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For other uses, see Propeller (disambiguation).
This article is about propellers in marine and other applications. For propellers on aircraft, see
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Propeller on a modern mid-sized merchant vessel
A propeller is a type of fan that transmits power by converting rotational motion into thrust. A
pressure difference is produced between the forward and rear surfaces of the airfoil-shaped
blade, and air or water is accelerated behind the blade. Propeller dynamics can be modeled by
both Bernoulli's principle and Newton's third law. A propeller is often colloquially known as
screw both in aviation and maritime.
o 3.1 Additional designs
o 3.2 Transverse axis propellers
o 3.3 History of ship and submarine screw propellers
o 3.4 Marine propeller cavitation
o 3.5 Forces acting on an aerofoil
o 3.6 Propeller thrust
3.6.1 Single blade
3.6.2 Thrust and torque
o 3.7 Actual performance
o 3.8 Types of marine propellers
3.8.1 Controllable pitch propeller
3.8.2 Skewback propeller
3.8.3 Modular propeller
o 3.9 Protection of small engines
4 See also
o 4.1 Propeller characteristics
o 4.2 Propeller phenomena
o 4.3 Propeller variations
o 4.4 Materials and Manufacture
6 External links
Ship propeller from 1843. Designed by C F Wahlgren based on one of John Ericsson propellers.
It was fitted to the steam ship Flygfisken built at the Motala dockyard.
The principle employed in using a screw propeller is used in sculling. It is part of the skill of
propelling a Venetian gondola but was used in a less refined way in other parts of Europe and
probably elsewhere. For example, propelling a canoe with a single paddle using a "j-stroke"
involves a related but not identical technique. In China, sculling, called "lu", was also used by
the 3rd century AD.
In sculling, a single blade is moved through an arc, from side to side taking care to keep
presenting the blade to the water at the effective angle. The innovation introduced with the screw
propeller was the extension of that arc through more than 360° by attaching the blade to a
rotating shaft. Propellers can have a single blade, but in practice there are nearly always more
than one so as to balance the forces involved.
The origin of the actual screw propeller starts with Archimedes, who used a screw to lift water
for irrigation and bailing boats, so famously that it became known as Archimedes' screw. It was
probably an application of spiral movement in space (spirals were a special study of Archimedes)
to a hollow segmented water-wheel used for irrigation by Egyptians for centuries. Leonardo da
Vinci adopted the principle to drive his theoretical helicopter, sketches of which involved a large
canvas screw overhead.
In 1784, J. P. Paucton proposed a gyrocopter-like aircraft using similar screws for both lift and
propulsion. At about the same time, James Watt proposed using screws to propel boats, although
he did not use them for his steam engines. This was not his own invention, though; Toogood and
Hays had patented it a century earlier, and it had become a common use as a means of propelling
boats since that time.
By 1827, Czech inventor Josef Ressel had invented a screw propeller which had multiple blades
fastened around a conical base; this new method of propulsion allowed steam ships to travel at
much greater speeds without using sails thereby making ocean travel faster (first tests with the
Austro-Hungarian Navy).
John Patch, a mariner in Yarmouth, Nova Scotia developed a two-bladed, fan-shaped propeller in
1832 and publicly demonstrated it in 1833, propelling a row boat across Yarmouth Harbour and
a small coastal schooner at Saint John, New Brunswick, but his patent application in the United
States was rejected until 1849 because he was not an American citizen. His efficient design
drew praise in American scientific circles but by this time there were multiple competing
versions of the marine propeller.
In 1835 Francis Pettit Smith discovered a new way of building propellers. Up to that time,
propellers were literally screws, of considerable length. But during the testing of a boat propelled
by one, the screw snapped off, leaving a fragment shaped much like a modern boat propeller.
The boat moved faster with the broken propeller. At about the same time, Frédéric Sauvage
and John Ericsson applied for patents on vaguely similar, although less efficient shortened-screw
propellers, leading to an apparently permanent controversy as to who the official inventor is
among those three men. Ericsson became widely famous when he built the Monitor, an armoured
battleship that in 1862 fought the Confederate States’ Virginia in an American Civil War sea
The superiority of screw against paddles was taken up by navies. Trials with Smith's SS
Archimedes, the first steam driven screw, led to the famous tug-of-war competition in 1845
between the screw-driven HMS Rattler and the paddle steamer HMS Alecto; the former pulling
the latter backward at 2.5 knots (4.6 km/h).
In the second half of the nineteenth century, several theories were developed. The momentum
theory or disk actuator theory—a theory describing a mathematical model of an ideal propeller—
was developed by W.J.M. Rankine (1865), Alfred George Greenhill (1888) and R.E. Froude
(1889). The propeller is modeled as an infinitely thin disc, inducing a constant velocity along the
axis of rotation. This disc creates a flow around the propeller. Under certain mathematical
premises of the fluid, there can be extracted a mathematical connection between power, radius of
the propeller, torque and induced velocity. Friction is not included.
The blade element theory (BET) is a mathematical process originally designed by William
Froude (1878), David W. Taylor (1893) and Stefan Drzewiecki to determine the behavior of
propellers. It involves breaking an airfoil down into several small parts then determining the
forces on them. These forces are then converted into accelerations, which can be integrated into
velocities and positions.
A World War I wooden aircraft propeller on a workbench.
Postage stamp, USA, 1923.
The twisted airfoil (aerofoil) shape of modern aircraft propellers was pioneered by the Wright
brothers. While both the blade element theory and the momentum theory had their supporters,
the Wright brothers were able to combine both theories. They found that a propeller is essentially
the same as a wing and so were able to use data collated from their earlier wind tunnel
experiments on wings. They also found that the relative angle of attack from the forward
movement of the aircraft was different for all points along the length of the blade, thus it was
necessary to introduce a twist along its length. Their original propeller blades are only about 5%
less efficient than the modern equivalent, some 100 years later.
Alberto Santos Dumont was another early pioneer, having designed propellers before the Wright
Brothers (albeit not as efficient) for his airships. He applied the knowledge he gained from
experiences with airships to make a propeller with a steel shaft and aluminium blades for his 14
bis biplane. Some of his designs used a bent aluminium sheet for blades, thus creating an airfoil
shape. These are heavily undercambered because of this and combined with the lack of a
lengthwise twist made them less efficient than the Wright propellers. Even so, this was perhaps
the first use of aluminium in the construction of an airscrew.
Main article: Propeller (aircraft)
Aircraft propellers convert rotary motion from piston engines or turboprops to provide
propulsive force. They may be fixed or variable pitch. Early aircraft propellers were carved by
hand from solid or laminated wood with later propellers being constructed from metal. The most
modern propeller designs use high-technology composite materials.
As well as being used for fixed wing aircraft, these propellers are also used for helicopters, and
other vehicles such as hovercraft, airboats and some trains (such as the Schienenzeppelin).
It has been suggested that this section be split into a new article titled propeller
Marine propeller nomenclature
1) Trailing edge 6) Leading edge
2) Face 7) Back
3) Fillet area 8) Propeller shaft
4) Hub or Boss 9) Stern tube bearing
5) Hub or Boss Cap 10) Stern tube
A propeller is the most common propulsor on ships, imparting momentum to a fluid which
causes a force to act on the ship.
The ideal efficiency of any size propeller (free-tip) is that of an actuator disc in an ideal fluid. An
actual marine propeller is made up of sections of helicoidal surfaces which act together
'screwing' through the water (hence the common reference to marine propellers as "screws").
Three, four, or five blades are most common in marine propellers, although designs which are
intended to operate at reduced noise will have more blades. The blades are attached to a boss
(hub), which should be as small as the needs of strength allow - with fixed pitch propellers the
blades and boss are usually a single casting.
An alternative design is the controllable pitch propeller (CPP, or CRP for controllable-reversible
pitch), where the blades are rotated normal to the drive shaft by additional machinery - usually
hydraulics - at the hub and control linkages running down the shaft. This allows the drive
machinery to operate at a constant speed while the propeller loading is changed to match
operating conditions. It also eliminates the need for a reversing gear and allows for more rapid
change to thrust, as the revolutions are constant. This type of propeller is most common on ships
such as tugs where there can be enormous differences in propeller loading when
towing compared to running free, a change which could cause conventional propellers to lock up
as insufficient torque is generated. The downsides of a CPP/CRP include: the large hub which
decreases the torque required to cause cavitation, the mechanical complexity which limits
transmission power and the extra blade shaping requirements forced upon the propeller designer.
