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

[johnconnor]

Guys I'm weak in heat and kinetic theory, so I'm going to 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!

 

[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.

 

[johnconnor]

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.

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

heat loss by water in pond = formation of ice.

[itex]\rho A h.l[/itex], 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...

 

[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

 

[asteriatic]

Thermal conductivity is a property that describes how well a material can conduct heat. It is measured in units of W/mK (watts per meter kelvin). In this problem, thermal conductivity is important because it describes how quickly heat can transfer through the ice layer.

To start, we can use the equation for the rate of change of heat transfer through a material:

dQ/dt = λA(dT/dx)

Where dQ/dt is the rate of heat transfer, λ is the thermal conductivity, A is the cross-sectional area of the material, and dT/dx is the temperature gradient across the material.

In this problem, we can assume that the cross-sectional area remains constant and the temperature gradient is equal to the temperature difference between the top surface of the ice and the bottom surface of the ice.

Using the specific latent heat of fusion, we can also say that the rate of change of heat transfer is equal to the rate of change of the thickness of the ice layer, multiplied by the latent heat of fusion.

dQ/dt = l(dh/dt)

Now, we can combine these two equations and solve for the rate of change of the thickness of the ice layer:

l(dh/dt) = λA(dθ/dx)

Solving for dh/dt, we get:

dh/dt = (λA/d)(dθ/dx)

Where d is the thickness of the ice layer.

To answer the second part of the question, if the temperature of the water at the bottom of the pond is 4°C, it will take longer for the bottom of the pond to freeze compared to the top surface. This is because there will be a smaller temperature difference between the water and the ice at the bottom, which will result in a slower rate of heat transfer. As a result, the rate of formation of ice will also be slower.

 

[asteriatic]

Hi there,

As a scientist, it is important to always have a strong understanding of the concepts and principles that are relevant to your field of study. In this case, it seems like you may need some additional support in the area of heat and kinetic theory. I would recommend reaching out to your teachers or peers for guidance and support in understanding these concepts. Additionally, there are many online resources and textbooks available that can help you build your knowledge in this area.

To address the specific question provided, here is a possible approach to solving it:

(i) The rate of increase of ice thickness can be determined by considering the amount of heat that is transferred from the water to the ice. This heat transfer is governed by the equation: Q = mL, where Q is the amount of heat transferred, m is the mass of the substance, and L is the specific latent heat of fusion. In this case, the mass of the ice being formed is ρh, where ρ is the density of ice and h is the thickness of the ice layer. Therefore, the rate of increase of h can be determined by dividing the amount of heat transferred by the time taken for the ice to form. This can be expressed as:

Rate of increase of h = Q/t = (mL)/t = (ρh)(l)/t

We can also express the amount of heat transferred in terms of the temperature difference between the water and the ice, and the thermal conductivity of ice. This can be expressed as:

Q = λA(θ-0) = λAθ, where A is the surface area of the ice layer and θ is the temperature difference between the ice and the water.

Therefore, the rate of increase of h can be further expressed as:

Rate of increase of h = (λAθ)/t = (λAθ)/((ρh)(l))

(ii) If the temperature of the water at the bottom of the pond is 4°C, this would result in a smaller temperature difference between the water and the ice. This would mean that less heat is transferred from the water to the ice, resulting in a slower rate of ice formation. Additionally, the thermal conductivity of ice may also play a role in the rate of ice formation. A higher thermal conductivity would result in a faster rate of heat transfer and therefore a faster rate of ice formation.

I hope this helps guide you in your understanding of this problem. It is important to continue seeking support and guidance