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Master thesis Trabajo Fin de Master Development of a model for computing tip loss corrections in wind turbines Author: Andrea Matiz Chicacausa Supervisors: PhD. Gerard Schepers PhD. Cristobal Cortes Master oficial en Energ´ıas Renovables y Eficiencia Energ´etica June 12, 2013 Resumen A la fecha, la comprensi´on de la relaci´on entre el comportamiento de la curva de potencia en una turbina e´olica y su n´umero de ´alabes parece ser insatisfactoria. Esto se debe en parte a que la teor´ıa Blade Element Momentum (BEM), que es la herramienta m´as usada para modelar el comportamiento aerodin´amico de las turbinas, asume un rotor con infinito n´umero de ´alabes [1, 2, 3, 4]. Esta suposici´on simplifica el an´alisis en la medida en que el rotor, a diferencia de las turbinas e´olicas reales, es descrito como un disco continuo que gira con una velocidad de rotaci´on determinada. A su vez, hay diferentes f´enomenos f´ısicos que suceden en el ´alabe y que no pueden ser captados por esta teor´ıa. Por ejemplo, la p´erdida de potencia en la punta se explica en parte por la no-uniformidad de flujo causada por la presencia de un n´umero finito de ´alabes en el rotor. Algunas correcciones realizadas para modelar la p´erdida de potencia en las puntas tratan de incluir como par´ametros importantes un n´umero finito de ´alabes y la relaci´on de velocidad en la punta λ. Los resultados de estas correcciones no son del todo satisfactorios y generan incertidumbres a la hora de calcular la distribuci´on de cargas sobre el ´alabe. Es razonable suponer que tambi´en hay una relaci´on entre la geometr´ıa del ´alabe y la p´erdida de potencia en la punta del mismo; para incluir dicho efecto creemos que el p´arametro que es necesario considerar es la estructura de la cuerda del ´alabe en la punta. La primera correcci´on para incorporar un n´umero finito de alabes fue presentada de forma anal´ıtica por Prandtl hacia la d´ecada de 1920. ´ El mostr´o que la p´erdida de potencia decae de manera exponencial en la cercan´ıa de la punta del ´alabe. Esta correcci´on tiene en cuenta no s´olo un n´umero finito de ´alabes en el rotor sino tambi´en la posici´on radial en el alabe, y corrige la predicci´on de las fuerzas aerodinamicas calculadas con BEM. En 2005, Wen Z. Shen y sus colegas desarrollaron una correcci´on que mejora los resultados obtenidos por el modelo de Prandtl [6]. Su propuesta, basada en el an´alisis emp´ırico de los datos de un proyecto, consiste en agregar un factor de correcci´on al modelo desarrollado por Prandtl, de manera que aparece el efecto de la velocidad de rotaci´on sobre las cargas en la punta. Dicho factor correctivo como funci´on de la distancia rdel eje del rotor es el siguiente: F(g) = 2 πcos−1exp −gB(R−r) 2Rsin φR,(1) donde φRes el ´angulo de flujo, ByRel n´umero y longitud de los ´alabes. Aqu´ı g es la funci´on exponencial introducida por Shen que depende de λy que fue hallada emp´ıricamente (si g= 1 se obtiene el factor de correcci´on de Prandtl). A pesar de mejorar notablemente la precisi´on en el c´alculo de las cargas, el modelo de Shen presenta algunas diferencias respecto a los casos de bajas λ. Con el objetivo de mejorar los resultados obtenidos por Prandtl y Shen este trabajo presenta un nuevo factor de correcci´on que incluye par´ametros que no han sido incluidos en los modelos existentes; por ejemplo, como ya se ha mencionado, la cuerda del ´alabe. Como resultado, la estructura correctiva de nuestro modelo es la siguiente: Fnew(g, s, σ) = F(g)m(s, σ),donde m(s, σ) = 1 −sc3exp(−c4σ) (2) y donde s=r/R,σes el solidity1yc3yc4son constantes a-dimensionales estimadas emp´ıricamente. Finalmente, vale le pena a˜nadir que, m´as all´a de estudiar y corregir los modelos de Prandtl y Shen, este trabajo pretende ser un aporte a la mejor comprensi´on de los fen´omenos que ocurren en las puntas de los ´alabes e´olicos. 1Esta variable se relaciona con la longitud de la cuerda cmediante la siguiente relaci´on: σ=Bc 2πr . 2 Contents Introduction 3 Statementofresults .............................. 4 Ackowledgements................................ 6 1. Tip loss effect 8 1.1. Effect of a finite number of blades . . . . . . . . . . . . . . . . . . . . 9 1.1.1. Non-uniformity of the flow through the rotor . . . . . . . . . . 10 1.2. Tip loss correction models . . . . . . . . . . . . . . . . . . . . . . . . 11 1.2.1. Prandtl tip loss factor . . . . . . . . . . . . . . . . . . . . . . 11 1.2.2. Other tip loss correction models . . . . . . . . . . . . . . . . . 