Wind power output formula

The power in wind is the rate of flow of kinetic energy of air passing through the area swept by the turbine rotor.

Derivation

Let $\rho$ = air density (kg/m³), $A$ = swept area (m²), $v$ = wind speed (m/s).

Kinetic energy of air of mass $m$:

$$E = \frac{1}{2} m v^2$$

Mass flow rate through area $A$:

$$\dot{m} = \rho A v$$

Power available in the wind (energy per second):

$$P_w = \frac{1}{2}\dot{m}v^2 = \frac{1}{2}\rho A v^3$$

A turbine cannot take all this power, because air must leave the rotor with some speed. The fraction extracted is the power coefficient $C_p$. By Betz's law, $C_p \le 16/27 = 0.593$.

Mechanical power on the rotor:

$$P_m = \frac{1}{2}\rho A v^3 C_p$$

Electrical output includes the gearbox and generator efficiencies:

$$P_e = \frac{1}{2}\rho A v^3 C_p\, \eta_g\, \eta_{gen}$$

For a horizontal-axis turbine with blade length (radius) $r$: $A = \pi r^2$.

Key points

• $P \propto v^3$: doubling wind speed gives 8 times the power.
• $P \propto r^2$: doubling blade length gives 4 times the power.
• $P \propto \rho$: output is less at high altitude (lower air density).
• Practical $C_p$ is about 0.35–0.45.