wind energy i. lesson 8. power losses at rotor blade
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Wind Energy I
Michael Hölling, WS 2010/2011 slide 1
power losses at the rotor blade
Wind Energy I
slideMichael Hölling, WS 2010/2011 2
Class content
4 Wind power
5 Wind turbines in general 6 Wind - blades
interaction
9 Π-theorem and Wind turbine characterization
8 Power losses at the rotor blade
10 Generator
11 Electrics / grid
3 Wind field characterization
2 Wind measurements
Wind Energy I
slideMichael Hölling, WS 2010/2011 3
Power coefficient
Optimized design of blades - why is the power coefficient not cp = 16/27 for the whole wind speed range ?
0 5 10 15 200.0
0.2
0.4
0.6
!
cp(!
)
cp = Betz limit
Wind Energy I
slideMichael Hölling, WS 2010/2011 4
Power coefficient
Real cp values change over the tip speed ratio !
Wind Energy I
slideMichael Hölling, WS 2010/2011 5
losses at the profile due to drag forces
Losses at the rotor will lead to rotor power coefficient cpr
.
β
α
u2
ures
urot
ω
Fl
Fd
Fres
plane of rotation
β
β
Power losses
Wind Energy I
slideMichael Hölling, WS 2010/2011 6
Losses at the rotor will lead to rotor power coefficient cpr
losses at the tip of the blades creates by tip vortices
Power losses
Wind Energy I
slideMichael Hölling, WS 2010/2011 7
Power losses
Determine rotor power coefficient cpr by including losses in addition to Betz limit - cprdrag and cprtip additional factors:
cprdrag =dProt
dProtideal
cprdrag = 1! 1!(")
· 32
· # · r
R
Calculations lead to:
Wind Energy I
slideMichael Hölling, WS 2010/2011 8
Power losses
Possible behavior of cprdrag over blade radius r for different ε and λ:
0 10 20 30 40 5030
40
50
60
70
r [m]
!(r
)
!(r)
0 10 20 30 40 50
0.8
0.9
1.0
r [m]
cpr d
rag(r
)
! = 4
0 10 20 30 40 50
0.8
0.9
1.0
r [m]
cpr d
rag(r
)
! = 4! = 7
0 10 20 30 40 50
0.8
0.9
1.0
r [m]
cpr d
rag(r
)
! = 4! = 7! = 10
Wind Energy I
slideMichael Hölling, WS 2010/2011 10
Power losses
r
For a ring-segment:
dPBetz =1627
· 12
· ! · u31 · (2 · " · r · dr! "# $
dA
)
For just the circumference of a circle:
dPBetz =1627
· 12
· ! · u31 · (2 · " · r · dr! "# $
dA
)
Wind Energy I
slideMichael Hölling, WS 2010/2011 11
Power losses
For a constant ε over the whole blade cprdrag is given by:
cprdrag = 1! !
"
0 5 10 15 200.0
0.2
0.4
0.6
0.8
1.0
!
cpr drag(!)
"(#)=20
0 5 10 15 200.0
0.2
0.4
0.6
0.8
1.0
!
cpr drag(!)
"(#)=20"(#)=40
0 5 10 15 200.0
0.2
0.4
0.6
0.8
1.0
!
cpr drag(!)
"(#)=20"(#)=40"(#)=60
Wind Energy I
slideMichael Hölling, WS 2010/2011 12
cl(r)
ures
Power coefficient cprtip due to tip losses are caused by balancing pressure differences at tip of the blade.
Power losses
Wind Energy I
slideMichael Hölling, WS 2010/2011 13
Power losses
Estimating tip losses cprtip by means of reduced diameter D’:
D! = D ! 0.44 · b
Projection of distance “a” between rotor blades into a plane perpendicular to the resulting velocity ures gives “b”.
urot
β
u2
ures
a
b.
D! = D
!
"1! 0.92
z ·#
!2 + 49
$
%
Wind Energy I
slideMichael Hölling, WS 2010/2011 14
Power losses
cprtip =
!
"1! 0.92
z ·#
!2 + 49
$
%2
0 5 10 15 200.0
0.2
0.4
0.6
0.8
1.0
!
cpr tip(!)
z=1
0 5 10 15 200.0
0.2
0.4
0.6
0.8
1.0
!
cpr tip(!)
z=1z=2
0 5 10 15 200.0
0.2
0.4
0.6
0.8
1.0
!
cpr tip(!)
z=1z=2z=3
Wind Energy I
slideMichael Hölling, WS 2010/2011 15
Rotor power coefficient
The total rotor power coefficient is a result from the Betz limit, losses due to drag and tip losses:
cpr = cpBetz · cprdrag · cprtip
0 5 10 15 200.0
0.2
0.4
0.6
!
cpr(!)
z=1,"(#)=40
0 5 10 15 200.0
0.2
0.4
0.6
!
cpr(!)
z=1,"(#)=40z=2,"(#)=40
0 5 10 15 200.0
0.2
0.4
0.6
!
cpr(!)
z=1,"(#)=40z=2,"(#)=40z=3,"(#)=40
Betz limit
Wind Energy I
slideMichael Hölling, WS 2010/2011 15
Rotor power coefficient
The total rotor power coefficient is a result from the Betz limit, losses due to drag and tip losses:
cpr = cpBetz · cprdrag · cprtip
0 5 10 15 200.0
0.2
0.4
0.6
!
cpr(!)
