Wind turbine design according to Betz and Schmitz

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

Stream Tube V

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] • •

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

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

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

Continuity (We assume incompressible flow)

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

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

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

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

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:

Power Coefficient Rankine-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:

Power Coefficient Rankine-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:

Maximum Power Coefficient Rankine-Froude theorem

Power Coefficient

The Betz Power

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

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