Sustainable Energy Science and Engineering CenterWind Energy - AerodynamicsSustainable Energy Science and Engineering CenterOne dimensional momentum theoryAssumptions:Incompressible, inviscid, steady state flowInfinite number of bladesUniform thrust over the rotor areaNon rotating wakeThe thrust T (equal and opposite to the force of the wind on the wind turbine) is given bySource: Wind Energy Explained by J.F Manwell, J.G. McGowan and A.L. Rogers, John Wiley, 2002. Wind Turbine AerodynamicsT=T=U1ρUA()1−U4ρUA()4Ý m =ρUA()1=ρUA()4T =Ý m U1−U4()= A2p2− p3()p1+12ρU12= p2+12ρU22p3+12ρU32= p4+12ρU42Sustainable Energy Science and Engineering CenterOne dimensional momentum theoryUsing the Bernoulli equation on either side of the rotor and assuming p1= p4Where a is the axial induction factor and U1a is referred to as the induced velocity at the rotor. The power output, P is given byT =12ρA2U12−U42()U2=U1+ U42a =U1−U2U1P =12ρA2U12−U42()U2PPo= Cp=P12ρU3ACpmax imum= 0.5926(1-a)U1(1-2a)U1P =12ρAU34a(1 − a)2U1= U;A2= ACp=P12ρAU3= 4a(1 − a)2dCpda= 0 ⇒ a =13Cpmax=1627= 0.5926a < 0.5Betz LimitSustainable Energy Science and Engineering CenterBetz Limit(2/3) U1(1/3) U1Maximum power production:The axial thrust on the disk at maximum power:T =12ρAU124a 1− a()[]CT=T12ρAU2=89Overall efficiency:ηoverall=Pout12ρAU3=ηmechCPPout=12ρAU3(ηmechCP)Sustainable Energy Science and Engineering CenterIdeal wind turbine with wake rotationAngular velocity of the rotor : ΩAngular velocity imparted to the flow stream: ωAngular induction factor: a` = ω/2ΩBlade tip speed : λ = ΩR/ULocal Speed ratio:λr= λr/RWake RotationWhen deriving the Betz limit, it was assumed that no rotation was imparted to the flow. Rotating rotor generates angular momentum, which can be related to rotor torque. The flow behind the rotor rotates in the opposite direction to the rotor, in reaction to the torque exerted by the flow on the rotor.Sustainable Energy Science and Engineering CenterΩΩ+ωp2− p3=ρΩ+12ω⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ωr2dT = p2− p3()dA =ρΩ+12ω⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ωr22πrdr()′ a =ω2Ωangular Induction factorThe induced velocity at the rotor consists of not only the axial component Uabut also a component in the rotor plane rΩa`Loss of Energy Due to Wake RotationSustainable Energy Science and Engineering CenterdT = 4′ a (1 +′ a )12ρΩ2r22πrdrThrust on an annular cross section due to linear momentum:dT = 4a(1 + a)12ρU22πrdrEquating the two expressions for thrust gives:a(1− a)′ a (1+′ a )=Ω2r2U2=λr2Where λris the local speed ratio. The tip speed ratio λ defined as the ratio of the blade tip speed to the free stream wind speed is given by λ=ΩrU=λrRrLoss of Energy Due to Wake RotationSustainable Energy Science and Engineering CenterLoss of Energy Due to Wake RotationTorque exerted on the rotor: Q = change in the angular momentum of the wakeThe power generated at each element becomes:The axial angular induction factors determine the magnitude and direction of the airflow at the rotor plane.The incremental contribution to the power coefficient, dCpfrom each annular ring is given by:dQ=dÝ m ωr()r=ρU22πrdr()r⎡⎤dP =ΩdQ =12ρAU38λ2′ a 1− a()λr3dλr⎡ ⎣ ⎢ ⎤ ⎦ ⎥ dCp=dP12ρAU2Cp=8λ2′ a (1 − a)λr30λ∫dλrSustainable Energy Science and Engineering Centerλ = 7.5Wake Rotation EffectSustainable Energy Science and Engineering CenterAirfoil TerminologyDU-93-W-210Airfoil used in some wind turbinesSustainable Energy Science and Engineering CenterForces on AirfoilImportant parameter: Reynolds number = UL/ν = 0.5 − 10 x 106Lift coefficient, l: span of the airfoil Drag CoefficientPitching moment Coefficient, A = lcCl=Ll12ρU2cCD=Dl12ρU2cCm=M12ρU2AcSustainable Energy Science and Engineering CenterAirfoil characteristicsCl= 2παCD=Cdo+(Cl2/πeAR)Sustainable Energy Science and Engineering CenterDU-93-W-210DU-93-W-210S809 airfoilAerodynamic CoefficientsSustainable Energy Science and Engineering CenterStall Characteristics of Turbine Blade α=30ºt=261 msα=25ºt=217 msα=20ºt=174 msSustainable Energy Science and Engineering CenterRelative Wind VelocityUrel= U2+Ωr()2= U 1+λ2λ=ΩrUFL= CL12ρAUrel2⎛ ⎝ ⎜ ⎞ ⎠ ⎟Sustainable Energy Science and Engineering CenterBurton et al. (2001)0.00.51.01.52.02.53.00.0 20.0 40.0 60.0LFA (deg)CnPARKED, 20 m/sPARKED, 30 m/sROTATINGSource: Pat Moriarty, NRELRotation Effect on CnSustainable Energy Science and Engineering CenterAssumptions: no aerodynamic interaction between elements and the forces on the blades are determined solely by the lift and drag characteristics of the airfoil shape of the bladeWind velocity at the rotor =U(1-a)+(Ωr+ωr/2)blade section velocity induced angular velocityBlade Element TheorySustainable Energy Science and Engineering CenterIf the rotor has n blades, the total normal force on the section at a distance r, from the center isThe differential torque due to tangential force operating at a distance r, from the center is given byUrel=U 1− a()sinφdFL= Cl12ρUrel2cdrdFD= Cd12ρUrel2cdrdFN= dFLcosφ+ dFDsinφdFT= dFLsinφ− dFDcosφdFN= n12ρUrel2Clcosφ+ Cdsinφ()cdrdQ = nrdFT= n12ρUrel2Clsinφ− Cdcosφ()crdrBlade Element TheorySustainable Energy Science and Engineering CenterThe relative velocity can be expressed as a function of free stream velocity and the resulting equations for normal force or thrust and torque can be written asWhere σ` is the local solidity, defined bydFN=′ σ πρU21− a()2sin2φClcosφ+ Cdsinφ()rdrdQ =′ σ πρU21− a()2sin2φClsinφ− Cdcosφ()r2dr′ σ =nc2πrBlade Element TheorySustainable Energy Science and Engineering CenterBlade Element TheoryIf we set Cd=0 - a reasonable assumption that simplifies the analysis and introduces negligible errors, we obtainCl= 4sinφcosφ−λrsinφ()′ σ sinφ+λrcosφ()a′ a =λrtanφa =11+4sin2φ′ σ Clcosφ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ ′ a =14cosφ′ σ Cl−1⎡ ⎣ ⎢ ⎤ ⎦ ⎥Sustainable Energy Science and Engineering CenterOptimum PerformanceLosses are due to tip speed ratio, airfoil drag and tip losses (a function of number of blades). The maximum achievable power coefficient for turbines with an optimum blade shape with finite number of blades and aerodynamic drag is given by an empirical formula developed from experimental data as follows:Cpmax=1627λλ+1.32 +λ− 820⎛ ⎝ ⎜ ⎞ ⎠ ⎟ n232⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