Geometric CSTR solution

The geometric interpretation of the CSTR allows for a convenient method for solving the CSTR equation instead of solving a nonlinear system of equations by standard numerical methods (i.e., Newton s method), we can find CSTR solutions by forming the vector v = C-Cf and then testing the rate vector at C for colinearity between r(C) and v. 

Figure 4.18 (a) Geometric interpretation of the CSTR. (b) CSTR solutions are collinear with the rate vector evaluated at that point and the feed. (See color plate section for the color representation of this figure.)... 

Let us now look at how to compute CSTR solutions using the geometric properties of the CSTR vector equation. Consider... 

We can use the geometric nature of both CSTRs and PFR to determine whether a point is achievable or not. This is accomplished by checking whether the resulting backward PFR or CSTR solutions intersect the current polytope boundary or not, which works in an opposite way to the complement method described in Section 8.4.3. 

The shrink-wrap method is generally more successful at handling complicated kinetics (where multiple steady states may exist) when compared to competing methods. Since CSTR solutions are found geometrically as opposed to through the solution of a potentially difficult system of nonlinear equations, the method is attractive for highly nonideal systems. 

Solution via the geometric interpretation follows a somewhat different methodology. We shall use the fact the mixing vector between the CSTR effluent concentration and the feed vector Cf is collinear with the rate vector evaluated at the exit concentration r(C). That is we have... 

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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