Renewable Wind Energy & Distributed Aerodynamics, Darrieus Lift vs Savonius Drag VAWT
Darrieus & Savonius VAWT Solidity, TSR & Torque Formula
Calculate Vertical-Axis Wind Turbine Blade Chord Solidity (N*c/R), Tip-Speed Ratio lambda, Rectangular Swept-Area Shaft Power (kW), and Aerodynamic Rotor Torque (N·m).
Why low-solidity Darrieus H-rotors capture 2x the wind power of Savonius S-buckets—yet cannot self-start without starter torque at TSR < 1!
**Vertical-Axis Wind Turbines (`VAWTs`)** accept omnidirectional turbulent rooftop/offshore wind without yaw drives across a **Rectangular Swept Area `A = 2 * R * H`** (`m²`, where `R` is Rotor Radius `m` and `H` is Vertical Blade Span `m`), splitting into two opposite aerodynamic regimes governed by **Rotor Chord Solidity `sigma = (N * c) / R`** (`N` blades of airfoil/bucket chord `c` m): `1)` **Darrieus Lift-Type `H`-Rotors (`sigma = 0.15–0.40`)** fly symmetrical `NACA 0018` blades at supersonic-to-wind **Tip-Speed Ratios `lambda = (omega * R) / V = (2 * PI() * R * (RPM / 60)) / V` of `3.0–4.5`** to reach **`C_p = 0.35–0.40`** (near-HAWT efficiency, but near-zero starting torque at `lambda < 1`!), whereas `2)` **Savonius Drag-Type `S`-Rotors (`sigma ~= 1.8–2.2`)** self-start effortlessly with high torque at **`lambda ~= 0.8–1.0`** at the cost of low peak **`C_p ~= 0.15–0.18`**! Standard sea-level air (`rho = 1.225` kg/m³) yields **Shaft Power `P = (0.5 * 1.225 * (2 * R * H) * V^3 * C_p) / 1,000`** (`kW`) and **Rotor Shaft Torque `tau = (P * 1,000) / (2 * PI() * (RPM / 60))`** (`N·m`)!
Verify on a `3`-Blade Straight-Bladed Giromill / `H`-Darrieus VAWT—Rotor_Radius `R` = 3.20 m (B2), Blade_Span `H` = 8.50 m (B3 -> `A = 54.4` m²), Blade_Count `N` = 3 (B4), Airfoil_Chord `c` = 0.36 m (B5), Wind_Speed `V` = 11.5 m/s (B6), Rotor_Speed = 125 RPM (B7), and Power_Coefficient `C_p` = 0.36 (B8): Solidity `sigma` equals `0.3375` (E2), Tip-Speed Ratio `lambda` equals `3.64` (E3), Shaft Power reaches `18.24` kW (E4), and Direct-Drive Generator Torque equals `1,393.7` N·m (E5)!
| VAWT Aerodynamic KPI (Sea-Level Air 1.225 kg/m³: Darrieus Lift vs Savonius Drag) | Formula | Verified Output |
|---|---|---|
| Rotor Chord Solidity sigma (N*c/R) | (Blade_Count_N * Chord_c_m) / Radius_R_m | 0.3375 Darrieus (2.0000 Savonius) |
| Tip-Speed Ratio lambda (TSR) | (2 * PI() * Radius_m * (RPM / 60)) / Wind_m_s | 3.64 lift-driven (0.95 drag-limited) |
| Aerodynamic Shaft Power (kW) | (0.5 * 1.225 * (2*R*H) * V^3 * Cp) / 1000 | 18.24 kW (0.55 kW compact Savonius) |
| Direct-Drive Rotor Torque (N·m) | (Shaft_Power_kW * 1000) / (2 * PI() * RPM / 60) | 1,393.7 N·m (76.7 N·m Savonius) |
VAWT Rectangular Swept-Area Power, Chord Solidity & Direct-Drive Torque Formula
Validate `Radius R` in `[0.2, 60]` m, `Span H` in `[0.3, 120]` m, integer `Blades N` in `[2, 6]`, `Chord c > 0.03` m, `Wind_Speed V` in `[2, 40]` m/s, `RPM > 1`, and physical Betz limit `0.05 <= C_p <= 0.5926`!
=LET(rr_m, B2, hb_m, B3, nb_c, B4, ch_m, B5, vw_s, B6, nr_p, B7, cp_f, B8, pk_w, (0.5 * 1.225 * (2 * rr_m * hb_m) * (vw_s ^ 3) * cp_f) / 1000, IF(OR(rr_m < 0.2, rr_m > 60, hb_m < 0.3, hb_m > 120, nb_c < 2, nb_c > 6, ch_m <= 0.03, vw_s < 2, vw_s > 40, nr_p <= 1, cp_f < 0.05, cp_f > 0.5926), "Invalid VAWT Input", ROUND((pk_w * 1000) / (2 * PI() * (nr_p / 60)), 1)))2-Scoop Savonius Drag Rotor (`R = 1.2` m, `H = 3` m, `c = 1.2` m, `Cp = 0.17`) Comparison Check
Switch to a self-starting `2`-bucket Savonius telecom-tower charger (`B2 = 1.2`, `B3 = 3`, `B4 = 2`, `B5 = 1.2`, `B6 = 9`, `B7 = 68`, `B8 = 0.17`): Solidity E2 jumps to `2.0000`, Tip-Speed Ratio E3 stays sub-unity (`0.95`), Power E4 equals `0.55` kW, and Torque E5 equals `76.7` N·m!
=IF(ISTEXT(E5), E5, ROUND((0.5 * 1.225 * (2 * B2 * B3) * (B6 ^ 3) * B8) / 1000, 2))Try the example
Use the original values or change them to test your understanding.