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the engine developed 530 indicated horse-power when making

138.6 revolutions per minute, the speed being 12.33 knots per hour.

The diameter of the four-bladed propeller was 7.33 feet, the pitch-

ratio was 1.5, the projected area-ratio about 0.4, and the apparent

slip 0.18. Assuming a wake of o.io, and a mechanical efficiency

of 0.9, the values of R and D are

_(i38.6)(477)*

/ ZJ j)} =2.70,

7-33(138.6)*

The area-ratio comes between 0.36 and 0.48; the comparison

can be made with either table. The latter gives R = 2.66 at

pitch-ratio 1.5 and slip 0.28, at which = 48.2.

Now since the apparent slip from the trial was 0.18, and the

nearest tabular value is 0.28 the wake factor chosen appears to

be too small. Solving equation (13), page 74, for wake,

i-w = (i -s) -r- (i si) =0.88, w = o.i2.

A repetition of this work with 0.12 for the wake factor gives

but a slight improvement. So we may conclude that the wake

may be taken as o.io or 0.12.

90 PROPELLERS

Now the diameter is directly proportional to the diameter

factor D, and as the computed result is about two per cent smaller

than the tabular value, we may further allow for peculiarities of

the propeller by subtracting two per cent from the value which we

get from the use of the tables.

The propeller was of cast iron with rather blunt edges and an

unfinished surface.

For example, the torpedo-boat Biddle on trial developed 4225

indicated horse-power on two screws, making 325.2 revolutions per

minute and had a speed of 30 knots and an apparent slip of 0.142.

The diameter of the propellers was 6.68 feet, the pitch-ratio 1.63,

and the projected area-ratio about 0.59.

An assumption of zero wake gives R = I.JO, and turning to the

three-bladed table for area-ratio 0.54, this comes for a pitch-ratio

of 1.63 at about 0.13 slip. The corresponding value of D is 51.0,

which is in close concordance with the tabular value.

Tow-boat Propellers. The conditions of service of a tow-boat

are peculiar and incompatible; running free the speed is fairly

high, 12 to 14 knots per hour; when towing the speed may be

half or less as much as when running free. A part of the duty is

to push large ships into position, the speed being then practically

zero. Tow-boats are relatively short, and the water lines may

be fairly full, but there is a good rise of floor, so that the block-

coefficient is low. The pitch-ratio of the propeller is about 1.5,

and the apparent slip running free may be about 0.20. The slip

when towing is likely to be 0.50 or more; when pushing a ship

into position the slip is nearly unity.

The propellers are four-bladed, made of cast iron, with straight

edges, and wide tips; the projected area-ratio is large. From

Froude's tests it appears that propellers with wide tips take about

the same power as those with oval contours, but that the efficiency

is two per cent less, or smaller. The propeller tables for four

blades and large area-ratios may be used directly or may be made

the basis of comparison with good practice by the method just

given. Little is known about towing, consequently the design is

made for running free.

Though deviating from common practice it is recommended

SMALL-BOAT PROPELLERS 91

that the rounded form of the standard projected contour be used

for tow-boats, and that the projected area be not made excessive;

an area-ratio of 0.48 or 0.60 will be found sufficient. If a quicker

running engine can be used better results will be obtained for

towing from the use of a moderate pitch-ratio, not more than

unity.

Steam-launches, especially for serving war-ships, have some of

the characteristics of tow-boats and may be designed in the same

way, except that the towing speed is relatively higher and the

area-ratio need not be so high.

Small-boat Propellers. The owner or prospective purchaser

of a small boat often is confronted with the questions, what engine

should be selected and what propeller should be chosen to go

with the engine? Knowing the length and beam of his boat,

the engine may be selected by aid of Keith's method on page 28,

which allows the determination of the speed approximately.

Unless there is reason to the contrary the propeller may have

three blades and a projected area-ratio of 0.27, that is, the table

on page 117 may be used. The wake can be assumed to be

w = o.i, except for racers which are likely to have zero wake.

Problem. Required the propeller for a boat to make 7 knots

per hour with a 10 horse-power engine which runs at 450 revo-

lutions per minute. This corresponds to the problem on page 28,

where it is computed that a 10 horse-power engine will give a speed

of 7 knots to a boat that is 32 feet long and has a beam of 8| feet.

This being a cruiser it may be assigned a wake of ten per cent,

w = o.io; consequently the speed of advance will be

V a = (iw)V = (10.10)7 = 6.3 knots.

Equation (23) on page 81 gives for the revolution factor,

21.2X1.78

- -

I0

The various powers required are interpolated in the tables on

pages 122 to 125. Having R we turn to page 117 for three-bladed

92 PEOPELLERS

propellers and find at pitch-ratio unity and real slip 0.30 the

value Z) = 59-2 corresponding to ^ = 3.78; the efficiency is 0.64.

Equation (25), page 82, gives for the diameter,

59-2 X^ 5 X6~3* 59.2X1.47X1.36 -

Bow-screws. Screws are properly placed at the stern so that

the wake gain may offset the thrust-deduction. A bow-screw

throws a stream of water against the bow and produces an augmen-

tation of resistance, and further it reduces the wake for the stern

screw. The only ships with bow-screws are double-ended ferry-

boats, and for them Col. E. A. Stevens advises that the propellers

be so designed that the sterr; screw shall be as efficient as possible

and that it be depended on for driving the boat. The forward

screw should be inefficient, in fact, it should act as little as possible.

