In a material processing experiment conducted aboard the space shuttle, a coated niobium sphere of l0-mm diameter is removed from a furnace at 900°C and cooled to a temperature of 300°C. Although properties of the niobium vary over this temperature range, constant values may be assumed to a reasonable approximation, with p = 8600 kg/m³, c = 290 J/kg ∙ K, and k = 63 W/m ∙ K.

(a) If cooling is implemented in a large evacuated chamber whose walls are at 25°C, determine the time required to reach the final temperature if the coating is polished and has an emissivity of ε = 0.1. How long would it take if the coating is oxidized and ε = 0.67

(b) To reduce the time required for cooling, consideration is given to immersion of the sphere in an inert gas stream for which T_∞ = 25°C and h = 200 W/m² ∙ K. Neglecting radiation, what is the time required for cooling?

(c) Considering the effect of both radiation and convection, what is the time required for cooling if h = 200 W/m² ∙ K and ε = 0.6? Explore the effect on the cooling time of independently varying h and ε.