A circular pond of radius 12 m has a layer of ice 0.25 m thick. On a spring day, the temperature of the water below the ice is 4.0°C and the temperature of the air above is 11°C. What is the rate of heat transfer through the ice, in kW?

To find the rate of heat transfer through the ice, we can use the formula for heat transfer through conduction:

Q = k * A * ΔT / d

Where:
Q is the rate of heat transfer
k is the thermal conductivity of the material (ice)
A is the surface area through which heat is transferred
ΔT is the temperature difference across the material
d is the thickness of the material

First, let's find the surface area of the circular pond. The formula for the surface area of a circle is:

A = π * r^2

Where:
A is the surface area
π is a mathematical constant approximately equal to 3.14159
r is the radius of the circular pond

Plugging in the values:
A = 3.14159 * 12^2
A = 452.389 m^2

The temperature difference across the ice is the difference between the air temperature and the water temperature:

ΔT = T_air - T_water

Plugging in the values:
ΔT = 11°C - 4.0°C
ΔT = 7.0°C

Now we need to find the thermal conductivity of ice. The thermal conductivity of ice is approximately 2.22 W/(m·K).

Plugging in the values into the formula for heat transfer:

Q = 2.22 * 452.389 * 7.0 / 0.25

Calculating the rate of heat transfer:

Q = 2210.5936 W

To convert this to kilowatts, we divide by 1000:

Q = 2.2106 kW

Therefore, the rate of heat transfer through the ice is approximately 2.2106 kW.