Oscillation behavior

Computer simulation of these equations is shown in Fig. 24-25. Real systems do have this type of oscillating behavior, but frequencies and amplitudes are erratic. 

For strongly structured microemulsions, g is negative, and the structure functions show a peak at nonzero wavevector q. As long as g < 2 /ca, inverse Fourier transform of S q) still reveals that the water-water correlation functions oscillate rather than decay monotonically. The lines in phase space where this oscillating behavior sets in are usually referred to as disorder lines, and those where the maximum of S q) moves away from zero as Lifshitz lines. ... 

Lattice models for bulk mixtures have mostly been designed to describe features which are characteristic of systems with low amphiphile content. In particular, models for ternary oil/water/amphiphile systems are challenged to reproduce the reduction of the interfacial tension between water and oil in the presence of amphiphiles, and the existence of a structured disordered phase (a microemulsion) which coexists with an oil-rich and a water-rich phase. We recall that a structured phase is one in which correlation functions show oscillating behavior. Ordered lamellar phases have also been studied, but they are much more influenced by lattice artefacts here than in the case of the chain models. 

As a consequence of implicit mass conservation, the gas-dynamic conservation equations, expressed in Lagrangean form, can describe contact discontinuities. To prevent oscillating behavior in places where shock phenomena are resolved in the... 

For chaotic or oscillating behavior the mechanism must contain an autocatalytic step ... 

A thorough insight into the comparative photoelectrochemical-photocorrosion behavior of CdX crystals has been motivated by the study of an unusual phenomenon consisting of oscillation of photocurrent with a period of about 1 Hz, which was observed at an n-type CdTe semiconductor electrode in a cesium sulfide solution [83], The oscillating behavior lasted for about 2 h and could be explained by the existence of a Te layer of variable width. The dependence of the oscillation features on potential, temperature, and light intensity was reported. Most striking was the non-linear behavior of the system as a function of light intensity. A comparison of CdTe to other related systems (CdS, CdSe) and solution compositions was performed. 

Boure, J. A., 1966, The Oscillator Behavior of Heated Channels, Part I II, French Rep. CEA-R 3049, Grenoble, France. (6)... 

Boure, J. A., and A. Ihaila, 1967, The Oscillator Behavior o f Heated Channels, EURATOM Rep., Proc. 

Kim, K. I., Ohtani, H. and Uehara, Y., Experimental study on oscillating behavior in a small-scale compartment fire, Fire Safety J., 1993, 20, 377-84. 

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. 

A much more interesting case of chaotic dynamics of the reactor can be obtained from the study of the self-oscillating behavior. Consider the simplified mathematical model (8) and suppose that the reactor is in steady state with a reactant concentration of Prom Eq.(8) the equilibrium point [x, y ] can be deduced as follows ... 

Eq.(18) has two complex roots with real part equal to zero, and consequently it is possible to deduce a relation between x and y. By substituting Eq.(18) into Eq.(12) one obtains a parametric equation xo = fi y )- Eliminating xo between xo = fi y ) and Eq.(13), the parametric equations of self-oscillating behavior are deduced ... 

Another interesting aspect of the self-oscillating behavior is the following one. If the values of xo,yo) are inside the lobe, an external periodic disturbance of the coolant flow rate can drive the reactor to chaotic behavior. 

The values of Km and T2d from Eq.(36) can be obtained from the transfer function of the linearized model at the equilibrium point, applying conventional methods from the linear control theory (see [1]). In order to investigate the self-oscillating behavior, one can determine the linearized system at the equilibrium point, and the corresponding complex eigenvalues with zero real part, when the parameters Km and of the PI controller are varied. For example, taking into account Eq.(34), the Jacobian matrix of the linearized system at dimensionless set point temperature xs is the following ... 

Assuming that S > 0, S 4 > 0 and S1S2 — S3 >0, the condition of self-oscillating behavior is given by the equation ... 

