A Wind Turbine Efficiency Limit Higher than the Lanchester (Betz) Limit

It is shown that the upper limit on the fraction of power that can be extracted from an airstream approaching a wind turbine is about 78% rather than the historical value of 59%. This higher limit is based on the assumption that the wind turbine cannot accelerate the flow at its exit plane above the freestream velocity. The derivation of the historical limit of 59% must either violate the angular momentum equation or the continuity equation.

💡 Research Summary

**
The paper revisits the classic theoretical limit on wind‑turbine aerodynamic efficiency, traditionally known as the Betz (or Lanchester‑Betz‑Joukowsky) limit of 59 % of the kinetic power in the incident wind. The author argues that the original derivation neglects the rotational (swirl) component of the flow and therefore violates either the continuity equation or the angular‑momentum balance. By explicitly incorporating the swirl velocity and imposing a physically motivated constraint that the magnitude of the velocity at the turbine exit plane cannot exceed the free‑stream velocity, a new set of pressure‑velocity relations is derived.

The analysis proceeds as follows. Starting from the momentum balance for the whole control volume (Eq. 3) and for a small control volume surrounding the rotor (Eq. 4), the author eliminates the reaction force and obtains a relation between pressure differences and velocities (Eq. 6). The classic Betz derivation then assumes (v_2 < v_1) (Eq. 9) and uses Bernoulli’s equation upstream and downstream of the rotor to eliminate pressure terms, leading to the familiar power coefficient (C_p = 16/27 \approx 0.593).

The author challenges the (v_2 < v_1) assumption by noting that a real turbine must generate torque, which inevitably creates a swirl component (v_{\theta,2}). Conservation of mass requires the axial component of the exit velocity to satisfy (v_{2,axial} = v_{1,axial}), while the total speed cannot be greater than the free‑stream speed because the turbine cannot act as a nozzle that accelerates the flow. This yields the condition (v_2 = v) (Eq. 11). Substituting this into Bernoulli’s equation gives a revised pressure drop (Eq. 12) that is consistent with both continuity and angular‑momentum balance.

Using the new pressure‑velocity relation, the author derives an alternative expression for the power (Eq. 17). Instead of differentiating analytically, the author solves numerically for the ratio (\gamma = v_1/v) that maximizes power, finding (\gamma \approx 0.818) (Eq. 18). The resulting maximum power is

\