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Approximate aerodynamic analysis for multi-blade darrieus wind turbines
Journal of Wind Engineering and Industrial Aerodynamics, 25 (1987) 131-150
131
Elsevier Science Publishers B.V., Amsterdam - - Printed in The Netherlands
APPROXIMATE AERODYNAMIC ANALYSIS FOR MULTIBLADE DARRIEUS WIND TURBINES
E. WILLIAM BEANS
Department of Mechanical Engineering, The University of Toledo, Toledo, 0H43606 (U.S.A.) (Received November 7, 1985; accepted in revised form April 22, 1986)
Summary An approximate analysis, which is based on the assumption of constant aerodynamic coefficients and the trigonometry of small angles, is presented. The analysis produces a performance model which consists of a series of closed-form equations. A result of the analysis is an interference factor which varies with the azimuthal angle. The model is valid for any number of blades up to six. The model predicts a power coefficient which agrees fairly well with experimental data and other models. The region of validity for the model is from the cut-in speed to rotor stall. The model predicts a variation in the upwind and downwind blade normal force. The model also predicts a power coefficient which is relatively constant with planform shape.
Nomenclature A
rotor area
a B b
local interference factor combined blade parameter blade length
CD Cf C CL CN Cn Cp Ct C~
airfoil drag coefficient local normal force coefficient cosine integral lift coefficient blade normal force coefficient local blade normal force coefficient power coefficient local tangential force coefficient blade coefficient-- SL + CD
c
chord
F h I J K m N
force rotor semi height integral constant constant
number of blades blade normal force
0167-6105/87/$03.50
© 1987 Elsevier Science Publishers B.V.
132 n
P p r r'
S SL V
v~ X Xo
normal force parameter power power parameter rotor reference radius localradius constant slope of the liftcoefficientline= dCL/dC~ wind velocity tangential velocity tip speed ratio tip speed ratio at cut-in
Greek symbols p 7 a 0 p
~n Zt ¢
co
angle of attack blade parameter radius ratio drag parameter
azimuthal angle density summation of normal terms summation of tangential terms blade angle angularvelocity
Subscripts c i m s 0
curved section index maximum or maximum condition straight section any azimuthal angle
1. Introduction
The aerodynamic analysis of a Darrieus rotor wind turbine is more complex than that for a horizontal axis wind turbine because: (1) the flow field varies with aximuthal angle, (2) the downwind blade passes through the wake of the upwind blade, and (3) the blades are curved. In De Vries' excellent summary of wind turbines [ 1 ], he classifies the analyses of vertical axis wind turbines into the following groups: (1) Momentum theory, which is a modification of Glauert's method [ 2 ]; (2) Multi-stream tube method, which is attributed to Strickland [ 3 ]; (3) Vortex method which is attributed to Holmes [ 4 ]. To solve these models for other than a straight blade requires the use of a computer. Hence, the importance of the various design parameters cannot be assessed. This paper presents an approximate aerodynamic analysis which follows the approach presented by Beans [ 5 ]. The analysis results in closed form solutions for the various performance factors. Since the solutions exist in closed form,
133
nl
I~
h
r
~I
Fig. 1. Blade geometry.
the importance of the various design parameters can be assessed. This makes the model extremely useful for preliminary design studies such as was done by Wilson and Walker [6 ] and for the performance of propellers [7 ]. The model does have limitswhich willbe discussed in thispaper. The results of the model are in fairagreement with wind tunnel data from Sharpe [8 ]. The approximate analysis is a variation of the m o m e n t u m or single streamtube approach. The analysis differsfrom other single streamtube analyses in that (1) drag is included in the determination of the interference factor, (2) closed form solution is obtained for the Darrieus rotor,and (3) the blade loading and output are not symmetric with azimuthal angle. According to De Vries [1 ], other m o m e n t u m theory approaches do not predict a variation with azimuthal angle. 2. Assumptions
The approximate analysis is based on the following assumptions: (1) the airfoilcan be characterized by a constant slope for the liftcurve and a constant value for the profiledrag; (2) the blade shape can be approximated by a circular arc and a straight line,see Fig. 1; (3) the blade chord is constant; (4) the turbine angle of attack can always be treated as a small angle; (5) the airfoilnever stalls; Assumption 1 is reasonable for airfoilsat small angles of attack. Assumptions 2 and 3 are made to simplify the analysis. The blade shape for a true Darrieus rotor isa troposkien which Sharpe [8 ] used. The circulararcmstraight
134 line shape is used by Strickland et al. [9]. Parashivoiu and Delaclaux [10] showed that the difference in performance between the troposkien and the circular arc--straight line shapes is small. Assumptions 4 and 5 are the key to the analysis as they were in ref. 5. Assumption 4 is valid up to 15 °. Assumption 5 is valid for the linear portion of the CL--C~ curve, which is about 15 ° for most airfoils. An angle of attack of 15°corresponds to a tip speed ratio (x) of 4. At tip speed ratios less than 4 the airfoil will stall. By envoking Assumption 5, the CL for the airfoil continues to increase as tip speed ratio decreases. Hence, the analysis overestimates the performance in the post-stall region. The analysis is invalid in the post-stall region as it was in ref. 5. A portion of the blade near the hubs is always in and out of stall at any tip speed ratio. Due to dynamic stall there is some evidence that CL does increase beyond the maximum static value. Francis and Keesee [ 11 ] have measured a dynamic C L ~ which js three times the static CLmax.Hence, there may be some validity to Assumption 5. Also, the inner portion of the rotor contributes little to the performance due to a diameter-to-the-fifth-power (D s) effect upon performance. 3. Development of equations The blade geometry for a Darrieus rotor is illustrated in Fig. 1. For any angle ¢~up to ~m ( Point m) the local rotor radius r' on the circular arc is
r'/r= l - y(1-cosO)
(1)
where 7 = re~r, rc is the radius of the circular arc and r is the reference radius of the rotor. At Point m, the rotor radius is
rm/r= l-- y(1--COS~)m)
(2)
Along the straight portion of the blade where ~m is constant, the local rotor radius is
r'/r=r.~/r-bsin~m/r
(3)
where b is the length along the blade. The boundary condition which relates the semi-height h to the maximum radius of the rotor is
(h/r)sin¢m = y + (1-- y) COSg~m
(4)
The relationship for the straight portion of the blade bs is
bs/r = ( rm/r ) /sin~m
(5 )
From eqns. ( 1 ) - - ( 3 ) , one can determine the swept area of the rotor upon which the various coefficients are based. The swept area A is
135 V
/ / ~
Blade 3
Cn3
~
~~ ~
~ ~ C
Blade1
///) k
Cn2
Blade2 Fig. 2. Blade elements.
