Advances in Chemical Engineering, Vol. 24 by James Wei (Ed.)

By James Wei (Ed.)

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This is impossible in the (exceptional) cases where the system considered is a merization one, in the sense discussed in Section 11. , in systems subjected to at least one linear constraint in addition to mass balance), phase splitting in one of the directions of the orthogonal complement is possible, and heterogeneous chemical equilibria may prevail. However, since the dimensionality of the orthogonal complement is likely to be small, the number of phases present at a heterogeneous chemical equilibrium point is also likely to be small.

One obtains so that, since is always positive, at any stationary point = x for all x’s for which X(x) is nonzero. This is clearly possible only if XA(x) = 6(x - x*). Clearly, the surviving component is the most refractory one, x = 0, and that component does not react. The result is related to the strong degeneracy of the xy form for b(x,y): If regarded as a continuous linear operator, xy transforms any function X(x) into x; we say xy has rank one. More general forms for b(x,y) furnish a nontrivial geometry of possible reaction pathways X(x, T ) , with nontrivial stationary points that may be stable or unstable; if more than one stable stationary point is possible, which one is reached depends on the initial distribution X(x,O).

28 R. OCONE AND G . ASTARITA analysis has been strengthened by Krambeck (1984b). Even stronger constraints on reaction pathways, based on the assumption that every individual term O,r, in Eq. (45) (and not only their sum) is nonpositive, have been established by Shinnar and Feng (1985). In this regard, see also the discussion in Section IV,B,4,b. While Eq. (70) holds in general for monomolecular systems, it holds only in a neighborhood of equilibriumfor the general case considered by Krambeck (1994): 6f = .

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