Wind Energy Systems MEMS 5705 Spring 2017 Lecture 9, Feb. 15 Lecture 10, Feb. 20

1. Wind Energy Systems MEMS 5705 Spring 2017 Lecture 9, Feb. 15 Lecture 10, Feb. 20 1

2. L9 and L10 1. Assignments. Project 1. due Monday, August 20. Homework: 3.3, 3.7 due February 27 2

3. 2. Class background and level of coverage 3. Key points of L7 & L8, WT aerodynamics under steady conditions 4. WT aerodynamics under steady conditions (or for steady state operations) pp. 91-96 5. Revisiting momentum theory 5.1 Range of validity and flow states 5.2 Empirical equations and test data 5.3 Factors decreasing the maximum power coefficient Cp, max 6. Momentum theory with wake rotation 7. Airfoils 7.1 Basics 7.2 HAWT airfoils 7.3 Flat plate 8. Background in mechanics and review

4. 2. Class background and level of coverage • From seniors to master’s students to doctoral students • 2. From EE to ME to AE to Energy Area • Occasionally, for some students, the coverage may seem a bit elementary and repetitious. This is unavoidable, please bear with the instructor.

5. 3. Key Points of L7 & L8

6. ABL and surface layer turbulence Of concern to wind turbines is turbulence in the lowest levels of the ABL, also referred to as, surface-layer turbulence. The ABL thickness is not a precisely defined quantity; it varies “from a few hundred meters to several kilometers.” For practical purposes, surface layer encompasses operational heights up to 200 ft, say 10% of the boundary layer, and the rest is called outside layer. 6

7. Surface-layer turbulence has been an actively researched area of the past 30 years by meteorologists, and engineers associated with dynamic loading on exposed structures and wind turbines. As for modeling, “ classical turbulence theory” is keyed to surface-layer conditions on the basis of phenomenological and analytical considerations as well as other guidelines such as continually updated Engineering Science Data Unit (ESDU) series. 7

8. Basically, the von Karman turbulence model with empirically adjusted parameters correlate well with test data and it is widely used. Therefore in this lecture, we follow the text and briefly describe turbulence intensity and von Karman power spectral density (PSD), which is a frequency-domain description of turbulence (Eq. 2.27, p. 43). This PSD is usually referred to as von Karman turbulence model. 8

9. In General, For the present, we follow the text: Wind speed in the x direction, which is perpendicular to the disk

10. Turbulence Intensity TI (2.28) page 40 over the static loads of the system. (2.23) p. 40 deterministic steady loads due to mean wind speed U-bar. 10

11. For the Weibull distribution (2.23) p. 40 11

12. Energy Pattern Factor Ke For Weibull distribution 2nd Edition (2.69) p. 61

13. Table 2.4 P. 61 Variation of parameters with Weibull k shape factor 13

14. Autocorrelation We consider a stochastic process {U(t)} and consider two time instances ‘t’ and ‘t + ’. Given {U(t)}, we cannot predict {U(t + )}. But {U(t)} and {U(t + )} are ‘related’ or correlated. This correlation is described by the autocorrelation function RUU(t,) RUU(t, t + ) = E[U(t) U(t + )] 14

15. Wind Speed measurements belong to a Stationary random process. That is, Ū(t) = constant, E[U2(t)] = constant Autocorrelation function depends only on time difference: RUU(t1 , t2) = RUU(t2 – t1) RUU() = E[U(t)U(t+)] 15

16. While the autocorrelation function characterizes turbulence as a function of time or time lag  in the time domain, the corresponding description in the frequency domain leads us to the power spectral density function. That is, the power spectral density function characterizes turbulence as a function of frequency in the frequency domain. 16

17. UU UU UU UU 17

18. 18

19. Eq. (2.27) p. 43, 2nd Edition (2.27) p. 43 19

20. 20

21. Predicted and Measured Longitudinal Turbulence PSD • Experiment • ___ Von Karman Rotational-coordinates Fixed-coordinates (Radius 24.5 m, Rotation Rate = 0.625 Hz) 21

22. 4. WT aerodynamics under steady conditions (or for steady state operations) pp. 91-96

23. In the treatment of aerodynamics U = U (deterministic) = Usteady+ Uunsteady If not stated otherwise, Usteady = U Uunsteady = Airfoil unsteady aerodynamics and unsteady wake (dynamic inflow)

