Thermal ignition in rectangular and triangular regions

Abstract

When cellulosic materials such as cotton, hay, sawdust or bagasse (sugar-cane residue) are stored in sufficiently large quantities they may self-heat with the possibility of spontaneous ignition. Mathematically, there is a bifurcation to the burning state if ignition occurs. It is important to know the critical values of the basic physical quantities, such as the ambient temperature or characteristic size of the self-heating sample, at which the bifurcation to the burning state takes place. The solution method for this class of problem depends strongly on the domain under consideration. Here we consider triangular and rectangular domains with the appropriate mixed boundary conditions. The governing PDEs for the time-dependent problem can be solved by the method of lines, with finite difference schemes used for the discretisation of the spatial derivatives. Any suitable ODE solver can be used for the time integration, so that stiff problems such as those that arise naturally in combustion problems are easily dealt with. In addition, with this approach the steady-state equations are readily extracted and hence the bifurcation structure describing the criticality of the material can be calculated without difficulty. We demonstrate the crucial role played by the boundary conditions in determining, for example, the location of the point of maximum heating.