Oscillatory regimes

In Chap. 3 the problems of single-phase flow are considered. Detailed data on flows of incompressible fluid and gas in smooth and rough micro-channels are presented. The chapter focuses on the transition from laminar to turbulent flow, and the thermal effects that cause oscillatory regimes. 

The behavior of liquid flow in micro-tubes and channels depends not only on the absolute value of the viscosity but also on its dependence on temperature. The nonlinear character of this dependence is a source of an important phenomenon - hydrodynamic thermal explosion, which is a sharp change of flow parameters at small temperature disturbances due to viscous dissipation. This is accompanied by radical changes of flow characteristics. Bastanjian et al. (1965) showed that under certain conditions the steady-state flow cannot exist, and an oscillatory regime begins. 

Under certain conditions the energy dissipation may lead to an oscillatory regime of laminar flow in micro-channels. The oscillatory flow regime occurs in microchannels at Reynolds numbers less that Recr- In this case the existence of velocity flucmations does not indicate change from laminar to turbulent flow. 

For an incompressible fluid, the density variation with temperature is negligible compared to the viscosity variation. Hence, the viscosity variation is a function of temperature only and can be a cause of radical transformation of flow and transition from stable flow to the oscillatory regime. The critical Reynolds number also depends significantly on the specific heat, Prandtl number and micro-channel radius. For flow of high-viscosity fluids in micro-channels of tq < 10 m the critical Reynolds number is less than 2,300. In this case the oscillatory regime occurs at values of Re < 2,300. 

A passivating oxide is formed under sufficiently anodic potentials in HF, too. However, there are decisive differences to the case of alkaline and fluoride-free acidic electrolytes. For the latter electrolyte the steady-state current density prior to passivation is zero and it is below 1 mA cnT2 for alkaline ones, while it ranges from mA cm-2 to A cm-2 in HF. Furthermore, in HF silicon oxide formation does not lead to passivation, because the anodic oxide is readily etched in HF. This gives rise to an anodic I-V curve specific to HF, it shows two current maxima and two minima and an oscillatory regime, as for example shown in Fig. 4.7. 

The transition from a stable steady-state solution observed at large p to the oscillatory regime assumes the existence of the critical value of the parameter pc, which defines the point of the kinetic phase transition as p > pc, the fluctuations of the order parameter are suppressed and the standard chemical kinetics (the mean-field theory) could be safely used. However, if p < pc, these fluctuations are very large and begin to dominate the process. Strictly speaking, the region p pc at p > pc is also fluctuation-controlled one since here the fluctuations of the order parameter are abnormally high. 

The solution of the first kind is stable and arises as the limit, t —> oo, of the non-stationary kinetic equations. Contrary, the solution of the second kind is unstable, i.e., the solution of non-stationary kinetic equations oscillates periodically in time. The joint density of similar particles remains monotonously increasing with coordinate r, unlike that for dissimilar particles. The autowave motion observed could be classified as the non-linear standing waves. Note however, that by nature these waves are not standing waves of concentrations in a real 3d space, but these are more the waves of the joint correlation functions, whose oscillation period does not coincide with that for concentrations. Speaking of the auto-oscillatory regime, we mean first of all the asymptotic solution, as t —> oo. For small t the transient regime holds depending on the initial conditions. 

The largest deviation is observed for d = 1 - see Fig. 8.13. Here irrespectively of the value of parameter k the oscillatory regime arises rather rapidly... 

These models consider the mechanisms of formation of oscillations a mechanism involving the phase transition of planes Pt(100) (hex) (lxl) and a mechanism with the formation of surface oxides Pd(l 10). The models demonstrate the oscillations of the rate of C02 formation and the concentrations of adsorbed reactants. These oscillations are accompanied by various wave processes on the lattice that models single crystalline surfaces. The effects of the size of the model lattice and the intensity of COads diffusion on the synchronization and the form of oscillations and surface waves are studied. It was shown that it is possible to obtain a wide spectrum of chemical waves (cellular and turbulent structures and spiral and ellipsoid waves) using the lattice models developed [283], Also, the influence of the internal parameters on the shapes of surface concentration waves obtained in simulations under the limited surface diffusion intensity conditions has been studied [284], The hysteresis in oscillatory behavior has been found under step-by-step variation of oxygen partial pressure. Two different oscillatory regimes could exist at one and the same parameters of the reaction. The parameters of oscillations (amplitude, period, and the... 

