Wind energy I. Lesson 4. Wind power

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Wind energy I. Lesson 4. Wind power

  1. 1. Wind Energy I Wind powerMichael Hölling, WS 2010/2011 slide 1
  2. 2. Wind Energy I Class content 5 Wind turbines in 6 Wind - blades general 2 Wind measurements interaction 7 Π-theorem 8 Wind turbine characterization 3 Wind field 9 Control strategies characterization 10 Generator 4 Wind power 11 Electrics / gridMichael Hölling, WS 2010/2011 slide 2
  3. 3. Wind Energy I Motivation How much power can be extracted from the wind ?Michael Hölling, WS 2010/2011 slide 3
  4. 4. Wind Energy I Betz limit Approach: free-stream air flow and conservation of mass flow A1, u1 A2, u2 A3, u3 ρ · A1 · u 1 = ρ · A2 · u 2 = ρ · A3 · u 3 d ⇒ · m = m = const. ˙ dtMichael Hölling, WS 2010/2011 slide 4
  5. 5. Wind Energy I Betz limit Extracted kinetic energy and extracted power A1, u1 A2, u2 A3, u3 1 Eext = m u1 − u3 2 2 2 1 u3 Pext = m u2 − u2 ˙ 1 optimal for max. Pext ???? 2 3 u1Michael Hölling, WS 2010/2011 slide 5
  6. 6. Wind Energy I Betz limit Substitution of u2 will lead to a function Pext(u1,u3) 1 Pext = · ρ · A2 · u 2 · u 1 − u 3 2 2 2 has to be substituted Thrust: T = m (u1 − u3 ) ˙ Corresponding power: Pthrust = m (u1 − u3 ) · u2 ˙ Equals mechanical power 1 that can be extracted: Pmech = Pext = m u2 − u2 ˙ 1 3 2 u1 + u2 ⇒ u2 = 2Michael Hölling, WS 2010/2011 slide 6
  7. 7. Wind Energy I Betz limit Extractable power as function of u1 and u3 1 1 u3 u2 u3 Pext = · ρ · A2 · u 1 · 3 · 1+ 3 3 − 2− 3 2 2 u1 u1 u1 wind power cp 0.6 cp max ?? 0.4 cp 0.2 0.0 0.0 0.5 1.0 u3/u1Michael Hölling, WS 2010/2011 slide 7
  8. 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 − x 2 3 2 1 !First derivative of cp(x): cp (x) = · 1 − 2x − 3x = 0 2 2For maximum second derivative of possible solution x1/2 1must smaller than zero: cp (x1/2 ) = · (−2 − 6x) < 0 2 1 16 Solution: x = ⇒ cpmax. (1/3) = ≈ 59% 3 27Michael Hölling, WS 2010/2011 slide 8
  9. 9. Wind Energy I Betz limit What happens at the rotor plane ?u1u2 2/3 u1u3 u(x) 1/3 u1 xMichael Hölling, WS 2010/2011 slide 9
  10. 10. Wind Energy I Betz limit What happens at the rotor plane ? p-2u1u2 2/3 u1u3 u(x) 1/3 u1 p(x) p(x)p1 p3 x Pressure drop at rotor plane p+2Michael Hölling, WS 2010/2011 slide 9
  11. 11. Wind Energy I Betz limit What does this results mean for wind energy converter ? There is an optimal rotational frequency that the blockage u3 1 results in = , characterized by tip speed ratio λ. u1 3Michael Hölling, WS 2010/2011 slide 10
  12. 12. Wind Energy I Betz limit What is the value for cp for a WEC over wind speed ? 1.2 1.2 P(u) / Pr 1.0 1.0 cp(u) / cpmax cp(u) / cpmaxP(u) / Pr 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0.0 0.0 0 10 20 30 u [m/s] Michael Hölling, WS 2010/2011 slide 11

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