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Question

A steel ball of 10 mm diameter at 1000 K is required to be cooled to 350 K by immersing it in a water environment at 300 K. The convective heat transfer coefficient is 1000 W/m2K. Thermal conductivity of steel is 40 W/mK. The time constant for the cooling process t is 16 s . The time required (in s) to reach the final temperature is

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Solution

Diameter of ball =10 mm
Tinitial=1000 K
Tfinal=350 K
T=300 K
h=1000 W/m2K
k=40 W/mK
Time constant=16 s
Time required=t s
TTTiT=ehAtρvc

But ρvchA=Time constant
=16 second
3503001000300=e116×t
0.07143=et/16
Taking loge both sides, we get
loge0.07143=t16logee
2.639=t16
or t=42.23 s

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