TECI_ICAL f[OTES

NATIONAL ADVISORY CO)_{IIiTTEE FOR AERONAUTICS

NO oZ_

PROPELLER DES iOl{

PRACTzC L APPLICA_ ION OF T_ B_AD_. _,L_I!_NT THEORY - I

By Fred E. Weick
Langley Memorial Aeronautical Laboratory

/
Reproduced by

NATIONAL TECHNICAL
INFORMATION SERVICE
U S Department of Commerce
Springfield VA 22151

Wa sh ing% on
 NATIONAL ADVISORY COMb!ITTEEFOR AEROZAUTiCS.

TECHITICAL YOTE _IO. 2_L_5.

PROPELLER DESIGN,

PI_%CTICAL APPLICAT!CN OF THE EI:ADE EL_{EI, TT TH_IORY- I.

By Fred E. Weick.

8ut_ma ry

This report is the first of a series of four on y_rooe!ier

design and contains a description of the blade element or _:_od-

ified Drzewiecke tneozy as used in tlhe _. -.o-,a_- - -_ '

U. S. Navy Department. Blade interference corr_ctions are used

which v:ere taken from R.& l_f. No. G39 of the British Ag_visory

Committee for Aero_o, utics. The s,irfoil characteristics used

were obtained from tests of" model propel lers, not from tests

of model _vings.

A short method is also shown iu which the forces on only

one blade element are considered in order to obtain the char-

acteristies of the whole propeller.

The methods described have proved satisfactory in use.

rOduced from @
st available cop,/.
Int roduct ion

The U. S. Navy method of aircraft propeller design is

based on data obtained from a combination of the blade element

theory, ,"_odel propeller tests in a _:rind t_r:nel, an& tests of

N.A.G.A. Tecnn cal Note No, <,o5

full scale propellers in flight. The data are plotted as curves

for propellers of standard. Navy -:_orm, makin_ the actual oper-

ations in designing a propeller very short and simple. For the

ana!ysis or des_ig_ of special propellers not conformin_ to the

standard, the modified blade element theory is used, with air-

foil section characteristics which give result, ant powers and

effieiencies checking the standard model da,ta.

Although the blade element theory is well kno_rfl to aeronau-

tical engineers, the accuracy of the results obtained through

its use depends upon corrections which can be obtained only

through experience • It "_

i_ the intention to present in thls

report a modified form of the theory with data which makes its

application sufficiently accurate' for practical use.

Discussion and Developmc.nt _i the T._eo_y

In the simple blade el!one,It or Drzewiccke theory the pro-

poller blade is considered as made up of a number of small ele-

ments.and, the forces acting on ca,oh are found. From the su<ma-

tion of the forces on the elements the resultan+ forces on the

whole blade are determined.

No account is taken in the simple theory of the inflow or

increase in s!io-stresm velocity which takes place in front of

the propeller. The fact _hat the blades interfere with e_ch.

other in a manner similar to the interference of the win_s of a

"_ A C.A Tecnnzc I Note I_o _,o5

multinlane with backward stag_or is slso neglected. These fac-

tors are Taken care of in this roport by blade interference

corrections to the lift and dra_ coefficients of the airfoil

sections obtained from R.& H No. G39 of the British Advisory

Committee for Aeronautics (Reference I).

Airfoil characteristics as found from 'tests on mode], wings

at low air velocities do not aDply to propeller sections which

are un&er entirely different conditions. Propeller airfoil

section characteristics have been calculated from model propel-

ler test data (Fig. 6), as will b@ explsined in Technical Note

!,_o. 236. These are used in calculations of propeller perform-

ance and 6jive mowers and efficiencies corresponding to the tip

speed of the model propellers (about SS0 ft./sec.). The nowers

are then modified according to the particular tip sneod of

the full sized propeller (Reference2).

