Page 71 - Fluid Power Engineering

Aerodynamics of W ind T urbine Blades 49

where a is the axial flow induction factor, v 1 is the wind speed at the
rotor, and v 0 is the wind speed upstream (see Eq. 2-29).
The final tangential speed of air is expressed in terms of the tan-
gential flow induction factor b:

v t = 2ωrb (4-7)

The tangential speed of air at the middle of the rotor is one-half of the
final tangential speed. The torque and power because of wind blowing
through the annulus ring is:

δQ = δ ˙mv t r = δ ˙m(2ωrb)r (4-8)
2
δP = δQω = 2δ ˙m(ωr) b (4-9)
λ r = ( ωr ) is called the local speed ratio, and λ = ( ωR ) is called the tip
v 0 v 0
speed ratio, where R is the length of the blade.
1

2 2 3 2
δP = 2 (ρv 0 (1 − a) 2πrδr) v λ b = ρv 2πrδr 4 (1 − a) λ b (4-10)
0 r
r
0
2
The term in the square parenthesis is the power contained in an an-
nulus of air upstream. Note that in the above equations a and b may
be functions of r. The efficiency of power capture in this annulus is
defined as:
δP 2
= 4 (1 − a) λ b (4-11)
3
1 ρv 2πrδr
ε = r
2 0
Maximum efficiency will require:
dε da 2 2
= 0 =− λ b + (1 − a) λ r
r
db db
(4-12)
da 1 − a
=
db b
The relationship between variables a and b in Eq. (4-12) will hold true
at maximum efficiency. Power coefficient function (C P ) can be written
using Eq. (4-10) and dividing by the total power contained in disk of
radius R.

2
δP 8r(1 − a)λ bδr
r
=
3
1 ρv πR 2 R 2
δC P =
2 0
(4-13)
2
δC P 8r(1 − a)λ b
r
=
δr R 2