Chapter 3, Problem 3.164P

In a material processing experiment conducted aboard the space shuttle, a coated niobium sphere of 10-min 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 $\rho =8600kg/m^3,c=290J/kg\cdot K,\text{and}k=63W/m\cdot K$.

(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 £ = 0.1. How long would it take if the coating is oxidized and $\epsilon =0.6$?

(b) To reduce the time required for cooling, consideration is given to immersion of the sphere in an inert gas stream for which $T_{\infty }=25°C\text{and}h=200W/m^2\cdot K$. 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 $h=200W/m^2\cdot K\text{and}\epsilon =0.6$? Explore the effect on the cooling time of independently varying h and e.