aerodynamic analysis of hawt by use of helical vortex theory

8/8/2019 Aerodynamic Analysis of HAWT by Use of Helical Vortex Theory

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Aerodynamic Analysis of a HorizontalAxis Wind Turbine by use ofHelical Vortex TheoryVolume I-Theoryiuaaa-c~- oaosq acnoam anzc nans%sx3 oa n f185 -299 16gOGX%CB2Ab 9XXS Q f t i G 9QRD9 8H DP GSZ Ol?~ ~ C & ~ ~ , ? LVODTCX 233EQBP, VCEa ISB 1: TPIEObH

P i m Z B c p o ~ t ( 2~9cdoXlaiw.) 88 p U a ~ l a snc ao~spapa01 CSSL o ln GWU I G O I GD. R. J@ng,T. G . Keith, Jr.,and A.Aliakbark!~ansfj&University 01 Toledo

D Q C Q ~ ~sr 3982

Prspavsd forNATIONAL AERONAUTICS AND SPACE ADMINISTRATIONLewis Research CsnlsvUndsr Gonfrac"lN6 3-5forU.S. DEPARTMENT OF ENERGYConservation and Renewable EnergyWind Energy Technology Division


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DOEINASNOQOS-1NASA CR-168054

Aerodynamic Analysis sf a HorizontalAxis Wind Turbine by use ofHelicalVortex Theory

D. 8. Jeb'ng, T. G.Keith, Jr.,and A. AliakbarkhanafjehUniversity of ToledoToledo, Ohio 436'76

December 1982

Prepared forNational Aeronautics and Space AdministrationLewis ResearchCst-rterClsveland, Ohio 44135Under CsntrasMNCC 3-5fo rU.S. DEPARTMENT OF ENERGYConservation and Renewable EnergyWind Energy Technology DivisionWashiplgton, D.G. 20545Under Interagency Agreement DE-A101-76FT20320


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AERBBYP4AMIC ANALYS IS OF A IIBR I ZONTAL AX I SWIND TURBINE BY USE OF HELICAL VORTEX THEORY

(VOLUME E TltEBRY)

byB e R e Jeng T. G . Kei th , J r .and

Report o f research conducted under aNASA Lewis Research Center CooperativeAgreement NCC 3-5NASA Technical Monitor: J. Savino

Department o f Mechanical Engi ne er i ngUni v e r s i t y o f T o l e d oToledo, Ohio 43606

December 1982


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CONTENTS

PageSUb!t4ARY . . e a . a . a a e c e a o . a a a e a ~ a o a a o a 1. . . . . . . . . . . . . . . . . . . . . . . . . . .. INTRBDUGTIBN 2NOMENCLATURE . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2. ANALYSIS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10. . . . . . . . . . . . . . . ..1 I n t ro duc t ion t o Vor tex Theory 10. . . . . . . . . . . . . . . . .. 2 Vortex Theory of P r o p e l l e r s 10. . . . . . . . . . . . . . . . . . . . ..3 General Assumptions 112.4 Caleulat'ican o f I nduced V e loc i t y . . . . . . . . . . . . . . . 13

. . . . . . . . . . . . . . . ..5 Appl i c a t i on of M or lya 's Theory 25. . . . . . . . . . . . . . . ..6 Governjng Equation Formulation 272.7 DeterrnJnation o f ' Induced Angle o f At tac k . . . . . . . . . . a 332.8 Power. Torq ue and Drag a on Wind Turbine . . . . . . . . . . . 35

3 RESULTS AND DISCUSSION e a a o 363.2 Bsuble-51 aded. MACA 23024 A i r f s i . W ind Turb ne Cal culat4 ons . 383.2 Sfng'le Blad ed Wind "Turbine C a lc ul a ti on s . . . . . . . . . a . 51

4. CONCLUSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63. . . . . . . . . . . . . . . . . . . . . . . . . . . .EFERENCES 64APPENDIX A: Revjew o f VarSous Methods s f Wind Turbine AerodynamJc. . . . . . . . . . . . . . .erformance Cal cul a t? n 66


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The theo rcst l ca l development o f a me thod o f ana lys4s f o r p r ed ic t i on o f th eaerodynamic performance o f h o r iz o n t a l a x i s wf nd tu rb ines i p re se nt ed i n t h i srep or t . The method i s based on th e assumpt ion th a t a h e l i c a l vortex emanatesfrom each blade element. C o l l e c t i v e l y t h e s e v s r t l c e s f o rm a v o r t e x s y s te mt h a t exte nd s i n f i n i t e l y f a r downstream o f t h e blade. V e l ~ c i t i e s nduced beyt h i s v o r t e x s ys te m a re f o u nd b y a p p l y i n g t h e B io t -S av a r t : w. Accord ing ly ,t h i s method avo ids th e use o f any in te r f er en ce fa c t or s ' t.tS1ich ar e used i n manyo f t h e momentum th eo ri es . What' : more, th e method can be used t o p re d i c t t hep er fo rma nc e o f w ind t u rb i n e s w i t h a small number o f b la des .

The wind t u r b i ne pe r formance o f a two -b laded ro t o r i s de te rmined andcompared t o e x i%tig experimental data and t o corresponding value s computedfrom the widely used PROP code. I t was found that the present method com-pared favo rab ly w i th exper imen ta l da ta espec ia l y f o r l w wind ve l oe i ie s .

The w ind t u r b i ne pe rfo rmance o f a s i ng l e- b la d e d r o t o r i s d e te rm in eds u b j e c t t o th e c o n d i t i o n t h a t t h e r e was no I i f t on th e counterwc4ght o rsuppor t span. Output power f o r a one-bladed wind t u r b i n e was compared t ot h a t o f a two-bl aded mach i neo


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2 . j PRODUCTBBOver tho past Pew years, i t er ss t i n t he rase o f w indmi l 1s t o produce power

has grown s i g r ~ li a n t ly . I n s up p or t s f t t l l s i te res t , cons iderab l e researchhas been di r ec te d toward th e measurement and pr ed ic t ion o f w ind turb ineperformance. The m a j o r i t y o f t h e t h e o r e t i c a l e f f o r t employed t hu s P ar i sbased on a var ie ty o f me thods tha t have 1ong been used i n th e c al e ul a t i o n o fp rope l1e r performance; c l as si c a l momentum th eo ry a1 sng w i t h a number of b l ade-element theor ies have been u t il zed w i t h va ry9 ng r'egree:, sf PIUCC~SS.

A computer code developed by W i :son and l.lay'~;ri e'5 %e ,a?$St*s~:eU I O ~6 r s I t yS&I],nd en t itl d PROP, i a t presr 8n"tws da3,t t~ - r:Lf:.r-:~i neth e aerodynamic performance o f ho r izo nta l ax is ~ . t l n d . ~ t - ~ \ B t ~ c s ~II c 4de wasdeveloped several years ago as a general purpose .I +a " I%l i ?~99r r ir2:tancecomputer program. Acc ordin gly, PROP has a certain anlbu .c af !?: 8 e- inf l e x i b i l i t y , Fo r example, th e code con ta ins fo u r b lade t i p - lo ss models(NASA, Prandt l , Geldste in , and no t i p - lo ss ) which can be incorporated in t oth e performance pre d ie t jon ; however, th e cho ice o f wkSck model t o use i s l e f tt o t h e u s e r 9 d dis cre tio n.

The PROP code i s based on a modi f i ed b lade e lement ana lys i s ca l l e d th eGlauert Vortex Theory [I].n e s s en t ia l a ssum ptio n o f t h i s t h e o r y I s t h a tv o r t i c e s o r i g i n a t e Prom t h e r o t a t i n g b la de s t o form a k e l ical vortex systemthat passes downstream. O f fundamental importance, t h i s t r a f l i g vo r texsystem induces ve loc i t ies which a9 t e r the f l ow a round the h l ades and i n tu r na f f e c t t h e fo r c es a c t i n g on t i le blades. It should be noted t h a t because s fth e compl ex i y i vol ved th e induced vel oc it ies a re no t determ5 ned d t re c t l y i nt h e G l auer t Vor tex Theory. Ins tead, ax ia l and ra d i a l in ter f ere nce fac to rs areea lel t l ated from which .[;he induced ve lo ci ty o f t he vor tex system on any bladeelement can be evaluated. The ca lcu l a t i o n o f these jn te r fe ren ce fac to rs Ss


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sPmpl i f i e d by assuming t h a t t h e r o t o r has an i n f i n i t e number o f $1 ades. Th i sassumption removes th e c omp lex i ty a ssoc iated w i t h th e p e r 9o d ic i 3y o f t h eb lade f l o w and p ermi t s d i r e c t app l i c a t i o n o f momentum theory f o r t he eva lu -a t i on of t he i n t e r f e r enc e v e l oc i i es . However, f o r s9 ng1e and dou bl e-b l adedr o to r s , o f p r l n c ipa? i n t e r e s t here , t h i s assumpt ion i s que s t i onab lem

