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Font size of 600 feet; the corresponding speed for the new ship will give

23.18 : 25 :: V6oo : VL, .'. L = joo feet (nearly).

The beam and draught as computed on page 7 are 76.1 feet
and 29.2 feet, and the displacement is about 28,600 tons.
The extended law of comparison gives

18000 : 28600 1:31050 : I.H.P., /. I.H.P. = 53300.

This problem may conveniently be solved by logarithms as
follows :

log 28600 = 4.4564
log 18000 = 4.2553

.2011
7

6)1.4077

.2346

log 3 1050 = 4.4921

log 53300 = 4.7267

CHANGE OF SPEED 13

Should the computation be for a smaller ship the order for
logarithmic work may be changed as follows. Suppose the dis-
placement were 12,000 tons; then

18000 : 12000 : : 31050 : I.H.P. /.I.H.P. = 19300

log 18000 = 4.2553 log 31050 = 4.4921

log 12000 = 4.0792 - 2 55

.1761 log 19400 = 4.2866

7

6)1.2327
.2055

Change of Speed. In using the laws of similitude it will fre-
quently happen that the desired speed will differ from that derived
from the type ship. If the difference is large another type ship
must be chosen especially when the speed is high. If the difference
between the desired speed and the corresponding speed is small
then we may allow, for the change of speed on the assumption that
the power varies according to the law:

The power for a ship is proportional to the cube of the speed.

Example. The power required to drive the Campania at 26
knots per hour will be approximately

^ 3 : 2~6 3 : 131050 : I.H.P. /.I.H.P. = 34900.

Change of Displacement. A ship is designed for a certain
normal displacement but frequently is loaded to a different dis-
placement and it is important to know what influence such a change
will have on the speed. This matter has no relation to the theory
of similitude because the ship at a different draught will have an
under-water body which is not similar to that at normal draught.
In particular the relation of beam to draught and the block-coef-
ficient will be different, and both of these features have an appre-
ciable effect on propulsion.

In much the same way it may be found that the design for a ship
is restricted in draught and cannot have the draught that the

14 PROPELLERS

laws of similitude would indicate, when used with the proportions
of a certain type ship. Also the lines may be fuller (or finer) and
the displacement may thus vary from that computed by the
laws of similitude.

The best method of finding the influence of displacement on
speed is by trials of ships at various draughts; such trials are
seldom made. When models are tried to determine power, they
are frequently towed at various draughts.

This subject is both difficult and uncertain but we may use the
following equation for allowing for small changes of draught or
displacement

(LH.P.)i : (LH.P.) 2 ::ZV : D 2

and the value of the exponent n may vary from f for large ships
of moderate speed to ^ for ships and boats at high speeds.

Problem. Let it be required to determine the dimensions and
power of a ship 700 feet long and having a displacement of 28,000
tons to make 25.5 knots per hour.

First let the problem be solved directly from comparison with
the Campania and afterwards allow for change of displacement
and speed.

The relative speed of a ship 700 feet long will be found by the
equation

A/6oo : V^oo 1:23.18 : V, :.V = 2\$ knots.

The displacement of a ship 700 feet long and similar to the
Campania as shown on page 5 will be 28,600 tons.

Such a ship at 25 knots per hour should have 53,300 I.H.P.
as computed on page 12; at 25.5 knots the power would be

^5 3 : ^5 3 : : 533oo : I.H.P., /. I.H.P. = 56600.

If the power is proportional to the two-thirds power of the
displacement the design for 28,000 tons will call for

28600 : 28000 1:56600 : I.H.P., /.I.H.P. = 55800.

SPEED-LENGTH-RATIO 1 5

Speed-length-ratio. The rule for corresponding speed shows
that intelligent comparison of speeds of ships must take account
of the lengths. For this purpose we may use the speed-length-
ratio expressed by the ratio

VL

in which V is the speed in knots per hour and L is the length in
feet. A study of the table on page 8 will show that the speed-
length-ratio is approximately as follows:

Ratio.

