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A general criterion for absence of diffusion control in an isothermal catalyst pellet
Chemical Engineering Science., 1968, Vol. 23, pp. 93-94.
Pergamon Press.
Printed in Great Britain.
A general criterion for absenceof diffusion control in an isothermal catalyst pellet (Received
23 June 1967)
IT HAS been shown by Weisz and Prater[l] that for zero to second order reaction kinetics, diffision control in intraparticle reaction can be avoided by keeping the Damkohler Group I less than a value of approximately unity.
when reaction is carried out using an external atmosphere of pure carbon dioxide. Austin and Walker[6] found that for this reaction where, at lOOO”C, k, = 5.06 x lo-’ atm’, A:,= 4.12 atm-‘. For reaction with pure CO, at 1 atm pressure and noting that 2(pOc,-_c,) = Pc,, the critical value of p0 is found to be 0.71 x lo-*, compared with a value of 3.03 x lo-* obtained by Petersen[Z] using the asymptotic method. The present criterion is seen to be more conservative than the Petersen criterion. Schneider and Mitschka[7J have shown, however, that the Petersen criterion for this Langmuir-Hinshelwood model is not sufficiently conservative, the above value corresponding to an effectiveness of’only 0.381 when an exact solution is obtained for the differential equation. When inhibition by product is weak, the criterion shows that the critical value of (pOmoves towards unity again. Let us consider, as an example, the vapor phase dehydration of ethanol to ether and water over an ion exchange catalyst. Rabel and Johanson@] found that this reaction could be described by the rate expression:
Petersen[2] and others [3,4] have shown the shortcomings of the Wiesz-Prater criterion in the presence of strong product inhibition. The purpose of this note is to suggest a general criterion for the absence of diffusion effects, regardless of the form of the kinetic rate expression r(C), for any material that can undergo catalytic decomposition, or reaction with a solid. We proceed by means of a method similar to that used by Anderson[S]. The rate is expanded in a Taylor series about the external concentration of reactant and second order and higher terms are neglected. The concentration profile is approximated by the simplest function having a minimum at the center of the particle, i.e. a parabolic profile. If it is required that the overall rate be within, say, 5 per cent of the rate obtained if the particle were completely accessible to the external concentration, it can be shown that the following criterion results: 1 Pn’ P.G’Z-<
k(A*-IX/K& r=
1 r(G) .c r’(C,).
(1)
where A, B, and C represent partial pressures of ethanol, water, and diethyl ether, respectively. Values of KAi Ks, Kc, and K,, at 120°C were found to be 3.4, 7.0, 0, and 25.2 respectively. Pure ethanol vapor at 1 atm was used as the external concentration. Noting that&-A = 2B, and B = C, we find the critical value ofp, to be 0.7. The criterion developed here for the absence of appreciable diffusion effects generalizes the Weisz-Prater criterion for kinetics other than simple power-type rate expressions, and involves differentiation rather than integration of the rate equation. It also appears to be generally more conservative than the Petersen criterion obtained by asymptotic solution of the differential mass balance.
The value 1 in the right hand side of (1) results from clearing the expression of fractions (0.75 becomes 1), without introducing serious error. This criterion is a simple generalization of the Weisz-Prater criterion, valid for all but zero order kinetics. Several examples follow. It is convenient here to denote the left hand side of (1) by woz. For first and second order kinetics, we findcp,,2 < 1and 0.5 respectively. Comparison in Table 1 between critical values of cpOobtained from (1) and those obtained from asymptotic solution of the differential mass balance by Petersen [2] shows satisfactory agreement. Reaction between carbon and carbon dioxide to form carbon monoxide provides an example of product inhibition
University Waterloo Ontario Canada
Table 1.
Order 1
2
Critical value ofcpo 1.0 0.71
(l+KJ+KBB+KcC)Z
Q” From Petersen (ideal pore)
R. R. HUDGINS
of Waterloo
NOTATION C G
1.0 0.80
93
concentration external concentration surface
at
catalyst
particle
Shorter Communications 9 k,
kz,k; KA, KB, Kc PC09Pco, r(C) P
effective diffusivity rate constant adsorption constant adsorption constants partial pressure of CO, Cq respectively rate of reaction as a function of concentration the measured catalytic reaction rate
r’
= dr/dC
Greek letters PO
pellet radius
m. r”
1 PO2 ES._._ p* C” 2
REFERENCES
[II WEISZ P. B. and PRATER C. D.,Adu. Catalysis 19546 143. PI PETERSEN E. E., Chem. Engng Sci. 1965 20 587. J. and CARBERRY J. J.,A. I. Ch. E. Jll966 12 20. [31 HUTCHINGS P. and MITSCHKA P., Chem. Ennnn Sci. 1966 21455. r41 SCHNEIDER is1 ANDERSON J. B., Chem. Engng Sci. 1963 18 147. AUSTIN
L. G. and WALKER P. L. Jr..A.I.Ch.E.JI
1963 9 303.
t;; SCHNEIDER P. and MITSCHKA P., Chem. Engng Sci. 1966 21726. PI KABEL R. L. and JOHANSON L. N., A. I. Ch. E. Jll962 8 621.
