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Font size they vary in somewhat the same way, so that the hull -efficiency is
likely to vary less than the elements from which it is derived.
Moreover, the hull-efficiency for large well-formed ships will not
be very different from unity, and unless we have direct evidence,
we may therefore commonly leave it aside in powering ships and
designing propellers.

Determination of Wake and Thrust-deduction. There are two
ways of investigating the factors for wake and thrust-deduction,
namely, by model experiments in the towing-tank and by the
analysis of progressive speed trials.

Model experiments in the towing-tank, as outlined, are made
by these three operations, or their equivalents:

(1) The model is towed with all appendages in place, but
without the propeller, to determine the resistance R at the speed V.

(2) The propeller is adjusted behind the model and is driven
at such a number of revolutions r as will develop a thrust T equal
to the pull of the model at the speed F; on account of the thrust-
deduction the pull is now greater than R.

(3) The propeller is run in the open water at the same number
of revolutions r, and the speed of the carriage V a is adjusted so
that the thrust shall be T as in the second operation.

The thrust-deduction is then found by the equation,

R

R i-r ' t ~ I r

76 PROPELLERS

The wake is

V-V a

Since it is difficult to secure the exact adjustments given above
it is customary to make a series of experiments for each condition
and to select the quantities derived from faired curves, the details
are a matter for the experimenter to adjust and need not be con-
sidered at length here.

The operations for finding wake and thrust-deduction are
purposely stated in the form which is convenient for calculation
rather than for experiment, in order to clarify the conceptions of
those properties and to emphasize the fact that they are the
properties of models; the corresponding properties for ships may
be inferred from those for models, but with considerable difficulty
and uncertainty.

In the first place it is difficult to get sufficiently certain and
exact information for ships even after careful and exhaustive trials;
but when the trials are satisfactory so far as they go, they are
necessarily incomplete. Thus, for reciprocating engines, it is
necessary to allow for the friction of the engines, of which but
little is known positively; for turbine steamers the shaft horse-
power is found directly, and in so far there is less uncertainty.
The feature in which trials are necessarily incomplete is the
power delivered by the propeller to the thrust-block.

Even so explicit a matter as pitch of the propeller may be
uncertain, either because the pitch may vary or because the
measurement of the pitch may have been slighted. Planed pro-
pellers are of course free from this difficulty.

When we undertake to infer the wake and thrust-deduction for
a ship from its model it is necessary to use the theory of similitude,
which is known to fail for the resistance and may be suspected for
the propeller. In particular it is known and allowance is made for
the fact that surface friction does not follow the laws of similitude.
In consequence the slip of a model propeller must be larger than
the slip of the ship's propeller; the apparent slips are known to
vary in this manner, and the real slips may vary more markedly.

FACTORS FOR WAKE AND THRUST-DEDUCTION 77

From these considerations it is clear that in order to make
towing- tank results of real value they must be a part of a system
including trials of the ships after construction. From SUC!L a
system certain factors can be determined by which it is possible to
infer with sufficient certainty for practical purposes what a ship
will do from tests on its model. Very commonly all the factors
are lumped into one called the coefficient of propulsion, denned
on page 22.

A statement of methods of making progressive speed trials
the observations to be taken, the precautions to be observed,
and the deductions from them will be found in the author's Naval
Architecture. Fortunately, a reasonably good approximation to
the wake of the ship is sufficient for the design of the
propeller.

Factors for Wake and Thrust-deduction. The factors which
are given for wake and thrust-deduction are mainly those reported
from time to time by R. E. Froude, which were deduced mainly
for war-ships, some of which are of obsolete types. Recently an
extensive series of experiments were reported by Mr. W. J. Luke
for twin- screws applied to a common form of merchant ship.

Both Froude and Luke report that the number and area of the
blades of a propeller have little effect on either wake or thrust-
deduction. Luke reports that increased diameter increases both
wake factor and thrust-deduction, but considers that the effect
is rather due to changes in clearance between the propeller and
and the hull than to the increased size.

The change of clearance between the propeller and the hull
has a great effect on both wake and thrust-deduction; insufficient
clearance is always to be avoided.

Pitch-ratio has an appreciable but not important effect on both
factors.

Change of speed of the model had practically no effect on thrust-
deduction, but the wake decreased appreciably with increasing
speed. For a speed-length-ratio

VL

78 PEOPELLERS

which is common for such a type of ship; the wake was about
0.17, and the thrust-deduction was about 0.16, so that the hull
efficiency was somewhat more than unity.

