A COMPACT, CLOSED-FORM SOLUTION FOR THE OPTIMUM, IDEAL WIND TURBINE. Dr. David A. Peters McDonnell Douglas Professor of Engineering

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1 A COMPACT, CLOSED-FORM SOLUTION FOR THE OPTIMUM, IDEAL WIND TURBINE D. David A. Petes McDonnell Douglas Pofesso of Engineeing Ramin Modaes Gaduate Reseach Assistant Depatment of Mechanical Engineeing & Mateials Science Campus Box 1185 Washington Univesity in St. Louis St. Louis, Missoui 6313 Apil 4, 1 Revised: Novembe 1, 1 Abstact The classical momentum solution fo the optimum induced-flow distibution of a wind tubine in the pesence of wake swil can be found in many textbooks. This standad deivation consists of two momentum balances (one fo axial momentum and one fo angula momentum) which ae combined into a fomula fo powe coefficient in tems of induction factos. Numeical pocedues then give the pope induction factos fo the optimum inflow distibution at any adial station; and this, in tun, gives the best possible powe coefficient fo an ideal wind tubine. The pesent development offes a moe staightfowad deivation of the optimum tubine. The final fomulas give the identical conditions fo the ideal wind tubine as do the classical solutions but with seveal impotant diffeences in the deivation and in the fom of the esults. Fist, only one momentum balance is equied (the othe being edundant). Second, the solution is povided in a compact, closed fom fo both the induction factos and the minimum powe athe than in tems of a numeical pocess. Thid, the solution eliminates the singulaities that ae pesent in cuent published solutions. Fouth, this new appoach also makes possible a closed-fom solution fo the optimum chod distibution in the pesence of wake otation. Keywods: wind, optimum, design, powe, inflow, efficiency 1

2 Notation a axial induction facto, u/u value of a at blade tip a' classical swil induction facto, v/ω b total induction facto, w/u value of b at blade tip B numbe of blades c blade chod, m blade dag coefficient blade lift coefficient, L/[cV /] tubine powe coefficient, P/[πR U 3 /] net powe coefficient tubine toque coefficient, Q/[πR 3 U /] tubine thust coefficient, T/[πR U /] total thust coefficient f new swil induction facto, v/u (a' = ) integal of powe due to dag integal of thust due to dag dl lift on annula ing of width d, N adial position, m non-dimensional adial position /R R tip adius, m P powe deliveed by wind tubine, N-m/sec Q oto toque, N-m u induced velocity in axial diection, m/sec U wind velocity, m/sec v induced velocity in swil diection, m/sec V total velocity elative to aifoil, m/sec (U of Ref. 1) w total induced velocity (paallel to total thust), m/sec x Ref. 1 change of vaiable, x=1-3a y pesent change of vaiable, y=3b-1 z intemediate vaiable, z=b /(1+ ) blade angle of attack, ad small quantity, << 1 blade inflow angle, ad nondimensional total flow, V/U local inflow atio, Ω/U tip speed atio, ΩR/U blade pitch angle, ad ai density, kg/m 3 local solidity, Bc/(π) (' of Ref. 1) oto angula velocity, ad/sec

3 Intoduction The fist objective of this pape is to demonstate that the solution fo the induction factos of an ideal optimum wind tubine (and fo the esultant optimum powe coefficient) can be deived in a moe diect manne than has been done in ealie deivations. The second objective of this pape is to show that as the esult of this moe diect deivation one can obtain a set of compact, closedfom expessions fo these optimum induction factos and powe coefficient. This is in contast to past deivations, which teminate with a numeical pocedue to complete the optimum oto (athe than ending in compact fomulas). Futhemoe, because of the moe diect appoach of a unified momentum theoy, singulaities inheent in ealie appoaches ae eliminated The thid objective of this pape is the evelation that this new, diect deivation (along with the moe compact expessions) is possible because of the fact that axial momentum theoy and swil momentum theoy ae edundant to each othe, which implies that only a single, unified momentum theoy is necessay fo the development. Finally, the fouth objective of this pape is to show how the optimum chod and twist distibutions fo a oto (including pofile dag) can also be found in closed fom without the necessity of the neglect of wake swil. It is hoped that these compact, closed-fom expessions will yield additional insights into the natue of an optimum wind tubine. Backgound It is a well-known fact that the Betz optimum fo an actuato disk (acting as a wind tubine oto) is that the disk slows the wind at the disk to /3 of its incoming value (which implies that it is slowed to 1/3 of the incoming value fa downsteam). This theoetically yields 16/7 (o 59.3%) of the incoming kinetic enegy conveted as useful enegy. Howeve, fo a disk that geneates lift and toque though lifting blades, the lift is not pependicula to the disk. The lift consequently ceates a swil velocity that impats kinetic enegy in the wake due to otation. As a esult, the distibution of induced flow fo optimum powe with swil is moe complicated than is the Betz solution; and the maximum possible powe is smalle than 16/7. The deivation of the optimum induced-flow distibution (including swil) and of the esultant optimum powe can be found in wind tubine texts, such as Refs. [1-4]. These deivations consist of seveal standad elements. Fist, a tanslational momentum theoy is pefomed in the axial diection in ode to elate oto thust to the axial induced flow, u. Second, an angula momentum theoy is used to find the elationship between oto toque and swil velocity, v. These momentum esults ae expessed in tems of an axial induction facto and an angula induction facto (a and a', espectively) in the following fom: u au, v a (1) a(1 a) = () a(1 a) The esulting powe pe unit adius can be expessed in tems of tip-speed atio to give: 3 dcp 8 a(1 a) d (3) 3

