Chapter 2 Blade Elements

2.1Blade Element Equation and Optimal Blade Product

It is a separate dependent problem to the above actuator optima to design the blades. For low solidity =Bc/ 2 r then on the scale of the chord c of each of the B blades each is immersed in a flow at infinity of W. Then the blade element drive is

dL sin= ½W2CL sindS

(2.1.1)

In all momentum theories, if Q is the volume flux, 2v is the swirl jump at the rotor, the tangential force is also

2vdQ=2vWsin dS (8) Remember the profile drag is considered separately from lift and the induced flow. Even when

the axial momentum equation involves a pressure term such as for a turbine in a duct ,this local

tangential balance always holds giving the universal blade element equation

¼CL(=v/W

(2.1.2)

For the standard non-pressure BEM ¼CL(=v/W =FJsin/W =Fsintan((2.1.3)

Exercise: Show the propeller equivalent is CL(=v/W =FJsin/W

=Fsintan((identical for symmetric CL)

In general CL() is a function of the ¾ chord or no-lift angle of attack = the blade pitch

angle and most existing simulations solve (2.1.3) for , using tabulations of CL and given

values of design and for off-design

Exercise Show (2.1.3) generalises [Manwell 2002] Hawt (sinxcosCL=4Fsincos-

xsin

With the simpler (2.1.3) (for any ) it is possible to analytically study the off-design

performance. Prior to stall C=CL/2l=sin, l the lift slope thickness factor. If =l/2=Bcl/4r,

the operating eqn is ¼CL=C=sin= Fsin tan((2.1.4)

For ≠ (2.1.4) linearises at high x as the tractable operating F(2.1.5)

At the BM optimum design point sin≈¼CL=Fsintan½ =F-Fcos (2.1.7)

A simple case is tangent blade with no pitch =0 so = I=W (2.1.5) Ftan(=

operating which at high x, F=1 and optimum design gives ≈⅓d so 4 x=2xl=Bkl≈4/3

where k=c/V is the reduced frequency, indicating spanwise constant design chord c. Then the

Prandtl loss of tip is just .69c.

In general the exact nondimensional net chord for the BM optimum design as

CLx=BklC=4xv/W =4(1-cos)cot3/2=4(2cos-1)sin /(2cos+1) (2.1.8)

which peaks at .555 at =34.5o or x¾ as graphed in Fig 2.1. This shape is the ideal planform

k(x) for Glauert’s design at constant CL0 at minimum 0 for lowest relative drag penalty at each x,

CD/CLtan for the design airfoil of a given thickness t chord c.

Generally high X (Hawt and Vawt) blades are thus found to be high aspect ratio (eg relative to

aircraft wings), causing at large scale critical root bending moments especially for the Hawt and

vibrational concerns (especially for the highly unsteady Vawts). Since the blade strength carries

as t2 or c2 and stiffness as t3 or c3, it is structurally advantageous to mininise B. However 3

bladed configurations are much smoother and inherently for the Vawt and for the rigid Hawt in

wind shear For the Hawt 2 blades can be continuous through the center of maximum bending

moment and they can be teetered on a chordwise axis there to dynamically balance the wind

shear moment through inertia and centrifugal stiffness. The teeter hinge can even be inclined to

pitch the blades versus azimuth to reduce the teeter moment. Counterbalanced (and teetering)

single blades have the structural benefit of greater thickness and chord but not quite twice

because their higher tip swept area loss as 1.84/BX optimises against the drag and wake losses

at higher X and so smaller Bk from Fig 3.1 Numerically at L/D=75 Fig 2.1.2 tabulates the

midspan k at optimum for various B.

Rather than further pursue this one if unavoidable shear case of wind variation, it is good

preparation to see first how ideally strong fixed pitch blades can be designed for the widest CP

peak in light of general wind variations rather than the highest peak of best CD/CL and so get the

most annual power of the wind spectrum ......

2.2 Robustly Optimal Blade Elements

For a blade element to ‘robustly’ remain optimal at its fixed c,r,, as the windspeed varies

the angle and so the optimal in proportion, the optimum(2.1.2) must not change with , so

its derivative in must also equate, eg for the BEM F=1 (2.1.7): sin(-)=1-cos derives

cos=sin(2.2.1) Dividing into sin=1-cos gives tan=tan½so =½=(2.2.2)

trisecting the optimal into ½ with =2sin Bkl=4 x8/3 (2.2.3)

In this always optimal zone around design CL= 4-4cos (2.1.4) applies so CL≈8/ 9x2

For small sheet (metal) blades, to avoid the CD of leading edge separation such thin circular

arcs should be set like jibs to lift only from camber and not from any chordal angle of attack at

design . To make this optimum robust, their trailing edges at twice the ¾ chord should be at

from W or just parallel to the motion.

2.3 Robust optimal attributes

The robust condition gives cP a very broad optimum. With it -

⅔ is zero and stationary in at the design d and its Taylor series begins

with the square of variations negatively as seen from the feathering

limit ==. Since the expansion of cP about its optimum for any likewise

begins with the square of - ⅔, the combined expansion is a weak quartic

O()4 whereas for non-robust blade angles and solidities it is a stronger

quadratic decrease O()2 , as Chap 3 will explore. But the cT is only a

quadratic peak with the peaking of a≈(, at a=1/3

The robust off-peak >⅔ , a<1/3 benignly helps avoid high downwind

induced drag

dE=2(Tsin) ds 2(1-a)a(T sin)2ds and structural loads away from

the main power design point. Eqn (A1) shows the thrust efficiency =dP/ T

dE or tan/tan for a Hawt, is the ratio of useful to total power removed

from the wind and so an upper limit on wind farm capture.

