Short-Cut Method for the Estimation of Isothermal Effectiveness Factors

Ind. Eng. Chem. Res. 2005, 44, 3947-3953

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Short-Cut Method for the Estimation of Isothermal Effectiveness Factors J. Alberto Ochoa-Tapia,* Francisco J. Valde´ s-Parada, and Jose´ de J. A Ä lvarez-Ramı´rez† Divisio´ n de Ciencias Ba´ sicas e Ingenierıá, Universidad Auto´ noma Metropolitana-Iztapalapa, Apartado Postal 55-534, Iztapalapa, D.F., P.C. 09340 Mexico

The computation of effectiveness factors (EFs) is commonly used for simplifying heterogeneous models of catalytic processes. To do this, the reaction rate in a catalyst pellet is expressed by its rate under surface/bulk conditions multiplied by a functional factor, namely, the EF. When many EF evaluations are required, it is desirable to dispose of short-cut methods to alleviate the computational burden. Existing short-cut methods are based on the availability of analytical solutions for a related linear, but nonhomogeneous, approximation to the nonlinear kinetics. However, the nonhomogeneous term can lead to spurious solutions, such as the presence of negative concentration values. This paper proposes a derivation of the EF short-cut method as an iteration procedure in the average concentration. On the other hand, aimed at avoiding negative concentration values, the proposed linear boundary-value problem is equipped with a nonactive region. In this way, the short-cut method is composed of two nonlinear equations on the average concentration and the position of the boundary between the active and nonactive regions. Numerical results show that such a modification increases the prediction capacity of EF short-cut methods for a practically acceptable Thiele modulus region. 1. Introduction The computation of effectiveness factors (EFs) is commonly used for simplifying heterogeneous models of catalytic processes. The underlying idea is to express the reaction rate in a catalyst pellet by its rate under surface/bulk conditions multiplied by a functional factor, namely, the EF. Although highly accurate numerical methods for the solution of the corresponding boundaryvalue problem have been reported elsewhere (see, for instance, work by Davis1), in some cases their use can be computationally expensive. Such is the case of optimization-based process design where the EF has to be evaluated many times for different sets of parameters. In other cases, one would like to have available simple-to-use methods for a preliminary evaluation of the process performance. In this way, there is an incentive to derive short-cut methods for the evaluation of EF for nonlinear kinetics. In a first approach, Wedel and Luss2 used perturbation series to derive a rational expression for the EF as a function of the Thiele modulus. Haynes3 proposed a modified Thiele modulus, obtained by either differentiation or integration of kinetic expressions, to obtain an approximate EF based on the expression for first-order kinetics. On the basis of this approximation idea, Marroquıń de la Rosa et al.4 obtained a short-cut method to estimate the EF from a modified Thiele modulus at the pellet surface conditions. Gottifredi and Gonzo5 used one collocation point approach to obtain an approximate expression for predicting concentration and temperature profiles. Comparison between approximate and numerical profiles for the * To whom correspondence should be addressed. Tel.: +52-55-57244648. Fax: +52-55-58044900. E-mail: [email protected]. † Also at Programa de Investigacioń en Matema´ticas Aplicadas y Computacioń, Instituto Mexicano del Petro´leo.

particular case of a Hougen-Watson-type kinetics showed a reasonable degree of accuracy. In a subsequent paper, Gottifredi and Froment6 extended this approximation idea to the case of a catalyst particle with coke formation. Recently, motivated by the linear driving force concept by Kim,7 Szukiewicz and Petrus8 used a linear approximation to the nonlinear kinetics at the surface conditions to obtain an estimate of the EF in terms of the average particle concentration 〈c〉. A distinctive characteristic of this approach, different from previous approaches, is that the average concentration is found as a solution of a nonlinear equation. Essentially, Szukiewicz and Petrus’ method can be briefly described as follows (see work by Szukiewicz9): (i) From the exact model, obtain an approximate model by linearly expanding the nonlinear kinetic term about the pellet surface conditions. This step leads to a linear boundaryvalue problem with nonhomogeneous kinetics, whose solution can be obtained with analytical methods. (ii) From the obtained analytical solution, compute the average concentration 〈c〉, which leads to a nonlinear equation to be solved numerically. (iii) The EF is estimated with the computed average concentration. It has been shown that Szukiewicz and Petrus’ method is able to handle multiple steady-state regions, which are retained within the nonlinear equation defining the average concentration. Additionally, contrary to the EF estimation methods reported so far, this approach leads to implicit EF estimates because the procedure involves the solution of a nonlinear equation on the average concentration. The computation of the EF involves solving a nonlinear reaction-diffusion equation where the nonlinearity is concentrated on the kinetic term. A salient feature of recently reported short-cut methods for estimating the EF is to propose an approximation to the nonlinear

