Reactions in Continuous Mixtures

Department of Chemical Engineering and Materials Science University of Minnesota Minneapolis, MN 55455 

To the best of my knowledge, the work of Prasad et al. contains the first mention of the use of two indices, for, in their description of coal liquefaction 

Here c(x, t)dx is the concentration of material with index in the slice (x, x + dx) whose rate constant is k(x) K(x, z) describes the interaction of the species. The authors obtain some striking results for uniform systems, as they call those for which K is independent of x (Astarita and Ocone, 1988 Astarita, 1989). Their second-order reaction would imply that each slice reacted with every other, K being a stoichiometric coefficient function. Only if K = S(z -x) would we have a continuum of independent parallel second-order reactions. In spite of the physical objections, the mathematical challenge of setting this up properly remains. Ho and Aris (1987) have shown how not to do it. Astarita and Ocone have shown how to do something a little different and probably more sensible physically. We shall see that it can be done quite generally by having a double-indexed mixture with parallel first-order reactions. The first-order kinetics ensures the individuality of the reactions and the distribution 

Since the dimensionless time for a first-order reaction is the product of the reaction time t and a first-order rate constant k, there is no reason why k(x)t should not be interpreted as k(x)t(x), that is, the reaction time may be distributed over the index space as well as the rate constant. Alternatively, with two indices k might be distributed over one and t over the other as k x)t(y). We can thus consider a continuum of reactions in a reactor with specified residence time distribution and this is entirely equivalent to the single reaction with the apparent kinetics of the continuum under the segregation hypothesis of residence time distribution theory, a topic that is in the elementary texts. Three indices would be required to distribute the reaction time with a doubly-distributed continuous mixture. 

To these three a priori reasons for considering the generalization to two indices a fourth may be added a posteriori. We shall encounter, in important specific cases, one or two of the rarer special functions associated with the confluent hypergeometric function. 

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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) = discrete component each satisfying the kinetic law given by G. We see that this is... 

Lumping Coupled Nonlinear Reactions in Continuous Mixtures, AIChE J. 35, 533... 

One of us (RA) is indebted to the PRF of the ACS for continued support of an ongoing investigation of reactions in continuous mixtures (PRF25133-AC7E). The figures and the calculations that lie behind them were done by Paolo Cicarelli. 

Gavalas, G. and Aris, R., 1966, On the theory of reactions in continuous mixtures. Phil Trans Roy Soc, A260 351. As far as I know this is the first use of continuous mixtures in the context of chemical reaction. They had been used in distillation by Amundson and Acrivos, 1955, Chem Eng Sci, 4 29, in froth flotation by Loveday, 1966, Inst Min and Metall Trans. C75 219 and in thermodynamics by de Donder, 1931, L Afflnite (Gauthier-Villars, Paris). 

Wei s later work with J. C. Kuo on lumping and that of Li and Rabitz at Princeton [Chem. Eng. Sci. 44,1413, (1959)] also opened up an important area of formal kinetics. See also Bischoff and Coxson, Lumping strategy. l.E.C. Res. 26,1329 2151, (1983). J. E. Bailey [Lumping analysis of reactions in continuous mixtures. Chem. Eng. J. 3, 5261 (1962)]. forged an important link between continuous mixtures and lumps. 

I will not attempt to summarize the present state of the theory of reactions in continuous mixtures. Progress has also been made in the understanding of nonlinear intrinsic kinetics by T. C. Ho (v. e.g., his papers in Kinetic and Thermodynamic Lumping of Multicomponent Mixtures. Ed. G. Astarita and S. I. Sandler. Elsevier. Amsterdam 1991) and in the applications in other areas... 

The topic of reactions in continuous mixtures is one that has occupied my attention since learning of second-order cracking. With the revival of interest that Astarita43 and Ocone s paper generated, I returned to it and got some further results in collaboration with Paolo Cicarelli. 

Aris, R., Reactions in continuous mixtures. AIChE J. 35,539 (1989b). 

Aris, R. Gavalas, G. R. (1966). On the theory of reactions in continuous mixtures. Phil. Trans. Roy. Soc., London, A260, 351-93. 

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