(17) Van Rysselberghe, P., Bull. Acad. Roy. Belgique, (Classe Sc.) 23. 416 (1937’1. Theory of Gases and Liquids,” Wiley, New York, 1964. (18)’Van Rysseiberghe, P., J . Phys. Chem. 41,787 (1937). (8) Miller, P. G., Chem. Rem. 60,15 (1960). (19) Van Rysselberghe, P., Sczeme 8 5 , 383 (1937). (9) Nebeker, E. B., Ph.D. thesis, California Institute of Tech(20) nology, Pasadena, Calif.! 1965; available through Dissertation . . Van Rvsselberghe, P., “Thermodynamics of Irreversible Processes,” Biaisdell Publishing Go., New York, 1963. Abstracts, Inc., Ann Arbor, Mich. (10) Nebeker, E. B., Pings, C. J., J . Phys. Chem. 96,2483 (1965). (11) Pings, C. J., Physica 29, 243 (1963). RECEIVED for review October 19, 1965 (12) Pings, C. J., Nebeker. E. B.. IND.END.CHEM.FUNDAMENTALS ACCEPTED April 4, 1966 4. 376 (1965). , (13) Prigogine, I., Defay, R., “Chemical Thermodynamics,” Excerpt from a thesis submitted by E. B. Nebeker to the California Longmans Green, New York, 1954. Institute of Technology in partial fulfillment of the requirements (14) Prigogine, I., Outer, P., Herbo, Cl., J. Phys. Colloid Chem. for the Ph.D. degree. Work supported in part by a grant from 52, 321 (1948). the Research Corp. and by Contract AF 49(638)-1273 with the (15) Rabinowitch,, E... Wood, W. C., Trans. Faraday Sac. 32, 540 Directorate of Chemical Sciences of the Air Force Office of Scien(i936). tific Research. (16) Seery, D. J., Britton, I]., J.Phys. Chem. 68, 2263 (1964).
(7) Hirschfelder, J. O., Curtiss, C. F., Bird, R. B., “Molecular
EFFECTIVENESS FACTOR FOR POROUS CATALYSTS Langmuir-Hinsheluiood Kinetic Expressions for Bimolecular Surface Reactions GEORGE W.
ROBERTS AND CHARLE’S N. SATTERFIELD
Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, Mass.
A generalized method of predicting the catalyst effectiveness #actor has been developed for kinetic expressions of the Langmuir-Hinshelwood (Hougen-Watson) type, for the case of two reactants that combine by a second-order surface process, where inhibiting effects of either reactants or products may be significant. The catalyst efFectiveness factor may exceed unity under some isothermal conditions. A region where the effectiveness factor is not uniquely determined by specifying the reaction conditions at the outside surface of the catalyst also exists and the implications of this multiplicity are discussed. Generalized charts for computation are presented and the method is illustrated by a typical calculation.
GENERALIZED
method (6) has been developed for predicting
A the effectiveness factor of a porous catalyst, when the intrinsic reaction kinetics are described by the type of Langmuir-Hinshelwood (Hougen-Watson) rate expression shown in Equation 1. 1 dnA r = ~ I P A / ( ~KAPA K B ~ B . . .) = -(1) VdT This can be derived by assuming that the rate-limiting step is either the surface reaction or the nondisassociative adsorption of a single reactant and allows for possible inhibition of the reaction rate by either reactants or products. This form will be referred to as Type I kinetics. T h e present paper considers the case of the rate expression shown in Equation 3, which is derivable assuming a single kind of adsorption site, and that the rate-limiting step is the reaction between two adsorbed reactant molecules. Rate equations of this form have been applied to the description of the kinetics of a substantial number of catalytic reactions, including, for example, the hydrogenation of codimer (70),the formation of phosgene ( 3 ) , and the dehydration of ethanol (2). There appears to have been no previous generalized treatment of effectiveness factors for reactions in which more than one reactant is involved.
+
+
+
-
T h e stoichiometric equation for this group of reactants is
...
A+bB+xX+yY+
(2)
This expression includes the case of two or more molecules of A reacting with one another, in which case B = A. The reaction rate is assumed to obey the equation
K,p,
+.