10.1021/ie040190c CCC: $30.25 © 2005 American Chemical Society Published on Web 04/21/2005

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Ind. Eng. Chem. Res., Vol. 44, No. 11, 2005

zone is borrowed from studies of a fractional order reaction in catalyst particles where a nonactive region within the particle may exist.10-12 Similar to the method reported by Szukiewicz and Petrus,8 the result is an implicit EF estimate where two nonlinear equations on the average concentration and the dead-zone position must be solved numerically. Illustrations by means of numerical simulations show that the incorporation of a dead zone adds an important improvement in the prediction of EF when compared with previously reported results.4,8 2. EF Estimation as an Iteration Procedure

Figure 1. EF vs Thiele modulus obtained with the numerical exact solution and the approximate solution using 〈c〉 (AS1) and using cs (AS3) in the linear expansion of R(c), for n ) 2,γ ) 0.5, and cs ) 1.

kinetics in order to obtain a linear boundary-value problem, whose analytical solutions can be found with existing methods. That is, the basic idea is to exploit the existence of analytical solutions for the approximate linear problem. Given this approximation framework, it is interesting to note that Szukiewicz and Petrus’ method is capable of providing nonlinear information on the EF from a linear method. As mentioned before, this characteristic is not shared by previously reported short-cut methods. Despite its advantages over other short-cut methods, the following criticisms to Szukiewicz and Petrus’ method can be made: (a) Although some improvements on the systematization of short-cut methods for EF estimation have been obtained, there are still some issues that deserve further study. For instance, it would be interesting to provide a systematic connection between the derived linear boundary-value problem and the resulting nonlinear algebraic equation for the average concentration 〈c〉. In principle, this would allow the extension of short-cut methods for more general situations. For instance, it would be interesting to explore the use of short-cut methods for estimating mass flux through a catalyst pellet boundary under dynamic conditions or the mass flux through a separation membrane, regardless of the internal concentration profiles. (b) Given that the linear expansion of the nonlinear kinetics contains a nonhomogeneous (i.e., constant) term, the analytical solution of the linear boundary value can yield negative concentrations into a pellet subdomain. It is clear that it is an unfeasible solution from the physical point of view. In turn, negative concentrations can introduce significant deviations in the estimation of the EF. The existence of solutions with negative concentrations was not explored by Szukiewicz and Petrus,8 which can explain the significant deviations (about 15%) of the estimated EF for Thiele modulus values larger than 4 (see Figure 1 in Szukiewicz and Petrus’ paper). The aim of this paper is to address these two points. Specifically, the first one is addressed by deriving the EF short-cut methods as an iteration procedure on the average concentration 〈c〉. On the other hand, negative concentration values in the analytical solution are avoided by introducing a pellet dead zone where chemical reaction is inactive. The idea of using a particle dead

In a normalized domain [0, 1], diffusion and reaction for a single chemical species in a porous spherical particle can be described by the dimensionless transport equation

d2c 2 dc + - Φ2R(c) ) 0 dx2 x dx

(1a)

with boundary conditions

dc/dx ) 0

at x ) 0

(1b)

and

c ) cs

at x ) 1

(1c)

where cs is the surface concentration, which is assumed to be known. The EF, denoted by η, is defined as follows:

∫01R[c(x)] x2 dx η) ∫01R(cs) x2 dx

(2)

