Thiele modulus definition

Figure 10 shows that Tj is a unique function of the Thiele modulus. When the modulus ( ) is small (- SdSl), the effectiveness factor is unity, which means that there is no effect of mass transport on the rate of the catalytic reaction. When ( ) is greater than about 1, the effectiveness factor is less than unity and the reaction rate is influenced by mass transport in the pores. When the modulus is large (- 10), the effectiveness factor is inversely proportional to the modulus, and the reaction rate (eq. 19) is proportional to k ( ), which, from the definition of ( ), implies that the rate and the observed reaction rate constant are proportional to (1 /R)(f9This result shows that both the rate constant, ie, a measure of the intrinsic activity of the catalyst, and the effective diffusion coefficient, ie, a measure of the resistance to transport of the reactant offered by the pore stmcture, influence the rate. It is not appropriate to say that the reaction is diffusion controlled it depends on both the diffusion and the chemical kinetics. In contrast, as shown by equation 3, a reaction in solution can be diffusion controlled, depending on D but not on k. 

We would be remiss if we did not indicate that some authors have defined yet another Thiele-type modulus ( ) for this problem as <. = 0s/3 and they have developed their analysis according to this parameter. In using plots of f] versus S9 one must be careful to determine which of the two alternative definitions has been adopted for s. Otherwise his value for rj may be considerably in error. A variety of other symbols for dimensionless groups akin to the Thiele modulus have been employed by different authors. 

Using this definition of the Thiele modulus, the reaction rate measurements for finely divided catalyst particles noted below, and the additional property values cited below, determine the effectiveness factor for 0.5 in. spherical catalyst pellets fabricated from these particles. Comment on the reasons for the discrepancy between the calculated value of rj and the ratio of the observed rate for 0.5 in. pellets to that for fine particles. 

The conclusions about asymptotic values of tj summarized in Tables 8.2 and 8.3, and the behavior of tj in relation to Figure 8.11, require a generalization of the definition of the Thiele modulus. The result for " in equation 8.520 is generalized with respect to particle geometry through Le, but is restricted to power-law kinetics. However, since... 

This study employed conventional diffusion-reaction theory, showing that with diffusion-limited reactions the internal effectiveness factor of a heterogeneous catalyst is inversely related to the Thiele modulus. Using a standard definition of the Thiele modulus [100], the observed reaction rate of an immobilized-enzyme reaction will vary with the square root of the immobilized-enzyme concentration in a diffusion-limited system. In this case, a plot of the reaction rate versus the enzyme loading in the catalyst formulation will be nonlinear. 

The definition of the electrochemical Thiele modulus [Eq. (9b)J characterizing the degree of electrocatalyst utilization is a prerequisite for properly tailoring the micromorphology of porous electrocatalytic electrode coatings and fuel cell electrodes, as it allows matching of the coating or catalyst particle dimension to the catalytic activity of the material ... 

A great deal of attention has been devoted to this topic because of the interesting and often solvable mathematical problems that it presents. Results of such calculations for isothermal zero-, first-, and second-order reactions in uniform cylindrical pores are summarized in Figure 17.6. The abscissa is a modified Thiele modulus whose basic definition is... 

The Prater number (3 - in contrast to eq. (14.25) is related to Ts and not to TG - and the Arrhenius number have a major influence on the development of the T and c profiles. The pore utilization factor qp is therefore dependent upon Arr, (3 and Thiele modulus . The correlation between these four pi-numbers is represented in Fig. 83. For T]p and the following definitions apply ... 

Using the definition of the Thiele modulus according to eq 27 finally yields... 

However, in many practical situations the problem exists that effective rate constants and activation energies have been derived on the basis of laboratory experiments. The question arises then as to whether or not these parameters arc influenced by transport effects. With the relations given so far, this question cannot be answered yet, since according to its definition by cq 27 the Thiele modulus is based on the intrinsic rate constant k. This problem can be solved by introducing a new modulus, which in contrast to only contains observable (effective) quantities, and thus can... 

Comparing eqs 56 and 27, and recalling the definition of the effectiveness factor according to cq 40, yields the following simple relationship between the Thiele modulus and the Weisz modulus ... 

