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a convenient speed which is measured by appropriate devices.

The propeller is driven at a convenient number of revolutions by

some motor with arrangements for measuring the power required

to drive it. The propeller pulls on the shaft and this force, which

corresponds to the thrust of the ship's propeller, is measured;

this force and the speed of the carriage give the data for the

calculation of the power exerted by the propeller. To determine

and allow for the friction of the driving gear and of the extruded

part of the shaft, a test is made without a propeller on the shaft

but with a filling piece shaped like the hub. After proper cor-

rections and computations have been made the results can be stated

in the form of the shaft horse-power required to drive the pro-

peller and the propeller horse-power exerted by the propeller.

The ratio of the propeller horse-power to the shaft horse-power

is the efficiency of the propeller.

The method of determining the friction by a test without a

propeller, but with a piece to replace the hub, has the effect of

slightly underestimating the shaft horse-power, and consequently

the efficiency is slightly overestimated; the effect is probably a

small fraction of one per cent.

It is customary to make three or more runs with the same

conditions; individual runs may vary as much as two or three

per cent; the variations from the average is about half that amount.

After a series of runs has been made with varying conditions,

the results are represented by a fair curve. As two or more con-

ditions may be subject to variation it is necessary to fair the

results by the method of cross curves. The probable error of

final results may be from half a per cent to one per cent.

Slip. Let p be the pitch of a propeller in feet and let r be the

revolutions per minute, then if it acted like a screw-gear working

04

PROPELLEES

in a fixed rack the speed would be pr feet per minute. Let the

speed of the carriage be V a knots per hour; then, since there

are 6080 feet in a knot, the speed of the carriage is

V a = 101. 3 V a ft. per min.

If this quantity is equal to pr it is considered that the screw-

propeller does act as though it ran in a fixed rack. But in general

the velocity of the carriage is less than pr, so that the relation is

expressed by the equation

#r(i-*) = 101.3 F; (8)

the quantity s is called the slip; it will hereafter be distinguished

as the real slip.

Virtual Pitch. The theory of internal propulsion indicates that

a propeller can exert thrust and apply power only by imparting

velocity to the water acted on. Now the slip is related to the

action of imparting velocity and increases with that action. A

natural inference would be that a propeller running without slip

would exert no thrust, and this is nearly true for thin-bladed

propellers which have the thickness equally distributed between

the face and the back of the blade. If, however, the pitch used

in calculating the real slip is that of the true helical face of the

blade, then such a propeller will show an appreciable, and some-

times a large thrust with zero slip. Now the real action of the

propeller blade on the water is an extremely complicated hydro-

dynamic problem, so that even qualitative conclusions must be

drawn with caution. However, we may gain some insight into

the matter under consideration if we consider that the action of a

thick blade is comparable to that of a very thin blade having the

form of the medial line, as shown in Fig. 29. Such a blade would

have increasing axial pitch and the final acceleration would appear

to be controlled by the pitch at the

^^^_ 7T^^\ after edge. Since both width and

^ ~~^s thickness vary from tip to hub we

FlG 2Q cannot well" assign a pitch on this

consideration, but we can readily see

why there is thrust at zero slip when the pitch is that of the

YAEIABLE PITCH 65

face. It has been proposed to assign to a propeller a virtual pitch

which should be computed on the assumption that the slip is zero

at zero thrust, by equation (8). It does not appear to be practical

to base the design of propellers on virtual pitch, but the conception

allows us to dispose of certain anomalies.

The question of virtual pitch and virtual slip is occasionally

important; for example, it is desirable that the bow screw of

a double-ended ferry-boat shall run idle and this can be

accomplished by providing that there shall be no virtual

slip. This condition is likely to obtain if the back of the

blade is rounded because it becomes the driving surface for the

bow screw.

Variable Pitch. If it be considered that a propeller blade

produces thrust by imparting acceleration to the water, it appears

desirable that the blade shall have increasing axial pitch; this

conception has exerted great influence especially on thoughtful

engineers.

Now it is shown by experiments that there is a reduction of

pressure ahead of the propeller and an increase aft of the pro-

peller, the whole disturbance extending over a distance three or

four times the diameter. The axial dimension of a propeller is

small compared with this region of disturbance and the acceleration

of the water while in contact with the propeller is only a fraction

of the whole acceleration.

A propeller blade with a true helical face and rounded back

may be considered to have increasing axial pitch; if the blade is

narrow and thick the increase is excessive, and for this and other

reasons the efficiency decreases with the thickness. There appears

to be a slight advantage in dividing the thickness between the

face and back of a propeller blade which has medium width. On

the other hand wide blades with true helical faces show better

efficiency with increasing thickness. Such blades if thin will

have some advantage from increasing axial pitch. Mr. S. W.

Barnaby says that very thin and wide blades may be crumpled

at the forward edge when the thrust per square inch is high. Such

blades may be designed with uniform pitch of the face at and

near the after edge and then the pitch may be slightly decreased

66

PROPELLERS

toward the forward edge; there is no good guide for such a dis-

tribution of pitch.

Pitch-ratio. The ratio of the pitch of a propeller to the

diameter is called the pitch-ratio. It is one of the determining

features of the design of a propeller.

