Annual energy of the 5-turbine row (Weibull k = 2 winds, mean speed below) for spacings 3–10 D, using the same wake model as the 3D view.
The science
What you see. A 100 m rotor turns a 90:1 gearbox (real tooth counts 50/10 × 54/12 × 48/12 are modelled and spin at their true relative speeds) and a generator; the grid receives what is left after each stage's losses. Click any part for its live speed and torque.
Wind power and Betz.
P_wind = ½ ρ A v³ A = πR²
P_rotor = P_wind · C_p(λ,β) λ = ωR / v
C_p ≤ 16/27 ≈ 0.593 (Betz limit)
Cp(λ,β) uses the common empirical fit of Heier for a 3-blade rotor (peak 0.48 at λ ≈ 8.1, β = 0). It is a typical shape, not the data of a real machine.
Momentum link. Cp = 4a(1−a)² gives the axial induction a; thrust coefficient Ct = 4a(1−a); far wake speed = (1−2a)v. Streamlines through the rotor expand by √((1−a)/(1−2a)).
Control. J dω/dt = Qaero − Qfriction − Tg. Below rated: Tg = Kω² with K = ½ρπR&sup5;Cp,max/λ³opt keeps λ at its optimum. Above rated: torque is held, a PI loop (plus wind feed-forward) pitches the blades to hold ω = ωrated. Rotor inertia J = 1.2×10&sup7; kg m², pitch rate limit 8°/s (illustrative).
Losses. Each stage loses fixed + linear + quadratic shares of rated power (gearbox ~2–3 %, generator ~3–4 %, converter ~2–3 %): illustrative typical values, not a specific product.
Yaw error. Only the wind component normal to the rotor does work, so power ∝ cos³γ (ideal; field data often show a smaller exponent). Wake steering is not modelled.
Wake (Jensen / Katic).
deficit = (1 − √(1−C_t)) · (R / (R + k x))² · (A_overlap/A_rotor)
k = 0.075 (onshore) v_in = v·(1 − √Σ deficit²)
A top-hat wake with no turbulence build-up, deflection or terrain effects. It captures the trend (more spacing, less loss) but real farm losses need CFD or measurements.
Simplifications, honestly. Steady wind (no turbulence, shear or veer), rigid blades and tower, no tower shadow, drivetrain compliance, grid faults or noise. Flow streaks and energy pulses are for understanding and are drawn faster than real speeds. Terrain is artistic. Numbers marked typical should be spot-checked against a specific turbine's data sheet.
Independent educational model. Made with Blender and Three.js.