14.1.1 The Content of this Chapter

14.1.2 Propeller Configurations

Propellers can be mounted in a number of ways to an airplane. Three common methods are shown in Figure 14-2; a tractor (A), pusher (B), and a configuration featuring the engine and propeller mounted on the wing in nacelles (C). Configuration (C) is a variation of (A) or (B). The advantages and disadvantages of these installation methods are listed in Table 14-1. Note that so-called “inline” configurations, which consist of a tractor and pusher, can simply be treated as a combination of configurations (A) and (B).

A few additional explanations are needed for the pusher propeller. As stated in Table 14-1 special regulatory requirements affect the certification of pusher aircraft and are stipulated in 14 CFR 23.905(e) through (f). Paragraph 23.905(e) states that:

“All areas of the airplane forward of the pusher propeller that are likely to accumulate and shed ice into the propeller disc during any operating condition must be suitably protected to prevent ice formation, or it must be shown that any ice shed into the propeller disc will not create a hazardous condition.”

Paragraph 23.905(f) states that:

“Each pusher propeller must be marked so that the disc is conspicuous under normal daylight ground conditions.”

Paragraph 23.905(g) states that:

“If the engine exhaust gases are discharged into the pusher propeller disc, it must be shown by tests, or analysis supported by tests, that the propeller is capable of continuous safe operation.”

And, finally, paragraph 23.905(h) states that:

“All engine cowling, access doors, and other removable items must be designed to ensure that they will not separate from the airplane and contact the pusher propeller.”

There are further requirements made in paragraph 23.925(b) Propeller clearance that state that:

“(b) Aft-mounted propellers. In addition to the clearances specified in paragraph (a) of this section, an airplane with an aft-mounted propeller must be designed such that the propeller will not contact the runway surface when the airplane is in the maximum pitch attitude attainable during normal takeoffs and landings.”

Excluding the last paragraph, these are extra requirements not demanded by the certification of tractor aircraft. Other drawbacks of the pusher propeller that need further explanation are the higher flyover noise and propeller corrosion. Experience has demonstrated that the ingestion of fuselage wake produces additional broadband noise,⁵ often adding several dB(A) to the airplane’s noise. Consequently, all other things being equal, pushers tend to be noisier than tractor configurations. A pusher propeller placed behind the engine exhaust will collect soot on the blades, forming acids that attack the propeller structure. Maintenance protocols for many pushers require the operator to wash the propeller after the last flight of the day as preventive maintenance. The competent aircraft designer keeps such drawbacks in mind when choosing a pusher over a tractor. As stated in Chapter 2, Aircraft conceptual layout, there are important consequences to a configuration selection, some of which have environmental, operational, and maintenance impacts. It is not wise to base the choice on looks only – operational practicality should lead such decisions.

14.1.3 Important Nomenclature

Any discussion about propellers involves jargon that the designer must be familiar with. For instance, it is likely that in a discussion about modern aircraft, one will hear pilots and mechanics alike describing a propeller as “an advanced composite five-blade, constant-speed, reversible, and fully featherable.” What does it mean? The following definitions help clarify these terms and must be kept in mind for the discussion that follows:

A fixed-pitch propeller is a propeller whose pitch angle (the incidence of the blades with respect to the plane of rotation) cannot be changed. Such propellers are comparatively inexpensive, light, and require very little maintenance. The lack of a mechanical system renders them pretty much failsafe; however, they will windmill in the case of an engine failure and, thus, increase drag when a high glide ratio is sorely needed. Generally, such propellers are designed for best efficiency at either climb (“climb prop”) or cruise (“cruise prop”).

A ground-adjustable propeller is a propeller whose pitch angle can be adjusted using simple tools while stopped on the ground only. Thus, the operator can change the pitch from, say, a “climb” to a “cruise” style propeller (see Section 14.1.7, Fixed versus constant-speed propellers) between flights. Such propellers have become popular among homebuilders and LSA aircraft.

A two-position propeller is one that allows the pilot to change the pitch of the blades mechanically in flight, but only allows two settings (versus multiple ones for the ground-adjustable propeller). This way, the pilot could select a climb setting for the take-off and subsequent climb, and then adjust it to a coarser cruise setting once cruise altitude is reached.

A controllable-pitch propeller is one that allows the pilot to change the pitch over a range of possible pitch angles during flight. This is almost always accomplished through the use of hydraulic boost built into the hub of the propeller. Such a system allows the pilot to adjust the propeller pitch for the most efficient thrust output possible.

A constant-speed propeller is a propeller that will automatically adjust its pitch to maintain a preset RPM, which otherwise is highly affected by airspeed. It does this through the use of a controlling mechanism attached to the engine, called a governor, which balances centripetal and hydraulic forces.

A feathering propeller is a controllable-pitch or constant-speed propeller that allows the pilot to align the blades perfectly with the forward speed of the aircraft in the case of an emergency, such as an engine failure. This prevents the propeller from windmilling and, thus, reduces the drag impact of the inoperative engine.

A reversing propeller is a controllable or a constant-speed propeller that allows the pilot to align the blades to angles beyond what is possible with a feathering propeller. Effectively, it allows the thrust force to be pointed in a direction opposite to the forward motion of the aircraft. This feature is used during landing to decelerate the airplane and even to allow the aircraft to “back out” out from an air terminal using its own power. It is commonly found on turboprop engine installations.

An idealized power lever quadrant console is shown in Figure 14-3. Common reverse operation of propellers involves settings called ground fine, reverse, and beta. Ground fine is a low pitch setting used when taxiing the airplane with the engine running at a “reasonably high” rotation rate. Maintaining such RPMs on some turboprops is recommended to prevent damaging oscillation that may occur at lower RPMs. The ground fine setting prevents too high thrust from being generated at those rotation rates and this is most helpful for low-speed ground operations.

The reverse setting, as stated earlier, generates thrust in the opposite direction and is helpful immediately after landing. The setting labeled “Taxi” in the figure is where the pilot would operate the power lever while taxiing. In this range, the RPM is constant, but the blade pitch angles are adjusted. Beta refers to the range of power lever positions that are used on the ground that include the reverse and taxi power settings.

A beta control prop is a propeller specifically designed to allow operation of the propeller as described above.

Material for propellers: the modern propeller is made from three different materials: wood, forged aluminum, or composites. All have their pros and cons, but only aluminum and composites should be used for high-performance aircraft. Wooden propellers are lighter and should not be overlooked as a solution for light airplanes that may have the CG too close to a forward limit (tractor) or aft limit (pusher).

A propeller extension can consist of a separate spacer piece (most commonly for fixed-pitch propellers) or an integral extension of the propeller hub (most common with constant-speed propellers). The purpose of the spacer or hub extension is to increase the spacing between the engine and propeller so that the geometry of the engine cover (cowling) may be made more aerodynamically efficient and, thus, reduce the drag penalty of the engine installation. This often leads to improved cooling. While heavier, the overall reduction in drag may easily surpass the penalty of the weight addition. Propeller extensions must be carefully installed because if the propeller becomes misaligned, a severe vibration may result. Ideally, it should have a centering boss to make the installation easier.

14.1.4 Propeller Geometry

A three-bladed propeller is shown in Figure 14-4, rotating about an axis we call the axis of rotation. The spinner is an aerodynamically shaped cover, whose purpose is to reduce the drag of the hub of the propeller and to protect it from the elements. The propeller blades are what generate the thrust of the device, denoted by T. The pressure differential between the front and aft face of the propeller blade results in a vortex that is shed from the tip of the blade and is carried back by the airflow going through the propeller. This forms the typical helical shape shown in the image. Note that only one vortex is shown, but two others are also formed by the other blades and are hidden for clarity.

