Non-dimensionalising combustion problems: Why the Frank-Kamenetskii variables lost favour

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

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].


  1. P.C. Bowes. Self-heating: evaluating and controlling the hazards. Elsevier, Amsterdam, 1984.
  2. 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.
  3. D.A. Frank-Kamenetskii. Diffusion and heat exchange in chemical kinetics. Princeton University Press, first edition, 1955.
  4. 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.
  5. 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.
  6. 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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