good one in this case, however. Derivation for Spherical Pellets. T h e conservation of mass and ecergy equations for a differential spherical shell of radius r and thickness dr are ( 4 ):
yields an equation containing temperature as the only dependent variable: 1
(r2
$)
- AHk [PI.
nX
DAAH
+
( T - T,L]? = 0
(10)
Introducing the dimensionless variables:
just as in thc isothermal case. However, the partial pressure gradient (dPA/dr)n will be different for nonisothermal particles and is related to the temperature gradient a t the surface, (dT/dr),. through Equation 5. Combination of Equations 15 and 5 and the introduction of the dimensionless variables, 0 and p , yield the final Lvorking equation: v
Equation 10 becomes 1 (where f; is the rate of formaiion of component i caused by reaction per unit volume and v i is the whole number stoichiometric coefficient of component i in the general chemical reaction, Zv,Mi = 0 ; vi is positive for products, negative for reactants. The value of the ratio, fi/vi, is the same for all components), The boundary conditions are : u-henr
dPi/dr = dT/dr = 0
=
d (11)
where 01 = R2k, P.4z-1/DA, a dimensionless parameter. The ratio of rcac.ionrate constants may be expressed as a function of dimensionless temperature by means of the Arrhenius equation:
(16)
= -3(d6jdp),=1.0
in which (dO,'dp),= 1.0as evaluated from Equations 13 or 14 will be a function of cy and p. The classical result of Thiele ( 7 4 ) is a special case of these equations corresponding to 0 = 0. Thus, for the first-order case, integration of Equation 13 produces :
e
=
sinh d sinh
1
i p
ap
(17)
+
and the effectiveness factor is obtained from Equations 16 and 17 as:
0
because of symmetry, and:
T=T,
r = R
at
and
Pi = Pi,
Sincefi is a function of both temperature and partial pressure, the energy balance and the material balances are coupled. The equation connecting them is derived as follows: Multiplication of Equation 1 by AH/:vi and addition to Equation 2 eliminate the term containing fi:
(3)
Integration from
Y
=
0 to r = r gives:
A second integration from 1' = r to r = R results in an expression relating partial pressure of component i to temperature and known surface values, P,, and T,:
sionless parameter. The error introduced by the approximation in Equation 12 is not particularly significant in calculating particle effectiveness. LVhile k / k g may be appreciably in error under fairly extreme reaction conditions. any effect on the effectivenms factor is an integrated one and much smaller in magnitude in view of the small effect which the reaction conditions in the central part of the pellet have on over-all rate for the entire pellet. This is especially true in the case of the second-order reaction. Finally, for the first-order reaction :
and for the second-order reaction :
The boundary conditions in dimensionless form are: Before proceeding further, specific reaction mechanisms must be chosen. Considering the stoichiometric and mechanistic equation: nA+B
then fB
=
k
P.kn
n = l , 2
v.t, =
--n
(6)
vg
=
+1 (7)
Substitution of these relationships into Equations 2 and 5 gives:
At
p =
0
Atp = 1.0
d6/dp
=
0
e=o
The dimensionless temperature profile is a function of only two parameters, a and 0. (If the simplifying assumption employed in Equation 12 were not used, a third parameter would necessarily appear.) The effectiveness factor of the catalyst particle is given as:
4; reduces to
the Thiele modulus, I t may appear that a paradox exists, in that Equation 17 may be used to calculate a finite temperature difference whereas Thiele presupposed an isothermal pellet. However, the same reaction rates are calculated whether ihe particle is isothermal or whether it is not but the activation energy is zero (rate constant independent of temperature). One is somewhat more internally consistent if one admits the necessary coexistence of temperature and concentration gradients, hence the temperature differences of Equation 17, and simply lets the rate constant be non-temperature dependent. Derivation for the Flat Plate. This geometry applies to all particle shapes in the limiting case of very high reaction rate (low penetration of reactants into the particle), and comparison with spherical pellet results also shows the effect of particle geometry. T h e balances representing conservation of mass and energy are derived for a differential slice of m.idt!i d.x located a t a distance Y from the mid-plane of the plate and become analogously to Equations 1 and 2 for the case of spherical geometry : and
R
\/k,lDa.
