design of optimum propeller

8/18/2019 Design of Optimum Propeller

1/7

J O U R N A L O F P R O P U L S I O N A N D

P O W E R

Vol. 10, No. 5, Sept.-Oct. 1994

Design of Optimum Propellers

Charles

N .

A d k in s *

Falls

Church,

Virginia

22042

and

Robert H. Liebeckt

Douglas Aircraft Company,

Long

Beach,

California

90846

Improvements have been made

in the

equa tions

and

computational procedures

for

design

of

propellers

and

wind turbines of maximum efficiency. These eliminate the small angle approximation and some of the light

loading

approximations prevalent

in the

classical design theory.

An

iterative scheme

is

introduced

for

accurate

calculation

of the

vortex displacement velocity

and the

flow angle distribution. Momentum losses

due to

radial

o w

can be estimated b y either th e Prandtl or Goldstein momentum loss function. The methods presented here

bring

into exact agreement

the

procedure

for

design

a nd

analysis. F urthermore

the

exactness

of

this agreement

makes

possible an empirical verification of the Betz condition that a constant-displacement velocity across the

wake provides

a

design

of

maximum propeller efficiency.

A

comparison w ith experimental results

is

also

presented.

Nomenclature

a =

axial interference factor

a ' = rotational interference factor

B = num ber of blades

b =

axial slipstream factor

C

d

= blade section drag coefficient

C, =

blade section lift coefficient

C

p

-

power

coefficient, P/pn

3

D

5

C

T

= thrust coefficient,

T/pn

2

D

4

C

x

=

torque force coefficient

C

y

=

thrust force coefficient

c

= blade section chord

D

=

propeller diameter,

2R

D' = drag force per unit radius

F

= Prandtl momentum loss factor

G = circulation function

/

= advance ratio, VlnD

K

= Goldstein momentum loss factor

L' = lift force per unit radius

n

= propeller

rp s

P

= power into propeller

P

c

= power coefficient,

2P/pV

3

7rR

2

Q

=

torque

R = propeller ti p radius

r

= radial coordinate

T =

thrust

T

c

=

thrust coefficient, 2T/pV

2

7rR

2

V =

freestream velocity

v'

= vortex displacement velocity

W = local total velocity

w

n

=

velocity norm al to the vortex sheet

w

t

= tangential (swirl) velocity

x = nond imen sional distance,

£lrlV

a

= angle of attack

j8

=

blade twist angle

F =

circulation

s = drag-to-lift ratio

Presented

as

P aper

83-0190

at the

A I A A

21st

A erospace

Sciences

M eeting,

Reno, N V, J a n .

10-13,

1983; received July 23 , 1992; re-

vision

received July

6 , 1993;

accepted

for

publication

D e c. 15 , 1993.

£ = displacement velocity ratio, v'lV

17 =

propeller

efficiency

A = speed

ratio,

V/flR

g =

nondimensional radius,

r/R = A J C

\

e

= nondimensional P randtl radius

£

0

=

nondimensional

hu b

radius

p =

fluid density

a =

local solidity,

Bc/2

/

nr

f>

— flow angle

< f)

t

=

flow angle

at the tip

f l

=

propeller

angular velocity

Superscript

' =

derivative with

respect

to

r

or

£,

unless otherwise

noted

Introduction

I

N

1936,

a

classic treatise

on

propeller theory

w as

authored

by H . Glauert.

1

In this work, a combination of mo men tum

theory and blade element theory, when

corrected

for mo-

mentum loss due to radial flow, provides a good method for

analysis of arbitrary designs even though contraction of the

propeller wake is neglected. A lthoug h the theory is developed

for

low disc loading (small thrust or pow er per unit disc area ,

it works quite well fo r moderate loading, and in light of its

simplicity, is

adequate

fo r

est imating performance even

fo r

high

disc loadings. The conditions und er which a design would

have minimum energy loss were stated

by A. Betz

2

as

early

as 1919; ho wever, no organized procedure for produ cing such

a design is evident in

Glauert's

w o r k .

