Chapter 5, Problem 5.75P

A sphere 30 mm in diameter initially at 800K is quenched in a large bath having a constant temperature of 320 K with a convection heat transfer coefficient of $75{\text{W/m}}^{\text{2}}\cdot \text{K}\text{.}$ The thermophysical properties of the sphere material are: $\rho =400{\text{kg/m}}^{\text{3}},c=1600\text{J/kg}\cdot \text{K,}$ and $k=1.7\text{W/m}\cdot \text{K}\text{.}$

1. Show, in a qualitative manner on $T-t$ coordinates, the temperatures at the center and at the surface of the sphere as a function of time.
2. Calculate the time required for the surface of the sphere to reach 415 K.
3. Determine the heat flux $\left({\text{W/m}}^{\text{2}}\right)$ at the outer surface of the sphere at the time determined in part (b).
4. Determine the energy (J) that has been lost by the sphere during the process of cooling to the surface temperature of 415 K.
5. At the time determined by part (b), the sphere is quickly removed from the bath and covered with perfect insulation, such that there is no heat loss from the surface of the sphere. What will be the temperature of the sphere after a long period of time has elapsed?
6. Compute and plot the center and surface temperature histories over the period $0\le t\le 150\text{s}\text{.}$ What effect does an increase in the convection coefficient to $h=200{\text{W/m}}^{\text{2}}\cdot \text{K}$ have on the foregoing temperature histories? For $h=75$ and $200{\text{W/m}}^{\text{2}}\cdot \text{K,}$ compute and plot the surface heat flux as a function of time for $\text{0}\le t\le 150\text{s}\text{.}$