First-order three-factor model

Let us compare computations of the effectiveness factor, using each of the three approximations we have described, with exact values from the complete dusty gas model. The calculations are performed for a first order reaction of the form A lOB in a spherical pellet. The stoichiometric coefficient 10 for the product is unrealistically large, but is chosen to emphasize any differences between the different approaches. 

From now on the two-parameter model is used because it is almost as accurate as the three-parameter model and it gives a better insight. For example, the curves which were drawn by Weisz and Hicks [2] for different values of a and s (Figure 6.4) reduce to one. This is illustrated in Figure 7.1 where the effectiveness factor is plotted versus An0 for several values of C, and for a first-order reaction occurring in a slab. Notice that all the curves in Figure 7.1 coincide in the low ij region, since ij is plotted versus An0. The formulae used for Ana now follow. 

As to the first question, lattice models do exhibit oil/water inlerfacial tensions that are reduced to various degrees from the value in the absence of amphiphile. For example, in the three-component model solved within mean-field theory, a reduction on the order of 30 was found in the oil/water interfacial tension at three-phase coexistence with the microemulsion [101]. When simulated so that fluctuations were included [102], the reduction increased to about a factor of 100, which is characteristic of a weak amphiphile. Other lattice models [103] have obtained reductions as large as a factor of 800, larger than that provided by even the strong amphiphile C6E3 [104]. 

The Ishigami test function is a three parameter model. It is in so far interesting as the second and third input factors have a Pearson Correlation Coefficient of zero. A variance-based analysis retrieves a 44% first order... 

As can be seen by inspection of the set of three first-order differential equations, Eqs. (6) to (8), the model profile depends nonlinearly on the set of 2 unknown parameters, namely the apex radius b and the shape factor p. Moreover, as an additional third parameter, the apex correction error, e, is also taken into account. 

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