By James Wei (Ed.)
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Extra resources for Advances in Chemical Engineering, Vol. 24
ASTARITA It is worthwhile to observe for future reference that, if indeed the equilibrium points are strong ones, det B is nonzero, and hence r(q) is invertible at equilibrium. We will also make the (likely) assumption that the systems considered are, in Coleman and Gurtin's (1967) terminology, strictly dissipative, that is, that Eq. (45) is satisfied as an equality only at equilibrium points. Since 8 is a function of q, Eq. (50) by itself does not exclude the possibility of the existence of more than one equilibrium value q*.
The original eigenvectors, which do not vanish, preserve the same eigenvalues in the transformed system. Wei and Kuo (1969) have shown a direct construction of the matrix K^ 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 N ^ X N^ matrix whose elements are the inverses of the number of species in each “lump,” that is, in each ^ pseudospecies. Then K^ = “
The critical warped time T~ at which the zero-order approximation breaks down is estimated as -In E ; that is, it is well in excess of unity. Even for linear kinetics, there is an induction time significantly longer than the inverse of the kinetic constant during which very few end products are formed (this is even more true for nonlinear kinetics of the type considered). The solution can be obtained formally at all levels of perturbation; the first-order level is of particular relevance because it yields (to within order E ) the total amount of end products formed up to the critical time.
Advances in Chemical Engineering, Vol. 24 by James Wei (Ed.)