14 1.2.3. Shen correction model . . . . . . . . . . . . . . . . . . . . . . 14 2. Data description and Analysis 17 2.1. Unsteady Aerodynamics Experiment (UAE) NREL-NASA Ames . . . 17 2.1.1. Test turbine description . . . . . . . . . . . . . . . . . . . . . 18 2.1.2. Dataprocessing.......................... 20 2.2. IEA: Enhanced Field Rotor Aerodynamics Database . . . . . . . . . . 21 2.2.1. National Renewable Energy Laboratory (NREL),USA . . . . . 21 2.2.2. RISO National Laboratory, Denmark . . . . . . . . . . . . . . 21 3. New Model 23 3.1. Angle of attack determination . . . . . . . . . . . . . . . . . . . . . . 23 3.2. InverseBEM ............................... 25 3.3. Newcorrectionmodel........................... 26 1 4. Results and conclusions 31 4.1. Results................................... 31 4.2. Conclusions ................................ 32 A. Theoretical background 36 A.1. Two dimensional aerodynamics . . . . . . . . . . . . . . . . . . . . . 36 A.2. Three dimensional analysis . . . . . . . . . . . . . . . . . . . . . . . . 37 A.3. One dimensional momentum theory . . . . . . . . . . . . . . . . . . . 37 A.3.1. Wake rotation and Vortex system . . . . . . . . . . . . . . . . 38 A.4. The blade element momentum Method (BEM) . . . . . . . . . . . . . 39 A.5. Prandtl tip loss factor . . . . . . . . . . . . . . . . . . . . . . . . . . 40 B. Two dimensional data for S809 Airfoil 42 C. Two dimensional coefficients for NACA Series 63nn-2nn 43 2 Introduction True be told, the understanding of the power behavior over the blade span (specially at the tip region) in a wind turbine seems to be unsatisfactory. One of the reasons to explain it is that the Blade Element Momentum (BEM) theory, which is the most common tool for computing aerodynamics loads and modeling the behavior of wind turbines, assumes a rotor with an infinite number of blades [1, 2]. The study of wind turbines as continuous discs at high rotational speeds simplifies the aerodynamical analysis and provides good results. However, the fact that real wind turbines have finite number of blades makes that some fluid particles pass through the rotor without any interaction with the blades, producing a non-uniformity on the flow leading into a variation of the induction factor around the rotor, then a loss of lift and hence a loss of power. In order to compute this power losses it is needed to know the azimuthal variation of the induction factor, but BEM is a limited method for this requirement. For this reason and to correct the assumption of infinite number of blades Prandtl in the decade of 1920 developed a correction factor. This Prandtl’s factor takes into account the number of blades and the relation between the local radius and the overall radius of the rotor [3, 4]; and at the same time expresses the ratio between the average induced factor on the rotor plane and the induced velocity at the blade position. Besides, he showed that the power loss goes according to an exponential decay in the vicinity of the tip [5]. Nevertheless, the results obtained with Prandtl correction over predict the aerodynamic loads over the tip of the blade. Therefore, we can think that there are other factors which, at least in principle, could be included in this model; for instance, the chord and the pitch angle of the blade. To date, various tip loss correction models based on original ideas proposed by Prandtl have been developed [6, 7, 8, 9]. 3 In 2005, Wen Z. Shen and coworkers showed the inaccuracy of the existing tip loss correction models when they predict the behavior of the aerodynamical forces (normal force) in the proximity of the tip [6]. Their proposal was based in introducing a new model in order not only to correct the mentioned disagreement but also to include in the model the tip speed ratio and enhance the effect of the number of blades. It is worth to note that while Prandtl’s correction is applied to the induced velocities which affect the loads indirectly. Shen’s model is applied directly to the loads. Albeit the correction introduced by means of this model displays a good behavior, there exists some disagreement under low tip speed ratios condition. We believe there is also a relation between the tip loss and the geometry of the blade. since, the relation between the blade and the rotor surface is given by Bc 2πr where Bis the number of blades, cis the chord of the blade and rthe local radius. This parameter is called in the literature the solidity. Motivated by the goal of obtaining a better understanding of the tip loss phenomena, we developed an