Generalized Internal Effectiveness Factor

The generalized internal effectiveness factor in the asymptotic region of strong diffusion effects is obtained from ... 

For all practical purposes, then, the generalized internal effectiveness factor is ... 

Much more rigorous tests of the importance of diffusional effects can be made if the intrinsic kinetics are known on the basis of the generalized internal effectiveness factor and reactor point effectiveness. Using the rate expression of Eq. 4.2 in Eq. 4.73 for 4>c 8 =... 

This section develops generalized internal effectiveness factors (Aris 1965 Bischoff 1965 Petersen 1965), generalized in the sense that they are applicable to arbitrary kinetics. For a slab-like, isothermal pellet, Eq. 4.21 reduces to ... 

For diffusion-affected reactions, the generalized internal effectiveness factor for fresh catalyst, tjq, can be used to arrive at the following overall pellet effectiveness for uniform chemical deactivation ... 

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 ... 

The general approach in the evaluation of nonisothermal internal effectiveness factors is analogous to that outlined for determining isothermal rj. However, since substantial temperature... 

So far, only a single reaction has been considered. While the reactor point effectiveness cannot be expressed explicitly for a reversible reaction, the internal effectiveness factor can readily be obtained analytically using the generalized modulus (see Problem 4.23). For complex multiple reactions, however, it is not possible to obtain analytical expressions for the global rates and one has to solve the conservation equations numerically. The numerical solution of nonlinear, coupled diffusion equations with split boundary conditions is by no means trivial and often presents convergence difficulties. In this section, the same approach is taken as was used for the reactor point effectiveness. This enables the global rates to be obtained in a straightforward manner and the diffusion equations to be solved as an initial value problem (Akella 1983). 

Obtain the internal effectiveness factor for species A. Use the generalized modulus approach. 

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. 

For regular solutions, the influence of the solvent is determined by molar volumes and internal pressure terms. Since the molar volumes do not vary greatly, the internal pressure factor is more important. If the internal pressures of solvent, reactants and activated complex are similar, the solvent will have little effect on the rate of reaction as compared to a solvent in which reaction behaves ideally. If the internal pressure of the solvent is close to that of reactants but appreciably different from that of the activated complex, the rate of reaction in this solvent will be low. On the other hand, if solvent has an internal pressure similar to that of activated complex, but different from one or both the reactants, rate of reaction in this solvent will be high. Since the activated complex has properties which approach the properties of the product, it may be concluded, in general, that the reaction in which the products are of higher internal pressure than the reactants, it is accelerated by solvent of high internal pressure. 

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

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). 

In the general case of immobilized enzymes not only the internal diffusion addressed above, but also diffusion through the film should be taken into account. Similarly to heterogeneous catalysis the catalyst effectiveness factor for slab geometry and low substrate concentrations (first order kinetics) is decribed by eq. (9.173) in a somewhat different form... 

The concept of effectiveness developed separately for external or internal transport resistances can be extended to an overall effectiveness factor for treating the general diffusion-reaction problem where both external and internal concentration and temperature gradients exist The overall effectiveness factor, D, is defined for relating the actual global rate to the intrinsic rate, that is, -Ra)p to (-Ra)6- To stun up the definitions for y, 7], and D,... 

The general problem of diffusion-reaction for the overall effectiveness factor D is rather complicated. However, the physical and chemical rate processes prevailing under practical conditions promote isothermal particles and negligible external mass transfer limitations. In other words, the key transport limitations are external heat transfer and internal mass transfer. External temperature gradients can be significant even when external mass transfer resistances are negligibly small. 

To suppress the internal mass transfer resistance in the pores of the solid material, small enough particles should be used. The role of internal mass transfer resistance can be evaluated using the concepts described in Chapter 5, that is, by evaluating the generalized Thiele modulus and the effectiveness factor. 

The effectiveness factor is defined as the ratio of the reaction rate in the presence of internal or pore diffusion to the reaction rate in the absence of pore diffusion. The value of the effectiveness factor is a measure of the extent of diffusional limitation. For isothermal reactions (generally true of most biochemical reactions), diffusional limitations are negligible when the effectiveness factor (t/) is close to unity. If Tj < 1, the reaction is diffusion limited. 

All of these models require the availability of the effective difiEiisivity of the solute within ELM globules. Generally, the effective diffusivity is estimated from the Jefferson-Witzell-Sibbett equation (2J,8JS). In this equation, the ELM globule is considered to be an assembly of spheres of the internal phase embedded within cubes of the membrane phase. Recently, Goswami et al. (76) pointed out that in order to fit cubic elements of the membrane phase, each with an embedded, internal phase droplet, within the spherical emulsion globule, the cubic elements must be distorted. They introduced a shape factor to account for the distortion. 

Observed rates in a number of trickle-bed reactors employed in hydrodesulfurization and hydrotreating of heavy residuals indicate that they operate in the regime free of major gas-liquid mass transfer limitations (jLfact that often the liquid reactants are nonvolatile or dilute at the operating conditions used the reaction is frequently liquid reactant limited and confined to the catalyst effectively wetted by liquid. Since porous packing, typically 1/32" to 1/8" (0.08 cm to 0.318 cm) extru-dates is most often employed it is clear that reaction rates may be affected both by Internal pore fill-up with liquid and by internal diffuslonal limitations. Catalyst effectiveness factors from 0.5 to 0.85 have been generally reported ... 

The effectiveness factor t) can be used to estimate the actual rate of reaction, including the effect of internal concentration gradients. For most nth-order reactions, t) can be estimated by calculating the generalized Thiele modulus [Pg.369]

In the case of internal diffusion, analytical solutions can be used direcdy only for simple kinetics, such as first or zero orders thus, approximations should be applied, since many reactions are not of zero or first order. Alternatively, utilization of the generalized diagrams of Aris [7] is possible (Fig. 10.19). Such diagrams relate for various kinetics effectiveness factors and the generafized Thiele modulus expressed by Eq. (10.121). 

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