Chapter 5, Problem 5.34P

A metal sphere of diameter D, which is at a uniform temperature $T_i,$ is suddenly removed from a furnace and suspended from a tine wire in a large room with air at a uniform temperature $T_{\infty }$ and the surrounding walls at a temperature $T_{\text{sur}}.$

1. Neglecting heat transfer by radiation, obtain an expression for the time required to cool the sphere to some temperature T.
2. Neglecting heat transfer by convection, obtain an expression for the time required to cool the sphere to the temperature T.
3. How would you go about determining the time required for the sphere to cool to the temperature T if both convection and radiation are of the same order of magnitude?
4. Consider an anodized aluminum sphere $\left(\epsilon =0.75\right)$ 50 mm in diameter, which is at an initial temperature of $T_i=800\text{K}\text{.}$ Both the air and surroundings are at 300K, and the convection coefficient is $10{\text{W/m}}^{\text{2}}\cdot \text{K}\text{.}$ For the conditions of parts (a), (b), and (c), determine the time required for the sphere to cool to 400 K. Plot the corresponding temperature histories. Repeat the calculations for a polished aluminum sphere $\left(\epsilon =0.1\right).$