For smaller motors there are self-pitching propellers. The blades freely move through an entire
circle on an axis at right angles to the shaft. This allows hydrodynamic and centrifugal forces to
'set' the angle the blades reach and so the pitch of the propeller.
A propeller that turns clockwise to produce forward thrust, when viewed from aft, is called right-
handed. One that turns anticlockwise is said to be left-handed. Larger vessels often have twin
screws to reduce heeling torque, counter-rotating propellers, the starboard screw is usually right-
handed and the port left-handed, this is called outward turning. The opposite case is called
inward turning. Another possibility is contra-rotating propellers, where two propellers rotate in
opposing directions on a single shaft, or on separate shafts on nearly the same axis. One example
of the latter is the CRP Azipod by the ABB Group. Contra-rotating propellers offer increased
efficiency by capturing the energy lost in the tangential velocities imparted to the fluid by the
forward propeller (known as "propeller swirl"). The flow field behind the aft propeller of a
contra-rotating set has very little "swirl", and this reduction in energy loss is seen as an increased
efficiency of the aft propeller.
 Additional designs
An azimuthing propeller is a vertical axis propeller.
The blade outline is defined either by a projection on a plane normal to the propeller shaft
(projected outline) or by setting the circumferential chord across the blade at a given radius
against radius (developed outline). The outline is usually symmetrical about a given radial line
termed the median. If the median is curved back relative to the direction of rotation the propeller
is said to have skew back. The skew is expressed in terms of circumferential displacement at the
blade tips. If the blade face in profile is not normal to the axis it is termed raked, expressed as a
percentage of total diameter.
Each blade's pitch and thickness varies with radius, early blades had a flat face and an arced back
(sometimes called a circular back as the arc was part of a circle), modern propeller blades have
aerofoil sections. The camber line is the line through the mid-thickness of a single blade. The
camber is the maximum difference between the camber line and the chord joining the trailing
and leading edges. The camber is expressed as a percentage of the chord.
The radius of maximum thickness is usually forward of the mid-chord point with the blades
thinning to a minimum at the tips. The thickness is set by the demands of strength and the ratio
of thickness to total diameter is called blade thickness fraction.
The ratio of pitch to diameter is called pitch ratio. Due to the complexities of modern propellers
a nominal pitch is given, usually a radius of 70% of the total is used.
Blade area is given as a ratio of the total area of the propeller disc, either as developed blade area
ratio or projected blade area ratio.
 Transverse axis propellers
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This section requires expansion.
Most propellers have their axis of rotation parallel to the fluid flow. There have however been
some attempts to power vehicles with the same principles behind vertical axis wind turbines,
where the rotation is perpendicular to fluid flow. Most attempts have been unsuccessful. Blades
that can vary their angle of attack during rotation have aerodynamics similar to flapping flight.
Flapping flight is still poorly understood and almost never seriously used in engineering because
of the strong coupling of lift, thrust and control forces.
The fanwing is one of the few types that has actually flown. It takes advantage of the trailing
edge of an airfoil to help encourage the circulation necessary for lift.
The Voith-Schneider propeller pictured below is another successful example, operating in water.
 History of ship and submarine screw propellers
A propeller from the Lusitania
James Watt of Scotland is generally credited with applying the first screw propeller to an engine,
an early steam engine, beginning the use of an hydrodynamic screw for propulsion.
Mechanical ship propulsion began with the steam ship. The first successful ship of this type is a
matter of debate; candidate inventors of the 18th century include William Symington, the
Marquis de Jouffroy, John Fitch and Robert Fulton, however William Symington's ship the
Charlotte Dundas is regarded as the world's "first practical steamboat". Paddlewheels as the
main motive source became standard on these early vessels (see Paddle steamer). Robert Fulton
had tested, and rejected, the screw propeller.
Sketch of hand-cranked vertical and horizontal screws used in Bushnell's Turtle, 1775
The screw (as opposed to paddlewheels) was introduced in the latter half of the 18th century.
David Bushnell's invention of the submarine (Turtle) in 1775 used hand-powered screws for
vertical and horizontal propulsion. The Bohemian engineer Josef Ressel designed and patented
the first practicable screw propeller in 1827. Francis Pettit Smith tested a similar one in 1836. In
1839, John Ericsson introduced practical screw propulsion into the United States. Mixed paddle
and propeller designs were still being used at this time (vide the 1858 Great Eastern).
The screw propeller replaced the paddles owing to its greater efficiency, compactness, less
complex power transmission system, and reduced susceptibility to damage (especially in battle)
Initial designs owed much to the ordinary screw from which their name derived - early propellers
consisted of only two blades and matched in profile the length of a single screw rotation. This
design was common, but inventors endlessly experimented with different profiles and greater
numbers of blades. The propeller screw design stabilized by the 1880s.
In the early days of steam power for ships, when both paddle wheels and screws were in use,
ships were often characterized by their type of propellers, leading to terms like screw steamer or
Propellers are referred to as "lift" devices, while paddles are "drag" devices.
Cavitation damage evident on the propeller of a personal watercraft.
 Marine propeller cavitation
Cavitation can occur if an attempt is made to transmit too much power through the screw, or if
the propeller is operating at a very high speed. Cavitation can occur in many ways on a propeller.
The two most common types of propeller cavitation are suction side surface cavitation and tip
Suction side surface cavitation forms when the propeller is operating at high rotational speeds or
under heavy load (high blade lift coefficient). The pressure on the upstream surface of the blade
(the "suction side") can drop below the vapour pressure of the water, resulting in the formation of
a pocket of vapour. Under such conditions, the change in pressure between the downstream
surface of the blade (the "pressure side") and the suction side is limited, and eventually reduced
as the extent of cavitation is increased. When most of the blade surface is covered by cavitation,
the pressure difference between the pressure side and suction side of the blade drops
considerably, and thrust produced by the propeller drops. This condition is called "thrust
breakdown". This effect wastes energy, makes the propeller "noisy" as the vapour bubbles
collapse, and most seriously, erodes the screw's surface due to localized shock waves against the
Tip vortex cavitation is caused by the extremely low pressures formed at the core of the tip
vortex. The tip vortex is caused by fluid wrapping around the tip of the propeller; from the
pressure side to the suction side. This video demonstrates tip vortex cavitation well. Tip vortex
cavitation typically occurs before suction side surface cavitation and is less damaging to the
blade, since this type of cavitation doesn't collapse on the blade, but some distance downstream.
Cavitation can be used as an advantage in design of very high performance propellers, in form of
the supercavitating propeller. In this case, the blade section is designed such that the pressure
side stays wetted while the suction side is completely covered by cavitation vapor. Because the
suction side is covered with vapor instead of water it encounters very low viscous friction,
making the supercavitating (SC) propeller comparably efficient at high speed. The shaping of SC
blade sections however, make it inefficient at low speeds, when the suction side of the blade is
wetted. (See also fluid dynamics).
A similar, but quite separate issue, is ventilation, which occurs when a propeller operating near
the surface draws air into the blades, causing a similar loss of power and shaft vibration, but
without the related potential blade surface damage caused by cavitation. Both effects can be
mitigated by increasing the submerged depth of the propeller: cavitation is reduced because the
hydrostatic pressure increases the margin to the vapor pressure, and ventilation because it is
further from surface waves and other air pockets that might be drawn into the slipstream.
14-ton propeller from Voroshilov a Kirov-class cruiser on display in Sevastopol
 Forces acting on an aerofoil
The force (F) experienced by an aerofoil blade is determined by its area (A), chord (c), velocity
(V) and the angle of the aerofoil to the flow, called angle of attack (α), where:
The force has two parts - that normal to the direction of flow is lift (L) and that in the direction of
flow is drag (D). Both are expressed non-dimensionally as:
Each coefficient is a function of the angle of attack and Reynolds' number. As the angle of attack
increases lift rises rapidly from the no lift angle before slowing its increase and then decreasing,
with a sharp drop as the stall angle is reached and flow is disrupted. Drag rises slowly at first and
as the rate of increase in lift falls and the angle of attack increases drag increases more sharply.
