Blade Element Calculations - Small Wind Turbines: Analysis, Design, and Application - page 96

Fig. 5.14 Normalised
circulation for k = 8, 10, and
12

1.1

= 8
= 10
= 12

1

N / = 8/9

0.9

0.8

0.7

0.6

0.5

0 0.2 0.4 0.6 0.8 1

radius, r/R

U 1 ¼ 1 NC
2pp 1

ð 5 : 8 Þ

where the second term on the right is subtracted for consistency with the con-
vention that the circulation is positive. Now assume that the velocity of the tip
vortices in the direction of the wind is the average of the wake velocity and the
wind speed. Thus

p 1 ¼ 1 þ U 1
2k

ð 5 : 9 Þ

which seems entirely reasonable, but is hard to prove, see [ 11 , 12 ], and is only
approximately true at high k. From ( 5.8 ) and ( 5.9 ),

r

1 NCk
p

U 1 ¼ ð 5 : 10 Þ

At the Betz-Joukowsky limit, the circulation is given by
NCk = p ¼ 8 = 9 ð 5 : 11 Þ

It is worth noting that ( 5.11 ) can be derived from the different starting point of
conservation of angular momentum, see Eq. 5.44 of Spera [ 6 ] and Exercise 5.2.
For a given N,( 5.11 ) gives the target C to achieve maximum performance for
any k. For comparison with the data in Fig. 5.4 , for example, Eq. 5.11 in com-
bination with ( 4.12 b) gives NC = 0.348, 0.278, and 0.232 near the tip for k = 8,
10, and 12 respectively. By comparing these values to those shown in Fig. 5.4
redrawn in Fig. 5.14 , it is clear that the best performance occurs at k = 10 for the
NACA 4412 section.
If ( 5.11 ) is combined with Eq. 4.12b , it is easy to show that

16p
9Nk 2 r

16p
9Nkk r

cC l

¼ ð 5 : 12a Þ