ACR No. LlH29 NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS WARTIME REPORT ORIGINALLY ISSUED September 19^^ as Advance Conf identialvEeport LU129 CEABTS FOB nETEBMHUIiG FROnHUEB EITICIBHCT By John L. Crigler and Herlwrt W. Talkln lAngley Memorial Aeronautical Laboratory Langley Field, Va. V^/ . N ACA ^ WASHINGTON NACA WARTIME REPORTS are reprints of papers originally issued to provide rapid distribution of advance research results to an authorized group requiring them for the war effort. They were pre- viously held under a security status but are now unclassified. Some of these reports were not tech- nically edited. All have been reproduced without change in order to expedite general distribution. " '^'''' bOCUMENTS DEPARTMENT Digitized by the Internet Arcliive in 2011 with funding from University of Florida, George A. Smathers Libraries with support from LYRASIS and the Sloan Foundation http://www.archive.org/details/chartsfordetermiOOIang 3 Q> I Hi HI YikCk kOR No. L4I29 NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS ADVANCE CONFIDE^ITIAL REPORT CHARTS FOR DETERMINING PROPELLER EPPICIEKGY 5y John L. Crigler and Herbert W. Talkin SUMMARY A short method of estimating propeller efficiency for a given operating condition is described. The efficiency is determined for any design condition by evaluating separately from charts the induced losses and the profile-drag losses. The estimated efficiency is compared ;vlth experimental results for a v/ide range of operating conditions and found to be in agreement near peak efficiency. The present analysis covers single-rotating pro- pellers of twoj three, four, six, and eight blades and includes charts showing the rotational-energy loss for the given operating condition in order to assist in esti- mating the gain in efficiency for dual-rotating propel- lers. The change in efficiency to be expected from body interference is discussed. Two examiples illustrating the use of the method are given in an appendix. INTRODUCTICI In reference 1, analytically determined propeller performance is compared with experimental results for propellers having four, six, and eight blades of con- ventional design. The calculated results '.ane in agreement with the experimental results over the com- plete range of blade angle investigated (25-* to 65'-' at 0,75 radius) . The calculations w-ere made by a strip- theory analysis by which the thrust and torque contri- butions for several elements along the radius are graphically integrated for each operating condition. The timie required to analyze a single operating condi- tion by the strip- theory method Is negligible compared with the time required to obtain experimental data for the same condition. The time required to analyze the COKPIDEKTIAL NACA ACR No. L4I29 complete range of operation 3.s considerable, Ipowever, and a shorter method is desirable. A method of eptimatine; the propeller performance supported by the, results of reference 1 is presented herein. By this method^ a large reduction in the time and effort required for propeller analysis is effected as compared v/lth the strip- theory method. The results obtained are in agreement vvith those from expcrim.ent. The basic propeller parameters are Interrelated in charts that aid in the selection of a propeller for a given design condition. The charts are useful in analyzing data for any propeller and aid in the deter- m.lnation of excessive losses. The induced power losses for a conventional roun^d- shank propeller are compared with the losses for the optimum load distribution. The induced losses are divided into axial- and rctational-energy, losses so that the maxiriiimi gains possible by the use of dual- rotating propellers instead of optimum single-rotating propellers can be estimated. The effect of profile drag is treated separately. Because drag losses are evaluated separately, increased losses due to compres- sibility can be estimated directl^r vhen airfoil data at high Mach mjimbers become available. Detailed applications of the m.ethod are illustrated by examples in the appendix. SYTffiOLS a axial-velocity interference factor a' rotational-velocity interference factor B number of propeller blades b chord of propeller blade elcm.ent Cj;, section drag coefficient ("'D-j/qS") C-^ section lift coefficient (l/qS) Cp povv'er coefficient (p/pn'-^D'~^ ) COl^IDENTIAL NACA ACR No. L4I29 GOJJPIDENTIAL 