DETERMINATION OF HORIZONTAL AXIS WIND TURBINE PERFORMANCE IN YAW BY USE OF SIMPLIFIED VORTEX THEORY

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1 5 th INTERNATIONAL MULTIDISCIPLINARY CONFERENCE DETERMINATION OF HORIZONTAL AXIS WIND TURBINE PERFORMANCE IN YAW BY USE OF SIMPLIFIED ORTEX THEORY Piotr Strzelczk, PhD Eng. Dept. Of Fluid Mechanics and Aerodnamics RZESZÓW UNIERSITY OF TECHNOLOGY Ul. W. Pola Rzeszów Poland Abstract The paper presents an application of non-iterative lifting line theor of Horizontal Ais Wind Turbine(HAWT) to determination of HAWT s performance. The presented model is developed on the basis of modified Witoszński s propeller theor presented in [4]. In this model an iterative determination of induced velocit field is avoided thanks to solution method similar to as in Witoszński s propeller theor. The Goldstein kappa factor or Prandtl tip loss factor are introduced to the equations of momentum and angular momentum for ideal HAWT. Ideal aial force (thrust) and ideal torque are determined using blade element method (BEM) (profile losses are neglected). This values of thrust and torque are compared to those obtained from momentum and torque equation. Thanks to approimation of lift force coefficient vs. angle of attack b sine curve one ma obtain a quadratic equation for aial velocit component An etended version of the model including nonuniform inflow is presented. In the theor a quasi-stead approach to blade element characteristics was applied. The calculations are compared with eperimental data obtained at Risφ 1 kw test turbine. Presented results show that, the method described in the paper underestimates performance for low speed winds, whereas for strong winds the power output is slightl overestimated. Ke words: Wind turbine, orte theor, HAWT, Nonuniform inflow, aerodnamics 1. SIMPLIFIED ORTEX THEORY OF HAWT ROTOR UNDER NONUNIFORM INFLOW CONDITIONS It is well known that in real conditions rotar wings (airplane and marine propellers, helicopter rotors and wind turbines) work in nonuniform inflow. In case of HAWT this nonuniformit is caused b man different conditions.in this, stud the following effects were considered: 1. Yaw angle or tilt angle;. Wind velocit profile; 475

2 It was also assumed that blades are infinitel rigid, so flapping motions, and blade torsion could be neglected. Momentum and angular momentum for circumferential varing velocit field can be epressed in following form: 1 4π ρ W BcC φ ψ ρ L sin rdrd Xr Fdrdψ (1.1.) ( cos ) 1 4π ρ W BcC φ drd ψ φ Lcos X W X rfdrdψ (1..) where: ξrr, r-current radius, c- blade chord, F-Prandtl tip-loss factor, R-propeller tip radius. The sectional lift coefficient is given b the formula: C L ( ) a χ sinα p (1.3.) a ( dc d α ) -lift curve slope in its linear portion for airfoil section. L From eqns. (.1.) and (..) we obtain: tan φ ( wcosφ ) (1.4.) plane of rotation - T Ω + / υ W α o φ dl dr dd zero-lift line β (o) Fig 1.1. elocit triangles and forces at the blade element at awed conditions. elocit triangle gives us second (see: Fig 1.1.) eqn. for tan φ : tan φ Ωr w sin φ cos ψ+ (1.5.) B comparison of (1.4.) and (1.5.) one ma establish a formula epressing dependence between aial velocit at actuator disc and tangential induced velocit: ( λ φ ) cos ξ λsin φ cos ψ Angular momentum equation (1.1.) ields: (1.6.) 476

3 WσC L F (1.7.) Wkere: σ Bc/( πξ) -local disc solidit Inserting (1.3.) and (1.6.) to (1.7.) and neglecting higher order term, proportional to square of swirl velocit one obtains quadratic equation for. The solution of this eqn. is: A λcosφ A λcosφ + + A ( ) ( ξ λsin φ cos ψ) tanβ (1.8.) Where blade element shape coefficient A takes form: ( ) ( sin cos ) cos σa χ ξ λ φ ψ β p A 4 σaχp ( ) F + sinβ 4 When is calculated one ma eas determine inflow angle: φ arc tan ( λcosφ ) Shaft power coefficient can be calculated as: (1.1.) (1.9.) ψ Ω c / υ dr ψ φ cos sin φ cosψ ψ9 sin φ ψ7 r ψ18 ψ Fig 1.. HAWT rotor at awed inflow conditions. Schematic. 477