*
.)’ =
--V1 dnA dT
(3)
When A is the only reactant,pB is replaced bypA in Equation 3, and KBPs disappears from the denominator. I t will be assumed that the catalyst mass is infinite in two directions and of thickness L in the third, and is exposed to a reactant gas on one face and sealed on the other. I t will further be assumed that the effective diffusivities of all species are constant but not necessarily equal, that the pellet is isothermal, and that the ideal gas laws are applicable. By the mathematical procedure used previously, the partial pressure, p i , of any species other than A can be related to pa. Substitution of such expressions into Equation 3 gives
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317
where i is any species other than A and v is taken to he negative for a reactant.
The quantity + L is the usual Thiele modulus for a secondorder reaction in flat-plate geometry. The effectiveness factor, q , of a catalyst mass is defined as the actual reaction rate divided by the rate that would result if internal gradients were absent. Thus
(16) When only one reactant is present-i.e., the numerator of Equation 3 is k2pAZ-x becomes zero. The equations defining w and K are the same as those used previously (6). The quantity w is dimensionless, x has the dimensions of pressure, K the dimensions of an adsorption constant, and k’ the dimensions of a rate constant. A positive value of w is implicitly assumed in the following derivation. For any reaction, a positive value of w will result if the reactant with the smallest value of (Dps/v) is chosen as component A. If this rule is followed, x will also always be zero or positive. A negative value of K , as shown previously, indicates strong inhibition of the reaction rate by adsorption of products (a positive value of K indicates inhibition by reactants). By substituting Equations 5 to 8 into 4, the rate of reaction may be expressed as r =
~ Y A ( P+Ax ) / ( l
+ KPA)’
(9)
When x = 0, the rate equation in 9 approaches zero-order as KpA becomes very large relative to unity, and approaches second order as KPA approaches zero. If Equation 9 is substituted into the differential equation describing simultaneous diffusion and reaction within the catalyst particle, Equation 2 of the previous paper, Equation 10 results.
Case A.
K
=
I n order to use Equation 17, the value of pa,, must be determined. When the effectiveness factor is low, is nearly zero and Equation 17 becomes
The accuracy of Equation 18 is discussed below. In general, the relationship between 2, and $ L is found by substituting Equations 13, 14, and 15 into Equation 12 and integrating the resulting expression. The result depends upon the relative values of Z, and E.
Z, > E/2
=0
When K = 0, inhibition effects by reacting species are nil, and the reaction rate follows a simple second-order relationship, Thiele’s derivation of the effectiveness factor for a second-order reaction (7) is valid only when the reactants are present in stoichiometric ratio. The following is an extension of Thiele’s derivation to a general second-order reaction, with the reactants present in any ratio. For K = 0, Equation 10 may be integrated subject to the boundary conditions PA
Combination of Equations 12, 13, 14, 15, and 16 yields
F is the elliptic integral of the first kind. Z, < E / 2
kF =
[
(dpA/dx) = 0
a t x = L (sealed surface)
(11) sin-‘{[
The result is
-
(2) (7)
318
+ 1.5 E + (2.25 E2 - 3 EZ, - 3 Zoz)1/2
(2la)
a t x = 0 (exposed surface)
PA,s
2 k’RT
2(2.25 E2 - 3 EZ, - 3 Zo2)1/2
3 2,
‘Iz
=
l&EC FUNDAMENTALS
+ + + + +
-
-
3 Z, 1.5 E (2.25 E 2 3 EZ, 3 2. Z, 1.5 E (2.25 E’ 3 EZ, -3 Zo2)1’2
-
Substitution of E = 0 into Equations 19, 20a, and 20b reduces them to the equations for a simple second-order reaction. The equivalent equations in Thiele’s original article, which are in the form of the inverse sine rather than the inverse cosine, contain some errors (8) although the curves shown in his figures are correct. Effectiveness factors were calculated from the above equations in the following manner. The value of E was specified and a value of Z, was assumed. The values of k p and 6~ were then calculated from the appropriate equations, and were used to calculate $ L . The effectiveness factor, q , was then calculated from Equation 17.
Case 8.
