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Robustly Optimal Fixed Pitch HAWT ( with Tip Correction & Drag )

Robustly Optimal Fixed Pitch HAWT ( with Tip Correction & Drag ) Exact trig solution of BEM optimal rotor New criterion of robust optimality Max annual power pitch& chord. s= 2 p sin  r/B  advance/ B . H = I /F, F a function of (R-r)/s .221 s .

erin-fletcher
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Robustly Optimal Fixed Pitch HAWT ( with Tip Correction & Drag )

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  1. RobustlyOptimal Fixed Pitch HAWT • (with Tip Correction & Drag ) • Exact trig solution of BEM optimal rotor • New criterion of robust optimality • Max annual power pitch& chord

  2. s= 2psinr/B advance/B. H=I/F, F a function of (R-r)/s.221s. ½WCL rx W= -dL /dA= 2I(Wz)the (BEM) eqn(1)

  3. dP= xTq (dL+dD)=2 xT(Wz) I (sin- cos tane) dA I=FH H=Nsin(-), Wz =Wsin, W=Ncos(-) dP/dA=xT N2F sin2(-)(sin2-½sin2 tane) 2cote= cot +cot (3-2) o=2/3+e/6 +O(e2)

  4. Robustly Optimal Blade Elements CL =4Fsintan(-) CL / 2= esin, -o=½o-¼e sin (o -r)=Fsintan(½o-¼e)  =½e =¼eBc/r cos(o -r)=d { Fsintan(½o-¼e)} / dorobust c o t  r = cot d + (1-½de/da) csc(d-½e)+ dF/ Fdo cos (pF/2)=e-f , f=p(R-r)/s=B(R-r)/ 2rsino cotrcot½d+½ecscdcotd-½de/dacsc(d) - 2 f cot (½pF) cotd/pF r½d/h,h=1+¼e/d-¼de/da-fcot(½pF)/ pF r ½d –e/8 Tip r2d /3 –e/8

  5. robust r sinr =Fsindtan(½d-¼e) perturb=rcot r effect on d/dvs robust do/d Fsintan(-)=sin(-) Fsinsec2(-)= u= vd/d v=Fsinsec2(-)-....+cos d/d =u/v(d/d) -(u/v)2 v/ u u/v=do/d=2/3, v =(cosa)=/ sina d(-o)/d-4/9tan(-)cos2(-)/sin2r4(ad-ar) h2/ 9tan(½d)

  6. Blade Pitch for highest net mean Annual power ddP/dA=-½ xTN2Fsind {6(-o)2+ ½d2/d2sin2d(ad-ao)2} mean 6{4(ad-ar)cot½h2/9}2+ ½d2/d2sin2d(ad-ao)2 Relate , x to T: synchronous generator X as T-1. p.d. rotary pump fixed torque has T2CT =T2Cp/X so XT2. Say xTg, g2 >0 Then with τ=T/Td , gτ sin 2. . peak p(τ)=3{1-4(τ-1)2}/2 betweenτ=.5 and1.5 (τ)2= .05 (ad-ar)8h4cot2½d+(ad-ao) d2/d2270sin2d/g2sin23d0

  7. WIND SHEAR as T0(1+ msin β) where m= k x /X p(β)= (1+ msinβ)3/a averaging over β, a=1+3m2/2. t0= T/T0 t0= msin β in τ0= T/T0aτ0=3 m2/2+3m4/8 and a(τ0)2 = ½ m2+9m4/8. τ 0= 1+τ01+3m2/2-15m4/8 and the true minimum variance about itis (τ0)2 - { τ0}2 ½ m2-15m4/8 , so {τ}2 ½ m2-27m4/8 ; rat k=1/3 1+k=2(1-k) at x/X=.7, about .017 versus .05 above; T i p , {τ}2 is exactly .029 . Such independent variances ADD. YAW the  variance is proportional to the mean 4th power of yaw angle. Need wind data and yawcontrol algorithm.

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