Computing Critical DSR Trajectories

1 Nature of a Critical DSR Profile The control of a DSR profile in concentration space is mathematically similar to the control of a rocket to the moon. The dynamics of both systems may be described by a system of ordinary differential equations, affected by the appropriate control inputs  

Both systems of ODEs may be integrated from an initial condition to obtain a solution trajectory that describes the path of the system in state space along an appropriate time horizon. Whereas in the rocket example, controllability is required everywhere along the solution path, for critical DSRs we will argue that the opposite must be enforced—critical DSRs must be completely uncontrollable on the AR boundary. 

A critical DSR trajectory cannot be completely controllable for all residence times to travel on the AR boundary. 

To motivate why, let us consider the physical significance of traversing on the AR boundary. If a reactor structure achieves effluent concentrations that form part of the true AR boundary, then it is not possible for another reactor structure to produce concentrations that could expand the set further—otherwise, these concentrations would form part of a larger region, and the original set would not belong to the tme AR. 

A critical DSR must hence operate in a highly irregular state. If a perturbation only serves to produce deviations into the region, then not all states can be realized by the system and the system must be uncontrollable. By the nature of the AR boundary, a critical DSR trajectory cannot be completely locally controllable. To determine an a policy that corresponds to a traversal on the AR boundary, the controllability ideas discussed earlier must be related. The same ideas that served as a neeessary condition for finding a unique 

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Computing critical DSR trajectories and critical CSTR points is based on a lack of controllability. 

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. 

F now, only tbe results of tbe 1be< em will be discussed. Details of how to compute crilical DSR trajectories mid critical CSTR solutions are described in Section 6.4. 

We wish to compute the controllability matrix E for this system, which will be used to determine the existence of a critical DSR trajectory. What does the matrix N look like for this system ... 

A CSTR that produces effluent concentrations on the AR boundary is termed a critical CSTR. From Section 6.3.3 it is known that PFRs form the final pafh fo the outermost limits of the AR boundary, whilst DSR trajectories and CSTR points form connectors to these trajectories. Similar to critical DSR trajectories, it is possible to mathematically describe conditions for critical CSTRs to exist. Since both CSTRs and DSRs act as connectors on the AR boundary, the underlying theory for computing critical CSTR solutions is closely related to the theory of critical DSRs. 

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). 

In order to reduce the computational workload, it is often necessary to only select a sample of all extreme points in conv(X) as feed points C[ for use in the DSR integrations. We often also designate specific points to act mixing points C w (j fee(j equilibrium points), since, from Chapter 6, it is known that critical DSR trajectories must be fed with sidestream concentrations that reside on the true AR boundary, which also cannot originate from protrusions (concentrations from PFR trajectories). A trade-off between constmction time and computational accuracy must often be estabhshed when implementing the RCC algorithm in practice. 

Null Space For many purposes in AR theory, it is useful to understand the set of concentrations that lie perpendicular (orthogonal) to S, which are spanned by the stoichiometric coefficient matrix A. For instance, the computation of critical DSR solution trajectories and CSTR effluent compositions that form part of the AR boundary require the computation of this space. It is therefore important that we briefly provide details of this topic here. It is simple to show from linear algebra that all points orthogonal to the space spanned by the columns of A are those that obey the following relation ... 

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. 

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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