(73:).
with the boundary conditions
Combination of Equations 8 and 9
664
INDUSTRIAL AND ENGINEERINGCHEMISTRY
dPi/dx = d T / d x
=
Pi
=
=
Pia and T
0
T,
at x = 0 at x =
L
N O N IS0 TI4 ERMA L CATALYSTS 10.00
10.00
I
1.00
.oo
F
F
0.10
0 IO
\I
EAHPA, DA
\
Y.
1111 I i i I
mTe2X BROKEN LINES INDICATE EXTRAPOLATED REGION
BROKEN LINES LNDICATE EXTRAPOLATED REGION
0.01
I
I .o
0.2
10.0
Figure 1. Values of a and E determine magnitude of effectiveness factor for first-order reaction within a spherical pellet
iManipulations similar to those used in the case of spherical geometry give, in dimensionless form :
Again, for the special case /3 = 0 the equations may be solved analytically, and one obtains Thiele's result: 17 =
tanh
l/o(
~-
6-
(23)
Results Method of Solution, Accuracy. The equations developed represent nonlinear boundary value problems in one dimension. Since partial pressure profiles and effectiveness factor may be calculated from temperature distribution, the only differential equation to be solved is the dimensionless energy balance. An analytical solution to a nonlinear, ordinary differential equation of this type is very difficult, if not impossible, to obtain. For this reason the equations were solved on an Electronic Associates' Model 16-31R PACE computer using a programming procedure reported elsewhere (75). The PACE computer is capable oi very precise and accurate solutions; component accuracies arcs on the order of *0.01% of full scak value. How-
Figure 2. For a second-order reaction within a spherical pellet, the effectiveness factor curves change shape
ever, accuracy must be examined iiidependently because it is a function of the type of equation being solved, electronic circuit used, and transformation scale factors, as well as being dependent on the approximation introduced by Equation 12. Analog computer solutions are compared with the analytic solutions of Thiele (74) and two numerical solutions of Schilson ( 7 7) in the table for spherical pellets and for first- and second-order, irreversible reactions. I n cases 1 through 4, cy values extend from the kinetically controlled region, in which the effectiveness factor is approximately unity, to the diffusion-controlled region, in which the effectiveness factor is
reduced to a small fraction. Throughout this range analytical and analog solutions agree very well. To illustrate his numerical procedure, Schilson worked several sample hypothetical problems using fictitious numerical values. Only two of his problems are within the range of variables likely to be encountered in practice and, therefore, included in the present calculations. The results are presented as cases 5 and 6 in the table. For case 5 , in which Schilson considered the temperature dependence of the Knudsen diffusion coefficient, the coefficient evaluated at the surface temperature is used in calculating cy.
Comparison of Computer with Analytical and Numerical Solutions for Reactions within a Spherical Pellet Shows Agreement Reace, 17 tion % ?
Case O r d r
1 2
1 1
3
1
4 5
1 1
a:
1.0 9.0 100 900 18.4
p
Calcd.
0.0 0.0 0.0 0.0 -27.9
0.149 0.0778 0.0100 0.00111 0.0531*
Analog Diff." Crtlcd.
Analog
Diff.a
0.149 0.0778 0.0100 0.00111 0.0540"
0.936 0.666 0.272 0.0975 0.747
0.3 0.6 0.7 0.8 5.4
0.0 0.0 0.0 0.0 1.7
0.939 0.672 0.270 0.0967 0.709
Ref.
(14) (14) (14) (14) (11,
Probl. 2)
6
2
13.1
-33.5
0.0327d 0.03226 1.5
0,461
0.435
5.6
(11.
Probl.
