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The effectiveness factor for an isothermal pellet with decreasing activity towards the pellet surface
Shorter Communications
1212
Chemical Engineering Sciences, 1976, Vol. 31, pp. 1212-1213. Pergamon Press.
Printed in Great Britain
The effectiveness factor for an isothermal pellet with decreasing activity towards the pellet surface (Received Non uniform catalyst activity may have a significant effect on the
performance of a porous catalyst. In some cases catalysts are prepared with a non uniform activity to improve selectivity or average rate of conversion as compared with a uniformly active catalyst with the same total amount of active material. In other cases the non-uniformity occurs unintentionally as the result of an improper catalyst preparation technique or it may be caused by poisoning of the catalyst by either the product or by the reactant. Kasaoka and Sakata[l], Stadman-Yazdi and Petersen]21 and Corbett and Luss[3] studied activity distributions increasing or decreasing towards the pellet surface but always with a non zero value at the surface. Corbett and Luss[3] pointed out that the asymptotic dependence of the effectiveness factor n on Thiele modulus 6 will eventually be the same for a uniformly active pellet and for a non-uniformly active pellet when the activity is non-zero at the pellet surface and Q is based on surface activity. Reactant poising mechanisms with large Thiele modulus for the poisoning reaction lead to activity distributions with negligible activity at the pellet surface and increasing activity towards the pellet center. If a finite section of the pellet is completely poisoned n will eventually become proportional to d-’ for a first order reaction, since the dead pellet section acts as an exterior film resistance. If the activity is zero at the pellet surface but non-zero at any point inside the pellet, it seems intuitively clear that log n vs log $ will have a slope somewhere between - 1 and -2 for large Q and a tirst order reaction. It becomes a matter of some interest for the study of poisoning mechanisms to find this asymptote, and it turns out that the result can be given in a neat, closed form for a first order reaction on a slab when the activity distribution is k = k,(l -z/L)” with k,, = activity at the pellet center.
March 1!976)
For n = 0, (4) simplifies to a well known expression
(5) For any finite a and d +m the two Bessel-functions cancel and the asymptote is r(a)(n
%-s=T(b)
t2)nb(n t 1) $2” .
(6)
The numerical constants depend on the definition of n and 4. In (3) the effectiveness is defined as the average value on the poisoned pellet with diffusion restriction, relative to the rate on the same pellet without diffusion restriction. For large d the position of the asymptote shifts towards very large values of d and it might be convenient to show n as a function of an average Thiele modulus: Al=
LJ($ where E=k,(l-x)“/(n+l) (7)
In (7) the asymptotes log n... vs log& emanate from approximately the same point on the log & axis for any finite value of n. The location of the asymptote is however, not very interesting since it depends on the scaling of n and 4. The general result of (6) or (7) is that n_. is proportional to I$ (or &) to the power - (2n t 2/n t 2). This result is obtained for any activity distribution that can be approximated by ko(l -x)’ when x is close to 1. THEVOLTZ-RRACTIONSCHEMEFOR NON UNIFORMCATALYST ACTIVITY
THR FIRST ORDER IRRRVIMIRLE RRACTION The
mass balance is:
&b2(1-x)“y =o Y =
c/co,
x = z/L,
d=LJ($)
activity = k = k,(l -x)” with n ~0. Substituting v = (1 - x) leads to an equation that can be soived for y(u) in terms of Bessel functions: y = ,“2[AZ,(2b&‘“b)
+ EZ_,(2b&‘“b)J
Z3= (q5b)bT(a)
Za(2b4) A= -Z_.(2b&B
a=- n+l
b=--&.
n+2
(2)
dZy koL’(1 -x) -D dx’
The result is similar to but slightly more complicated that the result obtained by Shadman-Yazdi and Petersen[Z] fork = kox”, a monotonously increasing activity towards the pellet surface.
Z,(2bd)
=-r(b)
r(a)(n
Becker and Wei] have recently investigated the effect of non-uniform catalyst distribution for a reaction with negative R, = (aR(y)/ay). Their system is the oxidation of CO on catalyst slabs and they compare a uniform catalyst distribution to three other distributions with catalyst (of three times higher activity) in the outer, the middle or the inner one third of the catalyst slab. Numerical calculations show that the highest effectiveness is obtained when the catalyst is placed close to the catalyst center, an intuitively reasonable result since the rate of reaction increases with decreasing concentration of reactant, and the dead catalyst close to the pellet surface, acting as a pure diffusion resistance, reduces the reactant concentration in the active part of the pellet. An obvious extension of their work is to investigate the performance of catalysts with non-uniform activity distribution of the type described in the previous section. Using the rate expression proposed by Voltz[S] one obtains:
+2)*(n
4’”
t 1) ’
(4)
(ltkc,y)‘=O.
