An approximate and simple formula concerning four bladed propellers of single screw cargo ships

(1)

RCHEF

AN APPROXIMATE AND

SIMk

FORMULA CONCERNING

FOUR-BLADED PROPELLERS OF

SINGLE SCREW CARGO SHIPS

B

SIR AMOS L. AYRE, K.B.E.,

Hon. Fellow

A Paper read before the North East Coast Institution of

Engineers and Shipbuilders in Newcastle upon Tyne

on

the 30th April,

1945, with

the

discussion and

correspondence upon it, and the Author 's reply thereto.

(Excerpt from the Insituiion Transactions, Vot. 61)

P1EwcASTLg UPON YNI

PUBLISHED BY THE NORTH EAST COAST INSTITUTION

OF ENGINEERS AND SHIPBUILDERS, BOLBEC HALL LOPON

H, & P. N. SPON, LIMITED, 57, RAYMARKET, S.W.I

'945

Au

U, VI

(2)

AN APPROXIMATE AND SIMPLE

FORMULA CONCERNING

FOUR-BLADED PROPELLERS OF

SINGLE SCREW CARGO SHIPS

_By

SIR AMOS L. AYRE, K.B.E,

hon. Fellow

A Paper read before the North East Coast Institutibn of

Engineers and Shipbuiiders in Newcastle pon Tyne

on

the 30th April,

1945, with the

dlscussion and

correspondence upon it, and the Author's reply thereto.

(Excerpt from the Institution Transactions, Vol. 61)

NEWCASTLE UPON TTh

PUBLISHED BY THE NORTH EAST COAST INSTITUTION

OF ENGINEERS AND SHIPBUILDERS, BOLBEC HALL LONDON

E. & F. N. SPON, LIMITED, 57, HAYMARKET, S.W.I

(3)

THE INSTITUTION IS NOT RESPONSIBLE FOR THE STATEMENTS MADE, NOR FOR TEE OPINIONS EXPRESSED, IN S PAPER, DISCUSSION AND AUTHOR'S RELY

W

k16

MADE ANb PRINTED IN GREAT BRITAIN

(4)

AN APPROXIMATE AND SIMPLE FORMULA

CONCERNING FOUR-BLADED PROPELLERS

OF SINGLE SCREW CARGO SHIPS

By Sm AMOS L. AYRE, K.B.E., Hon. Fellow.

30th April, 1945

Syropsis.A simple short formula for use in

approximating propeller

dimensions, etc., in the early stages of design, is introduced, i.e.,

D5

IO x p = s.h.p. at propeller,

where

D = diameter of propeller in feet, N = revolutions of propeller per minute, p = a coefficient varying with pitch-ratio.

AT a much earlier moment than the running of a propelled-model

experiment, it is usually necessary to approximate the main

dimen-sions of diameter and pitch and to relate the corresponding proposed rate

of propeller revolutions to conditions that fundamentally affect engine

design, e.g. piston speeds and pressures.

It is the preliminary design

work with which the following is concerned.

The derivation of the formula, as well as the. values of the variable

coefficient, p, now referred to, were based on full-size work, and, in its empirical

nature, is not otherwise related to the normal science as applied to propellers.

The paper is, therefore, restricted to the simple features that are referred

to as having been the means of derivation.

From analyses of progressive trials of vessels driven by steam reci-procating machinery it had long been observed that, over a fairly wide range

of the higher speeds, the power (i.h.p.) varied, very closely, as revolutions, per minute, cubed, (N8).

The eventual accumulation of data concerning a large number of

propellers having a pitch-ratio,

(s), of unityat one time termed "square"

propellers, and, in fact, somewhat popularshowed that, where there

was.

reasonable similarity in general design, i.e. in such matters as blade thickness, boss diameter, and wash-back at the trailing-edges, the powers absorbed by the

propellers, at a given rate of revolutions, varied as diameter to the fifth.

power, (D5).

-Apart from the sensitiveness of wash-back at the trailing-edgewhich, when of abnormal amount, would seem to have had an effect identical with

that produced by a reduction in pitchthe various other changes,

even

including those of blade outline and extent of surface within reasonable limits, did not point to appreciable influences.

Excluding abnormal cases, which usually were accompanied by an Admiralty coefficient either suspiciously high or low, no doubt resulting from

inaccurate measurement of power, it was fairly well established, _{as an average}

condition over a fairly wide range of cargo vessels, that, for propellers of the same pitch-ratio,

Sh.p. at propeller cc D5 x JsJ8

(5)

352 AN APPROXIMATE AND SIMPLE FORMULA CONCERNING

For the propellers of unity pitch-ratio, which covered a range of about 8' to 19' diameter, it was found that:

S.h.p. at propeiler

D5x N3

x (p = 33)

could readily be accepted. These propellers had segmental blade sections and the coefficient, p = 33, corresponded tothe material being of bronze (solid) and with normal scantlings for thatmaterial.

It remained to ascertain the variation

that resulted from the use of

other pitch-ratios, a slow process having regard to the need to collect the

neces-sary amount of dependablefull-size data. Eventually, however, it was possible

to conclude that for such

propellers, reduced to a solid bronze basis, and

having a moderate amount of wash-back at the. trailing-edges, the variation of p averaged as follows

P

D 60

80

1-00 1-20 l4O

l60

1-80

p

..

.4

240 -330 433 .550

65

-830

Having regard to the fact that these results were determined from data

covering a fairly, wide range .of ship form as well as speed, it had beenexpected

that, at any particular

pitch-ratio, the variation of p would have been more

appreciable than it was. Some of the variation undOubtedly arosefrom

inac-curate power measurement,and possibly from the use of factors formechanical and transmission efficiency (m.t.e.), which were somewhatarbitrarY, in addition'

to other circumstances

applying to the trial trips from which the values of

p were derived; butit was not possible, with the, amount Of reasonably depend-able full-size data availdepend-able, definitely to analyse this feature.

With the

growing amount of model propeller 'data now becoming available there should

be scope for this. It is quite likely that it,.may be found possible to determine a variation of p. relative to the block coefficient of the afterbody, and, perhaps,

associated with the length and breadth of the vessel. Iii any event, from the

full-size 'analyses, it would 'seemthat such variation may not be found to be of

substantial extent.

In making these analyses, it was necessary to anchor the ..vork to

what was, no doubt, an arbitrary

variation of mechanical and transmission

efficiency (m.t.e.i. Practically all the vessels dea.lt with were driven by steam

reciprocating machinery and for which, on the basis of modern installations, the following values at maximum poyver were used:

Designed max. i.h.p. 500 1,000 1,500 2,000 2,500 3,000

mte

810 855 880 890 896 900

Having regard to the' values Of p which were derived for the propellers of

smaller diameter: driven by the smaller power units, and those of.larger diameter driven by the larger power units, i.e. similar values of p forpropellers .that were,

to scale, physically

similar this variation of mt e was

found to be justified In the cases of engins aft, the above values were incre.sed by 005.

