Aerofólio S809 - 3 - s2 0 - b9781856177931000109 - main

Aerofólio S809 - 3 - s2 0 - b9781856177931000109 - main

(Parte 3 de 6)

From Newton’s laws of motion it is evident that the torque exerted on the turbine shaft must impart an equal and opposite torque on the airflow equal to the rate of change of the angular momentum of the flow. There is no rotation of the flow upstream of the blades or outside of the boundary stream tube.

According to Glauert, this rotational motion is to be ascribed partly to the system of trailing vortices and partly to the circulation around the blades. Due to the trailing vortices, the flow in the plane of the turbine blades will have an angular velocity a0Ω in the direction opposite to the blade rotation, and the circulation around the blades will cause equal and opposite angular velocities to the flows immediately upstream and downstream of the turbine blades. The sum of these angular velocity components, of

FIGURE 10.10 Schematic Drawing of the Vortex System Convecting Downstream of a Two-Bladed Wind Turbine Rotor

374 CHAPTER 10 Wind Turbines course, is zero upstream of the blades, because no rotation is possible until the flow reaches the vortex system generated by the blades. It follows from this that the angular velocity downstream of the blades is 2a0Ω and the interference flow, which acts on the blade elements, will have the angular velocity a0Ω. These deliberations will be of some importance when the velocity diagram for the turbine flow is considered (see Figure 10.1).

Glauert regarded the exact evaluation of the interference flow to be of great complexity because of the periodicity of the flow caused by the blades. He asserted that for most purposes it is sufficiently accurate to use circumferentially averaged values, equivalent to assuming that the thrust and the torque carried by the finite number of blades are replaced by uniform distributions of thrust and torque spread over the whole circumference at the same radius.

Consider such an elementary annulus of a HAWT of radius r from the axis of rotation and of radial thickness dr. Let dτ be the element of torque equal to the rate of decrease in angular momentum of the wind passing through the annulus. Thus,

(a) cx2



(a) Blade Element at Radius r Moving from Right to Left Showing the Various Velocity Components. The Relative

Velocity Impinging onto the Blade is w2 at Relative Flow Angle and Incidence Angle α. (b) The Various Force Components Acting on the Blade Section.

10.7 Blade Element Theory 375

In the actuator disc analysis the value of a (denoted by a) is a constant over the whole of the disc. With blade element theory the value of a is a function of the radius. This is a fact that must not be overlooked. A constant value of a could be obtained for a wind turbine design with blade element theory, but only by varying the chord and the pitch in some special way along the radius. This is not a useful design requirement.

Assuming the axial and tangential induction factors a and a0 are functions of r we obtain an expression for the power developed by the blades by multiplying the above expression by Ω and integrating from the hub rh to the tip radius R:

Forces Acting on a Blade Element

Consider now a turbine with Z blades of tip radius R each of chord l at radius r and rotating at angular speed Ω. The pitch angle of the blade at radius r is β measured from the zero lift line to the plane of rotation. The axial velocity of the wind at the blades is the same as the value determined from actuator disc theory, i.e., cx2 ¼ cx1(1 a), and is perpendicular to the plane of rotation. Figure 10.1 shows the blade element moving from right to left together with the velocity vectors relative to the blade chord line at radius r. The resultant of the relative velocity immediately upstream of the blades is, and this is shown as impinging onto the blade element at angle to the plane of rotation. It will be noticed that the tangential component of velocity contributing to w is the blade speed augmented by the interference flow velocity, a0Ωr. The following relations will be found useful in later algebraic manipulations:

Figure 10.1 shows the lift force L and the drag force D drawn (by convention) perpendicular and parallel to the relative velocity at entry, respectively. In the normal range of operation, D although rather small (1–2%) compared with L, is not to be entirely ignored. The resultant force, R, is seen as having a component in the direction of blade motion. This is the force contributing to the positive power output of the turbine.

