Phase plane discontinuous

In the ascending bubble problem (Example 8), the requirement that the pressure at the surface should be positive rules out the dramatic behavior near the singular curve ( = but, by suitably defining the value of the derivative at the discontinuity, solution paths of perilous stability can be found4 (Cf. [312]). In Fig. 13, the broken line A = 0 is manifestly unstable. However, the broken line B = 0 can be reached in a finite time by a solution starting above A = 0 in the lower part of the phase plane and below A - 0 in the upper part. If the value at the singularity, B = 0, which is otherwise undefined, is correctly chosen, the solution turns and follows the path B = 0. In practice, it falls victim to the peril, even when stable. 

We consider a reactor with a bed of solid catalyst moving in the direction opposite to the reacting fluid. The assumptions are that the reaction is irreversible and that adsorption equilibrium is maintained everywhere in the reactor. It is shown that discontinuous behavior may occur. The conditions necessary and sufficient for the development of the internal discontinuities are derived. We also develop a geometric construction useful in classification, analysis and prediction of discontinuous behavior. This construction is based on the study of the topological structure of the phase plane of the reactor and its modification, the input-output space. 

Having established the geometric properties of shocks and the structure of the phase plane, including appropriate functions needed in the analysis, we may look at the discontinuous solutions as a whole. 

In any attempt to establish a theory of surface tension, it is very natural to assume, as a first approximation, that the transition from the liquid phase to the gas phase occurs discontinuously at a mathematical plane. As stated in Section V, Fowler introduced this assumption and obtained Eq. V.9 for the surface tension. The nature of this assumption will now be studied more closely by calculating the normal pressure and the tangential pressure />x.14-21 The assumption is stated as... 

In contrast to closed discontinuous (batch) cultivation systems, in which the final concentrations depend on the initial concentrations, in an open system in continuous culture the final concentration is largely independently of the initial states, which phenomenon is called equifinality. An appropriate plot for demonstration of equifinality is the plot of so-called trajectories in the phase plane, as shown in the next section and in Fig. 6.13. 

The aspect ratio E/IE refers to the shape of the particles in the discontinuous phase. It is the average dimension of this phase parallel to the plane of the film E divided by the average dimension perpendicular to the film W. Plates in the plane of the film would have a high aspect ratio. Spheres or cubes would have an aspect ratio equal to 1. 

Fig. 54. Schematic phase diagrams for wetting and capillary condensation in the plane of variables temperature and chemical potential difference, (a) Refers to a case in which the semi-infinite system at gas-liquid condensation (ftaKX — d = 0) undergoes a second-order wetting transition at T = 7V The dash-dotted curves show the first-order (gas-liquid) capillary condensation at p = jt(I), T) which ends at a capillary critical point T v, for two choices of the thickness D. For all finite D the wetting transition then is rounded off. (b), (c) refer to a case where a first-order wetting transition exists, which means that ps remains finite as T - T and there jumps discontinuous towards infinity. Then for /iaKX - /i > 0 a transition may occur during which the thickness of the layer condensed at the wall(s) jumps from a small value to a larger value ( prewelting ). For thick capillaries, this transition also exists (c) but not for thin capillaries because then /Jcnn - (D,T) simply is loo large.

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