-Analyses 'Of a large number of trials, covering a wide range of-size of

power units, indicatçd that over

the raiie of '55% to 100%

of the maxifnum

power, the. .average condition was such that i.h.p. oc N3, and ,s.h.p. (at

pro-peller) produced over that range, by the use of the values

of mt e given m

the preceding paragraph, was very close to a variation m accordance with N8 that extended throughOUt the whole range down to zero (This is illustrated

in the accompanying

diagram.) Accepting this

straight-line variation of

s.h.p. to N3, and after, taking

opportunities to run some trials extended to.

(6)

H.PI

9

B

4

FOUR-BLADED PROPELLERS OF SINGLE SCREW CARGO

sms

353

N i00% .F I.H.P 90% 80% 70% 60% 50% 30% 10% 1 VI.. ,o 00

I.H.P, 5.H.R ç te. FOR STEAM REC PROCATII4G ST

D0IGNED MPXIMUM. 1.0001.14.4 Ar 100

A

9. 00 00 00 1838

0mriiu

_{r I}

-44

flL00I

3OO 404000 500000 404000 TC000b 400000 900000 LVII

'0 100

7 8 .p.cc

(7)

354 AN APPROxIMATE AND SIMPLE FORMULA CONCERNING.

the derived value of p depended, the following variation of m.t.e. for the range of power below 55 °/ of the maximum:

%ofdesignedmax.i.h.p.. 10 15 20 25 30 35 40 45 50 55

Max. m.t.e. x c=m.t.e. at lower

powers-=

_..

75 84 89 93 95 97 98 99 P995 l0O Whilst this variation may not be in close 'accordance with conventional thought on the subject of m.t.e., it can only be said that it represents the fair

average result of many analyses, covering large and small power units ; but it is of course, dependent on the acceptance of s.h.p.. N3 throughout the full range.

§1 2 In the case of vessels driven by. Diesel engines, s.h.p. at the propeller,

was in all cases, and for all sizes of units, taken as being b.h.p. x 97

§13. In the course of collecting the full-size data, much was learned con-cerning the work of conducting measured-mile trials.

This arose out of the

fact that sister vessels, under he same conditions of loading and weather,

and with identical propellers, sometimes produced appreciably different values of p. - Apart from errors in pOwer recording, the princij,aI causes of such differences were found to be:

Tide effect. Rudder effect..

Insufficient running on the straight after turning and on the

approach to the course.

-J'Tide effect was reasonably well' dealt- with by making three runs at each rate of revolution, using the mean of means of the various recOrded figures. In the .course of a progressive trial it is better to make three sets of three runs óach, than five sets of two runs each. Rudder effect largely depends on the helmsman, and is made worse if the vessel enters the mile before "settling

down" after the turn.

Sometimes the helm is constantly on the move up to

about 8 degrees port and starboard, and this in spite of the vessel, when in

other hands and under other conditions, being able to maintain a steady course.

Five degrees of helm, at the same rate of revolutions, requires 2% more power than when the helm is -at midships, whilst at 10 degrees an increase of about .6% is necessary. Cargo vessels of 10,000 tons displacement, having run a maximum distance of two miles beyond the course while at the middle of the

turn, and having. about a mile of straight approach, were found, at speeds

of 10-li knots, not to experience a full restoration of the rate of revolutions until apout half-way over'the course.

§14. For reasons similar to the conditions referred to in (i) and (ii) of the

previous paragraph,the value of p will be a few per cent. higher for the -service

condition, as compared with that during trial if this has been made under

fine weather conditions and precautions taken to avoid (i), (ii) and (iii). Pitching, in particular, will affect the value of p, and this will be experienced to a greater

extent in small vessels of the coaster class.

§15. _{In the case of cast iron propellers, in the new condition, and in which}

all features and dimensionsbut, of course, excluding blade thickness and

boss diameterare identical, the values ofp should, in the average case, be

increased by 6% beyond those given in §8.

16. FOr " built" propellers, it has not been possible to collect sufficient

data from which to determine the difference applying to p. But, it would seem that the effect of the larger diameter of boss, provided it is of smooth formation

and the fore end swept kindly into the boss of the propeller post, together with a suitable cone fitted to the after end, may be to reduce the value.of p by about

(8)

FOURBLADED PROPELLERS OF SINGLE SCREW 'CARGO SIflPS 355 l 7.. Aerofoil blade sections are responsible for a further variation.

Full-size examination Of such propellers, as far as it has been posible to make

Teasonable comparison, with those of segmental blade section, but Otherwise similar in design, points to the effect of aerofoiling being such as tO reduce p b' about 3 %-8 % over the range of 9' to 19' diameter respectively, and with

a tendency to larger reductions when the rate of revolutions is high.

§18.

Variation of pitch throughout the radius of the blade, calls for the

determination of a mean value in connexion with the present subject. The

following method, due to Mr. Kari, has been found to give results. in the form of mean pitchthat did not conflict with that of propellers of uniform pitch

B

= mean pitch.

§19. During the course of the analyses it was interesting to 'observe that, if recognition was given to the differences that arose owing to bronze and cast-iron, also segmental and aerofoil blade sections, as referred to earlier, some of the other varyng circumstances such as

"Small" diameter of propeller relative to breadth and fineness of

form of vessel

"Small " pitch-ratio

(iil) The type of fairing or fin on rudder post, and the amount of clearanàe

bótweén that and the trailing-edges of the propeller;

were features, the effect of which would sern to be taken into account by the

propulsive coefficient.

§20. Sufficient data concerning three-bladed propellers fitted to single

screw cargo vessels have not been available from which to determine any

amend-ment to the formula, or to the value of p, for that type.

§21.

The whole of the foregoing applies strictly to single-screw cargo

ships, butit may be of interest to record jthat when twin-screw propellers are

analysed by the use of the formula, the values of p that result can be said always

to be less than those derived from the same propeller used in the single-screw

vessel. Some cases worked out at as much as 40% less, which indicates a very wide range. Only in one case did it result that the derived value of p equalled that of the single-screw condition, and jt occurred in a very fine-lined vessel, the block coefficient being only 45, the propellers being placed very far aftthe

fore-and-aft centre of the bosses almost coinciding with the after perpendicular of the vesselandset outboard to what may be.described as a dangerous and very

much unprotected positiOn.

It may be reasonable to conclude, from such

analyses as have been made by the use of the formula in the' case of the twin-screw condition, that location of propellers in such vessels is a most funda-mental feature relative to their power absorption capability.

§22. Whilst it is realized that there are many short-comings in this paper (and the Author would apologize for not having taken the subject to a more

finished state, inability to do so arising out of the difficulty to find the necessary.

time) and recognizing how much of the conventional science of the subject is

at

Radius Pitchz S.M. S.M. x Radius' Pitch2 x S.M. x Radius'

20% 1

40% 4

60% . 2

80%

4.

(9)

356 FOURBLADED PROPELLERS OF SINGLE. SCREW CARGO SHIPS

either Omitted or slumped in the short: simple formula, it is, however felt that others may find some mterest in it, and, perhaps some value as has occurred within the Author's experience when it has been necessary to make quicidy an approxirnate, though reasonably, dependable, approach to propeller design in

the early stage. It is most likely that improvement can be sought in directions such as are referred to in §9.

