Optimal reactor structure

D. Hildebrandt and D. Glasser. The attainable region and optimal reactor structures. Chem. Eng.Sci., 45 2161, 1990. 

Hildehrandt. D.. and Glasser. D. "The Attainable Region and Optimal Reactor Structures." Proc. ISCRE Meet.. Toronto (1990). 

In AR theory, we have an interest in the reactor network synthesis problem, but it is not our chief interest. It is sometimes not possible to generate an optimal reactor structure using AR theory, but it is still possible to generate important information that may help to understand what the optimal reactor structure might be. Let us refer back to the following two points from Section 1.4.1.2 ... 

PFRs are the simplest of all three fundamental reactor types in AR theory. This is clear when the close link between the kinetics and PFR solutions trajectories can be shown. In Chapter 6, we will discuss why almost all optimal reactor structures must terminate with a PFR. 

The system is flexible in many respects— particular specification of the rate constants (kj, k2, and k3) results in different ARs, and thus different optimal reactor structures may be generated from a single set of reactions. In the following sections, three distinct scenarios shall be discussed that will facilitate how the AR may be constructed from the theory developed in the previous chapters. From the information generated by the AR, informed decisions can be made regarding the most appropriate reactor structure for a desired duty. 

Step 3 Interpreting the Boundary The region displayed in Figure 5.2(a) represents the AR for the Van de Vusse system, when aj = 2- The boundary of the region is composed entirely of a PFR from the feed a CSTR locus from point F does not serve to expand the region. Hence, we may conclude that the optimal reactor structure for this situation is simply a PFR from the feed. 

Geometrically, this is achieved by moving the horizontal line (the objective function) until it intersects a point on the boundary of the AR. Furthermore, to achieve this concentration, a PFR with a feed concentration specified by point F must be employed. This reactor structure is obtained by consulting the optimal reactor structures that form the boundary of the AR in Figure 5.3. Using the maximum Cg value, Figure 5.1 may be used to determine the residence time of the PFR required. In this scenario, a residence time of 0.76 s is needed to obtain this concentration. 

Figure 5.4 displays the results of the new objective function. The new objective function may be written as a straight-line equation as a function of c. The line has a gradient of 0.25 and an intercept of 0.15moI/L. Thus, the line intersects the boundary at two points, labeled Xj and x2, respectively, in Figure 5.4. Since these points coincide with the AR boundary and the objective function, they are the optimal operating points. Hence there are multiple optima for the given objective— there are in fact an infinite number of optimal operating points for this system, for all eoncentrations on the mixing line joining Xj and Xj also satisfy the objective function. Point X3 is a representative point that satisfies this relation. The following three optimal reactor structures may be formulated that satisfy the objective function ...

Figure 5.4 Optimizing for a new objective function, (a) Points intersecting the AR that satisfy the objective function, (b) The required optimal reactor structures that produce concentrations...

Observe that the optimal reactor structure has not changed in this instance even though the kinetics and associated AR have changed. However, this result is unique to the kinetics. Generally, a change in the kinetics may affect the AR and hence the optimal reactor stmcture associated with it. This behavior is shown next. 

For each expression, determine the intersection with the AR boundary and comment on the optimal reactor structure required. Answer ... 

Suppose that the optimal reactor structure in Figure 5.9(b), for scenario 3, was modified to include another CSTR in series after the PFR (to create a CSTR-PFR-CSTR structure). Using your understanding of the AR boundary, would this structure be able to extend the set of concentrations beyond what is achievable by the region in Figure 5.9(a) ... 

In Equation 5.7a, component A decomposes to form component B. If B is present, then components A and B may combine, in an autocatalytic reaction, to form additional B by Equation 5.7b. Component A may also decompose in a parallel reaction to form C via Equation 5.7c. For this example, we shall assume that component B is the desired product and seek to determine the optimal reactor structure that maximizes the concentration of B, Cg. 

The optimal reactor structure is still a CSTR from the feed followed by a PFR however, this region is significantly larger than that obtained initially, (c) Rate field for the isola system. 

With the full AR for the isola system now known, the associated optimal reactor structure may be determined. To achieve the entire set of points defined by the region in Figure 5.18(b), a CSTR followed by a PER is required. This gives the boundary points of the AR from which all interior points may be found via the appropriate mixing operations. 

Optimization of the system is easily performed now that the optimal reactor structure used to form the AR boundary is known. This discussion is best left to the following illustration for practice. 

Moreover, since the associated optimal reactor structures are also known for generating all points on the boundary of the AR, the particular reactor arrangement required to achieve P is also known. In this instance, the value of 0.617 mol/L in component B is obtained via a CSTR followed by a PFR. 