Usars o f ;he PROP code have f requent ly encountered problems wi th thec;nvlrat*clait~:cp :)f th e numer ical procedure i n th e preogram. I t has been Poundt h a t ~ a d 2 ; e r t a i n o p e r a ti n g c o n d it io n s o f a wind t u r b ine , p a r t i c u l a r l y t hos ef o r I arge t I p speed r a t i o s [ gene r a l l y s ma ll w ind ve loc i t i e s ) , t he 1ocal va lueo f t h e a x i a l i n t e r f e r e n c e f a c t o r c an exceed a t h e o r e t i c a l l y 1 m i t i n g v al uef o r a i l t i p - l o s s models. When t he ax i a l i n t e r f e r enc e fa cto ? l i e s i n t herange 1/2 6 a 6 1, t h i s s t a t e i s known as a t u r bu lenc e wake o r v o r t ex r i n gs t a te . Under s uc h a c ond i ti on , t h e v e l o c i t y i n t he f u l l y dev eloped wakew i l l then revers e d i r e c t i o n and f lo w back toward th e rotop. Therefore, th emomentum-blade element th eo ry ceases t o be val id , and i t i s t hen nec es sa ryt o m o d if y th e a n a l y si s by us in g G la ue rt ' s empira"ca1 data. These empiricalco rre ct io ns have been developed f o r th e momentum-bl ade element th eo ry andd is c us s ed i n some de t a i l i n [2,41, however, th eo re t ic a l performance e s t i -mat jons i n the reg ion a re suspect even when cor re c te d by empirical methods.

Exper ience has a1so revea led f rom comparison s f PROP code pred ic t i onsw i t h experimental da ta t h a t f o r a1 1 b l ade t i p - l os s models, PROP r e s u l t s t e ndt o underest ima te measured power outp ut b oth near "cu t- in " and rate d windspeeds.

Because o f t h e i n heren t 1 m i ta t i o ns w i h th e momentum-bl ade elemen ttheory and because o f t h e need t o develop an ana ly t Jc a l method f o r p r ed ic t i n gth e aerodynamic performance o f wf nd turb in es w i t h a small number o f blades,i t was decided th a t a di f fe re nt and more complete method o f anlys is was 4n


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order. The purpose of Lhi s report i s t o describ e the develoyxlient of themetf~od ?rtllcta Js t ~ r m b d v ~ t ~ ~ t o xake medel.

I n t h e a n a l y 8 1 ~ ~ile *induced ve!ocity ' i s dl'reetly caleulatsd byintegration of t h e B i at-Savart Iav under* the assumption t i l a t a f i 1ament oft Eae tra'i -i g vortices P s he1 i ca l it1 form, extends -f nf i n'i tely downstream ofthe rotor, and has a constant diameter. I t i s also assumed th a t the he1 icalvartsx system produced a t the blade ' i s travelqingd~wnstream i t h n constantvelocl'ty. This velocity i s equal t o the value a t the roto r blade, tee. , theinflow velocity through ro to r' s plane of rota tion . This imp1 ie s t h a t thei nt erac tjo ns between wake elements are ignored. The 1 fti ng 1 ne theory 4 sused t o model th e blades.

An analysls an d a computer program for predicting the aerodynamicperfornlance of a hcsr.e"zsntalwind turbine by using a vortex wake method hasalso been developed by R. He Miller, et a1 [3]. The difference between thatanalysis and the present one i s in the m~delingo f t h e circulation around th eblade and i n the vortex wake and J n the method of integrating the Biot-Savart1aw t o evaluate local induced ve locit ies . In [3], the bound circul at isn %orblade Js modeled as a step functl'on. Th f s means t h a t each blade i s divided1nlo a number s f spanwi se se ctions, wh-ich are, in general, of urequal l ength.Each blade se ction i s then represented by a bound line vortex sf constantstrength, Si nce jump d iscent5n u t ? es ex$s t t n eir cu l at is n between adjacentspanwi se blade sections, a t r a i l n g 1 ne vortex must be generated a t each ofthese points. These he1 i ~ d l ortlces a re furt her approxl'mated by $7. s~t.g.!r-:Ba?strajght line vertex segments, each spanning an ang le oP tQo o r 15* overwhich int eg ra tio n i s performed, En the current analysSs, a continuous vari-a t ion of ~ J r c u l a t t o n t1 th e spanwi se dJrectJon is used and the 4ntegratJon 4s


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NOMENCLATURE

axia1 interference factorro ta ti onal i nlerferonce fac to r1 f t curve s lope + 3rnagntitude of vecto r ( s - ros f ne serfes coefficientsdr'stance shown 'i n Fig, A-5l i f t coeffier 'ent for zero -Incidencectiordm a t r i x defined by Eq. (2-68)drag ~ o s f f i$et? i1 f t coef f i c ien tpower coefficient [P/(? l?pv~%' ,3)1dragparameter def i ned i n Eq. [A-36)general funstionsaxial force or drag on w ind turbJne

reducaf;9n f a s t a rfunction def ined i n Eq. (2-71)parameter defined as h = r tan$u n i t vector a7ong X a x i sinduetion factoru n i t vector a1sng Y ax i su n i t vector along Z axis


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ser4as Pndoxnon-dqmanst onal cr'rculats iondi str.il>utio nfunction1i f tmass f low r a t snumber o-f' bladespressuredownstream pressurepowerrated electrical powergenerated e l ec tri ca l powerpower produced by rotortorquenon-dimensi onal torquedista nc e a1 ang th e b l a deradius o f rotore f f ee t i ve rad iu sradius defined as R L = R cos$posftfsn vector from the origin t o a n element s f the

trai l ing vortex f l amentgap width between vortex sheets shown S n Fig. A-5vel oe ity of airf low through rotorradial component of velocltycir cumterenti a1 cemponent of vel sc4 t ymean vel oei t yw i nd vel ocl t ydownst eam vel oci t yf tiduced vel oc i t y


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vel oeei2:yof rf g id he1 i co* idaI vortex sheetnormal csmpsnsnt s f S nduced vel sc i tytangential component of i ndue ed vel scltyeonrponent o f i n d u e s d vel oc'ity i oa X direc t?oncomponent o f 9 ndue ed valoci ty i n Y dirgetjoncomponent s f induced vel oei ty J n Z directionresultant vel oc i l yu n d i s t u r b e d vel ocit yX axisY ax i srearward distance from the rotor , Z a x i sangle s f a t t acke f f e c t i v e angle s f a t t ackgeomatrqc angle o f attackinduced angle o f attackblade anglevortex strength

Jnlesval around a s i ngul a r i t yan g l e defined Jn Eq. (2-43)vector def ined i n Fjg. 2-2transformati on variab'le de fi ned in EQ, 2-74)pjt ch angle s f a he1 ix-ical vo rtax filam entazimut ha1 varS abl e of a he1 ica l vortex f i 1amentmeasured from 0kang le between the X - a x i s and t h e k th b lade


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vospeed r a t i o , ---RQrDspeed r a t i o , --vo~ l r 'speed r a t i o , ---VQ

Dl?t i p speed r a t i o , ---\Ponon-dimens f onal d-istances a1ong th e b l ads

parameter de f-ied by Fq.A-36)dens9 t yr o t ~ rol i d i t y [ N C / ( 2 m R ) ]1ocal sol id i t yparameters defined by Eq. (A-43) and Eq. (A-44)angle de f j ned by Eq. (2-26)vel ocPty potenti a l

con9ng angler o ta t iona l speedrota t ional speed impar ted t o t h e a i r


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From *ideal flsw t h ~ o ~ ~ gt may bc recallad t h a t vorlletity $ 5 dcf jncd 3st w i c e th e anger1clr v ~ l o @ ~ ! t y% an ele l~~aatf fluxid. A regiesn eontainjlag aootacantrated amount s f v s v t l e i t y 'is gal3 Pcd a V @ P % ~ X 13d a v8r t~ i~s sa id to6aave a s t r ~ t ~ g t ! ~qual t o the ctrcu?atfsta aroua~ei he vortex at3@& F S B ' B ~ ! I D I ~ Zt ! i lssr~ns E7LB aro a set ~f ~ ~ r ~ p o s ~ t 0 ~ f a sh a t apply t o vorticas and vortexno@[email protected]. tkcooqenstates @laat t a an 'inv'isctid flow wl th jn whblch t h eforces a re ssnsarvat9vs and the p296ure is only a funct lon s f BansPty,rotatPsn sf Flu id par t ie l es, wh4cR travel a1sng stream j lnas, i s p e r ~ an a t ~ t ,Fson th-is thaoren, s f t en reforred %sns t h e vortex eantfnua!i;gr theorem, i tbewmss sv-ident g h a t a vor'ex aSa"1~montcan~ot nd abruptly, and, un lass thefflanent farms a elasad pa t h , &it ust sxtsnd oPthar t o f n f %n4 tyor t o t h eP-ield that. the fJ 1anent generates. This velseity 4 s commsnly referred t o ast acl l~aduecdvc."PscOty. The '8"ndkaeed vsl hacity can be ealcesl a"cd froon theCiot -Savar t l a w %83as

