If Ts ≈ Tsur in Equation 1.9, the radiation heat transfer coefficient may be approximated as hr,a = 4εσT3 where T ≡ (Ts + Tsur)/2. We wish to assess the validity of this approximation by comparing values of hr and hr,a for the following conditions. In each case represent your results graphically and comment on the validity of the approximation.

(a) Consider a surface of either polished aluminum (B = 0.05) or black paint (B = 0.9), whose temperature may exceed that of the surroundings (Tsur = 25°C) by 10 to 100°C. Also compare your results with values of the coefficient associated with free convection in air (T∞ = Tsur), where h (W/m2. K) = 0.98∆T1/ 3.

(b) Consider initial conditions associated with placing a work piece at Ts = 25°C in a large furnace whose wall temperature may be varied over the range 100 < Tsur < 1000°C. According to the surface finish or coating, its emissivity may assume values of 0.05, 0.2, and 0.9. For each emissivity, plot the relative error (hr – hr,a}/hr, as a function of the furnace temperature.