His experience is that a blade with the thickness applied to the

back and with blades raked away from the hull will conform to

requirements both as the stern screw (driving) and the bow-screw

(idle). It does not appear certain whether the rake of the blade

is essential, though it is known that blades raked forward are

inefficient. Perhaps the most effective way of making the bow-

screw run idle would be to give it zero virtual slip in that position

This could best be accomplished from model experiments, but a

fair approximation can be had by dealing with the medial line (see

Fig. 29, page 64) at about the mid-length of the blade and pro-

viding that the pitch of this line at the following edge shall give

no slip when the screw acts as a bow-screw.

Number of Propellers. Though there is little direct informa-

tion on the subject it is probable that single screws are more

efficient than twin screws, and that there is a progressive dis-

advantage in using triple and quadruple screws. The differences

are not large and any type under favorable conditions may be more

efficient than others for which favorable conditions cannot be

secured. A single engine is, of course, simpler and cheaper than

two engines with the same power, and in like manner -two are

cheaper than three or four. For moderate powers and speeds a

INCLINATION OF SHAFTS CAVITATION 93

single screw will be chosen unless there are distinct advantages

such as handiness or greater security from breakdown, which justify

the greater expense. For example, all war-ships have two screws

or more; and turbine steamers have two, three, or four screws

for the better accommodation of the turbines. Large ships and

high-speed ships most commonly have twin-screws to get favorable

conditions for designing them.

Inclination of Shafts. In a general way the flow of water at

the stern of a -ship is upward and inward, that is, toward the

middle line. In order to get a flow parallel to the shaft of the

propeller, the shaft should be inclined in the same way. In a few

cases a shaft has been inclined upward in order to get the engine

lower down. Cable-laying steamers have had their twin-shafts

inclined in to get better maneuvering. But generally any inclina-

tion of the shaft has been in the wrong direction either to get

better immersion for a single screw or to spread twin screws.

Now the effect of the flow of an inclined stream past the pro-

peller is to vary the slip and consequently the thrust of a given

blade. The angle which the blade makes with the stream flow-

ing past it is always small, 5 being a fair estimate. It will

therefore appear that inclinations of the shaft outward or down-

ward are to be avoided, and that only small inclinations in such

directions should ever be allowed. The effect of inclination of

flow from the line of the shaft is to reduce the efficiency and to

cause vibrations. The effect on efficiency is not known further

than that propellers in towing-tanks show as good an efficiency

behind a model as they do in open water, but this is not conclusive

even for models. It is but too well known that propellers and

especially turbine propellers cause unpleasant vibrations. As

such propellers are set well clear of the hull it may be fair to

charge the vibration in part to inclination of flow. Some large

turbine steamers with four screws have had the out-board screws

changed from three to four blades.

Cavitation. When an attempt is made to apply an excessive

power to a quick-running propeller, the stream of water acted on

appears to break into eddies and the propeller cannot absorb the

power or deliver the thrust expected. This phenomenon appears

91 PKOPELLERS

to have been first identified by Mr. S. W. Barnaby on the torpedo-

boat destroyer Daring, and was called cavitation by him. The

propellers which showed this failure were of the Admiralty type

with a width about 0.2 of the diameter. After the blades were

made half again as wide and the pitch slightly increased the boat

made 29 knots, then an unprecedented speed.

Mr. Barnaby concluded that the phenomenon was due to an

attempt to produce too large a thrust for the area of the blades.

Having computed the mean thrust per square inch of the projected

blade area, he found that the stream broke when that pressure

became nj pounds, and that the difficulty was remedied by

increasing the area so as to avoid so large a thrust. He further

concluded that deeper immersion of the propeller would allow

somewhat greater mean thrust. Since that time Mr. Barnaby has

used his method with satisfaction for high-speed ships including

turbine steamers.

In a paper on the application of steam turbines to ship pro-

pulsion Mr. E. M. Speakman quoted the performance of a

number of steamers, giving among other things the thrust per

square inch of projected area and the peripheral speed of the tips of

the blades. He expressed the opinion that cavitation is liable

to occur when the thrust exceeds 12 pounds per square inch or

when the peripheral speed exceeds 12,000 feet per minute.

Unfortunately cavitation cannot be produced in the towing-

tank for normal propellers, and those instances in which it has

inadvertently occurred in practice have not been reported in such

a way as to form a satisfactory basis for a theory.

Having made the blades as thin and sharp as possible it will

be wise to restrict the peripheral speed to 12,000 feet per minute

and to limit the thrust per square inch by Mr. Barnaby's method

to 12 or 14 pounds per square inch.

To compute the thrust per square inch we may first find the

effective horse-power by multiplying the indicated horse-power by

the coefficient of propulsion from 0.5 to 0.65. The effective

horse-power may be multiplied by 33,000 to find the foot-pounds

per minute, and this quantity divided by the speed of the ship in

feet per minute (ioi.3F) will give the tow-rope resistance; this

CAVITATION 95

last quantity must be divided by i t to find the thrust of the

propeller; so that

33000 E.H.P.

Thrust =

in which V is the speed of the ship in knots and t is the thrust-

deduction (about o.i).