1)
computer result - calculated result X 100. * T , = 352' C.: T, = 390.4' C . calculated result T , = 352' C.; T, = 391.0' C. d T , = 327' C . ; To = 370.3' C. e 7'8 327" C ; Tc = 369.6' C . a
70Difference
=
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665
10.00
I0.00
I O 0
F
aio 0 IO
0.10
I I
'
0.01 0.2
EAHPAs Dn EBROKEN L I N E S INDICATE E X T R A P O L A T E D REGION
I .o
,
1 ~
I I
~
I ' I
~
! I l l
1
~
1
'
1
I
1
10.0
0.01 0.2 Figure 3. Effectiveness factors for first-order reaction within a "flat plate" differ from those for spherical geometry, except when E lies between - 2 and $2
I n cases 5 and 6, differences between analog computer solutions and numerical solutions are due not only to machine error but also to the approximation introduced into Equation 12 for the ratio of rate constants. This approximation is very good for small activation energies and/or small temperature differences across the pellet, but for the large differences in temperature combined with high activation energies as encountered in these examples (AT = 40' C., E = 30, and 18 kcal., respectively), the approximation is not particularly good, and percentage error in the effectiveness factor exceeds 5%. However, even this is not excessive in view of other uncertainties generally involved in reactor design. I n general, the present calculations may be assumed accurate whenever the effectiveness factor does not exceed that for isothermal conditions by more than a factor of about 2.0. At higher values, the simplifying assumption of Equarion 12 is no longer quantitatively valid because of the large temperature differences in the pellet under these conditions. There is no such limitation on the present calculations in the endothermic region, as the central portions of the catalyst contribute negligibly to over-all reaction rate under conditions of a large temperature drop. Ranges Covered. T h e Thiele modulus, which is identical to &, was varied over a wide range, extending from the completely kinetically controlled region well into the diffusion-controlled region.
666
IO 0
f i = R T f i Figure 4. Temperature a t the center of a spherical pellet for first-order reaction i s determined b y vclucs of Q and e
T h e characteristic dimension in this modulus was taken as equal to R for the spherical pellet and as 3L for the flat plate solution, as suggested by Aris (2). He noted that in the "isothermal" case, the effectiveness factor is nearly independent of particle shape if the characteristic dimension of the particle is defined as 3Vp/Sp, where V,, and S, are volume and external surface area, respectively, of a single particle. The introduction of a new parameter defined as:
is made so that the particle dimension appears in only one of the two parameters required to describe the physical situation. This parameter may be called a heat generation parameter, since it includes the activation energy, heat of reaction, pellet thermal conductivity, and surface temperature; it is negative for exothermic reactions and positive for endothermk reactions. To determine the range of parameter values which should be included in the calculations to cover essentially all reactions of practical interest, consider the numerical values arising in three typical kinds of reactions. M'heeler (77) considered a gas-phase cracking reaction at 500' C. and atmospheric pressure. For a molecule like cetane (C16H34)and a mean pore radius of 30 A., the effective Knudsen-diffusion coefficient is 2 X 10-8 gram mole per
INDUSTRIAL AND ENGINEERING CHEMISTRY
I O
second-cm.-atm. The thermal conductivity of silica-alumina cracking catalysts, as determined experimentally by Sehr (72),is 4 X lop3cal. per second-cm.-' C. A heat of reaction of +10 kcal. per gram mole and an activation energy of 40 kcal. per gram mole give E = 0.017. Love and Emmett (7) studied ammonia decomposition on an iron ammonia synthesis catalyst at 387' C. and atmospheric pressure. For one catalyst. calculated activation energy was 31.7 kcal. per gram-mole. Using a diffusion coefficient of 6.5 X 10-7 gram-mole per second-cm.-atm., a thermal conductivity of cal. per secondcm.-O C., and heat of reaction of +11 kcal. per gram mole of ammonia one obtains E = 0.26. The vapor-phase hydrogenation of benzene at 112' C. and 1 atm. on an alumina-supported platinum catalyst was studied by Amano and Parravano ( 7 ) who found the activation energy to be 12 kcal. per gram-mole. The heat of reaction, -50 kcal. per gram-mole, is unusually large. At 112' C. and for a pore radius of 30 A., the effective Knudsen-diffusion coefficient of hydrogen is 3.1 X 10-7 gram-mole per secondcm.-atm. Since the catalyst support is alumina, the value of the thermal conductivity for silica-alumina cracking catalysts may be used here. A value of B = 1.9 results. I n these three examples, the absolute value of E changed by two orders of magnitude. For most practical cases, the absolute magnitude of E will not
N ON I S O T H ERMAL CATALYSTS exceed that in the third example because of the large heat of reaction and the low temperature level of this reaction. However, to be sure that virtually all reactions of interest may be included, E was varied from - 5 to 5 in this work. The value of E will probably exceed these limits only if the larger diffusion coefficients of ordinary diffusion, through large pores, are substituted for the smaller Knudsen coefficients used in the present study. The entire validity of studies like the present rests on the assumption of constant diffusivities, however, and this assumption is likely to be grossly in error under conditions of ordinary gaseous diffusion. Obviously, there would be no point to extending the ranges over which E is varied unless the Stefan-Maxwell equations for multicomponent diffusion were employed simultaneously. Tabulated a n d Graphical Results. Effectiveness factors are shown graphically in Figures 1-3; curves of the dimensionless temperature at the center several are plotted in Figures 4-6; typical temperature profiles are given in Figure 7 . More complete profiles, as well as the tabulated results, are available elsewhere (75). Figures 1 and 3 clearly show that for the case of first-order reactions the
