The behaviour of Sal'nikov's reaction, P --> A --> B, in a spherical batch reactor with the diffusion of heat and matter.
The behaviour of a simple chemical reaction, occurring with the release of heat in a closed batch reactor, is considered for the situation when matter and heat are transported only by diffusive processes; thus, the reacting fluid has negligible velocity, so that heat transfer is by thermal conduction. The reaction is Sal'nikov's, which 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 of these steps is assumed to be thermoneutral, with zero activation energy, whilst the second is exothermic, with an appreciable activation energy. These features make Sal'nikov's reaction the simplest to display thermokinetic oscillations that characterise many, more complex schemes, e.g. cool flames in hydrocarbon combustion. This study involves identifying the regions of parameter space, in which these oscillations in the temperature and the concentration of the intermediate A occur, by means of numerical simulation. These regions are compared with previous analytical stability analyses in one-dimensional systems. It was found that oscillations occur over a much larger range of conditions in the case considered here, i.e. a reactor with spherical symmetry, than in the simple 1-D case, previously studied by Gray and Scott (P. Gray and S. K. Scott, Chemical Oscillations and Instabilities, Clarendon Press, Oxford, 1990, pp. 264-291). In addition, approximate analytical solutions for the temperature and concentration of A are presented for two limiting cases of non-oscillatory behaviour. These analytical solutions have been verified by comparison with full numerical solutions of the governing equations.