Fluid Mechanics and Thermodynamics of Turbomachinery, 5e

It is convenient to define an axial flow induction factor, a – (invariant with radius), for
the actuator disc,
Hence,
The power coefficient
(10.9)
(10.10)
For the unperturbed wind (i.e. velocity is c x1) with the same flow area as the disc (A2
= pR 2 ), the kinetic power available in the wind is
1 P c Ac Ac
2
1 3
= ( r )= r .
0
2
a = ( c -c
) c
P= 2a A c ( 1-a)
3
cx2 = cx1( 1-a)
r
A power coefficient Cp is defined as
Cp = P P0= 4a( 1 -a)
.
(10.11)
The maximum value of Cp is found by differentiating Cp with respect to a – , i.e. finally
dCp da
= 41 ( -a)
( 1-3a)= 0
which gives two roots, a – = 1/3 and 1.0. Using the first value, the maximum value of
the power coefficient is
C pmax = 16 27 = 0. 593.
(10.12)
This value of Cp is often referred to as the Betz limit, referring to the maximum
possible power coefficient of the turbine (with the prescribed flow conditions).
A useful measure of wind turbine performance is the ratio of the power coefficient
CP to the maximum power coefficient CPmax. This ratio, which may be called the relative
maximum power coefficient, is
z = 27 16C P.
x12 x1 2 2 x1
The axial force coefficient
The axial force coefficient is defined as
1 C X c A 2 r
= ( )
X 2 x1
x1 x2 x1
2 x1
( ) ( )
1
= mc -ccA
2
2 ˙
r
x1 x2 2 x1
x2 x1 x2 x1
2
= 4c
( c -c
) c
= 4a( 1-a)
(10.12a)
(10.13)
By differentiating this expression with respect to a – we can show that CX has a maximum
value of unity at a – = 0.5. Figure 10.6 shows the variation of both Cp and C X as functions
of the axial induction factor, a – .
Example 10.1. Determine the static pressure changes that take place
(i) across the actuator disc;
(ii) up to the disc from far upstream;
(iii) from the disc to far downstream.
2
2
2
2
Wind Turbines 333