Chapter 13 Aerodynamics Of Wind Turbines - Kimerius Aircraft
Chapter 13Aerodynamics of Wind Turbines
13.1Introduction Drawsimilarities between helicopter rotors and windturbines Classic analysis by Lock (1925) More complicated than helicopter rotors Ground boundary layers effectsAtmospheric turbulenceWind gustsThermal convection stratificationTower (another wind turbine) shadow Classicalblade element-momentum theory allows study ofthe effects of primary design variables (e.g. blade twist,planform, # of blades)Chapter 13: Aerodynamics of Wind Turbines
13.2Historyuse in Babylonia times, Persians (7th century), Europecentury) 17th century: Tower mills (twisted blades, tapered planformscontrol devices to point to the wind. 18th century: Dutch bring wind mills to US; pumping water 20th century: Used to generate power, especially in Europe. 2005: 2% of total energy demand (25GW) Typical configuration: HAWT: Horizontal Axis Wind Turbine;(lift machine) – see next figure; tall tower, alignment parallelto the wind, tower shadow 5-100m diameter (few KW-MW) VAWT: Vertical Axis Wind Turbine, not as common, lessefficient, do not scale up easily Relative cost of wind energy still high, but going down First(15thChapter 13: Aerodynamics of Wind Turbines
Chapter 13: Aerodynamics of Wind Turbines
Chapter 13: Aerodynamics of Wind Turbines
13.3Power in the Wind Thekinetic energy per time through a disk of area A isif process is 100% efficient For example: d 5m, 50% efficient, velocity 10m/s standardsee level Ithelps if diameter and wind speed are high If not perpendicular to wind velocity has cosγ factor, powerdrops by a cos3γ factorChapter 13: Aerodynamics of Wind Turbines
13.4Momentum Theory Analysis for a Wind Turbine Thrustnot known; momentum theory alone is not enough,need BEMTChapter 13: Aerodynamics of Wind Turbines
Windturbine extracts energy from the flow, thus velocitydecreases, thus slipstream expands downstream Conservation of mass Changein momentum can be related to thrust Expanding: Workdone on the air by the turbine per unit time is:Chapter 13: Aerodynamics of Wind Turbines
Turbinedoes negative work (windmill state) Power: Substituting13.5 in 13.7:w 2υi (or υi w/2); same as in helicopter rotors For model validity: V -w 0; thus V w 2υi Thus Thrustnot known; define induction ratio: a υi/V, or υi aV Larger a more flow is slowed as it passes the turbineChapter 13: Aerodynamics of Wind Turbines
13.4.1 FromPower and Thrust Coefficients13.7 and 13.3 Powercoefficient (different from a helicopter) Thus Shownin following figure (valid for 0 a 1/2)Chapter 13: Aerodynamics of Wind Turbines
Chapter 13: Aerodynamics of Wind Turbines
Forthrust using 13.3: Thrustcoefficient: Using 13.12 Alsoshown in previous figure; also Cp (1-a)CT Empirical results for a 0.5 are shown; e.g. Notethat:Chapter 13: Aerodynamics of Wind Turbines
13.4.2Theoretical Maximum Efficiency Differentiate13.11 wrt a Thus Betz-Lanchester limit; upper limit for power extractionNo viscous or other lossesTypically: CP 0.4 – 0.5 (66-83% of max)Chapter 13: Aerodynamics of Wind Turbines
13.5Representative Power Curve See following figure for 47m/0.66MW turbine“cut-in” speed to overcome frictionRated power – to make sure power can be absorbed byelectric generatorBlade pitch control (stall at high speeds) Furling (mechanical/aerodynamic brake) Chapter 13: Aerodynamics of Wind Turbines
Chapter 13: Aerodynamics of Wind Turbines
Definetip speed ratio (TSP)(similar to advance ratio µ for helicopter rotors) The corresponding CP is shown in the next figurePeak efficiency only at a fixed value of wind speed Anotheroption is a variable speed turbine (more commontoday) – see figure More efficient extraction at low speedsLower efficiency at high speedsChapter 13: Aerodynamics of Wind Turbines