The total thrust is to be divided by the allowable thrust to find

the projected area of all the blades; or conversely we may divide

by the projected area to find the thrust per square inch. Precision

is not important in this matter.

Example. Let it be required to investigate the propellers for

a turbine steamer that has a speed of 20 knots per hour, and a

shaft horse-power of 10,500, applied to three screws. The pro-

pellers have a diameter of 6f feet and make 450 revolutions per

minute.

Assuming a coefficient of propulsion from the shaft horse-power

of 0.6, the effective power per screw will be

10500X0.6^3 = 2100.

If the thrust-deduction is assumed to be o.i, the thrust will be

33000X2100

- = 38000 pounds.

101.3X20X0.9

A circle 6f feet or 80 inches in diameter has an area of 5026

square inches, and if the area-ratio per blade is 0.20, the area for

three blades will be

5026X0.2X3=3015 square inches;

and the thrust per square inch will be

3800^3015 = 12.6 pounds.

A circle 6| feet in diameter has a perimeter of 20.9 feet, so that

the peripheral speed of the tips of the blades will be

450 X 20.9 = 9400 feet per minute.

96 PROPELLERS

Theory of Mechanical Similitude. The conceptions of geomet-

rical similitude and some of the simpler conclusions from the theory

of mechanical similitude are so embedded in practical engineering

that the extensions to the cases quoted in this book will probably

be accepted by the casual reader without much hesitation. In

the presentation of a method for practical use rather than for

technical training, it was thought best to count on such an accept-

ance of the rules of similitude and to reserve a statement of the

theory for those who have leisure and interest for it. More

especially as the statement of the theory requires a careful

definition of the fundamental conceptions of mechanics.

Velocity. The rate of motion of a body is known as the velocity;

if the body moves uniformly, the velocity can be found by dividing

the space passed over by the time required to pass over it. If

the velocity is not uniform, the velocity is found by taking a small

distance along the path and dividing by the small time required.

Acceleration. The rate of increase of velocity is known as

acceleration. If the rate is uniform the acceleration can be

found by dividing the increase in velocity by the time required.

If the acceleration is not uniform it can be found by taking a

small increase in velocity and dividing by the small increase in time.

Force. The weight of a body is the force with which gravity

attracts it toward the earth. Statical forces can be measured

directly or indirectly by comparing with the weight of a standard

piece of metal; moving forces cannot be so measured but are

determined by comparison with the acceleration produced by

gravity.

To be precise we first determine the mass of a body by measuring

the acceleration produced by gravity on a piece of metal at a

certain place; the actual experiments are not so simple, but that

is a matter of detail. The mass of the body is now computed by

the equation,

weight w w

Mass = 7 , or

acceleration' g 32.16'

where g is taken as the mean acceleration of gravity at the surface

of the earth.

TABLE FOR MECHANICAL SIMILITUDE 97

One of the fundamental conceptions of mass is that it is

invariable, although weight and acceleration vary from place to

place.

If some other force than gravity acts on a body to produce

velocity it can be measured by the equation,

Force = mass X acceleration, or / = ma.

Table for Mechanical Similitude. There is given below a table

for mechanical similitude giving the functions to which various

properties are proportional.

In this table the fundamental units are those of length, time,

and mass.

Geometrically the areas of similar figures are proportional to

'the square of a linear dimension and the volumes are propor-

tional to the cube of a linear dimension.

TABLE FOR MECHANICAL SIMILITUDE.

Properties. Symbols. Functions.

Linear dimension /

Time /

Mass m

Surface A ft

Volume V P

Velocity v

v " I

Acceleration a ~ cc ~^

T* ml

Force / mace

Work W flee

t

W ml 2

Power P ~7*~F

Densit y d *

98 PROPELLERS

The definition of velocity gives at once the form of the func-

tion - -, which may be read as the length or space passed over

t

divided by the time required.

In like manner the first form of the function for the acceler-

ation comes from the definition; the second form is obtained by

substituting the function for the velocity. The second form is

correctly written as proportional to the first; it is not equal for a

numerical factor must be introduced which is \ for uniform acceler-

ation.

The measurement of force is represented by the function ma;

the second form introduces the quantity which is proportional to

the acceleration.

Work is defined as the product of a force by the distance

through which it acts. This gives the first form of the function,

in the table, and the second, is obtained by introducing the pro-

portional function for the force.

Power is the rate of doing work and is expressed by dividing

the work by the time in which it is done. The second form of

the function introduces the proportional function for force.

Density is the weight per unit of volume obtained by dividing

the total weight (or force) by the volume; which latter is propor-

tional to the cube of a linear dimension. The linear dimension

in the proportional function for force reduces / in the denominator

to the squaie.

In dealing with propulsion of ships the density of the water is

constant which gives

- = d = constant,

and the force (which is here weight or displacement in tons) is

proportional to the volume, so that

as has been assumed in the discussion of power.

RELATIVE SPEED EXTENDED LAW OF COMPARISON 99

Relative Speed. The condition of relative speed comes from

the assumption that the resistance shall be proportional to the

displacement, that is,

Remembering that resistance is a force and using the propor-

tional function,

ml

7"*

But at a given place the mass is proportional to the weight

or displacement which has been shown to be proportional to / 3 ,

so that the above proportion can be reduced to

or remembering that the first member is the proportional function

for velocity,

>cc/,

Writing this in the form of a proportion with V to the first

power to represent the speed of the ship in knots per hour,

Vi : F 2 ::VZI:

where the linear dimension chosen is the length of the ship in

feet.

