Effectiveness factor general reaction

The patterns shown in the figure are typical of an adiabatic reactor for exothermic reactions. The effectiveness factor generally decreases with reactor length, but more rapidly than shown in the figure. Because of the reaction equilibrium, the decrease is much more moderate in this case. The conversion and temperature usually increase rapidly near the inlet, but these increases are moderate toward the outlet, reaching plateaus as the reactant is depleted. 

The internal effectiveness factor is a function of the generalized Thiele modulus (see for instance Krishna and Sie (1994), Trambouze et al. (1988), and Fogler (1986). For a first-order reaction ... 

Figure 3.32. Interna effectiveness factor as a function of the generalized Thiele modulus for a first order reaction.

Microlevel. The starting point in multiphase reactor selection is the determination of the best particle size (catalyst particles, bubbles, and droplets). The size of catalyst particles should be such that utilization of the catalyst is as high as possible. A measure of catalyst utilization is the effectiveness factor q (see Sections 3.4.1 and 5.4.3) that is inversely related to the Thiele modulus (Eqn. 5.4-78). Generally, the effectiveness factor for Thiele moduli less than 0.5 are sufficiently high, exceeding 0.9. For the reaction under consideration, the particles size should be so small that these limits are met. 

One of the important differences between calciothermic and aluminothermic reduction of oxides concerns the interaction between the reduced metal and the reductant. Calcium does not form stable solid solutions or alloys with the reduced metals calcium contamination in the metal is, therefore, relatively small. Aluminum, on the other hand, readily forms solid solutions with the reduced metals, and the product generally contains appreciable quantities of residual aluminum. This is not a serious problem because in many cases either a certain aluminum content is desired in the reduced metal or the residual aluminum can be effectively removed in post-reduction purification operations. The extent of the contamination of a reduced metal with the reductant can be related to factors such as the reaction temperature, the standard free energy change associated with the reaction, and the slag composition. Let the following generalized reaction be considered ... 

All of these steps are rate processes and are temperature dependent. It is important to realize that very large temperature gradients may exist between active sites and the bulk gas phase. Usually, one step is slower than the others, and it is this rate-controlling step. The effectiveness factor is the ratio of the observed rate to that which would be obtained if the whole of the internal surface of the pellet were available to the reagents at the same concentrations as they have at the external surface. Generally, the higher the effectiveness factor, the higher the rate of reaction. 

This example illustrates calculation of the rate of a surface reaction from an intrinsic-rate law of the LH type in conjunction with determination of the effectiveness factor (rj) from the generalized Thiele modulus (G) and Figure 8.11 as an approximate representation of the 7]-Q relation. We first determine G, then 17, and finally (—rA)obs. 

In the case that the chemical reaction proceeds much faster than the diffusion of educts to the surface and into the pore system a starvation with regard to the mass transport of the educt is the result, diffusion through the surface layer and the pore system then become the rate limiting steps for the catalytic conversion. They generally lead to a different result in the activity compared to the catalytic materials measured under non-diffusion-limited conditions. Before solutions for overcoming this phenomenon are presented, two more additional terms shall be introduced the Thiele modulus and the effectiveness factor. 

The feasibility (i.e. direction and extent of the chemical change) and mechanism (i.e. study of rate and factors effecting it) are two important aspects, which should be distinguished clearly while considering a chemical reaction. There is no simple connection between the two aspects. For a general reaction... 

Vayenas, C. G. and S. Pavlou. 1987a. Optimal catalyst distribution and generalized effectiveness factors in pellets single reactions with arbitrary kinetics. Chem. Eng. Sci. 42(11) 2633-2645. 

The influence of all these factors, both individual and together, has been studied only in general, without any quantitative estimation of their effect on the reaction course. This may be explained by the complicated character of the polycondensation process and the strong mutual effect of the above factors. Nevertheless, the hydro-... 

