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Renewable Energy 55 (2013) 296e304

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Renewable Energy

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Optimum aerodynamic design for wind turbine blade with a Rankinevortex wake

Déborah Aline Tavares Dias do Rio Vaz a, Jerson Rogério Pinheiro Vaz a,*, André Luiz Amarante Mesquita a,João Tavares Pinho b, Antonio Cesar Pinho Brasil Junior c

aUniversidade Federal do Pará, Faculdade de Engenharia Mecânica, Av. Augusto Correa, s/n e Belém, PA 66075-900, BrazilbUniversidade Federal do Pará, Faculdade de Engenharia Elétrica, Av. Augusto Correa, s/n e Belém, PA 66075-900, BrazilcUniversidade de Brasilia, Faculdade de Tecnologia, Departamento de Engenharia Mecânica, Av. L3 Norte, Asa Norte, Brasília, Distrito Federal, Cep. 70.910-900, Brazil

a r t i c l e i n f o

Article history:Received 28 August 2012Accepted 8 December 2012Available online 30 January 2013

Keywords:Aerodynamic optimizationWind turbineWind energyBEM model

* Corresponding author. Tel.: þ55 91 8179 5591.E-mail address: jerson@ufpa.br (J.R.P. Vaz).

0960-1481/$ e see front matter Published by Elseviehttp://dx.doi.org/10.1016/j.renene.2012.12.027

a b s t r a c t

This paper presents a model to optimize the distribution of chord and twist angle of horizontal axis windturbine blades, taking into account the influence of the wake, by using a Rankine vortex. This model isapplied to both large and small wind turbines, aiming to improve the aerodynamics of the wind rotor,and particularly useful for the case of wind turbines operating at low tip-speed ratios. The proposedoptimization is based on maximizing the power coefficient, coupled with the general relationship be-tween the axial induction factor in the rotor plane and in the wake. The results show an increase in thechord and a slightly decrease in the twist angle distributions as compared to other classical optimizationmethods, resulting in an improved aerodynamic shape of the blade. An evaluation of the efficiency ofwind rotors designed with the proposed model is developed and compared other optimization models inthe literature, showing an improvement in the power coefficient of the wind turbine.

Published by Elsevier Ltd.

1. Introduction

In wind turbine design, the maximization of the power coeffi-cient is of fundamental importance in order to optimize theextraction of energy from the wind. This paper addresses theproblem of the aerodynamic optimization of a horizontal axis windturbine rotor, considering the search of optimum shape design ofthe blade. The optimum design of wind turbine can be achievedfrom three different approaches that describe the energy conver-sion in the turbine blades. The first one uses the classical BladeElement Model (BEM), which describes this energy conversion bymeans of force and moment balances in radial sections of theblades (Glauert theory [7,18]). The second approach is based invortex methods [1], and the last methodology uses the modernfluid flow simulation tools (CFD approach [14]).

Presently, an important effort has been devoted to the devel-opment of methodologies of optimization using advanced geneticalgorithm and evolutionary computation [1,4,6] coupled to CFDtools [14]. Global optimum design can be achieved by the use ofthose methodologies, where each blade analysis requires a CFD

r Ltd.

computation, which can make this process computationallyexpensive and thus time-consuming. The high computational costsfor these approaches have motivated proposals for modified algo-rithms or other alternatives for faster methodologies.

A reasonably and fast methodology can be proposed based onthe BEM method [18], which is the model most frequently used byscientific and wind power industry communities for design andanalysis of wind rotors. This method is essentially an integralmethod, with semi-empirical information from aerodynamicsforces in airfoil sections issued from two-dimensional airfoil flowmodel or experimental data.