Twisted Blades. Large propellers are commonly made with

separable blades, as shown by Fig. 25, page 58. They have the

advantage that the pitch can be changed by twisting the blades.

For this purpose the bolt holes in the flanges are elongated; filling

pieces are provided so that the blade may be held securely. The

development of the helix of Fig. 17, page 234, shows that the

angle ebf is given by the equation,

tan A=p-r- Kd c ,

where p is the pitch of the helix and d c is the diameter of the helix.

If the pitch is increased to p' the angle is increased, as shown by

the equation,

tan 4' =/*-*&.

By aid of this equation the following table was computed.

The diameter of the flange of a blade (Fig. 28, page 62) in inches

is to be multiplied by the factor given in the table, to find the

distance measured along the circumference of the flange, through

which the blade must be twisted in order to increase the pitch

ten per cent.

Factors for Twisted Blades. To increase the mean pitch ten

per cent:

Pitch-ratio.

Factor.

Pitch-ratio.

Factor.

Pitch-ratio.

Factor.

0.6

0.0142

0.9

0.0191

1.4

0.0232

0.65

0.0151

I .O

0.0202

1.6

0.0236

0.70

0.0160

I.I

O.O2I2

1.8

0.0239

0-75

0.0169

I .2

O.O22O

2.0

0.0243

o 80

o 0176

I 3

O.O227

For example, suppose the pitch-ratio is 1.2 and that it is desired

to increase it ten per cent to 1.32, then the factor being 0.0220,

a flange which is 40 inches in diameter should have a distance

40X0.0220 = 0.880 inch,

FACTOES FOB TWISTED BLADES 67

marked off on its edge; and if the flange is turned through that

distance the mean pitch will be increased ten per cent.

If the desired increase of pitch is less than ten per cent the

distance marked off on the edge of the flange can be proportionally

diminished. Thus, in the preceding example, the distance r may-

be made 0.440 of an inch to increase the pitch five per cent.

If the distance is marked off backwards the pitch will be

diminished nearly ten per cent, or a proportionally smaller amount

for a less distance.

It is not advisable to increase or decrease the pitch more than

ten per cent by this method, as it is approximate only and liable

to decrease the efficiency.

The table has been constructed to alter the mean pitch ten

per cent; the mean pitch being assumed to be that of the middle

of the length of the blade, that is, at 0.3 of the diameter from

the axis.

The construction of the table can be shown by computing one

of the factors; for example, that at pitch-ratio 1.2. The diameter

of the cylinder on which the helix at half-blade length lies is 0.6

of the diameter of the propeller,

/. d c =o.6d.

The equation on page 66 gives

tan A = p + xXo.6d = i. 2+0. 6x = o. 6367

for the angle at pitch-ratio 1.2, while at pitch-ratio 1.32 the

tangent becomes

tan ^4 ' = i. i^-j-xXo^d = 1.3 2-^0.6-11: = 0.7001.

The angles are therefore

.4=32 29'; ^' = 35 o'

and

A'-A=23i' = i S i'.

Now a circle one inch in diameter has a circumference of 3.1416,

and 151' will subtend an arc of

151 X3.i4i6-i- 60X360 = 0.0220

of an inch.

68 PROPELLERS

Since the angle of the helix is smaller near the tip of the blade

than near the hub, an increase of pitch by twisting the blade has

relatively larger effect near the tip; consequently twisting a blade

to increase the pitch gives the face an increasing radial pitch.

On the other hand, the application of thickness to the back only,

gives radially decreasing virtual pitch. One tendency counter-

acting the other, there is little harm in twisting the blade to

increase the pitch. On the contrary, it is undesirable to decrease

pitch by twisting the blade, a thing to be borne in mind in design-

ing and adjusting blades.

Rake of the Blade. The blades of a propeller are commonly

raked aft to give them clearance from the hull. They may be

raked aft as much as 15 without materially affecting the power

or efficiency of the propeller. Raking the blades forward reduces

the efficiency; fortunately there is no occasion for it. A raked

propeller blade is longer than one without rake, and if it be made

as thick it will weigh more. The worst effect, however, comes

from the bending moment due to the eccentricity of the centrif-

ugal force acting on the blade; quick-running propellers, like

those for turbine steamers, should have no rake.

Blade Contour. The oval blade contour is superior in efficiency

to the wide- tipped type; but considerable variation in the form of

the oval is allowable. The difference between the Admiralty type

and Taylor's blade is inappreciable. The standard projected

contour proposed falls within the limits of these two types, as

shown by the development of Fig. 24, and Taylor's experimental

results can be applied to it directly.

Thickness-ratio. In Fig. 15, page 45, the lines of the face

and back are extended to the axis; the ratio of the dimension od

to the diameter of the propeller is called the thickness-ratio. In

general, the thickness-ratio should be kept as small as may be

consistent with strength. In order to provide sufficient strength

the thickness must be greater for narrow blades, and as thick

narrow blades are inefficient, a good width of blade will usually

be chosen. But small propellers are commonly strong enough,

so that narrow thin blades of high efficiency may be used for

speed launches.

FORM OF BACK TESTS OF SIMILITUDE 69

Form of Back. As already indicated, the back of the blade,

as shown by a section parallel to the axis of the shaft, is com-

monly rounded to the arc of a circle. Sometimes the section is

parabolic or sinusoidal to give a sharp edge. Or the greatest

thickness may be nearer the after edge for the same purpose.