f
A/r'2=2~y2~m+7sin@m(1-y)-I-h/r[1 - 7(1--COS~m) ] ~
(6)
Equations (I)--(6) are used to determinethe various blade parameters. The bladeelementsof a multi-bladerotor at any azimuthalangle0 are shown in Fig. 2. The blades of the rotor are equallyspaced. The azimuthal angle for any blade is
Oi=O+2~(i--1)/m
i=ltom
(7)
where m is the number of blades. It can be seen from Fig. 2 that the tangential velocity component (Vi) and the turbine angle of attack (ai) for any blade are
136 Vi = t o w - V ( 1 - a ) s i n O i
(8)
and tan
V(1 - a ) cosCcosOi a i - - t o r ' - - V ( 1 - - a ) s i n O i ~ ai
(9)
Under Assumption 4, t a n a l = cei as indicated in eqn. (9). The normal and tangential force coefficients on any blade are C.i = CLiCOSCei+ CD since/= ( SL + CD ) cei
(10)
Cti ~- C L i s i n c e i -- CD COScei =---SL cei 2 -- C D
(11)
In writing eqns. ( 10 ) and (11 ), Assumptions 1 and 4 were employed. The terms SL and CD are the slope of the CL--ce curve (dCL/dce) and the profile drag for the airfoil, respectively. These aerodynamic characteristics are a function of the blade airfoil. Once they are selected, they are assumed constant over the entire length of the blade and at all azimuthal angles. The force coefficient of any blade in the direction of the wind is Cfi : C n i c O s ~ c o s O i -~- CtisinO i,
or Cfi = ( S L -~- C D ) cos~cos0i cei "~-SL cei 2sin0i
- CD sin0i
(12 )
using eqns. (10) and (11 ). The total force on rn blade elements in the direction of the wind is
cP[ c° V~ , Co V~~ 7 d b = ~ ( C~V: ~ db d E e = kcos2ce, cos2ce..,j \cos cei ]
(13)
Using eqns. (8), (9) and (12) and the assumption of small angles, one can write the summation as _~_ (S L ~-CD)
Z n ~-Z(Cfi Vi2/cos2cei)
V(1-a)cos 2
~ l { c o s 20i[ for' - V(1 - a) sin0i ] }
(14)
+ SL V 2 (1 - a) 2cos 2~ I ( sinOi cos 2Oi ) -
C D I [ for' -- V(1 - a) sin0i ] 2sin0i
The summation in eqn. (14) has been evaluated for up to five equally spaced blades. The result of this operation is presented below. m=l
I , = (SL--~CD) V(1-a)eor'cos2t~cos2O
+ CD [ 2 V ( 1 - a ) o)r'sin20i -
-
V 2 (1 - a) 2 ( cos 2~sinOcos0 + sin a O) ]
(15)
137
m=2
X, =2(SL +CD) V ( 1 - a ) o)r'cos2C~os20
(16)
+CD [ 4 V ( 1 - a ) a)r'] sin20 m=3
2,, = (3/2) (SL + CD) V(1 - a) o)r'cos2~
(17)
+ CD [ 3 V(1 - a) (or' + sin0cos 20] m=4
X, =2(SL +CD) V(1-a)o)r'cos20+4CD V(1-a)a)r'
(18)
m--5
2:, = (5/2) (SL +CD) V(1--a)ogr'cos~@+5CD V(1-a)ear'
(19)
It is now convenient to do an order of magnitude analysis. For most airfoils, SL ~ 6 rad -1 and CD ~0.01. Therefore, it is fair to assume that the CD terms in eqns. (15)--(19) are negligible compared to the (SL+CD) terms. This assumption simplifies the analysis without significantly affecting the results. Neglecting the CD terms reduces the summation to
m= l
I , =C~ V ( 1 - a ) w r ' cosC)cos20
(20)
rn=2
In =2CaV(1-a)o)r' cos2C~cos2fl
(21)
rn=3
X , = (3/2)C~V(1-a)a)r'cos2~
(22)
m=4
I,=2C~V(1-a)a)r'cos2~
(23)
m=m
X , = (rn/2)C~V(1-a)o)r'cos2¢
(24)
where
C~ =SL + CD The term CD is retained in the definition of C~ because it does not add to the complexity of the analysis. It is worth noting that the analysis was completed for a two-bladed rotor to compare eqns. (16) and (21). The difference in the results is 1%. From impulse theory [ 2 ] across the entire rotor, the force in the direction of the wind for m blade elements at any azimuthal angle is dFo = 4 a ( 1 - a ) (pV2/2) (2r'cos~) db
(25)
Equating eqns. (13) and ( 25 ), one obtains the relationship for the interference factor
m= 1 and 2 m>~3
a= (m/8) C~(c/r)xcosC)cos20
(26)
a= (rn/16)C~(c/r)xcos~
(27)
138 where x is the tip speed ratio. It is now convenient to define the following term which is constant for any blade airfoil fl= ( m C J 8 ) (c/r)
(28)
With this definition eqns. (26) and (27) become m = 1 and 2 m >i 3
a=flxcos¢)cos20
(29)
a = flxcosg)/2
( 30 )
Equations (29) and (30) are an important conclusion of the analysis. One can see from eqn. (29) that the interference factor varies with the azimuthal angle 0 and the blade angle ¢ for one- and two-bladed rotors. For three or more blades there is no azimuthal variation. The equations also show that for a constant blade angle, the interference factor is constant along the blade at any azimuthal angle. For a single blade element at any azimuthal angle, the incremental normal force on the element in the direction of the radius is cp 1/12cosCdb dNe =-~Cnl - cP I °) r ' - Vcoso~ ( 1 - a ) sin
( SL + CD ) OlCos~db
Substituting eqns. (8) and (9) into the above equation and assuming small angles dNe -- (SL + C D ) [ x ( r ' / r ) -- (1--a)sin0] (1-a)cos2¢)cosOcpV2db/2
(31)
Equation (31) is for the lead or No. 1 blade. All other blades behave in the same way, but out of phase by the spacing angle [ 2u ( i - 1 ) / m ]. The total normal force on the blade is obtained by integrating from ~ = 0 to ~m along the curved section and then along the blade for ~-- ~,~ = constant for the straight section. In performing this integration, the relationship for the interference factor is used which is appropriate to the number of blades in the rotor (eqn. 29 or 30). This force is positive in the inward radial direction and is parallel to the reference radius r of the rotor. The results of the integration are the functions nc = y [ x ( J 2 - B J 3 ) - sin0(C2 - 2 B C 3 +B2C4) ]
(32)
ns = (rm/r) (cos2 ~m/sin~,O (1 -Bcos~m) [ (x/2) (rm/r)
(33)
- sin0 (1 - BcosCim) ] and C s o : (16/m ) flCOSO( nc -}-ns) / ( A / r 2)
(34)
139
where
B =fix cos 20
for m = 1 or 2
(35a)
and
B=flx/2
for m>~3
(35b)
In writing eqns. ( 3 2 ) - - ( 3 4 ) , the following notation is used to define the various integrals and terms Ci =
'cosi~d~
(36)
Ji = ( 1 - 7 ) C i + ~ C i + l
(37)
o
and
The expressions for the integrals of the cosine power can be found in any mathematical handbook. The power output for a single blade element in a multi-blade rotor at any azimuthal angle is
dPo = (p o) r' Ctl V12 cdb/2 Using eqns. (8), (9) and (11 ), the above expression can be written as
dPe = (p V3 /2 ) { ( r' /r ) ( SL COS2@COS20 - CD sin2 0 ) (1-- Bcos~ ) 2 -- CD ( r' /r ) 2 [ ( r' /r ) x 2 - 2 x ( 1 - B c o s ~ ) sin0] }cdb
(38)
In writing eqn. (38), the appropriate form of eqn. (35) is used according to the number of blades in the rotor. Equation (38) is now integrated along the blade in the same m a n n e r as eqn. ( 31 ). The result of this integration is the power coefficient for a single blade. Cpl0 =
(16/m)flx(p~ +p~ +Pc +Pcl)/ (A/r 2)
(39)
T h e power functions in eqn. (39) are defined as Pc = ~ [ (1 - J) cos2 0 (J2 - 2 J 3 B + J 4 B 2)
(40)
- Jsin20(Jo - 2 J I B + J 2 B2) - J x 2 I s ] Pc~ = 2Jx~sin0 [(1 - ~ ) (Jo - B J ~ ) ÷ 7(J1 - B J 2 ) ]
(41)
p, = { [ (1 - J) cos20cos2 ¢~
(42)
-- ($x2/2)(rm/r)2}So and
-- ¢~sin2/~]
(1 - B c o s 2 @~) 2
140 Psi = [ ( 4J/3 ) xsin~ ( rm/r) (1 - Bcos 2~m) ] So
(43)
where
a=c.c In writing eqn. (43), it is convenient to define the following generalized constant, to be used later
Si = ( r,,,/r ) i cos i ~ / 2 s i n ~
(44)
In eqn. (40), the t e r m / 3 is defined as
I3 = f ( r ' / r ) 3 d ~ = ( l - y ) 2Jo + 2 y ( 1 - y ) J 1 + y 2 J 2
(45)
Equation (39) is analogous to eqn. (34), which is the normal force coefficient for a single blade. Equation (39) is for the lead or No. 1 blade. All other blades are the same, but out of phase by the blade spacing angle. The variables in eqns. (34) and (39) are the azimuthal angle and the tip speed ratio. The power output for a rotor of m blade elements at any azimuthal angle is
