Page 114 - Wind Energy Handbook

88 AERODYNAMICS OF HORIZONTAL-AXIS WIND TURBINES

angular momentum to the non-dimensional rate of change of axial momentum, is
not changed because it deals with the whole flow through the disc and so uses
average values. If drag is ignored for the present, Equation (3.59) becomes
ºìa9(1 a)
tan ö ¼ (3:81)
a(1 a) þ (a9ºì) 2

Hence
a
1
(1 a)ºìa9 ¼ f
a(1 a) þ (ºìa9) 2 a9
ºì 1 þ
f
which becomes

( f 1) a9
2 2
2
2 2
º ì a9 º ì (1 a)a9 þ a(1 a)1 ¼ 0 (3:82)
f f
A great simplification can be made to Equation (3.82) by ignoring the first term
because, clearly, it disappears for much of the blade, where f ¼ 1, and for the tip
2
region the value of a9 is very small. For tip speed ratios greater than 3 neglecting
the first term makes negligible difference to the result.
a
2 2
º ì a9 ¼ a 1 (3:83)
f

As before, Equation (3.57) still applies
d 1 a
a ¼
da9 a9

From Equation (3.83)

d 1 a
a9 ¼ 1 2
2 2
da º ì f
Consequently
a
2 2
(1 a)1 2 ¼ º ì a9
f
which combined with Equation (3.83) gives

2 1
2
a (f þ 1)a þ f ¼ 0
3 3
so