Thiele modulus basis

Equation 21.3-16 may be used to convert rate constants from a mass to a particle volume basis for calculation of the Thiele modulus (eg., equation 8.5-20b). In this chapter, (—rA) without further designation means ( -rA)m. 

Note that if ( -rA) or a corresponding rate constant kA is specified on a mass basis, kA must be multiplied by tbe particle density to obtain tbe form of the rate constant to be used in the Thiele modulus that is, fcA = (kA)p = pp(ka)m, which follows from equation 21.3-16. 

However, in many practical situations the problem exists that effective rate constants and activation energies have been derived on the basis of laboratory experiments. The question arises then as to whether or not these parameters arc influenced by transport effects. With the relations given so far, this question cannot be answered yet, since according to its definition by cq 27 the Thiele modulus is based on the intrinsic rate constant k. This problem can be solved by introducing a new modulus, which in contrast to only contains observable (effective) quantities, and thus can... 

A prominent trade-off in fixed-bed reactor design concerns the catalyst particle size. What is the basis for the choice of a certain particle size When the catalyst performance is to be optimized, the application of the Thiele model helps to provide an answer (Figure 7). The Thiele modulus accounts for the competition between the chemical reaction and the limitation of transport of reactants by diffusion in a porous catalyst particle. It is defined as the square root of the ratio of the characteristic diffusion time fo = L /D and the characteristic reaction time (r. For a... 

Contrary to the afore mentioned Authors, Chien considering various polyethylene and polypropylene catalytic systems, on the basis of the Thiele modulus criteria (Q 5 for a > 5 and Q 1 for a 1) recently concluded that Ziegler-Natta catalysis is not controlled by diffusion phenomena while Phillips catalysis is. 

Figure 6. Determination of catalytically active cell concentration, X,ei. on the basis of the effectiveness factor/Thiele modulus relation.

On the basis of the rate constant of the chemical reaction of COD conversion (kcoo) the effectiveness factor for pore diffusion can be calculated by the Thiele modulus 4> [59] ... 

This can be easily obtained for a flat geometry on the basis of the Thiele modulus as (Missen et al, 1999) ... 

The 4>s versus if plot has the same shape as that of (pi versus ij for a flat plate but is shifted by a factor of 3 on the same log-log plot, if Equations 2.60 and 2.63 are used as basis. Aris (29,38] has shown that, for a first-order reaction in various particle geometries, the plots between the Thiele modulus q> and the isothermal internal effectiveness factor ij become identical at high and low values of

size parameter in the Thiele modulus is defined on a common basis. The characteristic size parameter L is, therefore, defined as the ratio of the particle volume to the external surface area available for reactant penetration, which enables its use for any arbitrary particle shape ... 

Let us consider a situation in which the intraparticle Peclet number X=10 and the Thiele modulus is (j)=10.For a first order irreversible reaction we get E around 0.5.This means that in order to get the desired product specification the reactor should have a length two times longer than that calculated on the basis of the apparent diffusivity D. ... 

The value of the generalized Thiele modulus (p will be calculated at the bed inlet and the bed outlet using Eqn. (9-14). The characteristic dimension will be calculated from Eqn. (9-10), and the rate constant will be converted from a weight-of-catalyst basis to a volume-of-catalyst basis using the given particle density. Equation (9-15) or Figure 9-7 can then be used to estimate the effectiveness factor at each position. 

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