376 CHAPTER 10 Wind Turbines

From Figure 10.1 the force per unit blade length in the direction of motion is Y ¼ Lsin Dcos , ð10:22Þ and the force per unit blade length in the axial direction is X ¼ Lcos þ Dsin . ð10:23Þ

Lift and Drag Coefficients We can define the lift and drag coefficients as

where, by the convention employed for an isolated aerofoil, w is the incoming relative velocity and l is the blade chord. The coefficients CL and CD are functions of the angle of incidence, α ¼ β,a s defined in Figure 10.1, as well as the blade profile and blade Reynolds number. In this chapter the angle of incidence is understood to be measured from the zero lift line (see Chapter 5, Section 5.15,

Lift Coefficient of a Fan Aerofoil) for which the CL versus α curve goes through zero. It is important to note that Glauert (1935), when considering aerofoils of small camber and thickness, obtained a the- oretical expression for the lift coefficient,

The theoretical slope of the curve of lift coefficient against incidence is 2π per radian (for small values of α) or 0.1 per degree but, from experimental results, a good average generally accepted is 0.1 per degree within the pre-stall regime. This very useful result will be used extensively in calculating results later. However, measured values of the lift-curve slope reported by Abbott and von Doenhoff (1959) for a number of NACA four- and five-digit series and NACA 6-series wing sections, measured at a Reynolds number of 6 106, gave 0.1 per degree. But, these blade profiles were intended for aircraft wings, so some departure from the rule might be expected when the application is the wind turbine.

Again, within the pre-stall regime, values of CD are small and the ratio of CD/CL is usually about 0.01. Figure 10.12 shows typical variations of lift coefficient CL plotted against incidence α and drag coefficient CD plotted against CL for a wind turbine blade tested beyond the stall state. The blades of a wind turbine may occasionally have to operate in post-stall conditions when CD becomes large; then the drag term needs to be included in performance calculations. Details of stall modelling and formulae for CD and CL under post-stall conditions are given by Eggleston and Stoddard (1987). The correct choice of aerofoil sections is very important for achieving good performance. The design details and the resulting performance are clearly competitive and not much information is actually available in the public domain. The U.S. Department of Energy (DOE) developed a series of aerofoils specifically for wind turbine blades. These aerofoils were designed to provide the necessarily different performance characteristics from the blade root to the tip while accommodating the structural requirements. Substantially increased energy output (from 10 to 35%) from wind turbines with these

10.7 Blade Element Theory 377

new blades have been reported. The data are catalogued and is available to the U.S. wind industry.2 Many other countries have national associations, research organisations, and conferences relating to wind energy and contact details are listed by Ackermann and Söder (2002).

Connecting Actuator Disc Theory and Blade Element Theory The elementary axial force and elementary force exerted on one blade of length dr at radius r are

For a turbine having Z blades and using the definitions for CL and CD given by eqns. (10.24) and (10.25), we can write expressions for the elementary torque, power and thrust as

It is now possible to make a connection between actuator disc theory and blade element theory.

(Values of a and a0 are allowed to vary with radius in this analysis.) From eqn. (10.2), for an element of the flow, we obtain

FIGURE 10.12 Typical Performance Characteristics for a Wind Turbine Blade, CL versus a and CD versus CL

2See Section 10.1, HAWT Blade Section Criteria, for more details.

378 CHAPTER 10 Wind Turbines

Equating eqns. (10.29) and (10.30) and with some rearranging, we get a=ð1 aÞ¼ ZlðCL cos þ CD sin Þ=ð8πr sin2 Þ. ð10:31Þ Again, considering the tangential momentum, from eqn. (10.16a) the elementary torque is dτ ¼ð 2πrdrÞρcx2ðrcθÞ. Equating this with eqn. (10.27) and simplifying, we get

Tip–Speed Ratio A most important non-dimensional parameter for the rotors of HAWTs is the tip–speed ratio, defined as

This parameter controls the operating conditions of a turbine and strongly influences the values of the flow induction factors, a and a0. Using eqn. (10.38) in eqn. (10.21) we write the tangent of the relative flow angle as

Turbine Solidity

A primary non-dimensional parameter that characterises the geometry of a wind turbine is the blade solidity, σ. The solidity is defined as the ratio of the blade area to the disc area:

10.7 Blade Element Theory 379 where

AB ¼ Z lrðÞ dr ¼ 1

2 Rlav.

This is usually written as σ ¼ Zlav=ð2πRÞ, ð10:40Þ where lav is the mean blade chord.

Solving the Equations

The foregoing analysis provides a set of relations which can be solved by a process of iteration , enabling a and a0 to be determined for any specified pitch angle β, provided that convergence is possible. To obtain faster solutions, we will use the approximation that ε ffi 0 in the normal efficient range of operation (i.e., the pre-stall range). Equations (10.35a) and (10.36a) can now be written as

These equations are about as simple as it is possible to make them and they will be used to model some numerical solutions.