z=1,"(#)=40
0 5 10 15 200.0
0.2
0.4
0.6
!
cpr(!)
z=1,"(#)=40z=2,"(#)=40
0 5 10 15 200.0
0.2
0.4
0.6
!
cpr(!)
z=1,"(#)=40z=2,"(#)=40z=3,"(#)=40
0 5 10 15 200.0
0.2
0.4
0.6
!
cpr(!)
z=1,"(#)=40z=2,"(#)=40z=3,"(#)=40z=1,"(#)=60
0 5 10 15 200.0
0.2
0.4
0.6
!
cpr(!)
z=1,"(#)=40z=2,"(#)=40z=3,"(#)=40z=1,"(#)=60z=2,"(#)=60
0 5 10 15 200.0
0.2
0.4
0.6
!
cpr(!)
z=1,"(#)=40z=2,"(#)=40z=3,"(#)=40z=1,"(#)=60z=2,"(#)=60z=3,"(#)=60
Betz limit
Wind Energy I
slideMichael Hölling, WS 2010/2011 16
Rotor power coefficient
Maximum convertible power from wind based on Schmitz (and Gaulert) including conservation angular momentum:“Based on the conservation of angular momentum, if the rotor gains angular momentum from the linear wind stream, then there must be some compensation, which is in the form of an opposite rotating wake, so that the
overall angular momentum does not change.”
Wind Energy I
slideMichael Hölling, WS 2010/2011 17
Rotor power coefficient
Just to be complete, the maximum convertible power from wind based on Schmitz including angular momentum is given by:
PSchmitz =12
· ! · " · R2 · u31
! 1
04 · # ·
" r
R
#2·sin3
$23 · arctan
$R
!·r%%
sin2$arctan
$R
!·r%% · d
" r
R
#
& '( )cpSchmitz
0 5 10 15 200.0
0.2
0.4
0.6
!
cpSchmitz
cpSchmitz
Wind Energy I
slideMichael Hölling, WS 2010/2011 18
Rotor power coefficient
The total rotor power coefficient is a result from the Schmitz limit (losses due to conservation of angular momentum), losses due to drag and tip losses: cpr = cpSchmitz · cprdrag · cprtip
0 5 10 15 200.0
0.2
0.4
0.6
!
cpr
cpSchmitz
cpSchmitz, z=1,"(#)=60
0 5 10 15 200.0
0.2
0.4
0.6
!
cpr
cpSchmitz
cpSchmitz, z=1,"(#)=60
cpSchmitz, z=2,"(#)=60
0 5 10 15 200.0
0.2
0.4
0.6
!
cpr
cpSchmitz
cpSchmitz, z=1,"(#)=60
cpSchmitz, z=2,"(#)=60
cpSchmitz, z=3,"(#)=60
Wind Energy I
slideMichael Hölling, WS 2010/2011 18
Rotor power coefficient
The total rotor power coefficient is a result from the Schmitz limit (losses due to conservation of angular momentum), losses due to drag and tip losses: cpr = cpSchmitz · cprdrag · cprtip
0 5 10 15 200.0
0.2
0.4
0.6
!
cpr
cpSchmitz
cpSchmitz, z=1,"(#)=60
0 5 10 15 200.0
0.2
0.4
0.6
!
cpr
cpSchmitz
cpSchmitz, z=1,"(#)=60
cpSchmitz, z=2,"(#)=60
0 5 10 15 200.0
0.2
0.4
0.6
!
cpr
cpSchmitz
cpSchmitz, z=1,"(#)=60
cpSchmitz, z=2,"(#)=60
cpSchmitz, z=3,"(#)=60
0 5 10 15 200.0
0.2
0.4
0.6
!
cpr
cpSchmitz
cpSchmitz, z=1,"(#)=60
cpSchmitz, z=2,"(#)=60
cpSchmitz, z=3,"(#)=60
cpBetz, z=1,"(#)=60
0 5 10 15 200.0
0.2
0.4
0.6
!
cpr
cpSchmitz
cpSchmitz, z=1,"(#)=60
cpSchmitz, z=2,"(#)=60
cpSchmitz, z=3,"(#)=60
cpBetz, z=1,"(#)=60
cpBetz, z=2,"(#)=60
0 5 10 15 200.0
0.2
0.4
0.6
!
cpr
cpSchmitz
cpSchmitz, z=1,"(#)=60
cpSchmitz, z=2,"(#)=60
cpSchmitz, z=3,"(#)=60
cpBetz, z=1,"(#)=60
cpBetz, z=2,"(#)=60
cpBetz, z=3,"(#)=60
Wind Energy I
slideMichael Hölling, WS 2010/2011 19
0 5 10 15 200.0
0.2
0.4
0.6
!
cp
r cpSchmitz
cpSchmitz, z=1,"(#)=60
cpSchmitz, z=2,"(#)=60
cpSchmitz, z=3,"(#)=60
Rotor power coefficient
Even with all used approximations the calculated curves show the characteristics of real cpr curves:- number of blades effects maximum - number of blades effect λopt for maximum cpr
Wind Energy I
slideMichael Hölling, WS 2010/2011 20
Blade optimization - Schmitz
Chord length optimization based on Schmitz limit in comparison to Betz limit:
Wind Energy I
slideMichael Hölling, WS 2010/2011 21
Blade optimization - Schmitz
blade twist optimization based on Schmitz limit in comparison to Betz limit::