3 Cq torque coefficient (o/pn^D^) Cqi thrust coefficient ('T/pn^D^) D propeller diameter Dq drag of propeller blade element for infinite aspect ratio Ea_ axial energy pier unit time in slipstream Eji rotational energy per miit time in slipstream P Goldstein correction factor for finite number of blades J advance -diameter ratio (v/nD) L lift of blade section n propeller rotational speed, revolutions per second P input power to propeller Pq power disk-loading coefficient (P/qSV) Q torque of propeller q dynamic pressure of air stream. R tip radius r radius to any blade element S disk area of propeller T thrust of propeller Uq axial velocity in plane of propeller (propeller removed) V axial velocity of propeller X radial location of blade element ( r/R) dCq/dx element torque coefficient f — ^— —-J pn^D^ CO^IDENTIAL 4 C0]^J?IDE1'ITIAL KACA ACR !Io . L4I29 / /'dT/d.y\ d.CT/d.x element thrust coefficient { — fe-— r Vpn^D^/ Cq angle of attack of blade element for infinite aspect ratio 3 propeller blade angle at C.75 radius T] propeller or elemsnt efficiency p mass density of air a propeller eleiaent solidity (3b/2TTr) aCr propeller ele^.cnt load coefficient ^ angle of resultant velocity to plane of rotation ^p - ap,N Subscripts; 0.7R at 0.7 radius D due to drag i for zero drao; PORNaJLAS The derivation of the foritiulas for ele^.ert thrust and torque calculations is slven in reference 2, from which the element thrust coefficient is dT - -^ R --—f^- f^'L "^ 7^ - Cd sm i^.) (i) and the element torque coefficient is —,—^ = — ^-.,— t: .:: — — ( Gt 3in f + C-j cos p ) (2) ax Id R <:,4v^2^' ^ -' ' ^ ^/ co:ifide::tial F.ACA ACR No. L4I29 COIIPIDENTIAL The expression for the axial-veloclty interference factor a is 1 + a 4F sin (5) The values of the correction factor F as used in the present report are given in figure 1. The values of P for two-, three-, and four-blade propellers are taken from reference 3. The values of F for six- and eight-blade propellers v/ere extrapolated from these data by the method developed in reference 3. For calculations showing the effect of drag on propeller performance, the following formulas v/ere obtained from equations (1) to (3) for Cj^ = 0: dCr = -aCD ~J\/j^ + (ttx)^ dx ^' 4 = -'^Cd ^ -^ (4) ^4 sin p ax J sin (ttx) (5) Equations (4) and (5) are derived for zero loading without inflow and consequently are not exact for a finite loading. The formulas show, however, that the error in estimating the loss due to drag is negligible for light loadings which occur near peak propeller efficiency. CONFIDENTIAL COI^IDENTIAL NACA ACR No. L4I29 The formulas for the rotational-energy and axial- energy losses from reference 4 are P Co L dx rrl.O a'r:^^- dx (5) Xjq t n-^ » uCrn wnere dCn a' - — ^ ^ k2j:c-(1 + B.)F and dC^/d:K -1 + \/l -f- 4 y TTX J These formulas from reference 4 have heen modified herein by inclusion of the correction factor P. IvETHOD Charts for Induced Losses The ba.sic propeller performance charts are presented in figure 2 for tv/o-^ three--, four-^, six-, and eight- blade propellers. The ordinate s represent values of the efficiency for propellers operating v.'ithout drag and the 1 ' rrnV"^ abscissas represent values of — rm = '^\l~~^. — • Against ,.'p y SP V '"I these scales J curves of constant element load coefficient CONFIDENTIAL NACA ACR No. L4I29 CONFIDENTIAL 7 {oCt'\ are crossed by long-dash lines of constant ^ ^>0.7R ^ V/nD and short-dash lines of constant power coeffi- cient Cp.. The a.p^-curves show the variation of (aCj) ^ with V/nD for constant Cp. and are included ^ ' , c R ■'•1 for convenience of computation for constant-speed pro- pellers. The curves were obtained from calculated optini'urn torque and thrust distributions graphically integrated from, the tip of the blade to x = 0,2. These perform.ance charts are the same as the propeller selection charts in reference 5 except that the drag losses are not included. In the present report, the value of the solidity at 0.7R, <^o.7R' -^ taken as a convenient m.easure of the propeller solidity. The value of /'aC-rV „_ is corre- spondingly taken as a m.easure of the power absorbed. The activity factor has frequently been taken as an index of the power- absorbing qualities of a propeller. For the Hamilton Standard 3155-6 propeller reported in reference 6 (for which comparisons are m.ade in the present paper) , the activity factor Is 90 (per blade) and cJq r?p is C.0345B; that is, for propellers of this design, the 7R activity factor is 2500 — ;=; — , This number is approxi- mately the same for all conventional propellers. If the exact relationship is desired, however, the activity factor A.F. m.ay be obtained from , „ 100000 r^ """'^b 3 ^ A.F. = -■^;g— ^x^ dx Jx=0.2 Breakdovm of Propeller Po?