On the other hand, cast blades sometimes have considerable

thickness at the edge. Propellers that are likely to work in float-

ing ice may have blunt edges. Thick edges are likely to lose

five per cent in efficiency if not more. '

In much the same way the tip of a cast blade is given con-

siderable thickness, as shown by Fig. 15, page 45. The longi-

tudinal section of the blade may then have a straight back, as

shown in the same figure. Sometimes the straight line of the back

is drawn from e to d, and then the blade near the tip has a uniform

thickness to favor the casting; this gives a hollow line near the

tip. There is reason to believe that the greatest stress due to

bending is found about 0.2 of the diameter from the axis. If

this be accepted the greatest thickness should be located there,

and the thickness might then be made uniform to the hub.

Tests of Similitude. In order to investigate the application of

the laws of similitude to propellers Mr. Taylor tested propellers

having diameters of 8, 12, 16, 20, and 24 inches. All had the shaft

1 6 inches below the water level; the largest size consequently

had the tip immersed only 4 inches, and the surface was appreci-

ably disturbed, while the usual size of experimental propellers

(16 inches in diameter) had an immersion of 8 inches, and showed

no surface disturbance.

In general, the larger propellers absorbed relatively less power

and had less efficiency than the small ones. The differences are

not large and may be charged in part to the varying immersion.

Mr. Taylor is of the opinion that the tests are favorable to the

assumption that propellers follow the laws of mechanical simili-

tude. Now the experimental propellers had three pitch-ratios,

0.6, i.o, and 1.5; those having the largest pitch-ratios showed

but little variation, and those having the smallest had not much

variation. But the propeller having the pitch-ratio unity showed

an appreciable variation, which may possibly aid in explaining

70 PROPELLERS

certain discrepancies between full-sized propellers and their models.

Those propellers showed a loss of efficiency, the efficiency decreas-

ing regularly from the 8-inch to the 24-inch sizes, the total

difference being from three to five per cent. The 24-inch pro-

pellers required two per cent more slip than the 8-inch propellers

in order to absorb the corresponding power. There is evidence

that in some cases full-sized propellers show both less efficiency

and less power absorbed than would be inferred from model experi-

ments by the law of similitude. A few tests on full-sized propellers

that would bear on this question would be very valuable.

Interaction of Propeller and Ship. Thus far the propeller has

been considered to act on undisturbed water, as a model does

when carried on a frame in the towing-tank. When a propeller

is placed behind a ship it acts on water which is disturbed by the

ship, and, on the other hand, it disturbs the natural flow of water

which closes in after the ship. This leads to the consideration of

the wake and what is known as thrust deduction.

The Wake. A ship propelled by sails or towed in undisturbed

water, sets in motion a stream in the same direction; this stream

or wake may be attributed mainly to the friction of the water on

the skin of the ship. But near the stern there are other actions

that may make the water move in the same direction and influence

the wake at that place, namely, the stream-line flow and the

effect of the transverse wave; also in some cases the wake may

be affected by eddies. We may therefore consider that the wake

may be attributed to

(1) Surface friction;

(2) Stream lines;

(3) Transverse wave;

(4) Eddies.

The predominant element in forming the wake is the surface

friction; this can be seen from the fact that for all except very

fast boats, the power to overcome frictional resistance is more

than half the net horse-power, often it is two-thirds or more.

This frictional wake is more intense near the middle and near

the surface, diminishing sidewise and downward.

The whole subject of stream-lines whether considered theoret-

THE WAKE 71

ically or practically is difficult and illusive. But both consider-

ations show clearly that the pressure is higher near the bow and

near the stern; in consequence there is formed the bow- wave

and the stern-wave, each of which is about a quarter of a wave-

length abaft the generating cause, which cause is the excess of

pressure just mentioned. Now, just as in flow of water through

a pipe an increase of pressure at the same level is due to the slack-

ening of velocity. The water near the stern (which flows past the

ship as the ship is driven through it) flows at a less relative velocity

than the average, and consequently moves along with the ship,

and contributes to making the wake. This influence is sensible

near the ship but at a distance of a quarter of the ship's length is

probably insensible.

Mention has been made of the transverse waves of the

bow-system and the dependence of their location on the speed

of the ship. When the crest of a transverse wave comes directly

over a propeller, the water affected by the wave has a forward

motion that extends to a considerable depth, gradually dying out.

To illustrate the possible effect of such a wave on the wake it

may be stated that a wave 200 feet long and which has a speed

of 19 knots per hour, will have a velocity at the crest of 1.5 knots

per hour, provided that the height of the wave from hollow to

crest is 5 feet. This height is only one-fortieth of the length and

is not excessive for the conditions found in practice. The speed

dies away with increase in depth; at a depth of 5 feet the speed

is 1.28 knots, at 10 feet it is i.io knots, and at 20 feet it is 0.80

knot; a rough average gives six per cent for the wake due to the

wave in question. A shallow draught boat might have more than

five per cent wake due to a crest of the transverse wave.

Conversely if there is a hollow of a transverse wave over a

propeller the wake may be decreased six per cent or more in the

case described above. Reports of zero wake or even of a negative

wake are given by reliable authorities when there is a hollow

over the propeller.

