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Wind Energy I
Michael Hölling, WS 2010/2011 slide 1
Wind power
Wind Energy I
slideMichael Hölling, WS 2010/2011 2
Class content
4 Wind power
5 Wind turbines in general 6 Wind - blades
interaction
7 Π-theorem
8 Wind turbine characterization
9 Control strategies
10 Generator
11 Electrics / grid
3 Wind field characterization
2 Wind measurements
Wind Energy I
slideMichael Hölling, WS 2010/2011 3
Motivation
How much power can be extracted from the wind ?
Wind Energy I
slideMichael Hölling, WS 2010/2011 4
Betz limit
Approach: free-stream air flow and conservation of mass flow
A1, u1 A2, u2
A3, u3
! · A1 · u1 = ! · A2 · u2 = ! · A3 · u3
! d
dt· m = m = const.
Wind Energy I
slideMichael Hölling, WS 2010/2011 5
Betz limit
Extracted kinetic energy and extracted power
A1, u1 A2, u2
A3, u3
Eext =12m
!u2
1 ! u23
"
Pext =12m
!u2
1 ! u23
" u3
u1optimal for max. Pext ????
Wind Energy I
slideMichael Hölling, WS 2010/2011 6
Substitution of u2 will lead to a function Pext(u1,u3)
Betz limit
T = m (u1 ! u3)Thrust:
Corresponding power: Pthrust = m (u1 ! u3) · u2
Pext =12
· ! · A2 · u2 ·!u2
1 ! u23
"
has to be substituted
Equals mechanical powerthat can be extracted: Pmech = Pext =
12m
!u2
1 ! u23
"
! u2 =u1 + u2
2
Wind Energy I
slideMichael Hölling, WS 2010/2011 7
Betz limit
Extractable power as function of u1 and u3
Pext =12
· ! · A2 · u31
! "# $wind power
·%12
·&
1 +u3
u1! u2
3
u21
! u33
u31
'(
! "# $cp
0.0 0.5 1.00.0
0.2
0.4
0.6
u3/u1
cp
cpmax ??
Wind Energy I
slideMichael Hölling, WS 2010/2011 8
Betz limit
Find the maximum cp by taking the first derivative
cp(x) =12
·!1 + x! x2 ! x3
"Substitute u3/u1 with x:
First derivative of cp(x): c!p(x) =
12
·!1! 2x! 3x2
" != 0
For maximum second derivative of possible solution x1/2
must smaller than zero: c!!p(x1/2) =
12
· (!2! 6x) < 0
Solution: x =13! cpmax.(1/3) =
1627" 59%
Wind Energy I
slideMichael Hölling, WS 2010/2011 9
Betz limit
x
u1 2/3 u1
1/3 u1 u3
u2 u(x)
What happens at the rotor plane ?
Wind Energy I
slideMichael Hölling, WS 2010/2011 9
Betz limit
x
u1 2/3 u1
1/3 u1 u3
u2 u(x)
p(x)p(x)p1
p+2
p-2
p3
What happens at the rotor plane ?
Pressure drop at rotor plane
Wind Energy I
slideMichael Hölling, WS 2010/2011 10
Betz limit
What does this results mean for wind energy converter ?
There is an optimal rotational frequency that the blockage
results in , characterized by tip speed ratio λ.u3
u1=
13
Wind Energy I
slideMichael Hölling, WS 2010/2011 11
Betz limit
What is the value for cp for a WEC over wind speed ?
0 10 20 300.0
0.2
0.4
0.6
0.8
1.0
1.2
u [m/s]
P(u
) /
Pr
P(u) / Pr
0 10 20 300.0
0.2
0.4
0.6
0.8
1.0
1.2
0.0
0.2
0.4
0.6
0.8
1.0
1.2
u [m/s]
P(u
) /
Pr
P(u) / Pr
cp(u) / cpmax
cp(u
) /
cp
max