Compare tangent =0 ==⅓d ==2 and robust optimal =2

== element designs... For the tangent = cancels in eqn (15), reducing

it to - ==. So if is doubled to 6, say by halving x, the tangent

gets = =5 . But for the robust the quadratic (15) must be solved yielding

-=1.55, which "under-induces" less versus the ideal =⅔. The cP as (1-

/)(/)2 (Eqn 4) are reduced by .78 and .96 respectively, the loss ratio

slightly exceeding the ratio of the above (- ⅔)2 . Yet the difference in stall

margin in favour of the robust =4.45 has increased to 5-3.45=1.55

from And if is reduced to 2, robustly a=1-√½=.29 safely and slightly

lower than optimal ⅓ with cP ratio .99, but for the tangent cP ratio .84 at a=½.

This is the threshold of ‘turbulent’ [1] ‘over-induced’ reverse flow in the

wake with heavy E at low cP and so very low . The only price of the robust

design is the higher profile drag penalty from the higher chord, which Sec 5

will adjust for. Figure 2 combines results from Sec3-5, showing how the

robust operating curve kisses the high side of the optimum =⅔ line at

the design point.

At the speed ratio for vanishing lift of approximately 3Xd, T times the

Hawt rotor net ‘drag’ equals the power lost to profile drag, approximately the

integral of ½CD(3T x)3Bc dr. For uniform (outer) robust chord, the rotor drag

coefficient based on blade area and true wind is thus CD (3Xd)3/4 vs. the drag

coefficient of about 1 on the blade area if the rotor is locked against rotation.

Since these are equal at low Xd of only 2.5 and freewheeling will have the

greater drag moment arm and also large centrifugal forces, it is still best to

brake the robust optimal Hawt in high winds.

4. Robust Hawt design Small blades are very strong for their weight and can furl to regulate or

simply withstand the steady and productive loading of a near robust broad cP

peak, and they benefit most from its indifference to wind fluctuations of

longer timescale than their small r/T. Reducing the chord below the robust

value to reduce drag decreases the zero rpm stalled blade torque needed for

self starting. Too tangent blades may also fatigue with their high rpm in

unproductive turbulent over-inducing in high winds as most off-grid loads are

too soft to prevent the tip speed ratio increasing significantly with windspeed.

At most the interference should peak at ½, the threshold of reversed wake

flow, at some ½ below the design. The small angle BEM eqn (16)

()=(- gives this minimum ‘just safe’ =½ =.39 d vs robust =⅔d

or / <.41 . Such hyperbolae in Fig 2 are asymptotic to the negative

axis.

Exercise: Prove for small angles (2.1.6)gives this minimum

‘just safe’ = 1/2 = 0.39d versus robust = 2d/3. Whereas grid-connected Hawts can use synchronous generators to start

and then to hold their rpm very constant. The robust advantage in off-design

yawed running in windshifts is hard to quantify, though likely very

significant with their active yaw power consumption. But their large size

requires careful design not just for maximum annual power but also for

minimum peak gravity and high wind blade bending moment. A near robust

(tip) solidity produces too much absolute blade bending moment, torque, and

power above the design wind, so the outer chord c, pitch , and section stall

( ¾ chord) angle of attack s is reduced not only to lower the profile drag but

to deliberately prompt stall at high T and constant near the tip. [7] The

constant rpm means such more tangent tips are safe in overinducing in lighter

less powerful winds below design. But part of the remarkable success of large

Hawts is due to the broadness of a near robust optimum in mid-blade.

Note that to maintain optimal by active pitch for less-than-robust chord blades requires

counter-intuitively increasing their pitch as is decreased. For greater than robust chord blades

more sensibly optimal would follow , for greater freedom from stall at the price of higher

drag penalty. In practice the performance peak with (robust) fixed pitch is sufficiently broad and

active pitch is only used for feathering the blades to depower faster in high winds. Tangent blades require high xd not to stall at design, (especially in Vawts). For design at the

same fraction of stall then the robust xd is half, and then the robust chord from 4x8/3 (2.2.3)

is 4 times the tangent chord from 4x ≈4/3, but the drag penalty ½x3V3BcCD is halved. This is

offset for Hawts by the higher swirl loss at smaller xd

2.4 Drag corrections to robust values

Since (1.7.5 ) gives optimally≈ ⅔+/6 and -≈ ½¼, then differentiating

sin(-)=sintan½¼≈cos⅔ at set and gives

cos(-)=costan-½-¼d/d) sinsec2-

Dividing gives

cot =cot + (1-½d/d) csc(½) (2.4.2)

Taylor expanding the second term gives

cot cot ½ +½ csccot-½d/d csc (2.4.3)

Taylor expanding cot -1 of the right side gives

½ tan2 ½¼ d/d tan(½)+ (2.4.4)

showing drag to little perturb the trisection, especially if is at a minimum 0 which reduces the

first correction and makes the second nil. Then at high x, r ≈⅓ r ≈⅔ -/12

so not only does drag slightly reduce the optimal solidity but equally the (robust)optimal angle of

attack. Near the minimum at o , o½z(o)2 where z=d2/d2

so then solving (2.4.4)

r {1-¼z tan(½)} ½o tan2 ½¼zotan(½)+ z(o)2 /16}

Though the quadratic could also be formed and solved.

