Heat conduction and phase changes

In summary, the homework statement is asking for a differential equation for the thickness of ice vs time. The problem is that the ice layer melts and flows upwards, which complicates the problem.
  • #1
ClassicalMechanist
21
1

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.
 
Last edited:
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  • #2
Can anyone help me with this?
 
  • #3
ClassicalMechanist said:

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.
 

Related to Heat conduction and phase changes

1. What is heat conduction?

Heat conduction is the transfer of thermal energy from a region of higher temperature to a region of lower temperature through a material or substance, without any visible movement of the material itself.

2. How does heat conduction occur?

Heat conduction occurs through the transfer of kinetic energy between molecules in a substance, as hotter molecules collide with cooler molecules and transfer energy to them.

3. What factors affect heat conduction?

The rate of heat conduction is affected by the thermal conductivity of the material, the temperature difference between the two regions, the distance over which the heat is transferred, and the surface area of the material.

4. What are the three modes of heat conduction?

The three modes of heat conduction are conduction, convection, and radiation. Conduction is the transfer of heat through a solid material, convection is the transfer of heat through a fluid, and radiation is the transfer of heat through electromagnetic waves.

5. What are phase changes and how do they relate to heat conduction?

Phase changes refer to the physical changes that occur in a substance as it transitions from one state of matter to another (e.g. solid to liquid). These changes require a transfer of energy, which can be in the form of heat. Heat conduction plays a role in these phase changes by transferring thermal energy to or from the substance, causing it to change its state.

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