dPe = w d T = (p/2 ) w r ' ICti Vi2 cdb Using eqns. (8), (9) and (11) and the assumption of small angles, one can express the above summation for m = 2
V 2I t = ICti Vi 2 = V 2 [ 2 ( SL cos 28cos 2~-- CD sin 2~) ( 1 -- a) 2 --2CDX2(r'/r) 2]
( 46 )
and for m >/3
Y 2 I t = V2[m/2(Sacos2f)-CD) ( l - - a ) 2 --mCDx2(r'/r) 2]
(47)
In terms of the above summation the incremental power is
dPe = (flV3/2)•tx(r'/r)cdb
(48)
In writing eqn. (48), eqns. (29) and (45) or eqns. (30) and (46) are used, whichever are appropriate for the number of blades in the rotor. Integrating eqn. (48) in the same m a n n e r as eqn. (31) or (38)
Cpe = 16fix (Pc +Ps) / ( A / r 2 )
(49)
For a rotor with two blades the relationships for the power functions are given by eqns. (40) and (42). For three or more blades, the power functions are Pc =Y{ ( 1 - 6 ) [ J 2 - 2J3 (flx/2 ) + J4 (flx/2 ) 2 ] - J [ J o - J 1 (fix~2) + J2(flx/2 ) 2] -Jx213 } and
(50)
141
p,={[(1-J)cos2~bm-Jl(1-flxcos~m/2)2-(Jx2/2)(rm/r)2}So
(51)
The average power coefficient for the rotor is
Cp = l fCpodO- n ( -16fix ~ 2 ) [fpcdO+ fpsdO]
(52)
It is seen from eqns. (50) and (51) that the power functions for a rotor with three or more blades is not a function of the azimuthal angle. Hence, the integration need only be performed on eqns. (40) and ( 42 ). After the integration, the relationship for the average power coefficient is
Cp = [ 16flx/ (A/r2) ] [Ko -1(1 (fix) +1(2 (fix) 2 -KaJx 2]
(53)
The values for the constants in eqn. (53) are, for m = 2 Ko = [ ( 1 - 6 ) ( 7 J 2 +$2) -J(TJo +So) ]/2
(54a)
K,=[3(I-J)(TJs+Sa)-J(TJ,+S,)]/4
(54b)
K2=[5(1-J)(TJ4+S4)-J(TJ2+S2)]/16
(54c)
K3 =T& + (So/2)(rmlr) ~
(54d)
and for m >t 3 Ko = [ ( l - J ) (TJ~ +S2) -J(TJo +So) ]/2
(55a)
K, = [ (l-J)(7,]3 +S3) -J(TJ, +S,) 1/2
(55b)
K~=[(I-J)(TJ4+S4)-J(FJ2+S2)]/8
(55c)
K3 =7/3 + (So/2) (rm/r) 2
(55d)
The variable in eqn. (53) is the tip speed ratio. 4. Use of the m o d e l
To use the model to determine the various performance factors,the aerodynamic characteristicsCDO and SL are selectedfirst.These characteristicscan be obtained for the selected airfoilfrom a source such as ref.12. The rotor geometry is established by selectingthe number of blades (m), the chord ratio (c/r) and the height ratio (h/r). The various geometric parameters for the rotor can be determined from eqns. (2), (4) and (6). The blade parameters C~, J and fl can be determined from their respective definitions and eqn. (28). For the established blade and geometric parameters, the values of the inte-
142 grals are obtained from eqns. (36), (37), (44), (45) and (54) or (55). The blade parameters and the integrals are functions of the aerodynamic characteristics and the geometric parameters. If the airfoil and the rotor planform are not changed, these values will not change. The blade normal force coefficient can be determined at any tip speed ratio and azimuthal angle from eqn. (34) and the auxiliary functions in eqns. ( 32 ), (33) and ( 35 ). The power coefficient as a function of the independent variables, x and 8, is obtained for a single blade from eqn. (39) and for a twobladed rotor from eqn. (49) and the. auxiliary functions of eqns. ( 40 ) m ( 43 ). The average power coefficient Cp is obtained from eqn. (53) and the auxiliary functions in eqns. (54) or (55) as a function of the tip speed ratio. Since the model consists of a series of closed form equations, the various design constants can be evaluated and the performance factors, such as the average power coefficient, can be determined. The advantage of a closed form solution is the ease with which one can assess the effect of the various design parameters. For this reason and because of the complexity of the model, it is best to program the model for a personal computer. The use of the model to assess the effect of the various design parameters will be illustrated in a later section. 5. Comparison with experimental results
Wind tunnel test data for a Darrieus rotor are presented in Fig. 3. The test data in Fig. 3 are from Sharpe [8]. The Sharpe rotor has two blades and a constant chord ratio of 0.1185. The blade shape is a troposkien. A troposkien is a shape which minimizes the bending stress. The troposkien has no aerodynamic advantage. The average power coefficient which was predicted from the approximate model is presented in Fig. 3 for a NACA 0012 and an NACA 0021. The aerodynamic characteristics for the model were obtained from ref. 12 for a halfrough airfoil. The values used are: NACA 0012, CDO= 0.0078, SL = 6.068; NACA 0021, CD0= 0.0096, SL = 5.626. One can see from Fig. 3 that the model agrees fairly well with the test data for the NACA 0012 up to the point of m a x i m u m Cp. At this point the airfoil is stalled and it is known t h a t the model is not valid for a stalled airfoil. The model does predict a decrease in performance for the NACA 0021. The decrease is not as much as observed in the wind tunnel. Figure 3 also gives the theoretical performance for a NACA 0012 as determined by Sharpe [8]. This theoretical performance is for a multiple streamtube model with dynamic stall and flow curvature [13]. It can be seen from Fig. 3 that the approximate model agrees fairly well with the model used by Sharpe for tip speed ratios greater than 4. For tip speed ratios less t h a n 4 the airfoil is stalled and the approximate model is not valid.
143 0.4,--
~
NACA 0012, Ref. 8 2 Blades 0.1185
c~= 0.3
QO~ACA0021[
&
NACA 0012 .~
0.2
2 2
O1
u.
/$'
/
NACA
0021j
O.O Mu] tiple Streamtube Model, NAC 0012, Ref 8 -O.1
r
•
2
1
I
3
,4
00 % ] 5
6
7
Tip Speed Ratio, X
Fig. 3. Comparison of model for a rotor with two blades.
Figure 4 is a comparison of the approximate model with other models and experimental data for a Darrieus rotor with three blades. The experimental data are from Sandia as presented in ref. 14. The theoretical models are forms of the multiple streamtube method. It is seen from Fig. 4 that the approximate model again agrees fairly well in its region of validity with the experimental data and the other models. 6. Numerical examples
The advantage of the model is that it allows one to readily assess the effect of the rotor design parameters upon performance. An example of this is given in Fig. 3 where the effect of airfoil shape is illustrated. One can see that the maximum power coefficient and the cut-in tip speed ratio decrease with increasing airfoil thickness. This is due to an increase in profile drag with thickness. The cut-in tip speed ratio is defined as the tip speed ratio for Cp = 0 where x # 0. The average power coefficient at tip speeds greater than the value for maximum Cp, which is the region of validity for the model, is sensitive to the value of profile drag CD. The coefficient of drag for a rotor is always a difficult parameter to assess by other than experimental means. The rotor drag includes more than the airfoil drag. The effect of profile drag on performance is presented in
144 0.6
ff
0.5=
f
0.4
/
n. 0.3
/;
0.2
.~
01
o~
0
Sandia 5m 3 Blades NACA 0015 c/r ffi 0.075 O Sandia Data, Refl4 m
/-Model
I
F
~Multiple Streamtube, Ref '4
-XN
l
~X
Double Multiple Streamtube, Ref 14
I
-0.1
/ 0
4
1
5
6
7
8
9
10
Tip Speed Ratio, X
Fig. 4. C o m p a r i s o n of m o d e l for a rotor w i t h three b h d e s .
0.4 I 2 Blades e/r = 0.1 SL = 6.0 i/rad
0.3 ~m = 60 °
J
0.2
0.1
/
CD = 010'0--/
/
0 0
!
4 Tip Speed Ratio, X
Fig. 5. Effect of profile drag.