Here, issues in relation to the trickle flow regime—isothermal operation and plug flow for the gas phase—will be dealt with. Also, it is assumed that the flowing liquid completely covers the outer surface particles (/w = 1 or aLS = au) so that the reaction can take place solely by the mass transfer of the reactant through the liquid-particle interface. Generally, the assumption of isothermal conditions and complete liquid coverage in trickle-bed processes is fully justified with the exception of very low liquid rates. Capillary forces normally draw the liquid into the pores of the particles. Therefore, the use of liquid-phase diffusivities is adequate in the evaluation of intraparticle mass transfer effects (effectiveness factors) (Smith, 1981). 

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... 

It is possible to combine the resistances of internal and external mass transfer through an overall effectiveness factor, for isothermal particles and first-order reaction. Two approaches can be applied. The general idea is that the catalyst can be divided into two parts its exterior surface and its interior surface. Therefore, the global reaction rates used here are per unit surface area of catalyst. 

First-order reactions without internal mass transfer limitations A number of reactions carried out at high temperatures are potentially mass-transfer limited. The surface reaction is so fast that the global rate is limited by the transfer of the reactants from the bulk to the exterior surface of the catalyst. Moreover, the reactants do not have the chance to travel within catalyst particles due to the use of nonporous catalysts or veiy fast reaction on the exterior surface of catalyst pellets. Consider a first-order reaction A - B or a general reaction of the form a A - bB - products, which is of first order with respect to A. For the following analysis, a zero expansion factor and an effectiveness factor equal to 1 are considered. 

Calculated reaction rates can be in the spatially ID model corrected using the generalized effectiveness factor (rf) approach for non-linear rate laws. The effect of internal diffusion limitations on the apparent reaction rate Reff is then lumped into the parameter evaluated in dependence on Dc>r, 8 and Rj (cf. Aris, 1975 Froment and Bischoff, 1979, 1990 Leclerc and Schweich, 1993). 

Values of effectiveness factors in washcoat layers with non-uniform thickness around the channel perimeter have been studied by Hayes et al. (2005). However, the applicability of (even the generalized) effectiveness factor approach is quite limited in complex systems with competing reactions, surface deposition of reaction components, non-linear rate laws and under transient operating conditions (e.g. periodically operated NSRC). Typically, the effectiveness factor method can be used for more accurate prediction of CO, H2 and HC oxidation light-off and conversions in DOC. 

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. 

The functional dependence of other parameters on the reaction rate also becomes modified when diffusion determines the overall rate. If we write the rate of reaction for an nth order reaction in terms of equation 3.8 and substitute the general expression obtained for the effectiveness factor at high values of, where rj is proportional to 1/ and is defined by equation 3.20, we obtain ... 

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 ... 

The design of a gas-liquid-solid reactor is very much dependent upon the size of the solid particles chosen for the reaction and the anticipated value of the effectiveness factor is one of the most important considerations. Generally, the smaller the particle size the closer the effectiveness factor will be to unity. Particles smaller than about 1 mm in diameter cannot, however, be used in the form of a fixed bed. There would be problems in supporting a bed of smaller particles the pressure drop would be too great and perhaps, above all, the possibility of the interstices between the particles becoming blocked too troublesome. There may, however, be other good reasons for choosing a fixed-bed type of reactor. 

In the case of gel entrapped biocatalysts, or where the biocatalyst has been immobilised in the pores of the carrier, then the reaction is unlikely to occur solely at the surface. Similarly, the consumption of substrate by a microbial film or floe would be expected to occur at some depth into the microbial mass. The situation is more complex than in the case of surface immobilisation since, in this case, transport and reaction occur in parallel. By analogy with the case of heterogeneous catalysis, which is discussed in Chapter 3, the flux of substrate is related to the rate of reaction by the use of an effectiveness factor rj. The rate of reaction is itself expressed in terms of the surface substrate concentration which in many instances will be very close to the bulk substrate concentration. In general, the flux of substrate will be given by ... 

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. 

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