As shown in lecture, the thickness of an ice sheet formed on the surface of a la

Question

As shown in lecture, the thickness of an ice sheet formed on the surface of a lake is proportional to the square root of the time if the heat of fusion of the water freezing on the underside of the ice sheet is conducted through the sheet.

Part A:

Assuming that the upper surface of the ice sheet is at -15?C and the bottom surface is at 0 ?C, calculate the time it will take to form an ice sheet 30cm thick.

Part B:

If the lake in part (c) is uniformly 50m deep, how long would it take to freeze all the water in the lake?

Part C:

Is this likely to occur in the continental United States? (Yes/no)

Explanation / Answer

Assume that the freezing of an additional layer is slow enough. Then the temperature profile can be assumed to be linear across the layer of ice. This quasistatic assumptions allows us to solve this problem without the need of solving the much more difficult moving boundary condition.

The amount of heat that leaves the top surface has two origins: the latent heat of freezing of the additional sheet of water at the bottom plus the heat from additional cooling of the existing ice.
This will give a heat balance equation from which a "thickness goes as sqrt(time) " formula emerges.

First, the linear temperature profile assumed is (meauring thickness x from the top surface downwards)

T = T0 + x delta(T)/d

The energy balance is for infinitesimal time delta(t) is

k (delta(T)/d) delta(t) = rho L delta(d) + integral_0^d rho sigma delta(T) dx

d is the thickness of the sheet (distance of growth)
With the linear profile this gives

d(t) ^2 = d(0)^2 + t * 2 k delta(T)/( rho L + 1/2 rho sigma delta(T) )

Here k is the conductivity of ice, L the latent heat of melting, rho the density of ice and delta(T) the temperature difference between top and bottom.

From this you can calcute the time it takes to grow an additional 2.10 mm on 24.6 cm.
Just solve for t with d(0) = 0.246 m and d(t) = 0.2481 m

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