This is the proportion of relative speeds; and these are the

speeds at which the resistances are proportional to the displace-

ments.

Extended Law of Comparison. The proportional function

for power gives

Replacing ^ by v 3 from the proportional function for velocity,

we have

Poc/V,

100 PROPELLERS

but the relative velocity is proportional to the square root of a

linear dimension, so that

or, writing the above in the form of a proportion with indicated

horse-power and the length of the ship,

(I.H.P.)i : (LH.P.) 2 ::i*:2*.

Since the displacement is proportional to the cube of a linear

dimension the proportion may bj

(I.H.P.)i : (I.H.P.)2 ::/>!*: 2*.

Sometimes the shaft horse-power (S.H.P.) is used instead of

the indicated horse-power.

Admiralty Coefficient. To show that the method of the

Admiralty coefficient is a variant of the extended law of com-

parison, the velocity is made proportional to the square root of

a linear dimension for then

Independent Estimate. Of the two parts that enter into the

independent estimate of power the second dependent on the wave-

making resistance conforms to the laws of similitude, but the

first, dependent on the surface friction, does not. The power to

overcome wave-making resistance has the form

D" I 2

0.00307 b -j- F 5 oc /-=/ 2 .

J-j /

The power to overcome frictional resistance has the form,

KEITH'S METHOD REVOLUTIONS OF PROPELLER 101

where n is less than two. If n were two the form would conform

to the law of similitude because then we would have

Since the experiments of Froude show conclusively thai: the

resistance of friction increases with a power of the speed less than

two, it is clear that the theory of similitude tends to overestimate

power for a larger vessel than the type, for speed-length-ratios less

than unity.

If we consider the entire equation for the independent extimate

0.00307 fSV n+1 +b

it appears that the first term increases as a power of V less than

the cube, while the second term increases as the fifth power. So

long as the first term is preponderent, the combined influence of

both terms may make the power increase as the cube of the speed,

as is assumed by the Admiralty coefficient.

For speed -length-ratios which approach unity, the second

term has large influence and the power increases faster than would

be indicated by the cube of the speed. If the speed-length-ratio

is greater than unity the exponent of the speed may be four or

even larger.

Keith's Method. The equation for finding speed of small

boats on page 28 may be reduced as follows:

~ B I '

which agrees with the condition for corresponding speed.

Revolutions of Propeller. From equation (8) on page 64 we

have

pr(i-s) = ioi.^Va,

where p is the pitch of the propeller in feet, r represents the revo-

lutions per minute and s is the real slip while V a is the speed of

102 PROPELLERS

advance of the propeller. If the slip is assumed to be constant,

then

Va

and since the pitch is a linear dimension and the speed varies

as the square root of a linear dimension, we have

i

roc -=..

Vl

Writing this as a proportion we have

i i

r s : r m :: ^ :

which is the proper proportion for the revolutions per minute of the

propellers of a ship and its model. Then if the model is one-

sixteenth as long as the ship its propeller should make four times

as many revolutions per minute. This relation does not hold in

passing from a type ship to a new design, for in that case the

number of revolutions depends on other conditions; for recip-

rocating engines the piston speed is usually constant, which,

for a larger ship requires fewer revolutions than the above pro-

portion would indicate.

Propeller Equations. For the propeller equation on pages 81

and 82, we may readily show conformity with the theory of

similitude now that the revolutions are found to vary inversely

as the square root of a linear dimension. As for equation (23),

we have

r*(S.H.P.)* (/*)* /*

Va*

that is -R is a numerical factor independent of the size of the

propeller.

Again equation (25) is

p t t t , ,

ENGINE POWER AND WEIGHT 103

for here D is a numerical factor; the diameter therefore is correctly

proportional to a linear dimension.

Engine Power and Weight. The power of a steam emgine is

computed by the equation

33000'

in which p is the mean effective pressure as determined by the

indicator, a is the area of the piston in square inches, and s is the

stroke in feet, while r is the revolutions per minute.

For a given type of engine the steam pressure and the piston

speed are likely to be the same, independent of the size; meaning

by the piston speed the quantity,

2sr constant.

This condition requires that the revolutions of an engine shall

be inversely proportional to the stroke. The power of the engine,

from the equations above, becomes proportional to

asrccd 2 s ccd 2 ,

where d is the diameter of the cylinder; that is, to the square of

a linear dimension. We may therefore write the proportion,

If the engines are of similar construction the weights will be

proportional to the cube of a linear dimension, so that

Wi :W 2 ::di 3 : d 2 3 :: (LH.P.)i* : (I.H.P.) 2 f .

But the theory of mechanical similitude makes the indicated

horse-power for a ship proportional to the seven-sixths power of

the displacement, so that the ordinary convention that the piston

speed shall be constant leads to the proportion,

104 PROPELLERS

This shows clearly the difficulty or impossibility of attaining

relatively high speeds with large ships.

It is worthy of note that the weight of the engine increases

faster than the power even when the revolutions are made inversely

as the square root of the length, as required by the theory of simil-

itude. In this case the equation for indicated horse-power gives

the proportion,

/

VL 2

Replacing the horse-power by the seven-halves power of a

linear dimension and transferring the VL from the second ratio

138.6 revolutions per minute, the speed being 12.33 knots per hour.

