Modified Thiele modulus

This equation is approximately valid for other orders and other geometries by substituting the modified Thiele modulus,... 

Graphs are shown for spheres and slabs separately, and with the modified Thiele modulus for all geometries and orders. 

Table I. Modified Thiele Modulus for Benzene Ethylation at 573°K...

A great deal of attention has been devoted to this topic because of the interesting and often solvable mathematical problems that it presents. Results of such calculations for isothermal zero-, first-, and second-order reactions in uniform cylindrical pores are summarized in Figure 17.6. The abscissa is a modified Thiele modulus whose basic definition is... 

The residence time is t = 5, the inlet is C0 = 1.2 and the outlet is C The modified Thiele modulus for this order of reaction is... 

The effectiveness factor tj may be given in terms of the modified Thiele modulus 0 which is defined by Atkinson 59 and Atkinson and Mavituna 2 as ... 

This type of reaction has been investigated by Smith and Amundsen [95] and Carberry [14]. Without derivation it may be stated that under isothermal conditions the same solution for the effectiveness factor is obtained as in the case of an irreversible reaction if a modified Thiele modulus rev is introduced [91] ... 

A general expression for the modified Thiele modulus for an infinite slab was derived by Bischoff [10] ... 

This is not seen easily in considering the defining Equations 6.22 and 6.23, so a mathematical proof of Equation 6.29 is given in Appendix A. Hence the modified Thiele modulus 0m now becomes... 

From Equation 6.30 we find that the proper modified Thiele modulus for arbitrary catalyst geometries, for this case, is given by... 

From Equation 6.30 the corresponding modified Thiele modulus is obtained as ... 

In summary, it can be said that the square root of the low rj Aris number An0 is a generalized modified Thiele modulus. If we compare this number with the modified Thiele moduli given in literature we find that serious deviations generally occur for kinetic expressions which are not first order. This is because most of the modified Thiele moduli given in literature are incorrect. 

Figure 4. Effectiveness factor as a flmction of the modified Thiele modulus. T=503 K.

In his calculations, Ruckenstein introduced a modified and proton-related Thiele modulus. He replaced the substrate-related by the pH-related [Eq. (7)] and used the effective diffusion coefficient of protons instead of substrates. It turned out that even at comparably low values of this modified Thiele modulus the reaction rates decrease and, instead of a bell-shaped profile, a curve which becomes flatter towards alkaline pH-values is obtained (Fig. 7, curve b). 

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