Wind turbine - 2D streamtube and power curve

How much power is in the wind

The kinetic power flowing through the rotor disc of area A = πD²/4 is P_wind = ½ ρ A U³, so doubling the wind gives eight times the power, and a bigger rotor gives power in proportion to its area. A turbine extracts only the fraction C_p: P = ½ ρ A U³ C_p(λ, β).

Betz limit

In actuator-disc theory the air slows by a fraction a at the rotor and 2a in the far wake, giving C_p = 4a(1-a)², which peaks at a = 1/3 with C_p = 16/27 = 0.593. No open rotor can beat this. Real rotors reach about 0.45 to 0.5 because of blade drag, tip losses and wake swirl. The streamtube drawn on the left uses the induction a implied by the current C_p: it widens behind the rotor because slower air must occupy more area.

Tip-speed ratio, pitch and control

The tip-speed ratio λ = ωR/U sets the angle at which the air meets the blades. C_p has a single best λ, so below rated wind the controller keeps λ at that optimum (variable speed, MPPT). Above rated wind it turns the blades (pitch β) to shed power and holds the output at the rated value, and it parks the machine above the cut-out speed. The C_p(λ, β) surface comes from a blade-element-momentum (BEM) calculation of the same tapered, twisted blade as in the 3D page (28 strips, simple airfoil polar, Prandtl tip loss), tabulated on a grid; it stays below the Glauert optimum, which in turn stays below Betz. Annual energy integrates the power curve against a Weibull wind distribution f(U) = (k/c)(U/c)^k-1 e^-(U/c)k.

Honest simplifications: a generic C_p fit rather than blade-element data for a specific machine, no electrical or drivetrain losses, no wake effects from neighbouring turbines, no wind shear or turbulence, a mean site wind of 7 m/s, and the same C_p fit regardless of rotor size.

• Raise the wind speed past the rated speed and watch the controller pitch the blades to hold the output flat.
• Switch to MANUAL and set the tip-speed ratio to 4: the power coefficient collapses.
• Double the rotor diameter at the same wind: the swept area, and the power, quadruple.
• Lower the Weibull shape k to see a gustier site and how it changes the annual energy and capacity factor.