Naca Report 924

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REPORT 924

APPLIC ATION OF THEODORSEN'S THEORY TO PROPELLER DESIGNBy JOHN L. CRIGLER

SUMMARYA theoretical analysis is presented for obtaining by use ofTheodorsen's propeller theory the load distribution along apropeller radius to gice the optimum propeller efficiency forany design condition. The efficiencies realized by designingfor the optimum load distribution are gicen in graphs, andthe opt-imum efficiency for any design condition may be readdirectly from the graph uriihout any laborious calculations,Eramples are included to illustrate the method of obtaining the

optimum lo ad d istrib utio ns jo r b oth sin gle -ro ta tin g a nd d ua l-rotating propellers.INTRODUCTION

Recent contributions to the theory of propellers have beenmade by Theodorsen in a series of reports (references 1 to 4).In the first report of the series (reference 1) a method basedon electrical analogy was devised for obtaining the idealcirculation functions for single-rotating propellers. Thesecirculation functions were shown to be in good agreementwith the theoretical calculations made by Goldstein inreference 5 for two- and four-blade single-rotating propellersand with the extrapolations to other numbers of bladesmade by Lock and Yeatman in reference 6. The electrical-analogy method of measuring these functions was alsoapplied to more difficult cases for which no theoreticalcalculations had previously been made; in particular, to thecase of dual-rotating propellers.Theodorsen in reference 1 introduced the concept of themass coefficient K, which is an integrated value of the circula-tion functions. The mass coefficient represents the effec-tive cross section of the column of the medium pushed bythe propeller divided by the projected-propeller-wake area.This mass coefficient is made use of in the development ofTheodorsen's theory. Inreference 4, expressions are givenfor computing the thrust, the energy loss, and the efficiencyof any propeller with ideal circulation distribution based onthe conditions in the final wake in terms of the mass coeffi-cient. It is of interest. to mention that the mass coefficientor mass of air operated on by the dual-rotating propeller ismuch greater than that affected by the single-rotatingpropeller for the same set of operating conditions. Thislarge difference in the mass coefficients for the two easesindicates that calculations for dual-rotating propellersbased on the ideal circulation functions for single-rotatingpropellers are inadequate.Theodorsen's theory, as previously mentioned, is based onthe conditions in the final wake. The present analysis

attempts to interrelate the conditions in the final wakepropeller and to give the information necessary to despropeller for any desired operating condition. For srotating propellers, the method yields the same results aconventional vortex theory with the Goldstein tip correcapplied. By the conventional vortex theory, however,necessary to determine the optimum blade-load distribuand then to make element strip-theory calculations into obtain the optimum efficiency for a given design condThis procedure has been followed in reference 7 for arange of operating conditions. By Theodorsen's theoryoptimum efficiency 1 1 can be obtained directly for any dcondition from its relationship to the mass coefficient witlaborious calculations. Thus, in the selection of a propfor any design condition, a close estimate of the efficcan be obtained before the design is made.The circulation functions and mass coefficients for therotating propeller WE'rE. ' obtained in reference 1 for thecase and refer to conditions in the ultimate wake.propellers were assumed to operate in the same plane.viously, this condition is unattainable in the designactual propeller, The degree to which the ideal case crealized in practice, or the applicability of the ideal functo a given case, require further considera tion and confirma

SY:\IBOLSBb

number of propeller bladeschord of propeller-blade elementsection drag coefficientsection lift coefficientideal power coefficient (c.+e)total-power coefficient (Pc+tr)thrust coefficient ( 1 ~ . )2 " p " t -Fnet thrust coefficient (c.-t..)diameter of propellerdrag of propeller sectiondiameter of wake heli;x surfaceideal energy loss in wake ( p F K . W ( ~ w+~ V )energy loss due to blade draginduced energy loss coefficient ( 1 ~ )2 " p l 3F

c.

