Category AERODYNAMICS, AERONAUTICS, AND FLIGHT MECHANICS

Figure 6.30a, 6.30b, and 6.30c presents a qualitative comparison of the turbojet, turbofan, and turboprop engines, each having the same core engine.

The specific fuel consumption for a turbojet or turbofan engine is expressed as a thrust specific fuel consumption (TSFC). In the English system of units, one states TSFC as pounds of fuel per hour per pound of thrust, so that TSFC actually has the dimensions of 1/time. Thus its numerical value is the same in the SI system as in the English system. In the SI system, TSFC is given as N/hr/N. The characteristics of the three engines are seen to be quite different with the turbofan, not surprisingly, lying between the turboprop and turbojet. The relative differences in these curves are explained mainly by the momentum and energy considerations undertaken previously for the pro­peller. “Disc loadings” for turbojet engines are of the order of 81,400 Pa (1700psf), while turbofans operate at approximately half of this loading and propellers at only approximately 4% of the disc loading for a turbojet. If we assume that the core engine is delivering the same power to each engine configuration then, from Equation 6.17, for static thrust one obtains,

Note that thrust for a turbojet engine is denoted by F instead of Г, since T is understood to refer to temperature when working with a gas turbine. Thus, with its appreciably lower disc loading, one would expect the static thrust of a turboprop to be significantly higher than the corresponding turbojet, possibly even more so than that shown in Figure 6.29a (taken from Ref. 6.6).

The rapid decrease in thrust with airspeed for the turboprop and the more gradual changes for the turbofan and turbojet engines are also explained in part by the relative disc loadings. Combining Equations 6.13, 6.14, and 6.15 gives

(6.77)

If the power to produce the thrust is assumed to be constant, then Equation

Comparative net thrust at sea level.

Comparative thrust specific fuel consumption.

Relative maximum continuous thrust comparison during climb.

6.77 can be written

(6.78)

where Fa is the static thrust and w0 is the static-induced velocity given by Equation 6.16. This implicit relationship between F/F0 and VI w0 can be easily solved iteratively using a programmable calculator. The solution is presented graphically in Figure 6.31. Thus, it is not V per se that determines the ratio of F to F0 but, instead, the ratio of V to w0. For a high disc loading with a concomitant w0, a given V will have a lesser elfect on F than for the case of a low disc loading.

Disc loading is not the total explanation for the relative dilferences in T as a function of V shown in Figure 6,29a. Consider a typical turbojet with a static disc loading of around 81,400 Pa (1700 psf). For this engine at sea level, w0 will equal approximately 180 m/s (600 fps). An airspeed of 400 kt in this case gives and

F

= 0.64

F0

However, Figure 6.30a shows only а 20% decrease in the thrust. This is because the gas generator power is not-constant but also increases with V

because of the increased mass flow and ram pressure. If we tacitly assume the power proportional to the product of F and TSFC then, from Figure 6.30b, one would predict the core engine power to have increased by about 10%. This results in a decreased value of VI w0 of 1.08, giving a new T/T0 of 0.66. However, T0 corresponds to the core engine power at 400 kt. Based on the original F0 corresponding to the core engine power at V = 0, F/F0 becomes 0.73. Figure 6.30a, 6.30b, and 6.30c is, of course, not too accurate and is really intended only to show relative differences. You may wish to apply Figure 6.31 to the performance curves of the JT9D-3 turbofan engine that follow. In this case, the predicted variation of F and V will be found to match closely the results given in the installation handbook.

I

In order to understand, at least qualitatively, why a particular configura­tion of a gas turbine engine performs as it does, let us consider a few basic principles. The ideal thermodynamic cycle for the gas turbine engine is shown in Figure 6.29a, where it is compared to the cycle for the piston engine. The Otto cycle, which approximates the piston engine thermodynamics, consists of an, isentropic compression of the gas followed by a rapid combustion at

Figure 6.29a A comparison between the Otto and Brayton cycles.

nearly constant volume. The gas then expands isentropically, forcing the piston ahead of it. Unlike the piston engine, the gas turbine engine involves a continuous flow of the working gas. The Brayton or constant pressure cycle, which approximates the actual gas turbine cycle, begins with an isentropic compression of the air from ambient conditions. Part of this compression occurs prior to the compressor stages as the air enters the engine inlet. Following the compression, burning occurs at constant pressure, resulting in increased volume and total temperature. The air then expands isentropically through the turbines and jet nozzle to the ambient static pressure. In a turboprop or turboshaft engine, nearly all of the expansion occurs within the turbines in order to drive the compressor and produce shaft power. In a turbojet engine, an appreciable amount of expansion occurs after the turbines in order to produce the high-momentum jet.

