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)/(Ap)=(Ktheta)/(pxL)dt` . or, `dt=(pL)/(Ktheta)xdX` . Thus, the time T taken for the whole mass of water to freeze is given by
`int_(0)^(T)dt=(pL)/(Ktheta)int_(0)^(h)xdx` . or, `T=(pLh^(2))/(2Ktheta)`