A well-formed steel ship should have no appreciable eddies,

and should therefore not be affected by eddying wake. But

there will be some eddying abaft propeller struts, and there may

72 PROPELLERS

be considerable effect from eddies near the webs for spectacle-

frames of twin-screw ships, if those webs are set at unfavorable

angles. There is in this case a partial compensation in that the

propellers appear to be able to extract some energy from the

eddies. Nevertheless, it is better to avoid such conditions unless

the designer has full information from model experiments or

otherwise.

A wooden ship with a wide stern-post shows a large and

unfavorable eddying effect on a propeller set close behind it.

If the stern-post cannot be narrowed then the propeller should

be set well clear of the stern-post and a fair-water should be fitted

to avoid eddies.

All these elements, namely, friction, stream-lines, waves, and

eddies, tend to give a varying velocity to the wake. The wake

will have higher velocity near the surface and near the axis of

the ship. Now a propeller imparts kinetic energy to the water

which is proportional to the square of the velocity imparted; in

dealing with the influence of wake on the propeller we should

therefore consider the squares of the effective accelerations pro-

duced by the propeller. But as such a method is impossible for

various reasons, the wake is treated as though it were a uniform

stream, which is equivalent to using the square of the mean acceler-

ation instead of the mean of the square. Consequently, the

efficiency of a propeller in a varying wake is likely to appear to

be higher than in the open water, and such an effect is reported

by Froude, but as the effect is small he recommends that wake

be treated as uniform.

The mean value attributed to the wake of a large well-formed

ship by Froude is ten per cent of the speed of the ship. The

wake factor is the ratio of the velocity of the wake to the velocity

of the ship, and is represented by w. Froude's mean value for w

is o.i; this is to be used for twin-screw ships; single-screw ships

are likely to have more wake.

There is very little known about the wakes of large ships

either as to the velocity or its distribution. The values reported

for wake have been derived from experiments in the towing-tank,

first on propellers in the open water and then on the same pro-

EEAL AND APPARENT SLIP 73

pellers properly placed behind models; the computations will be

explained later.

Real and Apparent Slip. The slip of the propeller as denned

on page 63 gives

where V a is the speed of the carriage in knots per hour, p is the

pitch in feet and r is the number of revolutions per minute.

The conditions for a propeller working in a uniform wake can

be inferred from what would happen if the water in the tank

could have a forward velocity imparted to it equal to the speed

of the carriage multiplied by the wake factor. Suppose that the

speed of the carriage is now V knots per hour and that the wake

factor is w\ the speed of the water would be wV knots per hour,

and the speed of the propeller through the water will be

V a V-wV = (i-w)V ..... (10)

knots per hour. This speed of the propeller through the water

may be called the velocity of advance. So far as the propeller

is concerned it will behave just as though it were driven through

still water from a carriage with the speed V a . For a given real

slip computed as before by equation (9) it will require the same

torque and will deliver the same thrust. The work delivered to

the propeller will be the same because the torque and revolutions

are unchanged; but the work delivered by the propeller will be

larger because the thrust will now act through

loi.jF* 101.3 F a -f-(i-w) ..... (n)

feet per minute.

Apparent Slip. If a ship is driven at a speed of V knots per

hour by a propeller having a pitch of p feet, and making r revolu-

tions per minute, the apparent slip is the quantity computed by

the equation

74 PROPELLERS

If the wake of the ship is assimilated to a uniform stream then

a propeller astern of the ship may be assumed to have a speed of

advance of

and its properties may be inferred from those of a model pro-

peller having the real slip computed from this speed of advance.

From equations (9) and (12) the relations of wake factor, real

slip, and apparent slip can be determined, and expressed by the

equation

i-s = (i-si)(i-w). . ..... (13)

It is to be remembered that Si is the apparent slip computed

from the speed of the ship, w is the wake factor, and s is the real

slip which depends on the speed of advance of the propeller through

the water.

Wake Gain. It is evident that there is a material gain in

placing the propeller astern, where it can get the advantage of

the wake. This comes from the fact that the thrust on the thrust-

block works at the speed of the ship; the thrust as previously

explained depends on the speed of advance. The gain from

working the propeller in the wake is

The wake gain is really due to the fact that the propeller is

able to extract from the wake a small part of the power expended

by the ship in making the wake. Though the advantage of

working in the wake is properly utilized, a greater advantage

comes from anything that will reduce the wake.

Thrust-deduction. If the screw-propeller could be placed a

considerable distance behind the ship, it might get the advantage

of working in the wake without disturbing the stream-lines about

the ship; but it is necessary for various reasons to place the pro-

peller well under the stern; consequently, the propeller disturbs

the stream-lines and reduces the pressure at the stern. This

reduction of pressure is equivalent to an increase in resistance, so

HULL-EFFICIENCY 75

that it takes more power to propel a ship than it would to tow it.