E

e83


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84 REPROT 924-NATIONAL ADVISORY COMMITl'EE FOR AERONAUTICSFK(x)l

projected area of helix (at infinity)circulation functionlift of propeller sectionpropeller rotational speed, revolutions per secondinput power to propellertip radiusradius to any blade elementthrust of propellerpower loss due to drag (nondimensional)axial power loss due to dragrotational power loss due to dragforward axial velocity of propelleraxial interference velocity (at propeller)average axial interference velocity (at propeller)resultant interference velocity (at propeller)rotational interference velocity (at propeller)average rotational interference velocity (behind eachpropeller)resultant velocity on the propeller at radius rlocal self-interference velocityrearward displacement velocity of helical vortexsurfaceratio of displacement velocity to forward velocity(wlv)radial location of blade element (rIR)angle of attack, degreesinduced angle of attack, degreesblade angle, degreesd . ( 1 V-FW)A a vance ratio ;:-:nrJ;

A g geometric advance ratio (VI-rrnD)" mass coefficient ( 2 1 1 K(x)x d x )faxial energy loss factor'1 propeller efficiency ( F c ~ ; )f}j ideal propeller efficiency (c,/p.)p mass density of airU propeller element solidity (Bb/27fr)uCI propeller element load coefficient.r circulation at radius x ( r e X ) 27f(~~W)W K(x) )c p angle of resultant velocity Wat plane of rotationA. =tan-1 F/nD'1'0 7fXW angular velocitySubscripts:F frontR rear0.7R at 0.7 radius

nPRrTt

wW

x

fJ

OPTIMUM PROPELLER DESIGNSINGLE. ROTATING PROPELLERS

Velocity diagram.- The velocity diagram for the single-rotating propeller is shown in figure 1. This figure is areproduction of figure 13 (reference 2) .with some additionaldesignations. The relationship between the axial inter-ference velocity at the radius.r, as given by the vortex theory,


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The resultant velocity isAPPLICATION OF THEODORSEN'S THEORY TO PROPELLER DESIGN

1V+,?wW= . - 1 .sin c p 2 W Sill c p=S~ c p ( V + i W cos! t P )

The interference velocities may be obtained from the geom-etry of the figure by

andT" T?' 1 .r=,Sill tP="2 W Sill r p cos c p

Optimum blade-load distribution.-The design problem ofan optimum propeller consists essentially in obtaining the'Value of the element load coefficient bCI at each radius of thepropeller blade. With the direction and magnitude of therelative velocity given at each station there remains only thechoice of a section to give efficiently such a lift at the appro-priat-e angle of attack. The 'Valueo f CI should be at or nearthe ideal lift coefficient for the section in order to giveminimum drag coefficient.The method developed in references 1 to 4 treats thevelocity W as an independent parameter upon which all theother quantities depend. This reversal of procedure is con-venient since all quantities are actually functions of w. Thevelocity w is related to the power coefficient P, of the pro-peller and also to the element load coefficient UCI' The rela-tion of w to rrel is developed herein and the relation ofw toPo , which must be obtained in order to use it for design, isgiven subsequently in the section" Procedure for PropellerDesign."The required ideal circulation r e x ) isgiven in reference 1 by

r e x ) 2'l1 '(~~W)W K(x)_(V+w)wK1)_ Bn ,x

In order to determine the element load coefficient bCI the rela-tion for the equality of the force on a vortex element and onan element of a lifting surface is given asprw=~ pW~c,b

where b is the chord of the element. Hence,r=~ lrc~ (4)

where W is given in equation (2), and thusr=~s~ c p ( V+~w cos2t P ) c~

(2)

(3)

(5)

Using equations (3) and (5) for I'gives at once the idenbe, (V +w)w K(x) 2 sin t PBn Tr+l!y '2 w cos t P

Introducing the nondimensional 'Velocity w= ~ ,Bb r+!wsolidity u=2- ' and tan tP= 2 2 (equation (1 gives'lI'r 'lI'rn

non dimensional relationl+w _ sin2t PUCI= 2 'wK (z) --( 1+~w ) ( 1 +~W cos !tP) cos tP

