Aerofólio S809 - 3 - s2 0 - b9781856177931000109 - main

Aerofólio S809 - 3 - s2 0 - b9781856177931000109 - main

(Parte 4 de 6)

Example 10.7 Determine the total axial force, the torque, the power, and the power coefficient of the wind turbine described in

Solution Evaluating the elements of axial force ΔX having previously determined the mid-ordinate values of a, a0, and to gain greater accuracy (the relevant data is shown in Table 10.3):

(deg) r R

FIGURE 10.13 Variation of Blade Pitch Angle β with Radius Ratio r/R for CL¼0.8, 1.0 and 1.2 (See Example 10.6 for Conditions)

10.8 The Blade Element Momentum Method 383

Hence, the power developed is P¼τΩ¼67.644 kW. The power coefficient is, see eqn. (10.1),i s

and the relative power coefficient is, see eqn. (10.12b),

384 CHAPTER 10 Wind Turbines

Example 10.8 The relationship between actuator disc theory and blade element theory can be more firmly established by evaluating the power again, this time using the actuator disc equations.

Solution To do this we need to determine the equivalent constant value for a. From eqn. (10.13),

Solving the quadratic equation, we get a ¼ 0:12704.

and this agrees fairly well with the value obtained in Example 10.7.

Note: The lift coefficient used in this example, admittedly modest, was selected purely to illustrate the method of calculation. For an initial design, the equations just developed would suffice but some further refinements can be added. An important refinement concerns the Prandtl correction for the number of blades.

Correcting for a Finite Number of Blades

So far, the analysis has ignored the effect of having a finite number of blades. The fact is that at a fixed point the flow fluctuates as a blade passes by. The induced velocities at the point are not constant with time. The overall effect is to reduce the net momentum exchange and the net power of the turbine. Some modification of the analysis is needed and this is done by applying a blade tip correction factor. Several solutions are available: (i) an exact one due to Goldstein (1929), represented by an infinite series of modified Bessel functions, and (i) a closed form approximation due to Prandtl and Tietjens (1957). Both methods give similar results and Prandtl’s method is the one usually preferred.

Prandtl’s Correction Factor

The mathematical details of Prandtl’s analysis are beyond the scope of this book, but the result is usually expressed as where, as shown in Figure 10.14, s is the pitchwise distance between the successive helical vortex sheets and d¼R r. From the geometry of the helices,

10.8 The Blade Element Momentum Method 385

This can be evaluated with sufficient accuracy and perhaps more conveniently with the approximation,

The circulation at the blade tips reduces to zero because of the vorticity shed from it, in the same way as at the tip of an aircraft wing. These expressions ensure that F becomes zero when r¼R but rapidly increases towards unity with decreasing radius.

The variation of F¼F(r/R) is shown in Figure 10.15 for J¼5 and Z¼2, 3, 4, and 6. It will be clear from the graph and the preceding equations that the greater the pitch s and the smaller the number of blades Z, the bigger will be the variation of F (from unity) at any radius ratio. In other words the amplitude of the velocity fluctuations will be increased.

Prandtl’s tip correction factor is applied directly to each blade element, modifying the elementary axial force, obtained from eqn. (10.13), to become and the elementary torque, eqn. (10.16b), dτ ¼ 4πρΩcx1ð1 aÞa0r3dr is modified to become

Following the reduction processes that led to eqns. (10.35) and (10.36), the last two numbered equations give the following results:

r R

FIGURE 10.14 Prandtl Tip Loss Model Showing the Distances Used in the Analysis

386 CHAPTER 10 Wind Turbines

The application of the Prandtl tip correction factor to the elementary axial force and elementary torque equations has some important implications concerning the overall flow and the interference factors. The basic meaning of eqn. (10.45) is i.e., the average axial induction factor in the far wake is 2aF when the correction factor is applied as opposed to 2a when it is not. Note also that, in the plane of the disc (or the blades), the average induction factor is aF, and that the axial velocity becomes

From this we see that at the tips of the blades cx2 ¼ cx1, because F is zero at that radius. Note: It was explained earlier that the limit of application of the theory occurs when a → 0.5, i.e., cx2 ¼ cx1(1 2a), and, as the earlier calculations have shown, a is usually greatest towards the blade tip. However, with the application of the tip correction factor F, the limit state becomes aF ¼ 0.5.

As F progressively reduces to zero as the blade tip is approached, the operational result gives, in effect, some additional leeway in the convergence of the iterative procedure discussed earlier.