Freighters 0.5 to 0.55

Passenger ships 0.7 to 0.8

Fast passenger ships 0.9 to i.o

Battleships 0.9 to i.o

Cruisers i.o to 1.2

Torpedo-boats and destroyers 1.8 to 2.0

Fast motor boats 2.5 to 5.0

In a rough way all craft having a speed-length-ratio under
unity may be classed as slow or moderate speed, and all with a
greater ratio, as fast.

Model Basins. In order to understand the methods of esti-
mating power which are called independent estimate and model
experiments it is necessary to know how model experiments are
made and how the results are used.

Model experiments are habitually and desirably made at
model basins or tanks; improvised methods in open water are
difficult and liable to be misleading. Such experiment stations
have costly and delicate apparatus, and experimenters must have
experience and discretion to get valuable results. But the funda-
mental conceptions are simple.

A model basin or tank is a canal 300 or 400 feet long, about
30 feet wide and 10 feet deep. The side walls of the canal carry
rails bedded on masonry. A carriage, like a traveling-crane, spans
the canal and travels on the rails. This carriage is driven electri-

16 PROPELLERS

cally much like a trolley car and can be started quickly and driven
at a uniform speed.

A model of the ship, 10 to 20 feet long, is cut to the lines of the
ship and is ballasted to float with the proper displacement and
trim. The model is towed from the carriage at various speeds and
the resistance or pull on the towing apparatus is measured. This
is known as the tow-rope resistance.

The first experiments of this sort were made by William Froude,
who also determined surface friction and proposed the method
of independent estimate.

Resistance. The force necessary to maintain a ship at uniform
speed is known as the resistance. When a ship is propelled by
its own machinery the resistance is affected by the methods of
propulsion and usually is greater than the tow-rope resistance.

As proposed by Froude, the two-rope resistance is separated
into surface or frictional resistance and residual resistance. The
residual resistance is further separated into wave-making resistance,
eddy-making resistance and steam-line resistance.

Frictional Resistance. It is customary to calculate the sur-
face or frictional resistance by the equation

in which R f is the force, in pounds, required to overcome the
surface resistance, S is the wetted surface in square feet and V
is the speed in knots per hour; / and n are quantities taken from
tables given on pages 17 and 18.

This equation is seldom used directly in practice but is used in
building up the method of independent estimate of power.

The first two tables were derived by Naval Constructor
D. W. Taylor from values published by R. E. Froude. The third
table is slightly modified and extended and used by Wm. Denny
and Bros. Tideman's table was derived by him from Wm.
Froude 's experiments.

FRICTIONAL EESISTANCE

17

FROUDE'S SURFACE FRICTION CONSTANTS.
Given by Taylor.

SURFACE-FRICTION CONSTANTS FOR PARAFFIN MODELS IN FRESH WATER. EXPONENT

Length, Feet.

Coefficient.

Length, Feet.

Coefficient.

Length, Feet.

Coefficient.

2 .O

0.01176

IO.O

0.00937

14.0

0.00883

3-0

0.01123

10.5

0.00928

14-5

0.00887

4.0

0.01083

II .0

0.00920

15-0

0.00873

5-o

0.01050

s

0.00914

16.0

. 00864

6.0

O.OIO22

12 .O

o . 00908

17.0

0.00855

7.0

O.OOQQ7

12.5

0.00901

18.0

0.00847

8.0

0.00973

13.0

0.00895

19.0

o . 00840

9.0

0.00953

13-5

o . 00889

20. o

0.00834

SURFACE-FRICTION CONSTANTS FOR PAINTED SHIPS IN SEA-WATER. EXPONENT

n = 1.825.

Length, Feet.

Coefficient.

Length, Feet.

Coefficient.

Length, Feet.

Coefficient.

