Wind Energy & Atmospheric Engineering
Hellmann Wind Shear Hub-Height Power Formula
Extrapolate meteorological mast wind speed to turbine hub height via Gustav Hellmann’s Power Law v_hub = v_ref * (z_hub / z_ref)^alpha and compute the Cubic Wind Power Density Multiplier (z_hub / z_ref)^(3*alpha).
See why a 50% hub-height wind-speed gain more than TRIPLES wind power density!
Surface friction slows wind near the ground across the atmospheric boundary layer! Gustav Hellmann’s Power-Law Wind Profile extrapolates wind speed v_ref measured at reference anemometer height z_ref up to wind-turbine hub height z_hub via v_hub = v_ref * (z_hub / z_ref)^alpha, where Hellmann Shear Exponent alpha ranges from ~0.10 over smooth offshore seas, 1/7 (~0.143) over open plains, to 0.25–0.30 over forested/suburban terrain! Because wind kinetic power per swept square meter scales with the CUBE of wind speed (P / A = 0.5 * rho * v^3), raising tower height multiplies available Wind Power Density by (z_hub / z_ref)^(3 * alpha)!
Verify on Met_Mast_Height z_ref = 16 m (B2), Turbine_Hub_Height z_hub = 81 m (81 / 16 = 5.0625x height ratio, B3), Reference_Wind_Speed v_ref = 6.00 m/s (B4), Hellmann_Exponent alpha = 0.25 (wooded countryside, B5), and Air_Density rho = 1.20 kg/m^3 (B6): Since (81 / 16)^0.25 = (3 / 2) = 1.5000 exact, Hub-Height Wind Speed is 6.00 * 1.50 = 9.00 m/s (E2); Cubic Wind Power Multiplier is 1.50^3 = 3.3750x (E3—+237.5% more power!); Reference Power Density at 16 m is 0.5 * 1.20 * 6^3 = 129.60 W/m^2 (E4); and Hub-Height Specific Wind Power Density reaches 0.5 * 1.20 * 9^3 = 437.40 W/m^2 (E5)! Test zero-shear uniform flow alpha = 0: Hub Speed stays 6.00 m/s and Power Multiplier equals 1.0000x!
| Wind Resource Extrapolation (z_ref=16m -> z_hub=81m, v_ref=6 m/s, alpha=0.25) | Formula | Verified Output |
|---|---|---|
| Hub-Height Wind Speed v_hub (m/s) | v_ref * (z_hub / z_ref)^alpha | 9.00 m/s (+50.0% Speed) |
| Cubic Power Density Multiplier | (z_hub / z_ref)^(3 * alpha) | 3.3750x (+237.5% Power) |
| Reference Power Density @ 16m | 0.5 * rho * v_ref^3 | 129.60 W/m^2 |
| Hub-Height Power Density @ 81m | 0.5 * rho * v_hub^3 | 437.40 W/m^2 |
Hellmann Hub-Height Wind Speed & Power Density Formula
Validate z_ref > 0, z_hub > 0, v_ref >= 0, 0 <= alpha <= 0.60, and rho > 0!
=LET(z_rf, B2, z_hb, B3, v_rf, B4, al_s, B5, rh_a, B6, IF(OR(z_rf <= 0, z_hb <= 0, v_rf < 0, al_s < 0, al_s > 0.6, rh_a <= 0), "Invalid Wind Shear Input", LET(v_hb, v_rf * (z_hb / z_rf) ^ al_s, ROUND(0.5 * rh_a * v_hb ^ 3, 2))))Cubic Wind Power Elevation Multiplier Formula
Compare against Betz’s 16/27 (59.26%) maximum rotor capture ceiling to size actual mechanical shaft power!
=IF(ISTEXT(E5), E5, ROUND((B3 / B2) ^ (3 * B5), 4))Try the example
Use the original values or change them to test your understanding.