Introduction

The rotary and radial types, once the workhorses of aviation, have given way to the “flat” horizontally opposed “boxer” engines. These reciprocating piston internal combustion engines, which drive a propeller, continue to be the powerhouses of modern general aviation. Their use in relatively low to moderate-performance airplane types is a testament to their relatively low cost, decent performance, and reliability.

An example of a Piper Cherokee airplane is shown in the photograph below, which has a 4-cylinder, 160 hp (119.3 kW) horizontally opposed “boxer” piston engine driving a fixed-pitch propeller. The advantages of this propulsion system are that it is reliable, affordable, and provides reasonable propulsive efficiency. However, the system is not readily scalable because the power-to-weight ratio of a piston engine decreases rapidly with increased power output, making it unattractive for larger and heavier aircraft; a turboprop engine will be preferred if a propeller is used in such cases.

Piston engines, including the higher efficiency afforded by types of diesel, may also be suitable for certain classes of drones and UAVs where the required flight range and/or endurance cannot be achieved using batteries alone, e.g., by using hybrid propulsion systems. One of the most common systems combines a gasoline (petrol) engine or a diesel engine with electric motors, which can be operated purely on diesel or electrical power but using both when needed. To this end, engineers must understand the basic principles of aircraft piston engine operation, including the effects of density altitude on their power output and specific fuel consumption, as well as the associated propeller performance under the conditions of flight.

Reciprocating Piston Engines

The first piston engine used for aircraft was a rotary engine, which saw extensive use up through the 1920s. In this type of arrangement, the cylinders are arranged in a radial configuration that “radiate” outward from a central crankshaft like the spokes of a wheel, as shown in the figure below. It also resembles a star when viewed from the front, so the type is sometimes called a star engine. In a rotary engine, the entire crankcase and its attached cylinders rotate around it as a unit along with the propeller. The pistons are connected to the crankshaft using a primary rod assembly, and the remaining pistons and their connecting rods are attached to rings around the primary rod. The air/fuel inlets are connected to a carburation system, and the exhaust outlets are connected to a radial manifold. The inlet and outlet valves are opened and closed using a cam. In some versions, a carburetor delivers the fuel/air mix into the crankcase, reaching the cylinders through transfer ports, so only exhaust valves are needed.

One advantage of the rotary engine design is good cooling because the cylinders rotate. However, a significant disadvantage is the large gyroscopic moments on the aircraft from the rotating engine mass. This characteristic led to several serious issues with the flight handling qualities of early aircraft, i.e., the ability to turn only in one direction.

The radial engine is another early reciprocating piston engine used to power many pre-1950 airplanes, as shown in the photograph below. The radial engine has fixed (non-rotating) cylinders, and the propeller is connected to the crankshaft. The inlets are connected to a carburation system, and the exhaust outlets are connected to a system of pipes. In the radial engine, the gyroscopic effects on the airplane are much lower than for the rotary style of the engine in which the cylinders rotate with the propeller. Consequently, radial engines were much preferred over the rotary type, which became obsolete for aircraft use after WW1.

Powerful radial engines with 12 or more cylinders have been built for aviation use, mainly for airplanes constructed through the 1950s, raising power output levels from 2,000 hp (1,491 kW) to 4,000 hp (2,983 kW) range. In addition, extra rows of radial cylinders can be added to increase the engine’s power. However, this approach rapidly drives up the engine’s weight and also results in cooling challenges for the downstream rows of cylinders. The most powerful piston aircraft engine ever built was the Lycoming XR-7755, which had 36 cylinders and a shaft power output of 5,000 hp (3,700 kW). However, with the advent of jet engines, which offered better performance and efficiency, the XR-7755 project was eventually discontinued, and no aircraft were ever powered by this engine in operational service.