8

0.01197

40

0.00981

180

o . 00904

9

0.01177

45

0.00971

200

o . 00902

10

O. OIl6l

50

. 00963

2 S

0.00897

12

0.01131

60

0.00950

300

0.00892

14

o. 01106

70

o . 00940

350

o . 00889

16

O.OIO86

80

0.00933

400

o . 00886

18

0.01069

90

0.00928

450

o . 00883

20

0.01055

IOO

0.00923

500

o . 00880

25

O.OIO29

1 20

0.00916

550

0.00877

30

O.OIOIO

140

0.00911

600

0.00874

35

0.00993

1 60

o . 00907

Given by Denny.

SURFACE-FRICTION CONSTANTS. EXPONENT, 1.825.

Length, Feet.

Coefficient. ;

Length, Feet.

Coefficient.

Length, Feet.

Coefficient.

40

. 00996

260

0.00870

550

. 00853

60

0.00957

280

o . 00868

600

O . 00850

80

o . 00933

300

. 00866

650

o . 00848

IOO

0.00917

320

O . 00864

700

0.00847

120

o . 00905

340

O . 00863

750

o . 00846

140

o . 00896

360

6.00862

800

o . 00844

1 60

o . 00889

38o

0.00861

850

0.00842

1 80

o . 00884

400

. 00860

900

0.00841

200

0.00879

420

0.00859

950

o . 00840

22O

0.00876

450

0.00858

IOOO

o . 00839

240

0.00872

500

0.00855

IS

PEOPELLERS

TIDEMAN'S SURFACE-FRICTION CONSTANTS.
Derived from Froude's Experiments.

SURFACE-FRICTION CONSTANTS FOR SHIPS IN SALT WATER OF 1. 026 DENSITY.

Copper or Zinc Sheathed.

Trrm "Rnttrm ("M a r\

Length
of Ship

Well Painted.

Sheathing Smooth and

Sheathing Rough and

in Feet.

in Good Condition.

/

n

/

n

/

n

10

0.01124

.8530

O.OIOOO

I-9I75

0.01400

.8700

2O

0.01075

.8490

o . 00990

I .9000

0.01350

.8610

30

0.01018

.8440

o . 00903

I . 8650

0.01310

.8530

40

o . 00998

8397

0.00978

I . 8400

0.01275

.8470

5

0.00991

8357

0.00976

I . 8300

0.01250

.8430

100

0.00970

.8290

o . 00966

1.8270

O.OI2OO

.8430

150

0.00957

.8290

0.00953

1.8270

O.On83

.8430

200

o . 00944

.8290

0.00943

I .8270

O.OII70

.8430

250

0.00933

.8290

o . 00936

I .8270

o. 01160

8430

300

0.00923

.8290

o . 00930

I .8270

O.OII52

. 8430

35

0.00916

.8290

0.00927

I .8270

O.OII45

.8430

400

0.00910

.8290

0.00926

1.8270

O.OII40

.8430

45

o . 00906

.8290

o 00926

I .8270

O.OII37

.8430

500

o . 00904

.8290

0.00926

I .8270

O.OII36

. 8430

Residual Resistance. The residual resistance is computed
from trials on ships or experiments on models, by subtracting the
surface or frictional resistance from the tow-rope resistance. A
convenient form for expressing residual resistance is

&Z> f F 4

(S)

where D, V, and L are the displacement in tons, the speed in
knots and the length in feet, and b is a numerical factor.

Long fine ships, like Atlantic liners may have = 0.35; moder-
ately fine ships may have 6 = 0.40; ships broad in proportion to
length but fine at ends, like war-ships, may have b = 0.45 ; freight
ships may have 6 = 0.45 to 0.5. The value of b is also likely to be
affected by speed especially when the speed-length-ratio is high.