2.5 Robust Optimal Tip

As the windspeed varies it changes all the x and along the blade proportionately including at

the tip. As increases the optimum =⅔ this increases the sheet spacing which decreases F()

which reduces the proportionate increase in angle of attack required by (2.1.2) to maintain F-

Fcos , so the robust pitch and chord factors are smaller at the tip....

Using small angle approximations for r and d appropriate at the tip’s large X for

( -r)=F½-¼ to robustly hold the optimum despite varying with

about d at set and requires

= F-½-¼d/d) F+ (-dF/d

Dividing equations by at gives

1/r=1/ + (1-½d/d)/ (½-¼) dF/ Fd (2.5.3)

Now cos (F/2)=e-f, so pdF= 2cot (½ F)df f=(R-r)/s=B(R-r)/ 2ro so df= - f d/o

1/r=1/ + (1-½d /d)/ (½-¼)+ 2f cot(½F) /Fo (2.5.4)

r ½o h, h=¼o - ¼d /d f cot(½ F) / F (2.5.5)

At the very tip in the limit of small f , 2f= (½ F)2 so the last term in (2.5.4) tends to -¼,

reflecting the ½ exponent of potential flow around an edge [Batchelor p412].

Then h ¾¼o - ¼d/d

Everywhere the robust r is given by Fsindtan(½o -¼sinr (2.5.5)

So the robust blade reduces its chord by hF very near the tip from the constant chord beyond

the tip region (when the drag correction is ignored.) and the robust blade tips twist towards

tangential (‘washin’) as in Fig 2.3

2.6 Design at Low x for Blade Root (and Pumping-Optional)

Note the limit CL=2 at x=0 =60 is the design point for a windmill to budge a horizontal

crank against a vertical rod to a piston pump. Then the blades must be so close together they

cannot be treated as isolated airfoils but as a cascade. Glauert [1935] proved elegantly the

momentum theory holds for the cascade with W taken as the mean of the inlet and outlet W, but

showed that the isolated CL must be corrected by a function K ( ,) similar for straight and

cambered blades. For instance from Figure 2.2 at =40 =1, K ≈.95 raising the required

isolated CL to 2.1.

At such large values CL deviates from Joukowski’s 2 sin due to boundary layer separation

leading to stall. The general robust rule is that (2.1.7): ¼CL= C =1-cos should be invariant

to small changes in with at fixed So differentiating dC/d= sin and dividing into the

original Cd/dC=CL d/dCL= tan ½ (2.6.1) The two sides of (2.6.1) are sketched in Figure

2.3 Clearly the robust angle of attack r will always be less than s , the angle at which CL

peaks. Accurate enough data for differentiation being lacking, a smooth transition is at =1

from C=CL /2 = to C=1)3 / 3(s -1)2 For C to peak at value p at =s,

p -1=2(s -p). The cubic (2.6.1) for robust r is solved for the 23018 p = 13 =s =15 1

=9 in Fig 3.2. So at x=0 the value of r tops at about .9 deg below stall.

To try to reach large enough CL at x=0 for maximum starting torque, waterpumping

windmill blades are designed at greater than robust so they overinduce at first as x is raised. If

d = s for absolute peak CL , then eqn (2.1.3) gives d/d =2/(2-cos ).(2.6.2) Then at x=0

drops at this stall rate d/d =4/3, twice the optimal rate, towards reversed flow by ≈or

x≈.32 but this will be overcome by the trend to feathering by say xrms=1.25 based on

Fig 2.4 which may explain the jog visible in the torque coefficient Co= O/ ½V3 R3 curve. The

starting Co of .70 falls 20% short of the BEM ideal ⅔co =3/2 =.866. This fair agreement

and the generally low CP curve of this empirically optimised design are strong experimental

support for the BEM prediction of heavy swirl losses at low speed ratio.

For direct cranking a single acting pump the mean torque is 1/ the maximum static starting

torque, so that in the same wind that it starts in, the multiblade equilibrates at an x past its

optimum with a CP of .28 and then as the wind increases more its CP gets even worse. Only

when the wind slowly decreases after starting can the peak CP of .32 be realised. By

counterweighting the crank arm at the top of the tower or by (unstably) pushing a well pump rod

of half the rising pipe area down the riser Fig 1.3, a double acting effect can be achieved. Then

the starting windspeed is lowered by 2 and its operating point is on the low x side of the peak

CP and will pass through it whilst the wind is building to the original starting value. (A cam or a

slotted crank (Fig 1.3) quick return to decrease or eliminate the counterweight runs up against

the pushing buckling instability on the inertia of the rod unless its return acceleration of the rod

is less than the gravitational. To avoid this even the standard crank small multiblades for use in

strong prairie winds are backgeared from rotor to crank, (whereas in a wind-generator high wind

areas lower the gearing required) . To avoid the manufacturing cost of the gearing and all the

standard blades, another approach is to use a slower turning larger rotor with fewer peripheral

blades [Kentfield ]) The peak to mean torque is even higher for cranking a single acting air

compressor but some have centrifugal unloaders that drain the line to the check valve entering

the tank.

Averaged over a typical wind spectrum the net effect is that windpumps have a mean useful

CP (fraction of the mean wind energy flux converted) of about .05 for single acting and about .09

for double acting and .12 for completely smooth (rotary positive displacement) . [Dixon 1979]

To improve the latter requires an automatic transmission of which the reciprocating equivalent is

a variable stroke mechanism.

