Application of the Gamma Distribution

The introductory example may be reworked using the Gamma distribution, since the special case given there is n = 1. Let c(x, 0) = Cogn( ) where C0 is the total initial concentration. Let the first order rate constant be k(x) = kx and make time dimensionless as kt. This reaction time or intensity of reaction—severity of reaction as the oil people have it—is really the Dam-kohler number, Da, for the reactor, with t the time of reaction if it is a batch reactor or the residence time if a PFTR. Thus 

It follows that the lumped kinetics is of apparent order 

Astarita and Ocone11 generalized the component kinetics from the first order to what they called uniform kinetics 

The uniformity lies in the fact that K is not a function of x and this allows the problem to be reduced to linearity by suitably warping time. Astarita12 shows that in this way any order of reaction may be achieved for the lump. Indeed Astarita and Aris13 showed that any kinetics could be imitated by choosing F and K adroitly. We shall return to uniform kinetics later, but, for the moment, will stick with the linear case. 

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APPLICATION OF THE GAMMA DISTRIBUTION, 107 A GENERAL THEOREM FOR SIMPLE LINEAR REACTOR MODELS, 108 APPLICATION TO A MODEL OF THE BUBBLING FLUIDIZED BED, 109 THE DAMKOHLER NUMBER, 111... 

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