projected area-ratio is
0.1648 Xo. 2 -7-0.4 = 0.08241.
Conversely, the width-ratio corresponding to any given area-
ratio may be found by multiplying by 0.4 and dividing by 0.1648.
Thus a blade having the area-ratio 0.08 will have the width-ratio
The blade-area computed by this method is very nearly
correct for propellers which have spherical hubs; if the hub is
barrel shaped and the blade is narrow there may be an error of
one per cent, a quantity which has no appreciable effect.
The total projected area-ratio for any propeller is found by
multiplying the area-ratio for one blade by the number of blades.
Factor for Blade-angle. In drawing the standard projected
blade contour it is convenient to lay off the angle eof, Fig. 12,
by aid of the dimension ef.
Turning to the circular blade contour of Fig. 13, we have for
ef=eo tanc0/ = o.5 tan 41 48' = 0.44721.
For any other blade the factor may be made to depend on
the width-ratio, or the projected area-ratio. By projection, the
width-ratios and the dimensions ef are proportional. But the
area-ratios are proportional to the width-ratios, so that the dimen-
sions ef are proportional to the area-ratios. Thus the area-ratio
0.08 has the width-ratio 0.1942 as computed. The factor for ej
ef= 0.4472X0.1942^0.4 = 0.21 71,
or /=o.4472Xo.8-7-o. 1648 = 0.2171.
Axial Dimension. Turning to Fig. n it will be remembered
that the blades there subtend 60, and have consequently one-
TO DRAW PROJECTIONS 43
sixth of a turn of the screw; the axial width shown by Fig. 10
is therefore one-sixth of the pitch. In the same way the axial
dimension of the blade in Fig. 12 will have the same ratio to the
pitch that the angle tot' has to 360. The laying-down table gives
the dimension ef, and ef divided by oe gives the tangent of the-
angle eof\ this is the half-angle and is to be divided by 180. There
the factor for the blade-angle is 0.2171 for an area-ratio of 0.08,
and the axial dimension factor is computed as follows:
0.2171-^0. 5 =0.4342 = tan 23 28' = tan 2347;
23.47-^-180 = 0.1304.
To Draw Projections. Since all the dimensions and propor-
tions can readily be computed for the standard projected contour,
the designer will follow his judgment and habit whether he will
make a drawing of the propeller or trust that to the makers.
The following method will be found rapid and accurate. After
the diameter and the projected area-ratio of the blade of a pro-
peller have been determined by methods to be given later, the
projections can be drawn as shown in Figs. 14, 15, and 16.
Let it be assumed that the propeller has four blades, a diameter
of 10 feet, a pitch of 20 feet, and a projected area-ratio of 0.075
for one blade. By interpolation in the laying-down table the
following dimensions can be found.
Width-ratio 0.1820; width 10X0.1820 = 1.820 ft. = 21.84 in.;
Axial factor 0.1230; axial dimension 20X0.1230 = 2.46 ft. = 29.52111.
In Fig. 14 the length is laid off equal to 5 ft., scale i in. =
i ft.; and the radius of the hub is made 0^ = 0.2X10^2 = 1 ft.
The line we is bisected at h and the width 21.84 m - is laid off
from x to y. An ellipse is drawn with we and xy as the axes.
The dimension ef is computed as follows: after the factor 0.2036
is found in the laying-down table,
ef= 10 X 0.2036 = 2.036 ft. = 24.43 i n ->
and is laid off on Fig. 14 and the line of is drawn; it is tangent
to the ellipse at t and locates the straight-edge ut of the blade
TO DRAW PROJECTIONS
contour. The line u f t f laid off on the other side of the blade com-
pletes the contour. In Fig. 14 the hub is drawn cylindrical,
as shown by the arc uu' '.
From the centre o the arc ge is drawn and divided accurately
into ten parts by spacing with dividers; the arc on the other side
of the blade gives a symmetrical construction and is therefore
The axial dimension uu' in Fig. 15 is laid off equal to 29.52 in.
(scale i in. = i ft.) and is divided accurately into twenty
equal parts and numbered consecutively from the right-hand
or forward edge. This propeller, like that shown by Fig. 10, is
right-handed and is represented as driving the ship toward the
right. The left-hand surface or face is to be a true helical surface.