2.7 The Ideal Contra-rotating Hawt (at any Tipspeed ratio)

A second opposed blade Hawt immediately behind the first should pre-empt the flung flow transitions to improve the starting torque and the running power. If there is no net angular momentum given to the fluid, there is no net torque (even with a stator) on the contra Hawt making mooring of an underwater current turbine easier. Counterspinning rotors attached to the generator’s casing and shaft effectively double its rpm and reduce its size and cost even with the extra slip rings and balancing. For a windpump, miter gearing between the two rotors can reduce the stroke rate to prevent downward accelerations greater than gravity that buckle the pumprod against the side of the well. The equal and opposite rpms reduce the gyroscopic loading and preclude braking as the heavy fanmill is ‘furled’ sideways to a high wind by the spring yielding on an offset tailvane. Given that a multiblade windpump must be designed at zero tipspeed to start a single acting pump and then only captures about 1/10 of the ideal Betz windpower in a typical wind regime.[Dixon], there is large room for improvement to pump more water with the wind. Here the standard Hawt interferences and blade elements are easy to solve and compare with the single Hawt

So, as in Fig 1 consider a row of blades moving across a wind V one way with just downwind a row of opposed blades moving the other way. In the HAWT a swirl power loss is possible so the circulation BK should be equal and opposite on its two row rings each of area dS. Then the induced crossflow/swirl is nil in front and behind, uniform in the gap at BK =2V and then V at the blades so ±2V are the tangential components of the row velocity changes 2I . Fig 1 shows the aft rotor straightens the streamlines bent crosswind by the forward rotor. Consider in general different rotor angular speeds fore 1 and aft 2 so the local speed ratios xi= ir/V are proportional at the same radius r.

Fig 1 Rotor and Flow Schematic with Velocity Vector Diagrams

The swirl-free wake must be at freestream pressure. The axial flow due to the axial I interference aV of the two rows second evolves on the larger diametrical scale of the rotor streamtube and so is assumed the same aV at the closely spaced rows. So in the velocity diagrams in Fig 1, the true flow F at the actuator discs is the same (1-a) V axial and vV left to right , but not the apparent winds VW relative to the oppositely moving blades. At x2= with the aft straightening effect, W has cancelling lateral components self wind x2 and induced - for magnitude W= 1-a angle =90 to tangential or purely axial ; The conversion (regain) of the gap tangential flow back into pressure is then so great that there is no net pressure across and thrust on the rear rotor. …

From (1.3.3) The pressure jumps at the rotors are p1=2(x1+v)v V2 and p2=2(x2-v)vV2 and also 2 V(1-a) aV = p (1.3.7) so adding

a(1-a)=(x1+x2) (1)

Note that 2vV=BK / 2 r or the same total circulation BK of the B blades in each row which if tangent is BcWsin or BcV(1-a) or the same total chord. Combining with (1) leads to Bc =2aV (2) for tangent blades, the same as 5.1 ) and ½ of (2.1.5). (So fixed tangent blades in the Schneider belt will achieve the equal and opposite circulations if their x is high enough to avoid

stall.)

Now by dP/dQ = vVdD = 2xVv V(1.3.2)

cp=dP/ dS /½V2=4(x1+x2) (1-a) =4a(1-a)2 (5)

as in 2D theory with peak the Betz limit 16/27 at a constant optimum of ⅓, since there is full regain of the transverse 2v and its kinetic energy. By (4) the optimal double HAWT then has constant net circulation

(x1+x2) =2/9 (6)

(as at large x for the optimal single HAWT). A natural though not necessary interpretation is to segment a= a1+a2 in (1.3.7) such that ai(1-a)=(xi+ ) so I 1∙W 1=I 2∙W 2=0 leading to the familiar vector velocity diagram of each row as in Fig 1 Then with constant a’s and so x1 front blade lateral wind component x1+v is a minimum of 2x1

when equal to 2v so its is maximum and W minimum there. Then back substituting (6) with a=⅓ gives for a rotor downwind of a stator x1=0, a2=0 when x2=x2

2= v2= 2/9. For equally ‘contra’-rotating a2=0 at x1

2 = x2=x22= v2= 1/9 a’=1 , ⅓= a1= x1=x2. where a’=v/x. These flow interpretative

factors are plotted in Fig 2 for this optimal ‘stator’ x1=0 against its rotor rpm x=x2 and for ‘contra’ x1=x2=x : a1=(1+a’)/6 and a2=(1-a’)/6 . This interpretation would allow the theory to be generalised to the case of unequal v in which residual swirl is dissipated for the Hawt.

Fig 2 Optimal rotor flow factors, all double HAWT values singular at x=0

The row W’s are different in magnitude W and especially direction , as computed for Fig 3.

Note at x=0 they both diverge to the same absolute left to right singular crossflow so fore tends to zero but aft tends to . For a stator at the front tan 1=3x2 whose increase with x will not allow robustly optimal design Whereas a rear stator behind a front rotor has the complementary (reflected about 2 = - Tan-13x1 so decreasing with x and at x1=v , b=⅔ and c=-⅓. In fact the stator being aft will be seen next to be preferred for any optimal blade element design...

Fig 3. Apparent Wind Angles at Optimum

Numerical calculations of the constant circulation expanding ring vortex tube wake are applicable here and show that whilst Cp is unchanged from the Betz limit, a tends to be few percent below average ⅓ in the center with cp slightly higher[7], due to the radial segmental assumption neglecting possible streamtube pressure forces.

