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Ind. Eng. Chem. Res. 2006, 45, 1542-1547
An Improved Short-Cut Method for Effectiveness Factor Estimation Francisco J. Valde´ s-Parada,*,† Jose´ A Ä lvarez-Ramı´rez,† J. Obet Marroquı´n de la Rosa,‡ and † J. Alberto Ochoa-Tapia Departamento de Ingenierı´a de Procesos e Hidra´ ulica, DiVisio´ n de Ciencias Ba´ sicas e Ingenierı´a, UniVersidad Auto´ noma Metropolitana-Iztapalapa, Apartado Postal 55-534, Me´ xico D.F., 09340 Me´ xico, and Instituto Mexicano del Petro´ leo, Eje central La´ zaro Ca´ rdenas 152, Me´ xico D.F., 07730 Me´ xico
Two main short-cut methods for catalyst particle effectiveness factor (EF) estimation that use kinetics linearization about surface and average concentration conditions have been recently considered in the literature. The former produces simple computations with acceptable estimations for low Thiele modulus values, although negative particle concentrations can be generated. The latter is intended to improve the particle concentration profiles but at the expense of increased computations. The aim of this paper is to propose a hybrid method that combines the advantages of the two approaches to obtain EF and concentration profile estimations that are better than those of the individual ones. This is done by taking a kinetics linearization at an artificial concentration condition resulting from a type of Crank-Nicholson scheme. From a simple error analysis, it is shown that the resulting short-cut procedure displays enhanced convergence properties, which is corroborated by means of numerical computations. 1. Introduction Some approaches have been proposed for short-cut computation of the catalyst particle effectiveness factor (EF). Wedel and Luss1 used perturbation series to derive a rational expression for the EF as a function of the Thiele modulus. Haynes2 proposed a modified Thiele modulus, obtained by either differentiation or integration of kinetic expressions, yielding an approximate EF based on a first-order kinetics expression. Marroquı´n de la Rosa et al.3 proposed a short-cut method by departing from a kinetics first-order Taylor series expansion at surface conditions (SCC). By incorporating the linear driving force concept,4 Szukiewicz and Petrus5 improved the surface concentration-based EF estimation scheme and reported acceptable results for small and moderate Thiele modulus values. Although the use of the SCC approach5 yields acceptable EF estimation for relatively small Thiele modulus, significant errors can be obtained for large Thiele modulus since the particle concentration profile becomes very sharp. In fact, such errors are caused by estimated concentration profiles with negative values, which lead to overestimated EF values. To address this problem, Ochoa-Tapia et al.6 proposed a short-cut method based on a kinetics linearization about average concentration conditions (ACC). To face the problem of estimated negative concentrations, the method was endowed with a heuristic deadzone artifact where the chemical reaction vanishes. This approach resulted in enhanced EF estimations with an error 0
(26)
In a strict sense, eq 19 states that the weighting parameter θ should be chosen as a function of the concentration profile c(x). This would imply an algorithm with a high computational complexity, since the parameter θ depends on the second derivative of the kinetics expression at both surface and average concentration conditions. To overcome this difficulty, the following heuristic rule is proposed:
θ)-
1 Φ
(27)
The rationale behind the above selection is to add weight to the ACC component for a large Thiele modulus. For a small Thiele modulus, the larger weight is assigned to the SCC component. In fact, for a small Thiele modulus, the particle
Ind. Eng. Chem. Res., Vol. 45, No. 4, 2006 1545
Figure 1. Concentration profiles and error estimation obtained between the numerical exact solution ()); the approximate solutions made by Ochoa-Tapia et al.6 [(- - -)AS1] and Marroquı´n de la Rosa et al.3 without using the dead-zone idea [(‚ ‚ ‚)AS3]; and this work [(s)AS5] for Φ ) 5, n ) 2, γ ) 1, and cs ) 1.
Figure 2. Effectiveness factor and error estimation as a function of the Thiele modulus obtained with the numerical exact solution ()), by Ochoa-Tapia et al.6 with [(- - -)AS2] and without [(‚ ‚ ‚)AS1] the dead-zone correction, and in this work [(s)AS5] for n ) 2, γ ) 0.5, and cs ) 1.
