Page 95 - Wind Energy Handbook

BLADE GEOMETRY 69

(see Equation 3.50a), giving
d 1 a
a ¼ (3:57)
da9 a9
From Equations (3.48) and (3.50a) a relationship between the flow induction factors
can be obtained. Dividing equations 3.48 and 3.50a

C l
tan ö 1
C d ºìa9(1 a)
¼ (3:58)
C l a(1 a) þ (a9ºì) 2
þ tan ö
C d
The flow angle ö is given by
1 a
tan ö ¼ (3:59)
ºì(1 þ a9)
Substituting Equation (3.59) into Equation (3.58) gives

C l 1 a
1
C d ºì(1 þ a9) ¼ ºìa9(1 a)
1 a 2
C l a(1 a) þ (a9ºì)
þ
C d ºì(1 þ a9)
Simplifying,

C l
(1 a) ºì(1 þ a9)
C d ºìa9(1 a)
¼
C l a(1 a) þ (a9ºì) 2
ºì(1 þ a9) þ (1 a)
C d

C l 2 C l
(1 a) ºì(1 þ a9) [a(1 a) þ (a9ºì) ] ¼ ºì(1 þ a9) þ (1 a) ºìa9(1 a)
C d C d
(3:60)

At this stage the process is made easier to follow if drag is ignored, Equation (3.60)
then reduces to

2 2
a(1 a) º ì a9 ¼ 0 (3:60a)
Differentiating Equation (3.60a) with respect to a9 gives

d 2 2
(1 2a) a º ì ¼ 0 (3:61)
da9

and substituting Equation (3.57) into (3.61)
2 2
(1 2a)(1 a) º ì a9 ¼ 0 (3:62)