Digital simulation of thermal reactions

Simulations of the equation for thermal expansion of a reacting gas have been carried out, exploring both the (possible) steady states and time-marching solutions. The critical Frank-Kamenetskii parameter δcr has been evaluated to seven decimal places for the slab, cylinder and spherical geometries and the role of the critical activation parameter ϵ was explored. It was found that there exist one or more mathematical steady states for any δ if ϵ>0, the curves for steady temperature at the center of the geometry plotted against δ tending to a straight line at large δ. Critical values of ϵ, the values above which this plot has a single solution for a given δ, have been computed to eight decimals. Time marching simulations showed that the Crank-Nicolson method, applied consistently, produces very accurate results, compared with the implementation in which the nonlinear term is rendered explicit. Where for a given δ there are several mathematical steady states, a time march usually settles on the lowest such state (if it settles at all), regardless of where the simulation is started, within the possible limits. The mathematical multiple steady states are not attained by time marching simulations, and are also physically unlikely.