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Proof no. 01 — proof, walked through

A perfect wind turbine still leaves 41% of the wind alone.

Not because of friction, or noise, or imperfect blades. Because of arithmetic. The ceiling is 16/27 — 59.26% — and here is exactly where it comes from.

26 August 2026~8 min3 levels of mathsAfter R. Zorich (2023)
Choose your levelThe words, the maths and the drawing all change. Press one. Switch any time — you keep your place.
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The set-up

Three speeds, and one thing you get to choose

The turbine is treated as a black box. We will not use a single fact about blades, aerofoils or gearboxes. Only this: something in that box takes energy out of the air, and it is bolted to a tower so it doesn’t blow away.

Three speeds matter. The wind arrives at V0. It passes through the machine at Vm. Far downwind, once the surrounding air has dragged the wake back up to speed, it ends at Vf.

The faint outline in the diagram is the streamtube: the parcel of air that actually goes through the swept area. Everything outside it is pushed around the machine and never contributes anything.

Betz’s law is the answer to one question: what should Vm be?

or press ←→
Through-speed0.66
Power captured59.2

Share of the wind's power captured

drag anywhere on the curve
Why the number matters

Five things the Betz ceiling actually tells you

The proof is a piece of arithmetic about momentum. Its consequences are commercial.

1. Real turbines are close to done. A good modern rotor runs at a power coefficient of about 0.45–0.50 in its sweet spot — roughly 76–84% of the theoretical maximum. There is no factor-of-two hiding in better blades. Anyone selling you a rotor that beats 59.3% is either measuring a different area or selling you something else.

2. The wind speed term is cubed, and that dominates everything. The power on offer is ½ρAV0³. A site with 20% more wind has 73% more power in it. Site selection beats engineering, every time.

3. Length beats cleverness. The other term is area, and area goes with the square of blade length. Doubling blade length quadruples the harvest. That, not efficiency, is why rotors have grown from 15 m to over 220 m in forty years.

4. Wakes are a real asset on the balance sheet. The whole proof turns on the fact that a turbine leaves slow air behind it. Field measurements put the velocity deficit at 15–25% still six and a half rotor diameters downwind, and full farm-wake recovery takes tens of diameters. That is why turbine spacing, farm layout and “wake steering” control are worth arguing about.

5. It is not the Carnot limit. Carnot is thermodynamics — entropy forbids you from doing better. Betz is bookkeeping — mass has to keep flowing, so you can’t take all its energy. Any open-flow harvester is bound by the same argument: tidal stream turbines live under the same ceiling. Put a duct or diffuser round the rotor and you can exceed 16/27 of the rotor’s own swept area, because you are now funnelling air from a larger capture area. The physics has not been beaten; the denominator has been changed.

The wind does not owe you all of its energy. It has to keep moving, because more wind is arriving behind it.

Small print

What the proof assumes

Every clean result is clean because something was assumed. These are the load-bearing ones.

A black boxNo knowledge of the mechanism is used — no blades, no aerofoils, no rotational speed. Only that something extracts energy and is bolted to a tower so it doesn’t move downwind.
Uniform through-flowEvery part of the swept area A passes air at the same speed Vm, aligned with the wind. Real rotors vary radially, and shed swirl.
Constant densityρ is fixed. Fine at wind speeds; air is effectively incompressible far below the speed of sound.
Steady, uniform windV0 is the same everywhere outside the machine’s influence. No shear, no gusts, no yaw misalignment, no turbulence intensity.
A tiny machine in a huge skyThe atmosphere is treated as effectively infinite, so the wake eventually recovers: Vf → V0 as A → 0. This is the step that lets the two halves of the thrust be equal.
Viscous wake mixingThis derivation leans on it — the surrounding wind drags the wake back up to speed, and that pull is half the thrust. The classical route instead assumes inviscid flow so it can use Bernoulli, and arrives at the same number by a different road.
Sources

Where this comes from

Zorich, R. (2023). Brief communication: Betz’s Law: the Zorich Derivation. Wind Energy Science Discussions, preprint wes-2023-55, CC BY 4.0. The derivation walked through here, equation numbering included.
Betz, A. (1926/1966). Wind-Energie und ihre Ausnutzung durch Windmühlen. The original, alongside contemporaries — Lanchester and Joukowsky reached it too, which is why you will also see it called the Betz–Joukowsky limit.
van Kuik, G. (2007); Okulov & van Kuik (2012). Historical accounts of the classical derivation and who got there first.
Aitken, M. et al. (2014). Lidar measurements of turbine wakes — velocity deficit still 15–25% at 6.5 rotor diameters downwind.
Dong, G. et al. (2022). Large-eddy simulation of farm wakes — 95% velocity recovery around 55 rotor diameters downstream.
End of proof