A lake surface is exposed to an atmosphere where the temperature is $$< 0^o$$C. If the thickness of the ice layer formed on the surface grows from $$2$$cm to $$4$$cm in $$1$$ hour. Find the atmospheric temperature, $$T_a$$.
(Thermal conductivity of ice $$K=4\times 10^{-3}$$ cal/cm/s/$$^oC$$, density of ice $$=0.9$$ gm/cc. Latent heat of fusion of ice$$=80$$ cal/gm. Neglect the change of density during the state change. Assume that the water below the ice has $$0^o$$ temperature everywhere).
A lake surface is exposed to an atmosphere where the temperature $$<0°C$$
Thickness of ice layer formed $$=2cm \quad to \quad 4cm$$
Time $$t=1h$$
Atmospheric temperature $$=T_a$$
Thermal conductivity of ice $$K=4\times 10^{-3}cal/cm/s°C$$
Density of the ice $$=0.9gm/cc$$
Latent heat of fusion of ice $$=80cal/gm$$
Neglect the change of density during the state change
Assume that the water below the ice has $$0°C$$ temperature everywhere.
Solution 'We shall use the following relation which determines the rate of change of the thickness of ice
$$\cfrac { dx }{ dt } =K\theta /_{ x }\\ \cfrac { dx }{ dt } =\cfrac { 4NO^{ -3 }\times \theta }{ 0.9\times 80\times 2 } \\ 4-2=\cfrac { 4\times 10^{ -3 }\times 10 }{ 90\times 80\times 2 } \\ \cfrac { 2\times 4\times 9 }{ 10^{ -3 }\times 1 } =\theta \\ \theta =8\times 9\times 10^{ 3 }\\ \quad =72\times 10^{ 3 }°C\\ \quad =72000°C$$