Fluid Mechanics and Thermodynamics of Turbomachinery, 5e

In the actuator disc analysis the value of a (denoted by a – ) is a constant over the whole
of the disc. With blade element theory the value of a is a function of the radius.
This is a fact which must not be overlooked. A constant value of a could be
obtained for a wind turbine design with blade element theory but only by varying the
chord and the pitch in some special way along the radius. This is not a useful design
requirement.
Assuming the axial and tangential induction factors a and a¢ are functions of r we
obtain an expression for the power developed by the blades by multiplying the above
expression by W and integrating from the hub rh to the tip radius R,
2
3
P= 4prW c ( 1-a)
a¢ r d r.
x1
Forces acting on a blade element
(10.17)
Consider now a turbine with Z blades of tip radius R each of chord l at radius r and
rotating at angular speed W. The pitch angle of the blade at radius r is b measured from
the zero lift line to the plane of rotation. The axial velocity of the wind at the blades is
the same as the value determined from actuator disc theory, i.e. cx2 = cx1(1 - a), and
is perpendicular to the plane of rotation.
Figure 10.11 shows the blade element moving from right to left together with the
velocity vectors relative to the blade chord line at radius r.
The resultant of the relative velocity immediately upstream of the blades is,
2 2 2 2
{ x1
}
05 .
w = c ( 1-a) + ( Wr)
( 1+
a¢
)
(10. 18)
and this is shown as impinging onto the blade element at angle j to the plane of
rotation. It will be noticed that the tangential component of velocity contributing to w
is the blade speed augmented by the interference flow velocity, a¢Wr. The following
relations will be found useful in later algebraic manipulations
sinj = c w = c ( -a)
w
rh
x2 x1
1
cosj = Wr( 1+
a¢ ) w
(10.19)
(10.20)
cx1
Ê 1-
a ˆ
tanj =
(10.21)
Wr
Ë 1+
a¢
¯
Figure 10.11 shows the lift force L and the drag force D drawn (by convention)
perpendicular and parallel to the relative velocity at entry respectively. In the normal
range of operation, D although rather small (1 to 2%) compared with L, is not to be
entirely ignored. The resultant force, R, is seen as having a component in the direction
of blade motion. This is the force contributing to the positive power output of the
turbine.
From Figure 10.11 the force per unit blade length in the direction of motion is
Y = Lsinj -Dcos
j,
and the force per unit blade length in the axial direction is
X = Lcosj+ Dsin
j.
Ú
R
Wind Turbines 341
(10.22)
(10.23)