concentration profile is almost flat, so that the surface concentration can represent very closely the conditions in the particle. The reason for this heuristic rule to have negative values comes from analyzing Figures 1 and 2 in Ochoa-Tapia et al.6 In those figures, the results from the exact solution were not between the predictions using the ACC or the SCC approaches. Thus, the predictions using θ ∈ (0,1) or θ > 1 will show no benefit at all; only values of θ < 0 will prove to be more useful. In this way, the heuristic rule eq 27 drastically reduces the complexity of the method, in comparison with using θ(c), making it suitable for easy application in practical cases. However, the use of eq 27 does not guarantee the reduction of the truncation error as given in eq 18; for this reason, each comparison is accompanied with the corresponding error percent plot defined as
Error % )
|ψexact - ψapprox| × 100% ψexact
4.2. Catalyst Particle with External Resistance. The second situation corresponds to a catalyst particle with external resistance effects. For the sake of simplicity, the model equations correspond to those of a two-phase, steady-state stirred-tank reactor (STR). The approximate reaction-diffusion equation for the dispersed phase is given by eq 1 using the approximation of eq 20; on the other hand, for the fluid in the STR, the corresponding equation is
1 (c - cf) + ψp(cs - cf) ) 0 τR in and the corresponding boundary conditions are
at x ) 1
(28)
where ψ is used to denote either the effectiveness factor or the dimensionless concentration. In Figures 1 and 2, the results obtained with the SCC, ACC, and HCC approaches are compared. Observe that the hybrid approach, together with the heuristic rule (eq 27), yields better EF and particle concentration profile estimations. Indeed, the maximum EF estimation error is 0) R ) dimensionless exact reaction rate term Rj ) dimensionless reaction rate term approximated by about the jCC, j ) A, H, S Tj ) truncation error relative to the jCC approach, j ) A, H, S x ) dimensionless radial position Subscripts A ) relative to the ACC approach H ) relative to the HCC approach s ) at the surface S ) relative to the SCC approach Greek Symbols
(38)
It should be remarked that our objective is to develop simpleto-use methods for the desired prediction. In this form, the above heuristic rule to assign θ weights the SCC and ACC in terms of the intraparticle, Φ-1, and the external, Bi-1, resistances satisfies our purposes. The results are shown in Figures 3 and 4, where improvements in the dispersed and fluid phases concentration predictions by the HCC approach can be observed, with the maximum absolute error percent for the fluid concentration in the STR being below 0.1%, in general. Even the concentration profile is very close to the exact one obtained from the numerical solution (finite-differences) of the nonlinear model. These results show that a combination of internal and external information in the particle can be necessary to obtain accurate EF estimates in the base of single-concentration models. 5. Conclusions In this paper, we propose a short-cut method that improves the EF estimations reported by other approaches. The proposed hybrid method consists of a reaction rate approximation resulting from a combination of approximations at surface and average concentration conditions previously reported. In this way, as the hybrid approach retains concentration information of both surface and internal particle conditions, the predictions have a lower error than those obtained with individual approaches. Therefore, we now have a sufficiently versatile method that allows the user to decide which approach is better for the desired estimation, i.e., if one is interested only in the estimation of the effluent concentration in a STR, the SCC (θ ) 1) approach is enough. However, if acceptable predictions of the reagent concentration inside the particle are desired, then the HCC (θ ) θ(Φ,Bi)) is recommended. In addition to the improved EF
Φ ) Thiele modulus γ ) parameter in the Langmuir-Hinshelwood kinetic equation (>0) η ) effectiveness factor Λ0 ) zero-order reaction term Λ1 ) modified Thiele modulus ψp ) dimensionless fluid-pellet exchange parameter θ ) weighting parameter used in the HCC approach τR ) dimensionless residence time Literature Cited (1) Wedel, S.; Luss, D. Rational approximation of the effectiveness factor. Chem. Eng. Commun. 1980, 7, 245-259. (2) Haynes, H. W., Jr. An explicit approximation for the effectiveness factor in a porous heterogeneous catalyst. Chem. Eng. Sci. 1986, 41, 412415. (3) Marroquı´n de la Rosa, J. O.; Viveros-Garcı´a, T.; Ochoa-Tapia, J. A. Evaluation of isothermal effectiveness factors for nonlinear kinetics using an approximate method. Ind. Eng. Chem. Res. 1998, 37, 3780-3781. (4) Kim, D. H. Linear driving force formulas for diffusion and reaction in porous catalysts. AIChE J. 1989, 35, 343-346. (5) Szukiewicz, M.; Petrus, R. Approximate model for diffusion in a porous pellet and effectiveness factor. Chem. Eng. Sci. 2004, 59, 479483. (6) Ochoa-Tapia, J. A., Valde´s-Parada, F. J.; A Ä lvarez-Ramı´rez, J. J. Short-Cut method for the estimation of isothermal effectiveness factors. Ind. Eng. Chem. Res. 2005, 44, 3947-3953. (7) Morales-Cabrera, M. A.; Pe´rez-Cisneros, E. S.; Ochoa-Tapia, J. A. Approximate method for the solution of facilitated transport problems in liquid membranes. Ind. Eng. Chem. Res. 2002, 41, 4626-4631. (8) Morales-Cabrera, M. A.; Pe´rez-Cisneros, E. S.; Ochoa-Tapia, J. A. An approximate solution for the CO2 facilitated transport in sodium bicarbonate aqueous solutions. J. Membr. Sci. 2005, 1-2, 98-107.
ReceiVed for reView July 13, 2005 ReVised manuscript receiVed December 16, 2005 Accepted December 21, 2005 IE050829S
