The air above the surface of a freshwater lake is at a temperature \(T_{A}\), while the water is at its freezing point \(T_{i}\), where \(T_{A}

Thermodynamics is the study of energy, temperature and how they affect matter. In this problem, we focus on how heat transfers from one substance to another. The temperature difference between the air and water drives this heat transfer process. We can observe this through the formation of ice, as heat is removed from the water to the air.

To understand thermodynamics better:

• Remember that heat flows from warm areas to cooler ones.
• There are three main ways heat can transfer: conduction, convection, and radiation. This problem deals specifically with conduction and convection.

In summary, thermodynamics explains the fundamental principles behind the energy shifts and temperature changes during ice formation.

Convection is the transfer of heat by the movement of fluids, which could be a gas or liquid. In our scenario, it's the air that helps transfer heat away from the surface of the ice.

Here’s how it works:

• The heat from the freezing water is conducted through the ice.
• Once it reaches the ice's surface, it transfers to the air by convection.

The heat transfer rate by convection is given by the formula: $$ q = h(T_{upper} - T_{A}) $$ where

• h is the convection heat transfer coefficient
• T_{upper} is the temperature at the surface of the ice
• T_{A} is the temperature of the air

Convection makes it possible for heat to be carried away from the ice much faster than conduction alone would allow.

Latent heat of fusion refers to the heat absorbed or released when a substance changes state, say from liquid to solid, without changing its temperature. In the context of this problem, the latent heat of fusion of ice is crucial.

Here’s why it’s important:

• When water freezes at 0°C, it releases heat. This heat is known as the latent heat.
• The amount of heat released during the formation of ice depends on the volume of the ice and the latent heat of fusion.

The formula for latent heat release is:
$$ dQ = \rho L A \frac{dy}{A}$$ where:

• \rho is the density of ice.
• L is the latent heat of fusion of ice.
• dy is an infinitesimal change in ice thickness.

This heat release term balances the heat conducted through ice as well as the heat convected into the air.

Fourier's Law is central to understanding heat conduction, which is how heat flows through materials like ice. It tells us that the heat transfer rate is proportional to the temperature gradient and the material’s conductivity.

According to Fourier's Law, $$ q = -K \frac{dT}{dy}$$ where:

• q is the heat flux through the material.
• K is the thermal conductivity of the material.
• \frac{dT}{dy} is the temperature gradient along the thickness y of the ice.

This principle helps us calculate how heat moves from the water to the ice and then through the ice. In the end, Fourier's Law enables us to link the ice thickness with the time it takes for it to form as shown in the final relationship:
$$ \frac{y}{h} + \frac{y^2}{2K} = \frac{(T_{i} - T_{A}) t}{\rho L}$$
This equation ties together all factors, showing the effectiveness of Fourier's Law in predicting outcomes of heat conduction in ice formation.