Thus, complex three-dimensional effects are not accounted for,but the method provides accurate good performance prediction fora large range of wind turbine operation. Besides that, the BEMtheory has resulted in good accuracy, allowing the chord and twistdistributions to be optimized for maximum power extraction withlow computational cost [10,11,13]. Currently, BEM is the basis fora great number of optimization methods using the evolutionarycomputation [1,3]. However, optimization methods based onimproved BEM models with variational or maximization principlescan also provide good solutions, with lower computational cost andadvantages in the implementation of the design procedure [8,15].Based on that, this work presents a mathematical model to opti-mize the distributions of chord and twist angle of horizontal axis

D.A. Tavares Dias do Rio Vaz et al. / Renewable Energy 55 (2013) 296e304 297

wind turbine blades, taking into account the influence of the wake,by using a Rankine vortex model, as proposed by Wilson and Lis-saman [18]. This model is applied to both large and small windturbines, aiming to improve the aerodynamics of the wind rotor.This is particularly useful for the case of wind turbines operating atlow tip-speed ratios, as for example the ones with multibladedrotors. The proposed optimization is based on maximizing the localpower coefficient, coupled to the general relationship between theaxial induction factor in the rotor plane and in thewake. The resultsshow a modification in the chord and twist angle distributions,resulting in an improved aerodynamic shape of the blade. Anevaluation of the efficiency of wind rotors designed with the pro-posed model is developed and compared with Glauert [7] andStewart [15] optimization models. The results also show animprovement in the extraction of wind energy due to the change inthe aerodynamic shape of the wind blade.

2. Mathematical model

2.1. Basic formulation

Wilson and Lissaman [18] present a one-dimensional mathe-matical model that considers the vortex wake caused by the windturbine, using a more general form than Glauert’s model [7] for thecalculation of the theoretical power coefficient. Fig. 1 illustrates thebehavior of the flow in a streamtube and the flow axial velocities.The induced velocities u and u1 in the rotor plane and in the wake,respectively, are written as:�

V0 � v ¼ uhð1� aÞV0V0 � v1 ¼ u1hð1� bÞV0

(1)

V0 is the velocity of undisturbed flow, v ¼ aV0, v1 ¼ bV0, a andb are the axial induction factors in the rotor plane and in the wake,respectively, and defined by

a ¼ V0 � uV0

(2)

b ¼ V0 � u1V0

(3)

The power coefficient, by applying the energy and momentumbalance, has the form [17,18,20]:

Cp ¼ b2ð1� aÞ2b� a

(4)

In this expression it is considered that the vortex in the wakebehaves as a free vortex. To relate the induction factors to the bladegeometry, the model described by Mesquita and Alves [12] is

Fig. 1. Simplified illustration of the velocities in the rotor plane and in the wake.

employed, where from the flow of a streamtube it is possible toderive expressions for the thrust, torque and power for the controlvolume shown in Fig. 1, and using the velocity diagram shown inFig. 2. In this figure L and D are the lift and drag forces, respectively.These relationships are:

b1� a

¼ Bc4pr

CnFsin2f

(5)

b0

1þ a0¼ Bc

4prCt

Fsinfcosf(6)

where a0 and b0 are the tangential induction factors at the rotor andfactor at the rotor wake, respectively, and defined by

a0 ¼ u

2U(7)

b0 ¼ u1

2U(8)

U is the angular speed of the wind turbine, w and w1 are therotor and wake angular velocities of the fluid, c is the local chord, ris the radial position in the rotor plane, f is the angle of flow, B is thenumber of blades, and Cn and Ct are the coefficients of the normaland tangential forces (Fn and Ft) to the rotor plane, given by

Cn ¼ Fn12rW2cdr

¼ CLcosfþ CDsinf (9)

Ct ¼ Ft12rW2cdr

¼ CLsinf� CDcosf (10)

whereW is the relative velocity, r is the air density, Cl and Cd are thelift and drag coefficients, which are usually obtained from windtunnel tests or numerical methods [5], and F is the Prandtl tip-lossfactor, as described in Ref. [17], which is defined as the ratio be-tween the bound circulation of all blades and the circulation of

Fig. 2. Velocity diagram for a rotor blade section.

D.A. Tavares Dias do Rio Vaz et al. / Renewable Energy 55 (2013) 296e304298

a rotor with an infinite number of blades. It is the most acceptedcorrection employed, and is usually taken as corresponding toa model of the flow for a finite number of blades. Strip theorycalculations made with the Prandtl model show good agreementwith calculation made through free wake vortex theory and withtest data [17]. Finally, it is pointed out, according to the analysisperformed by Wald [26], in the case of X < 2, that the Prandt tip-loss correction does not fit the exact solution of the circulation onthe potential flow in a propeller. X is the tip-speed ratio, defined by

X ¼ RUV0

(11)

where R is the rotor radius. This result suggests that amore detailedinvestigation for the validity of the use of the Prandt correction inthe case of X < 2 and for turbine flow is necessary. However, thisanalysis is beyond the scope of this paper and will be considered infuture works.