The selection of a propeller for a . given airplane installatmay be based on a method of evaluating a series of propelfor various operating conditions in order to determinemost suitable propeller. It is probable that several propelvarying in diameter, blade number, propeller operatiospeed, and direction of rotation are equally as efficientthe design condition so that other considerations may einto the propeller selection. However, the optimumciency for the propeller selected may be obtained fromcharts, and therefore the load distribution along the rathat will give this optimum efficiency remains todetermined.The value of rrel may be calculated for any radius fromrelation

where

DUAL-ROTATING PROPELLERS

In the design of dual-rotating propellers, it hascustomary to select two propellers designed for sirotation and to use them as a dual-rotating propeller,fact that the circulation functions and the mass coefficobtained by the electrical-analogy method (referenceare very much larger for the dual-rotating propeller thansum of the values for the two single-rotating propelindicates that the functions as used heretofore are not proThe electrical-analogy method represents the case oidealized dual-rotating propeller in which the two componeare in the same plane with the same load distribution oncomponent and with equal power absorption. Since acpropellers cannot conform to this ideal case, the applicabiof the ideal functions requires further confirmation. Netheless, the optimum distribution for the dual-rotatpropeller is essentially different from the single-rotatpropeller, and in this analysis the loading functions andmass coefficients as determined by the electrical-analomethod are assumed to apply to the optimum dual-rotarpropeller.


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86 REPORT 924-NATIONAL ADVISORY COMMITI'EE FOR AERONAUTICSInterference velocities for dual-rotating propellers,-Theaverage axial interference velocity far behind the propellerobtained from the momentum considerations is

2Va=KWwhere K is the mass coefficient and W is the axial displacementvelocity. This mean value is equally due to each of the twooppositely rotating propellers. The average _axial interfer-ence velocity due to each is therefore exactly

The average interference velocity at the propeller plane isone-half the value in the final wake and, therefore,

where ~ V I Z represents the average axial interference velocityat the propeller plane due to each component of the dual-rotating propeller. With the two propellers separated bya small axial distance, this velocity refers to a plane betweenthe two propellers. The interference velocity at the frontpropeller is smaller and at the rear propeller is larger than atthe plane between the propellers. In the following treat-ment, the propellers are considered to be very close togetherso that the axial interference velocity is the . same on bothpropellers.In the final wake, the mean value of the rotational inter-ference velocity for the ideal case is given hy

2Vr=OFor an infinite number of right and left blades equally loaded,rotational components would cancel exactly. However, theaverage rotational interference velocity immediately behindeach propeller may be considered as

In summary, the mean interference velocities acting on thefront propeller from the rear propeller areAxial: 1- 1- V =-I(W2 a 4Rotational:

Vr=OThe mean interference velocities acting on the rear propellerfrom the front propeller areAxial:

Rotational:

It is useful to recognize that the mean self-interfereneach propeller in its own plane isAxial:

Rotational: 14 ' , , - w tan < PVelocity diagram for the dual-rotating propellers,'velocity diagram for the dual-rotating propellers isin figure 2. As in the case for the single-rotating prothe axial displacement velocity at the propeller is eq

~ w. In figure 2 the vector AB gives tho mean axialference velocityiW of each propeller acting on thepropeller. The vector Be gives the mean rotationalference velocity lW tan t /J of the front propeller actinthe rear propeller. The total interference velocityon the front propeller from the rear propeller is thegiven by AB, and the total interference velocity actingrear propeller from the front propeller is equal to theAC. The local self-interference velocity of the frontpeller is given by W S F ' and the corresponding helix angiven by C P F . The local self-interference velocity of thpropeller is given by WSR ' and the correspondingangle is given by C P R . The angle < p y is slightly largerthe ideal helix angle < p given by tho displacement ve~ 1 .0 and < P H is slightly smaller than t / J . The design conof most interest is the one for which r, for each bladefront propeller is equal to rR for each blade of thepropeller. The number of blades on the front and

-----l:.wF

---1117 - -------->1FIGURE 2.-Veloclty dlagre.m tor duaJrotatln, proIM'lk>r.