-3. -Pwl-risve dw j J s th e *induced velseity due t o a vortex e l e m ~ n t ,o f l engtll ds-3.and strength y, and r i s the pos i tJon vector from the poin t a t which the

induced VEIc'it.y i being ealcul ated t o the eleiwnt o f the vortex.2.2 Vortex Theory of: Propellers

The vor tex @ ! ~ e ~ r yf propellers i s based on the concept t h a t eache l ~ m sn t f the bl ads can be treated a s a two-diinensirtnal airfoJ1 section


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s~b je6 t@aa GQB I )%C ? J ~86a? resul t a n t velocity W and t h a t the Iocdl 1 f t sf'th(? l ado olemetat, QL, (S s as8068ated wit11 the efreul a l l on r around thecontour o f the c?9r3f0'i'D. Tho ~ Q E ; Q ' B resul tant veloc'ity ' i s found from thevector sun o f "9aa f r e e stream v b ' P ~ ~ $ t y ,h e clrcufifsrsnt-!a? veloeIty 0s thes ~ c t 4on nd the t nduced 41sw vel oef ty. Tha Kutta-Jsukowski tt~eoremg93, f o reach blade elemsnd, may be written

where a 4s the Iscal dsns i ty of the a i r and d r 9s the elemental bladeI cagth. In general, th$circul ation r around a blade wi l l vary a1ong the

dl'blade sljara. A the cireulatiepn changes tlv the amount - r between thed rpoints s and r + d r along t ho blade length, i t f s l lows, frrsm t k a vortexeontinu-ity princi p9 e, t h a t a t r a i l i n g vortex f i 1ament o f the same strengthcrnanatss f rsm the b l a d e el emsnt and extends i f inttely Par downstream of theblade. Because LraP 1f r ~gvo rt t ees emanate from every po int al ong th e ro ta t ingblade, they form a vortex sheet o f approximately helical shape. The number ofsuch sheets i s equal t o t h e number s f blades, Due t o t he t ra i l ing vor texsheets there 'Is an fndueed velocity dist~*!bution, w i , i n tile plane s f rotor.ThSs induced velscity can be used to obtain the loca l r esu l tan t ve l~cf tyth a t ilnay i n turn b e used 9n th e Kratta-JoukovrskJ theorem t o determine the l i f ton a bl ads element.2.3 General Assumptions

The princfple assumptions on which th e present analysis of theaerodynamjc performance sf a horizontal axis wind turbine is based may bes ta ted a s follows:

I ) t h a t the a i r ?low -Is incompressoibleand invtscid,


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2) t h a t th e Tree stream vs l scq i t y Ps al~vayapara1 a1 t o th o rstot* shaf tcelater%*iincnd un.iforn a long the span,

33 t h a t t h e r s l a t i v e ve%oeJty0F a b la d e e l ~ m a n t o the a i rs t ream i si snl$h=a?do the ve loc-ity Lhat t h e blade el oment wouf d ttavs were *itplaced Pn a tws-d9merns9anal stream w i t h an - i den t i ca l r e la t iveve loc i t y ,

4) Lhat t h e t r a i l 3 ~ gor tex system i s ne l:eo ida l w i th constan t p i chand diameter, ex tend ing in f in i te ly fa r downs t ream o f th e blade, andmavf ng wS t h a cons tan t ve l oc i t y determined a t t he r o t o r (semi-r i i dwake), and

5) t h a t the hub bas n@e f f e c t on th e rotor Flow.Unl ike sther vor tex t ! ieor iss & , g o , Golds te in 6101 and Theodorson 6113

theor ies, t h e present method of a na lys f s , wh4eR WP 11 b% termed th e he1 calvor tex method, does not r e s t s f s t t h e e i r c u l a t i a n along t h e b la de t o t h a t s fan optimum dl'str'e'butfon co r ~espond ig t o a r i g i d he1 ca l vo r tex system movingbackward from the blade w i t h con stan t ve loc ity, However, l lee other vor texmethods, th e cur ven t ~ e t h s d oes assume t h a t sl ipstream expansion i s s fsecondary impotatance and thus may be neglected. S1i stream expansi on 5 s duet o the gradual decrease o f ax ia l v e l oc i t y beh%nd be r o to r \ v l~ ieh s caused byt h e decrease r"as pressure benind the ro tor . The decrease i n a i r p ressurebehind a w5nd turb ain e and th e Sncrease i n aPr pressure Immediately i n f r o n ts f r o t o r i s a d i r e c t r e a c t i o n o f t h e a x i a l force on the wind t u r b i n e , Ltshould be noted t h a t th e fi r s t , second and thS rd assumptions i n the preceding1Sst of assurnpljons a r e i d e n t i c a l t o those used i n G olds tein and Fheodersqntheor ies. Further, no hub . i s permissJb9e i n e ?t he r 05: these thear iss; a1 3i l e g r a t f s n s must begin a t t h e o rb i g i n and no t a t t he hub radjus.


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The physsica'9 avrangemen'c o f t h e problein a t hand i s -i l l tnst ratsd i n F i9.2-2. Tho cssrd' inate systems f o r t h e a~ a l ys ' i a re shown *in Fig. 2-2. The@av*tes*ian oosd-i t~atos , Y, and Z a r e f i x e d i3 a space and were chosen suchel la t the It-Y axes form a p la n e o f r o t a t i o n and t he Z-axis i s t i ~ s x?"% fr o t a t ion. Moreover, t h e Z coordinate i s t he d i s tance measured From th e r o t o rt o a segment o f t r a i l n g v o r te x p ara 1 s 1 t o t h e a x i s o f r o t a t i o n o f t he ro to r .The second coordinate system used i s a c y l n d r i c a l c oo rd in a te system (r*, ssz )which i s f i x e d t o th e r o ta t i ng blade. The r - a x i s i s d i r ec ted a long t he b ladespan, the 0-axis measures the azimuthal angle measured f r o m th e kt11 blad e andt h e z - ax i s i s t h e a x j s o f r o t at i on . To f a c i l t a t e t h e a n al ys is , t h e $ la d ewhose induced v e l oc i t y i s t o be calculated, i s assumed t o be cor'nc4dent withth e x - a x J s as shown I n Fig. 2-2.2.4 Ca lcu la t i on s f Induced Ve lo c i t y

As has been mentioned, i t s assumed th a t the vor tex P i ament L raJ I lngf rom a b l ade e lement extends i n f i n i t e l y f a r downstream o f t h e rotor .Assec .ia ted w i th t h i s vor tex i s a v e l o c i t y field, commonly re fe rr ed t o as t k ei duced vel ec1 y. Th i s ve l oc i y f i e l d can be s a l cul ated f rom th e Biot -Sa var tlw 31.e., equat ion (2 -1). Th is law i s s ta te d i n vec tor d i f fe re n t ia l fo rma s

+I n t h i s equat ion, dwj i s th e d f f fe re n t i a l ve1ocit;y induced a t a p o f n t r ' on3th e b lade due t o a segment, do , s f t h e t r a t l n g v o rt ex f i l am e n t w hich had

d rcsrl'ginated a t a p o i n t r on t h e k t h blade, Also dl?, o r -dr, i s the change i nd r


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Fig. 2 4 Physical model of the problem


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@Prsur'Bat'ionbetween t h e points r and r -1- dr along t h o blade, w91-ieb-r s aqua1t o thc cjrcul a l i on o f ttae t r a i l Jng vor tex f ram the b l ad s element dr.

