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Unformatted text preview: S ustainable E nergy S cience and E ngineering C enter Wind Energy  Aerodynamics S ustainable E nergy S cience and E ngineering C enter One dimensional momentum theory Assumptions: Incompressible, inviscid, steady state flow Infinite number of blades Uniform thrust over the rotor area Non rotating wake The thrust T (equal and opposite to the force of the wind on the wind turbine) is given by Source: Wind Energy Explained by J.F Manwell, J.G. McGowan and A.L. Rogers, John Wiley, 2002. Wind Turbine Aerodynamics T = T = U 1 ρ UA ( ) 1 − U 4 ρ UA ( ) 4 Ý m = ρ UA ( ) 1 = ρ UA ( ) 4 T = Ý m U 1 − U 4 ( ) = A 2 p 2 − p 3 ( ) p 1 + 1 2 ρ U 1 2 = p 2 + 1 2 ρ U 2 2 p 3 + 1 2 ρ U 3 2 = p 4 + 1 2 ρ U 4 2 S ustainable E nergy S cience and E ngineering C enter One dimensional momentum theory Using the Bernoulli equation on either side of the rotor and assuming p 1 = p 4 Where a is the axial induction factor and U 1 a is referred to as the induced velocity at the rotor. The power output, P is given by T = 1 2 ρ A 2 U 1 2 − U 4 2 ( ) U 2 = U 1 + U 4 2 a = U 1 − U 2 U 1 P = 1 2 ρ A 2 U 1 2 − U 4 2 ( ) U 2 P P o = C p = P 1 2 ρ U 3 A C p max imum = 0.5926 (1a)U 1 (12a)U 1 P = 1 2 ρ AU 3 4 a (1 − a ) 2 U 1 = U ; A 2 = A C p = P 1 2 ρ AU 3 = 4 a (1 − a ) 2 dC p da = ⇒ a = 1 3 C p max = 16 27 = 0.5926 a < 0.5 Betz Limit S ustainable E nergy S cience and E ngineering C enter Betz Limit (2/3) U 1 (1/3) U 1 Maximum power production: The axial thrust on the disk at maximum power: T = 1 2 ρ AU 1 2 4 a 1 − a ( ) [ ] C T = T 1 2 ρ AU 2 = 8 9 Overall efficiency: η overall = P out 1 2 ρ AU 3 = η mech C P P out = 1 2 ρ AU 3 ( η mech C P ) S ustainable E nergy S cience and E ngineering C enter Ideal wind turbine with wake rotation Angular velocity of the rotor : Ω Angular velocity imparted to the flow stream: ω Angular induction factor: a` = ω/2Ω Blade tip speed : λ = Ω R / U Local Speed ratio: λ r = λ r / R Wake Rotation When 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. S ustainable E nergy S cience and E ngineering C enter Ω Ω+ω p 2 − p 3 = ρ Ω + 1 2 ω ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ω r 2 dT = p 2 − p 3 ( ) dA = ρ Ω + 1 2 ω ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ω r 2 2 π rdr ( ) ′ a = ω 2 Ω angular Induction factor The induced velocity at the rotor consists of not only the axial component Ua but also a component in the rotor plane r Ω a` Loss of Energy Due to Wake Rotation S ustainable E nergy S cience and E ngineering C enter dT = 4 ′ a (1 + ′ a ) 1 2 ρ Ω 2 r 2 2 π rdr Thrust on an annular cross section due to linear momentum: dT = 4 a (1 + a ) 1 2 ρ U 2 2 π rdr Equating the two expressions for thrust gives: a (1 − a ) ′ a (1 + ′ a ) = Ω 2 r 2 U 2 = λ r 2 Where...
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This note was uploaded on 10/22/2011 for the course EML 4450 taught by Professor Greska during the Fall '06 term at FSU.
 Fall '06
 Greska

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