Renewable Energy Engineering, Wind Rotor Aerodynamics & Acoustic Control
Wind Turbine Tip-Speed Ratio (TSR) & Rotor Solidity Formula
Calculate Horizontal-Axis Wind Turbine Blade Tip Linear Velocity (m/s), Dimensionless Tip-Speed Ratio λ (TSR), Aerodynamic Rotor Solidity σ, and Onshore 80 m/s Aeroacoustic Tip-Noise Curtailment Gate.
Balance optimal Cp Tip-Speed Ratio (~7–8) against onshore aeroacoustic 80 m/s tip-noise limits!
Modern 3-blade variable-speed horizontal-axis wind turbines adjust generator torque below rated wind speed so Blade Tip Linear Velocity V_tip = (π * D * RPM) / 60 tracks an optimal dimensionless Tip-Speed Ratio λ = V_tip / U_wind ≈ 7.0 to 8.0 where aerodynamic power coefficient Cp peaks! Optimal λ is inversely coupled to Rotor Solidity σ = (B * c_mean) / (π * R) (total blade planform area divided by swept rotor area, ~5% on sleek 3-blade utility rotors vs >30% on high-torque multi-blade farm windmills)! Onshore permitting, however, caps V_tip near ~80 m/s because trailing-edge aeroacoustic noise scales with the FIFTH power of tip velocity (V_tip^5)!
Verify on a 4 MW Onshore Utility Turbine—Rotor_Diameter_D = 120.0 m (Radius R = 60 m, B2), Rotor_Speed = 12.0 RPM (B3 -> Blade Tip Speed = (π * 120 * 12) / 60 = 75.40 m/s in E2), Hub_Wind_Speed = 10.0 m/s (B4 -> Tip-Speed Ratio λ = 75.3982 / 10 = 7.5398 in E3), Blade_Count = 3 (B5), and Mean_Chord_c = 3.20 m (B6 -> Rotor Solidity σ = (3 * 3.2) / (π * 60) = 0.0509 or 5.09% in E4): Tip speed stays safely below 80 m/s ("Compliant Onshore Tip Speed (<=80 m/s)" in E5)!
| Wind Rotor Aerodynamic Metric (D = 120 m, 12 RPM, U = 10 m/s, 3 Blades, c = 3.2 m) | Formula | Verified Output |
|---|---|---|
| Blade Tip Velocity V_tip (m/s) | (PI() * D_m * RPM) / 60 | 75.40 m/s (87.96 m/s @ 14 RPM) |
| Tip-Speed Ratio λ (TSR) | V_tip_ms / Wind_Speed_ms | 7.5398 |
| Rotor Solidity σ | (Blade_Count * Mean_Chord) / (PI() * (D/2)) | 0.0509 (5.09% Swept Area) |
| Onshore Noise Gate (<=80 m/s) | IF(V_tip <= 80, "Compliant...", "Acoustic...") | Compliant Onshore Tip Speed (<=80 m/s) |
Wind Turbine Tip-Speed Ratio (TSR) Formula
Validate Rotor_Diameter > 0, RPM > 0, Wind_Speed > 0, integer Blade_Count >= 1, and Mean_Chord > 0!
=LET(dm_r, B2, rp_m, B3, uw_s, B4, nb_c, B5, ch_m, B6, IF(OR(dm_r <= 0, rp_m <= 0, uw_s <= 0, nb_c < 1, ch_m <= 0), "Invalid Wind Rotor Input", LET(vt_p, (PI() * dm_r * rp_m) / 60, ROUND(vt_p / uw_s, 4))))Nighttime Aeroacoustic Overspeed Check (Set Rotor Speed = 14.0 RPM)
Raise Rotor_Speed B3 from 12 RPM to 14 RPM (typical offshore high-tip-speed rating): Blade Tip Velocity E2 climbs to 87.96 m/s, triggering "Acoustic Curtailment: Tip > 80 m/s" in E5 for onshore noise-sensitive sites!
=IF(ISTEXT(E2), E2, IF(E2 <= 80, "Compliant Onshore Tip Speed (<=80 m/s)", "Acoustic Curtailment: Tip > 80 m/s"))Try the example
Use the original values or change them to test your understanding.