Activation-energy asymptotics in ignition theory

Consideration of the history (to calculate quantities like ignition times) necessitates retention of time derivatives in the conservation equations. Just as in the previous section, to achieve the greatest simplicity we adopt a thermal theory, although in various applications that have been cited the full set of conservation equations has been considered. Let a reactive material occupy the region x 0, and to avoid complications assume that the material remains at rest and has a constant density p, although coordinate transformations readily enable this assumption to be removed. Let the material, initially at temperature Tq, be exposed to a constant heat flux q = —X dTJdx at x = 0 for all time t 0, where A is the constant thermal conductivity of the material. The time-dependent equation for conservation of energy for the material, analogous to equation (9), is 

FIGURE 8.4. Illustration of the solution to the problem of ignition by a constant heat flux. 

Ignition processes often are characterized by a gradual increase of temperature that is followed by a rapid increase over a very short time period. This behavior is exhibited in the present problem if a nondimensional measure of the activation energy E is large, as is true in the applications. Let L denote an ignition time, the time at which the rapid temperature increase occurs a more precise definition of L arises in the course of the development. In the present problem, during most of the time that t L, the material experiences only inert heat conduction because the heat-release term is exponentially small in the large parameter that measures E. The inert problem, with w = 0, has a known solution that can be derived by Laplace transforms, for example, and that can be written as 

A first approximation to the ignition time may be obtained by equating the heat-release term in equation (31) to either of the other two terms, as calculated from the inert solution, evaluated at x = 0. In view of equations (32) and (33), this gives 

To proceed with an improved theory, we may investigate the process of transition to ignition by stretching the time variable about t = t. To accomplish this, let 

For an improved analysis see J. W. Dold, Quart. J. Mech. Appl. Math. (1985), 

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