k k,, kc’, k l l Kg, K9’, K l o
= = = =
(3) Jellinek, K., Z. Anorg. Chem. 70, 93 (1911). (4) Kolthoff, I. M., Miller. C. S., J . A m . Chem. Sac. 63, 2818 (1941). (5) Lynn. S., Ph.D. thesis, California Institute of Technology,
literature Cited
(Gf-ginker, K. G.: Gordon. T. P.. Mason, D. M., Sakoida. R. K., Corcoran. \V. H.. J . Phys. Chem. 64, 573 (1960). (7) Schonbein. J.. J . Prakt. Chem. 61, 193 (1852). (8) Schutzenberger. P.. Compt. Rrnd. 69, 196 (1869). (9) Semenoff, N. N., “Chemical Kinetics and Chain Reactions,” Oxford Clarendon Press. New York; 1935.
r)nijuctl,,n
R
T 0
specific rate constant in appropriate units specific rate constants, liters;/mole sec. equilibrium constants in concentration units rate of reaction in induction period, moles/ liter sec. = gas constant, 1.987 cal.;/mole OK. = temperature, OK. = time, sec.
1954
(1) EerrnAk: V., Chem. Zfiesli 8, 714 (1954). (2) Fischer, O., DraEka, O., FischerovA, E., Collection Czech. Chem. Commun. 25, 323 (1960).
RECEIVED for review July 13, 1964 ACCEPTED March 1. 1965
EFFECTIVENESS FACTOR FOR
POROUS CATALYSTS Langmuir- Hiizshelwood Kinetic Expressions GEORGE W.
ROBERTS A N D C H A R L E S N. S A T T E R F I E L D
Department of Chemical Engineering. Massachiisetts Institute of Technolog). Cambridge. Mass.
A generalized method of predicting the catalyst effectiveness factor has been developed for kinetic expressions of the Langmuir-Hinshelwood type for the case of a single reactant in which adsorption of products or reactant may be significant. Generalized charts for computation are presented and the method i s illustrated by a typical set of data for the reaction of carbon dioxide with finely porous carbon. HE
relationships between the “effectiveness“ of a porous
Tcatalyst and the characteristics of the catalyst and of the reaction were first developed mathematically in the United States by Thiele (73).and have since been extended by IVheeler (78), LYeisz and coworkers (76: 77), and many others. T h e present state of development of the theory has been summarized in a recent book ( 7 7 ) . Given knoivledge of the effective diffusivity of the reacting species in the porous catalyst, a body of mathematics exists today that permits the effectiveness factor of the catalyst, q , to be calculated for a wide range of conditions. including various pellet geometries and reaction orders. various degrees of volume change on reaction, temperature gradients within the pellet, and various diffusionflux equations. Almost all of the present solutions are based on the assumption of an integer-po\ver kinetic equation-i.e., a zero-: first-: c)r second-order reaction. Only limited attention has been given to determining the effectiveness factor for cases in Lvhich the kinetics follo\vs a more complex expression, as represented for example, by the Langmuir-Hinshel\vood type of rate equation. To be sure, over a narroiv region of concentration, the Langmuir-Hinshelwood form may be well approximated by an integer-poivei equation. Ho\vever. if the diffusional resistance xvithin the pellet is high-viz., the effectiveness factor is lou-the reactant partial pressure may vary from its value a t the pellet surface dowm to a value approaching zero in the interior of the catalyst. If such is the case, the range of partial pressure \vi11 not, in general, be small and it is necessary to consider the effect of the more complex rate equation on the effectiveness factor. Several analyses of this type of problem have been made 288
l&EC
FUNDAMENTALS
previously. T h e procedure is basically the same as for integerpo\ver kinetic expressions. in that the differential equation for simultaneous diffusion and reaction of the reactant inside the catalyst pellet must be solved. T h e mathematics, however, are much more complex. Chu and Hougen (4) used a numerical technique to calculate values of the effectiveness factor for the reaction .4 + Q , \vhen the rate equation is of the form
r = kK.4.b~. (1
+ KAPA + KQPQ)
Their results are presented as plots of q us. a dimensionless parameter, M , for various values of the mole fraction of reactant a t the pellet surface. and for various values of K A P . All of their solutions are for KQ = 0-that is. no product adsorption-and constant total pressure throughout the pellet is assumed. If diffusion is in the Knudsen or transition region? the total pressure does vary through the pellet to a degree which may be significant in some real cases. This point has been illustrated in a recent article by Otani, \Vakao, and Smith (6). Furthermore, Chu and Hougen used three parameters to specify q , although it is shoivn below that, by a judicious choice of variables. two parameters suffice. Sumerical techniques have also been used to calculate 7 for specific reactions obeying rate equations of the LangmuirHinshelFvood type-namely, the oxidation of NO to NO2 ( d ) and the cracking of cumene to benzene and propylene (7). Several investigators have obtained closed-form solutions for the effectiveness factor by making various assumptions to simplify the mathematical treatment. .4kehata, Namkoong. Kubota, and Shindo ( 7 ) suggest expanding the kinetic equation in a Ta!dor series around the outside concentration, neglecting all but the first tkvo terms. S o comparison of this method with the more accurate numerical technique is avail-