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APPLICATION OF THEODORSE.."1"S THEORY TO PROPELLER DESIGNpropeller are considered equal and the rotational speeds, thesame. This condition gives the self-interference velocityon the front propeller equal to the self-interference velocityon the rear propeller and means that D and E must be atthe same horizontal level.As rfJ,and rfJBre needed in the design of the propeller, itis seen from figure 2 that the associated displacement velocity

on the front and rear propellers has been increased anddecreased, respectively, by the amount

The displacement velocity is thereforeFront:

~ 11) ( 1+~ tan2rfJ)Rear:

~ w (1-~tan2rfJ)From figure 2, the velocity WI' is shown to be given by the

relationship

and the angle rfJ,is given byy+~w (1+~ K tan%rfJ)tan rfJF=---~CIl-r---~

= n \ ; ~ [ 1+~it;( 1+~K tan2rfJ)]where rfJis given by the relationship

"V+~wtan rfJ= -C I l r

Similarly,TT" 1 ; : - +1 . +1HB=sin < P o " 4 I('IJ) sin rfJa 2 " / C Wtan rfJocos rfJa

y 3 .= - . - - + - / C W sm rfJosin rfJo 4v ( + 3 - . 2 )=in rfJa 1 " 4 IC W sin rfJa

and

V 1 [ 1 _ ( 1 ) ]-- l+-w 1--1[ tan2rfJnD Tor 2 2

(8)

(9)

(10)

(11)

Optimum blade-load distrlbution.-The optimum dbution of blade loading is obtained from the determinationthe element load coefficient be, at each radius from the fumental relation

where r has been given in equation (3) byr C V i : ) W K(x)

Eliminating r gives~Bbc,W (V~W)W K(x)

but 4 :!ca is the solidity of each component of the drotating propeller, if the number of blades in each comnent are assumed to be equal. Therefore,

For the front propeller, this equation may be solved byof equation (8)( ) _ '17 1 (1+W)w sin < P o K( )erc'F--D- xn Tor 1+1 - . 2 - 1 . . _" 4 KWSIn 'I'U

and for the rear propeller by use of equation (10)( ) _ V 1 (1+W) w sin 1 / 1 0 K C )erCIB--D- xn 1rX 1+3 - . 2 - 1 . .4 KWSlll 'l'a

USE OF DESIGN FOR.'\U."LASIn order to use the relation for erc" note that it containsonly the independent variable w but also the function Kand the angle rp . The parameter K(x) should be expre

as a function of V + D w which is based on the wake hn 0diameter. .A s was shown in reference 3, however, Do dionly slightly from the propeller diameter D and in the predesign procedure D is used instead of Do. The funcK(x) for single-rotating propellers is plotted against ~ tfigure 3. Similar plots for dual-rotating propellers wtaken from reference 1 and are presented in figure 4.

EQUATIONS FOR PERFORMANCE CALCULATIONSSINGLE-ROTATING PROPELLERS

In reference 4 the thrust has been given by

and the ideal energy loss in the wake has been given byE=pFICw ( ~ w + - ~ -V )


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88

K(J:) .5

_ ..REPORT 924-NATIONAL .ADVISORY COMMITTEE FOR AERONAUTICS

1 .0 J:.6 -r-B:\7 o r :.4 .8.8 ~~ .7 ~3 .45\ \ ~~ r---r-.8 I~\ - .--~.2 \ ~\\ \~ ~- f--.3.9

\ \ \ ~\ \ ~\~\ \ \ \ ~~ \ \ \ \\ \ \ ~\ .2 \ '~\.95 1 \ \ I'.\~ \ 1 \ \\\ \\

\ \ 1 \ \ ; \ s \ \ \ \\! \ \\ \ \ \''\\ \~ \ 1 \ ' \\ r\ \ I\\~ ~ \ \ '\ '\\ ~ "\ \ \' ,\~, \ \ \'~ _ .. ._ - - -_, .-\ \\

