Aris’ numbers

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 zeroth Aris number Ana is defined as the number which becomes equivalent to 1/ if if the effectiveness factor 17 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. 

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... 

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...

At this point the introduction of Aris numbers rather than Thiele moduli is stressed. One could say that a Thiele modulus corresponds to the square root of an Aris number. Hence a negative Aris number will correspond to an imaginary Thiele modulus. Because negative values of An, often occur (An0 can only be positive ) and because we want to avoid working with imaginary numbers, we abstain on the use of Thiele moduli. 

An, is always defined for any differentiable function R(CA). The low ij Aris number An0, however, is defined only when the function R(CA) fulfils the condition... 

Several formulae have been given for the calculation of the effectiveness factor as a function of one of the Aris numbers An or An, or as a function of a Thiele modulus. These formulae can become very complex and, for most kinetic expressions and catalyst geometries, it is impossible to derive analytical solutions for the effectiveness factor, so... 

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

Hie Aris numbers Anx and An0 can be applied to nonisothermal catalyst pellets. Three items will be discussed ... 

Figure 7.1 Schematic plot of the effectiveness factor versus the zeroth Aris number Ana for an exothermic nth-order reaction (n > 0 and > n). Notice the occurrence of a maximum for the effectiveness factor.

As elucidated in Appendices A and B the Aris numbers Art0 and Aw, can be calculated from Equation 6.29 and 6.38, provided the temperature dependency of the conversion rate is included in the formulae. This yields ... 

These formulae give the Aris numbers for arbitrary reaction kinetics and intraparticle temperature gradients. Thus the dependency of the conversion rate on the temperature can also be of any arbitrary form. 

The above suggests that the discussion of the Aris numbers for simple reactions also holds for nonisothermal pellets. For example, effectiveness factors larger than one are found if the number An, becomes negative. According to Equation 7.14 this is the case if... 

Figure 7.2 Effectiveness factor ij versus the zeroth Aris number An, for an exothermic zeroth-order reaction in an infinite slab. Lines for several values of (t a 0) are drawn.

Figure 7.4 Effectiveness factor r versus zeroth Aris number Anc for a bimolecular reaction with (1,1) kinetics occurring in an infinite slab, and for several values off).

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. 

Substitution of Equation 7.41 into 7.35 yields the following formula for the high rj Aris number ... 

Substitution of the Aris numbers Ant and Ana into Equations 7.47 and 7.48 then yields... 

Also worthy of note is that if the conversion rate depends on the concentrations of more than two components, the Aris numbers can be calculated using the same approach as outlined above. For example, it for the reaction... 

If component Q is also important for the conversion rate, the concentration Ca can be introduced in the formulae for. An, and An0 in a similar way as CP. Hence, it is concluded that calculating the effectiveness factor for multimolecular reactions is basically not very different than for simple reactions (if the proper Aris numbers are used). 

Using the dusty gas model [5] analytical solutions are derived to describe the internal pressure gradients and the dependence of the effective diffusion coefficient on the gas composition. Use of the binary flow model (BFM, Chapter 3) would also have yielded almost similar results to those discussed below. After discussion of the dusty gas model, results are then implemented in the Aris numbers. Finally, negligibility criteria are derived, this time for intraparticle pressure gradients. Calculations are given in appendices here we focus on the results. 

If the concentration CA influences the effective diffusion coefficient DtA, the Aris numbers can be calculated from... 

In Figure 7.7 the effectiveness factor ij is plotted versus the low y Aris number An0 for several values of y. We see that the effect of y is rather small. This is a consequence of the use of An0, which already corrects for the influence of y. 

---