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Wind turbine design according to Betz and Schmitz

Wind turbine design according to Betz and Schmitz . Power output from wind turbines: Energy production from wind turbines:. Energy and power from the wind. A. v. Stream Tube. V. Extracted Energy and Power. Where: E ex = Extracted Energy [J] E ex = Extracted Power [W] m = Mass [kg]

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Wind turbine design according to Betz and Schmitz

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  1. Wind turbine design according to Betz and Schmitz

  2. Power output from wind turbines: Energy production from wind turbines: Energy and power from the wind A v

  3. Stream Tube V

  4. Extracted Energy and Power Where: Eex = Extracted Energy [J] Eex = Extracted Power [W] m = Mass [kg] m = Mass flow rate [kg/s] v = Velocity [m/s] • •

  5. Extracted Energy and Power • If the wind was not retarded, no power would be extracted • If the retardation stops the mass flow rate, no power would be extracted • There must be a value of v3 for a maximum power extraction

  6. Extracted Energy and Power • The retardation of the wind cause a pressure difference over the wind turbine

  7. We assume the following: • There is a higher pressure right upstream the turbine (p-2) than the surrounding atmospheric pressure • There is a lower pressure right downstream the turbine (p+2) than the surrounding atmospheric pressure • Since the velocity is theoretically the same both upstream and downstream the turbine, the energy potential lies in the differential pressure. • The cross sections 1 and 3 are so far away from the turbine that the pressures are the same A3 A2 A1

  8. Continuity (We assume incompressible flow) A3 A2 A1

  9. Balance of forces: (Newton's 2. law) Because of the differential pressure over the turbine, it is now a force F = (p-2 – p+2)∙A2 acting on the swept area of the turbine. Impulse force Pressure force Impulse force

  10. Energy flux over the wind turbine: (We assume incompressible flow)

  11. Energy flux over the wind turbine: (We assume incompressible flow)

  12. Energy flux over the wind turbine: (We assume incompressible flow)

  13. Continuity: Balance of forces: Energy flux: If we substitute the pressure term; (p-2-p+2) from the equation for the balance of forces in to the equation for the energy flux, and at the same time use the continuity equation to change the area terms; A1 and A3 with A2 i we can find an equation for the velocity v2:

  14. Power CoefficientRankine-Froude theorem We define the Power Coefficient: In the following, we assume that the velocity v3 can be expressed as v3=x·v1, where x is a constant. We substitute: From continuity:

  15. Power CoefficientRankine-Froude theorem We insert the expressions for A1 and A3 in to the equation for the power coefficient. We will end up with the following equation:

  16. Maximum Power CoefficientRankine-Froude theorem Maximum power coefficient:

  17. Power Coefficient

  18. The Betz Power

  19. Thrust v2 T At maximum power coefficient we have the relation: x =1/3

  20. Example Find the thrust on a wind turbine with the following specifications: v1 = 20 m/s D = 100 m cT = 8/9

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