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M.I. Nelson and H.S. Sidhu. Bifurcation phenomena for an oxidation reaction in a continuously stirred tank reactor. I Adiabatic operation. Journal of Mathematical Chemistry, 31(2):155--186, 2002. Abstract available.
In this article I review work examining the Sal'nikov scheme in well-stirred systems.
The simplest non-isothermal model generating oscillatory behaviour is due to Sal'nikov [Sal'nikov 1948; Sal'nikov 1949]. Sal'nikov's aim was to explain the occurrence of cool flames observed in the oxidation of hydrocarbons. It is now accepted that the correct description of this phenomenon is the unified chain-thermal theory developed by Gray and Yang [Gray 1969; Yang & Gray 1969]. Sal'nikov's model retains interest because it provides an archetypal example of non-isothermal chemical oscillations that completely satisfies chemical principles whilst being simple enough to be understood easily and to be analysed deeply. Moreover, as described in section 2.4 it has been realised experimentally, thus showing that chemical complexity involving chain branching is not a prerequisite for the existence of thermokinetic oscillations.
Sal'nikov's scheme consists of an exothermic reaction proceeding by two consecutive first-order reactions. A precursor F generates a reactive species B which decomposes through an exothermic reaction to an inert product C. It is assumed that all the heat output is associated with the second reaction. This scheme is represented as
Sal'nikov investigated the dynamics of the single and double schemes using techniques from classical dynamical systems theory. He assumed that the reactions took place in a well-stirred batch reactor and that depletion of the precursor species was negligible. A region in parameter space was established in which the system is in a state of undamped oscillations since the relevant phase portrait consists of an unstable steady state surrounded by a stable limit cycle.
The single Sal'nikov scheme was examined by Gray and Roberts [1988a] using the techniques of singularity theory. They interpreted the boundary of the region enclosing cool flames as the locus of Hopf bifurcation points in the pressure-ambient temperature plane. A complete description of all the qualitatively distinct behaviour exhibited by the model was obtained, including two regions of parameter space not previously located. Subsequently they investigated two extensions of this work [Gray & Roberts, 1988b]. Firstly, the limiting asymptotic behaviour was obtained exactly as the dimensionless heat capacity tends to zero. Secondly, the effects of including fuel consumption was studied both numerically and in a second asymptotic-limit as the rate of fuel consumption tends to zero.
Simultaneously with the work of Gray and Roberts, the single Sal'nikov scheme was investigated by a group at the University of Leeds [P. Gray, 1988; Gray et al 1988; Kay & Scott 1988]. A drawback of this work is the use of a non-dimensionalisation scheme popularised by Frank-Kamenetskii. These variables have the disadvantage that a key experimental control parameter, the ambient temperature, appears in more than one dimensionless variable. It is therefore difficult to directly relate the results of bifurcation analysis to experimental data. The disadvantages of using the Frank-Kamenetskii variables are discussed elsewhere. The work in Leeds was not as comprehensive as that reported by Gray and Roberts.
Forbes  has provided a rigorous proof of the non-existence of oscillations in certain regions of parameter space. Although the Frank-Kamenetskii variables were used, the results are of a form whereby their dependence upon the ambient temperature-pressure can be extracted. Moreira and Yuquan , also using the Frank-Kamenetskii variables, established conditions under which the single Sal'nikov scheme has an unstable limit cycle inside a stable one. The proof requires rewriting the system in the form of a Liénard equation.
The main success of Sal'nikov's scheme is that it can generate thermokinetic oscillations. It is therefore natural to investigate the effect of periodic forcing. Forbes and Gray  considered sinusoidal forcing of the ambient temperature. They showed that such forcing can give rise to chaos, which results from either the Feigenbaum period-doubling route or from the Ruelle-Takens approach through quasi-periodicity. Delgado  considered forcing of the temperature derivative. In a rather cursory treatment, it is shown that complex and mixed mode oscillations are possible, chaotic behaviour was not found. Delgado  then considered periodic forcing of the temperature derivative in a system in which the reactor volume is proportional to the temperature. Under these assumptions a period-doubling route to chaos was found as the amplitude of the forcing term is decreased. It was subsequently demonstrated that unstable periodic orbits embedded in the strange attractor could be stabilised by a continuous delay time method [Delgado et al, 1995]. Delgado has also investigated a thermal engine driven by the single Sal'nikov oscillator . The heat released by the second reaction is used to move a piston which exchanges work between the system and its environment. The piston is shown to have a three-stage cycle comprising nearly isometric, isobaric, and adiabatic branches.
A preliminary investigation of this scheme was made by Forbes et al . The border in parameter space between regions possessing one and three equilibria was identified, as was the location of the Hopf locus. They proved that oscillatory behaviour is only possible within a certain region of the parameter space. The definitive analysis of this system was made by Gray and Forbes  who showed that it contains 16 qualitatively different phase portraits. A large number of degenerate bifurcations were identified and located, including a spectacularly degenerate Bogdanov-Takens bifurcation that generates three homoclinic bifurcation curves rather than the usual one. Many of these bifurcations are structurally unstable in the sense that their codimension exceeds the number of unfolding parameters. It was shown that the exotic behaviour only occurs if the activation energy of the first reaction is smaller than that of the second, as conjectured by Sal'nikov.
Sexton and Forbes  investigated the use of linear control in eliminating oscillations in batch reactor. They showed that oscillatory behaviour can not always be eliminated using negative feedback. In fact there are circumstances whereby negative feedback creates a worse situation than no feedback, in these conditions positive feed-back is required to eliminate limit cycles. In addition they exhibited a nonlinear feedback control which guarantees no oscillations.
Sal'nikov schemes typically generate relaxation oscillations of considerable stiffness and special numerical solution techniques are required to obtain periodic solutions. Forbes  described a shooting algorithm which gave results of great accuracy, automatically determined the stability of the solution, and was capable of computing unstable periodic orbits. Herges and Twizell  computed solutions using the first-order explicit Euler method and an implicit finite-difference method of the same order. Neither of these latter methods were able to compute unstable limit cycles.
The single Sal'nikov scheme has been realised experimentally in a semi-batch reactor [Coppersthwaite et al 1991; Gray 1988; Gray & Griffiths 1989; Griffiths et al 1988]. In this work the thermally-neutral temperature-independent decay of the precursor species was mimicked by the physical process of admitting the reactant into the batch reactor at a controlled constant rate via a calibrated capillary leak. The reactant itself, stored in an external reservoir connected to the capillary, constitutes the precursor species.
The dynamics of the decomposition of di-tert-butyl peroxide [Gray 1988; Gray & Griffiths 1989; Griffiths et al 1988] and the hydrogen-chlorine reaction [Coppersthwaite et al 1991] have been investigated. The behaviour in both these systems were shown to be satisfactorily explained by a single Sal'nikov scheme. Under the experimental conditions investigated the rate of decomposition of di-tert-butyl peroxide is first order, the exothermicity of the overall reaction is modified in the presence of oxygen but the rate determining step is not changed. This system is therefore analogous to a single Sal'nikov scheme. The chemistry of the hydrogen-chlorine reaction is more complicated, comprising five elementary reactions and five chemical species. By making appropriate approximations it is possible to reduce this to a single Sal'nikov scheme. The hydrogen-chlorine system has been investigated by Sidhu et al  who compared the location in parameter space in which oscillations are predicted to occur using the full chemical scheme against that predicted by the Sal'nikov model.
Nelson & Sidhu  modified the single Sal'nikov thermokinetic scheme by introducing an additional chemical species, making the second stage an oxidation reaction. Their chemical scheme is represented by