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--- lab:zephyr:rotors [2016/07/16 19:40] – [Power Estimation] chrono
+++ lab:zephyr:rotors [2023/04/19 14:18] (current) – [Power Estimation] chrono
@@ Line 1: / Line 1: @@
 ====== Rotors ======
-Compared to drag-only type rotors (Savonius), the lift-only type rotors (Darrieus) haven proven to be generally less suitable for low wind environments. However, the maximum speed of drag-only type rotors is always lower than a comparable lift-only type rotor, because a lift-only type rotor can rotate faster than the wind speed at the tips but with less torque. A drag-only type rotor can develop more torque, even at early stages in low wind conditions, but that would require a very specific and resource-intensive generator to accommodate for the very low rotational speed. A typical low end for a direct driven axial flux permanent magnet alternator with many poles is about 100 revolutions per minute. Everything under 100 rpm means huge additional resource investments into rare earth magnets and loads of copper (windings).
+Compared to drag-only type rotors (Savonius), lift-only type rotors (Darrieus) have been proven to be generally less suitable for low wind environments since they're difficult to start up. The maximum speed of drag-only type rotors is always lower than a comparable lift-only type rotor, because a lift-only type rotor can rotate faster than the wind speed at the tips but with less torque. However, the Gorlov rotor with a NACA 0015 airfoil may be a very well suited lift-type rotor for small-scale, low wind environments.
+A drag-only type rotor can develop more torque, even at early stages in low wind conditions, but that would require a very specific and resource-intensive generator to accommodate for the very low rotational speed. A typical low end for a direct driven axial flux permanent magnet alternator with many poles is about 100 revolutions per minute. Everything under 100 rpm means huge additional resource investments into rare earth magnets and loads of copper (windings).
 ===== VAWT Rotor Types =====
@@ Line 76: / Line 80: @@
 === Available power in the wind ===
-<x 20>
+<m>P_{k} \approx {{1}/{2}} * A * V^3 * \rho</m>
-P_{k} ≈ {{1}/{2}} ∗ A ∗ V^3 ∗ ρ
-</x>
 ^ Parameter ^ Unit ^ Detail ^
-^ <x 12>P_{k}</x> | Watt | Available power in the wind, as kinetic energy |
+^ <m 12>P_{k}</m> | Watt | Available power in the wind, as kinetic energy |
-^ <x 12>A</x> | m² | Swept area (turbine/sail etc.) |
+^ <m 12>A</m> | m² | Swept area (turbine/sail etc.) |
-^ <x 12>V</x> | m/s  | Wind speed |
+^ <m 12>V</m> | m/s  | Wind speed |
-^ <x 12>\rho</x> | kg/m³ | (rho) Air density (~1.225) |
+^ <m 12>\rho</m> | kg/m³ | Density of Air (rho) ~1.225 at 25°C |
 **Example: eXperimental Turbine Lenz-Rotor with 0.96 m² surface @ 4 m/s**
-<x 16>
+<m>{{1}/{2}} * 0.96 * 4^3 * 1.225 = 36.86 W</m>
-{{1}/{2}} ∗ 0.96 ∗ 4^3 ∗ 1.225 = 36.86 W
-</x>
+Example values at certain wind speeds:
 ^ Windspeed ^ 1 m/s ^ 2 m/s ^ 4 m/s ^ **8 m/s** ^ **16 m/s** ^
@@ Line 99: / Line 101: @@
 === Conversion Efficiency ===
-<x 20>
+<m>
-P_{r} ≈ P_{k} ∗ C_{P}
+P_{r} \approx P_{k} * C_{P}
-</x>
+</m>
-  * <x 12>P_{r}</x> -> Converted rotational energy in Watt
+  * <m 12>P_{r}</m> -> Converted rotational energy in Watt
-  * <x 12>P_{k}</x> -> Available power in the wind, as kinetic energy in Watt
+  * <m 12>P_{k}</m> -> Available power in the wind, as kinetic energy in Watt
-  * <x 12>C_{P}</x> -> Power coefficient
+  * <m 12>C_{P}</m> -> Power coefficient
 **Example: eXperimental Turbine Lenz-Rotor with 0.96 m² surface @ 4 m/s**
-<x 16>
+<m>
-.86 ∗ 0.25 = 9.21 W
+.86 * 0.25 = 9.21 W
-</x>
+</m>
 The power coefficient accounts for the efficiency of the turbine in converting the wind’s kinetic energy into rotational energy. According to [[https://en.wikipedia.org/wiki/Betz%27s_law|Betz's law]], no wind turbine can capture more than 16/27 (59.3%) of the available kinetic energy in wind. This theoretical maximum power limit is also known as Betz's coefficient (0.593) or Betz-Limit. However, most wind turbines operate at a power coefficient of less than 0.45:
@@ Line 119: / Line 121: @@
 | Decent VAWT | 0.30 |
 | Good VAWT | 0.35 |
-| Superb  VAWT | 0.40 |
+| Good HAWT | 0.40 |
-| Superb  HAWT | 0.45 |
+| Big Grid MW+ HAWT | 0.45 |
 === Torque ===
@@ Line 126: / Line 128: @@
 For turbines which use drag forces (not lift forces), the following equation can be used to estimate the amount of torque in the system, where R is the radius of turbine in meters((Brandmaier, et al. 2013)).
-<x 20>
+<m>
-τ ≈ {{1}/{2}} ∗ R ∗ A ∗ V^2 ∗ ρ
+\tau \approx {{1}/{2}} * R * A * V^2 * \rho
-</x>
+</m>
 === Tip Speed Ratio ===
@@ Line 135: / Line 137: @@
 wind speed((Deisadze, et al. 2013)).
-<x 20>
+<m>
-λ = {{ω ∗ R}/{V}}
+\lambda = {{\omega * R}/{V}}
-</x>
+</m>
 This equation shows the relationship between the tip speed ratio and the power
@@ Line 156: / Line 158: @@
 low. In comparison, the Reynolds number operating regime of most airfoils used for aircrafts ranges from **6.3e6 for a small Cessna** to **2.0e9 for a Boeing 747**.
-<x 20>
+<m>
-Re = {{V ∗ D ∗ \rho}/{\u}}
+Re = {{V * D * \rho}/{\nu}}
-</x>
+</m>
 ^ Parameter ^ Unit ^ Detail ^
-^ <x 12>V</x> | m/s | Incoming flow velocity |
+^ <m 12>V</m> | m/s | Incoming flow velocity |
-^ <x 12>D</x> | m | Turbine Diameter |
+^ <m 12>D</m> | m | Turbine Diameter |
-^ <x 12>\rho</x> | kg/m³ | Density of Air (rho) ~1.225 at 25°C) |
+^ <m 12>\rho</m> | kg/m³ | Density of Air (rho) ~1.225 at 25°C |
-^ <x 12>\u</x> | m²/s | Kinematic viscosity of Air (1.57e-5 at 25 °C) |
+^ <m 12>\nu</m> | m²/s | Kinematic viscosity of Air (nu) ~1.57e-5 at 25 °C |
 **Example: Helical Gorlov-Rotor with 35 cm radius @ 4 m/s**
-<x 16>
+<m>
-{{4 ∗ 0.7 ∗ 1.225}/{0.0000157}} ≈ 218471
+{{4 * 0.7 * 1.225}/{0.0000157}} \approx 218471
-</x>
+</m>

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