/er Losses In the calculation of propeller efficiency, it has been customary to compute the thrust and torque at a given value of v/nD for a fixed blade-angle setting. The analytical determination of propeller performance may be considerably simplified in m.any cases, however, by evaluating the several sources of power loss rather than by attempting the direct computation of thrust and torque. In the present paper, the efficiency is CONFIDENTIAL CONFIDENTIAL " NAG A ACR No. L4I29 deterffilned by deducting the sum of the power losses from unity. The total power losses are divided into induced losses and profile-drag losses; the Induced losses are subdivided into axial- and rotational-energy losses for use in evaluating the efficiency of dual-rotating pro- pellers. The hlade drag has no appi'eciable effect on the induced losses for norraal propellers but must be considered in obtaining the total power losses. This method of analysis has the advantage that compressibility corrections can be included wh?n the airfoil section characteristics at the operating Mach number become known. Kot ational-energy losse s . - The losses of efficiency due to rotational velocity are shewn in figure 3 for three-, four-, six-, and eight-blade propellers. The rotational-energ^/" loss for a given operating condition (constant P^ and V/nB) is seen to increase as the number of blades decreases. This increase in power loss arises from the increase in the tip loss as the num.ber of blades decreases. Calculations for a large nioiuber of load distributions show that overloading the inner radii is of secondary significance at values of operating V/nD < £.5. As an illustrative case, the calculated rotational energy for the four-blade Hamiilton Standard 3155-6 propeller of reference 6, vvhich has round blade shanks, is compared In figure 4 with that for a four-blade propeller com.puted from the charts of figure 3. The rotational energy for the Ham.ilton Standard 3155-6 propeller was com- puted for several blade angles up to 65" at 0.75R and the value of V/nD for peak propeller efficiency. The values of E-p/'P for the optimum, propeller were taken from- figure 3(b) at the same values of V/nD and Pq as for the Harrdlton Standard 3155-6 propeller. Figure 4 shows that no appreciable difference In rotational energy exists between the two load distributions at V/nD < 2.5 and that the losses differ by only 1.5 percent of the total power at V/nD = 4.5, which corresponds to a blade-angle setting of 55^ at 0.75R. The rotational- energy losses given in figure 3 are therefore close approximations to the expected losses for conventional round-shank propellers over the usual present-day operating range. Simxilar results are found even when airfoil sections are used over the inner radii v;hen V/nD < 2.5. It cannot be emphasized too strongly, however, that if cuffs are used to cover the round CONFIDENTIAL KACA ACR No. L4I29 CONFIDENTIAL shanks, the loss may become serious at high values of v/nD iinless the cuffs are set at an angle of attack between 0'^ and the optimum to give low loading over these sections. This dependence of the rotational energy in the slipstream, on the loading over the inner radii is apparent from equation (6) , which shows how the effect of overloading the inner radii increases in importance as the operating V/nD increases. Axial-energy losses .- The axial-energy loss for any operating condition may be obtained from the rela- tionship ^ - 1 r — P ~ 'i P where the induced efficiency t]^ is obtained from figure 2 and E^/P is obtained from figure 3. The axial-energy loss shows but little variation among propellers in present-day usage operating near peak efficiency. As an illustration, the axial-energy losses for a four-blade propeller obtained from the charts of figures 2 and 3 are compared in figure 5 with the calculated values for the four-blade Hamilton Standard 3155-6 propeller operating at peak propeller efficiency and for the Ideal propeller (actuator disk). The values of P^^ at v/nD for peak efficiency for the Hamilton