The growth of ice on a pond

An interesting application of thermal conductivity is the calculation of the rate of growth of ice on top of a pond.
Let the air temperature be q and the water temperature just below the ice be 0 ^oC. At a certain time let the thickness of the ice be x, and let it increase by a further thickness dx in a time dt. The latent heat released on melting has to be conducted away through the ice layer as the water freezes and therefore we have:

quantity of heat lost due to increase dx = rLAdx

where r is the density of ice, L the specific latent heat of fusion of water and A the area of the ice surface. Then, if x is the thickness of the ice after a time t,

rate of loss of heat = LArdx/dt = kqA/x
Therefore dx/dt = kq/Lrx

Integrating gives:

Thermal conductivity and kinetic theory

Consider three horizontal planes in the gas each of area A. The heat conducted downwards through A per second is then -kAdq/dx

However, each second a mass of gas m at a temperature q₁ crosses A moving downwards and a mass of gas m at a temperature q₂ crosses A moving upwards.

Now: m = rcA/6
q₁ = q + ldq/dz and q₂ = q + ldq/dz

Therefore, since heat = mcq , the net transfer of heat downwards is:
–[rcA/6][2dq/dq]C_v.

But this must equal –kAdq/dz and therefore:

k = 1/3rclC_v and since h = 1/3rcl