Question

Based on the following data, find the main dimensions and characteristics of a horizontal-axis wind turbine:

Power delivered at rated speed: 250 kW.
The wind turbine will be installed on flat terrain.
The roughness will be uniform and of the "cut grass" type, i.e. Z0 = 0.01 m.
The reference height is 10 m.
The reference speed is 10 m/s.
Mechanical losses of around 8% and a generator electrical efficiency of 85% will be assumed.
Three-blade wind turbine with upwind rotor.
Startup time: 170 to 240 seconds.
Nominal shutdown time: 25 seconds.
Emergency shutdown time: 20 seconds.
Survival speed: approx. 60 m/s (216 km/h).
Physical characteristics of the air:
Wind speed connection: 5 m/s.
Wind speed at rated power (design): 10-15 m/s.
Wind speed at disconnection: 25 m/s.
Air density, r = 1.25 kg/m³ (at sea level).
Air viscosity, n = 1.33 x 10-5 m²/s.

Answer 1

The power output of wind turbines can be calculated using a specific formula that includes the wind speed cubed, the swept area, air density, and the turbine's efficiency. The largest turbines can deliver significant power at design wind speeds, and the actual power delivered by any turbine varies notably with efficiency and wind speed.

Step-by-step explanation:

To calculate the power output of a wind turbine, we need to use the formula for wind power, which considers the area swept by the turbine blades and the cube of the wind speed, factoring in the air density and the efficiency of the turbine. The power is given by the equation P = 0.5 × Cp × r × A × V^3, where Cp is the power coefficient (efficiency), r is the air density, A is the swept area of the rotor (πR^2 for a circle), and V is the wind speed. Given this, we can answer the following example questions:

• For the largest wind turbines with rotor diameters of about 150 m and an efficiency of 50%, operating at a wind speed of 13 m/s, the power output can be calculated using the provided equation.
• A 100 m diameter wind turbine rated for a maximum wind speed of around 12 m/s and operating at a 30% capacity factor with 45% efficiency would have its average power output estimated accordingly.
• A more moderate-sized wind turbine with a radius of 10 m at an efficiency of 50% would produce different power outputs at wind speeds of 5 m/s, 10 m/s, 15 m/s, and 20 m/s, respectively, demonstrating the cubic relationship between wind speed and power output.
• The rotor diameter of a turbine can be deduced using the power output at a certain wind speed and the known efficiency from a graph similar to the one shown in Figure 12.7.
• For a small 4 m diameter 3-blade wind turbine atop a house in a breeze of 5 m/s, the expected power delivery can be calculated with the same power equation.