Optimal Diameter B-Series Propellers

br)

OPTIMAL DIAMETER

B-SERIES PROPELLERS

o

Lb.

y.

No'24j,.

h

.

I

r'

w'JJ

August 1*82

Ddft

M.M. Bernitsas

D. Ray

Ri1

_TE

AND

MARINE ENGINEERING

THE UNIVERSITY OF MICHIGAN

COLLEGE OF ENGINEERING

OPTIMAL DIAMETER B-SERIES PROPELLERS

DATUM'

by

M.M. Bernitsas

D. Ray

Bib!iotheek van de

fdeig

ScOW!

e

TehVe

4oc

De t

No. 245

August 1982

Department of Naval Architecture

and Marine

qineerinq

College of Engineering

The University of Michigan

Ann Arbor, Michigan

48109

ABSTRACT

This work is a continuation of the study on optimal revolution propellers

carried out in report 244 [3].

_{Topics and relations presented in [3] will not}

be repeated here.

_{The reader is expected to be familiar with the preliminary}

propeller design problem studied in reference [31.

The approach used in this paper is based on systematic series and in

par-ticular the Wageningen B-Series propellers [2,6].

_{In the B-Series the}

open-water propeller characteristics are expressed in terms of multiple regression

polynomials of advance coefficient, number of blades, blade area ratio and

pitch-diameter ratio.

Corrections to account for Reynolds number and blade

thickness effects are also given for the B-Series propellers in reference [61.

In this report one of the preliminary propeller design problems is

stud-ied (7], namely, the problem of calculation of the maximum efficiency

propel-1er and its operating condition for given RPM and hull speed.

_{This problem is}

studied in two ways by considering

(1) the hull-propeller system and

(2)

the machine-propeller system.

It is also shown that the two optimal systems

are equivalent if certain compatibility equation is satisfied.

The optimal diameter propeller characteristics, for the hull-propeller

and machine-propeller systems, are plotted for the entire range of validity of

the B-Series regression polynomials, the entire range of practical interest,

and for

Re = 2 x io6

The preliminary design propeller optimization problems analyzed in this

paper and in reference [3] are studied in NA474, a senior and first

year

graduate course on "Optimization and Numerical Methods in Marine Design."

ACKNOWLEDGEMENTS

This report was prepared in fulfillment of the requirements for the

design projects of Mr. Debasish Ray and Mr. Davinder Sood for the qraduate

course NA574 on "Computer-Aided Ship Desiqn" offered by Professor Michael M.

Bernitsas of the Department of Naval Architecture and Marine Engineering at

the University of Michigan in Winter 1981 and Winter 1982.

Computer funds were provided by the Department of Naval Architecture and

Marine Engineering.

Thanks are due to Mrs. Paula Bousley for the excellent

typing and editing of this report.

-V-TABLE OF CONTENTS

page

ABSTRACT

iii

ACKNOWLEDGEMENTS

y

LIST OF FIGURES

_ix

LIST OF TABLES'

_xiii

NOMENCLATURE

_XVii

INTRODUCTION

ND OUTLINE

HULL-PROPELLER SYSTEM OPTIMIZATION

5

1.1.

Example

_S

1.2.

Problem Formulation

₈

1.3.

General Solution

10

1.4.

_{Optimal Diameter Propellers for Re = 2 x 106}

₁₄

MACHINE-PROPELLER SYSTEM OPTIMIZATION

64

11.1.

Example

₆₄

11.2.

Problem Formulation

65

11.3.

General Solution

70

11.4.

Optimal Diameter Propellers for Re = 2 x 106

71

RELATION BETWEEN HULL-PROPELLER AND MACHINE-PROPELLER SYSTEMS..

122

III 1.

Example

_{1 22}

111.2.

General Relation of the Optimal Systems

123

CONCLUSIONS AND FURTHER WORK

125

REFERENCES

127

-Vii-1.

Example of Optimal Revolution Propeller:

Hull-Propeller

System

2.

Hull-Propeller System:

Optimal

2 Blades and

E/o

0.30, 0. 3.

Hull-Propeller System:

Optimal

2 Blades and

_AE/AO

= 0.35, 0.

4.

Hull-Propeller System:

Optimal

2 Blades and

_AE/AO

= 0.40, 0.

5.

Hull-Propeller System:

Optimal

2 Blades and

_RE/AO

0.45, 0.

6.

Hull-Propeller System:

Optimal

3 Blades and

AE/?.o = 0.30, 0.

7.

Hull-Propeller System:

Optimal

3 Blades and

_AE/AO

= 0.35, 0.

8.

Hull-Propeller System:

Optimal

3 Blades arid

_7EIAo

0.40, 0. 9.

Hull-Propeller System:

Optimal

3 Blades and

_AE/Pio

0.45, 0.

10.

Hull-Propeller System:

Optimal

4 Blades and

AE/AO 0.30, 0. 11.

Hull-Propeller System:

Optimal

4 Blades and PF/AO = 0.35, 0.

12.

Hull-Propeller System:

Optimal

4 Blades and

_E/Ao

0.40, 0.

13.

Hull-Propeller System:

Optimal

4 Blades and

E/Ao =

0.45, 0.

14.

Hull-Propeller System:

Optimal

5 Blades and

RE/Ao =

0.30, 0.

15.

Hull-Propeller System:

Optimal

5 Blades and

AE/

=

0.35, 0.

16.

Hull-Propeller System:

Optimal

5 Blades and ¡/k = 0.40, 0.

17.

Hull-Propeller System:

Optimal

5 Blades and

AE/Ao =

0.45, 0.

Hull-Propeller System:

Optimal

6 Blades and

/o = 0.30, 0.

Hull-Propeller System:

Optimal

6 Blades and

AE/AQ

= 0.35, 0.

Hull-Propeller System:

Optimal

6 Blades and

AE/P.O

= 0.40, 0.

LIST OF FIGURES

Diameter

50, 0.70,

Diameter

55, 0.75,

Diameter

60, 0.80,

Diameter

65, 0.85,

Diameter

50, 0.70,

Diameter

55, 0.75,

Diameter

60, 0.80,

Diameter

65, 0.85,

Diameter

50, 0.70,

Diameter

55, 0.75,

Diameter

60, 0.80,

Diameter

65, 0.85,

Diame ter 50, 0.70,

Diameter

55, 0.75, Diame ter

60, 0.80,

Diameter

65, 0.85,

Diameter

50, 0.70,

Diameter

55, 0.75,

Diameter

60, 0.80,

page

6

Propeller with

0.90

16

Propeller with

0.95 18

Propeller with

1.00 20

Propeller with

1.05 22

Propeller with

0.90 24

Propeller with

0.95 26

Propeller with

1.00 28

Propeller with

1.05 30

Propeller with

0.90

32

Propeller with

0.95 34

Propeller with

1.00 36

Propeller with

1.05 38

Propeller with

0.90 40

Propeller with

0.95 42

Propeller with

1.00 44

Propeller with

1.05 46

Propeller with

0.90 48

Propeller with

0.95 50

Propeller with

1.00 52

21.

Hull-Propeller System:

Optimal Diameter Propeller with

6 Blades and PIE/Ao = 0.45, 0.65, 0.85, 1.05

54

Hull-Propeller System:

7 Blades and AE/A, =

Hull-Propeller System:

7 Blades and AE/AO

=

Hull-Propeller System:

7 Blades and AE/AQ

=

Hull-Propeller System:

7 Blades and AE/AO

₌

Machine-Propeller System:

with 2 Blades and AE/A,

Machine-Propeller System:

with 2 Blades and AE/A

Machine-Propeller System:

with 2 Blades and AE/AO

Machine-Propeller System:

with 2 Blades and

_E/

Machine-Propeller System:

with 3 Blades and AE/.O

Machine-Propeller System:

with 3 Blades and AE/O

Machine-Propeller System:

with 3 Blades and AE/AO

Machine-Propeller System:

with 3 Blades and AE/O

Machine-Propeller System:

with 4 Blades and AE/AO

Machine-Propeller System:

with 4 Blades and AE/AO

Machine-Propeller System:

with 4 Blades and AE/AO

Machine-Propeller System:

with 4 Blades and AE/AO

Optimal Diameter

0.30, 0.50, 0.70,

Optimal Diameter

0.35, 0.55, 0.75,

Optimal Diameter

0.40, 0.60, 0.80,

Optimal Diameter

0.45, 0.65, 0.85,

-X--Propeller with

0.90

Propeller with

0.95

Propeller with

i 00

Propeller with

05

page

56 58 60 62

Optimal Diameter Propeller

= 0.30, 0.50, 0.70, 0.90

74

Optimal Diameter Propeller

= 0.35, 0.55, 0.75, 0.95

76

Optimal Diameter Propeller

= 0.40, 0.60, 0.80, 1.00

78

Optimal Diameter Propeller

= 0.45, 0.65, 0.85, 1.05

80

Optimal Diameter Propeller

= 0.30, 0.50, 0.70, 0.90

82

Optimal Diameter Propeller

= 0.35, 0.55, 0.75, 0.95

84

Optimal Diameter Propeller

= 0.40, 0.60, 0.80, 1.00

86

Optimal Diameter Propeller

= 0.45, 0.65, 0.85, 1.05

88

Optimal Diameter Propeller

= 0.30, 0.50, 0.70, 0.90

90

Optimal Diameter Propeller

= 0.35, 0.55, 0.75, 0.95

92

Optimal Diameter Propeller

= 0.40, 0.60, 0.80, 1.00

94

Optimal Diameter Propeller

= 0.45, 0.65, 0.85, 1.05

96

39.

Machine-Propeller System:

Optimal Diameter Propeller

with 5 Blades and AE/A, = 0.30, 0.50, 0.70, 0.90

98

26.