Chapter 13: Aerodynamics of Wind Turbines
Chapter 13: Aerodynamics of Wind Turbines
13.6Elementary Wind Models Factorsaffecting power: wind speed, tower height,likelihood of gusts Standard atmosphere (section 5.2) Forheight Power law href 10m, m 1/6 or 1/7Logarithmic lawChapter 13: Aerodynamics of Wind Turbines
Chapter 13: Aerodynamics of Wind Turbines
zo values: Stochasticvariations of speedChapter 13: Aerodynamics of Wind Turbines
Turbulence intensity IuT time interval (e.g. 10secs)Iu : 0.1U-0.2U, function of h, larger at low wind speeds Average Goodpower generated (probability distribution p(V )wind models are neededChapter 13: Aerodynamics of Wind Turbines
13.7Blade Element Model for the Wind Turbine Similarto helicopter rotor in descend Angle of attack α θ φ Assume swirl components are low Incremental lift and drag are Perpendicular Contributions Ifand parallel to the rotor disk:to thrust, torque, powerprecone angle multiply thrust by cosβpChapter 13: Aerodynamics of Wind Turbines
Chapter 13: Aerodynamics of Wind Turbines
Substituting Assumingexpresions from13.27small angles: Nondimensionalizing:Chapter 13: Aerodynamics of Wind Turbines
r y/R, σ Nbc/πR, as before X Ωy/V local section speed ratio Where Thrustcoefficient Powercoefficient Inflowangleevaluate CT and CP need υi and local Cl Cd, a a(V,θ,υi) 2D lift drag coefficients can be assumed Similar to helicopters (sec. 3.3) ToChapter 13: Aerodynamics of Wind Turbines
13.8Blade Element Momentum Theory BEMTa hybrid method - BEM plus momentum on annuli(use 2D lift/drag) - allows evaluation of the induction factor Works OK when wind perpendicular turbine blade Allows understanding of the effects of geometricparameters Similar to helicopters (section 3.2) Mass flow on annulus of turbine disk Usingdifferential form of 13.12: Non-dimensionalizing:Chapter 13: Aerodynamics of Wind Turbines
Non-dimensionalizing: Thisintegrates to 13.14 Using 13.38 Thesecond term is small: Assumingno stallChapter 13: Aerodynamics of Wind Turbines
Thelift coefficient becomes: Using Afterthis (and assuming αo can be absorbed into θ)some algebra Thus:Chapter 13: Aerodynamics of Wind Turbines
Fundamentalequation of BEMT allows calculation of a andvi as a function of r for a given blade pitch, twist, chord,airfoil section (Clα, aο) Similarity with eq. 3.61 (climbing rotor) Clα can be approximated with 2π; valid when 0 a 0.5 Rotor thrust and power can be found by integration: Power Incan be expandedthe ideal case a is uniform, and Cd CdoChapter 13: Aerodynamics of Wind Turbines
Inthe ideal case a is uniform, and Cd Cdo Thefirst term is the induced power from simple momentumtheory Second term (profile power) depends on σ, Cd, tip speed Eq. 13.50: For uniform a; θr constant, ideal twist (section3.3.3); lowest induced losses on the turbine. Magnitude of twist depends on tip velocity (figs 13.10-13.11) –a low blade pitch works for a wide range of wind speeds a 1/3condition for max energy (valid for 0 a 0.5)Chapter 13: Aerodynamics of Wind Turbines
Chapter 13: Aerodynamics of Wind Turbines
Chapter 13: Aerodynamics of Wind Turbines
13.8.1Effect of the number of blades Seefig 13.12 Increasing Nb (or s) does not affect max efficiency Affects tip speed ratio where maximum efficiency isobtainedChapter 13: Aerodynamics of Wind Turbines
Chapter 13: Aerodynamics of Wind Turbines
13.8.2Effect of Viscous Drag Seefig. 13.13 Lower drag higher maximum efficiency Usually 0.01 Cdo 0.02Chapter 13: Aerodynamics of Wind Turbines
Chapter 13: Aerodynamics of Wind Turbines
13.8.3Tip-Loss Effects Prandtl’s Where If“tip-loss” correction factor F (section 3.3.10)f:root is included: Putinto eq. 13.50 (BEMT)Chapter 13: Aerodynamics of Wind Turbines
