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--- lab:zephyr:rotors [2016-07-14 21:41] – 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 72: / Line 76: @@
 ==== Power Estimation ====
-The following equation provides a means to estimate the approximate amount of kinetic power
+The following equations provide a means to estimate the approximate amount of kinetic and electric power converted by a wind turbine:
-converted by a wind turbine
-<x 20>
+=== Available power in the wind ===
-P ≈ {{1}/{2}} ∗ A ∗ V^3 ∗ ρ ∗ C_{P}
-</x>
-  * <x 12>P</x> -> Convertable power in the wind, as kinetic energy in Watt
+<m>P_{k} \approx {{1}/{2}} * A * V^3 * \rho</m>
-  * <x 12>A</x> -> Swept area (turbine/sail etc.) in m²
-  * <x 12>V</x> -> Wind speed in m/s
-  * <x 12>\rho</x> -> (rho) Air density (about 1.2 Kg/m³)
-  * <x 12>C_{P}</x> -> Power coefficient
-Ideally, power scales linearly with the area swept out by the turbine blades, and cubically with the speed of the wind as it contacts the blades. However, these relationships have some variation depending on the design of each particular turbine.
+^ Parameter ^ Unit ^ Detail ^
+^ <m 12>P_{k}</m> | Watt | Available power in the wind, as kinetic energy |
+^ <m 12>A</m> | m² | Swept area (turbine/sail etc.) |
+^ <m 12>V</m> | m/s  | Wind speed |
+^ <m 12>\rho</m> | kg/m³ | Density of Air (rho) ~1.225 at 25°C |
-The power coefficient at the end of the equation accounts for the efficiency of the turbine in converting the wind’s kinetic energy into mechanical 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 known as Betz's coefficient (0.593), or Betz limit. However, most wind turbines operate at a power coefficient of less than 0.45:
+**Example: eXperimental Turbine Lenz-Rotor with 0.96 m² surface @ 4 m/s**
+<m>{{1}/{2}} * 0.96 * 4^3 * 1.225 = 36.86 W</m>
+Example values at certain wind speeds:
+^ Windspeed ^ 1 m/s ^ 2 m/s ^ 4 m/s ^ **8 m/s** ^ **16 m/s** ^
+^ Power | 0.57 W| 4.60 W| **36.68 W** | **294.91 W** | **2.36 kW** |
+Basically, power scales linearly with the area swept out by the turbine blades and cubically with the speed of the wind as it sweeps the blades. However, these relationships have some variation depending on the design of each particular turbine.
+=== Conversion Efficiency ===
+<m>
+P_{r} \approx P_{k} * C_{P}
+</m>
+  * <m 12>P_{r}</m> -> Converted rotational energy in Watt
+  * <m 12>P_{k}</m> -> Available power in the wind, as kinetic energy in Watt
+  * <m 12>C_{P}</m> -> Power coefficient
+**Example: eXperimental Turbine Lenz-Rotor with 0.96 m² surface @ 4 m/s**
+<m>
+.86 * 0.25 = 9.21 W
+</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:
+^ Turbine Type ^ Power Coefficient ^
 | Simple drag VAWT | 0.20 |
 | 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 |
-**Example: eXperimental Turbine Lenz-Rotor with 0.96 m² surface @ 4 m/s**
+=== Torque ===
-<x 20>
+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)).
-.96 * 4^3 * {{1}/{2}} \rho = 36.86 W
-</x>
-^ Windspeed ^ 1 m/s ^ 2 m/s ^ 4 m/s ^ **8 m/s** ^ **16 m/s** ^
+<m>
-^ Power | 0.57 W| 4.60 W| **36.68 W** | **294.91 W** | **2.36 kW** |
+\tau \approx {{1}/{2}} * R * A * V^2 * \rho
+</m>
+=== Tip Speed Ratio ===
+The tip speed ratio (λ) defines the relationship between blade tip speed and incident
+wind speed((Deisadze, et al. 2013)).
+<m>
+\lambda = {{\omega * R}/{V}}
+</m>
+This equation shows the relationship between the tip speed ratio and the power
+coefficient for various blade types. For each type, there is a unique curve, and therefore a
+unique optimal tip speed ratio which corresponds to the maximum power coefficient that
+can be achieved.
+For example, a Savonius rotor will produce a maximum power coefficient
+of about 0.31 at a tip speed ratio of about 0.9. However, a Darrieus rotor produces a
+maximum power coefficient of around 0.35 at a much higher tip speed ratio of around 5.8.
+To be most efficient, a blade and rotor should be designed to perform near its optimal tip
+speed ratio at wind speeds it is likely to encounter((Ragheb and Ragheb, Wind Turbines Theory - The Betz Equation and Optimal Rotor Tip Speed Ratio 2011)).
+=== Reynolds Number ===
+The Reynolds number range for small-scale gorlov VAWTs is quite
+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**.
+<m>
+Re = {{V * D * \rho}/{\nu}}
+</m>
+^ Parameter ^ Unit ^ Detail ^
+^ <m 12>V</m> | m/s | Incoming flow velocity |
+^ <m 12>D</m> | m | Turbine Diameter |
+^ <m 12>\rho</m> | kg/m³ | Density of Air (rho) ~1.225 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**
+<m>
+{{4 * 0.7 * 1.225}/{0.0000157}} \approx 218471
+</m>
-You can watch these calculations in action, applied to reference wind speed measurements on the [[https://apollo.open-resource.org/flight-control/vfcc/#/dashboard/db/wind-power-simulation|Wind Power]] VFCC Dashboard
+You can watch these calculations in action, applied to reference wind speed measurements on the [[https://apollo.open-resource.org/flight-control/vfcc/#/dashboard/db/wind-power-simulation|Wind Power]] VFCC Dashboard.
-A tuned VAWT probably has a best-case efficiency of 40%, while a simple drag-based turbine with no optimization nor special aerodynamics may have an efficiency of about 20%.
+A tuned VAWT probably has a best-case efficiency of 35%, while a simple drag-based turbine with no optimization nor special aerodynamics may have an efficiency of about 20%.
 {{tag>zephyr wind energy research vawt turbine rotor c-rotor h-rotor}}

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