" +Vectors s , , nd tlae d iQfarent i 1 vortex f.i 1anent 3 ength , n, a r estlawn J I I Fig. 2-2, and may be expressed as

I n tiaesa expressions 0 9 s t h e azimuthal angul ar var iable o f th e he1 i x measuredfrom the I:th blade and f s t h e p i t c h o f the helix. B o t h a r e shown i n Fig.2 t e t t j n g h = r tan$ f o r sfmp lie i ty and not ing th at @ i s r e la te d t o th ei n f l o w ve loc i ty U and t h e r o t a t i o n a l veloci ty s f ro tor , Q, by

i t follaws t h a t

Thus, equations (2-5) an d (2-6) may be wri t ten

Using these equations, t he cross product in the numerator o f equation (2-3)


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can be evaluated as follows

A 1so,

+L e t t i n g A = I f - I for simp1 icity, no t i ng that d r j can be expressed interms s f 4 ts components as


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Fig. 2-3 V2ba2Bty diagram sho~vlargnduced ve i~c i ty nd its components


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Lr2 rr' cos(o + o k ) ] u 8 + h Lr cos(o + ok) +dwn =-5i I:-1 A3(1 ,112)?/2I f the coning angle Y , shown in Fig. 2-4, is taken J n t s account, t h ecomponants o f t h e indanced velocity may be expressed as

d r 00z7 dr [rz - rr' cos(0 + e,)] u' + h [ r cos(0 + ~ 1 ~ )dwn =-l ~ b = 6 6Y A3(1+ v ' 2 ) 1 / P

re sin(0 + 0 k ) - r ' ]d0


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Fig. 2-4 Effect of coning angle onMade element forces


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Nsn-d~imensfsnalCzit ig he $*!stanceparamstsrs by t h e r o t o r r ad iu s R

and combbi i ng w i th equat i on (2 -19) y i e l ds

A1 so,

where Xo, t he speed ra t i o , i s def i ned as

By ap pl yin g equatfons (2-33) through (2-38) to equatfans (2-31) and (2-32),t h e d i f f e r e n t i a l components of induced ve l oc i t y can be r ew r i t t en as


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Phs t o t a l csmponak~tso f induced vel o e i t i e s can be obtained by i t e g r a t i n gequat ions (2-39) and (2-40) along th e blade span from t h e non-dimcnsSsna1hub radius, QheDb, o t he non-dimsnsional t i p ra diu s, I. Ilowevar, M ~ r i y aproved t h a t wt i s always zero a t any point on t h e blads. I-lence, t h e t o t a lfndueed v e l o c f ty a t p a j n t re o f th e blade due Lo a l l t r a j l ng v o r t i c e so r - i g i na t f ng Prom t h e bl adc i s ,

E t can bs seen t h a t d i f f i c u l t y a r i s e s when t h e j n t e g r a l i n eq u at io n (2-41) 5savaluated a t the p sOn t 6 = 5 ' when Q ok = 8, In th is case, th e i n t e g r a lbecsriles *if i t s i t e l y l a rge , and because o f t h i s sJ n gu l a r i y , t h e .lurtegral i 11equat ion (2-39) can not be accurately determined i n the ne ighborhood o f t h i spa r t i cu?ar po int . However, t h e probl em can be res o l ved by app ly ing Moviya' sTheory 663 which i s d es cri be d i n th e next section.


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Woriya g43 has developad a simple method for determining thedsv~nwash~ 1 aeOty. Thi s was aecompl ished by in tr od uc in g an a dd it io na lparametea" t o t h ~nalys9s. Tl~ l ' s a c t o r * i s defined as t he r a t i o o f t he fnducedve loc i t y , due t o a he l f ca l vor tex, t o th e induced veloc. i ty , due t o a s t r a i g h tvortex, o f the same strength iae,

where dwn and dwn1 are i nduced ve loc i t i es c rea ted by he1 cal and st ra ightvo r t l c s s r sspsc tl ve l y . The f ac to r I s app rop r ja te l y termed an i nduc t i onf a c t o r and may be considered t o be a continuous function.

I nduced ve loc i t y a t p o i n t in ' o f the b lade due t o a he1 cal vor tex sheetemanatf ng from p o i n t r o f t he blade can be obtained from equation (2-39).The ve loc i t y i nduced a t po i n t r ' of th e b lade due t o a s t ra igh t vor tex sheetemanating f rom p o in t r of t h e b lad@ and ex ten din g i f i n i e l y f a r downstreamo f th e $1 ade can be obta ined by app ly ing the B io t -Savar t 1 w. The f o l s w in gapply f o r a s t r a i g h t v o r t e x l i n e


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=I= =>I n theso rslat-ions, G 9.3 tias angle Eaotr5sen t h e vec to r (s - r ' ) and ttx-I=normal t o dq. Introduc.ing equations (2-43), (2-44) nod (2-45) .into equation

'ii(2-2) ratad .ikat@grattiflgrom 8 o t h e 4nQucad vel oc i y a t t h e p s ist an'2'~f t h e blade due t o a s t r a i g h t vs~tsx One t h a t emanates frsa po4nt r om?h eblade and LraJIs 9nfJnjGely f a r dawnstseam s f t h e b l ade can be expressed as

I n terms of non-dimens-ional ze d radial d i st a nc e s t h i s e q u a ti on may bew r i t a n

I f t he b lade I s r o t a t e d by a coning angle I#,equat ion (3-47) becomes

By csnbtn ing equat ions (2-42) and (2-431, t h e element o f t h e normal 4nducedvelocity due t o a helical vortex sheet orjgJnatSng a t p o i n t r on t h e blademay be written as


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An fnductPon f a c t o r wk4ch * is a f u n e t h n sf 6 , 5 ' and A, only i s d eriv ed ,usPng aquations (2-39) and (2-4-91, s

I t should be p o i n te d ou t t h a t the i n te gr a l i n t h e above equat ion becomesi n f i n i t e l y l a rg e as % approaches 5 ' and at 0 -- '3k 0. Thus, i t wou ld appeart h a t t h e j n d u c t io n f a c t o r ca n n o t be determined wi th any accuracy f o r a smal lsegment i n t h e neighborhood o f 6 ' . However, s in c e th e i n du c ti o n fa c t o r i s acont inuous func~Son, by v i r t u e o f i t s def - i r i t io i l , -it ecn be apprsximated byi n t e r p o l a t i o n i n the neighborhosd of 6 = E ' , Numerical va lues f o r thei duc t ion Pactor were detsrmi ned by us in g Laguerre-Gaussi an quadratures.D e t a i l s o f t h a t i n t e g r a t i o n may be found i n vo lume I 1 o f t h i s r ep o rt . Havingdets rm8ined he d4s t r i bu t i on o f t he i n duc t i on f ac t o r a l ong t he b lade , t hei nduc ed v e l oc i t y a t any po i n t 6' on th e b lade can be obta ined by i n t e g r a t i n gequat ion (2-49) rom t h e hub t o th e blade t i p , i.e.,

YhS s i n t e g r a l c an be numer i c a ll y ev a lua ted p r ov i d irig an appro pr iate Punct i onf o r c i r c u l a t i s n s i s 'in hand. Th i s s t ep w i l l be per fo rmed i n t h e nex t twosect.isns.

2.6 Gsvern4ng Egu ati n Formul a t is nThe govern ing equat ion can be fo rmula ted f rom the fac t t ha t i f t h e b l ad e

s e c t i o n i s p o s i t i o n e d a t a g e om etric a n gle o f a t ta c k , ag, r e l a t i v e t o t heu n d is t ur b e d in co min g f l u i d v e l o c i t y W ' , t s s e t t i n g r e1a t i ve t o t h e r e s u l t a n t


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vc loc r ' t y U, 'is ng dSmineisbed the downwash angle, Fig (2-5). Thus, t h i ssqeaation may be \we i t tan as

ag (x e - a j (2-52)where ae i s e f f e c t i v e angle o f a t t a c k a nd ai, t h e induced angle of at tack.The i nduced angle o f a t t ac k may be ob2;a.i ned Prom Ff g. (2 - 5 ) as

-1 rJn".I = t a n -The f a c t t h a t t h e i nd uc ed velocfty, and therefore t h e induced ang le o f a t tack ,-i ftsel a negative rauint~eraccounts f o r t h e negat ive sei n J n equat ion(2-52). The geomstr ic angle o f at ta ck can be obtained from Fig. (2-5) as

-1 Aag = Can (-1 - 6E

where M i s the p i t c h a ng le s f the blade element.The induced angle 04 attack may be expanded i n a power ser ies i n terms

wn *ns f - nd, since t he va lue o f - "s usua l l y small, t h e series may be\I ' W 't runca ted a f te r one te rm i m e e 3

Comb-inf rmg equatl'sns (2-51) and (2-551, t h e +induced angle o f a t t a c k at pol'nt


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Fig. :2-5 WePocity diagram ah~wing he effect of induced veiocity


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Oa6ceYe66C-@i+0QJPms(90>*Pi,wPPaGI20Qsaow.%QC0aeaCC0OF--@0%0Lti

s

ta

*=

3=0w

a3

skP1'U

IhALPVk

mIswI4

w


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Now i n t rodue i t i g (2-61) n t o (2-56) y-ields

From equat ions (2-591, (2-60) and (2-61), and approximat ing the re1a t ivev e l o c i t y y , W, by t h e und is tu rbed ve loc i t y W ' ( i n f i r s t i te r a t io n ) , t hef o l l owf ng equat ion may be w r i t t e n for any point 5 ' a l sng the b lade

D5 - Chubf b s i nL (ae) 50 1 - chubm-1

v 0a ~ ~ d n- P S ~ ~ s e do r t h e F i r s t jterats'caw.SEC!al s cquat i on can be regarded as a re1at io n between ae and the unknown circu-l a t i on caaPP-ieients Am. Thus, by s u b s t i u t l ng equat ion (2-63) i n t o equat ion(2-62) Tcsu. aey,and making use o f equation (2-62) f o r a i , a single elrpressiona"sobtniroed vilafch alllows the unknown Am t o ba datern~inad. f.loweverr, i t st ~ o t c?lwl?ysp o s s ib l e t o o b t a i n an expl-iea't equa t ion f o r the see t ion l i f tcac.FF-licden-;; i n terms s f t h e ssc t i o n ang le o f a t t ack 4von two-dimawsionalair-70.21d a t a o I n t h a t event, t h e scs%ut-is~~ust be obtained numerical ly.