For a given strength of circulation (τ), Lift = L = ρVτ. The effect of the flow over and the
circulation around the aerofoil is to reduce the velocity over the face and increase it over the
back of the blade. If the reduction in pressure is too much in relation to the ambient pressure of
the fluid, cavitation occurs, bubbles form in the low pressure area and are moved towards the
blade's trailing edge where they collapse as the pressure increases, this reduces propeller
efficiency and increases noise. The forces generated by the bubble collapse can cause permanent
damage to the surfaces of the blade.
 Propeller thrust
 Single blade
Taking an arbitrary radial section of a blade at r, if revolutions are N then the rotational velocity
is . If the blade was a complete screw it would advance through a solid at the rate of NP,
where P is the pitch of the blade. In water the advance speed is rather lower, , the difference,
or slip ratio, is:
where is the advance coefficient, and is the pitch ratio.
The forces of lift and drag on the blade, dA, where force normal to the surface is dL:
These forces contribute to thrust, T, on the blade:
From this total thrust can be obtained by integrating this expression along the blade. The
transverse force is found in a similar manner:
Substituting for and multiplying by r, gives torque as:
which can be integrated as before.
The total thrust power of the propeller is proportional to and the shaft power to . So
efficiency is . The blade efficiency is in the ratio between thrust and torque:
showing that the blade efficiency is determined by its momentum and its qualities in the form of
angles and , where is the ratio of the drag and lift coefficients.
This analysis is simplified and ignores a number of significant factors including interference
between the blades and the influence of tip vortices.
 Thrust and torque
The thrust, T, and torque, Q, depend on the propeller's diameter, D, revolutions, N, and rate of
advance, Va, together with the character of the fluid in which the propeller is operating and
gravity. These factors create the following non-dimensional relationship:
where f1 is a function of the advance coefficient, f2 is a function of the Reynolds' number, and f3
is a function of the Froude number. Both f2 and f3 are likely to be small in comparison to f1
under normal operating conditions, so the expression can be reduced to:
For two identical propellers the expression for both will be the same. So with the propellers
T1,T2, and using the same subscripts to indicate each propeller:
For both Froude number and advance coefficient:
where λ is the ratio of the linear dimensions.
Thrust and velocity, at the same Froude number, give thrust power:
 Actual performance
When a propeller is added to a ship its performance is altered; there is the mechanical losses in
the transmission of power; a general increase in total resistance; and the hull also impedes and
renders non-uniform the flow through the propeller. The ratio between a propeller's efficiency
attached to a ship ( ) and in open water ( ) is termed relative rotative efficiency.
The overall propulsive efficiency (an extension of effective power ( )) is developed from the
propulsive coefficient ( ), which is derived from the installed shaft power ( ) modified by the
effective power for the hull with appendages ( ), the propeller's thrust power ( ), and the
relative rotative efficiency.
P'E/PT = hull efficiency = ηH
PT/P'D = propeller efficiency = ηO
P'D/PD = relative rotative efficiency = ηR
PD/PS = shaft transmission efficiency
Producing the following:
The terms contained within the brackets are commonly grouped as the quasi-propulsive
coefficient ( , ). The is produced from small-scale experiments and is modified with
a load factor for full size ships.
Wake is the interaction between the ship and the water with its own velocity relative to the ship.
The wake has three parts: the velocity of the water around the hull; the boundary layer between
the water dragged by the hull and the surrounding flow; and the waves created by the movement
of the ship. The first two parts will reduce the velocity of water into the propeller, the third will
either increase or decrease the velocity depending on whether the waves create a crest or trough
at the propeller.
 Types of marine propellers
 Controllable pitch propeller
A controllable pitch propeller
One type of marine propeller is the controllable pitch propeller. This propeller has several
advantages with ships. These advantages include: the least drag depending on the speed used, the
ability to move the sea vessel backwards, and the ability to use the "vane"-stance, which gives
the least water resistance when not using the propeller (e.g. when the sails are used instead).
 Skewback propeller
An advanced type of propeller used on German Type 212 submarines is called a skewback
propeller. As in the scimitar blades used on some aircraft, the blade tips of a skewback propeller
are swept back against the direction of rotation. In addition, the blades are tilted rearward along
the longitudinal axis, giving the propeller an overall cup-shaped appearance. This design
preserves thrust efficiency while reducing cavitation, and thus makes for a quiet, stealthy
See also: astern propulsion
 Modular propeller
A modular propeller provides more control over the boats performance. There is no need to
change an entire prop, when there is an opportunity to only change the pitch or the damaged
blades. Being able to adjust pitch will allow for boaters to have better performance while in
different altitudes, water sports, and/or cruising.
 Protection of small engines
A failed rubber bushing in an outboard's propeller
For smaller engines, such as outboards, where the propeller is exposed to the risk of collision
with heavy objects, the propeller often includes a device which is designed to fail when over
loaded; the device or the whole propeller is sacrificed so that the more expensive transmission
and engine are not damaged.
Typically in smaller (less than 10 hp/7.5 kW) and older engines, a narrow shear pin through the
drive shaft and propeller hub transmits the power of the engine at normal loads. The pin is
designed to shear when the propeller is put under a load that could damage the engine. After the
pin is sheared the engine is unable to provide propulsive power to the boat until an undamaged
shear pin is fitted. Note that some shear pins used to have shear grooves machined into them.
Nowadays the grooves tend to be omitted. The result of this oversight is that the torque required
to shear the pin rises as the cutting edges of the propeller bushing and shaft become blunted.
Eventually the gears will strip instead.
In larger and more modern engines, a rubber bushing transmits the torque of the drive shaft to the
propeller's hub. Under a damaging load the friction of the bushing in the hub is overcome and the
rotating propeller slips on the shaft preventing overloading of the engine's components. After
such an event the rubber bushing itself may be damaged. If so, it may continue to transmit
reduced power at low revolutions but may provide no power, due to reduced friction, at high
revolutions. Also the rubber bushing may perish over time leading to its failure under loads
below its designed failure load.
Whether a rubber bushing can be replaced or repaired depends upon the propeller; some cannot.
Some can but need special equipment to insert the oversized bushing for an interference fit.
Others can be replaced easily.
The "special equipment" usually consists of a tapered funnel, some kind of press and rubber
lubricant (soap). Often the bushing can be drawn into place with nothing more complex than a
couple of nuts, washers and "allscrew" (threaded bar). If one does not have access to a lathe an
improvised funnel can be made from steel tube and car body filler! (as the filler is only subject to
compressive forces it is able to do a good job) A more serious problem with this type of propeller
is a "frozen-on" spline bushing which makes propeller removal impossible. In such cases the
propeller has to be heated in order to deliberately destroy the rubber insert. Once the propeller
proper is removed, the splined tube can be cut away with a grinder. A new spline bushing is of
course required. To prevent the problem recurring the splines can be coated with anti-seize anti-
In some modern propellers, a hard polymer insert called a drive sleeve replaces the rubber
bushing. The splined or other non-circular cross section of the sleeve inserted between the shaft
and propeller hub transmits the engine torque to the propeller, rather than friction. The polymer
is weaker than the components of the propeller and engine so it fails before they do when the
propeller is overloaded. This fails completely under excessive load but can easily be replaced.
From Wikipedia, the free encyclopedia
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See also: Propeller
The feathered propellers of an RAF Hercules C.4
Aircraft propellers or airscrews convert rotary motion from piston engines or turboprops to
provide propulsive force. They may be fixed or variable pitch. Early aircraft propellers were
carved by hand from solid or laminated wood with later propellers being constructed from metal.
The most modern propeller designs use high-technology composite materials.
The propeller is usually attached to the crankshaft of a piston engine, either directly or through a
reduction unit. Light aircraft engines often do not require the complexity of gearing but on larger
engines and turboprop aircraft it is essential.