Those

equations which

ar e

given

by

Betz make extensive

use of

small-angle approx-

imations and relations applicable only to light loading con-

ditions.

Theodorsen

3

showed that the Betz condition for min-

imum energy loss can be applied to heavy loading as well.

I n

1979, E.

Larrabee

4

resurrected the design equations and

presented

a

straightforward procedure

fo r

optimum design.

However ,

there

are still some problems:

first,

small angle

approximations are used; second, the solution for the dis-

placement velocity is accurate only fo r vanishin gly small val-

ues (light loading), although an approximate correction is

suggested for moderate loading; and third,

there

are viscous

terms missing in the expressions for the induced velocities.


2/7

ADKINS

AND LIEBECK: DESIGN OF OPTIMUM PROPELLERS

67 7

Th e purpose of this article is to

correct these

difficulties

and bring the design method into exact agreement with th e

analysis.

It is

then possible

to

verify empirically

t he

optimality

of

the design.

This

work was

initiated

at

M c D o n n e ll D o u g la s

in 1980

in

response

to a

requirement

for simple

estimates

of

propeller performance.

In-house

m ethods, i f they

existed,

h ad

been irretrievably archived. A n early version was presented

as

AIAA Paper

83-0190. Continuous

requests

fo r copies of

th e paper plus some

ref inements

to the method have moti-

vated

its

publication

in the

Journal.

M omentum Equations

D etailed axial and general momentum theory is

described

by Glauert ,

1

and only a

brief summary

is

given

here

to em-

phasize

several important features. Consider

a

fluid e l e m e n t

of mass dm, far upstream

moving

toward th e propeller disc

in

a

thin, annular stream tube

with velocity V . It arrives at

the disc with

increased

velocity, V (l +

0 ), where

a is the

axial interference

factor. A t the disc, dm exists in the an nulus

27rr dr , and the mass

rate

per unit radius passing through th e

disc is 2irrpV(l + a ), neglecting radial

flow.

The

e lement

dm

moves downstream into the far wake, increasing

speed

to the

value

V (l

4 -

& ),

where

b

is the axial

slipstream

factor.

A x ia l

m o m e n t u m theory determines b to be

exactly

20 , whereas the

general theory

(which includes rotation

of the

flow) deter-

mines b to be

approximately

2a .

U s in g

th e

axial

approxi-

mation, which is generally

accepted,

the overall

change

in

m o m e n t u m

of the

e lement

is 2VaF

dm

w h e r e F, th e

m o m e n -

tu m

loss factor,

accounts

fo r

radial

flow of the

fluid.

The

thrust

p er unit radius

T',

acting on the

annulus

can now be

expressed as

rr

T = — = 27rrpV(l

+ a)(2VaF)

(la)

By

similar arguments, the

torque

pe r unit

radius

Q ' is

given

by

Q'lr = 2irrpV(l + a)(2Slra'F)

(Ib)

Flow geometry

about

a

blade

element at the disc is shown in

Fig.

1,

where W

acts on the blade element

with

a, and acts

on the disc at < j > . F

goes

from

about

1 at the hub (where th e

radial flow

is

typically negligible),

to 0 at the

tip,

and is not

unlike

th e

spanwise

loading of a

wing.

The

functional form

of this factor was first estimated by P r a n d t l

1

-

2

and a more

accurate, though

more complex, form

w as

determined

by

Goldstein

5

an d Lock.

6

8

Circulation

Equations

A t each radial position along the blade,

infinitesimal

vor-

tices are shed and

move

aft as a

helicoidal vortex sheet. Since

these vortices

follow

th e

direction

of local

flow,

the helix angle

of

the spiral surface is > , shown in

F i g . 1.

Th e

Betz condition

for

minimum energy

loss,

neglecting

contraction

of the wake,

requires

the

vortex sheet

to be a

regular screw surface;

i.e.,

r t a n

< t>

must b e a constant

i n d e p e n d e n t

o f radius. Theodorsen

3

AXIS

VORTEX FILAMENT

AFTER

TIME

INCREMENT,

A t

VORTEX FILAMENT (t = 0)

Fig.