A P f r s t appraximat fon s f t h e %olut*isn o r ae m a y be obtained by assuminga l d near re1a t i on betwee11 section l i t c o e f f icf ent and th e s e e t ion e f f e c t i v eangle o f a t tack ie.,

CL - aoae + bs


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By introducing GCI "; qux2.lr 2 4nto seqeoaLPon (2-59) and rearrang9ng, t h esec t iona l s.l.'.Tccp vvoe c~s s : ^~ o f a t tack *isTound t o be

Combf n4ng cquati ons [2-521, (2-54) and (2-65) y i e l d s

T:JW u~3.i$-Ig ~q~c? ' ; i ~ S 2-61) and (2-62) w i t h equation (2-66) rcsul t s 9 n

Equation (2-67) Is app l i sd a t M l o ca t i o n s of 5 ' . T h u s , a system of Mequat ions a re abtatned which may be s ol v ed f o r t h e Am. S u b s t i t u t i n g t h e Ami n t o equat ion (2-61) 'and introducing the results into equatlon (2-65),produces an approximate value o f ae a1ong the blade.

A bet ter approximation than the above l inear form may be obtained by usingtwo-d4mensPonal a i r f o i l da ta and empl oy4ng the f o l l m i ng i t e ra t ion p rocess :1) Use l o ca l ae obtained from equation (2-65) f o r M loca t ions a long the b lade

t o e al cu ?a t e t h e s e ct i on a l CL f r o m two-dimensional af rfsil data.12 ) C al cu l a t e the 1ocal c i r e u ? a t i o n r : r =- cGICL.2

3 ) Using equat ion (2-61) and values of r obtained in the second s tep , thePol lowi ng m at ri x eq ua tio n may be e s t a bl i s h ed t h a t can be used t o determine


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The second integral on r-i ht hand sf cRe 0s" equat jon (2-761, s t4 11conta ins th e s f ngu l a r i ty ; however, i t may be given an approximatetreatment as fol lowc,. The * ;nduction Pastor i n equat ion (2-50) i s suinnledover a1I blades. Thus, t h a Pastor may be w r i t t e n

I - Ip-I ;?+=., (2-72)where t he subs cr ip ts denote th e cor responding klades. ConsJderirig t h ep a r t cul ar case s f twe-b l aded r o t o r , -1 ,e., N=2, and us ing equat ion (2-71))t h e s econd i n t e g r a l i n e q ua t io n (2-70) can be r e w r i t t e n as

I t should be noted on l y t h a t t he f i r s t i n te g ra l i n th e above con ta ins t h es i n g u l a r f t y , The reason f o r t h i s i s th a t t h e 1 s t b l a de co inc ides wi th theX a x i s and c o n t a i n s t h e p o i n t 6 = 6' and 8 = 01: = 0. Making a change o fv a r i a b l e

11 = E - E la l l o w s th Ps i n t e g r a l t o b e r e w r i t t e n as

The in teg ra l now can be expanded as f o l l o w s


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By sxpanding s i n e and cosine tei-i: ": a power ser ies and drspplng terms o fo r d e r n2 and la rg er and no t i ng C,'. 2 6 1 can be approximated as unf ty (*in t h er e g io n o f t h e s f ngul a r i t y the !;>I .Sea%vsrbort appears t o b e stra"rgkt , thus byd e f i n i t i o n (9~1) in t h e fntpi i3~;17Z Tcstsgratisn, i t ca n lac shown t h a t

5 ' 9 rl - %ub I 5' - Ehubfim) ( s i n (mn)(--1 - &hub 2 - thub 1 - chub 1-6 (2-76)The second integral i n equa9t*isn(2-73) i n wE~i.eh = x can be d i r e c t l ycalc ulat ed, and hence, the induced ang le 0-C' a t tack may be determined,

2.8 Power, T o q u e and Drag On a Wind Turb ineOnce t h e i e r a t i o n ! successful ly conc luded and the e f f e c t i v e ang le

o f a t t ac k c a l c u l ated, r o t o r fo rc es and momentums and potvQr can be ca l eu l atedby app ly ! ng the f o l 1owi ng equ ation s


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Equations (2-771, (2-78) and (2-79) can eas i ly be obtained from the geometryof Fig. 2-6 and noting t h a t CL and CB, the section 1i f t and drag toe-FPicl"entsresp&et*ively, re defined as f 9 j

dl-CL =1 2p~Zcdr

3. RESULTS AND DISCUSSION--A dl'gital computer program was wrjtten t o perform the analysis of the

precedfng section. The flow diagram and the algorithm of performing t h ejteratf on procedures described in the previous section and a 1i st of thecomputer programs will be presented in Volume I1 of this report. Resultsfrom several numerical investi g a t i ons usb!ig the program w 11 be presentedfor bo th single and double bladed machi nes. Computed performance da t a willbe compared, where possible, t o existing experimental da t a and t o resultsobtained from the PROP code Tor the same operating conditionso


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Fig. 2-6 Force components on a blade element


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8,1, Dstsbl eaCil a d a t , VACA 23024 A t \-Pot!1 WB nd Teaa9Eslne C n l eul a t onsI n tEaOs s e ~ t i ~ n ,lao power outpt~t f a dduble=b4 aded r o t o r winti t u r b i n e

w i l l be presented far four p i t c h angles and tw o d 4 f f e ~ e n t otat*isnal speeds.Tho ada p r ~ f fP E? U S O ~ba a11 CQ ' ~ ~ 1t4 on6 9 s ~n NhCA 23024 a*!r f P 1. Sh J spi l r t ' i~ t41k blade se@t!8n was ef1086n baeauso *it1s the blade soet.ion used .int h e e a r l y NASA, 1args, w e ind turbines. PerqtP cnt bl ad c .infsrmat on a'sd* ispl yed t-i Table I. The dimens.iont3 o f t h e $1ado are g ivan %in 4 g. 3-1,and the aerodynamic blada data, 9 . em, t h e 11 f t and drag coef f IcBentcurves, are shown i n Figs. 3-2 and 3-3. These curves ware taken from [12%.Furth er, because port'rons 04 t h e wind t u r b i n e blades may o pe ra te i n t h eon sta l le d eg ion, " whfck occur at b lada angles s f a t tack app~oxPrnately 13' andl arger, %heaJrfof1 data had t o be supplemented wi th ad di t ion al in format ion,Th is high angle o f at tack info rma t ion was taken f rom C a l m HL w i l l be s h w n'8 a te r i n t h f s sec t* ion th a t be t t e r understand ing o f t h i s oper a t i ig regiovr Psu r g e n t l y t ~ ~ a d e df improvements i n performance p r e d i c t i o n are t o be real ized.'Ed: shsuld also be ment ionad that suv*fa~e Q U ~ ~ ~ ~ S Sas been accounted for" JnFigs. 3-2 and 3-3, and th a t t he curves correspond t o a "half rough" (averagebetween rough and smooth cu rve s) values. The Reynolds number i s t h a tc a l c u l a t e d a t t h e wid span s f the blade, hanee, no account o f the Reynoldsnumber var iat ion a long the b lade Js included. Subsequent ev al ua tio ns can,however, s a s i l y account f o r bo th b l ads surface roughness and Reynolds numbereffects sdmply by i n c l ud i n g t h i s data as p a r t o f th e computer inpu t. A t t h epresent t ime, these ef fe ct s were cons4 dered secondary and t h e r e f s re no e f f o r twas expended t o imp1 ement them i n th e computer program.

C i rcul a t ion d i str4b u t ion a1sng the b la d e must f i r s t be deterwi ied i no r d er t o eal cu l a te w i nd turbine performance. T h i z i vol ves eva l uat f on o f theassumed ser je s f o r th e c i r c u l a t i o n as descr ibed i n t h e previous sect ion.


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A sampla of @6aevar3atJori o f t h S s parametar aga ins t nsn-d*imenseiona9 adius 'isp l olted , in F.1 go 3-4 for the opcrati ng cond i t ions sp~e~ iied.

