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On a cold winter day, the atmosheric temperature is -theta (on celsium scale) which is below `0^(@)C. A cylindrical drum oof 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 laternt heat of fusion is L. Neglect expansion of water on freezing.

Answer» 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)`


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