It is customary to represent the increased power required to over-

come this action by aid of a factor,

i-t'

(iS)

Hull-efficiency. The ratio of the wake gain to the factor for

thrust-deduction

i-t

iw

(16)

is called the hull-efficiency. Now, while both wake and thrust-

deduction may be appreciably different for a ship and its model,

The propeller is driven at a convenient number of revolutions by

some motor with arrangements for measuring the power required

to drive it. The propeller pulls on the shaft and this force, which

corresponds to the thrust of the ship's propeller, is measured;

this force and the speed of the carriage give the data for the

calculation of the power exerted by the propeller. To determine

and allow for the friction of the driving gear and of the extruded

part of the shaft, a test is made without a propeller on the shaft

but with a filling piece shaped like the hub. After proper cor-

rections and computations have been made the results can be stated

in the form of the shaft horse-power required to drive the pro-

peller and the propeller horse-power exerted by the propeller.

The ratio of the propeller horse-power to the shaft horse-power

is the efficiency of the propeller.

The method of determining the friction by a test without a

propeller, but with a piece to replace the hub, has the effect of

slightly underestimating the shaft horse-power, and consequently

the efficiency is slightly overestimated; the effect is probably a

small fraction of one per cent.

It is customary to make three or more runs with the same

conditions; individual runs may vary as much as two or three

per cent; the variations from the average is about half that amount.

After a series of runs has been made with varying conditions,

the results are represented by a fair curve. As two or more con-

ditions may be subject to variation it is necessary to fair the

results by the method of cross curves. The probable error of

final results may be from half a per cent to one per cent.

Slip. Let p be the pitch of a propeller in feet and let r be the

revolutions per minute, then if it acted like a screw-gear working

04

PROPELLEES

in a fixed rack the speed would be pr feet per minute. Let the

speed of the carriage be V a knots per hour; then, since there

are 6080 feet in a knot, the speed of the carriage is

V a = 101. 3 V a ft. per min.

If this quantity is equal to pr it is considered that the screw-

propeller does act as though it ran in a fixed rack. But in general

the velocity of the carriage is less than pr, so that the relation is

expressed by the equation

#r(i-*) = 101.3 F; (8)

the quantity s is called the slip; it will hereafter be distinguished

as the real slip.

Virtual Pitch. The theory of internal propulsion indicates that

a propeller can exert thrust and apply power only by imparting

velocity to the water acted on. Now the slip is related to the

action of imparting velocity and increases with that action. A

natural inference would be that a propeller running without slip

would exert no thrust, and this is nearly true for thin-bladed

propellers which have the thickness equally distributed between

the face and the back of the blade. If, however, the pitch used

in calculating the real slip is that of the true helical face of the

blade, then such a propeller will show an appreciable, and some-

times a large thrust with zero slip. Now the real action of the

propeller blade on the water is an extremely complicated hydro-

dynamic problem, so that even qualitative conclusions must be

drawn with caution. However, we may gain some insight into

the matter under consideration if we consider that the action of a

thick blade is comparable to that of a very thin blade having the

form of the medial line, as shown in Fig. 29. Such a blade would

have increasing axial pitch and the final acceleration would appear

to be controlled by the pitch at the

^^^_ 7T^^\ after edge. Since both width and

^ ~~^s thickness vary from tip to hub we

FlG 2Q cannot well" assign a pitch on this

consideration, but we can readily see

why there is thrust at zero slip when the pitch is that of the

YAEIABLE PITCH 65

face. It has been proposed to assign to a propeller a virtual pitch

which should be computed on the assumption that the slip is zero

at zero thrust, by equation (8). It does not appear to be practical

to base the design of propellers on virtual pitch, but the conception

allows us to dispose of certain anomalies.

The question of virtual pitch and virtual slip is occasionally

important; for example, it is desirable that the bow screw of

a double-ended ferry-boat shall run idle and this can be

accomplished by providing that there shall be no virtual

slip. This condition is likely to obtain if the back of the

blade is rounded because it becomes the driving surface for the

bow screw.

Variable Pitch. If it be considered that a propeller blade

produces thrust by imparting acceleration to the water, it appears

desirable that the blade shall have increasing axial pitch; this

conception has exerted great influence especially on thoughtful

engineers.

Now it is shown by experiments that there is a reduction of

pressure ahead of the propeller and an increase aft of the pro-

peller, the whole disturbance extending over a distance three or

four times the diameter. The axial dimension of a propeller is

small compared with this region of disturbance and the acceleration

of the water while in contact with the propeller is only a fraction

of the whole acceleration.

A propeller blade with a true helical face and rounded back

may be considered to have increasing axial pitch; if the blade is

narrow and thick the increase is excessive, and for this and other

reasons the efficiency decreases with the thickness. There appears

to be a slight advantage in dividing the thickness between the

face and back of a propeller blade which has medium width. On

the other hand wide blades with true helical faces show better

efficiency with increasing thickness. Such blades if thin will

have some advantage from increasing axial pitch. Mr. S. W.

Barnaby says that very thin and wide blades may be crumpled

at the forward edge when the thrust per square inch is high. Such

blades may be designed with uniform pitch of the face at and

near the after edge and then the pitch may be slightly decreased

66

PROPELLERS

toward the forward edge; there is no good guide for such a dis-

tribution of pitch.

Pitch-ratio. The ratio of the pitch of a propeller to the

diameter is called the pitch-ratio. It is one of the determining

features of the design of a propeller.

