Consider a cylindrical reactor supporting a solid$-$state endothermic reaction that absorbs heat volumetrically $($q$)$ within the annulus spanning r$2$ and r$3($see schematic below$).$ This reacting region is supported on its inner and outer surfaces by two metal tubes each having a thermal conductivity km$.$ The temperature at r$1$ is maintained at Tw within the inner cylinder. The outer surface of the reactor $($r $=$ r$4)$ is subject to convection from a fluid at temperature T $\backslash$infty and convection coefficient h$.$ The thermal conductivity of the reacting region is kr where kr