Thiele modulus interpretation

No practical catalyst pellet can be described by the geometry of the wafer, yet it will be shown that eqn. (6), albeit with a slightly different interpretation of the Thiele modulus, 0, is of practical value. For a cylindrical catalyst pellet (a shape often used in practice) of radius r and sealed at its flat ends... 

Thiele modulus. This can be interpreted by current diffusion-reaction theory-The reaction occurs mainly inside the pores of zeolites at lower Thiele modulus. Thus, the pore entrance sites are not important and the effects of pore entrance deactivation on the product selectivity and effectiveness are not obvious. With the increase of Thiele modulus, the reaction is gradually diffusion-controled, the pore entrance sites become important, and the effects of pore entrance deactivation become distinct. 

The analysis of the effects of transport on catalysis has focused on a comparison of the availability of reacting species by diffusion to the rate of reaction on the catalytic sites. High-surface-area catalysts are usually porous. Comparison of transport to reaction rates has usually been based on Knudsen diffusion (by constricted collision with the pore walls) as the dominant mode of transport. DeBoer has noted that for small pores surface diffusion may dominate transport (192). Thiele modulus calculations may therefore not be valid if they are applied to systems where surface diffusion can be significant. This may mean that the direct participation of spillover species in catalysis becomes more important if the catalysts are more microporous. Generalized interpretations of catalyst effectiveness may need to be modified for systems where one of the reactants can spill over and diffuse across the catalyst surface. 

The usual interpretation of the parameter P, referred to here as the deposition modulus, is that it is the square of the ratio of the characteristic time for diffusion to the characteristic time for surface deposition. In this view it is equivalent to the square of the Thiele modulus commonly appearing in analyses of porous-bed catalysis. Another useful interpretation of this parameter is that it is the ratio of two rates - the rate of deposition on the preform fiber surfaces, Ss a, to the maximum rate of diffusive transport, pDDf/a. Thus when P is small, the actual rate of diffusive transport will be less than this maximum, and the mean gradient of the reactant fraction will be smaller than the maximum value off/a. Under any of these interpretations, small values of P are associated with high uniformity of both the reactant fraction and coating thickness. 

Under strong diffusion effects, the effectiveness factor is inversely proportional to Thiele modulus. When the reaction has a generic n-order, one should take care in interpreting the experimental results, since, according to Equation 18.29, the reaction order is not the same as the intrinsic rate. The observed rate is as follows ... 

In the case of a zero-order reaction (r = feof) fhe Thiele modulus jS has been modified to suit the interpretation for biofilm kinetics. It is then the reciprocal of the normal Thiele modulus for a zero-order reaction,... 

The dimensionless substrate concentration profile in a porous membrane, where the biocatalyst is entrapped, depends on the Thiele modulus heterogeneous catalyst, since it allows for the estimation of the penetration depth within the support and can also be used to identify the mechanism that controls the process rate. The Thiele modulus can be interpreted as a ratio between a diffusion time (SVD if.s) and a kinetic time (A a/ max) or, equivalently, as a ratio between a characteristic kinetic rate Vmax/ m AS = AS and a merely diffusive transport mechanism, characterized by (D s s/S) x (A5/5). When (/) 1 kinetics is the limiting step, the process rate coincides with the reaction rate and the concentration profile is uniform, completely penetrating the support. 

In reactions occurring within porous catalysts the reactants have to adsorb and diffrise to and from the active centres. Typically this Affusion-sorption behaviour is measured under inert conditions at temperatures well below that at which reactions occur. It is preferable to study catalysts under reaction conditions. Such data can be interpreted by reaction-diffusion-adsorption models, the so-called "Thiele modulus" reproach [1]. The model parameters can be determined by using steady state or transient measurements. 

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