Parallel Reaction in a Doubly Distributed Continuum

Ho and Aris (1987) argued that any formulation of reaction in continuous mixtures must satisfy the single-component identity (SCI), namely that it should reduce to the kinetics of a single component when the mixture is pure. This is true of Eq. 29, for with/(x) = S(x - x0), U(t) = V(x0t). The corresponding H(x, y) = S(jc - Xo)g(y) is not pure in the second index, so to speak. The requirement should be enlarged to a discrete component each satisfying the kinetic law given by G. We see that this is 

Astarita kinetics satisfy the SCI, but not necessarily the discrete-component identity (DCI). For example, putting g(x, 0) = f(x) = 2utS(x -, ) in the uniform kinetic law 

When n = 2 this represents the second-order reaction scheme aikAt + Ak - products with certain constraints on the stoichiometry and kinetic constants. When there is a single component this reduces to a second-order reaction. This cooperative element in the Astarita kinetics is, of course, no defect—indeed it may be its strength. 

We can have parallel nth-order reactions (n 1) from the y continuum by taking 

If a = a, an integer, the last integral is Ea-j(ablT), E being the exponential integral 

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