(8)
c. is the surface concentration of CO, and when K.c. is large compared to 1 the order of reaction is effectively - 1 while as the reaction is first order in CO when y = c/c. is small. For large values of d = Ld(kJD) y is smaUin most of the pellet and it is interesting to see whether the asymptotes n(d) obtained for pure first order kinetics and various a are also obtained here. A variable transformation u = @‘“+,(l -x)
(9)
Shorter Communications
1213
0.6
0 -0.7 -0.6 -0.5
-0.4 -0.3 -0.2 -0.1
0
0.1
0.2 'OQ@ld
Fig. 1. Effectiveness factor for isothermal CO oxidation on a slab. The rate expression is given by eqn (8) and results for different n in the activity distribution k = k,,(l- x)’ are shown. The slope of the asymptote is - (2n +2)/n t 2. is introduced in (8) to give
The initial step is empirically chosen as Au(u = 0) = O.l(&)““+’
with boundary conditions (11) The solution of (10)is a unique function of s, the slope (dyldu) at u = 0. For a given negative value of s (10)is solved by forward integration from u = 0 until a value u = u,, is reached for which (dyldu) = 0. This determines I$ and q: $ = uon+*‘2; 1) =
-2
(I+ Kd
4
sp+* =_ S(l+
&c”)2~-*n+2’n+z. (12)
Tracing 4 and n as functions of a suitable parameter was used already by Weisz and Hicks[6]. They used the y value at x = 0 as parameter but this is not applicable here due to the function (1 -x)” in (8). The basic method does however work equally well for plane parallel symmetry when s is used as parameter in (9). Each profile was found by collocation applied to each of a number of subintervals of variable length Au. In each subinterval N = 3-4 collocation points-zeros of PN(‘.‘),the Nth order Jacobi polynomial with a = 1 and g = 0, is used. This is the Radau collocation method described in [7] Chaps. 4 and 5) and in 181. The steplength Au is chosen such that (13) at the left hand endpoint (u, y) of each subinterval. ton leave from Instituttet for Kemiteknik, Danmarks tekniske Hdjskole, Lyngby, Denmark.
(14)
where $’ is the 4 value obtained for the previous s-value. The final adjustment of u to make (dy/du) = 0 at u0is made by a Newton Raphson technique. A few results are shown in Fig. 1 as log n vs log & where & = $~/(l+K,c,). The maximum n value increases with increasing n and at n = 4 it is about 1.5 times larger than the maximumfor n = 0. The n = 4 result is practically the same as that obtained by Becker and Wei when they stack the active material in the inner one-third of the pellet. It is interesting to note that the asymptotes conform precisely to the results obtained for a first order reaction. This appears directly during the calculations, since s can never be chosen below a certain, n-dependent value if y shall stay positive throughout the integration from u = 0 to uO.The form of (12) shows that this is equivalent to 94” +“”+*+ constant for large 4, which is exactly the result of the previous analysis for first order reactions. Deparfment of Chemical Engineering University of Houston Houston TX 77004 U.S.A.
JOHN VILLADSENt
REFERENCES
HI Kasaoka S. and Sakata Y ., J. Chem. Engng Japan 19681138. PI Shadman-Yazdi F. and Petersen E. E., Chem. Engng Sci. 1972 21 227.
131Corbett W. E. and Luss D., Chem. Engng Sci. 197429 1473. 141Becker E. R. and Wei T., Preprints IV ISCRE Heidelberg 1976. ISI Voltz S. E., Morgan C. R., Liederman D. and Jacob S. hf., Ind. Engng. Chem. Res. Dev. 1973 12 294.
161Weisz P. B. and Hicks J. S., Chem. Engng Sci. 1%2 17265. 171Villadsen J. and Michelsen M. L., Solution of DiJFerential Equation Models by Polynomial Approximation. PrenticeHall, New York 1977. 181Michelsen M. L. and Vi&&en J., Chem. Engng Sci. 197227 751.
