Wind Turbine Physics: Power Output & the Betz Limit

simulator intermediate ~10 min
Loading simulation...
~1.4 MW with 80m rotor at 10 m/s wind

A wind turbine with 40m blade radius at 10 m/s wind speed and Cp=0.45 extracts approximately 1.4 MW. The Betz limit caps maximum extraction at 59.3% of available wind power, regardless of turbine design.

Formula

Available wind power: P = ½ρAv³
Betz limit: P_max = (16/27) × ½ρAv³ ≈ 0.593 × P
Swept area: A = πr²

Power from Moving Air

Wind turbines convert the kinetic energy of moving air into electricity. The available power in wind scales with the cube of wind speed — doubling the wind speed means eight times more power. This cubic relationship makes wind speed the single most important factor in turbine siting. A site with average winds of 8 m/s has nearly twice the energy potential of one with 6.5 m/s winds.

The Betz Limit: Nature's Speed Limit

In 1919, German physicist Albert Betz proved that no turbine can extract more than 16/27 (59.3%) of the kinetic energy in wind. The reasoning is elegant: if a turbine extracted all the energy, the air behind it would stop, blocking incoming air. The optimal extraction slows air to 1/3 of its original speed, leaving enough flow to clear the rotor disk. This limit is as fundamental to wind energy as the Carnot limit is to heat engines.

Modern Turbine Engineering

Today's utility-scale turbines have blade radii exceeding 80 meters, swept areas larger than two football fields, and rated capacities of 10-15 MW. They achieve power coefficients of 0.45-0.50, capturing about 80-85% of the Betz limit. Variable-pitch blades and sophisticated control systems optimize performance across a wide range of wind speeds, from cut-in (~3 m/s) to cut-out (~25 m/s).

Exploring the Physics

This simulation lets you vary wind speed, blade radius, and power coefficient to see how turbine output changes. Watch how the cubic wind speed relationship dominates all other factors. Push the power coefficient toward the Betz limit and see how little room remains for engineering improvement. The animated turbine visualization shows blade rotation speed scaling with wind conditions.

FAQ

What is the Betz limit?

The Betz limit (59.3%) is the theoretical maximum fraction of kinetic energy a wind turbine can extract from wind. Derived by Albert Betz in 1919, it arises because the turbine must let some air pass through — if it extracted all energy, air would stop and no new air would arrive.

Why does wind power scale with the cube of wind speed?

Wind power P = ½ρAv³. Kinetic energy is ½mv², and the mass flow rate through the rotor is ρAv. Multiplying gives v³ dependence. This means doubling wind speed yields 8× more power — making site selection critical.

What is a good power coefficient for a modern turbine?

Modern utility-scale turbines achieve power coefficients (Cp) of 0.40-0.50 at optimal wind speeds. The best designs reach Cp ≈ 0.50, which is about 85% of the Betz limit. Small turbines typically achieve Cp = 0.25-0.35.

What is the tip speed ratio?

The tip speed ratio (TSR) is the speed of the blade tip divided by the wind speed. Modern three-blade turbines operate optimally at TSR ≈ 6-8. Too low and the blades miss wind; too high and the rotor acts as a solid disk, deflecting air around it.

Sources

Embed

<iframe src="https://homo-deus.com/lab/energy-systems/wind-turbine/embed" width="100%" height="400" frameborder="0"></iframe>
View source on GitHub