The residual resistance for ships that have small external

STREAM-LINE RESISTANCE 19

appendages is mainly wave-making resistance and is frequently
called by that name. It probably follows the laws of mechanical
similitude (at least approximately) and may be used with fair
confidence when properly derived from tests or experiments.
For ships having a speed-length-ratio less than unity the wave-
making resistance is not large (relatively) and may be used as a
valuable check on other methods even though the factor b is
uncertain.

The residual resistance is seldom used in practice, but forms
an element of the method of independent estimate of power; all
the reservations for residual resistance apply to that elment of the
method of independent estimate.

Stream-line Resistance. The passage of a ship through the
water deflects it to the sides and it closes in again astern of the ship.
This action is accompanied by the formation of a system of waves
which travel along with the ship. The crests of the waves may
be broken especially near the bow of the ship; but on the whole
the water appears to flow past the ship in an unbroken stream.
The curved path followed by a drop of water in the stream, is known
as a stream line. The hydrostatic pressure of water in a stream
line varies much as it would in a pipe through which water is
flowing, decreasing as the velocity increases and vice-versa. There
is therefore a variation in pressure along the side of the ship. If
on the whole the variation of pressure of the whole stream of
water which appears to flow past the ship gives an unbalanced
resultant pressure, then there is stream-line resistance.

Both theory and experiment lead us to think that stream-
line resistance is small for a well formed ship. In practice no
attempt is made to compute stream-line resistance separately.
Care should be taken that bilge-keels and other external appendages
do not interfere with stream-line flow, and cause undue resistances
from formation of eddies or otherwise.

Stream Lines about Ships. To give an idea of forms of stream
lines past the hulls of ships Figs, i and 2 are given on page 20.
The first represents a cruiser with a block-coefficient of 0.53 and
a speed-length-ratio of i.i, while the second represents a collier
with a speed-length-ratio of 0.7 and a block-coefficient of 0.72.

20

PROPELLERS

FIG. i. Stream Lines about a Cruiser,

FIG. 2. Stream Lines about a Collier.

EDDY-MAKING RESISTANCE 21

Eddy-making Resistance. A well formed ship of proper pro-
portions has little if any eddy-making resistance, unless it has
external appendages, like propeller-shaft struts, or spectacle-
frames. Bilge-keels if they cut across stream lines and especially
if extended toward the ends of the ship may cause large eddy-
making resistances. Well arranged bilge-keels may give a resist-
ance equal to two or three per cent of the resistance without bilge-
keels; this is little more than the resistance computed by Froude's
method for their surface.

Merchant ships with two or more shafts, usually have the
propeller shafts carried by spectacle-frames. With outward
turning screws the fins for such frames should droop at an angle
of about 22^. So arranged the resistance may be 2 or 3 per
cent of the resistance of the bare hull. At improper angles the
resistance of such fins may be 10 or 12 per cent.

War-ships and yachts commonly have the propeller shafts
carried by brackets which may increase the resistance as much
as 10 per cent. The resistances of such appendages are habitually
investigated at model basins when precision is desired.

Wave-making Resistance. It has been stated of stream-
line resistance and of eddy-making resistance that they individu-
ally are small for well formed ships, consequently the residual
resistance can be charged mainly if not entirely to wave-making
resistance. From theoretical considerations it can readily be shown
that the power required to maintain the system of waves which
travels along with a ship at high speed is large enough to account
for most if not all the residual resistance but a useful quantitative
value cannot be assigned to residual resistance in this way. It
is customary to derive the form for calculating this resistance from
theoretical considerations but to base the computation on comparison
with tests on ships and experiments are made as previously stated.

Total Resistance. Summing up the surface resistance as
expressed by equation (4), page 16 and the residual resistance
as given by equation (5), page 18, we have for the total tow-
rope resistance in pounds

bV* (6)

22 PROPELLERS

To repeat, V is the speed of the ship in knots per hour, D is the
displacement in tons, 5 is the wetted surface in square feet, and
L is the length in feet; /, n, and b are factors for which values are
given on pages 17 and 18.