6
7
11
12
145 Fig. 5. The airfoilis a nominal N A C A 0015. The profiledrag values are representative of a smooth, half-rough and rough airfoil. The effectof the aerodynamic characteristicson m a x i m u m performance and cut-in speed is illustratedin Figs. 6 and 7 for a two-bladed rotor. For a given design the point of m a x i m u m power coefficientcan be obtained from eqn. (53) by differentiation.It can be seen from Fig. 6 that the value of the slope of the liftline (SL) has a small effect on performance. The value of m a x i m u m Cp decreases 2 % for a 10% decrease in SL. The effectof SL upon Xo and xm is much less. Figure 7 illustratesthat the effectof drag is more significantand, therefore, is more important since drag is difficultto determine. This is particularlytrue for the cut-in speed (Xo). As stated previously and by Sharpe [8], the shape of the Cp--x curve above the m a x i m u m Cp point isprimarily a function of rotor drag. The effectofplanform shape is illustratedin Figs.8 and 9. In Fig. 8 the Cp---x relation is presented for various m a x i m u m blade angles (~m). A blade angle of 45 ° describes a triangular rotor planform, which is one limiting case. A blade angle of 90 ° describes a circulararc planform, which isthe other limitingcase. The results in Fig. 8 indicate that performance is not a strong function of the rotor planform. This conclusion supports those made by Parashiviou and Delaclaux [10]. The m a x i m u m power coefficient,the tip speed ratio at m a x i m u m Cp (x~) and the cut-in tip speed ratio (Xo) are presented in Fig. 9 as a function of the m a x i m u m blade angle (¢~). The tip speed ratiosx~ and x0 both decrease with increasing blade angle. This is because rotor solidityincreases.A n interesting observation from Fig. 9 is that the m a x i m u m power coefficientattains a mini m u m value at a blade angle of 56 o. This blade angle describes a planform which is basicallya troposkien. The effectof solidityin terms of the chord ratio for a two- and a three-blade rotor is presented in Fig. 10. As the chord ratio increases, the interference factor and the m a x i m u m power coefficientincrease (see eqns. 26 and 27). The tip speed ratiofor Cpm decreases with increasingchord ratio.This occurs because the parameter fix is nearly constant for all designs at the point of m a x i m u m power coefficient.The same can be said for the cut-in tip speed ratio,see eqn. (53). The azimuthal variations of the blade normal force coefficient and the power coefficient are illustrated in Figs. 11 and 12. These figures were generated from eqns. ( 34 ) and (39) for a constant tip speed ratio. These variations are useful in the static and dynamic analysis of the blade. The normal force coefficient is positive in the radial outward direction. Hence, a negative value indicates an inward force. A negative coefficient of power is the result of blade drag. The single-blade normal force coefficient in Figs. 11 and 12 has a variation in the upwind and downwind condition. This is an advantage of the approxi-
146 I
X
o
7
E
>
2 Blades CD = 0.OO8
~o
J 0.37
c/r = 0 . 1 era = 6 0 °
x~ 6
.o
o
C "~
5
pm
~..~ ~
0.36
J
.~
c~ 0.35
4
X m
-- 0.34
3 5.4
5.8
5.6
6.0
Slope of Lift Line,
6.2
SL
Fig. 6. Effect of the slope of the lift line on maximum performance. 8
X° .o ×
6
o;
k Blades SL = 6.0
$/~
o
0.36
2 .2
I/tad
= 0.i
m = 60 ° C
0.37
pm
.2
u 4
0.35
Xm
3i 0.006
0.34 0.008
0.010
0.012
0.014
C o e f f i c i e n t of Drag, C D
Fig. 7. Effect of profile drag on maximum performance.
mate model. According to De Vries [ 1 ], other momentum theory models do not predict such a variation. The variation predicted by the approximate model is believed due to the inclusion of drag in the analysis.
147 0.4
I
2 Blades C D = 0.008 S L = 6.0
0.3
~2
/
,
I/tad
c/r = i.1
\f+::
4~
0.2
f
/
~m Om
/
O.
0
1
2
3
6
4 Tip Speed Ratio,
7
8
X
Fig. 8. Effect of blade planform. 0.6
24 C D = 0.008 S L = 6.0
I/rad
_ . ~
~
0.5
20
i0\
/
J
~
3 Blades
0.4 C
, 2 Blades
0.3
••2
8
0.2
Blades
X o, 3 B l a d e s . - ~
~
~
~
0.1
4
¢ 0
0.02
0.04
0.06 Chord Ratio,
0.08
0.I0
c/r
Fig. 9. Effect of blade planform on maximum performance.
0.12
0.14 °
.i
148
X
o
7 2 Blades (.9
C D = 0.008
>~o
SL :
6
6.0
i/re d
0.38 c/r
x
= 0.I
J 2
o r~
0.37
5 c
~9
D.36
4 X
3
3.35 45
60 Maximum
75 Blade
90
A n g l e , ~]~m
Fig. I0. Effect of chord ratio on m a x i m u m
performance.
The coefficient of power presented in Fig. 11 is for the total rotor. The singleblade coefficient of power is essentially half this value. Although the coefficient of power variation in Fig. 11 is symmetric, it is by no means a simple relationship. The single-blade coefficient of power is presented in Fig. 12 for a rotor with three blades. This variation is also symmetric and appears to be a simple sine curve. It can be seen from Fig. 11 that the coefficient of power for a rotor with two blades has a cyclic variation. For a rotor with three or more blades the coefficient of power is constant with azimuthal angle. This behavior is analogous to the use of multiple cylinders in a reciprocating engine to dampen out torque variations. 7. Conclusions
A model consisting of closed form equations has been developed for estimating the performance of Darrieus wind turbines. The analysis which leads to the model is based on the assumptions of constant aerodynamic coefficients and the trigonometry of small angles. The analysis is a combination of disk impulse and blade element theories and results in an interference factor which varies with the azimuthal angle. The model predicts performance which agrees fairly well with experimental data for rotors with two and three blades from cut-in to the point of rotor stall.
149 0.6
u
0.4
c) j
0.2
~
X = 3.0
60 ~120
-o.2
Azimut::: Angle, @
I\1/ 240; 370
360
\
? ~
2XCp@I
-o.4
-0.6
Fig. 11. Azimuthal performance for a rotor with two blades. 0.6Xffi3.0 mffi 3
,a
o
:
pg:
0
? ~
- 0 •l
~m ==6 ; -0.!
i
Fig. 12. Azimuthal performance for a rotor with three blades.
The level of agreement in this region for the model is about the same as that for more complex models. The performance of the wind turbine in this region is sensitive to rotor drag, which is difficult to determine. Agreement in the stalled region of the rotor was not expected since the model uses a non-stalling airfoil. Hence, the region of validity for the model is from
150 the point of m a x i m u m power coefficient to cut-in speed. This is the region of interest for the aerodynamic pe r f or m a nc e of wind turbines. In the stalled region the model overestimates the m a x i m u m coefficient of power and underestimates the tip speed at which it occurs. It is known t h a t the point of m a x i m u m power and hence m a x i m u m load occurs at a speed below the m a x i m u m Cp point. Therefore, using the model to estimate the loads at m a x i m u m Cp will be conservative. T h e closed form of the model is useful for parametric design studies. T h e effect of such p a r a m e t e r s as chord ratio and n u m b e r of blades can readily be determined. B o t h the average and cyclic p e r f o r m a n c e are easily assessed. T h e model predicts a variation in the upwind and downwind blade normal force. Such a result is useful in determining blade loads. T h e model predicts a minim u m in the m a x i m u m power coefficient as a function of blade planform. T hi s m i n i m u m occurs at a blade angle of 56 ° which is basically the troposkien. T h e m i n i m u m is only 2% less t h a n the m a x i m u m value. Blade shape can therefore be selected based on strength r at her t h a n aerodynamics.
References 1 O. De Vries, Fluid dynamic aspects of wind energyconversion,AGARD-AG-243,July 1979. 2 H. Glauert, The Elements of Airfoil and AirscrewTheory, MacMillan, New York, 1943. 3 J.H. Strick!~md,The Darrieusturbine, a performanceprediction methodusing multiplestream tubes, SAND 75-0431, Sandia Laboratories, Albuquerque, NM, Oct. 1975. 4 O. Holmes,A contribution to the aerodynamictheory of the vertical-axiswind turbine, Proc. Int. Syrup. on Wind Systems, St. John's College, Cambridge, U.K., Sept. 1976. 5 E.W.Beans, Approximateaerodynamicanalysis for horizontal axis wind turbines, J. Energy, 7 (1983) 243. 6 R.E.Wilsonand S.N. Walker, A fortran program for the determination of performance,loads and stability derivatives of wind turbines, Oregon State University, GI-41840, Oct. 1974. 7 D.O.Domnash, S.S. Sherby and T.F. Connally, Airplane Aerodynamics,3rd edn., Pitman, New York, 1961, Chap. 7. 8 D.J. Sharpe, Wind tunnel performance tests on small Darrieus wind turbine models, SERI/CP-217-2902, Proc. Wind Power '85, San Francisco, CA, Aug. 1985. 9 J.H. Strickland, B.T. Webster and T. Nguygen,A vortex model of the Darrieus turbine: an analytical and experimental study, SAND 79-7058, Sandia Laboratories, Albuquerque, NM, Feb. 1980. 10 I. Parashivoiu and F. Delaclaux, Double multiple streamtube model with recent improvement, J. Energy, 7 (1983) 250. 11 M.S.Francis and J.E. Keesee,Airfoildynamicstall performancewith large-amplitudemotion, AIAA J., 23 (1985) 1653. 12 Ira H. Abbott and Albert E. Van Doenhoff, Theory of Wing Sections, Dover Publications, New York, 1985. 13 D.J. Sharpe, Refinements and developments of the multiple streamtube theory for the aerodynamic performance of verticalaxis wind turbines,Proc. 6th B.W.E.A. Wind Energy Conf., Reading, U.K., Mar. 1984. 14 I.Parashivoiu, Double-multiple streamtube model for Darrieus wind turbines,Wind Turbine Dynamics, N A S A Conf. Publ. 2185, Cleveland, OH, Feb. 1981, p. 19.
