wind-energy

name	wind-energy
description	Wind energy — Betz limit, blade element momentum (BEM) theory, power curve, tip speed ratio, airfoil selection, wind turbine loads, capacity factor, site assessment, IEC 61400.
metadata	{"priority":7,"promptSignals":{"phrases":["wind energy","wind turbine","Betz","blade element","BEM","tip speed ratio","power curve","capacity factor","wind resource","IEC 61400"],"minScore":3}}

Wind Energy — Complete Skill

Wind Power Fundamentals

Available Power in Wind

P_wind = ½ × ρ × A × V³ ρ = air density [kg/m³] ≈ 1.225 kg/m³ (sea level, 15°C) A = rotor swept area = π R² [m²] V = wind speed [m/s]

Air density variation with altitude: ρ = ρ₀ × (1 - z/44,300)^4.256 (standard atmosphere) Correction for temperature: ρ = p/(R_air × T) = 353/T [kg/m³] at sea level, T in K

Betz Limit (Actuator Disk Theory)

Optimal power extraction limited to 16/27 (59.3%) of available wind power. Proof: Conservation of momentum through actuator disk: u_wake = (1-2a)V (a = axial induction factor) Power: P = 4a(1-a)² × ½ρAV³ C_P = 4a(1-a)² → maximum at a = 1/3 → C_P,Betz = 16/27 = 0.593

Real turbines: C_P ≈ 0.35-0.50 (blade aerodynamic losses, mechanical losses) Tip-speed ratio at max C_P: λ_opt ≈ 6-9 (modern 3-bladed HAWT)

Tip Speed Ratio

Definition

λ = Ω R / V = blade tip speed / wind speed Ω = rotor angular velocity [rad/s], R = rotor radius [m]

Optimal TSR by turbine type: 1-blade: λ ≈ 10-15 (rarely used — vibration) 2-blade: λ ≈ 8-12 3-blade HAWT: λ ≈ 6-9 (most commercial) Darrieus VAWT: λ ≈ 4-6 Savonius: λ ≈ 0.8-1.5 (low, drag-type)

Tip speed: V_tip = λ × V_rated; typical 60-90 m/s for onshore; higher noise above 60 m/s

Blade Element Momentum (BEM) Theory

Blade Divided into Annular Elements

Each ring (radial width dr) analyzed independently. Local wind speed relative to blade: W = √[(V(1-a))² + (Ωr(1+a'))²] a = axial induction factor, a' = tangential induction factor

Inflow angle φ: tan(φ) = V(1-a) / (Ωr(1+a')) Pitch + twist: α_eff = φ - θ (geometric angle of attack = inflow angle - pitch - twist)

Airfoil Forces

Lift per unit span: dL = ½ ρ W² c C_L [N/m] Drag per unit span: dD = ½ ρ W² c C_D [N/m] c = chord length [m], C_L and C_D from airfoil polars (XFOIL, wind tunnel data)

Normal and tangential forces (to rotor plane): dF_N = dL cosφ + dD sinφ (thrust direction) dF_T = dL sinφ - dD cosφ (torque direction)

Power from annular element: dP = Ω r dF_T × B (B = number of blades) dT = dF_N × B (thrust contribution)

Prandtl Tip Loss Correction

F = (2/π) × arccos(exp(-B(R-r)/(2r sinφ))) Applied: replace uniform induction with: F × 4a(1-a) = σ C_N/sin²φ σ = local solidity = Bc/(2πr)

Iterative BEM solution:

1. Initialize a = a' = 0
2. Compute φ, α_eff, C_L, C_D
3. Update a and a' from momentum equations
4. Repeat until convergence (typically < 10 iterations)

Turbine Performance

Power Curve

Wind speed regions: V < V_cut_in (3-4 m/s): turbine stopped V_cut_in < V < V_rated (10-15 m/s): P ∝ V³ approximately; C_P ≈ constant (blade pitch = 0) V = V_rated: rated power P_rated (limited by generator/converter) V_rated < V < V_cut_out (20-25 m/s): pitch control reduces C_P to maintain rated power V > V_cut_out: shutdown (storm protection)

