Question

In a $10$ m deep lake, the bottom is at a constant temperature of $4^{o}C$. The air temperature is constant at $-4^{o}C$. The thermal conductivity of ice is $3$ times that of water. Neglecting the expansion of water on freezing, the maximum thickness of ice will be

• A. 7.5 m
• B. 6 m
• C. 5 m
• D. 2.5 m

Answer 1

The correct option is A 7.5 m

As it is mention that bottom of the lake is maintained at temperature of $4^{0C}$ and upper portion is at ${-4}^{0C}$ and depth of lake is $10m$,so let us assume an intermediate temperature of $0^{0C}$ will lie at distance $X$ from top end so it's distance from bottom end will be $10-X$.

now we know that heat produced is given as $Q=\frac{KA\Delta T}{L}$

now for(water) bottom end $Q_1=\frac{\left(1\right)A(4-0)}{10-X}$-----(1)

and for(ice) top end $Q_2=\frac{\left(3\right)A(0-(-4\left)\right)}{X}$--------(2)

it is given that $K_{ice}=3\times K_{water}$

so equating (1) & (2) we get,

$4X=120-12X$

$X=7.5$