Derivation of the Frank-Kamenetskii equation

An equation holding between the rate of heat transfer per unit volume per unit time by the thermal conduction in a stationary medium, i.e., in a solid, in reality, and, the rate of increase in temperature of the solid is given in the following form. 

This is referred to as the Fourier equation regarding the thermal conduction. 

In this connection, when the shape of the solid chemical takes the onedimensional axial symmetry, the term, A T, is expressed as 

when a solid chemical of the TD type is placed in the atmosphere maintained at a temperature below T, which is shown in Fig. 2 in Section 1.2, a spatially gradient distribution of temperature is effected in the solid chemical. However, the temperature as a whole does not vary with time in other words, the spatially gradient distribution of temperature effected in the self-heating solid chemical placed in the atmosphere maintained at a temperature below T is stationary. The stationary equation of the thermal explosion theory, Eq. (22), is thus obtained by considering the value of the derivative with regard to the time to be zero in Eq. (20). This approach is called the stationary theory of the thermal explosion [7].  

For the argument made in this paragraph, refer to Section 2.3 as well. 

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