HKDSE • Elective 3: Energy and Use of Energy • Wind energy

Wind Turbine Power

The kinetic energy of the air passing through the rotor each second gives the wind power $P = \tfrac12 \rho A v^3$. Drag the wind-speed slider and watch the power grow with the cube of $v$; lengthen the blades to grow the swept area $A = \pi r^2$.

Parameters

Betz limit: no turbine can extract more than 59% of the wind's kinetic energy.

Presets

Live readings

Swept area $A = \pi r^2$ 5027 m²
Air mass per second $\rho A v$ 49.3 t/s
Wind power $\tfrac12 \rho A v^3$ 1.58 MW
Electrical output $\eta P$ 0.63 MW
Energy in 24 h 15.1 MWh
If $v$ doubles ($\times 8$) 5.05 MW

HKDSE formula

$$ P = \tfrac12 \rho A v^3 $$
$\rho$: air density (kg m⁻³)
$A$: area swept by the blades, $\pi r^2$ (m²)
$v$: wind speed (m s⁻¹)

Derivation: mass of air arriving per second $= \rho A v$; its kinetic energy per second $= \tfrac12(\rho A v)v^2$.

3D wind farm
Drag to rotate • pinch / scroll to zoom
Wind velocity $v$ Swept area $A$ Electrical output

Power against wind speed (cubic law)

Physics note

Because $P \propto v^3$, a site with twice the average wind speed offers eight times the power — this is why turbines are placed on ridges and offshore. Because $P \propto r^2$, doubling the blade length gives four times the power, which explains the trend towards ever-larger turbines. A real turbine extracts only a fraction $\eta$ of the wind's kinetic energy (Betz limit ≈ 59%), because the air must keep moving to leave the rotor.