# Thermal ignition in rectangular and triangular regions

## Author

**M.J. Sexton**, **C. Macaskill** and **B.F. Gray**

## Status

Research Report 2000-10

To appear in Journal of the Australian Mathematical Society Series B
Electronic

Date: 24 September 1999, revised 15 May 2000

## 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.

## Key phrases

thermal ignition. method of lines. two-dimensional domains.

## AMS Subject Classification (1991)

Primary:

Secondary: 35K57, 65M20, 80A25

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