Region vertex

Fig. XV-11. Electron micrograph of a freeze fracture replica of a region inside a mul-tivesicular liposome. Note the tetrahedral coordination nearly every vertex has three edges, and each face is connected to three others. The average number of edges per face is 5.1. (From Ref. 77.)...

The vertex of the complete separation region, evaluated for k = 0.5 s is characterized by Yu - -86 and Ym 4 -75. The corresponding minimum value of Yi is Yi -7.135 -H 3.86 - 4.75 = 6.245, which is still higher than the critical value 7 = 6.045. We conclude that, considering negligible mass transfer, within the whole separation region built under these conditions, the Yi value does not affect the SMB performance. It should be pointed out that the presence of mass transfer resistance can influence the critical value for 7 and the form of the separation region [32]. 

The vertex of a separation region points out the better operating conditions, since it is the point where the purity criteria are fulfilled with a higher feed flow rate (and so lower eluent flow rate). Hence, in the operating conditions specified by the vertex point, both solvent consumption and adsorbent productivity are optimized. Comparing the vertex points obtained for the two values of mass transfer coefficient, we conclude that the mass transfer resistance influences the better SMB operating conditions. Moreover, this influence is emphasized when a higher purity requirement is desired [28]. 

The four-color problem (prove that four colors are sufficient to color any map in the plane or on a sphere so that no two adjacent regions have the same color) is another problem in which it is possible to associate a vertex with each region of a map and join the vertices if their corresponding regions have a com-/) mon boundary that is more than one point. Graph... 

The fact that the extremum of a linear program always occurs at a vertex of the feasible region is the single most important property of linear programs. It is true for any number of variables (i.e., more than two dimensions) and forms the basis for the simplex method for solving linear programs (not to be confused with the simplex method discussed in Section 6.1.4). 

SLP convergence is much slower, however, when the point it is converging toward is not a vertex. To illustrate, we replace the objective of the example with x + 2y. This rotates the objective contour counterclockwise, so when it is shifted upward, the optimum is at x = (2.2, 4.4), where only one constraint, jc2 + y2 < 25, is active. Because the number of degrees of freedom at x is 2 — 1 = 1, this point is not a vertex. Figure 8.10 shows the feasible region of the SLP subproblem starting at (2, 5), using step bounds of 1.0 for both Ax and Ay. 

Since hydrogen ions do not take part in reaction (10) the unproton-ated dimer occurs in the same pH region as [HV04]2 (Fig. 3). In the solid state the dimer consists of two V04 tetrahedra sharing a vertex (Fig. 4) and in solution it is assumed to have the same structure (30,... 

As the feed concentration increases the basis of the triangle and the position of the vertex shifts downwards to the left. The complete separation region becomes narrower and concomitantly also less robust. This implies that when the concentration of the feed is increased, the flow rate ratios in Sects. 2 and 3, as well as the difference (m3 - m2) decrease in consequence (see also Fig. 5). Material balances show that the maximum productivity increases with the feed concentration and asymptotically approaches a maximum value. Hence, when feed concentration increases, productivity improves, but robustness becomes poorer. So the optimum value for the feed concentration of an SMB tends to be defined by a compromise between the opposite needs of productivity and robustness [25,27]. 

Many mixtures exhibit edge effects such that the behavior of the formulation shows drastic changes when one or more of the components is omitted from the mixture [Anderson and McLean (1974)]. Thus, if simple empirical models such as Equations 12.90 and 12.91 are to be used to model the system, it is often best to work in regions that have all components present. Such systems can be prepared with so-called pseudo-components [Cornell (1990)] as shown in the lower two panels of Figure 12.33. The pseudo-components correspond to the vertexes in these designs and are seen to be mixtures that are relatively rich in one of the components. In practice, the pseudo-components can be prepared first, and then the other mixtures in the design can be prepared from these pseudo-components. 

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. 

A primary disadvantage of the flexibility index is the need to solve 2 (N)LPs corresponding to the vertex directions when the critical point must be a vertex of the hyperrectangle 0(F). To decrease the required computational effort, Saboo et al. (1985) define a different index, the resilience index (RI), to measure the size of feasible region R. 

The solution 0C of max-min problem (16) is the critical point which limits the RI that is, it is the point where the largest inscribed polytope meets the feasible region R. In general, 8C need not correspond to a vertex of the polytope (e.g., for some nonconvex feasible regions R). However, to date no general algorithm has been developed to find nonvertex critical points which limit the RI. 

A sufficient condition that the RI be determined by a vertex critical point is that the feasible region R be convex. (Of course, a special case of this is when all the feasibility constraints are linear see Section III,B.) Unfortunately, when flow rates or heat transfer coefficients are included in the uncertainty range, the feasible region can be nonconvex (see Examples 1 and 2 and Section III,C,3). Thus, current algorithms for calculating the RI are limited to temperature uncertainties only. 

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