A pipe is used for transporting hot fluid in which the inner surface is at
\(150^{\circ} \mathrm{C}\). The pipe has a wall thickness of \(5 \mathrm{~mm}\)
and an inner diameter of \(15 \mathrm{~cm}\). The pipe wall has a variable
thermal conductivity given as \(k(T)=k_{0}(1+\beta T)\), where $k_{0}=8.5
\mathrm{~W} / \mathrm{m} \cdot \mathrm{K}, \beta=0.001 \mathrm{~K}^{-1}$, and
\(T\) is in \(\mathrm{K}\). The pipe is situated in surroundings of freezing air
at \(0^{\circ} \mathrm{C}\) with a convection heat transfer coefficient of $60
\mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$ on the pipe's outer surface.
Solar radiation is incident on the pipe's outer surface at a rate of $100
\mathrm{~W} / \mathrm{m}^{2}$, and both the emissivity and solar absorptivity
of the outer surface are \(0.9\). Determine the outer surface temperature of the
pipe.