Wind power Part 3: Technology San Jose State University FX Rongère February 2009

Wind Turbine Aerodynamics Energy balance over the stream tube Inlet: index 1 Outlet: index 2 Turbine: index T

Betz’s Simplified Approach The turbine is perfect: there is no internal entropy generation Then, when air considered as an incompressible fluid:

Betz’s Simplified Approach Energy balance equation is reduced to: In addition, we assume that the wind speed stay axial through the turbine: Balance of Forces: The power of the torque absorb by the turbine is equal to the power taken from the air flow

Momentum equation on the system: Betz’s Simplified Approach Then: And:

Introducing a the interference factor: Betz’s Simplified Approach Then:

The power absorbed by the turbine is maximum if: Betz’s Simplified Approach Then Betz’s law

Other factors Other factors limit the performance of wind turbines Wake rotation Aerodynamic drag Finite number of blades and interaction between them Blade tip losses

Wake rotation In fact, the air is rotating downstream the rotor by reaction to the torque applied to the rotor

Maximum conversion rate with wake rotation: Glauert’s law To characterize this effect we introduce: Angular velocity of the rotor : Ω T Angular velocity imparted to the flow: ω Angular induction factor: a’ = ω/2Ω T Axial interference factor: Blade tip speed : λ = Ω T R/v 1 We can then show that the transfer power is maximum if:

Then the maximum conversion rate is given by the following equation: With a(λ) defined by: Betz’s law corresponds to: and: Maximum conversion rate with wake rotation: Glauert’s law

By drawing Cp Max we can show that: Larger is the blade tip ratio higher is the conversion rate For blade tip ratio greater than 4 conversion rate is close to Betz’s law The optimal axial interference factor is close to 1/3 for large enough blade tip ratio Maximum conversion rate with wake rotation: Glauert’s law λλ Angular induction factor

Aerodynamic drag The aerodynamics of a blade is defined by: L : Lift force v : Wind speed on the blade c : Chord of the blade D : Drag force l : Span of the blade α : Angle of attack

Lift and Drag Lift and Drag are related: Lift transfers power Drag generates losses Bearings Viscous Friction Noise About % of aerodynamic inefficiencies

Design optimization Blade shape may be optimized using laboratory test and numerical modeling See: Wind Energy, Explained by J.F. Manwell, J.G. McGowan and A.L.Rogers, John Wiley, 2002, for detailed explanations See also: Riso DTU – National Laboratory for Renewable Energy

Interaction between blades Blade tip speed is limited by the interaction between the blades. Typically, if the rotation is too fast (λ>λMax) then the flow for each is perturbed by the previous one Multiblade Wind pump Three Blade Wind Turbine Two Blade Wind Turbine Vertical axis Wind Turbine (Darrieus) λ

Blade tip losses Local rotating airflow at the tip of the blades due to the difference of pressure between the faces of the blades Similar to aircraft wings Thesis at DTU has shown a gain of 1.65% but with an increase in thrust of 0.65% Performance glider with winglets Wingtip loss of Boeing 737 Source: Mads Døssing Vortex Lattice Modelling of Winglets on Wind Turbine Blades Risø-R-1621(EN) Aug. 2007

Actual Wind Turbine Conversion Rate Example: Liberty of Clipper Windpower Power curve

Actual Turbine Conversion rate Power curve and Conversion rate

Optimization Parameters Cut in Cut out Efficiency Max Power

Generated Energy Generated Energy = Wind Speed Distribution x Turbine Power Curve

Capacity Factor Lower Capacity factor allows higher energy capture Generally preferred CF is between 30% and 40% Characteristics of the tested turbine

Wind Turbine Optimization

Wind and Power Distribution CF=24% Captured Energy =33%

Wind and Power Distribution v m = 9m/s CF=44% Captured Energy =22%

Companies to follow GE: Clipper Windpower: Vestas: Repower: Gamesa: Suzlon: Mitsubishi: