Higher Dimensional AR Theory

Furthermore, when AR constructions were carried out in previous chapters, they were described without adequate justification of why the resulting region represented the true AR. In order for these properties to be validated, and also in preparation for higher dimensional examples to be discussed (in Chapter 7), a more detailed AR theory, generalized to n-dimensional spaces, is required. This theory also assists in understanding of the kinds of reactor structures that should be expected when higher dimensional systems are considered in general. 

Our goal then, is to provide discussions for a reasonable (but not mathematically rigorous) understanding of the relationships between reaction and mixing, and how these relate to optimal reactor networks in n-dimensional space. In all 

When reaction and mixing are the only two processes available within a system, the AR may be constructed via combinations of CSTRs, PFRs, DSRs and mixing alone. No other reactor types are required to form the AR. 

It is theoretically possible (but often not practically feasible) to compute concrete conditions (mathematical expressions) that dictate how PFRs, CSTRs, and DSRs should be operated on the AR boundary. 

Much of the content in this chapter is taken from important contributions by Martin Feinberg (Feinberg, 1999, 2(X)0a, 2000b Feinberg and Hildebrandt, 1997). As will be shown, results from four papers by Feinberg, in particular, broadly define the major findings of AR theory in concentration space—at the time of writing, these results have yet to be expanded to wider state spaces, such as mass fraction space. It is for this reason that primary focus will be placed on AR constructions in concentration space alone wherein density is assumed constant. 

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In this chapter, we wish to apply aspects of higher dimensional AR theory to a number of problems. These problems are organized into the following three sections ... 

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