A cylinder of diameter \(D=10 \mathrm{~mm}\) and temperature \(T_{1}=\) \(100^{\circ} \mathrm{C}\) is centrally located in an extruded square coating of bakelite of dimension \(w=11 \mathrm{~mm}\). The ambient temperature is \(T_{\infty}=25^{\circ} \mathrm{C}\) and a convection coefficient of value \(h=100 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\) exists at the outer surfaces of the bakelite. Determine the heat transfer rate per unit length of the cylinder. Plot the heat transfer rate per length over the range \(10 \mathrm{~mm} \leq w \leq 100 \mathrm{~mm}\). Explain the dependence of the heat transfer rate on the bakelite thickness that is evident in your plot. Also derive the critical value of \(w\) that maximizes heat transfer from the cylinder.