Fig. 37. Gray-scale representation of a calculated spatio-temporal evolution of the potential at a ring electrode (top) and time series of the global current (bottom) in the oscillatory regime of the N-NDR oscillator. (The calculation was done for reduction currents hence, the largest current is in the peak of the oscillation.)...

For the CO oxidation reaction on Pt, in particular, several fundamental questions still remain unanswered. Quite recently doubts and questions have been raised even about the existence of an oscillatory regime for this reaction system. Cutlip and Kenney [l2] have been unable to observe any oscillations during the oxidation of CO over a 0.5% Pt/y-Al203 catalyst in a recycle reactor. Their study, however, utilized low feed compositions of CO (0.5-3%)... 

The Oscillatory Behavior. We have so far focused our attention on three questions (a) Does an oscillatory regime exist for this reaction system (b) What is the effect of hydrocarbon impurities and (c) What is the effect of catalyst deactivation on the dynamic behavior ... 

Operating in the oscillatory regime produced higher yields than did the steady state for the CO oxidation reaction over Pt (22) and Pd (23), and for the endothermic methylamine decomposition over Pt wires (24). In these cases comparison between oscillatory and steady states was possible because the unstable states coexisted with the respective stable states of the systems. 

Multi-product batch plant Two cases (a) minimization of both investment and number of different sizes for each unit operation, and (b) minimization of investment, number of different sizes for each unit operation and number of campaigns to reach steady state or oscillatory regime. Multi-Objective GA (MOGA) Both design and retrofit problems were smdied. Dedieu et al. (2003)... 

It is interesting to note that in an oscillatory regime the concentration dependence of Flos is dominated by the decay length d2 in the exponent (see Equations 5.216 and 5.217). Roughly speaking, for a given distance h, the oscillatory disjoining pressure rtoj, increases five times when ( ) is increased 10%. ... 

Equation (16) was originally derived to model the reduction of In " from SCN solution on the HMDE. The bifurcation behavior of this system is summarized in the two-parameter bifurcation diagram in Fig. 29. Most remarkably, the two distinct MMO sequences of the model also show up in the experiment. Farey sequences were observed close to the Hopf bifurcation at low values of the series resistance, whereas at the high resistance end of the oscillatory regime, periodic-chaotic mixed-mode sequences were found. Owing to this good agreement of the bifurcation... 

The experimental results are compiled in Section III.2, which starts with a short description of the methods used to visualize (potential) patterns at electrode surfaces. First wave phenomena in the bistable regime and then in the oscillatory regime are reviewed, with the focal point being on how they fit into the theoretical picture developed in Section III.l. 

Otterstedt et a/.also studied waves in the oscillatory regime during Co dissolution. The oscillations possess a relaxationhke character, which is typical for oscillations between the active and the passive state of metal dissolution reactions. They are characterized by long, quasi-stationary periods of vanishing current density, followed by a sharp... 

Oxidation of methane in the presence of such a binary oxide-metal catalyst proceeds in an oscillatory regime, and both temperature and concentration oscillations take place. Oscillations arise at the temperature at which the rate of reaction over the oxide component becomes noticeable ( 500°C). As temperature increases, the oscillation amplitude passes through a maximum. The oscillatory behavior disappears when complete conversion of oxygen is reached. In other words, the range of temperatures in which the oscillations are observed covers the range of oxygen conversions from 0 to 100%. 

The cooperative effects observed during methane oxidation over a binary oxide-metal system are due to the formation of active intermediates (free methyl radicals) over the oxide component, their escape from the grains of oxide, and transformation into the final products (including CO and H2) over the metal component, which proceeds in a non-steady-state oscillatory regime. 

Reactions on metals, including many oxidation processes, are known to proceed in a way very different from stochastic collision types, which can be described by mass action (or acting surfaces ). The number of systems in which collective effects or topochemical type processes (via nucleation and growth of nuclei) are proved to determine the kinetic behavior is increasing. Despite the extensive literature on reactions in oscillatory regimes and spatially-structured reactions on surfaces (Gorodetskii et al., 2005 Latkin et al., 2003 Peskov et al., 2003), such facts have not yet found an adequate reflection in the area under consideration. 

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