Effectiveness factors complex kinetics

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... 

For a more detailed analysis of measured transport restrictions and reaction kinetics, a more complex reactor simulation tool developed at Haldor Topsoe was used. The model used for sulphuric acid catalyst assumes plug flow and integrates differential mass and heat balances through the reactor length [16], The bulk effectiveness factor for the catalyst pellets is determined by solution of differential equations for catalytic reaction coupled with mass and heat transport through the porous catalyst pellet and with a film model for external transport restrictions. The model was used both for optimization of particle size and development of intrinsic rate expressions. Even more complex models including radial profiles or dynamic terms may also be used when appropriate. 

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. 

The concept of an effectiveness factor is useful in estimating the reaction rate per catalyst pellet (volume or mass). It is, however, mainly useful for simple reactions and simple kinetics. When there are complex reaction pathways, the concept of effectiveness factor is no longer easily applicable, and species and energy balance equations inside the particle may have to be solved to obtain the reaction rates per unit volume of... 

The above analysis and Fig. 19-25 provide a theoretical foundation similar to the Thiele-modulus effectiveness factor relationship for fluid-solid systems. However, there are no generalized closed-form expressions of E for the more general case ofa complex reaction network, and its value has to be determined by solving the complete diffusion-reaction equations for known intrinsic mechanism and kinetics, or alternatively estimated experimentally. 

Before discussing the details of the product energy distributions for specific reactions, we shall try to identify the important factors that determine the disposal of energy in chemical reactions. Broadly, energy disposal is determined by the nature of the interactions (the reaction potential-energy surface), by dynamical or kinetic effects and by the initial states of the reagents. For a particular reaction, one effect may dominate or the system may be governed by a complex interaction of all three effects. 

In reality however, situations also exist where a more complex form of the rate expression has to be applied. Among the numerous possible types of kinetic expressions two important cases will be discussed here in more detail, namely rate laws for reversible reactions and rate laws of the Langmuir-Hinshelwood type. Basically, the purpose of this is to point out additional effects concerning the dependence of the effectiveness factor upon the operating conditions which result from a more complex form of the rate expression. Moreover, without going too much into the details, it is intended at least to demonstrate to what extent the mathematical effort required for an analytical solution of the governing mass and enthalpy conservation equations is increased, and how much a clear presentation of the results is hindered whenever complex kinetic expressions are necessary. 

Several formulae have been given for the calculation of the effectiveness factor as a function of one of the Aris numbers An or An, or as a function of a Thiele modulus. These formulae can become very complex and, for most kinetic expressions and catalyst geometries, it is impossible to derive analytical solutions for the effectiveness factor, so... 

Mathematical models [216] for calculating these effectiveness factors involve simultaneous differential equations, which on account of the complex kinetics of ammonia synthesis cannot be solved analytically. Exact numerical integration procedures, as adopted by various research groups [157], [217]-[219], are rather troublesome and time consuming even for a fast computer. A simplification [220] can be used which can be integrated analytically when the ammonia kinetics are approximated by a pseudo-first-order reaction [214], [215], [221], according to the Equation (21) ... 

Steric factors in the ion-exchanger phase present a complexation depressing effect by restricting rotational and translational motion. This reduces the number of collisions leading to association. The rate of the reverse reaction, hydrolytic dissociation of associated species is, however, not necessarily decreased because of lowered or less ordered hydration of the species in the ion-exchanger phase. In any case, the kinetic aspects seem to exercise a net negative effect on the complexation tendencies. The increase in ion interaction due to the lower medium dielectric constant more than compensates for the above. As a consequence, the lowering in effective collision frequency may become important only when very weak complexes are involved. 

The charge of the critical complex AB naturally is equal to the algebraic sum of the charges of A and B. If, for example, A is a univalent ion and B is a neutral molecule, the critical complex will have a valence of 1. The kinetic activity factor therefore will be independent of the electrolyte content between rather wide limits since /a and fx will vary approximately in the same manner. At larger ionic strengths a more distinct salt effect will be observed since individual differences between A and AB will enter and because /b will no longer be equal to unity. 

Although for simple cases of isothermal non-porous pellets with linear kinetics the concept of the effectiveness factor is not practically important it is quite useful to present it since the dehnition and the principles involved are the same as for the more complex cases which will be discussed later. 

In most investigations concerning the reactor modelling, simple pseudohomogeneous (t = 1) reactor models were used. The effect of external and internal mass and heat transfer resistances on the effectiveness factors using realistic complex reaction network has not been widely investigated. The simple linear kinetics proposed by... 

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