Question

Consider the situation of the previous problem. Assume that the temperature of the water at the bottom of the lake remains constant at 4°C as the ice forms on the surface (the heat required to maintain the temperature of the bottom layer may come from the bed of the lake). The depth of the lake is 1.0 m. Show that the thickness of the ice formed attains a steady state maximum value. Find this value. The thermal conductivity of water = 0.50 W m−1°C−1. Take other relevant data from the previous problem.

Answer 1

Explanation:

Step 1:

Given,

Thermal conductivity of water = $0.50 \mathrm{Wm}^{-1 \circ} \mathrm{C}^{-1}$

Depth of the lake =$1.0 \mathrm{m}$

Let ice is formed at ‘x’ m below from top of lake

Step 2:

At steady state,

Rate of flow of heat from ice to this point= rate of flow of heat from water

Temperature of top layer=$-10^{\circ} \mathrm{C}$

Temperature of water at bottom of lake $=4^{\circ} \mathrm{C}$

Temperature at point where ice is formed =$=0^{\circ} \mathrm{C}$

Step 3:

$\left(\frac{\Delta Q}{\Delta t}\right)_{i c e}=\left(\frac{\Delta Q}{\Delta t}\right)_{w a t e r}$

$\frac{K_{i c e} A(0+10)}{x}=\frac{K_{\text {water}} A(4)}{1-x}$ (area is equal)

By solving the above equation we get  $x=17 / 19$

$x \approx 89 c m$