Self-oscillation reaction

Kinetic study of the self-oscillating reaction observed in a potassium iodate-hydrogen peroxide-cysteine-sulfuric acid (acid medium) system was carried out [57], It is found that according to an adequate model the feedback mechanism is associated with autocatalytic reaction... 

The sensitivity analysis method was employed successfully for studying the mechatrisms of a great number of complex chemical transformations, such as reactions on combustion [33-36], pyrolysis [37,38], the self-oscillation reaction of Belousov-Zhabotinsky [39,40], atmospheric chemistry [41-44], as well as in many other fields of natural science, such as physics, economy, sociology, population science, etc. [7,45,46]. 

Sensitivity analysis method. To reduce the redundant steps in the kinetic model of chemical reactions, preference is mostly given to the method of sensitivity analysis, aheady described in Chapter 2. It was successfully applied in different areas of chemisty when studying the kinetic models of combustion reactions [21-28], cracking [28,29] atmospheric processes [30-33], self-oscillation reaction of Belousov-Zhabotinsky [52,53], and biological systems [54], In addition to the given referenees one can find solutions of similar problems in [55-59],... 

Recently there has been an increasing interest in self-oscillatory phenomena and also in formation of spatio-temporal structure, accompanied by the rapid development of theory concerning dynamics of such systems under nonlinear, nonequilibrium conditions. The discovery of model chemical reactions to produce self-oscillations and spatio-temporal structures has accelerated the studies on nonlinear dynamics in chemistry. The Belousov-Zhabotinskii(B-Z) reaction is the most famous among such types of oscillatory chemical reactions, and has been studied most frequently during the past couple of decades [1,2]. The B-Z reaction has attracted much interest from scientists with various discipline, because in this reaction, the rhythmic change between oxidation and reduction states can be easily observed in a test tube. As the reproducibility of the amplitude, period and some other experimental measures is rather high under a found condition, the mechanism of the B-Z reaction has been almost fully understood until now. The most important step in the induction of oscillations is the existence of auto-catalytic process in the reaction network. 

It is well known that self-oscillation theory concerns the branching of periodic solutions of a system of differential equations at an equilibrium point. From Poincare, Andronov [4] up to the classical paper by Hopf [12], [18], non-linear oscillators have been considered in many contexts. An example of the classical electrical non-oscillator of van der Pol can be found in the paper of Cartwright [7]. Poore and later Uppal [32] were the first researchers who applied the theory of nonlinear oscillators to an irreversible exothermic reaction A B in a CSTR. Afterwards, several examples of self-oscillation (Andronov-PoincarA Hopf bifurcation) have been studied in CSTR and tubular reactors. Another... 

The experiments and the simulation of CSTR models have revealed a complex dynamic behavior that can be predicted by the classical Andronov-Poincare-Hopf theory, including limit cycles, multiple limit cycles, quasi-periodic oscillations, transitions to chaotic dynamic and chaotic behavior. Examples of self-oscillation for reacting systems can be found in [4], [17], [18], [22], [23], [29], [30], [32], [33], [36]. The paper of Mankin and Hudson [17] where a CSTR with a simple reaction A B takes place, shows that it is possible to drive the reactor to chaos by perturbing the cooling temperature. In the paper by Perez, Font and Montava [22], it has been shown that a CSTR can be driven to chaos by perturbing the coolant flow rate. It has been also deduced, by means of numerical simulation, that periodic, quasi-periodic and chaotic behaviors can appear. 

More recently, the problem of self-oscillation and chaotic behavior of a CSTR with a control system has been considered in others papers and books [2], [3], [8], [9], [13], [14], [20], [21], [27]. In the previously cited papers, the control strategy varies from simple PID to robust asymptotic stabilization. In these papers, the transition from self-oscillating to chaotic behavior is investigated, showing that there are different routes to chaos from period doubling to the existence of a Shilnikov homoclinic orbit [25], [26]. It is interesting to remark that in an uncontrolled CSTR with a simple irreversible reaction A B it does not appear any homoclinic orbit with a saddle point. Consequently, Melnikov method cannot be applied to corroborate the existence of chaotic dynamic [34]. 

