Question

Considering that wind turbines are rated for the maximumtolerable wind speed around \(12 \mathrm{~m} / \mathrm{s}\), and tend to operate at about \(30 \%\) capacity factor, how much average power \(^{44}\) would a \(100 \mathrm{~m}\) diameter turbine operating at \(45 \%\) efficiency be expected to produce?

Short answer

Answer: The expected average power output of the wind turbine is approximately 372.1 kW.

Step-by-step solution

Calculate the area of the turbine

To get the area of the circular turbine, we can use the formula: \(A = \pi r^2\) where \(A\) is the area, \(r\) is the radius, and \(\pi\) is a constant equal to approximately 3.14159. The diameter is given as 100 m, so the radius is half the diameter, which is 50 m. Plugging the value of the radius into the formula, we get: \(A = \pi (50 \mathrm{~m})^2 = 2500 \pi \mathrm{~m^2}\)

Calculate the wind power density

The formula to calculate wind power density is: \(P = \frac{1}{2} \rho A v^3\) where \(P\) is the power, \(\rho\) is the air density (approximately 1.225 kg/m³ for sea level and 15°C), \(A\) is the area, and \(v\) is the wind speed. Using the given maximum tolerable wind speed (12 m/s), we can calculate the power: \(P = \frac{1}{2} (1.225 \mathrm{~kg/m^3})(2500 \pi \mathrm{~m^2})(12 \mathrm{~m/s})^3\) \(P \approx 2.756 \times 10^6 \mathrm{~W}\)

Apply the capacity factor

Multiply the power by the capacity factor (30%) to get the power produced at average wind speed: \(P_{avg} = P \times capacity~factor\) \(P_{avg} = 2.756 \times 10^6 \mathrm{~W} \times 0.3\) \(P_{avg} \approx 8.268 \times 10^5 \mathrm{~W}\)

Apply the efficiency

Multiply the power produced at average wind speed by the efficiency (45%) to get the expected average power output: \(P_{expected} = P_{avg} \times efficiency\) \(P_{expected} = 8.268 \times 10^5 \mathrm{~W} \times 0.45\) \(P_{expected} \approx 3.721 \times 10^5 \mathrm{~W}\) So, a 100 m diameter wind turbine operating at 45% efficiency is expected to produce approximately \(3.721 \times 10^5 \mathrm{~W}\), or 372.1 kW, of average power.

Key concepts

Wind Power Density

Understanding wind power density is essential to comprehend the potential power generation of wind turbines. It is a measure of the amount of power available for conversion by a wind turbine from the wind at a particular location and height. To calculate wind power density, we apply the formula:
\[ P = \frac{1}{2} \rho A v^3 \]
where \( P \) is the wind power density in watts per square meter (W/m²), \( \rho \) represents air density in kilograms per cubic meter (kg/m³), \( A \) is the area swept by the turbine blades in square meters (m²), and \( v \) is the wind velocity in meters per second (m/s). Air density varies with altitude and temperature, while wind velocity can change based on local weather conditions and landscape features. The cubic relationship between wind velocity and power means that small increases in wind speed can result in significant increases in power density. When assessing a location for wind energy potential, high wind power density indicates a more favorable environment for wind turbines.

Wind Turbine Efficiency

Wind turbine efficiency is a measure of how well a wind turbine converts the kinetic energy in the wind into electrical energy. The efficiency of a turbine depends on its design, including blade shape and generator technology, as well as the wind speed. The Betz limit, which dictates that no more than 59.3% of the wind's kinetic energy can be converted by a wind turbine, establishes an upper bound on efficiency. In our example, the wind turbine is operating at 45% efficiency, which means it is converting 45% of the kinetic energy passing through the swept area of its blades into electricity. It's important to note that efficiency can vary with wind speeds and that turbines are designed to be most efficient within a particular range of wind speeds—often, just below the rated peak speed. Turbine efficiency plays a significant role in determining the output of the turbine and is crucial for economic and environmental evaluations of wind power projects.

Capacity Factor

The capacity factor of a wind turbine is a ratio that compares its actual output over a certain period with the theoretical maximum output if it operated at full capacity nonstop during the same period. It is expressed as a percentage and depends not just on the turbine's design and efficiency, but also on wind availability, which is impacted by location and seasonality.
A 30% capacity factor, as mentioned in our exercise, implies that the turbine generated 30% of what it could potentially produce had it been running at its maximum rated output all the time. The capacity factor is an important metric for planning and economic assessment, as it reflects how often and how well a turbine can be expected to operate in the real world. For instance, a higher capacity factor indicates that the turbine location experiences consistently strong winds, boosting energy production and, consequently, the return on investment for the wind farm. Understanding the capacity factor helps in creating realistic projections for energy production and in determining the viability of a wind power project.