Computing Critical CSTR Points

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. 

1 Rocket Example Revisited Consider what might occur if there is now free choice to specify the rocket s initial position and state from Section 6.4.3. If the rocket is launched from a location that coincides with a controllable state (in other words, if the initial state of the rocket is not associated with the E matrix having less than full rank), then it is possible to steer the rocket locally for all small inputs. Conversely, if an initial state is chosen that corresponds to an uncontrollable state (where the E matrix does not have full rank), the rocket will not be locally controllable with the given control inputs. 

Similarly, if the initial condition for a DSR is one that coincides with a critical DSR solution trajectory, then the DSR is uncontrollable for all inputs, and it must lie on the AR boundary. Since CSTRs operate at discrete points in space for a set feed and t, these might be used as a way of transitioning from an initial feed state to one on the AR boundary. CSTRs thus facilitate jumping between different points in space. 

Critical CSTRs are determined by the same condition used to compute critical DSRs. A critical CSTR cannot be locally controllable on the AR boundary. 

Feinberg (1999,2000b) proves that in order for a CSTR to lie on the AR boundary, the following condition must hold  

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Hence since is a scalar, it can be eliminated from the expression without interfering with the particular level set defined by Det[M2] = 0. The condition Det[M2] = 0 is equivalent to Det[Mi] = 0, and hence Equation 6.18 is equivalent to Equation 6.19. Equation 6.19 may therefore also be used to determine the condition for a critical CSTR. Since Equation 6.19 does not depend on a feed point Cf, this is the preferred method for computing critical CSTRs. ... 

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

This is a useful result for it is possible to compute the critical CSTR condition without specification of the feed streams. This comes about as a direct result of the geometric properties of the CSTR—the rate vector evaluated at the CSTR effluent composition is coUinear with the mixing point C - Cf, or... 

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. 

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 ... 

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). 

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. 

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