able. Rozovskii and Shchekin (70). using a series of approximations, derived a formula for the effectiveness factor that might be expected to be valid a t low values of 7 ) if the reactants and products all have the same diffusivities. Again, no aiialysia of the accurac!- of this method is available, but the result almost certainly breaks down a t high effectiveness factors. since zero partial pressure of reactant a t the center of the pellet is implicitly assumed. I n addition, the linearization technique of Schilson and Amundson (72) could be applied to Langmuir-Hinshelwciod rate expressions. l ' h e purpose of the present Lvork was: (1) to develop a general method of predicting the effectiveness factor for kinetic expressions of the Langmuir-Hinshelwood type. involving as feiv restrictive assumptions as practicable, a n d (2) to compare the efrectiveness factors as calculated for LangmuirHinshelwood expressions with those for integer-power equations. in order to assess the degree of error incurred by the common procedure of assuming an integer-power relationship for the kinetics. Mathematical Derivation
T h e case to be developed in this paper is the particular kinetic relationship represented by Equation 1. This expression includes reactions in which 4 decomposes or isomerizes by a first-order process. or rtaction of A \t.ith B in which the concentration of B does not appear in the numerator, b u t may appear in the denominator. For the reaction of A a n d B, such an rxpression might result, for example, if adsorption of A on the catalyst is the rate-controlling process. A subsequent paper \vi11 consider the reaction of A and B, when the numerator contains the partial pressures of both reactants. T h e general chemical equation describing reactions under consideration is A
+ bB 4- . . .
+
XX
+ JY + . . .
T h e rate equation is taken to be
Substituting the resulting expressions into Equation 1 gives 7
= kPA
(1
+
-
(Kzut
DA
D2)1
I
c K,[P,,s +
+
(Pa
S~IDA:D*)ll (5)
T h e value of w will normally be positive, but in the case of a reaction having a reactant in addition to A. a negative value of w could result if the second reactant had a very large value of K i and a very small value of Dp, Y. Results of the present computations cannot be used for negative values of w. Further let k' = k w and let K
=
[ K A - DA
(7) i
(Kivi/Di)]/w
(8)
Since w is dimensionless, K has the dimensions of a n adsorption constant, and k' has the dimensions of a rate constant. 4 s the values of K A and the various Kl become so small that the reaction approaches simple first-order, K approaches zero ; a negative value of K indicates that the sum of the groups KvDA Dlfor the products is greater than that for the reactants. Qualitatively. a negative value of K indicates inhibition by reaction products. Using these definitions, Equation 5 reduces to r = k'Pa/(l
+
(9)
K'bA)
Let a modified Thiele modulus Q . be ~ ~defined by
When K = 0, the reaction is simple first-order, a n d $.+I becomes identical to the familiar Thiele modulus. pL, defined by Equation 15. Equation 9 may be substituted into Equation 2 a n d integrated. Substitution of Equation 10 into the result gives
d(KPA)! d ( x / L ) = where index i is used to denote any reaction product or reactant other than A . Slab geometry is assumed-i.e., that the catalyst is infinite in t\vo directions, exposed to the gas stream on one face and sealed a t the other. T h e thickness of the slab is L. I t is further assumed that the effective diffusivities of all species are constant bui. not necessarily equal. that the pellet is isothermal, a n d that i.he ideal gas la\vs are applicable. .4 material balance on component A , over a differential thickness Lvithin the catalyst, gives d2cA = DA
( ) d"pA -1
RT
Da dx2
7 = = I
dx2
Taking the stoichiometric coefficient, Y : of a reactant other than .A to be negative, a similar balance on any other species gives (3) Equations 2 a n d 3 ma) be combined and integrated subject to the boundary conditions of 4 to yield a n expression in terms of P A .