In general, intersection points on the AR boundary are the most desirable and convenient operating points, even if the objective function intersects the AR along interior points. The particular value of P = 0.2 mol/L intercepts the AR boundary at a point that suggests the optimal reactor structure for this value of P is obtained by a CSTR with bypass of feed material to the CSTR effluent concentration. 

Example Optimal Reactor Structure for Minimum Residence Time... 

We wish to determine the optimal reactor structure corresponding to the lowest possible residence time achieved by the system. 

In Figure 5.29, a number of different cases for the desired value of c, along with associated optimal structures, are shown. The optimal reactor structure varies depending on the final desired product concentration. Although the stmcture is specific for each objective, all structures are part of a generalized optimal reactor structure related to the AR. 

Figure 5.29 Different optimal reactor structures obtained depending on the desired exit concentration, (a) A CSTR with bypass of the feed, (b) a CSTR operated at point X in Figure 5.28 (C = 10 mol/L), and (c) a CSTR followed by a PFR.

The particular choice of payback period might also influence the optimal reactor structure necessary to achieve it. Longer payback periods will require a CSTR with bypass as the optimal reactor structure (this structure occurs when c concentrations are greater than point X= lO.Omol/L). Table 5.6 provides a summary of the intersection points for the payback periods. 

The boundary of the AR exhibits a simple structure, irrespective of the kinetics used. As a demonstration, refer back to Chapter 5 and count the number of occurrences where the optimal reactor structure terminated with a PFR. (You will find that all examples considered in Chapter 5 resulted in a final optimal reactor structure terminating with a PFR.) Theorem 1 helps prove that this behavior is not a coincidence, for the final approach to all optimal reactor structures on the boundary of the AR terminates with a PFR. 

Figure 6.7 Different proposed optimal reactor structures, (a) PFRs in parallel, (b) CSTR with feed bypass, (c) CSTR-PFR, (d) DSR-PFR with feed bypass (e) PFR-CSTR, and (f) DSR-PFR.

Structures (c) and (e) are both similar, as both structures involve combinations of a CSTR and PFR in series. It is known that the final approach to the extreme points of the AR take place as a result of the union of PFR trajectories, and thus we should expect that final fundamental reactor type of any optimal reactor structure on the AR boundary is a PFR. We may conclude that structure (e) does not produce an effluent concentration that is an exposed point on the AR boundary (although the effluent concentration may still lie on the AR boundary, the point will not be exposed). The CSTR feeding the PFR in (c) must therefore produce a concentration that is a point on the AR boundary. 

We wish to compute the AR and the corresponding optimal reactor structure for the set of reactions belonging to the Van de Vusse system. The full system involves three independent reactions, assumed to obey mass action kinetics, and is given as follows ... 

A root of A(C) exists near r 36.7 s. The value of A(C) also appears to approach zero as the value of t approaches large values. This suggests that the equilibrium CSTR point also acts as a critical CSTR point to the Van de Vusse system. The curve in Figure 7.4 implies that there are exactly two concentrations that lie on the AR boundary. The remaining concentrations do not lie on the boundary and are thus not associated with an optimal reactor structure from Chapter 6. PFR trajectories initiated from the CSTR locus not associated with the two critical CSTR solutions therefore also do not form part of the manifold of PFR trajectories... 

Figure 7.6 Optimal reactor structures associated with the three-dimensional Van de Vnsse system. Note that both structures terminate with a PFR.

In fact, if it is desired to only achieve the concentration given at point I, then there is no need to generate the entire AR boundary structure and thus structure 1 in Figure 7.6 need not be utilized. Nevertheless, it is still important to understand the optimal reactor structures that form the AR boundary as the optimization of several objective functions may require the use of more than one optimal structure. 

Nevertheless, since two optimization scenarios have been investigated that each required different optimal structures, an understanding of the AR boundary structures that form the entire region has been beneficial. Under a traditional optimization scenario, two separate optimizations would need to be performed in order to obtain an answer (and also provided that the correct optimal reactor structure is identified in the optimization as well). Hence, not only have we determined the limits of achievability, but we also have an understanding of what reactor structures are required for different optimization scenarios. 

Now that we have a full understanding of AR theory, we are in a better position to answer this question. Let us determine the point of maximum toluene concentration, and provide the optimal reactor structure required to achieve the point. 

Note also that the point of maximum toluene concentration corresponds to a position on the AR boundary that requires a critical DSR followed by a PFR in series to reach it. This is hence the optimal reactor structure needed to obtain the desired toluene concentration. 

Figure 7.15 (a) Optimal reactor structures in Cg-Cp-Cg space and (b) filled AR for the BTX system in Cg-Cj.-Cg space. (See color plate section for the color representation of this figure.)... 

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