+
6
=
R
effectiveness factor may be several times larger than that calculated by neglecting heat effects, as the over-all reaction rate is greatly increased. The temperature rise increases the reaction rate constant sufficiently to offset the effect of decreased partial pressures created by the diffusional resistance, a t least until very high values of the Thiele modulus are reached. At small values of the Thiele modulus, the heat generation parameter and the Thiele modulus have little effect upon reaction rate. I n this region, the effectiveness factor approaches unity because the particle offers negligible resistance to heat and mass transfer. At intermediate values of the Thiele modulus in the exothermic region, the effectiveness factor may pass through a maximum a t a given value of the heat generation parameter, as the Thiele modulus changes. Therefore, for a given reaction occurring in a specific catalyst at set operating conditions, there may be an optimum pellet size. Pellet size becomes more critical as the e value increases. Because of the sharply peaked nature of the maxima, operation a t this optimum condition presents difficulties. For example, small changes in the rate constant caused by catalyst fouling or changes in fluid tem-
perature may cause great decreases in over-all reaction rate. Obviously, design conditions must be well removed from the regions of these maxima. Additionally, the large temperature rises occurring within the pellet under these conditions may, of course, cause degradation or undesirable side reactions: if operation under such conditions really was feasible, one would normally operate with a higher surface temperature and a more nearly isothermal particle to take advantage of higher reaction rates everywhere in the pellet and not just a t its center. At large values of the Thiele modulus, the effectiveness factor is inversely proportional to the modulus; thus, the rate per particle is proportional to external surface area. In this region, chemical reaction is confined to a thin shell a t the pellet surface. The curves of dimensionless center temperature bear a striking resemblance to those of the effectiveness factor. At small values of the Thiele modulus, neither the heat generation parameter nor the Thiele modulus has much effect upon the dimensionless center temperature. At intermediate Thiele modulus values, maxima may appear in the exothermic region; it is here that large temperature rises may occur within the
m
Figure 5. Center temperature and maximum temperature rise for second-order reaction within a spherical pellet are obtained from these curves
Figure 6. Midplane temperature for first-order reaction within a “flat plate”
VOL. 53, NO. 8
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667
x 0 16
LY 0 14
@ 0 12
0 10
EAHL2k,P,,” for flat plate _____ RgT,2h = dimensionless parameter = @,‘a = effectiveness factor = dimensionless temperature T, - T for sphere = AHR2k,PA& “/A T, - T = __-___ for flax plate AHL2k,PA,”/ X = thermal conductivity of pellet, B.t.u./(sec.) (ft.) (” F.) = whole number stoichiometric coefficient of component in general chemical reaction, 2 v L M 2 = 0. Positive for products, negative for reactants = dimensionless distance = 7, R for sphere, x / L for flat plate
-
CD
E
0 08
7
0 0 06
0 04
A Y
0
02
06
04
08
P
P
Figure 7. Typical dimensionless temperature profiles are shown as functions of a: for first-order reaction within a wherical pellzt when E = - 1 (left) and E = - 3 (right)
p
Subscripts c
i pcllct. At large values of the Thiele modulus, all the curves approach the curve for E = 0 ; in this region the central portion of the pellet is isothermal since the chemical reaction occurs only near the surface of the particle. The curves of the effectiveness factor for a spherical pellet, Figure 1, and those for a flat plate, Figure 3, qualitatively have the same shape. In the endothermic region, the agreement is within about 7 076 ; however, in the exothermic region the maxima for the flat plate are larger, more peaked, and occur at larger values of the Thiele modulus than those for the sphere. Thus, particle shape has little effect only if the heat generation parameter lies between + 2 and -2, unless ct is very large (values of the abscissa near 10). A comparison of Figures 1 and 2 or of 4 and 5 shoivs that reaction order has a very pronounced influence on both effectiveness factor and temperature profile within the particle. For a secondorder reaction, the spread of the curves is very small, and no maxima occur in the exothermic region even at the largest value of E studied. This difference is due to the fact that the presence of heat effects is diminished by the quadratic dependence of the reaction rate on the partial pressure of the reactant: maxima would only be expected a t values of the heat generation parameter well below - 5. I n general, heat effects become less important as reaction order increases; however, because of the large differences between the effectiveness factor curves for first- and second-order reactions, it is
668
normal distance from mid-plane of plate, ft. = dimensionless parameter; = R2k, PAsrL-’/DAfor sphere = Lzk,PA.n-l/DA for flat plate = dimensionless parameter; - -E AHR2kaPAan _ 8, TS2X for sphere =
difficult to interpolate or extrapolate for fractional-order reactions or to make predictions for reactions following a Langmuir type of rate expression. Figure 7 shows complete dimensionless temperature profiles (for two values of the heat generation parameter) for the case of a first-order reaction in a sphere.
