Concentration oscillations time periodicity

Oscillations of concentrations of some intermediates in the BZ reaction occur in a rather wide range of initial concentrations of the reagents. The reaction may be carried out in a closed system, for example in a graduated cylinder provided with a stirrer, or in an open system — in a flow reactor with stirring. 

Some of the reagents may be replaced with others. A different cerium salt, e.g. Ce(NH4)4(S04)4, can be used, NaBr03 may be replaced with KBr03, Mn(III)/Mn(II) may be employed instead of Ce(IV)/Ce(III), malonic acid may be replaced with citric acid. 

Ferroin is a redox indicator visualizing changes in concentration of the Ce(IV)/Ce(III) system, rather stable under reaction conditions blue colour corresponds to an excess of the Fe(III) ions, red colour corresponds to an excess of the Fe(II) ions. Proportions of the ferroin components are given in Table 6.2. 

Yellow (blue) colour corresponds to an excess of the Ce(IV) ions while the colourless (red) solution corresponds to an excess of the Ce(III) ions. Changes in concentrations may be followed directly or measured poten-tiometrically (Br, Ce(IV)/Ce(III)) or colorimetrically (without an addition of ferroin) — Ce(IV) absorbs radiation of a wavelength about 340 nm. The observations, and particularly the quantitative measurements, allows us to distinguish four fundamental phases of oscillations of concentrations, see Fig. 92. 

The red solution rapidly turns to blue and then slowly becomes violet. The violet colour slowly passes into red and, after some time, a rapid change in colour to blue takes place, the colour changing in part of the cylinder, e.g. at its bottom, and then rapidly spreading throughout the stirred solution. 

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Furthermore, it is concluded that under determined conditions (e.g., an optimal combination of periodic variations of selected parameters, such as forced concentration-oscillations and periodic variation of the liquid flow rate at the same time) could be used in order to achieve further significant performance and especially selectivity improvements for complex reactions systems in trickle-bed reactors. 

Figure 4.26 presents the results obtained for a CSTR with Q = 6 mL/h, [A]o = 0.3 mM, and a cycle time of 5 min. In this case the time period of the output signals represented by Si, S2, and B is reduced to 10 min. Results for the case when a PFR is employed with the same parameter values are presented in Figure 4.27. Here the oscillations disappear and all the concentration profiles reach a constant value after the transient time.

The signal for the 41 amu transient, a measure of the time-dependent rise and fall of BrCH2CH2CH2, rises (xi = 2.5 ps) and then decays (X2 = 7.5 ps), and it shows the same periodic coherent modulation, with a characteristic oscillation time Xc = 680 fs, phased shifted by n radians The local peaks of signal intensity proportionate to the BrCH2CH2CH2 radical concentration match the local troughs of signal decay for the 202 amu periodic modulation they are 180° out of phase. 

This type of a pattern of singular points is called a centre - Fig. 2.3. A centre arises in a conservative system indeed, eliminating time from (2.1.28), (2.1.29), one arrives at an equation on the phase plane with separable variables which can be easily integrated. The relevant phase trajectories are closed the model describes the undamped concentration oscillations. Every trajectory has its own period T > 2-k/ujq defined by the initial conditions. It means that the Lotka-Volterra model is able to describe the continuous frequency spectrum oj < u>o, corresponding to the infinite number of periodical trajectories. Unlike the Lotka model (2.1.21), this model is not rough since... 

A numerical solution of the basic equations demonstrated their ability to reproduce concentration oscillations. At the same time, for the systems possessing three and more intermediate products the standard method to prove existence of periodical solutions, using a phase portrait of a system (Section 2.1.1) fails. An additional reduction in a number of differential equations, e.g., using an idea that one of concentrations, say, [BrOj-], serves as a rapid variable and thus the relevant kinetic equation (8.1.5) could be solved as the stationary [10], cannot be always justified due to uncertainty in the kinetic coefficients hi. 

Therefore, oscillations of K (t) result in the transition of the concentration motion from one stable trajectory into another, having also another oscillation period. That is, the concentration dynamics in the Lotka-Volterra model acts as a noise. Since along with the particular time dependence K — K(t) related to the standing wave regime, it depends also effectively on the current concentrations (which introduces the damping into the concentration motion), the concentration passages from one trajectory onto another have the deterministic character. It results in the limited amplitudes of concentration oscillations. The phase portrait demonstrates existence of the distinctive range of the allowed periods of the concentration oscillations. 

For a given set of parameters the period of concentration oscillations (or its average for a periodic motion) exceeds greatly the period of the correlation motion. For the slow concentration motion not only the period of the standing wave oscillations but also their amplitudes and, consequently, the amplitude in the K (t) oscillations depend on the current concentrations Na(t) and Nb(t). In other words, the oscillations of the reaction rate are modulated by the concentration motion. Respectively, the influence of the time dependence K K(t) upon the concentration dynamics has irregular, aperiodic character. A noise component modulates the autowave component (the standing waves) but the latter, in its turn, due to back-coupling causes transition to new noise trajectories. What we get as a result is aperiodic motion (chaos). The mutual influence of the concentration and correlation motions and vice versa is illustrated in Fig. 8.2, where time developments of both the concentrations and reaction rates are plotted. 

