Effectiveness factor Hougen-Watson kinetics

Exemplary results of modeling processes inside the catalytic layer are presented in Fig. 9. The solid lines show the dependency of the overall effectiveness factor on the relative distribution of the catalyst between the comers and the side regions. The two cases represent two levels of the first-order rate constants, with the faster reaction in case (b). As expected, the effectiveness factor of the first reaction drops as more catalyst is deposited in the comers. The effectiveness factor for the second reaction increases in case (a) but decreases in case (b). The latter behavior is caused by depletion of B deep inside the catalytic layer. What might be surprising is the rather modest dependency of the effectiveness factor on the washcoat distribution. The explanation is that internal diffusion is not important for slow reactions, while for fast reactions the available external surface area becomes the key quantity, and this depends only slightly on the washcoat distribution for thin layers. The dependence of the effectiveness factor on the distribution becomes more pronounced for consecutive reactions described by Langmuir-Hinshelwood-Hougen-Watson kinetics [26]. 

Figure 3. Effectiveness factor versus Thiele modulus at various times of poisoning (Hougen-Watson kinetic model for benzene hydrogenation).

Effectiveness factor versus an inappropriate Thiele modulus in a slab Hougen-Watson kinetics. . . . . , , . 384. 

Since 1 a is only a function of spatial coordinate r, the partial derivative in (19-38) is replaced by a total derivative, and the dimensionless concentration gradient evaluated at the external surface (i.e., ] = 1) is a constant that can be removed from the surface integral in the numerator of the effectiveness factor. In terms of the Hougen-Watson kinetic model and the dimensional scaling factor for chemical reaction that agree with the Langmuir-Hinshelwood mechanism described at the beginning of this chapter ... 

In terms of surface-averaged mass transfer across the external surface, the effectiveness factor for Hougen-Watson kinetics in flat-slab catalysts is... 

Many theoretical embellishments have been made to the basic model of pore diffusion as presented here. Effectiveness factors have been derived for reaction orders other than first and for Hougen and Watson kinetics. These require a numerical solution of Equation (10.3). Shape and tortuosity factors have been introduced to treat pores that have geometries other than the idealized cylinders considered here. The Knudsen diffusivity or a combination of Knudsen and bulk diffusivities has been used for very small pores. While these studies have theoretical importance and may help explain some observations, they are not yet developed well enough for predictive use. Our knowledge of the internal structure of a porous catalyst is still rather rudimentary and imposes a basic limitation on theoretical predictions. We will give a brief account of Knudsen diffusion. 

Effectiveness Factors for Hougen-Watson Rate Expressions. The discussion thus far and the vast majority of the literature dealing with effectiveness factors for porous catalysts are based on the assumption of an integer-power reaction rate expression (i.e., zero-, first-, or second-order kinetics). In Chapter 6, however, we stressed the fact that heterogeneous catalytic reactions are more often characterized by more complex rate expressions of the Hougen-Watson type. Over a narrow range of... 

In other instances, reaction kinetic data provide an insight into the rate-controlling steps but not the reaction mechanism see, for example, Hougen and Watson s analysis of the kinetics of the hydrogenation of mixed isooctenes (16). Analysis of kinetic data can, however, yield a convenient analytical insight into the relative catalyst activities, and the effects of such factors as catalyst age, temperature, and feed-gas impurities on the catalyst. 

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