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].
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