Problem $(2)$: A catalytic reaction is being carried out at constant pressure in a packed bed between coaxial cylindrical walls with inner radius $r_0$ and outer radius $r_1.$ The entire inner wall is at uniform temperature $T_0,$ and it can be assumed that there is no heat transfer through this surface. The reaction releases heat at a uniform volumetric rate $S_c$ throughout the reactor. The effective thermal conductivity of the reactor contents $k_{\text{e}\text{f}\text{f}}$ is to be treated as a constant throughout.

By a shell energy balance, derive a second$-$order differential equation that describes the temperature profiles, assuming that the temperature gradients in the axial direction can be neglected. What boundary conditions must be used?

Rewrite the differential equation and boundary conditions in terms of the dimensionless radial coordinate and dimensionless temperature defined as

$=\frac{r}{r_0}$;$,=\frac{T-T_0}{S_c\frac{r_0^2}{4}{k}_{\text{e}\text{f}\text{f}}}$

Integrate the dimensionless differential equation to get the radial temperature profile.

Develop expressions for the temperature at the outer wall and for the volumetric average temperature of the catalyst bed.