,\ '\~

" , \ K~'," 1\ \,\ ,,\ "~ -,~ \ '\'0 .. .~, - . " ,t-,[\ " , ~ ",~ ~ -, ,r- , r ' : : ~, '.. . ~," . . .~r-, . . . . . . . . . . r-,~ ~~ r-, . . . . . . . . . . . . . ~~


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APPLICATION OF THEODORSEN'S THEORY TO PROPELLER DESIGN 8

1 .0

./

;rI:,{\ :rt-- I--1-1~~ ~ .6~t-- 0 . 4 ~ \. . 9 A ~ .45 ~3 \ 1..~~, \ \ ~ '\ ~

~\\ 1\\ .3 \ ~ l\.95.s5\~ 1 \ \ \;~

'--- \ \ 1 \ \\ \ ' !\' f . . \ \ \ \l\\\\ \' \ 1 \ 1 \ \2l~, \ \ f\ "l\ \ . \ \ \ \~\' \ \\ \~~ \ \ \ ~ ~\ \ \ \;~'\ \ \ '\ ~\ " ' x 1 \ \I-- . 1 \ \ \ \~ ~ ~ '\ 1'\. '\ ~ ~\\ \ \ \ ~ I\~ \ \ '\ \ "\ ~, \ 1\ "\ r -~ '\ ~~ \ '\ " '\ ~,\. ~ "- ' "\ \ -'~ ,r0~ 1 \ ~ ';;


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901 .0

REPORT 924-NATIONAL ADVISORY COMMITTEE FOR AERONAUTICSI.a

:9JJ.l-;2-'~8 .3--

~.4"-.5" 1-1--- -_.- 1---.6-- ~7 .7--~\\\'~0 .8-- \ , \ ' \ ~\\ \\~ -~r--1 \ \ \ '0 ~. ;g-- [\ \'\~~'" '\ 1\ [\\~'\ ~\ ~ ~.95- (H - ; i i i )

The efficiencyis given by

These formulas are all that are necessary forotating propellers. The performance of tho dualpropeller is computed by the same formulas.DUAL-ROTATING PROPELLERS

The thrust of the front propeller is given bydT,=~ p(2'l11')ll' ,2(crcl)' cos c p , dr

and with (crc!), from equation (12) and W,from equT,=p VSW(l+W) ~ 1 1 ( 1 +i K I D sinl C P O ) iil~t


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I.APPLICATION OF THEODORSEN'S THEORY TO PROPELLER DESIGN

1.0

K ( o c )

0 I~; ~3":4/ ~ ~5':f \\t'0 ~\'\~ ~.E 1 \ 1\ [\\~ :\\ r - - . \ .'\~I \\\~ ~6

.9';'- \ r\ \- , ~ ~\ \ [\ ~ ~ ~5 -, \ -, I'\. ~ ~. .95'r\ [\ ."\ " - t- . ~ ~,\ -, -,


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92 REPORT 924-NATIONAL ADVISORY COMMITTEJl) FOR AERONAUTICaThe component power losses are then, to the same degreeof approximation in nondimensional form,

Rotational:

Axial:S o l (Fedta= -------;: dxo sin 'I'

For the dual-rotating propeller operating at the design con-ditions, the terms containing w are small, as is the case withthe single-rotating propeller, and a close approximation tothe drag loss is obtained if these terms are neglected. Fur-thermore, if it is assumed that the average of the resultantvelocity tV for the dual combination is equal to W for thesingle propeller, equations (20) to (22) may be used forthe dual-rotating propellers. Of course, for conditions otherthan the design condition, especially for very heavy loadings,exact drag-loss calculations require that the exact equationsbe used for either single-rotating or dual-rotating propellers.In summary, the equations for obtaining the propellerperformance are given by the quantities C,! e, and P, andthe drag-loss factors are given by t, and t.The net thrust power is

The power input is

The efficiency is

where from equation (16)Pc=2K1i5(I+W) (I+~ w )

