Catalysts, general diffusion factor

In many processes of interest to the hydrocarbon processing industry the size and shape of the catalyst has been chosen as a compromise between catalyst effectiveness and pressure drop. Hence, with effectiveness factors for the main reaction somewhat below 1, intraparticle pore diffusion is generally a factor to be reckoned with. Its effect is not easily quantified since the processing of a practical feedstock involves the conversion of a large variety of molecules with widely different reaction rates and therefore the translation of catalyst performance data obtained with crushed particles to that of the actual catalyst may be difficult and of questionable validity. 

As discussed later, this factor is more commonly of concern in flow reaction systems. The catalysts generally used with batch reactors are fine powders with which these diffusion limitations are minimal. 

There are a number of factors which may influence the activity or selectivity of a polymer-immobilized catalyst. Substrate diffusion is but one. This article has reviewed the mathematical formalism for interpreting reaction rate data. The same approach that has been employed extensively in heterogeneous systems is applicable to polymer-immobilized systems. The formalism requires an understanding of the extent of substrate partitioning, the appropriate intrinsic kinetic expression and a value for the substrate s diffusion coefficient. A simple method for estimating diffusion coefficients was discussed as were general criteria for establishing the presence of substrate transport limitations. Application of these principles should permit one to identify experimental conditions which will result in the intrinsic reaction rate data needed to probe the catalytic properties of immobilized catalysts. 

Here, we consider the general case of a porous catalyst, where the internal diffusion effect is included in the effectiveness factor (//,). 

As mentioned earlier, if the rate of a catalytic reaction is proportional to the surface area, then a catalyst with the highest possible area is most desirable and that is generally achieved by its porous structure. However, the reactants have to diffuse into the pores within the catalyst particle, and as a result a concentration gradient appears between the pore mouth and the interior of the catalyst. Consequently, the concentration at the exterior surface of the catalyst particle does not apply to die whole surface area and the pore diffusion limits the overall rate of reaction. The effectiveness factor tjs is used to account for diffusion and reaction in porous catalysts and is defined as... 

For liquid-phase catalytic or enzymatic reactions, catalysts or enzymes are used as homogeneous solutes in the hquid, or as sohd particles suspended in the hquid phase. In the latter case, (i) the particles per se may be catalysts (ii) the catalysts or enzymes are uniformly distributed within inert particles or (hi) the catalysts or enzymes exist at the surface of pores, inside the particles. In such heterogeneous catalytic or enzymatic systems, a variety of factors that include the mass transfer of reactants and products, heat effects accompanying the reactions, and/or some surface phenomena, may affect the apparent reaction rates. For example, in situation (iii) above, the reactants must move to the catalytic reaction sites within catalyst particles by various mechanisms of diffusion through the pores. In general, the apparent rates of reactions with catalyst or enzymatic particles are lower than the intrinsic reaction rates this is due to the various mass transfer resistances, as is discussed below. 

In assessing whether a reactor is influenced by intraparticle mass transfer effects WeiSZ and Prater 24 developed a criterion for isothermal reactions based upon the observation that the effectiveness factor approaches unity when the generalised Thiele modulus is of the order of unity. It has been showneffectiveness factor for all catalyst geometries and reaction orders (except zero order) tends to unity when the generalised Thiele modulus falls below a value of one. Since tj is about unity when 0 < ll for zero-order reactions, a quite general criterion for diffusion control of simple isothermal reactions not affected by product inhibition is < 1. Since the Thiele modulus (see equation 3.19) contains the specific rate constant for chemical reaction, which is often unknown, a more useful criterion is obtained by substituting l v/CAm (for a first-order reaction) for k to give ... 

Note that the results of our simulation via the pseudohomogeneous model tracks the actual plant very closely. However, since the effectiveness factors r]i were included in a lumped empirical fashion in the kinetic parameters, this model is not suitable for other reactors. A heterogeneous model, using intrinsic kinetics and a rigorous description of the diffusion and conduction, as well as the reactions in the catalyst pellet will be more reliable in general and can be used to extract intrinsic kinetic parameters from the industrial data. 

As a consequence of diffusion there is a reduction in the reaction rate as we progress inside the catalyst with a result that the overall rate is much less than would be achieved if the reactant were at a concentration as supplied at the outer surface. Thus the catalyst regions are not effectively used and the concept of effectiveness is introduced. Effectiveness is defined as the average reaction rate, i.e., with diffusion, divided by the reaction rate if the rate of reaction is evaluated at the boundary condition value at X = 1. The effectiveness factor can be generally given by... 

When the effective reaction rate is controlled by pore diffusion, then the asymptotic solution of the catalyst effectiveness factor as a function of the generalized Thiele modulus can be utilized (cq 108). This (approximate) relationship has been derived in Section 6.2.3.1. It is valid for arbitrary order of reaction and arbitrary pellet shape. 

Here it is assumed that it is possible to use the concept of an effective diffusion coefficient without making too large an error. Hence the effect of micro properties will not be studied here and it is assumed the value of De is known. The discussion is restricted to the impact of the macro properties and reaction properties on the effectiveness factor. Furthermore only simple reactions are discussed. Generalized formulae are provided that enable calculation of effectiveness factor for varying properties of the catalyst or the reacting system. 

The problem of the optimal particle shape and size is crucial for packed bed reactor design. Generally, the larger the particle diameter, the cheaper the catalyst. This is not usually a significant factor in process design - more important are the internal and external diffusion effects, the pressure drop, the heat transfer to the reactor walls and a uniform fluid flow. 

The number of metal zeolites and their application to the epoxidation of olefins rose in parallel from the late 1980s. TS-2, Ti,Al-P, Ti-P, Ti-MWW and, rarely, Ti-MOR are catalysts that have been studied in some detail [7-9, 35, 77-84]. TS-2 behaves, according to the few studies published, similarly to TS-1. The greater spaciousness of pores in Ti-Beta zeolites and of external cups in Ti-MWW allows the epoxidation, under mild conditions, of olefins unable to diffuse in TS-1 and TS-2, such as methylcyclohexenes, cyclododecene, norbornene, camphene and methyl oleate [80-83]. Steric constraints still prevail over electronic factors, however, as in medium pore Ti-zeolites, even in the epoxidation of linear olefins (Table 18.9). It is generally believed that active sites and epoxidation mechanisms are not significantly different from those of TS-1. 

In contrast with the PPR, bed utilization in a fixed-bed reactor is essentially complete. Mass transfer outside the bed is generally not a limiting factor, for the main resistance is in most cases the intraparticle diffusion, which gives rise to incomplete utilization of the catalyst particle (see Section 1II.B.2). 

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