empirical model, which is presented in this thesis and aims to improve Shen’s model. We address the issue of understanding the role played in the tip loss phenomenon by the solidity σ. Statement of results Having discussed the motivations of the present work, we will summarize the main results obtained during the development of this master thesis. Before all, it is worth emphasizing that the present model is an empirical one based in experimental observation. We found that a good performance is obtained when we include a function of an exponential type multiplying Shen’s model. Therefore, our results suggest that a new function m(s, σ) depending on the solidity must be included in the following way: Fnew(g, s, σ) = F(g)m(s, σ),where F(g) = 2 πcos−1exp −gB(R−r) 2Rsin φR and where φRis the flow angle at the tip and Ris the overall radius of the blade. Here gis the function introduced by Shen, which depends on the number of blades 4 and the tip speed ratio. The model developed in this work results in a new function which looks like an exponential decay of the following type showed in the next figure. m(s, σ)=1−sc3exp(−c4σ), where s=r/R is the blade span, c3and c4are dimensionless constants found empirically from data and equal to 8 and 34.2 respectively. Correcting factor m(s, σ). Notice that it is almost everywhere equal to 1 except when sis close to 1 and the chord takes small values. We summarize the plan of this document by means of the following map shown in next figure. Our framework is the BEM theory, as said. We address the problem of the tip loss as a consequence of the finite number of blades and then we study different proposals for modeling it. In order to develop a new model we have to process the data obtained from the different projects, this means to compute angle of attack when is needed, then filter, clean and averaging data. Finally, we tested the new model with experimental data. Before finishing this introductory chapter, we give the customary description of the contents of the present work. - In chapter 1 we review the basic facts about tip loss effect. The main purpose of this chapter is to explain the tip loss phenomena. Finally, at the end of this chapter we provide a brief survey of the tip loss correction models to date. - Chapter 2 is devoted to the description of the data and the different experimental projects used in the empirical model development. We also offer a preliminary sight of the normal force behavior over the blade for some of the turbines used in this work. The goal of this chapter is to show the characteristics of the turbines over which the new model is based on. 5 Tip loss diagram. - In chapter 3 we present the computations done to process the data in order to develop the new model, and finally we show the new model. - In the last chapter we test the accuracy of the model and offer some conclusions and recommendations for future work. - We finish this document with three appendices. The first one with a theoretical survey of the aerodynamics basis. The other two present two dimensional data for the airfoils used NREL S809 and NACA n-series. Ackowledgements To finalize this chapter I would like to express my deepest gratitude to ECN for supporting my stage here in The Netherlands. Several people made of this time an exciting and enriching experience. Starting with Dr. Gerard Schepers who carried out with the uneasy task of supervise my work and dedicate much of his time to discuss, teach and correct my project. In addition, I would like to thank to all my the colleagues at Wind Energy technologies unit, specially, Dr. Koen Boorsma and Dr. Francesco Grasso, their time and comments made possible succeed in important parts of this work. Besides the support that I had from ECN, Universidad de Zaragoza was also an important part in the development of this project. Dr. Cristobal Cortes who dedicate time to read and correct my work and 6 CHAPTER 1. TIP LOSS from the rotor blades which are modeled as lifting lines. This software is used to probe the Prandtl tip loss factor behavior for different velocities. Figure 1.4 show the comparison between Prandtl’s factor computed by AWSM for different wind speed 10, 15 and 24 m/s and by means of equation (A.9). It is very interesting to note, over the tip blade region, that for wind speed of 10 m/s Prandtl tip loss factor computed analytically shows an under prediction of F, unlike for wind speed of 15 m/s where both computations show a good agreement; finally, for higher velocities, as is the case of 24 m/s Prandtl method compute