(10)

DISCUSSION ON AN APPROXIMATE AND

SIMPLE FORMULA CONCERNING

FOUR-BLADED PROPELLERS OF SINGLE

SCREW CARGO SHIPS *

The PRESIDENT (Sir Stmssap.s HUNTER):

The Institution is very indebted to Sir

Amos for his short but very valuable paper

it is rOlative to 'a Lubject which appears to have been always One of his hobbies.

I have had the pleasure of knowing and working with him for many years and,

whenever propellers have been mentioned,

he has always been only too pleased to

discuss them in detail and in a progressive spirit.

It appears to me that the' more that

be-comes known abOut propellers, the more

in-tricate the subject; for example, I imagine

one hears more about "singing" propellers

now than one used to and no one is quite

certain whether a bronze propeller will be a

singer" or a "non-singer" until loaded

trials,'are run. It may be that the British

Shipbuilding Researcl Association have this subject in their extensive programme and we are pleased to see the Chairman of the Research Council of B.S.R.A. present

tonight.

I hope that those taking part in the dis-cussion will not be too critical about Sir

Amos's formula and remember that he covers himself in the' title of the paper,

namely, that it is "An Approximate and

Simple FOrmula" for a certain class of

merchant ship 'and a' type of propeller for

that, and also that it relates to the early

stages in the design of a. modern propeller.

Mr. JOHN NEJILL, Fellow:

This paper reminds me of the somewhat

V.

the curves .of C2 to a base of,-given in the

case of the e.h.p. formula. Superimposed

on such curves it might be possible to give

some indicatipn of efficiency, for even in the early stages of the design it is some'

times advisable to be able to assess not

only the power absorption of various

screws but also their relative efficiencies.

Altogether, it seems we 'have a suitable

basis on which to build an approximate

formula which in its simplest 'formula may

be subject to errors of the order of 10% or say 3% on the speed of revolution, but from the mass of data availablO 'it should be possible to narrow down, by 'suitable

corrections, to a formula which may -prove

in practice to be just as valuable as the e.h.p. formula given to us by Sir Amos

in his. earlier papers.

Mr. WILFRID AYRE, Member: if I were to congratulate the Author for strbmitting this simple formula, he might

well think I was fifteen years too

late. I, of course, have had opportunity

during the whole of that time, partly by

clOse collaboration with him, of using the formula to advantage, and to have analysed - full-scale data with, the object of deriving

appropriate correctives to take account of

changes in propeller "working conditions, such as wake 'and slip, which are influenced

by ship dimensions, hull form, propeller

diameter and weather effect. The formula,

in my view, is scientifically sound; it con'

(11)

(12)

which presumably is the one covered in

Sir Amos's formula.

- Open-water propeller tests indicate that

the power absorbed by the screw varies as the diameter to the fifth power; yet.in the open-water experiments propellers of vary-ing diameters are tested at a constant speed of advance which is synonymous with the

speed of the testing carriage. Hence, in

the open-water test, an increase in propeller

diameter is not accompanied by a change

in the speed of advance.

In the behind-ship condition, whether we

approach it from a model self-propulsion

point of view or from that of full-scale

experience, an increase in propeller diameter

is accompanied. by a reduction in mean

wake fraction, and the latter in its turn

implies an increase in speed of advance,

a reduction in real slip, and hence a reduc-tion in the torque-absorbing capabilities of the screw. Hence, if in the open-water

test the power absorbed by the screw varies

as the diameter to the fifth power, then in the behind-ship condition the power

ab-sorbed, by the screw must vary as the

diameter taken to a power less than five. I have spent a great deal of time in trying to ascertain what the difference might be between the two conditions, and analysis

of self-propulsion, data as well as full-scale experience suggests that the diameter index should be 9/2 rather than 5. The diameter

index of 5 used by Sir Amos would,

how-ever, be correct for similar diameter draught ratios because, in that case, discs of different diameters would occupy the same position relative to the wake pattern behind the hull. As Sir Amos suggests the application of his

formula to "type ships" in which the ratio screw diameter/draught will obviously be fairly constant, I would suggest that the

diameter index of 5 should be retained, but

a correction should be introduced for

departures from a standard diameter/

draught ratio which may correspond with

values of p given by Sir Amos.

Sir Amos holds the view that blade out-line and the extent of surface, within

reason-able limits, do not appear to influence the power-absorbing capabilities of a screw. I think Sir Amos's conclusions regarding effect ofsurface can be accepted for .pitch

ratio of about 09 with small variation

above and below that figure.

At pitch ratios below 09 an increase in

surface leads to an increase in revolutions, and at pitch ratios above 0'9 an increase in surface leads to a reduction in propeller

re-volutions, the latter being more or less in

line with indications given to us by'Froude. Maximum effect of change in blade surface is, of 'course, experienced at pitch ratios of well below or above 09.

In so far as effect of change in blade shape

is concerned, this is closely related to that

of blade thickness, area, and quite a number

of other. factors, and is not quite as

neg-ligible as Sir 'Amos suggests, because

e

FOUR-BLADED PROPELLERS OF SINGLE SCREW CARGO SHIPS D195

camber ratios become involved.

I am

inclined to agree, however, with Sir Amos

that provided that the formula is applied to "type ships" this aspect will be taken

care of in a global manner by the empirical coefficient entering his formula.

Before concluding my remarks, I should like to refer to §18, in which Sir Amos was good enough to mention the method evolved

by me some time ago for ascertaining the' mean pitch of variable-pitch screws.' As

presented by Sir Amos the method applies to blades possessing a similar relative

dis-tribution of area from root to tip, but'

occasion may arise when it may be necessary to compare variable-pitch screws which do. not possess this similarity of blade outline,

and a

minor modification is

requir-ed to take this into account. This

modification consists in multiplying the product of Simpson multiplier and radius

in Column A, by blade width at the corres-ponding radius. Thereafter, the modified

product in Column A is to be multiplied

by Pitch squared and introduced into

Column B. The mean pitch ,will then be

given by square root of B/A as indicated by Sir Amos, but it will take into account any variation in blade shape.

I think it is necessary to mention here

that the mean pitch obtained in this nanner

is essentially "thrust mean pitch." That

is to say, it is the pitch that could be used for purposes of comparing screws on the

'thrust-identity principle.

Dr. E. V. TELFER, Associate Member:

I have found Sir Ams's paper most

stimulating. It also evidently stimulated

my friend Dr. Tutin, who suggested to me

that although the presentation adopted in the paper was dimensionless, the law of comparison would really require presenta-' tion in the form S.H.P./D3 to a base of

NVD.

The analogy between the hull effective

horse-power divided by displacement of the

seven-sixths power, plotted to a base of

VVL, and the propeller problem suggests thatabetter propeller speed base would be NV'HF, where HF iS the propeller face

pitch, since pitch as length is in the direction

of advance while D is across it. In order

to steady the ordinate value of relative

power (SHP/D3) it can be divided by any

appropriate- function of 'the base, one

obvious function being the cube, i.e.

N3HF1, thus giving a power coefficient

SHP/N3.D3HF', which plotted to an

NVH (or any) base gives

a constant

ordinate for constant slip. ,Such a present-ation is especially suitable for self-propelled model results.

