Surface hardening of metallic machine components, such as ball and roller bearings, is carried out in a heat treatment process where the surface temperature is increased to a desirable level without altering the internal temperature substantially. In one such operation, steel ball (spherical) bearings with a diameter of 25 mm, initially at 25°C, are immersed in a very high temperature (1000°C) molten electrolyte-solution bath and quickly removed when the temperature at a depth of 1 mm from the outer surface of the steel ball attains 725°C. If the convective heat transfer coefficient between the steel ball and the bath liquid is assumed to be 4.95 kW/(m2 K), determine the time required for attaining the surface hardening condition.
GIVEN
Steel ball bearing diameter (d)= 25 mm=0.025 m
Initial temperature (To) = 25°C
Temperature of electrolyte solution (T?) = 1000°C
The heat transfer coefficient (hc) = 4.95 kW/(m2 K) = 4950 W/(m2 K)
Radius of the surface where temperature of 7250C is attained= 11.5 mm
FIND
Time required for attaining temperature of 7250C at depth of 1 mm from outer surface.
ASSUMPTIONS
constant bath temperature, thermal properties and heat transfer coefficient
SKETCH
The Biot number for the steel ball based on r0 is
Therefore, the internal resistance is significant Approximate solutions can be used to solve the problem.
For Bi=1.44, r/r0=11.5/12.5= 0.92 we have
Using approximate solution, we have
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