Twisted Blades. Large propellers are commonly made with

separable blades, as shown by Fig. 25, page 58. They have the

advantage that the pitch can be changed by twisting the blades.

For this purpose the bolt holes in the flanges are elongated; filling

pieces are provided so that the blade may be held securely. The

development of the helix of Fig. 17, page 234, shows that the

angle ebf is given by the equation,

tan A=p-r- Kd c ,

where p is the pitch of the helix and d c is the diameter of the helix.

If the pitch is increased to p' the angle is increased, as shown by

the equation,

tan 4' =/*-*&.

By aid of this equation the following table was computed.

The diameter of the flange of a blade (Fig. 28, page 62) in inches

is to be multiplied by the factor given in the table, to find the

distance measured along the circumference of the flange, through

which the blade must be twisted in order to increase the pitch

ten per cent.

Factors for Twisted Blades. To increase the mean pitch ten

per cent:

Pitch-ratio.

Factor.

Pitch-ratio.

Factor.

Pitch-ratio.

Factor.

0.6

0.0142

0.9

0.0191

1.4

0.0232

0.65

0.0151

I .O

0.0202

1.6

0.0236

0.70

0.0160

I.I

O.O2I2

1.8

0.0239

0-75

0.0169

I .2

O.O22O

2.0

0.0243

o 80

o 0176

I 3

O.O227

For example, suppose the pitch-ratio is 1.2 and that it is desired

to increase it ten per cent to 1.32, then the factor being 0.0220,

a flange which is 40 inches in diameter should have a distance

40X0.0220 = 0.880 inch,

FACTOES FOB TWISTED BLADES 67

marked off on its edge; and if the flange is turned through that

distance the mean pitch will be increased ten per cent.

If the desired increase of pitch is less than ten per cent the

distance marked off on the edge of the flange can be proportionally

diminished. Thus, in the preceding example, the distance r may-

be made 0.440 of an inch to increase the pitch five per cent.

If the distance is marked off backwards the pitch will be

diminished nearly ten per cent, or a proportionally smaller amount

for a less distance.

It is not advisable to increase or decrease the pitch more than

ten per cent by this method, as it is approximate only and liable

to decrease the efficiency.

The table has been constructed to alter the mean pitch ten

per cent; the mean pitch being assumed to be that of the middle

of the length of the blade, that is, at 0.3 of the diameter from

the axis.

The construction of the table can be shown by computing one

of the factors; for example, that at pitch-ratio 1.2. The diameter

of the cylinder on which the helix at half-blade length lies is 0.6

of the diameter of the propeller,

/. d c =o.6d.

The equation on page 66 gives

tan A = p + xXo.6d = i. 2+0. 6x = o. 6367

for the angle at pitch-ratio 1.2, while at pitch-ratio 1.32 the

tangent becomes

tan ^4 ' = i. i^-j-xXo^d = 1.3 2-^0.6-11: = 0.7001.

The angles are therefore

.4=32 29'; ^' = 35 o'

and

A'-A=23i' = i S i'.

Now a circle one inch in diameter has a circumference of 3.1416,

and 151' will subtend an arc of

151 X3.i4i6-i- 60X360 = 0.0220

of an inch.

68 PROPELLERS

Since the angle of the helix is smaller near the tip of the blade

than near the hub, an increase of pitch by twisting the blade has

relatively larger effect near the tip; consequently twisting a blade

to increase the pitch gives the face an increasing radial pitch.

On the other hand, the application of thickness to the back only,

gives radially decreasing virtual pitch. One tendency counter-

acting the other, there is little harm in twisting the blade to

increase the pitch. On the contrary, it is undesirable to decrease

pitch by twisting the blade, a thing to be borne in mind in design-

ing and adjusting blades.

Rake of the Blade. The blades of a propeller are commonly

raked aft to give them clearance from the hull. They may be

raked aft as much as 15 without materially affecting the power

or efficiency of the propeller. Raking the blades forward reduces

the efficiency; fortunately there is no occasion for it. A raked

propeller blade is longer than one without rake, and if it be made

as thick it will weigh more. The worst effect, however, comes

from the bending moment due to the eccentricity of the centrif-

ugal force acting on the blade; quick-running propellers, like

those for turbine steamers, should have no rake.

Blade Contour. The oval blade contour is superior in efficiency

to the wide- tipped type; but considerable variation in the form of

the oval is allowable. The difference between the Admiralty type

and Taylor's blade is inappreciable. The standard projected

contour proposed falls within the limits of these two types, as

shown by the development of Fig. 24, and Taylor's experimental

results can be applied to it directly.

Thickness-ratio. In Fig. 15, page 45, the lines of the face

and back are extended to the axis; the ratio of the dimension od

to the diameter of the propeller is called the thickness-ratio. In

general, the thickness-ratio should be kept as small as may be

consistent with strength. In order to provide sufficient strength

the thickness must be greater for narrow blades, and as thick

narrow blades are inefficient, a good width of blade will usually

be chosen. But small propellers are commonly strong enough,

so that narrow thin blades of high efficiency may be used for

speed launches.

FORM OF BACK TESTS OF SIMILITUDE 69

Form of Back. As already indicated, the back of the blade,

as shown by a section parallel to the axis of the shaft, is com-

monly rounded to the arc of a circle. Sometimes the section is

parabolic or sinusoidal to give a sharp edge. Or the greatest

thickness may be nearer the after edge for the same purpose.

