How thick is the frozen layer of ice on top of a pond?

[nik jain]

Homework Statement

The temperature at the bottom of a pond of depth L IS 4°C. The temperature of the air , just above the layer of ice frozen at the pond surface is -2°C.
THE THERMAL CONDUCTIVITY OF ICE IS THREE TIMES THAT OF WATER

The thickness(X) of the frozen layer of ice must be-

Homework Equations

Q= k*A*dQ/dx

The Attempt at a Solution

K(ice)A(T-{-2})/X = K(WATER)A(4-T)/L where T is the temp. just below the ice layer
3K(WATER)A(T-{-2})/X = K(WATER)A(4-T)/L
FROM WHERE TO GET THE ANOTHER EQUATION

 

[gneill]

If the pond is in equilibrium so that the ice is neither growing nor melting, what can you say about the temperature at the water/ice interface?

 

[nik jain]

So, how to solve this problem ?

 

[gneill]

So, how to solve this problem ?

Assume equilibrium. Write equations for heat flow and temperature that reflect the equilibrium conditions (which in this case are particular temperatures that need to be met at various places).

You can probably ignore the area aspect of the problem and just assume a linear dependence on the thickness for the heat resistance of the materials; since you won't need any particular values for the heat resistivities, just call the resistivity of ice $\rho$ so that a layer of thickness T will have heat resistance $R_{ice} = \rho$T. What then would be the resistivity of water, and its resistance for a layer W thick?

 

[technician]

I think it is safe to take the temperature at the ice/water junction to be 0 celcius
The equation you need is
dQ/dT = kA(dθ/dx)

 

[nik jain]

How can we take the temp. at the junction to be 0°C

But by the ans. given in the booklet , the temp. at the junction is coming -1°C.

 

[gneill]

How can we take the temp. at the junction to be 0°C

But by the ans. given in the booklet , the temp. at the junction is coming -1°C.

If the temperature at the water/ice interface was less than the freezing point for water then the ice would still be growing. If the temperature at the water/ice interface was greater than the freezing point for water then the ice would still be melting. Neither of those situations would be an equilibrium condition.

 

[nik jain]

THANKS to you all for the help..