On the Critical Conditions for Thermal Explosion in Slab Geometries: A Frank-Kamenetskii Approach and Chebyshev Approximation

Abstract

This study examines the thermal explosion behaviour in slab geometry using the classical Frank-Kamenetskii theory. The steady-state heat conduction equation with the Arrhenius reaction rate is reduced to a dimensionless form resulting in a nonlinear boundary value problem. The critical Frank- Kamenetskii parameter δc of slab geometry is determined  both analytically and numerically using the Chebyshev spectral collocation method. The study shows that beyond a certain threshold value, a steady-state solution does not exist, indicating the occurence of thermal runaway. We derive exact relationships of the temperature profile with respect to the spatial coordinate and compute the value of the critical parameter. The Chebyshev method producec results that agree closely with the analytically computed value to a high degree of accuracy. Several graphs are presented to illustrate the analytical and numerical results. The findings demonstrate that slab geometry plays an important role in the safe design of chemical reactors to avoid thermal explosions, and that they are distinctly different in critical behavior compared to cylindrical and spherical geometries.