A frontal projection of the three-bladed propeller is shown in Figure 14-5, where R is the blade radius, r is the radius to an arbitrary blade station, and Ω is the rotation rate, typically in radians per second or minutes.

The blade of a propeller is really a cantilevered wing that moves in a circular path rather than along a straight one. Just like an airplane’s wing, the planform of the propeller blade has a profound impact on the magnitude of the thrust force created, as well as at what “cost.” What constitutes “cost” is the amount of power required to rotate it, as well as side effects such as noise. Figure 14-6 shows a picture of a typical blade planform and its cross-sectional airfoils, taken from NACA TR-339 [1]. The blade planform, along with geometric properties such as twist and airfoil camber, is of crucial importance to optimize a propeller. These affect not only the capability of the blade to generate thrust, but also its strength and natural frequencies.

14.1.5 Geometric Propeller Pitch

Consider the propeller in Figure 14-7, whose diameter is D and radius is R. As the propeller rotates through a full circle, its tip rotates through an arc length (circumference) of C = πD = 2πR. As the propeller rotates it “screws” itself forward a certain distance P for each full rotation. Now consider the abstract case where the propeller is assumed to rotate slowly through some highly viscous fluid. If this were possible, the propeller would move through this fluid like a metal screw moves into a piece of wood. The distance it would cover in one full revolution is called the geometric pitch or pitch distance, P_D, of the propeller. It is commonly specified in terms of inches of pitch. Thus a propeller designated as a 42-inch pitch prop would move 42 inches forward in one revolution (using the metal screw through wood analogy). The angle the helix makes to the rotation plane is called the geometric pitch angle and is denoted by β.

14.1.6 Windmilling Propellers

Windmilling occurs in airplanes when power to the propeller is cut and it rotates solely due to the aerodynamic lift generated by the blades. A generalized schematic of this phenomenon is shown in Figure 14-12. First consider the left image. In normal operation the prop is driven by the engine at an angular velocity of Ω and is moving at airspeed V₀ through air. At a particular blade station r, the speed of the blade element is Ω·r, which yields the resulting airspeed V_R and α shown. The lift, L, generated by the entire blade is shown in blue and its component parallel to the rotation axis is the thrust, T. The term w is the induced velocity, which depends on the shape of the streamtube that goes through the propeller disc.

In the right schematic, the propeller is rotating at an angular velocity Ω and some forward airspeed V₀, both of which may be different from that of the left figure. This time, the speed components yield an α from the opposite side, producing lift, L, that points in the opposite direction. Again, thrust is generated, albeit in the opposite direction (and thus it should really be called drag).

Typical recovery procedures in airplanes following a non-catastrophic engine failure⁷ require its nose to be lowered until a specified forward speed (here denoted by V₀) is achieved. This is typically the airplane’s best glide speed (see Section 19.2.8, Airspeed of minimum thrust required, V_TRmin). As the angular velocity of the propeller gradually reduces, the AOA eventually becomes negative, as shown in the right image of Figure 14-12. This starts the propeller windmilling (or auto-rotating) and at this point drag begins to increase and can have detrimental effects on the L/D ratio. This effect must be considered in the aircraft design and the publication of a Pilots Operating Handbook (POH).

The reversed thrust (drag) generated by the windmilling propeller is higher than for the stationary propeller. The greater the airspeed, the greater is the propeller drag. Some pilots attempt to improve the glide ratio by stopping the autorotation. This requires the airplane to be slowed down so the propeller rotation is halted by the internal friction of the engine. More often than not this will require airspeeds in the neighborhood of the airplane’s stalling speed and, consequently, such maneuvers should not be attempted at low altitudes. Propeller-powered multiengine aircraft are always equipped with feathering propellers, which circumvent this problem by changing the pitch of the propellers such that windmilling is no longer possible and the pitch places the blades in a low-drag position, greatly reducing the impact on drag. A method to estimate the drag of windmilling (and stopped) propellers is presented in Section 15.5.13 Drag of windmilling and stopped propellers.

14.1.7 Fixed Versus Constant-Speed Propellers

Propeller efficiency is discussed in greater length in Section 14.3.9, Propulsive efficiency, but it is an indicator as to how much engine power is being converted into propulsive power (thrust × airspeed). Thus, a particular propeller may be 0.80 efficient at a specific condition. This means that 80% of the engine power (BHP) is being converted into propulsive power. As we have seen before, the AOA of the blade varies with airspeed (as shown in the lower part of Figure 14-11) and the propeller’s thrust will do so as well. It follows that propeller efficiency is a function of airspeed.

Generally, propellers for aircraft are designed for a particular airspeed or range of airspeeds specified by the airframe manufacturer. In this case the following rules of thumb apply:

A fixed-pitch propeller is one in which the blade pitch angles are permanently fixed. Such propellers are simple, light, and inexpensive. Their drawback is that their best efficiency is achieved at a particular airspeed only, so the designer must decide whether to emphasize climb or cruise performance and select a prop that favors either. Usually, two types of fixed-pitch propellers are available. The fixed-pitch “climb” propeller is designed to reach it maximum propeller efficiency at a relatively low airspeed. This makes it very suitable for use in airplanes where climb performance is of importance, such as trainers and bush-planes. The fixed-pitch “cruise” propeller is designed to reach its maximum efficiency at a higher airspeed, making it suitable for airplanes where higher cruising speed is of greater importance than climb performance.

Pilots who fly aircraft with fixed-pitch propellers notice that, if initially at cruise, the RPM of the engine will reduce if climb is initiated and increase if descent is initiated. In the former case, the climb inevitably slows down the aircraft and the propeller blades begin to experience a higher AOA. Thus they generate more drag, which, in turn, means higher torque is being generated and this slows the engine RPM. If we assume the pilot does not change the power setting, the RPM will simply drop as well and the prop will now be operating at a changed efficiency. In the latter case the opposite happens. The blades will experience lower AOA as the aircraft speeds up and generate less drag requiring less torque. The engine experiences this as reduced load so its RPM increases, again, changing the efficiency. If the aircraft is using a cruise prop, the change will be toward less efficiency, since it may very well have been operating at peak performance. If it is a climb prop, it would become more efficient as the airplane slows down, but less if it increases the airspeed.

An airplane specifically designed and operated as a cruiser will always feature a cruise-style propeller. Airspeed variations do therefore detrimentally affect its performance. This begs the question, is it possible to feature a prop that will not be so subject to such off-peak deviations? In other words: is it possible to design a prop so it tends to maintain the RPM and therefore its peak efficiency at the condition? The answer is yes. In fact, such propellers have existed for a long time and are called constant-speed propellers. However, such propellers went through an evolutionary phase, as in the 1920s manufactures, such as Pierre Levasseur, introduced manual control to adjust the blade pitch in flight [2] and that way “convert” a T-O prop into a climb prop, and a climb prop into a cruise one.

The variation in RPM with airspeed still occurred and this led to the development of an automatic controller, called a propeller governor, or simply a governor. It is designed to use accurate deviations from the setting selected by the pilot to maintain peak efficiency at all times. Figure 14-13 shows how the propeller efficiency typically varies with airspeed for three different but commercially available propeller types. The green and blue curves represent the efficiency of propellers whose pitch is fixed.

A detailed description of how the governor works is beyond the scope of this section; however, a brief introduction will be given. Once the desired RPM (using the cockpit RPM control) has been established the governor will automatically adjust the pitch of the propeller blades as needed to maintain that RPM. This is accomplished by means of throttle setting and flyweights. The governor mechanically detects RPM via the flyweights and uses it in conjunction with the throttle setting to maintain the required oil pressure inside a pressurized oil reservoir. Consider an airplane whose pitch angle deviates from a level flight such that an increase in RPM results (for instance if the nose pitches down). This implies an increase in airspeed and would cause the governor to increase the propeller pitch which would reduce the RPM. By the same token, a decrease in RPM would indicate a reduction in airspeed (for instance if the nose pitches up). The governor would then reduce the propeller pitch angle until the original RPM was again established. A section view of a hub of a constant-speed propeller is shown in Figure 14-14.

