Ball bearings are to be hardened by quenching them in a water bath at a temperature of 37°C. You are asked to devise a continuous process in which the balls 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, however, is not 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 has to be removed from the water bath in order to maintain a uniform temperature if 100,000 balls of 2.5-cm-diameter are to be quenched per hour.
GIVEN
FIND
(a) An approximate relation giving the minimum allowable cooling time in the water as a function of the ball
radius for balls upto 1.0 cm in diameter
(b) The cooling time, in seconds, required for a ball having a 2.5-cm-diameter
(c) 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
ASSUMPTIONS
The ball bearings are 1% carbon steel
SKETCH
(a) For 1.0-cm-diameter balls
Therefore, a lumped capacity method can be used for balls less than 1 cm in diameter. The time temperature
history of the ball is given by
Solving for the minimum cooling time
(b) For balls having a diameter of 2.5 cm
The internal resistance is not significant.
Therefore, a lumped capacity method can be used for balls less than 1 cm in diameter. The time temperature history of the ball is given by
Solving for the minimum cooling time
(c) We have for approximate solutions:
for sphere for Bi=0.17
From approximate solution for sphere
The amount of heat needed to quench 100,000 balls per hour is
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