Aris number first

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]. 

The first Aris number Anx is defined as the number which becomes equal to 1 -j/2 if An0 goes to zero. Hence ... 

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... 

Table 6.1 Effectiveness factor 17 as a function of the zeroth Aris number An, for first-order kinetics an infinitely long slab, an infinitely long cylinder and a sphere...

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. 

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

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. 

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]. 

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

This is illustrated in Figure 7.4 where the effectiveness factor is plotted versus the low ij Aris number An0 for a bimolecular reaction with (1,1) kinetics, and for several values of/ . P lies between 0 and 1, calculations were made with a numerical method. Again all curves coincide in the low tj region, because rj is plotted versus An0. For p = 0, the excess of component B is very large and the reaction becomes first order in component A. For p = 1, A and B match stoichiometrically and the reaction becomes pseudosecond order in component A (and B for that matter). Hence the rj-An0 graphs for simple first- and second-order reactions are the boundaries when varying p. 

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.

For simple reactions only the first point is important, and the following formulae (Equations 7.81 and 7.82) are found for the Aris numbers ... 

First, construct the slab. Changing the geometry from a hollow cylinder to a slab changes the geometry factor from T = 1.11 to T=2/3. To arrive at the same Aris numbers we must change the values of 2 en . For the slab those values follow from ... 

Derivation of a Formula for the First Aris Number Anx for Simple Reactions... 

Therefore, if the effectiveness factor for first-order reactions t]l is known as a function of the zeroth Aris number An0, the effectiveness factor in the high rj region can be estimated from ... 

This last number follows from the zeroth Aris number An0 for first-order reactions, by replacing the reaction rate constant k with (see the formulae in Table 6.4)... 

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... 

Hence, by definition, the first Aris number An1 is given by... 

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... 

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