Oscillations in a CSTR

Emulsion polymerizations in a CSTR can exhibit oscillations in the extent of conversion and the interfacial tension [68]. 

The study of nonlinear dynamics has been limited for too long to inorganic systems. We hope this chapter will spark enough interest to make one read and observe how polymeric systems can exhibit many of the same temporal and spatial phenomena observed in systems such as the BZ reaction and some surprising new ones. Polymer systems may provide the first commercial applications of this field, and will certainly provide interesting results in the future. 

Pojman, ).A. and Tran-Cong-Miyata, Q. (eds) (2003) Nonlinear Dynamics in Polymeric Systems, ACS Symposium Series No. 869, American Chemical Society, Washington, DC. 

Epstein, I.R. and Pojman, J.A. (1998) An Introduction to Nonlinear Chemical Dynamics Oscillations, Waves, Patterns and Chaos, Oxford University Press, New York. 

Strogatz, S.H. (1994) Nonlinear Dynamics and Chaos, Addison-Wesley, Reading, MA. 

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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. 

The system used in the Landolt clock reaction, IOj/SOl , when treated with [Fe(CN)6] , shows oscillations in a CSTR. A modification of the mechanism proposed previously resolves a number of problems. A slightly different system, I03/HS03/S20i produces a small number of high-amplitude pH oscillations (up to 2 units of pH) over a narrow range of conditions in a closed system. ... 

A bistability between an oscillatory branch and a flow branch has been reported for the gallic acid system by using the stirring rate as control parameter, indicating complex role of temperature and stirring rate as the control parameters [23], Bistability in an uncatalysed oscillator in a CSTR has been observed in initial [BrO jg [Br ]o concentration space. It occurs between an oscillatory and a flow branch. Hysterisis is also found to occur. 

Synthesis of Chemical Oscillations 79 Table 4.1 Chlorite-Based Chemical Oscillators in a CSTR... 

We are now ready to generalize our analysis to a simple design or search algorithm for new chemical systems that show Turing patterns. We start with a system that oscillates in a CSTR, preferably at relatively low flow rates, since high flow rates are not practical in a gel reactor. All model and experimental systems... 

A chlorite-thiosulfate system showed oscillations in a CSTR, and is the first chlorite-based oscillator involving no iodine-containing species. ... 

The new bromate oscillators really date from the experimental fulfillment of BAR-ELI s [23] prediction that a system consisting of bromate, bromide and cerous (or manganous) ions would show sustained oscillations in a CSTR. This "minimal bromate oscillator" was soon found by ORBAN et al. [24] and later independently by GEISELER [25]. Figure 1 shows the excellent agreement between the calculations based on the NFT mechanism [23,26] and the actual experimental conditions for oscillation. 

These results were sufficiently encouraging that a sensitivity analysis for this model was done. We chose to compare the calculated period of oscillation in a CSTR reactor to experimentally measured periods as a function of cobalt, bromide and benzaldehyde concentrations. 

Oscillating chemiluminescence has been observed in the H2O2/KSCN/ CuS04/NaOH/luminol system there are two types of oscillation, and the low-intensity mode may involve the reaction of superoxide with luminol. A simplified mechanism for the methylene-blue/HS /02 oscillation in a CSTR has been proposed. Experimental and modeling structures of oscillation in the C102/l2/malonic acid system show that the oscillations are not due to autocatalysis but to self-inhibition in the ClOf/I reaction. A study of the oxidation of hexacyanoferrate(II) by bromine is concluded to involve the formation of Brf as an intermediate, formed in the first step. The aerobic oxidation of NADH catalyzed by horse radish peroxidase... 

The existence of chaotic oscillations has been documented in a variety of chemical systems. Some of tire earliest observations of chemical chaos have been on biochemical systems like tire peroxidase-oxidase reaction [12] and on tire well known Belousov-Zhabotinskii (BZ) [13] reaction. The BZ reaction is tire Ce-ion-catalyzed oxidation of citric or malonic acid by bromate ion. Early investigations of the BZ reaction used tire teclmiques of dynamical systems tlieory outlined above to document tire existence of chaos in tliis reaction. Apparent chaos in tire BZ reaction was found by Hudson et a] [14] aiid tire data were analysed by Tomita and Tsuda [15] using a return-map metliod. Chaos was confinned in tire BZ reaction carried out in a CSTR by Roux et a] [16, E7] and by Hudson and... 

There are many variations on this theme. Fed-batch and continuous emulsion polymerizations are common. Continuous polymerization in a CSTR is dynamically unstable when free emulsifier is present. Oscillations with periods of several hours will result, but these can be avoided by feeding the CSTR with seed particles made in a batch or tubular reactor. 

The steady-state design equations (i.e., Equations (14.1)-(14.3) with the accumulation terms zero) can be solved to find one or more steady states. However, the solution provides no direct information about stability. On the other hand, if a transient solution reaches a steady state, then that steady state is stable and physically achievable from the initial composition used in the calculations. If the same steady state is found for all possible initial compositions, then that steady state is unique and globally stable. This is the usual case for isothermal reactions in a CSTR. Example 14.2 and Problem 14.6 show that isothermal systems can have multiple steady states or may never achieve a steady state, but the chemistry of these examples is contrived. Multiple steady states are more common in nonisothermal reactors, although at least one steady state is usually stable. Systems with stable steady states may oscillate or be chaotic for some initial conditions. Example 14.9 gives an experimentally verified example. 