2 Theory and design of aircraft propellers
o 2.1 Forces acting on a propeller
3 Propeller control
o 3.1 Variable pitch
o 3.2 Feathering
o 3.3 Reverse pitch
4 Contra-rotating propellers
5 Counter-rotating propellers
6 Aircraft fans
7 See also
o 8.1 Bibliography
9 External links
The twisted airfoil (aerofoil) shape of modern aircraft propellers was pioneered by the Wright
brothers. They found that a propeller is essentially the same as a wing and so were able to use
data collated from their earlier wind tunnel experiments on wings. They also found that the
relative angle of attack from the forward movement of the aircraft was different for all points
along the length of the blade, thus it was necessary to introduce a twist along its length. Their
original propeller blades were only about 5% less efficient than the modern equivalent, some 100
Alberto Santos Dumont was another early pioneer, having designed propellers before the Wright
Brothers (albeit not as efficient) for his airships. He applied the knowledge he gained from
experiences with airships to make a propeller with a steel shaft and aluminium blades for his 14
bis biplane. Some of his designs used a bent aluminium sheet for blades, thus creating an airfoil
shape. They were heavily undercambered, and this plus the absence of lengthwise twist twist
made them less efficient than the Wright propellers. Even so, this was perhaps the first use of
aluminium in the construction of an airscrew.
 Theory and design of aircraft propellers
A well-designed propeller typically has an efficiency of around 80% when operating in the best
regime. Changes to a propeller's efficiency are produced by a number of factors, notably
adjustments to the helix angle(θ), the angle between the resultant relative velocity and the blade
rotation direction, and to blade pitch (where θ = Φ + α) . Very small pitch and helix angles give a
good performance against resistance but provide little thrust, while larger angles have the
opposite effect. The best helix angle is when the blade is acting as a wing producing much more
lift than drag.
A propeller's efficiency is determined by
Propellers are similar in aerofoil section to a low-drag wing and as such are poor in operation
when at other than their optimum angle of attack. Therefore a method is needed to alter the
blades' pitch angle as engine speed and aircraft velocity are changed.
The three-bladed propeller of a light aircraft: the Vans RV-7A
A further consideration is the number and the shape of the blades used. Increasing the aspect
ratio of the blades reduces drag but the amount of thrust produced depends on blade area, so
using high-aspect blades can result in an excessive propeller diameter. A further balance is that
using a smaller number of blades reduces interference effects between the blades, but to have
sufficient blade area to transmit the available power within a set diameter means a compromise is
needed. Increasing the number of blades also decreases the amount of work each blade is
required to perform, limiting the local Mach number - a significant performance limit on
A propeller's performance suffers as the blade speed nears the transonic. As the relative air speed
at any section of a propeller is a vector sum of the aircraft speed and the tangential speed due to
rotation, a propeller blade tip will reach transonic speed well before the aircraft does. When the
airflow over the tip of the blade reaches its critical speed, drag and torque resistance increase
rapidly and shock waves form creating a sharp increase in noise. Aircraft with conventional
propellers, therefore, do not usually fly faster than Mach 0.6. There have been propeller aircraft
which attained up to the Mach 0.8 range, but the low propeller efficiency at this speed makes
such applications rare.
There have been efforts to develop propellers for aircraft at high subsonic speeds. The 'fix' is
similar to that of transonic wing design. The maximum relative velocity is kept as low as
possible by careful control of pitch to allow the blades to have large helix angles; thin blade
sections are used and the blades are swept back in a scimitar shape (Scimitar propeller); a large
number of blades are used to reduce work per blade and so circulation strength; contra-rotation is
used. The propellers designed are more efficient than turbo-fans and their cruising speed (Mach
0.7–0.85) is suitable for airliners, but the noise generated is tremendous (see the Antonov An-70
and Tupolev Tu-95 for examples of such a design).
 Forces acting on a propeller
Five forces act on the blades of an aircraft propeller in motion, they are:
Thrust bending force
Thrust loads on the blades act to bend them forward.
Centrifugal twisting force
Acts to twist the blades to a low or fine pitch angle.
Aerodynamic twisting force
As the centre of pressure of a propeller blade is forward of its centreline the blade is twisted
towards a coarse pitch position.
The force felt by the blades acting to pull them away from the hub when turning.
Torque bending force
Air resistance acting against the blades, combined with inertial effects causes propeller blades
to bend away from the direction of rotation.
 Propeller control
 Variable pitch
Cut-away view of a Hamilton Standard propeller. This type of propeller was used on many American
fighters, bombers and transport aircraft of WWII.
The purpose of varying pitch angle with a variable pitch propeller is to maintain an optimal angle
of attack (maximum lift to drag ratio) on the propeller blades as aircraft speed varies. Early pitch
control settings were pilot operated, either two-position or manually variable. Following World
War I, automatic propellers were developed to maintain an optimum angle of attack. This was
done by balancing the centripetal twisting moment on the blades and a set of counterweights
against a spring and the aerodynamic forces on the blade. Automatic props had the advantage of
being simple, lightweight, and requiring no external control, but a particular propeller's
performance was difficult to match with that of the aircraft's powerplant. An improvement on the
automatic type was the constant-speed propeller. Constant-speed propellers allow the pilot to
select a rotational speed for maximum engine power or maximum efficiency, and a propeller
governor acts as a closed-loop controller to vary propeller pitch angle as required to maintain the
selected engine speed. In most aircraft this system is hydraulic, with engine oil serving as the
hydraulic fluid. However, electrically controlled propellers were developed during World War II
and saw extensive use on military aircraft, and have recently seen a revival in use on homebuilt
A propeller blade in feathered position
On some variable-pitch propellers, the blades can be rotated parallel to the airflow to reduce drag
in case of an engine failure. This is called feathering. Feathering propellers were developed for
military fighter aircraft prior to World War II, as a fighter is more likely to experience an engine
failure due to the inherent danger of combat. On single-engined aircraft, whether a powered
glider or turbine powered aircraft, the effect is to increase the gliding distance. On a multi-engine
aircraft, feathering the propeller on a failed engine reduces drag, allowing the flight to continue
with the remaining powerplant.
Most feathering systems for reciprocating engines sense a drop in oil pressure and move the
blades toward the feather position, and require the pilot to pull the propeller control back to
disengage the high-pitch stop pins before the engine reaches idle RPM. Turboprop control
systems usually utilize a negative torque sensor in the reduction gearbox which moves the blades
toward feather when the engine is no longer providing power to the propeller. Depending on
design, the pilot may have to push a button to override the high-pitch stops and complete the
feathering process, or the feathering process may be totally automatic.
 Reverse pitch
Contra-rotating propellers of a modified P-51 Mustang fitted with a Rolls-Royce Griffon
In some aircraft, such as the C-130 Hercules, the pilot can manually override the constant-speed
mechanism to reverse the blade pitch angle, and thus the thrust of the engine (although the
rotation of the engine itself does not reverse). This is used to help slow the plane down after
landing in order to save wear on the brakes and tires, but in some cases also allows the aircraft to
back up on its own. This is known as "Beta Range" operation. See also Thrust reversal.
 Contra-rotating propellers
Main article: Contra-rotating propellers
Contra-rotating propellers use a second propeller rotating in the opposite direction immediately
'downstream' of the main propeller so as to recover energy lost in the swirling motion of the air
in the propeller slipstream. Contra-rotation also increases power without increasing propeller
diameter and provides a counter to the torque effect of high-power piston engine as well as the
gyroscopic precession effects, and of the slipstream swirl. However on small aircraft the added
cost, complexity, weight and noise of the system rarely make it worthwhile.
 Counter-rotating propellers
Main article: Counter-rotating propellers
Counter-rotating propellers are sometimes used on twin-, and other multi-engine, propeller-
driven aircraft. The propellers of these wing-mounted engines turn in opposite directions from
those on the other wing. Generally, the propellers on both engines of most conventional twin-
engined aircraft spin clockwise (as viewed from the rear of the aircraft). Counter-rotating
propellers generally spin clockwise on the left engine, and counter-clockwise on the right. The
advantage of counter-rotating propellers is to balance out the effects of torque and p-factor,
eliminating the problem of the critical engine.
