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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 0 C 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, ρ is the density and latent heat of fusion is L. Neglect expansion of water on freezing


A

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B

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C

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D

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Solution

The correct option is A


Suppose, the ice starts forming at time t=0 and a thickness x is formed at time t. The amount of heat flown from the water to the surrounding in the time interval t to t + dt is

ΔQ=KAθxdt

The mass of the ice formed due to the loss of this amount of heat is

dm=ΔQL=KAθxLdt

The thickness dx of ice formed in time dt is

dx=dmAρ=KθρxLdt

or, dt=ρLKθxdx

Thus, the time T taken for the whole mass of water to freeze is given by

T0dt=ρLKθh0xdx

or, T=ρLh22Kθ


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