2 Definition of

displacement

velocity v in the

propel ler wake.

develops

th e

Betz condition

fo r

heavy loading

by

including

th e

contraction

of the wake. He shows

that sufficiently

fa r

downstream

in the contracted

w a k e ,

th e vortex

sheet

must

be the

same

regular screw

surface

for a

propeller

of m i n i m u m

induced energy

loss. This

opt imum

vor tex sheet acts

as an

A rchimedean screw, pumping

fluid

af t

between

rigid spiral

surfaces.

A t th e

blade station,

r, the

total lift

per uni t radius is

given

by

L >

=

f=

BpWT

(2)

and in the wake, the circulation in the

corresponding

a n n u l u s

is

BY

= 27rrFw

f

3

Setting

th e

circulation F

in Eq. (2) equal to that in Eq. (3)

will ult imately determine that

circulation distribution

F(r)

that

minimizes

the induced p o w e r of the

propeller.

I n order to obtain F(r), it is necessary to

relate

v v

r

to a

more

measurable quant i ty.

Figure

2

shows

the wake vortex f i lament

at station r and the definition of the var ious

velocity

com-

ponents there. T he motion of the fluid must be normal to the

local

vortex sheet, and this

normal velocity

is

w

n

.

T h e r e f o r e ,

the tangential velocity is given by

W

t

= w

n

sin < >

However, for a

coordinate

system

fixed to the propeller disc,

th e

axial

velocity of the vortex

filament would

be

v '

= H ^ / C O S />

where the

increase

in magnitude of

v' over w

n

is due to ro-

tation of the f i lament.

This

is

analogous

to a

b a r b e r pole

w h e r e

it appears

that the

stripes

are translating in

spite

of the

fact

that only

a

rotat ional

velocity exists. It

will become

clear that

it

is convenient to use v' , and the

corresponding

displacement

velocity rat io, £ =

v'lV.

Th e

tangential velocity

is

then

W

t

= V£

sin

c f>

cos

4 >

and the circulation of Eq. (3) can be expressed as

F

27rV

2

£G/(Bty

(4)

G = F x cos

/>

sin

< >

(5 )


3/7

678

ADKINS AND LIEBECK:

DESIGN

OF OPTIMUM PROPELLERS

DISC

P L A N E -

Fig.

3 Force diagram for a

b l ade

element.

T he circulation

equations

for thrust

T

, and torque Q', per

unit radius

can be

written

by inspection of Fig. 3 as

T

=

L'

cos

0

-

D'

sin 4 >

= L'

cos

0(1

-

e

tan

< £ )

(6a)

Q'lr = L' sin < £ + D' cos

0

= L' sin

< / > (

+ e/tan

(6b)

where

e is the drag-to-lift-ratio of the

blade element. N e x t,

using

Eq.

(2), L'

can be

replaced

by

F(r) which,

in turn, is

related

to conditions in the w a k e b y E q . (3).

Based

on the

flow

in the

w a k e ,

F(r) is

given

by

Eqs.

(4) and

(5),

an d T

an d

Q' lr

are

reduced

to

being functions

of

f>

and the

dis-

placement velocity,

£ =

v'lV.

Th e

local flow angle

f> will

clearly be a function of the

radius;

however , at this stage of

th e

analysis,

th e

opt imum distribution £(r)

is not yet

deter-

mined.

Several diagrams and an excellent

photograph

of the

vortex

sheet can be found in a 1980 work by Larrabee.

9

Condition for M inimum Energy

Loss

A t

this

point , a departure

from

Larrabee's

4

design

proce-

dure is made, and the momentum equations, Eqs. (1), an d

the circulating

equations,

Eqs.

(6), are required to be

equiv-

alent.