4 P π 1 3 X W σ( C sin φ C cosφ) ξ dξdψ (1.11.) C L D ξ Rotor drag coefficient is given b formula: D h π 1 X W σ( C cosφ+ C sin φ) ξdξdψ (1.1.) π C L D ξ h where: ξ h -non-dimensional radius of hub, C D -blade section drag coefficient (two dimensional airfoil), XωR/ w is the tip speed ratio, Xλ -1. COMPARISON OF THE THEORY WITH EXPERIMENTAL DATA To compare presented theor with eperiment, 1 kw Risφ test turbine eperimental data were chosen. One of the blades of the HAWT was instrumented, to provide measurement of agles of attack, as well forces acting on selected blade segments. There were possibilit to obtain long time series( up to 6 sec.) of the data like: angle of attack at r.71r, wind speed, rotor aw angle, power output, rotor speed, and normal as well as tangential forces acting on three selected blade segments. The detailed description of the facilit as well as eperimental data ma be found in [3] This facilit was chosen because of airfoil characteristics for wide range of angles of attack are available [3]. It was also important that mechanical and electrical power curves were given. The static airfoil data were corrected for stall dela due to rotation. The correction was made b use of empirical stall dela model proposed b J.L. Tangler and M. S. Selig [5]. Figure.1. shows mechanical power generated b constant-speed (n47.5 r.p.m.) rotor for various wind velocities. Comparison of eperimental curve with results obtained for tilted rotor are in good agreement with eperimental data. However, for low wind velocities the presented theor underestimates power, but maimum power value is slightl overestimated.it is worth noting that for low wind velocities there is no significant difference between power calculated for tilted rotor and rotor in aial inflow. The appreciable difference appears onl for high wind velocities (over 1 m/s). 478

5 1 P [kw] aial inflow coditions rotor tilt angle: 5 [deg] eperimental data w [m/s] Fig..1. Shaft power curve as a function of wind velocit α 71% AOA time series calculations eperiment t [s] Fig... Measured and calculated angle of attack time series w 8.4 m/s (1) α 71% AOA time series eperiment calculated static stall a.o.a. t [s] Fig..3. Measured and calculated angle of attack time series w 14.1 m/s () 479

6 Database included to report [3] has provided oportunit to compare calculation results with field rotor measurements. The results of calculations in confrontation with eperimental data are depicted in figures...3. As far as wind profile is concerned, it was assumed that power law (.1.) is fulfilled: ( ) ( hub ) [ ] a H H H ub (.1.) w w h For the presented calculations the value a.6 has been chosen (terrain covered with numerous small obstacles). 3. FINAL REMARKS AND CONCLUSIONS The results presented above show good agreement with eperimental data. However, for angle of attack series one ma see that there is a time shift between calculations and eperimental data. This ma be caused b the fact that five-hole Pitot probe used for measurements was forwarding the blade in azimuth [3]. It was found that the model of HAWT has a limitation ensued from phsical conditions: for constand-speed Horizontal Ais Wind Turbines the relative induced velocit ( induction factor ) a( - w )/ w increases when wind velocit decreases. Calaculations has shown the model fails when for an blade element the induction factor reaches value about a.6. Of course, the eact limiting value of a depends on geometr of blade element. However, for this regime appeares a specific work-state called in helicopter aerodnamics vorte ring state. Hence, simple theoretical model should not be appllied. 4.REFERENCES 1. Glauert H. The Elements of Airfoil and Airscrew Theor, Cambridge Universit Press 1948;. Prosnak W. J. Calculation of the propeller performance, Technika Lotnicza (The Aeronautical Technolog) nr 5/1954 (str ) in Polish; 3. Schepers J.G. et al. Final Report of IEA Anne XI: Field Rotor Aerodnamics,ECN- C--97-7, June 1997; 4. Strzelczk P. Modification of the Witosznski Propeller Theor: Influence of Finite Number of Blades Prace Insttutu Lotnictwa 3/1996 (146) published b The Institute of Aviation in Warsaw (pp ) in Polish, with English and Russian abstracts; 5. Tangler J. L., Selig M. S. An Evaluation of an Empirical Model for Stall Dela due to Rotation for HAWTS NREL/CP , Presented at Windpower 97 Austin, Teas June 15-18, 1997; 6. Witoszński Cz. Selected Papers, PWN Warszawa 1957 (pp.19-45) in Polish; 48