Modern piston aero-engines for aircraft are usually flat and horizontally opposed, as shown in the figure below. The horizontally opposed engine configuration is also known as a “boxer” or “flat” engine. In this design, the cylinders are arranged in two banks on opposite sides of the crankshaft, forming a flat and wide configuration, which allows for a more streamlined and aerodynamic profile. This is important for minimizing drag and also allows for efficient air cooling. Several manufacturers, including Lycoming, Teledyne Continental, and Rotax, produce piston engines suitable for aircraft use. They are manufactured with 4, 6, or 8 cylinders with power ratings from about 125 hp (147 kW) to 600 hp (441 kW). They also come in normally aspirated and supercharged (i.e., forced induction) forms and carbureted or fuel-injected variants; fuel injection is used for higher-performance engines.

Superchargers and turbochargers are forced induction systems used in aircraft engines to increase the intake air pressure, allowing for better combustion and improved engine performance. Supercharging or turbocharging (or both) may be used to increase the power of a piston engine and maintain its power output to higher flight altitudes; this is a good solution for higher-performance aircraft despite the penalty of some extra engine weight and maintenance costs.

As shown in the figure below, superchargers are mechanically driven by the engine, typically through a belt connected to an accessory drive from the crankshaft. The engine’s exhaust gases drive turbochargers, consisting of a turbine and a compressor connected by a shaft. The exhaust gas spins the turbine, which, in turn, rotates the compressor and forces more air into the engine and, hence, into the fuel management system. Turbochargers are generally more efficient than superchargers because they harness waste energy from the exhaust gases. Superchargers and turbochargers have been used to maintain the power output from aero-engines. On some of the earlier high-powered radial engines, they were both combined in a system known as a “turbo-supercharger.”

While modern piston aero-engines are mechanically reliable and robust, one concern is that they are relatively heavy compared to their power output. Their power-to-weight ratios are relatively low, which are only about 0.2 hp/lb (0.33 kW/kg) to 0.4 hp/lb (0.66 kW/kg). Therefore, when such engines are required to produce higher power levels, they can become prohibitively heavy for use on an aircraft. This reason is why a turboshaft engine is usually necessary to drive a propeller (i.e., a turboprop) after a specific power requirement for flight is reached. A turboprop has a much better power-to-weight ratio of about 0.8 hp/lb (1.32 kW/kg) to 1.2 hp/lb (1.97 kW/kg). While turboshaft engines have a higher capital and maintenance cost per power unit, their better specific fuel consumption and reliability make them very attractive for larger propeller-driven aircraft.

The principle of operation of a reciprocating piston internal combustion engine is based on the Otto cycle, as shown in the figure below. The up and down movement of the piston is synchronized with the opening and closing of the two valves (intake and exhaust) by using a cam, which allows the sequential entry of the fuel/air mixture, followed by the compression and combustion process, and then the exit of the exhaust gases.

In summary, the operation of the engine consists of four cycles (or strokes), as shown in the animation below, namely:

1. In the intake stroke, the piston moves down the cylinder, and the cam opens the intake valve. A carburetor or fuel injection system draws the air and fuel mixture into the cylinder.
2. The compression stroke is where the intake and exhaust valves are shut. The upward-moving piston then compresses the fuel/air mixture to the point that it will support combustion.
3. The power stroke is where a spark plug ignites the compressed mixture. The resulting flame front and expanding gases (called deflagration) progressively force the piston downward in the cylinder to drive the crankshaft.
4. The exhaust stroke is where the exhaust valve opens, and the upward-moving piston forces the combustion products out from the cylinder before the entire four-stroke process starts again.

A piston engine may also work on the principle of the Diesel cycle, where the much higher compression in the cylinder raises the temperatures sufficiently to cause the fuel to burn without using a spark plug. Another advantage of a diesel engine is its better thermal efficiency and lower specific fuel consumption. In many countries, diesel fuel is also less expensive than gasoline.

Effects on Power

For a piston engine, the power from the engine to the crankshaft (the so-called shaft brake power) is given by

(1)  

where is the total displacement or swept volume, is the mean effective pressure (pressure in the cylinders), and rpm is the crankshaft revolutions per minute. The swept volume by the piston as it moves up and down inside the cylinder equals the displacement of one cylinder, so the total engine displacement is that value times the number of cylinders, e.g., 5.9 liters (5,900 cc) or 360 cubic inches.

