A wind turbine only captures a fraction Cp of the kinetic power passing through its swept disc: P = ½·ρ·A·U³·Cp. Cp itself is not a constant — it depends on the tip-speed ratio λ = ωR/U (how fast the blade tips move relative to the wind) and the blade pitch angle β (how much the blades are feathered out of the wind). This lab uses the standard analytical approximation for Cp(λ,β) used in wind-energy control engineering:
1/λi = 1/(λ+0.08β) − 0.035/(β³+1)
Cp(λ,β) = 0.5176·(116/λi − 0.4β − 5)·e^(−21/λi) + 0.0068·λ
Move the wind-speed and rotor-speed sliders and watch the operating dot slide along the Cp(λ) curve — there is one λ that maximizes Cp for a given pitch (around λ≈8 at β=0°), which is exactly what "Snap to optimal λ" searches for numerically. Increasing pitch β feathers the blades and lowers the whole curve, which is how real turbines spill excess power in high wind instead of over-speeding. No real turbine can beat the Betz limit Cp=16/27≈0.593 (dashed line) — that is the theoretical ceiling for any device extracting energy from an open flow, derived from momentum conservation, independent of blade design.
- Rotor radius is fixed at 50 m (utility-scale, matching the farm the 3D version shows) and air density at sea-level 1.225 kg/m³.
- Blade angular speed on screen is the real ω = rpm·2π/60 computed from your rotor-speed slider — not a decorative loop.
- Power history strip shows the actual P(t) trace as you move the sliders, so cause and effect stay visible.