Oscillation of concentration

Only a very few experimental studies have been made for detection of mnlti-plicities of steady states to check on theoretical predictions. The studies of multiplicities and of oscillations of concentrations have similar mathematical bases. Comprehensive reviews of these topics are by Schmitz (Adv. Chem. Sen, 148, 156, ACS [1975]), Razon and Schmitz (Chem. Eng. Sci., 42, 1,005-1,047 [1987]), Morbidelli, Vamia, and Aris (in Carberry and Varma, eds.. Chemical Reaction and Reacton Engineering, Dekker, 1987, pp. 975-1,054). 

Oscillations of concentrations of intermediates can occur when one substance in a sequence of reactions is either an activator or an inhibitor for a reaction step that occurs earlier in the sequence. Such activation or inhibition can control intermediate forming steps and will be responsible for producing oscillation in the concentration of the intermediate. The important conditions required to generate oscillations in a chemical system are ... 

Figure 13.8. Oscillations of concentrations and the limit cycle obtained from the Lengyel-Epstein model with the parameters cr = 2.0, a = 30.0, b= 11.0 > bM, h = 20, which indicates time (a) U, /versus time, the bold line is for V, (b) limit cycle /versus U with the initial conditions of 1/(0) = 0, and V(0) = 0, (c) U, Y versus time, the bold line is for V, (d) limit cycle V versus U with the initial conditions of 1/(0) = 0, and V(0) = 0 with the parameters

Kovalenko et al. (1980) reported the relation between malonic acid and BrO, concentrations for the oscillatory regions of the system Br03 —CH2(COOH)2— Mn(II) (or Ferroin). In addition, the induction period, the period and the amplitude of the oscillations of concentrations of Mn(III), were determined as a function of a) malonic acid for various Br03 concentrations and b) Br03 for various malonic acid concentrations. 

In the non-linear systems (5.2), a second type of attractor — a closed curve (limit cycle) is also possible. For example, the system of van der Pol equations (representing oscillations of current in electrical circuits and oscillations of concentrations, or more precisely the differences between the concentrations and their stationary values, in chemical systems)... 

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. 

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 first condition at the surface relates the concentration gradients to the current. This is the continuity of fluxes at the electrode surface, and it states that Ox is changed to Red according to Eq. (4.24), which is the law of conservation of matter. Far from the electrode at r oo there are no oscillations of concentrations of redox species and the phasors become zero. 

Let us consider now diffusion inside a sphere neglecting the diffusion gradient outside the sphere. Such a case might be observed for hydrogen absorption or Li intercalation into spherical particles. Diffusion inside the sphere can go only to the sphere center and is called finite-length internal spherical diffusion. In the steady state in which the impedance measurements are carried out, dc concentration inside the sphere is uniform, and no dc current is flowing. Ac perturbation causes oscillations of concentration at the sphere surface, which diffuse inside the sphere. In such a case, two boundary conditions in Eq. (4.94) are changed ... 

We now wish to test the above gel-reaction-diflfusion approach (Eqs 9.7-9.10) using the BZ reaction, which consists in the oxidation of malonic acid by bromate ions in acidic medium. The reaction proceeds only when catalyzed by a suitable metal ion. In the experiments, the chosen catalyst is the ruthenium tris(2,2 -bipyridine) that intervenes through its oxydo-reduction couple (Ru(bpy)3 /Ru(bpy)j ). In a batch reactor or a CSTR, this reaction exhibits well-documented periodic oscillations of concentrations in some region of the parameter space. 

Figure 9.3 Self-oscillations induced by the system described by Eq. (9.20). (a) full lines oscillations of the sphere s radius as a function of time dotted lines oscillations of concentration v at the sphere center, (b) profiles of concentration u as a function of...

The transient behavior of the reactor for finite perturbations was studied by computer simulation of Eqs (1.1)-(1.4). The curves 1 and 2 in the Fig. 6 demonstrate the transition from the lower stable state to stabilized intermediate one after beginning of the proportional control operation at C = 0. The curve 1 corresponds to the case of ideal control (f =0). The curve 2 corresponds to nonideal control with parameters d and Cj corresponding to the stability domain in Fig, 5 One can see that the curve 1 is monotonous and the curve 2 is not monotonous. In the case of nonideal control the damping oscillations of concentration and temperature near the intermediate steady values take place with the amplitude being dependent on the time lag value. The curve 3 corresponds to the case when the point (, d) belongs to the... 

By computer similation the transient behavior on nonlinear system is studied. It is shown that for zero time lag the stabilized intermediate steady state is achieved without oscillations. Under imperfect control operation damping oscillations of concentration profile and temperature near intermediate state can exist the amplitude being dependent on the time-lag value. For certain values of flow-control parameters sustained oscillations of concentration profile and temperature in the reactor are possible. 

Figure 6.2 Oscillations of concentrations (solid lines) in response to a periodic perturbation of pressure with frequency co are represented by broken lines. The periodic response of the reactant concentration is a function of the relaxation time T. If the frequency of the perturbation is high (o> 1/t) the reaction cannot follow the pressure change and will not be observed. The traces (a) and b) correspond to reactions with 1/T = c and 1/T=4

According to the data (Baranov, 1989 and Mikhaylenko, 2008), the additional constraint is unstable stability of work of the scrubbers, manifested in essential change of parameters of work at infinitesimal changes of entrance conditions. In particular, the basic deficiency of hydrocyclonic apparatuses is defined by considerable ehange of parameters of separation at small oscillations of concentration and composition of the firm phase in apparatus. 

Belykh, V. N. [1980] Qualitative methods of the theory of nonlinear oscillations of concentrated systems, Gorky State Univerosity press Gorky. 

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