a higher value of Fcompare with AWSM computations. It is worth noting how for low tip speed ratio the tip loss correction decrease significatevly. This results are obtained from MEXICO project and also showed in [11]. Figure 1.4: Prandtl’s tip loss factor comparisson [4]. Results showed are taken from Model Rotor Experiments in Controlled Conditions (MEXICO) was a project carried out in 2006 by European Union where 10 institutes from 6 countries cooperated in doing experiments on a full scaled, 3 bladed wind turbine of 4,5 m diameter placed in The Netherlands. The result of this project was a large database of combined blade pressure distributions, loads and flow field measurements [12]. 13 CHAPTER 1. TIP LOSS 1.2.2. Other tip loss correction models After Prandtl postulated his correction, several models have been developed in order to correct the tip loss effect, such as Wilson and Lissaman, de Vries, Mikkelsen et al, and more recently Shen, which are discussed in brief below [13]. Wilson and Lissaman [7] suggested that the mass flow through the rotor disc should be corrected in the same manner as the induced velocity is corrected with Prandtl’s factor. Based on Prandtl theory taking the wake behavior as a succession of discs, and also considering as the ratio between the total induced velocity and the average of the velocity between the discs times the distance between them. They used the concept of circulation to reformulate the tip loss correction. As circulation is only caused by lift, they only take into account this force. Thus, the tangential interference factor is the same as for the Prandtl correction but, for the axial induction factor the mass flux and the induced velocity are corrected. However, this leads to a formulation in which the orthogonality condition between the induced velocity and the relative velocity at the blade element is not satisfied. In order to satisfy this condition, de Vries [8] introduces a model that corrects the mass flux in the tangential momentum equation and in the axial momentum equation. Later on, Mikkelsen et al. [9] suggested a technique in which the induced velocities are first computed by the Navier-Stokes solver, after which they are corrected by means of the Prandtl tip loss function in the rotor before applying airfoil data. 1.2.3. Shen correction model Shen et al. [6] presented an improvement with a more accurate tip loss correction. Their work is based on the statement that for a real rotor the axial velocity at the tip, where the tip vortex is generated and convected into a wake, is not zero. Thus, the inflow angle at the tip is not zero, either. They proposed to calculate their correction factor as the result of dividing the force coefficient by the two-dimensional data of the airfoil at the tip region in this way the correction is done directly on the loads. The model was done by comparing the normal forces computed by the different tip corrections with the NREL experimental data and for different wind speeds, as it is shown in figures 1.5 and 1.6. The factor they found is very similar to that used by Prandtl but with the 14 CHAPTER 1. TIP LOSS Figure 1.5: Comparison of normal forces computed with tip corrections and NREL experimental rotor [6]. addition of two new coefficients that take into account the effect of the tip speed ratio and enhanced the effect of the number of blades, by means of including a different correction factor called gnamely g= exp[−c1(Bλ −c2)],(1.2) where the coefficients c1and c2were computed experimentally, deriving and comparing the distribution of normal force near to the tip and then fitting the curve. This results in c1= 0,125 and c2= 21. This factor is introduced into the function Fin the following way: F(g) = 2 πcos−1exp −gB(R−r) 2Rsin φR.(1.3) Looking at the data, one realizes that there is a much better achievement of the the results obtained by means of using this model. However, the agreement with the experimental data can be improved. 15 CHAPTER 1. TIP LOSS Figure 1.6: Comparison of normal forces computed with Shen tip correction and NREL experimental rotor [6]. 16 Chapter 2 Data description and Analysis Aimed to develop an empirical model, our work is based on data from different experiments, namely: the unsteady Aerodynamics Experiment (UAE) and the Enhanced field Rotor Aerodynamics Database (IEA). These experimental tests were done using three different turbines under different operational conditions. In this way, the present work will be supported by a wide range of parameters resulting in a corrected model ables to compute the tip loss effect and to correct the load distribution prediction over the blade. This chapter describes the main characteristics of each project as well as the turbine used and the data obtained. 