The total power is also given byP=~ P V31l"R2P CT

It should be remembered that the _calculation is based ona given w. This procedure may seem unjustifiable sincethis parameter is not given by the specification but is theend result of a calculation based on the original data. Theinduced loss does not depend on the total-power coefficientP e T but actually depends only on Pe, and the quantity wcannot be obtained from the total-power coefficient. How-ever, the value of P e T in most cases exceeds P, by not morethan 2 percent or

(21)

FIGURES USED IN PROPELLER DDlIGN

(22)

Since P, in equation (16) is based on wand the dof the final wake and since the value of PeT in equatiis based on the propeller diameter which is slightlythan the diameter of the final wake, a very dose apption to w is usually given by equation (10). 'Therefo

P eT: : .:


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PPLICATION OF THEODORSEN'S THEORY TO PROPELLER DESIGN

K

~~ ~.[ \ - : " t',1' , \ s : r,\\ 1 \ 1\ r,\ r -,

'. 1 \ - , r- , ';\ i\ - 1 " - r- ,\ \ \ r\. r-,\ '\ 1 \ .",\ 1 \ \ \


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94 REPORT 924-NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS/,00

,93

- -

~

' " '\ - ,-;\. - , -, - . - , , , ~~

~~ ~" ~ '-."q,,~*~ 'I---~.-- ....,:._-1-!--'/.0 .2 .3 .4 .5

.99

.98

.97

.96

.95

.94

.92

.S/

.80c"lk

FIGt:REB.-Propeller e.fI Ic lency against r,fc (reference


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APPLICATION OF THEODORSE.!.""IT'BHEORY TO PROPELLER DESIGN/.00

.!38

r.~ '\ ' " '\r - ,


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96 REPORT 924-NATIONAL ADYISORY COMMITTEE FOR AERONAUTICS

.3

._r--\ \

. \ \\\ \ Ii \'\ \ \ \ \\ \ 1 \ \ .\ \ . \ / " \ \'\ \ \ \ \i\ \ 1 \ \ --1---\ \ _ \ \ \ \ :\~/"\

\ \ \ \ \ \\ ~" - "\ K 1\ \ \ . . . .,\, \ r - - .1\ \ v I\. "\ r, \ E " 1 ' - . .\ 1\ ," "f'... ,. ( 1 + 4 i i i ) (l+~W c o s '< / > ) c o s tP


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APPLICATIO!Il OF THEODORSEN'S THEORY TO PROPELLER DESIGN

-, ~~ 1",9 -,!\ " \ ' " -,. . . . . . .\,\ \" ~\' "

- , E lK.~ \ \ , "\ '\.,\ \ , -,

\ \ '\ , '\ '\."7 \ \ i \ . K'- '\ ,~ \ -, -,\ \ \ I K 'E -, -,\ \ '\. ,6 \ \ \ " ~ . .,\ , j'..,\ \ - . ,


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98 REPORT 924-NATIONAL ADVISORY COMMITl'EE FOR AERONAUTICSand B is given in equation (11) by

.1

~0'.r-,....- . . . .\ " " r - , . . . . .r. . . . . . . .\ " '- . . . . . . . . . ~, r- , "\

" 1 " - K . r . . . . . . ." 1'"


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PPLICATION OF TBEODORSEN'S THEORY TO PROPELLER DESIGNthe same as given in reference 7 for the Hamilton StandardPropeller No. 3155-6 which has Clark Y sections and aregiven in the following table. Itis assumed that a spinnercovers the inner 0.2 of the radius. The distribution ofaC, with x and of sin c p with x have been included in the table:

C ol sin"c rC~ ~:!: crl:1 wsin.:!: sin.~

0.2 0.0967 0.1934. 0.400 0.968 0. 01600 0.00064.3 1 0 M .2108 .100 .932 .00697 .00061.4 .1044 .20SS .020 .889 .00188 .00030.~ .OO~ .1904 .010 .SW .00113 .00028.6 .08M .1TIO .008 .790 .00104 .00137.7 .cne .1432 .007 .742 0009 li . 0004G.8 .05M .11OS .006 .691! .fX1JT7 .OO()W.9 .0364 .0728 .006 .M2 .00060 .00049Performing the integrations and substituting in the for-

mulas gives for rotational-drag-loss coefficient' ) 5 0 1.0 aCtlt=~ -.-rd

T X . ' 0.2 sin c poJ=0.516 (0.000348)=0.0014

and for the axial-drag-loss coefficient

S o LO aCtlt..= 2 ------:i : x dx0.2 sm e=2(0.00213) =0.0043

The induced thrust coefficient has been given by equa-tion (14) as

C ,=2KW [ 1+W (4+~)=2(0.201)(0.155) [1+0.155 (~+ 0.29)J=0.0700

and the induced power coefficient by equation (16) asPc=2KW ( I+W ) ( 1+~W )=2(0.201) (0.155)(1.155)(1.045)=0.0754

The induced efficiency is

= 0.0700=0 9290.0754 .With drag included, the total thrust is given by

=0.0700-0.0043=0.0657and

= 0.0754+0.0014=0.0768

The efficiency is71=C'T=.0657=0 855P C T 0.0768 .

Thus it is seen that the blade drag of the magnitude giin the preceding table reduces the propeller efficiency f92.9 percent to 85.5 percent for the propeller operatconditions given.

CONCLUDING REMARKS.A . comparison of Theodorsen's propeller theory with

conventional vortex theory shows that the optimum ldistribution along the blade for single-rotating propelobtained by the two theories is essentially identical and aresult the optimum efficiencies are the same for a gioperating condition. Theodorsen's theory has thead vantahowever, that the optimum efficiency for any design cdition can be obtained quickly and accurately by the usthe mass coefficient Ie without any laborious calculationsbefore the final design is made.The distribution of the circ-ulation function K(x) foridealized dual-rotating propeller is radically different frthe existing values for the single-rotating propeller thave been previously used for the dual-rotating propelAlso, the mass coefficient Ie for the dual-rotating propeis larger than the sum of the values for two single-rotatipropellers. These quantities, which are not available frmathematical computations but are obtained fromelectrical-analogy method of Theodorsen, are used herfor obtaining the optimum load distribution along the blfor the dual-rotating propeller.

LANGLEY AERONAUTICAL LAB 0 R.-\.TORY,NATIONAL ADVISORY COlll1rl'TEE FOR AERONAUTICS,LANGLEY FIELD, VA., Marcb. 15, 1948.

REFERENCES1. Theodorsen, Theodore: The Theory of Propellers. I-Determina

tion of the Circulation Function and the :.\Ia.ss CoefficientDual-Rotating Propellers. NACA Rep. 775, 1944.

2. Theodorsen, Theodore: The Theory of Propellers. II-2.\IethodCalculating the Axial Interference Velocity. NACA Rep.1944.

3. Theodorsen, Theodore: The Theory of Propellers. III-The Sstream Contraction with Numerical Values for Two-BladeFour-Blade Propellers. NACA Rep. 777, 1944.

4. Theodorsen, Theodore: The Theory of Propellers. IV-ThrusEnergy, and Efficiency Formulas for Single- and Dual-RotatiPropellers with Ideal Circulation Distribution. NACA Rep.1944-

5. Goldstein, Sydney: On the Vortes Theory of Screw PropellProc, Roy. Soc. (London), ser, A, vol, 123, no. 792, April 6, 1pp. 440-465.

6. Lock, C. N. H., and Yeatman, D.: Tables for Use in an ImprovMethod ofAirscrew Strip Theory Calculation. R. & : lLNo. 1British A.R.C., 1935.

7. Crigler, John L., and Talkin, Herbert W.: Charts for DeterminPropeller Efficiency. NACA ACR lAI29, 1944.

8. Glauert, H.: Airplane Propellers. The Vortex Theory. Vol. IVAerodynamic Theory, div. L, ch. VI, sec. I, W. F. Durand,Julius Springer (Berlin), 1935, p, 231.

Naca Report 924

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