This

condition results in the interference factors being

related

to

£

by the

equations

a = (£/2)cos

2

(£(l - £ tan < / > ) (7a)

a' '= (£/2*)cos /> sin

) (7b)

where Eqs. (4) and (5)

have

been used to express L' in terms

of £, and the terms in

epsilon correctly describe

th e viscous

contribution. E q u a t i o n s

(7),

together

with

th e geometry of

Fig. 1,

lead

to the important simple relation

tan < £ =

£/2)/x

(1 + £/2)X/(

(8)

Here,

A i s a

constant ,

an d

£ varies from

£

0

at the hub to unity

at

th e

edge

of the disc. The relation between the two non-

dimensional

distances

and the

constant

speed

ratio

is

=

=

(r/R)/\

= f/A

R ecalling

th e Betz

2

condit ion,

r

ta n < > = const, Eq. (8)

proves that for the vortex sheet to be a regular screw surface,

Constraint

Eq uations

For design, it is necessary to

specify either

7, delivered by

the propeller or the power P, delivered to the propeller. The

nondimensional thrust and power coefficients

used

for design

are

T

c

=

2T/(pV

2

irR

2

)

(9a)

P

c

=

2P/(pV

3

7rR

2

)

=

2Q(l/(pV

3

>7TR

2

)

(9b)

an d

using

these

definitions,

Eq. (6) can be written as

T'

c

=

I (£

- Itf ( lOa)

P'

C

=J[{

+ J ( l O b )

where the pr imes denote derivatives with respect to f, and

/; =

4£G(1

-

e

tan

0) (lla)

1

2

=

A ( / ;/ 2 £ ) ( l + e/tan < / > ) s i n < j> cos < > (lib)

/;

4fG(l

+e/tan 0) (lie)

J '

2

=

(/;/2)(l

-

s

tan

< / > ) co s

2

0

(lid)

Since £ is

constant

for an opt imum design, a specified thrust

gives

the constraint equations

£

=

( A /2/

2

) - [(A/2/,)

2

-

T

C

/I

2

]»

2

(12)

P C = J^

+

/

2

f

2

(13)

S imilar ly, if power is specified, the constraint relations ar e

£ = ~(JJ2J

2

)

+

[(V2/

2

)

2

+ PJJ '̂

2

(14)

T

c

=

U

-

I

2

£

2

(15)

where the integration has been

carried

out over the region

f =

& t o f

= 1.

Blad e G eometry

For the element

dr

of a

single blade

at radial station r, let

c be the chord an d C

l

th e local

lift coefficient.

Then, th e lift

pe r

unit

radius of one

blade

is

=

P

w r

(16)

w h e r e F is given by Eq.

(4).

I t follows directly

that

We

= 47rXGVR£/(C

l

B)

A ssume for the

m o m e n t that

£ is know n;

then

th e

local

value

of

c j ) is known

from

E q.

(8),

and the

above relation

is a

function only

of the

local lift

coefficient.

Since

the

local Rey-

nolds number is

We divided

by the

kinematic viscosity,

E q.

(16) plus

a choice fo r C/

will

determine th e

R eynolds

n u m b e r

an d £, from th e airfoil section data. The total velocity is

then

determined by

Fig.

1 as

W = V(l + fl)/sin

(17)

w h e r e a is given by E q.

(7),

and the

chord

is

then

k n o w n

from

E q. (16). If the

choice

fo r C/ causes £ to be a m i n i m u m , then

viscous as well as m o m e n t u m losses will in most cases be

minimized, and overall propeller efficiency will be the highest

possible

value. For

preliminary

considerations,

it is usually

sufficient

to

choose

one C

h

the design C

h

for

determining

blade geometry. ( A n y

C

l

specification is

permissible

as long


4/7

ADKINS

AND

LIEBECK:

DESIGN OF

OPTIMUM PROPELLERS

679

at

th e

edge

of the

disc,

and the tip

chord

is therefore

always

zero for a finite lift coefficient.

Design

Procedure

Either F or K, relation for the momentum loss function can

be selected. For the sake of simplicity,

only

th e P r a n d t l re -

lation

is

described

as

where

F = (2/77)arc

f=

(18)

(19)

an d < / > , is the flow angle at the tip.

