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The independent estimate should be made to depend on
trials of ships, the value of b being derived from an analysis of
such trials, and may then be used with confidence. In the par-
ticular application, the factor b is probably underestimated because
the speed-length-ratio is high for the Campania.

Under the most favorable conditions the determination of
power from experiments on a model is liable to give a discrepancy
from the power actually found on the ship after trial. Fortu-
nately the discrepancy, which may be as large as ten per cent,
is liable to be on the safe side. Designers who have sufficient
information can usually estimate and allow for the discrepancy.

Methods for Small Boats. Any of the methods of estimating
power may be applied to small boats when there is sufficient
information. Used with discretion the Admiralty coefficient will
probably be found most direct and convenient. Some designers have
been very successful in working up from smaller to larger boats
by the theory of similitude. The experimental model should yield
good results provided good sized models can be towed at sufficiently
high speeds; in this case models less than the standard 10 or 20
feet may be used.

Fortunately an exact estimate of power is often of less import-
ance for a small boat than for a ship, and a failure to realize
speed is of less financial importance.

In some cases the owner or prospective purchaser will do well
to invert the usual procedure, and having selected such a hull
and engine, as he finds proper, may try to estimate the speed
to be expected.



28



PROPELLERS



Keith's Method. The following method of estimating the
speed of a boat is due to Mr. H. H. W. Keith, instructor at the
Massachusetts Institute of Technology; it has the peculiar merit
that it uses only such dimensions as are commonly known for
all boats and does not involve the displacement. The speed
is computed by the equation

JTxp

B '

L is length over all in feet;

B is the extreme beam in feet;

P is the brake horse-power of the engine or engines.

The coefficient C is to be selected from the following table:





Ratio,


C






L^-B.


Miles.


Knots.


Cruiser
Run about


3 to 5
5 to 7


9 to ii
8 to 10


8 to 9 . 5
7 to 8 . 5


High speed




8 to Q


7 to 8 .











If the constant is taken from the column headed miles, then
the speed is given by the equation in miles per hour; if from
the column headed knots, then in knots per hour.

Problem. Required the speed which will be given by a 10
horse-power engine to a cruiser having a length of 32 feet and a
beam of 8| feet.

The ratio of length to beam is

32-8.5 = 3.8.

This comes about half way between 3 and 5, so we may take
the value of C half-way between 8 and 9.5 in the column for knots
per hour, that is, C = 8.7- The equation gives

^32X10



8-5



7 knots per hour.



Had the constant been taken from the column for miles, its
value would have been 10 and the speed would be 8.0 miles per
hour.



WAVE INTERFERENCE 29

. As will be shown later, the equation conforms to the law of
similitude and may therefore be used with confidence for similar
boats at corresponding speeds provided that C is computed from
a type boat; considerable divergence from type, and speed, will
have comparatively little effect on the constant.

Example. A boat 27 feet long over all and with 4 feet beam,,
which makes 14.5 miles per hour with 10 horse-power will have

4X14.5



Wave Interference. Attention has been called to the speed-
length-ratio as a criterion of the relative speed of a ship or a boat,
and it was said that in a general way ships and boats having a
speed-length-ratio less than unity were relatively slow, while
fast craft have a speed-length-ratio greater than unity.

In order to show how this division between fast and slow
craft comes about and to emphasize the difficulty' of determining
power for high speeds a brief discussion will be given of the
system of waves which travels along.with the ship and the phe-
nomena of wave interference.

A ship at high speed is accompanied by a system of waves
which move with the same speed as the ship so that an individual
feature, such as a particular wave crest, keeps the same position
relatively to the ship. The most characteristic feature is the
diagonal bow wave followed by a series of similar waves gradually
spreading out in width and decreasing in height. These diagonal
waves run away from the ship and have no part in wave interfer-
ence.

Along the side of the ship and in the wake are a series of
transverse waves of which the diagonal waves are the terminators.
These transverse waves are approximately trochoidal in form and
the length measured from crest to crest corresponds with the
length computed by the theory of trochoidal waves for- the speed
'of the ship.

In order to bring out clearly the relation of lengths and speeds
of trochoidal waves and their relation to propulsion the following
table has been computed:



30



PROPELLERS



SPEEDS OF TROCHOIDAL WAVES.



Length of Wave,
Feet.


Square Roots of
Lengths.


Time,
Seconds.


Speeds.


