Isothermal limit cycles

Self-sustained Oscillations. Under certain conditions, isothermal limit cycles in gaseous concentrations over catalysts are observed. These are probably caused by interaction of steps on the surface. Sometimes heat and mass transfer effects intervene, leading to temperature oscillations also. Since this subject has recently been reviewed (42, 43) only a few recent papers will be mentioned here. 

Cutlip and Kenney (44) have observed isothermal limit cycles in the oxidation of CO over 0.5% Pt/Al203 in a gradientless reactor only in the presence of added 1-butene. Without butene there were no oscillations although regions of multiple steady states exist. Dwyer (22) has followed the surface CO infrared adsorption band and found that it was in phase with the gas-phase concentration. Kurtanjek et al. (45) have studied hydrogen oxidation over Ni and have also taken the logical step of following the surface concentration. Contact potential difference was used to follow the oxidation state of the nickel surface. Under some conditions, oscillations were observed on the surface when none were detected in the gas phase. Recently, Sheintuch (46) has made additional studies of CO oxidation over Pt foil. 

Fig. 3.8. Representation of the onset, growth, and death of oscillations in the isothermal autocatalytic model as /z varies for reaction with the uncatalysed step included, showing emergence of the stable limit cycle at and its disappearance at n. ...

The behaviour exhibited by this model is relatively simple. There is only ever one limit cycle. This is born at one bifurcation point, grows as the system traverses the range of unstable stationary states, and then disappears at the second bifurcation point. Thus there is a qualitative similarity between the present model and the isothermal autocatalysis of the previous chapter. The limit cycle is always stable and no oscillatory solutions are found outside the region of instability. 

The analyses applied to the simplest two-variable autocatalytic system in the previous sections can obviously be brought to bear on other systems. Much effort has been expended on the first-order non-isothermal model of chapter 7, and very similar ranges of complexity are found. Up to 35 phase portraits have been predicted for the full system with the Arrhenius temperature dependence and forced cooling, with different combinations of one or three stationary states and up to three limit cycles of varying stability. 

This work is centred around the study of the response to periodic forcing of systems that, when autonomous, had a stable limit cycle surrounding an unstable steady state in their phase plane. For the sake of simplicity—and since many of the fundamental phenomena are the same—we studied two-dimensional systems. We chose two examples of isothermal reactor models the first is an autocatalytic homogeneous Brusselator (Glansdorff and Prigog-ine, 1971) ... 

The effects of forced oscillations in the partial pressure of a reactant is studied in a simple isothermal, bimolecular surface reaction model in which two vacant sites are required for reaction. The forced oscillations are conducted in a region of parameter space where an autonomous limit cycle is observed, and the response of the system is characterized with the aid of the stroboscopic map where a two-parameter bifurcation diagram for the map is constructed by using the amplitude and frequency of the forcing as bifurcation parameters. The various responses include subharmonic, quasi-peri-odic, and chaotic solutions. In addition, bistability between one or more of these responses has been observed. Bifurcation features of the stroboscopic map for this system include folds in the sides of some resonance horns, period doubling, Hopf bifurcations including hard resonances, homoclinic tangles, and several different codimension-two bifurcations. 

The model that will be used for forced oscillation studies is one which was first proposed by Takoudis et al. (1981) as a simple example of an isothermal surface reaction without coverage dependent parameters in which limit cycles can occur. The bimolecular reaction between species A and B is presumed to occur as a Langmuir-Hinshelwood bimolecular process except that two adjacent vacant sites on the surface are required for the reaction to take place. 

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. 

A simple Langmuir-Hinshelwood model explains quantitatively the steady-state behavior (4) but it fails to explain the oscillatory phenomena that were observed. The origin of the limit cycles is not clear. Rate oscillations have not been reported previously for silver catalyzed oxidations. Oxidation of ethylene, propylene and ethylene oxide on the same silver surface and under the same temperature, space velocity and air-fuel ratio conditions did not give rise to oscillations. It thus appears that the oscillations are related specifically to the nature of chemisorbed propylene oxide. This is also supported by the lack of any correlation between the limits of oscillatory behavior and the surface oxygen activity as opposed to the isothermal oscillations of the platinum catalyzed ethylene oxidation where the SEP measurements showed that periodic phenomena occur only between specific values of the surface oxygen activity (6,9). 

To study a class of mechanisms for isothermal heterogeneous catalysis in a CSTR, Morton and Goodman (1981-1) analyzed the stability and bifurcation of simple models. The limit cycle solutions of the governing mass balance equations were shown to exist. An elementary step model with the stoichiometry of CO oxidation was shown to exhibit oscillations at suitable parameter values. By computer simulation limit cycles were obtained. 

