Heat conduction and phase changes

[ClassicalMechanist]

Homework Statement

Suppose we have a lake and a layer of ice on top such that the bottom of the lake is maintained at a constant temperature T_{bot} which is above the freezing point of water, and top of the ice is maintained at the air temperature T_{air} which is below the freezing point of water. As heat flows vertically, the layer of ice thickens , and we would like to find a differential equation for the thickness of ice vs time.

Homework Equations

dQ/dt=k*A(T_h-T_c)/d
Q=mL

The Attempt at a Solution

To keep things simple we neglect convection effects (when is this a reasonable assumption?).

So there are 3 processes going on here: conduction through water, phase change water to ice at water-ice interface, conduction through ice layer to the atmosphere.

Call the temperature at the water-ice interface T_1 (so T_1=0). Consider what happens in a column of area A. We have three equations

heat current through water: dQ1/dt=k_water*A*(T_bot-T_1)/d, Q1 is heat that passes through water

rate of ice formation: dQ2/dt=dm/dt*L=rho_ice*A*dh/dt*L, Q2 is heat that passes through small layer of water dh below water-ice interface.

heat current through ice: dQ3/dt=k_ice*A*(T_1-T_air)/h, Q3 is heat that passes through ice

h=height of ice, d=depth of water. I need to combine these three equations to get a differential equation in h. I guess I need to account for the change in the water depth as well, as the water gets converted to ice. The mass of water and ice in the column is conserved: initial mass=h*rho_ice+d*rho_water, so we have d in terms of h.

I am just confused as to how to combine the three equations. Consider the small layer of water dh below the water-ice interface where ice will form. During a time interval dt, the incoming heat is dQ1, the outgoing heat is dQ3, and dQ2 of the incoming heat is used for ice formation. So dQ3=dQ1-dQ2. Is this correct? I might have messed up the signs.

Edit: I think I forgot to account for the heat which conducts upwards through the water melting the bottom of the ice layer as it forms. I'm not sure about this.

 

[ClassicalMechanist]

Can anyone help me with this?

 

[rude man]

Homework Statement

Suppose we have a lake and a layer of ice on top such that the bottom of the lake is maintained at a constant temperature T_{bot} which is above the freezing point of water, and top of the ice is maintained at the air temperature T_{air} which is below the freezing point of water. As heat flows vertically, the layer of ice thickens , and we would like to find a differential equation for the thickness of ice vs time.

Sorry no one has replied to date. I would have thought it to be a fun challenge to some of our more talented physicists ..

So I will put in my 2c worth:
I considered a thin layer dx of water just below the ice at some point along the depth. I made x(t) the height of the water layer at time t, and d = total depth of lake. I set
(heat leaving the layer as it changed from water to ice) + (heat flowing from the water-ice layer to the surface)
= (heat supplied by the bottom of the lake).
This got me a 1st order nonlinear ODE but it can be solved by separation of variables. However, the solution is pretty messy.
Your initial condition is of course x=d at t=0. It seems this is pretty much what you tried so you'd have to give us your math details if you want to compare it to what I got.