Oscillatory behaviour

Figure Al.7.3. Schematic illustration showing side views of (a) a biilk-tenninated surface, (b) a relaxed surface with oscillatory behaviour, and (c) a reconstructed surface.

The reaction involving chlorite and iodide ions in the presence of malonic acid, the CIMA reaction, is another that supports oscillatory behaviour in a batch system (the chlorite-iodide reaction being a classic clock system the CIMA system also shows reaction-diffusion wave behaviour similar to the BZ reaction, see section A3.14.4). The initial reactants, chlorite and iodide are rapidly consumed, producing CIO2 and I2 which subsequently play the role of reactants . If the system is assembled from these species initially, we have the CDIMA reaction. The chemistry of this oscillator is driven by the following overall processes, with the empirical rate laws as given ... 

The existence of an upper and a lower limit to the range of oscillatory behaviour is more typical of observed behaviour m chemical systems. 

The next problem to consider is how chaotic attractors evolve from tire steady state or oscillatory behaviour of chemical systems. There are, effectively, an infinite number of routes to chaos [25]. However, only some of tliese have been examined carefully. In tire simplest models tliey depend on a single control or bifurcation parameter. In more complicated models or in experimental systems, variations along a suitable curve in the control parameter space allow at least a partial observation of tliese well known routes. For chemical systems we describe period doubling, mixed-mode oscillations, intennittency, and tire quasi-periodic route to chaos. 

E. V. Albano. Critical and oscillatory behaviour of a dimer-monomer catalyzed reaction process. Phys Rev E 57 6840-6843, 1998. 

If the second term on the right-hand side of the equation is omitted, the latter is transformed into Eq. (2.76). As the omission is possible only for t < tj, Fourier transformation of the reduced equation holds for co-tj 1 only. Consequently, the equality (2.75) is of asymptotic character, and may not be utilized to find full g(co) or its Fourier-transform Kj(t) at any times. When it was nevertheless used in [117], the rotational correlation function turned out to be alternating in sign. The oscillatory behaviour of Kj(t) occured not only in a compressed gas, but also at normal pressure, when Kj(t) should vanish monotonically, if not exponentially. The origin of these non-physical oscillations is easily... 

A striking feature of the effect of current on the CO oxidation oscillations is shown in Fig. 8.33. It can be seen that the frequency of oscillations is a linear function of the applied current. This holds not only for intrinsically oscillatory states but also for those which do not exhibit oscillations under open-circuit conditions, such as the ones shown on Fig. 8.31. This behaviour is consistent with earlier models developed to describe the oscillatory behaviour of Pt-catalyzed oxidations under atmospheric pressure conditions which are due to surface Pt02 formation35 as analyzed in detail elsewhere.33... 

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. 

Study the simple, open-loop (KC = 0) and closed-loop responses (KC = -1 to 5, TSET = TDIM, and 300 to 350 K) and the resulting yields of B. Confirm the oscillatory behaviour and find appropriate values of KC and TSET to give maximum stable and maximum oscillatory yield. For the open-loop response, show that the stability of operation of the CSTR is dependent on the operating variables by carrying out a series of simulations with varying Tq in the range 300 to 350 K. 

Modify the program to generate a dimensionless, phase-plane display of BDIM versus ADIM, repeating the studies of Exercise 1. Note that the oscillatory behaviour tends to form stable limit cycles in which the average yield of B can be increased over steady-state operation. 

For a single continuous reactor, the model predicted the expected oscillatory behaviour. The oscillations disappeared when a seeded feed stream was used. Figure 5c shows a single CSTR behaviour when different start-up conditions are applied. The solid line corresponds to the reactor starting up full of water. The expected overshoot, when the reactor starts full of the emulsion recipe, is correctly predicted by the model and furthermore the model numerical predictions (conversion — 25%, diameter - 1500 A) are in a reasonable range. 

A three-dimensional variation of the theme is offered by the oscillatory behaviour of anodic copper dissolution into a NaCl/KSCN electrolyte mixture.27 This is a complex process involving solid states CuSCN (pKsp = 14.32) and Cu20 or CuOH (pKsp = 14), and CuCl (pKsp = 5.92), and ionic species Cu+, CuCl22, CuCl3, and Cl. Among other plausible schemes,... 

