Page 87 - Fluid Power Engineering

Advanced Aerodynamics of W ind T urbine Blades 65

The lift and drag forces on a blade element δr of chord length c
and turbine with N blades are:
1 2
δL = ρv C L cNδr (5-3)
res
2
1 2
δD = ρv C D cNδr (5-4)
res
2
In the above equations, cNδr is the effective area or blade solidity in
an annulus of thickness δr. The direction of δL is perpendicular to
vector v res ; δD is parallel to v res . Force along the axial direction is:

1 2
δL cos β + δD sin β = ρv Nc(C L cos β + C D sin β)δr (5-5)
res
2
Torque is the tangential force multiplied by radius:
1 2
δQ = r(δL sin −δD cos β) = ρv Nc(C L sin β − C D cos β)rδr (5-6)
res
2
Assuming that the axial force is due to change in axial momentum of
1
air, that is, there is no radial interaction, then:
1 2 1 2
ρv Nc(C L cos β + C D sin β)δr = δ ˙m2av 0 + ρ(2bωr) 2πrδr (5-7)
res
2 2
In the above equation, axial force caused by pressure drop due to
wake rotation is added (last term on the right-hand side of Eq. 5-7).
The force because of pressure drop is in Eq. (4-18). The value of δ ˙m
is in Eq. (4-6). The following substitutions are made to simplify the
equations:

C x = (C L cos β + C D sin β) (5-8)
C y = (C L sin β − C D cos β) (5-9)
(5-10)
Tip speed ratio, λ = ωR/v 0
μ = r/R (5-11)
Nc
Blade solidity, σ = (5-12)
2πr
The axial force Eq. (5-7) becomes:

2

v res 2
σC x = 4(a(1 − a) + (bμλ) ) (5-13)
v 0
1 − a 2
2
σC x = 4(a(1 − a) + (bμλ) ) (5-14)
sin β