An approximate determination of the wake of a model may be

Single-screw ships

w = o.2o+| (block-coefficient -55)-
Twin-screw ships

w = 0.10+^ (block-coefficient 0.55).

The wake of a large ship is likely to be less than the amounts
given by these equations, perhaps as much as ten per cent. An
allowance of ten per cent would make the first term o.io instead
of 0.20 for single-screws and would reduce that term to zero for
twin-screws .

Mechanical Efficiency. A marine engine may be expected to
lose from 10 to 15 per cent of its power in friction, variously
distributed at the pistons, crank-pins, main-bearings, thrust-
block, and elsewhere; the power required to drive the air-pump
from the main engine is variously estimated from 3 to 7 per cent.
The mechanical efficiency may consequently be estimated from
0.8 to 0.9. Experiments with torsion meters from a few engines
in good condition with independent air-pumps have shown efficien-
cies from 0.9 to 0.92; though there are difficulties in applying
torsion meters to reciprocating engines, it is fair to assume that
engines may have an efficiency of 0.9 under favorable conditions.
There appears to be no reason why this factor should be affected
by size, but rather that it depends on the construction and con-
dition of the engine.

Effective Horse-power. The simplest and perhaps the most
useful information that can now be derived from a towing-tank is
the resistance of the hull with appendages. Let the resistance of
the ship as computed from model experiments, be represented by R
in pounds. Then if the speed of the ship in knots per hour is V

COEFFICIENT OF PROPULSION 79

the speed in feet per minute will be 101.37; the effective horse-
power will then be denned as

E.H.P. =R X 101.3 V -*- 33000 = 0.00307^7. . ^(17)

If the resistance is estimated in some other way than by direct
experiment on the model, the same form may be used to compute
the effective horse-power.

Coefficient of Propulsion. The coefficient of propulsion is taken
as the ratio of the effective horse-power to the indicated horse-
power,

Coefficient of propulsion = E.H.P. -^LH.P.

For turbine steamers the shaft horse-power may be substituted
for the indicated horse-power, bearing in mind that the mechanical
efficiency does not enter into the coefficient.

The connection between the effective horse-power and the
indicated horse-power can be built up in the following manner:

If e m is the mechanical efficiency the power delivered to the
shaft will be

S.H.P. = ^XLH.P ....... (18)

The shaft horse-power multiplied by the efficiency of the pro-
peller e p will give the power charged to the propeller. But the
propeller gains from the wake, so that the power applied to the
thrust-block is

^,XS.H.PX; ....... (19)

On the other hand, the interference of the propeller with the
stream-lines increases the resistance and consequently the power
required for propulsion is

(20)

The expressions (19) and (20) must be the same, so that finally,

E.H.P. i-/

Coefficient propulsion = T H p~ = e e p _ > ( 2I )

SO PROPELLERS

that is, the coefficient of propulsion is the continued product of the
mechanical efficiency, the efficiency of the propeller, and the hull-
efficiency.

If the hull-efficiency is assumed to be unity and if the efficiency
of the propeller is assumed to vary from 0.5 to 0.7, while the
mechanical efficiency is taken from 0.8 to 0.9, the coefficient of
propulsion may vary from

0.8X0.5=0.4 to 0.9X0.7=0.6.

The factor is commonly taken as 0.5 to 0.55 for well-formed ships;
this should usually give a margin for contingencies.

Method of Reporting Experiments. The Model Basin at
Washington undertakes tests of models of propellers for private
parties, under certain restrictions, and as the results are reported
in a particular way, it is proper to present it here. Usually the
information is in the form of curves plotted or real slips as abscissae
and gives the efficiencies at various slips, and also the factor A
for computing the shaft horse-power by the following equation,

S.H.P.=,4 3 ; (22)

1000

where d is the diameter of the propeller in feet and V a is the speed
of advance in knots per hour, while A is a factor that varies with
the slip.

A model to one-fifth natural size of the propeller of the U. S.
Revenue Cutter Manning was tested at the Basin with the results
given in the following table:

MODEL EXPERIMENTS ON "MANNING" PROPELLER.

