An Improved Approach for Performance Prediction of ...

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An Improved Approach for Performance Prediction of HAWT using the Strip Theory André L. Amarante Mesquita and Alex S. G. Alves GTDEM – Turbomachinery Group, Department of Mechanical Engineering, Federal University of Pará, Campus Universitário do Guamá, 66.0 75-90 0 Belém, PA, Brazil,

ABSTRACT T his w ork review s briefly the basis of the strip theory or Blade Element/ Mom entum m ethod and its most used approaches for design and performance prediction of H orizontal-A xis W ind Turbines – H AW T, and proposes a new improved approach for the w hole ex tension of the Tip Speed R atio – TSR region. T he results are compared to others found in the litera ture and to available ex perimental data. It is concluded that the proposed m odel provides similar results to the compared models to regions of TSR g reater than 2, as is the case of the electricity -generating fast-running turbines, and ex tends the validity of these methods to regions of TSR smaller than 2, g enerally the case of multi-bladed HAW T em ployed for w ater pumping.

1. INTRODUCTION The strip theory or Blade Element/ Mom entum – BE M m ethod is the usual tool for design and perform a nce prediction of H orizontal-A x is W ind Turbi nes – H AW T ( Gla uert 1935a -b, E g gleston a nd Stoddard 1987, H ansen a nd Butterfield 19 93, W ilson 19 94 ) . T he m ain assumptions in this theory are that the blade can be analyzed as a number of independent streamtubes, the spanwise flow is negligible, the flow is considered axisy metric, and usually the cascade effect is not tak en into account. In each streamtube, considering the spanwise flow negligible, the induced velocity is calculated by performing the conservation of the m omentum, and the aerodynamic forces are found through two-dimensional airfoil data available in literature. H owever, the BE M m ethod presents some limitations. Glauert 1926 pointed out that the BE M hy pothesis is no longer valid for high induced velocity regions, and proposed an em pirical ex pression for this range. A lso, the tip losses, not predicted by the m ethod, a re computed by em ploying some empirical correlation, as very w ell reported by W ilson et al. 1995 and W ilson and Lissaman 1974. Based on this approach, m uch w ork w as developed for H AW T design and perform ance prediction (G riffiths 1977, Shepherd 1984, G aletuse 1986, Rijs and Sm ulders 1990, Neogi 1995, Siddig 1992, A lves et al 1998). The value of the results from the strip theory for horizontal-axis wind turbine analy sis is g reatly dependent on the precision of the lift and drag coefficients. G enerally, during an opera tional period of a wind turbine, due to the variations in the w ind speed, the angle of attack can attend hig h values, and, for this condition (post-stall region), the aerody namic data is lacking. The available airfoil data is often limited in the range of a ngle of a ttack, as in A bbott and Von D oenhoff 1959. W hen the rotor blades operate in the post-stall region, the BE M m ethod underestimates the evaluated rotor power w hen compared to the ex periments of

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I M P R OV E D A P P R OA CH

FOR

P E R F OR M A NCE P R E DICT IO N

OF

H AW T

U S I NG ST R I P

T H E O RY

Musial et al. 1990. Some efforts are being made to ex tend these data in view of wind energy applications. F or ex ample, Ostowari and Naik 1984 have carried out studies w ith NACA 4415 airfoils for angle of attack ranging from –108 to 1008 . H owever, this kind of information is not g enerally available, and em pirical correction in the available aerody namic data is needed. Viterna a nd Corrig an 1981 proposed an empirical m odel to adapt the a vailable aerody namic data for wind energ y purposes. They have developed an em pirical approach to ex trapolate the tw o-dimensional airfoil data for an angle of attack of up to 90 degrees. F or the sm all tip-speed ra tio operation zone, generally the case of m ulti-bladed HAW T em ployed for w ater pumping, w here the tip-speed ratio can attend values smaller than 2, the usual strip theory m odels found in the literature is no longer valid. F or this region, the large tangential velocities in the w ak e behind the rotor are the reason for this behaviour (W ilson and Lissama n 1974). This paper presents a study a bout the aerody namics of HAW T using the strip theory including corrections for tip-loss, cascade, post-stall and turbulent w ak e effects. The m ain equations of the usual strip theory model are briefly reviewed, and their limitations and adva ntag es discussed. Nex t, a new im proved a pproach for this model is proposed and im plemented, w hich ex tends its validity to the region of tip-speed ratio smaller than 2, allowing the analy sis of sm all w ater pumping w ind energ y conversion sy stem s. Differently from som e w ork s that use alternativ e interactive procedures for this region (Rijs and Smulders 1990, Neogi 1995), the proposed m odel uses a n ex pression resulting from the momentum theory w ith rotational w ak e to obtain the main equations of the solution sy stem. The development of this ex pression is show n in W ilson and Lissam an 1974, but it w as not used to obtain the m odel equations, since the objective, for these authors, w as the treatment of rotors w ith tip-speed ratio g reater than 2, as is the case of the electricity-generating fastrunning turbines. The present w ork , for the first time, a t least for the authors’ k nowledge, presents the development of the sy stem equations of a m odel that is valid for the w hole ex tension of the tip-speed ratio ra nge.

2. STRIP THEORY EQUATIONS Considering the hy pothesis of the strip theory described previously, by applying the forces balance developed over a rotor blade section airfoil (see fig ure 1), a nd mom entum, mom ent of momentum and mass conservation principle on an individual stream tube, the annulus flow equations are obtained a s follow s:

Figure 1

Velocity diagram for a rotor blade section

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a 1-4 a9 1+4

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( )( ( )( s

=

r

CLcosf + CDsenf sen2f

4

s

=

CLcosf + CDsenf

r

4

tgf =

senf cosf 1

(1 – a)

x

(1+a’)

) )

[1]

[2]

[3]

w ith

V0-u

a=

[4]

V0 v

a’ =

[5]

V rV

x=

[6]

V0 RV

X=

[7]

V0

w here a a nd a’ are, respectively, the axial and tangential induction factor at the rotor; u and w are, respectively, the a xial flow velocity and the flow angular velocity at the rotor plane; V 0 is the free strea m w ind velocity ; V is the rotor angular velocity ; s r=Bc/ 2p r is the local solidity; B is the number of blades; r is the radial coordinate; c is the blade chord; C L and C D are the sectional lift and drag coefficients; f is the angle between the plane of rotation and the relative velocity, W ; X is the tip-speed ratio; x is the local tip-speed ratio and R is the rotor radius. It is important to note that these equations w ere derived by using the usual hy pothesis that

a=

b 2

and a’ =

w ith

b=

V0 – u1

b’ =

V0 v

1

b’ 2

[8]

[9]

[10]

V

w here b and b’, are, respectively, the axial and tangential induction factor a t the rotor w ak e,

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and u1 a nd v

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1 are,

P E R F OR M A NCE P R E DICT IO N

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OF

H AW T

U S I NG ST R I P

T H E O RY

respectively, the axial flow velocity and flow angular velocity at the rotor

w ak e. By using these equations a nd an iterative procedure it is possible to compute torque, thrust and power for each blade section. By integration along the radial coordinate torque and power developed by the w ind rotor are obtained. Thus, the power coefficient can be obtained by

8 Cp = 2 X

e

X

a (1 – a)(tgf – e) (1+etgf )

o

x2dx

[11]

H owever, some points deserve consideration: i) the relations [8] are valid only for X ³ 2, that is the case for fast w ind turbines as m entioned previously. Nevertheless, the w a k e rotation effect is very important for slow rotors w ith low X values (X