F r o m

Eq. (8)

tan < t >

t

= A(l +

£ / 2 ) (20)

so that a choice for £ determines the

function

F as well as

f>

by

ta n

=

(tan

< £ ,

(21)

which i s simply t he

condition that

th e

vortex sheet

in the

w a k e

is a rigid screw surface (r tan

< £

= const). For an

initial

value,

£ =

0

will

suffice.

The design is initiated with the

specified

conditions of power

(o r

thrust), hub and tip radius, rotational rate, freestream

velocity,

number of blades, and a

finite n u m b e r

of stations

at which

blade

geometry is to be

determined.

Also, the design

lift

coefficient—one

fo r

each station

if it is not

constant—

must be specified. The design then proceeds in the following

steps:

1)

Select

an

initial estimate

for

£

(£ = 0 will

work) .

2) D e t er m i ne the

values

for

F

an d < j) at each blade station

by Eqs. (18-21).

3 )

Determine

th e

product

W e,

a n d R e y n o ld s n u m b e r

from

Eq. (16).

4)

Determine e

and

a from

airfoil

section

data.

5 ) If e is to be

minimized,

change C, and repeat

Steps

3

an d 4 until this is accomplished at each station.

6 )

D e t e rm i n e

a a nd

a ' from

E q. (7), an d

W f ro m

E q.

(17).

7) Compute the chord from step 3, and the blade

twist

)3 = a + f > .

8 ) Determine the four derivatives in / and

from Eq.

(11)

and numerically

integrate

these from £ =

£

0

to

£

= 1.

9)

D e t er m i n e £

an d

P

c

from

Eqs.

(12) an d (13), or

£

an d

T

c

from Eqs. (14) an d (15).

10) If this new value fo r

£

is not sufficiently

close

to the

ol d one

(e.g., within 0.1%) start

over at step 2 using the

ne w

£.

11 ) D e t er m i n e propeller

efficiency

as TJ P

C

, an d other fea-

tures such

as

solidity.

The above steps converge rapidly, seldom taking m o r e than

three or four cycles. A n

accurate

description of viscous

losses

ca n

be obtained by creating another design

with

e equal to

zero

and

noting

the

difference

in

propeller efficiency.

Analysis of Arbitrary Designs

The analysis method is

outlined here

in

order

to

discuss

problems

of convergence for off design and for square-tipped

propellers in general, and to point out two minor errors in

Glauert's w o r k . Figure 4, which is

simply

an alternate version

of Fig. 3,

shows

th e relation between t he propeller

force coef-

ficients, C

y

an d C

x

, and the

airfoil coefficients,

C, an d C

d

. Th e

equations are

C

y

= C, co s f> - C

d

si n < £ = C,(cos < > - e sin < / > )

DISC P L A N E -

Fig. 4 Force

coefficients

for propeller blade element analysis.

and the relations for the thrust

7"

an d torque

Q '

per unit

radius are then

T

=

( )pW

2

BcC

y

Q'lr =

( )

P

W

2

Bc C

x

(22a)

(22b)

Again,

it is required

that

the

loading Eqs.

(22) be

exactly

equal to the

momentum

result Eqs.

(1).

With the use of the

flow

geometry in Fig. 1,

this requires

the interference factors

to be

where

and c r is given by

a

= o-KI(F -

< r K )

a '

=

)

K '

= C

x

/(4 co s 0sin < / > )

e r = Bc/(2m)

(23a)

(23b)

(24a)

(24b)

Equations (23)

correct

th e placement of the factor

F

used by

Glauert in his equations (5.5) of Chapter V I I a s identified by

Larrabee.

4

The

relation

for the flow angle is

obtained

from

Fig.

1 and

Eqs.

(23)

as

ta n < t>

=

[V(l +

a)]/[nr(l

- a')]

(25)

F or

determining the

function,

F, in

Eq. (18), Glauert suggests

the relation sin $, = f sin 4 > be used in E q. (19). It is rec-

ommended that Eq.

(21)

be used instead, i.e.,

tan < / > , = £ ta n < >

which

is

exact

for the

analysis

of an

optimally designed pro-

peller at the design point.