131
Elsevier Science Publishers B.V., Amsterdam - - Printed in The Netherlands
APPROXIMATE AERODYNAMIC ANALYSIS FOR MULTIBLADE DARRIEUS WIND TURBINES
E. WILLIAM BEANS
Department of Mechanical Engineering, The University of Toledo, Toledo, 0H43606 (U.S.A.) (Received November 7, 1985; accepted in revised form April 22, 1986)
Summary An approximate analysis, which is based on the assumption of constant aerodynamic coefficients and the trigonometry of small angles, is presented. The analysis produces a performance model which consists of a series of closed-form equations. A result of the analysis is an interference factor which varies with the azimuthal angle. The model is valid for any number of blades up to six. The model predicts a power coefficient which agrees fairly well with experimental data and other models. The region of validity for the model is from the cut-in speed to rotor stall. The model predicts a variation in the upwind and downwind blade normal force. The model also predicts a power coefficient which is relatively constant with planform shape.
Nomenclature A
rotor area
a B b
local interference factor combined blade parameter blade length
CD Cf C CL CN Cn Cp Ct C~
airfoil drag coefficient local normal force coefficient cosine integral lift coefficient blade normal force coefficient local blade normal force coefficient power coefficient local tangential force coefficient blade coefficient-- SL + CD
c
chord
F h I J K m N
force rotor semi height integral constant constant
number of blades blade normal force
0167-6105/87/$03.50
© 1987 Elsevier Science Publishers B.V.
132 n
P p r r'
S SL V
v~ X Xo
normal force parameter power power parameter rotor reference radius localradius constant slope of the liftcoefficientline= dCL/dC~ wind velocity tangential velocity tip speed ratio tip speed ratio at cut-in
Greek symbols p 7 a 0 p
~n Zt ¢
co
angle of attack blade parameter radius ratio drag parameter
azimuthal angle density summation of normal terms summation of tangential terms blade angle angularvelocity
Subscripts c i m s 0
curved section index maximum or maximum condition straight section any azimuthal angle
1. Introduction
The aerodynamic analysis of a Darrieus rotor wind turbine is more complex than that for a horizontal axis wind turbine because: (1) the flow field varies with aximuthal angle, (2) the downwind blade passes through the wake of the upwind blade, and (3) the blades are curved. In De Vries' excellent summary of wind turbines [ 1 ], he classifies the analyses of vertical axis wind turbines into the following groups: (1) Momentum theory, which is a modification of Glauert's method [ 2 ]; (2) Multi-stream tube method, which is attributed to Strickland [ 3 ]; (3) Vortex method which is attributed to Holmes [ 4 ]. To solve these models for other than a straight blade requires the use of a computer. Hence, the importance of the various design parameters cannot be assessed. This paper presents an approximate aerodynamic analysis which follows the approach presented by Beans [ 5 ]. The analysis results in closed form solutions for the various performance factors. Since the solutions exist in closed form,
133
nl
I~
h
r
~I
Fig. 1. Blade geometry.
the importance of the various design parameters can be assessed. This makes the model extremely useful for preliminary design studies such as was done by Wilson and Walker [6 ] and for the performance of propellers [7 ]. The model does have limitswhich willbe discussed in thispaper. The results of the model are in fairagreement with wind tunnel data from Sharpe [8 ]. The approximate analysis is a variation of the m o m e n t u m or single streamtube approach. The analysis differsfrom other single streamtube analyses in that (1) drag is included in the determination of the interference factor, (2) closed form solution is obtained for the Darrieus rotor,and (3) the blade loading and output are not symmetric with azimuthal angle. According to De Vries [1 ], other m o m e n t u m theory approaches do not predict a variation with azimuthal angle. 2. Assumptions
The approximate analysis is based on the following assumptions: (1) the airfoilcan be characterized by a constant slope for the liftcurve and a constant value for the profiledrag; (2) the blade shape can be approximated by a circular arc and a straight line,see Fig. 1; (3) the blade chord is constant; (4) the turbine angle of attack can always be treated as a small angle; (5) the airfoilnever stalls; Assumption 1 is reasonable for airfoilsat small angles of attack. Assumptions 2 and 3 are made to simplify the analysis. The blade shape for a true Darrieus rotor isa troposkien which Sharpe [8 ] used. The circulararcmstraight
134 line shape is used by Strickland et al. [9]. Parashivoiu and Delaclaux [10] showed that the difference in performance between the troposkien and the circular arc--straight line shapes is small. Assumptions 4 and 5 are the key to the analysis as they were in ref. 5. Assumption 4 is valid up to 15 °. Assumption 5 is valid for the linear portion of the CL--C~ curve, which is about 15 ° for most airfoils. An angle of attack of 15°corresponds to a tip speed ratio (x) of 4. At tip speed ratios less than 4 the airfoil will stall. By envoking Assumption 5, the CL for the airfoil continues to increase as tip speed ratio decreases. Hence, the analysis overestimates the performance in the post-stall region. The analysis is invalid in the post-stall region as it was in ref. 5. A portion of the blade near the hubs is always in and out of stall at any tip speed ratio. Due to dynamic stall there is some evidence that CL does increase beyond the maximum static value. Francis and Keesee [ 11 ] have measured a dynamic C L ~ which js three times the static CLmax.Hence, there may be some validity to Assumption 5. Also, the inner portion of the rotor contributes little to the performance due to a diameter-to-the-fifth-power (D s) effect upon performance. 3. Development of equations The blade geometry for a Darrieus rotor is illustrated in Fig. 1. For any angle ¢~up to ~m ( Point m) the local rotor radius r' on the circular arc is
r'/r= l - y(1-cosO)
(1)
where 7 = re~r, rc is the radius of the circular arc and r is the reference radius of the rotor. At Point m, the rotor radius is
rm/r= l-- y(1--COS~)m)
(2)
Along the straight portion of the blade where ~m is constant, the local rotor radius is
r'/r=r.~/r-bsin~m/r
(3)
where b is the length along the blade. The boundary condition which relates the semi-height h to the maximum radius of the rotor is
(h/r)sin¢m = y + (1-- y) COSg~m
(4)
The relationship for the straight portion of the blade bs is
bs/r = ( rm/r ) /sin~m
(5 )
From eqns. ( 1 ) - - ( 3 ) , one can determine the swept area of the rotor upon which the various coefficients are based. The swept area A is
135 V
/ / ~
Blade 3
Cn3
~
~~ ~
~ ~ C
Blade1
///) k
Cn2
Blade2 Fig. 2. Blade elements.