Chemical Engineering Science, 1968, Vol. 23, pp. 94-96.
Pergamon Press.
Printed in Great Britain.
On the use of criteria for determining transport limitations within heterogeneous
catalysts
(Received 27 July 1967) 1~ IS important during the early phases of a systematic study of a heterogeneously catalyzed reaction to determine whether the investigation is concerned primarily with catalytic kinetics or an interaction among catalytic kinetic and transport phenomena. The resolution of this problem is not simple either theoretically or experimentally, particularly if the reactions taking place upon the solid catalytic surface are represented by more than one independent stoichiometric equation. Several methods have been suggested which depend upon a comparison of the behavior of a model of the porous catalyst-reactant system and preliminary experimental measurements of the overall reaction rate. These methods lead to criteria for the recognition of transport limitations within porous heterogeneous catalysts. The purpose of these criteria is to permit the catalytic chemist to utilize physical data on the catalyst and preliminary kinetic information to estimate the importance of transport limitations on the the overall rates measured. There appears to be some confusion with respect to this intended-purpose, as evidenced by recent naoers bv Schneider and Mitschka r 11 and Hudgins f21. The ku&s criticism is that the generai criterion of?Petersen [ 31 is not “conservative” enough to apply to the carbon-carbon dioxide reaction for which a Langmuir-Hinshelwood model has been used to describe product inhibition. However, because the misunderstanding appears to be greater than the details of this single case, it seems appropriate to review
briefly the methods most often used to develop criteria and the proper application of each. The obvious and direct way to determine the regions of kinetic control and transport influence is to solve exactly the equations describing the model in terms of the parameters of the system as was done originally by Thiele[4], Zelodvich [5], and later expanded upon by Wheeler[B] for the isothermal reactions. The regions are apparent from the solution. However, for the general case the function representing the kinetic rate expression is nonlinear, and general solutions of the equations are obtained only by numerical or analog techniques. There are two methods now commonly adopted for analyzing the general equations which, in principle, still allow analytical methods: the perturbation and asymptotic methods. Although both methods permit the analysis of nonisothermal as well-as isothermal cases, only the latter cases will be considered here for the sake of simplicity and brevity. The reader is referred to the literature for-the application to nonisotherma1 cases, viz: Anderson [7] and Tinkler and Pigford [8] for the perturbation method and Petersen [9] for the asymptotic method. For isothermal cases, the effectiveness or efficiency factor, 8’. is a function of the Thiele modulus, h, which in turn, is a measure of the capacity of the system to react kinetically to the capacity of the system to diffuse material into the pores
94
Pergamon Press.
Printed in Great Britain.
A general criterion for absenceof diffusion control in an isothermal catalyst pellet (Received
23 June 1967)
IT HAS been shown by Weisz and Prater[l] that for zero to second order reaction kinetics, diffision control in intraparticle reaction can be avoided by keeping the Damkohler Group I less than a value of approximately unity.
when reaction is carried out using an external atmosphere of pure carbon dioxide. Austin and Walker[6] found that for this reaction where, at lOOO”C, k, = 5.06 x lo-’ atm’, A:,= 4.12 atm-‘. For reaction with pure CO, at 1 atm pressure and noting that 2(pOc,-_c,) = Pc,, the critical value of p0 is found to be 0.71 x lo-*, compared with a value of 3.03 x lo-* obtained by Petersen[Z] using the asymptotic method. The present criterion is seen to be more conservative than the Petersen criterion. Schneider and Mitschka[7J have shown, however, that the Petersen criterion for this Langmuir-Hinshelwood model is not sufficiently conservative, the above value corresponding to an effectiveness of’only 0.381 when an exact solution is obtained for the differential equation. When inhibition by product is weak, the criterion shows that the critical value of (pOmoves towards unity again. Let us consider, as an example, the vapor phase dehydration of ethanol to ether and water over an ion exchange catalyst. Rabel and Johanson@] found that this reaction could be described by the rate expression:
Petersen[2] and others [3,4] have shown the shortcomings of the Wiesz-Prater criterion in the presence of strong product inhibition. The purpose of this note is to suggest a general criterion for the absence of diffusion effects, regardless of the form of the kinetic rate expression r(C), for any material that can undergo catalytic decomposition, or reaction with a solid. We proceed by means of a method similar to that used by Anderson[S]. The rate is expanded in a Taylor series about the external concentration of reactant and second order and higher terms are neglected. The concentration profile is approximated by the simplest function having a minimum at the center of the particle, i.e. a parabolic profile. If it is required that the overall rate be within, say, 5 per cent of the rate obtained if the particle were completely accessible to the external concentration, it can be shown that the following criterion results: 1 Pn’ P.G’Z-<
k(A*-IX/K& r=
1 r(G) .c r’(C,).
