A manufacturing process involves heating long copper rods, which are coated with a thin film, in a large fur- nace whose walls are maintained at an elevated tem- perature Tw. The furnace contains quiescent nitrogen gas at 1-atm pressure and a temperature of T00 = Tw. The film is a diffuse surface with a spectral emissivity of sA = 0.9 for A 2 µ,m and sA = 0.4 for A > 2 µ,m.
(a) Consider conditions for which a rod of diameter D and initial temperature Ti 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 appropri- ate variables.
(b) If Tw = T00 = 1500 K, Ti = 300 K, and D = 10 mm, what is the initial rate of change of the rod temperature? Confirm the validity of the lumped capaci- tance approximation.
(c) Compute and plot the variation of the rod temperature with time during the heating process.