Question

On a cold winter day, the atmospheric temperature is -$\theta$ (on Celsius scale) which is below 0${}^{\circ }$C. A cylindrical drum of height h made of a bad conductor is completely filled with water at 0 ${}^{\circ }$ 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, $\rho$ is the density and latent heat of fusion is L. Neglect expansion of water on freezing

• A. $\frac{\rho L{h}^2}{2K\theta }$
• B. $\frac{\rho L{h}^2}{K\theta }$
• C. $\frac{\rho Lh}{K\theta }$
• D. $\frac{\rho Lh}{2K\theta }$

Answer 1

The correct option is A

$\frac{\rho L{h}^2}{2K\theta }$

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

$\Delta Q=\frac{KA\theta }{x}dt$

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

$dm=\frac{\Delta Q}{L}=\frac{KA\theta }{xL}dt$

The thickness dx of ice formed in time dt is

$dx=\frac{dm}{A\rho }=\frac{K\theta }{\rho xL}dt$

or, $dt=\frac{\rho L}{K\theta }xdx$

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

${\int }_0^{T}dt=\frac{\rho L}{K\theta }{\int }_0^{h}xdx$

or, $T=\frac{\rho L{h}^2}{2K\theta }$