The Thermal Explosion Revisited

TL;DR

This paper extends the Frank–Kamenetsky thermal explosion model by allowing only a fraction $\alpha$ of the cylinder wall to conduct heat, with periodic segmentation into $N$ conducting/insulating arcs. The steady, large-activation-energy model $\Delta u + \lambda^2 e^u = 0$ is solved numerically using Newton–Raphson with careful extrapolation to locate the critical parameter $\lambda_{cr}$. Key findings include an axisymmetric core that persists for large $N$, a thin nonaxisymmetric boundary layer near the wall that drives the onset of explosion, and a power-law scaling $\lambda_{cr}^2 = S(N) \alpha^{t(N)}$ for small $\alpha$. The results explain why the critical radius is only weakly sensitive to $\alpha$ at large $N$ and quantify how boundary-patterning controls stability, with implications for design and safety in reactive-gas vessels.

Abstract

The classical problem of the thermal explosion in a long cylindrical vessel is modified so that only a fraction $\a$ of its wall is ideally thermally conducting while the remaining fraction $1-\a$ is thermally isolated. Partial isolation of the wall naturally reduces the critical radius of the vessel. Most interesting is the case when the structure of the boundary is a periodic one, so that the alternating conductive $\a$ and isolated $1-\a$ parts of the boundary occupy together the segments $2π/N$ ($N$ is the number of segments) of the boundary. A numerical investigation is performed. It is shown that at small $\a$ and large $N$ the critical radius obeys a scaling law with the coefficients depending upon $N$. For large $N$ is obtained that in the central core of the vessel the temperature distribution is axisymmetric. In the boundary layer near the wall having the thickness $\approx 2πr_0/N$ ($r_0$--the radius of the vessel) the temperature distribution varies sharply in the peripheral direction. The temperature distribution in the axisymmetric core at the critical value of the vessel radius is subcritical