The onset of thermal runaway in partially insulated or cooled reactors

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Abstract
The conditions for the onset of thermal runaway in partially insulated or cooled reactors are investigated. The temperature in the reactor is taken to satisfy a nonlinear elliptic equation and the reaction is modelled by an Arrhenius heat generation term with finite activation energy. To determine the onset of thermal runaway, the method of matched asymptotic expansions is used to derive expressions for the critical Frank-Kamenetskii parameter λ c (ε) for reactors containing either a small cooling rod or having a small cooling patch on their boundary. The theory used to determine λ c (ε) is an extension of the results of Ward and Keller (1991). These previous results of Ward and Keller are also extended to the case of finite activation energies by using a numerical scheme to evaluate the coefficients in the asymptotic results for λ c (ε). In some special cases, the asymptotic expansions for λ c (ε) are compared with numerical results for λ c (ε), and clear agreement is found.

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