The analysis

procedure requires

an

iterative solution

for

the

flow angle < >

at each radial position,

£.

A n

initial estimate

for

< >

can be

obtained

from Eq. (8) by

setting £

equal to zero.

Since j3 is k n o w n , the value for a in Fig. 3 is /3 — < £ , and the

airfoil

coefficients

are known from th e

section

data. The

Reynolds n u m b e r

is determined

from

the

known

chord and

W,

which is obtained

from

Fig. 1 and Eq.

(23a),

and the new

estimate

for

< £

is then

found

from E q. (25). A direct

substi-

tution of the new f> for the old value will cause adequate


5/7

680

ADKINS

AND LIEBECK:

DESIGN

OF

OPTIMUM

PROPELLERS

nonoptimum

designs, some recursive combination of the old

and new

values

fo r

f>

is required to cause adequate conver-

gence.

U n d e r some

conditions (usually

near

th e tip),

con-

vergence

may not be

possible

at all due to large values fo r

the interference factors,

a and a' , in Eq. (23). Since Fis zero

at the tip and

a

is not for a

square

tip

propeller,

th e value

for

a

is 1 and a ' is +1. S uch values are physically

impossible

since

the slipstream

factors

are approxim ately twice the values

at

th e

rotor

plane.

W ilson

an d Lissaman

10

suggest empirical

relations for resolving this

problem, whereas Viterna

an d

Janetzke

11

give empirical arguments fo r

clipping

the magni-

tude of a an d a ' at the value 0 .7 (a/F at the tip is finite at the

design

point

for an

optimum propeller).

F or

analysis,

the conventional thrust and

power coefficients

ar e

C

T

= TI(pn

2

D

4

)

C

p

=

P/(pn

3

D

5

)

Using

Eqs. (22) and (24), th e differential forms with respect

to f ar e given by

C

T

=


6/7

ADKINS

AND LIEBECK: DESIGN OF OPTIMUM

PROPELLERS 68 1

Tab le 1

Propeller

design

solution

5

8958

1 2917

1 6875

2 833

2 4792

2 875

3424

46 5

4269

3569

2796

1913

58 3125

41 8645

32 2669

22 2978

18 7971

15 9619

13 8552

54 8118

38 3637

28 7661

22 7927

18 7971

15 9619

13 3552

4449

81 4

9834

1 295

974

783

348

644

8 4

89

938

968

a

633

365

219

142

98

72

Input : brake horsepower = 70, 2

blades;

hub diam = 1 ft, tip

diam

=

5.75

f t ; b l ade sect io n : N A CA

4415, C,

=

0.7,

velocity

= 110

mph, rpm

=

24001.

O utput: thrust = 207.61 I b, 77 = 0.86996.

N o t e : a an d

a'

have been se t eq u al to

zero

at the tip.

T a b le 2

Propeller

analysis solution

5

8958

1 2917

1 6875

2 833

2 4792

2 875

54 8116

38 3638

28 7661

22 7927

18 7971

15 9619

12 5862

c

7

7

7

7

7

7

7

4449

81 4

9834

1 295

974

783

348

644

8 4

89

938

968

a

633

365

219

142

98

72

Input : propeller geometry

from Table

1;

r, C, and /3 at the

same radial locations; velocity =110 m p h ,

rpm = 2400.

O u t pu t :

brake horsepower

= 70, thrust =

207.61 Ib ,

17 =

0.86996.

DISC PLANE

Fig. 6

Force

coefficients for w indmill bla de element.

0.08

0 06

0.04

0 02

0.8

0.6

0.30

0 40 0 50 0 60 0 70 0 80 0 90

Fig. 7 Example of

propeller

performance.

1 00

require an additional layer of

iteration

to

achieve

a specified

design

thrust or power. I n

light

of the favorable agreement

between

the present theory and the exp erimental results

given

later in

this

article, it is argued that such an increase in com-

plexity

is not

justified.