The diameter of the four-bladed propeller was 7.33 feet, the pitch-

ratio was 1.5, the projected area-ratio about 0.4, and the apparent

slip 0.18. Assuming a wake of o.io, and a mechanical efficiency

of 0.9, the values of R and D are

_(i38.6)(477)*

/ ZJ j)} =2.70,

7-33(138.6)*

The area-ratio comes between 0.36 and 0.48; the comparison

can be made with either table. The latter gives R = 2.66 at

pitch-ratio 1.5 and slip 0.28, at which = 48.2.

Now since the apparent slip from the trial was 0.18, and the

nearest tabular value is 0.28 the wake factor chosen appears to

be too small. Solving equation (13), page 74, for wake,

i-w = (i -s) -r- (i si) =0.88, w = o.i2.

A repetition of this work with 0.12 for the wake factor gives

but a slight improvement. So we may conclude that the wake

may be taken as o.io or 0.12.

90 PROPELLERS

Now the diameter is directly proportional to the diameter

factor D, and as the computed result is about two per cent smaller

than the tabular value, we may further allow for peculiarities of

the propeller by subtracting two per cent from the value which we

get from the use of the tables.

The propeller was of cast iron with rather blunt edges and an

unfinished surface.

For example, the torpedo-boat Biddle on trial developed 4225

indicated horse-power on two screws, making 325.2 revolutions per

minute and had a speed of 30 knots and an apparent slip of 0.142.

The diameter of the propellers was 6.68 feet, the pitch-ratio 1.63,

and the projected area-ratio about 0.59.

An assumption of zero wake gives R = I.JO, and turning to the

three-bladed table for area-ratio 0.54, this comes for a pitch-ratio

of 1.63 at about 0.13 slip. The corresponding value of D is 51.0,

which is in close concordance with the tabular value.

Tow-boat Propellers. The conditions of service of a tow-boat

are peculiar and incompatible; running free the speed is fairly

high, 12 to 14 knots per hour; when towing the speed may be

half or less as much as when running free. A part of the duty is

to push large ships into position, the speed being then practically

zero. Tow-boats are relatively short, and the water lines may

be fairly full, but there is a good rise of floor, so that the block-

coefficient is low. The pitch-ratio of the propeller is about 1.5,

and the apparent slip running free may be about 0.20. The slip

when towing is likely to be 0.50 or more; when pushing a ship

into position the slip is nearly unity.

The propellers are four-bladed, made of cast iron, with straight

edges, and wide tips; the projected area-ratio is large. From

Froude's tests it appears that propellers with wide tips take about

the same power as those with oval contours, but that the efficiency

is two per cent less, or smaller. The propeller tables for four

blades and large area-ratios may be used directly or may be made

the basis of comparison with good practice by the method just

given. Little is known about towing, consequently the design is

made for running free.

Though deviating from common practice it is recommended

SMALL-BOAT PROPELLERS 91

that the rounded form of the standard projected contour be used

for tow-boats, and that the projected area be not made excessive;

an area-ratio of 0.48 or 0.60 will be found sufficient. If a quicker

running engine can be used better results will be obtained for

towing from the use of a moderate pitch-ratio, not more than

unity.

Steam-launches, especially for serving war-ships, have some of

the characteristics of tow-boats and may be designed in the same

way, except that the towing speed is relatively higher and the

area-ratio need not be so high.

Small-boat Propellers. The owner or prospective purchaser

of a small boat often is confronted with the questions, what engine

should be selected and what propeller should be chosen to go

with the engine? Knowing the length and beam of his boat,

the engine may be selected by aid of Keith's method on page 28,

which allows the determination of the speed approximately.

Unless there is reason to the contrary the propeller may have

three blades and a projected area-ratio of 0.27, that is, the table

on page 117 may be used. The wake can be assumed to be

w = o.i, except for racers which are likely to have zero wake.

Problem. Required the propeller for a boat to make 7 knots

per hour with a 10 horse-power engine which runs at 450 revo-

lutions per minute. This corresponds to the problem on page 28,

where it is computed that a 10 horse-power engine will give a speed

of 7 knots to a boat that is 32 feet long and has a beam of 8| feet.

This being a cruiser it may be assigned a wake of ten per cent,

w = o.io; consequently the speed of advance will be

V a = (iw)V = (10.10)7 = 6.3 knots.

Equation (23) on page 81 gives for the revolution factor,

21.2X1.78

- -

I0

The various powers required are interpolated in the tables on

pages 122 to 125. Having R we turn to page 117 for three-bladed

92 PEOPELLERS

propellers and find at pitch-ratio unity and real slip 0.30 the

value Z) = 59-2 corresponding to ^ = 3.78; the efficiency is 0.64.

Equation (25), page 82, gives for the diameter,

59-2 X^ 5 X6~3* 59.2X1.47X1.36 -

Bow-screws. Screws are properly placed at the stern so that

the wake gain may offset the thrust-deduction. A bow-screw

throws a stream of water against the bow and produces an augmen-

tation of resistance, and further it reduces the wake for the stern

screw. The only ships with bow-screws are double-ended ferry-

boats, and for them Col. E. A. Stevens advises that the propellers

be so designed that the sterr; screw shall be as efficient as possible

and that it be depended on for driving the boat. The forward

screw should be inefficient, in fact, it should act as little as possible.

