Aerofólio S809 - 3 - s2 0 - b9781856177931000109 - main

Aerofólio S809 - 3 - s2 0 - b9781856177931000109 - main

(Parte 2 de 6)

Even small increases in wind speed can be important. It is shown later that the power extracted from the wind varies as the cube of the wind speed. Using this example the increase in the power extracted would be over 2% as a result ofincreasingthe hub heightfrom50m to 80m. Ofcourse there isa penalty as costs are likely to be greater for the stronger tower structure required.

Storing Energy

Because of the intermittency of wind and the unavailability at times of the required energy it is often claimed by opponents of wind turbines that it is better to rely on other sources of power. Clearly, some form of energy storage can be devised. In Spain, more than 13.8 GW of wind power capacity has been installed, providing about 10% of that country’s electicity needs, according to Renewable Energy World (September –October, 2009).A t Iberdrola, Spain, a pumped storage scheme (852 MW) is now being used to store the excess wind turbine energy and three further pumped storage plants are likely to be built with a total capacity of 1.64 GW.


In the following pages the aerodynamic theory of the HAWT is gradually developed, starting with the simple one-dimensional momentum analysis of the actuator disc and followed by the more detailed analysis of the blade element theory. The flow state just upstream of the rotor plane forms the so-called inflow condition for the rotor blades and from which the aerodynamic forces acting on the blades can be determined. The well-known blade element momentum (BEM) method is outlined and used extensively. A number of worked examples are included at each stage of development to illustrate the application of the theory. Detailed calculations using the BEM method were made to show the influence of various factors, such as the tip–speed ratio and blade number on performance. Further development of the theory includes the application of Prandtl’s tip loss correction factor, which corrects for a finite number of blades. Glauert’s optimisation analysis is developed and used to determine the ideal blade shape for a given lift coefficient and to show how the optimum rotor power coefficient is influenced by the choice of tip–speed ratio.



The concept of the actuator disc was used in Chapter 6 as a method of determining the three-dimensional flows in compressor and turbine blade rows. Betz (1926) in his seminal work on the flow through windmill blades used a much simpler version of the actuator disc. As a start to understanding the power production process of the turbine consider the flow model shown in Figure 10.5 wheret he rotoro ft he

364 CHAPTER 10 Wind Turbines

HAWT is replaced by an actuator disc. It is necessary to make a number of simplifying assumptions concerning the flow but, fortunately, the analysis yields useful approximate results.

Theory of the Actuator Disc The following assumptions are made:

(i) steady uniform flow upstream of the disc; (i) uniform and steady velocity at the disc; (i) no flow rotation produced by the disc; (iv) the flow passing through the disc is contained both upstream and downstream by the boundary stream tube; (v) the flow is incompressible.

Because the actuator disc offers a resistance to the flow the velocity of the air is reduced as it approaches the disc and there will be a corresponding increase in pressure. The flow crossing through the disc experiences a sudden drop in pressure below the ambient pressure. This discontinuity in pressure at the disc characterises the actuator. Downstream of the disc there is a gradual recovery of the pressure to the ambient value. We define the axial velocities of the flow far upstream (x → ∞), at the disc (x¼0) and far down- stream (x → ∞)a s cx1, cx2 and cx3, respectively. From the continuity equation the mass flow is where ρ¼air density and A2¼area of disc. The axial force acting on the disc is

X ¼ _mðcx1 cx3Þð 10:2Þ and the corresponding power extracted by the turbine or actuator disc is

Stream tube

Plane of disc

FIGURE 10.5 Actuator Disc and Boundary Stream Tube Model

10.4 Actuator Disc Approach 365

The rate of energy loss by the wind must then be

Assuming no other energy losses, we can equate the power lost by the wind to the power gained by the turbine rotor or actuator:


This is the proof developed by Betz (1926) to show that the velocity of the flow in the plane of the actuator disc is the mean of the velocities far upstream and far downstream of the disc. We should emphasise again that wake mixing, which must physically occur far downstream of the disc, has so far been ignored.