Wilson and Lissaman [18] showed that it is possible to establisha general relationship between the axial induction factors a andb from the application of the continuity, momentum and energyequations for the induced velocities in the streamtube shown inFig. 1. Thus, considering the hypothesis that the wake behaves likea free vortex,Wilson and Lissaman [18] demonstrated that the axialinduction factor in the rotor plane has a non-linear relationshipwith the axial inducing factor in the wake for low tip-speed ratios,X, especially for values X < 2, as shown in Fig. 3. This relationship isdefined by:

a ¼ b2

�1� b2ð1� aÞ

4X2ðb� aÞ�

(12)

Mesquita and Alves [12] showed that Eq. (12) can be rearrangedin the form of a complete cubic equation in b, where only one of theroots shows a consistent behavior with the physical constraints ofthe problem. It should be noted that Eq. (12) can also be solvednumerically, with good results as shown in Ref. [16]. In this case theanalytical solution to Eq. (12) is

b ¼ �12ðSþ TÞ � 1

3a1 � i

12

ffiffiffi3

pðS� TÞ (13)

where

S ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiZ þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiQ3 þ Z2

q3

r(14)

0 0.1 0.2 0.3 0.4 0.50

0.2

0.4

0.6

0.8

1

a

b

X = 0.25X = 0.50X = 0.75X = 1.00X = 2.00X = 3.00X = ∞

Fig. 3. Relation b/a for some values of X.

T ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiZ �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiQ3 þ Z2

q3

r(15)

Q ¼ 3a2 � a219

(16)

Z ¼ 9a1a2 � 27a3 � 2a3154

(17)

a1 ¼ 4X2

a� 1(18)

a2 ¼ 12aX2

1� a(19)

a3 ¼ 8a2X2

a� 1(20)

2.2. Effect of the wake rotation

As described in some experimental studies found in the liter-ature, for example Whale et al. [21], Ebert and Wood [22] and Huet al. [9], the flow structure in the wake of a wind rotor is complexwith 3D effects and unsteady behavior. However, as already com-mented, the BEM method provides good results for engineeringdesign purpose, and these results are improved when the wakerotation is considered. In order to improve their model, Wilson andLissaman [18] showed that it is necessary to introduce a correctionin Eq. (4), because the free vortex hypothesis causes infinite ve-locities on the wake near the axis of the wind turbine. This fact canbe seen in Fig. 4, where the power coefficient tends to high valueswhen X < 2. Also, Cp can take values greater than 1 when X is verysmall. However, for small values of a and X > 2 there are physicallyconsistent values for Cp.

In order to obtain a physically consistent solution, Wilson and Lis-saman [18] proposed the use of a Rankine vortex instead of the irro-tational vortex to represent thewake,which solves the problemof theinfinite velocities near the turbine axis. This approach is implementedby the introductionof aparameterNhU=wmax in theexpressionof thepower coefficient, which assumes the following form [18]:

Cp ¼ bð1� aÞ2b� a

½2Naþ ð1� NÞb� (21)

Fig. 4. Power coefficient as a function of axial induction factors a and b.

Fig. 6. Power coefficient as a function of axial induction factors a and b for N ¼ 2.

D.A. Tavares Dias do Rio Vaz et al. / Renewable Energy 55 (2013) 296e304 299

The parameter N represents the influence of the Rankine vortex,and wmax is the maximum speed of the vortex wake. The deter-mination of wmax is a limitation of this methodology and exper-imental data are needed, as, for example, the experiments carriedout by Hu et al. [9], Whale et al. [21] and Ebert and Wood [22].However, unfortunately, in the currently available literature there isa lack of measurements for the hub vortex, particularly for X < 2.