The s_bols used in the followin_ devclonment are _iven in

Table I. The lift in hounds on a section of the procellar

b!adc of length Ar s nd of _t_idth b is given by _hc exores-

b (Vr) C'! Ar
sion P ...... 2, , where V r is the resultant velocity of the

I Reproduced from _
section (Fly. 5). lbest available copy.

The -kotal forc_ on the section due to both lift and drag

is O b(V__ KAr and is in the direction of K in Fi_. 5, _:bere

2

K - 0_ .... As cos_-¢) is in all cases betw,,_en .995 and

cos --c)
I, d-_e ex-oression for the force on the blade element may ice
N.A-C;A. Technical _ote _o• ,_o5 4

taken as
2

The thrust is the component o:f this force in the direction

of the propeller axis or

b(Vr) _ c'L oos(¢'+7) ar (1)

dT= p
2

where tan 7 is taken as tan c p!us D/L of the airfoil sec-

tion at the corrected angle of attack _' . This is an approxi-

mation but is correct within the limits of accuracy of the de-

V and expressing b and r in

sign. Substituting Vr- sin@

terms of the diameter, equation (I) becomes

aT = pV2D _ × C'L. ×.b × cos(¢+_) d &rZ.

C' L × b
Let K_) "-=o and T c = Kp cos(@+7).
•- :_ sirff¢ × D

and the to_, I thrust for the propel-

Then dT = p V_D _ Tc¢_ _/

let (of B blades) is

T = P V2D2 B f_r_ _'Cd (D_ •

6

In like manner the expression for torque is

Q = -,
_ V_D_ B/_ Qc d _X_ ....
wL_eze Qc = KD × _r × sin@ +_) .
o _D /
.
The horseDowers are then found from the expressions:

T c and Qc are found for each section, and the inte_:rations in

N.A.C.A . Tecnnlc
_ " aI Note 1io. :335 5

equations (Z) and (4) are performed graphically with the aid of

a planimeter.

Single Section Method of Analysis.

It vJas _qoticed that for wood propellers of Navy sta_dard

blade foxTn (Fig. II) the torque and thrust grading curves used

in the abovc _raphical integration always reached their maxima

at approximately 75 per cent of the radius, also these curves

were approximately similar for all pi%ch ratios and slips.

This suggested that if a constant relation could be found be-

tween the ordinate Qc, at the 75 per cent radius and the to-

tal area, o
this one station only and multiply by a constant factor to _e%

the value of O
jr Qc d (_,;, and o
/ TO d rh could be found in

a similar manner.

The torque and thrust grading curves for a series of stand-

ard wood propellers were therefore calculated by mcans of the

full blade element theory using blade interference corrections

and airfoil characteristics a,s found in the _cCook Field. high

speed wind tunnel. The series included various pitch ratios,

sligs , blade widths and thicknesses. As the curves are all ap-

proximately horizontal az the 75 per cent station, any slight

shifting of the peak has _ractically no effect on the valuc of

the ordinate.

The actual ratios of Qc for 75 pe_ cent radi_is divided

 N.A.C.A. Technical Note No. 835 6

by / I/_Qc d #2g_ have a constant value of .272 within -o_.

_ For
•o KD,'
• , &_ 0 0,
thl_st, the ratio is constant within the same limits at °_'_

If the torque and thrust coefficients at 75 per cent radius

x are desi_ated as Q'c and _' _,_ ,_ively, the expressions

for horsepower become

•272 p VaD 3 BQ I c N
Q. HP. =
5255

.256P _D _ B T'O '

T HP =
• ' 550

T _ • v
- •156 x _,_ x q'c

$he expres-
As" TTc = Kp cos(@+_) and Q" = .375 Kp sin(@+7),

sion for efificiency car. be further reduced to

V
.416 x
_1 :
tan@ +'Y)

This short method of analyzing the 75 per cent radius sta,tion

only can be applied with _ood results to any conventional wood

propeller having a plan form tapering toward the tip, and ap-

proximately uniform pitch, but is strictly applicable to none

but standard Navy wood propellers.