2.1. Unsteady Aerodynamics Experiment (UAE) NREL-NASA Ames This project was conducted by the National Renewable Energy Laboratory of The United States of America and its main goal was to create a large database in order to quantify the behavior of a horizontal-axis wind turbines under different conditions. This experiment sought to acquire aerodynamic and structural measurements on a full-scaled wind turbine located in a wind tunnel in order to obtain an environment free from large inflow disturbances. Several quantities, such as blade surface pressures as well as the inflow dynamic pressure, were measured at five span locations over the blade. The instrumentation used, data collection and processing will not be explained in this thesis because they are not relevant issues for the main topic 17 CHAPTER 2. DATA DESCRIPTION AND ANALYSIS but they can be found in [14]. 2.1.1. Test turbine description The turbine used during this project was stall regulated, with full-span pitch control and a power rating of 20 kW. The tower was designed to obtain a hub height of 12.2 m. It was two bladed with twisted and tapered blades as shown in figure 2.1, with a 10 m diameter. The airfoil used was the S809. The two-dimensional data of this airfoil are shown in Appendix B. Figure 2.1: Blade dimensions [14] For the goals of this thesis, just one sequence was taken in order to evaluate the tip loss effect. This sequence is called in the UEA report [14] Test sequence H upwind baseline where the test was carried out under conditions representative of a field operation. Test sequence H is a 30-second campaign under upwind conditions. The wind speed ranged from 5 m/s to 25 m/s. The blade tip pitch was 3 ◦. The rotation speed was fixed at 72 RPM and the blade span was 5,029 m. The blade shown in figure 2.1 was divided in 5 stations (at 30%,46%, 63%, 80% and 95% spanwise); in each station two measurements were taken: the dynamic pressure and the stagnation-point dynamic pressure. These measurements allow to derive a normalization pressure that was used to normalize each of blade surface pressures and in this way to compute the aerodynamic coefficients, such as the normal and the tangential forces, Cnand Ct, and the thrust and torque coefficients CT Q and CT H . 18 CHAPTER 2. DATA DESCRIPTION AND ANALYSIS To calculate the aerodynamic coefficients shown in figure 2.2 for each span station, the pressure distribution for rotating blade was integrated in order to compute the aerodynamic coefficients. The torque and thrust coefficient were computed as function of Cn,Ct, the blade pitch angle βand local twist angle φ, to wit, CT Q =Cnsin(φ+β) + Ctcos(φ+β) (2.1) CT H =Cncos(φ+β)−Ctsin(φ+β).(2.2) In figure 2.2 are shown the angles involved in the aerodynamic analysis, such as the angle of attack α, the local inflow angle φ, and the pitch angle βas well as the aerodynamic coefficients (normal, tangential and thrust). Figure 2.2: Local flow angle measurement convention [14] To measure the angle of attack α, it was attempted to measure first the local inflow angle. Nevertheless, this measurement has to be corrected due to the effect of the upwash from the bound vorticity. Due to this the inverse BEM method has to be used to compute α. Besides these aerodynamic coefficients, other kind of data were measured, among others, rotation velocity, generator power, tunnel velocity, tunnel air density. 19 CHAPTER 2. DATA DESCRIPTION AND ANALYSIS Figure 2.3: Probe to measure the local flow angle. 2.1.2. Data processing The test was done for wind speed from 5 m/s to 25 m/s in time series of 10 min with a fix rotational speed of 72 rpm. From these data, was derived the figure 2.4 which shows the normal force coefficient Cnevolution over the blade span in order to have a first sight of the rotor behavior. Figure 2.4: Normal force Coefficient vs. blade span-wise for different wind speeds in m/s. The important fact that has to be noted in this figure is the clear reduction of the Cnover the outermost part of the blade (approx. 90%). It is also interesting the increasing of this value at the root of the blade (that means for 30% blade span) this is explained due to three dimensional effects as stall delay. It is worth mention that not all of these tests characterized the performance of a wind turbine, since they were done under fix rotational speed, generating high values of angle of attack 20 CHAPTER 2. DATA DESCRIPTION AND ANALYSIS that under normal operational conditions are not achieved. 