Thiele modulus general form

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. 

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... 

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. 

Here, the generalized form of the Thiele modulus is used, in order to be independent of the reaction kinetics, which is defined such that for the limit [Pg.393]

In the past, a number of attempts have been made to generalize the definition of the Thiele modulus. Aris [6] noticed that all the Thiele moduli for first-order reactions were of the form ... 

An early normalization of the Thiele modulus for an isothermal pellet and arbitrary kinetics was given by R. B. Bird, W. E. Stewart, and E. N. Lightfoot on pages 335-41 of their Notes on Transport Phenomena, the precursor to their well-known Transport Phenomena (New York John Wiley Sons, Inc., 1960). Slightly more general forms—all of them equivalent—have been given independently and almost simultaneously in ... 

The discussion above leaves the impression that there in a separate Thiele modulus and corresponding effectiveness factor for every specific form of reaction kinetics. Unfortunately, this is indeed so, and for most complex kinetic laws the mathematics is not tractable to direct analytical solution. Is it reasonable to see if we can pose the problem in more general form to get around this If so, how can this be done ... 

Lee and Reilly (1981) defined a more rigorous form of the Thiele modulus based on the generalized modulus of Bischolf and Aris (see Chapter 7) which is particularly useful in analyzing the role of diffusion in deactivation. Their analysis shows that, in reactant-independent deactivation, the presence of a strong dilfusional limitation lowers the rate of deactivation to half the diffusion-free value. Thus, surprisingly, diffusion seems to have a favorable effect on the performance of a deactivating immobilized enzyme catalyst. 

The Weisz-Prater criterion makes use of observable quantities like -Ra)p, the measured global rate (kmol/kg-s) dp, the particle diameter (m) pp, the particle density (kg/m ) Dg, the effective mass diffusivity (m /s) and the surface concentration of reactant (kmol/m ). The intrinsic reaction rate constant ky need not be known in order to use the Weisz-Prater criterion. If external mass transfer effects are eliminated, CAb can be used, and the effective diffusivity can be estimated using catalyst and fluid physical properties. The criterion can be extended to other reaction orders and multiple reactions by using the generalized Thiele modulus, and various functional forms are quoted in the literature [17, 26, 28]. 

Generalized Thiele modulus for a slab-formed catalyst particle... 

---