Standard 3155-6 propeller were taken from reference 6. The axial-energy losses for the optlm.um propeller load distribution and for the ideal propeller were determined at the same values of V/nD and P^ as for the Hamilton Standard 3155-6 propeller. The axial- energy losses for the optim^om distribution of loading (figs. 2 and 3) and for the loading obtained with the Ham.ilton Standard 5155-6 propeller are nearly equal and are about 1 percent higher than for the Ideal propeller over the range Investigated. A part of this increase in axial-energy loss is due to the finite nxomber of blades and therefore becomes less as the number of blades increases. A sm.all part of the difference occurs because the load distribution differs from that of an actuator disk. The axial-energy loss for optlm.um distri- bution, obtained with the aid of figures 2 and 3, Is therefore sufficiently accurate for application to conventional propellers. CONFIDENTIAL 10 CONFIDENTIAL ' NACA ACR No. L4I29 31ade -dre.g losses .- The effect of blade drag on the characteristics of lij^'htly loaded propellers (near peak efficiency) can be estimated from equations (4) and (5) . These equations v;ere obtained by eliminating the axial Inflow and putting Ct, = in equations (1) and (2), The equations are not exact but, near peak efficiency for modern high-speed propellers, the omission of the inflow factor a causes a negligible change in the calculated propeller efficiency. Equations (4) ana (5) shov; that, for a given radius and value of V/nL), the elem.ent thrust and torque coefficients due to drag are directly proportional to the drag coefficient. The profile-drag coefficients for -infinite aspect ratio for several sections along the Hanilton Standard 3155-6 blade are sho'An against lift coefficient in figure 6. These data were taken from reference 7. The profile-drag coefficients change v/ith lift coefficient but, since the change is very small for a v.-iie range of lift coefficient, average vali:;es v/ere used in the calcu- lations for operation near peak efficiency. The profile- drag coefficient increases rapidly near the stalling angle of the section and the average values are accord- ingly not representative for such conditions. Figures 7 and 8 show the effect of drag on the thrust and torque coefficients, respectively, for several values of v/nD. The values of the section profile-drag coefficient shown in figures 7 and 8 were used in the calculations for the Hamilton Standard 2155-6 propeller. Curves of the differential-thrust and differential-torque corrections due to drag, for the drag coefficients shown, are plotted against the racial location x, and the integrated corrections are also included in figures 7 and S. These integrated values apply for one blade and the correction is directly pro- portional to the number of blades and to the blade chord. The element thrust coefficient and the element torque coefficient due to drag at a given value cf V/nD are directly proportional to G-q, and a change in Ctj at any radius is represented by a proportional change in the ordinate of the differential-thrust and differential- torque curves. For this reason, the m.ethod of analysis is adaptable to any blade section for v.'hich the airfoil characteristics are available. The suggestion is also Tiade that the loss in efficiency due to drag at high speeds at which the blade section drag becom;e3 large CCNFIDZITTIAL KACA ACR No. L4I29 CONFIDENTIAL 11 can be predicted v/hen the drag coefficients at high Mach niffiibers become available. The calculations show that the drag correction to the thrust is chiefly due to the high drag of the inner sections (see fig. 7) and that a change in the drag coefficient of the principal working part of the blade due to a change in the lift coefficient near peak efficiency (see fig. 6) results in a negligible change in the correc- tion to the thrust coefficient. The effect of drag on the efficiency envelope and on the integrated power coefficient for operation at peak efficiency in unobstructed air flovi^ is shown in figure 9. The values of t]s and the corresponding Cp. without drag were taken from figure 2 for four-blade propellers at IOCt'^,^ „_ = 0.07. For optiin'Oin distri- bution of loading along the radius for the solidity of the four-blade riamilton Standard 5155-6 propeller, this value of /''^CtV „„ corresponds to Ct^ „_, = 0.51. The solid lines in figure 9 give the maxir.iur.i efficiency and the corresponding power coefficient? against v/nD for optimum distribution for a four-blade frictionless propeller at ('^Cr\ = 0.07. 