Analysis of the Early Stage of Thermal Runaway

It is shown that a variable grouping containing a logarithmic time-dependent factor is required for the development of a coordinate-perturbation expansion which realistically describes the spatially varying thermal-runaway process. The resulting solution is in good agreement with careful numerical computations. It describes a self-focussing temperature growth, the form and behaviour of which are remarkably independent of both the conditions leading to thermal runaway and the topology of the thermal-runaway region. The detailed solution also reveals an underlying structure in the temperature development, in which a strongly supercritical thermal runaway (where the relative effects of conduction are initially small) is found to be very much like a less strongly supercritical, but more highly-developed, thermal runaway. In this development the local rate of self-heating accelerates dramatically while, in comparison, the conductive impediment to the temperature growth diminishes towards zero. Attempts to develop a solution using a variable grouping without the logarithmic factor are shown to produce results which are unsuitable for describing inhomogeneous thermal runaway. Some numerically computed results are presented, detailing the Ignition Kernel formed as a result of supercritical thermal runaway in a fixed-temperature symmetric container.