Example of Optimal Diameter Propeller:

Machine-Propeller

Machine-Propeller System:

with 5 Blades and

Machine-Propeller System:

with 5 Blades and AE/AO

Machine-Propeller System:

with 5 Blades and AE/o

Machine-Propeller System:

with 6 Blades and AE/AO

Machine-Propeller System:

with 6 Blades and AF/O

Machine-Propeller System:

with 6 Blades and AE/AO

Machine-Propeller System:

with 6 Blades and AE/

Machine-Propeller System:

with 7 Blades and AE/

Machine-Propeller System:

with 7 Blades and AF/T

Machine-Propeller System:

with 7 Blades and AE/

Machine-Propeller System:

with 7 Blades and

paqe

Optimal Diameter Propeller

= 0.35, 0.55, 0.75, 0.95

100

Optimal Diameter Propeller

= 0.40, 0.60, 0.80, 1.00

102

ptimal Diameter Propeller

= 0.45, 0.65, 0.85, 1.05

104

Optimal Diameter Propeller

= 0.30, 0.50, 0.70, 0.90

106

Optimal Diameter Propeller

= 0.35, 0.55, 0.75, 0.95

108

Optimal Diameter Propeller

= 0.40, 0.60, 0.80, 1.00

110

Optimal Diameter Propeller

= 0.45, 0.65, 0.85, 1.05

112

Optimal Diameter Propeller

= 0.30, 0.50, 0.70, 0.90

114

Optimal Diameter Propeller

= 0.35, 0.55, 0.75, 0.95

116

Optimal Diameter Propeller

= 0.40, 0.60, 0.80, 1.00

118

Optimal Diameter Propeller

LIST OF TABLES

paqe

Hull-Propeller System:

Optimal Diameter Propeller

with 2 Blades and AE/AO = 0.30, 0.50, 0.70, 0.90

17

Hull-Propeller System:

Optimal Diameter Propeller

with 2 Blades and AE/AO = 0.35, 0.55, 0.75, 0.95

19

Hull-Propeller System:

Optimal Diameter Propeller

with 2 Blades and AE/Ao = 0.40, 0.60, 0.80, 1.00

21

Hull-Propeller System:

Optimal Diameter Propeller

with 2 Blades and AE/AO = 0.45, 0.65, 0.85, 1.05

23

Hull-Propeller System:

Optimal Diameter Propeller

with 3 Blades and AE/AO = 0.30, 0.50, 0.70, 0.90

25

Hull-Propeller System:

Optimal Diameter Propeller

with 3 Blades and AE/AO = 0.35, 0.55, 0.75, 0.95

27

Hull-Propeller System:

Optimal Diameter Propeller

with 3 Blades and AE/AO = 0.40, 0.60, 0.80, 1.00

29

Hull-Propeller System:

Optimal Diameter Propeller

with 3 Blades and AE/AO = 0.45, 0.65, 0.85, 1.05

31

Hull-Propeller System:

Optimal Diameter Propeller

with 4 Blades and AE/AO = 0.30, 0.50, 0.70, 0.90

33

Hull-Propeller System:

Optimal Diameter Propeller

with 4 Blades and AE/AO = 0.35, 0.55, 0.75, 0.95

35

Hull-Propeller System:

Optimal Diameter Propeller

with 4 Blades and AE/AO = 0.40, 0.60, 0.80, 1.00

37

Hull-Propeller System:

Optimal Diameter Propeller

with 4 Blades and AB/AO = 0.45, 0.65, 0.85, 1.05

39

Hull-Propeller System:

Optimal Diameter Propeller

with 5 Blades and AE/AO = 0.30, 0.50, 0.70, 0.90

41

Hull-Propeller System:

Optimal Diameter Propeller

with 5 Blades and Ap/A0

0.35, 0.55, 0.75, 0.95

43

Hull-Propeller System:

Optimal Diameter Propeller

with 5 Blades and AE/AO = 0.40, 0.60, 0.80, 1.00

45

Hull-Propeller System:

Optimal Diameter Propeller

with 5 Blades and AE/AO = 0.45, 0.65, 0.85, 1.05

47

Hull-Propeller System:

Optimal Diameter Propeller

with 6 Blades and AE/AO = 0.30, 0.50, 0.70, 0.90

49

Hull-Propeller System:

Optimal Diameter Propeller

with 6 Blades and Ap/A0 = 0.35, 0.55, 0.75, 0.95

51

Hull-Propeller System:

Optimal Diameter Propeller

with 6 Blades and Ap/A0 = 0.40, 0.60, 0.80, 1.00

53

Hull-Propeller System:

Optimal Diameter Propeller

Machine

with

Machine

with

Machine

with

Machine

with

Machine

with

Machine

with

Machine

with

Machine

with

Machine

with

Machine

with

Machine

with

Machine

with

Machine

with

Machine

with

Machine

with

Machine

with

Propeller System:

2 Blades and

E/o

Propeller System:

2 Blades and

_RE/AO

Propeller System:

2 Blades and

_AE/AO

Propeller System:

2 Blades and

_E/AO

Propeller System:

3 Blades and

Propeller System:

3 Blades arid _AE/AO

Propeller System:

3 Blades and

_AE/PO

Propeller System:

3 Blades and

_AF/AO

Propeller System:

4 Blades and

_AE/AO

Propeller System:

4 Blades and

_AE/A0

Propeller System:

4 Blades and

_AE/lO

Propeller System:

4 Blades and

_AE/AO

Propeller System:

5 Blades and

_AF/AO

Propeller System:

5 Blades and AE/A

Propeller System:

5 Blades and

_AE/lO

Propeller System:

5 Blades and

_AE/Po

paqe

Hull-Propeller System:

Optimal Diameter Propeller

with 7 Blades and

_AE/

=

0.30, 0.50, 0.70, 0.90

57

Hull-Propeller System:

Optimal Diameter Propeller

with 7 Blades and

_E/

=

0.35, 0.55, 0.75, 0.95

59

Hull-Propeller System:

Optimal Diameter Propeller

with 7 Blades and

_PE/AO

= 0.40, 0.60, 0.80, 1.00

61

Hull-Propeller System:

Optimal Diameter Propeller

with 7 Blades and

Ap/A0

= 0.45, 0.65, 0.85, 1.05

63

Optimal Diameter Propeller

= 0.30, 0.50, 0.70, 0.90

75

Optimal Diameter Propeller

= 0.35, 0.55, 0.75, 0.95

77

Optimal Diameter Propeller

= 0.40, 0.60, 0.80, 1.00

79

Optimal Diameter Propeller

= 0.45, 0.65, 0.85, 1.05

81

Optimal Diameter Propeller

= 0.30, 0.50, 0.70, 0.90

83

Optimal Diameter Propeller

= 0.35, 0.55, 0.75, 0.95

85

Optimal Diameter Propeller

= 0.40, 0.60, 0.80, 1.00

87

Optimal Diameter Propeller

= 0.45, 0.65, 0.85, 1.05

89

Optimal Diameter Propeller

= 0.30, 0.50, 0.70, 0.90

91

Optimal Diameter Propeller

= 0.35, 0.55, 0.75, 0.95

93

Optimal Diameter Propeller

= 0.40, 0.60, 0.80, 1.00

95

Optimal Diameter Propeller

= 0.45, 0.65, 0.85, 1.05

97

Optimal Diameter Propeller

= 0.30, 0.50, 0.70, 0.90

99

Optimal Diameter Propeller

= 0.35, 0.55, 0.75, 0.95

101

Optimal Diameter Propeller

= 0.40, 0.60, 0.80, 1.00

103

Optimal Diameter Propeller

Machine

with

Machine

with

Machine

with

Machine

with

Machine

with

Machine

with

Machine

with

Machine

with

Propeller System:

6

Blades and AE/Ao

Propeller System:

6 Blades and AE/.Q

Propeller System:

6

Blades and

_E/To

Propeller System:

6

Blades and AE/A

Propeller System:

7 Blades and AE/PQ

Propeller System:

7 Blades and AE/Ao

Propeller System:

7 Blades and AE/A0

Propeller System:

7 Blades and AE/AO

Optimal Diameter Propeller

= 0.30,

0.50,

0.70, 0.90

107

Optimal Diameter Propeller

= 0.35,

0.55, 0.75, 0.95

109

Optimal Diameter Propeller

= 0.40, 0.60, 0.80, 1.00

111

Optimal Diameter Propeller

= 0.45, 0.65, 0.85, 1.05

113

Optimal Diameter Propeller

= 0.30,

0.50, 0.70,

0.90

115

Optimal Diameter Propeller

= 0.35, 0.55, 0.75,

0.95

117

Optimal Diameter Propeller

= 0.40, 0.60, 0.80, 1.00

119

Optimal Diameter Propeller

= 0.45, 0.65, 0.85, 1.05

121

NOMENCLATURE

expanded blade area ratio

Cj

constraint number i

CQ

constant in machine-propeller system

CT

constant in hull-propeller system

D

propeller diameter

DHP

delivered horse power

EHP

effective horse power

J

advance coefficient

KQ

torque coefficient

KT

thrust coefficient

n

propeller revolutions per second

P/D

pitch-diameter ratio

Q

propeller torque in open water

QB

propeller torque behind hull

Re

Reynolds number

R

constraint number i

RPM

propeller revolutions per minute

total towing hull resistance

t

thrust deduction fraction

T

propeller thrust

t/c

thickness to chord ratio for propeller blades

V

ship speed

VA

speed of advance

w

Taylor wake fraction

Greek Sytnbols

propeller efficiency behind hull

rID

propulsive efficiency

hull efficiency

no

open-water propeller efficiency

relative rotative efficiency

A

Lagrange multiplier

INTRODUCTION PND OUTLINE

Preliminary propeller design problems are described in detail in

Princi-ples of Naval Architecture [4].

One of these problems, namely the

identifica-tion of the maximum efficiency propeller and its operating condiidentifica-tion for qiven

propeller RPM and hull speed, is studied in this paper.

The approach used is

based on systematic series and in particular the B-Series.

The properties and

limitations of these series are described in references [2],

[3] and [4].

In preliminary propeller design certain basic propeller characteristics

must be selected initially.

For instance, number of propeller blades,

expand-ed area ratio, diameter, RPM and blade thickness must be selectexpand-ed on the basis

of propeller strength, cavitation and vibration analyses.

This procedure is

briefly explained in references [3] and [4].

Once the basic propeller characteristics are selected the optimization

problem described above can be formulated and solved.

The propeller can be

considered as part of the hull-propeller or the machine-propeller system.

Either system can be optimized to yield the maximum efficiency propeller and

its operating condition.

The hull-propeller system optimization problem is formulated and solved

in section I.

_{Data for optimal diameter propellers are derived and plotted}

for the complete range of practical interest.

The machine-propeller

optimiza-tion problem is formulated and solved in secoptimiza-tion II and a similar set of

re-sults and graphs are produced.

Obviously both systems are part of the

hull-machine-propeller system and the optimization results must be related.