Figure 3-5 was constructed i n order t o compare predicted wind turbinepower output t o exSst+ingaxper.imenta1 data. The experimental d a t a was takenProm NASA Mod-0 t e s t r e su l t s described f m 643, 651 and &12]. Predicted pswaroutput values were computed for a b lade f i x e d a t a pitch angle of 0.Experimental outgvt d a t a were obtained for a blade pftch angle of 0" for windspeeds below 13 mph; a t wind speeds larger t h a n 18 mph, the blades werep.itched i n order t o ma4ntajn the ratad power level of the machine.

I t should be mentioned t h a t t h e pswar output i n Fig. 3-5 i s not rotorpower. Enstead, an em pirjca l re la tion , taken from 653, relat ing the rotorpower t o the a l t e rna to r power has been used. This r e la t i~ n s h l p. e c e u ~ ts ~ rtfae losses in th e power trananiss*ion (gearing, fric tl's n, etc.) and, of course,i s neat appl !cable t o al 1 wind turbins a'nstallatisns. The relat ion is given by

p~ c2 0.95 (PR - 0.075P~) (3-1 1wherePG = generated electrical porrer, kWPK = power produced by the rotor, kWPE = rated electrical pswer, kW

From FJg. 3-5, lit can bee seen t h a t i n general there *isexcel le nt agree-mcnt between pr~djcted nd mean values of o u t p u t power. FurEhern~re, hepresent method p red i c t s a "cut-in" wind speed o f 8 mph which agrees very wellw i t h the actual value reported i n &43.

EJgurs 3-6 was drawn so as t o compare predicted alternator power valuessf the present method to corresponding values found by using the PROP computercode. AS may be seen, four models of the PROP code have been used: t h ePra nd tl model, tile Goldster'n model, th e no-tip-loss model an d th e NASA model.The theoretfeal background for both the Prandtl and Goldstein models i s


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T a b l e 2 O p a ~ a t i n gCondit ion s f t h e Ut98 t y Pole

I Bl ade Geometry---ercent Root Cut-But

I ---Blade Radiuss ft.Coning Angle, Deg.

_Sol j d f t y- - - p l y

Th f ckness t o Chord h t f o- 7 p - W

IA?rf0.i NACA 230-24

PJtsh Angle, Deg* I - 1 , 0 , + 1 , + 3I

M rfof 1 SurfaceOpera%$g, RPM.LIBI.- ---Blade 7w-ist


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Fig. 3-1 Chord distribution along th e blade


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Fig. 3-3 SECTION DRAG CBEFFTCIENT US . SECTION RNGLE OF ATTACK


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WPWDwEtwrav @.B68TI? SPEED

- - -Fig. 3-4 CfRCULATXON DISTRIBU"l'0N ALONG THE BLADE


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bdacelle Wind S p e e d h p h jpig. 3-6 Alternator Power V.S. Wind Speed ;Zero Pitch Angle


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b r i e f y treated in Appendix A; the no49 p-loss rradel i s s el T descsipti ve as i tdoas not account for any blade end effects; the NASA model exploits thesimp1 9city o f the no-tip-loss model by accounting for blado end effects solelyby a reduclfsn (3%) o f t h e blade radius -- t h i s met1108 i s als o call ed theeffective radius model.

Reference t o Fig. 3-6 shows t h a t results determPned by the present methoda r e anywhere from 10 t o 15% higher t h a n those predicted by the PROP code Porwind speeds ranging from 8 t o 28 mph. These results when combined with thoseof Fig. 3-5 reveal that PROP code methods underestinate the power o u t p u t ofwin d turbines in the wind speed range of in te re st : 'qcut-in " t o rated,Comparison o f PROP predictions w i t h expei8irnental d a t a was also performed in

and $510 I t was found t h a t th e PRQP code methods underpredicted poweroutput: and overpredicted "'cut-in" wind speeds. Cl ~ a r l y ,hese Pi ndings arei n tstal agreement w i t h the current findings.

When exfsting experimental d a t a For blade pitch angles other than 0" werecompared to predicted results from both current and PROP models, i t was Poundt h a t agreement i s not as good as t h a t displayed -in Fig. 3-5. A t present,questions have been raised as t o the re1 iab il it y of the experimental dataE133. Thus, those comparisons w i 9 9 not be shown* However, i t was be1 ievedt h a t cofi~ parison etween PROP prediction an d currlent method eslculations ford i f f e r en t b la d e pitch angles may be of some value. This comparison i spresented !n the form o f Tables I I , 111 and I V where cornputed power output i nkw i s shown against wind speeds for the five methods of prediction a t bladeangles o f - l o s91' and respec tiv ely . I t must a1 so be noted t h a t the powerva% es presented are ro to r power values i *e ., they have not been corrected byu s f n g equatlon (3-1). Moreover, the d a t a are presented in tabular formbecause of the difficulty in graphing results t h a t are relat ively closetogether over various portions of the wind spectrum.


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Tab l e I I . Rotor Power Vs . W4nd SpeedP i t ch Angle: -1'

PRANDTMODEL-.Jk&--1.76.3

1% 832. 361.992.3116.7133.4141.0135.9123.4100.468,728.4-18.0-67. 3-119.2

-O TI lP-LOSSMQDEtA!kw)-

150 kw Rated Power33 RPM2 Blades= 3.8"23024 Airfoil


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Table 111. Rotor Psvrer Vs . nd SpeedPitch Angle: +lo

150 kw Rated Power33 RPM2 Blades$ = 3 . 8 O

23024 A i r f o i l


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Tab la IV. Rotor Power Vs. Wind SpeedPi tch Angle: 9.3'

150 kw Rated Power33 RPM2 Blades

= 3.8'23024 A i r f o i l


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Comparing data contained 4n Tables I I do % V r e v e a l s t h a t ths pred ic tedmaximum power output ificreases as the p i t c h a n g le i s i nc re as ed f ro m -I0 t o4-3'. F igure 3-7 was drawn i n o r d e r t o b e t t e r u nd er st an d t h s distinctionbetween p i tc h angle set t ings . I t can be seen that the largest va lues o f t h ee f f e c t i v e a n g l e sF at tack, ne, occur fo r nega t i ve ly p i t ched w ind tu rb in eblades and tha.t the smal lest a, occurDs a t p o s i t i v e p i c h angles. To o bt a in ani d e a as t o how t h e e f f e c t i v e a n gl e o f a t t a c k i s d i s t r i b u t e d a lo ng t h e b la de ,a s calculated by the present method, ore was p l o t t e d f o r f i v e l o c a t io n s a lo ngthe blade (3/8 span, k/2 span, 5/8 span, 3/4 span and 7/8 span) against windv e l o c i t y i n F ig s. 3-8 t o 3-11, Each f i g u r e co rresponds t o a p a r t t cu la r b ladep i t c h ang le . Re fe rence t o the fou r f i gu res revea ls tha t a t any b lade1ocat ion, a i n e ~ e a s e s s th e wind ve lo c i t y i nc reases and th a t the inc rease i sp r a c t i c a l l y l i n e a r . !4oreover, i t can be seen hy cornparison that at a givenblade I oeation and wind speed, ore decreases as th e p i t c h angle i s changed f rom-1' t o +3". Thus, b lades w i th nega t i ve p i te h angles a re " s ta l l e d " a t lowerw in d v e l o c i t i e s t h a n a r e p o s i t i v e ly p i t c h e d b l ades.

Un fo r tuna te ly the na tu re s f t h i s s t a l l c o n d i t i o n on r o t a t i n g bla de s i snot we11 u nd er sto od a t p r es en t. I n p a r t i c u l a r,th e e f f e c t s o f t h e c e n t r i f u g a lf o r c e on t he boundary layer near t he t i p o f t h e b lade may, i n fac t , de lay thepresence of s ta l l . It should be n o t e d t h a t t h e s t a l l c o n d i t i o n c o u l d b eavoided, i n th e oper at ing wind speed range o f t he wind turb ine, by appropr i -a t e l y t w i s t i n g t h e b l a d e a l sng i t s span. B l ade tw is t ang les may be found byap pl yin g the computat ional scheme developed i n the preceding section.3.2 S i ngle-51 aded Wind Turb ine Ca lcu la tio ns

C u r r e n t l y a l a r g e 5MW h o r i z o n t a l a x i s w in d t u r b i n e i s b e in g de sig ne d f o ruse Jn Germany. What makes t h i s w s ~ k o tab le i s th a t the machine has on lyone b l a d e t h a t i s b al an ce d by a la rg e counterwejght on a stubby arm as


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a ) Positive pitch

6 Negative pitch

Fig. 3-7 Diagram s51owing the flow and brade angles at a blade elenwit


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F ig * 3-8 PLOT OF SECTION BtbGLE OF CiTTCICK U S* MIND WELQCITV Rt DIFFERENT&O@hf%ONS Q t O KG THE BLADE


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Fig. 4-9 PLOT OF SECTPOW ANGLE OF 6TTACK USo WIND VELOCITY AT DIFFERENTLOCfiTIQNS ALONG THE BthDS


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f i g * 3-30 PLOT OF SECTfON ANGkE OF ATThCK US. WfND VELOCITY AT DIFFERENTLOCATIONS hLBMG THE BLfiDE


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Fig* 3-11 PLOWOF SECTfQN ANGLE OF 8TTCICK US. WIND VELOCITY AT DPFFERENTLQCRTIONS ALONG THE B t UDE


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d e p ic t e d i n F ig. 3-12. It i s be1 eved th a t f o r ve ry l a r ge w ind turb. ines theuse o f a s i n g l e blade reduces ca p i ta l co sts w i thout muck redu ct ion i naarodynamic e f f i c i e n c y .

