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Chapter 12: Problem 12

The largest wind turbines have rotor diameters \(^{40}\) around \(150 \mathrm{~m}\). Using a sensible efficiency of \(50 \%\), what power does such a jumbo turbine deliver at a maximum design wind speed of \(13 \mathrm{~m} / \mathrm{s} ?\)

Short Answer

Expert verified
Answer: The power delivered by the wind turbine at a wind speed of 13 m/s and an efficiency of 50% is approximately 6214 kW.

Step by step solution

01

Write the formula for wind power

The formula for wind power is given by: P = (1/2) * ρ * A * v^3 * η, where P is the power delivered by the turbine, ρ is the air density, A is the area swept by the rotor, v is the wind speed, and η is the efficiency of the turbine. We are given η = 0.5 (50%), and we will estimate ρ = 1.225 kg/m^3 as the air density at sea level in a standard atmosphere.
02

Calculate the area swept by the rotor

The area swept by the rotor, A, can be calculated using the formula: A = π * (D/2)^2, where D is the diameter of the rotor. In this case, D = 150 m, so: A = π * (150/2)^2 = 17671.5 m^2.
03

Use the given wind speed

We are given the maximum design wind speed as v = 13 m/s.
04

Calculate the power delivered by the turbine

Now, we can use the wind power formula to calculate the power delivered by the turbine: P = (1/2) * 1.225 kg/m^3 * 17671.5 m^2 * (13 m/s)^3 * 0.5 P ≈ 6214 kW So, at a wind speed of 13 m/s and an efficiency of 50%, such a large wind turbine delivers a power of approximately 6214 kW.

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Most popular questions from this chapter

A hard slap might consist of about \(1 \mathrm{~kg}\) of mass moving at \(10 \mathrm{~m} / \mathrm{s}\). How much kinetic energy is this, and how much warmer would \(10 \mathrm{~g}\) of \(\mathrm{skin}^{33}\) get if the skin has the heat capacity properties of water, as in the definition of a calorie (Sec. 5.5; p. 73 and Sec. \(6.2\); p. 85 are relevant)?

Referring to Figure 12.7, examine performance at \(5 \mathrm{~m} / \mathrm{s}\) and at \(10 \mathrm{~m} / \mathrm{s}\), picking a representative power for each in the middle of the cluster of black points, and assigning a power value from the left-hand axis. What is the ratio of power values you read off the plot, and how does this compare to theoretical expectations for the ratio going like the cube of velocity?

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Traveling down the road, you carefully watch a three-bladed wind turbine, determining that it takes two seconds to make a full revolution. Assuming it's operating near the peak of its efficiency curve \(^{41}\) according to Figure \(12.4\), how fast do you infer the wind speed to be if the blade length \(^{42}\) appears to be \(15 \mathrm{~m}\) long?
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