FIGURE 10.15 Variation of Prandtl Correction Factor F with Radius Ratio for Blade Number Z¼2, 3, 4, and 6

10.8 The Blade Element Momentum Method 387

Performance Calculations with Tip Correction Included

In accordance with the previous approximation (to reduce the amount of work needed), ε is ascribed the value zero, simplifying the preceding equations for determining a and a0 to

When using the BEM method an extra step is required in Table 10.1, between steps 1 and 2, to calcu- late F, and it is necessary to calculate a new value of CL for each iteration that, consequently, changes the value of the blade loading coefficient λ as the calculation progresses.

Example 10.9 This example repeats the calculations of Example 10.7 using the same blade specification [i.e., the pitch angle β ¼ β(r)] but now it includes the Prandtl correction factor. The results of the iterations to determine a, a0, , and CL and used as data for the summations are shown in Table 10.5. The details of the calculation for one mid-ordinate radius (r/R ¼ 0.95) are shown first to clarify the process.

388 CHAPTER 10 Wind Turbines

Evaluating the elements of the torque using eqn. (10.42),w here,i n Table 10.6 ,V ar.2 ¼ [(1 þ a0)/

Hence, P¼τΩ¼57.960 kW, CP¼0.324, and ζ¼0.547. These calculations, summarised in Table 10.7, demonstrate that quite substantial reductions occur in both the axial force and power output as a result of including the Prandtl tip loss correction factor.


Clearly, with so many geometric design and operational variables to consider, it is not easy to give general rules about the way performance of a wind turbine will be effected by the values of parameters other than (perhaps) running large numbers of computer calculations. The variables for the turbine include the number of blades, blade solidity, blade taper and twist, as well as tip–speed ratio.

Blade Planform

In all the preceding worked examples a constant value of chord size was used, mainly to simplify proceedings. The actual planform used for the blades of most HAWTs is tapered, the degree of taper is

Table 10.7 Summary of Results

Axial force, kN Power, kW CP ζ

10.9 Rotor Configurations 389 chosen for structural, economic, and, to some degree, aesthetic reasons. If the planform is known or one can be specified, the calculation procedure developed previously, i.e., the BEM method, can be easily modified to include the variation of blade chord as a function of radius.

In a following section, Glauert’s analysis is extended to determine the variation of the rotor blade planform under optimum conditions.

Effect of Varying the Number of Blades

A first estimate of overall performance (power output and axial force) based on actuator disc theory was given earlier. The choice of the number of blades needed is one of the first items to be considered. Wind turbines have been built with anything from 1 to 40 blades. The vast majority of HAWTs, with high tip–speed ratios, have either two or three blades. For purposes such as water pumping, rotors with low tip–speed ratios (giving high starting torques) employ a large number of blades. The chief considerations to be made in deciding on the blade number, Z, are the design tip–speed ratio, J, the effect on the power coefficient, CP, as well as other factors such as weight, cost, structural dynamics, and fatigue life, which we cannot consider in this short chapter.

Tangler (2000) has reviewed the evolution of the rotor and the design of blades for HAWTs, commenting that, for large commercial machines, the upwind, three-bladed rotor is the industry accepted standard. Most large machines built since the mid-1990s are of this configuration. The blade number choice appears to be guided mainly by inviscid calculatio ns presented by Rohrback and Worobel

(1977) and Miller, Dugundji et al. (1978). Figure 10.16 shows the effect on the power coefficient CP of blade, number for a range of tip–speed ratio, J. It is clear, on the basis of these results, that there is a significant increase in CP in going from one blade to two blades, rather less gain in going from two to three blades, and so on for higher numbers of blades. In reality, the apparent gains in CP would be quickly cancelled when blade frictional losses are included with more than two or three blades.

Tangler (2000) indicated that considerations of rotor noise and aesthetics strongly support the choice of three blades rather than two or even one. Also, for a given rotor diameter and solidity, a three-bladed rotor will have two thirds the blade loading of a two-bladed rotor resulting in lower impulsive noise generation.

Effect of Varying Tip–Speed Ratio

The tip–speed ratio J is generally regarded as a parameter of some importance in the design performance of a wind turbine. So far, all the examples have been determined with one value of J and it is worth finding out how performance changes with other values of the tip–speed ratio. Using the procedure outlined in Example 10.6, assuming zero drag (ε¼0) and ignoring the correction for a finite number of blades, the overall performance (axial force and power) has been calculated for CL¼0.6, 0.8, and 1.0 (with l ¼ 1.0) for a range of J values. Figure 10.17 shows the variation of the axial force coefficient CX plotted against J for the three values of CL and Figure 10.18 the corresponding values of the power coefficient CP plotted against J. A point of particular interest is that when CX is replotted as CX/(JCL) all three sets of results collapse onto one straight line, as shown in Figure 10.19. The main interest in the axial force would be its effect on the bearings and on the supporting structure of the turbine rotor. A detailed discussion of the effects of both steady and unsteady loads acting on the rotor blades and supporting structure of HAWTs is given by Garrad (1990).

