Effectiveness factor Aris numbers

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... 

Whether or not multiple steady states will appear, and how large the deviation of the effectiveness factors between both stable operating points will be, is determined by the values of the Prater and Arrhenius numbers. Effectiveness factors above unity generally occur when p > 0 (exothermal reactions). However, for the usual range of the Arrhenius number (y = 10-30), multiple steady states are possible only at larger Prater numbers (see Fig 13). For further details on multiple steady states, the interested reader may consult the monograph by Aris [6] or the works of Luss [69, 70]. 

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 ... 

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...

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... 

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 .

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.

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. 

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). 

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. 

The effectiveness factor must be calculated, with and without neglecting the heat effects. According to Equations 7.13 and 7.14, for the Aris numbers... 

Since An < 0, approximation 6.59 cannot be used. To calculate the effectiveness factor exactly involves solving partial differential equations, which is very time consuming. The effectiveness factor is therefore estimated as follows construct an infinite slab in such a way, that for an exothermic zeroth-order reaction, it has the same Aris numbers as given above. Since the Aris numbers are generalized the hollow cylinder under consideration and the constructed slab will have almost the same effectiveness factor. Calculation of the effectiveness factor for a slab is relatively easy. Hence an estimate for the effectiveness factor for the hollow cylinder is obtained relatively easily. 

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 ... 

In the previous sections the concept of the effectiveness factor has been discussed. In this section it is discussed in further detail with the aim of extending the concept to industrially important complex reaction networks. The effectiveness factor is the most widely used man-made factor to account (in a condensed, one number manner) for the effect of different diffusional resistances on the actual (or apparent) rate of reaction for gas-solid catalytic systems. Although the use of the effectiveness factor concept in the simulation of catalytic reactors taxes the solution by extra computations, nevertheless it is a very useful tool to account for the complex interaction between the diffusion and reaction processes taking place within the system. Most of the published work (e.g. Weisz and Hicks, 1962 Aris, 1975a,b) deals with the effectiveness factor for the simple irreversible reaction,... 

The effect of (T on the performance index of the reactor is important since it affects Sherwood (Sh) and Nusselt (Nu) numbers. The values of kg and h are computed from the j-factor correlations given by Aris... 

A number of ARIs have reached phase II or III clinical trials but only three are presently in clinical development (Fidorestat, Lidorestat and Ranirestat). In general most studies have been disappointing either due to lack of efficacy or evolvement of severe side effects. Some studies including recent long-term studies have shown effect with moderate to minor improvements in electrophysiological (i.e. 1 m/s in NCV) and symptomatic end points [18,19]. However, ARFs are not presently established as a treatment modality for diabetic neuropathy, and critical factors for success of ARFs in the future will depend on development of non-toxic compounds with sufficient nerve tissue selectivity and affinity to ensure tissue penetration to normalize sorbitol generation in the nerve. 

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