Detailed dynamometer testing is typically conducted to assess an engine’s performance accurately, considering various parameters and conditions. Also, the specific configuration and characteristics of the engine, as well as any forced induction systems (like superchargers or turbochargers), can influence the overall power output. The name “brake” power comes from the fact that the power is measured using a brake type of dynamometer, which provides a resistance or braking torque at the engine shaft. Remember that a torque, , is the product of a force times a distance, so it has units of work. Power is the rate of doing work, so the power, , at the shaft is the product of the torque and angular velocity of the shaft, i.e., .

Power Limitations

It will be apparent from Eq. 1 that the power out from the engine can be increased by:

1. Increasing the swept volume, i.e., by increasing the cylinder bore, stroke, number of cylinders, or all of these things.
2. Increasing the pressure in the cylinder by the appropriate design of the combustion chamber and/or the piston shape or by turbocharging the air entering the cylinders.
3. Running the engine at a higher rpm. However, in part, rpm will be limited by keeping the propeller’s tip Mach number below the speed of sound.

There is a practical limit to all of these things, partly by the allowable mechanical stresses and temperatures in the engine to prevent failure. To a large extent, the design of a piston engine comes down to the selection of high-strength and/or high-temperature metals and appropriate metallurgy to give the engine good operational reliability and durability, especially for the exhaust valves. On aviation engines, which run at high power settings and high average temperatures, the exhaust valves are often sodium-filled, which improves thermal conduction away from the valve stems and seats and keeps the engine cooler. The maximum attainable rpm of the engine will also be limited by the propeller tip speed, which should be kept below the speed of sound (Mach 1) to maintain its propulsive efficiency and keep noise levels down. Propellers with high tip speeds that approach the speed of sound are also always very noisy.

Altitude Effects

As the flight altitude of the airplane changes, so does the engine power available, as shown in the figure below.  Lower air density affects the amount of oxygen available for combustion in the engine cylinders. Because the power output of an engine is directly related to the amount of oxygen available for combustion, a decrease in air density leads to reduced engine performance. Engines also rely on the pressure difference between the outside air and the combustion chamber to facilitate the intake of air and the expulsion of exhaust gases. Reduced atmospheric pressure at higher altitudes can impact the efficiency of these processes, affecting engine performance. While lower temperatures can be beneficial for engine cooling, they also influence combustion efficiency. Cooler air can result in better air-fuel mixture density, but it may also affect fuel vaporization, potentially impacting combustion efficiency. So, the power output decreases or lapses with increasing density altitude.

An approximation for the effects of density altitude on the power output of a normally aspirated (non-supercharged) piston engine is to assume that

(2)  

where is the power available at altitude and is the power available at mean sea level conditions. Remember that the density ratio, , of the air, which is a surrogate measure of the oxygen content, can be calculated using the ISA model from measurements of pressure altitude and outside air temperature at that altitude. An empirical correction for a normally aspirated engine that is often used in practice is

(3)  

Supercharging

It can also be seen that supercharging can maintain the rated power of a piston engine at much higher altitudes. This outcome is obtained because a supercharger increases the pressure and density of the air supplied to the engine intake, i.e., boosting the manifold pressure and the oxygen content of the inducted air. The consequence is that more fuel can be burned, thereby increasing the power available from the engine at lower altitudes as well as maintaining that power at higher altitudes.

There may be a further boost in engine power as the flight speed increases. The pressure of the air entering the engine will generally increase by an amount proportional to the dynamic pressure, i.e., . Therefore, the power from the engine increases somewhat from a ram air effect. Ram air can be significant for some aircraft, especially those flying at airspeeds above 250 kts. However, pressure losses in the ducting between the air intake and the engine tend to reduce the significance of this potentially beneficial effect. Nowadays, reciprocating engines are used for lower-speed general aviation aircraft, so the ram air effect can usually be ignored as far as it might affect the engine performance. From a design perspective, relying on ram air effects at any airspeed would be inadvisable when sizing an engine to an airframe.

