1. Wind turbine design according to Betz and Schmitz

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

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 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:

15. 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:

16. Maximum Power Coefficient Rankine-Froude theorem

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