Oscillation steady-state

Fig. 6. Dynamical phase diagram of the ascorbic acid/copper(II)/oxygen system in a CSTR in the kf — [Cu2+]0 plane. Fixed reactor concentrations [H2Asc]0 = 5.0x10 4M [H2SO4]0 = 6.0 x 10-5 M [Na2SO4]0 = 0.04M. Symbols O, steady state , oscillations , bistability. The asterisk ( ) marks the Takens-Bogdanov point. Strizhak, P. E. Basylchuk, A. B. Demjanchyk, I. Fecher, F. Shcneider, F. W. Munster, A. F. Phys. Chem. Chem. Phys. 2000, 2, 4721. Reproduced by permission of The Royal Society of Chemistry on behalf of the PCCP Owner Societies.

This exponential decay rate for R in a stationary system will now be compared with that for a system in which X oscillates due to oscillations in a or 0. First, if the oscillations are driven solely by the anabolic term a and the rate of catabolism 0 remains time-independent, inspection of equations (6-10) shows that, for the steady state oscillation, relations (11) and (12) hold true. That is, the rate of removal of labeled compounds remains independent of the oscillations in a and X. On the other hand, if the rate of reaction through which the flux of R is occurring is made to oscillate, i.e., if 0(t) oscillates, will be a function of this oscillation. If... 

Multiple Steady States, Oscillations, and Chaotic Behavior There are reaction systems whose steady-state behavior depends on... 

If a surface reaction involving multisite adsorption exhibits a maximum with respect to concentration, slow reactant transport through the surface boundary layer can yield up to three steady states. The existence of a maximum is necessary but not sufficient for having multiplicity. The latter depends on the electrode potential, which can alter the shape and the position of the maximum, and on the magnitude of the mass transfer coefficient relative to the surface rate constant (418). Thus, as the potential becomes more negative for a reduction, the multiplicity region can be reached and oscillations may develop between two stable steady states. Oscillations could also arise from other simultaneous reactions such as oxide formation... 

Figure 14.5 shows a comparison between experimental results and the model. The startup transient has an initial overshoot followed by an apparent approach to steady state. Oscillations begin after a phenomenally long delay, t > 10 , and the system... 

The surface tension of the continuous phase of a polymer emulsion may be used as a measure of the free onulsifier concentration. ITie term free onulsifier is used here to denote surfactant which is dissolved in the aqueous phase rather than being adsorbed on to polymer particles or monomer droplets, or aggregated into micelles. The free emulsifier concentration is widely considered to be a critical variable in the phenomenon of steady-state oscillation in a CSTR and in preventing coagulation during polymoization. 

Stability of steady states, oscillations, and other peculiarities. The mathematical analysis given below supports this conclusion. 

The scattered intensity of the unperturbed system (Iq) is compensated, and the field-induced light-scattering variations (A/ = /g - Iq) with time are observed directly on the oscilloscope. The dc component of the effect. A/ is determined taking for the middle of the modulated steady-state signal, the ac component - by the amplitude of the steady-state oscillations. The results presented in this chapter were obtained by the use of white unpolarized light. 

Thus the steady state oscillator experiment determines Fn si - -Sn) as a function of only one transform variable at a time. It is easily seen that the result (16) corresponds to the assumption... 

This change in the PLASLA system is schematically illustrated in Fig. 16. At the initial point, the front wave is generated by laser ablation, grows after that, and is united with the stimulated wave to be transformed into a steady-state oscillation. Thus, by the study in this section, it is found that the waves generated in PLASLA have the systematic characteristics. 

Much research in this field is in the area of "learning models". This is because of the several phenomena that affect the modelling of chemical reactors such as instabilities, multiple steady states, oscillations, etc. The purpose of controlling is faced with several such as catalyst activity, may not be possible. 

Before discussing mechanical models or other mathematical representation of viscoelastic behavior, it is very important to note that the preceding section deals only with observed behavior or the experimental response of polymers under laboratory conditions. That is, the viscoelastic properties are defined from observations of real behavior and need not be defined by a particular mathematical model. Mathematical models are developed for the simple purpose of understanding and describing observed behavior. Also, as will be evident later, other loading modes such as constant strain rate and steady state oscillation, etc. can be used to determine viscoelastic properties. 

A constant strain rate test may be used to determine the relaxation modulus and a constant stress-rate test may be used to find the creep compliance. Steady state oscillation tests may also be used to determine the viscoelastic properties of polymers. These details and the interrelation between various test approaches are given in Chapter 5. 

Dynamic Properties - Steady State Oscillation Testing... 

Viscoelastic properties are often determined with steady state oscillation or vibratory tests using small tensile (compressive) bars, thin cylinders or flat strips in torsion, beams in bending, etc. The approach is usually referred to as dynamic mechanical analysis (DMA) testing or sometimes dynamic mechanical thermal analysis (DMTA). The latter term is more appropriate as properties are often determined and expressed in terms of temperature... 

Describe the molecular mechanisms associated with the regions of response in a steady state oscillation test. 

FIGURE 17.5 Steady-state oscillations during the emulsion polymerization of methyl methacrylate in a single CSTR. (a) Single CSTR (b) tubular reactor followed by a single CSTR. Reprinted (adapted) with permission from Nomura M, Harada M. On the optimal reactor type and operations for continuous emulsion polymerization. In Bassett DR, Hamielec AE, editors. Emulsion Polymers and Emulsion Polymeriz0ion. Washington, DC p 121-144. 1981 American Chemical Society. 

As a direct consequence of the particular role of Dynamics, as such,in the study of non-equilibrium behaviour of chemical systems, two classes of models are to be considered, depending on which aspect one is insisting on. Formal models, of mathematical or chemical-like nature, are designed to exhibit specific dynamical behaviours, without too much concern about chemical significance. Their aim is to provide examples of evolution equations of chemical reacting systems, as described by mass action kinetics, that are able to produce those exotic behaviours, such as bistability or multistability, between various types of attractors, like steady states, oscillations or deterministic chaos. A typical historical model of that kind is the "Brusselator ... 

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