Jaisinghani and Ray (40) also predicted the existence of three steady states for the free-radical polymerization of methyl methacrylate under autothermal operation. As their analysis could only locate unstable limit cycles, they concluded that stable oscillations for this system were unlikely. However, they speculated that other monomer-initiator combinations could exhibit more interesting dynamic phenomena. Since at that time there had been no evidence of experimental work for this class of problems, their theoretical analysis provided the foundation for future experimental work aimed at validating the predicted phenomena. Later studies include the investigations of Balaraman et al. (43) for the continuous bulk copolymerization of styrene and acrylonitrile, and Kuchanov et al. (44) who demonstrated the existence of sustained oscillations for bulk copolymerization under non-isothermal conditions. Hamer, Akramov and Ray (45) were first to predict stable limit cycles for non-isothermal solution homopolymerization and copolymerization in a CSTR. Parameter space plots and dynamic simulations were presented for methyl methacrylate and vinyl acetate homopolymerization, as well as for their copolymerization. The copolymerization system exhibited a new bifurcation diagram observed for the first time where three Hopf bifurcations were located, leading to stable and unstable periodic branches over a small parameter range. Schmidt, Clinch and Ray (46) provided the first experimental evidence of multiple steady states for non-isothermal solution polymerization. Their... 

The prototype, cubic autocatalytic reaction (A + 2B 3B) forms the basis of a simple homogeneous system displaying a rich variety of complex behaviour. Even under well-stirred, isothermal open conditions (the CSTR) we may find multi stability, hysteresis, extinction and ignition. Allowing for the finite lifetime of the catalyst (B inert products) adds another dimension. The dependence of the stationary-states on residence-time now yields isolas and mushrooms. Sustained oscillations (stable limit cycles) are also possible. There are strong analogies between this simple system and the exothermic, first-order reaction in a CSTR. 

Much of this Volume deals with the transition phenomena observed in isothermal or temperature dependent reaction sequences, involving appropriate cooperative interactions like autocatalysis, and functioning far from equilibrium. Classical bifurcation phenomena involving the loss of stability of a uniform steady state and the evolution to a limit cycle or a space pattern, abrupt overshoots associated to ignition and explosion, or transition to chaotic dynamics are some characteristic examples. 

One unique but normally undesirable feature of continuous emulsion polymerization carried out in a stirred tank reactor is reactor dynamics. For example, sustained oscillations (limit cycles) in the number of latex particles per unit volume of water, monomer conversion, and concentration of free surfactant have been observed in continuous emulsion polymerization systems operated at isothermal conditions [52-55], as illustrated in Figure 7.4a. Particle nucleation phenomena and gel effect are primarily responsible for the observed reactor instabilities. Several mathematical models that quantitatively predict the reaction kinetics (including the reactor dynamics) involved in continuous emulsion polymerization can be found in references 56-58. Tauer and Muller [59] developed a kinetic model for the emulsion polymerization of vinyl chloride in a continuous stirred tank reactor. The results show that the sustained oscillations depend on the rates of particle growth and coalescence. Furthermore, multiple steady states have been experienced in continuous emulsion polymerization carried out in a stirred tank reactor, and this phenomenon is attributed to the gel effect [60,61]. All these factors inevitably result in severe problems of process control and product quality. 

A necessary chemical condition for the appearance of limit cycles and selfroscillations is either nonlinearity of mechanistic steps or complexity functions of the catalyst with respect to substrates, including inhibitors. Sufficient chemical conditions are more difficult to establish even if under isothermal conditions. The following types of nonlinearity were identified leading to self-osciUations presence of nonlinear steps when two intermediates are reacting with each other and an autocatalytic step or nonlinearity with respect to the mass balance (in CSTR reactors). 

The electrochemical behavior of the C70 solvent-cast films was similar to that of the C60 films, in that four reduction waves were observed, but some significant differences were also evident. The peak splitting for the first reduction/oxidation cycle was larger, and only abont 25% of the C70 was rednced on the first cycle. The prolate spheroidal shape of C70 is manifested in the II-A isotherm of C70 monolayers. Two transitions were observed that gave limiting radii consistent with a transition upon compression from a state with the long molecnlar axes parallel to the water snrface to a state with the long molecnlar axes per-pendicnlar to the water surface. 

For every pair of sections, BA and DN, of the actual path we have corresponding (isothermal) pairs, BC and DE, which are parts of an approximate Carnot cycle. In the limit of an infinite mrmber of infinitesimally small cycles, sections BA and DN can be considered isothermal at temperatures T equal to the respective temperatures T for sections BC and DE. Hence we can write... 

However, the temperature range of the Rankine cycle is severely limited by the nature of the working fluid— water. Adding superheat in an attempt to circumvent this will remove the cycle from isothermal heat addition. Increasing the temperature range without superheating leads to excessive moisture content in the vapor turbines, resulting in blade erosion. 

If an infinite number of intercoolers, compressors, reheaters, and turbines are added to a basic ideal Brayton cycle, the intercooling and multicompression processes approach an isothermal process. Similarly, the reheat and multiexpansion processes approach another isothermal process. This limiting Brayton cycle becomes an Ericsson cycle. 

Shendalman and Mitchell (6) have developed an equilibrium model which assumes that the adsorption isotherm is linear. The model predicts the limiting product composition for varying operating conditions as well as the product composition as a function of cycle number for the initial transient period. 

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