14.1.8 Propulsive or Thrust Power

Propulsive power is the power required to move a vehicle at a specific speed using specific force or thrust. Power is defined by:

Since work is defined as the application of a force over a specific distance (force × distance) we get:

We therefore define propulsive or thrust power as follows:

(14-13)

where the subscript R stands for required as this term represents the power required in the performance analysis of later sections.

14.2.1 Angular Momentum and Gyroscopic Effects

Angular momentum and gyroscopic effects play an important role in stability and control theory and, thus, must be taken into account in the design process. Consider the propeller to the left in Figure 14-15, which rotates at a constant angular velocity Ω_x (computed using Equation (14-6)). As it rotates about the x-axis (the axis of rotation), an angular momentum, h_SRx, (SR stand for spinning rotor) is being generated. In conventional single-engine aircraft this momentum must be reacted by the airplane, otherwise it would itself begin to spin in the opposite direction. For instance, the angular momentum of a propeller that rotates clockwise, as seen from behind (right blade is moving down) will tend to rotate the airplane in the opposite direction – to the left. Generally, in such airplanes, this is easily suppressed by a very slight aileron deflection, since the change in lift due to aileron deflection acts through a long arm. The magnitude of the angular momentum is given by:

(14-14)

where

Some multiengine aircraft feature an even number of engines (two, four, six, etc.) and propellers that rotate in opposite directions about the plane of symmetry. As long as the engines all rotate at the same RPM, this cancels the angular moments.

Now consider the same propeller to the right in Figure 14-15, which shows the original propeller in the process of rotating to a new orientation. This change in orientation is called gyroscopic precession. As the motion takes place, additional gyroscopic couples, M_y and M_z, are generated. These moments are restoring since their action is to keep the propeller in the original orientation. Note that as soon as the rotation ceases, so will those moments. If the angular velocity of the precession is given by , then the gyroscopic moments can be found from:

(14-15)

The components p, q, and r can be interpreted as the rotation rates of an airplane about its x-, y-, and z-axes, respectively. Note that for the propeller in Figure 14-15, the gyroscopic moments would be given by:

(14-16)

14.2.2 Slipstream Effects

A typical single-engine aircraft is shown in Figure 14-16. It features a standard propeller whose rotation is clockwise as seen from the pilot, who sits behind it (clockwise motion of this nature is common among American propeller manufacturers, whereas counter-clockwise is common in Europe). The corkscrew-shaped blade tip vortex is sketched as it encloses the fuselage of the airplane. The tip vortex is caused by the pressure differential between the forward (low pressure) and rearward (high pressure) sides of the propeller. This pressure differential causes air on the back side of the propeller to flow toward the tip, while the opposite happens on the forward face. The opposite radial flow directions on the two sides cause the formation of this vortex at the tip, exactly as it does on a regular wing. The vortex is indicative of the downwash left by the propeller blade. The lower image of Figure 14-16 shows the vortex in the area of the vertical tail, subject to this downwash. It shows that the downwash component, which is perpendicular to the wake left by the propeller blade, changes its angle of attack, generating additional lift on it in the process. The resulting yawing moment for this aircraft (assuming a clockwise propeller rotation) tends to move the nose to the left. It can be suppressed by a rudder deflected trailing edge right (step on the right rudder), or by a cambered airfoil (camber on the left side), or by a small angle-of-incidence adjustment (leading edge left). All of these suppress the yaw by generating an opposing lift on the vertical tail. The magnitude of the yawing moment depends on engine power, RPM, and the airspeed of the aircraft.

14.2.3 Propeller Normal and Side Force

14.2.4 Asymmetric Yaw Effect

Another effect can be understood from Figure 14-17. The lower image shows the lift distribution over the propeller disc is highly asymmetric. Consequently, the right area of the disc generates more thrust than the left one. This effect generates a couple that tends to turn the nose to the left (again assuming a clockwise propeller). The pilot must step on the right rudder to suppress it. The magnitude of this couple depends on the engine power, RPM, and forward airspeed. On the other hand, the suppressing moment, which is generated by a rudder deflection, is dependent on the airspeed of the aircraft. And when the airspeed is low, the suppressing moment is low. However, the asymmetric moment may not necessarily be as low since it is highly dependent on the radial velocity of the propeller, which depends on the RPM. This highlights a potential critical flight condition for the aircraft – high power, low airspeed, high AOA, which is typical for an initial climb after take-off or during a balked landing maneuver. It is imperative the aircraft designer is aware of this effect and sizes the vertical tail to handle it.

14.2.5 Asymmetric Yaw Effect for a Twin-Engine Aircraft

The asymmetric yaw effect has serious implications for multiengine propeller aircraft. To explain this effect, consider the twin-engine aircraft shown in Figure 14-19. The left image shows the aircraft with both engines operating normally and both propellers rotating clockwise as seen from behind (note that everything said here is reversed for propellers that rotate counter-clockwise).⁹ As is unavoidable, both propellers are subject to asymmetric thrust loading that places the center of thrust (denoted by T) to the right of the rotation axis. The thrust of the left engine acts over a distance d_L and the thrust of the right engine acts over a distance d_R, which, as can be seen, is larger than d_L. Therefore, a relatively small moment M is generated about the CG and this must be reacted by introducing a small rudder deflection, δ_{r trim}. For the clockwise propeller rotation shown, the moment M would result in a nose-left tendency. And while relatively small, the pilot will detect it by observing that the “ball” on the Turn and Bank indicator swings to the right of center. The pilot will respond by pressing the right rudder pedal¹⁰ or, more appropriately, by trimming the airplane nose-right using the rudder trim in an attempt to center the Turn and Bank indicator ball. Some aircraft feature a yaw-string attached in front of the windscreen and this provides the pilot with additional help by allowing him to visually assess the severity of the yaw. Note that the drag of each nacelle is denoted by D_NAC, and both act over a distance d_NAC.

Now consider the right image in Figure 14-19, which shows a scenario in which the left (critical) engine has failed. This situation is referred to as flying with one engine inoperative (OEI) and can present a very serious situation to the pilot. An inspection of Figure 14-19 shows that a failure of the left engine results in a substantially greater moment than if the right engine fails. For this reason, the left engine is called the critical engine. In other words, an inoperative left engine presents a greater problem to the pilot than a failed right engine. The asymmetric thrust will cause a powerful moment, M, that forces the aircraft to yaw toward the dead engine. This may be compounded by the fact that the propeller may now be windmilling with the associated increase in drag (ΔD_windmill) of the dead engine. The drag may be reduced by feathering the propeller (see Section 14.1.3, Important nomenclature), which is a mechanical feature offered on most twin-engine aircraft.

The standard procedure in this situation is for the pilot to eliminate the yaw by stepping on the rudder pedal on the same side as the functional engine. This is followed by banking the airplane between 2° and 5° toward the functional engine (see Figure 14-20). Once this has been established, the pilot will “feather” the dead engine.

Let’s consider these actions in more detail. Pilots are trained to identify which side the dead engine is by stepping on the rudder pedals. It takes considerably less force to press the pedal on the same side as the dead engine compared to the one on the side of the functional engine.¹¹ This is caused by the floating tendency of the rudder, but without control inputs the condition causes the airplane to yaw (and bank) toward the dead engine. For the aircraft of Figure 14-19, before the pilots makes corrective control inputs, it will yaw so its right side is windward. The rudder will thus float trailing edge left and this is why the left rudder pedal feels dead.