Engine Performance Charts

The engine manufacturers provide detailed charts to allow engineers to calculate shaft power available at any combination of altitude and temperature, an example being shown in the figure below. Notice that there are two sides to this chart, the left side being the mean sea-level (MSL) power output performance and the right being the performance at altitude.

The instructions on the chart explain how it is used to determine the brake power of the engine. This chart differs from what pilots would use in flight, but engineers use it to estimate the available engine power under different flight conditions. All of the needed measurements to determine power output can be made using the chart. Measurements of the pressure altitude, engine rpm, manifold pressure, and outside air temperature can be made using standard cockpit instruments, which is very useful from a flight test perspective.

The power available at the engine shaft (the brake horsepower or bhp) can be determined given measurements of the following:

1. Pressure altitude can be measured directly on the altimeter by setting the reference pressure in the Kollsman window to MSL standard conditions of 29.92 inches of Hg.
2. Air temperature would be measured in flight using an appropriately calibrated outside air temperature (OAT) gauge.
3. Engine rpm, which would be measured using a tachometer. While a tachometer is part of the standard cockpit instruments, an optical tachometer that counts the passage of the propeller blades is more accurate.
4. Manifold pressure can be measured on the manifold pressure gauge, which is also part of the standard cockpit instruments.

The process starts by entering the left chart (MSL performance) at the bottom using the manifold pressure measurement, then reading up to point B on the lines of constant engine rpm. Notice that interpolation will generally be required. Reading across to the right to the axis and to point C gives the engine brake power at MSL standard conditions.

The next part of the process is establishing the engine performance at altitude, which is done using the right-side chart. After carrying point C onto the left chart, a straight line is connected between points C and A, point A being at the appropriate point on the rpm and manifold pressure map. To find the power available for a given pressure altitude, it is necessary to move along the line AC. Reading to the left axis will give the brake power output at altitude under standard temperature conditions. Finally, there is a minor correction for non-standard temperatures (the formula is shown on the chart) to give the final brake power output at point F. Notice again that interpolation will be required throughout this process.

Brake Specific Fuel Consumption (BSFC)

The efficiency of a piston engine is measured in terms of its power-specific or brake-specific fuel consumption (BSFC), which is often given the symbol . The BSFC measures the fuel used (in units of mass or weight) per unit of power supplied (in hp or kW) per unit time of engine operation (usually one hour). BSFC is used as a measure of the fuel efficiency of any engine that burns fuel and produces rotational or shaft power.

The BSFC is defined as

(4)  

The units of BSFC are typically in lb hp hr in the U.S. customary system, or kg kW hr or grams per kilowatt-hour (g/kWh) in the SI system. Notice that the unit of mass (kilogram or grams) is used in the SI units of BSFC, an anomaly of the SI system. However, the time unit is hours in both cases.

For a piston engine used on an aircraft, the values of BSFC are typically in the range of 0.4 to 0.6 lb hp hr (0.24 to 0.37 kg/kWh), as shown in the figure below. These values are approximate and can vary based on factors such as engine design, operating conditions (e.g., cruising, takeoff, climb), and the specific model of the engine. Additionally, advancements in engine technology may lead to improved fuel efficiency compared to older engines.

There is usually some dependence of BSFC on flight altitude, with the values increasing somewhat. Notice that the best (lowest) BSFC is obtained when the engine is operating at or near its rated rpm and power output. Today, the best performing and highest efficiency piston engines are supercharged types of diesel, which have better thermal efficiencies and BSFC values in the 0.45 to 0.7 lb hp hr range (0.27 to 0.43 kg/kWh).

Propeller Performance

As shown in the photograph below, a modern propeller is a remarkable accomplishment of aeronautical engineering. While the basic engineering principles used for aircraft propeller design have remained the same over a century, numerous detailed improvements have led to substantial gains in propulsive efficiency and operational reliability.