His experience is that a blade with the thickness applied to the

back and with blades raked away from the hull will conform to

requirements both as the stern screw (driving) and the bow-screw

(idle). It does not appear certain whether the rake of the blade

is essential, though it is known that blades raked forward are

inefficient. Perhaps the most effective way of making the bow-

screw run idle would be to give it zero virtual slip in that position

This could best be accomplished from model experiments, but a

fair approximation can be had by dealing with the medial line (see

Fig. 29, page 64) at about the mid-length of the blade and pro-

viding that the pitch of this line at the following edge shall give

no slip when the screw acts as a bow-screw.

Number of Propellers. Though there is little direct informa-

tion on the subject it is probable that single screws are more

efficient than twin screws, and that there is a progressive dis-

advantage in using triple and quadruple screws. The differences

are not large and any type under favorable conditions may be more

efficient than others for which favorable conditions cannot be

secured. A single engine is, of course, simpler and cheaper than

two engines with the same power, and in like manner -two are

cheaper than three or four. For moderate powers and speeds a

INCLINATION OF SHAFTS CAVITATION 93

single screw will be chosen unless there are distinct advantages

such as handiness or greater security from breakdown, which justify

the greater expense. For example, all war-ships have two screws

or more; and turbine steamers have two, three, or four screws

for the better accommodation of the turbines. Large ships and

high-speed ships most commonly have twin-screws to get favorable

conditions for designing them.

Inclination of Shafts. In a general way the flow of water at

the stern of a -ship is upward and inward, that is, toward the

middle line. In order to get a flow parallel to the shaft of the

propeller, the shaft should be inclined in the same way. In a few

cases a shaft has been inclined upward in order to get the engine

lower down. Cable-laying steamers have had their twin-shafts

inclined in to get better maneuvering. But generally any inclina-

tion of the shaft has been in the wrong direction either to get

better immersion for a single screw or to spread twin screws.

Now the effect of the flow of an inclined stream past the pro-

peller is to vary the slip and consequently the thrust of a given

blade. The angle which the blade makes with the stream flow-

ing past it is always small, 5 being a fair estimate. It will

therefore appear that inclinations of the shaft outward or down-

ward are to be avoided, and that only small inclinations in such

directions should ever be allowed. The effect of inclination of

flow from the line of the shaft is to reduce the efficiency and to

cause vibrations. The effect on efficiency is not known further

than that propellers in towing-tanks show as good an efficiency

behind a model as they do in open water, but this is not conclusive

even for models. It is but too well known that propellers and

especially turbine propellers cause unpleasant vibrations. As

such propellers are set well clear of the hull it may be fair to

charge the vibration in part to inclination of flow. Some large

turbine steamers with four screws have had the out-board screws

changed from three to four blades.

Cavitation. When an attempt is made to apply an excessive

power to a quick-running propeller, the stream of water acted on

appears to break into eddies and the propeller cannot absorb the

power or deliver the thrust expected. This phenomenon appears

91 PKOPELLERS

to have been first identified by Mr. S. W. Barnaby on the torpedo-

boat destroyer Daring, and was called cavitation by him. The

propellers which showed this failure were of the Admiralty type

with a width about 0.2 of the diameter. After the blades were

made half again as wide and the pitch slightly increased the boat

made 29 knots, then an unprecedented speed.

Mr. Barnaby concluded that the phenomenon was due to an

attempt to produce too large a thrust for the area of the blades.

Having computed the mean thrust per square inch of the projected

blade area, he found that the stream broke when that pressure

became nj pounds, and that the difficulty was remedied by

increasing the area so as to avoid so large a thrust. He further

concluded that deeper immersion of the propeller would allow

somewhat greater mean thrust. Since that time Mr. Barnaby has

used his method with satisfaction for high-speed ships including

turbine steamers.

In a paper on the application of steam turbines to ship pro-

pulsion Mr. E. M. Speakman quoted the performance of a

number of steamers, giving among other things the thrust per

square inch of projected area and the peripheral speed of the tips of

the blades. He expressed the opinion that cavitation is liable

to occur when the thrust exceeds 12 pounds per square inch or

when the peripheral speed exceeds 12,000 feet per minute.

Unfortunately cavitation cannot be produced in the towing-

tank for normal propellers, and those instances in which it has

inadvertently occurred in practice have not been reported in such

a way as to form a satisfactory basis for a theory.

Having made the blades as thin and sharp as possible it will

be wise to restrict the peripheral speed to 12,000 feet per minute

and to limit the thrust per square inch by Mr. Barnaby's method

to 12 or 14 pounds per square inch.

To compute the thrust per square inch we may first find the

effective horse-power by multiplying the indicated horse-power by

the coefficient of propulsion from 0.5 to 0.65. The effective

horse-power may be multiplied by 33,000 to find the foot-pounds

per minute, and this quantity divided by the speed of the ship in

feet per minute (ioi.3F) will give the tow-rope resistance; this

CAVITATION 95

last quantity must be divided by i t to find the thrust of the

propeller; so that

33000 E.H.P.

Thrust =

in which V is the speed of the ship in knots and t is the thrust-

deduction (about o.i).