As an illustration, the oscillatory behaviour of anodic copper dissolution discussed in Section III.3, with P elements shown in Table 5, is chosen. The eigenvalue theorem yields three relationships (k= 1,2,3) ... 

The observed CT-VPP-REDOR data confirms the expected general behaviour and can be best simulated assuming a second moment of 5.8 X lO rad s. Interestingly, in CT-VPDP-REDOR, this oscillatory behaviour can be observed even at low maximum AS / So values, which can be adjusted via the 0-pulse length. This is shown in Figure 12, in which the slices taken from a CT-VPDP-REDOR experiment on the glycine sample for five different 6 pulse lengths are collected. 

The link between the oscillatory behaviour of the structural energy curves and the moments of the local density of states can be made explicit by writing the bond order of a given bond as a many-atom expansion about that bond (Pettifor (1989)). Considering for simplicity the case of s orbitals, on a lattice where all sites are equivalent, the bond order can be expressed exactly (Aoki (1993)) as... 

Fig. 1.1. Typical experimental records of oscillatory behaviour in the Belousov-Zhabotinskii reaction (a) platinum electrode which responds primarily to the Ce3+/Ce4+ couple (b) bromide-sensitive electrode measuring In [Br ].

We are thus, in many instances, more interested in the transient behaviour early in a reaction than we are in the more easily studied final or equilibrium state. With this in mind, we shall be concerned in our early chapters with simple models of chemical reaction that can satisfy all thermodynamic requirements and yet still show oscillatory behaviour of the kind described above in a well-stirred closed system under isothermal or non-isothermal conditions. 

In open, or flow, reactors chemical equilibrium need never be approached. The reaction is kept away from that state by the continuous inflow of fresh reactants and a matching outflow of product/reactant mixture. The reaction achieves a stationary state , where the rates at which all the participating species are being produced are exactly matched by their net inflow or outflow. This stationary-state composition will depend on the reaction rate constants, the inflow concentrations of all the species, and the average time a molecule spends in the reactor—the mean residence time or its inverse, the flow rate. Any oscillatory behaviour may now, under appropriate operating conditions, be sustained indefinitely, becoming a stable response even in the strictest mathematical sense. 

If, however, we actually integrate the reaction rate equations numerically using the rate constants in Table 1.1 we find that the system does not always stick to, or even stay close to, these pseudo-steady loci. The actual behaviour is shown in Fig. 1.10. There is a short initial period during which d and b grow from zero to their appropriate pseudo-steady values. After this the evolution of the intermediate concentrations is well approximated by (1.41) and (1.42), but only for a while. After a certain time, the system moves spontaneously away from the pseudo-steady curves and oscillatory behaviour develops. We may think of the. steady state as being unstable or, in some sense repulsive , during this period in contrast to its stability or attractiveness beforehand. Thus we have met a bifurcation to oscillatory responses . The oscillations... 

Let us return to Fig. 1.12(c), where there are multiple intersections of the reaction rate and flow curves R and L. The details are shown on a larger scale in Fig. 1.15. Can we make any comments about the stability of each of the stationary states corresponding to the different intersections What, indeed, do we mean by stability in this case We have already seen one sort of instability in 1.6, where the pseudo-steady-state evolution gave way to oscillatory behaviour. Here we ask a slightly different question (although the possibility of transition to oscillatory states will also arise as we elaborate on the model). If the system is sitting at a particular stationary state, what will be the effect of a very small perturbation Will the perturbation die away, so the system returns to the same stationary state, or will it grow, so the system moves to a different stationary state If the former situation holds, the stationary state is stable in the latter case it would be unstable. 

We first examine the final state of the reaction, i.e. the chemical equilibrium composition. This is not of great relevance to oscillatory behaviour but is an important first check that the model is chemically reasonable . Equilibrium arises when all three rates of change become zero simultaneously. Equations (2.1)—(2.3) have a unique point satisfying this condition, as required chemically, given by... 

Can we understand this instability and, if so, can we predict when it will occur We would like to map out the experimental conditions under which oscillatory behaviour can be expected in terms of the rate constants k0 etc. and the concentration of the reactant. In fact we can do even better than this. We will see that much can be said about the details of the oscillations when they start, how they grow in period and amplitude, how long they will last, how and when they die out, and how many we can expect to see. Some typical results are given in Table 2.2. 

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