Thermal explosions, criticality and the disappearance of criticality in systems with distributed temperatures III. Asymptotic analysis of the disappearance of criticality (transition) at the extreme of small Biot number

Abstract

Although numerical attacks on the problem of transition (loss of criticality) in thermally igniting systems with generalized resistance to heat transfer are always possible, they are particular to the cases studied. Very general circumstances may be treated analytically by perturbation methods (asymptotic expansion). Asymptotic expressions of considerable precision starting from the Semenov extreme (( Bi ) = 0) can be made in terms of Biot number ( Bi ). They apply to any geometry, and they are presented here for the infinite slab, the infinite cylinder and the sphere as expressions in terms of ( Bi ) for transitional values of the Frank-Kamenetskii or Semenov parameters, for transitional values of centre temperatures θ 0 and for transitional values of reduced ambient temperatures є . The method can cope with any temperature-dependence of rate coefficient f ( θ , є ). Numerical comparison is given for the case corresponding to Arrhenius kinetics, namely f ( θ , ε ) = exp [ θ /(1 + єθ )].