Figs. 5 and 6 show the behavior of Eq. (21) for N equal to 1 and 2,respectively. The value of the maximum power coefficient of a rotoris heavily influenced by the parameter N in the regionwhere X < 2.It is observed that for X > 2, and consequently b ¼ 2a, there isno influence of N in the calculation of the theoretical power coef-ficient of an ideal wind turbine, because Eq. (21) reduces toCp ¼ 4a(1 � a)2, showing that in this case the maximum powercoefficient is 59.26% (Betz’s limit [2]). Note that for N ¼ 0, Eq. (21)reduces to Eq. (4). In this work, a sensibility analysis for theparameter N is performed in order to obtain more information onthe use of this formulation.

For this same correction, other alternatives for the vortex modelare possible. A more general model is the Vatistas model given by(Leishaman [23]):

vq ¼ Gr

2p�r2nc þ rn

�1=n (22)

where vq is the tangential velocity component, G is the vortex cir-culation strength, rc is the vortex core radius, n is an arbitraryinteger, and r is the local radial coordinate for the vortex tangentialvelocity profile. For n ¼ 1 the Scully vortex is reproduced, for n ¼ 2one obtains the Bagai-Leishaman vortex model, and for n/N thenVatistas model reduces to the Rankine vortex model.

Young [19] has performed an extensive review on existingexperimental works available in the literature and, using a kineticenergy conservation approach, concludes that all forms of theVatistas vortex model, included the Rankine vortex profile, matchreasonably well with the experimental data for the tip vortex. Thisresult is comprehensive because in the model implementation thevortex strength or the core size is often adjusted on a trial and errorbases with experimental data. However, as already commented, forthe hub vortex the available experimental data are scarce.

On the theoretical analysis, this problem was analyzed by someauthors after the pioneer work from Wilson and Lissaman [18] inthe middle of the 70’s. Wood [24] uses Vatistas vortex with n ¼ 1and concludes that this model does not have a significant effect onthe basic analysis that leads to the Betz limit [2], provided that thecore radius of the hub vortex is sufficiently small and the tip-speed

Fig. 5. Power coefficient as a function of axial induction factors a and b for N ¼ 1.

ratio is sufficiently high. For low tip-speed ratios the results becomenot physically possible with infinite values for the power coefficientwhen the tip-speed ratio decreases to zero. Recently, Sørensen andKuik [25] proposed a model that remedies this problem byincluding the contribution from the lateral pressure and frictionforces in the axial momentum theorem, with result that the powercoefficient never exceeds the Betz limit [2] and tends to zero at zerotip-speed ratio. Since the friction force and pressure on the lateralboundary on the control volume are not known a priori, the authorsproposed a model, in which this force is proportional to the wakeexpansion area and to the tip pressure drop multiplied by a smallcoefficient giving the net influence of the integrated pressure actingon the lateral boundary of control volume. The determination ofthis coefficient is yet open in this methodology.

Finally, it is important to analyze the influence of the wakerotation in the radial pressure gradient. Sørensen and Kuik [25]showed that the radial pressure gradient in the far wake is given by:

vp1vr1

¼ r

�1r1

v2q � u1vu1vr1

(23)

where p1 and r1 are the pressure and the local radial coordinate inthe far wake, respectively. This gradient has to deliver the cen-tripetal acceleration rv2

q=r1. This is satisfied when vu1/vr1 ¼ 0.

Consequently, u1 is constant in the fully developedwake. This resultallows the determination of a relationship between axial inductionfactors in the rotor plane and in the wake, respectively. Therefore,according to Sørensen and Kuik [25], applying a combination of themomentum, energy and mass balance equation gives:

q ¼ b2ð1� aÞ2Xðb� aÞ (24)

where q ¼ �G=ð2pRV0Þ. In this case, the axial momentum balanceis:

uV0

¼ 1� a ¼ q2 þ 2Xq2b

(25)

Combining Eq. (24) with Eq. (25), one obtains Eq. (12). For thepower coefficient, Sørensen and Kuik [25] showed that:

Cp ¼ 2Xqð1� aÞ (26)

Substituting Eq. (24) in Eq. (26), one obtains Eq. (4). These re-sults, show that the study developed by Sørensen and Kuik [25],about momentum theory at low tip-speed ratio, agree with the

D.A. Tavares Dias do Rio Vaz et al. / Renewable Energy 55 (2013) 296e304300

mathematical relations obtained by Wilson e Lissaman [18] for theaxial induction factors and power coefficient, which are used in thepresent work.