Figure 13 shows the variation of Km for various values of T2d and the corresponding frequencies of self-oscillation. Figure 14 shows the oscillation behavior of the reactor with the value Xg = 0.0398 and T2d = 0.5, Ktd = 19.6. 

Note that Figure 13 can be used to compare the parameters of the controller when they are obtained from the Ziegler-Nichols or Cohen-Coom rules. On the other hand, at Figure 14 it can be observed that the outlet dimensionless flow rate and the reactor volume reaches the steady state whereas the dimensionless reactor temperature remains in self-oscillation. The knowledge of the self-oscillation regime in a CSTR is important, both from theoretical and experimental point of view, because there is experimental evidence that the self-oscillation behavior can be useful in an industrial environment. 

Rauhut and coworkers were the first to obtain rate constants from emission kinetic studies and to verify the dependence of kobsi and kobsi on the concentration of the base catalyst and on hydrogen peroxide, respectively. Schowen and coworkers , using TCPO, H2O2 and DPA, with triethylamine as catalyst, observed an oscillatory behavior in emission experiments and proposed a mechanism involving the formation of two HEIs (involved in parallel chemiluminescent reactions) to explain it. Other authors have also observed a similar oscillating behavior but have explained it as a complex... 

A simple application of the multiple-oscillator theory is to fit measured reflectance data for MgO in the Reststrahlen region. In Section 9.1 we considered the electronic excitations of MgO, whereas we now turn our attention to its lattice vibrations. A glance at the far-infrared reflectance spectrum of MgO in Fig. 9.7 shows that it does not completely exhibit one-oscillator behavior there is an additional shoulder on the high-frequency side of the main reflectance peak, which signals a weaker, but still appreciable, second oscillator. The solid curves in Fig. 9.7 show the results of a two-oscillator calculation using (9.25) the reflectance data were taken from Jasperse et al. (1966), who give the following parameters for MgO at 295°K ... 

Figure 8.4 Adiabatic CSTR With the solid straight line, there is no clearly defined working point, which results in an oscillating behavior. With the dashed line, there are multiple solutions (points A, B, C).

As it is known from the literature, the integrated induced density of the local field oscillating behavior, tends to a constant value Q resulting from the incomplete screening of Si [126]. Q is calculated from the static dielectric constant es ... 

When limiting our attention to purely spherical pre-stress we find analytical forms for the solutions of Bessel or Modified Bessel equations in dependence on the coupling coefficient Ksf. The obtained density profiles may show an oscillating behavior we prove the conjecture that oscillating profiles are unstable as well as the non-oscillating ones which correspond to sufficiently high absolute values of Ksf. 

Photomodulation amperometry of the DHAs (21-24)a is shown in Figure 20. Because of the increased acceptor strength in 22b, the 2,4-dinitrophenyl derivative 22a exhibits oscillating behavior at an electrode potential less negative than that required for the constitutional isomer 21a (Figure 20 21a, 22a). On the other hand, the 4-cyanophenyl derivative 23a displays increased sensitivity, which seems to be the result of the higher quantum yield of the photoreaction from 23a to 23b (Figure... 

I. Atwater, C. M. Dawson, A. M. Scott, G. Eddlestone, and E. Rojas The nature of the oscillating behavior in electrical activity from pancreatic jS-cell. Harm. Metab. Res. 1980,10 100-107. 

However, the reason of the appearance of negative impedance is always a chemical/electrochemical process. In most cases the blocking (inactivation) of the electrode (metal) surface is the pivotal (autoinhibition) step in the mechanism behind the emergence of the oscillating behavior. The blocking can be a consequence of adsorption of ions or molecules, chemisorption of molecular fragments, deposition of metals, salts or other compounds, formation of oxide layer etc. In all cases several coupled, consecutive, and simultaneous processes occur. The oscillating behavior appears only at a certain set of parameters (concentrations of the electro-chemically active species, the nature and the concen-... 

The latter assumption also relates to the strong nonlinearity of elementary steps in the scheme. The solution of the resulted kinetic equations for the overall scheme leads indeed to auto oscillating behavior of the system within the certain parameter range. 

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