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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



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


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

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



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:















I .O












I .2




o 80

o 0176

I 3


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,


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'


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.


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. 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


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-


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 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


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

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-


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


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

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


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,



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




is called the hull-efficiency. Now, while both wake and thrust-
deduction may be appreciably different for a ship and its model,

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