Real slip o.o 0.02 0.04 0.06 0.08 o.io 0.12 0.14

Value of A 1.74 1.96 2.20 2.48 2.80 3.13 3.48 3.86

Efficiency 0.587 0.615 0.640 0.654 0.665 0-673 0.678 0.683

Real slip 0.16 0.18 0.20 0.22 0.24 0.26 0.28 0.30

Valueof^l 4-29 4-79 5-34 5-93 6.60 7.75 8.17 9.05

Efficiency 0.682 0.680 0.677 0.672 0.668 0.660 0.652 0.644

The Manning on trial had an apparent slip of 13.5 per cent
at 1 6 knots per hour, and special experiments indicated that the

PROPELLER TABLES 81

wake was 7 per cent. By equation (13), page 74, the real slip
was

S=I (l O.I35)(l 0.07) =0.20.

The above table gives at 0.20 real-slip A =5.34. The diameter
of the propeller was n feet, and consequently equation (22) gives
for the shaft horse-power,

1000

Va = (l w)V (l O.07)l6

from equation (10) on page 73.

From the indicated horse-power on trial the shaft horse-power
was estimated to be about eight per cent less than the amount
computed as above. Discrepancies of this nature under the most
favorable circumstances between computations from model experi-
ments and data from trials, are not unusual. Reasons for the
discrepancies can often be assigned and allowances can sometimes
be made which will reduce or remove apparent discrepancies.
But experienced designers who are familiar with model experi-
ments usually prefer to let the discrepancy stand and to allow for
it en bloc when they have occasion to predict trial results from
experiments. There is good reason for taking the small wake
factor 0.07 for the Manning; were it proper to take the more
common value of o.io, the discrepancy would appear to be dis-
posed of.

The form of report of experiments on propeller models is
convenient for comparison with trials of the ship, and its pro-
peller; it is not convenient for the selection of a propeller for a
particular service.

Propeller Tables. The tables at the end of this book will be
found convenient for determining the dimensions and proportions
of propellers; they may ordinarily be used without interpolation.

To enter the tables first compute the revolution factor R by
the equation,

r*(S.H.P.)

* - - T/T ; ....... (23)

82 PEOPELLERS

r = revolutions of the engine per minute;
S.H.P. = the shaft horse-power, to be estimated from the indicated

horse-power when necessary;

F = velocity of advance of the propeller to be estimated by the
following equation,

(24)

V = speed of the ship in knots per hour;
factor.

Fortunately, a considerable variation of either power or wake
factor will have relatively small effect.

Having computed R, enter any of the tables for two, three,
or four-bladed propellers and find the value of the diameter factor
D corresponding. Then compute the diameter by the equation,

D = tabular value corresponding to R of equation (23) ;
S.H.P. = shaft horse-power;

Fa = speed of advance of propeller;
r = revolutions of engine.

// is to be borne in mind that, there are two places to be pointed
off in tabular values of R and one place in D.

Problem. Required the dimensions for a propeller for a ship
which is driven at 16 knots by an engine which develops 3000
horse-power at 100 revolutions per minute.

Taking 0.9 for the mechanical efficiency gives for the shaft
horse-power,

0.9X3000 = 2700.

The speed of advance of the screw with a wake of o.i will be

V a = V(i w) = 16(1 o.i) = 14.4.

PROPELLER TABLES 83

The revolution factor will therefore be
(100)1(2700)* 10X7.21

K = - - f - ri - = - - - =2.^7-
(14.4)* 28.05

The four-bladed table, page 112, area-ratio 0.36 gives ,0 = 51.4
at 1.3 pitch-ratio and 0.2 slip. Consequently the diameter is

j .-_-..
21.54

# = 1.3X13.9 = 18.1 ft.

The apparent slip is computed by the equation
i Si = (i \$)-5-(i =w) = (10.2) -T-(I o.i) =0.889; /. \$1=0.11.

The powers required for solution of this problem are most
readily obtained by interpolation in the tables on pages 122 and 123,
after which the numerical computation can be made by aid of a
slice rule.

If preferred -the solution may be made by logarithms as follows:

log 100 = 2.0000 log 2700 = 3.4314 log 14. 4 = 1.1584

_1 _ i _ s

i. oooo 0.8578 4)5.7920
0.8578 1.4480

1.8578

1 . 4480

log 2. 57 =0.4098

log 2700 = 3.4314 log 100 = 2.0000
log 14. 4 = 1. 1584 2

6)4.5898 3)4.0000

0-7650 1-3333

log 51. 4 = 1. 7110

2.4760
1-3333

log 13. 9 = 1. 1427

84 PROPELLERS

Problem. Required the dimensions of twin-screw propellers
for a ship to be driven at 20 knots by two engines each developing
8000 horse-power at 90 revolutions per minute. Here

S.H.P. = 8000X0.9 = 7200,

V a = 20X0.9 = 18,

R= -^- =2.36.