The point a on Fig. 14 is the intersection of the No. 5 radial
line with the elliptical contour. On Fig. 15 this point is projected
onto the No. 5 ordinate; the symmetrical point a' of Fig. 14 is
projected onto the No. 15 ordinate. In like manner a sufficient
number of points like a may be located and the contour may be
drawn through them. It is now evident why the angle eog is laid
off and divided with precision. The point / is accurately located
by drawing st at 0.16667 of the diameter from the centre; in
os = 10X0.16667 = 1.667 ft. = 20 in.
The straight line edges ut and u't' of Fig. 14 appear as boundary
elements ut and u't' in Fig. 15. Since the hub is shown as a
cylinder the root line of the blade is shown as a helical curve
uwu f ' 7 a point like z is found by projecting z of Fig. 14 onto the
corresponding ordinate; in the case shown the ordinate is No. 5.
The contour of a blade at right angles to that just described
is shown by u"a"oa'"u' n on Fig. 15. The point a" is located
on the fifth ordinate by making 50" equal to ba of Fig. 14; the
point a'" is symmetrical with a" on the fifteenth ordinate. The
bounding elements are u"t" and u' f 't'" and the root line u"ou' n
is a part of a helix.
The drawing of the propeller for the inlormation of the designer
and the pattern-maker should be accurately drawn to a large
INTERSECTION AT HUB-PLANE SECTION 47
scale, if not full size. All lines should be drawn with a steel
straight-edge; the axis of the ellipse and the ordinates should be
laid off at right angles by a geometric method instead of depending
on a triangle. The division of eg of Fig. 14 and of the axial
dimension of Fig. 15 should be by spacing or some equally
accurate method. A line through e parallel to the axis of the
shaft should be laid off accurately and the divisions of the axial
dimension should be transferred to it, so that the ordinates may
be accurately located. The projection of a point, like a from
Fig. 14 to Fig. 15, should be made by measurement; thus 50
should be laid off equal to ob.
The thickness of the blade, which is all applied to the back
of the blade, is indicated by the line eid. The thickness od at the
axis divided by the diameter of the propeller is known as the
thickness ratio. In this case it is made equal to 0.02 of the diameter,
so that the thickness is 0.2 of a foot or 2.4 inches. The thickness
at the tip is 0.005 f tne diameter, which in this case is 0.6 of
an inch. Bronze blades are commonly made thinner at the tip;
the thickness at the hub is greater for narrower blades. Cast-
iron blades are much thicker.
Intersection at Hub. For simplicity the hub is represented
to be cylindrical and its intersection by the face of the blade
is a helix. The hub is always a surface of revolution so that
the intersection by an element of the face can be located by aid
of a plane through it and the axis, which plane is to be revolved
into the plane of the paper. The actual construction may be left
to the draughtsman who will work to a large scale. In practice
the blade joins the hub with rounded fillets cut by the pattern-
Plane Section. To show the form of the back of the blade
and for the instruction of the pattern-maker, it is customary to
give a number of sections like those shown on Fig. 14, where
mjhj'm'i is a plane section and vkhv'i is a developed cylindrical
section, to be explained in the next section.
A plane section perpendicular to the line oe, Fig. 14, cuts the
contour at xhy and in Fig. 15 at mhm' '; the points m and m' are
>rojected to / and I', and show the section of the blade contour
u"t"ot'"u" f , cut by a similarly placed plane parallel to the plane of
the paper and at the distance ob above it. The plane section
Ikok'l' is shown in its correct form; it will be found to be slightly
curved. To construct a point like k, draw the element ok 5 of
the helical surface on Fig. 14 and note where it cuts the line xhy
at the point &; this gives the correct transverse location of this
point. On Fig. 15 draw the corresponding element 5& and make
$k equal to hk of Fig. 14. The symmetrical point k' is located
by making 15^' equal to 5^. Having a sufficient number of points
like k and k r the section Ikok'l' can be drawn and transferred to
Fig. 14. The thickness of the blade is laid off equal to hi and
the back is drawn as the arc of a circle.
For cast-iron blades the edge cannot be so thin as this con-
struction gives; so some thickness is given at the edge and then
the back is rounded to the arc of a circle.
