Frank-Kamenetskii-rate expression

As implied in the previous section, the Russian investigators Zeldovich, Frank-Kamenetskii, and Semenov derived an expression for the laminar flame speed by an important extension of the very simplified Mallard-Le Chatelier approach. Their basic equation included diffusion of species as well as heat. Since their initial insight was that flame propagation was fundamentally a thermal mechanism, they were not concerned with the diffusion of radicals and its effect on the reaction rate. They were concerned with the energy transported by the diffusion of species. 

This problem was addressed and solved by Frank-Kamenetskii [6], who established the heat balance of a solid with a characteristic dimension r, an initial temperature T0 equal to the surrounding temperature, and containing a uniform heat source with a heat release rate q expressed in W m The object is to determine under which conditions a steady state, that is, a constant temperature profile with time, can be established. We further assume that there is no resistance to heat transfer at the wall, that is, there is no temperature gradient at the wall. The second Fourier Law can be written as (Figure 13.2)... 

Va cannot be solved explicitly from (3.1-7) for arbitrary n, so that no equation equivalent to (3.1-3) or (3.1-6) is obtained, may be obtained by iterative methods [see Frank-Kamenetskii, 1969]. Thus, in general, consecutive rate processes of different order cannot easily be combined into an overall expression, but can be handled by numerical techniques. 

---