Engine valves ( $k=48 \mathrm{~W} / \mathrm{m} \cdot{ }^{\circ} \mathrm{C}, c_p=440 \mathrm{~J} / \mathrm{kg} \cdot{ }^{\circ} \mathrm{C}$, and $\rho=7840 \mathrm{~kg} / \mathrm{m}^3$ ) are heated to $800^{\circ} \mathrm{C}$ in the heat treatment section of a valve manufacturing facility. The valves are then quenched in a large oil bath at an average temperature of $50^{\circ} \mathrm{C}$. The heat transfer coefficient in the oil bath is $800 \mathrm{~W} / \mathrm{m}^2$. ${ }^{\circ} \mathrm{C}$. The valves have a cylindrical stem with a diameter of 8 mm and a length of 10 cm . The valve head and the stem may be assumed to be of equal surface area, and the volume of the valve head can be taken to be 80 percent of the volume of steam. Determine how long it will take for the valve temperature to drop to (a) $400^{\circ} \mathrm{C}$, (b) $200^{\circ} \mathrm{C}$, and (c) $51^{\circ} \mathrm{C}$, and (d) the maximum heat transfer from a single valve.