24. In the treatment of turbulence U = U (random or stochastic) U = Ū + u Ū = mean wind speed u = turbulence or random variation about the mean

25. 5. Revisiting Momentum Theory

26. +ve direction U-w U U - vi (axial) induction factor 26

27. aU 2aU Ū = U1 = U U U Stream tube boundary Actuator disk U1= U; U2= U3= U(1-a); U4= U(1-2a) A2 = A3 = A = disk area = U2 U3 U1 U4 1 2 3 4 aU = induced velocity a = (axial) induction factor (pp. 93-97) Actuator disk model of a wind turbine: U = mean air velocity, 1,2,3,4 indicate locations F(on disk) F(on fluid) 2 3 4

28. A2 = A3 = A = disk area = Axial Thrust ….(3.16), p.95 ….(3.17)

29. Power output = P …..(3.12), p. 94 Power coefficient ….(3.14 ), p. 95

30. 5.1 Range of validity and flow states

31. Stream tube boundary Actuator disk U2 U3 U= U1 U4 1 2 3 4 (p. 93) Actuator disk model of a wind turbine: U, mean wind velocity, 1,2,3,4 indicate locations U4= (1-2a) U1=(1-2a) U For flow has slowed to zero velocity! For momentum theory is not applicable

32. , a ….(3.16), p.95 ….(3.14) , p.95 ….(3.10), p. 94 U4= (1-2a)U

33. Flow States U2 U

34. 5.2 Empirical equations and test data

35. For applications, momentum theory is considered invalid for a ≥ 0.4. We accept the following empirical formula: Ct=4(a-1)a+2……..………….(A) (Ref: G. Leishman, Principles of Helicopter Aerodynamics, Cambridge 2005,ch. 13, Aerodynamics of Wind Turbines, p. 730)

36. (Eq. A) (Ref: G. Leishman, ibid.)

37. In the literature [D. M. Eggleston and F.S. Stoddard, Wind Turbine Engineering Design, Van Nostrand Reinhold, 1987, p.33, and D.Spera (ed.) Wind Turbine Technology, ASME, 1994, P. 231] the following equation is suggested! Ct=4a|(a-1)| “This is obviously incorrect.” (Leishman, Ibid)

38. 5.3 Factors decreasing the maximum power coefficient Cp, max

39. This is decreased by • Wake rotation behind (downstream of) the rotor • Tip loss and number of blades • Aerodynamic drag • Following the text (p. 96), we will first study wake rotation (#3.3, p. 96) and then take up airfoil and general concepts of aerodynamics (#3.4, p. 101). Given this background, we address in later lectures tip loss (under advanced topics) and number of blades and aerodynamic drag.

40. 6. Momentum theory with wake rotation

41. Ideal HAWT with Wake Rotation,# 3.3, pp. 96-101

42. Stream tube model of flow behind rotating wind blade. Picture of stream tube with wake rotation.

43. , Fig. 3.4, p. 97

44. R U(1-a) U(1-2a) U r dr Annular section of width dr at radius r of HAWT of radius R

45. While the flow imparts a torque to the disk, the disk in turn imparts an equal and opposite torque to the flow. Therefore, if the disk rotates with angular velocity Ω, the flow rotates in the opposite direction, say with angular velocity ω. Now, consider an observer moving with the disk with angular velocity Ω. The observer sees that the flow behind the rotor is moving with angular velocity Ω - (-ω) or Ω + ω in the opposite direction.

46. In the literature (e.g. H. Glauert, 1935) it is shown that we can apply Bernoulli’s energy equation with respect to the observer, before and after the disk: (3.20-21) p. 97

47. Define a' = angular induction factor ……(3.22) p.97

48. a’r Ω Wind Direction aU U aU U U(1-a) U Relative Velocity rΩ Plane of Rotation Flow velocity diagram at an annulus in a HAWT rotor disk

49. An important statement in the text (p. 97) “Note that when wake rotation is included in the analysis, the induced velocity at the rotor consists of not only the axial component, Ua, But also a component in the rotor plane rΩa'.”

50. a’r Ω Wind Direction aU For future reference: U(1-a) U Relative Velocity rΩ Plane of Rotation Flow velocity diagram at an annulus in a HAWT rotor disk