For many years combustion problems, regardless of their source of origin, were non-dimensionalised using the variables popularised by Frank-Kamenetskii [Frank-Kamenetskii 1955]. This uses a dimensionless temperature rise over the ambient given by

\theta = \frac{E}{RT^{2}_{a}}\left(T-T_{a}\right).

The simplest spatially non-uniform combustion problem then contains two non-dimensionalised parameters, a reduced activation energy ($\epsilon$) and the Frank-Kamenetskii parameter ($\delta$), often called the Danköhler number in the chemical engineering literature, defined by

\epsilon & = & \frac{RT_{a}}{E}, \\ \delta & = & \frac{\rho Qa_{0}^{2}E}{kRT_{a}^{2}}\cdot A\exp\left[-\frac{E}{RT_{a}}\right].

The power provided by this choice of variables is that many problems have an analytic solution when the pre-exponential approximation ($\epsilon = 0$) is made [Bowes 1984].

In the mid 1980s it was realised that the Frank-Kamenetskii non-dimensionalisation complicates the role played by the ambient temperature in defining the combustion problem; in many case this is the most important experimental control parameter. Notice that it appears in both the reduced activation energy and the Frank-Kamenetskii parameter. Consequently varying one of these parameters, whilst keeping the other fixed, does not reflect experimental methodology; whereby ambient temperature is varied as the bifurcation parameter. Therefore the Frank-Kamenetskii variables are inappropriate when ambient temperature is viewed as the main bifurcation parameter. A further complicating factor is that ambient temperature also appears in the definition of the non-dimensionalised temperature-scale.

Accordingly a non-dimensionalisation was
introduced [Gray and Roberts 1988,
Gray and Wake 1988,
Burnell `et al` 1989]
in which
a non-dimensionalised ambient temperature is retained as a
distinguished bifurcation parameter. This is discussed in
more detail elsewhere
[Gray and Wake 1988,
Burnell `et al` 1989,
Gray `et al` 1991].
In particular, it was shown that the use of the exponential approximation
($\epsilon =0$) leads to large errors in the calculation
of critical initial temperatures
[Gray and Wake 1988] and that
the classical non-dimensionalisation is sufficiently nonlinear
to conceal some of the differences in behaviour shown when
different bifurcation parameters are varied
[Gray `et al` 1991].

**References**

- P.C. Bowes.
`Self-heating: evaluating and controlling the hazards`. Elsevier, Amsterdam, 1984. - J.G. Burnell, J.G. Graham-Eagle, B.F. Gray, and
G.C. Wake.
Determination of critical ambient temperature for thermal
ignition.
`IMA Journal of Applied Mathematics`,**42**:147--154, 1989. - D.A. Frank-Kamenetskii.
`Diffusion and heat exchange in chemical kinetics`. Princeton University Press, first edition, 1955. - B.F. Gray, J.H. Merkin, and G.C. Wake.
Disjoint bifurcation diagrams in combustion systems.
`Mathematical and Computer Modelling`,**15**(11):25--33, 1991. - B.F. Gray and M.J. Roberts.
Analysis of chemical kinetic systems over the entire parameter space
I. The Sal'nikov thermokinetic oscillator.
`Proceedings of the Royal Society A`,**416**: 391--402, 1988. - B.F. Gray and G.C. Wake.
On the determination of critical ambient temperatures and
critial initial temperatures for thermal ignition.
`Combustion and Flame`,**71**:101--104, 1988.

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