First Aris Number An

FI gure 6 Effectiveness factor ij versus the first Aris number An, for first-order kinetics in an infinitely long slab, an infinitely long cylinder and a sphere. 

As such, the high 17 Aris number is very important from a practical point of view. For chemical reactors in industry, the effectiveness factor will typically range from 

70% to 90%. In general, effectiveness factors below 70% are economically unfeasible because the surplus of reactor volume which is needed (larger than (l/0.7)-l = 43%) bears too heavily upon the investment costs. However, for effectiveness factors lager than 90% it is generally economically worthwhile to increase the size of the catalyst pellets, thereby decreasing the pressure drop across the reactor and thus the compression costs, but also decreasing the effectiveness factor somewhat. In the hot spot area somewhat lower effectiveness factor are often accepted, but if they become too low a different reactor type has to be selected, that can cope with smaller catalyst particles (e.g. a fluid bed). 

It can be shown (Appendix B) that the first Aris number equals  

This holds for arbitrary reaction kinetics and arbitrary catalyst geometries. For example, for n-th order kinetics this yields 

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For any ring-shaped catalyst pellet with known values of and A the value of T can be obtained from Figure 6.10. The value of the first Aris number An can then be calculated with Equation 6.38. 

Since the first Aris number An, is defined as the number which becomes equivalent to l-i/2 for values of t] close to one, it follows that An, can be calculated from... 

Table 6.4 Formulae for both the low ij Aris number An0 and high j/ Aris number An, for arbitrary kinetics, nth-order and first-order kinetics...

Figure 7.7 Effectiveness facto j/ versus zeroth Aris number An for a simple, first-order reaction occurring in an infinite slab. The figure was drawn for nondiluted gases lines for several values of y are given.

The theory is based upon two newly defined numbers, which we call Aris numbers. This recognizes that, to our knowledge, Aris was the first who substantially contributed to a generalized theory for effectiveness factors, by postulating his shape-generalized Thiele modulus [6]. Aris also wrote a book which gives an excellent survey of all that has been done in this field [31]. 

An0 is the Aris number that brings together all the ij curves in the low 17 region. This is illustrated in Figure 6.5 where rj is plotted versus An0 for first-order kinetics in an infinitely long slab, infinitely long cylinder and sphere see Table 6.1. An0, as such, brings... 

Hie Aris numbers An0 and An, are much alike. This is illustrated in Table 6.4 where the formulae for An0 and An, are given for arbitrary kinetics, for n-th order kinetics and for first-order kinetics. In practice, reaction kinetics do not differ too much from first-order kinetics, and hence the values of An0 and An, will remain very close to each other (as also the geometry factor T is close to one). In that case both Aris numbers will be roughly equal to the square power of the shape-generalized Thiele modulus of Aris [6]. 

Figure 6.18 Effectiveness factor t] versus zeroth Aris number An0 for first-order kinetics in an infinite slab and several values of .

Quantitative analytical treatments of the effects of mass transfer and reaction within a porous structure were apparently first carried out by Thiele (20) in the United States, Dam-kohler (21) in Germany, and Zeldovitch (22) in Russia, all working independently and reporting their results between 1937 and 1939. Since these early publications, a number of different research groups have extended and further developed the analysis. Of particular note are the efforts of Wheeler (23-24), Weisz (25-28), Wicke (29-32), and Aris (33-36). In recent years, several individuals have also extended the treatment to include enzymes immobilized in porous media or within permselective membranes. The important consequence of these analyses is the development of a technique that can be used to analyze quantitatively the factors that determine the effectiveness with which the surface area of a porous catalyst is used. For this purpose we define an effectiveness factor rj for a catalyst particle as... 

If the number of components is very large, a mixture can be regarded as continuous and sharp distinctions between individual components are not made. Methods for dealing with stoichiometry, thermodynamics and kinetics for continuous mixtures are discussed by Aris and Gavalas [33]. An indication is given that rules for grouping in such mixtures depend on the nature of the reaction scheme. Wei and Kuo [34] considered ways in which species in a multicomponent reaction mixture could be lumped when the reaction network was composed of first-... 

Blanding (10) first proposed the second order cracking kinetics for FCC. Krambeck (11) theoretically demonstrated that conversion in systems with a large number of parallel reactions can be approximated by simple second order kinetics. More recently, Ho and Aris (12) have developed a further mathematical treatment of this concept. An inhibition term was incorporated into the second order cracking kinetics for gas oil conversion to account for competitive adsorption. The initial cracking rate is then given by ... 

The structure of liquids can be analyzed by the calculated radial distribution function (RDF), which defines the solvation shells. In Fig. 16.1, the calculated RDF of the liquid Aris shown, and in Table 16.1, the structure is compared with the experimental results. Four solvation shells are well defined. The spherical integration of these peaks defines the coordination number, or the number of atoms in each solvation shell. The first shell that starts at 3.20A has a maximum at 3.75A, and ends at 5.35 A, has an average of 13 Ar atoms. Therefore, in the first solvation shell, there is a reference Ar atom surrounded by other neighboring 13 Ar atoms. All the maxima of the RDF, shown in Table 16.1, are in good agreement with the experimental results obtained by Eisenstein and Gingrich [29], using X-ray diffraction in the liquid Ar in the same condition of temperature and pressure. The calculated... 

First, a single wire problem will be considered as an introduction. It brings out essential features of the stability problem and is similar in many respects to the more familiar stability problem with a CSTR. A simple and yet realistic case of negligible internal heat transfer and external mass transfer resistances will also be treated. The general problem will then be analyzed in a limited form for the case of unit Lewis number (Le) appearing in Eq. 8.27. This considerably simplifies the problem and yet the result can be extended to a certain extent to the general case of arbitrary Lewis numbers. Readers interested in more details on stability can refer to the book by Aris (1975) and the review by Luss (1977), from which the subsequent sections are derived. 

Process modeling is both an art and a science. Creativity is required to make simplifying assumptions that result in an appropriate model. The model should incorporate all of the important dynamic behavior while being no more complex than is necessary. Thus, less important phenomena are omitted in order to keep the number of model equations, variables, and parameters at reasonable levels. The failure to choose an appropriate set of simplifying assumptions invariably leads to either (1) rigorous but excessively complicated models or (2) overly simplistic models. Both extremes should be avoided. Fortunately, modeling is also a science, and predictions of process behavior from alternative models can be compared, both qualitatively and quantitatively. This chapter provides an introduction to the subject of theoretical dynamic models and shows how they can be developed from first principles such as conservation laws. Additional information is available in the books by Bequette (1998), Aris (1999), and Cameron and Hangos (2001). 

The mixed solvent models, where the first solvation sphere is accounted for by including a number of solvent molecules, implicitly include the solute-solvent cavity/ dispersion terms, although the corresponding terms between the solvent molecules and the continuum are usually neglected. Once discrete solvent molecules are included, however, the problem of configuration sampling ari.ses. Nevertheless, in many cases the first solvation shell is by far the most important, and mixed models may yield substantially better results than pure continuum models, at the price of an increase in computational cost. 

In 2005, Ary a and co-workers reported on the stereocon-trolled solid-phase synthesis of a 90-membered library of indoline-alkaloid-like polycycles 41 and 42." These compounds are structurally related to a number of alkaloids such as vindoline, tabersonine, and yohimbine, and they could lead to compounds with similar or improved biological properties. Thus, chiral aminoindoline 36 was loaded via a silyl linker unit 37, deprotected, and reacted with an acid chloride to introduce the first point of diversity and to give 38. Treatment with piperidine removed the Fmoc group, and concomitant cyclization provided tricyclic... 

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