EML4450L21

EML4450L21 - S ustainable E nergy S cience and E ngineering...

Info iconThis preview shows pages 1–8. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 UA ( ) 1 U 4 UA ( ) 4 m = UA ( ) 1 = UA ( ) 4 T = m U 1 U 4 ( ) = A 2 p 2 p 3 ( ) p 1 + 1 2 U 1 2 = p 2 + 1 2 U 2 2 p 3 + 1 2 U 3 2 = p 4 + 1 2 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 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 A 2 U 1 2 U 4 2 ( ) U 2 P P o = C p = P 1 2 U 3 A C p max imum = 0.5926 (1-a)U 1 (1-2a)U 1 P = 1 2 AU 3 4 a (1 a ) 2 U 1 = U ; A 2 = A C p = P 1 2 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 AU 1 2 4 a 1 a ( ) [ ] C T = T 1 2 AU 2 = 8 9 Overall efficiency: overall = P out 1 2 AU 3 = mech C P P out = 1 2 AU 3 ( 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 : Angular velocity imparted to the flow stream: Angular induction factor: a` = /2 Blade tip speed : = R / U Local Speed ratio: r = 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 + p 2 p 3 = + 1 2 r 2 dT = p 2 p 3 ( ) dA = + 1 2 r 2 2 rdr ( ) a = 2 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 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 2 r 2 2 rdr Thrust on an annular cross section due to linear momentum: dT = 4 a (1 + a ) 1 2 U 2 2 rdr Equating the two expressions for thrust gives: a (1 a ) a (1 + a ) = 2 r 2 U 2 = r 2 Where...
View Full Document

Page1 / 34

EML4450L21 - S ustainable E nergy S cience and E ngineering...

This preview shows document pages 1 - 8. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online