1212
Chemical Engineering Sciences, 1976, Vol. 31, pp. 1212-1213. Pergamon Press.
Printed in Great Britain
The effectiveness factor for an isothermal pellet with decreasing activity towards the pellet surface (Received Non uniform catalyst activity may have a significant effect on the
performance of a porous catalyst. In some cases catalysts are prepared with a non uniform activity to improve selectivity or average rate of conversion as compared with a uniformly active catalyst with the same total amount of active material. In other cases the non-uniformity occurs unintentionally as the result of an improper catalyst preparation technique or it may be caused by poisoning of the catalyst by either the product or by the reactant. Kasaoka and Sakata[l], Stadman-Yazdi and Petersen]21 and Corbett and Luss[3] studied activity distributions increasing or decreasing towards the pellet surface but always with a non zero value at the surface. Corbett and Luss[3] pointed out that the asymptotic dependence of the effectiveness factor n on Thiele modulus 6 will eventually be the same for a uniformly active pellet and for a non-uniformly active pellet when the activity is non-zero at the pellet surface and Q is based on surface activity. Reactant poising mechanisms with large Thiele modulus for the poisoning reaction lead to activity distributions with negligible activity at the pellet surface and increasing activity towards the pellet center. If a finite section of the pellet is completely poisoned n will eventually become proportional to d-’ for a first order reaction, since the dead pellet section acts as an exterior film resistance. If the activity is zero at the pellet surface but non-zero at any point inside the pellet, it seems intuitively clear that log n vs log $ will have a slope somewhere between - 1 and -2 for large Q and a tirst order reaction. It becomes a matter of some interest for the study of poisoning mechanisms to find this asymptote, and it turns out that the result can be given in a neat, closed form for a first order reaction on a slab when the activity distribution is k = k,(l -z/L)” with k,, = activity at the pellet center.
March 1!976)
For n = 0, (4) simplifies to a well known expression
(5) For any finite a and d +m the two Bessel-functions cancel and the asymptote is r(a)(n
%-s=T(b)
t2)nb(n t 1) $2” .
(6)
The numerical constants depend on the definition of n and 4. In (3) the effectiveness is defined as the average value on the poisoned pellet with diffusion restriction, relative to the rate on the same pellet without diffusion restriction. For large d the position of the asymptote shifts towards very large values of d and it might be convenient to show n as a function of an average Thiele modulus: Al=
LJ($ where E=k,(l-x)“/(n+l) (7)
In (7) the asymptotes log n... vs log& emanate from approximately the same point on the log & axis for any finite value of n. The location of the asymptote is however, not very interesting since it depends on the scaling of n and 4. The general result of (6) or (7) is that n_. is proportional to I$ (or &) to the power - (2n t 2/n t 2). This result is obtained for any activity distribution that can be approximated by ko(l -x)’ when x is close to 1. THEVOLTZ-RRACTIONSCHEMEFOR NON UNIFORMCATALYST ACTIVITY
THR FIRST ORDER IRRRVIMIRLE RRACTION The
mass balance is:
&b2(1-x)“y =o Y =
c/co,
x = z/L,
d=LJ($)
activity = k = k,(l -x)” with n ~0. Substituting v = (1 - x) leads to an equation that can be soived for y(u) in terms of Bessel functions: y = ,“2[AZ,(2b&‘“b)
+ EZ_,(2b&‘“b)J
Z3= (q5b)bT(a)
Za(2b4) A= -Z_.(2b&B
a=- n+l
b=--&.
n+2
(2)
dZy koL’(1 -x) -D dx’
The result is similar to but slightly more complicated that the result obtained by Shadman-Yazdi and Petersen[Z] fork = kox”, a monotonously increasing activity towards the pellet surface.
Z,(2bd)
=-r(b)
r(a)(n
Becker and Wei] have recently investigated the effect of non-uniform catalyst distribution for a reaction with negative R, = (aR(y)/ay). Their system is the oxidation of CO on catalyst slabs and they compare a uniform catalyst distribution to three other distributions with catalyst (of three times higher activity) in the outer, the middle or the inner one third of the catalyst slab. Numerical calculations show that the highest effectiveness is obtained when the catalyst is placed close to the catalyst center, an intuitively reasonable result since the rate of reaction increases with decreasing concentration of reactant, and the dead catalyst close to the pellet surface, acting as a pure diffusion resistance, reduces the reactant concentration in the active part of the pellet. An obvious extension of their work is to investigate the performance of catalysts with non-uniform activity distribution of the type described in the previous section. Using the rate expression proposed by Voltz[S] one obtains:
+2)*(n
4’”
t 1) ’
(4)
(ltkc,y)‘=O.