Design Features

Many propellers are currently used, from those that use just two blades to those with four or more blades, some with fixed pitch, and others with variable pitch. In addition, some propellers may have swept blades, a design feature used to reduce compressibility drag loss at the tips of the propellers when the aircraft operates at higher flight speeds. As engine power increases, more blades (or more blade area) are needed to deliver the power to the air. In some cases, to prevent the propeller diameter from becoming too large, counter-rotating propellers may be used to absorb high amounts of available power from the engine shaft.

Propellers with larger numbers of blades also tend to be relatively more efficient (when compared based on the same thrust and total blade area). However, the propeller’s net efficiency depends on other factors, including its operating rpm, tip Mach number, diameter, and blade pitch. Today, there is an increasing emphasis on obtaining lower noise from propellers, which has driven modern propeller designs (even on general aviation airplanes) to have more blades and reduced diameters with lower tip Mach numbers.

Propeller blades are significantly twisted along their span, i.e., they have a form of washout. As shown in the figure below, the local pitch angle changes from a relatively high value at the root (next to the hub) and progressively decreases in value from section to section when moving out to the blade tip. The net twist varies for different propellers, but washout angles may be as large as 40 degrees over the blade span; of course, these angles are much larger than those used on a wing. The primary purpose of using blade twist is to get the local angles of attack at each section along the span of the blade to be low enough such that it operates close to the aerodynamic conditions where the section is most aerodynamically efficient.

Early propellers were made of laminated wood but needed more strength and durability for the much higher-powered engines produced during and after WW2. Aluminum alloy blades solved this problem and performed much better because they could be made thinner and rotate at higher speeds before suffering losses from compressibility effects.

Metal blades also allowed for variable pitch or constant speed propeller designs for high-performance aircraft. A speed governor automatically adjusts the blade pitch to maintain the propeller at a constant rotational speed. A fine (low) blade pitch is used for takeoff and landing, and a coarse (high) blade pitch is used for cruise flight. Therefore, an aircraft with a variable pitch or constant speed propeller can maintain good levels of propulsive efficiency over a broader range of airspeeds. However, such a capability requires a mechanism to adjust the blade pitch during flight, which incurs weight and cost.

The most recent advances in propeller technology have used composite propeller blades made from Kevlar or carbon fiber, allowing thinner, swept blades to be made. In addition to being lighter, modern propeller designs have shown substantial gains in propulsive efficiency and noise reduction. Wooden propellers are still made today and often used on lower-performance aircraft because they are lightweight and relatively inexpensive.

Operational Principles

The basic operational principle of a propeller has been previously introduced, along with some results for its propulsive efficiency based on applying the conservation laws of aerodynamic flows in integral form. As shown in the figure below, each rotating blade is designed to produce a lift force, so the propeller creates a net thrust. It takes torque to spin the blades to overcome their drag, so power must be delivered to the shaft.

Another way to explain the propeller’s performance is by using the blade element theory. Developing a computer program to calculate the performance of a propeller using the blade element method is a relatively straightforward task. The principle here is to calculate the angle of attack and the corresponding lift and drag at each propeller blade section, as shown in the figure below. All blades are twisted along their span from root to tip (which is done to give good aerodynamic efficiency), so the local value of blade pitch differs from point to point along the span.

Notice that the relative flow angles and, hence, the angle of attack of the blade element are obtained by using the vector addition of the relative velocity components, i.e., the vector sum of the rotational velocity, , of the blade element as it spins about the shaft and the free-stream velocity, , where the span position from the rotational axis is denoted by ,

The local pitch of the blade is , so the angle of attack of any blade section is then

(5)  

The lift coefficient on the blade element then follows as the product of the angle of attack of the blade and the local lift-curve slope of the airfoil section, i.e.,

(6)  

where is measured from the zero-lift angle and recognizing that the lift-curve slope will depend somewhat on the shape of the blade section used, as well as the local incident Mach number at the section and to some extent Reynolds number too. The lift on a blade element will be

(7)  

where is the local chord of the blade ( is the area of the blade element) and the resultant local velocity is

(8)  

When the lift on all the sections of the propeller blade is obtained, then the net thrust of the propeller can be obtained by resolution of the local lift vectors in the direction of the thrust component followed by spanwise integration, i.e.,

(9)  

where is the number of propeller blades, and is the propeller’s radius. The component of the drag as it affects the thrust can be ignored. Also, it is reasonable to assume that there is axisymmetry in the problem. Hence, the angles of attack at any given blade station are the same for any rotational angular position of the propeller blades. The integration process is performed numerically.

