Fluid Mechanics and Thermodynamics of Turbomachinery, 5e

340 Fluid Mechanics, Thermodynamics of Turbomachinery
f
f
b
a
b
U(1+a¢)
a
reaches the vortex system generated by the blades. It follows from this that the angular
velocity downstream of the blades is 2a¢W and the interference flow, which acts on the
blade elements, will have the angular velocity a¢W. These deliberations will be of some
importance when the velocity diagram for the turbine flow is considered (see Figure
10.11).
Glauert regarded the exact evaluation of the interference flow to be of great complexity
because of the periodicity of the flow caused by the blades. He asserted that for
most purposes it is sufficiently accurate to use circumferentially averaged values, equivalent
to assuming that the thrust and the torque carried by the finite number of blades
are replaced by uniform distributions of thrust and torque spread over the whole circumference
at the same radius.
Consider such an elementary annulus of a HAWT of radius r from the axis of rotation
and of radial thickness dr. Let dt be the element of torque equal to the rate of
decrease in angular momentum of the wind passing through the annulus. Thus,
2
dt = ( dm)◊ 2a¢ Wr =( 2prdrrc )◊ 2a¢
Wr
(10.16)
3
or dt = 4prWc ( 1-a)
a¢ r dr
(10.16a)
x1
w 2+
D
L
X
C x2
C q = 0
Y
R
(b)
x2
90°
(a)
2
w 2–
U(1+a¢)
C 2
C x2
C q = 2rWa¢
FIG. 10.11. (a) Blade element at radius r moving from right to left showing the various
velocity components. The relative velocity impinging onto the blade is w 2+ at relative
flow angle j and incidence angle a. (b) Showing the various force components acting
on the blade section.