K # 0
When K # 0, integration of Equation IO, subject to the boundary conditions in Equation 11, yields
Figures 1, 2, and 3 are plots of 7 us. dM for selected values of for Figure 1, E = 0 ; for Figure 2 , E = 1; for Figure 3, E = 10. Since careful choice of A and B ensures that x will always be positive, only positive values of E have been considered. Comparison of the present results with effectiveness factors for other orders is facilitated by plotting q against the parameter QL, which is defined by Equation 27. Equation 28 gives Q L in terms of the particular rate equation under consideration here. KPA,s;
@L
where +M is a modified Thiele modulus defined by
(observed reaction rate/gross catalyst volume)
This is the definition of + M used previously [Equation IO, (S)], except that the defining equation for k’ is slightly different in the present case, and the term IKI is included. Inclusion of IKI is necessary to make dM both dimensionless and real. In Equation 22, CY is given by a = -IKl
(24)
K
(L’/DACA,J X
+
The value of a,therefore, is either 1 or - 1. Substitution of Equation 22 into Equation 16 gives the following expression for the effectiveness factor.
In order to use Equation 25, the value of K P A , , must be determined. When q is low,, P A , , is nearly zero and Equation 25 becomes
(27)
The value of aLcan be calculated directly from experimentally observable quantities, using Equation 27. I t is necessary to know the correct form of the rate equation only later in the calculation of q , when the value of K p A , S must be known or estimated in order to use the q - Q L plots. Figures 4, 5, and 6 are plots of q us. aLfor selected values of K P A , s . For Figure 4, E = 0 ; for Figure 5, E = 1 ; for Figure 6 , E = 10. The arrows on each line of constant K / I A , ~in Figures 1 to 6 indicate the point a t which Equations 18 and 26 are accurate or @L greater than those to about 1%. For values of corresponding to the arrow, Equation 18 or Equation 26, whichever is appropriate, is an excellent approximation. For orientation, the zero-order curve is shown on Figures 4, 5, and 6 , and the second-order (for E = 0) curve is shown on Figures 4 and 5 . The second-order curve would be essentially coincident with the @ A , * = -0.40 curve on Figure 6 . The curve for a first-order reaction is not shown on any of the figures. However, on Figure 4,it would lie slightly above the KPA,s = +l.O line; on Figure 5 it would fall about halfway between K P A , S = 0 and K P A , S = +l.O; and on Figure 6 it would be essentially coincident with K P A , s = 0. As implied in the discussion of Equation 9, the second- and zero-order curves may be regarded as specific members of the family of curves when E = 0-Le., in Figure 4. When E is not zero, none of the integer-power curves is a specific member of the family. Discussion
Equations 18 and 26 show that q becomes inversely proportional to 6 at low values of q. The accuracy of Equation 26 depends on the actual value of K P ~ , ~ Values . of KfiA,,, were calculated by numerical integration of Equation 22, using a digital computer. Details are given by Roberts ( 5 ) . Results
The effectiveness factor of a catalyst mass, for the present model, is a function of three dimensionless groups, + M , K p A , , , and either lK/xor E . The following results have been presented in terms of the parameter E, rather than iKI x, because E has a clearer physical meaning. E, defined by Equation 13, is related to the stoichiometric excess of B over A in the gas outside the catalyst mass, and indeed equals the stoichiometric. excess if Q B = D A . For example, if the bulk stream contains a 100% excess of B over A, the ratio of PB to P A in the bulk is 2vB. With DB = D A , Equation 13 gives E = 1. With a 20070 excess of B over A and with DB = D A , E = 2.
Two characteristics of the curves on Figures 2 and 3 deserve comment. In the first place, effectiveness factors greater than unity result over a range of values of dAW,when K p A s s = 10 or 100. This is a consequence of the fact that the rate equation, Equation 9, possesses a maximum under certain conditions. In terms of the Langmuir-Hinshelwood model, this maximum results from a competition between the two reactants for sites on the catalyst surface. If A is strongly adsorbed relative to B, an increase inp, will displace B from catalytic sites if the partial pressure of A is high. Displacement of B from the surface tends to decrease the reaction rate. By differentiating Equation 9, it can be shown that effectiveness factors greater than unity will result when K p A , s is greater than [ ( E 2 ) / E ] , if x is always positive. This is in agreement with the results shown in Figures 1 to 3. Secondly, for & = 10, K p A , s = 10 or 100, and for E = 1, K P A , , = 100, a range of +dw exists for which q is a multiplevalued function of d M . Multiple-valued effectiveness factors with values greater than unity have previously been calculated to occur for first- and second-order reactions occurring in nonisothermal systems (7, 9, 7 7). However, these effects have not
+
VOL. 5
NO. 3
AUGUST 1966
319
Modified Thiele Modulus, + M Figure 1.