(1)
where A, B, and C represent partial pressures of ethanol, water, and diethyl ether, respectively. Values of KAi Ks, Kc, and K,, at 120°C were found to be 3.4, 7.0, 0, and 25.2 respectively. Pure ethanol vapor at 1 atm was used as the external concentration. Noting that&-A = 2B, and B = C, we find the critical value ofp, to be 0.7. The criterion developed here for the absence of appreciable diffusion effects generalizes the Weisz-Prater criterion for kinetics other than simple power-type rate expressions, and involves differentiation rather than integration of the rate equation. It also appears to be generally more conservative than the Petersen criterion obtained by asymptotic solution of the differential mass balance.
The value 1 in the right hand side of (1) results from clearing the expression of fractions (0.75 becomes 1), without introducing serious error. This criterion is a simple generalization of the Weisz-Prater criterion, valid for all but zero order kinetics. Several examples follow. It is convenient here to denote the left hand side of (1) by woz. For first and second order kinetics, we findcp,,2 < 1and 0.5 respectively. Comparison in Table 1 between critical values of cpOobtained from (1) and those obtained from asymptotic solution of the differential mass balance by Petersen [2] shows satisfactory agreement. Reaction between carbon and carbon dioxide to form carbon monoxide provides an example of product inhibition
University Waterloo Ontario Canada
Table 1.
Order 1
2
Critical value ofcpo 1.0 0.71
(l+KJ+KBB+KcC)Z
Q” From Petersen (ideal pore)
R. R. HUDGINS
of Waterloo
NOTATION C G
1.0 0.80
93
concentration external concentration surface
at
catalyst
particle
Shorter Communications 9 k,
kz,k; KA, KB, Kc PC09Pco, r(C) P
effective diffusivity rate constant adsorption constant adsorption constants partial pressure of CO, Cq respectively rate of reaction as a function of concentration the measured catalytic reaction rate
r’
= dr/dC
Greek letters PO
pellet radius
m. r”
1 PO2 ES._._ p* C” 2
REFERENCES
[II WEISZ P. B. and PRATER C. D.,Adu. Catalysis 19546 143. PI PETERSEN E. E., Chem. Engng Sci. 1965 20 587. J. and CARBERRY J. J.,A. I. Ch. E. Jll966 12 20. [31 HUTCHINGS P. and MITSCHKA P., Chem. Ennnn Sci. 1966 21455. r41 SCHNEIDER is1 ANDERSON J. B., Chem. Engng Sci. 1963 18 147. AUSTIN
L. G. and WALKER P. L. Jr..A.I.Ch.E.JI
1963 9 303.
t;; SCHNEIDER P. and MITSCHKA P., Chem. Engng Sci. 1966 21726. PI KABEL R. L. and JOHANSON L. N., A. I. Ch. E. Jll962 8 621.
Chemical Engineering Science, 1968, Vol. 23, pp. 94-96.
Pergamon Press.
Printed in Great Britain.
On the use of criteria for determining transport limitations within heterogeneous
catalysts
(Received 27 July 1967) 1~ IS important during the early phases of a systematic study of a heterogeneously catalyzed reaction to determine whether the investigation is concerned primarily with catalytic kinetics or an interaction among catalytic kinetic and transport phenomena. The resolution of this problem is not simple either theoretically or experimentally, particularly if the reactions taking place upon the solid catalytic surface are represented by more than one independent stoichiometric equation. Several methods have been suggested which depend upon a comparison of the behavior of a model of the porous catalyst-reactant system and preliminary experimental measurements of the overall reaction rate. These methods lead to criteria for the recognition of transport limitations within porous heterogeneous catalysts. The purpose of these criteria is to permit the catalytic chemist to utilize physical data on the catalyst and preliminary kinetic information to estimate the importance of transport limitations on the the overall rates measured. There appears to be some confusion with respect to this intended-purpose, as evidenced by recent naoers bv Schneider and Mitschka r 11 and Hudgins f21. The ku&s criticism is that the generai criterion of?Petersen [ 31 is not “conservative” enough to apply to the carbon-carbon dioxide reaction for which a Langmuir-Hinshelwood model has been used to describe product inhibition. However, because the misunderstanding appears to be greater than the details of this single case, it seems appropriate to review
briefly the methods most often used to develop criteria and the proper application of each. The obvious and direct way to determine the regions of kinetic control and transport influence is to solve exactly the equations describing the model in terms of the parameters of the system as was done originally by Thiele[4], Zelodvich [5], and later expanded upon by Wheeler[B] for the isothermal reactions. The regions are apparent from the solution. However, for the general case the function representing the kinetic rate expression is nonlinear, and general solutions of the equations are obtained only by numerical or analog techniques. There are two methods now commonly adopted for analyzing the general equations which, in principle, still allow analytical methods: the perturbation and asymptotic methods. Although both methods permit the analysis of nonisothermal as well-as isothermal cases, only the latter cases will be considered here for the sake of simplicity and brevity. The reader is referred to the literature for-the application to nonisotherma1 cases, viz: Anderson [7] and Tinkler and Pigford [8] for the perturbation method and Petersen [9] for the asymptotic method. For isothermal cases, the effectiveness or efficiency factor, 8’. is a function of the Thiele modulus, h, which in turn, is a measure of the capacity of the system to react kinetically to the capacity of the system to diffuse material into the pores
94