4 Equation () can then be used to find a' in tems of a, the latte of which can then be placed into Eq. (3). Noting that the deivative with espect to a of Eq. (3) must equal to zeo allows one (afte consideable algeba) to obtain the optimality condition: (1 a)(4a 1) (1 3 a) a =, (4) (1 3 a) (4a 1) The optimality condition in Eq. (4) can be witten as a cubic in a: 3 16a 4 a (9 3 ) a 1 (5) Although closed-fom solutions exist fo cubic equations, Ref. [5], the algeba is not paticulaly conducive to finding a closed fom fo the pope oot of Eq. (5). Thus, the solution to Eq. (5) is taditionally done numeically. The final inflow angle then follows diectly once the induction paametes (a and a') ae detemined: 1 a a 1 1 tan = tan (6) (1 a) a In ode to find the total optimum induced powe, Ref. [1] makes two additional changes of vaiable in the integal of Eq. (3). Fist, a change is made fom to a; and then a change is made fom a to x = 1-3a. The esultant integal fo yields an expession that must be evaluated at the uppe and lowe bounds to find ; CP x x x x x x x ln( ) (7) 79 5 x x 1 3a whee is the value of a at the tip, (i.e., at ). It should be noted that the above has singulaities both in the limit as goes to zeo (the blade oot) and goes to infinity. Nevetheless, Eq. (7) gives a numeical method fo finding the optimum tubine based on momentum theoy. Altenative Appoach What we offe hee is an altenate deivation of the paametes fo an optimum tubine along with a esulting closed-fom solution both fo the optimum induced flow and fo the total powe coefficient. The new fomulation diffes in fou ways fom ealie expessions: 1.) only a single momentum balance and single induction facto is equied;.) the optimum induction facto is found in a compact, closed fom; 3.) the total optimum powe is also obtained as a single, closed-fom expession; and 4.) the singulaities of ealie methods ae emoved. Although the esulting optimum tubine paametes ae of couse the same with the pesent method as with the pevious 4

5 method, the moe compact esults gives additional insight into the natue of the optimum wind tubine. The geomety of the flow is cucial to the new appoach fo finding the optimum induced flow distibution. Figue 1 shows the vaious flow velocities as seen at the aifoil. The figue follows the convention of Glauet, Ref. [6]. It is impotant to note that both momentum consideations and votex-tube theoy show that the induced flow w and the lift vecto L must be along the same line (but in opposite diections) and that this line must be pependicula to the total flow elative to the blade. This implies that the induced flow w completely detemines the local inflow angle. To be moe specific, one can note fom classical momentum developments [i.e., Ref. [1], Eqs. (3.1) (3.8)] that the combined vecto of swil velocity and axial velocity fom an induced-flow vecto that is exactly paallel to the local lift vecto (but in the opposite diection). Refeence [4] shows that the same esult follows when one consides votex-tube theoy with voticity that is diected along the wake helix. In fact, Glauet in Ref. [6] assumes that this must be the case fo the optimum oto. The physical basis fo this simple esult is that of Newton s laws of motion. Fo evey action, thee is an equal-and-opposite eaction; and the foce vecto must be popotional to the time ate of change of the momentum vecto. Thus, it is not supising that the induced flow and foce must be opposite but paallel. Once this facto is ecognized, one can see fom the geomety of Fig. 1 that because the lift must be pependicula to the total flow vecto at the blade (the Biot-Savat Law) it follows that the induced flow must be pependicula to the local votex sheet. Based on this obsevation, one can obtain a simple elationship between the oiginal total flow and the total flow with induced flow based on the Pythagoean theoem. V = U + w (8) Note that Fig. 1 includes seveal diffeent tiangles fom which one can fomulate,, and. Based on this tigonomety, many useful identities can be found. Fo example, if one daws fom the meeting point of and (at the bottom of the figue) a line that ends pependicula to the velocity vecto, then the length of that line can be expessed as. This equation can then be squaed and solved fo eithe o in tems of the flow vaiables (a vey useful esult). U U w w sin( ) (9) U U U w Uw cos( ) (1) The above elationships may also be witten in tems of non-dimensional paametes (including the oveall induction facto, b = w/u): 1 b (11) 5

