Number of Independent Reactor Structures

From Section 6.2.1.3, the dimension of the AR is equal to the dimension of S, it follows that the maximum number of independent reactor structures is directly related to the number of independent reactions taking part in the system. Moreover, this analysis may be determined in the absence of reaction kinetics and a feed point— the results are a consequence of the system reaction stoichiometry and Caratheodory s theorem only. 

It is possible to determine, beforehand, the maximum number of independent reactor structures needed to generate the AR by computing the rank of the stoichiometric coefficient matrix A. The maximum number of parallel structures is related to the number of independent reactions occurring in the system. 

A useful consequence of the dimension of the AR may be used to relate the maximum number of parallel structures needed to generate the AR, which is achieved by use of Caratheodory s theorem (Carathdodory, 1911 Eckhoff, 1993). Feinberg (2000a) shows that for an AR constructed in IR, the following limits, in terms of parallel reactor structures, may be enforced  

Here d is a positive integer representing the dimension of S, and hence the dimension of the AR. Note also that these results refer to a maximum (upper bound) on the number of reactor structures required. A particular system kinetics and feed point may require less than the maximum in order to generate the AR boundary in practice. A convex PFR trajectory in is an example of such a system. Specific kinetics are described in Chitra and Govind (1985). 

It follows that if the AR resides in a d-dimensional subspace in R , then one need only consider maximum of d parallel reactor structures in order to generate the AR boundary. Note further these results make no assumption about the complexity of each structure—an AR in R may have as its optimal boundary structure a single PFR for 

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