Effectiveness factor os a function of the modified Thiele modulus, $M, for selected values of
5'0r ;'I
KPA,,
E = O
0 0
I n I an C 3
.-9
. I -
V
z2 0.10
L
0.05 0.02 0.0050
310
Figure 2.
10.0
1.0
0.10
100.0
Modified Thiele Modulus, +M Effectiveness factor as a function of the modified Thiele modulus,
$M,
for selected values of KpA,.
E = 1.0
previously been reported for isothermal catalyst reactions. A consequence of the multiplicity in the 7 - 4~ curve is that the steady-state reaction rate may not be uniquely determined by specifying the conditions outside the catalyst pellets. The direction from which steady state is approached may instead determine which effectiveness factor is eventually realized. Further, operation may be unstable in certain regimes; the intermediate values of q in the triple-valued regions of Figures 2 and 3 are probably not attainable in steady-state operation. 320
l&EC FUNDAMENTALS
Figures 4, 5, and 6 show that when K,6a,s is negative, the curves can lie considerably below even the curve for a secondorder reaction. This confirms the previous conclusion (6) that strong inhibition of the reaction ratc. by reaction products causes a diffusional retardation to set in under milder conditions than would be predicted by the s1ab:geometry analogy to Weisz's criteria (72). The 7 - QPLcurve for the present rate expression, Equation 3, always lies below the curve for the previously considered rate expression, Equation 1, if Kpa,,
1
0.010 0.002
u 0.
Modified Thiele Modulus, + M Figure 3.
Effectiveness factor as a function of the modified Thiele modulus,
4 ~for ,
selected values of
KPA,~
E = 10.0
1.00
LL
3 0.10 C
a
.->
0.01
u
I .o
0.10
10.0
QL
Figure 4.
Effectiveness factor as a function of the modulus @L E = O VOL. 5
NO. 3
AUGUST 1966
321
10.0
1
I
1.0 L-
O
c
0
z
cn cn
W K
W
.-c> 0 w-al
0.10
I 0.010 0.0010
I
I
1 I I l l 1
1
0.010
1 1 I I l l
I 1 1 1 1 1 1 10.0
I
I 1 1 1 1 1
I
I
1.0
0.10
IOO.0
@L Figure 5.
K
+-F:
0
Effectiveness factor as a function of the modulus +L E = 1.0
1.0
I
I I I I Ill
.-c
0 W
YLc
w 0.10
IlIk
-
-
0
W K > W
I I I
-
-
LL v)
I
L t
tu
o~o'oo.o 0 IO
I
0 IO
1
I I I I I I
0.10
I
I
I I I Ill
1.0
I
1
I I I Ill
@L Figure 6.
Effectiveness factor as a function of the modulus @L E = 10.0
322
I h E C FUNDAMENTALS
10.0
100.0
I
I IIIII
I
I
I
I 1 1 1 1 1
KpA,*=O, E510.0 K P ~ , ~ = OE=I.O ,
F 1.0 L
0
c
0 0
LL ln 0,
c
0,
.-5
e
0
5 0.10 aJ
I
I
I'IIIII
1
I
I I I Ill1
1
1 I I I I I
10.0
Figure 7.
100.0
The q vs. @L relationship for representative combinations of Kp,,, and E
is the same in both cases and is negative. This is illustrated = by the dashed curve on Figure 4 for Type I kinetics, -0.90. Figure 5 and 6 also show that when @ A , , is large and E is greater than zero, the tf - @L curve lies above that for a zeroorder reaction. However, in these cases, the diffusional retardation persists to lower values of @L than it does for a zeroorder reaction. The dashed portions of the curves in Figures 5 and G show the region where operation is likely to be unstable-Le., the region corresponding to the intermediate value of the effectiveness factor in the triple-valued region. The dashed lines are used only to join the stable portions of each curve, and do not define the tf - @ L relationship in the unstable region. In some portions of the unstable region, 7 is a multiple-valued function of aL,but is alwayj a unique function of B L in the stable portion of the curve. Figure 7 is a crossplot of the results in Figures 4, 5, and 6, and illustrates the effect of the modified stoichiometric excess, E, on the effectiveness factor a t constant K ~ A , ~For . Kpa,. = -0.90, the E = 0 curve lies lowest, the curve for E = 10 lies highest, with E = 1 intermediate. However, the three curves lie so close together that they are almost indistinguishable. When KpA,, = 0, the order of the curves is the same; the spread is somewhat greater but still rather small. For KpA,, = 100, curves for E = 0, 0.10, 1, and 10 are shown. The spread between these curves is large, but the order is the same as previously. The fact that the curves for high values of E lie above those for low values of E is consistent with the interpretation of E as a stoichiometric excess. When E is large, p B is in excess throughout the pellet and the product pApB declines more slowly than it would if B were present in stoichiometric amount.