2.3. Aerodynamic optimization of wind turbine blade

In this work the aerodynamic optimization is obtained bymaximizing the power coefficient given by Eq. (21), making dCp/da ¼ 0, and resulting in:

2dbda

Na3 þ 2�2bN � db

da½bðN � 1Þ þ N�

a2 þ

�2b

ddðN � 1Þ

þ b2�1� 7N þ db

daðN � 1Þ

�aþ b2

�1� db

daðN � 1Þ þ N

�þ 2b3ðN � 1Þ ¼ 0

(27)

Dividing Eq. (27) by the term 2Ndb/da gives

a3 þ d1a2 þ d2aþ d3 ¼ 0 (28)

where

d1 ¼2bN � db

da½bðN � 1Þ þ N�

Ndbda

(29)

d2 ¼2b

dbda

ðN � 1Þ þ b2�1� 7N þ db

daðN � 1Þ

�

2Ndbda

(30)

d3 ¼b2

�1� db

daðN � 1Þ þ N

�þ 2b3ðN � 1Þ

2Ndbda

(31)

where db/da is obtained by differentiating Eq. (12).

dbda

¼ 8a2X2 � b�16aX2 � b

�bð1� bÞ þ 8X2 �

4a2X2 þ b�� 8aX2 þ b

�3að1� aÞ � 2bð1� aÞ þ 4X2

�(32)

Note that Eq. (27) is a complete cubic equation in a, where onlyone root presents a physically consistent solution, in a similar wayas demonstrated byMesquita e Alvez [12]. The solution of Eq. (27) isgiven by:

aopt ¼ �12ðS* þ T*Þ � 1

3d1 � i

12

ffiffiffi3

pðS* � T*Þ (33)

where

S* ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiZ* þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiQ3*þ Z2

*

q3

r(34)

T* ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiZ* �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiQ3*þ Z2

*

q3

r(35)

Q* ¼ 3d2 � d21 (36)

9

Z* ¼ 9d1d2 � 27d3 � 2d31 (37)

54

Therefore, with the calculated value of it is possible to developan iterative procedure for the optimum aerodynamic design ofwind blades. The optimum chord is calculated using Eq. (5), whichin this case can be rewritten as:

copt ¼ 4prbFsin2f

BCn�1� aopt

� (38)

For the calculation of a0, the relationship described in Ref. [8] isused, which is defined as:

a0opt ¼ aopttanfx

(39)

Eq. (39) is valid even for values of X < 2, as this relationship isobtained in the rotor plane and not in the wake. Once a0opt is cal-culated, b0 is described in Mesquita and Alves [12], and Vaz et al.[16], and given by:

b0opt ¼ ba0optaopt

(40)

The optimum twist angle is obtained from aopt the velocity di-agram shown in Fig. 2, by Eq. (41). The velocities diagram wasobtained according to the classical theory of turbomachinery.

bopt ¼ fopt � a (41)

where fopt is given by Eq. (42)

fopt ¼ tan�1

" �1� aopt

��1þ a0opt

�x

#(42)

The iterative procedure for the calculation of optimum chordand twist angle at each section of the blade is detailed as

Algorithm.

3. Results and discussion

3.1. Airfoil data

The results were obtained using the aerodynamics charac-teristics of the NACA 0012 symmetrical airfoil, obtained exper-imentally by Sheldahl and Klimas [27]. The NACA 0012 is one ofthe most popular symmetrical airfoils, so there are moreexperimental data available in the literature. Fig. 7 shows thelift and drag coefficients for a Reynolds number of 1.6 � 105. Inthis case, the optimizations have been developed for an angle of

0 5 10 150

0.2

0.4

0.6

0.8

1

CL

α0 5 10 15

0

0.05

0.1

0.15

0.2

CD

α

Fig. 7. (a) Lift coefficient and (b) drag coefficient of the NACA 0012 [27].

D.A. Tavares Dias do Rio Vaz et al. / Renewable Energy 55 (2013) 296e304 301

attack of 8�. This angle is chosen for maximum CL/CD, in thiscase around 44. The NACA 0012 airfoil is used here just for thepurpose of assessing the behavior of the proposed model, sinceit is not the objective of the work to evaluate airfoilsefficiencies.