In the present chapter, steady state, self-oscillating and chaotic behavior of an exothermic CSTR without control and with PI control is considered. The mathematical models have been explained in part one, so it is possible to use a simplified model and a more complex model taking into account the presence of inert. When the reactor works without any control system, and with a simple first order irreversible reaction, it will be shown that there are intervals of the inlet flow temperature and concentration from which a small region or lobe can appears. This lobe is not a basin of attraction or a strange attractor. It represents a zone in the parameters-plane inlet stream flow temperature-concentration where the reactor has self-oscillating behavior, without any periodic external disturbance. 

From the study presented in this chapter, it has been demonstrated that a CSTR in which an exothermic first order irreversible reaction takes place, can work with steady-state, self-oscillating or chaotic dynamic. By using dimensionless variables, and taking into account an external periodic disturbance in the inlet stream temperature and coolant flow rate, it has been shown that chaotic dynamic may appear. This behavior has been analyzed from the Lyapunov exponents and the power spectrum. 

McKarnin, M. A., Schmidt, L. D., and Aris, R. (1988). Forced oscillations of a self-oscillating bimolecular surface reaction model. Proc. R. Soc., A417, 363-88. 

A comparative study was done by Kevrekidis and published as I. G. Kevrekidis, L. D. Schmidt, and R. Aris. Some common features of periodically forced reacting systems. Chem. Eng. Sci. 41,1263-1276 (1986). See also two papers by the same authors Resonance in periodically forced processes Chem. Eng. Sci. 41, 905-911 (1986) The stirred tank forced. Chem. Eng. Sci. 41,1549-1560 (1986). A full study of the Schmidt-Takoudis vacant site mechanism is to be found in M. A. McKamin, L. D. Schmidt, and R. Aris. Autonomous bifurcations of a simple bimolecular surface-reaction model. Proc. R. Soc. Lond. A 415,363-387 (1988) Forced oscillations of a self-oscillating bimolecular surface reaction model. Proc. R. Soc. Lond. A 415,363-388 (1988). 

L. Forced Oscillations of a Self-Oscillating Bimolecular Surface Reaction Model... 

FORCED OSCILLATIONS OF A SELF-OSCILLATING BIMOLECULAR SURFACE REACTION MODEL... 

Slin ko, M. G. Slin ko, M. M. 1978 Self-oscillations of heterogeneous catalytic reaction rates. Catal. Rev. Sci. Engng 17,119-153. 

Forced oscillations of a self-oscillating bimo-lecular surface reaction model (with M.A. McKamin and L.D. Schmidt). Proc. Roy. Soc. A417, 363-388 (1988). (Reprint L)... 

Self-oscillations have also been revealed for heterogeneous catalytic reactions. Hugo and Jakubith [7] and Wicke and co-workers [8] found self-oscillations for CO oxidation on platinum. In the period 1973-1975, M.G. Slinko and co-workers studied self-oscillations in hydrogen oxidation on nickel [9,10]. 

As far as the models accounting for these conceptions are concerned, their construction and investigation have just started. The development of these models is sure to be retarded by the absence of data on the detailed reaction mechanism and its parameters. The exception is ref. 147, where the authors construct an unsteady-state homogeneous-heterogeneous reaction model and analyze it with respect to the cyclohexane oxidation on zeolites. The study was aimed at the experimental interpretation of the self-oscillations found. The model constructed is in accordance with the law of mass action. 

In conclusion of the discussion of reaction dynamics in closed systems, it can be suggested that the principal problems here have been solved closed systems "have been closed . The case is different for open systems. Progress in their study has been extensive. A large number of publications are devoted to the analysis of various dynamic peculiarities (multiplicity of steady states, self-oscillations, stochastic self-oscillations) in various open systems. It can hardly be said that most problems here are completely clear. 

We apply these conditions to distinguish simple catalytic mechanisms ensuring self-oscillations of reaction rates (see Chap. 5). 

Investigations with the graphs of non-linear mechanisms had been stimulated by an actual problem of chemical kinetics to examine a complex dynamic behaviour. This problem was formulated as follows for what mechanisms or, for a given mechanism, in what region of the parameters can a multiplicity of steady-states and self-oscillations of the reaction rates be observed Neither of the above formalisms (of both enzyme kinetics and the steady-state reaction theory) could answer this question. Hence it was necessary to construct a mainly new formalism using bipartite graphs. It was this formalism that was elaborated in the 1970s. 

Self-oscillations of the rate in high-vacuum experiments have been found only for the platinum-catalyzed reactions (NO + CO) [61] and (CO + 02) [88]. 

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