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lab:zephyr:rotors [2016-07-25 13:46] – [Power Estimation] chronolab:zephyr:rotors [2023-04-19 14:18] (current) – [Power Estimation] chrono
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 === Available power in the wind === === 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 ^ ^ 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 | Density of Air (rho) ~1.225 at 25°C |+^ <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** **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** ^ ^ Windspeed ^ 1 m/s ^ 2 m/s ^ 4 m/s ^ **8 m/s** ^ **16 m/s** ^
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 === Conversion Efficiency === === 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** **Example: eXperimental Turbine Lenz-Rotor with 0.96 m² surface @ 4 m/s**
  
-<x 16> +<m> 
-36.86 ∗ 0.25 = 9.21 W +36.86 * 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: 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:
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 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)). 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 === === Tip Speed Ratio ===
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 wind speed((Deisadze, et al. 2013)). 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 This equation shows the relationship between the tip speed ratio and the power
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 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**. 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}/{\nu}} +Re = {{V * D * \rho}/{\nu}} 
-</x>+</m>
  
 ^ Parameter ^ Unit ^ Detail ^ ^ 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>\nu</x> | m²/s | Kinematic viscosity of Air (nu) ~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** **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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