Very commonly the curvature of the line mjhj'mf is ignored in
drawing plane sections of a blade because it is slight. The curva-
ture is, however, important and must be allowed for, when sections
are made to be employed for sweeping up the mould of a propeller
on the floor of the foundry.
Developed Cylindrical Section. Suppose that a cylindrical
surface is constructed by revolving the line mhm', Fig 15, about
the axis of the shaft; it will cut the surface of the blade in a helix
shown by nhn' and by the arc nhn' of Fig. 14. If the cylinder
is developed into a plane the helix becomes a straight line. The
development of the cylinder can be made in Fig. 14 by laying
off the line hr equal in length to the arc hn f . The fore-and-aft
dimension of the helix hn f of Fig. 15 is pn f . If this be laid off at
hq, Fig. 14, the diagonal qr will give the half-width of the developed
helicoidal section. This dimension is laid off at hv and hv', and
the back is drawn through v, v' ', and i\ for this purpose an arc of
a circle may be used, though this is not quite correct if the plane
section is constructed with the back rounded to the arc of a circle.
Sections like those discussed in this and in the previous section
are drawn at intervals for the instruction of the pattern-maker;
the choice of section depends on how the pattern is made. The
draughtsman should have a practical knowledge of the making
BLADES WITH A RAKE 49
of propeller patterns; there should be a competent person charged
with the responsibility for the correct making of patterns and
for maintaining them in correct form.
Blades with a Rake. Fig. 16 shows the projection of the
propeller of Fig. 14, but with 15 rake aft. The ordinates are
now drawn with that inclination; the radius is measured per-
pendicular to the axis. In order to locate the helicoidal elements
the helix e'ee" must be constructed and then the elements like
o,e' and 2o,e" can be drawn. The points a and a' of Fig. 14
may now be projected onto the proper elements at a and a' on
Fig. 1 6. The contour of the edge of the blade u"t"a"a"'t"'u" f
can be drawn by the usual method of projections from Fig. 14
and the contour utaa't'u'-, then the point a" can be located on
the vertical line aa" at a distance b"a" below the axis, this distance
being equal to ba of Fi<*. 14. The thickness is laid off at right
angles to the line io,e.
The cylindrical section vkhv'i of Fig. 14 will be constructed
as before, and will differ only in that the dimension hi will be
slightly larger, because it is measured on a line inclined to the
axis 10, e of the blade.
As for the form of the plane section, it will depend on how it
is taken. If the plane is parallel to the axis of the shaft, the sec-
tion will differ very little from that shown in Fig. 14, and that con-
struction can be accepted for pattern-making or for sweeping up
blades in the foundry; the sections in the foundry must in such
case be set vertical, the blade being inclined at the angle of the
rake from the horizontal. But if the section is perpendicular to
the element io,e as shown by nhn f of Fig. 16, the form will be
materially different; it can be drawn by the ordinary methods of
descriptive geometry, but the construction is omitted to avoid
Helicoidal Area. The true or helicoidal area of the blade of
a propeller can be determined by aid of developed cylindrical
sections, such as that which gives the line vhv' of Fig. 14; a number
of such lines can be constructed at intervals from w to e, and a
contour or bounding line can be drawn; the area of that figure
will be the true area of the face of the blade. When the design
DEVELOPED CONTOUR 51
of a propeller is based on the projected area-ratio there is little
reason for dealing with the area of the blade.
Developed Contour. The surface of a screw-propeller is a ruled
surface which cannot be developed, but there are conventional
methods of constructing a plane figure which has nearly the same
surface as a blade. These methods are called developing the
blade, and the figure is called the developed contour.
The development of the blade of a propeller, and the inverse
process of constructing the projections from the developed contour
have an importance, because (i) certain propeller theories are
based on the developed contour, (2) nearly all the experimental
propellers tested in model basins have been designed from the
developed contours, (3) and the results of such experiments sys-
tematized in tables and diagrams are stated in the same terms.
In consequence engineers and designers are accustomed to working
with the developed contour, and for that reason, if no other,
the methods of drawing developed contour must be understood.
