Optimum characteristics equations for the K-J propeller design charts, based on the Wageningen B-screw series

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101

OPTIMUM CHARACTERISTICS EQUATIONS FOR THE 'K-J' PROPELLER DESIGN CHARTS,

BASED ON THE WAGENINGEN B-SCREW SERIES

by

K. Yosifoyi, Z. Zlatev^ and A. Staneva^

The numerical method described in [1] is oriented

to creation of Papmel's design charts [3] in the form

presented in Figure 1 and Figure 2. The algorithm can

easily be adapted to creation of design charts of Steinen

[ 4 ] , Schoenherr [5] and Danckwardt [6] as well,

since all these are of one and the same type, the basic

difference between them consisting in the form of the

non-dimensional parameters used to define the

so-called 'optimum revolutions line - n^^^' and ' optimum

diameter hne - D '.

opt

This fact is illustrated well i n Table 1, where the

formulae for the separate systems of parameters and

the relationships with the corresponding parameters of

Papmel are given. The formulae for the separate

sys-tems of design parameters are written using the standard

ITTC symbols.

The two kinds of Papmel's diagrams shown in the

Figures 1 and 2: X^. - J' called also 'hull' diagram,

and 'KQ - /'-'machine' diagram, correspond to the

so-called 'Naval architecture' or 'thrust' and 'Marine

en-gineering' or 'power' approaches for ship propeUers

design, used i n practice. The solving of the two main

design problems: determination of the optimum

revo-K ^ - J D E S I G N C H A R T

B ^ . A O S C R E W S E R I E S . Z = / . , A E / A O = 0.^0

1.6

Figure 1.

1. Introduction

I t is well known that the propeller design charts

based on results of systematic open-water model tests,

allow the direct obtaining of optimum solutions. As

compared to the existing theoretical methods, the

cal-culations in this case are noted for their simplicity and

satisfactory accurary. These basic advantages determine

the preferable usage of design charts f o r prelimmary

propeller design and prediction of ship performance.

In order to avoid the inconvenience connected with

the traditionally appHed manual plotting and usage of

diagrams, in the recent years a numerical method and a

program package for automated calculation and plott¬

ing of 'K - J' design charts and analytical

representa-tion of their optimum characteristics were developed

at BSHC [ 1 ] , [ 2 ] . The method works on the basis of

preliminarily obtained polynomial regression

equa-tions for the open-water characteristics of a given

systematic propeller series.

1) Ph.D., Senior Research Scientist. Deputy Director Research Activities of the Bulgarian Ship Hydrodynamics Centre ( B S H C ) .

2) Research Scientist, B S H C . 3) Research Scientist, B S H C .

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102

TABLE a . Typ& of Design Charts Author opt AT a C CJ opt opt opt K - J Papmel

1

—

1

11.94J' . h 4 \ "'A J 11.94/ steinen K - 4 1

~7

Schoenherr V -

4

P" K m i 11.94 . . 2 DanckiJardt I T

^ A

Z.4SS B - i Tayloi-33.09 ,1

REMARK : The design chart of Steinen is based on a " , - J " system of dimensionless parameters,where :

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103 lutions at specified screw propeller diameter and the

optimum diameter at specified propeller shaft

revo-lutions, is done by means of the optimum efficiency

lines - and D^^^. The analytical presentation of

these series' optimum lines is done by means of

equa-tions for the characteristics determining them - / , P/D

and

7?^

as functions of the corresponding Papmel's

design parameters isTrf, Kti, Kd' or

X J

' (see Table 1).

In the present paper equations for the optimum

cha-racteristics of the design X^, - / ' and 'K^ ~ J' charts

created at BSHC [7] for all known Wageningen B-screw

series are presented. The equarions initially obtained

for the four-bladed B series, together with results of

the investigations carried out in order to assess the

ac-curacy of the numerical method and comparison with

Sabit's similar work [8] were reported at the Jubilee

Scientific Session dedicated to the 10th anniversary of

BSHC [ 9 ] .

2. Description of the method for calculation and

ana-lytical presentation of the n^^^ and D^^^ design lines

2.1 Structure of the diagrams

On each of the TsTy, - / ' and 'K^ - J' diagrams the

following families of curves are plotted:

Curves of the hydrodynamic coefficients in the

fol-lowing two forms:

A. {K^, K } . = KJ) P/D = const.

B.

C. {K^,KQ}=fiJ)

Auxihary parabolic curves:

diagram

= const.

J' Kj, = —J^ , for = const. ;

K„ = —J"^ , for K = const. ;

diagram:

K

11.94 . „,

^, , for A , = const.

11.94 K

'4

fi , for A'^i = .const.

D. Lines ?i

opt

and £)^pj consisting of points belonging

to the B and C families and determined by the con

dition:

17^

=max. , for K^,K^ = const., or

foris:^, = const.

The information which these lines contain can be

presented in the following parametric form:

' ^ , - / ' d i a g r a m : ( / ; P / D ; , ^ ) J ^ ^ ' ' ^ ^ ' ' ° " V ;

*(A: ), f o r ö

'Ag - /'diagram: (J;P/D;-nJ =

opt

* ( ^ ; , ) > f o r ^ o p t

and at fixed values of the rest of the series' parameters.

For greater clarity and convenience during the work

with the diagrams, the C family curves are drawn only

in the regions around the and D^^^ lines, and serve

as a reference scale for reading the values of the design

parameters.

2.2 A method for calculation of the « , and D , lines

opt opt

The problem of calculating the optimum

characte-ristics of the and/)^pj Hnes of the 'hull' and

'ma-chine' diagrams is in principle reduced to searching for

maximum of the expression:

'Kj,(J,P/D) J

= max

J.P/D

KQ{J,PID-) 27rJ

(1)

= c. = const.

where • = {K^ \ I A j | r } is the corresponding

design parameter and {c.} , z = 1, . . . , « is a

predeter-mined set of values. I t is required that the values K^.,

KQ are presented analytically as functions of / , P/D

and the rest of the series' parameters {A^/A^, Rn, z,

etc.). For instance, in the case of Wageningen B-screw

series, we have [ 10], [ 11 ] :

K^, KQ =P(J, P/D, A^/A^^, z, logRn) ,

where P is a multi-dimensional polynomial operator.

The sets of values of / , P/D and , obtained after

solving (1), together with the given values of the

de-sign parameter define numerically the corresponding

"opt ° ^ ^ o p t

line.

A detailed description of the méthod f o r solution

of this problem and the computer program developed

is given in Ref. [ 1 ] .

The optimum characteristics values obtained are

used as input to. the regression analysis program [ 2 ] ,

from where the required analytical form of the series

optimuin characteristics is obtained.

3. Optimum characteristics equations

The equations for the optimum characteristics of

the Wageingen B-screw series, given in the present

paper, are obtained on the following f o r m :

/ P/D, = I I I ^ (A- )'-(lg RnYiA^ I A. f (2)

(=0 rOk=o

where A..^. are regression coefficients, and AT^.is the

cor-responding Papmel's design parameter. A t fixed blade

number z, the number of sets of equations of the type

(2) is equal to four: two sets for the n^^^ and D^^^ lines

for each of the 'hull' and 'machine' diagrams. Since z

varies from 2 to 7, the total number of these sets is 24.

The range of vaHdity of the polynomial equations

(2) is as follows:

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104 - Reynoldsnumber -i?« = 2.10'^ ^ I . I O '

- blade area ratio - according to the table given below

z

2 0 . 3 0 0 . 3 8 3 0 . 3 5 0 . 8 0

4

0 . 4 0 1 . 0 0 5 0 . 4 5 1 . 0 5 6 0 . 5 0 0 . 8 0 7 0 . 5 5 0 . 8 5

3.1. Equations for the 'hull' diagram optimum

cha-racteristics

The coefficients and powers of thh obtained

regres-sion equations for the optimum characteristics / , P/D

and T}^ of the n^^^ and D^^^ lines for the different z

va-lues are given in Tables Ila - Vila.

