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

Let the point upto which ice is formed is at a distance of x m from the top of the lake.
Under steady state, the rate of flow of heat from ice to this point should be equal to the rate flow of heat from water to this point.
Temperature of the top layer of ice = −10°C
Temperature of water at the bottom of the lake = 4°C
Temperature at the point upto which ice is formed = 0°C
$$\begin{gathered} {\left(\frac{\mathrm{\Delta }Q}{\mathrm{\Delta }t}\right)}_{\mathrm{ice}}={\left(\frac{\mathrm{\Delta }Q}{\mathrm{\Delta }t}\right)}_{\mathrm{water}} \\ \frac{K_{\mathrm{ice}}}{{\displaystyle \frac{x}{\mathrm{A}\times 10}}}=\frac{K_{\mathrm{water}}}{{\displaystyle \frac{1-x}{\mathrm{A}\times 4}}} \\ \frac{1.7\times 10}{x}=\frac{0.5\times 4}{1-x} \\ \frac{17}{x}=\frac{2}{1-x} \\ 17-17x=2x \\ \Rightarrow x=\frac{17}{19}=0.89\mathrm{m} \\ \Rightarrow x=89\mathrm{cm} \end{gathered}$$