6 1 b b v sin( ) (1) w 1 1 b b u cos( ) (13) 1 w Theefoe, the key paamete fo optimum powe is the total induction facto of the induced flow, b. The above development offes the necessay geometic elationships to allow a deivation of the optimal wind tubine based on a single, unified momentum balance of loads vesus induced flow, as shown below. Momentum Theoy Refeence [7] poves that momentum theoy can be applied diectly to a tilted lift vecto to give the same induced velocity that would be obtained fom votex-tube theoy. Refeence [8] poves that an actuato-disk theoy also gives the same answe as votex theoy (i.e., the exact answe) when applied to a tilted lift vecto and to a tilted induced flow vecto. Theefoe, in contast to the pevious deivations which invoke both axial and angula momentum balances, it is only necessay to look at one momentum balance fo the entie lift and induced flow. That momentum balance, when witten fo an annula ing, is: dl ( d) w U wcos( ) (14) This single momentum equation is all that is necessay because the swil and axial components of w ae automatically included in the geomety of Fig. 1. A sepaate axial o swil balance is edundant. To obtain powe fom Eq. (14), one can wite: dp ( d) w U wcos( ) sin( ) (15) The non-dimensional coefficient can then be witten in tems of : 6 dc 8 b 1bcos( ) sin( ) d (16) P At this point, thee ae two appoaches that can be taken. In the fist one, tigonometic identities can be used to solve fo in tems of followed by a deivative to find the maximum powe. This gives an equivalent esult to Eq. (5) fo the optimality condition. A second appoach is to substitute and fom Eqs. (1-13) and then take a diect deivative. Afte consideable algeba, this esults in a diffeent cubic equation than the one in Eq. (5),

7 3 16z 4z 9z (17) 1 whee. This cubic equation yields a closed-fom esult based on Ref. [5] that lends itself to a compact fom fo the optimum value of b. 1 b cos cos (18) b 1 1 cos cos (19) cos cos 3 sin cos () b Equations (18-) ae all equivalent expessions fo the optimum b. The axial and swil inductions ae: and. Complete Expessions With the above closed-fom expession fo b, the entie optimum blade can be educed to simple equations. At any given adial position, one immediately knows the appopiate. Fom that, Eq. () gives the optimum b. One can also solve fo the optimum given b. In paticula, fom Eq. (17), one obtains: 1b 1b 3b 1 (1) Based on the optimum b, one can also wite elationships between b and the inflow angle. 1 b 1 cos( ) () 1b 3b1 sin( ) (3) b 1b cos( ) (4) b Since one can elate b to and b to, one can also elate to. 7

