What if a turbine caught every bit of the wind?

The wind pushes a turbine's big blades around. The turbine turns that push into electricity.

But some wind always blows on past. What if the turbine caught every bit of it?

A wind turbine works by taking energy out of moving air. The air behind the blades moves more slowly than the air in front, because the turbine has taken some of its energy.

So here's a puzzle: if a turbine could slow the wind all the way to a stop, wouldn't it catch all of the wind's energy? In this lab, you push a turbine's power coefficient, the share of the wind's power it catches, as high as it can possibly go, and see why the answer is never 100%.

Wind power through a rotor disc scales as ½ρAv³, and a turbine converts a fraction Cp of it. Engineers improve blades, controls and generators to raise Cp, so it is natural to ask how high it could go.

This lab compares a turbine with Cp = 0.45 against the theoretical maximum, Cp = 16/27 ≈ 0.593, at a steady 10 m/s with 40 m blades. The model will not let Cp exceed that value; the question is why nature sets a ceiling below 1.

Olly's turbine has very long blades. The wind blows at a steady speed.

The turbine catches almost half of the wind's power.

In the model, a turbine with 40-meter blades sits in a steady 10-meter-per-second wind. The air passing through the circle its blades sweep carries about 3.08 megawatts of power.

With a power coefficient of 0.45, the turbine turns 45% of that into electricity: about 1.39 megawatts. Running all day in that wind, it would make about 33.3 megawatt-hours. The Energy Education site at the University of Calgary says real turbines commonly reach 35 to 45%, so this is a good turbine.

At v = 10 m/s, r = 40 m and ρ = 1.225 kg/m³, ½ρπr²v³ ≈ 3.08 MW. With Cp = 0.45 the model produces 1.39 MW, well below the 15 MW generator rating (it would reach that only at about 22.1 m/s), and about 33.3 MWh over 24 hours of steady wind.

Energy Education (University of Calgary) gives common real-world efficiencies of 35 to 45%. In the model Cp lumps together everything: aerodynamics plus gearbox, generator and cable losses.

Now Olly makes the blades as good as they can ever be.

The turbine catches more wind. But it stops at about 59%. It can never catch it all!

Now slide the power coefficient up as far as it goes: 0.593. The electricity rises from 1.39 to about 1.82 megawatts, roughly a third more, and a day's energy from 33.3 to about 43.8 megawatt-hours.

But try to go further and you can't. The model stops at 0.593, because physics says no turbine with open blades can catch more than about 59% of the wind's power. That ceiling is called the Betz limit. 'Every bit of the wind' isn't possible.

At Cp = 0.593 the output reaches the Betz maximum the model also reports: 1.82 MW from 3.08 MW of wind, 59% caught, about 43.8 MWh a day. That is about 32% more than at Cp = 0.45, and the 15 MW rating would now be reached at about 20.2 m/s instead of 22.1 m/s.

The model clamps Cp at 16/27. That is the model applying the Betz limit as a rule; why the limit exists is a question about how air must flow through and past the rotor.

To keep the blades turning, the wind has to keep blowing past them.

If the turbine stopped all the wind, new wind could not get through. Then the blades would stop too!

A turbine catches energy by slowing the wind. But the slowed air still has to move out of the way behind the blades, so fresh wind can arrive.

The Energy Education site explains it this way: if a turbine caught 100%, the wind would have to stop completely at the blades. Then that air would block the path, no new air could come in, and the turbine would stop spinning. Slow the wind too little and you waste it; slow it too much and you block it. The best balance gives about 59%.

Energy Education (University of Calgary) describes the reasoning: turbines extract energy by slowing the passing wind, and 100% efficiency would require the air to stop at the rotor, which would prevent further air from arriving, so the rotor would stop. Betz, in 1919, concluded the optimum lies at 59.3%.

The model takes that limit as a given rather than deriving it. Its other simplifications: one steady wind speed across the disc, a fixed Cp (real Cp varies with wind speed and blade angle), a 3 m/s cut-in, a 25 m/s cut-out and constant air density.

Real turbines don't quite reach the limit. But engineers keep making them better.

Their blades work like airplane wings, using the wind to spin around.

Real turbines can't reach the Betz limit. The U.S. Department of Energy explains that their blades work like an airplane wing: air flowing over them creates lift, which spins the rotor and turns a generator. Some energy is always lost along the way, in the air around the blades, the gears and the generator.

So engineers chase other gains too. The Department of Energy notes that longer blades sweep more area, capture more wind and produce more electricity, even if the share they catch stays the same.

The U.S. Department of Energy describes rotor blades acting like an airplane wing: the pressure difference creates lift and drag, lift exceeds drag, and the rotor turns a generator directly or through a gearbox that speeds up rotation and allows a physically smaller generator. Each stage loses some energy, which is why Energy Education puts common efficiencies at 35 to 45%, below 59.3%.

Because Betz caps the fraction, not the total, the bigger levers are area and wind speed: in the model, power grows with r² and v³. The DOE notes that larger rotor diameters sweep more area, capture more wind and produce more electricity, and that land-based turbines range from 100 kilowatts to several megawatts.