PA
= Pa.s;
PI
=
dpa dx = dp, d~
P i , qa t
=
x =
0 at x
where Pa,o is the partial pressure of A a t the sealed face. T h e effectiveness factor, 7, of the catalyst pellet is defined as the actual reaction rate (which equals the rate of diffusion of A into the slab a t the gas-stream face) divided b)- the rate that ivould result if internal concentration gradients \vere absent. I n terms of present nomenclature
=
0 (exposed surface)
This equation is not useful in itself, since is not known. However! PA .o can be determined by numerical integration of Equation 11: subject to the boundary conditions given in 4. Equation 11 shows that K,bA,o is a function only of oY and K P ~ , therefore, ~ ; from Equation 12. 7 is also a function of these two parameters only. \t-hen the effectiveness factor is OM.? the reactant partial pressure a t the sealed face, P A , * , approaches zero and Equation 12 reduces to
L (sealed surface) VOL. 4
NO. 3
AUGUST
1965
289
1.0 1.00 0.50
0.1c
F: ;0.100 c
u
t
I
Flr?tI O r d e r 4
1
L?
0.05
m Lo m c m
...2
E
r
0.010
w
1 1
I l l
0.005
0.0010 0.010
0.05 0.10
0.50 1.00 5.00 10.0 Modified Thiele Modulus, .#M
50
100
Ratio pA.o/pA,. as a function of modified Figure 1 . Thiele modulus, +.$f, for various values of K P . ~ . ~
Figure 2. Effectiveness factor, q, as a function of modified Thiele modulus, +.$f, for various values of K ~ A , ,
Values of pa,o were calculated by numerical integration of Equation 11, using a digital computer. Details of the numerical technique are given by Roberts ( 9 ) .
(74) and has been used extensively by Weisz and coworkers
(76, 77). T h e new dimensionless modulus, @L =
(observed reaction rate, gross catalyst volume)
T h e results of the numerical calculations are shown in Figures 1, 2, and 3. Figure 1 is a graph of (pA,,,/pA,,) us. q M for each of a number of values of K p A , , ranging from -0.98 to +50.0. T h e lines for KpA,8 of -0.10 and + O . l O are essentially coincident with the first-order line. T h e lines for KpA,, of -0.98 and -0.95 are coincident with that for -0.90 as shown on Figure 1. T h e lines for a constant value of Kp,,, are drawn of finite length. T h e left-hand terminus of a liqe occurs a t the value of @u corresponding to 7 greater than 0.95. T h e righthand terminus occurs a t a value of +,lf for which (< - 7 ) is less than 0.005. Thus, for values of 4.,< exceeding the righthand terminus, Equation 13 is very accurate. For values of KpA,, greater than +50.0 Equation 13 is also accurate to within 5y0,except for values of 7 greater than 0.95. T h e ends of the K p A , , = -0.95 line occur a t G.,~ values of 1.00 and 0.02; that of the -0.98 line a t q.rr values of 1.00 and 0.008. for the same Figure 2 shows the effectiveness factor, 7 , LIS. range of values of KpA,, that is represented in Figure 1 . Equation 1 3 is accurate to within about 0.005 for the region to the right of the dashed line in Figure 2. T o compute the effectiveness factor, k' a n d KpA,, must first be calculated. T h e values of k' and DA are then used to calculate $.M. T h e and K p A , , may then be used to locate 7 on Figure values of 2 . If interpolation between lines of constant Kp,,, is necessary, the calculation of several values of 6 will help fix the position of the desired line. I n some cases it may be more accurate to interpolate a value of from Figure 1 , and use Equation 1 2 to calculate 7 , but the relative accuracy of these procedures has not been checked. Interpolation is not necessary if the point falls to the right of the dashed line; Equation 13 can be applied with good accuracy in this case. Comparison of Effectiveness Factors Using Different Kinetic Expressions. An illuminating way to compare effectiveness factors for Langmuir-Hinshelwood kinetics with those for integer-power equations is by relating 7 to a modulus containing only quantities which can be either observed or predicted. This approach was first suggested by TVagner I&EC
FUNDAMENTALS
as
( L 2 / D A C ~ , ,X )
Results
290
aL, is defined
(14)
For any integer-power rate equation of order n, Equation 14 reduces to @L
=
adL2
where
and for the type of Langmuir-Hinshelwood rate equation considered here @L =
[email protected](1
+
KPA.8)
aL, as
defined by Equation 14: is for slab geometry and differs from the modulus a, which has been used extensively by \.t-eisz and coworkers for analyses in spherical geometry. T h e latter is given by Equation 16.
