Proper species lumping

The model includes fundamental hydrocarbon conversion kinetics developed on fresh catalysts (referred to as start-of-cycle kinetics) and also the fundamental relationships that modify the fresh-catalyst kinetics to account for the complex effects of catalyst aging (deactivation kinetics). The successful development of this model was accomplished by reducing the problem complexity. The key was to properly define lumped chemical species and a minimum number of chemical reaction pathways between these lumps. A thorough understanding of the chemistry, thermodynamics, and catalyst... 

Now one may want to impose another constraint, namely, that each of the original N species appears in only one of the N" species hence, every column of L has to be a unit vector of the type 0, 0,.. ., 1,.. ., 0. This is called proper lumping. Lumping that is not proper, however, is possible see Section II,C. 

Wei and Kuo (1969) have shown a direct construction of the matrix from the knowledge of the eigenvectors and eigenvalues of K. A simpler construction (for proper lumping) was proposed by Coxson and Bischoff (1987b) Let S be the diagonal A" X A matrix whose elements are the inverses of the number of species in each lump, that is, in each pseudospecies. Then K = [[,L ],S]. 

It is often difficult to predict the color of the gum arabic solutions on the basis of the color of the dry tears or powder. The size and condition of the lumps and powder affect judgment considerably. The smaller the size and the more frosted, the lighter will their color appear. A dark gum when finely powdered loses its color. When the gum is finely powdered or its surface is crazed, it presents so many minute facets at all angles that practically all the light is reflected and scattered before it has traversed more than the outermost layers of the substance. Proper comparison of color should be made in solutions of a definite concentration. According to Hamy (7), the rotatory power of solutions of gum from Accu ia verek is negative, that from other species of acacia is positive. 

When the species adding to the a,P-unsaturated compound is itself an enolate, the reaction is properly called the Michael reaction, but related additions are often also lumped under this heading (Fig. 19.82). 

In lumping methods, the reactants are grouped into a smaller number of pseudo-species so that the kinetic equations are easier to solve, and fewer parameters need to be experimentally determined. In proper lumping each reactant appears in only one lump. Thus, proper lumps are kinetically independent of each other. Empirically determined lumps are nearly always proper. In improper lumping, on the other hand, some components are allocated to more than one lump. Proper lumping schemes are more useful and provide more insights, but proper lumps may not exist for a system or may be difficult to derive improper lumps are easier to derive and more common. 

In other cases, the continuous view is convenient only because of the lack of detailed information. If the exact composition and reactions are unknown, so that the use of a model with discrete species, though appropriate, is not feasible, the continuous view may be a convenient description. We will show how all the reactants may be combined into a single, proper lump, when only the disappearance of that lump must be tracked. The continuum view was first applied to first-order, reversible, parallel chemical reactions. ... 

---