Thiele modulus general

Generalized Thiele modulus general notation for Thiele modulus. 

Thiele modulus (—) generalized Thiele modulus (—) diameter (m)... 

The results indicate that the use of Eq. (3.3.33), which is exact for first-order kinetics, for other reaction orders may introduce a serious error. This is in contrast to the result for the heterogeneous catalysis in a porous pellet or on a solid surface, in which the difference between the effectiveness factors for different reaction orders is much smaller, when plotted against the Thiele modulus generalized for different reaction order [25, 26]. 

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. 

The analyses of simultaneous reaction and mass transfer in this geometry are similar mathematically to those of the straight cylindrical pore model considered previously, because both are essentally one-dimensional models. In the general case, the Thiele modulus for semiinfinite, flat-plate problems becomes... 

In terms of the generalized Thiele modulus of equation 12.3.77, the last equation becomes... 

The asymptotic solution ( - large) for tj is [2/(n + l)]1/2/, of which the result given by 8.5-14c is a special case for a first-order reaction. The general result can thus be used to normalize the Thiele modulus for order so that the results for strong pore-diffusion resistance all fall on the same limiting straight line of slope - 1 in Figure 8.11. The normalized Thiele modulus for this purpose is... 

The conclusions about asymptotic values of tj summarized in Tables 8.2 and 8.3, and the behavior of tj in relation to Figure 8.11, require a generalization of the definition of the Thiele modulus. The result for " in equation 8.520 is generalized with respect to particle geometry through Le, but is restricted to power-law kinetics. However, since... 

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. 

Generalizati on. It has been observed that all plots of effectiveness against Thiele modulus are similar and that a single plot can represent the nine main cases fairly adequately by defining a generalized modulus as... 

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 modulus defined by eqn. (10) then has the advantage that the asymptotes to t (0) are approximately coincident for a variety of particle shapes and reaction orders, with the specific exception of a zero-order reaction (n = 0), for which t = 1 when 0 < 1 and 77 = 1/0 when 0 > 1. The curve of 77 as a function of 0 is thus quite general for practical catalyst pellets. Figure 2 illustrates the form of For 0 > 3, it is found that 77 = 1/0 to an accuracy within 0.5%, while the approximation is within 3.5% for 0 > 2. The errors involved in using the generalised curve to estimate 77 are probably no greater than the errors perpetrated by estimating values of parameters in the Thiele modulus. 

Arbitrary Reaction Kinetics. If the Thiele modulus is generalized as follows [see Froment and Bischoff (1962)]... 

While we have contact with this problem, however, we will look at the asymptotics of reactions in a slab of catalyst. In Chapter 3 we used the special form of the equations for diffusion and reaction to get rather general expressions for the Thiele modulus and effectiveness in terms of the center concentration v... 

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 ... 

For different geometries and for kinetics of arbitrary order a general expression for the calculation of the Thiele modulus and the relationship between tj and has been developed (Aris, 1957) [Eq. (5.64)]. 

To assess the generality of the decomposition solution method we have applied it to several arbitrary reaction orders, for example, n = 1.73, 0.67, -0.5, -1.0 etc.10,11 These reaction orders have been chosen to represent typical cases of kinetics in heterogeneous catalysis and electrocatalysis where adsorption phenomena play a major role. Values of effectiveness of a plane catalyst pellet for the different reaction orders are shown in Figure 5. Clearly all of the data for positive reaction orders show the expected trend of a decrease in effectiveness with increase in Thiele modulus. Effectiveness values determined for reaction... 

Unlike the Thiele modulus related to a reaction order n for porous catalyst pallets, co and co will maintain keep their form in this nonlinear case, due to the dimensionless concentrations used in the general B-V equations (Eqs. 41 and 84). The approximate general solutions of Eq. (18) can be used for Eqs. (86) and (87) as long as the reaction order n and Thiele modulus are replaced by q and co or co respectively. The expressions for the concentration A and the effectiveness factor for a r/th-ordcr reaction by using m term approximating can be obtained from Eqs. (25) and (26). 

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. 

Figure 6. Effectiveness factor rj as a function of the generalized Thiele modulus f for different pellet geometries. Influence of intraparticle diffusion on the effective reaction rate (isothermal, first order, irreversible reaction).

Figure 6 shows the effectiveness factor for any of the three different pellet shapes as a function of the generalized Thiele modulus p. It is obvious that for larger Thiele moduli (i.e. p > 3) all curves can be described with acceptable accuracy by a common asymptote t] — 1 / p. The largest deviation between the solutions for the individual shapes occurs around p x 1. However, even for the extremely different geometries of the flat plate and the sphere, the deviation of the efficiency... 

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