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Ind. Eng. Chem. Res. 1998, 37, 3780-3781
RESEARCH NOTES Evaluation of Isothermal Effectiveness Factors for Nonlinear Kinetics Using an Approximate Method J. O. Marroquı´n de la Rosa,† T. Viveros Garcı´a,‡ and J. A. Ochoa Tapia*,‡ Instituto Mexicano del Petro´ leo, Eje Central La´ zaro Ca´ rdenas 152, Me´ xico, D.F., C.P. 07730, Mexico, and Universidad Auto´ noma MetropolitanasIztapalapa, Avenida Michoaca´ n y la Purı´sima, Col. Vicentina, Me´ xico, D.F., C.P. 09340 Mexico
A method based on the Taylor series expansion of the reaction rate has been used to evaluate the isothermal effectiveness factors of catalytic pellets. The prediction of the effectiveness factor is below an error of 14% for the second-order reaction used as an example. The method can be applied to any type of kinetic expression. Introduction An approximation method to evaluate the isothermal effectiveness factors of catalytic pellets has been used. The approximation in the method is based on the Taylor series expansion of the reaction rate expression above the surface particle condition for the key component. The method yields an analytical expression for the effectiveness factor equivalent to the classical first-order irreversible reaction expression. The general analytical result for any type of kinetics contains only one parameter analogous to the Thiele modulus. We report the application of the method to second-order equations. The effectiveness factors predicted in this work are compared with those obtained from the numerical solution of the exact boundary value problem. The error is negligible at low Thiele moduli, and as this parameter is increased, the error grows, reaching an asymptotic value at high Thiele moduli. For the example presented, the error is below 14%.
reaction rate of the key component is indicated by Rk. The reference values are the ones at the surface of the slab, and they are denoted by the subscript s. The solution of eq 1 is subject to the following boundary conditions:
Yk ) 1, at X ) 1
The linearization of the reaction rate term above the surface concentrations and the use of the transport equations for each of the i species (i * key component) yield the following equation for the key component:
d2Yk dX2
- φ2Yk ) -(νkΦ2 + φ2)
∆i
d2Yi dX2
) -νiRk,
for 0 < X < 1
(1)
Here νi is the stoichiometric coefficient, and ∆i is the ratio of the effective diffusivity of species i to the diffusivity of the key component. The dimensionless
|
∂Rk
∑j ∂Y
φ2 ) -Φ2
νj
j X)1∆j
(4)
can be interpreted as a modified Thiele modulus since it is the coefficient of the homogeneous reaction term in eq 3. The solution of the boundary value problem defined by eqs 2 and 3 yields the concentration profile of the key component that can be used, together with the definition for the effectiveness factor (Satterfield, 1970), to obtain
ηa ) * To whom all correspondence should be addressed. Email:
[email protected]. Telephone: (52) (5) 7244648. Fax: (52) (5) 7244900. † Instituto Mexicano del Petro ´ leo. ‡ Universidad Auto ´ noma MetropolitanasIztapalapa.
(3)
Here Φ is the Thiele modulus as defined by Wedel and Luss (1980) and the parameter φ defined by
Development At steady state, the dimensionless concentration Yi for unidimensional diffusion and the reaction process for species Ai, for a one-reaction system occurring in a porous slab of thickness 2 xs is governed by the following transport equation:
dYk ) 0 at X ) 0 (2) dX
and
tanh(φ) φ
(5)
The effectiveness factor given by eq 5 has the same form as those derived for the first-order irreversible kinetics (Satterfield, 1970). The result can be applied to any
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Ind. Eng. Chem. Res., Vol. 37, No. 9, 1998 3781
ness factor as a function of the Thiele modulus. Curve I was obtained by the numerical solution of the problem defined by eq 2 and the boundary conditions at the center and surface of the particle. Curve II is the prediction from eq 5. For Φ smaller than 1 the error is not bigger than 1%, and in the whole domain the error is not bigger than 14%. The error % of the approximate prediction is defined as having the numerical result as the reference. The method has been applied to other cases successfully. In fact, the error of the effectiveness factor predictions for those cases is smaller than that found in the example presented in this paper. Figure 1. Comparison of the effectiveness factor predicted from the numerical solution of the nonlinear problem (curve I: s) and that of the method of this work (curve II: - -).
kinetic expression whose effect is taken into account in the modulus. Results To illustrate the capabilities of the method, it will be applied to a reaction with a second-order nonlinear kinetic expression with a limiting reactant. The example is based on the liquid-phase oxidation of aqueous dilute solutions of formic acid reported by Baldi et al. (1974). The stoichiometry is represented by
2A + B f 2C + 2D It is assumed that the reaction follows the next kinetic relationship
rB ) kCACB
(6)
Species B is selected as the key component, and the stoichiometric coefficients are νB ) -1 and νA ) -2. The dimensionless reaction rate is
RB )
YAYB YAs
x
2+
CBs CAs∆A
Effectiveness factors of slab shape isothermal catalytic pellets can be estimated using an approximated method based in the linearization of the reaction term of the diffusion-reaction equation above the surface of the pellet. The linearization is the only assumption involved in the development after the steady-state isothermal transport problem is stated. The resulting effectiveness factor expressions require the evaluation of only one parameter: a modified Thiele modulus. For the cases presented, the error of the predictions is low enough and comparable to that obtained by other methods that require much more mathematical manipulations or numerical solutions (Bischoff, 1965; Wedel and Luss, 1980). The method can be applied without major problems to other particle shapes and reaction rate expressions. Acknowledgment The authors thank the financial support of Instituto Mexicano del Petro´leo and Consejo Nacional de Ciencia y Tecnologı´a. Literature Cited
(7)
The use of this equation together with eq 4 yields the following relationship for the modified Thiele modulus in terms of the Thiele modulus:
φ)Φ
Conclusions
(8)
This parameter can be used with eq 5 to obtain the approximate effectiveness factor. In Figure 1 we show the predictions for the effective-
Baldi, G.; Goto, S.; Chow, C. K.; Smith, J. M. Catalytic Oxidation of Formic Acid in Water. Intraparticle Diffusion in Liquid-Filled Pores. Ind. Eng. Chem. Process Des. Dev. 1974, 13, 447-452. Bischoff, K. B. Effectiveness Factor for General Reaction Rate Form. AIChE J. 1965, 11, 351-355. Satterfield, C. N. Mass Transfer in Heterogeneous Catalysis; MIT Press: Cambridge, England, 1970. Wedel, S.; Luss, D. A Rational Approximation of the Effectiveness Factor. Chem. Eng. Commun. 1980, 7, 245-259.
Received for review September 23, 1997 Revised manuscript received June 8, 1998 Accepted June 9, 1998 IE9706774
