Fluid Mechanics and Thermodynamics of Turbomachinery, 5e
Fluid Mechanics and Thermodynamics of Turbomachinery, 5e
Fluid Mechanics and Thermodynamics of Turbomachinery, 5e
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342 <strong>Fluid</strong> <strong>Mechanics</strong>, <strong>Thermodynamics</strong> <strong>of</strong> <strong>Turbomachinery</strong><br />
Lift <strong>and</strong> drag coefficients<br />
We can define the lift <strong>and</strong> drag coefficients as<br />
( )= ( )<br />
1 2<br />
CLa L rw<br />
l<br />
( )= ( )<br />
1 2<br />
CDa D rw<br />
l<br />
2<br />
2<br />
(10.24)<br />
(10.25)<br />
where, by the convention employed for an isolated aer<strong>of</strong>oil, w is the incoming relative<br />
velocity <strong>and</strong> l is the blade chord. The coefficients CL <strong>and</strong> CD are functions <strong>of</strong> the angle<br />
<strong>of</strong> incidence, a = j - b, as defined in Figure 10.11, as well as the blade pr<strong>of</strong>ile <strong>and</strong><br />
blade Reynolds number. In this chapter the angle <strong>of</strong> incidence is understood to be measured<br />
from the zero lift line (see Chapter 5, in connection with “Lift coefficient <strong>of</strong> a fan<br />
aer<strong>of</strong>oil”) for which the CL vs a curve goes through zero. It is important to note that<br />
Glauert (1935, 1976), when considering aer<strong>of</strong>oils <strong>of</strong> small camber <strong>and</strong> thickness,<br />
obtained a theoretical expression for the lift coefficient,<br />
CL = 2psina (10.26)<br />
The theoretical slope <strong>of</strong> the curve <strong>of</strong> lift coefficient against incidence is 2p per radian<br />
(for small values <strong>of</strong> a) or 0.11 per degree but, from experimental results, a good average<br />
generally accepted is 0.1 per degree within the pre-stall regime. This very useful result<br />
will be used extensively in calculating results later. However, measured values <strong>of</strong> the<br />
lift-curve slope reported by Abbott <strong>and</strong> von Doenh<strong>of</strong>f (1959) for a number <strong>of</strong> NACA<br />
four- <strong>and</strong> five-digit series <strong>and</strong> NACA 6-series wing sections, measured at a Reynolds<br />
number <strong>of</strong> 6 ¥ 10 6 , gave 0.11 per degree. But, these blade pr<strong>of</strong>iles were intended for<br />
aircraft wings, so some departure from the rule might be expected when the application<br />
is the wind turbine.<br />
Again, within the pre-stall regime, values <strong>of</strong> CD are small <strong>and</strong> the ratio <strong>of</strong> CD/CL is<br />
usually about 0.01. Figure 10.12 shows typical variations <strong>of</strong> lift coefficient CL plotted<br />
against incidence a <strong>and</strong> drag coefficient CD plotted against CL for a wind turbine blade<br />
tested beyond the stall state. The blades <strong>of</strong> a wind turbine may occasionally have to<br />
C L<br />
1.2<br />
0.8<br />
0.4<br />
0.015<br />
0.010<br />
0.005<br />
0 8 16 –0.8 –0.4 0 0.4 0.8<br />
C D<br />
a (a ≥ 0) C L<br />
FIG. 10.12. Typical performance characteristics for a wind turbine blade, C L vs a <strong>and</strong><br />
C D vs C L.<br />
1.2
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