Steady-state equation of thermal explosion in a distributed activation energy medium: numerical solution and approximations

Abstract

This work presents a mathematical model of thermal explosion in a medium described by a Gaussian distribution of reactivity, along with the determination of critical values for model parameters and their dependence on the distribution dispersion. The numerical solution of boundary value problems for steady-state temperature distribution in a reaction medium (a sweep method along with the iterative refinement of a source function, a half-interval method to find the critical value of the Frank-Kamenetskii parameter) was used. The grid convergence was investigated for the used difference scheme; the first order of accuracy was observed as a result of numerical evaluation of the critical value of the Frank-Kamenetskii parameter. Calculations were carried out with accuracy to three decimal places. Numerical methods were implemented as programs in the MATLAB environment. Numerical approximations were obtained for solutions of the thermal explosion equation characterised by distributed activation energy in the quasi-steady-state approximation. It was shown that the critical value of the Frank-Kamenetskii parameter is associated with the dispersion of the distribution and the Arrhenius parameter by a simple approximate analytical formula, confirmed by comparing with numerical estimates. Since the dependence of the critical value of the Frank-Kamenetskii parameter on the dispersion is described by a Gaussian function, the reaction medium becomes thermally unstable even at small values of the distribution dispersion. Calculations showed that a significant dispersion of reactivity (on the order of tenths of the average) can be observed only for chemical reactions characterised by low sensitivity to temperature (i.e. a small heat effect or low activation energy). Approximate formulas for critical conditions were also obtained for asymmetrical distribution functions. The analysis allows the proposed mathematical model to be used for assessing the thermal stability of reactive media having distributed reactivity (for example, natural materials, polymers, heterogeneous catalytic systems, etc.).