3.2. Results for low tip-speed ratios (X < 2)

First, to check the behavior of algorithm implementation and itsconvergence, simulations were performed for small values of X.Fig. 8a presents the evolution of the computation of the theoreticalpower coefficient (Eq. (21)) as a function of the number of itera-tions. The numerical method converges to a number of iterationslarger than 25. For a number of iterations less than 25, the powercoefficient tends to exceed the Betz limit [2], and should not beapplied. Fig. 8b shows that the induction factors in the rotor planeand in the wake do not present a linear behavior for X < 2; in otherwords, bs2a and b0s2a0.

The results were obtained for X¼ 1.57. In this case, for N¼ 1 andN ¼ 2 the power coefficient tends to less than 59.26%. This effectwas expected, because the Rankine vortex causes a reduction in thetheoretical power coefficient for X< 2, as shown in Figs. 5 and 6. ForN ¼ 0 and X < 2, the Betz limit [2] can be exceeded, as shown inFig. 4, which, according to the actuator disk theory, is not possible.Therefore, the result for N ¼ 0 in Fig. 8a, shows the power coeffi-cient to be 59.75%, which is slightly larger than the Betz limit [2],due to the fact that the free vortex hypothesis causes infinite ve-locities on the wake near the axis of the wind turbine [18]. In an

0 10 20 30 400

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Cp

Number of iterations

N = 0N = 1N = 2

Betz Limit

Fig. 8. Convergence of the method: (a) Power coefficient. (b) Induction factors in the r

attempt to correct this problem the Rankine vortex can beemployed, as described byWilson and Lissaman [18] and presentedin Figs. 5 and 6. These results show that the efficiency of a windturbine is heavily influenced by wake rotation at low tip-speedratio operation.

Despite the difficulty to find detailed experimental results forthe vortices hub, some works give very useful information aboutthe vorticity field in the region. These data could provide someindication on the validity of the Rankine vortex assumption andabout the parameter N from this model. In order to estimate theparameter N it is necessary to calculate the maximum fluid angularvelocity, wmax. In the case of the flow behind the wind rotor thevector vorticity, z

!is given by

z! ¼

�1rvrvqvr

� 1rvvrvq

e!z ¼ zz e

!z ¼ 2wz e

!z (43)

By definition, the axial component of the vorticity field, zz, istwice the fluid angular velocity, wz. With the experimental flowvelocity data it is possible to estimate the vorticity and, con-sequently, the fluid velocity flow and its maximum,wmax. Note thatfor an axisymmetric flow with free vortex model the vorticity iszero.

Recently, Hu et al. [9] carried out an experimental study inorder to characterize the dynamic wind loads and evolution ofthe unsteady vortex and turbulent flow structures in the nearwake of a horizontal axis wind turbine model placed in an at-mospheric boundary layer wind tunnel. The measurements were

5 10 15 200

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Indu

ctio

n Fa

ctor

Number of iterations

bb’aa’

a = 0.3352

b’ = 1.3633

b = 0.7375

a’ = 0.6196

otor plane and in the wake for the optimized procedure for X ¼ 1:57 and N ¼ 0.

Table 1Estimates for N from measurements of Hu et al. [9].

X U zz max wmax N

3.0 94.5 50 25 3.83.5 110.2 50 25 4.44.0 126.0 50 25 5.04.5 141.7 50 25 5.7

Table 2Estimates for N from measurements of Whale et al. [21].

X U zn max zz max wmax N

3.0 8.6 0.8 6.7 3.3 2.64.0 11.4 0.8 6.7 3.3 3.45.0 14.3 0.8 6.7 3.3 4.36.0 17.1 0.8 6.7 3.3 5.18.0 22.9 0.8 6.7 3.3 6.9

Table 3Power coefficient and output power for the designed wind turbines ðX ¼ 1:57Þ.

