Find the rate of heat flow in the radial direction through a spherical shell of inner radius \(r_{\mathrm{i}}\) and outer radius \(r_{\mathrm{o}}\) for the case where the thermal conductivity varies as a linear function of temperature,

\[k=k_{0}(1+\beta T)\]

The inside shell is at temperature \(T_{\mathrm{i}}\) and the outside at temperature \(T_{0}\). Compare this with a case where the thermal conductivity is treated as a constant evaluated at the average temperature.