| 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:
- Initialize a = a' = 0
- Compute φ, α_eff, C_L, C_D
- Update a and a' from momentum equations
- 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 [%].
