Fluid Mechanics and Thermodynamics of Turbomachinery, 5e

360 Fluid Mechanics, Thermodynamics of Turbomachinery
of the product of the chord l and the lift coefficient C L (for C D = 0) to be determined.
By ascribing a value of CL at a given radius the corresponding value of l can be determined,
or conversely.
Example 10.10. A three-bladed HAWT, with a 30m tip diameter, is to be designed
for optimum conditions with a constant lift coefficient CL of unity along the span and
with a tip–speed ratio J = 5.0. Determine a suitable chord distribution along the blade,
from a radius of 3m to the blade tip, satisfying these conditions.
Solution. It is obviously easier to input values of j in order to determine the values
of the other parameters than attempting the reverse process. To illustrate the procedure,
choose j = 10deg so we determine jl = 0.0567, using eqn. (10.63). From eqn. (10.59)
we determine l = 3.73 and then find j = 3.733. Now
As
r
j J
c
r
= =
x R
r j
Ê ˆ
Ë ¯ =
W
1
\ = 3 = 1119 . m.
j J r
=
R
Ê ˆ
l
Ë ¯
5
15
ZlCL
J ZlCL l
= =
8prR8p 8p
after substituting J = 5, R = 15m, Z = 3, CL = 1.0. Thus,
l = 8p ¥ 0. 0567 = 1. 425 m
r
and Table 10.9 shows the optimum blade chord and radius values.
Figure 10.20 shows the calculated variation of blade chord with radius. The rapid
increase in chord as the radius is reduced would suggest that the blade designer would
ignore optimum conditions at some point and accept a slightly reduced performance.
A typical blade planform (for the Micon 65/13 HAWT, Tangler et al. 1990) is also
included in this figure for comparison.
The power output at optimum conditions
Equation (10.17) expresses the power output under general conditions, i.e. when the
rotational interference factor a¢ is retained in the analysis. From this equation the power
coefficient can be written as
TABLE 10.9. Values of blade chord and radius (optimum
conditions)
j (deg) j 4jl r (m) l (m)
30 1.00 0.536 3.0 3.368
20 1.73 0.418 5.19 2.626
15 2.42 0.329 7.26 2.067
10 3.733 0.2268 11.2 1.433
7.556 5 0.1733 15 1.089