f
A/r'2=2~y2~m+7sin@m(1-y)-I-h/r[1 - 7(1--COS~m) ] ~
(6)
Equations (I)--(6) are used to determinethe various blade parameters. The bladeelementsof a multi-bladerotor at any azimuthalangle0 are shown in Fig. 2. The blades of the rotor are equallyspaced. The azimuthal angle for any blade is
Oi=O+2~(i--1)/m
i=ltom
(7)
where m is the number of blades. It can be seen from Fig. 2 that the tangential velocity component (Vi) and the turbine angle of attack (ai) for any blade are
136 Vi = t o w - V ( 1 - a ) s i n O i
(8)
and tan
V(1 - a ) cosCcosOi a i - - t o r ' - - V ( 1 - - a ) s i n O i ~ ai
(9)
Under Assumption 4, t a n a l = cei as indicated in eqn. (9). The normal and tangential force coefficients on any blade are C.i = CLiCOSCei+ CD since/= ( SL + CD ) cei
(10)
Cti ~- C L i s i n c e i -- CD COScei =---SL cei 2 -- C D
(11)
In writing eqns. ( 10 ) and (11 ), Assumptions 1 and 4 were employed. The terms SL and CD are the slope of the CL--ce curve (dCL/dce) and the profile drag for the airfoil, respectively. These aerodynamic characteristics are a function of the blade airfoil. Once they are selected, they are assumed constant over the entire length of the blade and at all azimuthal angles. The force coefficient of any blade in the direction of the wind is Cfi : C n i c O s ~ c o s O i -~- CtisinO i,
or Cfi = ( S L -~- C D ) cos~cos0i cei "~-SL cei 2sin0i
- CD sin0i
(12 )
using eqns. (10) and (11 ). The total force on rn blade elements in the direction of the wind is
cP[ c° V~ , Co V~~ 7 d b = ~ ( C~V: ~ db d E e = kcos2ce, cos2ce..,j \cos cei ]
(13)
Using eqns. (8), (9) and (12) and the assumption of small angles, one can write the summation as _~_ (S L ~-CD)
Z n ~-Z(Cfi Vi2/cos2cei)
V(1-a)cos 2
~ l { c o s 20i[ for' - V(1 - a) sin0i ] }
(14)
+ SL V 2 (1 - a) 2cos 2~ I ( sinOi cos 2Oi ) -
C D I [ for' -- V(1 - a) sin0i ] 2sin0i
The summation in eqn. (14) has been evaluated for up to five equally spaced blades. The result of this operation is presented below. m=l
I , = (SL--~CD) V(1-a)eor'cos2t~cos2O
+ CD [ 2 V ( 1 - a ) o)r'sin20i -
-
V 2 (1 - a) 2 ( cos 2~sinOcos0 + sin a O) ]
(15)
137
m=2
X, =2(SL +CD) V ( 1 - a ) o)r'cos2C~os20
(16)
+CD [ 4 V ( 1 - a ) a)r'] sin20 m=3
2,, = (3/2) (SL + CD) V(1 - a) o)r'cos2~
(17)
+ CD [ 3 V(1 - a) (or' + sin0cos 20] m=4
X, =2(SL +CD) V(1-a)o)r'cos20+4CD V(1-a)a)r'
(18)
m--5
2:, = (5/2) (SL +CD) V(1--a)ogr'cos~@+5CD V(1-a)ear'
(19)
It is now convenient to do an order of magnitude analysis. For most airfoils, SL ~ 6 rad -1 and CD ~0.01. Therefore, it is fair to assume that the CD terms in eqns. (15)--(19) are negligible compared to the (SL+CD) terms. This assumption simplifies the analysis without significantly affecting the results. Neglecting the CD terms reduces the summation to
m= l
I , =C~ V ( 1 - a ) w r ' cosC)cos20
(20)
rn=2
In =2CaV(1-a)o)r' cos2C~cos2fl
(21)
rn=3
X , = (3/2)C~V(1-a)a)r'cos2~
(22)
m=4
I,=2C~V(1-a)a)r'cos2~
(23)
m=m
X , = (rn/2)C~V(1-a)o)r'cos2¢
(24)
where
C~ =SL + CD The term CD is retained in the definition of C~ because it does not add to the complexity of the analysis. It is worth noting that the analysis was completed for a two-bladed rotor to compare eqns. (16) and (21). The difference in the results is 1%. From impulse theory [ 2 ] across the entire rotor, the force in the direction of the wind for m blade elements at any azimuthal angle is dFo = 4 a ( 1 - a ) (pV2/2) (2r'cos~) db
(25)
Equating eqns. (13) and ( 25 ), one obtains the relationship for the interference factor
m= 1 and 2 m>~3
a= (m/8) C~(c/r)xcosC)cos20
(26)
a= (rn/16)C~(c/r)xcos~
(27)
138 where x is the tip speed ratio. It is now convenient to define the following term which is constant for any blade airfoil fl= ( m C J 8 ) (c/r)
(28)
With this definition eqns. (26) and (27) become m = 1 and 2 m >i 3
a=flxcos¢)cos20
(29)
a = flxcosg)/2
( 30 )
Equations (29) and (30) are an important conclusion of the analysis. One can see from eqn. (29) that the interference factor varies with the azimuthal angle 0 and the blade angle ¢ for one- and two-bladed rotors. For three or more blades there is no azimuthal variation. The equations also show that for a constant blade angle, the interference factor is constant along the blade at any azimuthal angle. For a single blade element at any azimuthal angle, the incremental normal force on the element in the direction of the radius is cp 1/12cosCdb dNe =-~Cnl - cP I °) r ' - Vcoso~ ( 1 - a ) sin
( SL + CD ) OlCos~db
Substituting eqns. (8) and (9) into the above equation and assuming small angles dNe -- (SL + C D ) [ x ( r ' / r ) -- (1--a)sin0] (1-a)cos2¢)cosOcpV2db/2
(31)
Equation (31) is for the lead or No. 1 blade. All other blades behave in the same way, but out of phase by the spacing angle [ 2u ( i - 1 ) / m ]. The total normal force on the blade is obtained by integrating from ~ = 0 to ~m along the curved section and then along the blade for ~-- ~,~ = constant for the straight section. In performing this integration, the relationship for the interference factor is used which is appropriate to the number of blades in the rotor (eqn. 29 or 30). This force is positive in the inward radial direction and is parallel to the reference radius r of the rotor. The results of the integration are the functions nc = y [ x ( J 2 - B J 3 ) - sin0(C2 - 2 B C 3 +B2C4) ]
(32)
ns = (rm/r) (cos2 ~m/sin~,O (1 -Bcos~m) [ (x/2) (rm/r)
(33)
- sin0 (1 - BcosCim) ] and C s o : (16/m ) flCOSO( nc -}-ns) / ( A / r 2)
(34)
139
where
B =fix cos 20
for m = 1 or 2
(35a)
and
B=flx/2
for m>~3
(35b)
In writing eqns. ( 3 2 ) - - ( 3 4 ) , the following notation is used to define the various integrals and terms Ci =
'cosi~d~
(36)
Ji = ( 1 - 7 ) C i + ~ C i + l
(37)
o
and
The expressions for the integrals of the cosine power can be found in any mathematical handbook. The power output for a single blade element in a multi-blade rotor at any azimuthal angle is
dPo = (p o) r' Ctl V12 cdb/2 Using eqns. (8), (9) and (11 ), the above expression can be written as
dPe = (p V3 /2 ) { ( r' /r ) ( SL COS2@COS20 - CD sin2 0 ) (1-- Bcos~ ) 2 -- CD ( r' /r ) 2 [ ( r' /r ) x 2 - 2 x ( 1 - B c o s ~ ) sin0] }cdb
(38)
In writing eqn. (38), the appropriate form of eqn. (35) is used according to the number of blades in the rotor. Equation (38) is now integrated along the blade in the same m a n n e r as eqn. ( 31 ). The result of this integration is the power coefficient for a single blade. Cpl0 =
(16/m)flx(p~ +p~ +Pc +Pcl)/ (A/r 2)
(39)
T h e power functions in eqn. (39) are defined as Pc = ~ [ (1 - J) cos2 0 (J2 - 2 J 3 B + J 4 B 2)
(40)
- Jsin20(Jo - 2 J I B + J 2 B2) - J x 2 I s ] Pc~ = 2Jx~sin0 [(1 - ~ ) (Jo - B J ~ ) ÷ 7(J1 - B J 2 ) ]
(41)
p, = { [ (1 - J) cos20cos2 ¢~
(42)
-- ($x2/2)(rm/r)2}So and
-- ¢~sin2/~]
(1 - B c o s 2 @~) 2
140 Psi = [ ( 4J/3 ) xsin~ ( rm/r) (1 - Bcos 2~m) ] So
(43)
where
a=c.c In writing eqn. (43), it is convenient to define the following generalized constant, to be used later
Si = ( r,,,/r ) i cos i ~ / 2 s i n ~
(44)
In eqn. (40), the t e r m / 3 is defined as
I3 = f ( r ' / r ) 3 d ~ = ( l - y ) 2Jo + 2 y ( 1 - y ) J 1 + y 2 J 2
(45)
Equation (39) is analogous to eqn. (34), which is the normal force coefficient for a single blade. Equation (39) is for the lead or No. 1 blade. All other blades are the same, but out of phase by the blade spacing angle. The variables in eqns. (34) and (39) are the azimuthal angle and the tip speed ratio. The power output for a rotor of m blade elements at any azimuthal angle is
