Self oscillations

Zaikin A N and Zhabotinsky A M 1970 Concentration wave propagation in two-dimensional liquid-phase self-oscillating system Nature 225 535-7... 

The simplest arrangement for a linear accelerator is shown in Fig. 5. Here a single source, either a self-oscillating magnetron or klystron amplifier with appropriate drive stages, feeds power into a single length of accelerator wave-... 

Proskuryakov, K. N., 1965, Self Oscillation in a Single Steam Generating Duct, Thermal Eng. (USSR) 12(3) 96 100. (5)... 

The next question is, is the efficiency being lost in the switch If so, there could be many reasons for that. A switch could be lossy simply because its drive is inadequate. Early self-oscillating converters (ringing choke oscillators) were extremely lossy because the drive would slowly droop to the point where it just couldn t sustain itself and then the switch would turn OFF. Modern self-oscillating converters have improved tremendously on this, and you can even find full-fledged multi-output PC power supplies that don t have a single... 

A. N. Malakhov, Fluctuations in Self-Oscillating Systems, Science, Moscow, 1968, in Russian. 

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. 

For low HF concentrations in the order of 0.1%, the behavior of the interface is not oscillation, but rather resonant if the potential is set to a fixed value and time is allowed for stabilization, a steady-state constant current is finally reached. Addition of a series resistor in the order of 1 kD crrf2 leads to sustained potentiostatic oscillations [Ch5], For higher HF concentrations of about 2-5% aqueous HF, the system is self-oscillating, if the series resistivity of the electrolyte itself is not electronically compensated. For even higher concentrations the periodicity is lost and... 

M. Perez, R. Font, and M.A. Montava. Regular self-oscillating and chaotic dynamics of a continuos stirred tank reactor. Comput. Chem. Eng., 26 889-901, 2002. 

M. Perez and P. Albertos. Self-oscillating and chaotic behaviour of a PI-controlled CSTR with control valve saturation. J. Process Control, 14 51-59, 2004. 

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. 

Self-oscillation and Chaotic Behavior of a CSTR Without Feedback Control... 

Equations (4) and (8) can be used to simulate the reactor at point P3 of Figure 5 in [1]. Remember that point P2 is unstable, so if the initial conditions are those corresponding to this point, it is easy to show [16], [28], the reactor evolves to points P or P3. Then, two forcing actions on the reactor are considered 1) when the coolant flow rate and the inlet stream temperature are varied as sine waves, and 2) reactor being in self-oscillating mode, an external disturbance in the coolant flow rate can drive it to chaotic behavior. 

It is well known that a nonlinear system with an external periodic disturbance can reach chaotic dynamics. In a CSTR, it has been shown that the variation of the coolant temperature, from a basic self-oscillation state makes the reactor to change from periodic behavior to chaotic one [17]. On the other hand, in [22], it has been shown that it is possible to reach chaotic behavior from an external sine wave disturbance of the coolant flow rate. Note that a periodic disturbance can appear, for instance, when the parameters of the PID controller which manipulates the coolant flow rate are being tuned by using the Ziegler-Nichols rules. The chaotic behavior is difficult to obtain from normal... 

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 ... 

In order to obtain the self-oscillation zone at the plane xo — yo consider the Jacobian of the linearized system (8) ... 

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 ... 

Dimensionless constant Ktd and self-oscillating frequency for different con-integral action T2d... 

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. 

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. 

From the results presented in this chapter, more advanced studies from the bifurcation theory can be planed. For example, inside the lobe, the behavior of the reactor is self-oscillating, i.e. an Andronov-Poincare-Hopf bifurcation can be researched from the calculation of the first Lyapunov value, in order to know if a weak focus may appear, or the conditions which give a Bogdanov-Takens bifurcation etc. Finally, it is interesting to remark that the previously analyzed phenomena should be known by the control engineer in order to either avoid them or use them, depending on the process type. 

To avoid this handicap, Boersch and coworkers 2) used coupled resonators. The first active laser cavitiy generates the radiation whose absorption is to be measured. The probe is placed in a second cavity, which is coupled to the first one and which is undamped by an active medium just below the threshold for self-oscillation. This arrangement enables changes in the refractive index as small as An 10 ° or absorption coefficients down to a 10 to be detected. 

Monitors range in size from 250-4,000 gpm (950-15,140 Ipm), with 500 and 1,000 gpm (1,900-3,800 Ipm) the most common sizes. In process areas that are not protected by water spray, monitors should be located so that each major piece of equipment can be covered by two monitors. Elevated monitors have application where it is necessary to deliver large volumes of water to areas that cannot be reached by ground level monitors or would be unsafe for manual firefighting. Elevated monitors can be fixed for remote use, self-oscillating or remote control from a safe location. 

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