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
`Delta Q=(KA theta)/(x) dt`

The mass of the ice formed due to the loss of this amount of heat is
`dm=(Delta Q)/(L)=(KA theta)/(x L) dt`
The thickness dx of ice formed in time dt is
`dx=(dm)/(a rho)=(K theta)/( rho xL) dt or, dt=(rho L)/(K theta)x dx`
Thus, the time taken for the whole mass of water to freeze is given by
`underset(0) overset(T )(int) dt=(rho L)/(K theta) underset(h)overset(0) (int) x dx or, T=(rho L h^2)/( 2 K theta)`