An Alternative Proof of Betz’s Result

The air passing across the disc undergoes an overall change in velocity (cx1 cx3) and a corresponding rate of change of momentum equal to the mass flow rate multiplied by this velocity change. The force causing this momentum change is equal to the difference in pressure across the disc times the area of the disc. Thus,

The pressure difference Δp is obtained by separate applications of Bernoulli’s equation to the two flow regimes of the stream tube. Referring to region 1–2i n Figure 10.5,

By taking the difference of the two equations we obtain ρðc2

Equating eqns. (10.6) and (10.7) we arrive at the result previously found,

366 CHAPTER 10 Wind Turbines

The axial flow induction factor is a. By combining eqns. (10.1) and (10.3), and from eqn. (10.5) we can obtain

It is convenient to define an axial flow induction factor, a (invariant with radius), for the actuator disc:

The Power Coefficient

For the unperturbed wind (i.e., velocity is cx1)w itht he same flow area as the disc (A2 ¼ πR2), the kinetic power available in the wind is

The maximum value of Cp is found by differentiating Cp with respect to a, i.e., finally dCp=da ¼ 4ð1 aÞð1 3aÞ¼ 0, which gives two roots, a ¼ 1=3 and 1.0. Using the first value, the maximum value of the power coefficient is

This value of Cp is often referred to as the Betz limit, referring to the maximum possible power coefficient of the turbine (with the prescribed flow conditions).

A useful measure of wind turbine performance is the ratio of the power coefficient Cp to the maximum power coefficient Cpmax. This ratio, which may be called the relative maximum power coefficient,i s

10.4 Actuator Disc Approach 367

The Axial Force Coefficient The axial force coefficient is defined as

By differentiating this expression with respect to a we can show that CX has a maximum value of unity at a ¼ 0:5. Figure 10.6 shows the variation of both Cp and CX as functions of the axial induction factor, a.

Example 10.1 Determine the static pressure changes that take place

(i) across the actuator disc; (i) up to the disc from far upstream; (i) from the disc to far downstream.

The pressure immediately before the disc is p2+. The pressure immediately after the disc is p2 .

Cp Cx

FIGURE 10.6 Variation of Power Coefficient CP and Axial Force Coefficient CX as Functions of the Axial Induction Factora

368 CHAPTER 10 Wind Turbines

Also, we have

Equating for power and simplifying, we get

This is the pressure change across the disc divided by the upstream dynamic pressure. For the flow field from far upstream of the disc,


For the flow field from the disc to far downstream,

and, noting that p3 ¼p1, we finally obtain

Figure 10.7 indicates approximately the way the pressure varies before and after the actuator disc.

FIGURE 10.7 Schematic of the Pressure Variation Before and After the Plane of the Actuator Disc

10.4 Actuator Disc Approach 369

Example 10.2

Determine the radii of the unmixed slipstream at the disc (R2) and far downstream of the disc (R3) compared with the radius far upstream (R1).

Solution Continuity requires that

Example 10.3 Using the preceding expressions for an actuator disc, determine the power output of a HAWT of 30 m tip diameter in a steady wind blowing at

(i) 7.5 m/s; (i) 10 m/s.

Assume that the air density is 1.2 kg/m3 and that a ¼ 1=3.

These two results give some indication of the power available in the wind.

Correcting for High Values ofa

It is of some interest to examine the theoretical implications of what happens at high values of a and compare this with what is found experimentally. From the actuator disc analysis we found that the velocity in the wake far downstream was determined by cx3 ¼ cx1ð1 – 2aÞ, and this becomes zero when a ¼ 0:5. In other words the actuator disc model has already failed as there can be no flow

370 CHAPTER 10 Wind Turbines when a ¼ 0:5. It is as if a large flat plate had been put into the flow, completely replacing the rotor. Some opinion has it that the theoretical model does not hold true for values of a even as low as 0.4. So, it becomes necessary to resort to empirical methods to include physical reality.

Figure 10.8 shows experimental values of CX for heavily loaded turbines plotted against a, taken from various sources, together with the theoretical curve of CX versus a given by eqn. (10.13). The part of this curve in the range 0:5 < a <1:0, shown by a broken line, is invalid as already explained. The experi- ments revealed that the vortex structure of the flow downstream disintegrates and that wake mixing with the surrounding air takes place. Various authors including Glauert (1935), Wilson and Walker (1976), and Anderson (1980), have presented curves to fit the data points in the regime a >0:5: Anderson obtained a simple “best fit” of the data with a straight line drawn from a point denoted by CXA located at a ¼ 1:0 to a tangent point T, the transition point, on the theoretical curve located at a ¼ aT. It is easy to show, by differentiation of the curve CX ¼ 4að1 aÞ then fitting a straight line, with the equation,


where aT ¼0.3262. Sharpe (1990) noted that, for most practical, existent HAWTs, the value of a rarely exceeds 0.6.