2.2. IEA: Enhanced Field Rotor Aerodynamics Database This project was developed by different organizations from six different countries, collaborating in aerodynamic experimental programs on full scale horizontal axis wind turbines at field conditions. The aerodynamic quantities were measured likewise UEA NASA Ames project and as a result of this cooperation was created a database of measured data from each participant. A total of five full scale test programs were coordinated but, for the purposes of this thesis, just two tests will be used. The complete performance of these projects will not be wide explained in this thesis but more information can be found in [15]. 2.2.1. National Renewable Energy Laboratory (NREL),USA Measurements for local flow angle and aerodynamic coefficients were taken at 34%, 51%, 67%, 81% and 91% blade span. The rotor is of 10 m diameter, threebladed and downwind positioned, the blade span was 4.521 m and the profile used NREL S809 which two dimensional data will be found in Appendix B This test was done in yawed and non-yawed sequences for different wind speeds and rotational speed. The Cnbehavior over the blade is shown in figure 2.5. It is easy to appreciate that the behavior under yawed and non-yawed conditions is pretty similar and it just depends of the wind speed. Moreover, as it was expected, over the tip of the blade the Cnvalue decrease. 2.2.2. RISO National Laboratory, Denmark The wind turbine tested is 100 kW, three-bladed turbine, 19 m diameter turbine, with twisted and tapered blades. The hub height was 29.3 m, blade length: 8,2 m, profile blade length: 6.8 m root chord: 1.09 m, tip chord: 0.45 m and blade profile NACA 63n-2nn series which two-dimensional data will be found in Appendix C. The measures of this project were mainly the aerodynamic forces over three segments of the blade (37%, 68% and 98%). These measures were taken under 21 CHAPTER 2. DATA DESCRIPTION AND ANALYSIS (a) (b) Figure 2.5: NREL–Cnbehavior over the blade for different wind speeds in (a) yawed condition and (b) non-yawed condition (a) (b) Figure 2.6: RISØ–Cnbehavior over the blade for different wind speeds in (a) yawed condition and (b) non-yawed condition yawed and non-yawed conditions and for tip speed ratios between 2 and 5. The change of the Cnbehavior is shown in figure 2.6 where is possible to see the change over the blade span. It is very important to note that in this case the blade geometry is not the same over its span because, as it was mention before, this blade is tapped which means that the airfoil use is not the same over the entire blade span. In figure 2.6 is shown test series under yawed and non-yawed conditions for different wind speeds. 22 CHAPTER 3. NEW MODEL (a) Correcting factor m(s, σ)(b) Profile of the function Fnew(g, s, σ) Figure 3.4: New model behavior where s=r Rand σis the solidity ( Bc 2πr ). Making use of a curve fit, we obtain that c3= 8 and c4= 34.2 are the coefficients which reduce the observed error. Figure 3.5: Profile of the function Fnew(g, s, σ), Fand m(s, σ) for a rotor with threeblade rotor and an inflow angle of 18◦and a solidity of 0,15. It is clear from figure 3.4 that our extremal cases are: m(1,0) = 0 and m(s, σ → ∞) = 1. These cases can be read in two ways. If the solidity is zero, there is no chord (there is no blade at all) and therefore there is no power to loss. If the solidity tends to infinity the number of blades or the chord is infinitely large; in both cases there is no tip loss. These are clearly reasonable assumptions. The model we have developed was constructed in order to correct the model by Shen only in the region in which it displays incorrect values, that is, when the 29 CHAPTER 3. NEW MODEL loads are computed near to the tip of the blade. In the other cases, the model does not play any role. This situation is better shown in the figure 3.5 where our model corrects the Shen’s profile only when the blade span is bigger than 75% and lower than 98%. In the inner region the model reduces the already mentioned over prediction of the loads resulted from applying the Shen model. 