 Aircraft fans
Main articles: Propfan and Ducted fan
A fan is a propeller with a large number of blades. A fan therefore produces a lot of thrust for a
given diameter but the closeness of the blades means that each strongly affects the flow around
the others. If the flow is supersonic, this interference can be beneficial if the flow can be
compressed through a series of shock waves rather than one. By placing the fan within a shaped
duct, specific flow patterns can be created depending on flight speed and engine performance. As
air enters the duct, its speed is reduced while its pressure and temperature increase. If the aircraft
is at a high subsonic speed this creates two advantages: the air enters the fan at a lower Mach
speed; and the higher temperature increases the local speed of sound. While there is a loss in
efficiency as the fan is drawing on a smaller area of the free stream and so using less air, this is
balanced by the ducted fan retaining efficiency at higher speeds where conventional propeller
efficiency would be poor. A ducted fan or propeller also has certain benefits at lower speeds but
the duct needs to be shaped in a different manner than one for higher speed flight. More air is
taken in and the fan therefore operates at an efficiency equivalent to a larger un-ducted propeller.
Noise is also reduced by the ducting and should a blade become detached the duct would contain
the damage. However the duct adds weight, cost, complexity and (to a certain degree) drag.
From Wikipedia, the free encyclopedia
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Diameter of the propeller.
In aeronautics, the advance ratio at which a propeller is operating is the ratio between the
distance the propeller moves forward during one revolution, and the diameter of the propeller.
When a propeller-driven aircraft is moving at high airspeed the advance ratio of its propeller(s) is
a high number; and when it is moving at low airspeed the advance ratio is a low number. For a
propeller, the advance ratio serves the same purpose as angle of attack serves for an airfoil or
The advance ratio J is given by:
V is the true airspeed of the aircraft
n is the propeller's rotational speed in revolutions per unit of time
D is the propeller's diameter
The performance of different propellers should be compared for the same advance ratio.
Propeller Aircraft Performance and The
The Bootstrap Approach: Background
The Bootstrap Approach
The Bootstrap Approach is a parametric performance method. You take the airplane out and fly
it, for an hour or two, doing very specific climbs, glides, and a level speed run. Those routine
maneuvers must be done at known weight and altitude, but it doesn’t matter what that weight or
that altitude is. That gives you the data, after you get back down, to calculate the parameters
making up the "Bootstrap Data Plate" (BDP). In all, the BDP consists of nine parameters. When
you want to know the airplane’s performance – say angle of climb – under specific
circumstances (weight and altitude), you simply take the appropriate Bootstrap formula,
substitute in the BDP parameters for that airplane, and the weight and altitude figures, and do the
calculations. Out comes the airplane’s performance number. Now let’s take a look at where the
Bootstrap Approach comes from.
Of the four forces acting on the airplane – thrust, drag, lift, and weight – thrust is the most
difficult to measure or predict. That is why most books about aircraft performance simply
assume that propeller efficiency h is some constant. Commonly cited values are = 80% and =
85%. Then thrust T = P, where P is the engine power. Unfortunately, propeller efficiency is in
fact not constant; it varies with air speed and RPM or, more precisely, with the dimensionless
ratio of those two variables:
where J is the "propeller advance ratio." As the propeller rotates through one circle the airplane
advances a distance V/n. J is then the ratio of that advance distance to the propeller’s diameter d.
Figure 1 is an example of how propeller efficiency varies with advance ratio.
Figure 1. Efficiency graph for McCauley 7557 propeller on some Cessna 172s.
The basic Bootstrap Approach, strangely enough, makes no assumption about propeller
efficiency. It has an alternate way, which we now explain, for coming up with thrust. Because a
section of propeller blade at distance r from the hub moves (in a sense) in two directions at once
– longitudinally with velocity V and to the side with speed 2nr – there are two distinct propeller
"coefficients," one (CT) having to do with thrust and the other (CP) having to do with absorbed
power. (In the third direction, along the length of the propeller blade, we assume the propeller is
rigid enough that it doesn’t move at all.) The propeller thrust coefficient is
and the propeller power coefficient is
Figure 2 shows, for the same propeller as in Figure 1, how these coefficients vary with advance
ratio J. Propeller efficiency can be obtained, knowing the two coefficients, from
Figure 2. All the important information about a propeller’s function can be obtained from its
thrust and power coefficient functions.
The Bootstrap Approach uses a little known but close approximate relation between these two
coefficients: that the so-called "propeller polar," defined as CT/J2 plotted against CP/J2, is linear.
That means that for any reasonable propeller there are two numbers m and b so that
The Bootstrap Approach depends upon our finding those parameters m and b, and a few others,
experimentally, by means of flight tests. For the same propeller as above, Figure 3 shows the
propeller polar and the best fit line approximating it.
The Bootstrap Data Plate
To predict the airplane’s performance using the Bootstrap method, a so-called "Bootstrap Data
Plate" or BDP, consisting of nine numbers, must first be ascertained. Table 4 is a sample BDP
for a particular Cessna 172 airplane.
Bootstrap Data Plate Item Value Units Aircraft
Wing area, S 174 ft2 Airframe
Wing aspect ratio, A 7.38 Airframe
Rated MSL torque, M0 311.2 ft-lbf Engine
Altitude drop-off 0.12 Engine
Propeller diameter, d 6.25 ft Propeller
Parasite drag coefficient, 0.037 Airframe
Airplane efficiency factor, e 0.72 Airframe
Propeller polar slope, m 1.70 Propeller
Propeller polar intercept, b –0.0564 Propeller
Table 4. Bootstrap Data Plate for a particular Cessna 172.
Figure 3. For most propellers, the best fit line to its polar diagram has a goodness-of-fit
parameter R2 = 0.95 or better.
Where do these nine BDP items come from? Five come from the Pilots Operating Handbook
(POH) or common knowledge. Those are:
1. Reference wing area S = 174 ft2;
2. Wing aspect ratio A = B2/S = 7.38 (B = wing span = 35.83 ft);
3. Mean sea level (MSL) full–throttle rated torque M0 = P0/2n0 (P0 rated power, n0
rated propeller revolutions per second). For this Cessna 172, P0 = 160 HP =
88,000 ft–lbf/sec and n0 = RPM/60 = 2700/60 = 45 rps. Hence M0 = 311.2 ft–lbf.
But in most of our formulas, though it makes them a little longer, we’ll retain P0
4. The proportional mechanical power loss independent of altitude, C, which can
almost always be taken as 0.12. This governs full-throttle torque at altitude
through the power drop-off factor (Greek capital ‘Phi’):
Relative atmospheric density (Greek small 'sigma') where is atmospheric density and
standard density = 0.002377 slug/ft3.The time-honored form (Gagg and Farrar, 1934) for this
drop off factor is
5. Propeller diameter d = 6.25 ft.
To simplify later calculations, it’s convenient to assume a “standard weight” for the airplane. For
our sample Cessna 172 we choose W0 = 2400 lbf, maximum certified gross weight. Standard
relative air density is taken to be unity.
Glide test for Drag Parameters
Of the four remaining "harder-to-get" BPD items, two typify drag and two characterize thrust.
The drag numbers are the usual:
6. Parasite drag coefficient, CD0; and
7. Airplane efficiency factor, e.
Getting CD0 and e by the usual method, linear regression analysis of many glides, is overkill.
Instead, simply find, by trial and error, the speed for best glide Vbg and its corresponding glide
angle bg (Greek small ‘gamma’) at one known aircraft weight W in an atmosphere of known
relative density. Let us take W = 2200 lbf and h = 5000 ft. That latter makes = 0.86167 and
() = 0.84281. (For convenience of the checking reader, we carry more decimal places than
makes strict sense).
Consider that we time glides from 5100 ft to 4900 ft; h = 200 ft. Glide angle (in calm wind) is
shallowest when product VT×t, true air speed times elapsed time, is greatest. To find that
maximizing V, one can just as well use calibrated air speed Vc. Best glide angle is later
The relation between true and calibrated air speeds is:
For our sample Cessna, take VCbg = 68.9 KCAS = 116.29 ft/sec and t = 16.96 sec From Eq. (9),
VTbg = 74.3 KTAS = 125.3 ft/sec. From Eq. (8), bg = 5.40 deg.