In

section III it is shown that if the two systems are compatible the

optimiza-tion results in secoptimiza-tion I and II are identical and the two systems

are

-1-and

p

AE

t

-, - , Z,Re ,- )

D A0 c

where

KT

is the thrust coefficient

T

KT

-pn2D

(4)

KQ

is the torque coefficient

KQ =

Q

pn2D5

-2-lent.

Finally, recommendations for extension of this work and generalization

of the propeller optimization problem are given.

The hull-machine-propeller system equations are listed and discussed in

reference (3].

Here we list only the basic equations that will be used in the

formulation and solution of the optimization problems in sections I and II.

The open-water propeller characteristics in B-Series are qiven in the

following form:

p

AE

t

KT=KT(J

, -

, - ,

Z, Re, -

) D

A0

c p

AE

t

KQ = KQ(J ,

-

, - ,

Z , Re , - ) D c (5)

T

is the open-water propeller efficiency

J

KT

, (6)

2ii KQ

T

is the propeller thrust,

Q

is the propeller torque,

-3-n

is the number of propeller revolutions per second,

D

is the propeller diameter,

J

is the advance coefficient.

VA

J=-

(7)

DHP = 2-irnQB , (11)

nD

VA

is the speed of advance,

P/D

is the pitch-diameter ratio,

AE/AO

is the blade area ratio,

Z

_{is the number of propeller blades,}

Re

is the Reynolds number at a characteristic radius (O.75D/2), and

t/c

is the ratio of the maximum propeller blade thickness to the

length of the chord at a characteristic radius (O.75D/2).

The hull, machine and propeller characteristics are related throuqh

T

Q

and

VA with the following formulas.

The effective horsepower,

El-lP , is

EHP=RTV

,

(8)

where

RT

is the total towing hull resistance at constant speed

V and is

given by equation (9)

RT = (1-t)T

, ₍₉₎

where

t

is the thrust deduction factor.

The speed of advance,

VA , is

VA = V(1-w)

(10)

where

w

is the Taylor wake fraction.

The hull efficiency,

RTV

1-t

11H = =

-TVA

1-w

11H

is

-4-where

_QB

is the propeller torque behind the ship.

Using equations (1) to (11) we can define the following efficiencies:

The open-water propeller efficiency,

r , is

TVA

J

KT

no =

2TrnQ 2g

(12)

The propeller efficiency behind the hull,

_11B is

11g

-2lrnQB

(13)

The ratio of the efficiency behind the hull to that in open water is

called relative rotative efficiency,

r ,

and is given by equation (19)

hg Q

11R . (14)

fl QB

(15)

Finally, we can define the propulsive efficiency,

_Tb as

EHP

Tb =

= 11H

11g = 11H

11R Tb (16)

To compute

fl ,

we must choose a propeller and find its operatinq

condi-tion.

The values of

_11H

and

î

depend on the particulars of the

hull-machine system and are given in PNA t41.

The above relations show that

can be maximized by considering the

propeller as part of the hull-propeller system (see section I) or the

machine-propeller system (see section II).

I.

HULL-PROPELLER SYSTEM OPTIMIZATION

In preliminary propeller design we select the number of propeller blades,

the blade area ratio

AE/AO

and

t/c

using the methods outlined in [31.

Then we can find the maximum efficiency propeller for given RPM, hull speed

and either hull or machine relevant data.

This implies computation of the

propeller diameter,

D ,

pitch to diameter ratio,

P/D

,

and the operating

condition of the propeller.

The procedure used in this section to optimize the hull-propeller system

follows the practice in reference (3) and will not be described in detail.

-5-I 1.

Example

Consider a standard Series-60 hull with the following particulars [3,4).

CB = 0.65

(I-1 V

- = 0.8

(I-2) L

- = 7.25

_(I-3) B B

- = 2.50

_(I-4) T

w = 0.252

_(I-5)

t = 0.155

(I-6)

= 1.018

and

(I-7)

L = 400 ft

_(I-S)

For the above ship we get

=

c >-co

()d

z

w

L) u-Li

LUd

f-u-

-Wt=

w

L)

;.00

FIGURE 1.

EXAMPLE OF OPTIMAL-DIAMETER PROPELLER:

HULL-PROPELLER SYSTEM.

WACENINCEN B-SERIES PROPELLERS

FOR 5 BLADES

AE/AO= 0.650

P10=0.50 TO 1.40

C) C) C) I.)

4

'I

070

LI

II74tA 11

0.20

0.40 0.5

0.60

° 0.80 0.91.00 1.0 1.1120 1.2 1.40 1.4

ADVANCE COEFF(J)

(D

d

D

LL u-LU

D

L)

w

coD

W

D

F-C)

and

RT = 61,900 lbs

where the total towing resistance has been computed using Series-60 data, the

ATTC line and a allowance coefficient of 0.0004.

Further we assume that the

ship is propelled by a single screw with blade area ratio

AE/Ao = 0.65

, (I-11)

operating at

RPM = 77

. (I-12)

The above data were selected in such a way that this example be

compat-ible to those in reference [31.

For these data we have

T

RT

and

-7-KT

Rpn2

- =

_{- .361 (I-15)} J&+

(1-t)(1-w)V

Equation (I-15) is plotted in Figure 1.

For each value of

P/D ,

that is

for each of the ten propellers whose characteristics are plotted in Figure 1,

the operating point is found at the cross-section of equation (I-15) and the

KT

versus

J

curve.

The propeller efficiency is then found and the

Tb

curve is plotted revealing a maximum efficiency propeller with the followinq

particulars

J = 0.85

, (I-16)

P/D = 1.10

, (I-17) and rk

_{= 0.70}

. (I-18)

VA

_V(1-w)

= _(I-14)

nD

nD

KT =

pn2D

(1-t)pn2D4

(I-13)

-8-This point corresponds to one of the points in Figure

17 for

KT/J

given by (I-15), for

AE/Ao = 0.65

and Reynolds number equal to 2 x

106.

For

higher values of

Re

the propeller curves must be corrected as explained in

references [2,3].

In the following section, the problem solved in the above example is

for-mulated in a general mathematical form which is solved with the aid of a

digi-tal computer for the range of practical interest of the

Krr/J'

values.

The

results are plotted in Figures 2 to 25 and are presented in Tables

1 to 24.

1.2.

Problem Formulation

The problem solved in the previous example can be stated in general as

follows.

Find the maximum efficiency B-Series propeller for a sinqle screw

ship

given that

z = m

(I-19) = (T-20)

D =

(T-21)

EHP

(I-22)

y = y

(123)

w = w

(T-24)

t =

(T-25)

where

m , a , ô , c , y

are known constants and the values of

w

and

t

can be found from available graphs and data

[4].

This problem can be formulated in the following standard mathematical

optimization form.

Problem Pl

p

AE

t

maximize

r0 = -

- ,

Z , Re , - ) D

A0

C J

KT

211K9

subject to:

Z=m

AE

- = a

A0

n=RPM/60r

EHPRTVE

V=v

WW

tt

VA

3=-nD

Rg:

VA = V(1-w)

T

KT =

Q

1(9=

25

RT = T(1-t)

p

AE

t

KT =

-

- ,

Z , Re ,

- )

given by the B-Series

D A0 c

P

AE

t

K9 = KQ(J ,

-

- ,

Z , Re ,

- )

given by the B-Series

D A0 c

R15;R16:

2 < Z c ₇

R17;R18:

0.30 < -

1.05

A0

P

R19;R20:

0.50 < - < 1.40

D

This is a nonlinear programming problem with continuous and discrete

variables aiming at the maximization of

r

given by equation (I-26) subject

to 20 equality and inequality constraints.

The design variables are 18, namely,

Z , J ,

P/D

, n , D ,

AE/AO

Re , t/c , EHP _,

RT ,

V ,

w , t , VA ,

KT ,

KQ ,

T

and Q .

Note

that in this problem, relations between

Q

OB

and

DF-IP

are not required.

These quantities can be computed using equations (11), (14) and (16) in the

introductory section.

The above problem is reduced and solved in section 1.3.

1.3.

General Solution

-10-Several of the constraints in the optimization probleui are equality

con-straints and can be used to eliminate an equal number of design variables and

reduce the problem.

Following the practice in reference t3] we can make the

following observations:

a. R1

can be used to eliminate

Z .

This means that the optimization

problem should be solved only for a given blade number.

b.

R2

sets the value of the blade area ratio equal to

cx .

Consequently

-

11-Activity of R1 and R2 basically indicates that the optimization problem will

be solved for the propellers of one fiqure in reference [21 at a time as was

done in the example in section 1.1.

Constraint R3 can be used to eliminate variable

n

Constraint R4 can be used to eliminate variable

EHP

Constraint R5 can be used to eliminate variable

V

R6 defines

w

from available graphs in reference [4].

R7 defines

t

using data in reference [4].

Rg can be used to eliminate

_VA

R1

can be used to express

T

in terms of

_RT

and

t

which can be

eliminated from the problem using equality constraints R4, R5 and R7.

Equality constraint R10 can be used to express

T

in terms of

Kip

n

and

D

Equality constraint R11 can be used to express

Q

as a function of

K2 , n and D

Thus at the end of the first step of reduction of the optimization

prob-lem Pl the design variables are

J

,

P/D

,

D

,

Re

, t/c ,

KT

and

K2 and

the problem becomes:

Problem P2

JKT

maximize

ri0 - -

_(II-26)

2 wK2

subject to:

equality constraints R8, R13 and R14

and

_{inequality constraints R19 and R20.}

Obviously of the 7 design variables only 4 are independent due to the

three equality constraints, R8, R13 and R14.

Thus we can further reduce the

problem as follows:

a.

Choose the standard

tic

design value of the B-Series for

t/c

Should a different value of

t/c

he required by propeller blade

strength analysis, the factors defined by NSMB [61 must be used to

correct

no

KT

and K0 .

Thus t/c

can be defined and eliminated

f rom the problem.

The exact values of

t/c

can be computed once the

propeller has been selected, its optimal operatinq condition has been

found and the strength computations have been completed.

If the

dif-ferences are unacceptable the method recommended in reference [31

should be used to improve the results.

h.

Choose

J , P/D and Re

as the independent variables of the

problem making

n ,

KT

and

K0 dependent.

These can be defined by

equations R8, R12 and R13 respectively.