The region in which effectiveness factors greater than unity can theoretically occur in isothermal catalyst pellets is probably actually encountered in a number of hydrogenation reactions. Since hydrogen has a relatively high diffusivity and is usually present in excess over the stoichiometric concentration, values of E in the range of 10 to 100 are common. High values of KpA,s imply relatively high adsorptivity and/or high partial pressure of the compound being hydrogenated. Adsorption coefficients for the species being hydrogenated that are high relative to Hzor to the products are fairly common -for example, that of ethylene in the hydrogenation of ethylene to ethane. However, hydrogenation reactions are exothermic and significant temperature gradients through the catalyst pellet can also cause effectiveness factors to exceed unity. Nevertheless, information now available on the thermal conductivity of various porous substances is sufficient to provide some judgment as to which explanation is the more probable in a given situation. Any general treatment of nonisothermal behavior combined with Langmuir-Hinshelwood kinetics would be complex and involve so many parameters that the results would probably be difficult to use or interpret. Appendix
Illustrative Example. kinetics of the reaction
Tschernitz et al. (70)studied the
+ Hz
i-CSHla (cod imer)
+
i-CBHle
over a nickel-on-kieselguhr catalyst, between 200' and 325' C. and 1- and 3.5-atm. total pressure. Their data were best described by a rate equation of the form r = kzpHpu/(l
+ KHPH+ KuPu + KsPa)'
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323
In the above expression, H refers to hydrogen, U to codimer, and S to iso-octane. At 200’ C., the following values were found: K H = 0.383 atm.-’, Ks = 0.489 atm.-l, and Ku = 0.580 atm.-l. Consider run 3d of this study, which was made a t an average temperature of 200’ C., and with the following partial pressures: f i ~ = , ~ 2.450 atm., pv,s = 0.530 atm., and p s q S = 0.515 atm. The observed rate of reaction was 0.0320 gram mole/gram catalyst, hour. The catalyst was in the form of cylinders 0.292 cm. in diameter and 0.269 cm. long; the bulk density was 1.39 grams per cc. In accordance with the rule for choosing component A, let A be U (codimer). For the purpose of calculating K , assume that the effective diffusivity is inversely proportional to the square root of the molecular weight. Then
@L =
(22.3 x 10-4) (1.37 X
x
(1.24 x 10-5) = 0.0577 X (0.0350)
The diffusional retardation is insignificant, and q is essentially unity, even if the estimate of the effective diffusivity is substantially in error. Acknowledgment
The machine computations were performed a t the Massachusetts Institute of Technology Computation Center. We also acknowledge the financial support of the National Science Foundation to George W. Roberts in the form of a fellowship during the period of this study. Nomenclature
Any consistent set of units may be used. Those specified below are used in (6). C
Parameter w is calculated from Equation 5 w = 1
and
K
+ 0.383 [2.450 - 0.134 X 0.5301 + 0.489[0.515 + 1.01 X 0.5301 = 2.42
K is calculated from Equation 6
= [0.580
+ 0.134 X 0.383 - 1.01 X 0.489]/2.42 = 0.137/2.42 = 0.0565 atm.-l
KPA,s
= 0.0565 X 0.530 = 0.0300
Parameter E is calculated from Equation 13
E = [(2.450/0.134)
- 0.530]/0.530
= 33.6
Examination of the values of K ~ J A and , ~ E shows that, for run 3d, the effectiveness factor could not have been greater than unity. Further, since K P A , S is essentially zero and the value of E is very large, the q - @L relationship will be very close to that for a first-order reaction. The effectiveness factor can therefore be accurately estimated from the K P A , s = 0 line on Figure 6. First, however, the value of @L must be estimated from Equation 27. The quantity L will be approximated as the ratio of the volume to the external surface of the catalyst particles. This approximation has been discussed (6). Thus,
L
0.0472 cm.;