In principle, the computation of η can be made by computing the approximate solution, say cj(x), x ∈ [0, 1], via an accurate numerical method like Galerkin or finite differences, and subsequently using it in eq 2 to obtain the estimated EF, say η j , via numerical quadratures.1 However, as discussed in the Introduction, repeated evaluations of the EF can impose an important computational burden, e.g., an optimization-based design of chemical reactors. This has motivated the search for short-cut methods for a quick and easy-to-use evaluation of the EF. Most reported short-cut methods are based on a linear expansion of the reaction term R(c) about surface conditions:4,6 R(c) ≈ R(cs) + R′(cs) (c - cs) where R′(cs) denotes the derivative dR(c)/dc. This expansion, together with the boundary conditions (1b) and (1c), leads to a linear boundary-value problem that can be solved analytically to obtain a solution cL(x;cs) (the subindex L is used to denote a solution based on the linear expansion). Subsequently, this solution is to be used in eq 2 to obtain an EF estimate, namely,

η j)

∫01R[cL(x;cs)] x2 dx

3 R(cs)

The EF refers to a sort of average reaction-diffusion measure into the particle. However, in the derivation of the EF estimate, the nonlinear reaction term is linearly expanded about surface conditions. It seems that, given that the EF is an average reaction measure

Ind. Eng. Chem. Res., Vol. 44, No. 11, 2005 3949

(see eq 2), an expansion of R(c) about the average concentration 〈c〉, which is defined as

∫01c(x) x2 dx

〈c〉 ) 3

R(c) ≈ R(〈c〉0) + R′(〈c〉0) (c - 〈c〉0) Step 2. The corresponding reaction-diffusion equation is given by

(4)

In this way, eq 4, together with eqs 1b and 1c, become a linear boundary-value problem whose analytical solution cL(x;〈c〉0) can be obtained with standard methods. Step 3. In contrast with the reported methods where the solution cL(x;cs) depends only on the known surface concentration cs, in this case the solution cL(x;〈c〉0) depends on a guess 〈c〉0 of the unknown average concentration 〈c〉. Hence, a correctness test for the solution cL(x;〈c〉0) can be made by computing the corresponding average concentration by means of the definition (3). That is, the computed average concentration 〈c〉*, based on a guess 〈c〉0, is

〈c〉* ) φ(〈c〉0) where def

φ(〈c〉0) ) 3

∫01cL(x;〈c〉0) x2 dx

(5)

In the event that 〈c〉* ) 〈c〉0, one has that the guess 〈c〉0 is correct with respect to the linear expansion of R(c). In this case, the estimated EF is computed as

η j)

3 R(cs)

∫

def

N(〈c〉) ) 〈c〉 - φ(〈c〉)

(3)

could yield better EF estimates than an expansion about the surface concentration cs. In the following, we will propose a procedure to obtain EF estimates based on a linear expansion of R(c) about the unknown average concentration 〈c〉. The procedure is composed of the following steps: Step 1. Let 〈c〉0 be a guess of the actual average concentration 〈c〉. Compute the linear expansion of R(c) about 〈c〉0 conditions:

d2c 2 dc - Φ2[R(〈c〉0) + R′(〈c〉0) (c - 〈c〉0)] + 2 x dx dx

notice that the fixed point 〈cj〉 corresponds to positive roots of the nonlinear equation

1

c (x;〈c〉0) x2 dx 0 L

Step 4. If 〈c〉* * 〈c〉0, then the average concentration guess 〈c〉0 is incorrect. In such a case, one should consider another average concentration guess. For instance, one can take 〈c〉* as the new average concentration guess and repeat this procedure until |〈c〉* - 〈c〉0| is sufficiently small. This repetition procedure is essentially a fixed-point iteration on the average concentration 〈c〉. In fact, it can be written as

〈c〉j ) φ(〈c〉j-1), for j ) 1, 2, ..., and starting with the initial guess 〈c〉0 This iteration will converge to a fixed point, say 〈cj〉, as long as the contraction condition |φ(〈cj〉)| < 1 is fulfilled and 〈c〉0 is sufficiently close to 〈cj〉. If |φ(〈cj〉)| g 1, convergence 〈c〉j f 〈cj〉 cannot be guaranteed. However,

(6)

