4. 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 = 25 C, and the convection coefficient is h = 15 W/m-K (a) Derive an expression for the exposed surface temperature To as a function of the prescribed thermal and geometrical parameters. The rod has an exposed length Lo and its tip is well insulated. (b) Will a rod with Lo = 200 mm meet the specified operating limit? If not, what design parameters would you change? Consider another material, increasing the thickness of the insulation, and increasing the rod length. Also, consider how you might attach the base of the rod to the furnace wall as a means to reduce To. ASSUMPTIONS: (1) Steady-state conditions, (2) One-dimensional conduction in rod, (3) Negligible thermal contact resistance between the rod and hot hot furnace wall, (4) Insulated section of rod, Lins, experiences no lateral heat losses, (5) Convection coefficient uniform over the exposed portion of the rod, Lo, (6) Adiabatic tip condition for the rod and (7) Negligible radiation exchange between rod and its surroundings. -Tw Air Tooth Hot furnace- wall Lins Insulation Lo