Optimization of CSTR systems

The economic optimum configuration and operating conditions of CSTR systems depend on a number of parameters. Capital investment in equipment is important, but energy costs and product yield are usually dominant considerations. Of course, the major theme of this book is that these steady-state economics need to be balanced with dynamic controllability 

There is an old saying There are many ways to skin the cat. Likewise, there are many different configurations of equipment and operating conditions that can take the same feedstreams and produce exactly the same products. These flowsheets will differ in topology and operating conditions, and they will differ in capital investment and operating costs. The engineer has to determine which is best.  

We have already given some qualitative consideration of this question for several of the reaction types. For example, with the simple irreversible, exothermic A — B reaction conducted in a single CSTR, the highest possible temperature-minimized reactor size and hence capital investment. But small reactors have small heat transfer areas, so dynamic control problems may limit the selection of reactor temperature. 

However, even for a given reactor temperature, we would expect that the use of several CSTRs in series should be more economical up to a point. Eventually the point of diminishing returns will make the addition of another vessel unattractive. 

As an alternative to multiple CSTRs we might consider the use of a single CSTR followed by a distillation column. The per-pass conversion of reactant can be low, giving a reactor effluent with considerable reactant. Then this mixture is separated in a distillation column that recycles the unreacted component back to the reactor. 

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The data are from Ref. 31. The objective for optimization is the maximization of the effluent concentration of component B. The performance limit of the system is identihed with each stochastic run requiring an average of only 120 CPU sec on an HP 9000-C100 workstation. Numerous designs are obtained from the stochastic search that perform close to the performance target, mostly variations of series arrangements of PFRs and CSTRs. A detailed discussion of this and other studies is given in Ref. 31. 

At the other extreme, it may be argued that the traditional low-dimensional models of reactors (such as the CSTR, PFR, etc.) should be abandoned in favor of the detailed models of these systems and numerical solution of the full convection-diffusion reaction (CDR) equations using computational fluid dynamics (CFD). While this approach is certainly feasible (at least for singlephase systems) due to the recent availability of computational power and tools, it may be computationally prohibitive, especially for multi-phase systems with complex chemistry. It is also not practical when design, control and optimization of the reactor or the process is of main interest. The two main drawbacks/criticisms of this approach are (i) It leads to discrete models of very high dimension that are difficult to incorporate into design and control schemes. 

More recent efforts (primarily at the simulation level) on the optimization of styrene-related systems include Cavalcanti and Pinto [4], suspension reactor for styrene-acrylonitrile, and Hwang et al. [5], thermal copolymerization in a continuously stirred tank reactor (CSTR). 

While these optimization-based approaches have yielded very useful results for reactor networks, they have a number of limitations. First, proper problem definition for reactor networks is difficult, given the uncertainties in the process and the need to consider the interaction of other process subsystems. Second, all of the above-mentioned studies formulated nonconvex optimization problems for the optimal network structure and relied on local optimization tools to solve them. As a result, only locally optimal solutions could be guaranteed. Given the likelihood of extreme nonlinear behavior, such as bifurcations and multiple steady states, even locally optimal solutions can be quite poor. In addition, superstructure approaches are usually plagued by the question of completeness of the network, as well as the possibility that a better network may have been overlooked by a limited superstructure. This problem is exacerbated by reaction systems with many networks that have identical performance characteristics. (For instance, a single PFR can be approximated by a large train of CSTRs.) In most cases, the simpler network is clearly more desirable. 

The performance bound from delay-limited control analysis (Section II.B.2) may be computed for a number of series CSTR systems and compared to the performance obtained by optimizing PI controllers. For simplicity and generality, this section assumes the pH nonlinearity to be canceled at the controllers. 

Chapter 21. Chapter 7 in Shinskey [Ref. 3] is again an excellent reference for the practical considerations guiding the design of feedforward and ratio control systems. It also discusses the use of feedforward schemes for optimizing control of processing systems. Good tutorial references are the books by Smith [Ref. 2], Murrill [Ref. 8], and Luyben [Ref. 9]. The last one has a simple but instructive example on the nonlinear feedforward control of a CSTR. 

The following example illustrates a simple case of optimal operation of a multistage CSTR to minimize the total volume. We continue to assume a constant-density system with isothermal operation. 

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