Thiele generalized

The solution was first obtained independently by Wertheim [32] and Thiele [33] using Laplace transfonns. Subsequently, Baxter [34] obtained the same solutions by a Wiener-Hopf factorization teclmique. This method has been generalized to charged hard spheres. 

The first quantum mechanical improvement to MNDO was made by Thiel and Voityuk [19] when they introduced the formalism for adding d-orbitals to the basis set in MNDO/d. This formalism has since been used to add d-orbitals to PM3 to give PM3-tm and to PM3 and AMI to give PM3(d) and AMl(d), respectively (aU three are available commercially but have not been published at the time of writing). Voityuk and Rosch have published parameters for molybdenum for AMl(d) [20] and AMI has been extended to use d-orbitals for Si, P, S and Q. in AMI [21]. Although PM3, for instance, was parameterized with special emphasis on hypervalent compounds but with only an s,p-basis set, methods such as MNDO/d or AMI, that use d-orbitals for the elements Si-Cl are generally more reliable. 

McCabe-Thiele diagrams for nonlinear and more practical systems with pertinent inequaUty constraints are illustrated in Figures 11 and 12. The convex isotherms are generally observed for 2eohtic adsorbents, particularly in hydrocarbon separation systems, whereas the concave isotherms are observed for ion-exchange resins used in sugar separations. 

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. 

When Herman Mark first evaluated the crystal structure of rubber (with E. A. Hauser) and cellulose (with J. R. Katz) in 1924 and 1925, it was generally accepted that these materials were low molecular weight or monomeric. The unusual properties of these substances, now known to be related to high molecular weight, were then attributed to aggolomeration or "association" of the low molecular weight precursors. A common explanation for the associations were secondary forces such as Johannes Thiele s partial valences. 

The cascade can be drawn graphically in the McCabe-Thiele form, but the equilibria are so distorted that the exercise is not very valuable. This is a general finding in industrial practice. Graphical methods are adequate to give an indication of the number of stages likely to be needed for a particular duty. They fail when separations between similar species are sought. 

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)]... 

A. Koslowski, M. E. Beck, and W. Thiel. Implementation of a general multireference configuration interaction procedure with analytic gradients in a semiempirical context using the graphical unitary group approach, J. Comput. Chem., 24 714-726 (2003). 

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