As is known, in the 'thrust' approach for given

thrust T and advance speed there are two main

pro-blems, namely:

- determining the propeller optimum revolutions ?^^^^

at specified diameter D ;

- determining the optimum diameter D^^^ at given

revolutions.

The solving of these two problems on the basis Of

the regression equations obtained presents no d i f f i

-culties and could be easily realized by means of a

com-puter. The values of the design parameters and

are calculated by the corresponding formulae from

Table I .

3.2. Equations for the 'machine'diagram optimum

cha-racteristics

The results are presented in Tables l i b - V l l b ,

cor-responding to the same z values. I n this case the

solv-ing of the two optimization problems is analogous to

that in item 3.1., there being one difference: instead

of the necessary thrust, here the dehvered power

is given.

The values of the design parameters K'^ and IC are

calculated by the corresponding formulae from Table I .

I t must be mentioned that both the method of

cal-culating the optimum characteristics and the computer

programs developed on its basis were carefully tested

for a long period of time during which test calculations

of different cases and comparisons with known

dia-grams, obtained graphically, were performed. The

re-sults of these tests showed the satisfactory accuracy

of the method itself as well as that of the software

products developed.

Some of the results of the tests and comparisons are

given in [ 9 ] .

4 .

Conclusion

The equations for the optimum characteristics of

the Wageningen B-screw series, presented in this paper,

complement the similar equations published by Sabit

[ 8 ] , since they are vaUd for all known series with blade

number from 2 to 7 and afford the possibility for

ac-counting the Reynolds number's influence. A certain

advantage is the use of a unified 'K - J' system of

co-efficients in solving the design problems. The

funda-mental character of this system allows, when

neces-sary, an easy transition from Papmel's design

para-meters K^, A j and K'^ to the parapara-meters used i n

Steinen's, Schoenherr's and Danckwardt's charts as

well as to the Taylor's dimensional B -

S

parameters

[ 1 2 ] , according to the relations given in Table I .

The equations obtained for the optimum

characte-ristics of the Wageningen B-screw series are

success-fully used in BSHC practice in the prehmmary choice

of propellers' optimum parameters and ship propulsion

analysis by means of the computerized express-method

developed for this purpose.

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105 regression coefficients

blade area ratio of screw propeller

Taylor's design parameters

diameter of screw propeller

screw propeller disk area

advance coefficient

thrust coefficient

torque coefficient

Papmel's design parameters for the 'n^pj'

and 'Z)^p,' lines of 'hull' diagram

Papmel's design parameters for the '«^pj'

and 'D^pj' lines of 'machine' diagram

Schoenherr's design parameters

number of revolutions per second

pitch ratio of screw propeller

delivered power

torque

Reynolds number

thrust

Danckwardt's design parameters

velocity of advance

number of blades

propeller open-water efficiency

mass density of water

Steinen's design parameters

References

1. Y o s i f o v , K . , Staneva, A . , Zlatev, Z. and Zhekov, Z . , ' A u t o -mated method f o r creating o f Papmel's type propeller de-sign charts' (in russian), Proc. 10th Anniversary o f BSHC Jubilee Scientific Session, vol. I . , BSHC, 1 9 - 2 4 Nov. 1 9 8 1 , Varna, Bulgaria.

2. Yosifov, K . , L y u t o v , N . , Zlatev, Z . and Ivanov, N . , ' A u t o -mated system f o r data acquisition, on-line processing and presentation o f results f r o m systematic model tests o f ship screw series i n open water' ( i n russian), Proc. 10th A n n i -versary o f BSHC Jubilee Scientific Session, v o l . 1, BSHC,

19 - 24 N o v , 1 9 8 1 , Varna, Bulgaria.

3. Papmel, E.E., 'Practical design o f the screw propeller' (in russian), Leningrad, 1936.

4 . Steinen, C. von den, 'Praktische Schraubendiagramme', Werft-Reederei-Hafen, B d . 4, 1923.

5. Schoenherr, K . , 'Propulsion and propellers'. Principles o f Naval Architecture, V o l . I I , Editors H . E . Rossell and L . B . Chapman, S N A M E , N . Y , 1949.

6. Danckwardt, E., 'Berechnungsdiagramme f i i r S c h i f f -schrauben', SBT, B d . 6, ( 1 9 5 6 ) , St., B d . 3 ( 1 9 5 6 ) . 7. 'Propeller design charts', BSHC , 1 9 8 3 .

8. Sabit, A.S., ' O p t i m u m efficiency equations o f the N . S . M . B . propeller series f o u r and five blades'. I n t . Shipbuilding Pro-gress, V o l . 23, Nov. 1976.

9. Y o s i f o v , K . , Zlatev, Z. and Staneva, A . , ' O p t i m u m charac-teristics equations f o r the Wageningen B ^ ^ ^ screw series', Proc. 10th Anniversary o f BSHC Jubilee Scientific Session, V o l . I , Varna, 1 9 8 1 .

10. Lammeren, W.P.A., Manen, J.D. van and Oosterveld, M.W.C., 'The Wageningen B-screw series', S N A M E , V o l . 77, 1969. 1 1 . Oosterveld,M.W.C.and

Oossanen,P.van,'Furthercomputer-analyzed data o f the Wageningen B-screw series'. I n t . Ship-building Progress, V o l . 22, N o . 2 5 1 , 1975.

12. T a y l o r , D.W., 'The speed and power o f ships', Washington, 1933.

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( a ) "Kj - J " d i a g r a m TABLE I I C o e f f i c i e n t s a n d T e r m s f o r J , P / D , ^ „ o f O p t i m u m E f f i c i e n c y E q u a t i o n s f o r t h e W a g e n i n g e n 3 - S c r e w S e r i e s , z = 2 n „ p ^ X P ; D , r j „ = Z A ^ p , J10) . l K d ) . ( l g R n ) ° ( A ^ / A / * a.6,c k a B c a.6.c; k a B c 0.4261881 0 1 0 0 0 , 7 2 4 5 8 1 7 ' 0 1 0 0 - 0 , 1 3 0 1 3 3 3 - 1 2 1 0 0,399889'! - 2 2 G - 0 . 4 4 0 1 1 5 2 - 5 3 3 0 - 0 . 3 0 4 1 8 0 9 -A 2 4 0 - 0 . 1 8 9 7 0 1 3 - 1 1 1 1 - 0 , 5 9 5 6 3 6 0 - 3 0 1 1 - 0 . 5 0 6 6 4 2 4 - 7 6 4 0 - 0 , 3 3 2 7 3 4 9 _9 0 1 0 0,3215752 - 2 6

_g

1 0,1225610 - 2 1 0 1 - 0 . 3 4 7 7 5 3 3 - 4 4 3 0 0,1247325 _o • A 1 - 0 , 1 5 9 9 2 2 7 2 4 1 - 0 , 3 9 5 0 5 1 7 - 5 0 i 1 - 0 , 1 4 0 2 4 7 6 - 1 2 1 1 - 0 , 1 2 3 7 2 8 3 - 2 3 3 1 3 0.3925600 -3 3 2 0 -1 _{- 0 , 3 3 1 0 4 1 8} _{- 1} ₁ ₁ ₀ 0,3779680 - 5 6 3 1 - 0 , 3 9 6 2 1 2 2 - 1 3 0 1 0.1040976 - 3 3 3 1 - 0, 5 7 7 5 9 6 8 _ i ₀ ₀ - 0 . 2 3 0 0 8 4 9 - 2 5 0 0 0,8227161 - 1 4 0 1 - 0, 6 5 6 3 3 2 6 - 3 6 1 1 - 0 , 1 1 3 7 5 0 2 —'~J 6 1 0 0,8755175 5 2 0 0,1013621 - 2 0 2 - 0 . 7 0 9 6 6 0 4 - 4 •Q 2 0 0,1013621 0,1513391 - 2 <l 0 0 0,1627751