Glauert Stewart N ¼ 0 N ¼ 1 N ¼ 2

Cp(%) 29.17 29.22 35.00 34.78 34.28P(W) 278 278 334 331 327

D.A. Tavares Dias do Rio Vaz et al. / Renewable Energy 55 (2013) 296e304302

performed by a PIV (Particle Image Velocimetry) system. Usingthese results, it is possible to calculate the parameter N in theRankine vortex model employed in this work. In Hu’s experi-ments the diameter of the turbine rotor model was 254 mm,which is made of twisted blades, and the mean wind speed at thehub height of the wind turbine model was set to 4.0 m/s. In theseexperiments, it is always reported that the strength of the vor-tices varies with the tip-speed ratio, the maximum value for theaxial vortices component was around 50 s�1. The experimentswere performed for tip-speed ratios of 3.0, 3.5, 4.0, and 4.5. Withthese data and using Eq. (43), Table 1 shows the estimation of theparameter N as a function of the tip-speed ratio. From this resultit can be observed that the parameter N decreases withdecreasing tip-speed ratio. Note that the value for maximumaxial vorticity is approximated because the results from Hu et al.[9] were presented in the form of color maps and not ina quantitative way.

Whale et al. [21] carried out wake velocity and vorticitymeasurements in an untwisted two-blade wind rotor modelwith diameter of 175 mm using also the PIV technique at tip-speed ratios in the range X ¼ 3e8. The experiments were per-formed in a water channel with V0 ¼ 0.25 m/s, and the axialvorticity maps were presented in a non-dimensional vorticitydefined by

zn ¼ 4zzdV0

(44)

0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.05

0.1

0.15

0.2

0.25

0.3

r/R

c/R

Stewart (1976)Glauert (1926)N = 0N = 1N = 2

Fig. 9. (a) Chord distributions. (b

where d ¼ 7.5 mm is the spacing of the experimental velocity grid.Using the experimental data obtained by Whale et al. [21], Table 2shows the estimation of the parameter N as a function of the tip-speed ratio. It is verified that the parameter N presents the samebehavior and the same order of magnitude for both experimentaldata obtained by Hu et al. [9] and Whale et al. [21]. From thisanalysis it is suggested that the value for the parameterN in the caseof X < 2 could be less than 3. However, these experimental data arenot detailed enough for a precise analysis, and more measurementsfor the vorticity field in the wake hub region are necessary.

A second group of simulations is presented to evaluate the per-formanceof themodel for a smallwind turbinewithmultiple blades,typically used inpumping systems,with3mdiameter and0.9mhubdiameter, constant rotation of 60 rpm, average wind speed of 6 m/sand 12 blades, giving a tip-speed ratio of 1.57. Fig. 9 shows the chordand twist angle distributions. Note that the aerodynamic shape isdifferent from the results obtainedwith Glauert [7] and Stewart [15]optimizations models. This occurs due to the fact that the modeltakes into account the maximum power coefficient point in thecalculation process of the chord and twist angle for any value of X.Table 3 shows the power coefficient and output power for the windturbines designed under the same operating conditions.

In order to verify the performance as a function of the windspeed, Fig. 10 shows that the power coefficient and the outputpower (P) are improved, around 18.54% for a wind speed of 6.5 m/s,when compared to the Glauert [7] and Stewart [15] optimizationsmodels (see Fig. 10b). The results shown in Table 3 can be observedin Fig. 10a for X ¼ 1.57. For the wind speed of 6 m/s, for which thewind turbine was designed, the results with the proposed modelpresent better efficiency.

3.3. Results for high tip-speed ratios (X > 2)

The proposed model can also be applied for high tip-speed ra-tios. Fig. 11 shows that the power coefficient tends to the Betz limit[2] for any value of N. This occurs because the tip-speed ratio ishigher than 2, since for X < 2 the model tends to power coefficientvalues lower than the Betz limit. This result shows that Eq. (33)locally maximizes the power coefficient, and satisfies the Betz [2]condition. The results observed in Fig. 11 were obtained for

0.3 0.4 0.5 0.6 0.7 0.8 0.9 110

15

20

25

30

35

40

r/R

β −

(o )

Stewart (1976)Glauert (1926)N = 0N = 1N = 2

) Twist angle distributions.

1 1.2 1.4 1.6 1.80.1

0.15

0.2

0.25

0.3

0.35

0.4

Cp

X

Stewart (1976)Glauert (1926)N = 0N = 1N = 2

5 5.5 6 6.5 7150

200

250

300

350

P −

(W

)

V0 − (m/s)

Stewart (1976)Glauert (1926)N = 0N = 1N = 2

Fig. 10. (a) Power coefficient as a function of tip-speed ratio. (b) Output power as a function of speed.