Banking toward the functional engine establishes a turning tendency that reduces the rudder deflection required to maintain heading. Increasing banking may even eliminate the need for rudder input, but usually at the cost of the already impaired performance. There is a limit to how much banking should be applied. If the bank angle is too great the rate-of-climb could easily be detrimentally affected and could prevent the aircraft from maintaining altitude. As a rule of thumb, in order to achieve the maximum rate-of-climb in the OEI configuration (V_YSE) a bank angle of 2° is recommended if the non-critical engine fails, and 3° if the critical engine fails. A bank angle in excess of 5° almost certainly results in rate-of-descent (loss of altitude) for underpowered twins.

Another situation may confront the pilot operating a twin in an OEI configuration. If the airplane is slowed down below a certain airspeed, control authority will be lost and the rudder will be incapable of opposing the yaw. This airspeed is called minimum control airspeed, denoted by V_MC. If the airspeed falls below this airspeed, the aircraft will roll upside down, possibly causing an inverted spin, if not impacting the ground first. At any rate, it is a likely fatal scenario for all the occupants. For this reason it is imperative the pilot maintains high enough airspeed at all times, and slows down only after reducing power and pointing the nose down. The minimum control airspeed (V_MC) is usually higher than the airplane’s stalling speed (V_S) and rotating to take-off should not be performed until after it is reached and exceeded.

This scenario is remedied by installing a right engine/propeller combination that rotates counter-clockwise (or opposite to that of the left engine). Figure 14-21 shows such a configuration. Interestingly, an examination of aviation history reveals this is in fact rarely done. This is called a counter-rotating configuration.¹² Albeit safer, the primary drawback of that configuration is that it is more expensive to produce, as logistically it must feature two engines and propellers that are dissimilar because they rotate in opposite directions. An operation of such an aircraft results in greater expenses as engine parts are no longer interchangeable. This can pose a peculiar situation for an operator, which might have parts for the left engine on hand, while none are in stock for a failed right engine. Such scenarios should not be overlooked when selecting an engine configuration as this could severely impact the finances of an operator. Instead, the designer should size the control system and surfaces to handle the critical engine. For this reason it is imperative the aircraft designer is fully aware of this condition.

It is of interest to note that the Wright Flyer featured two counter-rotating propellers, although this was due to the nature of the single engine drive train. It is almost certain that the notion of asymmetric thrust would not have entered the minds of the pioneers of aviation at the time. Among other aircraft that feature this option are the De Havilland DH-103 Hornet, which was a derivative of the DH-98 Mosquito. The Lockheed P-38 Lightning featured counter-rotating propellers, although, surprisingly, both propellers rotated inversely of what is shown in Figure 14-21. This suggests that a cancellation of propeller torque was driving that design decision and not asymmetric yaw. An identical arrangement is found on the Heinkel He-177 Greif. Among other aircraft is a series of Piper aircraft, in order of models, the Piper PA-31 Navajo Chieftain, PA-34 Seneca, PA-39 Twin Commanche, the cancelled PA-40 Arapaho, and the PA-44 Seminole. Other aircraft include the Cessna T303 Crusader, Beech Model 76 Duchess, and Diamond DA-42 Twinstar, all of which feature counter-rotating piston engines. All of these are designed with the correct propeller rotation shown in Figure 14-21.

It is interesting to note that large airplanes typically don’t feature propellers that rotate in opposite direction. Even large aircraft, such as the four-engine Lockheed P-3C Orion and C-130 Hercules, feature propellers that all rotate in a clockwise direction. This is an example of a design decision that focuses on reducing maintenance costs in lieu of safety. Of course, for the above aircraft, an easy counter-argument is to be made that safety is in fact not compromised, but maintained by requiring rigorous pilot training, as well as providing hydraulically actuated control surfaces, and very powerful engines – an argument that is hard to dispute. Many larger aircraft provide a rudder boost system; a pneumatically powered actuators that force the rudder to the proper deflection angle to counter the asymmetry [4]. Such systems detect which engine has failed and react accordingly – providing substantial safety to the operation of the aircraft. In addition, the operational history of these aircraft supports this notion – there are not many accidents in current times that can be attributed to asymmetric thrust on larger propeller-powered aircraft. However, for small twins this is not the case, as will now be demonstrated.

It is of great importance that the designer understands the implications of selecting a twin-engine configuration, in particular if the engine power is relatively low. In addition to increased acquisition and maintenance cost, the selection has important safety and performance implications associated with it. From a development standpoint, such configurations generate more drag than a single-engine aircraft of the same power (assuming a conventional on-wing engine layout). This results from the presence of two nacelles on the wing in addition to the fuselage. This impact on drag can be reduced by a twin-engine configuration such as that of the Cessna 337 Skymaster and Adam A500 aircraft. However, what is more serious is the performance degradation that results if one engine fails. If a spacious cabin and high cruising speed are desired, a solution such as is offered by recent small GA aircraft should be considered. Aircraft such as the French SOCATA TBM-700 and TBM-850 and the Swiss Pilatus PC-12 are powered by a single Pratt & Whitney PT-6 turbine engine and are capable of cruising speeds in excess of 300 KTAS at 30,000 ft. This beats any of the above twin-engine piston aircraft (excluding the P-38).

To discuss the effect of engine power on the performance of a generic twin-engine piston aircraft consider Figure 14-22. The aircraft portrayed is in the small Piper Apache size class of aircraft. The figure shows that in normal operation, the aircraft climbs 1478 fpm at S-L and offers a maximum speed of 173 KCAS. Upon losing an engine the best rate of climb will drop to 239 fpm and its maximum speed to 97 KCAS. The analysis assumes a drag degradation associated with maintaining a straight heading with OEI by adding 50 drag counts (ΔC_{D OEI}) to the original minimum drag. This is reasonable based on the airplane being flown cross-controlled in a yawed configuration. The problem with this turn of events is that the low rate of climb does not allow a lot of room for mistakes. In light of the low-power sensitivity, an inexperienced or frantic pilot may easily maneuver the airplane off the peak conditions, causing it to lose altitude.

14.2.6 Blockage Effects

There are two kinds of blockage effect. First is a body placed in the streamtube ahead of the propeller disc (a pusher configuration). The other is a body placed in the streamtube behind the disc (tractor configuration). An idealized streamtube going through a propeller disc is shown in Figure 14-46. Placing an obstruction such as a fuselage (or a nacelle) into the streamtube will distort it and prevent the formation of a proper vena contracta. This will reduce the acceleration of the flow in the streamtube and, thus, the resulting thrust. Figure 14-23 shows an ideal (“Unblocked”) streamtube superimposed on what the actual (“Blocked”) streamtube might look like.

An estimation of the impact of blockage effects requires the flow around the obstruction to be incorporated into the wake model (for instance, see the wake of Section 14.5, Rankine-Froude momentum theory). The most reliable approach (excluding empirical methods) is to use the method of superposition depicted in Figure 14-24. A version of this approach is detailed in NACA TM-492, Influence of Fuselage on Propeller Design, and NASA CR-4199, An Analysis for High Speed Propeller-Nacelle Aerodynamic Performance Prediction. The mathematics of these methods are beyond the scope of this book. The manufacturers of propellers do consider blockage effects in the design of their propellers, although the methods used are usually proprietary. The aircraft designer can help mitigate blockage problems by considering a hub extension for piston-engine aircraft and select the largest spinner possible to cover the inboard and least efficient portion of the blade. This allows the cowling or nacelle to be sculpted in a manner that provides less obstruction to the flow. Ideally, the cowling or nacelle should be designed to be as axi-symmetric as possible, as this will yield higher installed propeller efficiency.