On the other hand, cast blades sometimes have considerable

thickness at the edge. Propellers that are likely to work in float-

ing ice may have blunt edges. Thick edges are likely to lose

five per cent in efficiency if not more. '

In much the same way the tip of a cast blade is given con-

siderable thickness, as shown by Fig. 15, page 45. The longi-

tudinal section of the blade may then have a straight back, as

shown in the same figure. Sometimes the straight line of the back

is drawn from e to d, and then the blade near the tip has a uniform

thickness to favor the casting; this gives a hollow line near the

tip. There is reason to believe that the greatest stress due to

bending is found about 0.2 of the diameter from the axis. If

this be accepted the greatest thickness should be located there,

and the thickness might then be made uniform to the hub.

Tests of Similitude. In order to investigate the application of

the laws of similitude to propellers Mr. Taylor tested propellers

having diameters of 8, 12, 16, 20, and 24 inches. All had the shaft

1 6 inches below the water level; the largest size consequently

had the tip immersed only 4 inches, and the surface was appreci-

ably disturbed, while the usual size of experimental propellers

(16 inches in diameter) had an immersion of 8 inches, and showed

no surface disturbance.

In general, the larger propellers absorbed relatively less power

and had less efficiency than the small ones. The differences are

not large and may be charged in part to the varying immersion.

Mr. Taylor is of the opinion that the tests are favorable to the

assumption that propellers follow the laws of mechanical simili-

tude. Now the experimental propellers had three pitch-ratios,

0.6, i.o, and 1.5; those having the largest pitch-ratios showed

but little variation, and those having the smallest had not much

variation. But the propeller having the pitch-ratio unity showed

an appreciable variation, which may possibly aid in explaining

70 PROPELLERS

certain discrepancies between full-sized propellers and their models.

Those propellers showed a loss of efficiency, the efficiency decreas-

ing regularly from the 8-inch to the 24-inch sizes, the total

difference being from three to five per cent. The 24-inch pro-

pellers required two per cent more slip than the 8-inch propellers

in order to absorb the corresponding power. There is evidence

that in some cases full-sized propellers show both less efficiency

and less power absorbed than would be inferred from model experi-

ments by the law of similitude. A few tests on full-sized propellers

that would bear on this question would be very valuable.

Interaction of Propeller and Ship. Thus far the propeller has

been considered to act on undisturbed water, as a model does

when carried on a frame in the towing-tank. When a propeller

is placed behind a ship it acts on water which is disturbed by the

ship, and, on the other hand, it disturbs the natural flow of water

which closes in after the ship. This leads to the consideration of

the wake and what is known as thrust deduction.

The Wake. A ship propelled by sails or towed in undisturbed

water, sets in motion a stream in the same direction; this stream

or wake may be attributed mainly to the friction of the water on

the skin of the ship. But near the stern there are other actions

that may make the water move in the same direction and influence

the wake at that place, namely, the stream-line flow and the

effect of the transverse wave; also in some cases the wake may

be affected by eddies. We may therefore consider that the wake

may be attributed to

(1) Surface friction;

(2) Stream lines;

(3) Transverse wave;

(4) Eddies.

The predominant element in forming the wake is the surface

friction; this can be seen from the fact that for all except very

fast boats, the power to overcome frictional resistance is more

than half the net horse-power, often it is two-thirds or more.

This frictional wake is more intense near the middle and near

the surface, diminishing sidewise and downward.

The whole subject of stream-lines whether considered theoret-

THE WAKE 71

ically or practically is difficult and illusive. But both consider-

ations show clearly that the pressure is higher near the bow and

near the stern; in consequence there is formed the bow- wave

and the stern-wave, each of which is about a quarter of a wave-

length abaft the generating cause, which cause is the excess of

pressure just mentioned. Now, just as in flow of water through

a pipe an increase of pressure at the same level is due to the slack-

ening of velocity. The water near the stern (which flows past the

ship as the ship is driven through it) flows at a less relative velocity

than the average, and consequently moves along with the ship,

and contributes to making the wake. This influence is sensible

near the ship but at a distance of a quarter of the ship's length is

probably insensible.

Mention has been made of the transverse waves of the

bow-system and the dependence of their location on the speed

of the ship. When the crest of a transverse wave comes directly

over a propeller, the water affected by the wave has a forward

motion that extends to a considerable depth, gradually dying out.

To illustrate the possible effect of such a wave on the wake it

may be stated that a wave 200 feet long and which has a speed

of 19 knots per hour, will have a velocity at the crest of 1.5 knots

per hour, provided that the height of the wave from hollow to

crest is 5 feet. This height is only one-fortieth of the length and

is not excessive for the conditions found in practice. The speed

dies away with increase in depth; at a depth of 5 feet the speed

is 1.28 knots, at 10 feet it is i.io knots, and at 20 feet it is 0.80

knot; a rough average gives six per cent for the wake due to the

wave in question. A shallow draught boat might have more than

five per cent wake due to a crest of the transverse wave.

Conversely if there is a hollow of a transverse wave over a

propeller the wake may be decreased six per cent or more in the

case described above. Reports of zero wake or even of a negative

wake are given by reliable authorities when there is a hollow

over the propeller.