Iterations:assume F 1 initially, calculate a and F from eqs13.58 and 13.55; 3-10 iterations From 13.58 sqrt needs to be positive Flimits the range of wind speeds and operating conditions f If a 0.5 eq. 13.15 can be used (empirical) Equating momentum and blade element: Aftersome algebra:Chapter 13: Aerodynamics of Wind Turbines
Solution: Validfor 0.5 a 1.0 OK for engineering purposes Representative solutions are shown in figure 13.14Chapter 13: Aerodynamics of Wind Turbines
Chapter 13: Aerodynamics of Wind Turbines
13.8.4Tip Losses and Other Viscous Losses Fig.13.15 shows effects of tip losses and viscous losses –decrease power by 15% Fig13.16 shows the axial induction factor as a function oftip speed ratio for several blade pitch angles. Larger a at high XAt low wind speeds possibility of two directions (empiricalrelationship between Ct and a)At high wind speeds possible stallChapter 13: Aerodynamics of Wind Turbines
Chapter 13: Aerodynamics of Wind Turbines
Chapter 13: Aerodynamics of Wind Turbines
13.8.5Effects of Stall Occursat high speed (low X) and/or high blade pitch angles Can be incorporated in BEMT using a look-up table Stall causes power to drop more sharply with increasingwind speed Figure in next page shows resultsChapter 13: Aerodynamics of Wind Turbines
Chapter 13: Aerodynamics of Wind Turbines
Airfoilsfor Wind Turbines Familiesof airfoils developed for wind turbines NACA 4-digit, NASA LS-1 airfoils, NREL airfoils (see fig.) Low Reynolds #s Re # sensitivity, look up tables or equations Sensitivity to surface roughness, see fig. - fig. 7.37Increased drag, decreased Clmax, “soft” stall Forvariable pitches turbines high Clmax is important (seealso section 7.9) Pitch control adjusts AoA for best energy extraction Forfixed pitch airfoils begin stall at low Cl values, butmaintain the lift for a wide range of AoA (e.g. S809) This trend carries over for unsteady conditions too.Chapter 13: Aerodynamics of Wind Turbines
Chapter 13: Aerodynamics of Wind Turbines
Chapter 13: Aerodynamics of Wind Turbines
13.10Yawed Flow Operation Yawmisalignment to wind flow leads to skewed wake. Inflow gradient across the wind turbine Strong // to the direction of the wake skewAlso in the other direction because of asymmetry ofaerodynamic loads Vortex theory is needed (BEMT axisymmetric); or assume:γ is the yaw angleaγ is the corrected value for a (induction ratio)A good approach is to correct the value of a and iterate(momentum balances)Ks and Kc can be found from inflow measurements, or where:Chapter 13: Aerodynamics of Wind Turbines
13.11Vortex Wake considerations BEMTgood preliminary predictions, insight into parameters(e.g. blade planform, twist) Yaw is 3D 3D wake complicated structures (see next fig.) Distortions may also be due wind gradient. BEMmodifications (e.g. using inflow models section 3.5.2) Vortex methods (section 10.7) incompressible potential flow vorticity concentrated in vortexfilamentsvortex filament strengths induced velocities are obtained fromBiot-Savart lawhigher cost – more accurate. Prescribedvortex: position specified apriori aided byexperiments; steady-state Free vortex: elements are allowed to convect freely Thousands of elements needed (high cost), time accurateChapter 13: Aerodynamics of Wind Turbines
Chapter 13: Aerodynamics of Wind Turbines
Chapter 13: Aerodynamics of Wind Turbines
Timeaccurate predictions for a 2-bladed HAWT areshown in the next figure (13.22) Athigh tip speed ratio (low wind speeds) vortex ring state(part a) Lowering tip speed ratio: turbulent wake state (part b) Lowering tip speeds ratio further: wake convers morequickly downstream CPis shown nextChapter 13: Aerodynamics of Wind Turbines
Chapter 13: Aerodynamics of Wind Turbines
