A Compact, Closed Form Solution for the Optimum, Ideal Wind Turbines

Transcription

1 Department of Mechanical Engineering & Material Sciences A Compact, Closed Form Solution for the Optimum, Ideal Wind Turbines David A. Peters McDonnell Douglas Professor of Engineering dap@wustl.edu Ramin Modarres Graduate Research Assistant ramin.modarres@wustl.edu 1

2 References Manwell, J. F., McGowan, J. G., and Rogers, A. L., Wind Energy Explained: Theory, Design and Application Second Edition, John Wiley and Sons, West Sussix, 009, pp , Johnson, Gary L., Wind Energy Systems, Prentice-Hall, Englewood Cliffs, NJ, 1985, pp Eggleston, David M. and Stoddard, Forrest S., Wind Turbine Engineering Design, Van Nostrand Reinhold Company, New York, 1987, pp Garcia-Sanz, Mario and Houpis, Constantine H., Wind Energy Systems, CRC Press, Boca Raton, FL, 01, pp Abramowitz, Milton and Stegun, Irene A., Handbook of Mathematical Functions, Dover Publications, Inc., New York, NY, 1970, p. 17. Glauert, H., Aerodynamic theory: A General Review of Progress, Vol. IV, Chapter Division L., Airplane Propellers. Dover Publications, Inc. New York, NY, 1963, pp Barocela, Edward, "The Effect of Wake Curvature on Dynamic Inflow for Lifting Rotors," Master of Science Thesis, Washington University in St. Louis, May Makinen, Stephen M., Applying Dynamic Wake Models to Large Swirl Velocities for Optimum Propellers, Doctor of Science Thesis, Washington University in St. Louis, May 005.

3 Rotor Inflow Geometry 3

4 Background u au, v ar a(1 a) = r a(1 a) 4

5 Background 3 r dcp 8 a(1 a) d r a a a (1 3 a) (4a1) (1 )(4 1) (1 3 ) r =, a 3 16a 4 a (9 3 r ) a r 1 0 5

6 Background 1 a a 1 1 tan = tan r r (1 a) a 6

7 Background ln( ) 79 5 x CP x x x x x x x0.5 x 1 3a 0 7

8 Alternative Approach V = U + r w sin( ) U U r w rw U r cos( ) U r U r w Uw r 8

9 Alternate Approach 1 r b sin( ) 1 b rb v 1 w r r cos( ) r 1 b b u 1 w r r 9

10 Momentum Theory dl ( rdr) w U wcos( ) dp ( rdr) w U wcos( ) r sin( ) dc 8 b 1bcos( ) sin( ) rdr P r Where: 3 16z 4z 9z 0 1 r z = b / 1 + λ r 1 10

11 Momentum Theory b r 1 r 1 cos cos r r 1 cos cos b r r 1 11 r 1 cos cos 3 sin cos b 3 1r 3 1r The axial and swirl induction factors: a = bcos(φ) f = bsin(φ) 11

12 Complete Expressions r 1b 1b 3b 1 b 1 1 cos( ) 1

13 Complete Expressions sin( ) 1b 3b1 b cos( ) 1 b b 13

14 Complete Expressions 1 1 cos 1 r r r cos ( ) sin ( ) cos( ) 1 = sin( ) 1 cos( ) tan 3 14

15 Complete Expressions a 1b cos( ) bcos( ) = 1 cos( ) b 1a f bsin( ) sin( ) a 1 cos( ) r r r Total flow at the blade in terms of b: b 1 1 r b b 3b 1 15

16 Wake Induction Parameters as a Function of Local Speed Ratio 16

17 Optimized Inflow Angle as a Function of Local Speed Ratio 17

18 Optimized Inflow Angle as a Function of sin of Initial Inflow Ratio 18

19 Optimum Power Coefficient P (1 b) (1 b) r d r (1 ) (1 ) dc b b rdr r d r 6b 1 3b 1 b db C P b(1 b)(1 b) (3b 1) b 0 Where: 1 < b 0 < 1 3 db 19

20 Optimum Power Coefficient y1 7 y0 41 y0 y y 4 y0 4 CP 3 9 y y dy Where: y 1 = 1 and 0< y 0 < 1 0 0

21 Optimum Power Coefficient 1 y ln( ) (1 ) (1 ) 3 y y y 16 1 y y CP 1 y y 3 7 y (1 y) 1 4 1

22 Torque Coefficient P = QΩ C Q CP = = C P 3 3y 4 y 1y 1 y ln( ) (1 ) (1 ) 3 y y y y y CQ 1 y y y (1 y) 1 4

23 Thrust Coefficient dt rdr w U wcos( ) cos( ) dc 8b 1bcos( ) cos( ) rdr = T 1b d r r C T 1 = 0.5 b b b b 3b 1 db 3

24 Thrust Coefficient 1 ln 1 8 y y y = 1 4 CT y y y y 91 1 y 4 4

25 Power Coefficient as a Function of Tip Speed Ratio 5

26 Torque Coefficient as a Function of Tip Speed Ratio 6

27 Thrust Coefficient as a Function of Tip Speed Ratio 7

28 Optimal Chord and Pitch c 8 r BC l 1cos( ) Ref.1 Bc dl w U wcos( ) rdr U r w Cldr 8 r 1 bcos( ) BcC (1 b ) l r 8

29 Optimal Chord and Pitch c 3b 1 BC 1 l b BCl r 4r 16rb 1 Bc 8 1 r C C l l 9

30 Optimum Chord Distribution 30

31 Effect of Profile Drag C d Cl cos( ) Cd sin( ) Cl cos( ) 1 tan( ) Cl C d Cl sin( ) Cd cos( ) Cl sin( ) 1 cot( ) Cl 31

32 Effect of Profile Drag b b b 1 Thrust Integral IT db 3 3b 1 b b b b b Power Integral IP db 3b 1 b 3

33 Effect of Profile Drag 4y y41 y y 3 IT y y y y 4y 4 31 ln y y 4y 3 33

34 Effect of Profile Drag I 1 y 1 ln 1 (1 y) y y y T y y 4 y 9 y 48y 6 3 y y y y y 8 y y( y 4) 64 y y 38y 16y 9 7 y y( y 4) 3 3 y y4 3 34

35 Effect of Profile Drag 4y IP 13 4 y y y y y y 9 3 y41 y y 9y ( y ) y 4y 103 8ln

36 Effect of Profile Drag I P IP (1 ) C d C C [ Eq. 41 ] I Cl C d C C [ Eq. 35 ] I Cl TT T T PN P P 36

37 Thrust Coefficient as a function of Tip Speed Ratio Including the Effect of Profile Drag 37

38 Power Coefficient as a function of Tip Speed Ratio Including the Effect of Profile Drag 38

39 Summary & Conclusions An alternate derivation is provided for the parameters of an optimum, ideal wind turbine, Unlike previous derivations, only a single momentum theory is used (in the direction of the local lift) so that there are no separate accounts of axial and angular momentum. The results, also unlike previous results, are found in closed form for all variables and the singularities of previous numerical solutions are eliminated explicitly. Although the final parameters for the optimum turbine are no different from those of conventional approaches, the closed-form nature of the results yields insight into the properties of the optimum turbine 39

40 Summary & Conclusions Finally, because of the single momentum balance, it is quite straightforward also to write a closed-form expression for the optimum blade chord distribution. The true optimum does not become singular at the blade root, but rather approaches a combination of solidity and pitch angle that avoids blade-toblade interference. 40