Effectiveness factor asymptotic forms

The concentration and temperature Tg will, for example, be conditions of reactant concentration and temperature in the bulk gas at some point within a catalytic reactor. Because both c g and Tg will vary with position in a reactor in which there is significant conversion, eqns. (1) and (15) have to be coupled with equations describing the reactor environment (see Sect. 6) for the purpose of commerical reactor design. Because of the nonlinearity of the equations, the problem can only be solved in this form by numerical techniques [5, 6]. However, an approximation may be made which gives an asymptotically exact solution [7] or, alternatively, the exponential function of temperature may be expanded to give equations which can be solved analytically [8, 9]. A convenient solution to the problem may be presented in the form of families of curves for the effectiveness factor as a function of the Thiele modulus. Figure 3 shows these curves for the case of a first-order irreversible reaction occurring in spherical catalyst particles. Two additional independent dimensionless paramters are introduced into the problem and these are defined as... 

Petersen [12] points out that this criterion is invalid for more complex chemical reactions whose rate is retarded by products. In such cases, the observed kinetic rate expression should be substituted into the material balance equation for the particular geometry of particle concerned. An asymptotic solution to the material balance equation then gives the correct form of the effectiveness factor. The results indicate that the inequality (23) is applicable only at high partial pressures of product. For low partial pressures of product (often the condition in an experimental differential tubular reactor), the criterion will depend on the magnitude of the constants in the kinetic rate equation. 

A simplification is often employed for effectiveness factor calculations in the asymptotic limit of strong intraparticle diffusion resistance (12,13). In this situation, an alternative form of the key component mass balance can be written as follows ... 

Bueche suggests that this factor be applied to modify the structural factor to correct for entanglement effects. Thus, only the asymptotic form of Eq. (3.5) is utilized, the predictions at intermediate molecular weight being scarded. Equating the two asymptotic forms at the critical chain length Z (not to be confused with Z ) results in expressions for F that can be written in a form identical to Eq. (2.6) in terms of the parameter X = s )p/M) Z/v as ... 

With the technique described above, the asymptotic values for the effectiveness factors can be comfortably determined for the slab form of catalyst particles, since the integral, jr dy is usually rather simple to evaluate. 

The effectiveness factors versus Thiele modulus for the three geometries are plotted in Figure 7.8. Although the functional forms listed in Table 7.3 appear quite different, we see in Figure 7,8 that these solutions are quite similar. The effectiveness factor for the slab is largest, the cylinder is intermediate, and the sphere is the smallest at all values of Thiele modulus. The three curves have identical small and large 4> asymptotes. The maximum difference between the effectiveness factors of the sphere and the slab q is about 1.6%, and occurs at = 1.6. For 4> < 0.5 and > 7, the difference between all three effectiveness factors, is. less than 5%. 

The so-called black absorber method may be used for the determination offs- This uses the saturation effect offs when the effective thickness, Ta (see Eq. (25.30)), approaches infinity. In this case, the asymptotic form of the spectral peak amplitude can be used to obtain fs. The measurement of the /factor of the absorber, is based on Eq. (25.33) and the corresponding nonlinear dependence of the intensity, amplitude, and width of the Mdssbauer spectrum peak on the effective thickness (see later in Fig. 25.17). This latter is the so-called blackness effect. For the determination of the peak intensity and the peak amplitude should be measmed for at least two absorbers of different thicknesses (Vertes and Nagy 1990). 

Figure 3.4 Asymptotic forms of the effectiveness factor (7) as a function of the Thiele modulus ( ) for a slab catalyst with a first-order exothermic reaction (0 = 1/3 and y = 27) for Bim/Bii, =6. Numerical values in the intermediate region are given in Ref [80].

Multiplying the skeleton form function by this factor, we produce a decrease in the quantity q2H(q). In this manner, the observed fact that q2H(q) is constant in the asymptotic domain >0.3nm-1 results from the compensation of two opposite effects. 

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