Chapter 13: Aerodynamics of Wind Turbines
calculations for 30o yaw (after startwith no yaw) are shown in the next figure. Time-accurate Initiallyblades move into their own wake After 10 revs periodic solution againChapter 13: Aerodynamics of Wind Turbines
Chapter 13: Aerodynamics of Wind Turbines
Timehistory of Cp is shown next Powerdrops 35% initially (cos3γ factor, see eq. 13.2) Then equilibrium – some recovery Dynamic inflow theory-section 10.9 (unsteady aerodynamiclag of the inflow over turbine disk in response to changes inblade pitch or turbine thrust)Chapter 13: Aerodynamics of Wind Turbines
Chapter 13: Aerodynamics of Wind Turbines
13.12 Unsteady Aerodynamic Effects on Wind TurbinesChapter 13: Aerodynamics of Wind Turbines
Yawmisalignment is important Blade flapping effects Due to low speeds (20rpm for large 800 rpm forsmall turbines) relativelylow noise Significant changes in AoA due to changes in wind speed Reduced frequencies can be high Reducedfrequency for yaw Typicalvalue R/c 10, k 0.1 (for K 0.05 quasisteady); Specific effects are:Chapter 13: Aerodynamics of Wind Turbines
1.Varying wind speed Modest 2.wind fluctuations change AoA significantly.output transientsunsteady forcesLag of inflow (wake adjustments 10 revsVelocity gradients in the wind (fig. 13.20) Nonuniform 3.Non-steady velocity fluctuations in yawed flow Large 4.AoA - unsteadinessexcursions from axial flowSignificant unsteady effectsCannot assume small disturbancesUnsteady wake induction effect Lagin inflow developmentChapter 13: Aerodynamics of Wind Turbines
5.Local sweep effects Localsweep angle can be large when turbine is yawed Stall might occur 6.Tower shadow effects Importantfor downstream and upstream turbines k 0.2 (high) – airfoil with gust 13.12.1Tower Shadow Canbe seen in fig. 13.20 Velocity deficiency (up to 30%) approaching turbine disk 2-D airfoil, with V 1, and disturbance (normal to the chord)w 0.08 0.02cos(5ψ), 144o ψ 216o Fig. below shows difference of unsteady and quasi-steadyapproachesChapter 13: Aerodynamics of Wind Turbines
Chapter 13: Aerodynamics of Wind Turbines
13.12.2Dynamic Stall and Stall Delay Stallleads to large unsteady airloads (can cause structurardamage) Unsteady airloads exist even without stall; unsteady flowwithout stall needs to be known first Semi-empirical models for stall (e.g. Leishman Beddoes) Figure below shows that stalls creates a lot of turbulence Stall effects Unsteady pressure gradient reduction effects delay in 3-Dboundary layer developmentCoupled influences of centrifugal and Coriolis effects on theboundary layer in rotating flow Coriolisacceleration forces can act to alleviate adversepressure gradients and may delay the onset of flowseparation and stallChapter 13: Aerodynamics of Wind Turbines
Chapter 13: Aerodynamics of Wind Turbines
Themodified boundary layer equations are x:chordwise, y: spanwise, z normal to the blade W2x terms: centipetal accelerations 2Wu: Corriolis accelerationChapter 13: Aerodynamics of Wind Turbines
Effectsof radial flow more important in wind turbine blades(higher values of Cl) There is some experimental and CFD evidence (stall delaydue to 3D effects) 13.13Advanced Aerodynamic Modeling Issues BEMT structural dynamic analysis can be used CFD has some problems to overcome: separation (dynamicstall), vortical wake (difficulty preserving concentratedvorticity)Chapter 13: Aerodynamics of Wind Turbines
Chapter 13: Aerodynamics of Wind Turbines. Chapter 13: Aerodynamics of Wind Turbines. Chapter 13: Aerodynamics of Wind Turbines. Time accurate predictions for a 2-bladed HAWT are shown in the next figure (13.22) At high tip speed ratio (low wind speeds) vortex ring state (part a)
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