I was attracted to this particular power

constant nearly twenty yeers ago ; and for'

Dr. Tutin develops his argument in the preseit

(13)

Dl 96 AN APPROXIMATE AND SIMPLE FORMULA CONCERNING

the same reasons that Sir Amos has prepared

his own interesting work, i.e. to get down

to the "near enough" very early in the

powering problem. The attack was

differ-ent. Schaifran's model data were analysed to find the most constant form of constant,

and this was used to derive full-scale,

co-efficients. Applied to Sir Amos's work, a

surprisingly constant coefficient, p, is

obtained, the coefficient of 033 now apply-ing for bronze screws over the whole

practi-cal range of pitch-ratio with as 'cldse an accuracy as the full-scale data are likely

to supply.

The surprising constancy of the factor naturally causes speculation as to its

sig-nificance. One of the most interesting

sidelights i given by considering the related .significances of the Taylor wake propeller coefficient described by Admiral Taylor in his 1925 I.N.A. paper. Taylor's work also

simplifies by the use of the D3?ZHF1*

corn-bination. His Wp or K/(l w), now equals

1,000 P. V.

Wp=K/(lw)=

Sm.D3 HF' (N/l00)4. .1

in which Sm, the model power absorption coefficient now is substantially constant

over the practical range of face-pitch ratio,

whereas it was a variable in the Taylor

presentation. Sm has a value of 212 for 3-bladed propellers and 227 for 4-blade.

If we now separate out what may be called

the Ayre coefficient from the above, we

have

:---

P

K Sm

PD3HF1j'N3lO8l_w

X1ØX HFN 100 V K Sm '°

_{lO(lw) (lSa)}

For single-screw ships, a good average

value of w is 028 with Sa=5% K=l0,

Sm=227 (a Taylor value), the value of p

is 033, which is "where we came in.". This shows that we could write

p (1Sa)=A/(1---w)=B for the same ship, and thus for trial conditions at least,

we are able to reduce some of the

dis-crepancies usually found, since an increase

of p by external causes would also be associated with an increase in Sa ; and

p (1 Sa) could be expected to remain

constant under not excessively different

weather conditions. The factor A can be determined from model or fullscale data

and we could represent the whole expression in the form

p.(lSa)A

. 4

W

_p(1Sa)

and thus study full-scale work data. if

these wake values are divided by the

dimen-sions parameter which I gave in my -1936

BE(

3D\

paper, l:e. ,j,-jz{, '-m) where E is the

height abov the Ieel to the propeller centre, Tthe draught, the remaining symbols having

their present significance, the wake

para-3

meter so found should be principally a

function of hull fullness. Model torque

results give a value of 204 for this function

while model thrust results suggest the greater. value- of 2SçS, in which 4 is

the full prismatic. Co-operative research is

required to determine closer values of the

full-scale wake function and its dependence

upon other parameters such as the shape

of the frame sections, immedialely in front

of the propeller and so on. Sir Amos's

work should assist to this end..

In view of the range of vessels covered by

Sir Amos's data it is nót.surprising to find that a constant p factor over the practical pitch ratio range really Lies along the line

of maximum attainable, efficiencies. The

p factor is the lesson of successful experience.

There are a lot of loose ends here which

seem to suggest that a new Dyson approach

would now have some chance of being

reasonably successful. 'Both the p factor

and (1 Sa) are functions of the propulsive

coefficient. Sir Amos has also done some

work on this ; and in discussing Barnaby's 1943 I.N.A. paper on propulsive efficiency, Sir Amos gave an expression for propulsive

coefficient which, although at first sight

unacceptable, is nevertheless very intriguing.

Sir Amos distinguishes between small and large cargo vessels ; and with ', the

propulsive coefficient, he gives

V

8/C2C.

a

//

. 5

a, being 328 fOr small ships and 394 for

large. The expression is clearly

dimen-sional and at first sight somewhat

unin-telligible. After, however, a little

re-grouping and some reduction, it would

appear that expression 5 can be replaced by

3Ic2v

CF

=O-85

\f

_N . . 6

and without restriction as to size should be able to interpolate Sir Amos's data of

well-designed 'single-screw vessels run on

measured-mile trials at speeds suitable to

their block coefficient.

Since, however, we now have the means of calculating e.h.p. from C2 and s.h.p.

from the new p value, the propulsive

co-efficient is also given directly by

108 V3 ' 64

- pC2 N3 D3 >< D? HF1

which is an accurate expression, subject to

p being- constant, and thus really requires

the propeller to be runningat its maximum attainable efficiency. This restriction can

be removed by the introduction of

ex-pression 3 into exex-pression 7.

Thus sincep=A/(1w) (1Sa), we have

96 A64 HF1

(14)

or with a suitable value of A,

450

r

V 1

X

Ij%'/lOO.Dj

L4

Thus, if for a self-propelled model

experi-ment the values of

, (Iw) and the

constant term corresponding to 450 are

determined, say, from ship self-propulsiOn point to a knot below (C2 varying

accord-ingly), formula 9 in association with the

torque wake parameter

3BEI

3D

wKq

L'Th

enables a very wide generalization of a single

experiment to be made and becomes par-ticularly valuable to a firm building type

ships.

I must resist the temptation to proceed

further along the road which Sir Amos beckons us and would conclude with a

reference to two further points of detail

discussed by Sir Amos. in connexion with mean-pitch determination it can be said

that almost any method that requires

a radius weighting will give a good

approximation to. mean pitch. I prefer the

method given in my 1940 N.E.C. Inst.

paper to that given by Mr. Kari since power does not vary as pitch squared, although the

assumption does not matter much.

If a

better approximation is really required the mean pitch HF given by

HF=[HIfR W/ RW]*

would be experimentally acceptable.

The problem of mechanical efficiency

hardly enters into the design' stage but where data at low engine loads are being used for general analysis work, it is simpler to think

jn terms of mean referred pressure, which

for constant. jp, &ies vary as revolutions

squared plus an initial constant of the

order of 3 34 lb./in2 for steam

instal-lations. At any load the mechanical

efficiency is thus given by the ratio of brake to indicated mean pressure the former being

3 to 34 lb./in2 less than the latter. In

paragraph 12, it is implicity assumed that

the mechanical efficiency remains constant

from 55% power upwards. I doubt

whether this assumption can be generally

substantiated. It largely arises from the diffi-culty of working with i.h.p. a N3 compared with the simpler m.e.p. a N2 plus a constant.

This method is dealt with in my 1926

N.E.C.Inst. paper and has proved consist-ently reliable in practice.

Mr.' R. HINCHLIFFE, M.B.E., Fellow: The paper we have heard tonight deals with problems whicii frequently confront the naval architect in an early stage of a

design. and for their solution Sir Amos

suggests a simple formula which he

des-cribes as being "of an empirical nature,

4a

not related to the normal science as applied

to propellers." It is,.however, interesting

to note that in the now increasingly used

"J.K." system of non-dimensional

pro-peller coefficients the torque coefficient Kq can be written in the form

O952 F.H.F. x 108

Kq

N3d5

Where F.H.P.=Propeller Horse Power. N=Revolutions per minute.

d=Propeller diameter in ft.

While the Author's formula can be written

F.H.P x 10°

N2xd5

From which it follows that p= l0504 Kq. It can also be shown that p is a function of

the torque coefficient CQ. used in the Schaffrau notation and of Bp of the Taylor notation.