8 1 1 1 cos (5) cos ( ) sin ( ) cos( ) 1 = (6) sin( ) 1 cos( ) tan 3 Thus, Eqs. (-6) ae a complete, closed-fom set of expessions of any of the optimum paametes in tems of any of the othe two. Inteestingly, Eq. (6) is deived in Ref. [1] in the context of the optimum blade chod distibution and can be found in thei Eq. (3.9.15). It is also inteesting to compae these optimum paametes to the classic induction factos (a and a') of conventional wind-tubine aeodynamics. This is easily done based on the geomety. 1b cos( ) abcos( ) = (7) 1 cos( ) b1 a (8) f bsin( ) sin( ) a 1 cos( ) (9) Due to in the denominato, a' is singula at =. The new induction paamete f, howeve, is wellbehaved. Based on the above, one can wite a compact expession fo the total flow at the blade in tems of b. b 1 b 1 b (3) 3b 1 This completes the expessions fo the closed-fom optimum blade paametes. Note that, at the oot, =, we have b = 1/ and = 6. Anothe inteesting point to inteogate is (the point at which the oiginal inflow angle = 45 ). Thee, we have b = 1/ (1+ ) =.366 and = 3. In the limit as appoaches, we have b = 1/3 and =. Optimum Powe Coefficient With the optimum blade paametes in compact fom, it now emains to compute the optimum (i.e., maximum) powe coefficient that goes with these paametes. Based on Eq. (16), the incemental optimum powe is given by: P (1 b) (1 b) d (1 ) (1 ) (31) dc b b d 8

9 The task is to integate this in closed fom fom = to = 1 (o, altenatively, fom to. Since we have no closed-fom expession fo b in tems of, ou fomula fo in tems of b, Eq.(1) makes a b-integation moe tactable. A diffeential of Eq. (1) leads to: d 6b 1 3b 1 b db (3) Theefoe, we can wite a fomula fo the total in tems of a b integal. C P.5 1 b(1 b)(1 b) db (3b 1) b (33) whee.5 is the value of b at the blade oot, and is the value of b at the blade tip, ( ). Because of the singulaity in the denominato, it is advisable to make a change of vaiable to y = 3b 1. The tem can also be expessed in tems of and, theefoe, in tems of. The esultant integal fo is: 1 4 y (34) y y y y CP y y dy whee = 1/ and < < 1/. Equation (34) can be integated in closed fom. Because each integal tem involves eithe ln( ) o, one can facto out ( ) o, equivalently, facto out (1 ). In fact, the cube can be factoed out,. In addition, the on the outside of the integal cancels all tems in the denominato. This then ceates a tem of the fom which emoves the singulaities fom the final expession. The closed-fom esult fo becomes: y 1 y ln( ) (1 ) (1 ) 3 y y y 16 1 y y CP 1 y y 3 7 y (1 y) 1 4 (35) Fo convenience, we have dopped the subscipt on such that y = 1. Note that, fo y = (1 )/, the tem [ln(1-) + + /]/ 3 appoaches -1/3, such that the fomula is well-behaved. Similaly, at y = (i.e., appoaching ), ln() appoaches zeo such that the fomula gives the Betz limit, 16/7. Since is known in closed fom in tems of, Eq.(35) is the fist closed-fom expession fo optimum that has been published. 9

10 Toque and Thust Coefficients The powe coefficient in Eq. (35) goes to zeo as goes to zeo, this eflects the fact that, in the limit as Ω goes to zeo, thee can be no powe geneated. Howeve, thee can be a toque in the limit as Ω appoaches zeo. Since P = QΩ, the toque coefficient comes immediately fom the powe coefficient. It follows fom Eq.(35) that: C 3 3y P CQ= = CP (36) 4 y1 y 1 y ln( ) (1 ) (1 ) 3 y y y y y CQ 1 y y (37) y (1 y) 1 4 Thus, appoaches.8653 as appoaches zeo. On the othe hand, as appoaches infinity, goes to zeo. In a simila manne to the computation of powe coefficient, one can also compute a closed-fom expession fo the thust coefficient of the optimal oto. Fom momentum theoy, Eq. (14), the elationship fo thust is: dt d w U wcos( ) cos( ) (38) Fom this, the elemental thust coefficient fo the optimum oto is: 1b d dct 8b1 bcos( ) cos( ) d = (39) With the change of vaiable into a b integal, we have: b b b C = T 3b 1 db (4) b As with the powe integal, a change of vaiable to y = 3b 1 yields a closed-fom expession fo the thust of an optimum oto. 1