dPe = w d T = (p/2 ) w r ' ICti Vi2 cdb Using eqns. (8), (9) and (11) and the assumption of small angles, one can express the above summation for m = 2
V 2I t = ICti Vi 2 = V 2 [ 2 ( SL cos 28cos 2~-- CD sin 2~) ( 1 -- a) 2 --2CDX2(r'/r) 2]
( 46 )
and for m >/3
Y 2 I t = V2[m/2(Sacos2f)-CD) ( l - - a ) 2 --mCDx2(r'/r) 2]
(47)
In terms of the above summation the incremental power is
dPe = (flV3/2)•tx(r'/r)cdb
(48)
In writing eqn. (48), eqns. (29) and (45) or eqns. (30) and (46) are used, whichever are appropriate for the number of blades in the rotor. Integrating eqn. (48) in the same m a n n e r as eqn. (31) or (38)
Cpe = 16fix (Pc +Ps) / ( A / r 2 )
(49)
For a rotor with two blades the relationships for the power functions are given by eqns. (40) and (42). For three or more blades, the power functions are Pc =Y{ ( 1 - 6 ) [ J 2 - 2J3 (flx/2 ) + J4 (flx/2 ) 2 ] - J [ J o - J 1 (fix~2) + J2(flx/2 ) 2] -Jx213 } and
(50)
141
p,={[(1-J)cos2~bm-Jl(1-flxcos~m/2)2-(Jx2/2)(rm/r)2}So
(51)
The average power coefficient for the rotor is
Cp = l fCpodO- n ( -16fix ~ 2 ) [fpcdO+ fpsdO]
(52)
It is seen from eqns. (50) and (51) that the power functions for a rotor with three or more blades is not a function of the azimuthal angle. Hence, the integration need only be performed on eqns. (40) and ( 42 ). After the integration, the relationship for the average power coefficient is
Cp = [ 16flx/ (A/r2) ] [Ko -1(1 (fix) +1(2 (fix) 2 -KaJx 2]
(53)
The values for the constants in eqn. (53) are, for m = 2 Ko = [ ( 1 - 6 ) ( 7 J 2 +$2) -J(TJo +So) ]/2
(54a)
K,=[3(I-J)(TJs+Sa)-J(TJ,+S,)]/4
(54b)
K2=[5(1-J)(TJ4+S4)-J(TJ2+S2)]/16
(54c)
K3 =T& + (So/2)(rmlr) ~
(54d)
and for m >t 3 Ko = [ ( l - J ) (TJ~ +S2) -J(TJo +So) ]/2
(55a)
K, = [ (l-J)(7,]3 +S3) -J(TJ, +S,) 1/2
(55b)
K~=[(I-J)(TJ4+S4)-J(FJ2+S2)]/8
(55c)
K3 =7/3 + (So/2) (rm/r) 2
(55d)
The variable in eqn. (53) is the tip speed ratio. 4. Use of the m o d e l
To use the model to determine the various performance factors,the aerodynamic characteristicsCDO and SL are selectedfirst.These characteristicscan be obtained for the selected airfoilfrom a source such as ref.12. The rotor geometry is established by selectingthe number of blades (m), the chord ratio (c/r) and the height ratio (h/r). The various geometric parameters for the rotor can be determined from eqns. (2), (4) and (6). The blade parameters C~, J and fl can be determined from their respective definitions and eqn. (28). For the established blade and geometric parameters, the values of the inte-
142 grals are obtained from eqns. (36), (37), (44), (45) and (54) or (55). The blade parameters and the integrals are functions of the aerodynamic characteristics and the geometric parameters. If the airfoil and the rotor planform are not changed, these values will not change. The blade normal force coefficient can be determined at any tip speed ratio and azimuthal angle from eqn. (34) and the auxiliary functions in eqns. ( 32 ), (33) and ( 35 ). The power coefficient as a function of the independent variables, x and 8, is obtained for a single blade from eqn. (39) and for a twobladed rotor from eqn. (49) and the. auxiliary functions of eqns. ( 40 ) m ( 43 ). The average power coefficient Cp is obtained from eqn. (53) and the auxiliary functions in eqns. (54) or (55) as a function of the tip speed ratio. Since the model consists of a series of closed form equations, the various design constants can be evaluated and the performance factors, such as the average power coefficient, can be determined. The advantage of a closed form solution is the ease with which one can assess the effect of the various design parameters. For this reason and because of the complexity of the model, it is best to program the model for a personal computer. The use of the model to assess the effect of the various design parameters will be illustrated in a later section. 5. Comparison with experimental results
Wind tunnel test data for a Darrieus rotor are presented in Fig. 3. The test data in Fig. 3 are from Sharpe [8]. The Sharpe rotor has two blades and a constant chord ratio of 0.1185. The blade shape is a troposkien. A troposkien is a shape which minimizes the bending stress. The troposkien has no aerodynamic advantage. The average power coefficient which was predicted from the approximate model is presented in Fig. 3 for a NACA 0012 and an NACA 0021. The aerodynamic characteristics for the model were obtained from ref. 12 for a halfrough airfoil. The values used are: NACA 0012, CDO= 0.0078, SL = 6.068; NACA 0021, CD0= 0.0096, SL = 5.626. One can see from Fig. 3 that the model agrees fairly well with the test data for the NACA 0012 up to the point of m a x i m u m Cp. At this point the airfoil is stalled and it is known t h a t the model is not valid for a stalled airfoil. The model does predict a decrease in performance for the NACA 0021. The decrease is not as much as observed in the wind tunnel. Figure 3 also gives the theoretical performance for a NACA 0012 as determined by Sharpe [8]. This theoretical performance is for a multiple streamtube model with dynamic stall and flow curvature [13]. It can be seen from Fig. 3 that the approximate model agrees fairly well with the model used by Sharpe for tip speed ratios greater than 4. For tip speed ratios less t h a n 4 the airfoil is stalled and the approximate model is not valid.
143 0.4,--
~
NACA 0012, Ref. 8 2 Blades 0.1185
c~= 0.3
QO~ACA0021[
&
NACA 0012 .~
0.2
2 2
O1
u.
/$'
/
NACA
0021j
O.O Mu] tiple Streamtube Model, NAC 0012, Ref 8 -O.1
r
•
2
1
I
3
,4
00 % ] 5
6
7
Tip Speed Ratio, X
Fig. 3. Comparison of model for a rotor with two blades.
Figure 4 is a comparison of the approximate model with other models and experimental data for a Darrieus rotor with three blades. The experimental data are from Sandia as presented in ref. 14. The theoretical models are forms of the multiple streamtube method. It is seen from Fig. 4 that the approximate model again agrees fairly well in its region of validity with the experimental data and the other models. 6. Numerical examples
The advantage of the model is that it allows one to readily assess the effect of the rotor design parameters upon performance. An example of this is given in Fig. 3 where the effect of airfoil shape is illustrated. One can see that the maximum power coefficient and the cut-in tip speed ratio decrease with increasing airfoil thickness. This is due to an increase in profile drag with thickness. The cut-in tip speed ratio is defined as the tip speed ratio for Cp = 0 where x # 0. The average power coefficient at tip speeds greater than the value for maximum Cp, which is the region of validity for the model, is sensitive to the value of profile drag CD. The coefficient of drag for a rotor is always a difficult parameter to assess by other than experimental means. The rotor drag includes more than the airfoil drag. The effect of profile drag on performance is presented in
144 0.6
ff
0.5=
f
0.4
/
n. 0.3
/;
0.2
.~
01
o~
0
Sandia 5m 3 Blades NACA 0015 c/r ffi 0.075 O Sandia Data, Refl4 m
/-Model
I
F
~Multiple Streamtube, Ref '4
-XN
l
~X
Double Multiple Streamtube, Ref 14
I
-0.1
/ 0
4
1
5
6
7
8
9
10
Tip Speed Ratio, X
Fig. 4. C o m p a r i s o n of m o d e l for a rotor w i t h three b h d e s .
0.4 I 2 Blades e/r = 0.1 SL = 6.0 i/rad
0.3 ~m = 60 °
J
0.2
0.1
/
CD = 010'0--/
/
0 0
!
4 Tip Speed Ratio, X
Fig. 5. Effect of profile drag.