'The short-dash lines ^ V L/0.7R show T] and Cp as modified by blade drag integrated from. 0.20R to the tip. The curve for x] for blade drag integrated from 0.45R to the tip is shown by the long-dash line. The Cp-curve is not shown for the latter case but falls between the other Cp-curves. The introduction of blade drag of the magnitude shown in figures 7 and 8 has little effect on the total power absorbed during operation near peak efficiency, regard- less of V/nD; the effect of the drag on the integrated thrust and hence on the efficiency, however, is im.portant and increases rapidly with v/nD, For exainple, the loss in efficiency for the entire blade varies from. 5.5 per- TT Tr cent at ry- =1.0 to 23.0 percent at -;^- = G.O. On the other hand, the loss in efficiency due to the drag of the principal working sections - that is, from^ 0.45R to the tip - is relatively unimportant. The loss in effi- ciency for this portion of the blade (see long-dash line,, fig. 9) is thus seen to vary from 2.5 percent at — =- = 1,0 to 4.0 oercent at — r = 5.0; these losses nD '■ nD COI^IDSNTIAL 12 CONFIDENTIAL NAG A ACK No. L4I29 represent the upper Units for increases in efficiency that majr he achieved by reducing the profile drag for the working sections of the blade operating at Cj, = 0.51 virith optirnmii load dlstrihutlcn. The thick inner sections of conventional round-shank propellers, Vrfhich are used for structural reasons, are therefore the chief source of blade-drag loss, especially at high values of v/nD. This loss in efficiency due to drag may be greatly reduced bj?' covering the inner portion of the shank with a spinner and the outer portion v.'ith cuffs, if the cuff angle is properly set. It is emphasized that cuffs iv.ay result in a loss in efficiency instead of a gain unless set at the proper angle of attack. Overloading of the inner radii results in a large increase in the power in rotational energy for single-rotating propellers, and the blade-drag loss also becomes large for blade sections operating near the stall. The losses due to the thick inner sections are also reduced when the propeller is operating in front of a blunt body, such as an NACA co'wling, because of the low velocity over these sections. Calculations of thrust and torque coefficients have therefore been made at the same values of v/nB and for the same distribution of the element drag coefficients as in figures 7 and 8 but with the inner portion of the blade assumed to be operating in a region of low-velocity air as in front of a conventional air-cooled installation. The velocity distribution in the plane of the propeller, as used in these calculations, is given in figure 10, for v/hich the data are taken from reference 4. The region of low- velocity air depends on the ratio of the nacelle diameter to the propeller diameter, the conductance of the engine, and the distance of the propeller In front of the cowllrig. The ratio of the nacelle diameter to the propeller diameter in the setup in reference 4 viras 0.417. The effect of operation in front of an open-nose cowling (velocity dis- tribution of fig. 10) is shovm in figures 11 and 12 for a four-blade propeller. Thrust and power coefficients due to the distribution of drag in figures 7 and 8 are shown for free-stream operation by short-dash lines in figure 11; the long-dash lines represent operation in front of the NACA covirling. The value of the power coefficient due to drag ^ACp^ is seen to vary but little vifith V/nD whereas ^AC'ji')^^ rises rapidly. CONFIDENTIAL NACA ACR No. L4I29 COKPIDB^ITIAL 13 Squatioris (4) and (5) also show this effect. The effect on the efficiency of operating a propeller In the velocity distribution of figure 10 is shown in figure 12 and com- pared with the efficiency computed for operation in free stream. The increase in propeller efficiency resulting from the presence of the cow].ing varies from 1.0 percent at — rr = 1.0 to 9.0 percent at — ?:: = 6.0. In the calcu- nD ^ nD lations for the curves shown in figure 12, the only variation considered is the drag of the blade shanks. The low-velocity air causes an increase in lift over the outer sections of the shank and, accordingly, an increase in the