11 1 ln 1 8 y y y = 1 4 CT y y y (41) y y 4 Note that, fo small, it follows that y = (1 )/, whee is a small quantity. In the limit as appoaches zeo, [ln(1 ) + ]/ = -1/. Also, as appoaches, y =, the limit of ln() =. Thus, at = (y=1/), we have =.75; and, as appoaches infinity, y=, we have = 8/9. Optimal Chod and Pitch The above deivation gives the induction facto fo maximum powe output. It is natual to ask as to the chod distibution and pitch angles that would give this optimum induced flow and powe unde the assumption that each blade section is opeating at the angle of attack that gives the maximum lift-to-dag atio fo that section. Section 3.9, Eq. (3.16), of Ref. [1] gives without poof the optimum chod distibution (including swil). In tems of ou vaiables, the esult is. 8 c BC l 1 cos( ) Ref.1 (4) The equivalent esult can be obtained unde the pesent appoach with the use of single induction facto. In paticula, because the lift fom momentum theoy (as well as the esultant induced flow) lies along the same axis as does the lift fom blade-element theoy, we may wite: Bc dl wu w d U w C d cos( ) l (43) The balance of momentum and blade-element lift (in tems of nondimensional vaiables) is: 8 1 bcos( ) BcC (1 b ) (44) The paametes in Eq. (44) ae known fom Eqs. ( 3). This allows us to solve fo the optimum chod in tems of known quantities. l 4 3b 1 16b 1 c BCl b BCl 1 (45) Equations (1-6) can be used to shown that Eqs. (4) and (45) ae identical. A compaison of Eq. (45) with simila equations that neglect swil (i.e., Eq. (3.79) of Ref. [1]) shows that the two expessions fo chod agee fo lage, which is the case fo which swil is negligible. Howeve, 11

12 esults without swil give an infinite solidity nea the blade oot ( = ); wheeas the fomula that includes the effect of swil is well-behaved in Eq. (45). Thee ae two inteesting aspects of Eq. (45). Fist, nea the blade oot, the local solidity that esults fom the optimum chod is: Bc 8 1 C C l l Fo a typical of 1., this implies a solidity of which would seem to be physically impossible (the aea of blades exceeds the aea of the annula ing). Howeve, nea the oot, = 6 and. Theefoe, pojection of the blade chod onto the oto disk would exactly equal the available aea; and the oto would not intefee blade-to-blade. Anothe inteesting aspect of Eq. (45) is that the ideal blade has a maximum chod that is appoximately located at. Fo example, fo, the maximum would come oughly at 14% distance fom the oto cente and would give a local solidity of aound 1. still fee fom blade intefeence. This also is the location at which, and the local inflow angle is 3. The optimum pitch angle follows diectly fom the elationship that the angle of attack is given by. Since is known in closed fom fom Eq. (5), and since the angle of attack fo maximum is known fo each tubine aifoil, if follows that the optimum pitch angle is known and is given by. In summay, the use of a single momentum balance with all else following fom geomety gives a closed-fo solution fo the optimum oto and allows computation of the optimum chod and pitch angle. (46) Effect of Pofile Dag It is quite staightfowad to detemine the effect of pofile dag on the thust, powe, and efficiency of the optimum wind tubine. This is possible because the above deivation of the optimum oto is based on a momentum theoy and a blade element theoy that both assume lift pependicula to the votex sheet. Since the pofile dag is by definition along an axis paallel to the votex sheet, it is easily included in the blade loads. What futhe simplifies the computation is the fact that pofile dag does not affect momentum theoy. This is because only the ciculatoy lift tails voticity that ceates induced flow. Pofile dag may heat the ai and poduce a shea laye behind each blade; but these effects ae negligible in tems of thei influence on induced flow. Thus, since it is specified that the local dag is pependicula to the local lift (and has no effect on the momentum induced flow), we may wite the net elemental thust coefficient (due to both lift and dag) as a quantity that is popotional to: C d Cl cos( ) Cd sin( ) Cl cos( ) 1 tan( ) (47) Cl Similaly, the elemental powe coefficient is popotional to: C d Cl sin( ) Cd cos( ) Cl sin( ) 1 cot( ) (48) Cl 1

13 It follows that the existing integals fo and can be augmented with integals that multiply ( ) in ode to obtain the desied effect of dag on thust and powe. In ode to obtain insight to the effect of pofile dag, we conside the case in which all aifoil sections have the same maximum lift-to-dag-atio along the blade span. (Of couse, this does not imply that each section has the same lift and dag.) Although a poduction wind tubine geneally would not have all aifoils opeating at the same lift-to-dag atio, hee we ae consideing only the ideal tubine. Thus, it is instuctive to conside an ideal tubine with the same aifoil geomety at all sections (and thus the same optimum lift-to-dag atio). This allows a closed-fom expession fo the effect of dag on the ideal optimum. It is not, stictly-speaking, the optimum fo a case with dag. Howeve, since the effect of pofile dag is assumed a coection facto, one would expect the optimum induction factos not to change dastically due to the pesence of dag. Thus, this appoach should yield impotant insight. Accoding to Eqs. (47) and (48), the integal fo thust o powe can be multiplied by eithe o, espectively, in ode to obtain the integals that ae to go with in and, espectively. The integal fo due to is of the fom: b b b 3 1 Thust Integal IT db 3b 1 b 3 (49) The integal fo the effect of dag on is of the fom: b 1 b 1 b 1 b Powe Integal IP db 5 3b 1 b 5 These integals have been woked out in closed fom. The thust integal is: 4y IT 4 ln (51) y y y y y 3 y41 y y y y 4y 3 The singula tem at y =.5 ( = ) can be factoed, giving: 13

