Concave boundary

Capillary rarefaction. The continuous fluid phase filling the porous space between deformed bubbles has a common concave boundary with these bubbles. It follows that the local pressure Pi in the liquid phase is less than the pressure Pg in the gaseous phase and these variables are related by... 

Figure 5.31. The joining of section trajectories under minimum reflux for the direct split of (a) the ideal mixture, (b) the nonideal mixture with the convex boundary of the attraction region RegJ , and (c) the nonideal mixture with the concave boundary of the attraction region RegJjj.

Exploitation of Boundary Curvature A second approach to boundaiy crossing exploits boundaiy curvature in order to produce compositions in different distillation regions. When distillation boundaries exhibit extreme curvature, it may be possible to design a column such that the distillate and bottoms are on the same residue curve in one distillation region, while the feed (which is not required to lie on the column-composition profile) is in another distillation region. In order for such a column to meet material-balance constraints (i.e., bottom, distillate, feed on a straight hne), the feed must be located in a region where the boundary is concave. 

Peaking and Non-isothermal Polymerizations. Biesenberger a (3) have studied the theory of "thermal ignition" applied to chain addition polymerization and worked out computational and experimental cases for batch styrene polymerization with various catalysts. They define thermal ignition as the condition where the reaction temperature increases rapidly with time and the rate of increase in temperature also increases with time (concave upward curve). Their theory, computations, and experiments were for well stirred batch reactors with constant heat transfer coefficients. Their work is of interest for understanding the boundaries of stability for abnormal situations like catalyst mischarge or control malfunctions. In practice, however, the criterion for stability in low conversion... 

Remark 3 Convex and concave functions may not be continuous everywhere but the points of discontinuity have to be on the boundary of S. 

First, the boundary condition q = q0 at t = 0 is applicable. One can then test t0 by determining its value graphically, numerically, or by regression so that it creates a linear relationship when q versus ln(t + t0) is plotted. Thus, one is assuming an instantaneous pre-Elovichian process (Aharoni and Ungarish, 1976). However, studies have shown that a plot of q versus ln(t + t0) is usually concave toward the q axis. Moreover, to linearize such a plot indicates that experimental data are Elovichian when in fact they may not be. 

The feed must be sufficiently close to the distillation boundary, but on the concave side. 

A coexistence equilibrium corresponds to a solution (w, v) of (5.2), where u and V are positive in some region and satisfy the boundary conditions. By the results of the previous section, we expect that such a solution exists provided each single-population equilibrium is unstable to invasion by its rival. The stability of these equilibria can be considered (see [S9]), but we do not require the formalities for our brief treatment here. Both u and v must be positive and concave on the interval 0 < x < 1. There is a family of solutions of (5.2) given by... 

At this point, the utility of this property with respect to (P2) deserves attention. A careful look at P2 reveals that the shaded region in the projected space (for example, the X -X space) is exactly the projection on the X -X space of the feasible region of P2. The concave PFR projection defines the concentrations in segregated flow, and the interior is a convex combination of all boundary points created by the residence time distribution function. This gives a new interpretation to the residence time distribution as a convex combiner. For any convex objective function to be maximized, the solution to the segregated flow model will always lead to a boundary point of the AR. 

Thus on the surface g(x2, x ), the spinodal is the boundary which separates that part of the surface which is convex-convex from that which is convex-concave. 

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