Вітрогенерація: Від Межі Бетта до Енергії Мережі

Calculating the Power of Wind

The kinetic energy passing through a rotor's swept area A at wind speed v arrives as power P = ½·ρ·A·v³, where ρ is air density (~1.225 kg/m³). The cubic dependence on speed dominates everything about turbine design: doubling wind speed multiplies available power by eight, so even a 10% increase in average wind speed — from better siting or a taller hub — yields roughly 33% more annual energy.

P_wind = ½ · ρ · A · v³           A = π·R²
Example: R = 60m, v = 12 m/s → A = 11,310 m², P ≈ 11.95 MW

The Betz Limit: 59.3% and No More

Albert Betz proved in 1919 that no turbine can extract more than 16/27 ≈ 59.3% of the wind's kinetic energy. Extract it all and the air behind the rotor would stop dead, blocking further air from flowing through at all — a contradiction that caps the achievable extraction fraction.

Modern turbines reach a power coefficient C_P of about 0.45-0.50, or 75-85% of the theoretical Betz maximum, with the shortfall lost to blade drag, tip vortices, and wake rotation.

a = induction factor,  C_P = 4a(1−a)²,  maximised at a = 1/3
C_P,max = 4·(1/3)·(2/3)² = 16/27 ≈ 0.593
Modern turbines: C_P ≈ 0.45–0.50

Tip-speed ratio and staying at the sweet spot

Blades are true aerofoils, twisted from 30° at the root to about 2° at the tip so the angle of attack stays near its optimum along the entire span despite the changing apparent wind. Efficiency peaks at a specific tip-speed ratio λ = ΩR/v — around 6-8 for a three-blade design — so controllers continuously adjust rotor speed or blade pitch to chase that target as wind speed changes. Above rated wind speed, pitching the blades deliberately spills lift to cap power and protect the generator; noise and structural loads limit blade tips to roughly 80-90 m/s regardless of turbine size.

Часті запитання

Спробуйте наживо

Усе, що вище, працює прямо у вашому браузері — відкрийте the simulation і змінюйте параметри під час роботи. Нічого не встановлюється, нічого не завантажується на сервер, уся модель живе в одній вкладці.