The heat that is added to the flow per unit weight of gas is given by

<?i„=Cp(T3-T2) (6.66)

while the heat rejected is

<?ои,= Ср(Г4-Г1) (6.67)

Cp is the specific heat at constant pressure, as used previously in Chapter Five. The work output per unit weight of gas equals the added heat minus that which is rejected. The thermal efficiency equals the work output divided by the added heat. Thus,

Г4-Т,

T3-T2

Let r denote the compression ratio, p2lpі (or p3/p4). Since compression and expansion are both assumed to be isentropic,

Tl = Tl= ,(•>■—і )h

T T4

Thus, Equation 6.68, in terms of the compression ratio, can be written as

As stated previously, the compression ratio r is achieved partly in the inlet (ram pressure), and the remainder is achieved through the compressor. The pressure increase across the compressor, at a constant rpm, as a first approximation, is proportional to the mass density, pc, just ahead of the compressor.

Ap oc p<

If r0 denotes the value of r for static sea level operation,

P =Po(r0- 1) —

p0 is, of course, the standard sea level value of mass density.

For isentropic compression in the inlet up to the compressor, the ambient mass density and pc are related by

Pc. Г1 + (y — 1 l2)MjyKy~n

p® L 1 + (y – 1/2)M2 J

where М» is the free-stream Mach number and M is the local Mach number just ahead of the compressor. The pressure ratio, r, thus becomes

r^pc + Ap

P cc

or

pc/p® is given by

Thus r finally becomes

r = [/(M, М„)]’йт-,) + Г-~г^- [/(M, Moo)]I/(l,_,)

V

where

1 + (y — H2)Mj 1 + (y – 1/2)M2

Equation 6.75 is substituted into Equation 6.70 an expression for the thermal efficiency results that is a function of 5, r0, and M.

The effect of pressure ratio, altitude, and free-stream Mach number on the ideal thermal efficiency is shown in Figure 6.29b. This figure assumes the ratio of M to M„ just before the compressor to equal approximately zero. This is a fairly reasonable assumption, since values of this ratio up to at least 0.4 affect t) by less than 1%. Figure 6.29b shows the effect of varying one parameter at a time while keeping the other two parameters at their normal values. Increasing from zero to 0.8 is seen to result in a 7% improvement in 7j. The efficiency also improves with altitude, increasing by approximately 6% in going from sea level to 40,000 ft (12,200 m). Doubling the pressure ratio, r0, from 10 to 20 results in a 16% improvement in tj.

With regard to the production of thrust, r/ does not tell the whole story, tj is simply a measure of how efficiently the air passing through the engine is

being used. The heat added to the flow, per unit weight, is given by

Qm = vCp(T3 – T2)

Thus, for the same efficiency, if T3 is increased or the mass flow increased, the thrust will be increased.

An engine rating specifies the thrust that an engine can (or is allowed) to develop in a particular operating mode. For commercial certification, these ratings are defined as follows.

Takeoff (Wet) This is the maximum thrust available for takeoff for engines that use water injection. The rating is selected by actuating the water injection system and setting the aircraft throttle to obtain the computed “wet” takeoff thrust. The rating is restricted to takeoff, is time limited to 5 min, and has altitude and ambient air or water temperature limitations. Takeoff (Dry) This is the maximum thrust available without the use of water injection. The rating is selected by setting the aircraft throttle to obtain the computed takeoff (dry) thrust for the prevailing conditions of ambient temperature and barometric pressure. The rating is time limited to 5 min and is to be used only for takeoff and, as required, for reverse thrust operations during landing.

Maximum Continuous This rating is the maximum thrust that may be used continuously, and is intended only for emergency use at the discretion of the pilot.

Maximum Climb Maximum climb thrust is the maximum thrust approved for normal climb. On some engines, maximum continuous and maximum

climb thrusts are the same. For commercial engines, the term formerly used, normal rated thrust, has been replaced by the more appropriate term, maximum climb thrust.

Maximum Cruise This is the maximum thrust approved for cruising.

