Chapter 5, Problem 5.71P

In heat treating to harden steel ball bearings $\left(c=500\text{J/kg}\cdot \text{K,}\rho =7800{\text{kg/m}}^{\text{3}},k=50\text{W/m}\cdot \text{K}\right),$ it is desirable to increase the surface temperature for a short time without significantly warming the interior of the ball. This type of heating can be accomplished by sudden immersion of the ball in a molten salt bath with $T_{\infty }=1300\text{K}$ and $h=5000{\text{W/m}}^{\text{2}}\cdot \text{K}\text{.}$ Assume that any location within the ball whose temperature exceeds 1000 K will be hardened. Estimate the time required to harden the outer millimeter of a ball of diameter 20 mm, if its initial temperature is 300 K.