Wind Turbine Calculator

The Science Behind Wind Power

A wind turbine works by converting the kinetic energy of the wind into electrical energy. The amount of power it can generate is determined by a few critical factors, all captured in a fundamental physics formula:

Power (P) = ½ × ρ × A × v³ × η

This formula might look complex, but each part represents a simple concept:

• P
  : The final power output, measured in Watts.
• ρ (Rho)
  : The density of the air. At sea level, this is a fairly constant 1.225 kg/m³.
• A
  : The "swept area" of the turbine blades. This is the circular area the blades cover as they rotate, calculated as πr² (where r is the length of one blade). A larger swept area captures more wind.
• v³ (Velocity Cubed)
  : The wind speed, cubed. This is the most impactful part of the equation. It means that if the wind speed doubles, the available power increases eightfold (2×2×2 = 8). This is why even small increases in average wind speed at a site are so valuable.
• η (Eta)
  : The overall efficiency of the system. This accounts for aerodynamic, mechanical, and electrical losses.

Practical Example: A Utility-Scale Turbine

Let's calculate the potential output for a typical onshore wind turbine on a windy day.

• Blade Length (Radius): 50 meters (a common size).
• Wind Speed: 12 m/s (about 27 mph, a strong, steady wind).
• Overall Efficiency: 45% (a realistic value for a modern turbine).

The calculator first determines the swept area:

Area = π × (50m)² ≈ 7,854 m²

Then, it applies the full power formula:

P = 0.5 × 1.225 × 7854 × (12)³ × 0.45

P ≈ 3,749,800 Watts or ~3.75 MW

The Limit of Perfection: Betz's Law

A crucial concept in wind energy is Betz's Law. It states that no turbine can ever capture more than 59.3% of the kinetic energy in the wind. Why? Because if a turbine were 100% efficient, it would stop the wind completely, and no air would flow through the blades. To keep the air moving, some energy must be left behind.

This 59.3% is the absolute aerodynamic maximum. When you factor in friction in the gearbox, generator losses, and other inefficiencies, the "overall efficiency" of the best modern turbines lands between 40% and 50%.

Frequently Asked Questions (FAQ)

Q: What is a typical wind speed?

A: Wind speeds vary greatly by location and altitude. A good onshore site for a wind farm might have an average annual wind speed of 6.5 m/s (14.5 mph) or more. Offshore sites are often much higher, averaging over 9 m/s (20 mph).

Q: What are "cut-in" and "cut-out" speeds?

A: Turbines don't operate at all wind speeds. The "cut-in" speed is the minimum speed needed to start generating power (usually 3-4 m/s). The "cut-out" speed is the maximum speed at which the turbine will shut down to prevent damage (usually around 25 m/s or 55 mph).

Q: Why are wind turbines so tall?

A: Wind is generally faster and less turbulent at higher altitudes, away from ground-level obstructions like trees and buildings. Taller towers allow the turbine to access this more powerful and consistent wind, dramatically increasing its energy production.

Q: Does the calculator show how much energy a turbine makes in a day?

A: No, this calculator shows the instantaneous power (in Watts) at a specific wind speed. To find the energy produced over a day (in kWh), you would need to know the average power output over that 24-hour period, which requires detailed wind data for the location.