In this way, a more robust way to compute 〈cj〉 is by solving the nonlinear equation N(〈c〉) ) 0 by means of, e.g., the Newton-Raphson method. Once a positive root 〈cj〉 is found, the estimated EF is given by

η j)

3 R(cs)

∫01R[cL(x;〈cj〉)] x2 dx

which can be carried out by means of numerical quadratures. The expressions for cL(x;〈c〉0), N(〈c〉), and the estimated EF η j are given in the Appendix. Regarding the results obtained in this section, the following comments are in order: (i) it is interesting to note that the EF estimate reported by Szukiewicz and Petrus6 depends also on the root 〈cj〉 of a nonlinear equation. In such a case, the estimate 〈cj〉 satisfies the following nonlinear equation:

3Ψ(cs - 〈c〉) - Φ2R(〈c〉) ) 0

(7)

where Ψ is a function of the surface concentration cs. Equation 7 is obtained by taking the asymptotic time limit of a differential equation on 〈c〉, and its derivation was made by solving the dynamical version of the reaction-diffusion equation. (i) In our case, the nonlinear equation (see the Appendix) is obtained by posing an iteration procedure directly on the average concentration. (ii) Being that N(〈c〉) is a nonlinear equation, the existence of several positive roots is not forbidden. Hence, similar to the method presented by Szukiewicz and Petrus,8 the short-cut method described above has the potential of retaining steady-state multiplicity information based solely on the average concentration 〈c〉. To illustrate the ability of the short-cut method to estimate the EF, consider the Langmuir-Hinshelwood kinetic equation

R(c) )

(1 + γ)nc , (1 + γc)n

for γ, n > 0

As in the work by Szukiewicz and Petrus,8 calculations were performed for Φ ∈ [0, 10] because of its practical use. In the sequel, EF estimation based on a finite-differences solution of the nonlinear boundaryvalue problem (1) with Simpson quadratures for computing the integral (2) will be considered as the “exact” solution. Figure 1 shows the results for n ) 2, γ ) 0.5, and cs ) 1. It should be mentioned that the results presented in Figure 1 are representative because similar results are obtained for other parameter values. It is noted that the best EF estimates are obtained with the method based on the average concentration rather than those obtained based on the surface concentration. As explained before, this could be expected because the EF is a measure of the average reaction activity into the particle. Poorer EF estimates are obtained with the surface concentration because the linear expansion about it carries information only on a surface neighborhood. In this way, as observed in Figure 1, EF estimation based on a surface concentration gives acceptable results only for sufficiently small Thiele modulus Φ, for

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Because the dead zone position xdz is not available a priori, it becomes a new unknown for the short-cut EF estimation formulation. In this case, an EF estimate can be obtained by modifying the steps described in the above section. In fact, the procedure can be described as follows: Step 1′. Let 〈c〉0 be a guess of the actual average concentration 〈c〉. Compute the linear expansion of R(c) about 〈c〉0 conditions:

R(c) ≈ R(〈c〉0) + R′(〈c〉0)(c - 〈c〉0) Step 2′. Let xdz,0 be a guess of the dead zone position xdz. The corresponding reaction-diffusion equation is given by Figure 2. Concentration profiles obtained with the numerical exact solution, cL(x;〈c〉) and cL(x;cs), for Φ ) 5, n ) 2, γ ) 1, and cs ) 1.

which the reaction activity is concentrated in a relatively small particle region close to the surface. 3. Short-Cut EF Estimation with Dead Zone Correction In the short-cut methods described in the above section, the nonlinear term R(c) is linearly expanded about average concentration conditions; namely, R(c) ≈ def

L(c,〈c〉) ) [R(〈c〉) - R′(〈c〉0) 〈c〉] + R(〈c〉) c. The first term can be seen as a zero-order kinetics and the second as a first-order kinetics. In this way, the linear expansion L(c,〈c〉) is nonhomogeneous (i.e., L(0,〈c〉) * 0). The presence of the nonhomogeneous term R(〈c〉) - R′(〈c〉) 〈c〉 in the linear reaction-diffusion boundary-value problem (4) can lead to meaningless negative concentration values, which, by virtue of the EF definition (eq 2), can generate an important bias in the EF estimation. The same situation can be found for linear expansions about surface conditions. Figure 2 shows the concentration profiles cL(x;〈c〉) and cL(x;cs) for Φ ) 5, γ ) 1, n ) 2, and cs ) 1. It is noted that both profiles display negative concentration values for positions far from the surface, although cL(x;cs) presents more deviations from the exact solution. Similar to the approach taken for fractional-order reactions,8-10 a possible remedy to avoid negative concentration values is to consider only subdomains where the concentration takes positive values. This is done by moving the zero-flux boundary condition (1b) to a point xdz ∈ (0, 1) where cL(x;〈c〉) ) 0. That is, the boundary condition (1b) is substituted by the following one:

dc/dx ) 0

at x ) xdz

Notice that the above conditions do not represent an exact physical situation. Rather, they represent a fictitious position where, for the sake of profile approximation, the concentration profile is almost zero. If the real concentration profile is monotonically decreasing, the dead zone position xdz describes the boundary of a region where the concentration is practically zero. As a matter of fact, xdz is proposed to correct significant deviations of the approximate profile cL(x;〈c〉) from nonnegativeness. It should be stated that, although we are not concerned in this work with the concentration profile, the dead zone correction produced, in general, enhances EF estimations.

d2c 2 dc + - Φ2[R(〈c〉0) + R′(〈c〉0) (c - 〈c〉0)] ) 0 dx2 x dx (8a) with boundary conditions

dc/dx ) 0

at x ) xdz,0

(8b)

at x ) 1

(8c)

and

c ) cs

In this way, the linear boundary-value problem (8) has an analytical solution cL(x;〈c〉0,xdz,0). Step 3′. The computed average concentration 〈c〉*, based on the guess (〈c〉0, xdz,0) is

〈c〉* ) φ(〈c〉0,xdz,0) where 〈c〉* is computed from eq 5. Although 〈c〉* ) 〈c〉0, one is not sure that xdz,0 corresponds to the actual deadzone position. In this way, a criterion to check the correctness of xdz,0 is required. At the dead zone position xdz, we have imposed the zero-flux condition (8b). However, by itself such a condition does not remove the possibility of finding negative concentration values for positions x ∈ (xdz, 1). Hence, negative concentration values are avoided if the additional condition

c)0

at x ) xdz

is included. Therefore, the estimated dead zone position xdz,* based on the guess (〈c〉0, xdz,0) is computed as the root of the nonlinear equation

cL(xdz,*;〈c〉0,xdz,0) ) 0 In the event that the equalities 〈c〉* ) 〈c〉0 and xdz,* ) xdz,0 are satisfied, the estimated EF is given by

η j)

∫01R[cL(x;〈c〉0,xdz,0)] x2 dx

3 R(cs)

(9)

Step 4′. If 〈c〉* * 〈c〉0 and/or xdz,* * xdz,0, then the average concentration guess 〈c〉0 and the dead zone position xdz,0 guess are incorrect. In such a case, one can consider 〈c〉* and xdz,* as the new guess and repeat this procedure until |〈c〉* - 〈c〉0| and |xdz,* - xdz,0| are sufficiently small. As in the procedure described in the above section, this repetition procedure is essentially a fixed-point iteration on the average concentration 〈c〉


Figure 3. Concentration profiles obtained with the numerical exact solution, cL(x;〈c〉,xdz) and cL(x;cs,xdz), for Φ ) 5, n ) 2, γ ) 1, and cs ) 1.

Figure 4. EF vs Thiele modulus obtained with the numerical exact solution and the approximate solution using cL(x;〈c〉) (AS1) and cL(x;〈c〉,xdz) (AS2), for n ) 2, γ ) 0.5, and cs ) 1.

and the dead zone position xdz. In turn, such an iteration corresponds to finding positive roots of the two nonlinear equations

N1(〈c〉,xdz) ) 〈c〉 - φ1(〈c〉,xdz) N2(〈c〉,xdz) ) φ2(〈c〉,xdz)

(10)

where def

φ2(〈c〉,xdz) ) cL(xdz;〈c〉,xdz) In this way, the estimates 〈cj〉 and xjdz are obtained by solving the nonlinear equations N1(〈c〉,xdz) ) 0 and N2(〈c〉,xdz) ) 0 by means of, e.g., the Newton-Raphson method. The corresponding estimated EF is given by

