Steady-state oscillations and

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

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

If the roots are pure imaginary numbers, the form of the response is purely oscillatory, and the magnitude will neither increase nor decay. The response, thus, remains in the neighbourhood of the steady-state solution and forms stable oscillations or limit cycles. 

Exotic Kinetics Oscillating Reactions in the Troposphere (J. Phys. Chem. A 2001, 105, 11212-11219. "Steady State Instability and Oscillation in Simplified Models of Tropospheric Chemistry")... 

For y/3 1, one steady state exists and the regime is globally stable for all values of the Lewis number Lw. For 1 < y/3 < (y/3) and for sufficiently low values of Lewis number the system is again globally stable. Evidently for these conditions only one steady state occurs. For Lw > Lw, undamped oscillations exist. For supercritical values of y/3, y/3 > (y/3), and < a single steady state is stable or unstable according to the value of Lewis number. In the domain l>min < < max>... 

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... 

There is a voluminous literature on steady-state multiplicity, oscillations (and chaos), and derivation of bifurcation points that define the conditions that lead to onset of these phenomena. For example, see Morbidelli et al. [ Reactor Steady-State Multiplicity and Stability, in Chemical Reaction and Reactor Engineering, Carberry and Varrria (eds), Marcel Dekker, 1987], Luss [ Steady State Multiplicity and Uniqueness... 

The oxidation of propylene oxide on porous polycrystalline Ag films supported on stabilized zirconia was studied in a CSTR at temperatures between 240 and 400°C and atmospheric total pressure. The technique of solid electrolyte potentiometry (SEP) was used to monitor the chemical potential of oxygen adsorbed on the catalyst surface. The steady state kinetic and potentiometric results are consistent with a Langmuir-Hinshelwood mechanism. However over a wide range of temperature and gaseous composition both the reaction rate and the surface oxygen activity were found to exhibit self-sustained isothermal oscillations. The limit cycles can be understood assuming that adsorbed propylene oxide undergoes both oxidation to CO2 and H2O as well as conversion to an adsorbed polymeric residue. A dynamic model based on the above assumption explains qualitatively the experimental observations. 

Stable, limit cycle. The latter occurs in the Salnikov case and the modified bifurcation diagram is shown in Fig. 5.11(b). The stable limit cycle born at the lower Hopf point overshoots the upper Hopf point but is extinguished by colliding with the unstable limit cycle born at the upper Hopf point which also grows in amplitude as )jl is increased. Over a, typically narrow range, then there are two limit cycles, one unstable and one stable around the (stable) steady-state point. If we start with the system at some large value of /r, so we settle onto the steady-state locus, and then decrease the parameter, we will first swap to oscillations at the Hopf point /r - At this point there is a stable limit cycle available as the system departs from the now unstable steady-state, but this stable limit cycle is not born at this point and so already has a relatively large amplitude. We would expect to... 

Figure 4. Three different parameter regions. Left damping without oscillations (n = 2, 7 = 10). Centre damping with oscillations (n = 2, 7 = 1). Right damping with oscillations that violate the state operator properties (n = 0, 7 = 1). The other parameters are T = V = 1. In the high-temperature limit (11 — 00), there are always oscillations but their amplitude is smaller than the steady-state excitation and, consequently, the probability never goes negative.

Adsorption of more than one species and complex electrocatalytic surface reactions [Eqs. (10) and (16)] may result in nonunique steady-state operation and in current or potential oscillations (31, 78, 417). We examined recently conditions for isothermal multiple steady states at planar, and porous, flow-by or flow-through electrocatalysts (418). 

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... 

Equation (12.68) applies only to a steady-state condition, and this rarely happens. Some holes pass neither vapor nor liquid. There is considerable sloshing, froth pounding, oscillation, and so on. A modified Equation (12.68) is shown in Figure 12.35. The lines of the figure may be approximated by... 

Mass transfer in combination with even quite "normal" reaction kinetics can produce a wealth of phenomena including multiple steady states, instabilities, and oscillations. An example is the behavior of nonisothermal catalyst particles outlined in Section 9.5.2. Such phenomena are covered in detail in standard texts on reaction engineering, to which the reader is referred. The examination in this section will remain restricted to effects produced by vagaries of multistep or multiple simultaneous reactions. 

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... 

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