Feet per Second.


Knots.


IO


3-l6 f


1.40


7-15


4.2


2O


4-47


1.98


IO.I


6.0


40


6-32


2.80


14-3


8-5


60


7-74


3-42


17-5


10.4


80


8.94


3-95


2O. 2


12.0


100


IO.OO


4.42


22.6


13-4


150


12.25


S-4i


27.7


I6. 4


2OO


14.14


6.25


32.0


I9.O


300


17.32


7.66


39-2


23.2


400


20.00


8.84


45-3


26.8


500


22.36


9.89


50.6


30.0


600


24.50


10.83


55-4


32-8


700


26.46


ii .69


59-9


35-5


800


28.28


12.51


64.0


37-9


900 .


30.00


13.26


67.9


40.2


1000


31.62


13.98


7i-5


42.3



The first column gives the* length of the wave in feet and the
second gives the square roots of the lengths. The third column
gives the time required for a wave to run its own length. From
the lengths and times, the speeds of the waves may readily be
computed either in feet per second or in knots per hour. For
example, the waves which accompany a ship at a speed of 19 knots
per hour, are 200 feet long from crest to crest.

Thus far attention has been given to the bow waves only, but
a similar system is formed at the stern, consisting of transverse
waves with diagonal terminators. At a low speed the bow system
and the stern system are practically separate, because the bow
system is so spread out and diminished in height by the time it
gets to the stern that it has then little effect.

When the speed of the ship increases so that the speed-length-
ratio approaches unity, the bow waves preserve a considerable
height at the stern and into the wake, where they combine with
the waves of the stern system and wave interference becomes an
important feature in the resistance of the ship. The nature of this
phenomenon is most clearly seen from a study of its worst condition
when the first well formed transverse bow wave crest comes in
coincidence with the first stern-crest.

The bow wave at high speeds is irregular in form and is likely
to be broken so that the location of its crest cannot be well



WAVE INTERFERENCE 31

determined; it appears to be somewhat less than a quarter of a
wave-length from the stem of the ship. The first stern wave is
formed about a quarter of a wave-length from the stern-post;
it is difficult to locate because it is not well developed at slow
speeds, and at high speeds it is affected by the bow system. The
distance from the bow wave to the stern wave is something more
than the length of the ship between perpendiculars; it is esti-
mated to be 1.05 to 1.15 of the length of the ship, and this is
called the wave-making length.

The first well developed transverse bow wave crest is a wave-
length from the bow crest. When the speed of the ship is such
that the length of the trochoidal wave corresponding to that
speed is equal to the wave-making length, then the bow wave,
and stern wave coincide, resulting in the formation of a very high
transverse wave in the wake of the ship.

Suppose that a torpedo-boat 182 feet long is running at 19
knots per hour; its wave-making length may be assumed to be
1. 10, so that the length of the accompanying waves will be

182X1.10 = 200 feet.

The first well formed transverse bow-wave crest will coincide
with the stern-wave crest and the boat will be in the worst condition
for efficiency of propulsion. Up to a speed-length-ratio of unity,
which in this case gives a speed of

Vi82 = 13.5 knots,

the power increases nearly as the cube of the speed; above that
speed the power increases faster than would be indicated by the
rule of cubes, and when the boat gets to 16 knots (half way from
that speed to the worst speed) the power is likely to increase as
the fourth power of the speed.

If the boat is driven faster than the worst speed the bow-wave
crest draws astern of the stern-wave crest and the combination
of the wave systems gives a more favorable condition. The most
favorable condition would occur when the bow crest reaches
the first hollow of the stern system, for then it would tend to
suppress the formation of waves in the wake. Complete extinction
of waves cannot however be expected.



32 PROPELLERS

In order to find the most favorable speed, we may note that
the bow-wave and stern-wave systems will be half a wave-length
apart and that they are separated by the wave-making length
of the ship; that is to say the length of the waves will be twice
the wave-making length of the ship. In the case chosen for illus-
tration, twice 200 gives 400, for which the speed of the waves
is 26.8 knots per hour. This boat may perhaps make 30
knots, which is well above the most favorable speed.

The conditions are laid down in an explicit manner because
the phenomena thus appear to be simple; in reality they are not
so simple and things cannot be expected to happen just as com-
puted. But a complex system of phenomena may be compre-
hended better after a partial and simple statement has been made.

