Betz limit wind turbine
A wind turbine extracts kinetic energy by slowing the wind. But there is a fundamental limit: extract too little and the wind passes unused; extract too much and you block the airflow entirely. The optimum, derived by Albert Betz in 1919, is exactly 16/27 ≈ 59.3% of the incoming wind power. Adjust the extraction factor and watch the stream tube, velocity field, and power output respond in real time.
Cp = ½(1 + v2/v1)(1 − (v2/v1)²) | Cp,max = 16/27 ≈ 0.593
The Betz limit
In 1919, German physicist Albert Betz proved that no wind turbine can capture more than 16/27 (approximately 59.3%) of the kinetic energy in wind. This isn’t an engineering limitation — it’s a fundamental law of physics, analogous to the Carnot limit for heat engines. The proof follows from conservation of mass and momentum applied to a control volume around the turbine.
The stream tube model
Actuator disk theory models the turbine as a thin permeable disk. Wind approaches at velocity v1, slows to vd = v1(1 − a) at the disk, and continues to slow to v2 = v1(1 − 2a) far downstream, where a is the axial induction factor. Conservation of mass requires the stream tube to expand: as air slows, it must spread to maintain the same mass flow rate. The disk captures the momentum difference between incoming and outgoing air.
Why the optimum is at a = 1/3
The power coefficient is Cp = 4a(1 − a)². Taking the derivative and setting it to zero gives a = 1/3, which yields Cp = 16/27. At this point the downstream velocity is exactly 1/3 of the upstream velocity. Extract less (small a) and most energy passes through unused. Extract more (large a) and you increasingly block the flow — air diverts around the turbine rather than through it. At a = 0.5, the downstream velocity reaches zero, which is physically impossible for a real turbine.
Real turbines
Modern wind turbines achieve power coefficients of about 0.45–0.50, reaching roughly 75–85% of the theoretical Betz limit. Losses come from tip vortices, wake rotation, blade drag, and generator inefficiency. The three-bladed horizontal-axis design emerged as the practical optimum through decades of engineering refinement, though the Betz limit itself is agnostic about turbine design.