Flat Rating

Engines that must be operated at “part throttle” at standard ambient conditions to avoid exceeding a rated thrust are referred to as “flat-rated” engines. This refers to the shape of the thrust versus the ambient temperature curve. For example, the General Electric Company’s CF6-6 high bypass turbofan engine is flat rated up to an ambient temperature of 31 °С at sea level, or 16 °С higher than a standard day. Thus, its thrust as a function of ambient temperature varies, as shown in Figure 6.28. At full throttle, the thrust is seen to decrease with increasing temperature. Therefore, by flat rating an engine out to a temperature higher than standard, one is able to maintain rated thrust on a hot day.

Basically, the gas turbine engine consists of a compressor, a combustion chamber, and a turbine. The combination of these basic components is referred to as the gas generator or core engine. Other components are then added to make the complete engine. It is beyond the scope of this text to delve into the details of gas turbine engine design. However, the various types of gas turbine engines will be described, and their operating characteristics will be discussed in some detail.

Beginning with the core engine, the turbojet engine pictured in Figure 6.23 is obtained by adding an engine air inlet and a jet nozzle. As the air

enters the inlet, it is diffused and compressed slightly. It then passes through a number of blade rows that are alternately rotating and stationary. The collection of rotating blades is referred to as the rotor; the assembly of stationary blades is called the stator. This particular compressor configuration is known as an axial-flow compressor and is the type used on all of today’s larger gas turbine engines. Early gas turbine engines, such as Whittle’s engine, employed a centrifugal compressor, as shown in Figure 6.24. Here, the air enters a rotating blade row near the center and is turned radially outward. As the air flows out through the rotating blade passage, it acquires a tangential velocity component and is compressed. A scroll or radial diffuser collects the compressed air and delivers it to the combustion chamber. Centrifugal com­pressors were used on the early turbojet engines simply because their design was better understood at the time. As jet engine development progressed, centrifugal compressors were abandoned in favor of the more efficient axial – flow compressors. The axial-flow compressor also presents a smaller frontal area than its centrifugal counterpart and is capable of achieving a higher pressure ratio.

Smaller sizes of gas turbine engines still favor the centrifugal com­pressor. Figure 6.25 is a cutaway drawing of the Garrett ТРЕ 331/T76 turboprop engine. The compressor section of this engine consists of two stages of radial impellers made of forged titanium.

•(a) <b)

Figure 6.24 Typical centrifugal flow compressor impellers, (a) Single-entry im­peller. (b) Double-entry impeller. (Courtesy General Electric Co.)

After the compressed air leaves the compressor section, it enters the combustor, or burner, section. Atomized fuel is sprayed through fuel nozzles and the resulting air-fuel mixture is burned. Typically, the ratio of air to fuel by weight is about 60:1. However, only approximately 25% of the air is used to support combustion. The remainder bypasses the fuel nozzles and mixes downstream of the burner to cool the hot gases before they enter the turbine.

The mixed air, still very hot (about 1100 °С), expands through the turbine stages, which are, composed of rotating and stationary blade rows. The turbines extract energy from the moving gases, thereby furnishing the power required to drive the compressor. Nearly 75% of the combustion energy is required to drive the compressor. The remaining 25% represents the kinetic energy of the exhaust, which provides the thrust. For example, in the General Electric CF6-6 turbofan engine [180,000 N (40,0001b) thrust class], the turbine develops approximately 65,600 kW (88,000 shp) to drive the high – and low – pressure compressors.

Variations of the gas turbine engine are presented in Figure 6.26. In a turboprop or turboshaft engine, nearly all of the energy of the hot gases is extracted by the turbines, leaving only a small residual thrust. The extracted energy in excess of that required to drive the compressor is then used to provide shaft power to turn the propeller or a power-output shaft in general. Turboshaft engines power most of today’s helicopters and are used exten­sively by the electric utilities to satisfy peak power load demands.

A “spool” refers to one or more compressor and turbine stages con­nected to the same shaft and thus rotating at the same speed. Gas turbine engines generally use one or two spools and are referred to as single or dual compressor engines. A turboshaft engine may incorporate a free turbine that is independent of any compressor stage and is used solely to drive the shaft. Since the rotational speed of a turbine wheel is of the order of 10,000 rpm, a
reduction gear is required between the turbine shaft and the power output shaft.

A turboprop produces a small amount of jet thrust in addition to the shaft power that it develops; these engines are rated statically in terms of an equivalent shaft horsepower (eshp). This rating is obtained by assuming that 1 shp produces 2.5 lb of thrust. For example, the dash 11 model of the engine shown in Figure 6.25 has ratings of 1000 shp and 1045 eshp. From the definition of eshp, this engine therefore produces a static thrust from the turbine exhaust of approximately 1131b.

