Chapter 5, Problem 5.24P

In a material processing experiment conducted aboard the space shuttle. a coated niobium sphere of diameter is removed from a furnace at $900°\text{C}$ and cooled to a temperature of $300°\text{C}\text{.}$ Although properties of the niobium vary over this temperature range, constant values may be assumed to a reasonable approximation, with $\rho =8600{\text{kg/m}}^{\text{3}},c=290\text{J/kg}\cdot \text{K,}$ and $k=63\text{W/m}\cdot \text{K}\text{.}$

1. If cooling is implemented in a large evacuated chamber whose walls are at $25°\text{C,}$ determine the time required to reach the final temperature if the coating is polished and has an emissivity of $\epsilon =\mathrm{0.1.}$ How long would it take if the coating is oxidized and $\epsilon =0.6?$
2. To reduce the time required for cooling. consideration is given to immersion of the sphere in an inert gas stream for which $T_{\infty }=25°\text{C}$ and $h=200{\text{W/m}}^{\text{2}}\cdot \text{K}\text{.}$ Neglecting radiation, what is the time required for cooling?
3. Considering the effect of both radiation and convection, what is the time required for cooling if $h=200{\text{W/m}}^{\text{2}}\cdot \text{K}$ and $\epsilon =0.6?$ Explore the effect on the cooling time of independently varying h and $\epsilon .$