Validity discriminant description

The validity discriminant discussed in this section is the descendant of an earlier cluster validity measure used by Gunderson ( ) to assess the quality of cluster configurations obtained in an application of the Fuzzy ISODATA algorithms. It is closely related to a method suggested by Sneath ( ) for testing the distinctness, i.e. separation, of two clusters, and also borrows from the ideas of Fisher s linear discriminant theory (see chapt. 4, Duda and Hart,(2 0). The validity discriminant attempts to measure the separation between the classes of a cluster configuration usually, but not necessarily, obtained by application of the FCV algorithms. A brief description follows ... 

Heterogeneously catalyzed reactions are usually studied under steady-state conditions. There are some disadvantages to this method. Kinetic equations found in steady-state experiments may be inappropriate for a quantitative description of the dynamic reactor behavior with a characteristic time of the order of or lower than the chemical response time (l/kA for a first-order reaction). For rapid transient processes the relationship between the concentrations in the fluid and solid phases is different from those in the steady-state, due to the finite rate of the adsorption-desorption processes. A second disadvantage is that these experiments do not provide information on adsorption-desorption processes and on the formation of intermediates on the surface, which is needed for the validation of kinetic models. For complex reaction systems, where a large number of rival reaction models and potential model candidates exist, this give rise to difficulties in model discrimination. 

An orthogonal collocation method for elliptic partial differential equations is presented and used to solve the equations resulting from a two-phase two-dimensional description of a packed bed. Comparisons are made between the computational results and experimental results obtained from earlier work. Some qualitative discrimination between rival correlations for the two-phase model parameters is possible on the basis of these comparisons. The validity of the numerical method is shown by applying it to a one-phase packed-bed model for which an analytical solution is available problems arising from a discontinuity in the wall boundary condition and from the semi-infinite domain of the differential operator are discussed. 

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