A manufacturing process involves heating long copper rods, which are coated with a thin film, in a large furnace whose walls are maintained at an elevated temperature Tw. The furnace contains quiescent nitrogen gas at l-atm pressure and a temperature of T∞ = Tw. The film is a diffuse surface with a spectral emissivity of ελ. = 0.9 for λ < 2μm and ελ = 0.4 for λ > 2μm.

(a) Consider conditions for which a rod of diameter D and initial temperature T i is inserted in the furnace, such that its axis is horizontal. Assuming validity of the lumped capacitance approximation, derive an equation that could be used to determine the rate of change of the rod temperature at the time of insertion. Express your result in terms of appropriate variables.

(b) If Tw = T∞ = 1500 K, Ti = 300K, and D = 10 mm, what is the initial rate of change of the rod temperature? Confirm the validity of the lumped capacitance approximation.

(c) Compute and plot the variation of the rod temperature with time during the heating process.