Wind energy I. Lesson 8. Power losses at rotor blade

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


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


Power coefficient

Real cp values change over the tip speed ratio !

Wind Energy I


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


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


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


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


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


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


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


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


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


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



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



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



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



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



0 5 10 15 200.0

0.2

0.4

0.6

!

cpr

cpSchmitz




Wind Energy I



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


0 5 10 15 200.0

0.2

0.4

0.6

!

cpr

cpSchmitz



0 5 10 15 200.0

0.2

0.4

0.6

!

cpr

cpSchmitz




0 5 10 15 200.0

0.2

0.4

0.6

!

cpr

cpSchmitz




cpBetz, z=1,"(#)=60

0 5 10 15 200.0

0.2

0.4

0.6

!

cpr

cpSchmitz




cpBetz, z=1,"(#)=60

cpBetz, z=2,"(#)=60

0 5 10 15 200.0

0.2

0.4

0.6

!

cpr

cpSchmitz




cpBetz, z=1,"(#)=60

cpBetz, z=2,"(#)=60

cpBetz, z=3,"(#)=60

Wind Energy I


0 5 10 15 200.0

0.2

0.4

0.6

!

cp

r cpSchmitz





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


Blade optimization - Schmitz

Chord length optimization based on Schmitz limit in comparison to Betz limit:

Wind Energy I


Blade optimization - Schmitz

blade twist optimization based on Schmitz limit in comparison to Betz limit::

Wind energy I. Lesson 8. Power losses at rotor blade

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Transcript of Wind energy I. Lesson 8. Power losses at rotor blade