Application of attainable region theory to batch reactors
Introduction
The problem of determining optimal operating policies in batch operations is a popular one, and has been given a great deal of attention in the literature, particularly due to the importance of batch reaction in a large number of industrially relevant processes. Batch reactors are used extensively in the production of a variety of pharmaceutical (Davies and Gloor, 1971) and biological(Cheong et al., 2007, Modak and Lim, 1992, Najafpour et al., 2004, Senthuran et al., 1997) products, as well as in the wastewater (Woolard and Irvine, 1995, Zwiener et al., 2002) and polymerization (Zeman and Amundson, 1965) industries, and are typically considered to be rather more versatile than that of equivalent continuous processes (Bonvin, 1998). Batch reactors also find use in the small-scale production of highly specialized products, negating the benefit otherwise obtained by large-scale production. Moreover, batch reaction may be the only viable method of producing certain products, such as when experimental work is performed at lab-scale and pilot-scale.
It follows that optimization plays an important function in the design and operation of these processes. Much attention has been placed on the development of methods that seek to improve batch performance, particularly with regards to the determination of optimal operating conditions and scheduling strategies. Although research into determining efficient batch structures exists (Allgor et al., 1996, Aziz and Mujtaba, 2002, Capon-Garcia et al., 2011, Mendez et al., 2006), a brief search of the current literature would suggest that this area may not appear to be as popular as traditional batch optimization. Even then, current methods for determining optimal batch structures often rely on traditional optimization methods (Allgor and Barton, 1999, Allgor et al., 1999, Allgor et al., 1996).
For the past two decades, the papers of Feinberg and Hildebrandt (1997), Feinberg (1999, 2000a, 2000b, Glasser et al. (1987), Glasser and Hildebrandt (1997) have motivated the use of a novel method of identifying optimal reactor configurations, termed attainable region (AR) analysis. Determination of optimal reactor structures using the AR is unique since synthesis is achieved via a geometric interpretation of reactor configurations. This approach has shown to be an effective alternative method to optimal reactor synthesis, particularly with regard to systems involving multiple side reactions or reactions with complicated kinetics. Nearly all applications of the AR have been performed on continuous reactors however; the single paper of Davis et al. (2008) currently remains the only use of AR theory to batch systems.
Our aim in this work will thus be towards improving batch reactor structures with specific use of the AR and the associated benefits of this approach. This is done as follows: for a given set of feed conditions and kinetics, a candidate AR may be generated and interpreted, initially, in the form of a continuous reactor structure. Once generated, the appropriate conversions to the batch setting may be performed, providing an equivalent recommended batch reactor structure. A key objective in this work has been towards reusing the results and insight discovered from a continuous AR structure, such that the same outputs might be obtained with an equivalent batch system.
We will begin by first providing a brief overview of AR theory, and drawing a relevant set of results associated with continuous equipment; this is done in Section 2. Section 3 investigates how equivalent structures in batch may be derived from continuous equipment. The correct choice and exact combination of continuous equipment is guided by AR analysis, and thus the development of the AR is central to the approach presented in this work. Finally, examples and concluding remarks are given in 4 Examples, 5 Conclusions, respectively.
Section snippets
Motivation
The purpose of this section will be towards providing a brief overview of AR theory for continuous equipment. The results obtained will allow for the development of optimal batch structures. The geometric nature of AR analysis has been discussed in numerous papers before (Davis et al., 2008, Feinberg and Hildebrandt, 1997, Feinberg, 1999, Feinberg, 2000a, Feinberg, 2000b, Glasser and Hildebrandt, 1997, Glasser et al., 1987, Godorr et al., 1994, Godorr et al., 1999, Hildebrandt and Glasser, 1990
A note on continuous and batch operation
In the upcoming discussion, we will want to show how the states realized in the three fundamental continuous reactors used in AR theory can be achieved by batch equipment. Moreover, we would like to demonstrate that the necessary transformations from continuous to batch can be used in conjunction with the AR to help guide the design of optimal batch structures. Before we begin our discussion on the similarities between batch and continuous reactors however, we would like to highlight a clear
Identification of optimal structure
Consider the following autocatalytic system (Brooks, 1988)with k1=1 L2/mol2/h and k2=1 h−1. From the discussion of Section 2, the system contains two independent reactions and hence the entire system can be constructed in alone. The resulting rate vector is then expressed in terms of components A and B, and given by the following kinetics
Our objective here will be towards finding the appropriate batch sequence that maximizes the
Conclusions
At the outset, an important goal of this paper has been towards developing a systematic method of converting optimal continuous structures, obtained via conventional AR analysis, to an equivalent batch structure. Due to the one-to-one nature between continuous and batch reactive equipment, it is a fairly simple task to convert continuous structures to batch. Indeed, it is possible to derive an equivalent batch structure by first computing the AR in the continuous setting, and then applying the
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