(observed reaction rateigross catalyst volume)
(16)
Aris (2) sho\ved that the functions of 7 11s. q for a first-order reaction in a sphere, a slab, a n d a cylinder of infinite length lie very close together when the Thiele modulus, @! is based on a characteristic dimension equal to the ratio of the volume of porous mass to the outside surface through which reactants have access. T h e use of slab geometry greatly simplifies the mathematics in any analysis of effectiveness factors, so it has been tempting to apply the results of calculations on slab geometry to make predictions for other shapes by defining the characteristic dimension in the above fashion. However, the effect on the 7 - 4 relationship of changing geometrical shape has not been generally established for kinetic expressions other than first-order. For the infinite slab sealed on one face, the characteristic dimension of .4ris is exactly equal to L , while for a sphere it is equal to R,i'3. Thus, if the effect of pellet geometry on v could be essentially removed by using this characteristic dimension, the present results on slab geometry could be
~
compared directly with previous analyses on spheres by replacing ar,with 9. \\'eisz (77) has suggrsted that the criterion for the absence of difEusion effects is a value of '7 exceeding 0.95. For a second5 0.3; order reaction in a sphrre. this corresponds to about for first order. to Qs 5; 1.0; and for zero order. to @\ 5 6 . T'hr corresponding-values of become 0.033:0.1 1, and 0.66. respectively. Since the range from -zrro. to second order seemed to cover most cases of interest. \Vrisz siiggrstrd that. 2 6. diffusion eftpcts \vi11 definitely be present. and if if Q T 5 0.30. difTurion efTectr are insignificant. Figure 3 is a graph of 7 2 ' 1 . for various values of K p A , s and for first-. second-. and zero-order reactions. l ' h e curves for first- and zero-order expression5 may be regarded as specific members of the famil)-. T h a t for z t ~ oorder corresponds to a value of Kp.4,a approaching infinity and that for first order: to a value approaching zero. T h a t for a second-order expression. ho\vever. cuts across the family of curves. A negative value of Kp, , s indicates significant product adsorption effects. as sho\vn in the discussion of Equation 8. T h e minimum possible value of Kp;PA,s is -1. Using this plot, 7 can be estimated directly for any experimental run \\ithout a trial-anderror procedure. T h e error involved in the use of a n integerpower approximation can also be estimated for any case from Figure 3. Several conclusions can be drawn from Figure 3.