_g

1 0 1 0.9805634 - 3 2 2 0 0,6859140 C 0 0 0 0,4272359 0 1 0 0 , 4 9 5 7 1 8 ^ 0 1 0 0 0,4506893 0 0 0 0 - 0 . 1 3 1 4 0 1 1 0 1 1 0 0,2150319 - 2 1 1 0 - 0 , 6 5 3 9 2 9 1 - 1 3 0 0 0,9850057-0.1155018 6 4 0.8238480 - 4 0 3 1 0,9850057-0.1155018 C 6 Ö Ï - 0 . 1 0 8 7 5 3 9 — I 3 0 1 - 0 , 2 9 5 4 2 1 3 —o 5 3 0 0.1435443 - 1 1

z

0 - 0. 3 8 4 4 2 7 2 ~\j 0 2 1 - 0 . 5 1 1 2 1 4 2 _c 2 3 1 0.1737563 -A 5 A 1 - 0 , 4 5 9 4 3 2 1 - 4 2 0 C.2259018

Q

1 J 1 - 0 , 1 4 3 1 7 2 5 - 3 1 A 1 0,283 4215 -2 6 2 0 0,9164843 0,6637177 - 2 3 T 0 - 0 . 1 2 6 7 9 7 5 - 1 6 1 1 0,9164843 0,6637177 - 1 0 1 - 0 , 1 4 6 1 4 5 9 0 5 0 1 - 0 , 1 5 8 0 6 0 0 - 1 4 1 1 - 0 . 9 3 5 3 6 2 3 - 2 6 1 0 0,9294148 - 3 4

_;

1 0,1362344 0 4 0 1 0,5764219 - A 2 3 0 - 0, 6 3 5 5 2 4 5 - 1 1 1 -0,1187984 - 3 2 4 1 - 0 , 1 4 2 5 8 0 4 - 1 0 1 0 - 0 , 2 8 7 4 3 7 8 - 1 0 1 0 - 0 . 6 5 1 5 3 9 0 • - 5 3 4 0 • 0,6364388 - 1 2 0 0 - 0 , 1 3 5 5 0 1 2 - 2 0 1 0 - 0 . 2 7 0 2 7 3 4 ••-3 0 1 0 0,8249955 0 1 0 0 1,4046980 0 1 C 0 - 0 , 3 6 0 8 1 9 2 0 2 0 0 0,167490S 0 3 0 1 - 0 , 9 7 6 0 0 6 1 - 3 3 1 0 - 0, 1 2 4 7 5 1 7 - 2 6 1 1 - 0 , 4 0 5 6 3 2 5 0 1 .0 1 - 0 , 1 0 0 0 0 3 7 - 3 6 ' 0 0.1035400 - 6 4 4 0 0,3366114 ^A 1 4 0 0.5243339 - 3 4 0 1 - 0 , 5 8 0 5 7 0 0 - - 1 5 0 1

%

0,1208249 - 4 0 A 1 - 0 , 7 2 5 3 1 2 7 0 2 0 0

%

- 0 . 2 8 6 1 5 5 1 - 5 5

2

1 - 1 , 2 0 0 2 7 4 0 0 1 0 1 ^ , 1 6 0 2 5 2 1 - 7 S A 1 0,1442035

_c

1 1 1 0,3149657 - 4 6 1 1 0,2111882 - 1 6 0 1 0;7289268 - 1 3 0 0 - 0 , 1 4 9 2 7 0 8 - 2 2 3 1 0,2223767 - 1 1 1 0 0,1396137 0 3 0 0 - 0 , 1 0 6 7 5 6 2 -2 5 0 i 0.5232799 - 3 3 3 1 - 0 . 5 0 6 8 0 1 5 -2 4 Q 0 0.7806027 - 1 2 0 1 D o p t : 3 - ° ' ° ' 1 o = 2 W ° ' - l ' < n ) ° " S R n ) ^ l A ^ / A / ( b ) "K Q - J " d i a g r a m " o p t a,P/D.rl,= S A, c , 1 0 j' ' l K ^ ) ° | l g R n ) V ^ * a . t > . c k G EJ c 0,3289731 _-[ ₁ ₀ 0 - 0 , 7 0 3 9 6 1 3 Q — O O 1 0 0.9145457 - 1 0 • 0 0 0,3437545 - 4 3 1 0 - 0 , 2 5 0 4 7 0 1 - 5 A 0 0 - 0 , 7 4 5 1 0 1 5 - 2 Ö 1 0 0,7974492 - 1 1 0 1 - 0 , 4 3 7 4 1 6 9 - 5 1 4 -

_g

0,7441584 - 5 1 A ₁ 3 - 0 , 8 7 7 7 5 2 0 _7 4 2 - 0 , 3 7 7 9 5 5 8 Q S 0 1 0,1645353 2 3

_g

0,1437374 -S 5 2 0 0,6951720 1 1 0 0,3101603 —6 Ï 0 1 - 0 , 1 1 4 9 5 9 9 4 1 ₁ - 0, 1 4 7 3 1 5 5 - Ï 1 1 1 0,2044502 3 1 0,4585124

_c

0 0 Q 0.5395983 1 0 G 0,8374175 - 4 1 i 0 - 0, 3 2 1 6 0 4 4 - 2 2 0 n 0,1148970 —"i 0 1 0 0.1451993 —1 2 0 1 0,1205068 1 2 1 - 0 , 4 9 0 3 1 1 4 - 3 1 2 n - 0 , 3 5 9 2 8 9 9 - 2 2 1 1 0,2209724 2 2 1 1 D _0,^A51545 _{- 3} ₂ T 0 1 _{- 0 . 1 6 8 3 2 4 5} _{—6 2} ₀ 0.1097236 —"i 1 1 1 0.7779925 0 C 0 "] - 0 . 1 2 1 8 0 4 3 0 3 1 1 - 0. 1 0 0 9 1 7 1 0 1 G 0.2609130 ^-1 5 0 1 - 0 . 5 3 9 2 2 8 6 - 7 4 1 1 0.1014769 - 1 0 1 0 0.1287816 0 1 0 0 - 0 , 1 1 6 3 8 2 0 - 1 •2 0 0 0.7031317 - 3 3 0 0 - 0 , 4 5 5 4 5 3 5 - 1 1 0 1 0,2900218 O ₂ ₁ ₁ - 0 , 2 6 0 2 3 1 5 -A 4 0 0 I o 0,5173269 - 6 5 0 0 I o - 0 , 4 3 6 8 9 4 2 —.J 3 1 1 - 0 , 4 1 9 6 0 9 4 - 8 5 0 0 0,9459027 - 3 1 1 0 - 0 , 1 9 9 2 3 3 3 - 3 G 3 1 0,1875376 - 8 5 0 1 0.4749176 -A 1 3 1 - 0 . 3 9 6 1 0 7 6 - 5 2 3 1 0.3755340 - 1 0 0 0 0.7228755 - 3 3 A ₁ Dopt ( l O l ' ^ I K n f t l g R n ; k a 5 c 0.2404730 0 1 G 0 - 0 , 4 0 0 8 3 4 5 . - 2 2 1 0 0.4210846 - 3 5 0 1 - 0 , 9 9 9 8 0 3 0 - 2 0 0 0 - 0 . 7 3 9 5 0 4 3 - 3 1 1 0 0,9040768 - 0 . 5 7 3 5 5 9 2 — O 3 1 0 0,9040768 - 0 . 5 7 3 5 5 9 2 - 4 4 1 - 0 , 6 5 0 0 8 8 0 — 0 '

z

0 0 - 0 , 5 3 1 4 7 3 0 —A 5 •] ₁

:

0,2994353 - 8 c 4 1 0.1760945 0 r] 0 1 - 0 , 2 8 9 3 7 5 9 - 1 0 1 1 0,6354407 0 1 pi 0,7005751 - 1 1

_c

0 0,5747591 0 0 Q 1 0.1580955 - 1 Ï 1 0 1 0.3939063 0 1 0 1 i - 0 . 1 2 3 3 7 9 1 0 c 0 1 - 0 . 8 6 4 9 2 3 3 - 1 1 1 • 0.3909760 - ' • 6 2 0 - 0 , 1 5 1 9 3 0 0 -A ₆ ₁ ₁ ' - 0 . 3 4 5 4 8 1 4 - 1 0 1 0 0.5503990 _2 0 2 1 ' D _0,4723151 _{- 5} ₅ ₂ ₀ - 0 , 4 5 2 5 6 1 9 2 4 0 1 _0,1072159 _{- 3} ₆ ₀ 1 - 0 . 3 6 5 0 5 6 4 -A 5 1 0 0.1117839 —^ o 2 3 1 0.3270652 0 0 G 0 0.1114379 0 -] •1 ₀ ; 0.4597383 _3 3 2 1 0.1622940 —7 6 A 1 - 0 . 3 5 5 5 2 7 0 _>3 6 A G - 0 . 4 6 9 4 9 8 1 - 1 0 Ï 0 -0,5449330- - 5 4 3 1 0.2321820 - 4 c 0 G - 0 , 1 9 1 9 1 0 4 — o 1 3 0 - 0 , 5 3 5 4 6 3 4 - 1 0 0 1 - 0 , 1 8 5 1 9 3 5 - 1 : 1 0 0,4560272 - 2 3 0 0 0,1245801 - 3 0 4 1 - 0 , 4 2 4 7 2 1 2 _ 1 ; 1 1 - 0 , 7 7 3 7 5 9 0 - 3 4 0 1 —0,63o-50S7 —6 4 4 ' - 0 , 2 6 3 "1 4 V - 2 •] ₂ [1

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TABLE I I I C o e f f i c i e n t s a n d T e r m s f o r J , P / D , n„ o f O p t i m u m E f f i c i e n c y E q u a t i o n s f o r t h e W a g e n i n g e n B - S c r e w S e r i e s , z=3 ( a ) " K y - J " d i a g r a m ( b ) "Kf, - J " d i a g r a m " o p t X P / D . t , „ = Z A „ , 6 , c l 1 0 ) V d f ; l g R n ) ^ I A £ ' A o f A 0.5,0 k n & c

0.4607739

0 1 0 0

-0.1073401

-3

2 2 0

0,3058801

-1

1 0

A

-0,5738678

- 1

1 1 3

0,5534945

-A

_{d 3 0}

-0,1889090

-•1 2

1 1

-0,2500275

~1 3 0

2 0.4944170

0 1 0 3

-0,4704305

-5 0

A

₀

0,2028408

- y

_{5 0 0}

0,7526680

~4 A

₃

-0,2401106

-3

5 2 0

0,5102754

- 1

4 0 4

-0,5230738

— 4

_{3 4}

A

-0,7522627

1

O Ó

0.1335329

-5 6

1 0.5937434

n

0 0 0

0,1577377

₅

1 0 0

1,0761630

0 0 0

A

-0.1712749

-2

0 3

_^

0,1280068

-1

1

0 0,5516502

—1

_{3 0 0}

0,3326072

Q 1 0 4

-0,3606730

_2 1 2 2

0,1540500

0 o

O

0 4

D

_-0,5554471

_-2

_{2 2 2}

-0,2518935

0 0

1 4

0,2407330

O

0 4 2

-0,1023531

-1

3 f

0 0.1067409

0 0

1 3

0.5200193

-A

_{2 4}

₁

0,2185840

-3

6 0 0

0,8450977

0 1 0 0

-0,3497918

0

9

_{0 0}

-0,1935492

-2

3 1 0

-0,4110455

0 1 0

1 0,2703350

-5

0

A

₀

0,1261515

-2

4

Ó

1 0,3866236

-0,7832063

-6

i

A

₀

0,3866236

-0,7832063

-2

0 0

1 0,5033725

^1

3 0 0

0,5711273

- 1

1 1

"j

-0,3398351

-3

5 1 1

0,3071406

-2

1 1 0

0,8855547

- 1

Ll

₄

-0,3497963

-2

•!

_{2 2}

0,24785''2

- 1

C

1 ^ o p t ' A o f A o.S.c k Q & c

0,5554780

0 0 0

-0,1052947

-A A

-0,1003172

Ö Q d

0,1861682

-4

2 4

1 0,9 105952

—6

-0,8449988

-1

0 Ö

^

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(8)

TABLE I V C o e f f i c i e n t s a n d T e r m s f o r J , P / D , " o o f O p t i m u m E f f i c i e n c y E q u a t i o n s f o r t h e W a g e n i n g e n B - S c r e w S e r i e s , z = 4 ( a ) "KJ - J " d i a g r a m ( b ) " K g - J " d i a g r a m n o p t : 3 . p m . . ^ „ = : A ^ , . [ 1 0 l ' ? l K d f l l g R n ) ^ l A , J A p f Aa.B.c k Q b c 0,5254555 0 1 0 0 0.4471005 - 1 0 0 3 - 0 . 1 1 4 8 9 1 8 - 1 1 1 0 0.1282142 0 1 0 3 - 0 . 2 1 3 2 8 0 6 _o 2 2 1 - 0 . 3 6 6 5 9 1 5 - 3 3 2 1 0.8675564 - 4 2 3 0 3 _0,2739732 _-2 ₄ ₁ ₂ 0,1229769 - 2 6 0 c - 0 , 7 1 7 2 3 1 5 _2 0 1 -.3,7789898 —3 5 1 ₁ 0,6968981 0 0 0 0 0,6480457 -1 1 0 1 0,2976181 -2 1 3 2 -0,1146778 - 3 1 A - 0 , 1 7 5 2 6 2 9 -.•1 2 L Ö - 0 , 3 7 0 8 0 3 5 Ö 0 Ó A 0,2506078 0 1 0 Ó P, - 0 , 3 1 2 1 7 7 0 - 1 0 1 0

'•

- 0 . 2 7 7 5 6 6 2 -1 1 2 1 0,4324128 - 2 0 2 4 1.0227366 0 0 0 3 0,5348510 - 1 O «J 0 0 0.8384501 _ 9 ₁ ₂ ₀ 0,1791173 0 1 0 3 - 0 , 9 8 5 6 4 4 0 - 1 0 1 3 - 0 , 4 1 5 7 1 8 1 - 2 3 1 0 0.7656435 0 1 0 0 - 0 . 4 1 2 9 9 8 3 0 2 0 0 - 0 , 4 4 3 5 1 1 8 - 2 3 1 0 - 0 , 9 4 0 1 5 7 3 - 1 1 0 2 0,2974512 - 5 0 4 0 0,8949327 - 2 2 1 2 0,3742755 - 6 4 4 ₀ Ho - 0 , 1 5 3 4 4 8 3 - 1 0 Ó 1 - 0 , 3 2 1 2 9 4 5 - 2 4 0 0 0,5551380 - 5 1 A 1 0,1645377 - 1 1 1 0 0,1091100 0 3 0 - 0 , 1 5 0 6 3 9 4 - 2 5 0 *S 0,1281112 - 1 0 0 4 0,5191277 2 A 3 • o p t : 3 , P / D . - j , = S . A q E , < . nO)''(KnP(lgRn)^A , ; A / Aa,Ei,c k a 5 c 0,5984091 0 1 0 0 0,2404530 - 1 1 1 0 - 0 , 4 3 1 5 7 0 3 -1 1 0 A - 0 , 1 5 7 9 9 3 9 _-1 1 2 i 0,5360214 -1 4 0 0 - 0 , 5 3 5 6 5 6 1 - 3 2 n 1 0,7223284 - 1 1 1 2 3 _0,8052241 _{- 4} ₁ 4 1 - 0 , 3 4 5 0 6 8 0 _A 0 .1 ₄ 0,2558930 - 4 0 4 - 0 , 7 4 3 6 0 1 1 -A 0 3 0 - 0 , 2 2 8 6 0 3 8 - i 3 1 Q 0,3339343 - 2 2 n <.