L2
22.3 X 10-4 sq. cm.
CA,sis given by CA,s
= ’As = 0.530/82.06 X 4 7 3 ~
RT
1.37 X 10-5 gram mole/cc. The observed reaction rate per gross catalyst volume is given by (0.0320/3600) X 1.39 = 1.24 X 10-5 gram mole/(cc. catalyst) (sec.) Estimation of D A is difficult since the physical characteristics of the catalyst-Le., surface area, pore-size distribution, porosity, etc.-were not recorded. For the purpose of illustration, it will be assumed that molecular diffusion of iso-octene in hydrogen is occurring; that the porosity, 8, is about 0.50; and that the catalyst has a tortuosity, T, of 2. The molecular diffusivity of hydrogen-iso-octene was estimated to be 0.140 sq. cm. per second, using the method of Gilliland, as given in (4). Therefore, D A
324
2
0‘50 = 0.0350 sq. cm. per second
I&EC FUNDAMENTALS
= concentration, gram moles/cc. D = effective diffusivity, based on total cross-sectional area of catalyst, sq. cm./sec. E = modified Stoichiometric excess, defined by Equation 13 F = elliptic integral of the first kind K = parameter defined by Equation 6 K j = adsorption constant for j t h species in the LangmuirHinshelwood rate expression, (atm.)-l; j = i or A k p = parameter in elliptic integral, see Equations 20a and 21a k l = reaction-rate constant for Type I reaction (see Equation l ) , gram moles/(cc.) (sec.) (atm.) kz = reaction-rate constant for Type I1 reaction (see Equation 3), gram moles/(cc.) (sec.) (atm.)Z k’ = modified rate constant, defined by Equation 7, gram moles/(cc.) (sec.)(atm.)Z k” = second-order rate constant (see Equation 15), (cc.)/ (sec.) (gram mole) L = thickness of catalyst mass, cm. n = number of moles P = partial pressure, atm. R = gas constant, (atm.)(cc.)/(gram mole) (” K.) r = reaction rate, (gram moles)/(cc.) (sec.) T = absolute temperature, K. t = time, sec. v = volume, cc. x = Cartesian dimension, cm. z = parameter defined by Equation 14
GREEKLETTERS = parameter defined by Equation 24 LY q = effectiveness factor, see Equations 16, 17, and 25 fi = approximate effectiveness factor, see Equations 18 and 26 8 = porosity of catalyst, dimensionless v t = stoichiometric coefficient of ith component, taken to be negative for a reactant T = “tortuosity” of catalyst aL = modulus defined by Equation 27 & = parameter in elliptic integral, see Equations 20b and 21b 4L = Thiele modulus for second-order reaction, defined by Equation 15 +M = modified Thiele modulus, defined by Equation 23 x = parameter defined by Equation 8 w = parameter defined by Equation 5 SUBSCRIPTS A = index denoting reactant A (A always has a stoichiometric coefficient of unity) B = index denoting reactant B F = refers to elliptic integral i = index denoting any species other than A = sealed surface, x = L o s = exposed surface, x = 0 literature Cited (1) Carberry, J. J., A.Z.Ch.E. J . 7, 350 (1961). ( 2 ) Kabel, R. L., Johanson, L. N., Zbid., 8,621 (1962).
( 3 ) Potter, C., Baron, S., Chem. Eng. Progr. 47, 473 (1951). (4) Reid, R. C., Sherwood, T. K., “Properties of Gases and Liquids,” p. 268, McGraw-Hill, New York, 1958. (5) Roberts, G. W., 5jc.D. thesis, Department of Chemical Engineering, Massachusetts Institute of Technology, 1965. ( 6 ) Roberts, G. W., Satterfield, C. N., IND.ENG. CHEM.FUNDAMENTALS 4, 288 (19653. ( 7 ) Thiele, E. W.,Znd. E.qg. Chem. 31, 916 (1939). ( 8 ) Thiele, E. W., Notre Dame University, South Bend, Ind.,
personal communication, April 28, 1964. ( 9 ) Tinkler, 3. D., Metzner, A. B., Znd. Eng. Chem. 53, 663 (1961).