η j)

3 R(cs)

∫01R[cL(x;〈cj〉,xjdz)] x2 dx

computed with numerical quadratures. It is noted that the dead zone introduces an additional degree of freedom to estimate the EF. To illustrate the effect of the dead zone correction, let us consider the Langmuir-Hinshelwood kinetic equation used in the above section. Figure 3 presents the concentration profiles for the same conditions of Figure 2. Notice that negative concentration values are no longer present, demonstrating the usefulness of the dead zone correction in removing spurious solutions even for the case where the nonlinear kinetics is expanded about particle surface conditions. Figures 4 and 5 show the estimated EF for the average and surface concentration cases, respectively. It is shown that the dead zone correction reduces significantly the EF estimation by about 2030%. On the other hand, although the aim of this paper is not to carry out an exhaustive comparison of the EF estimates provided by the different reported methods, it is interesting to note that EF estimates based on the average concentration outperform the results obtained with surface conditions expansion. This is because the average concentration expansion of R(c) retains more information on the reaction-diffusion process into the particle than does the surface condition expansion. However, the above observations should not lead to

Figure 5. EF vs Thiele modulus obtained with the numerical exact solution and the approximate solution using cL(x;cs) (AS3) and cL(x;cs,xdz) (AS4), for n ) 2, γ ) 0.5, and cs ) 1.

general conclusions because the obtained results can depend on a specific rate expression, parameters, etc. 4. Computational Complexity Initially, short-cut methods were thought to give easyto-use formulas for a fast estimation of EF, maybe with a hand calculator. In this way, for instance, Marroquıń de la Rosa et al.4 obtained a formula for spherical pellets

η j)

[

]

1 3 1 m tanh(m) m

(11)

where m is the so-called modified Thiele modulus given by

m2 ) -Φ2R′(cs) It is clear that the expression (11) can be easily computed with available hand-held calculators. This means that the EF estimates involve a minimal computational complexity. However, during the derivation of eq 11, no correction for spurious concentration profiles was taken into account. Our results, in conjunction with Szukiewicz and Petrus’ results,8 show that better EF estimates can involve more complex expressions and that numerical solution of at least one nonlinear equa-

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tion should be addressed. It is apparent that, given the expressions of such nonlinear equations (see the Appendix), the computational complexity of the proposed EF estimate increases significantly. As a matter of fact, this is correct. The derivation of the expressions N1(〈c〉,xdz) and N2(〈c〉,xdz) is quite elaborated because it requires knowledge of analytical boundary-value solutions for linear reaction-diffusion systems. We think this apparent drawback is, at the same time, an advantage because it leads to a closer understanding of the diffusion-reaction phenomena for catalyst systems. On the other hand, it is clear that finding the roots of N1(〈c〉,xdz) and N2(〈c〉,xdz) can hardly be made in a hand-held calculator. However, there is a plethora of portable computational equipment where the implementation of, e.g., the Newton-Raphson method, can be easily implemented. One is tempted to suspect that finding the roots of N1(〈c〉,xdz) and N2(〈c〉,xdz) is as computationally expensive as implementing a finite-differences method to compute the EF. This is true for isolated EF estimations. However, when the EF is located within a repeated procedure (e.g., in optimization of the reacting equipment), the advantages of short-cut methods, such as those proposed in this paper, are evident. In fact, while a finite-differences method would involve the repeated solution of a large set of nonlinear algebraic equations (one equation for each finite-differences node), shortcut methods require only the solution of one, as in Szukiewicz and Petrus’ approach,8 or two, as in the present approach, algebraic equations to obtain a good EF estimate. For instance, to obtain an EF estimation with less than 5% error with respect to numerical EF computation, a short-cut method, including the one proposed in this work, requires about 20% computation time required by a finite-differences method with a mesh size of 50. Within an intensively repeated EF computation scheme, such as optimization procedures, this time savings would yield a better simulation performance. In this form, it is concluded that the computational complexity of short-cut methods is still substantially smaller when compared with an EF estimation based on “exact” methods.