The two required drag parameters are obtained from:
Substituting our numbers into Eqs. (10) and (11) gives us CD0 = 0.0370 and e = 0.720. Those
numbers (especially CD0) would have been different if we had run the glide tests with some flaps
Climb and Level Flight Tests for Thrust Parameters
Our last two BDP items are:
8. Slope of the linear propeller polar, m;
9. Intercept of the linear propeller polar, b.
Of several alternative flight test regimens for evaluating m and b, we choose: trial-and-error
climbs to find speed for best angle of climb, Vx, and subsequently b, followed by a test for
maximum level flight speed, VM, and then m.
Vx is the full–throttle partner of Vbg. The latter is the most nearly positive (smallest negative)
glide angle you can achieve. Accordingly, when product V×t is smallest one has found Vx. For
our sample Cessna 172, assume VCx = 60.5 KCAS = 102.1 ft/sec. The true value is then VTx = Vx
= 65.2 KTAS = 110.0 ft/sec. The Bootstrap formula which kinds polar intercept b is :
Substituting our sample values into Eq. (12) gives b = –0.0564.
We conclude our flight tests with a full-speed level run (still at 5000 ft, still at 2200 lbf) and find
VCM = 104.8 KCAS = 176.9 ft/sec. In the true terms needed in our formulas, VTM = VM = 112.9
KTAS = 190.6 ft/sec. The Bootstrap formula for polar slope m is:
Substituting our values into Eq. (13) gives us m = 1.70. The Bootstrap Data Plate of Table 4 is
The Bootstrap Approach: Formulas and Graphs
Composite Bootstrap Parameters Ease Calculation
Almost everything about the airplane’s full-throttle steady-state (non-accelerating) flight
performance depends on the nine BDP items plus three operational variables: weight W, bank
angle and relative atmospheric density . But only certain combinations of the nine BDP
numbers (combinations called E, F, G, and H) actually occur in Bootstrap formulas for V-speeds
or for thrust, drag, or rate or angle of climb or descent. In the V-speed formulas, in fact, only
certain combinations of combinations (those to be called K, Q, R, and U) occur.
Our flight tests to determine the four harder-to-get BDP parameters were done at 5000 ft and at
W = 2200 lbf. But the parameter values we got did not depend on those choices; BDP parameters
only depend on the particular airplane and its flap/gear configuration. Flight tests at some other
altitude or at some other weight would have given, within experimental error, the same BDP
But performance numbers themselves – rates of climb, V speeds, etc. – obviously do depend on
gross weight and on density altitude. Again for brevity’s sake, we will consider this airplane’s
behavior at one particular weight (maximum gross weight W0 = 2400 lbf) and at two particular
density altitudes (MSL and 10,000 ). These choices let us compare our performance numbers – at
least many of them – with the airplane’s flight manual, the POH. Looking ahead to that
comparison, let us evaluate all the above composite Bootstrap parameters for those two cases.
See Table 5
The composite definitions and their dependence on weight, bank angle and air density are:
Table 5. Bootstrap
Variable or Composite Case 1, MSL Case 2, 10,000 ft composite parameters for
two operational situations.
W 2400 lbf 2400 lbf
Gliding V Speeds
The V speeds we are
E 531.9 373.8 concerned with, in the
F –0.0052368 –0.0038673 are:
G 0.0076516 0.0056505 VM, maximum level
H 1,668,535 2,259,424 Vm, minimum level
K –0.0128884 –0.0095178
Vy, speed for best rate
Q –41,270.6 –39,277.5 of climb;
Vx, speed for best angle
R –129,460,301 –237,389,461 of climb (in calm air);
Vbg, speed for best
U 218,064,595 399,861,861 (longest) glide (in calm
Vmd, speed for minimum gliding descent rate.
Bootstrap formulas for these V speeds, each expressed as a true air speed in ft/sec, are:
Since the three full-throttle V speeds VM, Vy, and Vx depend on only two composite parameters,
Q and R, there must be a connection between them. It is:
In addition, it turns out that Vx is the geometric mean between VM and Vm. Performance
specifications for most manufacturers’ airplanes will not closely agree with Eq. (27) because the
quoted values of VM are too optimistic.
Additional Flight Performance Quantities
The Bootstrap Approach is not limited to predicting V speeds. There are also formulas for full-
throttle power available Pa = TV, power required Pr = DV, excess power Pxs = (T–D)V, thrust T,
drag D, rate of climb R/C, and flight path angle . In the gliding case, rate of sink R/S and glide
path angle can be obtained from the powered forms by setting E = F = 0 and replacing K by –G.
We’ve shown all the left-hand-side variables as functions of only true air speed V. But in all
except two cases gross weight W, relative air density , and bank angle also matter. It would
have been more instructive, if somewhat pedantic, to have written D(V), for instance, as
Where do all these formulas come from? In most cases, from standard "power-available/power-
required" analysis, which you can find in almost any textbook treating the aircraft performance
subject. The one big exception is Eq. (31) and other equations springing from it. Eq. (31) is the
Bootstrap Approach’s "hole card," giving us a good approximation to thrust without our having
to know propeller efficiency. That Bootstrap expression for thrust is quite easy to get. Here’s
The Bootstrap Formula for Full-Throttle Thrust
First, solve Eq. (2) for thrust:
Next, rewrite Eq. (5) as:
Then substitute Eq. (36) into Eq. (35), using the definition of CP (Eq. (3)) and the definition of J
(Eq. (1)), to get:
One additional "physical" fact is needed. For an internal combustion engine at given altitude,
throttle position determines torque output, irrespective of load and resulting RPM, to a good
approximation. So "full throttle" means "full torque," or at least as "full" as ambient density
altitude allows under the direction of Eqs. (6) and (7). Finally use Eqs. (14) and (15), the
definitions of Bootstrap composite parameters E and F, to get Eq. (31).
Now, let’s look at some graphs.
Bootstrap Performance Graphs
There’s no better way to learn the ins and outs of airplane performance than by looking carefully
at various performance graphs. The graphs drive home the definitions of the various V speeds,
the airplane’s "optimal" speeds.
Figure 4. Thrust, drag, and their difference (excess thrust), as functions of air speed, for a Cessna 172 at sea level,
flaps up, weighing 2400 pounds.
But keep in mind that almost everything about an airplane’s performance depends on its weight,
its altitude, its flaps/gear configuration, and whether or not its wings are level. For brevity, we’ll
stick to wings level. But, as you look over the graphs, you should ask yourself such questions as:
How would this graph look if it were for a more (or less) powerful airplane? What if the airplane
were heavier (or lighter)? At higher (or lower) altitude? Flaps down? Banked 40 degrees?
Figure 4 starts us off with graphs of thrust and drag (Eqs. (31) and (32)) and their difference,
excess thrust Txs. The speed at which excess thrust is a maximum is always the speed for best
climb angle, Vx. The speed at which drag is a minimum is the speed for best (longest) glide in
calm air, Vbg. The thrust and drag curves cross at two places, on the right at maximum level
flight speed VM and, to the left, at minimum level flight speed Vm. Vm is not marked because, in
fact, you can’t get there. Stall speed VS – not a Bootstrap-calculated V speed – is higher than Vm
except at high altitude.
But not everything depends on all three operational variables W, , and . Thrust, for instance, is
independent of W. But at higher altitude, lower , the thrust or power available curve will be a
lower one. How the drag curve behaves with changes in weight and altitude is harder to see;
altitude affects Bootstrap composite parameter G and both altitude and weight affect H. With a
spread sheet program (Lotus 1-2-3 or Quattro Pro or Excel) you can easily take the Bootstrap
formulas, assume reasonable aircraft parameters, and construct graphs for all kinds of scenarios.
Figure 5 shows most of the important performance V speeds and where they come from. VM and
Vm are speeds at which the Pa and Pr graphs cross. Or, alternatively, where Pxs has its zeroes.