At pitch-ratio 1.5 and real slip 0.24 in the table for three-bladed
propellers area-ratio 0.27, page 117, this corresponds to ^ = 50.5,
and

50.5(7200X18)*
d= i - = 17.9 feet,

^ = 1.5X17.9 = 26.85 feet,
10.24

=0.156.

Choice of Conditions. There is apparently a wide range of
choice given the designer by the tables on page 1 1 1 et seq. ; though
conditions are limited in practice and sometimes narrowly, the
designer usually has a considerable range which may at first seem
confusing. There are, however, a number of conditions that can
be stated simply to guide choice.

Number of Blades. Large single-screw ships habitually have

Ships with two, three or four screws usually have three-bladed
propellers. Sometimes two propellers out of four have three
blades and the other two have four.

Small craft of all sorts commonly have three-bladed pro-
pellers. Sometimes they have two-bladed propellers.

Area-ratio. The projected area-ratio for one blade may com-
monly be taken as 0.09; three-bladed propellers then have a
total area-ratio 0.27 and four-bladed propellers have 0.36.

BEST EFFICIENCY PITCH-RATIO AND SLIP 85

If there is danger of cavitation (a term to be explained later)
larger area-ratios are selected.

Narrow blades are useful mainly for small craft and_may
give comparatively high efficiency.

Best Efficiency. The best efficiency is indicated in the table
by printing values of R in full-faced type. Values of R as com-
puted by equation (23), should be located as near such full-faced
type as possible, but moderate deviations have little effect on
efficiency.

Pitch-ratio and Slip. In the presentation of this method of
selecting a propeller for a given purpose, pitch-ratio and slip
appear to enter incidentally or as matters of secondary import-
ance. In reality they are of first importance and the experienced
designer has a very good idea of the conditions desirable for his
problem. Fortunately, the method here proposed will usually

For large ships the pitch-ratio will range from i.o to 1.5 and
the apparent slip from o.io to 0.20; both pitch-ratio and slip
increasing with the speed-length-ratio. Turbine steamers suffer
from the necessity of using a high number of revolutions and a
small pitch-ratio, the latter being commonly 0.7 to 0.8.

Efficiency. The efficiency in the neighborhood of the full-
faced type ranges from 0.45 to 0.75, increasing with the pitch-
ratio, and being larger for narrow blades and for propellers with
few blades (three or two). But the variation for a given type
of propeller is not large and can be known approximately in

Small Diameter. The most common restriction on the design
of a propeller is the necessity to use a small diameter with a ship
of a given draught. This is the main reason for using four-bladed
propellers for single-screw ships. For the same reason wide
blades may be chosen, but they give little advantage except as a
means of avoiding cavitation.

Having selected the number of blades and the area-ratio,
special conditions, such as small diameter, can be sought by using
other parts of the table remote from the full-faced type. Thus
if we should on page 112 take a pitch-ratio of 1.50 and a slip of

86 PROPELLERS

0.26, the value of D becomes 48.3 and the propeller diameter will
be 13.1 instead of 13.9 as computed in the problem ^on page 83;
the efficiency is now 0.67 instead of 0.68.

If these several devices fail to give a propeller small enough
for the conditions, then the revolutions of the propeller must be
increased and the problem stated anew.

Precautions. In the use of the tables for propellers it must
be borne in mind that they apply to carefully made propellers,
with true smooth surfaces and sharp edges. If any of these
features are lacking, allowance must be made, which can best
be done by comparison of results from such propellers with the
known properties of the experimental propellers.

Degree of Accuracy. The degree of accuracy to be attributed
to Taylor's experiments has already been stated to be somewhat
better than one per cent in power; and as the power varies as
the square of the diameter the diameter factors may conversely
be given an accuracy of about one-half of one per cent for the model
experiments. But attention has been called to the possible inaccu-
racy of the law of similitude as applying to propellers, which
may amount to one or two per cent when the large propellers are
made with the care and precision of the models. Rough, blunt-
edged propellers may absorb somewhat more power than well-
made propellers; they will show a marked loss of efficiency in
some cases of three to five per cent or more.

