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Question

On a cold winter day, the atmospheric temperature is θ (on Celsius scale) which is below 0C. A cylindrical drum of height h made of a bad conductor is completely filled with water at 0C and is kept outside without any lid. Calculate the time taken for the whole mass of water to freeze. Thermal conductivity of ice is K and its latent heat of fusion is L. Neglect expansion of water on freezing.

A
ρLh22Kθ
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B
3ρLh22Kθ
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C
ρLh24Kθ
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D
5ρLh22Kθ
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Solution

The correct option is A ρLh22Kθ
Let the outside temperature be θoC.

Since it's given that the walls of the container are poorly conducting, the heat lost by the water is due to radiation.

Since the water is at 0oC, it freezes when it releases heat equal to latent heat of fusion.

At any time, let x be the level of water below the suface of the container.


dQdt=dmdt.Lf
where dmdt is the rate at which ice freezes.

Thermal current from the water to the surroundings flows through the frozen ice.

ith=kA(θ0)x where x is the thickness of the ice layer above water level.

ith=kAθx=dQdt=dmdt.Lf

kAθx=ρAdxdt.Lf

h0xdx=kAθρAlt0dt

x22=kθρlt

t=ρlh22kθ

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