Thus the problem reduces to

P3.

C2:

Kç = K0(J ,

- ,

Re)

given by the B-Series

D

KT

EJ-IP

n2

3.

(1t)p(1w)V5

-

-12-given by the B-Series

p(1-t)(1-w) v5

= CT

derived from constraints R8 and R10 using R3, R4, R5, R6, R7, Rg and

R12 and where

CT

is a constant.

Problem P3

JKT

maximize

no -2 irK0

sublect to:

C1:

KT=KT(J,-,Re)

p

C4;C5: 0.50 - ' 1.40

Problem P3 will be solved usinq the method of Lagranqe multipliers and

rejecting any optimum which violates inequality constraints C4 and C5.

There-fore P3 can be written as:

Problem P4

p j

KT(J,P/D,Re)

maximize

F(J

, - , Re , A) =

+ A (Kr.(J,P/D,Re) - CTJ)

D

2î K0(J,P/D,Re)

(I-29)

subject to:

p 0.50 - 1.40 _(I-30) D

where

A

is a Laqranqe multiplier.

To find the stationary points of

F

we set ali first partial derivatives

equal to zero:

_J

3(P/D) 2ïr P'

_J

(Re) 2ir

K0

2

KT

K0 KT

-Re Re -

13-KT

K0

-F i

KT

J J J

=--+-J

2rK2

2ïr

_K02

KT

K2 - KT

(P/D) 3(P/D)

KT

f

A = O

aRe

- 4CTJ3) = O

(I-31) aKT i-A

-0

a(P/D) (I-32) (I-33)

-14-=

- CTJ4 = O

(I-34)

Equations (I-31) to (I-34) can be solved for

J ,

P/D

, Re and À . The

value of

X

can he eliminated from equations (I-31) and (I-32), and (I-32)

and (I-33) to give:

3KT

_/

Kç Kç

_/

- IJ-

-5K01

+

I4KTJ 1=0

a(P/D)

j

(P/D) a

j

(I-35)

K0

_KT

KQ

KT - (I-36) (P/D) Re aRe 3(P/D)

KT = CTJ4

(I-37)

Equations (I-35) to (I-37) can be solved for

J , P/D and Re . The

solution is a single stationary point which gives the maximum efficiency

propeller and its operating condition sub-sect to constraints R1 to R18.

In the next section a special case of the above qeneral problem, namely

for

Re = 2 x 106 ,

is solved.

1.4.

Optimal Diameter Propellers for Re = 2

X

106

The

KT

,

K0

and r

reqression polynomials, as qiven in reference

[6], are corrected for Reynolds number effects only for

Re > 2

X

106

. In

this section the optimization problem is solved for

Re = 2 x 106

for the

entire range of practical values of

CT

and for the ranqe of validity of the

B-Series.

For values of

Re > 2 x i06

the method recommended in [3] should be used

to correct the results of optimization.

in Tables

1 to 24.

The optimal diameter propellers as computed from the optimization of the

hull-propeller system are plotted in these figures for

0.13 ( CT

457

(I-38)

and the ranges specified by constraints R15 to R20; that is for all the

pro-pellers whose

KT ,

KQ

and r

curves are plotted in reference [2J.

For

low values of

_CT

and near the extremes of the ranges of

Z and

AE/AO

no

results are given.

This does not imply that there is no solution.

It means

that in problem P4 inequality (I-30) is violated by the optimum of the

uncon-strained problem defined by equation (I-29).

Actually there is no solution to

equation (I-35) for

P/D

in the range specified by (I-30).

That is the

Lagranqe multiplier method is not valid.

In all these cases the optimum is

constraint bound at

p

- = 1.40

(I-39)

D

and the problem becomes trivial.

-15-z

w

Û-D

FIGURE 2.

WAGENINGEN BSERIES PROPELLERS

CURVE FOR OPTIMUM DIRMETER PROPELLERS

FOR 2 BLROES

RE/RO = 0.30.0.50,0.70.0.90

D

Q-(N c -16- r-S. -S .5

"N

«11'

.11 "SS

1/3

PIT)

0.30

...

0.50

....

0. 70

0.90

b.60

LOO 1.40

1.80

2.20

26

(KT/J4)1/4

TABLE i

Kp 1/4

()

WAGENINGEN B-SERIES PROPELLER DATA FOR 2 BLADE OPTIMUM DIAMETER PROPELLERS

AE/AO = 0.30 P/D J ETA-O AE/AO = 0.50 P/D J ETA-O AE/AO = 0.70 P/D J ETA-O AE/AO = 0.90 P/D J ETA-O o 60 0.98906 0.88022 0.86227 1.06015 0.90212 0.73677 0.64 0.95547 0.83540 0.85351 1.00703 0.84780 0.73267 o 68 0.92500 0.79453 0.84329 0.96406 0.80137 0.72734 0.72 0.89766 0.75735 0.83206 0.92695 0.76023 0.72088 0.76 0.87187 0.72290 0.82015 0.89531 0.72378 0.71349 1.09297 0.80328 0.63189 0 .80 0.84844 0.69133 0.80780 0.86719 0.69077 0.70531 1.03125 0.75453 0.62329 0.84 0.82656 0.66211 0.79517 0.84219 0.66079 0.69652 0.98750 0.71539 0.61499 0.88 0.80703 0.63535 0.78240 0.81953 0.63332 0.68727 0.95234 0.68161 0.60670 0.92 0.78828 0.61033 0.76959 0.79961 0.60830 0.67766 0.92305 0.65180 0.59834 0.96 0.77187 0.58743 0.75679 0.78125 0.58513 0.66783 0.89726 0.62479 0.58989 .00 0.75625 0.56597 0.74407 0.76484 0.56379 0.65783 0.87500 0.60038 0.58137 .04 0.74141 0.54587 0.73150 0.74922 0.54377 0.64777 0.85469 0.57783 0.57280 .08 0.72812 0.52727 0.71908 0.73516 0.52524 0.63769 0.83672 0.55716 0.56422 1.06797 0.63208 0.53236 1.12 0.71562 0.50983 0.70686 0.72226 0.50797 0.62763 0.82031 0.53799 0.55566 1.02344 0.60265 0.52270 1. 16₂₀ 0.70352 0.49333 0.69486 0.71016 0.49174 0.61764 0.80547 0.52018 0.54714 0.99219 0.57840 0.51352 0.69219 0.47786 0.68308 0.69922 0.47661 0.60776 0.79180 0.50355 0.53870 0.96641 0.55686 0.50469 I . 24₂₈ 0.68203 0.46343 0.67154 0.68867 0.46228 0.59800 0.77930 0.48802 0.53036 0.94492 0.53749 0.49015 0.67266 0.44988 0.66025 0.67891 0.44880 0.58840 0.76758 0.47340 0.52211 0.92578 0.51964 0.48787 .32 0.66328 0.43693 0.64921 0.66992 0.43613 0.57895 0.75664 0.45963 0.51400 0.90898 0.50320 0.47981 36 0.65469 0.42474 0.63841 0.66133 0.42410 0.56967 0.74648 0.44665 0.50601 0.89375 0.48789 0.47198 .40 0.64687 0.41329 0.62788 0.65351 0.41279 0.56058 0.73711 0.43444 0.49818 0.88008 0.47363 0.46435 .44₄₈ 0.63906 0.40233 0.61761 0.64609 0.40204 0.55166 0.72851 0.42294 0.49050 0.86719 0.46016 0.45692 0.63203 0.39202 0.60759 0.63906 0.39183 0.54296 0.72031 0.41199 0.48296 0.85560 0.44758 0.44968 .52₅₆ 0.62539 0.38220 0.59780 0.63242 0.38212 0.53443 0.71250 0.40157 0.47559 0.84492 0.43569 0.44263 0.61875 0.37279 0.58826 0.62578 0.37279 0.52612 0.70508 0.39166 0.46838 0.83476 0.42441 0.43577 I .60 0.61289 0.36393 0.57897 0.62031 0.36410 0.51797 0.69824 0.38227 0.46132 0.82539 0.41376 0.42908 I .64 0.60703 0.35541 0.56992 0.61445 0.35564 0.51003 0.69180 0.37333 0.45443 0.81660 0.40364 0.42256 I .68 0.60156 0.34730 0.56110 0.60937 0.34768 0.50228 0.68555 0.36476 0.44770 0.80840 0.39402 0.41621 I .72 0.59648 0.33957 0.55249 0.60391 0.33992 0.49472 0.67969 0.35659 0.44112 0.80059 0.38485 0.41003 I .76_.80 0.59141 0.33213 0.54411 0.59922 0.33262 0.48735 0.67422 0.34880 0,43470 0.79316 0.37610 0.40400 0.58672 0.32505 0.53595 0.59453 0.32556 0.48015 0.66875 0.34128 0.42842 0.78633 0.36777 0.39812 I .84 0.58203 0.31820 0.52797 0.59023 0.31885 0.47313 0.66367 0.33411 0.42230 0.77969 0.35979 0.39240 1.88 0.57812 0.31176 0.52021 0.58594 0.31235 0.46628 0.65898 0.32727 0.41633 0.77344 0.35215 0.38682 .92 0.57344 0.30538 0.51265 0.58203 0.30616 0.45961 0.65430 0.32065 0.41050 0.76758 0.34486 0.38138 I . 96 0.56953 0.29938 0.50529 0.57812 0.30017 0.45308 0.65000 0.31434 0.40481 0.76172 0.33780 0.37608 2 .00 0.56601 0.29364 0.49810 0.57461 0.29446 0.44672 0.64570 0.30823 0.39925 0.75644 0.33110 0.37092 2.04 0.56250 0.28809 0.49109 0.57109 0.28893 0.44053 0.64180 0.30239 0.39383 0.75117 0.32461 0.36587 2 .08 0.55859 0.28265 0.48425 0.56758 0.28358 0.43448 0.63789 0.29674 0.38854 0.74648 0.31844 0.36096 2.12 0.55547 0.27753 0.47759 0.56445 0.27847 0.42858 0.63437 0.29134 0.38337 0.74160 0.31242 0.35616 2.16 0.55234 0.27256 0.47108 0.56133 0.27351 0.42282 0.63086 0.28611 0.37833 0.73711 0.30666 0.35149 2 . 20 0.54922 0.26775 0.46474 0.55820 0.26870 0.41719 0.62734 0.28104 0.37341 0.73281 0.30112 0.34693 2 .24 0.54609 0.26309 0.45856