Power for High Speeds. In any case the determination of
the power required for propelling a ship at high speed is difficult
and uncertain. Of the several methods of estimating power the
theory of similitude is probably the best as the form for high
speed must follow acknowledged good models. The problem is
usually to get a higher speed with a larger boat, and will be solved
by making the length proportional to the square of the speed;
the power is then made proportional to the seven-sixths power
of the displacement as explained on page 12.

It is desirable that a type ship shall be tried at various
speeds from about half speed to full speed, the power for each
speed being determined. This forms a progressive speed trial.
By the aid of the theory of similitude the probable results of the
progressive speed trials may be predicted in advance and represented
by curves, and then as the trials progress the results may be
computed and compared with the predicted results. Unusual
and unfavorable features may be detected immediately and trials
may be repeated or discontinued. Skilled builders find that results
may usually be predicted with certainty.

Model basin experiments are very useful especially when new
forms or conditions are to be provided for, especially in avoiding
unfavorable combinations.

The Admiralty coefficient involves the theory of similitude and
may be used with confidence for corresponding speeds and is a



SCEEW PROPELLERS



33



fair guide for speeds that do not differ widely from that speed.
Up to a speed-length-ratio of unity the coefficient changes but
slowly with the speed. But as the worst speed is approached
the Admiralty coefficient is to be used with caution. Well above
the worst speed and to the most favorable speed the coefficient
changes slowly and may be used to advantage.

The independent estimate may be used up to and somewhat
above a speed-length-ratio unity, but not for high speeds.

Screw Propellers. The only kind of propelling agent which
will be considered in this book is the helical or screw propeller.

A true screw or helical surface is generated by a line which
moves forward uniformly and revolves uniformly with one point
in contact with a line called the axis.




FIGS. 3 and 4.

Fig. 3 represents one turn of a screw with a thin thread; the
end projection is shown by Fig. 4. A quarter turn of the helix
is shown by abc, a'b'c' of Figs. 3
and 4; the same figures are isolated
in Fig. 5.

Sometimes the generatrix is in-
clined to the axis as shown by Fig.
6; the screw is then said to have
a rake. The rake is usually aft to
carry the blades of the propeller
clear from the hull.




FIG



A helicoidal surface can be generated by a curved line like oa
in Fig. 7. Special forms of screws with such peculiarities are
made to conform to certain notions that sometimes are fanciful.



34



PROPELLERS



Pitch of a screw is the distance parallel to the axis between the
successive threads. Variable pitches have been used for pro-
pellers and must be clearly understood.



FIG. 6.



FIG. 7.



The pitch of a propeller blade may increase axially or radially.
Fig. 8 shows a half- turn of a screw with axially increasing pitch.
The generatrix revolves uniformly around the axis, but advances





FIG. 8.



FIG. 9.



with increasing velocity from p towards n. A propeller with such a
form is expected to accelerate the water gradually. There is some
advantage from increasing axial pitch with very wide blades.



PROJECTIONS OF A PROPELLER



35



. Fig. 9 shows half a turn of a screw with increasing radial
pitch. The point c moves uniformly, generating a true helix, and
the point p also moves uniformly but more slowly; the inter-
mediate curves like a'c' are true helices. There does not appear
to be any advantage from this device.

Projections of a Propeller. Figs. 10 and n give projections
of a four-bladed propeller with uniform pitch and no rake. It is
represented as driving a ship toward the right. The left or after
face of the propeller is a true screw, the blade thickness being put





FIG. io.



FIG. n.



on the back. There is a practical advantage in making the face
a true mathematical surface which can easily be constructed and
verified. It is customary also to consider the pitch of the face
of the propeller only, although the form of the back is as important.
The projection on a transverse plane, shown by Fig. n, shows
four blades each subtending 60, that is, one-sixth of a turn of the
screw. The contour o'r's't'u'o' shows the form which the blade
would have if the helical surface were complete, with square
corners; orstu, Fig. io, is the projection of the same contour on a
longitudinal plane.



36 PROPELLERS

To avoid vibration the corners are cut away, sometimes to a
large extent; in Figs. 10 and n the corners are slightly rounded
so that the helical forms shall not be obscured, and for the same
reason the hub is a straight cylinder. The helical face intersects
the hub in helices; the intersection of the back is more complex
on account of thickness. In practice the hub is barrel-shaped' or
partly spherical. Propellers with straight edges and slightly
rounded corners are but little used, because they have poor
efficiency.