0 10 20 30 400

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Cp

Number of iterations

N = 0N = 1N = 2

Betz Limit

5 10 15 200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Indu

ctio

n Fa

ctor

Number of iterations

bb’aa’

b = 0.667

a = 0.333

b’ = 0.410

a’ = 0.207

Fig. 11. (a) Power coefficient as a function of the number of iterations and (b) induction factors in the rotor plane and in the wake for the optimized procedure for X ¼ 4:18 andN ¼ 0.

0 0.2 0.4 0.6 0.8 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

r/R

c/R

Stewart (1976)Glauert (1926)N = 0N = 1N = 2

0 0.2 0.4 0.6 0.8 10

5

10

15

20

25

30

35

40

r/R

β −

(o )

Stewart (1976)Glauert (1926)N = 0N = 1N = 2

Fig. 12. (a) Chord distributions. (b) Twist angle distributions.

Table 4Power coefficient and output power for the designed wind turbines ðX ¼ 4:18Þ.

Glauert Stewart N ¼ 0 N ¼ 1 N ¼ 2

Cp(%) 37.23 36.10 39.57 39.57 39.57P(W) 631 612 671 671 671

D.A. Tavares Dias do Rio Vaz et al. / Renewable Energy 55 (2013) 296e304 303

X¼ 4.18, and show that b and b0 converge to themodel proposed byGlauert [7], where the induction factors b and b0 present a linearbehavior (b ¼ 2a and b0 ¼ 2a0; see Fig. 11b).

The results shown in Fig. 12 were obtained for X ¼ 4.18, consid-ering a small wind turbine with 4 m diameter and 0.4 m hubdiameter, constant rotation of 120 rpm, averagewind speed of 6 m/sand 3 blades. Table 4 shows the power coefficient and output powerfor thewind turbines designed under the same operating conditions.

2.5 3 3.5 4 4.5 5

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Cp

X

Stewart (1976)Glauert (1926)N = 0N = 1N = 2

5 5.5 6 6.5 7350

400

450

500

550

600

650

700

P −

(W

)

V0 − (m/s)

Stewart (1976)Glauert (1926)N = 0N = 1N = 2

Fig. 13. (a) Power coefficient as a function of tip-speed ratio. (b) Output power as a function of speed.

D.A. Tavares Dias do Rio Vaz et al. / Renewable Energy 55 (2013) 296e304304

The output power obtained using the rotor designed with theproposed model is around 12.73% higher than those obtained withthe Glauert [7] and Stewart [15] optimizations models, for a windspeed of 6.5m/s (see Fig.13b). The curves forN¼ 0, N¼ 1, andN¼ 2coincide. This result was expected, because for high tip-speed ratiosthe model present the same response.

4. Conclusions

The proposed mathematical model represents a simple alter-native tool for the optimal design of wind rotors, especially for tur-bineswithmultibladed rotors, where the proposed optimization canimprove the efficiency for both high and low tip-speed ratios, in thiscase, an improvementof 18%was obtained forX< 2,when comparedwith the classical models. The main advantage of the proposedmodel is to take into account the effects of the wake rotation usinga Rankine vortex model. The method shows better efficiency whencompared to the Glauert’s [7] and Stewart’s [15] optimizations, sat-isfying the condition described by Betz [2], where the maximumenergy to be extracted from the flow is 59.26% of its original energy.For low tip-speed ratios the behavior of the axial induction factors inthe wake is completely non-linear and needs to be considered. Thisfact considers the slow operation of a turbine, commonly used inwater pumping systems. The comparisons show that the developedmodel improves the aerodynamics of thewind rotor. However, somelimitations need to be considered, such as the difficulty in accuratelydetermining the maximum rotation of the wake vortex formedbehind the rotor, which indicates thatmore detailed experiments onthe wake structure for the hub region are necessary, as well as thevalidity of the Prandt tip-loss correction,wheremore investigation isalso needed. For high tip-speed ratios, the presented model showsthat the influence of the wake rotation is negligible.

Acknowledgments

The authors would like to thank CNPq (480135/2010e0), PRO-PESP/UFPA (EDITAL PARD 04/2011), GEDAE, INCT e EREEA andELETRONORTE for financial support.

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