14 1 y 1 ln 1 (1 y) y y y I T y y 4 y 9 y 48 y 6 3 y y 17 18y y y 8 y y( y 4) 64 + y y 38y 16y 97 y y( y 4) 3 3 y y4 (5) The powe integal becomes: 4y IP 13 4 (53) y y y y y y 9 3 y41 y y 9y ( y ) y 4y 13 8ln 4 8 It tuns out that this integal is nealy linea with and can be appoximated by: IP 16 7 (54) O to make the fomula a little moe accuate, one can use: I P (1 ) (55) The maximum eo of Eq. (54) is.4 (at =.7) and that of Eq. (55) is. (at = 1.1). These eos ae, espectively, about 6% and 3% of the local integal. Fo all values of, Eq. (55) gives lowe eos than does Eq. (54). Fo example, the elative eos at = 5. ae.11 and.7 espectively (.37% and.4%); and the eos at =. ae.16 and.4 (1% and 3%). Fom this, the integals fo the total thust coefficient and the net powe coefficient follow diectly: C d CTT CT [ Eq. 41 ] IT (56) Cl 14

15 C d CPN CP[ Eq. 35 ] IP (57) Cl Numeical Results The pesent closed-fom esults ae consistent with past esults, but it is nonetheless infomative to plot these optimum paametes based on the fomulas heein. Figue pesents the thee induction factos (b, a, f) vesus the local speed atio,. The total induction facto b vaies fom.5 at to.33 fo >> 1. The axial induction facto a vaies fom.5 to.33 ove the same ange. The swil induction facto f shows a wide ange of values, beginning at and going to zeo. Figue 3 shows the total inflow angle though this same ange. The optimum angle is 6 fo small, dops to 3 at = 1, and appoaches as becomes lage. Figue 4 plots the same data against 1/(1 + ) which is the squae of the sine of the initial aiflow angle (befoe the optimum induced flow is added). Thus, small inflow atios ae at the ight ( tending to infinity) and lage inflow atios on the left ( tending to zeo). Note that this cuve is symmetic about = 1 and = 3. This is a consequence of the closed-fom esult in Eq. (5). Figue 4, theefoe, is a univesal cuve that shows how the optimum inflow ation vaies fom º to 9º ove the entie inflow-atio ange. A tubine with a given tip adius and oot cut-out would have optimum values of the inflow angle as found fom the appopiate ange of 1/(1 + ) in the figue. Figue 5 shows a compaison between the appoximate optimum chod fomula (in which wake swil is neglected) and the exact, closed-fom esult pesented hee. Results ae fo = 7, B=3, and. Note that the oot solidity is infinite fo the appoximate method but is well-behaved fo the tue optimum. Figue 6 pesents the fom the closed-fom esult in Eq. (35). We have veified that this esult agees exactly with the numeical solution in Ref. [1]. This figue gives an undestanding as to why typical, poduction wind tubines have tip-speed atios between 5 and 7. Values of that magnitude ae needed to appoach ideal efficiency, but lage values do not yield much exta powe. Figue 7 shows the toque coefficient fo the same case. Note that, fo a stopped oto, =, thee can still be a toque due to the lift on the blades in the fee-steam. As oto tip-speed becomes lage, less and less toque is equied to poduce the same optimum powe; and appoaches to zeo. Figue 8 gives the closed-fom thust coefficient fo the optimum oto. It vaies between 3/4 and 8/9 in a monotonic fashion. Figues 9 and 1 give the optimum powe and thust including the effect of pofile dag fo the cases =.,.1,., and.4 and = 1.. Notice that, with pofile dag, a given value of implies an optimum tip-speed atio fo maximum powe. The maximum fo =.4, is.475 (at =.85). Fo =., the maximum =.514 (at = 3.85). Fo =.1, the maximum =.541 (at = 5.1). Fom this, one can infe an appoximate fomula fo the that gives maximum. C l C l C l (58) Cd Cd Cd 15