Independent Estimate. Now one knot per hour is

6080-^60 = 101.3

feet per minute; consequently the work required to tow a ship
can be found by multiplying the resistance as given by equation
(6) by 101.3 F, where V is the speed in knots per hour. To find
the horse-power, we may divide the work so computed by 33,000.
Consequently the horse-power required to tow the ship is

...

33000

Replacing R by its value in equation (6) we have for the net horse-
power.

E.H.P.=o.oo307(/5T" +1 +&^ F 5 ). (7)

\ ** /

F = speed in knots per hour, D = displacement in tons, S = wetted
surface in square feet, and L is the length between perpendiculars
in feet; for/, n and b, see pages 17 and 18.

Coefficient of Propulsion. The effective horse-power is that
required to tow the ship. To find the power which must be
developed by a steam-engine we must allow for the friction of the
engine, the efficiency of the propeller, and for the interaction
between hull and propeller. It is customary to lump all these in
a single factor called the coefficient of propulsion, which varies
from 0.45 to 0.65; that is to say the effective horse-power is
only 0.45 to 0.65 of the indicated horse-power.

For turbine steamers and for internal combustion engines
the shaft horse-power is reported and used in design. The coefficient
of propulsion in this case is the ratio of the effective horse-power
to the shaft horse-power. For turbine steamers the ratio is likely
to be from 0.45 to 0.65, because the propellers chosen for such ships

MECHANICAL EFFICIENCY 23

have a poor efficiency. For ships and boats driven by internal
combustion engines the ratio may run from 0.5 to 0.7; it does not
appear to be so well known.

Mechanical Efficiency. The mechanical efficiency of a steam-
engine is the ratio of the power delivered to the propeller shaft
to the power shown by the steam-engine indicator. This ratio
depends on the workmanship and condition of the engine and shaft
and may vary from 0.8 to 0.9. The larger value may be used for
engines known to be in good condition.

Efficiency of Propeller. The efficiency of propellers may be
estimated from tables on pages in to 121, allowing for imperfec-
tions when necessary. For reciprocating engines under favorable
conditions it may be taken as 0.65, for preliminary designs; for
steam turbines it is liable to be as small as 0.50.

Hull-efficency. The propeller from choice and necessity is
placed at the stern of the ship where it works in the wake or stream
of water set in motion by the ship. It can abstract some energy
from the wake, a gain of five or ten per cent being possible from
this action. On the other hand it disturbs the stream lines and
the flow of water toward the propeller causes a reduction of pres-
sure at the stern; it is popularly considered to produce a suction
on the stern and thus to increase the resistance. This effect
known as thrust deduction may amount to five or ten per cent.
The wake gain and thrust deduction tend to counteract each other.
To allow for this combined action it is customary to use a factor
called hull efficiency which may vary from 0.9 to unity. For large
well formed ships it is commonly taken as unity. A more com-
plete statement of wake, thrust-deduction and hull-efficiency
will be found on page 74.

The propulsion coefficient is the product of the mechanical
efficiency, the propeller efficiency and the hull efficiency. If the
mechanical efficiency is 0.9, the propeller efficiency 0.65 and the
hull-efficiency is unity, then the coefficient of propulsion will be

0.65X0.9X1 =0.6.

Problem. Recurring to the problem first stated on page 6
we may compute the indicated horse-power for a ship to make 25

24 PROPELLERS

knots per hour by the independent estimate as follows. Basing
the design on the Campania (page 8) we may first find the length
from the corresponding speed

23.18 : 25 :: V6oo : Vz,, .*. = 700 (nearly).

The other dimensions as computed on page 7 should be
beam 76.1 feet, and draught 29.2 feet. The displacement is
found by the proportion

600 : 700 :: 18000 : D, .'. D = 28600 tons.
The wetted surface may be computed by the proportion

600 : 700 :: 49620 :-S, .'. 5 = 67540 sq. ft.