Finally, it should be stressed that our results and the previously reported ones2-5,8 seem to indicate that the EF estimation is not a hard problem when compared to profile approximations. In fact, recent results13 have shown that an accurate EF approximation can be made with neural networks provided a suitable training with a sufficiently large number of data. Such an accurate approximation is due to the smoothness of the EF functions that allows an easy data interpolation. Appendix For the sake of clarity in presentation, the expressions for the concentration profile for the linear boundaryvalue problem, the nonlinear equations to be solved and the resulting EF are given in this appendix. 1. EF Expressions without Dead Zone. Let

F(〈c〉) ) Λ2(〈c〉)/Λ1(〈c〉) Λ1(〈c〉) ) Φ2R′(〈c〉) Λ2(〈c〉) ) Φ2[R(〈c〉) - R′(〈c〉) 〈c〉] In summary, the above procedure leads to the following expression for the solution cL(x;〈c〉0) of the boundaryvalue problem (4):

cL(x;〈c〉0) )

]

so that the nonlinear equation N(〈c〉) required to compute 〈cj〉 is the following one (see eqs 5 and 6):

N(〈c〉) ) 〈c〉 3[〈c〉Λ1(〈c〉)+Λ2(〈c〉)] [xΛ1(〈c〉)coth(xΛ1(〈c〉)) - 1]-F(〈c〉) Λ1(〈c〉)2 The roots 〈cj〉 of this equation are subsequently used in eq A.1, instead of the guess 〈c〉0 to compute the estimated EF using numerical quadratures on the expression

5. Conclusions A simple systematic framework to derive short-cut methods for EF estimations has been proposed in this paper. The main idea is to see the EF estimation as a model-reduction process, where the original model is reduced into a simple concentration (e.g., average concentration) model. Because the projected concentration is unknown, the model-reduction results in an iterative process, which can be regarded as the solution of a nonlinear equation on the unknown projected concentration. This model-reduction procedure can lead to (meaningless) spurious solutions containing negative concentration values. It has been shown that, by introduction of a dead zone correction where chemical reaction is inactive, such spurious solutions are removed. Numerical results have shown that the dead zone correction can improve significantly the EF estimates. It should be mentioned that the short-cut methodology presented in this paper can be extended for other more involved conditions, such as dynamical models and catalyst particles into a fluid (i.e., first-order surface boundary condition). Results in these issues are under progress and will be reported in a future work.

[

cs + F(〈c〉0) sinh(xΛ1(〈c〉0)x) - F(〈c〉0) x sinh(xΛ1(〈c〉0)) (A.1)

η j)

∫01cL(x;〈cj〉) x2 dx

3 R(cs)

2. EF Expressions with Dead Zone. In this case, the solution cL(x;〈c〉0,xdz,0) is given by

cs + F(〈c〉0) x sinh[xΛ1(〈c〉0)x] + ω j (〈c〉0,xdz,0)cosh[xΛ1(〈c〉0)x]

cL(x;〈c〉0,xdz,0) )

{

}

sinh[xΛ1(〈c〉0)] + ω j (〈c〉0,xdz,0)cosh[xΛ1(〈c〉0)] F(〈c〉0) (A.2)

where

ω j (〈c〉0,xdz,0) )

xΛ1(〈c〉0)xdz,0 - tanh[xΛ1(〈c〉0)xdz,0] 1 - xΛ1(〈c〉0)xdz,0 tanh[xΛ1(〈c〉0)xdz,0]

Let

j (〈c〉,xdz) m1(〈c〉,xdz) ) 1 - xΛ1(〈c〉)xdzω


and

* ) computed property in the iteration procedure

m2(〈c〉,xdz) ) ω j (〈c〉,xdz)xΛ1(〈c〉) - 1 +

Greek Symbols

coth[xΛ1(〈c〉)][xΛ1(〈c〉) - ω j (〈c〉,xdz)] + 1 {sinh[xΛ1(〈c〉)xdz]m1(〈c〉,xdz) + sinh[xΛ1(〈c〉)]

cosh[xΛ1(〈c〉)xdz]m1(〈c〉,xdz)}

Hence, the nonlinear equations N1(〈c〉,xdz) and N2(〈c〉,xdz) (see eq 10) are described as follows:

N1(〈c〉,xdz) ) 〈c〉 3[csΛ1(〈c〉) + Λ2(〈c〉) m2(〈c〉,xdz)] Λ1(〈c〉) {1 + ω j (〈c〉,xdz) coth[xΛ1(〈c〉)]} 2

-F(〈c〉)(1 - xdz3)

and

cs + F(〈c〉) x sinh[xΛ1(〈c〉)x] + ω j (〈c〉,xdz) cosh[xΛ1(〈c〉)x]

N2(〈c〉,xdz) )

{

sinh[xΛ1(〈c〉)] + ω j (〈c〉,xdz) cosh[xΛ1(〈c〉)]

}

- F(〈c〉)

The roots 〈cj〉 and xjdz of these equations are subsequently used in eq A.2 to obtain the estimated EF via numerical quadratures on eq 9. Nomenclature c ) dimensionless concentration of the reactive in the particle 〈c〉 ) average concentration as defined in eq 3 L ) linear expansion of the reaction rate expression m ) modified Thiele modulus n ) parameter in the Langmuir-Hinshelwood kinetic equation (>0) N ) function defined in eq 6 R ) dimensionless reaction rate term x ) dimensionless radial position

Φ ) Thiele modulus γ ) parameter in the Langmuir-Hinshelwood kinetic equation (>0) η ) effectiveness factor φ ) function defined in eq 5 Ψ ) function defined in eq 7

Literature Cited (1) Davis, M. Numerical Methods and Modeling for Chemical Engineers; Wiley: New York, 1984. (2) Wedel, S.; Luss, D. Rational approximation of the effectiveness factor. Chem. Eng. Commun. 1980, 7, 245-259. (3) Haynes, H. W., Jr. An explicit approximation for the effectiveness factor in a porous heterogeneous catalyst. Chem. Eng. Sci. 1986, 41, 412-415. (4) Marroquıń de la Rosa, J. O.; Viveros-Garcıá, T.; OchoaTapia, J. A. Evaluation of isothermal effectiveness factors for nonlinear kinetics using an approximate method. Ind. Eng. Chem. Res. 1998, 37, 3780-3781. (5) Gottifredi, J. C.; Gonzo, E. E. An approximate expression for predicting concentration and temperature profiles inside a catalyst pellet. Chem. Eng. Sci. 1996, 51, 835-837. (6) Gottifredi, J. C.; Froment, G. F. A semi-analytical solution for concentration profiles inside a catalyst particle in the presence of coke formation. Chem. Eng. Sci. 1997, 52, 1883-1891. (7) Kim, D. H. Linear driving force formulas for diffusion and reaction in porous catalysts. AIChE J. 1989, 35, 343-346. (8) Szukiewicz, M.; Petrus, R. Approximate model for diffusion in a porous pellet and effectiveness factor. Chem. Eng. Sci. 2004, 59, 479-483. (9) Szukiewicz, M. An approximate model for diffusion and reaction in a porous pellet. Chem. Eng. Sci. 2002, 57, 1451-1457. (10) Aris, R. The Mathematical Theory of Diffusion and Reaction in Permeable Catalyst, Vol. 1. Theory of Steady-State; Calderon Press: Oxford, U.K., 1975. (11) Temkin, M. I. Fractional-order reaction in a spherical porous catalyst particle. Kinet. Catal. 1982, 22, 844-849. (12) Garcıá-Ochoa, F.; Romero, A. The dead-zone in a catalyst particle for fractional-order reactions. AIChE J. 1988, 11, 19161918. (13) Parisi, D. R.; Chocroń, M.; Amadeo, N. E.; Laborde, M. A. Approximation by neural network of the effectiveness factor in a catalyst with deactivation. Chem. Eng. Technol. 2002, 25, 11831186.

Subscripts dz ) dead zone L ) solution based on the linear expansion s ) at the surface 0 ) initial guess

Received for review June 25, 2004 Revised manuscript received November 17, 2004 Accepted March 18, 2005 IE040190C

Short-Cut Method for the Estimation of Isothermal Effectiveness Factors

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