14.2.7 Hub and Tip Effects

The presence of the hub (root) and the tip of the blade will have a large effect on the distribution of section lift coefficients, as this is where lift must necessarily go to zero. The blade of a wind-turbine is shown in Figure 14-25 (hub is omitted from the view), and shows how the lift diminishes near the root (left) and tip (right). The effect is less noticeable near the root, because the radial speed of the air is less there. However, the tip shows how the section lift coefficients drop rapidly. Naturally, the twist of the blade and camber of the airfoil will influence this as well, but the figure illustrates that the phenomenon will affect thrust. For this reason, hub and tip effects must be taken into account when analyzing the thrust of the propeller. The so-called Prandtl’s tip and hub loss correction is presented in Section 14.6.4, Step-by-step: Prandtl’s tip and hub loss corrections, where it is used to correct analysis made by the blade element method. The correction method uses the geometric features of the propeller to approximate the reduction of the section lift coefficients and is relatively simple to apply.

14.2.8 Effects of High Tip Speed

The primary effect of a high tip speed is the risk of the formation of shock waves on the blade, which leads to a reduction in the propeller efficiency, increase in torque (requires more power to turn it), and an increase in the noise emitted by the propeller. The pitch, thickness, and curvature of the blade airfoil accelerate air well beyond what the tip speed might indicate. These effects are best avoided by selecting rotational tip speeds in the M ≈ 0.6 range for wooden propellers and in the M ≈ 0.75–0.8 range for metal and composite propellers.

14.2.9 Skewed Wake Effects – A·q Loads

A skewed wake is one for which the streamtube inflow through the propeller disc is not normal to its plane (see Figure 14-26). Such a wake is encountered any time a propeller-powered aircraft climbs or yaws. In general, there is no significant effect on the propeller performance, and usually the effect is most severe during transient maneuvers. However, the skewed wake has important effects on noise and propeller loads, especially for turboprops. Let’s consider these effects in some detail.

Consider the airplane in Figure 14-26, which presents a condition in which the propeller thrust line is not aligned with the flight path. Consequently, the streamtube is at an angle A to the thrust line (or normal to the propeller disc plane). As the propeller rotates at an angular speed Ω, each down-turning blade will encounter an increase in its AOA and each up-turning blade will see a reduction in its AOA, as explained in the discussion about the normal force. This will load up each blade and subject it to cyclic loading at the frequency described by Equation (14-8). This is commonly referred to as 1P cyclic loading, where 1P stands for once per revolution. A byproduct of this is increased noise.

As stated above, the phenomena increases the propeller’s 1P loads, which is the main source of loads for a turboprop. These loads are directly proportional to the product of the angle A and the dynamic pressure of the free-stream, q. For this reason they are often referred to as “A·q” loads. For piston-propeller configurations, the primary load of the propeller is caused by the sudden acceleration resulting from the shock of the cylinders firing and the subsequent torsional deformation of the crankshaft. These loads are much greater than the A·q loads.

However, for turboprops (and for electric aircraft) the A·q loads are critical. For this reason, it is imperative the designer of turboprop and electric aircraft performs careful mission performance analyses to determine the AOA of the vehicle at a representative cruise condition and then align the thrustline so it is mostly parallel to the flight path (the inflow angle A = 0°). Clearly, this cannot be achieved over the entire airspeed range, so propeller manufacturers recommend a “balanced” approach. That is, the product A·q is determined at the maximum and minimum operating weights (see Figure 14-27). Then, the thrustline is aligned for zero inflow angle between the two minimums, guaranteeing the least A·q loads possible. Of course low load is desirable: the less the load, the lighter the propeller. Failure to follow these guidelines may result in the design failing article 14 CFR 23.907 Propeller vibration and fatigue, which may cause a major redesign effort, with the associated development costs and program delays.

14.2.10 Propeller Noise

Noise associated with the operation of aircraft remains a topic of (often) hotly contested debate in society. One consequence of this debate has been a substantial reduction in aircraft noise over the last 25 years or so, in particular for jet aircraft. This has been achieved by a combination of new regulatory policies and increased research effort. Noise pollution generated by propellers has also been brought to more tolerable levels, although many aircraft types, primarily high-performance piston aircraft, still feature engines that operate at high RPMs at full power, resulting in objectionable noise pollution. Like all other aspects of aircraft design, noise level is a tradeoff with cost, weight, performance and other parameters. What remains is that noise pollution from propellers can be significant and should be seriously considered by the designer from the start of the design project. It is best to work with a propeller manufacturer and use their expertise to design and select propellers that are both quiet and efficient. Current manufacturers of propellers have responded with propeller blade designs that feature airfoils of lower thickness-to-chord ratio and swept planform geometry near the tip. An example of such a design is the scimitar-shaped propeller in Figure 14-28. Such blades are intended to reduce the magnitude of low-pressure peaks and delay the onset of normal shocks that increases the noise. Additionally, the trend has been toward an increased number of blades, as this allows the propeller diameter to be reduced, which, in turn, reduces the tip airspeed. Engine operating RPMs have also been reduced.

Propeller aircraft must comply with the requirements of 14 CFR Part 36 Appendix G [5]. Among a number of stipulations, this requires the aircraft to fly over a microphone placed some 8200 ft (2500 m) from brake release and the noise to be measured. The measured noise level must not exceed that shown in Figure 14-29.

The aircraft designer should not forget that two aircraft that use exactly the same engine and propeller combination can have two different measured noise levels. All other things being equal, the aircraft with the larger wing area and wing span will lift off using a shorter ground run, and it will climb faster at a lower airspeed (V_Y; see Chapter 14, Performance - climb), resulting in a steeper climb gradient. In short, it will climb to a higher altitude in the given distance of 8200 ft, resulting in a lower measured noise level. If there is a concern the aircraft may not or marginally comply with 14 CFR Part 36, the designer should perform a sensitivity study of wing geometry and noise levels.

Reference [6] presents a simple method to theoretically predict the near-field and far-field noise of propellers.

14.3.1 Tips for Selecting a Suitable Propeller

Although the aircraft designer should benefit from the experience of the propeller manufacturer, he should also be armed with a number of parameters. Among those are:

When considering thrust per number of blades, the fewer the blades the more efficient is the propeller. However, this should not be taken to mean that a prop with a large number of blades can’t have propeller efficiency equal to or higher than a comparable two-bladed propeller.

14.3.2 Rapid Estimation of Required Prop Diameter

The conceptual design of propeller-powered aircraft often requires the prop diameter to be quickly estimated. This is crucial in determining various characteristics, ranging from thrust for performance analysis to propeller noise and geometric constraints, such as the likelihood of a ground-strike. This section considers a couple of such rapid estimation methods. Stinton [3] presents a method to calculate the diameter of wooden propellers in inches, based on the intended cruising speed in KTAS, engine power, and RPM.

14.3.3 Rapid Estimation of Required Propeller Pitch

For a fixed-pitched propeller, the required pitch can be determined from Tables 14-3, 14-4, and 14-5. The first table represents the ideal, but physically impossible propeller with an efficiency of 100%, while the other two show pitch for 90% and 80% efficient respectively. The table is assembled using Equation (14-12) and is read as shown by the shaded cells. A 100% efficient propeller rotating at 2700 RPM must feature a 56 inch pitch in order to achieve 120 KTAS. The tables are intended to give the designer an idea of the propeller pitch required.

