AR construction algorithms

In this chapter, we wish to touch on a number of automated AR construction schemes. At the time of writing, research in AR theory has witnessed a shift toward the development of numerical AR constmction algorithms, with less emphasis placed on general AR theory. These developments have arisen primarily out of a practical need to determine candidate regions for complex, higher dimensional problems, which are not easily computed by hand, but which are still important for practical problems of interest. AR construction methods provide a numerical basis wherefrom theoretical predictions may be compared with in the search for a sufficiency condition. Inasmuch as how 

Our intention in this chapter is not to provide detailed descriptions of current AR construction algorithms. Rather, this chapter serves to provide an overview of the different methods available and, importantly, the underlying concepts that make AR construction possible. 

Automated AR computation is a popular research field at present, and advances continue to be made. We hope that this chapter will not only make the current AR construction literature more understandable but also promote new ideas and approaches that further advance the field. 

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For three dimensions, the AR construction algorithm is similar to the one described above—with the added possibility that we can find a (one-dimensional) connector on the AR that is described by a DSR. Glasser et al. (1992) defined conditions under which DSRs appear on the AR along with a direct method for finding the feed addition rate, q. While the conditions for DSRs appear to occur infrequently, examples have been constructed in the space of conversion, temperature, and residence time where the DSR was a prominent part of the AR. Nevertheless, Hildebrandt and co-workers conclude that most ARs will consist only of CSTR and PFR surfaces. In dealing with n-dimensional problems, Hildebrandt and Feinberg noted that the AR boundary is defined by line segments and PFR trajectories, with at most n structures needed to define a point on the AR boundary and n + 1 structures needed to define an interior point of the AR. Thus, for three-dimensional problems, at most three parallel structures (PFRs, CSTRs, DSRs) are needed to define any AR boundary point. 

The form of this expression is similar to the CSTR equation (Equation 4.8). Certainly, if the substitution r = 1/a is made, then the resulting expression is identical to the CSTR equation with feed point given by C = Cf. It is therefore possible for both CSTR and PFR behavior to be achieved in a DSR under the correct conditions. The generalized DSR expression is thus useful in approximating CSTR and PFR concentrations. Although these approximations might appear impractical, particularly for CSTRs, they find use as a theoretical tool. This dual-natured behavior of DSRs is also useful in many AR construction algorithms. 

Introduction Given a system of reactions, it is possible to compute bounds in concentration space wherein all feasible concentrations must lie. This space is typically much larger than the space of achievable concentrations (the AR). Consequently, we can use this space as an upper bound on the set of feasible concentrations that the AR must reside in. We call this space the stoichiometric subspace and denote it by the set S. Determining S is also useful for AR construction algorithms, which are discussed in Chapter 8. 

Three-dimensional Van de Vusse kinetics has been used extensively in AR research papers in the past. Since the system is well understood, AR practitioners often use the system as an acid test for many AR construction algorithms and hypotheses. Understanding this system therefore assists in understanding many research investigations that employ the system, and future research in the field of AR theory is easier to undertake if we are able to understand past work. 

Figure 7.10 The AR for the BTX system (a) computed by an automated AR construction algorithm. The region obtained is in agreement with the theoretical prediction given by (b), although there is still a moderate difference between the two regions.

ARs may also be generated for systems where a control parameter (often temperature) is employed. This theory is not part of the scope of this book, although we do provide a brief discussion of the relevant theory here for interest. The following discussion has ideas borrowed from Godorr et al. (1999), and all of the following discussions apply to alone. For higher dimensional problems, the use of an automated AR construction algorithm is often used instead. 

AR construction algorithms are similar to many numerical methods in engineering in that there are often different methods for solving the same problem. There is no definitive AR construction method that is suitable for all problem types, and it is the duty of the designer to pick the best method for the problem at hand. In general, AR construction methods may be classified into two categories ... 

The difference between inside-out and outside-in methods will be described in Section 8.3.2. In Figure 8.7, a schematic classifying the general types of AR construction algorithms available is given for reference. 

Chapter 8 In this chapter, we describe a number of AR construction algorithms that may be implemented on a computer. 

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