Windmills

A ll

of the analyses described in this

article

ar e

directly

1.6

1.4

1 .2

1 .0

I

0 8

0 6

0 4

0 2

7 DESIGN

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

r

/

R

T.P

Fig.

8 C

distributions

for

example propeller.

0 20

C P C T

0.15

0.05

REF:

NACA

TN 1834,

PROP

MODEL 5

1.0

0.8

Fig. 9 Comparison of theory and experiment .

geometry for a windmill is shown in Fig. 6 , w h e r e the primary

distinction is that the blade section is inverted (a s compared

with a propeller), and the local angle of attack is measured

from

below

the local velocity

vector. Corresponding

relations

fo r

th e

angles

ar e


7/7

68 2

ADKINS AND

LIEBECK:

DESIGN OF OPTIMUM

PROPELLERS

0. 3

0.1

J = 0.914

THEORY WITH PRANDTL F

- — — THEORY WITH GOLDSTEIN K

0 NACATEST

REF: NACA

TN

1834,

PROP MODEL 5

0 0.2 0.4 0.6 0.8 1.0

r/R np

Fig.

10

Comparison

of

propeller analyses thrust coefficient.

as shown in

Figs.

6 and 1,

respectively.

I n

these figures, C,

fo r

th e windmill is negative

with

respect to that for the pro-

peller, an d this

sign

change

together

with

the angle definition

will

convert

t he propeller metho ds to the windmill application.

For the design

case,

th e

input P

c

value should be negative,

and the resulting values of v ' (and th e interference factors a

an d

a ')

an d T

c

will

also be negative. (Thrust is of

less interest

fo r

a windmill since it typically represents th e tower

load

an d

is not a main performance parameter.) Similarly, the analysis

results for a

windmill

rotor

will

yield

negative

values fo r both

P

c

an d T

c

.

Examples

A s a sample calculation, the design of a propeller for a light

airplane is

considered.

T he

design conditions

and the

resulting

design are described in

Table

1 which

gives

fo r each radial

station: blade chord, blade pitch angle, local

flow

angle, local

R eynolds

n u m b e r ,

and the interference coefficients a an d a'.

This

propeller geometry has,

in

turn, been analyzed

at the

design con dition

and the

result

is

given

in

Table

2 .

A g r ee m e n t

is virtually

exact.

A nalysis over a

range

of

values

of the ad-

vance ratio, J = V/(nD),

provides

the typical propeller per-

formance

plots

which

are shown in

Fig.

7, and Fig. 8 gives

the blade lift coefficient distribution over a range of /s where

the design condition is the C, — 0.7 and is a

constant

line at

J

=

0.7.

A calibration of the method is given by

comparing

its

the-

oretical prediction with experimental results. Reid

12

has eval-

uated several

conventional propellers

extensively by experi-

ment, and one of

these

ha s

been

chosen fo r

comparison.

Figure 9 gives C

p

,

C

T

, a nd

77

vs

/

fo r

both Reid's

experiments

an d

the

corresponding

theoretical

prediction. The

agreement

here is quite good, with

most

of the disparity occurring after

the

blade

is

stalled. This

propeller uses

N A C A

16-series

air-

foils, and no poststall data were

available.

Figure 10 gives the comparison of the blade

thrust

coeffi-

cient distribution

as

measured

by

R e i d

an d

calculated

by the

method.

Tw o

theoretical

results are shown: one

using F ,

an d

th e

other using

th e

more complex

(from

a

calculation point

of view) K .

I n

principle,

th e

accuracy

of the

method should

be better with th e Goldstein factor for a propeller with few

blades—this

example had three blades—and the two

factors

should give similar results as the n u m b e r of

blades

is in-

creased.

The results of Fig. 10 confirm

this

t r en d , and the

overall

comparison for

both

factors is regarded as quite good.

Conclusions

and

Recommendations

The propeller theory of Glauert has been extended to im-

prove the design of optimal propellers an d

refine

the calcu-

lation of the performance of arbitrary propellers.