,
1. Unless K F , , ~ > 0.10, the 7 us. Q L curve is essentially coincident with the first-order line. Moreover, if KPA,x > 50.0, the 7 - aL curve is essentially coincident with the zeroorder line. 2. .411 curves for positive values of Kp,,, are bracketed by the first- and zero-order lines. 3. T h e actual valuer, of Q L for bvhich second-, first-, and zero-order reactions have 7 = 0.95 are Q L = 0.075. 0.15,and 2.1. respectively. \Vhen the actual values of @ L for these reaction orders are compared with the values of Q L that \vere calculated by transforming from as to @I,. good agreement results only for first-order reactions. This comparison is sho\vn in Table I . I n other words. the difference betlveen the numbers in columns 2 and 3 is a measure of the effect of geometry on the ~
Table I. Values of R m t i i o n Ordrr
0 1 2
@ I , Corresponding to Effectiveness Factor of 0.95 +I, for Slab +I f o r Sfihves 2 1 0 66 0 15 0 11 o 075 0 033
functional relationship between effectiveness factor and modified l'hiele modulus for each of three integer reaction for the three shapes orders. Althoiigh the functions of '7 1 s . studied by Aris fall close together for first-order reactions. Table I shokvs that they depart from one another substantially for reactions of other orders; for a given vaiue of QI, the true effectivenes. factor in a spherr \vi11 be less than that calculated for slab geometry, but the amount of the difference is only knokvn for the simple cases above. 4. For $lab geometry. and probably for other simple shapes, strong product adsorption (large negative values of Kpa , s ) results in '7 versus Q 1 ~curves jvhich are low relative to zero-, first-: or second-order curves. .\s a n example, '7 = 0.95 at = 0.085for a second-order reaction. For strong product adsorption-for example. \vith h;bA,, = -0.95--~ becomes 0.95 a t Q L = about 0.008. Therefore. \\'eisz' criterion that diffusional effects \vi11 be absent if a, 5 0.3 must be extended to considerabl! IoLver values of Qs if strong product adsorption is involved-e.g.. diffusional limitations \sill be encountered a t substantially lo\ver rates of reaction than heretofore predicted. Appendix
Illustrative Example. An example of a system in \vhich the rate of reaction is retarded by a product is the reaction of carbon dioxide \vith solid carbon Xvhich is retarded by carbon monoxide. T h e effect is particularly marked a t relatively lo\ver temperatures. L-se of the method developed in this paper for calculating the effectiveness factor may be illustrated by taking a set of data for one run from LValker. Rusinko. and Austin ( I T ) . 'I hey studied the reaction of COSwith spectroscopic carbon. a fincly porous material, a t temprratures ranging from 950' to 1305' C.. and a t various CO? partial pressures. I'he mathematical relaticnships for this reaction are
@L
Figure 3.
Eftectiveness factor, r ~ ,as a function of modulus a,, VOL. 4
NO. 3
AUGUST
1965
291
the same as those for decomposition or isomerization of a single reactant on a porous catalyst, except that the porosity and hence the effective diffusivity will increase as reaction proceeds. T h e reaction d a t a of Walker et al., however, are for only the first 11% of reaction, so the change in diffusivity during a run is relatively insignificant. T h e carbon was cylindrical in shape, 2 inches in height, and '/z inch in diameter. A '/*-inch hole was cut in the center, and a mullite rod was inserted in this hole. T h e top and bottom faces of the cylinder were sealed off with mullite plates so that access to the interior of the carbon was available only through the lateral exterior surface, and diffusion was truly radial. T h e initial weight of the carbon annulus was about 8.8 grams. Figure 22 (75, p. 197), shows that a t a COS partial pressure of 0.75 atm. and a t temperature of 1000' C., the rate of reaction was 0.125 g r a m of carbon per hour. Presumably, the partial pressure of carbon monoxide a t the exterior of the carbon was zero during the run, and it will be assumed that nitrogen, which was present in the feed stream, does not enter into the rate equation. Rates were measured over about the first 11% of burnoff, so that a t the midpoint of this interval Rate/gram
=
0.125/(8.8 X 0.945) 0,1251'8.32
=
(1
- 0)
=
= 5.04
= 0.013
X
8$2(
=
C A , s = 0.75j(82.06)(1273) = 7.16 X l o p 6 mole/cc T h e dimension L , given by the ratio of volume to surface, can be approximated by
L2 OCA ,s
0.80 X 63 X 2)
d 2 8 44 = 0.80 w =
( 2 4 - 101) w
=
-99
w
X 10-7 mole/(cc.) (sec.)
T h e external concentration of carbon dioxide is
~
= (2.4 -
=
3600
0.013 X 4.20 = 0.0545 sq. cm./sec.
@L =
(Dco, Dco)
K
From Figure 16 (75) the effective diffusivity D = 0.013 sq. c m per second a t NTP. Diffusion apparently occurs in the transition region between Knudsen and bulk diffusion. T h e authors suggested that D is proportional to about but i n a more recent study on a similar graphite electrode, for COz counterdiffusing through helium between 30" and about 400" C. a t a total pressure of 1 a t m . , Nichols (5) reported the temperature exponent to be about 0.98. Using this value
T h e value of
If it is assumed that the diffusivity is approximately proportional to the square root of molecular weight
(0.015) (1.45) 12
D (1000" C.)
