Critical a policy

This is significant from a controllability perspective. Suppose that a critical DSR and associated critical a policy are available, which produce a DSR trajectory that traverses on the AR boundary. Introduction of a perturbation to the a poUcy would result in a deviation away from the expected optimal solution trajectory. Since the DSR is critical, its trajectory must lie entirely on the AR boundary. Perturbation impUes the perturbed DSR trajectory would never result in one that could expand the AR boundary, for if this were satisfied, the original DSR would not be a critical DSR to begin with. Permissible trajeetories resulting from the perturbation would only serve to travel into the region. This situation is depicted in Figure 6.13. 

Critical a Policies It is now possible to describe the computation of critical DSR a policies. For nonlinear systems, the use of Lie brackets is required to determine controllability. It will be helpful, nevertheless, to remember the... 

Expanding the Lie bracket expressions and applying the properties of determinants, one may deduce an expression for the equation a in terms of r and v. Feinberg (1999) shows that although there is no explicit reference to a in the above expression, one may reason that the critical a policy be given by... 

The procedure for computing a critical a policy for four independent reactions follows that for d=3 however, the determinant function now involves a second iterated Lie bracket zl l. Thus, application of Equation 6.13 for / = 4 gives ... 

Thus for higher dimensions, an explicit expression for the critical a policy is no longer available. 

Equation 6.16 expresses the critical a policy for a DSR on the boundary of the AR in terms of the rate function r(C) and mixing point C°. This condition is specific to IR. In the presence of a suitable mixing point and reaction kinetics. Equation 6.16 may be used to compute a critical a policy. 

CONCEPT Critical a policies in three - V dimensions—the vDelR condition... 

If we wish to compute critical a policies for three-dimensional systems (three independent reactions), we can use the following equations ... 

EXAMPLE 12 Critical a policy for three-dimensional Van de Vusse kinetics... 

Rate constants have already been substituted into the rate expressions for convenience (they are kj = 1 s kj = 1 s, and kj = 10 L/(mol.s)). Determine an expression for the critical a policy if the DSR side-stream mixing composition is given by C° = [l,0,0]Tmol/L. 

Figure 6.14 shows the DSR trajectory using the critical a expression supplied above. The AR has also been plotted for comparison. The trajectory lies on the AR boundary at every point along its path. Notice that although there are four components in the system, computation of the critical a policy by Equation 6.16 requires vectors in The fourth component may be computed by mass balance. 

For systems in R, we can use the vDelR condition as a shortcut method to determining critical a policies in a critical DSR. 

Critical DSRs Let us now investigate the role of DSRs in the formation of the AR boundary. The form of the critical a policy for the DSR is generalized, and therefore potentially various a policies may be computed that all conform to the controllability criteria for critical DSRs. As noted in Chapter 6, sidestream concentrations C° used in a critical DSR must originate from points on the AR boundary. This assists in refining the set of compatible concentrations in a DSR. Since the feed point is always available, we often set C° = Cf for convenience. 

A critical DSR profile for the system may now be computed. Since the system under investigation is a three-dimensional problem, the following two possibilities are available for computing the critical a policy for the system ... 

Equation 7.9 is the critical a policy for the BTX system when C° = Cf = [1.0,0.5,0.0] mol/L is employed. This can be substituted into the DSR equation to compute critical DSR trajectories that serve to further expand the region. 

To generate this critical DSR trajectory, the DSR expression is integrated together with the critical a policy, given by Equation 7.9, using the CSTR equilibrium point as the initial condition to the DSR. From point C, the DSR trajectory may be computed and plotted, which is also displayed in Figure 7.12(a). 

For DSRs, the vDelR condition for critical a policies must still apply, although the conditions must be amended to include the temperature parameter as well. 

The system under investigation is identical to the system established in S ection 7.2.1. As a result, the AR for the batch system will be constructed in the same space and feed point (Cf = [CAf, CBf, Cof] = [1,0, 0] mol/L). We are also able to utilize the optimal reactor structures, developed in Section 7.2.1, and convert them to batch structures without the need to perform additional analysis. The critical a policy for the DSR, given in Section 7.2.1.5, will be used for the F/V policies in the batch system. 

From Chapter 6, critical a policies depend on the controllability matrix E of the system, which is a function of the concentration vector C. Hence, a is expressed in terms of C and not residence time t. Nevertheless, an explicit function for a in terms of t may be found once the optimal DSR concentration profile is determined, which is accomplished by substituting DSR concentrations, for a certain value of t, into Equation 7.3 and determining the value of a corresponding to the C - T pair. 

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. 

We also described how concrete equations for critical DSR and CSTRs may be computed. These expressions are complicated to compute analytically, which are derived from geometric controllability arguments developed by Feinberg (2000a, 2000b). These conditions are intricate, and thus it is often not possible to compute analytic solutions to the equations that describe critical reactors. For three-dimensional systems, a shortcut method involving the vDelR condition may be used to find critical a policies. Irrespective of the method used, the conditions for critical reactors are well defined, irrespective of the legitimacy of the kinetics studied, and thus these conditions must be enforced if we wish to attain points on the true AR boundary. 

The AR is composed of mixing lines and manifolds of PFR trajectories. The final approach to the extreme points of the AR boundary is achieved using PFR solution trajectories—if a desired operating point resides on the AR boundary, a PFR must be incorporated into the reactor structure in order to reach it, and thus PFRs are often the best terminating reactor to use in practice (for any kinetics and feed point). Only combinations of PFRs, CSTRs, and DSRs are required to form the AR. This result is true for all dimensions. Distinct expressions may be derived to compute critical a policies for the DSR profile and critical CSTR residence times. These expressions are intricate and complex in nature, which are ultimately based on the lack of controllability in a critical reactor. This idea is important in understanding the nature of the AR and how to achieve points on the true AR boundary. 

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