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

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

1. 1. Wind Energy I power losses at the rotor bladeMichael Hölling, WS 2010/2011 slide 1
2. 2. Wind Energy I Class content 5 Wind turbines in 6 Wind - blades general 2 Wind measurements interaction 8 Power losses at the rotor blade 9 Π-theorem and Wind 3 Wind field turbine characterization characterization 4 Wind power 10 Generator 11 Electrics / gridMichael Hölling, WS 2010/2011 slide 2
3. 3. Wind Energy I Power coefﬁcient Optimized design of blades - why is the power coefficient not cp = 16/27 for the whole wind speed range ? cp = Betz limit 0.6 0.4 cp(!) 0.2 0.0 0 5 10 15 20 !Michael Hölling, WS 2010/2011 slide 3
4. 4. Wind Energy I Power coefﬁcient Real cp values change over the tip speed ratio !Michael Hölling, WS 2010/2011 slide 4
5. 5. Wind Energy I Power losses Losses at the rotor will lead to rotor power coefficient cpr u2 losses at the profile due to drag forces urot plane of rotation β ures Fl β Fres α . Fd ω βMichael Hölling, WS 2010/2011 slide 5
6. 6. Wind Energy I Power losses Losses at the rotor will lead to rotor power coefficient cpr losses at the tip of the blades creates by tip vorticesMichael Hölling, WS 2010/2011 slide 6
7. 7. Wind Energy I Power losses Determine rotor power coefficient cpr by including losses in addition to Betz limit - cprdrag and cprtip additional factors: dProt cprdrag = dProtideal Calculations lead to: 1 3 λ·r cprdrag =1− · · (α) 2 RMichael Hölling, WS 2010/2011 slide 7
8. 8. Wind Energy I Power losses Possible behavior of cprdrag over blade radius r for different ε and λ: 70 1.0 !(r) !=4 !=7 60 ! = 10 0.9 cprdrag(r)!(r) 50 40 0.8 30 0 10 20 30 40 50 0 10 20 30 40 50 r [m] r [m] Michael Hölling, WS 2010/2011 slide 8
9. 9. Wind Energy I Power losses For a ring-segment: 16 1 dPBetz = · · ρ · u1 · (2 · π · r · dr) 3 27 2 dA r For just the circumference of a circle: 16 1 dPBetz = · · ρ · u1 · (2 · π · r · dr) 3 27 2 dAMichael Hölling, WS 2010/2011 slide 10
10. 10. Wind Energy I Power losses For a constant ε over the whole blade cprdrag is given by: λ cprdrag = 1 − 1.0 "(#)=20 0.8 "(#)=40 "(#)=60 cprdrag(!) 0.6 0.4 0.2 0.0 0 5 10 15 20 !Michael Hölling, WS 2010/2011 slide 11
11. 11. Wind Energy I Power losses Power coefficient cprtip due to tip losses are caused by balancing pressure differences at tip of the blade. cl (r) uresMichael Hölling, WS 2010/2011 slide 12
12. 12. 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”. u2 urot β ures   0.92 D = D 1 −  a . z· λ2 + 4 9 bMichael Hölling, WS 2010/2011 slide 13
13. 13. Wind Energy I Power losses  2 0.92 cprtip = 1 −  z· λ2 + 4 9 1.0 0.8 cprtip(!) 0.6 0.4 z=1 z=2 0.2 z=3 0.0 0 5 10 15 20 !Michael Hölling, WS 2010/2011 slide 14
14. 14. 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 Betz limit 0.6 z=1,"(#)=40 z=2,"(#)=40 z=3,"(#)=40 0.4 cpr(!) 0.2 0.0 0 5 10 15 20 !Michael Hölling, WS 2010/2011 slide 15
15. 15. 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 Betz limit 0.6 z=1,"(#)=40 z=2,"(#)=40 z=3,"(#)=40 0.4 z=1,"(#)=60 z=2,"(#)=60 cpr(!) z=3,"(#)=60 0.2 0.0 0 5 10 15 20 !Michael Hölling, WS 2010/2011 slide 15
16. 16. Wind Energy I 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. ”Michael Hölling, WS 2010/2011 slide 16
17. 17. Wind Energy I Rotor power coefﬁcient Just to be complete, the maximum convertible power from wind based on Schmitz including angular momentum is given by: 1 1 r 2 sin3 2 · arctan R r PSchmitz = · ρ · π · R 2 · u3 4·λ· · 3 λ·r ·d 2 1 0 R sin2 arctan R λ·r R cpSchmitz 0.6 cpSchmitz 0.4 cpSchmitz 0.2 0.0 0 5 10 15 20 !Michael Hölling, WS 2010/2011 slide 17
18. 18. Wind Energy I 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 0.6 cpSchmitz cpSchmitz, z=1,"(#)=60 cpSchmitz, z=2,"(#)=60 0.4 cpSchmitz, z=3,"(#)=60cpr 0.2 0.0 0 5 10 15 20 !Michael Hölling, WS 2010/2011 slide 18
19. 19. Wind Energy I 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 0.6 cpSchmitz cpSchmitz, z=1,"(#)=60 cpSchmitz, z=2,"(#)=60 0.4 cpSchmitz, z=3,"(#)=60cpr cpBetz, z=1,"(#)=60 0.2 cpBetz, z=2,"(#)=60 cpBetz, z=3,"(#)=60 0.0 0 5 10 15 20 !Michael Hölling, WS 2010/2011 slide 18
20. 20. Wind Energy I 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 0.6 0.4 cpSchmitz cpr cpSchmitz, z=1,"(#)=60 0.2 cpSchmitz, z=2,"(#)=60 cpSchmitz, z=3,"(#)=60 0.0 0 5 10 15 20 !Michael Hölling, WS 2010/2011 slide 19
21. 21. Wind Energy I Blade optimization - Schmitz Chord length optimization based on Schmitz limit in comparison to Betz limit:Michael Hölling, WS 2010/2011 slide 20
22. 22. Wind Energy I Blade optimization - Schmitz blade twist optimization based on Schmitz limit in comparison to Betz limit::Michael Hölling, WS 2010/2011 slide 21