A type 3 wind turbine has rated wind speed of 13 m/s. Coefficient of performance of this turbine is 0.3. Calculate the rated power density of the wind that is hitting the turbine. Calculate the mechanical power developed at the shaft connecting rotor and generator. Assume rotor diameter 100 m and air density 1.225 kg/m^3.
Engineering · College · Sun Jan 24 2021
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To calculate the rated power density of wind hitting the turbine, we can use the following equation for wind power density:
\[ P_{density} = \frac{1}{2} \times \rho \times V^3 \]
Where: - \( P_{density} \) is the power density in watts per square meter (W/m²), - \( \rho \) is the air density in kilograms per cubic meter (kg/m³), - \( V \) is the wind speed in meters per second (m/s).
Given: - \( \rho = 1.225 \) kg/m³, - Wind speed \( V = 13 \) m/s.
Plugging in the values, we get:
\[ P_{density} = \frac{1}{2} \times 1.225 \times 13^3 \] \[ P_{density} = 0.5 \times 1.225 \times 2197 \] \[ P_{density} = 1344.5875 \] W/m².
Now, to calculate the mechanical power developed at the shaft, we use the coefficient of performance (\( C_p \)) of the turbine, which represents the fraction of wind power that can be converted into mechanical power. The equation is:
\[ P_{mechanical} = P_{available} \times C_p \]
Where: - \( P_{mechanical} \) is the mechanical power developed at the shaft, - \( P_{available} \) is the power available in the wind that strikes the rotor area, - \( C_p \) is the coefficient of performance.
First, we need to compute the rotor area (\( A \)) since this determines how much wind the turbine intercepts:
\[ A = \frac{\pi D^2}{4} \]
Where \( D \) is the rotor diameter in meters. For our given diameter of 100 m:
\[ A = \frac{\pi \times 100^2}{4} \] \[ A = \frac{3.14159 \times 10000}{4} \] \[ A = 7853.975 \] m².
The power available in the wind that strikes this rotor area is:
\[ P_{available} = P_{density} \times A \] \[ P_{available} = 1344.5875 \times 7853.975 \] \[ P_{available} = 10568750.5125 \] W.
Finally, we compute the mechanical power developed at the shaft using the coefficient of performance \( C_p = 0.3 \):
\[ P_{mechanical} = 10568750.5125 \times 0.3 \] \[ P_{mechanical} = 3170625.15375 \] W, or approximately 3.17 MW.
So, the mechanical power developed at the shaft of the wind turbine, given the conditions, is approximately 3.17 megawatts.
Extra: Wind turbines convert kinetic energy from the wind into mechanical energy, and ultimately into electricity. The power that a wind turbine can produce depends on several factors including the wind speed, air density, rotor diameter, and its efficiency or coefficient of performance. The coefficient of performance is a key factor as it represents the efficiency with which a turbine converts the kinetic energy of the wind into mechanical energy.
The rotor diameter is also critical as it determines the area through which the turbine can capture the wind; larger rotors can intercept more wind, thus potentially generating more power. However, as the rotor size increases, the materials and construction requirements also become more demanding and costly.
Air density affects the amount of kinetic energy available in the wind; denser air carries more energy. This is why wind turbines are often situated in locations where wind speeds are higher and more consistent, such as offshore or on high ground.