Wind Turbine Power Yield, Betz Limit & Annual Energy (AEP

Renewable Energy Engineering: Aerodynamics of Wind Power & The Betz Limit

Wind turbines extract kinetic energy from moving air masses and convert it into mechanical torque and electrical grid power:

• The Cubic Velocity Law (v^3): Power in the wind scales with the cube of wind speed (P v^3). Doubling the wind speed from 5 m/s to 10 m/s increases available energy by making site wind resource assessment the most critical development factor.

• The Swept Area Quadratic Scaling (D^2): Doubling the rotor blade diameter quadruples the swept area), explaining why modern commercial offshore turbines feature massive 220+ meter rotors to harvest low-velocity winds.

• The Betz Limit (Albert Betz, 1919): Fluid dynamic conservation of mass and momentum proves that no rotor can capture more than **** of the kinetic energy in wind; attempting to extract 100% would cause the air velocity behind the turbine to drop to zero, blocking oncoming airflow.

How to Use This Calculator

Enter your Rotor Blade Diameter (D in meters) (e.g. 120 (Typical 3.5 MW commercial turbine: 100 to 140 m)), Hub-Height Wind Speed (v in m/s) (e.g. 9.0 (9 m/s ≈ 20.1 mph)), Air Density (ρ in kg/m³) (e.g. 1.225 (Standard sea-level air at 15°C)), Rotor Aerodynamic Power Coefficient (Cp ≤ 0.593) (e.g. 0.45 (Betz Limit = 0.593)), Drivetrain & Generator Electrical Efficiency (%) (e.g. 92), and Expected Annual Capacity Factor (%) (e.g. 38 (Onshore: 30%–45%, Offshore: 45%–55%)). Results update instantly as you type — adjust any value to compare scenarios.

The Formula

Aerodynamic Wind Power & Betz Law (Albert Betz, 1919) equations:

Rotor Swept Area: A = (π D^2) / (4) (m^2)

Available Kinetic Wind Power: P_wind = (1) / (2) ρ A v^3 (Watts)

Betz Limit Maximum Extraction: P_Betz = (16) / (27) P_wind ≈ 0.5926 P_wind

Electrical Power Output: P_elec = P_wind × C_p × η_gen = (1) / (2) ρ A v^3 C_p η_gen

Annual Energy Production (AEP) = P_elec (kW) × 8,760 hrs × Capacity Factor ÷ 1,000 (MWh/year)

Worked Example

For example, evaluating a commercial wind turbine with a 120 m rotor diameter in with 92.0% generator efficiency, and 38.0% capacity factor: Rotor swept area is ^24 = 11,309.7 m^

• Total incident wind power is 0.5 × 1.225 × 11,309.7 × (9)^3 = 5,050.2 kW (5.05 MW). Betz limit max is 2,992.7 kW. The resulting instantaneous electrical output is producing 6,960 MWh/year in AEP, powering 663 residential homes.

Understanding Your Result

Energy unit conversions: 1 kWh = 3,600,000 J = 3,412 BTU. US residential electricity: average 10,500 kWh/year (875 kWh/month) at approximately 0.16/kWh average. Energy Star appliances use 10–50% less energy than standard models. Solar panel output: a 400W panel produces approximately 1.5 kWh/day in average US locations. Power (watts) is the rate of energy use; energy (kWh or joules) is the total consumption over time — a 1,000W (1 kW) device running 1 hour uses 1 kWh.