Question
The air above the water surface in a pond is at temperature –θ°C as a result of which the water in the pond starts freezing. Find the time taken by the pond to freeze from thickness x1 to x2. Assume the required data.
Solution
Let us consider a pond of water of surface area A.
Let L be the latent heat, K be the thermal conductivity and ρ be the density of ice.
Let at any instant, a layer of thickness x has been already formed.
Let L be the latent heat, K be the thermal conductivity and ρ be the density of ice.
Let at any instant, a layer of thickness x has been already formed.
Let it take 'dt' time to further increase the thickness of ice by 'dx'.
Heat given up from water to freeze 'dx' layer of ice is,
This amount of heat is conducted from water to air through ice layer of thickness dx in time dt.
Thus heat conducted through ice is,
Equating (1) and (2)
Time taken to freeze the ice from thickness x1 to x2 can be obtained by integrating the equation (3).
i.e.,
Heat given up from water to freeze 'dx' layer of ice is,
This amount of heat is conducted from water to air through ice layer of thickness dx in time dt.
Thus heat conducted through ice is,
Equating (1) and (2)
Time taken to freeze the ice from thickness x1 to x2 can be obtained by integrating the equation (3).
i.e.,