I t will be assumed that these can be applied to the \vork of Austin et al. A value of KpA can now be calculated.
1.45 grams per cc.
T h e observed reaction rate per gross volume of carbon -
Kco? = 2.4 atm.-'
0.015 g r a m C/gram C hour
(0.64) (2.27)
pL =
T h e form of carbon most similar to that studied by \\'alker et a l . on which kinetic information is available, is probably electrode carbon. \ V u (79) reports valdes of the constants for electrode carbon over the temperature and pressure range of interest. \2'ith Figure 35 of ( 7 9 ) the values a t 1000O C. can be estimated as
=
Figure 13 (75) shows that the average porosity (cc./cc.) of this sample during the period of the burnoff is about 0.36. Taking 2.27 as the true density ( p t ) of carbon, the apparent density of the particle is then p =
T h e authors did not determine a rate equation for the reaction, but several other investigators have reported that on each of various types of carbon it is of the form
aLcan now
be calculated
(observed ratel'gross carbon volume) = (0.298)*(5.04 X lo-') (7.16 X
aL =
0.114
292
I&EC FUNDAMENTALS
(0.0545)
=
1
+ 0.80 X 0.75 X 63 X 2
=
77
Interpolating between Lhe curves for K p d , s = -0.95 a n d -0.98 in Figure 3, the effectiveness factor, 7 , for this run is about 0.35. If the temperature coefficient of the diffusivity were taken to be 1.3 as suggested by ivalker. aL would be 0.0715 and for 7 : about 0.5. Therefore, internal diffusion effects are significant. This conclusion has been confirmed in a subsequent paper by Austin and b'alker ( 3 ) . If the reaction \vere assumed to be of simple first order, the effectiveness factor would erroneously be calculated to be nearly unity-i.e.! diffusional effects would be thought to be insignificant. Even if a simple second-order reaction were assumed, the effectiveness factor would be taken to be about 0.92. Another study in which intrinsic reaction rate data are available on electrode carbon is that of Reif (8). Using his values for the constants in Equation A-1. the value of KpA,s= -0.970, which is very close to the value calculated from \\'u's data. .4ctually, IVu's correlation for coal coke gives a t this temperature a value of KpA,s = -0.965. identical to the value for electrode carbon. For the same observed rate of reaction. the effectiveness factors for these two types of carbon would therefore be very close to the one calculated above.
aL =
Acknowledgment
T h e machine computations necessary in the preparation of this paper were performed a t the Massachusetts Institute or Technology Computation Center. Acknowledgment is also made to the Sational Science Foundation for providing financial support to George W. Roberts in the form of a fellowship during the period of the project.
as qjL $.,f
w
= = = =
modulus defined by Equation 14 modulus defined by Equation 16 Thiele modulus, defined by Equation 15 modified Thiele modulus, defined by Equation 10 parameter defined by Equation 6
SUBSCRIPTS z o
s
= index denoting any chemical species other than = sealed surface, x = L = exposed surface, x = 0
A
Nomenclature
Any consistent set of units may be used. Those specified below are those used by Satterfield and Sherwood ( 7 7).
ENGLISH LETTERS A = geometrical surface area of catalyst mass, sq. cm. C , = concentration of ith species, g. moles!(cc.) D i = effective diffusivity of ith species, based on total cross section of catalyst mass. sq. cm./sec. I; = parameter defined by Equation 8, (atm.)-I K i = adsorption constant for ith species in LangmuirHinshelwood rate expression, (atm.) k = reaction-rate constant, g. moles l(cc.) (sec.) k' = modified rate constant (see Equation ?), (g. moles) (arm.) (cc.) isec.) L = thickness of catalyst mass. cm. p 1 = partial pressure of ith species, arm. R = gas constant, (atm.) ( c c . ) ! ( g . moles) (OK). R,? = radius of sphere. cm. I = reaction rate: (g. moles), (cc.) (sec.) 7 = absolute temperature, OK. V = volume: cc. .Y = Cartesian dimension, cm. I
GREEKLETTERS = effectiveness factor (see Equation 12) 17 4 = approximate effectiveness factor, defined by Equation 13 = void fraction of catalyst mass, (cc.)/(cc.) 6' v i = stoichiometric coefficient of ith component = apparent densi1.y of catalyst mass: g./(cc.) p p t = density of solid material in catalyst, g./(cc.)
