A very large wind turbine is used to extract power from the wind.The turbine has a diameter of 120 meters.What is the maximum power this turbine can extract from the wind.Assume a wind speed of 10 m/s and the density of air is 1.2 kg/m³. A)4.02 MW B)7.52 MW C)3.40 MW D)6.80 MW

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Quizplus · Accepted Answer

The maximum power that can be extracted from the wind by a wind turbine is given by the Betz limit, which states that no more than 59.3% of the kinetic energy in the wind can be captured. The power in the wind can be calculated using the formula: $P = \frac{1}{2} \cdot \rho \cdot A \cdot v^3$, where $P$ is the power in watts, $\rho$ is the air density in kg/m^3, $A$ is the area in square meters through which the wind is passing (the swept area of the turbine blades), and $v$ is the wind speed in m/s. The swept area $A$ can be calculated using the formula for the area of a circle, $A = \pi \cdot r^2$, where $r$ is the radius of the circle.Given:- Diameter of the turbine = 120 meters, so the radius $r = 120/2 = 60$ meters.- Wind speed $v = 10$ m/s.- Density of air $\rho = 1.2$ kg/m^3.First, calculate the swept area $A$:$$A = \pi \cdot r^2 = \pi \cdot (60)^2 = 3600\pi$$Then, calculate the power in the wind:$$P = \frac{1}{2} \cdot 1.2 \cdot 3600\pi \cdot (10)^3$$$$P = 0.5 \cdot 1.2 \cdot 3600 \cdot \pi \cdot 1000$$$$P = 2160000\pi$$To find the maximum power extractable (using the Betz limit of 59.3%):$$P_{max} = 0.593 \cdot 2160000\pi$$$$P_{max} = 1281676.8$$$$P_{max} \approx 4.02 MW$$Therefore, the maximum power this turbine can extract from the wind is approximately 4.02 MW.

A Very Large Wind Turbine Is Used to Extract Power