Stationary points local characteristics

For a distillation process not only the stationary point type, but also the behavior of the residue curve in the vicinity of the stationary point is of special importance. If the residue curves in the vicinity of the specific point are tangent to any straight line (singular line) (Fig. 1.4a, b, d, e, g, h), the location of this straight line is of great importance. A special point type and behavior of residue curves in its vicinity are called stationary point local characteristics. 

In principle, to study the local stability of a stationary point from a linear approximation is not difficult. Some difficulties are met only in those cases where the real parts of characteristic roots are equal to zero. More complicated is the study of its global stability (in the large) either in a particular preset region or throughout the whole phase space. In most cases the global stability can be proved by using the properly selected Lyapunov function (a so-called second Lyapunov method). Let us consider the function V(c) having first-order partial derivatives dY/dCf. The expression... 

In Safrit Westerberg (1997), a heuristic algorithm is based on the information about local characteristics of stationary points, and checked by the authors at large amounts of three-component mixtures and at some four-component mixtures. This algorithm takes into consideration azeotropes formed by any number of components. 

As mentioned above, the Newton process is a local procedure, i.e. convergence occurs only if the initial guess x° is chosen sufficiently closed to a stationary point (see theorem 3). When a minimization problem is under consideration, the applicability of the Newton method can essentially be extended if the characteristic of a minimizer is... 

In the vicinity of the true stationary point, the fixed-point iterations based on (12.3.2) will converge rapidly to the true optimizer with the characteristic second-order convergence rate of Newton s method discussed in Section 11.5.2. Further away from the optimizer, the Newton step may not necessarily lead us towards the tme optimizer of the function since the stationary point of the local surface may no longer resemble the true optimizer of the fimction. In such cases, we should not apply the Newton step (12.3.2) directly but instead determine our step based on some other strategy. Such a strategy is presented in Section 12.3.2. 

In this section, microdisc electrodes will be discussed since the disc is the most important geometry for microelectrodes (see Sect. 2.7). Note that discs are not uniformly accessible electrodes so the mass flux is not the same at different points of the electrode surface. For non-reversible processes, the applied potential controls the rate constant but not the surface concentrations, since these are defined by the local balance of electron transfer rates and mass transport rates at each point of the surface. This local balance is characteristic of a particular electrode geometry and will evolve along the voltammetric response. For this reason, it is difficult (if not impossible) to find analytical rigorous expressions for the current analogous to that presented above for spherical electrodes. To deal with this complex situation, different numerical or semi-analytical approaches have been followed [19-25]. The expression most employed for analyzing stationary responses at disc microelectrodes was derived by Oldham [20], and takes the following form when equal diffusion coefficients are assumed ... 

When the system is out of full thermodynamic equilibrium, its non-equilibrium state may be characteristic of it with gradients of some parameters and, therefore, with matter and/or energy flows. The description of the spontaneous evolution of the system via non equilibrium states and prediction of the properties of the system at, e.g., dynamic equilibrium is the subject of thermodynamics of irreversible (non-equilibrium) processes. The typical purposes here are to predict the presence of solitary or multiple local stationary states of the system, to analyze their properties and, in particular, stability. It is important that the potential instability of the open system far from thermodynamic equilibrium, in its dynamic equilibrium may result sometimes in the formation of specific rather organized dissipative structures as the final point of the evolution, while traditional classical thermodynamics does not describe such structures at all. The highly organized entities of this type are living organisms. 

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