Application

Blade Element Theory:

The application of the blade element -theory can best be

shown by tt_eans of an example. Consider a propeller 10 ft. in

diameter with a uniform geometric pitch of 7 ft. and blade

widths and thicknesses as shown in Table II, revolving at 1800

 7
N.A.C.A. Tecanlo_l Note No. 235

R.P.M. on an airplane having forward speed of 129 _,I.P.H.

For the section at 75 per cent of the radius the blade width

is S
._6 ft. and the camber or _s .107 "
i

Pitch _ 7 = .3975
tan blade angle CB = _°n r 2_ x 5.75

Hence the blade angle, ¢_ = 16`60.

For 129 _,I.P.H., V = 129 x 88 = 189 ft./see.

CO

V ....... = ]89 = .267

tan¢ = -2,_w rn 2_ × 3.75 x 30

Hence the path angle, ¢ = 15 .00.

The apparent angle of attack,

= ¢_ - ¢ :: 16.6 ° - 15.0 ° = 1.6 ° .

From Fig. 6, CL = .530 for h_b = .107 and m = 1.6 °.

Fig. 7 shows the blade interference correction to the lift

coefficient. The correction depends on the angle ¢, CL, the

lao u
radius, ol,,,dc width, and number of o!adeo. The throe
2 w r For
used in the form of a coefficient S, where S - Bb

2w x 3.75_ ]7.8. From Fig. 7, the correction

our ex_mp.'[ S= 2 x .66

to the lift coefficient CL, is .058, where ¢ = 15 ° ,

= .530, ancz S = 15. When 8 := 20, _C L = .032. Using

CL "

linear interpolation, when S = 17.8,

.-- .o5s - ( .o58- .o52) ( 7.8__so ,,

 B
N.A.C.A Technical Note No o+
+

The corrected lift coefficient is then

C, L = CL - _C L = ,550 - .044 = .486.

In like manner, from Fig, 8, the correction to the angle of oct-

15 h
tack is
++ -_
.:oun_ to be c = •
e4 -
( ' 6+- "
40) (\ -7++'°s
2,0- 15/ "

The corrected _+uole of attack

!
= C_-¢ = 1"6 ° - .5 ° = I.I °.

From Fi_. 6, the L/D for a standard propeller section of

hU = .107 at I.I ° is !6.8.

b
tan_ = .D + tan¢ = .0595 + .00S7 = .0682 and _ = 5.9 °
L

Kp = C'L x b x _ 1si_¢ .486

lO x x ; .66
x x 1_ _
.2588 = .259

r x sin(¢+_') = .259 x 5.75 x sin 18.9 0

Qc= ip x _ l---O-
= .239 x .375 x .5259

= . 0290

T O = Kp cos (¢+_) = .239 x .9461 : .226.

The va.luos of Qc and TO are found for each section (Tab!c iI)

and clotted against r/D (Fig. I0). Those are called the

torque and thrust grading curves. The area under the torquc

6fading curve is S _/_Qc d (rh

_/ :: .0081.

From equation (4) the torque is

= S I]_
0

= .00257 x 18_ x 10 a x 2 x .00805 : 1565 ft.lb.

.56u >< 1800 - 467.

Q. HP. = 5255 - 5_-------:.;55
i_.A.C.A. Technical Note No. 255 9

From equation (5) the thrust is

" =
o v_zfs / Tcd fZh
_D/
0 "

= .00237 × 189 _ × ! O_ × 2 × • 0620 = 1050 lb.

TV lo50 × s_s9 _ eel.

T.Y2. = _: 550

T.I_. 361 __ .772

The efficiency _ - B. P_ • 467

The torque horsepower found above is the power which the pro-

peller would absorb if it were operating alone (without body

interference) at a tip speed of about 250 ft./see. In this

case the propeller being i0 ft. in diameter and revolving at

1800 R.P.M., the tip speed =

wnD = .0524 iND : .0524 x 1800 × I0

= 944 ft./see.

