Oscillating behaviour

Figure 4.12 Typical oscillating behaviour of results obtained with the MP method...

Critical effects in CO oxidation over Pt catalysts were obtained [33, 34, 63-85] in various catalytic systems over wires, foils and gauzes, on single pellets and fixed beds, in isothermal and adiabatic reactors (differential and integral). The literature also reported the oscillating behaviour of the homogeneous oxidation of CO [86, 87]. 

Transient behaviour of lyotropic MCLCPs is similar to that of thermotropic MCLCPs as discussed in Sect. 15.7 (Fig. 15.47). An example is given in Fig. 16.35 for PpPTA in sulphuric acid (Doppert and Picken, 1987). It shows damped oscillating behaviour, which is in contradistinction to conventional polymers, but which is also found as a result of simple rheological models, like the Jeffreys model (see Chap. 15). This behaviour turns out to be a general feature of lyotropic MCLCPs. The oscillatory behaviour is easier to measure for lyotropic than for thermotropic systems, where it is less pronounced. 

For recording, a pencil lead protected in a brass tube from the shock of the explosion was used, and it recorded by sliding on the paper. To know about the effects of the swings on the recording pen, we observed the oscillating behaviour of the mortar with and without a pen. The results are shown in Fig.3.114 and Fig.3.115. We could see that the periodic time of the oscillation was 3.24 sec. under both conditions with and without a pen. In the case of the test without pen, it was found that the results obeyed the general reducing oscillation curve, where frictional resistance should be proportional to velocity. The equation was,... 

If l) t) shows regular behaviour, for instance periodicities, then 6 t) shows an oscillating behaviour. If 0 t) decays, this would be an indication that l) t) develops complicated structures by looking less and less like its starting configuration. For the wave function (1.4.3) it is easy to calculate 0 t). The result is ... 

In the present chapter current relaxation theories will be described first both damping of harmonically generated disturbances and relaxations to transient perturbations. Thereafter, experiments are described, based on the damping of capillary and longitudinal waves, oscillation behaviour of bubbles. Also transient relaxations with pendent drop and drop and bubble pressure measurements are shown. Finally, applications to different interfaces, using surfactants, surfactant mixtures, polymers and polymer/surfactant mixtures are discussed. 

Some Hopf bifiircations have been detected with that kinetic model (9-11), Fig.3 reports the trend of N2O outlet concentration as a function of time, resulted from the model simulation performed with a set of kinetic constant, ki, k2, ks, which expects an oscillating behaviour. The boundary condition for N2O inlet concentration has been chosen equal to 300 ppm. 

The results of the simulation carried out with Ml mechanism shows that it can successfully expect an oscillating behaviour. Fig.3 shows also that N2O and O2 outlet concentration... 

In propane oxidation, only weak kinetic oscillations were revealed by the spectrometer signal of COn at reactor outlet. The combustion of methane under similar operating conditions showed no oscillating behaviour. 

The mechanism of self-sustained oscillations is not completely understood at present, despite the remarkable number of studies reported in the literature [11, 16]. It has been shown that in the CO-O2 reaction over Pt catalysts, oscillations occur in the transition region between the high and low activity state, corresponding to predominantly CO- or O- covered surface [11]. The existence of such a region of sharp transition appears a basic requisite for oscillating behaviour to take place. This statement applies also to the Rh/AUOj catalyst of this study, for which a similar transition region in the CO oxidation has been revealed by kinetic experiments. 

Fig. 1. The principle of pumjvprobe spectroscopy by means of transient two-photon ionization A first fs-laser pulse electronically excites the particle into an ensemble of vibrational states creating a wave packet. Its temporal evolution is probed by a second probe pulse, which ionizes the excited particle as a function of the time-dependent Franck Condon-window (a) shows the principle for a bound-bound transition, where the oscillative behaviour of the wave packet will appear (b) shows it for a bound-free transition exhibiting the exponential decay of the fragmentizing particle, and (c) shows the process across a predissociated state, where the oscillating particle progressively leads into a fragmentation channel.

With an autocatalytic reaction the product acts as a catalyst for the reaction, which is thus accelerated. This can be true, for example, for decomposition processes (vid. Sect. 2.2). Autocatalytic reactions may exhibit oscillating behaviour. 

A distinct improvement of the overall oscillation behaviour becomes evident only after having decreased the bearing distance on the rotor. 

There is also P(to) = toa(to) where a(co) is the frequency dependent conductivity (see Chapter 25). In the case of a low conductivity ionic solid, P(co) is typically of an oscillating behaviour (a peak at = coq, more or less damped, see Fig. 11.2) and there are only very rare diffusive events which contribute to P(co = 0). In a liquid, the spectrum is centred at (u = 0, since all the particles diffuse. When a liquid becomes (more) viscous a pseudo-oscillating behaviour may be observed (c), while the oscillator damping in a superionic conductor may decrease the difference between the time of flight between two sites and the time of oscillation on a site (Fig. 30.2), leading to a quasi-liquid state ". In order to simplify the model, either the diffusive or the oscillatory behaviour is assumed to be predominant. The choice may depend on the supposition of relaxation time... 

Rigorous analysis of stability was performed by means of Liapunov s theorem, which was written concisely as the inequalities Eq. (1.19). This analysis showed that the initial and end points A and C are stable nodes while the middle B is usually an instable saddle point. This saddle point can sometimes transform into an unstable node or focus, thus allowing for birth of a limit cycle and self-oscillation behaviour. Unexpectedly, the mathematic analysis showed also the possibility of a stable intermediate state in some narrow region of parameters values (such that the maximum in Fig. 5.15 is not very far from the straight line W = ). This result differs from the intuitive physical considerations above. 

From the results of the attainable performance calculation it can be concluded that the structure 142 can achieve the best performance, if input saturation is not a problem. This structure allows very fast rise times and decoupling of the closed-loop system can be achieved. It can be assumed that also a decentralized controller achieves good performance. Also the structure 143 is attractive, however the third output shows a slightly oscillating behaviour. The other structures give slower responses and stronger couplings. 

The oscillation behaviour of the interfacial tension can be described by the complex elasticity E(ico) defined by the following equation... 

A continuous increase in the solidification velocity of a liquid diluted alloy gives rise thus to an oscillating behaviour in the interface stability. It is interesting also to note that the width of the four regions depends upon dopant concentration. An... 

Hong avoided the oscillating behaviour of the stresses ahead of an interface crack by using the isotropic interface layer, Wu, among others, investigated its influence on the interlaminar stress distribution. 

---