A turbojet engine equipped with an afterburner is pictured in Figure 6.26/. Since only 25% or so of the air is used to support combustion in the burner section, there is sufficient oxygen in the turbine exhaust to support additional burning in the afterburner. Both turbofans and turbojets can be equipped with afterburners to provide additional thrust for a limited period of time. Afterburning can more than double the thrust of a gas turbine engine, but at a proportionately greater increase in fuel consumption. Essentially, an afterburner is simply a huge stovepipe attached to the rear of an engine in lieu of a tail pipe and jet nozzle. Fuel is injected through a fuel nozzle arrange­ment called spray bars into the forward section of the afterburner and is

ignited. This additional heat further expands the exhaust, providing an in­creased exhaust velocity and, thereby, an increased thrust. The afterburner is equipped with flame holders downstream of the spray bars to prevent the flames from being blown out of the tail pipe. A flame holder consists of a blunt shape that provides a wake having a velocity that is less than the velocity for flame propagation. An adjustable nozzle is provided at the exit of the afterburner in order to match the exit area to the engine’ operating condition.

Two different types of turbofan engines are shown in Figure 6.26b and 6.26c; the forward fan with a short duct and the forward fan with a long duct. These engines are referred to as bypass engines, since part of the air entering the engine bypasses the gas generator to go through the fan. The ratio by weight of the air that passes through the fan (secondary flow) to the air that passes through the gas generator (primary flow) is called the bypass ratio. Early turbofan engines had bypass ratios of around 1:1; the latest engines have ratios of about 5:1. One such engine, Pratt & Whitney’s JT9D turbofan, is shown in Figure 6.27. Included on the figure are temperatures and absolute pressures throughout the engine for static operation at standard sea level conditions.

One might argue that the turbojet engine had its beginnings with the turbosupercharger, since the latter has an exhaust-driven turbine that drives a compressor to supply air to the engine. These are the essential ingredients of a turbojet engine. Dr. Sanford A. Moss is generally credited with developing the turbosupercharger, at least in this country. In 1918, Moss successfully tested his turbosupercharger atop Pikes Peak. Two years later, a La Pere biplane equipped with a turbosupercharger set a world altitude record of over 10,000 m.

In 1930, Frank Whittle (later to become Sir Frank Whittle) received a patent for a turbojet engine. Unfortunately, he was unable to gain support for the development of his design. It was not until 1935, when a young German aeronautical engineering student, Hans von Ohain, received a German patent on a jet engine, that development work began in earnest on the turbojet engine. On August 27, 1939 (some references say June 1939), the first turbojet engine was flown in a Heinkel He 178. This engine designed by von Ohain delivered 4900 N (11001b) of thrust. It was not until May 1941 that Whittle’s engine was flown in England.

German jet engine development progressed rapidly. By 1944, both BMW and Junkers turbojet engines were introduced into the Luftwaffe. One can imagine the astonishment of the allied aircrews upon first seeing propellerless airplanes zip by them at incredible speeds of over 500 mph.

On October 1, 1942, the first American jet-propelled airplane, the Bell Airacomet, was flown. This twin-engine airplane was powered by an Ameri­can copy of Whittle’s engine built by the General Electric Co. Designated the “1-А,” the engine weighed approximately 4450 N (10001b) with a thrust-to – weight ratio of 1.25. The first production American jet aircraft, the Lockheed F-80, first flew in January 1944. In production form, it was powered by the J33 engine, which delivered a thrust of approximately 17,800 N (4000 lb) at a weight of 8900 N (2000 lb).

General Electric’s J47 was the first turbojet power plant certified in the United States for commercial aviation in 1949. The world’s first commercial jet transport to fly, however, on July 27, 1949, was the British built de Havilland Comet powered by four de Havilland Ghost 50 Mkl turbojets. This eiigine, incorporating a centrifugal compressor, developed 19,800 N (44501b) of thrust. The Comet must be recognized as one of the most famous airplanes in history, because it truly ushered in the age of jet transportation. Un­fortunately, its career was short-lived after three of the nine that had been built broke up in the air. An exhaustive investigation showed the cause to be fuselage structural fatigue because of repeated pressurizations. Despite its tragic demise, the Comet proved the feasibility of commercial jet trans­portation and paved the way for Boeing’s successful 707. This airplane first

took to the air on July 15, 1954, powered by four Pratt & Whitney JT3 (military designation J57) turbojet engines. Each engine developed a static thrust of approximately 57,800 N (13,0001b) with a dry weight of 18,200 N (41001b).