Figure 12 shows the effectiveness factor as a function of the Wheeler-Weisz modulus for different reaction orders, indicating that criterion (33) holds for the generalized Thiele modulus. Due to the definition of L it is fairly independent of the catalyst geometry. 

It can be seen that the Thiele modulus may be regarded a measure for the ratio of the reaction rate to the rate of diffusion. However, many different definitions are used in the literature, as described below. 

In the past, a number of attempts have been made to generalize the definition of the Thiele modulus. Aris [6] noticed that all the Thiele moduli for first-order reactions were of the form ... 

This literature survey leaves the impression of an enormous but chaotic quantity of publications, because all the models and correlations focus on a single rather specific problem. A number of attempts have been made to integrate some of the models and correlation s, but without success. Also, the authors tend to use their own tailor-made definitions rather than generalized ones. As a result a large variety of definitions can be found for the same number (Doraiswamy and Sharma [4] use 20 different definitions for the Thiele modulus in their survey). This makes the results of different correlations hard to compare. Finally, some of the formulae which were given contradict each other, which is of course very confusing. 

According to the above definitions, the effectiveness factor for any of the above shapes can adequately describe simultaneous reaction and diffusion in a catalyst particle. The equation for the effectiveness factor in a slab is the simplest in Table 6.3.1 and will be used for all pellet shapes with the appropriate Thiele modulus ... 

Such a different conclusion can be understood by considering the difficulties connected to the experimental determination and the definition of Thiele modulus parameters, such as So and D. According to Chien, S means the catalyst primary particle size with a value of about 10 cm for a-TiCIj instead, in the Multigrain model, Sp seems to correspond to the size of the whole catalyst granule. 

It is evident that, in the cases of single-file diffusion and barrier-limited molecular exchange, a suitable definition of the Thiele modulus cannot be based on eq. 1. These cases may also be covered, however, if the Thiele modulus is expressed in terms of the... 

The simplest kinetics to assume would be a one-step reaction from an organometallic compound to metallic deposit, i.e. the A to B first order reaction. This is essentially the approach used by Sato et al [1], which led to a straight forward definition of the Thiele modulus and the pattern of metal deposition as the fantiliar... 

For the single-reaction cases, we performed dimensional analysis and found a dimensionless number, the Thiele modulus, which measures the rate of production divided by the rate of diffusion of some component. A complete analysis of the first-order reaction in a sphere suggested a general approach to calculate the production rate in a pellet in terms of the rate evaluated at the pellet exterior surface conditions. This motivated the definition of the pellet effectiveness factor, which is a function of the Thiele modulus. 

Figure 7.9 shows the effect of reaction order for n > 1 in a spherical pellet. As the reaction order increases, the effectiveness factor decreases. Notice that the definition of Thiele modulus in Equation 7.47 has achieved the desired goal of giving all reaction orders a common asymptote at high values of Figure 7.10 shows the effectiveness factor versus Thiele modulus for reaction orders less than unity. Notice the discontinuity in slope of the effectiveness factor versus Thiele modulus that occurs when the order is less than unity. Recall from... 

If the catalyst particles are not completely wetted by the liquid phase and the pores consequently not completely filled with liquid phase (static holdup gives some indication of whether this is the case or not), the situation is considerably more complex. In addition to being a function of the Thiele modulus, the catalytic effectiveness will now depend on the fraction of external wetting, rjcs, and the fraction of pore volume filled with liquid, rji. Dudokovic [M.P. Dudokovic, Amer. Inst. Chem. Eng. Jl., 23, 940 (1977)] proposed a reasonable approach that accounts for all three factors. If the reaction proceeds only on the catalyst surface effectively wetted by the liquid phase and components of the reaction mixture are nonvolatile, then one can in principle modify the definition of the Thiele modulus to... 

Primes or subscripts could be used to identify rate constants based on a unit mass of catalyst or on a unit volume of catalyst and those based on partial pressure or on concentration of reactant, but this notation would be cumbersome and perhaps confusing. In most cases, a simple k denotes the kinetic constant, but the units must be carefully checked for consistency. In the Thiele modulus, k must be expressed in sec, [Eq. (4.23)] but once 4> and r] are evaluated, other definitions of k can be used, as in Eq. (4.33). 

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