Figure 14.2 shows the numerical solution. Except for a continuous input of ten rabbits and one lynx per unit time, the parameter values and initial conditions are the same as used for Figure 2.6. The batch reactor has been converted to a CSTR. The oscillations in the CSTR are smaller and have a higher frequency than those in the batch reactor, but a steady state is not achieved.

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.

Automatic controllers can produce small oscillations of the controlled variable. The effect of sinsoidal variations in concentration, temperature or feed rate on the effluent concentration of a second order reaction in a CSTR will be examined. The unsteady material balance is... 

It is well known that self-oscillation theory concerns the branching of periodic solutions of a system of differential equations at an equilibrium point. From Poincare, Andronov [4] up to the classical paper by Hopf [12], [18], non-linear oscillators have been considered in many contexts. An example of the classical electrical non-oscillator of van der Pol can be found in the paper of Cartwright [7]. Poore and later Uppal [32] were the first researchers who applied the theory of nonlinear oscillators to an irreversible exothermic reaction A B in a CSTR. Afterwards, several examples of self-oscillation (Andronov-PoincarA Hopf bifurcation) have been studied in CSTR and tubular reactors. Another... 

It is well known that a nonlinear system with an external periodic disturbance can reach chaotic dynamics. In a CSTR, it has been shown that the variation of the coolant temperature, from a basic self-oscillation state makes the reactor to change from periodic behavior to chaotic one [17]. On the other hand, in [22], it has been shown that it is possible to reach chaotic behavior from an external sine wave disturbance of the coolant flow rate. Note that a periodic disturbance can appear, for instance, when the parameters of the PID controller which manipulates the coolant flow rate are being tuned by using the Ziegler-Nichols rules. The chaotic behavior is difficult to obtain from normal... 

Note that Figure 13 can be used to compare the parameters of the controller when they are obtained from the Ziegler-Nichols or Cohen-Coom rules. On the other hand, at Figure 14 it can be observed that the outlet dimensionless flow rate and the reactor volume reaches the steady state whereas the dimensionless reactor temperature remains in self-oscillation. The knowledge of the self-oscillation regime in a CSTR is important, both from theoretical and experimental point of view, because there is experimental evidence that the self-oscillation behavior can be useful in an industrial environment. 

As we have already commented, mappings of the type discussed above are not in any way easily related to a given set of reaction rate equations. Such mappings have, however, been used for chemical systems in a slightly different way. A quadratic map has been used to help interpret the oscillatory behaviour observed in the Belousov-Zhabotinskii reaction in a CSTR. There, the variable x is not a concentration but the amplitude of a given oscillation. Thus the map correlates the amplitude of one peak in terms of the amplitude of the previous excursion. 

Sincic and Bailey (1977) relaxed the assumption of only one stable attractor for a given set of operating conditions and showed examples of some possible exotic responses in a CSTR with periodically forced coolant temperature. They also probed the way in which multiple steady states or sustained oscillations in the dynamics of the unforced system affect its response to periodic forcing. Several theoretical and experimental papers have since extended these ideas (Hamer and Cormack, 1978 Cutlip, 1979 Stephanopoulos et al., 1979 Hegedus et al., 1980 Abdul-Kareem et al., 1980 Bennett, 1982 Goodman et al., 1981, 1982 Cutlip et al., 1983 Taylor and Geiseler, 1986 Mankin and Hudson, 1984 Kevrekidis et al., 1984). 

Another exotic feature of emulsion polymerization in a CSTR is that sustained oscillations are frequently observed in... 

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. 

We have therefore made a preliminary investigation of the effects of such disturbances using the model of Ganapathisubramanian and Noyes (3). This is a seventh order model for the Belousov-Zhabotinskii reaction in a CSTR. The equations and all necessary parameters are given in their paper. The model predicts a periodic 2-3 oscillatory region bracketed by a two peak and a three peak periodic oscillation (for constant feed rates). The transition points predicted by the model have been calculated to two or three significant figures by numerical simulation. The transition between 11(2) and n(2,3) occurs at... 

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 dynamic model is discussed as a first attempt to explain the experimentally observed oscillations in the rate of propylene oxide oxidation on porous silver films in a CSTR. The model assumes that the periodic phenomena originate from formation and fast combustion of surface polymeric structures of propylene oxide. The numerical simulations are generally in qualitative agreement with the experimental results. However, this is a zeroth order model and further experimental and theoretical work is required to improve the understanding of this complex system. The in situ use of IR Spectroscopy could elucidate some of the underlying chemistry on the catalyst surface and provide useful information about surface coverages. This information could then be used to either extract some of the surface kinetic parameters of... 

Oscillations in the number of polymer particles, the monomer conversion, and the molecular weight of the polymers produced, which are mainly observed in a CSTR, have attracted considerable interest. Therefore, many experimental and theoretical studies dealing with these oscillations have been published [328]. Recently,Nomura et al. [340] conducted an extensive experimental study on the oscillatory behavior of the continuous emulsion polymerization of VAc in a single CSTR. Several researchers have proposed mathematical models that quantitatively describe complete kinetics, including oscillatory behavior [341-343]. Tauer and Muller [344] proposed a simple mathematical model for the continuous emulsion polymerization of VCl to explain the sustained oscillations observed. Their numerical analysis showed that the oscillations depend on the rates of particle growth and coalescence. However, it still seems to be difficult to quantitatively describe the kinetic behavior (including oscillations) of the continuous emulsion polymerization of monomers, especially those with relatively high solubility in water. This is mainly because the kinetics and mech-... 

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