Example 10.5 Consider a three-bladed HAWT with a rotor 30 m diameter, operating with a tip–speed ratio J¼5.0. The blade chord is assumed to be constant at 1.0 m. Assuming that the drag coefficient is negligible compared with the lift coefficient, determine using an iterative method of calculation the appropriate values of the axial and tangential induction factors at r/R¼ 0.95 where the pitch angle β is 2°.

Solution It is best to start the calculation process by putting a¼ a0 ¼ 0. The values, of course, will change progressively as the iteration proceeds. Thus, using eqn. (10.39),

380 CHAPTER 10 Wind Turbines

Using these new values of a and a0 in eqn. (10.39), the calculation is repeated iteratively until convergence is achieved, usually taking another four or five cycles (but in calculations where if a > 0.3, many more iterations will be needed). Finally, and with sufficient accuracy,

Also, ¼9.582° and CL¼0.1 ( β)¼0.758. It may be advisable at this point for the student to devise a small computer program (if facilities are available) or use a programmable hand calculator for calculating further values of a and a0. (Even a simple scientific calculator will yield results, although more tediously.) An outline of the algorithm, called the BEM method, is given in Table 10.1, which is intended to become an important and useful time-saving tool. Further extension of this method will be possible as the theory is developed.


All the theory and important definitions to determine the force components on a blade element have been introduced and a first trial approach has been given to finding a solution in Example 10.5. The various steps of the classical BEM model from Glauert are formalised in Table 10.1 as an algorithm for evaluating a and a0 for each elementary control volume.

Spanwise Variation of Parameters

Along the blade span there is a significant variation in the blade pitch angle β, which is strongly linked to the value of J and to a lesser extent to the values of the lift coefficient CL and the blade chord l.

Table 10.1 BEM Method for Evaluating a and a0 Step Action Required

1 Initialise a and a0 with zero values 2 Evaluate the flow angle using eqn. (10.39) 3 Evaluate the local angle of incidence, α¼ β

4 Determine CL and CD from tables (if available) or from formula 5 Calculate a and a0

6 Check on convergence of a and a0, if not sufficient go to step 2, else go to step 7 7 Calculate local forces on the element

10.8 The Blade Element Momentum Method 381

The ways both CL and l vary with radius are at the discretion of the turbine designer. In the previous example the value of the pitch angle was specified and the lift coefficient was derived (together with other factors) from it. We can likewise specify the lift coefficient, keeping the incidence below the angle of stall and from it determine the angle of pitch. This procedure will be used in the next example to obtain the spanwise variation of β for the turbine blade. It is certainly true that for optimum performance the blade must be twisted along its length with the result that, near the root, there is a large pitch angle. The blade pitch angle will decrease with increasing radius so that, near the tip, it is close to zero and may even become slightly negative. The blade chord in the following examples has been kept constant to limit the number of choices. Of course, most turbines in operation have tapered blades whose design features depend upon blade strength as well as economic and aesthetic considerations.

Example 10.6 At hree-bladedH AWTw ith a3 0mt ipd iameteri st ob ed esigned to operate with a constantl iftc oefficient

CL¼0.8 along the span, with a tip–speed ratio J¼5.0. Assuming a constant chord of 1.0 m, determine, using an iterative method of calculation, the variation along the span (0.2 r/R 1.0) of the flow induction factors a and a0 and the pitch angle β.

Solution We begin the calculation at the tip, r¼15 m and, as before, take initial values for a and a0 of zero. Now,

The results of the computations along the complete span (0.2 r/R 1.0) for a and a0 are shown in Table 10.2.I t is very evident that the parameter a varies markedly with radius, unlike the actuator disc application where a was constant. The spanwise variation of the pitch angle β for CL¼0.8 (as well as for CL¼1.0 and 1.2 for comparison) is shown in Figure 10.13. The large variation of β along the span is not surprising and is linked to the choice of the value of J, the tip–speed ratio. The choice of inner radius ratio r/R ¼ 0.2 was arbitrary. However, the contribution to the power developed from choosing an even smaller radius would have been negligible.

Note: CL¼0.8 along the span.

382 CHAPTER 10 Wind Turbines

Evaluating the Torque and Axial Force The incremental axial force can be derived from eqns. (10.29) and (10.19) in the form

and the incremental torque can be derived from eqns. (10.27) and (10.20) as

In determining numerical solutions, these two equations have proved to be more reliable in use than some alternative forms that have been published. The two preceding equations will now be integrated numerically.

(Parte 3 de 6)