055547

0.26412 0.41171 0.62422 0.27618 0.36860 0.72871 0.29578 0.34247 2 . 28 0.54375 0.25868 0.45251 0.55273 0.25966 0.40635 0.62109 0.27148 0.36390 0.72461 0.29059 0.33812 2 .32 0.54062 0.25430 0.44662 0.55000 0.25533 0.40112 0.61797 0.26691 0.35932 0.72090 0.28562 0.33388 2 .36 0.53828 0.25015 0.44085 0.54766 0.25121 0.39602 0.61484 0.26246 0.35484 0.71719 0.28080 0.32973 2 .40 0.53516 0.24601 0.43523 0.54492 0.24712 0.39102 0.61211 0.25821 0.35046 0.71367 0.27614 0.32568 2.44 0.53281 0.24210 0.42973 0.54258 0.24322 0.38615 0.60937 0.25408 0.34618 0.71016 0.27162 0.32173 2 .48 0.53047 0.23831 0.42437 0.54023 0.23942 0.38139 0.60684 0.25010 0.34199 0.70684 0.26725 0.31787 2 .52 0.52812 0.23462 0.41913 0.53828 0.23580 0.37674 0.60430 0.24623 0.33791 0.70371 0.26304 0.31409 2 .56 0.52578 0.23103 0.41401 0.53594 0.23220 0.37219 0.60195 0.24249 0.33390 0.70059 0.25894 0.31040 2 .60 0.52422 0.22765 0.40900 0.53398 0.22877 0.36774 0.59961 0.23885 0.33000 0.69766 0.25497 0.30679

FIGURE 3.

WAGENINGEN B-SERIES PROPELLERS

CURVE FOR OPTIMUM DIAMETER PROPELLERS

FOR 2 BLADES

RE/AO = 0.35.0.55,0.75,0.95

18-N»

'S

...

:VI

0.95

0.60

1.00 1.40 1.80

2.20

26

(KT/J4)1/4

>-C)

z

LU Li

w

w

z

w

o-D

cD

D

o- r-(O

TABLE 2

KT

()

WAGENINGEN B-SERIES PROPELLER DATA FOR 2 BLADE OPTIMUM DIAMETER PROPELLERS

AE/AO = 0.35 Pio ti ETA-O AE/AO = 0.55 P/D J ETA-O AE/AO = 0.75 P/O 'J ETA-O AE/AO = 0.95 PIO ti ETA-O 0 60 0.98672 0.87500 0.83110 1.12656 0.93281 0.71108 0.64 0.95078 0.82954 0.82387 1.05469 0.86865 0.70624 0. 68 0. 9 1875 0.78845 0.81510 1.00156 0.81689 0.70094 0.72 0.88984 0.75102 0.80521 0.95937 0.77305 0.69487 0.76 0.86328 0.71670 0.79453 0.92344 0.73435 0.68801 0 80 0.83906 0.68520 0.78328 0.89297 0.70010 0.68045 1.15312 0.80187 0.61366 0.84 0.81719 0.65635 0.77163 0.86601 0.66909 0.67230 1.06094 0.74363 0.60373 0.88 0. 79727 0.62978 0.75976 0.84219 0.64096 0.66368 1.01172 0.70393 0.59465 0.92 0.77891 0.60520 0.74774 0.82070 0.61516 0.65470 0.97422 0.67054 0.58584 0.96 0.76172 0.58230 0.73568 0.80156 0.59155 0.64548 0.94375 0.64137 0.57714 .00 0.74609 0.56111 0.72363 0.78359 0.56951 0.63606 0.91797 0.61531 0.56851 .04 0.73164 0.54138 0.71167 0.76797 0.54936 0.62655 0.89492 0.59145 0.55992 I .08 0.71797 0.52286 0.69981 0.75312 0.53046 0.61700 0.87461 0.56966 0.55137 1.12 0.70547 0.50561 0.68811 0.73984 0.51294 0.60744 0.85664 0.54966 0.54289 I. IS 0.69375 0.48943 0.67658 0.72734 0.49648 0.59794 0.84023 0.53108 0.53448 20 0.68281 0.47423 0.66525 0.71562 0.48102 0.58852 0.82539 0.51383 0.52617 I . 24 0.67266 0.45995 0.65412 0.70508 0.46660 0.57920 0.81172 0.49770 0.51797 .28 0.66328 0.44653 0.64322 0.69531 0.45304 0.57001 0.79922 0.48263 0.50989 I . 32 0.65469 0.43394 0.63255 0.68594 0.44017 0.56097 0.78750 0.46843 0.50195 I . 36 0.64609 0.42187 0.62212 0.67695 0.42795 0.55208 0.77695 0.45515 0.49414 1 .40 0.63828 0.41052 0.61191 0.66875 0.41645 0.54335 0.76680 0.44254 0.48648 44 0.63047 0.39966 0.60195 0.66094 0.40554 0.53481 0.75742 0.43064 0.47897 .48 0. 62383 0.38953 0.59222 0.65391 0.39526 0.52644 0.74844 0.41933 0.47162 1 .52 0. 6 17 19 0.37981 0.58273 0.64726 0.38550 0.51825 0.74023 0.40867 0.46443 56 0.61094 0.37057 0.57347 0.64062 0.37610 0.51024 0.73242 0.39850 0.45738 .60 0.60469 0.36168 0.56444 0.63437 0.36716 0.50243 0.72539 0.38893 0.45050 1.64 0.59922 0.35332 0.55564 0.62891 0.35874 0.49477 0.71836 0.37971 0.44378 .68 0.59375 0.34528 0.54706 0.62344 0.35063 0.48731 0.71172 0.37092 0.43722 I .72 0. 58906 0.33769 0.53869 0.61836 0.34292 0.48003 0.70547 0.36255 0.43081 I .76 0.58398 0.33031 0.53054 0.61328 0.33547 0.47292 0.69961 0.35455 0.42455 .80 0.57930 0.32328 0.52260 0.60859 0.32837 0.46598 0.69414 0.34694 0.41844 I .84 0.57500 0.31656 0.51485 0.60391 0.32152 0.45922 0.68867 0.33957 0.41248 1.88 0.57070 0.31008 0.50729 0.59980 0.31503 0.45261 0.68379 0.33259 0.40666 1.92 0.56641 0.30383 0.49994 0.59570 0.30875 0.44617 0.67891 0.32584 0.40098 I .96 0.56250 0.29786 0.49276 0.59180 0.30272 0.43988 0.67422 0.31935 0.39543 2.00 0.55937 0.29224 0.48576 0.58789 0.29689 0.43374 0.66992 0.31316 0.39002 2 .04 0.55547 0.28666 0.47894 0.58437 0.29132 0.42776 0.66562 0.30717 0.38475 2 .08 0.55234 0.28140 0.47229 0.58105 0.28597 0.42192 0.66172 0.30143 0.37959 2.12 0.54922 0.27629 0.46579 0.57773 0.28078 0.41622 0.65781 0.29589 0.37456 2.16 0.54609 0.27136 0.45946 0.57461 0.27579 0.41066 0.65391 0.29050 0.36965 2 . 20 0.54297 0.26658 0.45328 0.57148 0.27095 0.40522 0.65039 0.28536 0.36486 2.24 0.53984 0.26194 0.44726 0.56875 0.26633 0.39993 0.64707 0.28040 0.36017 2.28 0.53711 0.25750 0.44137 0.56562 0.26177 0.39475 0.64375 0.27559 0.35560 2.32 0.53437 0.25320 0.43563 0.56328 0.25748 0.38969 0.64062 0.27096 0.35113 2 . 36 0.53203 0.24908 0.43002 0.56055 0.25325 0.38476 0.63750 0.26645 0.34677 2.40 0.52930 0.24502 0.42454 0.55781 0.24914 0.37994 0.63457 0.26211 0.34250 2.44 0.52695 0.24114 0.41919 0.55547 0.24521 0.37522 0.63164 0.25789 0.33834 2.48 0.52461 0.23737 0.41397 0.55312 0.24139 0.37062 0.62891 0.25382 0.33426 2.52 0.52266 0.23375 0.40886 0.55078 0.23767 0.36613 0.62617 0.24986 0.33028 2 . 5G 0.52031 0.23018 0.40387 0.54883 0.23411 0.36172 0.62383 0.24607 0.32639 2 .60 0.51797 0.22671 0.39900 0.54648 0.23059 0.35743 0.62109 0.24232 0.32258 1.05781 0.58348 0.50414 1.01953 0.55901 0.49496 0.99141 0.53830 0.48620 0.96875 0.51993 0.47780 0.94902 0.50309 0.46968 0.93203 0.48767 0.46184 0.91680 0.47332 0.45423 0.90273 0.45985 0.44685 0.89023 0.44729 0.43969 0.87851 0.43541 0.43273 0.86797 0.42426 0.42597 0.85781 0.41362 0.41939 0.84863 0.40361 0.41300 0.83984 0.39404 0.40678 0.83164 0.38494 0.40073 0.82383 0.37626 0.39484 0.81660 0.36799 0.38910 0.80976 0.36009 0.38352 0.80312 0.35250 0.37808 0.79687 0.34524 0.37278 0.79101 0.33829 0.36762 0.78535 0.33161 0.36260 0.78008 0.32521 0.35769 0.77500 0.31905 0.35292 0.76992 0.31308 0.34825 0.76523 0.30737 0.34371 0.76074 0.30186 0.33928 0.75644 0.29654 0.33495 0.75234 0.29142 0.33073 0.74824 0.28644 0.32661 0.74433 0.28163 0.32259 0.74062 0.27700 0.31866 0.73711 0.27252 0.31482 0.73359 0.26817 0.31107 0.73027 0.26396 0.30740 0.72715 0.25991 0.30382

>-L)

z

w

Ilc;

u-w

w

w

U

z

w

o-D

r

FIGURE 4.