By analogous arguments, the power required to spin the propeller will be

(10)  

where is the local profile drag that again depends on the airfoil shape used on the blade and the incident Mach number. Notice the inclusion of the moment arm to get to the torque (torque is a moment, so the product of a force times a distance or “arm”), and power is just the product of torque and angular velocity. In this case, however, the component of the lift (which is the induced drag) will contribute to the net drag of the section and, hence, the torque and power required to spin the blades.

If the section drag coefficient is known, then

(11)  

However, the challenge with a propeller is to properly represent the section because it depends on the Mach number, in this case, the helical Mach number, i.e.,

(12)  

where is the local speed of sound, i.e., ). Of course, suppose the helical Mach number becomes too large and exceeds the drag divergence Mach number of the airfoil sections. In that case, the propulsive efficiency of the propeller will decrease rapidly.

Variable Pitch Operation

The local angles of attack of the blade sections are critical to propeller operation. Regulating the pitch angles of the blades allows the efficient operation of the propeller to be optimized as a function of the flight condition, especially for takeoff and cruise, as shown in the figure below.

The highest section angles of attack attainable on the blades will be limited by the onset of stall, just as a regular wing is limited in its angle of attack capability by stall. However, because the local incident Mach number varies from the root to the tip of the propeller, and the stall angle of attack of an airfoil decreases with increasing Mach number, the allowable angle of attack before stall will decrease from root to tip. Therefore, for takeoff conditions, the blades need to be placed in a fine pitch, but as airspeed builds, the angle can be increased to maintain thrust and efficiency.

If the engine stops in flight, the propeller blades need to be “feathered” into the flow to reduce the otherwise high drag of the stationary blades. In this regard, strong springs are used on a constant-speed propeller so that the blades are automatically forced into the fully feathered position if the propeller stops. The drag of an un-feathered stationary propeller is considerable, so an airplane with a fixed-pitch propeller may have a poor glide ratio. The propeller blades may sometimes be set to a negative pitch to produce negative thrust and act as an aerodynamic brake during landing. However, this latter feature is usually only included in high-performance turboprop airplanes.

Propeller Performance Charts

Propeller performance curves are presented in the form of thrust coefficient, power coefficient, and propulsive efficiency, respectively, analogous to how airfoil and wing aerodynamic coefficients are used. All propellers have their characteristics quantified in this manner, and there is a separate set of curves for each reference blade pitch angle. By convention, the reference pitch angle is not the angle of attack of the blade sections of the propeller but the reference pitch at the 75% blade span (i.e., at ), which is a geometric quantity and can be measured. Sometimes, the pitch of a propeller is measured in units of length, which refers to the helical pitch that the reference blade section traces out during one revolution, i.e., like a screw thread. Hence, the old name of a propeller is known as an airscrew.

Thrust Coefficient

The thrust coefficient for a propeller is a non-dimensional thrust (which can be determined from dimensional analysis) and is defined as

(13)  

where is the thrust generated by the propeller, is the propeller’s diameter, and is the number of revolutions of the propeller per second. If the rotational angular velocity of the propeller is , then . The rotational speed is often measured in terms of revolutions per minute or rpm, so rpm equals . Normally, the results for the propeller characteristics are plotted as a function of the non-dimensional tip speed ratio or advance ratio, given the symbol , which is defined by

(14)  

Representative versus results for a propeller are shown in the figure below, with a separate curve for each reference blade pitch. Notice that results from the blade element method discussed previously (i.e., the “theory”) agree well with the measurements.