In Fig. 17 there is drawn half a turn of a helix gabch and the
development bf of half a turn of the helix beginning at b. A quarter
turn of the helical surface is shown by nabcp, comparable to the
quarter turn shown on Figs. 3 and 4. The line bf is tangent
to the helix at 6; the deviation of the tangent at s from the helix
at c, for an eighth of a turn is small; for less than an eighth the
deviation is insignificant. Propeller blades seldom if ever are so
wide as would be given by a quarter of a turn. x
The conventional development of the blade of a propeller
depends on the substitution of the straight line bs in place of the
helical arc be. The tangent bs is most conveniently located by
drawing the triangle tbu in which tu is computed by the proportion
be : ef::bt : tu.
But be = ^d, bt = %d, and ej=\p, where d is the diameter of the
propeller and p is the pitch. Substituting and solving for tu,
In Fig. 1 8 one-sixth turn of the helical surface is represented
approximately by nabcp, in which abc is the tangent line in place
of the true helical curve. Let a plane perpendicular to the plane
of the paper be passed through the cylinder at lbm\ it will cut
an elliptical section of the cylinder which can be rotated into the
plane of the paper, as shown on Fig. 19 by ea fr b'c"f. The elliptical
arc a"b r c" is considered to be the development of the helical arc
shown in projections by abc, Fig. 18, and a'b'c', Fig. 19. Two
other cylinders are represented by l\m\ and l^m^ in Fig. 18, with
approximate helical surfaces a\bc\ and azbcz', elliptical sections by
planes through the line l\m\ and hni2, are revolved into the plane
of the paper in Fig. 19, thus locating the elliptical arc* a\'c\ r
and 02 'W. A curved contour is drawn through a' r a\" , a 2 "ao
and another through c n c\'c^'c^ The points OQ and Co are located
oao = oco = bp (of Fig. 18).
The ellipses are all drawn from the foci o\ and 03, which may
be located in the usual way; that is, by drawing arcs from b'
with radii 0'0i and b'oz each equal to oe. Or since the triangles
FIG. 1 8.
/0/o of Fig. 1 8 and b'ooi of Fig. 19 are equal to each other, the
points 0i and 02 can be located by making
001=002 = 0/0 (Fig. 1 8)=,
because 6/0 in Fig. 18 corresponds to tu of Fig. 17.
Another and simpler method is to take the lines ac, aid, and
#2^2 of Fig. 1 8 and lay them off at kj, k\j\ and 2/2 on Fig. 20,
and then draw the contour aokjco for the developed contour of
the blade. The contour Ooa"ftVco is repeated for comparison.
In designing propellers the developed contour is frequently
drawn first and the projected contour is then constructed by
reversing the methods just explained.
As an example we may refer to Fig. 25, page 58, which is
given primarily to show the construction of a propeller with
separable blades. The developed contour is shown by the dotted
ellipse. OA is laid off equal to ^-f-2x to find the focus of the
elliptical section, the point A corresponding to 02 of Fig. 19.
Choosing a point B we draw through it a circular arc EB from
the centre O and an elliptical arc DB, with OB and AB for the
semi-minor and semi-major axes. Through the intersection D
STANDARD DEVELOPED CONTOUR
of the elliptical arc with the dotted contour, a horizontal line DF
is drawn, which cuts the circular arc at E\ this is a point of the
projected contour. A comparison with Fig. 19 will justify this
construction. A more precise method of locating points like F
will be given in the description of Fig. 25.
Standard Developed Contour. A form of developed contour
for propeller blades which was first proposed by Wm. Froude and
which is known as the Admiralty blade, is shown by Fig. 21. It
ADM I R ALT/
is an ellipse with the radius of the propeller as the major axis,
and the minor axis is 0.2 of the propeller diameter. The diameter
of the hub is 0.22 of the propeller diameter, and the contour is
shown cut off by a straight line. More correctly the development
of the root should be a curve depending on the form of the hub.
With the advent of high-power and high-speed ships, especially
turbine ships, the elliptical contour has been increased in width
till it approaches a circle.
Fig. 22 shows a contour proposed by Naval Constructor D. W.
Taylor, U.S.N., and used by him for many experimental propellers.
