A lake is covered by an ice layer of thickness x having thermal conductivity K₁ . On top of this ice is a uniform layer of snow of thickness y having thermal conductivity K₂ . The air above the snow is at a steady temperature of - ^ C . If is the density of water and L is its specific latent heat of fusion, the rate of increase of the ice layer thickness ( dx dt ) is given by

Step-by-step solution

The heat must conduct through both the ice layer and the snow layer in series. The thermal resistance of the ice layer is R₁ = x K₁ A and that of the snow layer is R₂ = y K₂ A . The equivalent thermal resistance of the composite slab is R_ eq = R₁ + R₂ = 1 A ( x K₁ + y K₂ ) . The temperature difference across the layers is = 0^ C - (- ^ C ) = . The rate of heat flow is dQ dt = R_ eq = A x K₁ + y K₂ . The rate at which heat is released due to the freezing of water is dQ dt = A L dx dt . Equating the two expressions