Turbine power output: P = C_P(λ, θ) × ½ρA V³ At rated: P_rated = C_P,rated × ½ρA V_rated³

Capacity Factor

CF = E_annual / (P_rated × 8760 hours) Typical: onshore wind CF = 25-45%; offshore CF = 40-60% Annual Energy Production (AEP): AEP = Σ P(V_i) × h(V_i) [kWh/year] h(V_i) = hours at wind speed V_i (from Weibull distribution of wind)

Weibull wind distribution: f(V) = (k/c)(V/c)^(k-1) × exp(-(V/c)^k) k = shape factor (2-3 for most sites), c = scale factor (related to mean wind speed) Mean wind speed: V̄ = c × Γ(1 + 1/k)

Structural Loads (IEC 61400-1)

Load Cases

Normal power production: fatigue analysis (millions of cycles over 20-yr lifetime) Extreme wind speed: 50-year gust V_50 = 1.4 V_50,3sec Emergency stop, grid loss, blade pitch failure: design load cases (DLCs)

Wind classes (IEC 61400-1): Class I: V_ref = 50 m/s (high wind sites) Class II: V_ref = 42.5 m/s (medium wind) Class III: V_ref = 37.5 m/s (low wind, e.g., India, China inland)

Turbulence intensity: TI = σ_V / V̄ (std dev of 10-min wind speed / mean) Category A: TI_15 = 0.16 (high turbulence); B: 0.14; C: 0.12 Turbulence → fatigue cycles; σ_V = I_ref × (0.75V + b) from Mann model

Blade Loading

Flapwise: primary load direction (out of rotor plane); driven by thrust Edgewise: in-plane; gravity dominates (cycles per revolution) Torsion: aerodynamic pitching moment + structural coupling

Blade natural frequency: avoid multiples of rotational frequency (1P, 2P, 3P etc.) Typical: f_1P = 0.1-0.3 Hz; blade f_edge > 1.1×3P (stiff rotor) or < 3P (soft rotor)

Site Assessment

Wind Resource Analysis

Measure 10-min average wind speed at hub height (typically 80-150m) 1-year minimum measurement; 10 years preferable (inter-annual variability) Vertical wind profile: V(z) = V_ref × (z/z_ref)^α (power law, α ≈ 1/7 for open terrain) OR V(z) = V_ref × ln(z/z₀) / ln(z_ref/z₀) (log law, z₀ = roughness length)

Roughness length z₀: Open water: 0.0002 m; flat open terrain: 0.03 m; farmland: 0.1 m; forest: 0.5 m; city: 1.0 m

Wake Effects

Upstream turbine creates velocity deficit in wake: Jensen model: V_wake = V₀ × (1 - 2a/(1 + k_w × x/R)²) k_w = wake decay coefficient (0.04 offshore, 0.06 onshore) x = downstream distance; a = axial induction factor

Park efficiency: AEP_wind_farm / AEP_isolated = 85-95% (depends on spacing and layout) Typical spacing: 5-7D perpendicular, 7-10D in wind direction

Key Turbine Data (Modern Commercial)

Rating	Rotor D	Hub H	Rated V	C_P	Platform
2 MW	80-90m	80m	12 m/s	0.45	Onshore
5 MW	120m	100m	13 m/s	0.47	Offshore fixed
12-15 MW	220-240m	150m	13 m/s	0.50	Offshore floating

Output

Provide: P_wind [kW] at given V, C_P vs. TSR curve, P_rated [kW] and V_rated [m/s], AEP [MWh/yr] from Weibull distribution, capacity factor [%], blade loading (thrust and torque), fatigue damage equivalent load, IEC wind class selection, wake loss estimate [%].