On a cold winter day, the atmospheric temperature is -theta (on celsius scale) which is below `0^(@)C`. 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` and its latent heat of fusion is `L`. Neglect expansion of water on freezing.

Explanation

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
`DeltaQ=(Katheta)/(x)dt`
The mass of the ice formed due to the loss of this amount of heat is
`dm=(DeltaQ)/(L)=(KAtheta)/(xL)dt`.
The thickness `dx` of ice formed in time `dt` is
`dx=(dm)/(Arho)=(Ktheta)/(rhoxL)dt`
or, `dt=(rhoL)/(Ktheta)xdx`
Thus, the time `T` taken for the whole mass of water to freeze is given by
`int_(0)^(T)dt=(rhoL)/(Ktheta)int_(0)^(h)xdx`
`T=(rhoLh^(2))/(2Ktheta)`