rotational-energy loss, which may result in a somewhat smaller gain in propeller efficiency than indi- cated by figure 12. In order to determine exactly the magnitude of the change in propeller efficiency, element calculations for each velocity distribution are neces- sary inasmuch as a change in velocity distribution pro- duces an effective change in pitch distribution. C Of/IP ARISON I'lTE EXPSRIIvISNT The experimental efficiency envelopes for four-, six-, and eight-blade Hamilton Standard 3155-6 propellers are compared with the calculated results integrated to 0.20R in figure 13. The experlm.ental results for the four- and six-blade propellers were taken from, refer- ence 6 and the results for the eight-blade propeller were taken from, reference 8. The single-rotating propellers v;ere miade up of tv\fo hubs mounted in tandem.. The spinner in the setup for all the experim.ental data covered the inner 0.189R of the front unit of one-half the blades and the inner 0.232R of the rear blades. The results are in agreement over the entire range investigated for each set of blades. The calculated efficiency is about 1 percent lower than the experi- mental efficiency at low values of V/nD and is higher than the experim.ental efficiency at very high values of V/nD; the calculated curve crosses the experimental curve at v/nD « 3.5. Part of the discrepancy is due to the use of the short method. C0NPID5KTIAL 14 CONFIDEI^'TIAL NACA ACR ITo . CONCLUSIONS A short method of estimating propeller efficiency for a given operating condition has been developed by a theoretical analysif?. The efficiency is d.etermined. by evaluating separately the Induced losses and the profile-drag losses. A comparison of the estimated efficiencies with experimental results indicated the follovv'lr.g conclusions: 1. The perfoi'manct for operation at values of the advance- diameter ratio equal to or less than 2.5 may be accurately predicted. 2. The approxim.ate performance of conventional round-shank propellers may be predicted to values of the advance-diameter ratio much higher than 2.5. 3. The upper limit of possible performance for other types of propeller (airfoil sections over the inner radii or the use of cuffs over the round shanks) is shown for values of the advance-diam.e ter ratio up to 6.0. 4. The cause of excessive losses m.ay be determined for any propeller design. 5. The maximum gain in efficiency to be realized with dual-rotating propellers over optimum single- rotating propellers is evaluated for a v;/ide range of operating condition. Langley Memorial Aeronautical Laboratory National Advisory Comjmittee for Aeronautics Langley Field, Va. CONPIDEKTIAI. NACA ACR No. L4I29 CONFIDENTIAL -^^ APPENDIX APPLICATION OF THE !.CETHOD The problem of determining the propeller efficiency for a given design condition by the methods of the present report may be resolved into tvra parts: (1) determination of the induced pov/er losses and (2) determination of the profile-drag losses. The induced power losses for a given design condition are available from figure 2 as 1 - tj^- . The induced losses obtained from figure 2 at v/nD < 2.5 are very close approxim.ations to those obtained with conventional propellers. This range of v/nD covers most current high-speed designs. The blade drag is handled separately and can therefore be used for high Mach n-ombers and high drags if the correct airfoil section character- istics are used for the corresponding Mach numbers. The profile-drag losses are obtained from figure 11, which shows the variation of the thrust and power coefficients due to drag for the values of the element drag coefficients shown in figures 7 and 8. These drag values are repre- sentative for the Hamilton Standard 3155-6 propeller operating near peak efficiency. The ordinates of the curves in figures 7 and 8 are directly proportional to the drag coefficients at each radius so that, if other drag values are used, new curves giving the thrust and povirer coefficients due to drag may be easily plotted. Since the total power losses are divided into induced power losses and profile-drag losses, the method aids in determining excessive losses for any design condition. Excessive losses may be due to the fact that the propeller is either too heavily loaded or too lightly loaded, to a poor load distribution along the blade, or to high blade drag due to comipresslbility. The use of the perform.ance charts determ_ines the lift coefficient at a representative station and thus the loading. Two examples are given to illustrate the use of the charts in the determination of the propeller efficiency for a given design condition. Example 1 In example 1, the propeller selected operates on a liquid-cooled Installation in the tractor position and all the sections are assumed to operate at free-stream velocity. No compressibility corrections are applied. CONFIDENTIAL 16 CONPIDEKTIAL II AC A ACR No. L4I29 The following design conditions are assioined: Power, hp = , 2000 Altitude, ft o 25,000 Velocity, mph 425 Rotational 3peed, rps . . . . 