A historical note of interest is the following quotation taken from a report by the Gas Turbine Committee of the U. S. National Academy of Sciences in 1940.

“… Even considering the improvements possible… the gas turbine could hardly be considered a feasible application to airplanes mainly because of the difficulty with the stringent weight requirements ”

This conclusion, made by a panel of eminent persons, including Dr. Theodore von Karman, is a sobering reminder to any engineer not to be too absolute.

This section deals mainly with the aerodynamic considerations of design­ing a new propeller. The optimum blade loading is prescribed by the Betz condition, which requires the trailing vortex system to lie along a helical surface in the ultimate wake. This condition will be met if

wr tan (ф + a,) = constant

= У + wo (6.60)

Wo is a fictitious velocity called the impact velocity. Given the design advance ratio, one can arbitrarily choose a value of wJwR. From the geometry of Figure 6.10, it follows that

sin (ф + ad cos (Ф + ai)

where

Substituting Equation 6.61 into Equation 6.42 leads to the product aCi as a function of x. One must then decide how to choose between a and Q. The procedure for doing so is not well defined. First, one must choose the number of blades. This may be done on the basis of experience or arbitrarily as a first step in a design iteration. Similarly, a radial distribution of thickness is chosen. Ultimately, stress calculations must be made. Based on these results, the thickness may be changed.

A very practical and completely nonaerodynamic consideration in the choice of an airfoil section for a propeller blade is the question of stress concentrations resulting from leading and trailing edge nicks and scratches, particularly leading edge nicks. To elaborate on this point, consider the two airfoil sections pictured in Figure 6.22. From a stress-concentration view­point, the symmetrical airfoil on the left is preferred since, in bending, stresses are directly proportional to the distance from the neutral axis. From an aerodynamic viewpoint, the cambered section is preferred. Hence, the engineer is faced once again with another compromise, a practice that characterizes much of the engineering profession.

M (b)

Figure 6.22 Susceptibility of airfoil shapes to leading and trailing edge stress concentrations, (a) Edges close to neutral axis. (b) Edges removed from neutral axis.

Having selected an airfoil family such as the NACA series-16 or the newer supercritical airfoil, one now chooses at each x a design C, that will avoid compressibility effects. The steps for doing this are:

1. Choose С/.

2. Calculate c from aCt.

3. Determine Mcr from tic and C(.

4. Compare Ma with the resultant local M.

5. If Mcr is less than M, decrease C( and repeat.

If Mach number is not a consideration in the design, then one can choose C(j to give the lowest Cd to C( ratio for the chosen airfoil family.

Having determined the radial distribution of c (and hence cr) and С/, the corresponding Cd values are calculated. These, together with (ф + a,), are substituted into Equations 6.32a and 6.32b to determine thrust and power. The entire design process is performed with different vv0 values until the desired value of Cr or Cp is achieved. Generally, increasing w0 will increase either of these coefficients.

Most propellers are designed to operate immediately in front of a fuselage or nacelle. The inflow velocity in this case is no longer a constant but is, indeed, a function of x, the dimensionless radial station. This three – dimensional flow field can be determined by the potential flow methods presented in Chapter Two. With V a function of x, the resultant flow angle, ф, becomes

or

For this case of a nonuniform, potential inflow, the Betz condition is not

V(x) + w0 = constant Instead, one should impose only

w0 = constant (6.65)

Equation 6.65 follows from superimposing the potential flow from the propeller on that produced by the body. In the ultimate wake V(x) will approach V0 and w0 will approach 2w0, so that the Betz condition is again satisfied.

Propeller manufacturers offer propellers covering a range of diameters, pitch values, and solidities. The choice of these parameters can depend on considerations other than aerodynamic efficiency. For example, to keep the noise level of a propeller low, one may have to employ wide blades with low tip speeds. As another example, the propeller diameter is sometimes limited by ground clearance considerations or by the distance from a nacelle to the fuselage. The dynamics of the propeller must also be matched to the engine. The natural frequency of the first bending mode of a blade should not coincide with an impulse frequency from the engine. For example, a horizon­tally opposed, six-cylinder engine has three torsional peaks per revolution. If a propeller being driven by this engine has a natural frequency close to 3/rev, it can lead to excessive vibration and fatigue stresses.

Aerodynamically, one strives to select a propeller that provides a high efficiency for cruise and a high static thrust for takeoff. These two require­ments are easier to satisfy with a variable pitch propeller. A fixed pitch propeller is usually a compromise between these two operating regimes.