A well-formed steel ship should have no appreciable eddies,

and should therefore not be affected by eddying wake. But

there will be some eddying abaft propeller struts, and there may

72 PROPELLERS

be considerable effect from eddies near the webs for spectacle-

frames of twin-screw ships, if those webs are set at unfavorable

angles. There is in this case a partial compensation in that the

propellers appear to be able to extract some energy from the

eddies. Nevertheless, it is better to avoid such conditions unless

the designer has full information from model experiments or

otherwise.

A wooden ship with a wide stern-post shows a large and

unfavorable eddying effect on a propeller set close behind it.

If the stern-post cannot be narrowed then the propeller should

be set well clear of the stern-post and a fair-water should be fitted

to avoid eddies.

All these elements, namely, friction, stream-lines, waves, and

eddies, tend to give a varying velocity to the wake. The wake

will have higher velocity near the surface and near the axis of

the ship. Now a propeller imparts kinetic energy to the water

which is proportional to the square of the velocity imparted; in

dealing with the influence of wake on the propeller we should

therefore consider the squares of the effective accelerations pro-

duced by the propeller. But as such a method is impossible for

various reasons, the wake is treated as though it were a uniform

stream, which is equivalent to using the square of the mean acceler-

ation instead of the mean of the square. Consequently, the

efficiency of a propeller in a varying wake is likely to appear to

be higher than in the open water, and such an effect is reported

by Froude, but as the effect is small he recommends that wake

be treated as uniform.

The mean value attributed to the wake of a large well-formed

ship by Froude is ten per cent of the speed of the ship. The

wake factor is the ratio of the velocity of the wake to the velocity

of the ship, and is represented by w. Froude's mean value for w

is o.i; this is to be used for twin-screw ships; single-screw ships

are likely to have more wake.

There is very little known about the wakes of large ships

either as to the velocity or its distribution. The values reported

for wake have been derived from experiments in the towing-tank,

first on propellers in the open water and then on the same pro-

EEAL AND APPARENT SLIP 73

pellers properly placed behind models; the computations will be

explained later.

Real and Apparent Slip. The slip of the propeller as denned

on page 63 gives

where V a is the speed of the carriage in knots per hour, p is the

pitch in feet and r is the number of revolutions per minute.

The conditions for a propeller working in a uniform wake can

be inferred from what would happen if the water in the tank

could have a forward velocity imparted to it equal to the speed

of the carriage multiplied by the wake factor. Suppose that the

speed of the carriage is now V knots per hour and that the wake

factor is w\ the speed of the water would be wV knots per hour,

and the speed of the propeller through the water will be

V a V-wV = (i-w)V ..... (10)

knots per hour. This speed of the propeller through the water

may be called the velocity of advance. So far as the propeller

is concerned it will behave just as though it were driven through

still water from a carriage with the speed V a . For a given real

slip computed as before by equation (9) it will require the same

torque and will deliver the same thrust. The work delivered to

the propeller will be the same because the torque and revolutions

are unchanged; but the work delivered by the propeller will be

larger because the thrust will now act through

loi.jF* 101.3 F a -f-(i-w) ..... (n)

feet per minute.

Apparent Slip. If a ship is driven at a speed of V knots per

hour by a propeller having a pitch of p feet, and making r revolu-

tions per minute, the apparent slip is the quantity computed by

the equation

74 PROPELLERS

If the wake of the ship is assimilated to a uniform stream then

a propeller astern of the ship may be assumed to have a speed of

advance of

and its properties may be inferred from those of a model pro-

peller having the real slip computed from this speed of advance.

From equations (9) and (12) the relations of wake factor, real

slip, and apparent slip can be determined, and expressed by the

equation

i-s = (i-si)(i-w). . ..... (13)

It is to be remembered that Si is the apparent slip computed

from the speed of the ship, w is the wake factor, and s is the real

slip which depends on the speed of advance of the propeller through

the water.

Wake Gain. It is evident that there is a material gain in

placing the propeller astern, where it can get the advantage of

the wake. This comes from the fact that the thrust on the thrust-

block works at the speed of the ship; the thrust as previously

explained depends on the speed of advance. The gain from

working the propeller in the wake is

The wake gain is really due to the fact that the propeller is

able to extract from the wake a small part of the power expended

by the ship in making the wake. Though the advantage of

working in the wake is properly utilized, a greater advantage

comes from anything that will reduce the wake.

Thrust-deduction. If the screw-propeller could be placed a

considerable distance behind the ship, it might get the advantage

of working in the wake without disturbing the stream-lines about

the ship; but it is necessary for various reasons to place the pro-

peller well under the stern; consequently, the propeller disturbs

the stream-lines and reduces the pressure at the stern. This

reduction of pressure is equivalent to an increase in resistance, so

HULL-EFFICIENCY 75

that it takes more power to propel a ship than it would to tow it.

It is customary to represent the increased power required to over-

come this action by aid of a factor,

i-t'

(iS)

Hull-efficiency. The ratio of the wake gain to the factor for

thrust-deduction

i-t

iw

(16)

is called the hull-efficiency. Now, while both wake and thrust-

deduction may be appreciably different for a ship and its model,