Abstract
Sal’nikov's chemical reaction in its simplest form 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 here 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 e.g., cool flames or a batch reactor. We first 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 number (Ra) to reach ∼ 103. The subsequent behaviour of the system depends on the interaction between convection, diffusion of heat and mass, and chemical kinetics. By examining the governing equations, we develop and evaluate scales for the characteristic velocity, the concentration of the intermediate A and the temperature rise during the progress of the reaction, for the two extreme cases when transport is dominated, in turn, by diffusion and then by natural convection. These scales depend on the characteristic timescales for the interacting phenomena of chemical reaction, diffusion and natural convection. Typically, the characteristic velocity in a relatively small reactor of radius 0.27 m is as large as 0.3 m s−1, when the temperature rise is ≈ 100 K near the centre of the vessel. Our theoretical predictions are verified by full numerical simulations. Oscillations of both the temperature and the concentration of the intermediate, A, are considered; the accompanying flow field proves to be toroidal, with the fluid ascending close to the reactor's axis, but descending adjacent to its walls. In addition, the effects of such process variables as the initial temperature of the batch reactor and its contents, the pressure and also the size of the reactor are all assessed, together with a consideration of what happens when the reaction proceeds in the liquid phase. In this case, because of the different physical properties of a liquid and a gas, natural convection is more intense than in the gas-phase and is quite likely to lead to turbulence and good mixing.