Given the results of a series of propeller tests, such as Figures 6.12 and

6.13, one can utilize these data to select the best propeller diameter and blade angle to match a given airplane-engine combination. One approach that is sometimes used is based on a Coefficient Cs, the speed power coefficient, defined by

(6.57)

Knowing Cp as a function of J, Cs can be calculated from

The advantage of Cs is that it does not contain the diameter in its definition.

Figure 6.21 presents / as a function of Cs for the same propeller for which Figures 6.12 and 6.13 hold. A maximum efficiency line is also shown in Figure 6.21. The use of this graph is best illustrated with an example. The problem will be to select the optimum diameter for this propeller if it is to be installed, on a Cherokee 180. Consider the selection of a propeller to absorb

75% of the maximum power of 180 bhp at 2500 rpm at standard sea level conditions. Using a value for / of 0.5 m2 (5.38 ft2) and an e of 0.6, CD can be calculated as a function of V. CT and CD are then related by (T = D).

Assuming a value for V of 130 mph leads to a Cs of 1.360. From the maximum efficiency line in Figure 6.21, a J of 0.76 and a /3 of 20° are obtained. These values in turn lead to a CT value of 0.0573, so obviously 130 mph will not be the trim speed for the optimum propeller at this power and rpm. By iteration, one obtains a trim speed of 132 mph and the following.

J = 0.76

В = 20°

CT = 0.0592 v = 0.84 D = 6.1 ft

The practicing aerodynamicist will normally have available both engine and propeller operating curves as supplied by the respective manufacturers. Using these curves together with a knowledge of the airplane’s aerodynamic characteristics, one is able to estimate the airplane’s performance. In order to illustrate the procedures that are followed in using a set of propeller charts, let us again use the Cherokee 180 as an example.

An estimated curve of efficiency as a function of advance ratio for the fixed pitch propeller used on the PA-28 is presented in Figure 6.15. This curve is applicable to the aircraft pictured in Figure 3.62 with the engine operating curves of Figure 6.3. This particular propeller has a diameter of 1.88 m (6.17 ft).

As an example in the use of the engine performance charts together with the graph of propeller efficiency, assume that in steady, level flight, the pilot of a PA-28 reads a manifold pressure of 24 in., an rpm of 2400, a pressure altitude of 3000 ft, an OAT of 65 °F, and an indicated airspeed of 127 mph. From this information, together with Figure 6.15, one can estimate the drag of the airplane at this indicated airspeed and density altitude.

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Advance ratio, J ~ V/nD

Figure 6.15 Estimated propeller efficiency for the Piper Cherokee PA-28.

From Figure 2.3 at 914 m (3000 ft),

£- = 0.90 Po

Thus, p = 91,163 Pa (1904 psf).

Furthermore, air obeys closely the equation of state for a perfect gas.

—= = constant = R

pT

In Equation 6.50 T is the absolute temperature.

Using standard sea level values for p0, p0, and T0, the preceding constant is seen to be

R = 287.1 (m/s)2/°R [1717 (fps)2/°K]

Thus, for this example, T = 292 °R (525 °K), so that

p = 1.087 kg/m3 (0.00211 slugs/ft3)

This corresponds to a tr of 0.888. Thus, from Figure 2.3, the density altitude is found to be 1006 m (3300 ft) and the true airspeed is calculated to be 60.4 m/s (135 mph or 198 fps).

The propeller advance ratio is defined by Equation 6.30.

J = 0.802

For this value of J, a propeller efficiency of 0.81 is read from Figure 6.15.

One can verify that the engine power for these operating conditions, from Figure 6.3, is equal to 138 bhp. Therefore, from Equation 6.33, knowing 17, P, and V, the propeller thrust can be calculated as

T-Vp T ~ V

= 310 lb (1380 N)

In steady, level flight, the propeller thrust and airplane drag must be equal. Thus, 3101b is the drag of the airplane at this particular density altitude and airspeed.

For analyzing a variable pitch propeller a set of curves for different blade pitch angles is required. These are given in Figures 6.16 and 6.17 for the propeller installed on the Piper PA-28R, the Cherokee Arrow. Here we are given both 17 and Cp as a function of J. To illustrate the use of such graphs, let us assume that they apply to the preceding example for the PA-28. Here, p = 0.00211 slugs/ft3, D = 6.17 ft, V = 198 fps, n = 40 rps, J = 0.802, and hp = 139. Thus,

P

Cp pn3D5 = 0.0633

From Figure 6.17 for the preceding cp and a J of 0.802, the blade pitch angle must be equal to 24°. Entering Figure 6.16 with this /3 and J results in an efficiency, Tj, of 0.83.