2S Propeller diameter, ft . 12 ITurri'ber of "blades Pour ^0.7R "" 2¥? ° ■ • • '^'-ISSO Activity factor 90 V/nD 2.25 Cp = —I— 0.342 pn-D"- The calculated values for operation in free air ai'e (ACp)^ (fig. 11) 0.006 Cp, =% - (-^Cp)^ 0.336 i/y'p^ - ^ s.e"/ (oCl)^^^^ (fig. 2(c)) 0.07 °Lo.7r' ^'--1 T]^ (fig. 2(c)) C.931 Sr/P (fig. 3(b)) .. C.C39 ■niCp. Ct. = —7-^ 0.1385 (ACrp)j^ (fig. 11) • -0.0103 Crp = C.'p. + (^0^) 0.1282 ^ = S nt • • • • 0-843 In order to determine whether the propeller selected is loaded properly, the value of Cj^ is first deter- mined in the selection chart in figure_2. Since the design conditions are given and l/jp',^- and v/nD have been predetermined, the value of (oC-.-V , , is read directly from the chart. The value of Cpq rj^ required OITPIDSriTIAL NACA ACR No. L4I29 CONFIDENTIAL 17 is found to be 0.51 and Indicatss a satisfactory design condition. This lift coefficient has been found to be that absorbed near peak propeller efficiency for the Hamilton Standard 3155-6 propeller for operation at the given v/nD. The power loss due to rotational velocity E r- ,/P, which is equal to 0.039, Is the maxiraum increase in efficiency to be expected froin the use of dual- rotating propellers of the same diameter and solidity. The induced efficiency is 0.G3 but the introduction of drag of the magnitude of that shown in figures 7 and 8 reduces the over-all propeller efficiency to 0.848. The use of l/\/Pc '^'ith drag included, instead of l/.J?^ . , results in nee;ligible changes in ri^- and i'0Ct\ Example 2 The only difference betv;een example 2 and example 1 is that the propeller in example 2 is miounted in front of an open-nose cowling (air-cooled installation) virith the inner sections of the propeller in retarded air flow. The design conditions, which are the same as in example 1, are Power, hr) 2000 Altitude* ft 25,000 Velocity, mph 425 Rotational speed, rps 23 Propeller diameter, ft , 12 Nximber of blades Pour- On 7 = ^ 0.1330 *-• ' 2TTr Activity factor 90 V/nD .' 2.25 Cp = — ^ 0.342 pn'-D^ The calculated values for operation in front of NACA cowling are (fig. 11) 0.006 (ACp) Cp. = Cp - (ACp)j^ 0.336 l/xfP7[ - 3-67 CONFIDENTIAL IS CONFIDENTIAL' NACA ACR No. L4I29 (aCT..)Q ^p^ (fig. 2(c)) 0.07 %.7r'-"^ ■• •- °-51 r]j_ (fig. 2(c)) . 0.931 Sp/P (fig. 3(b)) ..... 0.039 Ct., = — 7 — - c 0.1385 1 1 V/nD (AC^)_ (fig. 11) -0.0065 Ct = Ct, - (^Ct)j-) ■ . . . 0.1520 n ^i = ^ ZT^ C.S72 p ilU gonfide::tial MCA ACR No. L4I29 OONFIDEFTIAL 19 REFERENCES 1. Crlgler, John L.: Comparipon of Calculated and Experimental Propeller Characteristics for Four-, Six-, and Eight-Blade Single-Rotating Propellers. NACA ACR No. 4B04, 1944. 2. Glauert, H. : Airplane Pi^opellers. Vol. IV of Aero- dynamic Theory, div. L., '.'/. P. Durand, ed., Julius Springer (Berlin), 1935, pp. 169-360. 3. Lock, C. N. H. , and Yeatinan, D. : Tables for Use in an Improved Method of Airscrew Strip Theory Calculation. R. & M. No. 1674, British A.R.C., 1955. 4. Stickle, George W., and Ci-'jgler, Jolm L.: Propeller Analysis from Experimental Data. NACA Rep. No. 712, 1941. 5. Crigler. John L., and Talkln, Herbert W.; Propeller Selection from Aerodynamic Conside-T-ations . NACA ACR, July 1942. 6. Biermann, David, and Hartman, Edwin P.: Wind-Tunnel Tests of Four- and Slx-^Blade Single- and Dual- Rotating Tractor Propellers.. NACA Re".-> . No. 747, 19^x2. 7. Pinkerton, Robert M. , and Greenberg, Harry: Aei'ody- namic, Chars-cteristics of a Large Number of Airfoils Tested in the Vf^riable-Density^Wind Tunnel. NACA Rep. No. 628, 1933. 8o Biermann. David, aixd Gray, W, H« ; Wind-Tunnel Tests of Eight-Blade Single- and Di.ial -Rotating Propellers in the Tractor Position* NACA ARR, Nov. 1941. 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