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258 5. Some worked examples arising from physical problems

E5.24 Thermal runaway
A phenomenon that can be encountered in certain chemical reactions involves the
release of heat (exothermic) which increases the temperature, and the temperature nor-
mally controls the reaction rate. It is possible, therefore, to initiate a reaction, then heat
is released which raises the temperature and so increases the rate of reaction which
releases even more heat, and so on; this is called thermal runaway. In the most extreme
cases, there is no theoretical limit to the temperature, although physical reality inter-
venes e.g. the containing vessel might melt or the products explode. A standard model
used to describe this (Szekely, Sohn & Evans, 1976; see also Fowler, 1997, which
provides the basis for the discussion presented here) is the equation

where T is the temperature and is a parameter. In this example, we will examine the
nature of the steady-state temperature in one dimension i.e. the solution of

as for various The boundary condition is that the external temperature is
maintained; we will represent this by T = 0 on and so we seek a solution
for (and we will assume symmetry of the temperature distribution about
x = 0).
An important property of the solution of (5.138) can be derived by first writing

where satisfies the equation

with the boundary conditions This problem has the exact solution

where is the maximum temperature (attained at x = 0) defined by