If dS is the differential blade area in the rotor disc area dS , then the solidity is the number of blades B times their chord c and r the radius, ie Bc / 2πr and CL is the lift coefficient , then the blade element lift of each rotor gives (2.1.2)

¼CL(=v/W (9)

“the blade element momentum” equation. As x0 pure constant circulation designs have vW ∞ so tangential and CL4 . The former makes it physically impossible to overlap the blades to achieve solidities >1 to even approach the latter with the largest known section CL . Fig 1 shows stator CL = 4v/W = 4cos. At x=v=2/3 its CL=4/3=2.31 with a preferred front rotor CL =2⅔ =1.63 or with a rear rotor a too big CL=22=2.82 due to its absolute minimum W=⅔. But the empirical 1880 Bollees windpumps [8] had rear rotors with smaller

than their front stators.. For the contra at reversal x1=x2= v =⅓ , the max 1= /4 and W2=⅔ so aft CL =2 (2 fore) . Whereas for the single CL = 4v/W = 4(1-cos ) [7, Fig 1c], for the limit 2 as x0, just realisable at =60 and pitch say 45. Probably the most realistic if non-optimal x1=x2 scenario with the least bias vs. the optimal single is to hold the contra rear CL at 2 for x<⅓. (If CL is set at the stall value, a set aft rotor would operate at such constant CL as its X<⅓ varies.) From equations (2) minus (3), (b-c)(1-a)=2 (10)

At x=a=0 b=-c solving v/W =1/2 gives b=1/3 v=1/3=.577 at x=0 (4/3 times the single’s 3/4) . The fore CL rises to 2 with 1=60 ( 2= 120) at x=0 to become identical forward to the single’s innermost blade element. (In fact this modification to Fig 3 makes 1≈ s effectively for all x.) Then the x=0 CL=2 design drive (torque or cp/x) is 32/9=3.56 greater than the single optimum with none of the downwind thrust! To find the intermediate behaviour at CL=v/ W2=2

W22=(1-a) +(x-v)2=4v2 (11)

Expanding the (x-v)2 and then the net terms in and xv from (5) gives the quartic equation in a: 3a (1-a) =4x2{(1-a) + x2 - a(1-a)} (12)

which was solved numerically to find a virtually linear in x a≈2x/3 and then (5) gives v near linear and (10) gives b dipping slightly below ⅓ and then c as in Figure 4 which also compares the integrated Cp versus the single.

Figure 4 Design Power, CL and aft interference for aft CL=2 for x<⅓, optimal >

2.7.1 Constant Lift Coefficient Planforms

If k= c/V and C=CL/ 2 ptimally for x>⅓ again

CLx=BkC =4/ 9W (13)

for the pure optimum contra and double with a stator. The maximum of rotor CLx are at x= when aft W =⅔ with net lateral component v-x=0, and fore v+x is a minimum. Figure 5 of these Glauert ‘constant CL’ planforms shows the higher net and especially aft BklC’s at small x, due to the higher circulation and low aft W vs the single HAWT.

Figure 5 Optimal Blade ‘constant CL planforms’

Note that the stator blade has a high asymptote of constant BkC= CLx of 4/3 at large x because it does not see an increase in apparent wind W as x increases. Again the rotor is smaller if placed in front of the stator. Using static airfoil characteristics for CL and CD ignores the dynamic stall delay.

Correcting that with propellor boundary layer theory again falsely assumes that there is a

centripetal pressure gradient and curved torque reaction swirl behind a Hawt, making the

secondary flow inwards at small x. The author’s experiment (Appendix 1) saw no small x

secondary flow and no pressure gradient, so Appendix 2 is a corrected boundary layer calculation

with stronger stall delay predictions like the observed.

. In any case Chap 3 will show such planforms commonly designed like propellors at the CL with the least relative CD[5] generate less annual power in a typical wind regime with a typical load than robust designs

2.7.2 Contra Robust Chord & Pitch

The robust condition is that the blade element equation (9) CL =4v/W continues to be optimal at fixed blade location c,r, and and fixed pitch despite a small windspeed or load variation changing X and so x and so . So the derivative w.r.t =+ of the optimal equation CL =4v/W (9) also optimally holds i.e.

dCL /d=4d (v/W)/d (14)

So then dividing (9) by (14)

CLd/dCL= C d/dC = v/W d /d(v/W) (15)

The contra HAWT robust values are also plotted in Fig 7 from differentiating the optimal

d /d(v/W) numerically. Once the robust and so C is determined so is the robust Bk from

(11): CLx=BkC =4/ 9W. The robust breaks down with the peak in the forward at 45,

meaning small, then negative d which drives the fore robust angle of attack to zero and Bk to

infinity at the x=⅓ point. The same singularity due to vW ∞ 0 as x0 occurs with

robust design of constant circulation blades in Joukowski’s general momentum theory [2]. In

practice the front rotor would have to be designed at close to stall to restrain the solidity close

to unity, like the back and each like the single HAWT rotor. If the front design were made

identical to the single robust for all small x<1/3 and the aft designed at stall one would expect

the operating curves to be very close to the design curve in Fig 6 for small variations in X. In

general determination of the full operating curve of any set contra design will need to consider

the non-optimal possibility of unbalanced circulations.

Figure 7 Robust Blade Angle of Attack & Chord (planform)

Figure 8 shows the net contra robust pitches in radians. The difference in apparent wind angle

dominates making the lee pitch greater than the forward and a steep 77 by x=.325. Here robust

design is not feasible below x about .5

As seen above the contra-HAWT converges to the single HAWT at large speed ratio. Thus

the drag and tip effects modifying the outer optima will be the same as for the single[9].

