Thermal Explosions

While the term thermal explosion sounds very dramatic, it really just refers to a thermal event where the heat release (or absorption) occurs over a time period that is very small relative to the time scale of interest, and in a volume that is negligibly small compared to the surroundings in which the temperature distribution is to be calculated.  So, for example, the heat released by the combustion of  blasting powder in a small hole (time scale of milli-seconds) could be analyzed as a thermal explosion if the time scale of interest was on the order of seconds.  Also the decay heat generated by a pocket of radioactive rock (say, over 100 years) could be analyzed as a thermal explosion if the period of interest were, say, 10,000 years.

The transient temperature distribution resulting from heat conduction in a homogeneous medium due to a thermal explosion can be easily derived for the case where the surrounding medium is initially at a single uniform temperature.  The case of a point source energy release, with the temperature rise spreading radially with time may be the easiest to visualize.  For that case:

In this expression:

T =temperature rise (above the initial temperature of the medium) as a function of time (t) and distance (r) from the explosion

r =radial distance from the point of the thermal explosion

t =time from the explosion

ρ =density of the surrounding medium

c =specific heat capacity of the surrounding medium

α =thermal diffusivity of the surrounding medium

E₀=total heat released in the thermal explosion

Expressions for a line source (say, a buried wire which has a sudden electrical overload) and a plane source (say, heat generated by friction between two sliding tectonic plates) are also available.

 This figure shows the temperature rise distribution for a heat release of 6000 J from a point source using properties of sand for the surrounding medium.  Imagine a buried resistor dissipating 100 W for 1 minute.  You can see that as time passes, the temperature distribution gets wider and smaller.  This makes sense because there is only a fixed amount of energy (E₀) involved, and as the heat is conducted away from the original source, the temperature rise (above the initial temperature of the surrounding medium) gets smaller.

This figure shows the temperature as a function of time for fixed distances from the thermal explosion.  As you might surmise from looking at the first figure, at a given point the temperature will first rise, then drop off as the “heat wave” from the explosion passes over that point.

Finally, you might also imagine that a given temperature will be found close to the explosion at first, then move away as the temperature distribution flattens out, then move back in again as it continues getting flatter.  This can be seen in this final figure.

   

Reference

Grigull, U., Sandner, H., Heat Conduction, Hemisphere Publishing Corporation, 1984.