Automated AR construction

Many AR construction schemes still do not allow for the decoding of the optimal reactor structure associated with the region. Hence, it is still challenging to know what reactor structure is required to reach a particular state on the AR boundary once AR construction has been completed. 

the use of a hybrid construction scheme may provide an alternative method for a numerical sufficiency condition. Validation of the AR might then only be applicable to the specific system computed, but this is nevertheless valuable, for at least a theoretically reliable estimate of the AR may be established. 

Agarwal, V., Thotla, S., Mahajani, S.M., 2008. Attainable regions of reactive distiUation-Part I. Single reactant non-azeotropic systems. Chem. Eng. Sci. 63, 2946-2965. 

Alvarado-Morales, M., Hamid, M.K.A., Sin, G., Gernaey, K.V., Woodley, J.M., Gani, R., 2010. A model-based methodology for simultaneous design and control of a bioethanol production process. Comput. Chem. Eng. 34, 2043-2061. 

Eeinberg, M., 1987. Chemical reaction network structure and the stability of complex isothermal reactors—I. The deficiency zero and deficiency one theorems. Chem. Eng. Sci. 42, 2229-2268. 

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In Chapter 8, we shall discuss an automated AR construction algolthm that uses this property of the complement region to compute candidate regions. This algorithm can then be used to verify this statement... 

This chapter marks the end of Section I of the book. All of the information discussed up to this point provides a firm foundation of the basics of AR theory. In the following chapters, we shall extend on these ideas, and relate them to higher dimensional constructions. Automated AR construction methods and variable density systems will also be discussed, which will allow us to tackle even more realistic problems. 

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.

There are methods that exist that address these constraints however, these require additional theory to understand. In the absence of an explicit temperature expression, it is often easier to tackle the problem numerically, with the aid of an automated AR construction scheme. In Chapter 8, a number of AR construction methods are discussed that may be used for nonisothermal systems. Although these methods often do not suggest an optimal reactor structure, knowledge of the limits of achievability for a nonisothermal is often sufficient for setting design targets. 

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. 

We note again that these conditions are specific to IR only. Although similar results may be derived for higher dimensions, one often employs the use of an automated AR construction method instead. 

Yet, a number of complex, real-life problems are beyond the scope of traditional analytical methods, such as those methods discussed here. For this reason, we are often reliant on the results obtained by automated AR construction... 

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... 

Perhaps the earliest account of an automated AR construction method appears in the PhD thesis of McGregor (1998). In this work, AR construction is achieved not by a specific technique, but rather by a constructive, trial and error approach, given reaction kinetics and a feed point. In Figure 8.7, McGregor s recommendations are listed as one potential geometric method, although the recommendations are not discussed here. Interested readers are referred to McGregor (1998) for further details. 

We must therefore resort to automated (numerical) AR construction techniques to carry out our investigations. Yet, automated AR construction itself is not perfect, and all construction methods described in this chapter possess both strengths and weaknesses. Furthermore, all construction methods generally cannot decode the associated reactor structure at present. 

Automated AR construction has witnessed meaningful growth over the past two decades, although a bulletproof AR construction method— suitable for all design scenarios and complexities—may still be a long way away, and which may require significant advances both in fundamental theory and computational power before it is available. 

Since the system is three-dimensional in nature, the VdelR condition may be used to compute the critical a policy for the system. Since the system is no longer expressed in terms of concentrations, however, the specific critical a policy given by Equation 7.3 cannot be used. Rather, the Jacobian matrix expressed in terms of mass fractions dr(z) must be computed and then used to compute (p(z), as in Chapter 7. The computations are somewhat lengthy, and hence this approach will not be adopted here. Instead, the parallel complement automated AR construction method discussed in Chapter 8 shall be employed, providing a quick means to compute the AR for use in comparisons. 

Chapter 8 provided a general overview of automated AR construction techniques. These methods are necessary when very complex, higher dimensional, systems are investigated. AR constmction techniques have historically fallen into one of two categories inside-out and outside-in methods. Inside-out methods are additive in nature. These methods begin with a small achievable region (usually the feed... 

The absence of a sufficiency condition— which will signify when the true AR has been determined—presents a large theoretical challenge for both AR theorists and practitioners who employ AR theory to solve reactor synthesis problems. Without a sufficiency condition, there is no certainty that the regions produced are the true AR. This is true even if the region computed has been generated from an automated AR construction method, such as those described in Chapter 8. Only for systems of a simplified or unique nature (i.e., when a rate field contains completely convex PFR trajectories), or for systems that have been well studied (i.e.. Van de Vusse kinetics), is one confident that the true AR has been foimd. 

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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