Feasible intersections

For CSTRs, a backward CSTR solution is parameterized using a scalar, t, and checked using the same conditions described in section Computing Feasible Intersections . 

The intersection of section profiles as illustrated in Figure 12.17 can be used to test the feasibility of given product compositions and settings for the reflux ratio and reboil ratio. However, the profiles would change as the reflux ratio and reboil ratios change, and trial and error will be required. 

For the stripping and rectifying sections to become a feasible column, the two operation leaves must overlap. Figure 12.20 shows the system chloroform, benzene and acetone. The operation leaf for a distillate composition D intersects with the operation leaf for a bottoms composition... 

Bi in Figure 12.20. This means that there is some combination of settings for the reflux ratio and reboil ratio that will allow the section profiles to intersect and become a feasible column design. By contrast, bottoms composition B2 shows an operation leaf that does not intersect with the operation leaf of distillate D. This means that the two products D and B2 cannot be produced in the same column, and the design is infeasible. No settings of reboil ratio or reflux ratio can make the combination of B2 and D a feasible design. 

As long as this middle section operation leaf intersects with those for the top section (above the entrainer feed) and the bottom section (below the feed point for the feed mixture), the column design will be feasible. Note that there will always be a maximum reflux ratio, above which the separation will not be feasible because the profiles in the top and bottom sections will tend to follow residue curves, which cannot intersect. Also, the separation becomes poorer at high reflux ratios as a result of the entrainer being diluted by the reflux of lower boiling components. 

Figure 12.27 shows the section profiles for the three sections of a two-feed column. The three profiles intersect in Figure 12.27, and the column will, in principle, be feasible. 

The feasible region lies within the unshaded area of Figure 7.1 defined by the intersections of the half spaces satisfying the linear inequalities. The numbered points are called extreme points, comer points, or vertices of this set. If the constraints are linear, only a finite number of vertices exist. 

As shown in Fig. 1.2, to solve this problem we need only analytical geometry. The constraints (1.29) restrict the solution to a convex polyhedron in the positive quadrant of the coordinate system. Any point of this region satisfies the inequalities (1.29), and hence corresponds to a feasible vector or feasible solution. The function (1.30) to be maximized is represented by its contour lines. For a particular value of z there exists a feasible solution if and only if the contour line intersects the region. Increasing the value of z the contour line moves upward, and the optimal solution is a vertex of the polyhedron (vertex C in this example), unless the contour line will include an entire segment of the boundary. In any case, however, the problem can be solved by evaluating and comparing the objective function at the vertices of the polyhedron. 

Remark 3 Note that the set V represents the values of y for which the resulting problem (6.2) is feasible with respect to x. In others words, V denotes the values of y for which there exists a feasible jc X for h(x,y) — Q,g(x,y) < 0. Then, the intersection ofy and V, Y n V, represents the projection of the feasible region of (2) onto they-space. 

For each vertex of the convex hull of the observed colors, we compute the feasible maps. We then intersect all these maps, as the actual illuminant must lie somewhere inside the intersection of these sets. Therefore, each vertex of the convex hull of the observed gamut gives us additional constraints to reduce the set of possible illuminants that may have produced the observed image. Let Ain be the computed intersection. 

This observed gamut is shown in Figure 6.15. If we compute the intersection of the feasible linear maps, we obtain... 

We hope the reader has been convinced that it is technically feasible to describe a photochemical reaction coordinate, from energy absorption to photoproduct formation, by means of methods that are available in standard quantum chemistry packages such as Gaussian (e.g., OPT = Conical). The conceptual problems that need to be understood in order to apply quantum chemistry to photochemistry problems relate mainly to the characterization of the conical intersection funnel. We hope that the theoretical discussion of these problems and the examples given in the last section can provide the information necessary for the reader to attempt such computations. 

Despite the intriguing features of the concept and the demonstrated experimental feasibility [58, 71-73], few applications to real samples have been demonstrated so far. For reasons of the inherent complexity of the many junctions to be controlled and the repetitive character of the experiment, the effects of any non-idealities in the system will rapidly accumulate (e.g. a loss of sample material is frequently observed at each intersection, thus degrading the analytical performance). In order to demonstrate the usefulness of SCCE for real analytical applications, von Heeren et al. have shown that its performance can be improved when fluid flow is suppressed by filling the device with an entangled polymer solution [74]. The same authors have also used SCCE microstructures for MECC separations of biological samples [59] (see Sect. 3.3). 

For an irreversible reaction of the type A —> B + C, iso-conversion lines show the region of operating points, where a certain conversion can be obtained (Fig. 6.10). The feasible region of operating conditions for the SMBR can then be obtained as the intersection between the separation triangle and the region below the iso-conver-sion line corresponding to the desired value. 

Figure 11.7 shows a ternary system consisting of components A, B and a compound C. The presence of the reaction confines the feasible composition to the reaction equilibrium surface. Projection of the intersection between the solubility... 

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