In propellers of similar type the torque coefficient Kq is constant so long as real

slip and pitch ratio are constant. Sir Amos's

statement that for propellers of the same pitch ratio P.H.F a d2N3, is in conformity

with the law of comparison, so long as real slip remains constant. Slip remains

con-stant so long as thrust horse-power is

varying as speed cubed. For most cargo vessels of normal form real slip does not vary materially for speeds below a speed-length-ratio of about O7say 14 knots for

a 400-ft. ship.

From the above reasoning it would

appear reasonable to expect the coefficient

p would remain reasonably constant in a

given ship so long as the speed reached did not exceed the figure at which wave making

becomes important, but in passing from

ship to ship the value ofp will be materially

affected by the pitch ratio adopted, which is exactly what the Author's analysis has

demonstrated.

When a ship designer has decided upon

the shaft horse-power for a given design,

he usually finds himself called upon to

solve one- of two problems : to find the

diameter of pipeller that will give him

maximum efficiency when running at the

revolutions for which the machinery is

designed or alternatively, having fixed his propeller diameter, to decide upon the revolutions that will give him the most

efficient performance.

The usual procedure is first to decide

upon the type of screwsay a Taylor,

Schaifran or Troostand from the corres-ponding design charts the desired inform-ation is obtained so far as model results

can guide the designer. Applying any

correction factors for passing from model to

full-size screw which his experience suggests

are necessary, he obtains the required

data.

The present paper offers him an

alter-native and more rapid method of obtaining

his result provided that he can choose the

value of p that will give him the best

(15)

n198 AN APPROXIMATE AND SIMPLE FORMULA CONCERNING

efficiency. To do this he must arrive at

the best pitch ratio.

An analysis of Taylor's charts for

4-bladed propellers a his standard type, with

ogival blade sections, suggests that the

pitch ratio corresponding to maximum

efficiency was closely approximated by the

formuin :-Best pitch ratio =

.44 - O2l76

J P.H.P. *

197 23l

where V=Ship speed in knots. W=Taylor's wake fraction.

Obtaining the pitch ratio by one of these fonnuin, the p value can be lifted from Sir

Amos's figures and his formula used to

find the desired quantity.

Applying the above method to a number of imaginary ships, covering a wide range

of either diameter or revolutions, I found

that the results obtained from the Ayre formula agreed almost exactly with the

figures for maximum efficiency given by the

design chart, and differences being too.

small to affect efficiency to an appreciable extent. This agredment is somewhat

sur-prising when we remember that design

charts are built up from model experiments

while Sir Amos's figures were obtained

from full-sized ships, and appears to suggest that the difference in performance between

the model and the full-sized propeller are

not so great as some writers contend. The.

formulz quoted. above apply only to

pro-pellers with the Taylor type of blade,

constant pitch and ogival blade sections. Doubtless somewhat similar expressions

could be devised for screws with varying

pitch and aerofoil sections, though the problem is rendered more complicated by

the difficulty of deciding the pitch ratio that

is directly comparable with that obtained

when using the face pitch of constant-pitch

screws with ogival blade sections. The

problem is one: I can commend to our

younger members, but they must be pre-pared for some disappointments. Trial analysis is beset with pitfalls, owing to the many causes which may render the recorded data unreliable.

Sir Amos has called attention to three of these and my experience suggests that they

are probably the most frequent source of

error. Tidal effect, if complicated with

wind effect, is very difficult to eliminate

correctly. When running on the measured

mile with a cross and variable wind, only a quartermaster of superlative ability can prevent the course becoming serpentine. Insufficient length of straight run before entering the measured course, is, I fear, a

frequent fault. In the Firth of Clyde there

is, on the Arran coast, a two-mile course with a centre post; it is therefore possible

to take times for two consecutive mile runs

in the same direction. Practically all the

records I have seen where the times of the

two consecutive miles were separately recorded showed a less time for the second

than for the first

mile, suggesting. that

the ship's speed was still accelerating when

she entered the measured course, despite

the fact that the customary one mile straight run up had been carefully adhered to.

Owing to the uncertainty which tends to

surround trial records

the amount of

reliable data that can be collected by one

individual or even one firm is usually

insuffi-cient to enable general conclusions to be

drawn. If some system of pooling results

could be devised and accepted, we might in time have a clearer knowledge of the

many problems surrounding speed and

power prediction.. Meanwhile this

Institu-tion is greatly indebted to Sir Amos for placing before it the results of his large

experience; we can perhaps hope that his

example will induce others to do the same.

Mr. J. W. CORNEY, Associate Member:

For preliminary design purposes, the

formula evolved is certainly simple and

requires a reasonably correct approximation of only one coefficient p, its value depend-ing on the pitch ratio, type of blade,

percen-tage of power and, if required, service

conditions. The statement that no

appre-ciable correction to the coefficient is

necessary for variations in ship form or

speed over a wide range of cargo vessels

is surprising in view of the importance

attached to these factors in model-propeller

experiments.

As the paper deals with full-scale trials, it may be of interest to give here the results

of the application of the formula to the

actual performance of several single-screw turbine-driven cargo vessels.

Unfortuna-tely direct comparison with the Author's coefficient p is not possible on account of

insufficient data relating to loaded measured-mile trials. However, the loaded voyage performances of 11 cargo steaiiers have been analysed. These particular vessels

were selected because precautions had been

taken to obtain reliable measurements of

shaft horse-power during the voyage runs. They varied in displacement between 12,000 and 18,000 tons, displacement coefficients

69 to 76 and speed/lengths ratios .5 to

64. The propellers, of which seven were

of the "built" bronze, segmental type, two of the "solid" bronze aerofoil type

and two of the solid cast-iron aerofoil type,

varied in diameter between l75 ft. and l925 ft over a range of pitch ratio 8l

to l03.

To obtain the propeller horse-power,

the measured s.h.p. was in all cases

multi-plied by 97, the figure given in the paper

for Diesel engines.

The coefficients p were calculated for the

above propellers and plotted to a base of pich ratio; the Author's basic coefficients

(16)

were also added to the chart for comparison

purposes. Fig. 2 shows these curves.

It was found, that by disregarding one of the seven "built" segmental propellers a fair

curve could be drawn through the points of the other six which, incidentally were

within ± 2% of the faired curve. The

curve so obtained represented the values of

the coefficient for " built" bronze

seg-mental propellers for loaded voyages, under average weather conditions and was higher than the Author's basic curve for

measured-mile trials by 64% at 80 pitch ratio and

94% at 1-03 pitch ratio.

In §14 the Author has pointed out the necessity of increasing the value of p by a

few per cent. for service operation, with

weather similar to the measured-mile trials,

but I think he will agree that it is seldom

such ideal conditions of wind, sea, and

state of hull surfaces are ever reproduced

again during voyages. Consequently, the

foregoing higher values of p may, it is

thought, represent the average voyage con-ditions for these particular propellers,, but

the Author's views on this matter would

be appreciated.

With regard to the two solid bronze

aerofoil propellers, the p values were lower than the basic segmental type by 3% in the case of one and 11% in the other, but, apart

from showing a general reduction for

aerofoil sections, the results were not

otherwise considered conclusive.

