1. Wind Energy IWind powerMichael Hlling, WS 2010/2011 slide 1

2. Wind Energy IClass content5 Wind turbines in 6 Wind - bladesgeneral2 Wind measurementsinteraction7 -theorem 8 Wind turbine characterization3 Wind field9 Control strategiescharacterization 10 Generator4 Wind power 11 Electrics / gridMichael Hlling, WS 2010/2011 slide 2 3. Wind Energy I MotivationHow much power can be extracted from the wind ?Michael Hlling, WS 2010/2011 slide 3 4. Wind Energy IBetz limitApproach: free-stream air flow and conservation of mass flowA1, u1A2, u2 A3, u3 A1 u 1 = A2 u 2 = A3 u 3d m = m = const. dtMichael Hlling, WS 2010/2011slide 4 5. Wind Energy IBetz limit Extracted kinetic energy and extracted powerA1, u1A2, u2 A3, u31 Eext= m u1 u32221 u3Pext = m u2 u2 1 optimal for max. Pext ????23u1Michael Hlling, WS 2010/2011slide 5 6. Wind Energy I Betz limit Substitution of u2 will lead to a function Pext(u1,u3) 1 Pext = A2 u 2 u 1 u 3 2 2 2has to be substitutedThrust: T = m (u1 u3 )Corresponding power: Pthrust = m (u1 u3 ) u2 Equals mechanical power1that can be extracted: Pmech = Pext = m u2 u2 132 u1 + u2 u2 =2Michael Hlling, WS 2010/2011slide 6 7. Wind Energy I Betz limit Extractable power as function of u1 and u31 1u3u2 u3Pext = A2 u 1 3 1+33 2 32 2u1u1 u1 wind powercp 0.6cpmax ?? 0.4cp 0.2 0.0 0.00.5 1.0u3/u1Michael Hlling, WS 2010/2011 slide 7 8. Wind Energy I Betz limit Find the maximum cp by taking the first derivative 1Substitute u3/u1 with x: cp (x) = 1 + x x x2 3 2 1 !First derivative of cp(x): cp (x) = 1 2x 3x = 0 2 2For maximum second derivative of possible solution x1/21must smaller than zero: cp (x1/2 ) = (2 6x) < 02 1 16Solution: x = cpmax. (1/3) = 59% 3 27Michael Hlling, WS 2010/2011 slide 8 9. Wind Energy I Betz limitWhat happens at the rotor plane ?u1u2 2/3 u1u3 u(x) 1/3 u1xMichael Hlling, WS 2010/2011 slide 9 10. Wind Energy IBetz limitWhat happens at the rotor plane ?p-2u1u22/3 u1u3u(x)1/3 u1 p(x) p(x)p1 p3x Pressure drop at rotor planep+2Michael Hlling, WS 2010/2011slide 9 11. Wind Energy I Betz limit What does this results mean for wind energy converter ? There is an optimal rotational frequency that the blockageu31 results in= , characterized by tip speed ratio .u13Michael Hlling, WS 2010/2011 slide 10 12. Wind Energy I Betz limitWhat is the value for cp for a WEC over wind speed ?1.2 1.2P(u) / Pr1.0 1.0 cp(u) / cpmaxcp(u) / cpmaxP(u) / Pr0.8 0.80.6 0.60.4 0.40.2 0.20.0 0.0 0 10 20 30u [m/s] Michael Hlling, WS 2010/2011slide 11