Total Lumping Overall Kinetics

The problem addressed here can be stated as follows. Let c, t) be the concentration of the individual reactant and C(0=EjiiC,(i) be the total concentration of all reactants. The aim is to predict the dependence of C t) on feed properties and reactor type. It is also of interest to know if an overall kinetics R C) can be found for the reaction mixture as a whole. If so, the thus-found R C) can be included in the conservation equations for modeling the coupling between kinetics and transport processes in a reactor. The selectivity toward a specific class of reaction intermediates (e.g., gasoline) is also of interest in many situations. 

There have been four main lines of attacks to the problem posed here. The first addresses the question of whether one can find an overall kinetics 

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The idea of overall kinetics could be regarded as just an extreme case of lumping where the projection is onto a one-dimensional subspace, such as the unit vector. In this case the overall lump is just the total mass of the system or of some subset of the compounds. However, overall kinetics (and/or thermodynamics) are in no way exact or approximate lumping procedures, since the lump may well behave in a way that is totally different from that of the original system. 

Similar to Eq. (67), the first reaction (incorporating the enzyme phosphofructo-kinase) exhibits a Hill-type inhibition by its substrate ATP [126]. The overall ATP utilization v3 (ATP) is modeled by a saturable Michaelis Menten function. The system is specified by five kinetic parameters (with Gx lumped into Vm ), the Hill coefficient n, and the total concentration, 4 / = [ATP] + [ADP]. Note that the model is not intended to capture biological realism, rather it serves as a paradigmatic example to identify dynamic behavior in metabolic pathways. 

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