Power Coefficient

The power coefficient for the propeller is

(15)  

where would be the brake power, i.e., the power delivered to the propeller through the driving shaft. Normally, the torque, , would be measured, so then . The corresponding versus results for a propeller are shown in the figure below. Again, notice that the theory is in good agreement with the measurements.

Propulsive Efficiency

Finally, the propulsive efficiency of the propeller is defined as

(16)  

which is just a non-dimensional statement that the propeller’s efficiency is the ratio of the useful power to the input power. Using the definitions of and then

(17)  

The corresponding representative versus results for a propeller are shown in the figure below, again with one curve for each reference pitch angle.

Notice that for a given propeller operated with any given blade pitch and rotational speed , its propulsive efficiency increases with increasing forward airspeed to reach a maximum and then diminishes rapidly. Consequently, a propeller of a given (fixed) blade pitch cannot operate with high propulsive efficiency over a wide range of values of (or airspeed for a given rotational speed).

This latter outcome occurs because propeller blades are wings (rotating wings). All wings can only operate aerodynamically over relatively small ranges of the angle of attack, i.e., local blade section angles of attack between 2 to 14 degrees, depending on the local Mach number and Reynolds number. The local sectional angles of attack on the propeller depend not only on the rotational speed of the propeller and airspeed or advance ratio but also on how the propeller is twisted along its span. So, as airspeed changes, being assumed constant, then blade pitch must be increased to progressively maintain the angle of attack on the propeller. By gradually increasing the blade pitch, the best efficiency can be obtained over a much wider range of airspeeds, which is precisely the purpose of a continuously variable pitch or “constant speed” propeller.

Explanation of Propeller Performance

With this understanding of the flow at the blade section and the creation of thrust from the propeller, the various curves of , , and , as shown previously, can now be explained in greater detail.

At low values of , the corresponding angles of attack of the blade sections are relatively high. So, the blade sections produce relatively high lift but are close to the point of stall. The propeller still produces thrust but requires high power and is inefficient. As airspeed and increase, the blade sections operate at lower angles of attack and closer to their best section lift-to-drag ratios. As a result, the thrust is maintained, but the drag on the blades decreases, and so propulsive efficiency increases markedly.

The lowest angles of attack will produce little lift on the blades or thrust on the propeller. However, there is a range of airspeeds (assuming blade pitch does not change) for which good propulsive efficiency is obtained. Therefore, the best aerodynamic efficiencies will only be obtained when all blade sections (or most of them) operate at or near the angles of attack for their best lift-to-drag ratio, which is usually between 2 to 8 degrees, depending on the incident Mach number.

There is eventually a point at higher airspeeds (or high ) where the blade sections encounter diminished angles of attack and higher helical Mach numbers, simultaneously decreasing thrust and efficiency unless the blade pitch increases further. Eventually, the blade pitch cannot be mechanically increased to improve efficiency, so the efficiency drops off.

The results also explain the differences in propulsive efficiency of fixed-pitch versus constant-speed propellers, as shown in the figure below. Notice that if the propeller pitch is fixed, its propulsive efficiency increases slowly with airspeed, reaching a maximum and decreasing rapidly. The relatively low efficiency of a fixed-pitch propeller at low airspeeds means that the aircraft’s takeoff and climb performance will be relatively poor. Further increases in airspeed beyond the airspeed for peak efficiency will cause propeller efficiency to decrease precipitously. This behavior means that the usable power delivered to the airstream decreases, effectively setting an upper barrier to the airspeed achievable by the airplane.

The preceding situation is very different for a variable pitch or constant speed propeller, which can be set into a fine pitch for takeoff, giving good propulsive efficiency and low airspeed and giving the airplane markedly better takeoff and climb performance. As airspeed builds, the blade pitch can be increased to maintain a constant rpm schedule, so the propeller efficiency can now closely follow the envelope of peak efficiency. This reason is why airplanes with constant-speed propellers have much better overall flight performance and can cruise at much higher airspeeds, as shown in the figure below. A constant-speed propeller also maintains a steady load on the engine, which is essential for long engine life.