The total thrust is to be divided by the allowable thrust to find

the projected area of all the blades; or conversely we may divide

by the projected area to find the thrust per square inch. Precision

is not important in this matter.

Example. Let it be required to investigate the propellers for

a turbine steamer that has a speed of 20 knots per hour, and a

shaft horse-power of 10,500, applied to three screws. The pro-

pellers have a diameter of 6f feet and make 450 revolutions per

minute.

Assuming a coefficient of propulsion from the shaft horse-power

of 0.6, the effective power per screw will be

10500X0.6^3 = 2100.

If the thrust-deduction is assumed to be o.i, the thrust will be

33000X2100

- = 38000 pounds.

101.3X20X0.9

A circle 6f feet or 80 inches in diameter has an area of 5026

square inches, and if the area-ratio per blade is 0.20, the area for

three blades will be

5026X0.2X3=3015 square inches;

and the thrust per square inch will be

3800^3015 = 12.6 pounds.

A circle 6| feet in diameter has a perimeter of 20.9 feet, so that

the peripheral speed of the tips of the blades will be

450 X 20.9 = 9400 feet per minute.

96 PROPELLERS

Theory of Mechanical Similitude. The conceptions of geomet-

rical similitude and some of the simpler conclusions from the theory

of mechanical similitude are so embedded in practical engineering

that the extensions to the cases quoted in this book will probably

be accepted by the casual reader without much hesitation. In

the presentation of a method for practical use rather than for

technical training, it was thought best to count on such an accept-

ance of the rules of similitude and to reserve a statement of the

theory for those who have leisure and interest for it. More

especially as the statement of the theory requires a careful

definition of the fundamental conceptions of mechanics.

Velocity. The rate of motion of a body is known as the velocity;

if the body moves uniformly, the velocity can be found by dividing

the space passed over by the time required to pass over it. If

the velocity is not uniform, the velocity is found by taking a small

distance along the path and dividing by the small time required.

Acceleration. The rate of increase of velocity is known as

acceleration. If the rate is uniform the acceleration can be

found by dividing the increase in velocity by the time required.

If the acceleration is not uniform it can be found by taking a

small increase in velocity and dividing by the small increase in time.

Force. The weight of a body is the force with which gravity

attracts it toward the earth. Statical forces can be measured

directly or indirectly by comparing with the weight of a standard

piece of metal; moving forces cannot be so measured but are

determined by comparison with the acceleration produced by

gravity.

To be precise we first determine the mass of a body by measuring

the acceleration produced by gravity on a piece of metal at a

certain place; the actual experiments are not so simple, but that

is a matter of detail. The mass of the body is now computed by

the equation,

weight w w

Mass = 7 , or

acceleration' g 32.16'

where g is taken as the mean acceleration of gravity at the surface

of the earth.

TABLE FOR MECHANICAL SIMILITUDE 97

One of the fundamental conceptions of mass is that it is

invariable, although weight and acceleration vary from place to

place.

If some other force than gravity acts on a body to produce

velocity it can be measured by the equation,

Force = mass X acceleration, or / = ma.

Table for Mechanical Similitude. There is given below a table

for mechanical similitude giving the functions to which various

properties are proportional.

In this table the fundamental units are those of length, time,

and mass.

Geometrically the areas of similar figures are proportional to

'the square of a linear dimension and the volumes are propor-

tional to the cube of a linear dimension.

TABLE FOR MECHANICAL SIMILITUDE.

Properties. Symbols. Functions.

Linear dimension /

Time /

Mass m

Surface A ft

Volume V P

Velocity v

v " I

Acceleration a ~ cc ~^

T* ml

Force / mace

Work W flee

t

W ml 2

Power P ~7*~F

Densit y d *

98 PROPELLERS

The definition of velocity gives at once the form of the func-

tion - -, which may be read as the length or space passed over

t

divided by the time required.

In like manner the first form of the function for the acceler-

ation comes from the definition; the second form is obtained by

substituting the function for the velocity. The second form is

correctly written as proportional to the first; it is not equal for a

numerical factor must be introduced which is \ for uniform acceler-

ation.

The measurement of force is represented by the function ma;

the second form introduces the quantity which is proportional to

the acceleration.

Work is defined as the product of a force by the distance

through which it acts. This gives the first form of the function,

in the table, and the second, is obtained by introducing the pro-

portional function for the force.

Power is the rate of doing work and is expressed by dividing

the work by the time in which it is done. The second form of

the function introduces the proportional function for force.

Density is the weight per unit of volume obtained by dividing

the total weight (or force) by the volume; which latter is propor-

tional to the cube of a linear dimension. The linear dimension

in the proportional function for force reduces / in the denominator

to the squaie.

In dealing with propulsion of ships the density of the water is

constant which gives

- = d = constant,

and the force (which is here weight or displacement in tons) is

proportional to the volume, so that

as has been assumed in the discussion of power.

RELATIVE SPEED EXTENDED LAW OF COMPARISON 99

Relative Speed. The condition of relative speed comes from

the assumption that the resistance shall be proportional to the

displacement, that is,

Remembering that resistance is a force and using the propor-

tional function,

ml

7"*

But at a given place the mass is proportional to the weight

or displacement which has been shown to be proportional to / 3 ,

so that the above proportion can be reduced to

or remembering that the first member is the proportional function

for velocity,

>cc/,

Writing this in the form of a proportion with V to the first

power to represent the speed of the ship in knots per hour,

Vi : F 2 ::VZI:

where the linear dimension chosen is the length of the ship in

feet.