Figure 8 Net robust pitches of blade ¾ chord to blade path. cp’s

Finally the interferences and robust design for a rotor in front of a stator are shown in Figure 9.

Here the robust solidities rise above 1 a rough practical limit at about x=.75 again about 50%

greater than the maximum of 1 =.96 at x=2/3 at which robust design is singular. The

combination would need some practical design completion at lower x such as to a maximum CL

like the pure contra design above.

Figure 9 Interferences, robust design solidities and angle of attack for rotor in front of stator.

In summary, the contra-rotating HAWT is easy to optimise as the parallel vortex sheets show

the aft blades straighten the flow bent by the front row so that the rotor axial drags and

interferences are different for the fore and aft rotors. The swirl power loss of the single HAWT

is avoided as expected, though the singularity in torque at zero rpm cab be interpreted as a

singularity in the interference difference with the aft rotor restoring front excess interference.

The limiting of lift coefficient approaching stall is again easily incorporated into robust

optimal design. Robust optimality is impractical below a contra local speed ratio about ½ and

any optimality impractical below about ⅓ , but holding the product of lift coefficient by solidity

below 2 gives a realisable design with the elemental drive reaching 3.56 times the single

HAWT’s at zero speed ratio.

Robust optimal design is likewise possible for local speed ratios greater than about ¾ for

a front rotor and a rear stator with high solidity ⅔ at large x, whereas not for a front stator which

also requires a larger optimal (rear) rotor..

2. 6 Oscillating Windpump (Optional)

Even with contrarotating rotor, pump cyclic torque and matching complications remain and motivate looking for a different sort of wind actuator that is more naturally suited to this common water pumping (and likewise aircompressing) constant load and can achieve a better structural efficiency of CP to CT. Pumping a deep well is certainly easier by reciprocation than rotation, so changing to a reciprocating wind actuator is worth considering but with the many constraints on a workable windmill from the extreme variability of windspeed (if not direction) in mind. Thus many ideas for pitch mechanisms or even divergences to effect an oscillation have fundamental problems themselves starting and coping with different wind strengths especially storms. The need to avoid the shocks or inertial forces in mechanisms suggests a free harmonic oscillation such as flutter (Figure 2.7). Such an instability can be tailored to occur only in moderate winds, and not in high winds for inherent storm protection by dynamical feathering of the ratio of pitch amplitude to crosswind amplitude containing the peak thrust load.

The acceptable frequency range is likely to be much less than the rpm range of a rotary windpump. This necessitates variable stroke which conceptually matches the variable amplitude of an instability. A sufficiently non-linear conversion of crosswind amplitude into stroke allows the flutter to be excited by the small variation of light winds, its energy growing as amplitude squared at first but contained in moderate winds by the pump though with large peak pump

back forces approaching the end of the stroke. The single acting nature of the pump ensures the oscillator can always return to the starting position if the wind dies very suddenly within a period. Some insights into the CP and planform will be offered in Chap 6. A dynamic factor favouring low windspeed area applications is that any oscillating windmill is stress limited in its frequency. The live wind loads vary as V2, so should the structural mass but then the inertial loads would vary as 2V2 ie as V4, so V2 stronger. The inertial bending limit scales as V2 times the ratio of material density /fatigue stress showing an inherent limit on the design (optimum) flowspeed V independent of the machine size and fluid density (except that fluid density provides an effective lower limit on the material density due to the added mass inertial loads as well) Fortunately for better matched electric loads, the Hawt can be designed around the CP maximum and minimise, even eliminate exposure to the over-induced reverse flow high CT ....

Chapter 3 Annually Optimal Blade for Low Yaw Hawt

3.1 Robust design for high X Hawts .

Robustness allows the Hawt to remain optimal with small spatial variations in over

the blade path due to vertical gradients in wind speed, most significant for large wind turbines

on relatively short towers. If gusts vary slowly in r/V, the time scale for passage through the

rotor, the cP stay optimum with time, so especially for small windmills. In all it helps with the

slower wide excursions of windspeed with weather cycles of the jetstream, fronts and local daily

winds such as seabreezes.

Large grid-connected Hawts can use synchronous generators to start and to actively yaw (avoiding slip rings) and to hold their rpm very constant. Because of the yaw power penalty and need to restrict transient loadings, these Hawts run with high yaw variances, but 4.1 will indicate that blades should be on the tangent side of robust to lose the least power in yaw. A near robust (tip) solidity produces too much absolute bending moment above the design wind, so the outer chord c, pitch λ, and section stall (3/4 chord) angle of attack αs is reduced not only to lower the profile drag but more to deliberately prompt stall at high V and constant near the tip . The constant rpm means such more tangent tips are safe in overinducing in lighter, much less powerful winds below design. Small blades are very strong for their weight and can furl to regulate or simply withstand the steady and productive loading of a near robust broad

CP peak, and they benefit most from its indifference to wind fluctuations of longer time scale than their small r/V. Raising the outer chord above the robust chord increases the zero rpm stalled blade torque for self starting and lowers the maximum runaway X in high winds. Whereas lower drag and chord blades more tangent than the robust may fatigue with the very high rpm in unproductive turbulent over-inducing in high winds, as most off-grid loads are too weakly varying with to prevent decreasing with windspeed. So robust design is most appropriate to downwind passive yawing or small tailvane Hawts..... 3.1 Robust and Minimum drag to lift cp curves compared

Any variation of the windspeed will perturb x and so at each radius and so its operating

point ( away from the design. Then to maintain the lift optimum ( the design angle of

attack d should be the robust r . But the power at the design point is lowest for the smallest

CD/CL ratio o, which occurs at the best o.