The values of the coefficients for the

two solid cast-iron aerofoil propellers were -277 and 349 for respective pitch ratios

of -858 and 1-02. Correcting to the

Author's basis by means of the percentages

given, these values become 283 and -356

which are 74% and 5% higher for the

loaded-voyage conditions.

It is, of course, realized that the results

given may not be as reliable as those taken from measured-mile trials and, in fact,

cover only a small part of the wide field

*5 C0

?. _- _{FOR SOLID BROr'IZ}

--1.5.

0

I I 40

- -

+

-:. FO1

-VO A S - OW

-T

III

i:

-24 -20

80 -84 88

PITCH RATIO.

(17)

D200 AN APPROXIMATE' AND SIMPLE FORMULA CONCERNING

dealt with in this paper; but I think there

is sufficient' agreement to show that the Author's forniula can be 'used with confi-dence for quick and preliminary consider-ation of propeller dimensions and

revol-utiOns for singlescrew vessels of this type.

Mr. W. MUCKLE, Member:

I should like to say in the first place that

the form in which Sir Amos expresses the power can easily be shown to be

theôreti-cally correct; One can show that s.h.p. p N3 D6f(s)

where p is the density of the fluid and

f(s) is a function of slip

the formula proposed, the function of the ship is included in the coefficient p and,

since the fluid is always the same, the

density need not appear. My first reaction

to the fOrmula was that slip should appear

separately, but I know that this brings us

on to very dangerous ground since the

elusive quantity wake niust be introduced.

On the other hand, if -the propellers from

which the data have been derived are

actually working at the slip which gives the

best efficiency, there would appear to be

no point in introducing slip separately. I should like to ask Sir Amos,, therefore,

if the data given in §8 are used in deter, mining the diameter of a propeller which will give a certain power at given revolu-'

tibns, can one be sure that the propeller so obtained will be the most efficient one that

will satisfy the necessary conditions of

power and revolutions?

In deriving values of p from data for a few

ships which' were available to me, I have found variations of as much as 10 or 15%

froth the valves given in the paper. I wonder if Sir Amos would tell us how

much variatiOn he himself experienced in

deriving the mean values of p in §8. It is

worth noting in passing that an error of

10% in the value of p will only mean an

error of 2% on the diameter of the propeller

since diameter appears to the fifth power

in the expression.

Part of the variation that is likely to be

experienced in deriving values of p, I should

say, is due to m.t.e. having to be more or

less guessed. This would suggest that it would be worth while for the shipbuilder to take torsion-meter reüdings on trial for

all vessels and thrust readings, too, if

possible.

The points dealt with by the Author in

§13 are extremely interesting. Tide effect

is most important and I should like to ask

Sir Amos which method he. favours in

attempting to arrive at the true speed of the ship through the water. The method often

adopted, which makes assumptions

con-cerning the manner in which tidal speed

varying with time, leaves a great deal to

be derived. I personally prefer methods based on the revolutions and the time on

the mile. The effect of wind is another

factor which it is extremely difficult to take into account.

The third point mentioned by Sir Amos in §13, is one in which I am particularly

interested being at present engaged in

endeavouring tO derive a method of

calcu-lating the necessary distance which must be run in order to regain speed lost on the

turn.- So far I have found that my in-vestigation bears out Sir Amos's point. In a vessel of 12,500 tons displacement

having a speed of 11 knots, a distance of

about two nautical miles would be required

to regain the speed lost in making a turn having a tactical diameter of four lengths. A criterion which has presented itself for

comparing the distances which must be run before, coming on to the mile for

different vessels is

displacement of ship x trial speed

force causing acceleration.

The force causing acceleration I have

taken at a speed which is 1 knot below the

trial speed and this force is equal to the

difference between the propeller thrust at

constant revolutions and the ship resistance at this speed.

In this brief paper Sir Amos has Sug-.

gested a method of obtaining propeller

particulars based on full-size data.. No

doubt further investigations will show how

values of p may be' steadied up" and it

should be possible, sufficient data being

available, to derive a set of propeller design

charts obtained from actual full-size trial

results. - .

-VOTE OF THANKS

On the motion of the PRESIDENT (Sir

SuMMERs HuwraR) a vote of thanks was

accorded to the Author forhis paper.

(18)

FOUR-BLADED PROPELLERS OF SINGLE SCREW. CARGO SHIPS D20i CORRESPONDENCE

Dr. G. S BAKER, OB.E., Won. Fellow:

I have tried tO see what sort of basis one

could find' fdr this' formula which the

Author has produced and this note gives

the result. I have recently shown that for

all screws

Thrust = N' D4 x k sp . . (1)

where p here is effective pitch ratio, also

s(ls)

''

a+bs

where s is effective slip.

Writing TI', as s.h.p. at propeller

where V, is wake 'velocity at the stern

=pDN(ls), the expression for s.h.p.

reduces to :

-sh.p.= k p' (a+b s)N' D' . . (3)

This holds good over a slip range up 'to

about 40 per cent. The constantais given

by equation 14 in my recent paper on screw

efficiency, and broadly speaking contains kp2 in its denominator, and b is roughly independent of pitch ratio, so that we should expect the Author's formula to 'become

s.h.p. (constantH-p' slip term) N'D'

and actually the formula

s.h.p.=(0l+23p') N' D'.

. . (4)

fits his p values quite well except that it gives p=18 vice l64 at 6 pitch ratio, and

84 vice 83 at I 8 pitch ratio. It appears

that there is an underlying assumption here

that slip does not vary from ship to ship, which is only true for a restricted class of

cargo ships.,

Formula (3)' gives a broad explanation of why the Author's constant drops in passing

from single- to twin-screw ships. In the.

latter the effective working slip is very much

lower than in singles, butbdoes not alter

much, unless extreme blade widths are used. Hence the Author's p value is bound to drop in such yessels. Also it should be. noted that as ain formula (3) varies as the drag coefficient of the propeller, the constant

in formula (4) will vary with drag, i.e. it will drop with good aerofoil sections and rise with bad circular-back sections or bad bOss shape as 'fOund by the Author.

Mr. K. C. BARNABY:

Sir Amos Ayre has produced by methOds: of trial analysis a formula that agrees very well with theoretical consideratiOns. If

we take Taylor's power absorption

co-'Vs.h.p. x N

efficient Bp

-

we can write

V'=s.h.p. x N' - Bp' ' (I) Similarly from the diameter 'coefficient.

'ND

=

we can write

N'!)' --&' ...(2)

Equating (1) and (2) we get s.h.p. = N' D' x Bp' 6'.

Sir Amos Ayre's formula is

N3D'

s.h;p. _' ₁₀₈ X p

Bp'X 108.

so that his p must equal _8'

-or the square of the abs-orption coefficient

multiplied by 108 and divided by the fifth power of the diameter coefficient.

From suitable propellór charts we can take the value of 6' corresponding to any given Bp and pitch ratio for a propeller of

the type assumed. But, unfortunately, we

only know definitely one of' the three

variables, namely, pitch ratio. If, however,

w make the not unreasonable assumption

that the propellers of the vessels analysed were correctly designed, we can connect

Bp and'6 within fairly narrow limits. That

is, we can assume that the propeller was designed to work at nearly the optimum

slip corresponding to Bp value in question.

