Engineering
Wind Turbine Power & Betz Limit (16/27 = 59.26%) formula: 0.5 * rho * A * v^3 * Cp
Calculate swept rotor wind power, Albert Betz's theoretical 16/27 (59.26%) maximum extraction limit, and real-world Cp shaft output in kW.
Use an ideal rotor power model
For an isolated ideal actuator disk in steady incompressible unconfined flow, the Betz model gives a maximum power coefficient of 16/27 relative to upstream kinetic power through the rotor swept area. It is not a universal bound for every flow-concentrating geometry or arbitrary reference area.
The example assumes constant density and a chosen Cp=0.45, not a measured turbine power curve. Cubic wind-speed scaling holds with other factors fixed; real controls, cut-in, rated output and cut-out limit operation. Instantaneous modeled rotor power is not annual energy production or delivered electrical power.
Density and radius must be positive; wind speed is a nonnegative magnitude. Cp is constrained to [0,16/27] for this model. Generator and drivetrain losses are not included.
| Wind Rotor Parameter | Cell / Formula | 80m-Diameter Rotor (R = 40 m) | Power Output (kW) |
|---|---|---|---|
| Air Density & Radius | B2 = 1.225 kg/m^3, B3 = 40 m | Swept area PI()*40^2 = 5,026.5 m^2 | Wind B4 v = 10 m/s |
| Betz Limit Factor | ROUND(16 / 27, 4) | Theoretical aerodynamic ceiling | 0.5926 (59.26%) |
| Betz Max Power (10 m/s) | 0.5*rho*A*v^3*(16/27) / 1000 | 3,078.76 kW raw * 16/27 | 1,824.45 kW |
| Modeled rotor (Cp = 0.45) | B5 = 0.45 (or v = 5 m/s Betz) | At 10 m/s: 1,385.44 kW | At 5 m/s Betz: 228.06 kW (1/8!) |
Theoretical maximum Betz wind power in kilowatts (Cp = 16/27)
Divides Watts by 1,000 to report maximum extractable rotor power in kW.
=IF(COUNT(B2:B5)<>4,"Invalid inputs",IF(OR(B2<=0,B3<=0,B4<0,B5<0,B5>16/27),"Invalid inputs",ROUND((0.5 * B2 * PI() * (B3 ^ 2) * (B4 ^ 3) * (16 / 27)) / 1000, 2)))Modeled rotor power at a chosen Cp
B5 is an assumed aerodynamic coefficient for this calculation, not an electrical conversion efficiency.
=IF(COUNT(B2:B5)<>4,"Invalid inputs",IF(OR(B2<=0,B3<=0,B4<0,B5<0,B5>16/27),"Invalid inputs",ROUND((0.5 * B2 * PI() * (B3 ^ 2) * (B4 ^ 3) * B5) / 1000, 2)))Try the example
Use the original values or change them to test your understanding.