14.3.4 Estimation of Required Propeller Efficiency

Once a trustworthy drag model of an aircraft has been prepared, it is appropriate to evaluate the type of propeller that will be required based on its required propeller efficiency. This section shows how to do this in an effective manner. This involves the preparation of airspeed-thrust (see Figure 14-30) or airspeed-power (see Figure 14-31) maps. It does not matter which one is prepared, as long as one of them is. The maps show thrust and power in terms of propeller efficiency isobars. Then, important desirable performance characteristics can be added on a need-to basis. The map shows at a glance the propeller efficiency required to achieve those characteristics. It is easiest to explain such maps using an example. Here the SR22 is the obvious choice.

First consider Figure 14-30. The isobars are plotted using Equation (14-38) for constant values of η_p. The static thrust was calculated using Equation (14-63). Then the drag force is plotted based on the available drag model. The drag model for the SR22 was extracted using POH data in Example 15-18. It is then used to plot the drag force curve shown in the graph. The ROC curve is based on solving Equation (18-17) for thrust required to achieve a given ROC:

Here 1300 fpm was selected, as this is close to the capability of the SR22 at gross weight at S-L at V = 91 KCAS (recall that at S-L KCAS and KTAS are equal). This way, it is possible to explore the capability of its propeller. The graph of Figure 14-30 shows that a prop efficiency of around 0.85 is required to achieve a top speed of 186 KTAS at S-L. Similarly, in order to achieve the ROC at V_Y, the prop efficiency must be around 0.64. This sort of information can be very useful to the propeller manufacturer when helping to select or design a practical propeller.

Figure 14-31 is identical to Figure 14-30, except it uses power rather than force to relay the same information. Therefore, the propeller efficiency isobars are horizontal lines, which may make it a tad easier to evaluate the propeller efficiencies. Other than that, it gives exactly the same answers as the airspeed-thrust map.

14.3.5 Advance Ratio

The advance ratio is a measure of how far the propeller travels in unit time in terms of its diameter. The distance the prop travels per second is the vehicle’s speed through the medium and is given by V₀. For instance, if we measure an advance ratio of 0.5 for a 10 ft diameter propeller rotating at 600 RPM it means the propeller (i.e. the vehicle it is propelling) is moving at a forward speed of 50 ft/s.

14.3.6 Definition of Activity Factor

The activity factor (AF) represents the power absorption capability of all blade elements. Small aircraft have an AF ranging from 80-105, while larger aircraft have an AF around 130-170. The AF is used when reading propeller efficiency maps, which are special contour maps prepared by the propeller manufacturer that allow propeller efficiency to be read as a function of the advance ratio and power coefficient (see Section 14.3.7, Definition of power- and thrust-related coefficients):

where

See the list of variables at the end of this chapter for other variable details.

14.3.7 Definition of Power- and Thrust-Related Coefficients

The following concepts are definitions:

By combining Equations (14-26) and (14-28) we get:

See the list of variables at the end of this chapter for variable details.

14.3.8 Effect of Number of Blades on Power

The number of blades is fundamental to how the engine power is converted into propulsive power. Replacing propellers is a common occurrence in the aviation industry. Sometimes such changes involve a propeller that has a different number of blades than the original prop. For instance, consider a four-bladed propeller as a replacement for a two-bladed one. Assuming such a modification only involves a change in the number of blades, the addition will increase the torque required to swing the new propeller by a factor of two. This can be seen, for instance, in the formulation developed in Section 14.6, Blade element theory. However, since this propeller will be mounted on the same engine, the power available per blade will diminish. This will manifest itself in the engine being unable to rotate the propeller at the original RPM. With everything else being equal (e.g. propeller diameter and blade planform geometry being the same) less thrust will be generated and the performance of the airplane will suffer. The remedy is to reduce the torque generated per blade, as this will allow the propeller to be rotated at RPM closer to the original one. The easiest way to accomplish this is to reduce the diameter of the propeller. An advantage of such a modification is less propeller noise and weight. Also, the ground clearance of the prop will increase (assuming the number of blades is increased), rendering this an important option at the discretion of the designer struggling with potential prop strike issues.

It is of interest to estimate what propeller diameter is appropriate when changing the number of blades, regardless of whether the intent is to increase or reduce the number of blades. This can be done by solving the following expression for the new diameter, D_new, which is the diameter of the replacement prop:

(14-30)

where

The expression requires an iterative scheme for the solution. As an example, consider a 78 inch diameter two-bladed propeller used with some versions of the Cirrus SR22. Assume this prop is to be replaced with a three- or four-bladed propeller, whose blade areas are 90% of the original ones. Solving for D_new using the above method suggests prop diameters that are 90% and 82% of the original diameter, respectively. The recommended propeller diameter ratios are plotted in the two graphs shown in Figure 14-32. First consider the left graph, which shows how the ratio varies with the intended cruising speed, V₀. It shows that the dependency on the forward speed of the airplane is indeed negligible. The right graphs shows dependency on the blade area ratio and depicts how it, on the other hand, plays a fundamental role in the diameter requirements. Thus, a three-bladed propeller with a blade area ratio of 1.0 should have a diameter about 87% of the original one. A four-bladed propeller should be about 78% of the original diameter. The conclusion is that Equation (14-30) can be easily simplified by ignoring the contribution of V₀, yielding:

(14-31)

This expression allows simple arithmetic to take place with acceptable accuracy, as can be seen in Figure 14-32.

Finally, consider the special case in which the propeller chord, c, remains constant. In other words, the blades are reduced in length only. In this case it is possible to relate the blade area ratio to the diameter as shown below:

In this case, Equation (14-30) reduces to (note the change in power of the diameters):

(14-32)

The reader should not rely blindly on this analysis method. It is only presented here to give an idea as to how much the prop diameter could change. Propeller replacement should always involve the manufacturer as other details, not included here, may play an important role. Among these are planform shape, modified airfoils, and others. For instance, the TCDS for the Cirrus SR22 [8] reveals it can feature a 78 inch two-bladed or a 76 inch three-bladed prop, a diameter ratio closer to 0.974. An inspection of the propeller blades reveals a dissimilar planform shape.

14.3.9 Propulsive Efficiency

The power generated by an engine is not completely converted to a thrust force. It is inevitable that some losses occur in the process due to an imperfect conversion process. Consider an aircraft flying at an airspeed V generating thrust T by giving air a backward velocity w. This changes the kinetic energy of the affected air by an amount ½mw² per unit time¹⁶ and this change must equal the force acting on the mass of air, times its speed, T·w. The term T·w is the power consumed (or wasted) in driving the air backwards. The Froude efficiency is defined as the ratio of the power required to propel an object (useful power) to the sum of this power and that which is lost in the conversion:

The Froude efficiency represents inviscid losses that result from accelerating the propwash. It is also called the ideal efficiency (see Section 14.5.2, Ideal efficiency). However, there are also losses due to the viscous drag of the propeller blades. These losses further increase the inefficiency of the power-to-thrust conversion process. This effect is called viscous profile efficiency and is denoted by the term η_v. We now define the total propeller efficiency, η_p, as the product of the Froude and profile efficiency as follows:

A typical maximum value for η_v is about 0.8 to 0.9.

The propeller efficiency can also be defined as the fraction of engine power that gets converted into propulsive power (thrust × airspeed) to the total engine output power, P. A common expression of this ratio is:

14.3.10 Moment of Inertia of the Propeller

As has been showed before, the moment of inertia of the propeller is of interest to the aircraft designer. This can be estimated using the fundamental definition for the moment of inertia of a body about a given axis:

(14-36)

The term ρ·dV is the mass of an infinitesimal volume of the propeller blade, located a distance r from the reference axis (i.e. the axis of rotation of the propeller). While the planform of the propeller blade does not always yield an elegant closed-form solution of Equation (14-36), a reasonably accurate estimate of its moment of inertia can be made by idealizing each blade as a thin rod of a uniform cross-sectional area A (see Figure 14-35). If the radius of the prop is R, it weight is W_prop, and the number of blades is N_B, then its moment of inertia can be estimated from:

(14-37)

where g is the acceleration due to gravity.