(8)
c. is the surface concentration of CO, and when K.c. is large compared to 1 the order of reaction is effectively - 1 while as the reaction is first order in CO when y = c/c. is small. For large values of d = Ld(kJD) y is smaUin most of the pellet and it is interesting to see whether the asymptotes n(d) obtained for pure first order kinetics and various a are also obtained here. A variable transformation u = @‘“+,(l -x)
(9)
Shorter Communications
1213
0.6
0 -0.7 -0.6 -0.5
-0.4 -0.3 -0.2 -0.1
0
0.1
0.2 'OQ@ld
Fig. 1. Effectiveness factor for isothermal CO oxidation on a slab. The rate expression is given by eqn (8) and results for different n in the activity distribution k = k,,(l- x)’ are shown. The slope of the asymptote is - (2n +2)/n t 2. is introduced in (8) to give
The initial step is empirically chosen as Au(u = 0) = O.l(&)““+’
with boundary conditions (11) The solution of (10)is a unique function of s, the slope (dyldu) at u = 0. For a given negative value of s (10)is solved by forward integration from u = 0 until a value u = u,, is reached for which (dyldu) = 0. This determines I$ and q: $ = uon+*‘2; 1) =
-2
(I+ Kd
4
sp+* =_ S(l+
&c”)2~-*n+2’n+z. (12)
Tracing 4 and n as functions of a suitable parameter was used already by Weisz and Hicks[6]. They used the y value at x = 0 as parameter but this is not applicable here due to the function (1 -x)” in (8). The basic method does however work equally well for plane parallel symmetry when s is used as parameter in (9). Each profile was found by collocation applied to each of a number of subintervals of variable length Au. In each subinterval N = 3-4 collocation points-zeros of PN(‘.‘),the Nth order Jacobi polynomial with a = 1 and g = 0, is used. This is the Radau collocation method described in [7] Chaps. 4 and 5) and in 181. The steplength Au is chosen such that (13) at the left hand endpoint (u, y) of each subinterval. ton leave from Instituttet for Kemiteknik, Danmarks tekniske Hdjskole, Lyngby, Denmark.
(14)
where $’ is the 4 value obtained for the previous s-value. The final adjustment of u to make (dy/du) = 0 at u0is made by a Newton Raphson technique. A few results are shown in Fig. 1 as log n vs log & where & = $~/(l+K,c,). The maximum n value increases with increasing n and at n = 4 it is about 1.5 times larger than the maximumfor n = 0. The n = 4 result is practically the same as that obtained by Becker and Wei when they stack the active material in the inner one-third of the pellet. It is interesting to note that the asymptotes conform precisely to the results obtained for a first order reaction. This appears directly during the calculations, since s can never be chosen below a certain, n-dependent value if y shall stay positive throughout the integration from u = 0 to uO.The form of (12) shows that this is equivalent to 94” +“”+*+ constant for large 4, which is exactly the result of the previous analysis for first order reactions. Deparfment of Chemical Engineering University of Houston Houston TX 77004 U.S.A.
JOHN VILLADSENt
REFERENCES
HI Kasaoka S. and Sakata Y ., J. Chem. Engng Japan 19681138. PI Shadman-Yazdi F. and Petersen E. E., Chem. Engng Sci. 1972 21 227.
131Corbett W. E. and Luss D., Chem. Engng Sci. 197429 1473. 141Becker E. R. and Wei T., Preprints IV ISCRE Heidelberg 1976. ISI Voltz S. E., Morgan C. R., Liederman D. and Jacob S. hf., Ind. Engng. Chem. Res. Dev. 1973 12 294.
161Weisz P. B. and Hicks J. S., Chem. Engng Sci. 1%2 17265. 171Villadsen J. and Michelsen M. L., Solution of DiJFerential Equation Models by Polynomial Approximation. PrenticeHall, New York 1977. 181Michelsen M. L. and Vi&&en J., Chem. Engng Sci. 197227 751.