Thus a "square" propeller of unity pitch

ratio would only have been chosen if it had to work at a Bp of at least 12 and

probably rather more. If we assume a

Bp Of 14 the corresponding 6 can be taken

at 143 and this gives p 328; i.e.. a figure

practically identical' with Sir Amos Ayre's trial analysis value.

An almost equally good agreement can

be obtained by making sirnibir assumptions as shown in the accompanying table.

(Pitchratio) .'.

v'shpxN

06

60

08

32

i0

14 12 9

l4

70 16

50 l8

35

Bp=

= 6) .

..

₂₉₃ 212 ₁₄₃ 114 98 82 ' 685 Bp' xlO9

_..

_O167 ₀₂₃₉ _0328' ₄₂₁ ₅₄₂ ₆₇₄ ₈₁₂

CALCULATION OF 'p" FROM MODEL Exemmrs

(19)

Comparison with chart values will show

that up to about l0 pitch ratio the

pro-pellers 'used. must have been working at practically their optimum slip ratio. At higher pitch ratios the propellers are

working at a little above their optimum

slip values. This is to be expected and is

no doubt due to the effect of diameter

restriction which has necessitated fitting

propellers of slightly smaller diameter and

coarser pitch than those theoretically pre-ferable from model experiments. It is

interesting, however, to find such a very

close measure of agreement between theory

and practice and even at the highest pitch

ratio given the differences are very small. If, however, we are to accept p. as being

Bpx lOB

equal to and therefore directly

calculable from the results of 'model trials in open water, we have' an obvious difficulty

in accounting for the large differences

reported by Sir Amos Ayre between

single-and twin-screw vessels. The most probable

explanation is that the Author is comparing

ca

.

-U

_..1id

'F

ships of about the same total s:h.p. rather than the same s.h.p. per shaft. Then for

the same total power and the same Bp pitch ratio and 6 values the twin-screw vessel will be working at much higher r.p.m. (the usual case in practice) a reduced wake percentage

and probably an increased speed of

ad-vance.

We must then expet an increased blade area ratio to be necessary in order to give

the same blade loading on the greatly

reduced diaafeter. The reduced camber

ratio wil then lower the effective pitch,

increase 6 and reduce p. If, for example, the êfféctive pitch of about unity is reduced

by 5%, the increase in 6 is about 2%' and the decrease in p about 11%.

2g. 3

It would ' be of interest if the Author

could confirm that for his ships of about the same power the twin-screw r.p.m. are

considerably higher as suggested.

Mr. FRANK W. BENSON, Member I agree with the Author that in the early stages of design an approximate formula is very necessary for propeller

character-istics. It can also be said that this applies

equally well to other design data.

Mr. Peter Doig,' with whom 1 was associ-ated as far back as 1907, devised a formula

for the' optimum diameter of a propeller

and published a brochure entitled," The

Screw Propeller" * some years later. He

plotted the results of a great many ships and obtained the following expression for a four-bladed screw

:-D=48.5x$.

This can be converted to read :-D8xN3

x037l=p

which agrees very well *ith the Author's

figures at a pitch ratio which is an average of nearly all practical propellers.

Strictly speaking the value of p at a par-ticular pitch ratio should not be a constant, as shown by the Author on p. 352, but will

vary as slip. For this, reason it is to be

supposed that the coefficient p at each pitch

ratio is an average value for actual

pro-'pellers of the appropriate pitch ratio.

It is known that within limits Pitch

x Diameter is a constant 'and the Author's formula could be written

N3

_{x { p x}

_{(pitch ratio)1 } up 'and}

108

down from a pitch ratio of 10 but the values

'Association of Engineering and Shipbuilding Draughtsznen.

(20)

IIIuIIuIIIuIIuIIuIIIIIIIIIIIIlIIIIIIIIIlIi II

111111.1 IIIIIIIuI,IIIlIIIfIIIIIII 1111 I'.IiIIIIII

I:II!

UUIllllIIIIIIIIII

IRooST .. BLADED PROPELLER.

440. .uIIIIIIIIltuuIIIIIIIIIIIlIIIIuhIIIIIItl ti iiiii.i uiiiniiiiiiiiiiiiiiiiiii in 111111111 ii I liii I

t uiiiiiiiiiiuuiiii

UII!IIIIIIIIIIIIIIIIIIIIIIIlIIIIlIIIII 111111. II I!I: .1 IIIIIIIIIIIIIIIIIIIIIIIII;111 I111111!I :11 I 1:11

- IIIIIII!IiI!IIII

I( TPIII'I j C It f I P ITg R

IIIIii'IIIlIIIIIIIIIIIIlIIIIIIIIIIlIIIIIlIII liii 111111 IIIIIllhIIlIIIIlIIIIIlIII' III 1111111111111111

I 11IIIIlIIIli'I' i

lIIIlIIII1IIIllIIIIIIIIIIUIIIlII:IIIIIIIIIIIIiiIkI!;Il IiIiIiiIiiiiiiiiHiiiiiiIi IIIiIIII III' lll_l!l

-J 5, SI I V L illIllIllill 1 I I IIIIIIIIIIIIIIIIlIItIIIIII IIIi'IiI Ii III III I ii I I

!1IIUP iIIUI II

D AR 040 IPIIIIIIIIIIIiIIItIIIIIIIIIIIIIIIIIIIIIIIIiII' 1111111 IIIIIIIIIIIIIIIIIIHIIIII III II 'liii"' I IR ' I

IIilIIIIP'iIIFI

or IIIItIIItIIIIIIlli!IIIIIIlIlIIIIIIIIIII!lIII!IIF! II IIIIIIlIIIIIIIIlfflIIIIII'IllIIIIIir .!'I: 'I I

-IIIIIIIIIIIIIIIIIIIIh;!IIIIIIIIIlIIIIII 111111 liii' I IIIIIIII; uiiiiIIIIIIiIIII' iii 'H I J lIIIIllhIIIIIIIIIIIIIIIlIhtIIIIIIlIIIiiIIl I'll ' iIIIIIiiiIIIIIill'IIIII'Iii' .iii1d1, J' L IIIIIIIIIIIIIIIIIIIIIIIII:IIIIIIIiIII!!!!i ii 1L11.. ii' 'liii' I il!!!!

''i11IDi'ii' :;Iij1i1i II.J

i. * . IIIIIIIIIIIIIIIIl'!:;iiIIIIIIlIIII'IIIIIIDIII Iii :ii

'ii''

IIIIIIIIPIriIIIIIIIIILIIIIIIIIII' ii Ii'1: iiiiII(Ii il I 111111

iIIIIIIIIIIIIIIIII

III!II1I II 'j1 '1iI i -Ii

iiiiiii';i

I 'r U 110 OH III U '

IIIIIiII1IIIIIIIIIII

1111 ii!!HiHiiIIIIIIllhIIIIIIIIIIIfflIUD

V HH llU W 1]:

IIflhIIIIIjiIIIII

O. 3 4O 2. 6. J, 4 45 5 6 9. IC. U. 12. I& I P 19 2L 2 25. A PSI.., Fig.4 It. 2,. a, 0 a 'C w 'V. '1 /4 a _1d U C.)

(21)

of this modified p do not approximate to

the values given. In view of the foregoing

I prepared Fig. 3 from Taylor's paper of

"Average Results" * and it is seen that

p does vary approximately as (pitch ratio)* and does not have a c'onstant value at each

pitch ratio..

