# Thermal Conductivity - Forming ice under water

• unscientific
Clausius-Clapeyron Equation and found the latent heat of fusion of ice to be ##L = 3.44 \times 10^5 J kg^{-1}##. He then proceeded to find the rate of formation of ice using the heat flux equation and found it to be extremely slow at ##3.34 \times 10^{-7} m s^{-1}##. In steady state, he suggests that over half of the lake would be frozen, which seems unlikely given the slow rate of ice formation.
unscientific

## Homework Statement

Part (a): Derive Clausius-Clapeyron Equation. Find latent heat of fusion of ice.
Part (b): Find rate of formation of ice
Part (c): What is the maximum thickness of ice formed?

## The Attempt at a Solution

Part (a)
I have derived the relation. Using the values, I found that the latent heat of fusion of ice is ##L = 3.44 \times 10^5 J kg^{-1}## which seems right.

Part (b)

Using heat flux ##J = k (\frac{\partial T}{\partial z})##, consider in time ##\delta t## amount of ice ##\Delta m## is formed.

$$\Delta Q = (\Delta m)L = (JA) \Delta t$$
$$\rho (\Delta z)L = \frac{k(T_2-T_1)}{z} \Delta t$$
$$\frac{dz}{dt} = \frac{k(T_2-T_1)}{\rho L z}$$

Taking ##k = 2.3##, ##T_2-T_1 = 0.5##, ##\rho = 1000##, ##L = 3.44 \times 10^5##:

I find that ##\frac{dz}{dt} = 3.34 \times 10^{-7} m s^{-1}##, so ice forms awfully slow.

Part (c)

In steady state, ## \frac{dQ_1}{dt} = \frac{dQ_2}{dt}##.

Therefore, we equate the heat fluxes:

$$k_{water} \left(\frac{2}{1-z_f}\right)A = k_{ice}\frac{0.5}{z_f}A$$

$$z_f = \frac{k_{ice}}{4k_{water} + k_{ice}} = 0.51 m$$

It's hard to believe that over half the lake would be frozen, when the rate of formation of ice above (without the interference of the bottom of the lake) is merely ##3.34 \times 10^{-7} m s^{-1}##.

That's about 3 cm/day. That doesn't seem extremely low to me, considering the very low temperature driving force of 0.5 C.

Chet

1 person

## 1. What is thermal conductivity?

Thermal conductivity is the measure of a material's ability to conduct heat. It is the rate at which heat energy is transferred through a material.

## 2. How does thermal conductivity relate to forming ice under water?

Thermal conductivity plays a crucial role in the formation of ice under water. It determines how quickly heat from the surrounding water is transferred to the ice, causing it to form or melt.

## 3. What factors affect thermal conductivity of a material?

The thermal conductivity of a material is affected by its composition, density, and temperature. Generally, materials with high thermal conductivity have a high density and are good conductors of electricity.

## 4. Can thermal conductivity be measured?

Yes, thermal conductivity can be measured using various methods such as the hot wire method or the heat flow meter method. These methods involve applying a known heat source to the material and measuring the temperature difference across it.

## 5. How does thermal conductivity differ for different materials?

Thermal conductivity varies greatly among different materials. Metals, such as copper and aluminum, have high thermal conductivity, while insulating materials, like rubber or wood, have low thermal conductivity. Air and water have relatively low thermal conductivity compared to solids.

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