To t e s t t h i s premise, th e computer program was used t o compute th eaerodynamfc performance OF a p a r t icu l a r one-bl aded machf ne. I n th e ana lyses,t h e counterweight was taken t o be 2400 l b s and l o c a t e d a t a r a d i a l p o s i t i o n25 f e e t f r o m t he t u r b i n e r o t a t i o n a l axis . The counte rwel'ght and cyl ndr-icalsupport spar are shown i n Fig. 3-13. C a l c u l a ti o n o f c o u nt er we ig h t e f f e c t onthe performance was based on th e assumptions :hat the re i s no c i r c u l a t i s naround th e counterweight o r support spar and th a t t he drag eoeff*ieients f o rthese parts a re 1 and 0,s respect ive ly .

Perfornrance o f a one-bladed and a two-bladed rotor are compared i r r Figs.3-14 and 3-15. It can be seen tha t a t 22 mph bo th machines p r~ du ee h e i rmax*imum power. However, the one-bladed machine produces le ss tha n h a l f asmuch power as t h e two-$1 aded maehine. Furthermore, Fig. 3-15 reveal s th a tt h e Lluo-bladed machine i s also more e f f i c i en t . C lea rl y , ro to r power o f aone-bladed rotor can be increased, as shewn i n Fig. 3-96, by opera t ing a th igher ro ta t i on a l ve loc i t y . Th is , o f cou rse, would be accompl ished a t th ecost o f h igher s ta r t -up o r "cu t - i n " speed.

I n s p e c t i o n o f Figs. 3-14, 3-15, and 3-16 r e v ea l s t h a t t he y a r e n o ts u f f i c i e n t t o d e c i de whe the r a s i n g l e- b l aded r o t o r w in d t u r b i n e i s p r a c t i c a lor not. The Teas; b i l t y o f t h i s concep t s t r ong ly depends on t h e I ocal annualw ind d f s t r i b u t io n . W i k such w in d s t a t i s t i c a l d ata , l cal annual poweroutput of a one-bladed r o t o r cou ld be estimated. It should be mentfoned thatthe same annual power sf a two-bladed wind turbine may be o b t a i n e d f o r aone-b laded turb ine if i amete r sf t h i s tu rb in e i s approp r ia te l y i nc reased .


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Fig. 8-13 Counter-weight and support spar f o r one bladed rotor


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FQ. 2-84 ROTOR BOWER us. UIND VELOCITY eoa A FIXED PSTGWROTOR


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APPENDIX AReview o f Varjsus Methods of Mf nd Turbine Aerael,vnanic Performance

Ca?cul at ion


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I f Ap 4s the d$seotillnusus decrease fn the s t a t i c pressure across ther o t o r d t s k , than the ar ia1 f o r ce on r o t o r may be expressed as

F = AapAn expressl'sn f o r Ap can be obtaqt~bdby apply'ing Bsrnoanll s equatjan l 'nf r o t i t 05: and behind the r o t o r dJsk

Subtsactisn and rearrangement brings

Therefore, the a x i a l fo rce on the wind tu rb ine i s

U n i t i n g t h i s squat9011 with equation QA-1) r e s u lt s i n

Thus, t h e v e l o c i t y o f f low through the disk i s the arith me t ic average o f theupstrgarn and d~wnstreamvel sci 9es.

An axia l in te r fe ren ce fac tor, a, i s def ined as the f ra ct i on al change i nt h e freestream ve1ecSty that occurs ahead o f t h e disk, iee ,

V - Ua s -Hence, th e v e l o c i t y o f f l o w through the d isk may be w r i t t e n

bB = Vo(9 - a)


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Equatf ng squa t ions (A-2 9 and (A-3 1 and rearranging revea ls t h a t

And i t may be seen t h a t when the r o to r absorbs a l l t he energy o f the f l o w(V2 = O), t he Sn te r fe rence fac to r w i l l have a value o f 1/2.

The power absorbed by t he r o t o r i s equal t o the mass Plow ra te t imes thechange i n k f n d t i c energy o f t h e i nc om in g f l u i d , i . e . ,

Combtn4ng equat! ons (A-2) , (A-4 ) and (A-51, yields

I t can be seen t h a t f o r a g i ven w ind speed, ro to r s i ze and a i r dens i t y , thepower absorbed depends o n ly on t h e In te rf e re n c e fa ct or . Hence, maxjrnurn power

dPi s produced when- -- 0. It i s easy t o show t h a t t h i s c o n d i t i o n occurs f o rda1an a x i a l interference v a lu e s f - . Thus, from (A-6)3

and a power coef f ic ient , def ined as

has a maximum value o f 0.593.I f t h e r o t a t i o n i mp ar te d t o f l o w i s in clu de d, i n t h e a na ly si s, t h e t o t a l

reduc t ion J n trans1a t j sn a l k i n e t i c e n e r w o f t h e f l o w i sAK.E . = Power Ext rac ted 9 RotatSon K.E.


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Thcrcfore, in order t o obtain the maximum pswar Prom the stream, 4 t 4snecessary to keep th e rotational kinetic energy as small as possible.Mcareovsr, the oleinent of torque is equal t o th e angular momentum .imparted tot h e blade clement i n untt time. This equation fo r a blade element a t radialdistance r, *if QW 5 5 the rotational velocfty ~f the a ir , can be written as

dQ = 2n$,~~~ddr (A-9 9An angul ar i nterference facto r 9 s de fined as

where SJ i s t h e rotational v@loeityof the blade. Introducing equation(A-10) into equation (A-9) y9alds

dQ = 4 ~ 3 ~ ~8 Qd r (A-8 1

Usf ng t h i s equation a9ong with the previsf~sheory, a11ows the incrementalaxia l force, torque an d power t o be written respectively as

I t should be inenti~ned h a t momentum theory provides a goad indicatjon ofthe power coefficient, b u t f a i l s t o furnish the required design d a t a for therotor blades except for the 1imiting case o f an infinite number o f blades.


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Fig. A-2 Force and velocity diagram for blade eiemet theory


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I f the section 1 i f t and drag eaeff icients are known, the axial force andtorque can be determined by i ntegrat ing equations (A-16) and (A-17) a?eng theblade. It should be mentioned t h a t the blade characteristics are those of ana i r f o i l OF the same cro ss- sect i on i n a two-dimensional airflowo

The unsatisfactory feature of blade element theory i s t h a t i t re1 ies once rta in aSrfoil ch ar ac te ris tic s wh'l'ch vary w i t h aspect ratio an d which dependon the interference between th e adjacent blades of a rotor. Further, thistheory u~4iqgai rfo il data c~rsesponding o a moderate blade aspect ratio, hasbeen found t o properly represent observed behavior of a propeller, b u t f a i l st o p r~ v id e ccurate numerical res ult s [9Q. To improve th e accuracy o f bladeelement theory i t has been suggested, 693, t h a t the resultant velocity of theblade should be estimated i n terms of modified velocities such as those t h a twere determined by the momentum theory. The combined momentum- bl ade el ementtheory equations are similar to those of the Glauert Vortex theory, f161which will be discussed in the fol l owing section.