The behaviour of the correlation functions shown in Fig. 8.5 corresponds to the regime of unstable focus whose phase portrait was earlier plotted in Fig. 8.1. For a given choice of the parameter k = 0.9 the correlation dynamics has a stationary solution. Since a complete set of equations for this model has no stationary solution, the concentration oscillations with increasing amplitude arise in its turn, they create the passive standing waves in the correlation dynamics. These latter are characterized by the monotonous behaviour of the correlations functions of similar and dissimilar particles. Since both the amplitude and oscillation period of concentrations increase in time, the standing waves do not reveal a periodical motion. There are two kinds of particle distributions distinctive for these standing waves. Figure 8.5 at t = 295 demonstrates the structure at the maximal concentration... 

We have investigated the transitions among the types of oscillations which occur with the Belousov-Zhabotinskii reaction in a CSTR. There is a sequence of well-defined, reproducible oscillatory states with variations of the residence time [5]. Similar transitions can also occur with variation of some other parameter such as temperature or feed concentration. Most of the oscillations are periodic but chaotic behavior has been observed in three reproducible bands. The chaos is an irregular mixture of the periodic oscillations which bound it e.g., between periodic two peak oscillations and periodic three peak oscillations, chaotic behavior can occur which is an irregular mixture of two and three peaks. More recently Roux, Turner et. al. 

The Belousov-Zhabotinskii (BZ) reaction has been selected as an example illustrating diverse dynamical states observable in chemical systems. The BZ reagent is very convenient both for experimental and theoretical investigations, since the BZ reaction has many dynamical states of interest, which will be described below. In the BZ reaction one may observe the steady state, the time periodic state (concentration oscillations), the spatially periodic state, the stationary state (dissipative structures), the time and spatially periodic state (propagating chemical waves) and turbulent states (chaotic oscillations, stochastic spatial structures, stochastic chemical waves). 

The pi criterion is a sufficient condition for the existence of a periodic solution that is better than the neighboring optimal steady state solution of an optimal periodic control problem. Using the criterion, we would hke to know, for example, whether the time-averaged product concentration in a periodic process can be more than what the optimal steady state operation can provide. In other words, we would hke to check if oscillating the optimal steady state control with some frequency and time period improves on the steady state solution. 

In these experiments, the volume of the confined gel is constant its main role is to damp hydrodynamical motions that would otherwise perturb the chemical intrinsic patterns. More recently it has been shown experimentally that the coupling of a volume phase transition with a chemical oscillator can generate a self-oscillating gel (i, 4). More precisely, if one of the chemical species taking part in the chemical reaction modifies the threshold for the phase transition, then the time periodic variation of this concentration can generate autonomous swelling-deswelling cycles of the gel even in absence of any external stimuli (5, 6). This device thus provides a novel biomimetic material with potential biomedical and technical applications. 

Chemical reaction network is a typical example of complexity, where the reactants can interact in a variety of ways depending on the nature of interaction (chemical as well as non-chemical). Oscillatory reactions involve a number of steps, including positive and negative feedbacks. The complexity leads to periodic as well as aperiodic oscillations (multi-periodic, bursting/intermittency sequential oscillations separated by a time pause, relaxation and chaotic oscillations). The mechanism is usually determined by non-linear kinetics and computer modelling. Once the reaction mechanism has been postulated, the non-linear time-dependent kinetic equation can be formulated in terms of concentrations of different reactants, which would yield a multi-variable equation. Delay differential equations are sometimes used to characterize oscillatory behaviour as in economics (Chapter 14). 

Figure 5.61. Predicted behavior of a predator-prey system in single-stage CSTR culture with respect to mean residence time F and concentration of limiting substrate Sin for bacterial prey in the influent. A modification is represented by incorporating a multiple saturation predator rate of Equ. 5.206 (after lost et al, 1973, and Tsuchiya et al., 1972). The curves 1-5 separate the following regions of the operating diagram (a) total washout, (b) X2 washed out, (c) no oscillations, (d) damped oscillations, (Cj) periodic oscillations, and (Cn) sustained oscillations.

In order to really understand a system, we must study it under a variety of conditions, that is, for many different sets of control parameters. In this way, we will be able to observe whether bifurcations occur and to see how the responses of the system, such as steady-state concentrations or the period and amplitude of oscillations, vary with the parameters. Information of this type, which summarizes the results of a number of time series, is conveniently displayed in a constraint-response plot, in which a particular response, like a concentration, is plotted against a constraint parameter, like the flow rate. If the information is available, for example, from a calculation, unstable states can be plotted as well. Bifurcations appear at points where the solution changes character, and constraint-response plots are sometimes called bifurcation diagrams. An experimental example for a system that shows bistability between a steady and an oscillatory state is shown in Figure 2.13. 

Fig. 1. Data for the BZ reaction at three different residence times t (reactor volume/total flow rate) the reservoir concentrations, given in [21], were held fixed, (a) t 0.49 hour a periodic state with one oscillation per period. (b) t=0.90 hour chaos. (c) t=1.03 hour a periodic state with 2 oscillations per period. For each t the graphs show the time dependence of the bromide ion potential, the corresponding power spectrum, and a two dimensional projection of the phase portrait. The phase portraits for the periodic states in (a) and (c) are limit cycles, while the chaotic state in (b) is described by a strange attractor. (From [21].)...

Let s solve the set with Mathcad tools (Fig. 3.19), using the numerical integration method with an adaptive step. The constants ao, ai, 02, b, 62, bj, values have been chosen arbitrarily. The results show that there are periodical concentration oscillations, which decay over time. 

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