Energy2green Wind And Solar Power System
If wind, which circulates around a body, develops higher flow speeds along the upper surface than along the lower, an overpressure emerges at the upper surface and an underpressure at the lower. The result is a buoyancy force, according to Bernoulli:
The buoyancy force is calculated using the lift coefficient cL, the air density p, the apparent wind speed vA and the projected body area AP. Rotor blades of modern wind generators usually make use of the buoyancy force. The projected area
of a rotor blade is defined by the chord t and span that is approximately equal to the rotor radius r.
Section B-C
Figure 5.7 Apparent Wind Speed vA Resulting from the Real Wind Speed vW and Rotor Motion
Section B-C
Figure 5.7 Apparent Wind Speed vA Resulting from the Real Wind Speed vW and Rotor Motion
Drag forces, which have been described in the section about drag devices (see p191), also have effects on lift devices:
However, the buoyancy force on a drag device is much higher than the drag force. The ratio of both forces is called the lift-drag ratio e:
Some references also use the inverse ratio. Good rotor profiles can reach liftdrag ratios of up to 400.
The apparent wind speed:
used in the equations above is calculated from the real wind speed vW and the circumferential speed u (see Figure 5.7). With the tip speed ratio A = u/vw, the apparent wind speed becomes:
Figure 5.8 shows the ratio of the drag force FD to the buoyancy force FL. Vector addition of both forces provides the resultant force:
The resultant force can be subdivided into an axial component FRA and a tangential component FRt. The tangential component FRt of the resultant force causes the rotor to turn.
Chord
Figure 5.8 Ratio of the Forces for a Lift Device
Chord
Lift coefficient cL and drag coefficient cD vary significantly with the angle of attack a in For a < 10°, the following approximation can be used (Gasch and Twele, 2002):
Pitching the rotor blade, i.e. changing the pitch angle ûthat is shown in Figure 5.8, influences the angle of attack a and therefore the power coefficient cp. The maximum of the power coefficient decreases significantly at high pitch angles and occurs at lower tip speed ratios. Pitch-controlled turbines use this effect: during starting of the wind turbine, high pitch angles are chosen. Pitching of the rotor blades can also limit the power at very high wind speeds. The drag coefficient cD can be neglected at angles of attack lower than 15°. A lift device takes power Pt from the wind. With the power coefficient cp and the power content P0, the taken power is given by:
The resultant force causes torque M. With the torque becomes:
The torque M with the associated moment coefficient
Source: data from Vestas, 1997
Figure 5.9 Power Coefficient cP as a Function of the Tip Speed Ratio X for the Vestas V44-600-kW Wind Generator
Source: data from Vestas, 1997
Figure 5.9 Power Coefficient cP as a Function of the Tip Speed Ratio X for the Vestas V44-600-kW Wind Generator
Figure 5.10 Power Coefficients and Approximations using Third-degree Polynomials and A = n r2, can be also described as follows:
If the torque M or power P of a wind generator as a function of the wind speed vW is known, the power coefficient at constant speed can be calculated. Figure 5.9 shows the characteristics of the power coefficient and the tip speed
Table 5.6 Parameters for the Description of the Power Coefficient Curves in
Figure 5.10
Table 5.6 Parameters for the Description of the Power Coefficient Curves in
Figure 5.10
a3 |
a2 |
ai |
ao | |
Curve 1 |
0.00094 |
-0.0353 |
0.3841 |
-0.8714 |
Curve 2 |
0.00068 |
-0.0297 |
0.3531 |
-0.7905 |
Note: Curve 1 = lower curve; curve 2 = upper curve
Note: Curve 1 = lower curve; curve 2 = upper curve ratio of a 600-kW wind generator. The maximum power coefficient of 0.427 is much closer to the Betz power coefficient than that achieved by a drag device (see section on power content of wind, p188).
The calculation of the power coefficient curve is very difficult and is only possible when considering complex aerodynamic conditions along the rotor blades. Therefore, the dependence of the power coefficient on the tip speed ratio is usually estimated by measurement. A third-degree polynomial can approximately describe the curve of the power coefficient:
Figure 5.10 Power Coefficients and Approximations using Third-degree Polynomials
Tip spaed ratio X
Tip spaed ratio X
The coefficients a3 to a0 can be estimated with programs such as Matlab or MS-Excel from measurements. Figure 5.10 shows two real power coefficient curves and their approximation by third-degree polynomials. Table 5.6 shows the parameters of both curves.
The previous section explained in general how drag and lift devices can utilize wind power. This section describes technical solutions for this utilization. In the past, wind energy was mainly converted to mechanical energy; some modern wind pumping systems also use the mechanical energy directly.
However, today the generation of electricity is in higher demand; therefore, a wind rotor drives an electrical generator. Different concepts exist for the rotor design and are explained in the following sections.
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Renewable energy is energy that is generated from sunlight, rain, tides, geothermal heat and wind. These sources are naturally and constantly replenished, which is why they are deemed as renewable.