Frank-Kamenetskii transformation

Using the Frank-Kamenetskii (1969) transformation we present the relation in Eq. (3.27) in the following form (Zel dovich et al. 1985) ... 

Frank-Kamenetskii first considered the nonsteady heat conduction equation. However, since the gaseous mixture, liquid, or solid energetic fuel can undergo exothermic transformations, a chemical reaction rate term is included. This term specifies a continuously distributed source of heat throughout the containing vessel boundaries. The heat conduction equation for the vessel is then... 

We assume—at present this is only a hypothesis—that our theoretical investigations relate also to the combustion of powders, and especially to smokeless powders. Classical internal ballistics does not consider the question of the intimate mechanism and stages of the chemical reactions which comprise the essence of powder combustion. However, many of Belyaev s arguments and our—Frank-Kamenetskii and the author—work remain applicable. The presently proposed hypothetical scheme of powder combustion differs only in that, instead of reversible evaporation, we must speak of a primary (irreversible) reaction transforming the powder into a gas, but not into the final combustion products, followed by the reaction of combustion proper, accompanied by the release of the primary part of the reaction heat. 

The formula of Zel dovich and Frank-Kamenetskii [7], [34] may be obtained from the simplest result of von Karman by performing an asymptotic expansion of the integral 1 for large values of jS and retaining only the first term in the expansion. In view of equation (43), by transforming from the variable t to z = P ( T)/(l/a + t) in equation (64), we find that... 

ZeFdovich and Frank-Kamenetskii found that this condition can be transformed to... 

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