Page 109 - Wind Energy Handbook

THE EFFECTS OF A DISCRETE NUMBER OF BLADES 83

any radial motion other than the general expansion caused by the slowing down of
the flow. The fourth particle is totally unaffected by the fact that there is a finite
number of blades and follows the same progress as if it were passing through a
uniform actuator disc.
The axial flow induction factor varies, therefore, not only azimuthally but also
radially, is a function of both r and Ł. The azimuthally averaged value of a(r) ¼
a b (r)f(r), where f(r) is known as the tip-loss factor, has a value of unity inboard
(particle 4) and falls to zero at the edge of the rotor disc. The value a b (r) is the level
of axial flow induction factor that occurs locally at a blade element and is the
velocity with which the vortex sheet convects downstream. If a b (r) can be held
radially uniform then the vortex sheets will be radially flat, as shown in Figure 3.31,
but if a b (r) is not uniform the vortex sheets will warp.
In the application of the blade element–momentum theory it is argued that the
rate of change of axial momentum is determined by the azimuthally averaged value
of axial flow induction factor, whereas the blade forces are determined by the value
of the flow factor which occurs locally at the blade element, that experienced by the
first and second particles.
The mass flow rate through an annulus ¼ rU 1 (1 a b (r)f(r))2ðrär

The azimuthally averaged overall change of axial velocity ¼ 2a b (r)f(r)U 1
2
The rate of change of axial momentum ¼ 4ðrrU (1 a b (r) f(r))a b (r) f(r)är
1
1 1
2 2
The blade element forces ¼ rW NcC l and eW N c C d
2 2
where W, C l and C d are determined using a b (r).
The pressure force caused by the rotation of the wake is also calculated using an
azimuthally averaged value of the tangential flow induction factor 2a9 b (r) f(r).

3.8.3 Prandtl’s approximation for the tip-loss factor

The function for the tip-loss factor f(r) is shown in Figure 3.29 for a blade with
uniform circulation operating at a tip speed ratio of 6 and is not readily obtained by
analytical means for any desired tip speed ratio. Sidney Goldstein did analyse the
tip-loss problem for application to propellers in 1929 and achieved a solution in
terms of Bessel functions but neither that nor the Biot–Savart solution used above is
suitable for inclusion in the blade element–momentum theory. Fortunately, in 1919,
Ludwig Prandtl (reported by Betz, 1919) had already developed an ingenious
approximate solution which does yield a relatively simple analytical formula for
the tip-loss function.
Prandtl’s approximation was inspired by the fact that, being impermeable
(particles one and three in the above description pass around the outer edge of a
sheet but not through it), the vortex sheets could be replaced by material sheets
which, provided they move with the velocity dictated by the wake, would have no