CXA 1.6

FIGURE 10.8 Comparison of Theoretical Curve and Measured Values of CX

10.4 Actuator Disc Approach 371


Preliminary estimates of rotor diameter can easily be made using simple actuator disc theory. A number of factors need to be taken into account, i.e., the wind regime in which the turbine is to operate and the tip–speed ratio. Various losses must be allowed for, the main ones being the mechanical transmission including gearbox losses and the electrical generation losses. From the actuator disc theory the turbine aerodynamic power output is

Under theoretical ideal conditions the maximum value of Cp¼0.593. According to Eggleston and

Stoddard (1987), rotor Cp values as high as 0.45 have been reported. Such high, real values of Cp relate to very precise, smooth aerofoil blades and tip–speed ratios above 10. For most machines of good design a value of Cp from 0.3 to 0.35 would be possible. With a drive train efficiency, ηd, and an electrical generation efficiency, ηg, the output electrical power would be

ρA2Cpηgηdc3 x1.

Example 10.4 Determine the size of rotor required to generate 20 kW of electrical power in a steady wind of 7.5 m/s. It can be

Solution From this expression the disc area is

Hence, the diameter is 21.2 m.

10.6 POWER OUTPUT RANGE The kinetic power available in the wind is

where A2 is the disc area and cx1 is the velocity upstream of the disc. The ideal power generated by the turbine can therefore be expected to vary as the cube of the wind speed. Figure 10.9 shows the idealised power curve for a wind turbine, where the preceding cubic “law” applies between the so-called cut-in wind speed and the rated wind speed atwhich the maximum power is first reached.The cut-inspeed isthe lowest wind speed at which net (or positive) power is produced by the turbine. The rated wind speed generally corresponds to the point at which the efficiency of energy conversion is close to its maximum.

372 CHAPTER 10 Wind Turbines

At wind speeds greater than the rated value, for most wind turbines, the power output is maintained constant by aerodynamic controls [discussed in Section 10.13, Control Methods (Starting, Modulating, and Stopping)]. The cut-out wind speed is the maximum permitted wind speed which, if reached, causes the control system to activate braking, bringing the rotor to rest.



It has long been recognised that the work of Glauert (1935) in developing the fundamental theory of aerofoils and airscrews is among the great classics of aerodynamic theory. Glauert also generalised the theory to make it applicable to wind turbines and, with various modifications, it is still used in turbine design. It is often referred to as the momentum vortex blade element theory or more simply as the blade element method. However, the original work neglected an important aspect: the flow periodicity resulting from the turbine having a finite number of blades. Glauert assumed that elementary radial blade sections could be analysed independently, which is valid only for a rotor with an infinite number of blades. However, several approximate solutions are available (those of Prandtl and Tietjens, 1957, and Goldstein, 1929), which enable compensating corrections to be made for a finite number of blades. The simplest and most often used of these, called the Prandtl correction factor, will be considered later in this chapter. Another correction that is considered is empirical and applies only to heavily loaded turbines when the magnitude of the axial flow induction factor a exceeds the acceptable limit of the momentum theory. According to Sharpe (1990) the flow field of heavily loaded turbines is not well understood and the results of the empirical analysis mentioned are only approximate but better than those predicted by the momentum theory.

The Vortex System of an Aerofoil

To derive a better understanding of the aerodynamics of the HAWT than was obtained earlier from simple actuator disc theory, it is now necessary to consider the forces acting on the blades. We may

10 Wind speed, m/s

Cut-in wind speed

Cut-outwind speed Rated wind speed

FIGURE 10.9 Idealised Power Curve for a Wind Turbine

10.7 Blade Element Theory 373 regard each radial element of a blade as an aerofoil. The turbine is assumed to have a constant angular speed Ω and is situated in a uniform wind of velocity cx1 parallel to the axis of rotation. The lift force acting on each element must have an associated circulation (see, in section 3.4, Circulation and Lift) around the blade. In effect there is a line vortex (or a set of line vortices) along the aerofoil span. The line vortices that move with the aerofoil are called bound vortices of the aerofoil. As the circulation along the blade length can vary, trailing vortices will spring from the blade and will be convected downstream with the flow in approximately helical paths, as indicated for a two-bladed wind turbine in Figure 10.10. It will be observed that the helices, as drawn, gradually expand in radius as they move downstream (at the wake velocity) and the pitch between each sheet becomes smaller because of the deceleration of the flow (see Figure 10.5).

Torque, τ and the Tangential Flow Induction Factor, a0

(Parte 2 de 6)