Singular Point Analysis

For counter-current RD columns with a single chemical reaction taking place, the attainable bottom compositions x can be interpreted as singular points of the mass balances of the reactive reboiler depicted in Fig. 5.13a 

For a given single reaction, the pole n is single and is always outside the valid composition space since there should be at least one V with a sign opposite to Vj. For the special case with % = 0, the stoichiometric lines become parallel and the pole 7t approaches infinity. 

From (5.25) and (5.26), the locations of the singular points in the composition space can be determined as follows. At the bottom, the condition for singular points for a component i from (5.25) is 

For an arbitrary reference component k, the condition for singular points is given 

For all singular points with and without chemical reaction, (5.31) gives the necessary conditions. They only depend on the vapor-liquid equilibria and on the reaction stiochiometry, and are completed by (5.29), which reflects the influence of the chemical reaction kinetics. For the distillate composition, one ends up with the corresponding singular point condition 

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The determination of the singular points appearing in these maps yields important information about the attainable bottom product compositions in real counter-current columns. However, as shown by Chadda et al. [3], both the distillate and the bottom product compositions can be better obtained as singular points of a reactive enriching flash cascade or a stripping flash cascade, respectively. As will be shown, singular point analysis can also provide valuable information about the role of interfacial mass-transfer resistances in RD processes. 

In Eq (25), the integration kernel function K x)= rr x is smooth everywhere except the singularity point (x = 0). In a numerical analysis, the integration has to be evaluated in discrete form over a grid with the mesh size h... 

The local stability and character of the singular points can be determined by the usual analysis of the eigenvalues of the Jacobian matrix... 

Statement 2. Substitution into the concentration dynamics (equations (8.2.12) and (8.2.13)) of the reaction rate K — K(Na, Nb), dependent on the current concentrations, changes the nature of the singular point. In particular, a centre (neutral stability) could be replaced by stable or unstable focus. This conclusion comes easily from the topological analysis its illustrations are well-developed in biophysics (see, e.g., a book by Bazikin [30]). 

Higher order terms can be obtained by writing the inner and outer solutions as expansions in powers of e and solving the sets of equations obtained by comparing coefficients. This enzymatic example is treated extensively in [73] and a connection with the theory of materials with memory is made in [82]. The essence of the singular perturbation analysis, as this method is called, is that there are two (or more in some extensions) time (or spatial) scales involved. If the initial point lies in the domain of attraction of steady states of the fast variables and these are unique and stable, the state of the system will rapidly pass to the stable manifold of the slow variables and, one might... 

Bogdanov, R. I. 1975 Versal deformations of a singular point on the plane in the case of zero eigenvalues. Functional Analysis Appl. 9,144-145. 

The analysis of characteristic roots A determines not only the local stability (in the small) of a singular point for system (81), but also the character of motion near it, i.e. its type. Let us investigate the linear equation (82). For this purpose we will consider the following cases separately. 

The second and third terms handle two-body collisions while the fourth term is related to the three-body collision. The term second-order in R in the Fock expansion is also known, and Myers, et a/. [12] have verified that this term eliminates the discontinuity in the local energy at the origin. This article also contains an analysis of the behavior of the wave function in the vicinity of these singular points. 

For a more generalized analysis of the qualitative influence of membranes on the singular points, the reactive membrane separation process is now considered with a nondiagonal [/c]-matrix. The condition for a kinetic arheotropes is given by... 

As demonstrated by means of residue curve analysis, selective mass transfer through a membrane has a significant effect on the location of the singular points of a batch reactive separation process. The singular points are shifted, and thereby the topology of the residue curve maps can change dramatically. Depending on the structure of the matrix of effective membrane mass transfer coefficients, the attainable product compositions are shifted to a desired or to an undesired direction. 

