Frank-Kamenetskii theory of thermal ignition

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 

The stationary theory deals with time-independent equations of heat conduction with distributed sources of heat. Its solution gives the stationary temperature distribution in the reacting mixture. The initial conditions under which such a stationary distribution becomes impossible are the critical conditions for ignition. 

It is important to consider the definition of q. Defined as the amount of heat evolved by chemical reaction in a unit volume per unit time, q is the product of the terms involving the energy content of the fuel and its rate of reaction. The rate of the reaction can be written as Ze E,RT. Recall that Z in this example is different from the normal Arrhenius pre-exponential term in that it contains the concentration terms and therefore can be dependent on the mixture composition and the pressure. Thus, 

Since the stationary temperature distribution below the explosion limit is sought, in which case the temperature rise throughout the vessel must be small, it is best to introduce a new variable 

If the term in brackets is expanded and the higher-order terms are eliminated, this expression simplifies to 

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