14.4.1 Converting Piston BHP to Thrust

The performance analysis of piston engine aircraft requires brake horsepower (BHP) to be converted into thrust.¹⁸ In the UK system, the thrust is calculated using airspeed (in ft/s), engine power (in BHP), P_BHP, and propeller efficiency (dimensionless), by solving Equation (14-23) for the thrust as follows:

(14-38)

The factor 550 converts the BHP into ft lb_f/s. In the SI system the conversion is not necessary as power is given in kW. The expression indicates that as the airspeed V approaches zero the thrust will trend toward infinity. This is physically impossible. What happens in reality can be understood by noting that the magnitude of the velocity through the propeller disc requires a constant flow of air toward it. This, in turn, means V is never zero. Thrust at zero forward airspeed can be estimated using the momentum theory (Section 14.5) or blade element theory (Section 14.6).

14.4.2 Propeller Thrust at Low Airspeeds

This section presents two methods to estimate propeller thrust at low airspeeds, which is essential for the prediction of low-speed operation such as that of T-O performance. Equation (14-38) is extremely useful in the performance analysis of aircraft. However, it presents an important difficulty to the user: it requires the propeller efficiency, η_p, to be known, but, being a function of the airplane’s airspeed, propeller geometry, and RPM, it is a hard number to estimate. Therefore, a more accurate way to write Equation (14-38) is shown below:

(14-39)

where H is the altitude, D_p is the propeller diameter, RPM is the engine speed in revolutions per minute, MAP is the manifold pressure, and OAT is the outside air temperature. Naturally, this is far more cumbersome to write, which explains the short form shown in Equation (14-38).

The notion that η_p can be treated as a constant leads to a serious error in thrust calculations at low airspeeds because V is the denominator and it is becoming smaller, so the thrust grows unrealistically large. In performance analyses, low-speed performance cannot be ignored – for instance, it is needed to accurately predict take-off performance.

14.4.3 Step-by-step: Determining Thrust Using a Propeller Efficiency Table

Sometimes the propeller manufacturer provides the designer with a propeller efficiency table like the one shown in Figure 14-43. In it the propeller efficiency can be read based on the computed value of the advance ratio (left-most column) and power coefficient (top row). Using such a table presents a very convenient way to extract the propeller efficiency for use with Equation (14-38). To do this the user must calculate the power coefficient, C_P (defined in Section 14.3.7, Definition of power- and thrust-related coefficients), and advance ratio, J (defined in Section 14.3.5, Advance ratio), and use these to extract the prop efficiency. The sample table below has the advance ratio in rows, ranging from 0.20 to 1.20, and the C_P in columns, ranging from 0.02 to 0.13. Values of J or C_P that fall between the tabulated values must be extracted by double-interpolation.

Using such a table, the method of extracting thrust is as follows:

14.4.4 Estimating Thrust From Manufacturer’s Data

As said earlier, it is always wise for the aircraft designer to establish a good relationship with propeller manufacturers by committing to using their products. Such relationships may (1) encourage the propeller manufacturer to provide the designer with useful propeller performance information, and (2) if the business case is promising enough, even persuade the manufacturer to design a propeller to match the designer’s desired characteristics.

An example of typical data provided by a propeller manufacturer is shown in Figure 14-44. The data is for a two-bladed, 69 inch diameter propeller designed for an LSA aircraft. These propeller characteristics are plotted in Figure 14-45, with the appropriate curve fits. So, now we have the properties of the propeller formulation as a function of advance ratio.

14.4.5 Other Analytical Methods

Other analytical methods that allow one to extract thrust and other properties from propellers are extensive enough to be presented in their own sections. These are the Rankine-Froude momentum theory (Section 14.5) and Blade element theory (Section 14.6).

(1) The propeller is replaced by an infinitesimally thin actuator disc that offers no resistance to air passing through it.

(2) The disc is uniformly loaded and, therefore, experiences uniform flow passing through the actuator disc. Consequently, it is assumed to impart uniform acceleration to the air passing through it.

(3) A control volume surrounds the streamtube and sharply separates the flow going through it from the surrounding air.

(4) Flow outside the streamtube has constant stagnation pressure (which means no work is imparted on it).

(5) In the far-field, in front and behind the disc, streamlines are parallel and, there, the pressure inside the streamtube is equal to the far-field pressure.

(6) The propeller does not impart rotation to the flow.

(7) The theory assumes inviscid (no drag, no momentum diffusion) and incompressible flow.

14.5.1 Formulation

Consider Figure 14-46, which shows an idealized flow model for the actuator disc. The model consists of the four planes; , , , and . From Assumption (5), Plane is far enough ahead of the actuator disc for the streamlines to be parallel. Plane is far enough behind the disc for the streamlines to have become parallel again. Consequently, the static pressure at either plane is constant and equal to the far-field pressure P₀.

The front face of the control volume has surface area S. The flux of fluid entering through Plane is Q₀ = S·V₀. The flux exiting through Plane is Q₃ = (S − A₃)·V₀ + A₃·V₃. The change in flux between the front and aft surfaces is thus given by:

The non-zero value of the flux implies fluid must be entering the control volume from the sides. Now in order to determine the thrust it is necessary to apply the momentum theorem, repeated here in its conventional form;

(C-19)

where n = normal vector to the cross-sectional area

Note that the momentum flux of the fluid entering Plane is given by (the negative sign indicates it is entering the volume). Similarly, the momentum flux of the fluid entering Plane is given by (the positive sign indicates it is leaving the volume). Finally, the momentum flux of the fluid entering the sides is given by (the negative sign indicates it is entering the volume). This generates a force that points in the direction of the incoming flow (here renaming F as T for thrust):

Inserting our previous result for ΔQ leads to:

(14-51)

Considering Figure 14-46 it can be seen that T can also be computed from:

(14-52)

where

The changes in pressure and airspeed are depicted in Figure 14-47. It can be seen that the propeller energy is added as a rise in stagnation pressure. Its magnitude depends on the geometry of the propeller and the power delivered by the power plant.

We can calculate P₁ and P₂ using Bernoulli’s principle assuming streamlines exist up to the actuator disc and immediately behind it. We must note this principle is not applicable through the actuator itself, although it is reasonable to assume the velocity is continuous through the disc. Thus, we can determine P₁ using the flow conditions at Plane , and P₂ using the flow conditions at Plane as follows:

From which we find the pressure difference by subtracting Equation (i) from (ii).

The above equation is recognized as Froude’s Theorem [10] and is essential in propeller design. It states that the airspeed through the propeller is the average of the far-field airspeed (V₀) far ahead of it and the airspeed in the streamtube far behind (V₃). Let’s now define the following differences:

(viii)

where w is called the propeller-induced velocity. This allows V₂ to be redefined as follows:

(ix)

and V₃ as:

(x)

Inserting Equations (ix) and (x) into Equation (14-51) results in:

The power for this system is given by P = T·V or;

(14-56)

The above result shows the power required by the propeller is a combination of two terms; useful power and induced power:

We can determine the induced velocity, w, through the propeller disc from Equation (14-55):

This is a quadratic equation in terms of w and can be solved using Equation (E-12):

Simplifying and recognizing the negative sign in front of the radical represents a non-physical condition, we write this as:

(14-57)

Using the above analysis, we can consider two important cases; ideal efficiency and maximum static thrust.

