In a ball bearing production facility, steel balls that are each of 15 mm in diameter are annealed by first heating them to 8700C and then slowly cooling in air to 1250C. If the cooling air stream temperature is 600C, and it has a convective heat transfer coefficient of 35 W/(m2 K), determine the time required for the cooling.

Question

GIVEN

• Steel balls of diameter D=15 mm=0.015 mm

• Initial temperature of balls before cooling (To) =870°C

• Cooling air stream temperature (T?) = 60°C

• Convective heat transfer coefficient ( ch ) = 35 W/(m2 K)

FIND

• The time required for cooling ball bearing to 1250C.

ASSUMPTIONS

• Constant thermal conductivity

• End effects are negligible

SKETCH



PROPERTIES AND CONSTANTS

For steel ball:

Thermal conductivity (k) = 50 W/(m K) Specific heat (c) = 460 J/(kg K) Density (?) = 7865 kg/m3

Answer 1

For a spherical ball bearing we have Lc=D/6= 0.0025 m Biot number (Bi)= ch Lc/k=35*0.0025/50=0.00175 which is << 0.1 Thus, the ball bearing can be considered as lumped capacitance. Therefore, the internal resistance of the ball bearing is negligible. The temperature-time history of the ball bearing,