Wind 1

1. Wind 1

2. How Lift Based Turbines Extract Energy from Fluid Bernoulli’s Principle - air pressure on top is lower than air pressure on bottom because it has further to travel, creates lift Airfoil – could be the wing of an airplane or the blade of a wind turbine

3. Angle of Attack, Lift, and Drag Increasing angle of attack increases lift, but it also increases drag When angle of attack is too great, “stall” occurs where turbulence destroys the lift

4. Wind Turbines “Windmill”- used to grind grain into flour (or pump water in Holland) Can have be horizontal axis wind turbines (HAWT) or vertical axis wind turbines (VAWT) Groups of wind turbines are located in what is called either a “wind farm” or a “wind park” Important to note: very fast “energy payback” – it takes a few months for a wind turbine to generate (i.e. convert) as much energy as it took to manufacture it!

5. Lots of ideas, only a few good…

6. Vertical Axis Wind Turbines Darrieus rotor - the only vertical axis machine with any commercial success Wind hitting the vertical blades (airfoils) generates lift to create rotation • Advantages • No yaw (rotation about vertical axis) control needed to keep facing into wind • Heavy machinery located on the ground • Disadvantage • Blades are closer to ground where windspeeds are lower

7. Horizontal Axis Wind Turbines “Downwind” HAWT – a turbine with the blades behind (downwind from) the tower No yaw control needed- they naturally orient themselves in line with the wind Shadowing effect – when a blade swings behind the tower, the wind it encounters is briefly reduced and the blade flexes -Also causes noise

8. Horizontal Axis Wind Turbines “Upwind” HAWT – blades are in front of (upwind of) the tower Most modern wind turbines are this type Because blades are “upwind” of the tower • Require active yaw control to keep facing into wind • Operate more smoothly and deliver more power

9. Power in the Wind Consider the kinetic energy of a “packet” of air with mass m moving at velocity v Divide by time and get power The mass flow rate is

10. Power in the Wind Combining we get P (Watts) = power in the wind ρ (kg/m3)= air density (1.225kg/m3 at 15˚C and 1 atm) A (m2)= the cross-sectional area that wind passes through v (m/s)= windspeed normal to A (1 m/s = 2.237 mph)

11. Power in the Wind Power increases as (wind speed)3 Doubling the wind speed increases the power by eight 1h x 20mph wind is same energy as 8h x 10 mph wind… -i.e., most power from a turbine is produced at high wind speed for a short time…

12. Wind Power Classification Scheme

13. US Wind Resources http://www.windpoweringamerica.gov/pdfs/wind_maps/us_windmap.pdf

14. Power in the Wind (cont.) Power in the wind is also proportional to A For a conventional HAWT, A = (π/4)D2, so wind power is proportional to the blade diameter squared Cost is roughly proportional to blade diameter How do you think cost of wind power scales with turbine diameter?

15. Power Curve for Turbine Plateau Generator maxed out Cut out speed Park turbine to avoid damage Cut in speed Not enough energy to justify O&M costs

16. Maximum Rotor Efficiency At the extremes: • Downwind velocity is zero – turbine extracted all of the energy (for zero time…) • Downwind velocity is the same as the upwind velocity – turbine extracted no energy… Albert Betz 1919 Q: What is the ideal extraction of KE from wind so that the turbine extracts the maximum power

17. Maximum Rotor Efficiency Consider wind passing though turbine: as energy extracted, air slows down ṁ = mass flow rate of air within stream tube v = upwind undisturbed windspeed vd = downwind windspeed

18. Mass Flow Rate At the rotor with area A and, mass flow rate is If velocity through the rotor vb is the average of upwind velocity v and downwind velocity vd

19. Power Extracted by the Blades Then power relationship at the rotor could be Define new parameter l such that We can rewrite the power relationship as

20. Power Extracted by the Blades Power in the wind Rotor efficiency (CP)

21. Maximum Rotor Efficiency So what is the windspeed ratio λ which maximizes the rotor efficiency, CP ? • Plug into CP to find the maximum rotor efficiency: Maximum efficiency of 59.3% when air is slowed to 1/3 of its upstream speed! “Betz limit”

22. Maximum Rotor Efficiency

23. Number of Rotating Blades Windmills have multiple blades • need to provide high starting torque to overcome weight of the pumping rod • must be able to operate at low windspeeds to provide nearly continuous water pumping • a larger area of the rotor faces the wind Turbines with many blades must operate at lower rotational speeds – as speed increases, turbulence caused by one blade impacts other blades Most modern wind turbines have two or three blades

