Calculate: (a) time elapsed in cooling the surface of the sphere to 204°C and (b) time elapsed in cooling the center of the sphere to 204°C.

Question

A steel sphere with a diameter of 7.6 cm is to be hardened by first heating it to a uniform

temperature of 870°C and then quenching it in a large bath of water at a temperature of 38°C.

The following data apply



GIVEN



FIND

(a) Time elapsed in cooling the surface of the sphere to 204°C

(b) Time elapsed in cooling the center of the sphere to 204°C

ASSUMPTIONS

Constant water bath temperature, thermal properties, and transfer coefficient

SKETCH

Answer 1

The importance of the internal resistance can be determined from the Biot number



Therefore, the internal resistance is significant and an approximate solution will be used.

Using approximate solution we have



For Bi=0.52 From Table 3.1 for sphere we have



Solving for the time



(For the surface temperature to reach 204°C)

(b) For a center temperature of 204°C



Using approximate solution, we have



For the center r=0, the term becomes indeterminate, so using L Hospital’s rule we get.



For Bi=0.52 From sphere we have





Now, the time taken is calculated as



(For the center temperature to reach 204°C)