Temperature distributions, critical conditions and scaling for exothermic materials under different boundary conditions. Part 1.—Semenov and frank-kamenetskii extremes

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Abstract
An analytical solution to the temperature distribution within bodies of a wide range of shape and in which heat transfer is controlled by conduction is presented. This affords evaluation of the critical conditions for the onset of thermal runaway. The solution invokes a shape-factor j and a size factor R0(a characteristic radius), and through these the calculation of critical size as a function of ambient temperature is possible. Although not exact, the solutions have simple forms, are easy to apply, and reproduce all known exact results to better than 5% precision. More complicated boundary conditions, where heat transfer is not controlled solely by conduction, are considered in a second part. The uniform temperature approximation (Semenov boundary conditions) is exactly soluble, criticality being given by a simple function of the surface-to-volume ratio.

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