Question

A cylindrical drum of height $1\text{m}$ made of a bad conductor is completely filled with water at $0^{\circ }\text{C}$ and is kept outside without a lid. The temperature of the surroundings is $-{10}^{\circ }\text{C}$. Calculate the time taken for the entire mass of the water to freeze. (Thermal conductivity of ice $=k$, latent heat of fusion $L_f=L)$
[Neglect expansion of water on freezing, $\rho$ = density of water]

• A. $\frac{\rho L}{10k}$
• B. $\frac{\rho Lk}{10}$
• C. $\frac{\rho L}{20k}$
• D. $\frac{\rho L}{5k}$

Answer 1

The correct option is C $\frac{\rho L}{20k}$

At $t=t_0$, let the height of the ice formed from the top of the drum be $x$ and area of the top of the drum be $A$.
At $t=t_0+dt$, additional $dx$ length of ice will be formed.


The amount of heat flow from the water to the surroundings in the time interval $dt$ is
$dQ=\frac{kA[0-(-10\left)\right]}{x}\times dt$
$\Rightarrow \frac{dQ}{dt}=\frac{kA\times 10}{x}$ ....(i)
Let $dm$ amount of water be converted into ice in the same time interval.
Then, $dm=\rho Adx$
& $dQ=Ldm$
Divide both sides by $dt$
$\frac{dQ}{dt}=\frac{Ldm}{dt}$ ...(ii)
From equation (i) and (ii),
$\frac{Ldm}{dt}=\frac{kA\left(10\right)}{x}$
$\Rightarrow \frac{L\left(\rho Adx\right)}{dt}=\frac{kA\left(10\right)}{x}$
$\Rightarrow L\rho \frac{dx}{dt}=\frac{10k}{x}$
Integrating on both sides,
$\Rightarrow L\rho {\int }_0^{h}xdx=10k{\int }_{t=0}^{t_0}dt$
$\Rightarrow t_0=\frac{\rho L{h}^2}{20k}$
Given $h=1\text{m}$
Therefore, $t_0=\frac{\rho L}{20k}$