In combustion, Clarke's equation is a third-order nonlinear partial differential equation, first derived by John Frederick Clarke in 1978.
[1][2][3][4] The equation describes the thermal explosion process, including both effects of constant-volume and constant-pressure processes, as well as the effects of adiabatic and isothermal sound speeds.
[5] The equation reads as[6] or, alternatively[7] where
is the non-dimensional temperature perturbation,
is the specific heat ratio and
is the relevant Damköhler number.
describes the thermal explosion at constant pressure and the term
describes the thermal explosion at constant volume.
Similarly, the term
describes the wave propagation at adiabatic sound speed and the term
describes the wave propagation at isothermal sound speed.
Molecular transports are neglected in the derivation.
can be removed from the equation by the transformation
may also appear in the initial and boundary conditions.
Suppose a radially symmetric hot source is deposited instantaneously in a reacting mixture.
When the chemical time is comparable to the acoustic time, diffusion is neglected so that ignition is characterised by heat release by the chemical energy and cooling by the expansion waves.
This problem is governed by the Clarke's equation with
is the maximum initial temperature,
is the gas constant and
denote the distance from the center, measured in units of initial hot core size and
be the time, measured in units of acoustic time.
In this case, the initial and boundary conditions are given by[6] where
, respectively, corresponds to the planar, cylindrical and spherical problems.
Let us define a new variable which is the increment of
from its distant values.
Then, at small times, the asymptotic solution is given by As time progresses, a steady state is approached when
and a thermal explosion is found to occur when
, the solution in the first approximation is given by which shows that thermal explosion occurs at
For generalised form for the reaction term, one may write where
is arbitrary function representing the reaction term.