Zf it is operating in front of an average fuselage at 944 ft./

soc., tip speed, the Q.HP. •should be increased by about 15 per

cent to give the actual powe'r absorbed by the propeller (Refer-

ence 2) or B.HP. = 1.15 x 467 = 537.

Single Section Method:

In the short method Qc is found for the section at 75

Then since the ratio

_or cent radius only and called Q'c"

1,'2 {/._? \
e_ qe d _ has a constant value of .272,
_C
N.A.C.A. Tech_icai Note No. 255 I0

0
Q 0=
:,"%. @')-.....,.,× , .... _, y 029 = .0079

, o,,_ 189 _ x I_ x 2 × 0079 =--1342

q = p V2D3B .fl/_Qc d (_)= .00:,3, x
0

__@.N = 1542 x 1300 = 460

%.=_. = 58 55 o_.,55
-o

anti E,'BP. = 1.15 x 460 = 530.

V " 9
.4_e × FZ
_,_me efficiency - tan .tS.9 °
tan(¢ +W)

.4i6 x .6s : .765.

.34,24

Refe r@lqo @s

Liu!tip!ane Interference Ap_!icd to

Wood., R. }:tcXinnon
Air $orew Theory. Brit. Advisory Com.
- _'f ic!d.,
__r,:._ F . B. ' . _ r7 f
for Aeron. R.& _{. ,T_. 009 1919)
Bayke_', _l.

Promellcy Scale Effect and Body Inter-

o
,_. ,'r",i OK)
%,., - F',cd
- E.
felon c e. t_ A
_- • C.A. Tecnnlc,:,l llore
_,To. 225. (!s25)
 N.A • C.A" Technical Note _'_o.
_ 2 3 :_ ]:J.

TABLE I.

S ymbol s

D Diameter of propeller in feet.

p - Geometrical pitch of propeller in feet.

r - Roodius of any section of propeller, in feet (Fig. !).

b - Blade width at any section, in feet (Fi_. I).

e -- Maximum blade width, in feet (Fig. 1).

B "- Nurfotr of blades in propeller.

Q - Torque of propeller, in foot pounds.

T - Thrust of propeller, in pounds.

Qc - Torque coefficient of any section of propeller.

TO - Thrust coefficient of any section of propeller.

V - Vc!ocity of advance in ft./s@_c.

M.P.H. - Velocity of advance in mi./hr.

n - Revolutiors of propol!er per second.

I{ - Revolutions of propeller per minute.

B.B_°. - Brake horsepower of engine•

m
•._P. - Tnru_;t
,1 _
horsepov_er •

Q.HP. - Torque horsepower.

- Efficiency

Bl_,do an_le in dc_frces (Fig. 2).

- AY_g!e which path of blade makes with plane perpendic-

ular to propeller axis.

== arc tan V rn (Fig. 3).

2W
f.A,C,A. Technical Note No. $35 12

Apparent angle of attack of section = ¢_ - ¢ (Fig. 3).

£ - Interference correction for angle of attack (Fig, 5).

!
Corrected angle of attack : ct - c (Fig. 5),

CL - Abso!ut_e lift coefficient of aid'foil at ar_le of attack

(Fig. 6).
Interference correction for lift coefficient (Fig. 7).

C' L - Corrected lift coefficient = CL - _C L (Fig. 5).

Ratio of lift to drao4_ for the airfoil section.

hU - Ha::imtun upper cam?oer of section (Fi_. 4).

hu/h L St:_ndard ___u_ ratios (Fig. 9).

hL - !laximum lower camber of section (Fig, 4).

r

h/b- Cs_mber ratio of any section. __

- An_-le between resultant force on blade element and a

iine perpendicular to V r.

p -- 1{ass density of air. This may be t a: ne_ ' as .00857 for

sea level and standard a%mosphere.

Tip speed - The dista_me traveled by the tip of the propeller

in unit time in plane of rotation.