YJAGENINGEN B-SERIES PROPELLERS

CURVE FOR OPTIMUM DIAMETER PROPELLERS

FOR 2 BLADES

RE/AO = 0.40.0.60,0.80,1.00

CD -20-CD 0.40

0.60

0.80

L. OU b.6o 1.00

1.40

1.80

2.20

26

(KT/J4P 1/4

Ç\J

D

o-TABLE 3

KT

'

(-7-)

J.t

0.60

WAGENINGEN 8-SERIES PROPELLER DATA FOR 2 BLADE OPTIMUM DIAMETER PROPELLERS

AE/AO = 0.40 AE/AO = 0.60 AE/AO = 0.80

p/ J ETA-O _P/O J ETA-O P/U J

0.99687 0.87634 0.79881 ETA-O AE/AO 1.00 PIO J ETA-O 0.64 0.95781 0.82951 0.79309 1.13437 0.90456 0.68492 0.68 0.92344 0.78752 0.78578 1.05703 0.84092 0.67841 0.72 0.89219 0.74929 0.77726 1.00390 0.79153 0.67200 0.76 0.86484 0.71484 0.76781 0.96250 0.74995 0.66518 0.80 0.83984 0.68323 0.75769 0.92734 0.71330 0.65787 0.84 0.81719 0.65427

074708

0.89766 0.68083 0.65008 0.88 0.79687 0.62776 0.73613 0.87109 0.65130 0.64188 1.11719 0.74248 0.58760 0.92 0.77812

0.60320 072495

0.84805 0.62466 0.63336 1.04687 0.69681 0.57754 0.96 0.76094 0.58044 0.71364 0.82734 0.60021 0.62458 1.00429

066282

0.56813 1.00 0.74453 0.55913 0.70230 0.80859 0.57768 0.61564 0.97109 0.63370 0.55906 1.04 0.73047 0.53967 0.69097 0.79180 0.55693 0.60657 0.94375 0.60797 0.55020 1.08 0.71680 0.52126 0.67969 0.77617 0.53757 0.59746 0.91992 0.58462 0.54148 1.12

070430 0.50412

0.66853 0.76211 0.51964 0.58835 0.89922

056339

0.53292 1.16 0.69258 0.48804 0.65751 0.74922 0.50290 0.57926 0.88086 0.54387 0.52447 1.20 0.68203 0.47305 0.64664 0.73711 0.48718 0.57025 0.86406 0.52572 0.51617 1.24 0.67187 0.45886 0.63597 0.72578 0.47237 0.56132 0.84902 0.50891 0.50801 1.28 0.66250 0.44551 0.62548 0.71523 0.45846 0.55252 0.83516 0.49318 0.49998 1.32 0.65312 0.43275 0.61521 0.70586 0.44548 0.54385 0.82226 0.47841 0.49212 1.08281 0.55013 0.47938 1.36 0.64531 0.42097 0.60514 0.69687 0.43315 0.53531 0.81055 0.46459 0.48440 1.03437 0.52569 0.47060 1.40 0.63750 0.40968 0.59529 0.68828 0.42145 0.52694 0.79961 0.45155 0.47684 1.00547 0.50691 0.46227 1.44 0.62969 0.39885 0.58566 0.68008 0.41031 0.51872 0.78984 0.43935 0.46944 0.98281 0.49039 0.45430 1.48 0.62266 0.38867 0.57625 0.67266 0.39984 0.51067 0.78008 0.42766 0.46220 0.96367 0.47538 0.44662 1.52 0.61641 0.37910 0.56708 0.66562 0.38988 0.50279 0.77129 0.41667 0.45512 0.94687 0.46151 0.43922 1.56 0.61016 0.36990 0.55812 0.65898 0.38039 0.49508 0.76289 0.40620 0.44821 0.93242 0.44872 0.43205 1.60 0.60430 0.36114 0.54937 0.65273 0.37137 0.48755 0.75508 0.39628 0.44145 0.91914 0.43670 0.42512 1.64 0.59844 0.35272 0.54085 0.64687 0.36279 0.48018 0.74785 0.38686 0.43485 0.90703 0.42539 0.41839 1.68 0.59336 0.34479 0.53252 0.64101 0.35451 0.47299 0.74062 0.37781 0.42842 0.89609 0.41476 0.41188 1.72 0.58828 0.33716 0.52442 0.63594 0.34672 0.46597 0.73418 0.36925 0.42212 0.88555 0.40461 0.40555 1.76 0.58359 0.32988 0.51651 0.63086 0.33921 0.45912 0.72793 0.36105 0.41599 0.87617 0.39508 0.39941 1.80 0.57891 0.32286 0.50879 0.62578 0.33195 0.45242 0.72187 0.35317 0.41000 0.86719 0.38597 0.39344 1.84 0.57461 0.31617 0.50127 0.62109 0.32504 0.44590 0.71641 0.34570 0.40416 0.85898 0.37733 0.38764 1.88 0.57031 0.30972 0.49394 0.61680 0.31844 0.43952 0.71094 0.33848 0.39846 0.85098 0.36903 0.38200 1.92 0.56641 0.30355 0.48679 0.61250 0.31206 0.43330 0.70586 0.33159 0.39290 0.84355 0.36113 0.37651 1.96 0.56289 0.29768 0.47983 0.60859 0.30598 0.42724 0.70098 0.32496 0.38747 0.83652 0.35358 0.37118 2.00 0.55937 0.29199 0.47302 0.60469 0.30009 0.42132 0.69629 0.31859 0.38217 0.83008 0.34637 0.36598 2.04 0.55547 0.28643 0.46641 0.60078 0.29440 0.41554 0.69180 0.31246 0.37700 0.82363 0.33941 0.36092 2.08 0.55234 0.28118 0.45995 0.59766 0.28903 0.40990 0.68750 0.30657 0.37196 0.81758 0.33274 0.35600 2.12 0.54922 0.27609 0.45363 0.59414 0.28376 0.40439 0.68359 0.30093 0.36704 0.81191 0.32635 0.35120 2.16 0.54609 0.27117 0.44749 0.59062 0.27864 0.39902 0.67949 0.29543 0.36224 0.80644 0.32020 0.34652 2.20 0.54297 0.26639 0.44149 0.58750 0.27376 0.39377 0.67578 0.29017 0.35754 0.80117 0.31426 0.34197 2.24 0.54023 0.26183 0.43563 0.58476 0.26910 0.38865 0.67207 0.28507 0.35296 0.79609 0.30854 0.33752 2.28 0.53750 0.25740 0.42991 0.58164 0.26450 0.38365 0.66875 0.28019 0.34849 0.79141 0.30306 0.33319 2.32 0.53476 0.25310 0.42433 0.57891 0.26010 0.37877 0.66523 0.27541 0.34412 0.78691 0.29777 0.32896 2.36 0.53242 0.24900 0.41889 0.57617 0.25583 0.37399 0.66211 0.27084 0.33985 0.78242 0.29264 0.32484 2.40 0.52969 0.24494 0.41356 0.57383 0.25175 0.36933 0.65898 0.26641 0.33568 0.77832 0.28771 0.32082 2.44 0.52734 0.24106 0.40837 0.57109 0.24771 0.36478 0.65586 0.26208 0.33161 0.77422 0.28293 0.31688 2.48 0.52500 0.23729 0.40328 0.56875 0.24385 0.36033 0.65293 0.25792 0.32763 0.77031 0.27830 0.31305 2.52 0.52305 0.23369 0.39833 0.56641 0.24010 0.35598 0.65000 0.25387 0.32373 0.76641 0.27380 0.30930 2.56 0.52070 0.23013 0.39349 0.56406 0.23645 0.35173 0.64746 0.24999 0.31993 0.76289 0.26949 0.30564 2.60 0.51875 0.22671 0.38874 0.56211 0.23296 0.34757 0.64492 0.24622 0.31620 0.75937 0.26530 0.30206

TABLE i

Kp 1/4

()

