Betz Limit — Maximum Extractable Wind Power

Albert Betz published the limit in 1920, and the argument is so simple it is almost a trick. Put an imaginary disc in a stream and let it slow the air. The power extracted is the drop in kinetic energy multiplied by the mass flow — but slowing the air also reduces the mass flow through the disc, because the streamtube must widen to carry it. Take too little and you leave energy behind; take too much and the flow refuses to pass. Maximising the product gives an exit speed of exactly one third the free-stream speed, and an extraction of exactly $16/27 = 0.592593$.

No blade appears anywhere in that derivation. It is momentum and mass conservation and nothing else, which is why the limit applies to any device sitting in an open stream — shrouded rotors, vertical-axis machines, oscillating foils, bladeless resonators. Every few years a design is announced that "beats Betz", and it invariably turns out to be measuring $C_p$ against an area smaller than the flow it actually disturbs. A diffuser around a rotor genuinely raises power per unit of rotor area, and genuinely does not raise it per unit of the diffuser's own frontal area, which is the area the limit refers to.

The limit is a ceiling, not a target, and confusing the two is the standard error. Real utility rotors reach 0.35–0.45 at their design point and less everywhere else, because Betz's ideal disc has no drag, no tip vortices, no wake rotation and infinitely many blades. Glauert's later analysis, which accounts for the swirl the rotor puts into its own wake, lowers the practical ceiling further at low tip-speed ratios. A machine at 0.45 is doing very well indeed.

The cube law runs through this equation exactly as it runs through the others, so the ceiling itself is enormously sensitive to the wind speed used. And because power is proportional to area while area is proportional to the square of the diameter, doubling a rotor quadruples its Betz power at the same wind. That relationship — not any advance in $C_p$, which is already near its ceiling — is why turbines have grown from 15 m rotors to over 200 m in four decades.