literature Cited (1) Akehata, T., Namkoong, S., Kubota, H., Shindo; S.; Can. J . Chrm. Eng. 39, 127 (1961). (2) Ark. R., Chem. Eng. Sci. 6 , 262 (1957). (3) Austin, L. G.: Walker, P. L.. Jr., A.Z.Ch.E. J . 9, 303 (1963). (4) Chu. C., Hougen, 0. .4., Chem. Eng. Sci. 17, 167 (1962).
(5) Nichols, J. R.?Ph.D. thesis, The Pennsylvania State University, 1961. (6) Otani, S., Wakao, N., Smith, J . M . , A.I.Ch.E. J . 10, 130 (1964), (7)' Prafer, C. D., Lago. R. M.. Advan. Catalysis 8, 293 (1956). E., J . Phys. G e m . 56, 778 (1952). (8) Reif, -4. (9) Roberts. G. LV.. Sc.D. thesis, Department of Chemical Engineering, Massachusetts Institute of Technology, 1965. (10) Rozovskii, A. Ya., Shchekin, V. V.. Kinetics Cataiysis (USSR) 1, 313 (1960) : p. 286 of Consultants' Bureau English translation. (11) Satterfield. C. N.. Sherwood, T. K.; ''Rolr of Diffusion in Catalysis." Addison-Wesley. Reading. Mass., 1963. (12) Schilson. R. E.. Amundson, N. R., Chrm. Eng. Scz. 13, 226, 237 (1961). (13) Thiele: E. W.; Znd. Eng. Chem. 31, 916 (1939). (14) Wagner, C.. Z. Phyr. Chem. A193, 1 (1943). (15) Walker. P. L.. J r . : Rusinko. F.: Jr., Austin, L. G., Aduan. Catdysis 11, 134 (1959). (16) LVeisz. P. E.. Hicks, J . S.. Chem. Eng. Sci. 17, 265 (1962). ~ (1954). (17) LVeisz, P. B., Prater, C. D., Adam. C ~ t a l y6s, ~143 (18) Ll'heeler, A . H.. Zbzd.. 3, 249 (1951). (19) \Vu, P. C.. Sc.D. thesis, Massachusetts Institute of Technology. Cambridge, Mass.. 1949. RECEIVED for review August 17, 1964 ACCEPTED February 8, 1965 First in a series on effectiveness factor for porous catalysts.
DIFFUSIONAL EFFECTS IN GASISOLID REACTIONS J 0 H
N
S H E
N AND J
.
M
.
S M I T
H , University of
California, Dar?is, Car$,
The interaction of physical transfer processes and chemical reaction is considered for the reaction of a gas and spherical pellet of solid reactant. The conversion-time relationship is derived for both isothermal and nonisothermal conditions for which mass and energy transfer as well as reaction resistances are important, For isothermal conditions the conversion can be expressed in terms of a dimensiontess time and two parameters, while for nonisothermal systems three additional parameters are necessary. For an exothermic reaction, there is a region of unstable operation bounded at the upper temperature level by a stable diffusioncontrol regime and at the lower level by a stable kinetic-control regime. Approximate criteria are derived for the limits of instability. A numerical application is given showing the transition from the diffusion to kinetic regimes. HE rates of gas-solid reactions in which one reactant and T o n e product are solids are particularly susceptible to diffusion resistances. For nonisothermal, exothermic examples also. the interrelationship between mass and energy transfer can lead to interesting stability problems. T h e analysis that follows is restricted to a single. spherical pellet in a gas stream of constant composition. Initially the pellet consists of nonporous reactant B. As reaction proceeds, product W forms as a
porous solid around the shrinking core of B, according to the reaction A(g)
+ bB(s)
+
G(g)
+ rcb'(s)
(1 1
Further conversion occurs by diffusion of A through the product layer to the core of reactant. T h e reaction at the surface of B is assumed to be first-order and reversible, so that the local rate is given by VOL. 4
NO. 3
AUGUST
1965
293
