Environmental Engineering Reference
In-Depth Information
Fig. 5.15 Optimum chord
and twist compared to
Anderson et al. [ 5 ] blade
Equation (5.12a) with = 6
Equation (5.12b) with = 6
Actual blade
0.2
0.1
30
20
Equation (5.13a) with = 6
Equation (5.13b) with = 6
Actual blade
10
0
-10
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Radius, r/R
C l , is usually taken as C l,max , the value at which l/d is maximised since drag has
been ignored in obtaining ( 4.12b ). Equation 5.11 immediately gives the chord at
each radius once the number of blades and the tip speed ratio have been chosen.
Equation 5.12a agrees (at high k) with the more general, and more difficult to
derive, ''optimal chord'' Eq. 3.67a of Burton et al. [ 13 ].
16p
cC l ¼
r
ð 5 : 12b Þ
4 . 9 þ k r þ 2 = 9k ðÞ
2
9Nk
½
Ignoring a 0
on the grounds that it is small, Eq. 3.12 becomes
tan / 2 = 3k ðÞ
5 : 13a Þ
which determines /. Knowing the angle at which the chosen C l occurs, then gives
h P by Eq. 3.8 . Equation 5.13a is a high-k approximation to Eq. 3.68a of Burton
et al. [ 13 ]:
2
3k r þ 2 = k r
tan / ¼
ð 5 : 13b Þ
Once the designer has selected the aerofoil profile for the blade, the tip speed
ratio and tip radius, Eqs. 5.12 and 5.13 can be used to complete the blade aero-
dynamic design.
An example of the use of Eqs. 5.12 and 5.13 is shown in Fig. 5.15 which
compares the equations with the output from tcdist.m with the NACA 4412
section, U 0 = 10 m/sec, and k = 10. As shown above, power_calc.m gave the
best performance at k = 10 with C P over 0.46. The tip Re is just over 3 9 10 5 .
Figure 5.1 indicates that C l,max will occur at a = 6 and the data (not shown but
can be viewed in n4412.in ) gives C l,max & 0.90. The chord and twist of the actual
blade closely follow the theoretical optimum curves, with a discrepancy of about
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