The figure shows calculated the calculated cp(x) from (1.7.2) via (2.1.9)

≈for design (from 2.7.9) at xd= 2.7 =20.3, r =7 and o

=.173=10. for the static NACA 23018 [Maalawi, 2001] o=1/73=.78 =.0136 and

z=d2/d2.44. The robust cp becomes higher just below xd because its goes through o there

and the lift peak is broader and stalls at smaller x. At x>xd there is little net difference at first as

the o design lift power falls below the robust due to the narrower peak but the drag power loss

is lower. However then the lift power stays much higher far from the operating point. This is not

desirable at all for small Hawts where large x/xd occurs in high winds and the very high forces

associated with the very high absolute power level then are more destructive than productive.

(The tangent design is also plotted from ½d.The tangent cp is less but its forces are

even higher as it enters the reverse flow zone at x/xd=1.4.) Whereas the robust cp stays much

higher at x<xd

The drag perturbations to the optima and robust values made no perceptible difference in

the above calculations and will henceforth be ignored. The robust net peak not being at xd is due

to the variation of in the drag term. ...

The airfoil data on in Fig. 3.2 shows o are not very sharply defined and z lowered with the

roughness acquired by medium-sized Hawts by soiling from wind blown dust and sand from the

ground , and birds droppings and insect squashing. The variation in o also allows some bias of

blade airfoil section for an o close to r. To resolve the remaining gap between cp peak height

in o and breadth in r at each design x for the highest mean annual power requires some

preparation...

3.2 Wind distribution

Take the design wind Vd as the peak of the windpower density function w= (V/Vd)3 where

(V)dV is the frequency of winds between V and V+dV. All wind distributions are open-ended

with possible high winds far above the norm. The narrowest fit the Rayleigh distribution

(τ)=4τ exp (-2τ2) =d exp(-2τ2) / dτ with mean τ= V/Vd of .627 and a mean power flux of .47 of

½Vd 2. Note the near symmetry of w about Vd in Figure 3.1. The slight asymmetry to higher

gives the power mean V as 1.064 Vd . Logically the design point would be the mode of the

windpower distribution so that as much power comes from windspeeds above it as below it.

More tractable and in keeping with this near peak method is the normalised Taylor series

quadratic and symmetric about the peak w(τ)= 3{1-4(τ -1)2}/2 between zeroes at τ =.5 and 1.5

with width between half power points of Vd ½ whereas the true Rayleigh half width is about

1.19 more.

3.3 Variation of X with V for different loads

Consider how the temporal variation of V around Vd varies all the x and so For slow

wind variations and an ideal cubic rpm load such as a fluid dynamic churn for mixing or heat, X

is constant, but for rapid fluctuations, rotor inertia still causes an oscillation in all the x. For a

synchronous generator the grid frequency keeps the rpm always constant so then X varies as V -1.

For an ideal (resistanceless) permanent magnet generator charging an ideal battery the frequency

also stays constant. Whereas connected to an entirely resistive load, the power (and so CPd V 3) is

proportional to 2, so X varies a V 1/2 near the design peak CPd . A (clutched) positive

displacement rotary pump’s power against a fixed pressure head varies as so X varies as V2.

So consider in general xV 2 >0 even for cubic loads. Then with τ=V/Vd

½ (τ -1)sin 2 (τ -1)

3.4 Small Deviations from the Robust

Let -r=r-d and be small deviations from robust values r and r. Then for the same

sin(-) at =o one must have =rcotr. Differentiating the (non-optimal) BEM

(2.1.2) C Fsintan(-)

d/dFsinsec2( -{dC/d Fsinsec2(-)-Fcostan(-)+dF/dsintan(-)}

(3.4.1)

Call this u/v to be evaluated as ud/vd at the central design point =d. Now if and are

robust then it must be the always optimal d(/d urd/vrd= ⅔ so any variation must be due

to non-robust dC/d in the denominator vd.

ud/vd urd/vrd)2 vd / urd -4 dC/d urd where urd =Fsind/cos2½d

For small angles dC/d=cos rsinr Fsindtan½d sin2r so

(d/dd(-)/d-2sind sin2r

In the root zone where F=1 but C departs from sin C d/dC= tan ½d Then for the same

CL(d cot½d and dC/d C(cot2 ½d - d2C/d2Cso

= cot2 ½d - d2C/d2 C (3.4.3)

The last term gets big and dominant as the robust approaches stall at which d/d is as much

twice the optimal value from (2.6.2)

Exercise: Show in the small angle limit around design dcT/d=64 h2 81d2 (3.4.4)

3.5. Expansion of Mean Power

For a given the lift cp is stationary at the optimum( and varies with by (1.6.4) as

cP≈ cP0 -6x(1+x2) Fsino(o)2 (3.5.1)

The first term cP0 =2x(1+x2) Fsin3o does not depend on d nor affect its optimum. By sec 3.4

o ≈ -sindo so the lift power coefficient is about

cP0 - 8x(1+x2) Fsino sin2d 2o2 (3.5.2)

Then multiplying by V3 dV=wdV symmetric about Vd and integrating the mean lift cP (on mean

windpower .47 of ½Vd 3 ) leads as cP0 -4cPd ( (2o2 (3.5.3)

(where underlining indicates the windpower weighted average value) to at least O( because

any linear variation in 2x(1+x2) Fsinosin2d about cPd will average out to nil.