A well-designed propeller, or one carefully selected to match the engine and airplane on which it is to operate, can be expected to have a cruise efficiency of approximately 85%. At low speeds, however (e. g., during the takeoff roll), the efficiency is difficult to estimate. At zero forward speed, the efficiency of a propeller is zero by definition, even though its thrust is not zero. In fact, for the same shaft power, a variable pitch propeller will produce the most thrust at zero advance velocity (i. e., its static thrust is greater than the thrust produced in forward flight).

Figures 6.18 and 6.19 may be used to estimate the thrust attainable from a

variable pitch propeller at low forward speeds. The static thrust is first obtained from Figure 6.19 and then reduced by the factor from Figure 6.18 to give the thrust in forward flight. These curves apply only to a constant speed propeller, which will allow the engine to develop its rated power regardless of forward speed. As an example of the use of these figures, consider a propeller having a diameter of 6.2 ft, turning at 2700 rpm, and absorbing 200 hp. The power loading for this propeller is #

^ = 6.62 hp/ft2

Hence, from Figure 6.19, the static thrust to power loading should be

jr = 4-9

hp

resulting in a static thrust, T0,.for this propeller of 980 lb.

From Figure 6.18, the expected thrust at a speed of, say, 50mph (22.4 m/s) can be calculated as

= (0.715X980) = 700 lb

Approximate Useful Relationships for Propellers

Figures 6.18 and 6.19 were prepared using some approximations that are fairly accurate and convenient to use. Referring to Equations 6.31 and 6.32, assume that <j, Ch and Cd are constants, so that they can be removed from under the integral sign. In addition, it is assumed that a, and xh — 0 and ас, і. With these assumptions, CT and Cp can be written as

CT = -^r-Ci f x(A2 + x2)U2dx

Cp = JCt + ^ Q Г X2(A2 + x2)1’2 dx.
o Jo

Performing the integrations, CT and Cp become

CT = ~Qf()

CP = JCT + Cdg(X)

where

/(A) = (l + A2)3/2-A3

g(A) = [(1 + A2)1,2(2 + A2) – A4log1 + ^+— ]

/(A) and g(A) are given as a function of J in Figure 6.20. C, and Cd indicate average values of these quantities as defined by Equation 6.51.

# The term JCT in the expression for C„ simply represents the useful power. The remaining term in Cp is the profile power, or the power required to overcome the profile drag of the blades. The induced power is missing, since а, was assumed to be zero. Experience shows that the induced power is typically 12% higher than the ideal value given by Equations 6.14 and 6.15. Thus, in coefficient form,

Cp then becomes approximately

Cp = CTJ + CPi + ^ С^(Л) (6.54)

The average value of tr is referred to as propeller solidity and is equal to the ratio of blade area to disc area.

Propeller designs are sometimes identified by an “integrated design lift coefficient” and an activity factor. These are defined by

CLd = 3 f Cldx2 dx (6.55)

Jo

AF, mmf‘(£jx, dx (6 56)

The integrated design lift coefficient represents the average of the section design lift coefficients weighted by x2. The activity factor is simply another measure of the solidity. The higher the activity factor, the higher are the values of CT and Cp attainable by a propeller at a given integrated design CL.

Equation 6.54 represents about the best one can hope to achieve with a well-designed propeller operating at its design point. For the propeller shown in Figure 6.11, this corresponds to blade angles of around 15 to 25°. Beyond this range, the twist distribution along the blade departs too much from the optimum for these relationships to hold.

Other blade element theories differ from the momentum-blade element theory principally in the way in which the induced velocities are calculated.

Numerically based methods use vortex lattice models, either with a pres­cribed geometry or using vortices that are free to align themselves with the resultant flow. These numerical approaches generally require computer run times that are extremely long.

Goldstein’s classical vortex theory for predicting propeller performance is not much more difficult to apply than the momentum-blade element ap­proach. The vortex theory is described in detail in Reference 3.3 and is briefly outlined here.

Figure 6.10 shows the induced velocity at the propeller plane in more detail. The resultant velocity w is normal to Ve and is composed of a tangential component, w„ and an axial component, wa. w0 is a fictitious velocity to be discussed later. From the geometry, w, and wa are related by

V + Wg _ W, wr — Wt wa

This can be solved for wa as a function of wt.