The determination of feasible products is very important for conceptual process design and for the evaluation of competing process variants. In this chapter, methods have been discussed to identify feasible products as singular points of residue curve maps (RCM). RCM-analysis is a tool which is well established for nonreactive and reactive distillation processes. Here, it is shown how RCM can also be used for reactive membrane separation processes. 

Our analysis is based on the following theorem [3]. Suppose that the steady-state equation (1) has a singular point at which... 

The maximum number of solutions of equation (1) is r+1 next to such a singular point. Moreover, all the local bifurcation diagrams of the function F can be determined by the analysis of the simpler polynomial function G. 

The problem is reduced to finding the phase trajectories of the equation system (104) at the (g, 0)-plane at different y values (dimensionless reaction rate) and values of p (relationship of the rates of relaxation g and heat removal at T = Tq). Dependence of the solution on x and in the physically justified ranges of their variation (tj > I at q qi ij< 1) turns out to be relatively weak. The authors of ref 234 applied the well-known method of analysis of specific trajectories changing at the bifurcational values of parameters [237], In the general case, the system of equations (104) has four singular points. The inflammation condition has the form... 

A perturbation analysis of Equations 35 and 36 about this singular point shows that the solutions whose initial conditions are close to P, Z, oscillate sinusoidally about this singular point. Hence, no constant solution is possible. The prey and predator populations continually oscillate and are out of phase with each other. When the predator predominates, the prey is reduced, which in turn causes the predator to die for lack of food, which allows the prey to proliferate for lack of predator, which then causes the predator to grow because of the prey available as a food supply, and so on. The interesting feature is that these oscillations continue indefinitely. 

A perturbation analysis about this singular point yields a second order linear ordinary differential equation whose characteristic equation has the roots Ai and A2 where... 

The analysis of linearized sytem thus allows, when conditions (1)—(3) are met, us to find the shape of phase trajectories in the vicinity of stationary (singular) points. A further, more thorough examination must answer the question what happens to trajectories escaping from the neighbourhood of an unstable stationary point (unstable node, saddle, unstable focus). In a case of non-linear systems such trajectories do not have to escape to infinity. The behaviour of trajectories nearby an unstable stationary point will be examined in further subchapters using the catastrophe theory methods. 

Based on the foregoing analysis, stoichiometry (defined as the point at which the numbers of anions and cations equal a simple ratio based on the chemistry of the crystal) is a singular point that occurs at a very specific oxygen partial pressure. This immediately begs the question if stoichiometry is a singular point in a partial pressure domain, then why are some oxides labeled stoichiometric and others nonstoichiometric To answer the question, examine Table 6.1 in which a range of stoichiometries and chemical stability domains for a number of oxides are listed. The deviation from stoichiometry, defined by Ax, where Ax is the difference between the... 

This is called the residue of /(z) and plays a very significant role in complex analysis. If a function contains several singular points within the contour C, the contour can be shrunken to a series of small circles around the singularities Zn, as shown in Fig. 13.6. The residue theorem states that the value of the contour integral is given by... 

The Hamilton-Jacobi formalism, on the other hand, only holds for KPP kinetics, but in contrast to singular perturbation analysis there is no need to assume either weak or smooth heterogeneities. The local velocity approach is based on the assumption that for weak and smooth heterogeneities the velocity of the front is given by the local value of the reaction rate r and the diffusion coefficient D at each spatial point, i.e., the front velocity coincides with the instantaneous Fisher velocity V 2y/r x)D x). In general, this simple-minded approach is not consistent with results from the other analytical methods or with numerical solutions. 

A positive value of the Jacobian (1.19) in the bifurcation point is one of the conditions for oscillatory regime. This is an equivalent of the assertion that the self-oscillations are most probable in the systems where crossed feedbacks have opposite signs. There are some other methods to identify the self-oscillatory systems direct application of Hopf theorem, analysis of type of singular points, Bendixson criterion, reduction of the equation system to Lienard equation, and others. One can find details, for example, in Chap. 4 of [53], or elsewhere [65]. 

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