Question

Suppose that the outside temperature is $T$ and the depth of the lake is $h$. How much time will it take to freeze the entire lake. Latent heat of ice is $L$, thermal conductivity is $K$, and density of ice is $\rho$

• A. $t=\frac{\rho L{h}^2}{KT}$
• B. $t=\frac{\rho L{h}^2}{2KT}$
• C. $t=\frac{2\rho L{h}^2}{KT}$
• D. $t=\frac{\rho L{h}^2}{4KT}$

Answer 1

Answer: (B)

$t=\frac{\rho L{h}^2}{2KT}$

Let the air temperature be $T^{o}C$ and the water temperature just below ice is $0^{o}C$. At the certain time let the thickness of ice be $h$ and let the increase in its thickness further be $dh$ in a time $dt$. Latent heat released on melting has to be conducted away through ice layer as the water freezes and therefore we have :-

Quantity of heat lost due to increase $dh=\rho LAdh$

where $\rho$ is density, $L$ is specific heat of fusion of water, $A$ area of ice surface, $h$ thickness of ice at time $t$

Rate of heat loss $=\frac{LA\rho dh}{dt}=kT\frac{A}{h}\Rightarrow \frac{dh}{dt}=\frac{kT}{L\rho h}$

On integration,

Thickness of ice after time $t\Rightarrow h={\left(\frac{2kTt}{L\rho }\right)}^{1/2}\Rightarrow h^2=\frac{2kTt}{\rho L}$

Required time is $t=\frac{\rho L{h}^2}{2kT}$