6
7
11
12
145 Fig. 5. The airfoilis a nominal N A C A 0015. The profiledrag values are representative of a smooth, half-rough and rough airfoil. The effectof the aerodynamic characteristicson m a x i m u m performance and cut-in speed is illustratedin Figs. 6 and 7 for a two-bladed rotor. For a given design the point of m a x i m u m power coefficientcan be obtained from eqn. (53) by differentiation.It can be seen from Fig. 6 that the value of the slope of the liftline (SL) has a small effect on performance. The value of m a x i m u m Cp decreases 2 % for a 10% decrease in SL. The effectof SL upon Xo and xm is much less. Figure 7 illustratesthat the effectof drag is more significantand, therefore, is more important since drag is difficultto determine. This is particularlytrue for the cut-in speed (Xo). As stated previously and by Sharpe [8], the shape of the Cp--x curve above the m a x i m u m Cp point isprimarily a function of rotor drag. The effectofplanform shape is illustratedin Figs.8 and 9. In Fig. 8 the Cp---x relation is presented for various m a x i m u m blade angles (~m). A blade angle of 45 ° describes a triangular rotor planform, which is one limiting case. A blade angle of 90 ° describes a circulararc planform, which isthe other limitingcase. The results in Fig. 8 indicate that performance is not a strong function of the rotor planform. This conclusion supports those made by Parashiviou and Delaclaux [10]. The m a x i m u m power coefficient,the tip speed ratio at m a x i m u m Cp (x~) and the cut-in tip speed ratio (Xo) are presented in Fig. 9 as a function of the m a x i m u m blade angle (¢~). The tip speed ratiosx~ and x0 both decrease with increasing blade angle. This is because rotor solidityincreases.A n interesting observation from Fig. 9 is that the m a x i m u m power coefficientattains a mini m u m value at a blade angle of 56 o. This blade angle describes a planform which is basicallya troposkien. The effectof solidityin terms of the chord ratio for a two- and a three-blade rotor is presented in Fig. 10. As the chord ratio increases, the interference factor and the m a x i m u m power coefficientincrease (see eqns. 26 and 27). The tip speed ratiofor Cpm decreases with increasingchord ratio.This occurs because the parameter fix is nearly constant for all designs at the point of m a x i m u m power coefficient.The same can be said for the cut-in tip speed ratio,see eqn. (53). The azimuthal variations of the blade normal force coefficient and the power coefficient are illustrated in Figs. 11 and 12. These figures were generated from eqns. ( 34 ) and (39) for a constant tip speed ratio. These variations are useful in the static and dynamic analysis of the blade. The normal force coefficient is positive in the radial outward direction. Hence, a negative value indicates an inward force. A negative coefficient of power is the result of blade drag. The single-blade normal force coefficient in Figs. 11 and 12 has a variation in the upwind and downwind condition. This is an advantage of the approxi-
146 I
X
o
7
E
>
2 Blades CD = 0.OO8
~o
J 0.37
c/r = 0 . 1 era = 6 0 °
x~ 6
.o
o
C "~
5
pm
~..~ ~
0.36
J
.~
c~ 0.35
4
X m
-- 0.34
3 5.4
5.8
5.6
6.0
Slope of Lift Line,
6.2
SL
Fig. 6. Effect of the slope of the lift line on maximum performance. 8
X° .o ×
6
o;
k Blades SL = 6.0
$/~
o
0.36
2 .2
I/tad
= 0.i
m = 60 ° C
0.37
pm
.2
u 4
0.35
Xm
3i 0.006
0.34 0.008
0.010
0.012
0.014
C o e f f i c i e n t of Drag, C D
Fig. 7. Effect of profile drag on maximum performance.
mate model. According to De Vries [ 1 ], other momentum theory models do not predict such a variation. The variation predicted by the approximate model is believed due to the inclusion of drag in the analysis.
147 0.4
I
2 Blades C D = 0.008 S L = 6.0
0.3
~2
/
,
I/tad
c/r = i.1
\f+::
4~
0.2
f
/
~m Om
/
O.
0
1
2
3
6
4 Tip Speed Ratio,
7
8
X
Fig. 8. Effect of blade planform. 0.6
24 C D = 0.008 S L = 6.0
I/rad
_ . ~
~
0.5
20
i0\
/
J
~
3 Blades
0.4 C
, 2 Blades
0.3
••2
8
0.2
Blades
X o, 3 B l a d e s . - ~
~
~
~
0.1
4
¢ 0
0.02
0.04
0.06 Chord Ratio,
0.08
0.I0
c/r
Fig. 9. Effect of blade planform on maximum performance.
0.12
0.14 °
.i
148
X
o
7 2 Blades (.9
C D = 0.008
>~o
SL :
6
6.0
i/re d
0.38 c/r
x
= 0.I
J 2
o r~
0.37
5 c
~9
D.36
4 X
3
3.35 45
60 Maximum
75 Blade
90
A n g l e , ~]~m
Fig. I0. Effect of chord ratio on m a x i m u m
performance.
The coefficient of power presented in Fig. 11 is for the total rotor. The singleblade coefficient of power is essentially half this value. Although the coefficient of power variation in Fig. 11 is symmetric, it is by no means a simple relationship. The single-blade coefficient of power is presented in Fig. 12 for a rotor with three blades. This variation is also symmetric and appears to be a simple sine curve. It can be seen from Fig. 11 that the coefficient of power for a rotor with two blades has a cyclic variation. For a rotor with three or more blades the coefficient of power is constant with azimuthal angle. This behavior is analogous to the use of multiple cylinders in a reciprocating engine to dampen out torque variations. 7. Conclusions
A model consisting of closed form equations has been developed for estimating the performance of Darrieus wind turbines. The analysis which leads to the model is based on the assumptions of constant aerodynamic coefficients and the trigonometry of small angles. The analysis is a combination of disk impulse and blade element theories and results in an interference factor which varies with the azimuthal angle. The model predicts performance which agrees fairly well with experimental data for rotors with two and three blades from cut-in to the point of rotor stall.
149 0.6
u
0.4
c) j
0.2
~
X = 3.0
60 ~120
-o.2
Azimut::: Angle, @
I\1/ 240; 370
360
\
? ~
2XCp@I
-o.4
-0.6
Fig. 11. Azimuthal performance for a rotor with two blades. 0.6Xffi3.0 mffi 3
,a
o
:
pg:
0
? ~
- 0 •l
~m ==6 ; -0.!
i
Fig. 12. Azimuthal performance for a rotor with three blades.
The level of agreement in this region for the model is about the same as that for more complex models. The performance of the wind turbine in this region is sensitive to rotor drag, which is difficult to determine. Agreement in the stalled region of the rotor was not expected since the model uses a non-stalling airfoil. Hence, the region of validity for the model is from
150 the point of m a x i m u m power coefficient to cut-in speed. This is the region of interest for the aerodynamic pe r f or m a nc e of wind turbines. In the stalled region the model overestimates the m a x i m u m coefficient of power and underestimates the tip speed at which it occurs. It is known t h a t the point of m a x i m u m power and hence m a x i m u m load occurs at a speed below the m a x i m u m Cp point. Therefore, using the model to estimate the loads at m a x i m u m Cp will be conservative. T h e closed form of the model is useful for parametric design studies. T h e effect of such p a r a m e t e r s as chord ratio and n u m b e r of blades can readily be determined. B o t h the average and cyclic p e r f o r m a n c e are easily assessed. T h e model predicts a variation in the upwind and downwind blade normal force. Such a result is useful in determining blade loads. T h e model predicts a minim u m in the m a x i m u m power coefficient as a function of blade planform. T hi s m i n i m u m occurs at a blade angle of 56 ° which is basically the troposkien. T h e m i n i m u m is only 2% less t h a n the m a x i m u m value. Blade shape can therefore be selected based on strength r at her t h a n aerodynamics.
References 1 O. De Vries, Fluid dynamic aspects of wind energyconversion,AGARD-AG-243,July 1979. 2 H. Glauert, The Elements of Airfoil and AirscrewTheory, MacMillan, New York, 1943. 3 J.H. Strick!~md,The Darrieusturbine, a performanceprediction methodusing multiplestream tubes, SAND 75-0431, Sandia Laboratories, Albuquerque, NM, Oct. 1975. 4 O. Holmes,A contribution to the aerodynamictheory of the vertical-axiswind turbine, Proc. Int. Syrup. on Wind Systems, St. John's College, Cambridge, U.K., Sept. 1976. 5 E.W.Beans, Approximateaerodynamicanalysis for horizontal axis wind turbines, J. Energy, 7 (1983) 243. 6 R.E.Wilsonand S.N. Walker, A fortran program for the determination of performance,loads and stability derivatives of wind turbines, Oregon State University, GI-41840, Oct. 1974. 7 D.O.Domnash, S.S. Sherby and T.F. Connally, Airplane Aerodynamics,3rd edn., Pitman, New York, 1961, Chap. 7. 8 D.J. Sharpe, Wind tunnel performance tests on small Darrieus wind turbine models, SERI/CP-217-2902, Proc. Wind Power '85, San Francisco, CA, Aug. 1985. 9 J.H. Strickland, B.T. Webster and T. Nguygen,A vortex model of the Darrieus turbine: an analytical and experimental study, SAND 79-7058, Sandia Laboratories, Albuquerque, NM, Feb. 1980. 10 I. Parashivoiu and F. Delaclaux, Double multiple streamtube model with recent improvement, J. Energy, 7 (1983) 250. 11 M.S.Francis and J.E. Keesee,Airfoildynamicstall performancewith large-amplitudemotion, AIAA J., 23 (1985) 1653. 12 Ira H. Abbott and Albert E. Van Doenhoff, Theory of Wing Sections, Dover Publications, New York, 1985. 13 D.J. Sharpe, Refinements and developments of the multiple streamtube theory for the aerodynamic performance of verticalaxis wind turbines,Proc. 6th B.W.E.A. Wind Energy Conf., Reading, U.K., Mar. 1984. 14 I.Parashivoiu, Double-multiple streamtube model for Darrieus wind turbines,Wind Turbine Dynamics, N A S A Conf. Publ. 2185, Cleveland, OH, Feb. 1981, p. 19.