Self-oscillating behaviors

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 ... 

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. 

We have first verified that, in absence of chemicaf reactions, our dynamical equation for the gel reproduces the dramatic slowing downs in the transition rates which have been observed near the critical point or the spinodal limits of hydrogels. In the last section, we discuss an example of the instabilities that give rise to the self-oscillating behavior of the coupled gel-chemical system and we exhibit corresponding bifurcation diagrams. 

In situ real-time observation of the self-oscillating behaviors of vesicles was carried out by using blight-field optical microscopy with a time-lapse mode. The images provide confirmation that a rhythmical self-oscillation between association and dissociation of... 

FIGURE 11.2 Mechanism of self-oscillation and aspects of self-oscillating behavior on several scales. 

Figure 7.11 Self-oscillating behavior of immobilized polymer in the BZ substrate solution ([MA] = 0.1 M, [NaBrOs] =...

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