111. Gl auert Vortex TheoryGlauert Vortex Theory i s based on the assumption t h a t t r ai l ing

vortices emanate from the trai l ing edge s f rotating blades and form helicalvortex sheet s th a t pass downstream. The int erf eren ce velo city experienced bythe blades i s calculated as the induced velocity of t h i s vortex system on anyblade element. The ca lcul at io n of these induced velocities i s simplified byassuming t h a t the rotor has an Jn f in ite number of blades. This assumptionremoves the complexity associated w i t h the periodicity of rhe flow andpermits the momentum theory t o be dir ec tl y used t o evaluate the interferencevelocities. Fo r an element of blade shown i n Fig. A-3

W sin@= V o ( l a)W cosq = n r ( l + a ' )


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Fig. A - 3 Force and velocity diagram for GIlauert vortex theory


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Thus, the speed r a t i o i syo r 1 9 8 '

XO = - --a R l - awhere a and a' a r e a x i a l and r o t a t i o n a l i n t e r f e r e n c e f a c t o r s r e s p e c ti v e ly .The incrementa l a x i d l fo rc e and torque i n non-dimensiona l form can bewrJtten, usin g equat ions (8-16) and (A-17), as

where

and B, s o l i d i t y r a t i o , i s de fi ne d as

I n o r d e r t o s o l v e e q ua ti on s (A-21) and (A-22), +, and the jn te r fe rence f a c t o r smust f i r s t be determined. As an a i d t o t h i s pr80cess, i t may be assumed thatt h e r e i s an i n f i n i t e number o f b l ad es , thus, momentum theory equations can bed i r e c t l y a pp lie d. Tn add i t i on , i f t h e r o t a t i o n f n the s l i p s tr e a m i s n e g l ec t edt h e f o l l s w i n g r e l a t i o n s h i p is obtafned f r om (A-121, (A-21) and (A-23)

a ~ L ( C L C Q S ~ CD s i n4). = Llw- *l - a 4 s in2+


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Fig. A-4 Idealized vortex system

Fig. 64.- 5 Transformation representation


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regular gap S and extending i n f i n - i t e l y t o 'i;he left as s f io~nDn Fig. A-5, "Phag ap S rcpr~ssnts he normal d.istanee, a t th e boundary of th e sJ ipstrsam,bctwsen the successive vor tex sheets and can be wrf t ten as

whereA =-

Rsz

Prandtl showed that if : the rigid vortex system moves downward w i t h aT'velec i ty wa, the mean d~t:nf.sard v ~ l o c J t y n the 1 ins P P a a t a distance b from

t h e edge o f the 1 ines [ F l g. A-5) becomes

- 2 -1 -nb/SV =- g CO S (e7T 9LettingF g can be in terpreted as the fra@t%omf veloefty wo which i s imparted t o th ef 1ow o f s e et ion vfl. P m "ce a@orrespondr'ng ropel Per anal egy then F' i sdefined as

where


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,34 bus, F ' Ps a rcductlon f a c t o r wh4ck aust be app8 i e d t o the momentum @quationf o r tho flow a t radOus r, 5-issee i t represents t h e f a c t t h a t ofiilly a fbaactlosa,B\ of the f 1 44d between suseess4vc sheets o f t h e sl 9 pstream rccsJves theful l e f fect o f tl~eaa she9ts [93, AppPyJ t ~ g h i s c ~ r r e e t i s n o th e earliesa n a l y s is o f manenturn theory .i e., equations (A -22 ) and (A-13) produces

Nov cor W n i n g these equations wr'tla those obtajned i n the blade a1ement theory4 e ~ a , equateions (A-16) and (A-17) r e s u l t s i n

Furthermore, duo t o the radial f l o w near the boundary of t h e slipstreasn,t h e r e Os a d r ~ p f cOreul a t i s n which can be represented by an ecguiivalentrotor with inf.jnPte number o f blades bu t hav ing the same drag. Th isequivalent radius o f tkJs rotor 4s given i n 191 as


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Go'ldstcrin 6101 solved t he problem of potential fhw past a bodyeonsistirag o f a f in i te number of eoaceial he1 ic oid s s f in f i t i i t a I sngth , butf i n i t e r ad ius , moving th rough an inv ise jd f l u i d w f t h cons tan t velocity.Further, the f lu id mot ion is assumed t o be f r r o t a t i o n a l . The rssul t s a r eapp l ied t o th e cas e o f an ideal ai rscrew hav4ng a f i n i t e number of blades anda par l ieu1 ar disLr ibutOon of circ ul at ion al sng the blade fo r smal l va lueso f th rus t . Golds te in 9903 s p ec i f i cz l %yconsidered Getz's optimum condition[18] csrrespondit-rg t o a r igid kelicasid. The e s s e n t i a l result of' Goldsta*in'sc a lc u l a t i s n is th e de terminat ion sf th e s t r en g tk o f t h e he1 ic al vortex systemas a funct ion OP radius. I-? r is t h e c i r eu i a t i s n a ro und 5 c l o s e d c i r c u i tcut t ! ng a Rel icoi d a t radi us r and encl os i ng the p a r t o f t ie sur faee outs idet h a t r ad iu s , then r is equal t o t h e d i s co n t i n u i t y o f t h e v e lo c i t y p o t en t i a l

dracross t h e he1 ico ida l su r face a t r ad ius r, [9]. A1 so, -- s equal t~ thed rs t r en g th o f t h e vor tex sheet or equal t o the discontinuity i n the r a d i a lcomponent v e lo c i t y ur through the f lu id . The airscrew su rfa ce whl"ch wasconsidered by Goldstein f s def ined as

0 - Q z / V = O o r , 0 < r < R.. m (8-34)where r, 0 and s a r e cy l tndr ica l coord inates . The a x l s Z i s a1sny the ax$sof ' the airscrev surTace* Usi ng t he fol lowing coordinate t ransformation

and

Lap1 ace's equation, namely v2@ = 0, - in cy l indr ical cooimdinatescan be


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In reference [19], i t i s shown t h a t the velocity sf the a i r r e l a t i v e t o freestream velocity betweeti suecessPve sheets at a large distance behJnd therotor 3s normal t o the vortex sheets. NOW, if uZ and ue are th e axial andc rcumferent i a1 esmponents QP ve l oci t y, k e i vectsr ia1 sum CP sse t o thesurface f s equal t o the component velocity of the surface normal to i tself

where w0 -is th e vel aej y o? the surface and 4 is t he he1 x angle.Noting t h a t

and usaing t h i s equation together wf th equations (A -35 ) an d (A-361, i t can be


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shown that

I n a d d it io n t o s a t i s f y i n g equat ion (A-39), must bs a s ing le-va luedf u n c t j o n of p o s i t i o n a nd i t s d e r i v a t i v e m ust vanish when r i s i n f i n i t e .Furthermore, 0 m s t be continuous everywhere except a t the screw surface.Goldste in has solved t h i s prob'lern and the so lu t ion f o r t h e c l 'r c u l a t i o nd i s t r i b u t j o n along t h e b l a d e for any number o f b l a d e s i s g ive n i n

where I u n c t i o n s are t h e m o d i f ie d Bessel f unc t ions and t h e F f u n c t i o n s a r scomputed from

The T f u n c t i o n s appear ing i n e q u a l ion (A-41) can be calculated f r om


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so thatT p = 4$(1 - $1 1 (1 + ,12)4 (A-45)~4 = 16,2 t I - 1 4 ~ 2 21.~4 4,6 3 (I + ,217 (A-4 6Goldste in def ined a non-d imensional c i r cu la t f sn fa ct or as

Th is fac to r was ca lcu la ted and p l o t t e d CIQ].bock [%9% pp l ie d Go1 ds te in ' s soluta'on w i thou t r es t r Je t i on t o ana r b f ts a r y c i r c u l a t f on d i s t r i b u t io n i n o rd er t o o b ta in t h e i n te r fe renceveO sea"t9es i ns te a d o f f s l ming the standard form1a o f t he 61auert Vortextheory [f 61, He showed t h a t the a x i a l and eire um fe ren tia l components o f th ef nter ferenee velocity a t the r o t o r pP ane d i f fe rs f r om those sf GIauer t vor texcos29theory by a f a c t o r -, i f th e he l i x ang le i n bo th cases i s assumed t o beKt he sane. ApplyS ng t h t s fac to r , Lock modi f ied th e G l au ert V ortex method f o rt h e case s f a smal l number o f $1sdes.


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P. Tiate and S~9blilEo

Volksma 1 - Tk i a~ t y

- -O. Ferlomlng Qlf@anfza%nnmoand AddressUniversity o f Tsl edoDept. o f Mechani cal EngineeringToledo, Ohio 43686

Cleveland, Q h i s 44135.The theoretical development of a method of analysas for predict-ion of the aero-dynmic performance o-f' horizontal axis wing 4 t u ~ b i n e s s presented i n tha"s report.The method i s based on the assumption that a k~eliealvertex emanates Prom eachblade element. Collectively these vortices f vnm a vnrtsx system that extendsinf i n $ e ly far downstream of the blade. We1sclc2ies induced by t h i s vortex systemare found by app ly jng the Biot-Savart law. W c s ~ ~ d t n ~ l y ,his method avoids theuse o f any interference factors which are used i n many of the momentum theories.What's more, the method can be used to predict t h e performance of wind turbineswith a small number of blad es. The wlnd turbine performance o f a two-bladed rotoris determined and compared t o existing experlmantal data and Lo correspondingvalues computed from the widely used PROP code. I t was found th a t the presentmethod compared favorably w i t h experimental data especially for low wind veloci-t i es . The wind turblne perf~rmanceof a single-bladed rotor 3s determined subject

aerodynamic analysis of hawt by use of helical vortex theory

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