Numerical determination of critical conditions for thermal ignition : a thesis presented in partial fulfillment of the requirements for the degree of Ph.D. in Mathematics at Massey University

Abstract

The determination of ignition or thermal explosion in an oxidizing porous body of material is critically reexamined from a modern perspective using numerical methodologies. First, the classic stationary model is revisited to establish the proper reference frame for the steady-state solution space, and it is demonstrated how the resulting nonlinear two-point boundary value problem can be re-expressed as an initial value problem for a system of first order differential equations, which may be readily solved using standard algorithms. Then, the numerical procedure is implemented and thoroughly validated against previous computational results based on sophisticated path following techniques. Next, the transient non-stationary model is attacked, and the full nonlinear form of the reaction-diffusion equation, including a generalized convective boundary condition, is discretized and expressed as a system of linear algebraic equations. The numerical methodology is implemented as a computer algorithm, and validation computations are carried out as a prelude to a broad ranging evaluation of the assembly problem and identification of the watershed critical initial temperature conditions for thermal ignition. This numerical methodology is then used as the basis for studying the relationship between the shape of the critical initial temperature distribution and the corresponding spatial moments of its energy content integral and an attempt to forge a fundamental conjecture governing this relation. Finally, the effects of dynamic boundary conditions on the classic “storage” problem are investigated and the groundwork is laid for the development of an approximate solution methodology based on adaptation of the standard stationary model.