Oscillatory model

The model proposed on the basis of the experimental work of Colussi et al. (CGYN) is not consistent with the RWJ model, in some details (165). The most important difference lies in the interpretation of the role of bromide ion and of bromine that could be formed from bromide ion. Roelofs and co-workers assumed that Br is important only in the complex-formation and they did not take into consideration the formation and further reactions of Br2 in their oscillatory model (164). Colussi et al. suggested that Br2 may form at low concentration levels and verified experimentally that bromine added to the system during the oscillation reduces the concentration of Co(III), and makes the periods shorter. A detailed molecular mechanism was proposed in agreement with the observations on the overall reaction and sub-systems (165). 

The model has acted as a useful stimulus. It is the only oscillatory model involving not more than two intermediates and having elementary reaction steps with only first- or second-order kinetics, and also satisfying sudi basic diranical reasonableness as non-native concentrations. However, a chemically satisfactory identification of X and Y with species known to be involved has never been attained it is now widely recognized that the conservative oscillations, to which this model in its isothermal form corresponds, cannot be the basis for those actually observed. ... 

As mentioned above, interfacial films exhibit non-Newtonian flow, which can be treated in the same manner as for dispersions and polymer solutions. The steady-state flow can be described using Bingham plastic models. The viscoelastic behavior can be treated using stress relaxation or strain relaxation (creep) models as well as dynamic (oscillatory) models. The Bingham-fluid model of interfacial rheological behavior (27) assumes the presence of a surface yield stress, cy, i.e.. 

In an oxidized compound where chain generation occurs in the reaction with O2, reactions (2) and (3) occur simultaneously. The trimolecular reaction will predominate when A3[RH] > k2. The activation energy is 2 2 Ai—h 221 kJ/mol, and 3 = A//3 + A 3 = 2Z)r h 570 + A 3, where 3 is the additional energy barrier caused by the concerted bond cleavage and, according to the oscillatory model, for this reaction... 

One of the topics much under discussion in the early years of kinetics research was the nature of the two tautomeric forms, or where the proton actually resided. Laar [35] had proposed a so-caUed oscillatory model, where a hydrogen atom vibrates continuously between the two possible positions. Other early observations include dielectric effects the polarity of the solvent could help release a proton from one position, thus making a transfer possible [1]. Although a tautomerization reaction is not an ionization (Section 1.1), an ionization step does play a crucial role, and may in many cases be the rate-determining step. 

The next problem to consider is how chaotic attractors evolve from tire steady state or oscillatory behaviour of chemical systems. There are, effectively, an infinite number of routes to chaos [25]. However, only some of tliese have been examined carefully. In tire simplest models tliey depend on a single control or bifurcation parameter. In more complicated models or in experimental systems, variations along a suitable curve in the control parameter space allow at least a partial observation of tliese well known routes. For chemical systems we describe period doubling, mixed-mode oscillations, intennittency, and tire quasi-periodic route to chaos. 

Rajamani and Herbst (loc. cit.) compared control of an experimental pilot-mill circuit using feedback and optimal control. Feedback control resulted in oscillatory behavior. Optimal control settled rapidly to the final value, although there was more noise in the results. A more complete model should give even better results. 

Assume that oscillatory excitation torque of Tq sin cot is applied to the system in Figure 9-14. By definition, when the excitation frequency coincides with the torsional natural frequency of the model, all torques will balance and the system will be in a state of resonance. 

Recently, Vigil and Willmore [67] have reported mean field and lattice gas studies of the oscillatory dynamics of a variant of the ZGB model. In this example oscillations are also introduced, allowing the reversible adsorption of inert species. Furthermore, Sander and Ghaisas [69] have very recently reported simulations for the oxidation of CO on Pt in the presence of two forms of oxygen, namely chemisorbed atomic O and oxidized metal surface. These species, which are expected to be present for reaction under atmospheric pressure, are relevant for the onset of oscillatory behavior [69]. 

The HS model exhibits a rich variety of spatio-temporal patterns. During the oscillatory behavior, if the simulation starts with an empty grid in the hexagonal phase the only possible event is CO adsorption. Consequently, when a certain CO coverage is reached, the surface starts to convert into the 1 X 1 phase. Oxygen cannot adsorb yet, due to the lack of empty sites. 

A lattice gas model with adsorbate-induced surface reconstructions has also very recently been proposed by Kusovkov et al. [73]. This model also exhibits a rich oscillatory behavior. 

The relative fluctuations in Monte Carlo simulations are of the order of magnitude where N is the total number of molecules in the simulation. The observed error in kinetic simulations is about 1-2% when lO molecules are used. In the computer calculations described by Schaad, the grids of the technique shown here are replaced by computer memory, so the capacity of the memory is one limit on the maximum number of molecules. Other programs for stochastic simulation make use of different routes of calculation, and the number of molecules is not a limitation. Enzyme kinetics and very complex oscillatory reactions have been modeled. These simulations are valuable for establishing whether a postulated kinetic scheme is reasonable, for examining the appearance of extrema or induction periods, applicability of the steady-state approximation, and so on. Even the manual method is useful for such purposes. 

An entirely different approach to the correlation problem is taken in the plasma model (Bohm and Pines 1953, Pines 1954, 1955), in which the electrons in a metal are approximated by a free-electron gas moving in a uniform positive background. According to classical discharge theory, such a plasma is characterized by an oscillatory behavior having a frequency... 

A striking feature of the effect of current on the CO oxidation oscillations is shown in Fig. 8.33. It can be seen that the frequency of oscillations is a linear function of the applied current. This holds not only for intrinsically oscillatory states but also for those which do not exhibit oscillations under open-circuit conditions, such as the ones shown on Fig. 8.31. This behaviour is consistent with earlier models developed to describe the oscillatory behaviour of Pt-catalyzed oxidations under atmospheric pressure conditions which are due to surface Pt02 formation35 as analyzed in detail elsewhere.33... 

R. W. Thatcher, A. A. Omon-Arancibia, and J. W. Dold, Oscillatory flame edge propagation, isolated flame tubes and stability in a non-premixed counterflow. Combust. Theory Model. 6 487-502,2002. 

How relevant are these phenomena First, many oscillating reactions exist and play an important role in living matter. Biochemical oscillations and also the inorganic oscillatory Belousov-Zhabotinsky system are very complex reaction networks. Oscillating surface reactions though are much simpler and so offer convenient model systems to investigate the realm of non-equilibrium reactions on a fundamental level. Secondly, as mentioned above, the conditions under which nonlinear effects such as those caused by autocatalytic steps lead to uncontrollable situations, which should be avoided in practice. Hence, some knowledge about the subject is desired. Finally, the application of forced oscillations in some reactions may lead to better performance in favorable situations for example, when a catalytic system alternates between conditions where the catalyst deactivates due to carbon deposition and conditions where this deposit is reacted away. 

A simple model was proposed on the basis of the experimental findings, in which the solution pH plays a key role for the oscillatory instability [51]. The theoretical... 

Two open reservoirs with cross-sectional areas Ai, A2 and liquid depths hi and h2 respectively are connected by a pipeline of length L and diameter D. The resulting liquid levels form a decaying oscillatory response to final steady state. Ramirez (1976) provides a detailed model derivation for this problem and shows how the problem is solved by analogue computation. The parameters are the same as those used by Ramirez. 

Using the accurate theoretical model, the physical origin of the intriguing oscillatory structure of the DCS can be traced. One striking feature that emerges from the scattering calculations is the occurrence of very... 

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