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ClassicalMechanist

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## 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.

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