₆

0,1368944 - 3 0 3 - 0 , 1 4 1 9 1 1 8 - 4 1

_^

0 0,4844370 0 1 0 0 0,4401873 - 0 , 6 7 7 0 1 9 3 0 0 0 0 0,4401873 - 0 , 6 7 7 0 1 9 3 0 0 0 A 0,9958758 - 2 0 1 Ö 0,1355595 0 2 0 0 0,1265165 - 3 1 A ₁ 0,4009261 -1 •2 Ö A P, - 0 , 3 0 3 7 7 2 3 -A 0 A Ó D 1,0916294 Ó 0 0 3 - 0 , 1 3 8 3 4 4 5 - 1 1 2 1 - 0 , 1 9 3 8 1 6 5 0 0 0 1 0,1301724 0 1 0 3 - 0 , 2 0 1 0 6 4 6 - 1 0 1 1 0,2210511 -A 2 4 ₀ 0,3394758 - 1 6 0 - 0 , 9 9 8 9 5 9 1 - 2 5 1 0 1,1322024 0 1 0 0 - 0 , 5 8 0 3 0 9 4 0 2 0 0 0,8604841 - 3 0 Ó i j 0 0,1010791 0 0 0 A 0,1215773 0 4 0 Ö - 0 , 1 8 9 5 5 2 4 - 4 1 4 4 - 0 , 1 0 3 2 0 5 1 0 0 0 3 - 0 , 1 1 8 3 1 4 0 - 1 0 1 1 0,8771402 - 3 1 O ₁ - 0 . 3 7 4 5 5 6 7 -1 5 0 0 - 0 , 8 7 7 4 5 0 1 - 4 0 A 0 0,3290485 -3 6 1 0 - 0 , 1 3 2 9 9 5 5 0 1 0 3 0,5390386 - 1 2 0 4 - 0 , 3 3 8 2 7 9 8 - A ₂ A 1 - 0 , 1 5 3 0 2 3 0 - 1 1 1 0 " o p t : 3 , P / O . q , = S A ^ ^ J 1 0 ) ' ' ( K / d g R n f l A ^ ' A p f ° o p : 3 . W D , q „ = I A j , { , j . ( 1 0 ) ' ^ ( K n . P ( l g R n ) '

W

' A o l " * a.3,c k a B c A a,S,c k a B c 0.7729435 - 1 1 0 0 0.2484554 0 1 0 0 0.1739259 0 0 0 1 - 0 . 2 5 5 1 8 7 9 —4 2 lO 0 - 0 , 4 4 7 6 1 7 9 _c; 2 0 0,1381597 - 1 0 0 A - 0 , 2 5 4 0 9 3 0 - 1 0 1 1 0.1041314 - 3 2 Ö 0.2221071 - 1 1 0 - 0 . 1 5 1 4 5 1 2 _ 9 p 2 3 - 0 , 1 9 5 9 3 9 4 - 2 1 1 f 0.2705904 - 1 0 0 0 0.5235134 - A 0 3 0 - 0 . 1 5 5 7 3 2 9 - 1 1 0 A •3 _{- 0 , 2 5 2 1 0 0 4} _{- 3} ₄ _A ₀ _J _{- 0 . 1 9 2 1 8 5 7} _{- 2} 2 1 - 0 , 1 0 3 7 3 0 2 - 9

T

-] _0.3075596 _{- 2} ₂ _i" ₂ 0.2641646 - 3 0 0 2 • 0.3243079 —0 1 2 n 0,7179332 - 6 3 3 0 0.1201385 —4 2 A 1 0,7561360 _2 _{1 1} 2 -0.2476348 —0 3 1 4 - 0 , 7 3 1 2 9 1 9 - 6 A ₁ A _{- 0 , 8 9 6 7 5 7 2} -S Ö - 0 , 4 5 3 5 5 7 2 —2 1 1 4 0.7596752 -1 _C ₃ ₃ 0,3515023 -3 2 4 0 0.2470013 -1 0 0 0 0.5145109 0 0 0 0 0.1748369 0 1 0 0 - 0 . 1 1 1 8 4 6 5 0 "1 0 1 0.5208730 0 0 0 0 0.4784176 - 5 1 A 2 - 0 . 7 2 1 1 2 5 9 0 0 0 4 -0.5352970 0 0 0 4 0.5999874 -4 fl A 1 0.3907637 - 1 ₁ A _{Q '1AQ3792} _ i

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1 0 0.3385743 —3 G 3 1 0;3Ó223Ö5 _2 2 0 0 - 0 . 5 6 2 4 0 6 1 -2 1 1 1 - 0 . 1 2 7 3 3 1 1 - A 0 A 0 P/ 0.5159384 —3 3 0 0 1.2705472 0 0 Ö 3 D 0.9281620 0 _{0 0 3} P ; ^ _{- 0 . 1 0 4 9 8 4 8} _{0 0} ₁ ₁ 0.1320064 0 1 0 2 - 0 . 1 9 0 3 9 7 4 - 2 2 2 0 - 0 . 9 0 0 5 8 9 3 - 2 0 2 1 0.2233034 - 3 6 0 0 0.1436019 — 0 2 2 0 - 0 , 2 5 3 7 4 0 2 - 3 5 1 0 - 0 , 1 3 9 5 8 3 9 - 5 1 A 0 0,6324610 _3 3 2 0 - 0 , 5 3 1 4 3 3 8 - A 0 _i 0 0,2471137 —2 3 0 A - 0 , 1 2 0 0 5 0 1 - 2 2 ₁ 0 : - 0 , 4 3 7 0 3 6 9 - 3 2 2 1 - 0 , 4 8 2 0 5 8 8 - 2 1 1 3

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0,1242778 0 1 0 0 i 0,4123315 - 1 0 0 0 0.7531501 _ i ₀ ₀ ₀ _0,3555346 ₀ ₁ ₀ ₀ - 0 . 1 7 5 2 2 9 0 - 1 2 0 0 - 0 , 8 1 0 1 5 7 4 - 1 2 0 0 - 0 , 2 5 4 3 6 0 6 - 4 ₃ 1 0 • 0,4180808 - 2 0 2 0 - 0 , 2 3 4 6 7 5 0 - 1 0 0 2 : - 0 , 2 8 8 4 5 2 3 0 0 0 3 - 0 , 3 9 2 3 7 4 8 -3 4 A 0 0,7018972 - 2 3 0 0 0,4186035' - 4 Ó 3 2 0,1490119 - A 1 4 2 ' l o 0,1599013 - 2 3 0 0 I o - 0 , 1 1 5 5 0 5 0 - 7 6 0 q 1 0,4258094 - 7 A Q 0 I o 0,1875996 0 0 0 A 0,2728530 - 2 1 1 0 - 0 , 8 9 6 4 1 1 2 - 5 1 A A - 0 , 7 0 1 8 4 8 5 - A 4 0 0 - 0 , 3 5 2 5 4 6 4 - 3 0 3 Ö 0,1220182 - 5 0 0 0.9219241 - 2 0 1 1 - 0 , 5 0 5 6 9 6 4 - 2 1 0 ₁ 0,5903057 - 5 2 2 '1