24. Brush wind turbine

27. Tip-Speed Ratio (TSR) Efficiency is a function of how fast the rotor turns Define “Tip-Speed Ratio” (TSR) as ratio of speed of tip of blade to windspeed D = rotor diameter (m) v = upwind undisturbed windspeed (m/s) rpm = rotor speed, (revolutions/min)

28. Air moved this far Airfoil interacted with this much air, call it Xs

29. Optimal Tip Speed Ratio If ts<<tw then wind turbine is interacting with disturbed air → low efficiency If ts>>tw then turbine does not get to all useful air… → low efficiency Optimal is if ts≈tw

30. Optimal Tip Speed Ratio Then for a three bladed turbine, And for a two bladed turbine

31. Tip-Speed Ratio (TSR) Rotors with fewer blades reach their maximum efficiency at higher tip-speed ratios

32. Impact of Terrain on Windspeed We saw power depends on cube of windspeed: small change of wind speed can have large impact System design must consider effect of terrain friction on wind speed • Important in first few hundred meters above ground level • Smooth surfaces (like water) are better • Windspeeds are greater at higher elevations – tall towers are better • Forests and buildings slow the wind down a lot Can we quantify impact of terrain and height on wind speed?

33. Wind Speed Losses as Function of Terrain v = windspeed at height H v0 = windspeed at height H0 (H0 is usually 10 m) α = friction coefficient Open terrain, α ≈ 1/7 (0.147) City, α= 0.4; Calm water, α= 0.1 Note this is just an approximation, others exist (ex. von Karman’s log velocity profile)

34. Impact of Terrain on Wind Power Remember wind power goes as third power of wind speed.

35. Impact of Terrain on Wind Power In a town (a≈0.3), windspeed at 100 m is twice that at 10 m Areas with smoother surfaces have less variation with height

36. Rotor Stress Let’s calculate ratio of power at highest point to lowest point on wind turbine with hub at 50m, 30m diameter rotor, α = 0.2 65 m 50 m 35 m • Power in the wind at the top of the blades is 45% higher! • Can cause significant stress (failure) Picture may not be to scale

37. As you may expect, turbines interfere with each other…

38. Wind Farms It makes sense to install a large number of wind turbines in a wind farm or a wind park Benefits Able to get the most use out of a good wind site Reduced development costs Simplified connections to the transmission system Centralized access for operations and maintenance How many turbines should be installed at a site? What is a sufficient distance between wind turbines so that windspeed has recovered enough before it reaches the next turbine?

39. Wind Farms For closely spaced towers, efficiency of array becomes worse as more wind turbines are added

40. Wind Farms Previous figure considered square arrays (but square arrays don’t make much sense) Rectangular arrays with only a few long rows are better Recommended spacing is 3-5 rotor diameters between towers in a row and 5-9 diameters between rows Offsetting or staggering the rows is common Sites commonly have a prevailing wind direction

41. Average Power in the Wind How much energy can we expect from a wind turbine? Remember, power goes as cube of wind speed Therefore we need to know the average of the cube of wind speed… I.e., we can’t use average windspeed to find the average power in the wind

42. Windspeed probability density function (pdf) If we had a function f(v) that gave windspeed we could calculate average power in wind… People have examined statistics of windspeed over various locations A reasonable approximations is the Weibull distribution # of hours/year that the wind is between two windspeeds:

43. Idealized Site Windspeed Data

44. Weibull p.d.f. k=2 looks reasonable for wind

45. Wind Probability Density Functions Windspeed probability density function (p.d.f) Values between 0 and 1 Area under the curve is equal to 1

46. Weibull p.d.f. Weibull with k=2 has shape similar to windspeed distribution Often used as first guess when little is known about a particular site Fairly realistic for a wind turbine site: winds are mostly pretty strong, but there are some periods of low wind and high wind k = shape parameter c = scale parameter

47. Rayleigh p.d.f. (Weibull with k=2) Higher c implies higher average windspeeds

48. Real Data vs. Rayleigh Statistics (It is important to gather as much real wind data as possible!)

49. Average Windspeed using p.d.f. Now that we have a function that approximates wind speed… And for average v3

50. Average windspeed from Rayleigh p.d.f. For a Rayleigh p.d.f., there is a direct relationship between average wind speed v and scale parameter c (not surprising really) You can, of course, use this to extract a c for your site