Oscillations of simple exothermic reactions in a closed system. II. Exact Arrhenius kinetics

Kay, S. R.; Scott, Stephen K.

doi:10.1098/rspa.1988.0038

Cited by 27 publications

(9 citation statements)

References 2 publications

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“…In a later paper, Kay and Scott [11] analysed the exact equations with the full exponential non-linearity, and showed that there are parameter regions in which use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0334270000009103 two oscillatory solutions can exist simultaneously; one of these is stable and the other unstable.…”

Section: Rt Amentioning

confidence: 99%

One-dimensional pattern formation in a model of burning

Forbes

1993

J. Aust. Math. Soc. Series B, Appl. Math

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We discuss a model of a burning process, essentially due to Sal'nikov, in which a substrate undergoes a two-stage decay through some intermediate chemical to form a final product. The second stage of the process occurs at a temperature-sensitive rate, and is also responsible for the production of heat. The effects of thermal conduction are included, and the intermediate chemical is assumed to be capable of diffusion through the decomposing substrate. The governing equations thus form a reaction-diffusion system, and spatially inhomogeneous behaviour is therefore possible.This paper is concerned with stationary patterns of temperature and chemical concentration in the model. A numerical method for the solution of the governing equations is outlined, and makes use of a Fourier-series representation of the pattern. The question of the stability of these patterns is discussed in detail, and a linearised solution is presented, which is valid for patterns of very small amplitude. The results of accurate solutions to the fully non-linear equations are discussed, and compared with the predictions of the linearised theory. Parameter regions in which there exists genuine nonuniqueness of solutions are identified.

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Section: Rt Amentioning

confidence: 99%

One-dimensional pattern formation in a model of burning

Forbes

1993

J. Aust. Math. Soc. Series B, Appl. Math

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“…Previous theoretical studies of such a system have focussed largely on two limiting cases of behaviour, namely the well-mixed, spatially uniform case obtained with forced convection, (see e.g. Kay and Scott 1988;Gray and Roberts 1988;Forbes 1990;Gray and Scott 1990a) when the effects of diffusion can be neglected, and the case with purely diffusive transport of heat and mass (Gray and Scott 1990b).…”

Section: Introductionmentioning

confidence: 99%

A scaling analysis of Sal'nikov's reaction, P→A→B, in the presence of natural convection and the diffusion of heat and matter

2005

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1 Sal'nikov's chemical reaction is very simple; it consists of two consecutive first-order steps, producing a product B from a precursor P via an active intermediate A, in P → A → B. The first step is assumed to be thermoneutral, with zero activation energy, whilst the second step is exothermic and has a positive activation energy. These properties make this mechanism one of the simplest to display thermokinetic oscillations, as seen in cool flames. We consider a pure gas, P, undergoing Sal'nikov's reaction in a closed spherical vessel, whose walls are held at a constant temperature. Natural convection becomes significant once the temperature is high enough for the Rayleigh

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“…An approximate solution, based on integrated forms of the boundary-layer equations can be constructed. This leads to a system of equations similar to the Sal'nikov scheme, discussed in detail by Kay and Scott [21,22]. The corresponding steady states of this approximate solution have all the features of the present problem (saddle-node bifurcations and hysteresis points) as well as supercritical Hopf bifurcations, suggesting that these could also be a feature of the present initial-value problem (7).…”

Section: Time Dependent Problemmentioning

confidence: 79%

“…Finally we note that (22) implies that the critical points become large, of 0((_ 4 ), as --> . In particular…”

Section: Non Fuel Consumption Case (A = 0)mentioning

confidence: 99%

Free-convection stagnation-point boundary layers driven by catalytic surface reactions: I the steady states

Chaudhary

Merkin

1994

J Eng Math

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The free convection boundary-layer flow near a stagnation point driven by catalytic surface heating is considered. The case without fuel consumption is treated first, and it is shown that the steady state equations admit multiple solutions. Explicit expressions can be obtained for these solution branches and it is found that a hysteresis point occurs when the activation energy parameter e = 1/5. The effect of fuel consumption is seen to be characterised by the dimensionless parameter a and numerical results are obtained for a range of values of a and E, as well as Prandtl number or and Schmidt number S c . Multiple solutions are again observed and analytic expressions for the bifurcation points can be found when a-= S c . For o-# Sc these have to be determined numerically.

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Oscillations of simple exothermic reactions in a closed system. II. Exact Arrhenius kinetics

Cited by 27 publications

References 2 publications

One-dimensional pattern formation in a model of burning

One-dimensional pattern formation in a model of burning

A scaling analysis of Sal'nikov's reaction, P→A→B, in the presence of natural convection and the diffusion of heat and matter

Free-convection stagnation-point boundary layers driven by catalytic surface reactions: I the steady states

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