(9)

TABLE V ( a ) "K J - J " d i a g r a m C o e f f i c i e n t s a n d T e r m s f o r J , P / D , E q u a t i o n s f o r t h e W a g e n i n g e n E n ^ o f O p t i m u m E f f i c i e n c y - S c r e w S e r i e s , z = 5 ( b ) "K« - J " d i a g r a m " o p t J . P / D , n ^ i Z A g 5 ^ 0 ) ' ' l K ( j ) ° ( l g R n ) ? ( A E ; A o f A Q.6.C k 0 6 c 0.5332422 0 1 0 0 0 . 1 0 4 9 8 2 1 - 3

_p

3 0 0.9664782 - 3 0 C 3 0.7322604

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0 1 2 - 0 . 2 5 2 9 5 1 2

_{- T}

3 0 1 - 0 , 7 2 8 8 1 5 0

4

0 - 0 . 1 9 2 9 3 1 2 0 1 0 3 - 0 . 4 8 0 3 3 2 7 - 3 1 2 1 0.7828406 - 2 ₄ 0 0 - 0 . 5 0 2 9 5 3 2 -A 4 2 0.5653117 0 1 0 9 -C.337229S - 1 2 1 - 0 . 1 9 0 4 7 3 3 0 • 0 1 0.5559'i87 - 1 2 0 0 0.1043545 _2 2 3 2 - 0 , 2 1 2 1 9 0 3 -3 2 3 0.3535827 _ , i "1

c

0.5691525 0 0 Q 0 0.3587835 0.1961538 0 1 0 0 0.3587835 0.1961538 - 1

_c

1 0 - 0 . 2 3 9 5 1 7 9 0 0 0 q - 0 , 5 1 8 3 5 4 2 - 1 1 1 1 - 0 , 1 9 7 3 0 0 0 - 3

_c

2 1 - 0 . 6 6 5 5 2 1 9 - 3 1 3

\

0,8625717 0 0 0 2 u _{- 0 . 7 6 6 1 5 2 8} _-1 ₂ ₃ 0.1611131

_Ó

2 0 0 - 0 . 5 4 5 8 3 2 4 0.1064705 - 1 2 1 - 0 . 5 4 5 8 3 2 4 0.1064705 - 1 1 2 2 - 0 . 1 0 3 7 7 9 4 0 0 1

1'"

0,45848^9 - 2 £ 2 2 0,1638075 - 3 2 - 0 , 7 5 5 9 1 5 3 - 3 3 1 0 0,7465203 0 1 0 0 - 0 , 4 3 1 2 5 8 9 0 2 0 0 - 0 . 7 0 2 5 5 5 2 O O 1 0 - 0 . 3 5 8 7 9 2 4

_-T

1 0 3 0.1057080 -5

_A

4 0 0,3169880 - 5 0 4 0 - 0 . 2 3 7 7 0 2 8 - 1 2

_Ó

1 - 0 . 1 8 5 5 6 8 0 - 5 1 4 3 0.1254864 0 3 0 0 0,1927812 - 1 1 ₁ 0 - 0 . 4 2 7 7 2 7 4 - 2 5 0 1 - 0 , 2 7 3 7 0 4 8 - 1 0 0 2 - 0 , 5 6 4 7 4 5 3 - 5 1 4 0 0,1126060 - 1 2 ₁ 1 0,5678112 - 3 5 0 2 0,3044791 0 1 3 D o p t : 3 . P ' D . % = I A ^ E „ c ( 1 0 ) ' ' ( K n f t l g R n ) ' ' ( A E ' A o l ' Ao.!),c k a 6 c 0,7205813 0 - 0 G - 0 , 1 7 0 2 0 2 5 ₁ ₁ G - 0 , 4 7 6 9 7 0 5 - 2 0 0 dl 0,9998995 0 1 2 0.8277942 -A A O ₁ -0.397A9S1 —5 1 4 0.1557334 —o 3 2 0 - 0 , 2 3 7 9 3 8 3 - 5 2 4 41

:

0.2189303 2 O ₆ 0,3059576 1 ó 4 0.5293357 —£ 4 4 2 —0,3761922 _^ 3 ü - 0 , 5 5 2 3 2 7 1 - 4 2 1 0,6131527 —2 o j 1 -"0,3184312 _p 5 - 0 , 1 5 8 3 7 5 4

_{- T}

0- 4 0.8568055 0 1 0 0 0,3491824 0 0

_{c c}

- 0 , 2 4 0 4 4 0 5 - 2 'J 1 0 - 0 . 3 4 3 5 3 5 9 C 0

_c

- 0 , 2 0 6 3 5 3 0 1^ 1 1

_ï

- 0 . 3 4 2 7 2 4 2 _•] Q 1 1 - 0 , 1 9 0 8 0 3 0 —"j 2

_c

0 P; _0,38.57208 _{— I} ₃ _•j ₀ D _0,8724737 _ i ₁ ₁ ₃ 0,7585332 0 0 u 2 0,997A66A —2 1 2 1 - 0 , 1 0 9 3 8 8 2 r; 0 3 0,2529910 —1 6 0 0 - 0 , 3 0 3 3 3 7 2 — I 4 1 0 0.5013520 -1 2 1 n - 0 , 7 7 4 0 5 8 1 1 2 0,3930373 0 1 0 0 -0,4795.370 0 2 0 0 0.2573832 0 2 0 0 . 1 1 8 1 9 5 3 0 o' 4 0.7333303 _{- Ï} 5 0 _ó 0,2917360 - 2 0 2 2 - 0 , 3 4 7 3 9 5 5 - 1 6 0 0 0 , 2 8 7 9 0 4 4 - 3 1 3 1 - 0 , 7 8 8 4 9 2 1 6 1 A - 0 , 2 0 6 5 6 3 5

_Ó

0 0 3 - 0 , 2 3 4 5 4 1 0 —O 0 3 3 - 0 , 2 2 4 7 8 6 7 — ^ 0 3 0 - 0 , 6 7 9 0 5 5 9 - A

_A

₃ 0 - 0 . 3 5 7 8 3 7 0 - 2

_i

1 4 0.1683.353 - 3 6 2 0" " o p t : J , P / D . q o = Z A ^ e , ^ ( 1 0 ) ' ; i K d ) ° | l g R n ) ' ^ l A ^ / A o f '^a.6.0 k G c 0,S3922A5 - 1 1 G G -0,6373271 2 1 0 0,1702805 0 0 0 1 -0,2385259 -1 0 1 1 0.1352095 _A 3 2 0 -0,2712484 - 4 2 3 1 - 0 , 5 0 7 1 8 3 5 -5 4 A 0.7136523 -5 0 A

_c'

:

- 0.5132825 p ₂ ₀ 0.3122521 —2 "1 4 0.3245055 -5 2 2 - 0 . 4 3 4 3 2 1 2 - 3 2 r _Z 0.7501510 - 7 5 ï ó - 0, 6 7 7 7 5 0 4 -c 2 4 4 0,1368253 - 2 1 u 0,1736411 - 1 0 0 G 0,7200681 0 0 0 n 0,7557011 - 1 1 0 G -0,4931970 - 2 0 1 0 - 0 , 4 4 8 6 7 7 4 0 T; 0 4 -0,1733079 - 1 ï 1 1 - 0 . 1 0 1 8 5 7 3 -1 1 0 3 - 0 . 1584^55 _ 9 2 G 2 0.2639413 2 4 2 ü _{- 0 . 1 7 8 7 1 8 4} _{- ï} ₀ 4 0.1424793 1 1 2 0.3052455 - 4 3 0 0 0.9714305 ü 0 0 -0.519A3S3 -1 G 1 1 - 0 . 3 1 7 0 5 0 7 —5 1 4 - 0 . 1 6 0 0 3 2 1 2 0 0.4681043 - 5 1 A 0 0.1045435 G 1 0 0 0.7455531 0 0 n - 0 . 7 0 5 6 5 7 4 9 0 0 0,3316174 — 0 0 ₁ 0 - 0 . 5 8 7 7 7 0 5 - 2 0 0 3 0,8518131 -G A A Q - 0, 1 0 3 4 6 5 1 - 2 ó 1 2 1= -0,2186468 _3 3 Q 0 1= _{- 0 , 3 2 0 5 7 3 4} ₂ ₀ - 0 , 1 3 3 4 5 5 3 - 8 4 0 0,1585613 - 7 A 3 0 0,5847930 - 2 1 1 0 - 0 . 3 2 1 7 9 6 4 _2 1 0 3 0.1975365 _A 1 3 1 -0,1979931 - 2 2 1 g 0,2232209 - 7 5 3 —n 1816SG0 - 9 3 0 C e p , : : . P / D . r j o = 5 : A „ g J 1 0 f t K n f ? i l g R n ) ^ l A ^ A o f k a

fi

c 0,2552250 0 1 0 0 0,9045851 _ A 2 2 0 0,14947A9 ö 0 0 3 - 0 , 3 7 5 0 6 5 5 - 2 1 2 1 - 0 , 5 2 1 8 0 9 2 - A 2 / ₃ 0,2945789 - 1 0 0 0 0,2705096 — 1 1 1 3 0.1050257 n —C 2 0 3 - 0 Aa3A3-]i — 1 Q 1 3 0;5"2S3391 —5 2 A 1 0.3329618 _A_ ₆ _ö ₀ - 0 , 5 3 8 3 2 0 6 —0

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' 0.3117-336 - 2 2 - 0 . 5 6 7 7 0 3 3 —2 2 2 3 0.4073371 - 2 d 2 n 0.3163882 _9 1 0 0.2597547 0 1 G 0 0.5237395 0 0 0 0 - 0 . 2 3 0 9 1 3 2 -1 0 1 0 0.4583493 0 0 0 3 - 0 . 4 0 6 2 3 0 3 - 1 ₁ 1 - 0 . 4 1 0 3 4 1 1 — i ₁ 0 3 0,6167056 - 3 2 lil 0 P; _{- 0 , 1 1 5 9 9 4 1} — A 2 4 3 D _{- 0 , 6 4 0 0 0 7 2} _{- 1} ₀ _- ₃ 0,3598344 - 1 ' 0.4105739 — A _ó ₄ 3 -0.15A9239 —3 2 3 1 0,5982779 _ A 6 0 0 - 0 . 1 2 6 1 4 0 5 - 3 4 2 1 0.1330778 - 3 3 9 - 0 . 4 1 5 9 6 8 5 - 4 3 3 3 0.1409157 _ - i _{C 0} ₀ 0.3830541 0 1 0 0 - 0 . 1 0 7 3 8 1 3 f' 2 0 n 0.1733883 - 1 0 1 n 0.3746578 - 1 G 0 A 0.1543633 - 1 3 0 g - 0 . 1 8 4 7 7 9 2 - 2 0 2 2 - 0 . 2 5 3 3 9 3 9 - 4 1 4 1 - 0 . 3 9 4 8 9 8 0 ó 6 3 1 - 0 , 2 5 1 0 5 3 1 _ A ₀ ₄ ₀ 0,5503310 - 4 0 A ₁ - 0 , 1 5 7 3 4 1 2 - 2 2 Ö 4 0,7355626 - 5 2 2 2 0,2552245 - 3 1 3 1 -0,70.34355 - 1 g 0 3 -0,917-5973 - 3 4 n 0

(10)

TABLE V I

C o e f f i c i e n t s a n d T e r m s for J , p / ü , o f üptii,ium E f f i c i e n c y E q u a t i o n s for tiie W a g e n i n g e n B-Screw Seri"'., z^5 ( a ) - J " " o p t 3 , P . ' D , . ] „ = E A ^ J 1 0 ) ^ ( K j j f l l g R n k a B c 0.6306142 0 1 0 0 0.2711837 - 2 2 2 0 0.2522845 - 2 3 1 0.5553534 - 5 0 4 0 - 0 . 1 1 8 3 1 9 3 _ d 2 4 0 0.1194451 - f 0 0 3 - 0 . 9 2 6 1 3 5 1 -A 0 3 1 - 0 . 2 6 2 0 7 3 5 - 4 4 4 3 3 - 0 . 2 4 1 8 9 9 4 - 1 1 1 0 0.1130209 0 1 0 3 0.9819397 - 3 6 1 0 - 0 . 5 3 3 9 2 7 6 —2 3 1 0 - 0 , 1 5 8 3 0 3 7 - 4 ₆ ₃ 1 - 0 . 4 4 6 9 0 4 5 - 2 2 3 2 - 0 , 3 3 5 3 2 5 9 - 2 2 2 1 0,5673407 - 3 4 3 2 0,5144046 0 0 0 0 0,5388794 0 -i 0 0 0,1512496 -.1 Ó 1 0 - 0 , 3 2 7 9 1 6 7 _ i 3 1 1 - 0 , 1 7 3 0 8 1 4 Ö 0 0 o - 0 . 1 0 2 5 8 7 0 0 0 1 1 0.2281315 - 3 6 2 2 - 0 . 3 1 7 3 5 6 6 0 1 0 1 - 0 . 4 2 5 2 9 5 5 - 3 5 2 1 J _0.7731034 ₀ ₀ _C ₂ 0.5943654 - 1 0 1 2 0,4840510 - 2 6 0 0 0.9695528 -A _o A ₂ - 0 . 1 3 3 9 9 5 9 - 3 A 3 - 0 , 2 1 1 0 0 9 2 - 1 3

Ó

3 - 0 , 3 6 7 0 0 5 8 - 5 6 A 2 0,5557046 - 1 2

Ó

0 - 0 , 4 6 2 5 4 7 3 - 4 1 3 0 0,7876198 0 1 0 0 - 0 , 4 2 9 4 9 1 9 0 2 0 0 - 0 , 1 0 4 3 6 9 7 - 2 4 1 0 - 0 , 5 5 3 8 3 9 1 - 4 0 3 0 0,3572548 - 5 5 2 0 - 0 , 2 9 5 5 3 9 0 _2 0 0 0 0,4595948 - 6 6 3 0 To 0 4060019 - 6 3 4 1 To - 0 : 5 5 5 1 1 8 4 - 1 0 0 3 0.1297142 - 2 0 2 1 0.4768921 - 3 1 2 1 0.. 1148473 0 3 0 0 0.8968338 - 2 1 1 0 - 0 . 8 2 2 8 0 7 2 - 2 4 0 0 ( b ) - 0 " d i a g r a m Dopt 3 , P / D , . ^ ^ = I A ^ , 1 1 0 f t K „ l ° i g R n ) ? l A '^o.B.c k 0 B c 0.7652501 0 1 0 0 - 0 . 1 3 5 8 9 3 4 - 4 1 4 1 - 0 . 2 3 5 7 3 5 7 - 1 1 1 0 0,1000005 - 1 6 1 - 0 . 6 4 5 0 5 4 0 - 4 0 1 0 - 0 , 1 8 7 2 5 3 1 —1 1 1 0,1964523 - 1 2 1 3 0,1453374 - A A ₀ 3 0.1153906 - 1 2 i 0 - 0 , 1 1 4 3 6 2 2 O — 2 4 3 - 0 , 9 0 5 3 2 7 3 - 5 6 4 2 0,1225043' - 3 3 4 3 - 0 , 1 7 8 4 5 7 8 - 2 3 2 3 0,7893732 O u 1 0 0 0,4970097 0 0 0 0 - 0 , 1 5 0 1 7 8 2 0,2259815 - 1 0 1 0 - 0 , 1 5 0 1 7 8 2 0,2259815 0 3 1 1 0,7096509 _2 5 1 0,1806252 - 4 1 4 0 0,7471231 -3 A ₃ ₂ 0,3222735 0