Acknowledgment
D. E. Lamb provided advice and assistance pertaining to the programming and operation of the computer. Nomenclature
D E
f AH k
L
iM n
P R,
R r
S, T V,
INDUSTRIAL AND ENGlNEERlNG CHEMISTRY
= effective diffusion coefficient in
pellet, 1b.-mo1ejsec.-atm.-ft. activation energy of reaction, B.t.u./lb.-mole = rate of formation per unit volume, 1b.-mole ’cu. ft.-sec. = heat of reaction, negative for exothermic reaction, B.t.u. ’1b.mole = reaction rate constant, 1b.-mole/cu. ft.-sec.-atm.” = normal distance from mid-plane to surface of plate, ft. = chemical symbol for component = empirical order of reaction = partial pressure, atm. = gas constant, B.t.u./lb.-mole-’R. (indicated as script R o n figures) = pellet radius, ft. = radial distance from pellet center, f t. = external surface area of particle, sq. ft. = absolute temperature, R. = volume of catalyst particle, cu. ft. =
s
evaluated at center or mid-plane of pellet = component = evaluated a t pellet surface =
Literature Cited (1) Amano, A , Parravano, G., “Advances in Catalysis,” Vol. I X , pp. 716-26,
Academic Press, New York, 1951. (2) Ark, R., Chem. Eng. Sci. 6 , 262 (1957). (3) Bar-Ilan, M.. Resnick, W., IND.ENG. CHEM.49, 313 (1957).
(4) Bird, R.B., Stewart, W.E., L,i,ghtfoot, E. N., “Transport Phenomena, Wiley, New York, 1960. (5) Carberry, J. J., A.I.Ch.E. Journal F, 460 (1960). (6) Ibih., in.press. (7) Love, E(. S., Emmett, P. H., J . Am. Chem. Soc. 63, 329; 7 (1941). ,-.. .,. (8) Perry, J. H., “Chemical Enginfers’ Handbook,” 3rd ed., p. 547, McGrawHill, New York, 1950. (9) Pigford, R. L., Tinkler, 3. D.. Chern. Eng. Sci.,in press (1961). (IO) Prater, C. D., Ibid., 8, 284 (1958). (11) Schilson. R. E.. Ph.D. thesis, Lniv. of ’ ~~innesota,’Minneapolis.. Minn., 1957. (12) Sehr, R. A,: Chem. Eng. Sci. 9, 145 I1 9581. (13) Shu-lman, H. L., A.I.Ch.E. Journal 1, 253 (1953). (14) Thiele, E. W., IND.ENG.CHEM.31, 916 (1939). (15) Tinkler, J. D., M.Ch.E. thesis, Univ. of Delaware, Newark, Del., 1960. (16) Weisz, P. B., Chem. Eng. Soc., in press. (17) Wheeler, A., ”Advances in Catalysis,” Vol. 111, pp. 250-327, Academic Press, Kew York, 1951. (18) Wheeler, A,: “Catalysis,” Vol. 2 , pp. 105-65, Reinhold, New York, 1955. RECEIVED for review hIarch 15! 1961 ACCEPTED June 7,1961 139th Meeting, ACS, St. Louis, M o . , March 1961. Work supported by fellowship to J . C. Tinkler from National Science Foundation.
