Solve the following problem:
Q: Ball bearings are to be hardened by quenching them in a water bath at a temperature of 37°C. Suppose you are asked to devise a continuous process in which the balls could roll from a soaking oven at a uniform temperature of 870°C into the water, where they are carried away by a rubber conveyer belt. The rubber conveyor belt would, however, not be satisfactory if the surface temperature of the balls leaving the water is above 90°C. If the surface coefficient of heat transfer between the balls and the water may be assumed to be equal to 590 W/(m2 K),
(a) find an approximate relation giving the minimum allowable cooling time in the water as a function of the ball radius for balls up to 1.0-cm in diameter,
(b) calculate the cooling time, in seconds, required for a ball having a 2.5-cm diameter, and
(c) calculate the total amount of heat in watts which would have to be removed from the water bath in order to maintain its temperature uniform if 100,000 balls of 2.5-cm diameter are to be quenched per hour.
GIVEN
Ball bearings quenched in a water bath
Water bath temperature (T∞) = 37°C
Initial temperature of the balls (To) = 870°C
Final surface temperature of the balls (Tf) = 90°C
Heat transfer coefficient (hc ) = 590 W/(m2 K)
ASSUMPTIONS
The ball bearings are 1% carbonsteel