Tip speed- = wnD = .0584- !@ ft,/see.

<
N.A.C.A. Technical Note No. 335 !3

TABLE I I.

Analysis of Standard Navy Wood Propel!e:o.

Diameter = I0 ft. R.P._ff. : ]_800 I,,ioP, H: = ]_g9

Pitch : 7 ft. n : 30 r.p.s. V = 189 :ft•/sec.

:o/[{ .30 ,4-5 •60 .75 .9O

• 788 •833 .788 •660 •d-50

_p

h j/h • 20O .167 •]o3 •107 •090

hL/b • 058 .007

r (ft.) 1,5
,,m .-]
_r3 .Co5 ! 3:0 3,75 4.5
f
2w r 9.42 -1_.4.13 23.55 28.25
]8.84:
_w rn 383 4-34- 1565 707 1848

S - 2wr 6 8.5 12 17.8 31.4.

Bb

_D

•743 • 495 ,372 • 2975 • 8.'!8

e° 26.4. ° o0 . 4° ].6.6 ° 13,9 °

¢

V .4.46 .334 • 267 ,223

tan@= .... •668
TI :Oil

¢ 33.8 ° 2,4..0 ° 1P • 5 ° 15,0 ° 12 • 6 °

2.8 ° 2.4o 1.9 ° 1.6 ° 1,3 °

_=¢B-¢

,760 .802 •656 • 530 • 430

CL

%_CL •100 • 096 • 066 • 044 • <.JAO

•660 •706 • 590 • 486 •406

C IL : CL - _ C L
_0
£ 2.4 -° 1.8 ° !.0 o •5° a(O

,..,0
• _o •o .9 ° 1.1 ° I.i °

L/D 16.3 1G.8 t6.5

1 .5 i ]5.4- 1

• 0690I .0649 •0613 •0595 .0606

I
N.A.C.A. Technical Note No. 2,35 14

Table !I (Cont.)

-,a _"
A,_c,Iysis of Standard iTavy Wood Propeller

Diameter = I0 ft_ R.P.}I. = 1800 M.P,H. : 729

..... ,:

Pitch - 7 ft. n = 30 r.p.s. V = ]_89 ft./soo_

!
t_n £ o4!91 .0314 • 0175 :0087 ,0035

tan7 = D/L + tanc •1109 i • 0963 r0788 • 0 G82 .0641

5.5 ° i 4.5 ° 3r9 ° 3.7 °

7
G.3 ° t

39.5 ° 123.0° 18.9 ° : 16.5 °

!_o.1_ t
• 4.087 .3173 .2588 ,_18!
sin ¢ •5063

.0788 .OGGO •O45O

b/n • 0833 .0788 1

•GGO • ,...} O <J • 48G •406

,i •70G _"'" i

•O8 41 .2390 • 1920

1805 o_, n I
"n_¢
,,b×Ss I1
Kp=C'L_N

•6641 .49£_ • 5907 •32,39 .2,807

sin(c+'@
.,82,5 .3O .375 •45
r/n ,15

•0081 •02 O0 .0271 •0290 .02,42

• 7649 • 8704 .9205 .94-6t •9 598

cos(_+_)
•OG_3 .].573 • 2130 .22G0 •]8 50
To- -p cos (¢ +'Y)

i:
• L
\
t
N.A.C.A. Technical Note No. _35- Figs.l,2,3,4 & 5

_L

I
I
.__-----%-
I
.........b
i i
// o1
__. _/
R

FJ.g,l

i l, i _,_'_ _- .... _ ""_ _n

/ J t' _ I _ ___._
I t ....." i_T 1 _P
I ""- ..... _ _ b ......