Since rate of climb R/C = Pxs/W, speed for best rate of climb Vy is where Pxs peaks. When
gliding (thrust zero), Pxs = –Pr; therefore the minimum descent rate speed Vmd is at the low point
of the Pr curve. Speed for best glide Vbg and speed for best climb angle Vx take a bit more
analysis. Since Pxs/V = Txs, we see that the straight line tangent to the Pxs curve which has the
largest slope is the one hitting the Pxs curve at Vx. Similarly for Pr/V = D and Vbg.
Figure 5. Power available, power required, and excess power for the Cessna 172 at 7,500 ft, flaps up, 2400 pounds.
Figures 4 and 5 are graphs for the airplane at one weight and one altitude. Figure 6 gives a
broader view, in the vertical direction, showing how the main full-throttle V speeds differ, again
for the Cessna 172 weighing 2400 pounds, over the full range of accessible density altitudes.
Figure 6 shows that speed for minimum level flight Vm doesn’t come out from hiding behind the
skirts of stall speed VS until the altitude is way high, above 14,000 feet. Which means, in almost
all practical cases, never. There are conflicts between Figure 6 and the Cessna 172P POH. The
POH has Vy not down to as low as 70 KCAS until 12,000 ft; we get it there at only 7000 ft. And
the book has Vx increasing a little with altitude, about 4 KCAS over 10,000 ft, where we get that
it is constant in calibrated terms. Naturally, we believe we are right. Figure 7 delves into drag. As
Eq. (32) shows, total drag D is made up of two terms. DP, parasite drag, is proportional to the
square of the air speed. See Eq. (16) for G to see what makes up the proportionality constant.
The second term, induced drag Di (also known as "drag due to lift"), works much differently.
Induced drag is inversely proportional to V2. That means induced drag is higher the slower the
airplane goes through the air. See Eq. (17) for H to see what the "constant" depends on.
Figures 8 and 9 show how performance drops off with altitude for a Cessna 172 weighing 2400
pounds, flaps up. By 10,000 feet, considerably less than half the mean sea level (MSL) best rate
of climb or best angle of climb remains. This is a real problem for a relatively underpowered
airplane like the Cessna 172. The Civil Air Patrol, for instance, doesn’t allow its airplanes to
conduct mountain search and rescue missions at altitudes where their best rate of climb is less
than 300 ft/min. A fully-loaded Cessna 172 with the stock 160 HP engine could not qualify.
Perhaps a special "climb" propeller would let it meet that mark.
Figure 6. Full-throttle V speeds for Cessna 172, flaps up, 2400 pounds.
These graphs give you a reality check, samples showing how various factors influence propeller
airplane performance. The references contain many more such graphs and numerical
Figure 7. How drag varies with air speed for a Cessna 172, flaps up, 2400 lbf, at MSL.
Figure 8. Rate of climb graphs, at three density altitudes, for a Cessna 172 weighing 2400 pounds, flaps up. Notice
that calibrated air speeds for best rate of climb decrease with altitude.
Figure 9. Angle of climb for a Cessna 172, flaps up, 2400 pounds, at three density altitudes.
So the Bootstrap formulas are relatively simple and straightforward. But how accurate are they?
Comparison of Bootstrap and POH Performance Figures
That depends on whom you ask. Certainly the airplane’s POH won’t be far wrong, so let’s take a
look at it. Sifting through the Cessna 172P Pilots Operating Handbook (for the airplane without
speed fairings), and making occasional use of the air speed indicator calibration curve given
there, one can come up with the two columns headed ‘POH’ in Table 6. We’ve translated all the
speeds in that table into KCAS.
Performance Case 1, 2400 lb at MSL Case 2, 2400 lb at 10,000 Ft
Item TBA POH TBA POH
VM 115.3 121.0 90.8 98.8
Vy 75.8 76.0 67.5 71.0
R/C(Vy) 700.5 fpm 700.0 fpm 258.5 fpm 237.0 fpm
Table 6. Comparison of
Vx 63.2 62.0 63.2 66.0 Bootstrap (TBA) and
Vbg 72.0 66.0 72.0 66.0 predictions.
bg –5.40 deg –6.26 deg –5.40 deg –6.26 deg
There is a major
discrepancy where maximum level flight speed VM is concerned. We’ve flown a number of
Cessna 172s, but never one which would go 121 KTAS at sea level. The Cessna test pilot might
explain her higher speed is due to her brand new engine, the pristine leading edge of the wing, no
dents, .... Perhaps. In glide performance, on the contrary, our airplane did better than the factory-
fresh one. There we suspect (but only suspect) the Cessna Aircraft Company corporate attorneys
came into play. For liability reasons, they certainly wouldn’t want to claim a longer best glide
than might be demonstrated by a lawyer whose client’s engine had failed.
The only way to settle these questions of whose performance statements are more accurate is
through careful, well-instrumented, and un-manipulated test flights. Our point here is that the
Bootstrap Approach lets you fly the airplane for an hour or so, performing a few simple climbs
and glides and a level speed run, and then lets you calculate many interesting and apparently
accurate performance numbers for that actual airplane. Bugs, dents, tired engine and all.
Bootstrap Approach Extensions
Maneuvering flight (wings banked, turning) was implicitly included in the various Bootstrap
formulas we presented, but – for lack of time and space, and to avoid sensory overload – we did
not pursue or exemplify that flight performance realm. From those formulas one can get so-
called "steady maneuvering charts" which graphically demonstrate (for a given airplane at a
given weight, configuration, and altitude) the relations between air speed, bank angle, rate of
climb or descent, and either turn radius or turn rate. The charts also include a banked stall curve,
a buffer curve paralleling the stall curve, and a structural load factor limit line. Reference 11
gives details and explicit formulas for constructing steady maneuvering charts.
If you bank the airplane and try to maintain level flight, you must use back stick to increase
angle of attack. For level flight, the lift vector must be long enough that its vertical component
balances weight. But there are limits. If you bank too far, the wing will stall. Stall speed goes up
with bank angle. More surprising, for a given thrust and altitude, there is always a bank angle
beyond which the airplane cannot maintain level flight at any speed. There it is at its "absolute
banked ceiling." At that altitude banked values of Vx and Vy, as functions of bank angle, cross.
Vx then becomes larger than Vy. In many respects, banking the airplane is tantamount to
suddenly making it heavier.
There are important safety considerations, especially for underpowered trainers at high altitude,
in the banked absolute ceiling concept.
There is also a Bootstrap extension to partial-throttle operations. For that, you need one
additional but very simple flight test: cruise level at various air speeds (at any known gross
weight and density altitude) and record engine RPM. From that information one can construct
graphs of propulsive efficiency and of both propeller thrust and power coefficients. Moreover,
and more to the operational point, one can answer all such questions as the following: if I take
this airplane up to (say) 9000 feet, weighing 2150 pounds, flaps up, and want to put it into a 300
ft/min standard rate (3 degrees per second) descending turn, at 90 KCAS, what RPM should I
throttle back to? Being able to answer that kind of question is a major Bootstrap advance. The
Bootstrap partial-throttle extension also lets one get accurate cruise performance tables, with
speeds down to the quite low ones for best range Vbr and for best endurance Vbe. Those safety V
speeds are not and cannot be given by the standard GAMA (General Aviation Manufacturers
Association) format cruise tables. The Bootstrap partial-throttle theory also allows one to take a
portion of a cruise performance table, for one weight and altitude, and use scaling laws to
calculate corresponding cruise performance entries at any other weight and at any other altitude.
Bootstrap Approach Advantages
Because of its relatively simple analytic (formula-based) construction, the Bootstrap Approach
also lets us find values of the two "Earth-based" V speeds, Vbg and Vx, for any steady wind
conditions. That way one can find, for any given headwind, tailwind, updraft, downdraft, or
combinations, how much to slow down, or speed up, from the nominal calm air V speed values,
to ensure optimum glide or climb. Trial-and-error calculations are required, but with a modern
spread sheet program those are easily figured.