Proposed Standard Blade. It will appear that the design of a
propeller can conveniently be based on the projected area of the
blade, as shown by Fig. n, and for this purpose a standard pro-
jected contour is proposed, as it greatly simplifies the design. But
the acceptance of this standard contour is not essential provided
the contour selected does not vary in a marked manner from it.

Various theories of propellers have been based on a conven-
tional development of the blades, and standard developed con-
tours have been selected to go with the theories. But now that we
have enough experimental information to avoid any other theory
than that of mechanical similitude, we may save the labor of
drawing the conventional developments by the simple expedient
of choosing a standard projected contour.

The proposed standard blade contour is shown by Fig. 12.
It has a cylindrical hub 0.2 of the diameter of the propeller, to
correspond with the experimental propellers on which the pro-
peller-tables are based, and also it provides a hub large enough
for separable blades for three blades. The diameter of the hub
may be increased if necessary or it may be made as small as con-
venient for a solid propeller, without much effect on the action
of the propeller.

The remainder of the radius of the blade is to be taken as the
major axis of an ellipse, which ellipse, together with tangents
from the centre of the shaft, is to be taken as the projected con-
tour. The major axis of the ellipse is therefore 0.4 of the diameter
of the propeller. In Fig. 12 the projected contour is vrtet'r'.

Comparing this contour with that of a blade on Fig. n, it is
apparent that it differs in that the corners are very much cut



PROPOSED STANDARD BLADE



37



away, but the edges of the blade near the hub are elements of the
helical surface. This conception is important because it is the
basis of the method given later for drawing the projections of
the propeller. The projected area of a blade is the area of the
contour mtet'r' , Fig. 12, in square feet. The area-ratio of a blade
is the ratio of this projected area to the area of a circle having
the diameter of the propeller.




o

FIG. 12.

The projected width of a blade is measured at the minor axis
of the ellipse, that is at 0.3 of the diameter of the propeller from the
axis. The width may vary from about 0.2 to about 0.45 of the
diameter of the propeller. When the width is 0.4 of the diameter
the ellipse becomes a circle, as shown by Fig. 13; this circular
contour is a convenient basis for determining properties of the
propeller. If the width of the blade is more than 0.4 of the
diameter, the width becomes the major diameter of the projected
contour.



38



PROPELLERS



All the blades, of whatever width, that are obtained from the
standard contour have that kind of similarity that comes from
the choice of an ellipse for the contour. In particular the pro-
jected area of the blade is proportional to the width.

The straight edge of a blade, as rt Fig. 12, terminates at the
point of tangency with the ellipse, that is, at t. To locate / make




FIG. 13.

os equal to 0.1667 of the diameter and draw st perpendicular
to os till it intersects the ellipse. The computation of this quantity
and other convenient properties of the standard blade will be
explained later. They may be accepted now without discussion*
those who do not care to deal with minor theoretical points may
accept them permanently.

It is convenient to lay off the angle tos by drawing a perpen-
dicular at e to oe and laying off ef\ the computation for this factor
will be given later.



LAYING-DOWN TABLE



39



Laying-down Table. If the standard projected blade contour
is accepted, it is easy to compute and tabulate properties by aid
of which propellers may be drawn with facility and precision.



LAYING-DOWN TABLE FOR ONE BLADE.



Area




Factor




Area




Factor




Ratio.


Width


for


Axial


Ratio.


Width


for


Axial


One


Ratio.


Blade


Factor.


One


Ratio.


Blade


Factor.


Blade.




Angle.




Blade.




Angle.




O.o6


0.1456


0.1628


0. IO02


0.14


- 339 8


0.3799


. 2068


0.07


o. 1699


o. 1900


o. 1156


0-15


0.3640


0.4070


0.2175


0.08


0.1942


o. 2171


0.1304


o. 16


0-3883


0-4341


o. 2276


O.OQ


o. 2184


0.2442


0.1447


0.17


0.4126


0.4613


0.2372


0. IO


0.2427


0.2714


0-1583


0.18


0.4369


o . 4884


0.2463


O. II


o. 2670


0.2985


0.1714


o. 19


0.4611


0.51.56


0.2549


O. I 2


o. 2912


0.3256


0.1838


o. 20


0.4854


0.5427


o. 2630


0.13


0-3155


0.3527


o. 1956














The laying-down table gives the following items: (i) the pro-
jected area-ratio for one blade; (2) the width ratio at 0.3 of the
diameter; (3) the factor for laying down the blade-angle; (4) the
axial factor.