............. __ _ ., ..s ,
8 v :_

Fig.2 Fi_:3 Fig.4

Fie;. 5 theory.
Diagrams used in the devc_.lopm_nt of f;ie modified blade element
 N.A.C.A. Technical Note No.235 Fis.6

, /_.o_ h__ , 1 2o

• _,,,..":_ _3 "_ " \ \.1

- " ' !___,. _--'_-_ w w,,m_ \ '

. ' _J_'_"
i .',-_
7"-H--..__.X.4_.
_5_
"-' . " ]" ..
O.r,
1
I
_'- _ '// /////// / ' 1 I ._. ..:_"_.//://_;?>_ p-
?.
////
'/_YL////,/// I ./.</,//,
V_ __
<:.4"/_>,.sr'/
_o
_-7 _--_-_o-
-----;
_'7"--_7"-_
7-? _:__.77,<_ ....
.-".-- i -" 4 z
- - _
_ -7--7
";Tj
7->_T;:>,.t..../_
.... -; [P
,'...Z/ii
I
e, ' " "/-, .///AL/---J-_--./.-/-_ , /; _----_-----i----
• ,18- ./// / j//'}/ //....!. / !
•__/"/',/_</,J//")V:.-"
/.-.- _,_?//_J" I I I
•_s/i///'//</'..y.'4"' . !
• 14 / /' -/ /" /" // / "/ ' j I.
•" - / .// ////i.// ! ,
.__s
"/.//< z;.///i .. 1
• 12 /,,/.u/ /'// "/" I i ..... .

3.-.://V/
.___,_0./,//"/
, -
°t ¢<//
.o_- //

.OG _]o (p _o 2o _o o _;o _o o ?o i

An#'le of i#]oidenco . mr ,-
Fi,@.6 L/D and C L curves fro_ model Navy prop.d_ta exp,:,_d,&.
 N.A.C.A. Technical Note No.235 _.,

8 CL .30 --j

50 o

= - :301

I
i

• 2O
I
I

t
0
0 IO o 0 I0 ° 20 °

Fig.7 Lift correction,8 CL,

_ /'7-
N.A.O .A.
Tech.
No .235
No t,i_'._ •:° ,],Li , ,
IF:_.8
I ......

6° ,___ I :

\l ........ []-_ ' !1' '

.......... _--i , [ i

I
S= 9Trr
50 __
I

4 ° ....

¢ 3° ___ ......J_._k..
'._I_, X IX ./ ......

2°

1°

0 ,, , I r i

a?oo 300 4do 0° 10 o 20 ° 30 o

I

CL::I. I( S
Bb BI_

hO ' '
A--}.:°
............... i

_L__

__ .8<
......... ... J .....
.7
_, ( \\

.6

_0 10 ° 20 o 30o0 ° 10 ° ¢ 20°-/g,. 0° I0 ° 20 °
 q_
0

p_

0
It- Max. camber =I.0
CD

_0
_dge _- b Oq
Leading Oq

.o_ .l ._ .5 .4 .'5 ;6 ? .8 .9
I
i< Blade width(b} -

tation .1 i. i.5i.6
!

7 I "8 .9 T_R:!
Ordinate!ilO .A1 .59 !.79 .95 .998 t .99 t ,95 .87 [7_ I.SS .35 .0v"

Fig.9 Navy standard blade'section.R.A.F.No.6 Modified , flat face.

c_
 N.A.C .A. Technic%l Noto No.255
Fig.lO

sq.in. =.0620
'/o TO a(r/D):O.Olx6.20
,, 1/2
o Qc d(r/D) = 0.001×8.05 sg.in,=.O0805

.04 [

..... i

/ •

.O3

/I
71 --o_<.....
-9c

rc_ ......

0
c_ .08
r J
\ \

.01 i

l
- !
• ,,,
I
I
t
f
t I

0 .1 .3 .4 .5
r/D
Fig.lO Thrust and t o r que grading OUrVeS.

_ _0-
 _-3
O
O

C_

_4
o
1.00
.900 b/o .945 L
] oq
I

i _9_
I J
!
d-
J
i i
i
÷
i
i
.15 .30 ._5 .60 .75 .9O
Radius,r/R

Fig.ll Standard _avy plan form for wooden propellers.