30 Chapter 4 Results and conclusions Previously in this thesis, we have discussed the models existing to date for correcting the BEM theory in what it concerns to the tip losses. We have analyzed the data of three different wind turbines and concluded that an improvement can be carried out if we take in consideration the chord of the tip. Our main goal has been to produce an empirical description of this phenomena. This section is aimed to test the model in a different project. Therefore, the main goal of this chapter is to evaluate the performance of the new correction, comparing its behavior as well as the model developed by Shen et al. with the experimental data. At the end of this chapter, we discuss the conclusions of this work and offer some recommendations for future work. 4.1. Results The new correction model was tested and compared with the data from the test sequences with axial flow in NREL and RISØ projects, under different conditions as yaw, pitch angle, wind speed and tip speed ratio. The comparisons exhibit in figures 4.1, 4.2 and 4.3 show a remarkable performance of the proposed model, whose error with the experimental data is much lower than the error produced by Shen’s model. Figures 4.1 and 4.2 show the comparison for NREL data under yawed and nonyawed conditions, respectively. In the case of non-yawed conditions the new model does not improve the results obtained with Shen’s model. It is clearly remarked the cases when Shen’s results are accurate the new model does not have a good agreement with experimental data. In the case of yawed conditions campaign showed in 4.2 the new correction 31 CHAPTER 4. RESULTS AND CONCLUSIONS (a) U∞= 10.4(b) U∞= 12.9 (c) U∞= 17.1 Figure 4.1: Comparison of the Cnbehavior with the new correction with Shen correction and two-dimensional and experimental data. NREL-Non-yawed campaigns present a much better behavior improving Shen’s results, specially for the cases of higher wind speed, in other words, lower tip speed ratios. In the same way, for RISØ cases there is a very good agreement between the new correction and the experimental data for wind speeds of 7.2 m/s and 10.1 m/s. These results will be shown in figure 4.3 It is important to recall that due to this model was developed in an empirical way it is difficult to explain some of the results obtained as far there is not a physical model. 4.2. Conclusions In this work we have studied in depth the tip loss phenomena. We have investigated the results obtained from Shen’s correction which is nowadays the model most used and compare them with experimental values. Our main goal has been to introduce other parameters which could affect the tip loss effect. We found that, for instance, the geometry of the airfoil plays an important role in the physics in- 32 CHAPTER 4. RESULTS AND CONCLUSIONS volved in this effect and in this sense the chord is an important variable in the phenomenology of the power losing. There are three main conclusions: Our new model corrects some disagreements between the prediction of the model by Shen and the observed values, including the solidity (a function of the chord) as a new parameter. This results in a remarkable improvement of the load prediction especially for low tip speed ratios. Taking in consideration extremal cases (a length of the chord close to zero, for instance), it is possible to produce a profile of the connection between the chord and the tip loss. We have proposed an exponential decay law. In any case, it is clear that the geometry of the tip influences the airflow crossing the blades. In this sense, future corrections of the Prandtl, Shen and our model itself, has to deal with the task of comprehend this connection. Of course, a more physical analysis shall yield a complete mathematical description of the connection between the geometry of the chord and the tip loss phenomena, but we hope that this empirical analysis is a first step in the right direction. It would be interesting define the different parameters involved in this effect and the way that they affect the aerodynamical behavior of the wind turbine. 33 CHAPTER 4. RESULTS AND CONCLUSIONS (a) U∞= 7.0(b) U∞= 9.9 (c) U∞= 10.2(d) U∞= 10.3 (e) U∞= 12.6 Figure 4.2: Comparison of the Cnbehavior with the new correction with Shen correction and two-dimensional and experimental data. NREL– Yawed campaigns 34 CHAPTER 4. RESULTS AND CONCLUSIONS (a) U∞= 7.2(b) U∞= 8.4 (c) U∞= 10.1 Figure 4.3: Comparison of the Cnbehavior with the new correction with Shen correction and two-dimensional and experimental data. RISØ– Non-Yawed campaigns 35 Appendix A Theoretical background This chapter provides a brief theoretical survey in aerodynamics, concepts needed to develop the model for the tip loss effect correction. The first three sections introduce some basis doing a brief mention of the one-dimensional momentum theory, blade element