Alternate Derivation of the Critical Value of the Frank-Kamenetskii Parameter in Cylindrical Geometry

Abstract

Noether's theorem is used to determine first integrals admitted by a generalised LaneEmden equation of the second kind modelling a thermal explosion. These first integrals exist for rectangular and cylindrical geometry. For rectangular geometry the first integrals show the symmetry of the temperature gradients at the rectangular walls. For a cylindrical geometry the first integrals show the dependence of the critical parameter on the temperature gradient at the cylinder wall. The well known critical value for the Frank-Kamenetskii parameter, δ = 2, is obtained in a very natural way.

The critical condition for thermal explosion in an isoperibolic system

“…In the stationary approach, the condition under which a stationary state is impossible establishes the critical condition. For a given symmetry the boundary problem can be solved to obtain the critical value . In particular, if we neglect the reactant consumption and we assume an infinite activation energy, the critical condition depends only on the Frank‐Kamenetskii parameter (see Appendix ) where subscript cr stands for the parameter value at the threshold of ignition, and C ( 1D ) is a constant that depends on the system's geometry: for a spherical vessel, for an infinite cylinder and for a thin film.…”

“…For a given symmetry the boundary problem can be solved to obtain the critical value. 31,[62][63][64] In particular, if we neglect the reactant consumption and we assume an infinite activation energy, the critical condition depends only on the Frank-Kamenetskii parameter (see Appendix A) d cr 5C ð1DÞ (15) where subscript cr stands for the parameter value at the threshold of ignition, and C (1D) is a constant that depends on the system's geometry: C The nonstationary approximation consist in describing the evolution of a space-averaged temperature h and transformation degree a. It has been shown that this approach allows to reduce PDE systems that exhibit a low-dimensional dynamics to an equivalent ODE system.…”