(10) Tschernitz, J., Bornstein, S., Beckmann, R., Hougen, O., Trans. Am. Znst. Chem. Engrs. 42,883 (1946). (11) Weisz, P. B., Hicks, J. S., Chem. Eng. Sci. 17, 265 (1962). (12) Weisz, P. B., Prater, C. D., Advan. Catalysis 6, 143 (1954).
RECEIVED for review July 19, 1965 ACCEPTEDMay 2, 1966
EFFECT OF GEOMETRY ON CATALYST EFFECTIVENESS FACTOR Langmuir-Hinshei‘wood Kinetics CHRISTIAN W. KNUDSEN, GEORGE W.ROBERTS, CHARLES N. SATTERFIELD
AND
Department of Chemical Engineering, Massachusetts Znstitute of Technology, Cambridge, Mass.
A generalized chart compares the catalyst effectiveness factor for spherical and slab catalyst shapes for kinetic expres’sionsof the Langmuir-Hinshelwoodtype, covering the case of a single reactant in which adsorption of products or reactant may be significant. For a given value of the modulus @L, the maximum deviation between the value of the effectiveness factor for spherical and for slab geometry varies up to 34%, depending upon the kinetic expression.
method of calculating the catalyst effectiveq, for kinetic expressions of the LangmuirHinshelwood type has been presented (3) for the case of a single reactant A in which adsorption of products or reactant may be significant-Le., a reaction obeying the rate equation given by: GENERALIZED
A ness factor,
geometry, cuts across the family of curves within the range of of about -0.7 to -0.3 over the region of interest. The modulus @ L is defined by Equation 2: @’L=
L2
-X
DACA.~
(observed reaction rate/gross catalyst volume) (2)
I t was assumed that the effective diffusivities of all species are constant but not necessarily equal, that the pellet is isothermal, that the ideal gas laws are applicable, and that reaction occurred in slab geometry-i.e., that the catalyst is infinite in two directions, exposed to the gas stream on one face and sealed on the other. The present paper extends the earlier results to the case of spherical geometry, all other assumptions remaining as before. The mathematical derivation of this problem is described in the thesis of Knudsen (i?),which includes details of the numerical calculation method.. The Fortran program and program flowsheet used are also given in the thesis. The single case of a zero-order reaction was treated by an analytical procedure rather than by machine computation. The results are displayed in Figure 1 as a family of curves (shown as solid lines) of the effectiveness factor, q, us. the modified modulus, ( P L , for various values of the parameter K ~ A , . . For comparison, the family of dashed-line curves shows the results of earlier calculations for slab geometry. K $ A , ~can vary from to -1.0. A zero-order reaction corresponds to a value of K $ A , ~of. -tm , a simple first-order reaction to a value of zero, and reactions strongly inhibited by reaction products have negative values of K ~ A , A ~ . simple secondorder reaction is not EL member of the family but, for slab
+
The dimension L is the ratio of the volume of the catalyst particle to its outside surface. Some of the pertinent literature uses a modulus (Ps for spheres, which is also defined by Equation 2, except that the linear dimension is taken to be the radius of the sphere. Since for a sphere, L = R / 3 , @ L = @,s/9. In any mathematical treatment of effectiveness factors, the assumption of slab geometry as the catalyst shape simplifies the mathematical analysis, even though most catalysts would be much more closely approximated as a sphere. Aris (7) showed that the functions of q us. the Thiele modulus 6 for a first-order reaction in a sphere, a semi-infinite slab, and a cylinder of infinite length lie very close together when the characteristic dimension is defined as the ratio of the volume to outside surface through which reactant can diffuse. However, a similar comparison has not been made for kinetic expressions in general. Figure 1 shows that the q us. @ L functions become identical at very low and very high values of (PL-i.e., when q approaches unity and when q becomes very small in magnitude. Comparison of the two families of curves in Figure 1 shows that the functions for spherical and slab geometry lie closest together for negative values of K p A , * (strong product inhibition or highest order reactions) and show the greatest deviation for a zero-order reaction ( K ~ Aapproaching , infinity). For a specified value of @ L , the calculated effectiveness factor, q, is always greater in slab geometry than in spherical VOL. 5
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