The independent estimate is more flexible than either the
Admiralty coefficient, or the theory of similitude, but is most
successful when related to a type ship. In particular the factor
b for the residual resistance should properly be deduced from
speed trials of the type ship; but unfortunately it is not often
determined or quoted.

If the main dimensions are determined in some other way
or if they are modified from those derived from a type, then the
displacement may be computed from the block-coefficient, and
the wetted surface may be computed from equation (3), page
ii. The block-coefficient should be the same or nearly the
same as that for the type ship, and the length should vary but
little from that determined by the law of corresponding speed.
To complete the computation and exhibit the forms just quoted
we may find this displacement and wetted surface as follows :

Displacement = 0.644X700X76. i X 29. 2 -7-35 = 28600 tons.
The ratio of beam to draught is

76.1-7-29.2 = 2.6,

MODEL EXPERIMENTS 25

for which the factor C (page n) is 15.51, so that

Wetted surface = 1 5 . 5 1 A/2 8600 X 700 = 69400

which is somewhat in excess of two per cent more than the wetted
surface from the type ship by the theory of similitude; the former-
value (67,540) will be used in our computation.

The factor / and exponent n may be taken from Denny's
table on page 17, as

7=0.00847 n = 1.825.
Equation (7) applied to this case gives

E.H.P. = 0.00307 ( 0.00847 X 67540 X25 2 ' 5 +o.35 2
\ 700

=0.00307X0.00847X67540X8895

+0.00307 Xo.35 X935 X9766ooo-^ 700
= 15600+14000 = 29600.

The computation is best made by aid of the tables of powers
of displacements and speeds on pages 123 and 125. As a matter
of convenience in the solution of the next problem the friction
power and the residual power are computed separately and then

The coefficient of propulsion may be assumed to be 0.6 and
the indicated power may be estimated as

I.H.P. = 29600 -f- 0.6 = 49300.

Model Experiments. The fourth method for determining power
is by aid of model experiments in a towing basin. To illustrate
the method suppose that the tow-rope resistance for a paraffine
model 20 feet long is 12.8 pounds, when towed at the corresponding
speed of 4.23 knots. This speed is computed by the proportion

V7oo : A/20 1:25 : V m , .'. 7^ = 4.23 knots.

The theory of similitude gives for the wetted surface of the
model

- 2 - 2

700 : 20 1:67540 : S m , .'. 5 m = 55.i sq.ft.

26 PROPELLERS

The friction factor and the exponent taken from Froude's
table on page 17 are

7=0.00834 = i.Q4,
consequently the frictional resistance is

0.00834 X 55. i X4-23 1 ' 94 =0.00834X55. 1X16.41 = 7. 54 pounds.

Subtracting this frictional resistance from the total tow-rope
resistance of the model gives for the residual resistance

12.8 7.54 = 5.26 pounds.

The corresponding residual resistance for the ship will be
proportional to the displacement and the displacements are pro-
portional to the cubes of the length, so that

20 : 700 :: 5.26 : jR, /. ^ = 225500 pounds.

At twenty-five knots per hour the horse-power to overcome
this resistance will be

0.00307X225500X25 = 17310.

This residual power added to the frictional power previously
computed on page 25 will give for the total power

E.H.P. = 15600+17310 = 3290,

and with the coefficient of propulsion 0.6 the indicated power will be
I.H.P. = 32910 -i-o.6 = 54900.

Comparison of Methods. The four several methods of esti-
mating power given on pages 2, 12 and 24 may be compared as
follows :

METHODS FOR SMALL BOATS 27

Law of comparison 53,3

Independent estimate 49,300

Model experiment 54>9OO

In this particular application the Admiralty coefficient and the
law of comparison should give satisfactory results, because the
type ship is supposed to be followed closely in the design. In
passing from a smaller to a larger ship the tendency is to over-
estimate the power but not to a troublesome degree.

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