Wind is incident on the blades of a wind turbine. The radius of the blades is 12 m. Thefollowing data are available for the air immediately before and after impact with the blades.Before AfterDensity of air 1.20 kg m–3 1.32 kg m–3Wind speed 8.0 m s–1 4.0 m s–1(a) Determine the maximum power that can be extracted from the wind by this turbine.
Question
Wind is incident on the blades of a wind turbine. The radius of the blades is 12 m. Thefollowing data are available for the air immediately before and after impact with the blades.Before AfterDensity of air 1.20 kg m–3 1.32 kg m–3Wind speed 8.0 m s–1 4.0 m s–1(a) Determine the maximum power that can be extracted from the wind by this turbine.
Solution
To determine the maximum power that can be extracted from the wind by this turbine, we need to use the formula for the power of the wind, which is given by:
P = 0.5 * ρ * A * v³
where:
- P is the power
- ρ is the density of the air
- A is the area of the turbine blades
- v is the wind speed
First, we need to calculate the area of the turbine blades. The blades are circular, so we use the formula for the area of a circle, which is πr². The radius r is given as 12 m, so:
A = π * (12 m)² = 144π m²
Next, we need to calculate the wind speed before and after the impact with the blades. The wind speed before the impact is given as 8.0 m/s, and the wind speed after the impact is given as 4.0 m/s. The change in wind speed is the difference between these two values, which is:
Δv = 8.0 m/s - 4.0 m/s = 4.0 m/s
The power of the wind before the impact is:
P_before = 0.5 * 1.20 kg/m³ * 144π m² * (8.0 m/s)³ = 27648π W
And the power of the wind after the impact is:
P_after = 0.5 * 1.32 kg/m³ * 144π m² * (4.0 m/s)³ = 7373π W
The maximum power that can be extracted from the wind by this turbine is the difference between the power before and after the impact, which is:
P_max = P_before - P_after = 27648π W - 7373π W = 20275π W
So, the maximum power that can be extracted from the wind by this turbine is approximately 63717 W, or 63.7 kW.
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