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1. Wind Energy I
Wind power
Michael Hölling, WS 2010/2011 slide 1

2. Wind Energy I Class content
5 Wind turbines in
6 Wind - blades
general
2 Wind measurements interaction
7 Π-theorem
8 Wind turbine
characterization
3 Wind field 9 Control strategies
characterization
10 Generator
4 Wind power
11 Electrics / grid
Michael Hölling, WS 2010/2011 slide 2

3. Wind Energy I Motivation
How much power can be extracted from the wind ?
Michael Hölling, WS 2010/2011 slide 3

4. Wind Energy I Betz limit
Approach: free-stream air flow and conservation of mass flow
A1, u1
A2, u2
A3, u3
ρ · A1 · u 1 = ρ · A2 · u 2 = ρ · A3 · u 3
d
⇒ · m = m = const.
˙
dt
Michael Hölling, WS 2010/2011 slide 4

5. Wind Energy I Betz limit
Extracted kinetic energy and extracted power
A1, u1
A2, u2
A3, u3
1
Eext = m u1 − u3
2 2
2
1 u3
Pext = m u2 − u2
˙ 1 optimal for max. Pext ????
2 3 u1
Michael Hölling, WS 2010/2011 slide 5

6. Wind Energy I Betz limit
Substitution of u2 will lead to a function Pext(u1,u3)
1
Pext = · ρ · A2 · u 2 · u 1 − u 3
2 2
2
has to be substituted
Thrust: T = m (u1 − u3 )
˙
Corresponding power: Pthrust = m (u1 − u3 ) · u2
˙
Equals mechanical power 1
that can be extracted: Pmech = Pext = m u2 − u2
˙ 1 3
2
u1 + u2
⇒ u2 =
2
Michael Hölling, WS 2010/2011 slide 6

7. Wind Energy I Betz limit
Extractable power as function of u1 and u3
1 1 u3 u2 u3
Pext = · ρ · A2 · u 1 ·
3
· 1+ 3 3
− 2− 3
2 2 u1 u1 u1
wind power cp
0.6 cp
max ??
0.4
cp
0.2
0.0
0.0 0.5 1.0
u3/u1
Michael Hölling, WS 2010/2011 slide 7

8. Wind Energy I Betz limit
Find the maximum cp by taking the first derivative
1
Substitute u3/u1 with x: cp (x) = · 1 + x − x − x
2 3
2
1 !
First derivative of cp(x): cp (x) = · 1 − 2x − 3x = 0
2
2
For maximum second derivative of possible solution x1/2
1
must smaller than zero: cp (x1/2 ) = · (−2 − 6x) < 0
2
1 16
Solution: x = ⇒ cpmax. (1/3) = ≈ 59%
3 27
Michael Hölling, WS 2010/2011 slide 8

9. Wind Energy I Betz limit
What happens at the rotor plane ?
u1
u2 2/3 u1
u3 u(x) 1/3 u1
x
Michael Hölling, WS 2010/2011 slide 9

10. Wind Energy I Betz limit
What happens at the rotor plane ?
p-2
u1
u2 2/3 u1
u3 u(x) 1/3 u1
p(x) p(x)
p1 p3
x
Pressure drop at rotor plane
p+2
Michael Hölling, WS 2010/2011 slide 9

11. Wind Energy I Betz limit
What does this results mean for wind energy converter ?
There is an optimal rotational frequency that the blockage
u3 1
results in = , characterized by tip speed ratio λ.
u1 3
Michael Hölling, WS 2010/2011 slide 10

12. Wind Energy I Betz limit
What is the value for cp for a WEC over wind speed ?
1.2 1.2
P(u) / Pr
1.0 1.0 cp(u) / cpmax
cp(u) / cpmax
P(u) / Pr
0.8 0.8
0.6 0.6
0.4 0.4
0.2 0.2
0.0 0.0
0 10 20 30
u [m/s]
Michael Hölling, WS 2010/2011 slide 11