Question

The temperature at the bottom of a pond of depth $L = 15\;{\rm{m}}$ is $4\;^\circ {\rm{C}}$. The temperature of the air, just above the layer of ice frozen at the pond’s surface is $ - 2\;^\circ {\rm{C}}$ for the past many days. The thermal conductivity of the ice is three times that of water. Find the depth (in m) of unfrozen water below ice?

Answer 1

Hint:The heat transfer between the water of pond and surrounding will take place by the conduction. The heat transfer in conduction takes place from high temperature to low temperature of the object.

Complete Step by Step Answer:
The depth of the pond is 15 m, temperature at the bottom of the pond is $4\;^\circ {\rm{C}}$, temperature ice at the surface of pond is $ - 2\;^\circ {\rm{C}}$, thermal conductivity of ice is three times of water ${k_i} = 3{k_w}$.
Write the relation for heat transfer from the bottom of the pond to the ice-water junction.
${Q_1} = {k_w}A\left( {\dfrac{{{T_b} - {T_j}}}{x}} \right)$ (1)
Here, ${k_w}$ is the thermal conductivity of water, $A$ is the area, ${T_b}$ is the temperature at the bottom of the pond, ${T_j}$ is the temperature at the water and ice junction and its value is zero, $x$ is the depth of unfrozen water below ice.
Write the relation for heat transfer for the layer of water above the frozen ice.
${Q_2} = {k_i}A\left( {\dfrac{{{T_j} - {T_i}}}{{L - x}}} \right)$ (2)

At the junction of ice and water, heat transfer is the same for water and ice and the temperature of the junction is $0\;^\circ {\rm{C}}$. Equate the equation (1) and equation (2) to find the depth of unfrozen water below ice.
$\begin{array}{l}
{k_w}A\left( {\dfrac{{{T_b} - {T_j}}}{x}} \right) = {k_i}A\left( {\dfrac{{{T_j} - {T_i}}}{{L - x}}} \right)\\
{k_w}\left( {\dfrac{{{T_b} - {T_j}}}{x}} \right) = {k_i}\left( {\dfrac{{{T_j} - {T_i}}}{{L - x}}} \right)
\end{array}$

Substitute the value ${k_i}$ as $3{k_w}$, ${T_b}$ as $4\,^\circ {\rm{C}}$, ${T_j}$ as $0\;^\circ {\rm{C}}$, ${T_i}$ as $ - 2\;^\circ {\rm{C}}$, and $L$ as $15\;{\rm{m}}$ in the above equation.
$\begin{array}{l}
{k_w}\left( {\dfrac{{4\;^\circ {\rm{C}} - 0\;^\circ {\rm{C}}}}{x}} \right) = 3{k_w}\left( {\dfrac{{0\;^\circ {\rm{C}} - \left( { - 2\;^\circ {\rm{C}}} \right)}}{{15\;{\rm{m}} - x}}} \right)\\
30\;{\rm{m}} - 2x = 3x\\
x = 6\;{\rm{m}}
\end{array}$

Therefore, the depth of unfrozen water below ice is $6\;{\rm{m}}$.

Note: Make sure to remember the relation of the heat transfer in the conduction between different locations of the object.