The degree of precision of one per cent or better is to be
attributed to those parts of the tables which are derived directly
from Taylor's experiments; but certain parts of the tables have
been extrapolated and are subject to more uncertainty, amounting
perhaps, to two per cent. This reservation applies to the upper
left-hand corner of the four-bladed table area-ratio 0.28, and the
three-bladed table area-ratio 0.21, and to the greater part of four-

Characteristics. The general characteristics of propellers, as
shown by the tables, should be clearly held in mind by the designer.

In a given table, as, for example, that for three blades, area-
ratio 0.27, it will be seen that the diameter factor and consequently
the diameter is nearly constant for a given pitch-ratio, whatever

CHARACTERISTICS 87

the slip may be. There is some variation, usually a decrease as
the slip increases, followed by an increase; thus at pitch-ratio 1.2
the values of D and the efficiency vary as follows:

Slip 0.06 0.12 0.18 0.24 0.30

D 56.2 55-6 554 55-2 55-3

e 0.693 0.710 0.709 0.693 0.667

For a considerable range of slip the efficiency changes but little,
but there is an appreciable falling off for large slips. These con-
ditions vary somewhat for the various pitch-ratios.

The best efficiency for a given value of R will be found near
the full-faced figures; in some cases a higher efficiency may be
had for some other slip and pitch-ratio but corresponding to a
different value of R. In order to take advantage of the higher
efficiency it would be necessary to change the revolutions. For
example, at pitch-ratio 1.2 in the table referred to, the best effi-
ciency is found at slips 0.14 to 0.16, which correspond to ^ = 2.33
to R = 2. 43, but a better condition for that range in R can be
secured at pitch-ratio 1.4, slip 0.20.

The effect of area-ratio, that is, of width of blade, can be brought
out by assembling values for the properties corresponding to a
certain value of the revolution factor R. Thus in the three-bladed
table at pitch-ratio 1.2 and near the full-faced figures we may
select the following values :

Area-ratio... 0.21 0.27 0.36 0.45 0.54

Slip O.2O O.22 O.22 O.22 0.24

R 2.70 2.74 2.68 2.67 2.70

D 544 55-3 55-9 5 6 4 5 6 - 2

e 0.693 0.700 0.699 0.678 0.647

The values of R are the tabular values, but as the value of D
changes slowly those here set down can all be taken as corresponding
to the initial value R = 2.jo. It will be seen that there is a slight
increase in diameter as the area-ratio increases, and an appreciable
loss of efficiency for wide blades. The whole effect is, however,

88 PEOPELLERS

of secondary importance and we may conclude that the diameter
required is practically the same for all widths of blade.

The effect of using two, three, or four blades can be brought
out by the following abstract from tables having the same projected
area-ratio per blade, all at pitch-ratio 1.2:

No. of

Area-
ratio.

Slip.

R

D

e

4

0.36

O.22

2-93

52.9

0.668

3

0.27

O.22

2.74

55-3

0.700

2

0.18

0.22

2.66

57-9

0.716

Effect of Blade-thickness. The tables for designing propellers
are arranged to vary the thickness inversely as the width of the
blade, as should be the case for sake of strength. The assigned
thicknesses are likely to be minima except for small propellers, and
may be required to be increased for large propellers and for those
that deliver a relatively large thrust. The thickness of propellers
designed from the tables may be increased to half again as much
as that given, without appreciable effect. On the other hand,
there is an appreciable gain in efficiency from reducing the thick-
ness when this is possible, amounting to five per cent when the
thickness ratio can be reduced to 0.02. This gain in efficiency is
accompanied by a reduction in the power absorbed, so that there
will be little if any reduction in the diameter of the propeller to
drive a boat at a given speed.

Propellers designed by the tables will be but little affected by
changes of thickness that occur in practice.

Comparison with Tables. If the conditions of service of a ship
are such that the tables for propellers cannot be used directly,
they may still be used as a means of basing a design on the known
performance of a ship of the same type.

The essential feature in the use of the table which cannot be
determined directly from the trial of a ship is the wake, which is
used for calculating the speed of advance of the propeller, in the
equation,

COMPARISON WITH TABLES 89

We may assume a probable wake and solve for R and D by the
equations,

dr*

"(S.H.P.)W

the latter being from equation (25), page 82. These values may-
be compared with the proper table and if our first assumption of
wake appears unsatisfactory we can try again.

In some cases it may appear desirable to increase (or decrease)
the diameter factor D by a percentage in addition to seeking for
a probable wake factor.

For example, a trial of the police-boat Guardian showed that

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