c

0 3 _ 0 , 4 A 9 4 4 5 7 _{- 1} ₀ 1 1 0 , 1 0 2 5 i 9 5 - 2 0 2 2 - 0 , 2 9 3 9 1 8 1 - 4 ₃ ₄ 1 0.1067715 0 6 0 0 - 0 , 1 2 4 4 1 3 4 0 A ₁ ₁ - 0 , 2 6 3 0 5 5 2 —A 5 3 - 0 , 4 9 5 6 6 6 6 - 2 1 2 3 - 0 , 1 3 1 3 0 7 7 0 5 0 0 - 0 , 1 3 1 3 6 7 4 0 2 1 1 -0.1080-143 - 2 6 1 0 1.1312350 0 •j 0 0 - 0 . 7 4 0 3 3 4 4 0 2 0 0 - 0 . 2 1 7 9 1 7 5 _ 9 ₀ ₁ ₀ - 0 . 2 3 1 8 0 6 6 - 4 4 3 1 - 0 , 1 2 4 1 2 0 5 - 3 Ö A ₁ - 0 . 2 2 9 5 8 5 5 0 C Ö 0 - 0 , 1 6 2 2 3 5 2 - 1 0 0 3 Ho - 0 , 2 3 3 8 2 6 0 - 1 0 1 2 0,5719454 - 6 6 4 ₁ 0,.3536376 0 A _{Ö 0} 0,1251212 - 1 1 1 1 0,1289975 - 2 0 3 1 0,4672955 - 1 6 0 0 0,2344375 -A 0 4 2 - 0 . 1 4 8 2 1 9 3 - 1 1 -0 " o p t . • a, P / D , g ^ = E A ^ , . ( 1 0 ) ' ' l K d ) ° | l g R n ) ^ ( A f / A p f 0,5.0 k c 8 c 0.1005598 0 1 0 0 - 0 . 8 5 1 4 7 4 1 —3 2 1 0 0.3655449 -1 0 0 0 - 0 . 8 5 2 0 4 1 7 -3 0 1 0 0.1767.506 _4 5 2 0 - 0 . 6 0 8 7 5 1 4 _5 p J 0.3089954 - 6 3 2 0,1168335 0 0 0 3 3 0,1667115 -2 0 2 1 - 0 . 2 6 9 0 8 5 6 -2 /2 0 2 0.1346512 _ T ₄ 4 0 -- 0 . 1 3 4 4 5 2 3 - 5 4 0 0,2315559 - 7 0 0 - 0 , 6 0 8 2 4 2 0 - 6 4 2 0 0,7210392 -5 Ö 0 3 1 - 0 , 3 8 6 8 4 8 2 -2 0 2 2 0,2855215 0 3 - 0 , 7 3 4 5 9 5 5 - 6 3 3 0 0.3899241 0 0 0 0 0.5535354 -1 1 0 0 - 0 . 3 5 2 7 6 7 0 -1 0 1 0 0.1313132 -3 2 0 0 -0.6272941 -3 2 1 1 0,7855771 - 6 5 0 3 0,3596565 0 0 0 3 - 0 , 5 5 3 0 4 1 0 - 1 0 "i 0,8181453 -A _{3 1} - 0 . 1 0 5 2 4 1 1 -3 0 4 2 - 0 . 1 3 5 6 3 4 4 -A •! 4 2 0.4334136 - 7 5 0 0 -0.3A79772 _g 6 2 0

-0.2955791

- 6 2 3 - 0 . 1 0 4 0 7 4 6 - 5 4 _{2 3} 0.7790974 Ó •3 1 - 0 . 3 8 3 3 7 6 5 - 6 5 0 0 0,3520865 -5 3 3 3 0 l A z W R Q A ₀ ₁ n 0 O!5353447 - 1 0 0 0 - 0 , 2 3 4 9 7 0 2 - 1 9 ₀ ₀ 0,2594759 - 6 •3 1 0 - 0 , 3 5 5 8 0 6 4 — J 0 _{4 4} 0 0,2713064 -2 0 0 0 - 0 , 3 3 6 9 2 9 5 -5 0 4 0 - 0 . 1 9 6 2 0 2 1 — «J 4 0 0 0.1545823 -11 6 4 1 -0.1103269 -10 6 Q ₃ 0,8722501 — _{1 1} 0 0,7597330 -5 5 0 0 0,1200927 - 1 0 5 A ₂ - 0 , ^ 4 4 9 8 5 0 - 1 0 0 3 0,1042038 -2 0 2 1 -0.1246079 — T j _e n _Ó 0,57^4639 —c.i _{2 1} ^ o p t : 3 . P / D , r i „ = I A Q B ^ 1 0 ) ' ' l K r , f l l l g R n ^ i A E / A o f A G,Ö,0 k a S 0 0,2735311 0 1 0 0 - 0 , 8 0 7 0 9 0 0 - 3 2 2 0 - 0 , 2 0 4 0 4 4 4 — £j ₃ _{4 0} 0.9756063 - 2 3 4 1 0.7817073 - 2 3 ,1 9 -0.1526303 -1 4 2 0,1452569 - 3 3 3 0,5598400 - 2 3 0 2 3 - 0 , 1 5 2 5 9 7 9 - 3 0 4 2 0,7514405 - 1 0 0 1 0.7578748 - 5 1 A 0 0.4638177 -A ₆ _{0 0} 0.1477751 -3 0 4 3 - 0 . 7 7 3 2 7 3 9 -3 3 2 2 - 0 . 4 4 8 8 9 ' i 5 _o 0 3 2 0.4786724 -A ₀ _{4 1} - 0 . 4 8 5 5 2 0 3 _{— u}Ö ₄ _{1 0} 0.5176114 _—00 ₃ _{2 0} 0.2583117 0 1 _{0 0} 0,5358006 0 0 0 0 - 0 . 3 4 7 0 3 0 1 - 1 0 1 0 -0.7285307 - 3 3 2 1 - 0 . 4 8 5 6 2 8 0 - 6 6 3 3 0,2060787 _3 1 S 0 - 0 , 2 3 0 1 0 8 4 - 2 4 0 2 -0,1572992 - 7 6 4 C ° • 0.2351155 -3 4 1 3 ° • 0,1106257 - 4 5 1 0 0.5735830 -7 6 4 2 - 0 , 6 5 5 9 7 7 0 - 4 " 4 1 - 0 , 8 3 6 4 4 5 3 - 1 1 Ö 3 0,2655373 2 4 1 0,3985554 Ö 0 0 3 0,4502963 q 1 2 - 0 . 4 2 1 0 5 3 4

-

0 1 ? 0.2058008 - 1 0 0 0 0.4224120 0 1 0 0 -0.1357114 0 0 0 0,7463728 _2 Ó 1 0 0,5502530 6 1 1 0,2422526 _1 _{0 0} -0,3512793 - 5 2 4 2 - 0 , 1 4 7 6 7 0 2 0 0 0 3 0,2213500 - 1 0 1 2 0,1437852 - 2 2 1 1 - 0 , 1 8 5 4 7 2 7 _ 9 A _{0 0} - 0 , 3 2 5 8 4 1 7 -0,6504327 - 4 5 1 1 - 0 , 3 2 5 8 4 1 7 -0,6504327 _2 1 0 1

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