A rod of diameter D = 25 mm and thermal conductivity k = 60 W/m ⋅ K protrudes normally from a furnace wall that is at Tw = 200°C and is covered by insulation of thickness Lins = 200 mm. The rod is welded to the furnace wall and is used as a hanger for supporting instrumentation cables. To avoid damaging the cables, the temperature of the rod at its exposed surface, To, must be maintained below a specified operating limit of Tmax = 100°C. The ambient air temperature is T[infinity] = 25°C, and the convection coefficient is h = 15 W/m2 ⋅ K. Derive an expression for the exposed surface temperature as a function of axial location on the rod if it is 700 mm long and the tip adiabatic, total and plot the temperature as a function of length. Repeat your calculations for an insulation layer only 100 mm thick.