Its form is sufficiently determined by the ratios of the widths to
the maximum width. Fig. 23 is put in to show the relative form
of a straight-edged blade having about the same area, and slightly
In order to show the comparison of the proposed projected
contour with the Admiralty blade, two developments by the
conventional method are given by Fig. 24. The contour ertml
is the proposed circular projected contour, the radius cr being
0.2 of the propeller diameter so 'that the projected width is 0.4
of the diameter. The hub is 0.2 of the diameter and the contour
at the hub is completed by a circular arc. The contour errffm' is
drawn with the width equal to 0.2 of the diameter. The developed
contours are drawn by the method of page 53, for a pitch-ratio
of unity, that is, with the pitch equal to the diameter; a different
pitch-ratio would have but little effect on the conclusion that
AREA OF THE ADMIRALTY BLADE 57
can be drawn from the figure. The dotted ellipses are drawn
through the points h and hi on the line gh at the middle of the
radius; they are the developed contours of the corresponding
Admiralty blades. The developed contours shown by the full
lines are wider at the tips and narrower at the hub; the arealsf
somewhat less. Our design of propeller will be based on projected
area-ratio which will set aside questions of width and area, but
minor variations of either property have no appreciable influence.
Area of the Admiralty Blade. The importance that is attached
to the Admiralty blade makes it desirable to give ready means
of determining both the developed and the projected areas.
The developed contour being an ellipse its area will be pro-
portional to its width. If its width were half the diameter of
the propeller, its area, neglecting the hub, would be 0.25 that
of the disk or circle having the propeller diameter. The hub
may be assumed to take away a segment having a rise 0.2 of the
diameter of the circular contour; the segmental area is 0.1424 of
that of the circle; consequently the net area is
0.25(1 -0.1424) =0.214,
that of the disk. For any other width the area will be pro-
portional; then for a width 0.2 of the diameter of the propeller
the developed area is 0.0856 of the disk.
Barnaby gives the following rule for the projected area:
Projected area =
This rule will give approximate results for other oval projected
Construction Drawings. The construction drawings for a four-
bladed propeller with separable blades are shown by Figs. 25 to 28.
The projection on a transverse plane looking forward is shown by
Fig. 25, which gives also the developed contour on which the
design is based. As previously explained OA is laid off equal to
P-T-ZK and is the focus for the ellipses used in the development
of the blade; or in the construction of the projected contour.
A point B is chosen and through it an elliptical arc is drawn; also
a circular arc from the center 0. A horizontal line DF locates
the point E of the projected contour. Fig. 26 shows the projection
of two blades, without rake, on a plane, through the axis of the
shaft. 05 is equal to OA of the preceding figure and SR is equal
to OB', 01 gives the projection of EF in its true length. Drawing
// perpendicular to 05 gives O/, the proper half-breadth FE of
the contour EG. It also gives // the proper length of EF of
Fig. 25, and this is a more precise way of locating that point than
that previously given.
The points I and / are points on the contour of a blade which
presents its tip to the observer. The thickness of the blade and
other details are omitted to avoid complexity.
Fig. 27 gives the projections of a blade with a rake, together
with the effect of thickness of blade on the configuration of the
tip and the root. The blades are fastened to the hub by flanges
and bolts. Fig. 28 gives a longitudinal section of the blade and
hub and shows details of construction.
Propeller Experiments. The first systematic propeller experi-
ments were made by the Froudes, father and son, at the Admi-
ralty experimental tank, all being of the Admiralty type with the
width of blade equal to 0.2 of the diameter. Mr. R. E. Froude
has reported later experiments with various widths of blades.
Probably the most satisfactory tests are those by Naval Con-
structor D. W. Taylor, U.S.N., made at the model basin at
Washington. The tables in this book are derived from these
tests with the permission of Mr. Taylor. It has been shown that
the tests by the Froudes and by Mr. Taylor are in substantial
accord, so that both series of experiments may be claimed as the
basis of the tables given in this book.
The tables for three-bladed propellers are based directly on
an extensive set of experiments made on propellers of the Admi-
ralty type with various widths, thicknesses, and pitch-ratios.
The tables for four-bladed propellers were deduced from a com-
parison of tests on thin-bladed propellers of the type shown by
Fig. 22 (some with three and some with four blades) with the
tests on the Admiralty type. A table for two-bladed propellers
METHOD OF EXPERIMENTS 63
was deduced in like manner from tests of thin bladed propellers
on that type.
Method of Experiments. In making experiments in a model
basin, the model propeller is placed at the front end of a shaft
which is suspended from the towing carriage. The shaft atrthe
rear extends into a boat-shaped box which contains the driving
gear on the propeller shaft. The towing carriage is propelled at