14.5.2 Ideal Efficiency

Ideal efficiency can be defined as the ratio of useful power (Equation (xi)) to the total power per Equation (14-56):

(14-58)

We can introduce the fraction w/V₀ to Equation (14-57), yielding:

(xv)

where C_T is the coefficient of thrust (not to be confused with the thrust coefficient of Equation (14-27)) as follows:

(14-59)

Inserting this into Equation (xiv) leads to:

(14-60)

14.5.3 Maximum Static Thrust

Consider the special case when V₀ = 0. In this case, Equation (14-57) the velocity through the propeller reduces to:

(14-61)

Similarly, the power from Equation (14-56) becomes:

(14-62)

This yields the following expression for static thrust, T_STATIC:

(14-63)

As a rule of thumb, Equation (14-63) overestimates static thrust by some 15–20% for various reasons, such as blockage effects, the presence of the hub, which reduces the disk area, reduction of lift distribution near the tip and hub, and so on. Therefore, for design purposes, for typical aircraft, estimate T_STATIC by the following empirical correction:

(14-64)

14.5.4 Computer code: Estimation of Propeller Efficiency Using the Momentum Theorem

Propeller efficiency is required to predict the performance of propeller-powered aircraft. During the development of such aircraft, the designer must be able to predict with a reasonable level of accuracy the propeller efficiency. One way of doing this is to use the blade element theory (BET), which is presented in Section 14.6, Blade element theory. However, there are times, in particular during the early phases of development, during which it may simply seem too unwieldy. This is due to the fact that detailed information about the geometry of the propeller must be known in advance, but this may be impossible at that stage of development when perhaps the only thing known is that the airplane will be powered by a propeller. The following method, which uses the momentum theorem, is intended to remedy this problem. As such, it is far simpler to implement than the BET, although it is also best implemented using software such as a spreadsheet or mathematical script software like MATLAB. Here, a function written in Visual Basic for Applications (VBA), intended for use with Microsoft Excel, is presented.

The momentum theory can also be used to extract propeller efficiency from flight test data. To do this, details of the flight condition (airspeed, altitude, etc.) and power setting are recorded and reduced in the manner shown below. It must be kept in mind that initially, neither thrust, T, nor propeller efficiency, η_p, is known. The method requires the viscous profile efficiency, η_v (see Section 14.3.9, Propulsive efficiency) for the propeller to be known. Its value may be provided by the propeller manufacturer and typically ranges from 0.7 to 0.9, depending on propeller type (i.e. fixed or constant speed) and geometry. Sometimes it is given as a function of airspeed. From a certain point of view it can be considered the maximum value or ceiling of the propeller efficiency, η_p.

We begin the process by estimating token thrust using some initial value of the propeller efficiency, denoted by η_p. This is then used to estimate the induced airspeed in the propeller streamtube, which, in turn, is used to calculate the ideal propeller efficiency. This is then used to calculate the new propeller efficiency, η_pnew, as the product of the ideal and the viscous profile efficiencies. If the difference between the original and new values is larger than some selected level of accuracy (e.g. 0.0001) then the value of η_p is replaced with that of η_pnew and the process is repeated until the difference has reached the desired accuracy.

14.6.1 Formulation

Figure 14-50 shows a propeller of a radius R rotating with an angular velocity of Ω radians/sec. A representative blade element is shown whose chord is denoted by c(r) and width by dr. Its centerline is located at a distance r. A cross-sectional view of this blade element, detailing airspeed components and angles, is shown in Figure 14-51.

The following parameters are identified in Figure 14-51:

The differential lift and drag forces of the element, dL and dD, can now be written as follows:

(14-65)

(14-66)

where

The section lift coefficient of the element airfoil can also be written as follows:

(14-67)

where the helix angle is calculated from the relation:

(14-68)

14.6.2 Determination of α_i Using the Momentum Theory

Having demonstrated the importance of induced AOA, α_i, on propeller thrust and power (see Example 14-18), we will now develop a method to help determine the induced velocity, w. One way of accomplishing this is to use the actuator disc or momentum theory (see Section 14.5). Considering Figure 14-51 we see that α_i depends on w, which, in turn, depends on the blade loading. The method shown here is best implemented in a spreadsheet or a computer program. We will incorporate it into the spreadsheet just developed in the preceding examples.

Consider the propeller of Figure 14-61. A circular area (or annulus) of width dr located at blade station r is shown. The total area of the annulus is dA = 2πr·dr. The total thrust of the propeller using the Rankine-Froude momentum theory of Section 14.5 is given by Equation (14-29), repeated here for convenience:

(14-75)

where

Therefore, the thrust generated by the annulus according to the momentum theory is:

(i)

However, dT is also defined by the blade element theory using Equation (14-38). Using that expression and by combining Equations (14-34) for dL and (14-35) for dD and noting that dT is the product of the number of blades, N_B, we get:

Considering Figure 14-51 again shows that we can make the following approximations for V_E, w, and the angle ϕ+α_i:

Insert this into Equation (ii) to get:

It must be pointed out that the induced flow angle will reduce the value of the C_l and consequently change the magnitude of C_d. This, in turn, will change the loading on the blades and therefore affect the induced velocity.

Equating Equations (i) and (vii) leads to (rewrite c(r) = c for clarity):

Canceling terms and expanding:

The purpose of these manipulations is to ultimately retrieve the value of w at a given radial station r. Equation (viii) is best solved by an iterative scheme, such as Newton-Raphson (see Appendix E.6.19), but this also requires a specific function, call it f(w), and its derivative, i.e. f′(w).

Let’s define a function f(w) as follows:

(ix)

Rewriting it using the definition of V_E we get:

(14-76)

The derivative of f(w) is determined as follows:

Carrying the differentiation further:

(x)

Differentiate the right-most term using the chain rule (see Appendix E.6.2),

Differentiate the center term:

Completing differentiating the center term leads to:

Insert into Equation (x) and manipulate.

Finally, rewrite by using the effective resultant velocity, V_E:

(14-77)

Having defined f(w) and f′(w), we can now determine the root of f, i.e. the value of w that results in f(w) = 0. The value of w is the induced velocity at the blade element at blade station r. Using the Newton-Raphson iterative scheme (mentioned earlier) this would be implemented as follows.

14.6.3 Compressibility Corrections

The local velocity of the blade elements residing near the outboard part of a normally operating propeller is in the high subsonic range. Consequently, these elements should be corrected for compressibility effects. These effects call for corrections of the drag that differ from those for lift. The corrections are applied to the drag and lift coefficients.

14.6.4 Step-by-step: Prandtl’s Tip and Hub Loss Corrections [14]

The section lift coefficient at the tip of any lifting surface must necessarily go to zero. Unmodified BET ignores this effect and effectively assumes the blade element at the tip generates undiminished lift. This error was recognized by early scientists. Most prominent of those was Ludwig Prandtl (1875–1953), who developed a correction that is easily and conveniently incorporated into the BET. This correction method has been expanded to include corrections at the hub as well (which is another radial where the lift goes to zero). Applying the corrections is a three-step process.

14.6.5 Computer code: Determination of the Propeller Induced Velocity

The solution procedure here was implemented in Microsoft Excel. The program allows the iterative determination of w using the Newton-Raphson algorithm to be written in Visual Basic for Applications (VBA) and is shown below. The function F_of_w calculates the value of f(w) using a number of input variables, V₀, Ω, r, N_B, c(r), C_d, C_l, and w, using Equation (14-76).

The function InducedVelocity returns the final iterated w. Note that rather than calculating f′(w) using Equation (14-77) it uses a Taylor series finite difference scheme for the derivative (the variable FprimeofX0). It also counts the number of iterations and exits the do-loop if it takes more than 10 interations to converge. The code generally converges in fewer than 10 iterations. This function can be accessed from within Excel by a simple cell reference such as ‘=InducedVelocity(list of input cells)’.