Acknowledging the previous work of

Sir Amos Ayre in the field of the preliminary

estimation of power,: it was felt that the

matter coulq not be dismissed with the

foregoing comments. In Doig's

approxi-mate method of propeller design, the pitch

is calculated from a formula based oh the

slip observed in the trial results from which he obtained his diameter formula, thus

N2D2 Real slip per cent. == 20 + _{.52 v2}

If curves of a methodical 'series of pro-peller experiments are available in which

pitch ratios are

plotted on a

base of

Pxl000

. ND

V3XD2 and ordinates of they

can be combined with Doig's formula for slip

per cent., and the slip per cent, at which

the Author's p values occur can be

calculated.

Pxl,000

N2!)2

LetA=

_V3XD2

_{andS=.2+4852.}

Then P _{4,850 (S--2)} X N x D°

A

from which it follows thatp=1

_{141 (S 2)}

Scan be calculated from corresponding

to any value of A. Solving for the value of p given by the Author at pitch ratios of

0'8, l'O, and I '2 it was found that they occurred at slips of 365%, 27'5% and 245% respectively and, as already said,

were extremely sensitive to slip. The

three spots are shown on Fig. 3that at pitch ratio=0'8 falls on the curves, the

Other two spots fall above the curves.

-From this examination I feel that the

Author will have to give a more detailed

explanation of his formula and how to tise it, as the practitioner should know how any

formula in use by him is derived and also

its limitations.

In the particular case of propeller's I

devised a diagram Fig. 4 by which all the tharacteristics of a propeller can be

deter-mined by coefficients, just as easily

calcu-lated as those of the Author's formula.

The diagram was prepared from Troost's

Seriest of four bladed propellers of 0'4. D.A.R. and the curves are plotted on a

logarithmic, base. Slip % curves are not

shown but the slip can be obtained from ND.

To design a, propeller any diameter

* "Comparison of Model PropeUer Results in Three

Nations." Soc.N.,4. af MarE. 1924.

t N.E.C.Inst. 1938.

is assumed and A and C calculated and the

spot marked on the diagram at B.

If a

line is drawn through this spot at 45° or

parallel to the line Y Y any desired diameter

will

lie oh this

line. For instance the

diameter asstimed in calculating spot B

was 12 ft. Spot D will result in a more

efficient propeller and its diameter will be

12XCDI2X2OI'3ft

_{P.R. 074}

CR 184

Wake can also be found as follows :if F and N are known from trial results A

and C can be calculated using ships speed

uncorrected. - if there is wake this spot will

not fall on the pitch ratio line

corres-ponding to the actual propeller but if a

line is drawn through the spot parallel to X X to cut the actual pitch ratio line we shall

have two values of Cone corresponding

to the calculated spot and one to the

cutting point

C C

then

C =,u .(Froude).

Mr. H. BOCLER, Member:

I have not had opportunity to test the

Author's formula over the full range of p

given in the paper, but in the case of a

number of single-screw cargo ships having

propellers in the region of '7 to '85 pitch

ratio, I find the average values to conform

nearly to the Author's table, in §8 of the

paper. Since the formula for p involves I.) to the fifth power it follows that a fair

difference in p. other conditions than D' being the same,. does not mean much

difference in D and therefore the formula

gives a useful approximation to 1) for pre-liminary design purposes

On the other hand, the method of

present-ing propeller characteristics due to D. W.

Nd

Taylor in the form of variables ₌

and Bp4 =- and also used by Troost

in a paper read before the Institution in

1938, does not seem to me to entail much

more labour in application, if any, than

the formula now proposed, and Taylor's method has the advantages of indicating

the pitch ratio for optimum efficiency and

of making allowance for a wake suitable

to the block coefficient.

Professor L. C. BURRILL, Member:

This is a very interesting paper and the

practical implications of the full-scale

analysis work carried out by Sir Amos

Ayre are very far-reaching indeed.

In §2 the Author states that the formula

which he introduces has been deduced

en-tirely from the full-scale results and is not otherwise related to the normal science as

applied to propellers, whereas this formula

is, in fact, directly comparable with the

(22)

FOUR-BLADED PROPELLERS OF

analysis coefficient 'Aq as shown in Fig. 5

of my paper of August, 1943, read before

the Institute of Marine Engineers, i.e.,

4q4=l,000Kq=94 6N!'D5 x'10

and p _N3D5 x 108. So that Aq4=946p.

It is therefore a verji simple matter to

calculate the values of Aq from the values of p given on p. 352 of the Author's paper,

and these values will be found to lie very

close to the optimum pitch ratio line given

in Fig. 5 of the 1943 paper referred to

above. That is to saythe values deduced

by Sir Amos Ayre from this full-scale work

coincide almost exactly with the corres-ponding. values obtained drectly from

0

9

6O 100

systematic model propeller tests in open.

water.

In order to illustrate this comparison, I

have taken the liberty of reproducing Fig. 8 of the paper referred to above (Fig. 5) and

have added the values derived from Sir Amos Ayre's figures for pitch ratios of

6,

8, l0, I 2 and 14.

This shows very

clearly the relation of the figures derived from full-scale work to the corresponding

model-experiment values for optimum pitch

ratio, and hs the advantage of indicating

ere dimensioned in accordance with model-experiment results, but when due consider-ation is given to the fact that the full-scale

analysis in question was concemed with

actual measured powers and engine

revolu-tions, and that such matters as ship speed, analysis wake, etc.; do not enter into the discussion, I am of the opinion that this

work by Sir Amos Ayre does, in fact, form

a very striking confirmation of

model-experiment work, and normal design'

procedure.

80

70

z

50

a-0

300

Turning now to the question of three

bladed propellers, and assuming that, in

view of the above striking comparison, the

model test results may be used as a basis

for deriving the appropriate optimum values

of Aq and p, the following figures are

- obtained for three-bladed propellers having segmental blade sections

-

200

- C RVES o OPTIMUM CF F

iIP!INSU

TAyO.FNTS

ON SIR OS

4 .

EE1!LES SRoWPI THUS

OPTIMUM PITCH-RATIOS

AND

C0RRESP0NNG EFF1OENaES

AROFOIL & ROU'DBACK PRcELLEPS

(4-BLAbrb

ftreu)

C RVES o OPTIMUM CF F

iIP!INSU

TAyO.FNTS

ON SIR OS

4 .

EE1!LES SRoWPI THUS

OPTIMUM PITCH-RATIOS

AND

C0RRESP0NNG EFF1OENaES

AROFOIL & ROU'DBACK PRcELLEPS

(4-BLAbrb

ftreu)

Pitch Ratio .. OptimuthAq .. Corresponding p 60 Pitch Ratio .. 60 80 100 120 140 OptimuthAq ..

2l2 260'3l9 409

Corresponding p 224 275 338 .433 80 100 120

2l2 260'3l9

224 275 338 140 409 .433 .7 6