Since the Bootstrap Approach includes a good formula for propeller thrust, extension to the take
off maneuver is perfectly feasible. The same is true of the landing maneuver, including the
trickier portion bringing the airplane down to the runway from altitude 50 feet. Since several
different forces – rolling friction, braking, runway slope and contamination, ground effect – are
required for take off or landing analysis, we leave that subject to another time
An advantage of The Bootstrap Approach for manufacturers of small airplanes is that design
changes – say a different engine – only require, for new performance predictions, new BDP
items for that engine. The three subsystems (airframe, engine, and propeller) are relatively
independent. Even after the airplane’s design has been frozen, performance flight testing by
"standard" methods, according to a professional performance test pilot often hired by Cessna,
takes about eighty hours of flying and calculating. The Bootstrap Approach requires only two to
three hours. While at least a couple of the larger kitplane manufacturers (Skystar Aircraft and
RANS) do currently (1999) use the Bootstrap Approach, many more propeller aircraft
manufacturers should look into doing likewise. Several "mod shops" – businesses which install
STOL (short take off and landing) devices such as wing cuffs, gap seals, stall fences, and larger
engines and propellers – use the Bootstrap Approach as a sales tool and to demonstrate to their
customers that the modifications they bought will pay off with enhanced performance.
This article is only a primer on the aircraft performance subject. What about designing an
airplane? Knowing about performance comes first; the airplane is designed to perform a certain
job. There is also the vast subject of airplane stability and control, which we did not even touch.
But knowing about performance is also a prerequisite to that subject. The references suggest
where you might look further. Learning the ins and outs of aircraft performance will make you a
better pilot and make you a better engineer. There are few technical subjects more interesting, or
more fun, than aircraft performance. Calculate thoughtfully. Fly the same way!
General Works on Aircraft Performance, Stability, and Design
1. Roskam, J., and C.-T.E. Lan, Airplane Aerodynamics and Performance, DARcorporation,
Lawrence, Kan., 1997. A large book, currently the text in the aircraft performance course at the
U.S. Military Academy, covering both propeller and jet airplanes.
2. Hubin, W.N., The Science of Flight, Iowa State University Press, Ames, Iowa, 1992. The
author is a physicist at Kent State University and an experienced aerobatic pilot. Well illustrated.
Requires only algebra and elementary physics.
3. McCormick, B.W., Aerodynamics, Aeronautics, and Flight Mechanics, Wiley, New York,
1979. A calculus-level treatment for aspiring aeronautical engineers.
4. Hurt, H.H., Aerodynamics for Naval Aviators, U.S. Navy, 1960. A good full qualitative
(almost completely non-mathematical) treatment. Covers both propeller and jet airplanes.
5. Von Mises, R., Theory of Flight, Dover, New York, 1959. A somewhat old-fashioned
reference (for instance somewhat confusingly separates wing drag out from the remainder of the
airframe) but packed with wisdom and information. Written by the best educated person to ever
clamber into a cockpit. Treats only propeller airplanes. Uses calculus and differential equations
6. Hale, F.J., Aircraft Performance, Selection, and Design, Wiley, New York, 1984. Written by a
former U.S. Air Force jet combat pilot. Uses minimal calculus. Treats propeller airplanes and the
several types of jet aircraft.
7. Hiscocks, R.D., Design of Light Aircraft, published by the author (designer of the deHavilland
Beaver) and distributed by Murphy Aircraft Manufacturing, Ltd., Unit #1, 8155 Aitken Road,
Chilliwack, B.C., Canada V2R 4H5. Takes the reader step by step through the full range of light
airplane design techniques and considerations.
8. Perkins, C.D., and R.E. Hage, Aircraft Performance Stability and Control, Wiley, New York,
1949. A classic, still in print.
Theoretical Max Propeller Efficiency
Note: This essay is meant for people that already have a pretty firm understanding of
aeronautical engineering. I will not go into a lot of detail explaining things that are generally
known in the field.
I'm writing this essay, like I do a lot of my essays, because I couldn't find this information on the
net, so I had to develop it myself, and I thought I'd make that available to everybody. By the end
of this essay, I will have developed a method to calculate the theoretical maximum thrust that
can be produced by a propeller for a given diameter and a given power. Note that all equations
use consistent units. You will have to convert all nonconsistent units (ex: 1 HP = 550 ft-lb/s).
While working on a project designing a propeller at work, I wanted to know just how good I was
doing. Efficiency is one measure of how well a propeller is performing, but it's not necessarily a
good indication of how well the design is performing up to its potential. In aviation, propulsive
efficiency is defined as:
where η is efficiency, T is Thrust, V is Velocity, and Pavail is Power Available, or power going
into the propeller. Basically, power out divided by power in. This equation is very useful for
many cases, but you should see a problem in that as your velocity goes to zero, no matter how
much thrust you're producing, your efficiency goes to zero. So how do you know how good your
prop is doing at low speeds or statically? Well, there's another term that can be used, Figure of
Merit, which is Induced Power divided by Power Available, or how much of your power is going
into accelerating the air. A Figure of Merit of 1 is the best you're ever going to do. You can't
accelerate the air any more than that. So, Figure of Merit gives a pretty good indication of how
well the prop is doing at any airspeed. Figure of Merit is calculated as:
where F.O.M. is Figure of Merit, T is Thrust, VA is aircraft velocity, VI is induced velocity at the
propeller, and Pavail is Power Available.
Now, we'd like to turn this equation around to solve for thrust for a given Figure of Merit. The
only problem is that induced velocity is a function of thrust. So, let's figure out a way to solve for
induced velocity. The fluid flow analogy to F=ma is
where ρ is density, A is the propeller swept area, V is the velocity at the propeller, and ΔV is the
amount that the air is accelerated by the propeller, the difference between the freestream and
some point downstream of the rotor where no more acceleration takes place. It's pretty well
known that at the propeller, the air has accelerated one half of what it will do downstream (V I =
1/2 ΔV). So, at the propeller, the velocity is VA + 1/2 ΔV. Substituting this into the equation
Rearranging this equation into something that can be solved using the quadratic equation yields:
Solving with the quadratic equation gives:
And remembering that VI = 1/2 ΔV:
where the addition solution, not subtraction, gives the correct answer for most conditions at
which a propeller will be operating.
Now, let's go back to equation (1), and solve for thrust
Substitute that back into equation (3):
Now we've got the three equations that we'll need to solve for thrust for a given figure of merit.
Taking the equation for Figure of Merit (2), and solving for Power Available yields:
Substitute this into the equation for efficiency (1), and simplify to get:
Subsitute equation (5) into this to get:
Now, I'm sure we could go on further from here, solving for η, but that would be rather tedious,
and the equation would get even messier than it already is. So, from this point, I just built myself
an Excel spreadsheet, and used the Solver function to solve for η. Once you have that, it's a
trivial matter to go back to equation (4) and calculate your thrust.
Static Thrust is a special case, since both airspeed and efficiency will be zero. This makes the
final equation above that we had derived become invalid, since the denominator will be zero.
But, it is extremely easy to solve for induced velocity, and then thrust from there using the
F.O.M. equation. I will leave it up to the reader to go through the derivation, but the resulting
equation for thrust is:
These have proven to be very useful calculations for my job when designing propellers. They
show the theoretical limit of just how much thrust a propeller could be producing at any given
airspeed, giving a good bench mark for my designs. These calculations can also be useful for
getting a good understanding of propeller performance in general. The graphs below were made
using the equations derived above. They illustrate the limitations of propellers, as well as some
In the first graph, you can clearly see that even with the propeller operating as well as it could,
your efficiency is going to be lousy up until you get to a decent airspeed, but in the second graph,
you can see that even with increased efficiency, your thrust drops off as airspeed increases.
The first of the next two charts shows just how important propeller diameter is to static thrust.
For the same horsepower, the amount of static thrust that you can produce just keeps going up
with diameter. This increased thrust won't be nearly so marked throughout the whole flight,
however, as the second chart below illustrates. It can be seen that even with diameters different
by a factor of 2, both propellers produced nearly the same thrust at 200 mph. Remember the
equation for efficiency (1) above. It has nothing to do with diameter. The max possible
efficiency is 100%, and as seen in the first graph above, once you get up to higher airspeeds, you
can start approaching that efficiency even with a smaller propeller. These charts help to show
why helicopters use large rotors, and modern jet engines use high bypass designs- it's a whole lot
more efficient to get thrust by accelerating a large amount of air by a little bit, than to acclerate a
small amount of air by a lot.