390 CHAPTER 10 Wind Turbines

Power coefficient, C p

FIGURE 10.16 Theoretical Effect of Tip–Speed Ratio and Number of Blades on Power Coefficient Assuming Zero Drag

FIGURE 10.17

Variation of the Axial Force Coefficient CX versus Tip–Speed Ratio J for Three Values of the Lift Coefficient CL¼ 0.6, 0.8, and 1.0

10.9 Rotor Configurations 391

Power coefficient, C P

FIGURE 10.18 Variation of the Power Coefficient CP versus J for Three Values of the Lift Coefficient CL¼0.6, 0.8, and 1.0

FIGURE 10.19

Axial Force Coefficient Divided by JCL and Plotted versus J (This Collapses All Results Shown in Figure 10.17 onto a Straight Line)

392 CHAPTER 10 Wind Turbines

Note: The range of these calculated results is effectively limited by the non-convergence of the value of the axial flow induction factor a at, or near, the blade tip at high values of J. The largeness of the blade loading coefficient, λ¼ ZlCL/(8πr), is wholly responsible for this non-convergence of a.I n practical terms, λ can be reduced by decreasing CL or by reducing l (or by a combination of these). Also, use of the tip correction factor in calculations will extend the range of J for which convergence of a can be obtained. The effect of any of these measures will be to reduce the amount of power developed. However, in the examples throughout this chapter, to make valid comparisons of performance the values of lift coefficients and chord are fixed. It is of interest to note that the curves of the power coefficient CP all rise to about the same value, approximately 0.48, where the cutoff due to non-convergence occurs.

Rotor Optimum Design Criteria

Glauert’s momentum analysis provides a relatively simple yet accurate framework for the preliminary design of wind turbine rotors. An important aspect of the analysis not yet covered was his development of the concept of the “ideal windmill” that provides equations for the optimal rotor. In a nutshell, the analysis gives a preferred value of the product CLl for each rotor blade segment as a function of the local speed ratio j defined by j ¼ Ωr

By choosing a value for either CL or l enables a value for the other variable to be determined from the known optimum product CLl at every radius.

The analysis proceeds as follows. Assuming CD¼0, we divide eqn. (10.36b) by eqn. (10.35b) to obtain

Also, from eqns. (10.39) and (10.49), we have

We now substitute for tan in eqn. (10.50) to obtain

Thus, at any radius r, the value of j is constant for a fixed tip–speed ratio J, and the right-hand side is likewise constant. Looking again at eqn. (10.17), for specific values of cx1 and Ω, the power output is a maximum when the product (1 a)a0 is a maximum. Differentiating this product and setting the result to zero, we obtain

10.9 Rotor Configurations 393

From eqn. (10.52), after differentiating and some simplification, we find

Combining this equation with eqn. (10.52) we obtain

Solving this equation for a0,

Equations (10.53) and (10.5) can be used to determine the variation of the interference factors a and a0 with respect to the coordinate j along the turbine blade length. After combining eqn. (10.5) with (10.56) we obtain

Equation (10.57), derived for these ideal conditions, is valid only over a very narrow range of a, i.e., 14 < a < 13. It is important to keep in mind that optimum conditions are much more restrictive than general conditions. Table 10.8 gives the values of a0 and j for increments of a in this range (as well as and λ). It will be seen that for large values of j thei nterferencef actor a is only slightly less than 13 and a0 is very small. Conversely, for small values of j the interference factor a approaches the value 14 and a0 increases rapidly.

394 CHAPTER 10 Wind Turbines

The flow angle at optimum power conditions is found from eqns. (10.50) and (10.5),


Again, at optimum conditions, we can determine the blade loading coefficient λ in terms of the flow angle . Starting with eqn. (10.5), we substitute for a0 and a using eqns. (10.36b) and (10.35b). After some simplification we obtain

Solving this quadratic equation we obtain a relation for the optimum blade loading coefficient as a function of the flow angle ,

Returning to the general conditions, from eqn. (10.51) together with eqns. (10.35b) and (10.36b),w e obtain


Rewriting eqns. (10.35b) and (10.36b) in the form

and substituting into eqn. (10.60) we get

Reintroducing optimum conditions with eqn. (10.59),

(Parte 4 de 6)