WAGENINGEN B-SERIES PROPELLER DATA FOR 2 BLADE OPTIMUM DIAMETER PROPELLERS

AE/AO = 0.30 P/D J ETA-O AE/AO = 0.50 P/D J ETA-O AE/AO = 0.70 P/D J ETA-O AE/AO = 0.90 P/D J ETA-O o 60 0.98906 0.88022 0.86227 1.06015 0.90212 0.73677 0.64 0.95547 0.83540 0.85351 1.00703 0.84780 0.73267 o 68 0.92500 0.79453 0.84329 0.96406 0.80137 0.72734 0.72 0.89766 0.75735 0.83206 0.92695 0.76023 0.72088 0.76 0.87187 0.72290 0.82015 0.89531 0.72378 0.71349 1.09297 0.80328 0.63189 0 .80 0.84844 0.69133 0.80780 0.86719 0.69077 0.70531 1.03125 0.75453 0.62329 0.84 0.82656 0.66211 0.79517 0.84219 0.66079 0.69652 0.98750 0.71539 0.61499 0.88 0.80703 0.63535 0.78240 0.81953 0.63332 0.68727 0.95234 0.68161 0.60670 0.92 0.78828 0.61033 0.76959 0.79961 0.60830 0.67766 0.92305 0.65180 0.59834 0.96 0.77187 0.58743 0.75679 0.78125 0.58513 0.66783 0.89726 0.62479 0.58989 .00 0.75625 0.56597 0.74407 0.76484 0.56379 0.65783 0.87500 0.60038 0.58137 .04 0.74141 0.54587 0.73150 0.74922 0.54377 0.64777 0.85469 0.57783 0.57280 .08 0.72812 0.52727 0.71908 0.73516 0.52524 0.63769 0.83672 0.55716 0.56422 1.06797 0.63208 0.53236 1.12 0.71562 0.50983 0.70686 0.72226 0.50797 0.62763 0.82031 0.53799 0.55566 1.02344 0.60265 0.52270 1. 16₂₀ 0.70352 0.49333 0.69486 0.71016 0.49174 0.61764 0.80547 0.52018 0.54714 0.99219 0.57840 0.51352 0.69219 0.47786 0.68308 0.69922 0.47661 0.60776 0.79180 0.50355 0.53870 0.96641 0.55686 0.50469 I . 24₂₈ 0.68203 0.46343 0.67154 0.68867 0.46228 0.59800 0.77930 0.48802 0.53036 0.94492 0.53749 0.49015 0.67266 0.44988 0.66025 0.67891 0.44880 0.58840 0.76758 0.47340 0.52211 0.92578 0.51964 0.48787 .32 0.66328 0.43693 0.64921 0.66992 0.43613 0.57895 0.75664 0.45963 0.51400 0.90898 0.50320 0.47981 36 0.65469 0.42474 0.63841 0.66133 0.42410 0.56967 0.74648 0.44665 0.50601 0.89375 0.48789 0.47198 .40 0.64687 0.41329 0.62788 0.65351 0.41279 0.56058 0.73711 0.43444 0.49818 0.88008 0.47363 0.46435 .44₄₈ 0.63906 0.40233 0.61761 0.64609 0.40204 0.55166 0.72851 0.42294 0.49050 0.86719 0.46016 0.45692 0.63203 0.39202 0.60759 0.63906 0.39183 0.54296 0.72031 0.41199 0.48296 0.85560 0.44758 0.44968 .52₅₆ 0.62539 0.38220 0.59780 0.63242 0.38212 0.53443 0.71250 0.40157 0.47559 0.84492 0.43569 0.44263 0.61875 0.37279 0.58826 0.62578 0.37279 0.52612 0.70508 0.39166 0.46838 0.83476 0.42441 0.43577 I .60 0.61289 0.36393 0.57897 0.62031 0.36410 0.51797 0.69824 0.38227 0.46132 0.82539 0.41376 0.42908 I .64 0.60703 0.35541 0.56992 0.61445 0.35564 0.51003 0.69180 0.37333 0.45443 0.81660 0.40364 0.42256 I .68 0.60156 0.34730 0.56110 0.60937 0.34768 0.50228 0.68555 0.36476 0.44770 0.80840 0.39402 0.41621 I .72 0.59648 0.33957 0.55249 0.60391 0.33992 0.49472 0.67969 0.35659 0.44112 0.80059 0.38485 0.41003 I .76_.80 0.59141 0.33213 0.54411 0.59922 0.33262 0.48735 0.67422 0.34880 0,43470 0.79316 0.37610 0.40400 0.58672 0.32505 0.53595 0.59453 0.32556 0.48015 0.66875 0.34128 0.42842 0.78633 0.36777 0.39812 I .84 0.58203 0.31820 0.52797 0.59023 0.31885 0.47313 0.66367 0.33411 0.42230 0.77969 0.35979 0.39240 1.88 0.57812 0.31176 0.52021 0.58594 0.31235 0.46628 0.65898 0.32727 0.41633 0.77344 0.35215 0.38682 .92 0.57344 0.30538 0.51265 0.58203 0.30616 0.45961 0.65430 0.32065 0.41050 0.76758 0.34486 0.38138 I . 96 0.56953 0.29938 0.50529 0.57812 0.30017 0.45308 0.65000 0.31434 0.40481 0.76172 0.33780 0.37608 2 .00 0.56601 0.29364 0.49810 0.57461 0.29446 0.44672 0.64570 0.30823 0.39925 0.75644 0.33110 0.37092 2.04 0.56250 0.28809 0.49109 0.57109 0.28893 0.44053 0.64180 0.30239 0.39383 0.75117 0.32461 0.36587 2 .08 0.55859 0.28265 0.48425 0.56758 0.28358 0.43448 0.63789 0.29674 0.38854 0.74648 0.31844 0.36096 2.12 0.55547 0.27753 0.47759 0.56445 0.27847 0.42858 0.63437 0.29134 0.38337 0.74160 0.31242 0.35616 2.16 0.55234 0.27256 0.47108 0.56133 0.27351 0.42282 0.63086 0.28611 0.37833 0.73711 0.30666 0.35149 2 . 20 0.54922 0.26775 0.46474 0.55820 0.26870 0.41719 0.62734 0.28104 0.37341 0.73281 0.30112 0.34693 2 .24 0.54609 0.26309 0.45856

055547

0.26412 0.41171 0.62422 0.27618 0.36860 0.72871 0.29578 0.34247 2 . 28 0.54375 0.25868 0.45251 0.55273 0.25966 0.40635 0.62109 0.27148 0.36390 0.72461 0.29059 0.33812 2 .32 0.54062 0.25430 0.44662 0.55000 0.25533 0.40112 0.61797 0.26691 0.35932 0.72090 0.28562 0.33388 2 .36 0.53828 0.25015 0.44085 0.54766 0.25121 0.39602 0.61484 0.26246 0.35484 0.71719 0.28080 0.32973 2 .40 0.53516 0.24601 0.43523 0.54492 0.24712 0.39102 0.61211 0.25821 0.35046 0.71367 0.27614 0.32568 2.44 0.53281 0.24210 0.42973 0.54258 0.24322 0.38615 0.60937 0.25408 0.34618 0.71016 0.27162 0.32173 2 .48 0.53047 0.23831 0.42437 0.54023 0.23942 0.38139 0.60684 0.25010 0.34199 0.70684 0.26725 0.31787 2 .52 0.52812 0.23462 0.41913 0.53828 0.23580 0.37674 0.60430 0.24623 0.33791 0.70371 0.26304 0.31409 2 .56 0.52578 0.23103 0.41401 0.53594 0.23220 0.37219 0.60195 0.24249 0.33390 0.70059 0.25894 0.31040 2 .60 0.52422 0.22765 0.40900 0.53398 0.22877 0.36774 0.59961 0.23885 0.33000 0.69766 0.25497 0.30679

TABLE 2

KT

()

WAGENINGEN B-SERIES PROPELLER DATA FOR 2 BLADE OPTIMUM DIAMETER PROPELLERS

AE/AO = 0.35 Pio ti ETA-O AE/AO = 0.55 P/D J ETA-O AE/AO = 0.75 P/O 'J ETA-O AE/AO = 0.95 PIO ti ETA-O 0 60 0.98672 0.87500 0.83110 1.12656 0.93281 0.71108 0.64 0.95078 0.82954 0.82387 1.05469 0.86865 0.70624 0. 68 0. 9 1875 0.78845 0.81510 1.00156 0.81689 0.70094 0.72 0.88984 0.75102 0.80521 0.95937 0.77305 0.69487 0.76 0.86328 0.71670 0.79453 0.92344 0.73435 0.68801 0 80 0.83906 0.68520 0.78328 0.89297 0.70010 0.68045 1.15312 0.80187 0.61366 0.84 0.81719 0.65635 0.77163 0.86601 0.66909 0.67230 1.06094 0.74363 0.60373 0.88 0. 79727 0.62978 0.75976 0.84219 0.64096 0.66368 1.01172 0.70393 0.59465 0.92 0.77891 0.60520 0.74774 0.82070 0.61516 0.65470 0.97422 0.67054 0.58584 0.96 0.76172 0.58230 0.73568 0.80156 0.59155 0.64548 0.94375 0.64137 0.57714 .00 0.74609 0.56111 0.72363 0.78359 0.56951 0.63606 0.91797 0.61531 0.56851 .04 0.73164 0.54138 0.71167 0.76797 0.54936 0.62655 0.89492 0.59145 0.55992 I .08 0.71797 0.52286 0.69981 0.75312 0.53046 0.61700 0.87461 0.56966 0.55137 1.12 0.70547 0.50561 0.68811 0.73984 0.51294 0.60744 0.85664 0.54966 0.54289 I. IS 0.69375 0.48943 0.67658 0.72734 0.49648 0.59794 0.84023 0.53108 0.53448 20 0.68281 0.47423 0.66525 0.71562 0.48102 0.58852 0.82539 0.51383 0.52617 I . 24 0.67266 0.45995 0.65412 0.70508 0.46660 0.57920 0.81172 0.49770 0.51797 .28 0.66328 0.44653 0.64322 0.69531 0.45304 0.57001 0.79922 0.48263 0.50989 I . 32 0.65469 0.43394 0.63255 0.68594 0.44017 0.56097 0.78750 0.46843 0.50195 I . 36 0.64609 0.42187 0.62212 0.67695 0.42795 0.55208 0.77695 0.45515 0.49414 1 .40 0.63828 0.41052 0.61191 0.66875 0.41645 0.54335 0.76680 0.44254 0.48648 44 0.63047 0.39966 0.60195 0.66094 0.40554 0.53481 0.75742 0.43064 0.47897 .48 0. 62383 0.38953 0.59222 0.65391 0.39526 0.52644 0.74844 0.41933 0.47162 1 .52 0. 6 17 19 0.37981 0.58273 0.64726 0.38550 0.51825 0.74023 0.40867 0.46443 56 0.61094 0.37057 0.57347 0.64062 0.37610 0.51024 0.73242 0.39850 0.45738 .60 0.60469 0.36168 0.56444 0.63437 0.36716 0.50243 0.72539 0.38893 0.45050 1.64 0.59922 0.35332 0.55564 0.62891 0.35874 0.49477 0.71836 0.37971 0.44378 .68 0.59375 0.34528 0.54706 0.62344 0.35063 0.48731 0.71172 0.37092 0.43722 I .72 0. 58906 0.33769 0.53869 0.61836 0.34292 0.48003 0.70547 0.36255 0.43081 I .76 0.58398 0.33031 0.53054 0.61328 0.33547 0.47292 0.69961 0.35455 0.42455 .80 0.57930 0.32328 0.52260 0.60859 0.32837 0.46598 0.69414 0.34694 0.41844 I .84 0.57500 0.31656 0.51485 0.60391 0.32152 0.45922 0.68867 0.33957 0.41248 1.88 0.57070 0.31008 0.50729 0.59980 0.31503 0.45261 0.68379 0.33259 0.40666 1.92 0.56641 0.30383 0.49994 0.59570 0.30875 0.44617 0.67891 0.32584 0.40098 I .96 0.56250 0.29786 0.49276 0.59180 0.30272 0.43988 0.67422 0.31935 0.39543 2.00 0.55937 0.29224 0.48576 0.58789 0.29689 0.43374 0.66992 0.31316 0.39002 2 .04 0.55547 0.28666 0.47894 0.58437 0.29132 0.42776 0.66562 0.30717 0.38475 2 .08 0.55234 0.28140 0.47229 0.58105 0.28597 0.42192 0.66172 0.30143 0.37959 2.12 0.54922 0.27629 0.46579 0.57773 0.28078 0.41622 0.65781 0.29589 0.37456 2.16 0.54609 0.27136 0.45946 0.57461 0.27579 0.41066 0.65391 0.29050 0.36965 2 . 20 0.54297 0.26658 0.45328 0.57148 0.27095 0.40522 0.65039 0.28536 0.36486 2.24 0.53984 0.26194 0.44726 0.56875 0.26633 0.39993 0.64707 0.28040 0.36017 2.28 0.53711 0.25750 0.44137 0.56562 0.26177 0.39475 0.64375 0.27559 0.35560 2.32 0.53437 0.25320 0.43563 0.56328 0.25748 0.38969 0.64062 0.27096 0.35113 2 . 36 0.53203 0.24908 0.43002 0.56055 0.25325 0.38476 0.63750 0.26645 0.34677 2.40 0.52930 0.24502 0.42454 0.55781 0.24914 0.37994 0.63457 0.26211 0.34250 2.44 0.52695 0.24114 0.41919 0.55547 0.24521 0.37522 0.63164 0.25789 0.33834 2.48 0.52461 0.23737 0.41397 0.55312 0.24139 0.37062 0.62891 0.25382 0.33426 2.52 0.52266 0.23375 0.40886 0.55078 0.23767 0.36613 0.62617 0.24986 0.33028 2 . 5G 0.52031 0.23018 0.40387 0.54883 0.23411 0.36172 0.62383 0.24607 0.32639 2 .60 0.51797 0.22671 0.39900 0.54648 0.23059 0.35743 0.62109 0.24232 0.32258 1.05781 0.58348 0.50414 1.01953 0.55901 0.49496 0.99141 0.53830 0.48620 0.96875 0.51993 0.47780 0.94902 0.50309 0.46968 0.93203 0.48767 0.46184 0.91680 0.47332 0.45423 0.90273 0.45985 0.44685 0.89023 0.44729 0.43969 0.87851 0.43541 0.43273 0.86797 0.42426 0.42597 0.85781 0.41362 0.41939 0.84863 0.40361 0.41300 0.83984 0.39404 0.40678 0.83164 0.38494 0.40073 0.82383 0.37626 0.39484 0.81660 0.36799 0.38910 0.80976 0.36009 0.38352 0.80312 0.35250 0.37808 0.79687 0.34524 0.37278 0.79101 0.33829 0.36762 0.78535 0.33161 0.36260 0.78008 0.32521 0.35769 0.77500 0.31905 0.35292 0.76992 0.31308 0.34825 0.76523 0.30737 0.34371 0.76074 0.30186 0.33928 0.75644 0.29654 0.33495 0.75234 0.29142 0.33073 0.74824 0.28644 0.32661 0.74433 0.28163 0.32259 0.74062 0.27700 0.31866 0.73711 0.27252 0.31482 0.73359 0.26817 0.31107 0.73027 0.26396 0.30740 0.72715 0.25991 0.30382

