Numerical determination of critical conditions for thermal ignition

Abstract

Ignition or thermal explosion in an oxidizing porous body of material can be described by a dimensionless reaction–diffusion equation of the form $u_t=\nabla^2u+\lambda\exp(−1/u)$. Here such equations will be formulated in symmetrically shaped bounded regions, effectively reducing the mathematical formulation to that of one dimension. This is critically re-examined from a modern perspective using numerical methods. A computer algorithm is constructed and used to carry out a broad-ranging evaluation of the watershed critical initial temperature conditions for thermal ignition of nonuniform assemblies. It is then shown how the resulting mathematical structure for the ignition threshold curves can be correlated by a hyperbolic conic section with a high degree of accuracy over the full range of positive ambient temperature values. However, this sometimes over-predicts (which is bad) and sometimes under-predicts (which is good) the critical initial condition. The definition of additional dimensionless parameters is found to generate further simplification, leading to a universal correlating form capable of collapsing the entire solution space onto a single line in the plane of the new variables. In addition, this study considers the physically intuitive conjecture that spatial moments of the initial temperature profile ought to possess a direct mathematical link to the critical ignition threshold. As such, the $m$th-order spatial moment of the critical total energy content integrals is defined, and an empirical result is derived stating that certain orders of this moment should be insensitive to changes in ambient temperature and initial shape profile and may be considered functionally dependent on the dimensionless eigenvalue only, within some quantifiable error band. Spatial moment integrals, based on computed critical threshold conditions, are found to support this conjecture, with the best accuracy obtained for the second-order moments. doi:10.1017/S1446181109000224