Extensions

of

th e theory include 1) elimination of the small angle as-

sumptions in the optimal design theory; 2)

accurate calcula-

tion of the vortex displacement velocity which properly ac -

counts

for the blade

section drag;

and 3)

elimination

of the

small

angle assumptions in the P r a n d t l m o m e n t u m l o s s

func-

tion for

both

design and analysis. These

extensions

bring th e

design

an d

analysis procedures

to exact numerical agreement

within th e precision of

c o m p u t e r

analysis.

The pr imary approximation remaining in both

procedures

is the use of the axial m o m e n t u m equations which require th e

increase in w a k e velocities to be twice those at the

disc.

Under

certain

conditions

this approximation is not good an d gives

rise to the u n n a t u r a l

conditions

and

convergence

problems

described in the

analysis

section. Improvements might be made

by

replacing th e

axial m o m e n t u m equations with

relations

more closely aligned with th e general theory, particularly in

those differential

stream

tubes in

which

heavy

loading

ex -

ists. Such conditions appear to be more prevalent in the anal-

yses

at off-design

conditions

than in the

design

itself,

an d ,

when combined with poststall misknowledge, can lead to large

errors

in

analysis. However,

fo r design and analysis

within

the conventional operating regime, both procedures ar e sim-

ple,

accurate, and

reliable. This

method has

been

extended

by

P ag e

an d L ieb eck

13

to the design and analysis of dual-

rotation propellers.

A favorable

comparison between theory

an d experiment w as also observed.

References

l

G la u e rt ,

H.,

Airplane

Propellers,

Aerodynamic

Theory, edited

by W . D u r an d ,

D i v.

L ,

Vol.

5 , P e t e r Smith , Gloucester, M A ,

1976,

pp . 169-269.

2

Betz, A., with appendix by Pra n d t l ,

L.,

Screw

Propellers

with

M i n im u m E n e rg y Loss, Gottingen Reports, 1919, pp . 193-213.

3

Theodorsen, T ., Theory of Propel lers, M c G ra w -H il l, N ew York,

1948.

4

Lar r abee, E., Practical D esign of M in imu m Induced Loss Pro-

pellers,"

Society

of

A utomotive En gineers, Business

A ircraft

M e et -

ing

an d

Exposition,

W ic h ita , K S, A pril 1979.

5

Goldstein,

S.,

"O n the Vortex Theory of Screw

Propellers,

Pro-

ceedings of the Royal Society of London, Series

A,

Vol. 123, 1929,

pp .

440-465.

6

L o c k ,

C.,

The

A pplication

of Goldstein's A irscrew Theory to

D es ign, " British A er onautical

R esearch

Com m ittee , RM

1377, N o v.

1930.

7

L o c k , C.,

"An A pplication of P randtl 's Theory to an A irscrew,"

British

Aeronautical R es ear ch

Com m ittee , RM

1521,

A u g.

1932.

8

L o c k ,

C., Tables

for U se in an

E m pirical M ethod

of

A irscrew

Strip Theory C alculations,"

British A er onautical

R es ear ch

Commit-

tee, RM 1674, Oct.

1934.

9

Larrabee,

E.,

The Screw Propeller, Scientific American,

Vol.

243, No. 1, 1980, pp. 134-148.

10

W ilson,

R., and Lissaman, P., Applied A e ro d yn a mic s of

W i n d

P o wer M achines ," O r egon

S tate

U n i v ., N S F / R A / N - 7 4 - 1 1 3 ,

P B-

2318595/3,

Corvallis,

O R , July 1974.

H

Viterna, A., and J anetz ke, D., Theoretical an d

Experimental

Power from L arge Horizontal-A xis Wi n d

Turbines,

Proceedings from

the Large Horizontal-Axis Wind Turbine Conference, DOE/NASA-

L e R C ,

July 1981.

12

R e id , E. G., The Influence of Blade-W idth Distribution on

Propeller Char acter is t ics , " N A CA TN 1834,

M a rc h 1949.

13

Page, G . S., and Liebeck, R.

H.,

Analysis of D ual-R otation

Propellers,"

AIAA P a p e r 89-2216, A u g . 1989.

design of optimum propeller

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