# Year 12: Cambridge Physics Problem (Rate of increase of ice thickness)

1. Jun 7, 2012

### johnconnor

Guys I'm weak in heat and kinetic theory, so I'm gonna need extra guide and pointers from you guys to solve this and the coming questions. Thank you.

Question:
A pond of water at 0 degrees Celsius is freezing. The thickness of the ice layer is h and the top surface of the ice remains at a temperature θ (θ being < 0°C).

(i) Derive an equation for the rate of increase of h in terms of θ, l, h, λ, and ρ, where l is the specific latent heat of fusion, λ is the thermal conductivity and ρ is the density of ice.

(ii) Discuss how the rate of formation of ice would be affected if the temperature of the water in the pond was 0°C at the water-ice interface but increased with depth to 4°C at the bottom of the pond.

[Specific latent heat of fusion of ice, l = 3.3E5 J/kg; thermal conductivity of ice, λ = 2.3 W /mK; density of ice, ρ = 920 kg/m^3]

Attempt:
None? I don't think I learned thermal conductivity in CIE A Level (I couldn't find it in the syllabus list either), and I'm currently googling for more info on it. Pointers anyone? Thank you!

2. Jun 7, 2012

### pcm

http://en.wikipedia.org/wiki/Thermal_conductionFor the formation of layer of thickness dh calculate how much heat will be transferred from pond to surounding .then use 1D heat conduction formula to relate it to the given parameters.

3. Jun 8, 2012

### johnconnor

Attempt:
How much heat will be transferred to surroundings from pond?

heat loss by water in pond = formation of ice.

$\rho A h.l$, where A is the surface area of the pond.

I haven't even related the thermal conductivity to the equation! I honestly don't know what to do over here. Can anyone please provide a partial guide or additional pointers for me? Thank you...

4. Jun 8, 2012

### ehild

Heat is transferred from the pond to the surroundings through the ice layer, by heat conduction.

The rate of heat transferred through unit cross section of the ice layer is proportional to the temperature gradient inside the layer,

ΔQ/Δt=λΔT/Δx.

We can take that the surface of the ice layer is at the same temperature as the ambient, and the temperature changes linearly through the layer. If θa is the temperature of the ambient, θw is the temperature of the water in the pond, and h is the thickness of the ice

ΔT/Δx=(θa-θw)/h,

so the rate of heat loss by the water through the ice layer of area A is dQ/dt=λA(θa-θw)/h. The heat lost by the water will cause freezing some amount and increasing the thickness of the ice layer.

ehild