The width-ratio multiplied by the diameter of the propeller
gives the minor diameter of the ellipse in Fig. 12, which is laid
off at 0.3 cf the diameter from the axis.

The factor for the blade angle multiplied by the diameter of
the propeller gives the line ef, which may be used for drawing the
tangent jo.

The axial factor is to be multiplied by the pitch to find the
axial width of the blade as shown in Fig. 10.

As already stated, the projected area-ratio is the ratio of the
area of the contour wtet'r', Fig. 12, to the area of a circle having
the diameter of the propeller.

Deduction of Properties. The proportions and properties of
the standard projected contour are deduced exactly by geometrical
methods and may be accepted as correct, whether or not the
reader cares to follow the deductions which are given here.

All the properties are readily computed for the circular contour
shown by Fig. 13 and may be applied to any elliptical contour
like that shown by Fig. 1 2 by the principles of projection. Suppose



40



PROPELLERS



that Fig. 13 were drawn on transparent tracing paper and held
with the axis oe directly over and parallel to oe of Fig. 12; then
suppose that Fig. 13 were turned to an angle about the axis oe
till a given point in Fig. 13 (such as /) should come directly over
the point indicated by the same letter in Fig. 12; then any other
point (for example /) will also come over the point so lettered
in Fig. 12, and the circle will lie directly over the ellipse. The
circle is then said to be projected into the ellipse and in fact the
whole of Fig. 13 is said to be projected into Fig. 12. Since all
lines perpendicular to the axis, like ef, make the same angle they
will be foreshortened to the same degree when projected onto
Fig. 12. For example, if ef in Fig. 12 is half as long as the cor-
responding line in Fig. 13, so also will st be equal to half of the
corresponding line. Since this relation holds for the half width
of the ellipse taken at any distance from 0, then the area of the
ellipse in Fig. 12 is half that of the circle in Fig. 13.

The basis of comparison is the ratio of the semi-minor diameter
of the ellipse of Fig. 12 to the semi-diameter of the circle of Fig.
13. Consequently having the properties of Fig. 13 we may
get the corresponding properties of Fig. 12 or of any other elliptical
contour, by multiplying by that ratio.

Tangent Point. The straight edge of a blade ends at the
tangent point of the ellipse of Fig. 12 or the circle of Fig. 13. The
distance of the point s from o may readily be computed for Fig.
13 by drawing ct perpendicular to the tangent; then

oc : ct :: ct : cs.

a 2 (o. 2 ) 2



.-. 05=0.3-0.13333=0.16667.

But since all the ellipses are obtained from the circle by pro-
jection, this relation holds for all the elliptical contours.

Projected Blade Area. The half -blade of Fig. 13 can be divided
into three parts: (i) the circular sector ect, (2) the triangle oct,
and (3) the circular sector oqp, to be subtracted.



PROJECTED BLADE AREA 41

Begin by computing the angle cot from the equation,

sin cot = = =0.6667:
oc 0.3

c<rf = 4i48'37" = 4i.8i.
The sector ect has the angle



The area of a circle having the diameter 0.4 is 0.1257, and as the
area of a sector is proportional to its angle,

Area ect = o. 1 2 5 7 X 13 1 .8 -f- 360 = 0.04601 .
The triangle cot has the area

\oc Xst = \oc X ct sin oct

= ^ocXct sin (90 41 49')
.2 sin 48 n' =



The area of a circle of the radius o.i is 0.031416, and the
angle qop = cot = 4i. 81; consequently, the area of the sector
qop is

0.031416X41.81 ^-360 = 0.00365.

Adding the first two areas, and subtracting the third, and
then multiplying by two for both sides of the blade gives

2(0.04601+0.02236 0.00365) =0.1294,

which is the area sought for a circular blade; the area of a circle
having the diameter unity is 0.7854; consequently, the area-
ratio of the blade with circular projected contour is

o.i 294 -^-0.7854 = 0.1648.

This is an important quantity for the standard blade, because all
the properties of the blade are made to depend on it.



42 PROPELLERS

The projected area ratio for any projected width of blade is
found by multiplying the ratio just computed, by the width-ratio
and dividing by 0.4. Thus the width-ratio of Fig. 12 is 0.2; its


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