Sustained oscillation

Most chemically reacting systems tliat we encounter are not tliennodynamically controlled since reactions are often carried out under non-equilibrium conditions where flows of matter or energy prevent tire system from relaxing to equilibrium. Almost all biochemical reactions in living systems are of tliis type as are industrial processes carried out in open chemical reactors. In addition, tire transient dynamics of closed systems may occur on long time scales and resemble tire sustained behaviour of systems in non-equilibrium conditions. A reacting system may behave in unusual ways tliere may be more tlian one stable steady state, tire system may oscillate, sometimes witli a complicated pattern of oscillations, or even show chaotic variations of chemical concentrations. 

K. Fichthom, E. Gulari, R. Ziff". Self-sustained oscillations in a heterogeneous catalytic reaction A Monte Carlo simulation. Chem Eng Sci 44 1403-1411, 1989. 

In its applications, science was encountering gradually-increasing difficulties in view of the impossibility of explaining numerous oscillatory phenomena, particularly those connected with the so-called self-sustained oscillations (first, the oscillating arcs and gaseous discharges and still later, the electron tube oscillators). 

The oxidation of CO on Pt is one of the best studied catalytic systems. It proceeds via the reaction of chemisorbed CO and O. Despite its complexities, which include island formation, surface reconstruction and self-sustained oscillations, the reaction is a textbook example of a Langmuir-Hinshelwood mechanism the kinetics of which can be described qualitatively by a LHHW rate expression. This is shown in Figure 2.39 for the unpromoted Pt( 111) surface.112 For low Pco/po2 ratios the rate is first order in CO and negative order in 02, for high pco/po2 ratios the rate becomes negative order in CO and positive order in 02. Thus for low Pcc/po2 ratios the Pt(l 11) surface is covered predominantly by O, at high pco/po2 ratios the Pt surface is predominantly covered by CO. 

Figure 8.31. Induction of self-sustained rate and catalyst potential, or work function, oscillations by NEMCA during CO oxidation on Pt. Inlet composition Pco 0.47 kPa, Po2=16 kPa, T=297°C.33 Reprinted with permission from Academic Press.

Figure 8.33. Effect of applied constant current on the frequency of the self-sustained rate and Uwr oscillations during CO oxidation on Pt conditions as on Fig. 8.32 Filled circles on the frequency vs current diagram are oscillatory states of this figure open circles include states shown on Fig. 8.31.33 Reprinted with permission from Academic Press.

Autocatalysis can cause sustained oscillations in batch systems. This idea originally met with skepticism. Some chemists believed that sustained oscillations would violate the second law of thermodynamics, but this is not true. Oscillating batch systems certainly exist, although they must have some external energy source or else the oscillations will eventually subside. An important example of an oscillating system is the circadian rhythm in animals. A simple model of a chemical oscillator, called the Lotka-Volterra reaction, has the assumed mechanism ... 

The Lotka-Volterra reaction described in Section 2.5.4 has three initial conditions—one each for grass, rabbits, and lynx—all of which must be positive. There are three rate constants assuming the supply of grass is not depleted. Use dimensionless variables to reduce the number of independent parameters to four. Pick values for these that lead to a sustained oscillation. Then, vary the parameter governing the grass supply and determine how this affects the period and amplitude of the solution. 

Example 14.2 demonstrates that sustained oscillations are possible even in an isothermal flow system. This is hardly surprising since they are possible in a batch system provided there is an energy supply. 

D. Durox, T. Schuller, and S. Gandel. Self-sustained oscillations of a premixed impinging jet flame on a plate. Proc. Combust. Inst., 29 69-75, 2002. 

To visualize the situation one may take Y as a population of foxes, E as the foxes that have died, X as rabbits and A as carrots. Rabbits live very well on carrots and the population would grow exponentially if there are sufScient carrots. The fox population starts to grow when the rabbit population is high, untU there are more foxes than can be sustained by the rabbits. Famine sets in and the fox population diminishes, after which the rabbit population starts to grow again. In other words, X and Y are oscillating out of phase. It is essential that the mechanism contains autocatalytic steps, and that there is a continuous supply of reactant A, which keeps the system far away from equilibrium. 

Oscillations such as in Fig. 2.15 are quite regular and can be sustained for hours if the conditions are kept the same. Depending on the feed rate of the reactants, which determines how far the system deviates from equilibrium, the oscillations may become more complex, and develop into chaotic oscillations (see, for example, P.D. Cobden, J. Siera, and B.E. Nieuwenhuys, J. Vac. Sci. Technol. A10 (1992) 2487). 

The response of a controller to an error depends on its mode. In the proportional mode (P), the output signal is proportional to the detected error, e. Systems with proportional control often exhibit pronounced oscillations, and for sustained changes in load, the controlled variable attains a new equilibrium or steady-state position. The difference between this point and the set point is the offset. Proportional control always results in either an oscillatory behaviour or retains a constant offset error. 

Figure 3.17. Phase-plane representations of reactor stability. In the above diagrams the point -I- represents a possible steady-state solution, which (a) may be stable, (b) may be unstable or (c) about which the reactor produces sustained oscillations in temperature and concentration.

The parameters used in the program give a steady-state solution, representing, however, a non-stable operating point at which the reactor tends to produce natural, sustained oscillations in both reactor temperature and concentration. Proportional feedback control of the reactor temperature to regulate the coolant flow can, however, be used to stabilise the reactor. With positive feedback control, the controller action reinforces the natural oscillations and can cause complete instability of operation. 

The oscillation at a liquid liquid interface or a liquid membrane is the most popular oscillation system. Nakache and Dupeyrat [12 15] found the spontaneous oscillation of the potential difference between an aqueous solution, W, containing cetyltrimethylammo-nium chloride, CTA+CK, and nitrobenzene, NB, containing picric acid, H" Pic . They explained that the oscillation was caused by the difference between the rate of transfer of CTA controlled by the interfacial adsorption and that of Pic controlled by the diffusion, taking into consideration the dissociation of H Pic in NB. Yoshikawa and Matsubara [16] realized sustained oscillation of the potential difference and pH in a system similar to that of Nakache and Dupeyrat. They emphasized the change of the surface potential due to the formation and destruction of the monolayer of CTA" Pic at the interface. It is... 

There existed a certain range of applied current in order to get the sustained oscillation, and the upper limit of the range was affected by ucsTPhs and lowered with a decrease... 

When 1,2-dichloroethane, DCE (e = 10.37), or u-nitroanisole, o-NA (e = 5.23), was employed instead of nitrobenzene (e = 34.78), the induction time and period with o-NA were obviously less than those with NB or DCE. It should be noted that, in case of o-NA, the sustained oscillation was observable even with cxpatphe of less than 0.01 M, though the oscillation was not observed with NB or DCE. The result suggests that ocsXPhB is the essential factor for the appearance of the oscillation, taking into account that the ion pair formation is more serious in o-NA than in NB or DCE. 

When the applied current was much less than the maximum current and more than the diffusion-controlled current, the sustained oscillation was observed. 

Ziegler-Nichols Continuous Cycling (empirical tuning with closed loop test) Increase proportional gain of only a proportional controller until system sustains oscillation. Measure ultimate gain and ultimate period. Apply empirical design relations. 

Furthermore, conjugate poles on the imaginary axis are BIBO stable—a step input leads to a sustained oscillation that is bounded in time. But we do not consider this oscillatory steady state as stable, and hence we exclude the entire imaginary axis. In an advanced class, you should find more mathematical definitions of stability. 

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