This is the proportion of relative speeds; and these are the

speeds at which the resistances are proportional to the displace-

ments.

Extended Law of Comparison. The proportional function

for power gives

Replacing ^ by v 3 from the proportional function for velocity,

we have

Poc/V,

100 PROPELLERS

but the relative velocity is proportional to the square root of a

linear dimension, so that

or, writing the above in the form of a proportion with indicated

horse-power and the length of the ship,

(I.H.P.)i : (LH.P.) 2 ::i*:2*.

Since the displacement is proportional to the cube of a linear

dimension the proportion may bj

(I.H.P.)i : (I.H.P.)2 ::/>!*: 2*.

Sometimes the shaft horse-power (S.H.P.) is used instead of

the indicated horse-power.

Admiralty Coefficient. To show that the method of the

Admiralty coefficient is a variant of the extended law of com-

parison, the velocity is made proportional to the square root of

a linear dimension for then

Independent Estimate. Of the two parts that enter into the

independent estimate of power the second dependent on the wave-

making resistance conforms to the laws of similitude, but the

first, dependent on the surface friction, does not. The power to

overcome wave-making resistance has the form

D" I 2

0.00307 b -j- F 5 oc /-=/ 2 .

J-j /

The power to overcome frictional resistance has the form,

KEITH'S METHOD REVOLUTIONS OF PROPELLER 101

where n is less than two. If n were two the form would conform

to the law of similitude because then we would have

Since the experiments of Froude show conclusively thai: the

resistance of friction increases with a power of the speed less than

two, it is clear that the theory of similitude tends to overestimate

power for a larger vessel than the type, for speed-length-ratios less

than unity.

If we consider the entire equation for the independent extimate

0.00307 fSV n+1 +b

it appears that the first term increases as a power of V less than

the cube, while the second term increases as the fifth power. So

long as the first term is preponderent, the combined influence of

both terms may make the power increase as the cube of the speed,

as is assumed by the Admiralty coefficient.

For speed -length-ratios which approach unity, the second

term has large influence and the power increases faster than would

be indicated by the cube of the speed. If the speed-length-ratio

is greater than unity the exponent of the speed may be four or

even larger.

Keith's Method. The equation for finding speed of small

boats on page 28 may be reduced as follows:

~ B I '

which agrees with the condition for corresponding speed.

Revolutions of Propeller. From equation (8) on page 64 we

have

pr(i-s) = ioi.^Va,

where p is the pitch of the propeller in feet, r represents the revo-

lutions per minute and s is the real slip while V a is the speed of

102 PROPELLERS

advance of the propeller. If the slip is assumed to be constant,

then

Va

and since the pitch is a linear dimension and the speed varies

as the square root of a linear dimension, we have

i

roc -=..

Vl

Writing this as a proportion we have

i i

r s : r m :: ^ :

which is the proper proportion for the revolutions per minute of the

propellers of a ship and its model. Then if the model is one-

sixteenth as long as the ship its propeller should make four times

as many revolutions per minute. This relation does not hold in

passing from a type ship to a new design, for in that case the

number of revolutions depends on other conditions; for recip-

rocating engines the piston speed is usually constant, which,

for a larger ship requires fewer revolutions than the above pro-

portion would indicate.

Propeller Equations. For the propeller equation on pages 81

and 82, we may readily show conformity with the theory of

similitude now that the revolutions are found to vary inversely

as the square root of a linear dimension. As for equation (23),

we have

r*(S.H.P.)* (/*)* /*

Va*

that is -R is a numerical factor independent of the size of the

propeller.

Again equation (25) is

p t t t , ,

ENGINE POWER AND WEIGHT 103

for here D is a numerical factor; the diameter therefore is correctly

proportional to a linear dimension.

Engine Power and Weight. The power of a steam emgine is

computed by the equation

33000'

in which p is the mean effective pressure as determined by the

indicator, a is the area of the piston in square inches, and s is the

stroke in feet, while r is the revolutions per minute.

For a given type of engine the steam pressure and the piston

speed are likely to be the same, independent of the size; meaning

by the piston speed the quantity,

2sr constant.

This condition requires that the revolutions of an engine shall

be inversely proportional to the stroke. The power of the engine,

from the equations above, becomes proportional to

asrccd 2 s ccd 2 ,

where d is the diameter of the cylinder; that is, to the square of

a linear dimension. We may therefore write the proportion,

If the engines are of similar construction the weights will be

proportional to the cube of a linear dimension, so that

Wi :W 2 ::di 3 : d 2 3 :: (LH.P.)i* : (I.H.P.) 2 f .

But the theory of mechanical similitude makes the indicated

horse-power for a ship proportional to the seven-sixths power of

the displacement, so that the ordinary convention that the piston

speed shall be constant leads to the proportion,

104 PROPELLERS

This shows clearly the difficulty or impossibility of attaining

relatively high speeds with large ships.

It is worthy of note that the weight of the engine increases

faster than the power even when the revolutions are made inversely

as the square root of the length, as required by the theory of simil-

itude. In this case the equation for indicated horse-power gives

the proportion,

/

VL 2

Replacing the horse-power by the seven-halves power of a

linear dimension and transferring the VL from the second ratio