On a cold winter day the temperature of
atmosphere is – TºC. The cylindrical diagram
shown is made of insulating material and it
contains water at 0ºC. If L is latent heat of fusion
of ice, ρ is density of ice and KR is thermal
conductivity of ice, the time taken for total mass of
water to freeze is -

To determine the time taken for the total mass of water to freeze in the given scenario, we need to consider several factors, including the temperature difference, the properties of ice and water, and the geometry of the system. Let's break this down step by step.

Understanding the Variables

We have the following variables at play:

• T: The temperature of the atmosphere in degrees Celsius (–T°C).
• L: The latent heat of fusion of ice, which is the amount of energy required to freeze water.
• ρ: The density of ice, which affects the mass of ice formed.
• K_R: The thermal conductivity of ice, which determines how quickly heat can be conducted through the ice.

Heat Transfer Mechanism

When the water at 0°C is exposed to the cold atmosphere at –T°C, heat will flow from the water to the surrounding air. The rate of heat transfer can be described using Fourier's law of heat conduction:

Q = K_R * A * (T_water - T_air) / d

Here, Q is the heat transfer rate, A is the surface area of the water in contact with the air, and d is the thickness of the ice layer that forms as the water freezes.

Calculating the Total Heat Required

The total heat required to freeze the mass of water (m) can be calculated using the formula:

Q_total = m * L

Where m is the mass of water that needs to freeze. This mass can be expressed in terms of volume and density:

m = V * ρ_water

Time Calculation

To find the time (t) taken for the water to freeze, we can relate the total heat transfer to the rate of heat transfer:

t = Q_total / Q

Substituting the expressions we derived earlier:

t = (m * L) / (K_R * A * (0 - (–T)) / d)

Rearranging gives us:

t = (V * ρ_water * L * d) / (K_R * A * T)

Final Thoughts

This formula provides a clear way to calculate the time required for the water to freeze under the specified conditions. The key factors influencing this time are the temperature difference between the water and the air, the properties of the ice, and the geometry of the system. By plugging in the appropriate values for each variable, you can find the exact time it will take for the water to freeze completely.