The Thermal Explosion Confined by a Constant Temperature Boundary:I—The Induction Period Solution

The time history of a spatially varying thermal explosion in a vessel with constant wall temperature is considered. A one-step irreversible, high activation energy reaction of the Arrhenius type is assumed to occur in a rigid, nondiffusing, combustible material. The induction period equations are solved numerically for a system confined to a slot-like region. A stiff-equation integrator is employed to delineate the nature of the thermal runaway process. A well defined hot spot is observed to form in the vicinity of the symmetry line. A precise description of the hot spot development is given in terms of an asymptotic theory valid close to the explosion time. The solution is constructed in terms of a slowly varying conduction-controlled outer region surrounding a much smaller zone in which the relatively rapid chemical kinetics determine how the hot spot therein develops. This analytical solution describes the final phase of the spatially varying induction period thermal runaway process, which cannot be obtained by numerical means alone. It is found that the dimension of the hot spot depends upon the square root of the product of the material thermal diffusivity and the time increment from the explosion time value. The hot spot development, which is kinetically controlled, is entirely independent of the vessel size.