theory and vortex system. The last section is aimed to explain the blade element momentum model. A.1. Two dimensional aerodynamics A first approximation for understanding the flow passing over the blades is done by a two-dimensional analysis. The first statement for this analysis is that the flow is confined into a plane and then is projected outward a blade of infinite span. This allows to analyze and understand the basic principles behind the aerodynamic behavior in a wind turbine. P=~ F·~u (A.1) where Pis power, ~ Fis a force vector and ~u is the blade speed. The power comes from the force of the air concentrated at the rotor disc. As power depends proportionally to the force, one of the goals of wind turbine modeling is to increase and control its magnitude and distribution. The force vector is produced for the flow passing over the blades. It is decomposed in two components: one perpendicular and another parallel to the incoming flow direction. These components are lift and drag, respectively. To describe completely the force system in a blade it should be introduced a moment around a point in the airfoil which is positive when it tends to turn the airfoil clockwise. 36 APPENDIX A. THEORETICAL BACKGROUND In order to describe the characteristics of a wide range of wind turbines those forces are dimensionless resulting in the lift, drag and momentum coefficient, which at the same time derive in normal, tangential and torque coefficients. A.2. Three dimensional analysis Unlike the section above, in this case the blade is no longer a wing with infinite span. Now, it is a finite length wing with different kinds of airfoils as cross sections. The air flow splits in two sides of the airfoil is found, at the tip the streamlines flowing over and under the airfoil will be deflected inwards and outwards. Therefore, at the trailing edge there is a jump in the tangential component of the velocity [1] yielding to a sheet of vorticity in the wake behind the blade called free vortices. A.3. One dimensional momentum theory The one dimensional momentum theory is based on the assumption that the rotor is an ideal disc, which means that it is considered frictionless and without rotational component in the wake. Figure A.1: Actuator disc. 37 APPENDIX A. THEORETICAL BACKGROUND Shown in Figure A.1 is the actuator disc where it is possible to see the drop in the velocity at the rotor plane. Then, one can define the axial induction factor a, as the decrease in wind velocity between the free stream and the rotor plane. The induced velocity at the rotor is usually referred as U1ain this case the velocity at the rotor is a combination of the free stream and the induced velocity. Where the axial induction factor increase from zero (far away from the rotor)[21]. Due to the assumption of an ideal rotor, it is possible to derive relations between the velocities U∞, the thrust Tand the power P, [1] yielding: T= 2aρU2 ∞(1 −a)A(A.2) Pavail =1 2˙mU2 ∞=1 2ρU3 ∞A, (A.3) where ρis air density, ainduction factor, Athe rotor disc area, ˙mis the mass flow and Pavail is the power available in the air. A.3.1. Wake rotation and Vortex system In a real rotor there is rotation in the wake, contrary to what was assumed in previous section; then, the effect of the wake should be noted. In every blade of a wind turbine a vortex sheet is generated in the trailing edge. Due to the rotation of the blade the tip describes a circular path. Then, the free vortices generated on the tip form a helix in the wake behind the rotor. And the root vortices follow a linear path along the rotor axis, as it is shown in figure A.2.The generation of rotational kinetic energy in the wake results in a loss of energy extraction by the rotor. This energy in the wake will be higher if the generated torque is higher [3, 21]. Thus, turbines running at low speed ratios and high torque will experience more wake rotation losses. The inclusion of angular velocity component provided by the wind to the rotor can be express as an angular induction factor defined as a0=ω 2Ω. Thus, the induced velocity will consist in two components one axial Ua and one component in the rotor plane rΩa0. Deriving into a different equation of power applied to an infinitesimal control volume with thickness dr. dP = 4πρω2U∞a0(1 −a)r3dr. (A.4) 38 [14] Hand M.M. et al., Unsteady Aerodynamics Experiment Phase IV wind tunnel test configurations and available data campaigns, National Renewable Energy Laboratory, NREL/TP-500- 29955, 2001. [15] Schepers J. 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