The drag power loss cPD is stationary about o, cPD = - xcT{o½z(o)2} (3.5.4)

If the design were robust, cT would peak with a at design and vary insignificantly as O(.

From (3.3.4) cT≈64 h2 81d2. Now o=d-o +d where d depends upon

With fixed pitch =- so d=d=od+ -o≈ (⅔-2sindo2)

where the second term is smaller since the magnitude of =r-d will prove much less than

r and will be neglected appropriately in averaging

cPD ≈-xd (1+x/ xd)(cTd+64 h2 81d2{o½z[(do)+(⅔-2sindo)]2} (3.5.6)

by multiplying by fV3 dV=wdV symmetric about Vd and integrating for the mean drag cPD. loss.

The and x/ xd =-2sindterms average to nil and dropping obvious d subscripts , the

d dependent loss is roughly

64oxh2x/81x2+½zx{cT [(do)2+4(do)x/3x-82sino21.05(do)h222} (3.5.7)

plus 4cPLd (2o2 from the lift power (3.5.3) to get the d dependence of the net mean

loss from cP0.The sum is a quadratic in d so for the optimum, zero the linear expression in d (3.5.8):

8cPd-r)o2-64oxh2x/ 81x2+zxcT[d-o+2x/3x+42sino.53zx(2d-o-r)h222=0

With just the bold terms d is just a weighted average of o and r. The last bold term favours the arithmetic average. The first other term is positive and lowers d below r=0 so that the rise in the drag loss with x couples with a decline in a and factor ct for a net decrease in drag loss. The negative zxcT2x/3x term biases d above o so that o and o are reached above design where the drag penalty (3.5.4) is higher due to its leading x term. The penultimate positive term moves d lower again to minimise the excursion in and with (eg Fig 2.1). The net computed term is positive for a net bias away from tangency dropping in effect with increasing x.

3.6 Optimum Design Pitch in Rayleigh Wind

The normalised Rayleigh variance (τ -1)2w(τ) integrates to 1/20 with the under-

approximating Taylor expansion. (Note this is 1/10 of the square of the half width in τ. ) Dividing

(3.3.8) by cTd and the average value ( x/ x d=2sin dgives in the small angle

approximation for z=1 =cPd/ cTd =2/3 , .82h4d-r) xd + (d-oxdso that the robust weight

is twice at 2h4x=2.5 or at the very tip equal at 2x =3.

Fig 3.2 presents 2=1 calculations of (3.5.8) for the typical smooth (17%LS1 Fig 3.2) =.009

minimum with z=1 at zero lift reference o =10 reasonable vs 1 =9 in the C fit of sec 2.8

also used. Firstly the optimal local coefficient of performance curve cP was calculated from (8),

and then numerically integrated to get the rotor coefficient CPo= cPm dx2 / X2 which peaks when

CPo=cPo. The intersection of these curves helps to locate the peak CPo at X=5 =11.3 as its

maximum in X is very broad. Then the mean tip is 7.66 but to give this effective tip speed

ratio, the actual tip speed ratio must be increased relatively by .221s or for the standard B=3,

by 1.062 to 4.31.

The calculations show dominance of the robust r over the entire x range. Based on trisection o

would gain equal weight for x<≈1 =30 or , but that is the stall value s. Transitioning to

computing r from the lift curve fit at 1=9 at x≈2, and changing from (3.3.2) to (3.3.3), its

dominance then increases again as it climbs to a limit of 12.75 at x=0. The net bias away from

tangency is clear where r =0 but d is .6 less.

Conversely the sum of the two quadratic losses from the ideal highest and always optimal is

in turn dominated the reduction of the peak height by non-minimal drag/lift shown as cp versus

cpo. Inboard of the final tip rounding, the design net chord Bkl declines almost linearly at slope

d Bkl /dx =-.24. The lines above the tip X show the design angle of attack and chord with no tip.

The lines just below the tip X show the Prandtl F influence in raising r and the design

proportion above it due to the local weakening of the robust dominance with h4, so much so that

the design angle d reaches a minimum of .0103 or 4.9 at about x=3.25 and comes with .5 of

equalling for tangency at the very tip. Synchronously at g=-1 these help to relieve the high

wind tip bending moment by reducing its design stall margin.

3.7 Variance due to Wind Shear

The shear variation of V is weaker but additive to the above (power) variance of { V / V d }2=.04.

Consider a Hawt in a linearly sheared wind as V0(1+ cost) where = r/R to be V0(1+ ) at the

tip’s highest point azimuth t =π/2, and despite the loss of axisymmetry generalise the Hawt

momentum theory to hold at each streamline. The windpower distribution sketched in Figure 3.1

(and similarly a monochromatic tide) does not have a central peak in V. The mode of the

distribution is then a logical design choice but here it is easiest to just power average with

weight V03(1+ cos t)3 in the time domain by averaging over t 0 to 𝝅 with normalisation

V03(1+32/2) The power average wind appropriate as the design wind is then

Vd=V0(1+32/2+34/8)/ (1+3n2/2) ≈ V0(1+32/2-154/8)

Exercise then show½sin 2V/Vd and { V / V d }2 ½ 2-274/8 or

about .017 at representative r/R=.7 when ⅓ the top wind is twice the bottom vs .05 for the

tight approximation of the tight Rayleigh wind distribution

Such a variance about the mean is smaller than about any other choice of Vd. When

combined with the Rayleigh distribution, this broadens the peak. Independent variances such as

these are additive. Every other source of variation likewise adds further to the dominance of the

robust blade setting. But it will appear in the next chapter that the BM theory is not valid for

the yaw variation.....