TABLE 3

KT

'

(-7-)

J.t

0.60

WAGENINGEN 8-SERIES PROPELLER DATA FOR 2 BLADE OPTIMUM DIAMETER PROPELLERS

AE/AO = 0.40 AE/AO = 0.60 AE/AO = 0.80

p/ J ETA-O _P/O J ETA-O P/U J

0.99687 0.87634 0.79881 ETA-O AE/AO 1.00 PIO J ETA-O 0.64 0.95781 0.82951 0.79309 1.13437 0.90456 0.68492 0.68 0.92344 0.78752 0.78578 1.05703 0.84092 0.67841 0.72 0.89219 0.74929 0.77726 1.00390 0.79153 0.67200 0.76 0.86484 0.71484 0.76781 0.96250 0.74995 0.66518 0.80 0.83984 0.68323 0.75769 0.92734 0.71330 0.65787 0.84 0.81719 0.65427

074708

0.89766 0.68083 0.65008 0.88 0.79687 0.62776 0.73613 0.87109 0.65130 0.64188 1.11719 0.74248 0.58760 0.92 0.77812

0.60320 072495

0.84805 0.62466 0.63336 1.04687 0.69681 0.57754 0.96 0.76094 0.58044 0.71364 0.82734 0.60021 0.62458 1.00429

066282

0.56813 1.00 0.74453 0.55913 0.70230 0.80859 0.57768 0.61564 0.97109 0.63370 0.55906 1.04 0.73047 0.53967 0.69097 0.79180 0.55693 0.60657 0.94375 0.60797 0.55020 1.08 0.71680 0.52126 0.67969 0.77617 0.53757 0.59746 0.91992 0.58462 0.54148 1.12

070430 0.50412

0.66853 0.76211 0.51964 0.58835 0.89922

056339

0.53292 1.16 0.69258 0.48804 0.65751 0.74922 0.50290 0.57926 0.88086 0.54387 0.52447 1.20 0.68203 0.47305 0.64664 0.73711 0.48718 0.57025 0.86406 0.52572 0.51617 1.24 0.67187 0.45886 0.63597 0.72578 0.47237 0.56132 0.84902 0.50891 0.50801 1.28 0.66250 0.44551 0.62548 0.71523 0.45846 0.55252 0.83516 0.49318 0.49998 1.32 0.65312 0.43275 0.61521 0.70586 0.44548 0.54385 0.82226 0.47841 0.49212 1.08281 0.55013 0.47938 1.36 0.64531 0.42097 0.60514 0.69687 0.43315 0.53531 0.81055 0.46459 0.48440 1.03437 0.52569 0.47060 1.40 0.63750 0.40968 0.59529 0.68828 0.42145 0.52694 0.79961 0.45155 0.47684 1.00547 0.50691 0.46227 1.44 0.62969 0.39885 0.58566 0.68008 0.41031 0.51872 0.78984 0.43935 0.46944 0.98281 0.49039 0.45430 1.48 0.62266 0.38867 0.57625 0.67266 0.39984 0.51067 0.78008 0.42766 0.46220 0.96367 0.47538 0.44662 1.52 0.61641 0.37910 0.56708 0.66562 0.38988 0.50279 0.77129 0.41667 0.45512 0.94687 0.46151 0.43922 1.56 0.61016 0.36990 0.55812 0.65898 0.38039 0.49508 0.76289 0.40620 0.44821 0.93242 0.44872 0.43205 1.60 0.60430 0.36114 0.54937 0.65273 0.37137 0.48755 0.75508 0.39628 0.44145 0.91914 0.43670 0.42512 1.64 0.59844 0.35272 0.54085 0.64687 0.36279 0.48018 0.74785 0.38686 0.43485 0.90703 0.42539 0.41839 1.68 0.59336 0.34479 0.53252 0.64101 0.35451 0.47299 0.74062 0.37781 0.42842 0.89609 0.41476 0.41188 1.72 0.58828 0.33716 0.52442 0.63594 0.34672 0.46597 0.73418 0.36925 0.42212 0.88555 0.40461 0.40555 1.76 0.58359 0.32988 0.51651 0.63086 0.33921 0.45912 0.72793 0.36105 0.41599 0.87617 0.39508 0.39941 1.80 0.57891 0.32286 0.50879 0.62578 0.33195 0.45242 0.72187 0.35317 0.41000 0.86719 0.38597 0.39344 1.84 0.57461 0.31617 0.50127 0.62109 0.32504 0.44590 0.71641 0.34570 0.40416 0.85898 0.37733 0.38764 1.88 0.57031 0.30972 0.49394 0.61680 0.31844 0.43952 0.71094 0.33848 0.39846 0.85098 0.36903 0.38200 1.92 0.56641 0.30355 0.48679 0.61250 0.31206 0.43330 0.70586 0.33159 0.39290 0.84355 0.36113 0.37651 1.96 0.56289 0.29768 0.47983 0.60859 0.30598 0.42724 0.70098 0.32496 0.38747 0.83652 0.35358 0.37118 2.00 0.55937 0.29199 0.47302 0.60469 0.30009 0.42132 0.69629 0.31859 0.38217 0.83008 0.34637 0.36598 2.04 0.55547 0.28643 0.46641 0.60078 0.29440 0.41554 0.69180 0.31246 0.37700 0.82363 0.33941 0.36092 2.08 0.55234 0.28118 0.45995 0.59766 0.28903 0.40990 0.68750 0.30657 0.37196 0.81758 0.33274 0.35600 2.12 0.54922 0.27609 0.45363 0.59414 0.28376 0.40439 0.68359 0.30093 0.36704 0.81191 0.32635 0.35120 2.16 0.54609 0.27117 0.44749 0.59062 0.27864 0.39902 0.67949 0.29543 0.36224 0.80644 0.32020 0.34652 2.20 0.54297 0.26639 0.44149 0.58750 0.27376 0.39377 0.67578 0.29017 0.35754 0.80117 0.31426 0.34197 2.24 0.54023 0.26183 0.43563 0.58476 0.26910 0.38865 0.67207 0.28507 0.35296 0.79609 0.30854 0.33752 2.28 0.53750 0.25740 0.42991 0.58164 0.26450 0.38365 0.66875 0.28019 0.34849 0.79141 0.30306 0.33319 2.32 0.53476 0.25310 0.42433 0.57891 0.26010 0.37877 0.66523 0.27541 0.34412 0.78691 0.29777 0.32896 2.36 0.53242 0.24900 0.41889 0.57617 0.25583 0.37399 0.66211 0.27084 0.33985 0.78242 0.29264 0.32484 2.40 0.52969 0.24494 0.41356 0.57383 0.25175 0.36933 0.65898 0.26641 0.33568 0.77832 0.28771 0.32082 2.44 0.52734 0.24106 0.40837 0.57109 0.24771 0.36478 0.65586 0.26208 0.33161 0.77422 0.28293 0.31688 2.48 0.52500 0.23729 0.40328 0.56875 0.24385 0.36033 0.65293 0.25792 0.32763 0.77031 0.27830 0.31305 2.52 0.52305 0.23369 0.39833 0.56641 0.24010 0.35598 0.65000 0.25387 0.32373 0.76641 0.27380 0.30930 2.56 0.52070 0.23013 0.39349 0.56406 0.23645 0.35173 0.64746 0.24999 0.31993 0.76289 0.26949 0.30564 2.60 0.51875 0.22671 0.38874 0.56211 0.23296 0.34757 0.64492 0.24622 0.31620 0.75937 0.26530 0.30206

w

z

w

o

D

D

o

(N co

cb

-22-FIGURE 5.

iRGENINGEN B-SERIES PROPELLERS

-

CURVE FOR OPTIMUM DIAMETER PROPELLERS

FOR 2 BLADES

RE/ÑO = 0.45,0.65,0.85,1.05

o 0.45 0.65 0.85 sS' 1.05 1111

065

0.85 1.0

o1/:;J:uPP

---.

\\\\ \

--S. .. 1.05

---

V

1.00 1.40 1.80 2.20

26'

(KT/J*4)

1/4

co

dT

N

(N