16 This sheds futhe insight as to why typical wind tubines with high have tip speeds of the ode of 5 to 7. Note that the effect of the pofile dag becomes moe ponounced at high tip-speed atios. Summay and Conclusions An altenate deivation is povided fo the paametes of an optimum, ideal wind tubine, Unlike pevious deivations, only a single momentum theoy is used (in the diection of the local lift) so that thee ae no sepaate accounts of axial and angula momentum. The esults, also unlike pevious esults, ae found in closed fom fo all vaiables and the singulaities of pevious numeical solutions ae eliminated explicitly. Although the final paametes fo the optimum tubine ae no diffeent fom those of conventional appoaches, the closed-fom natue of the esults yields insight into the popeties of the optimum tubine. Finally, because of the single momentum balance, it is quite staightfowad also to wite a closed-fom expession fo the optimum blade chod distibution. The tue optimum does not become singula at the blade oot, but athe appoaches a combination of solidity and pitch angle that avoids blade-to-blade intefeence. Acknowledgements This wok was accomplished, in pat, due to new esults developed unde contact to the National Renewable Enegy Lab D. Patick Moiaty, Technical Monito and due to funding fom the U.S. Amy/NASA/Navy Cente of Excellence in Rotocaft Technology though a subcontact fom the Geogia Institute of Technology D. Robet A. Omiston, and D. Michael Rutkowski, technical monitos. 16

17 Refeences 1. Manwell, J. F., McGowan, J. G., and Roges, A. L., Wind Enegy Explained: Theoy, Design and Application Second Edition, John Wiley and Sons, West Sussix, 9, pp , Johnson, Gay L., Wind Enegy Systems, Pentice-Hall, Englewood Cliffs, NJ, 1985, pp Eggleston, David M. and Stoddad, Foest S., Wind Tubine Engineeing Design, Van Nostand Reinhold Company, New Yok, 1987, pp Gacia-Sanz, Maio and Houpis, Constantine H., Wind Enegy Systems, CRC Pess, Boca Raton, FL, 1, pp Abamowitz, Milton and Stegun, Iene A., Handbook of Mathematical Functions, Dove Publications, Inc., New Yok, NY, 197, p Glauet, H., Aeodynamic theoy: A Geneal Review of Pogess, Vol. IV, Chapte Division L., Aiplane Popelles. Dove Publications, Inc. New Yok, NY, 1963, pp Baocela, Edwad, "The Effect of Wake Cuvatue on Dynamic Inflow fo Lifting Rotos," Maste of Science Thesis, Washington Univesity in St. Louis, May Makinen, Stephen M., Applying Dynamic Wake Models to Lage Swil Velocities fo Optimum Popelles, Docto of Science Thesis, Washington Univesity in St. Louis, May 5. 17

18 Figues 18

19 .5 Figue. Wake Induction Paametes as a Function of Local Speed Ratio.45.4 a (Axial Induction Facto) f (Swil Induction Facto) b (Total Induction Facto).35.3 a,b,f Figue 3. Optimum Inflow Angle as a Function of Local Speed Ratio

20 6 Figue 4. Optimized Inflow Angle as a Function of sin of Initial Inflow Ratio / (1+ ).4 Figue 5. Optimium Chod, Theoy with Wake Rotation vs Theoy without Wake Rotation, = 7, C l = 1, B = 3 Nondimensionalized blade chod, c/r Theoy without Wake Rotation Theoy with Wake Rotation Nondimensionalized blade adius, /R

21 .6 Figue 6. Powe Coefficient as a Function of Tip Speed Ratio.5.4 C p Figue 7. Toque Coefficient as a Function of Tip Speed Ratio C Q

22 .9 Figue 8. Thust Coefficient as a Function of Tip Speed Ratio.85 C T Figue 9. Powe Coefficient as a Function of Tip Speed Ratio including the effect of pofile dag C PN.3..1 C d / C l =. C d / C l =.1 C d / C l =. C d / C l =

23 .9 Figue 1. Thust Coefficient as a Function of Tip Speed Ratio including the effect of pofile dag.85 C TT.8 C d / C l =. C d / C l =.1 C d / C l =. C d / C l =