= |[-V + W2 + 4 wt((or-wt)]

It is convenient to express all velocities in terms of Vr, so that the preceding equation becomes

t4b+V^tFt)] <«5>

Goldstein’s vortex theory relates w, to the bound circulation, Г, around any blade station by

.ВГ = 47jtkw, (6.36)

к is known as Goldstein’s kappa factor. This factor is not expressible in a closed form, but it is available in graphical form in the literature. An approximation of к is Prandtl’s tip loss factor, F, which becomes more exact as J becomes smaller or as the number of blades increases. Thus

ВГ = 4tttFw, (6.37)

where

F = fees" exp (6.38)

фт is the helix angle of the propeller’s helical trailing vortex system at the tip. For a lightly lpaded propeller,

фт = tan-1 A (6.39)

However, the lift must vanish at the tip of a propeller blade, which generally means that the local angle of attack at the tip must be zero. Thus, another expression for фт can be obtained by reference to Figure 6.6.

фт = /3 t (6.40)

For most cases, Equation 6.40 is preferred to Equation 6.39.

From the Kutta-Joukowski theorem,

L = pVT

Г = |cC(Ve (6.41)

Substituting Equation 6.41 into Equation 6.37, the result can be expressed as

o-C, £ = 8xF (6.42)

V j V t

Q can be calculated from

С/ = a 3 – tan-1 (6.43)

and VJVT from

Equations 6.35 to 6.44 can be solved iteratively for w, IVT■ C( and the other quantities can be found to evaluate Equation 6.32a and 6.32b.

If a, is assumed to be small, Reference 3.3 shows that a, can be solved directly by assuming that

The result is

«і * i(- X + /X2 + 4Y) (6.46)

where

X = tan4>+5-~—t ^ 8xF cos ф

Y _ <ra(g – ф)

8xF cos ф

As the flow passes through a propeller, the axial component of velocity increases gradually. An estimate of this variation of wa with axial distance, s, can be obtained from

w-‘w-(l+v? m) <6-47>

Woq is the value of wa at the plane of the propeller. Note that far ahead of the propeller (s = — °°), wa vanishes, while far behind the propeller (s = +°°), wa equals twice its value at the propeller plane.

The tangential component, w„ increases from zero just ahead of the propeller to 2w, just behind the propeller. This rapid change in w, through the propeller results in a curved flow field that effectively reduces the camber of the blade sections. Expressed as a reduction in the section angle of attack, Да, Reference 3.3 derives the following expression for Да. Assuming Да to be small,

As just given, Да is in radians.

wa and w, can be obtained approximately from

w, = Vra, sin (ф + a,)

wa = V&i cos (ф + a,)

As an example in the use of the vortex theory, consider the three-bladed propeller having the geometry shown in Figure 6.11. Wind tunnel testing, of this particular propeller, designated 5868-R6, Clark-Y section, three blades, is reported in Reference 6.3. These measurements are presented in Figures 6.12,

6.ІЗ, and 6.14.

This particular propeller has nearly a constant pitch from the 35% radius station out to the tip corresponding to a 15° blade angle at the 75% station. From the definition of the pitch,

p = 2irr tan /3

Thus, p/D. = xv tan /3. For an x of 0.75 and a j3 of 15°, plD = 0.631. Thus, for
a blade angle of 15° at x = 0.75, /3 at any other station will equal

0.631°

TOC

This propeller is a variable pitch propeller and the curves shown in Figures 6.12, 6.13, and 6.14 are for different values of /3 at the 75% station. If /З0.75 denotes this angle, then /3 will generally be given by

The preceding /3 is measured relative to the chord line of the Clark-Y airfoil. The angle between the zero lift line and the chord line for this airfoil is equal approximately to

a(o = 46 t/c°

where tic is the section thickness-to-chord ratio. Thus, to obtain /3 of the zero lift line, с*!,, is added to the geometric pitch angle given by Equation 6.47. Consider a value for /З0.75 and 35° and an x of 0.6. From Figure 6.7,

^ = 0.073 ’

«

– = 0.103

c

Using these values, a(o, /3, and a are calculated to be

«,„ = 4.7°

0- = 0.139 /3 = 38.5°

For this example, assume the propeller to be operating at an advance ratio of

1.4. J and A are related by

J = 7tA

Thus, for this example,

A = 0.446

For the Clark-Y airfoil,

a — 6.0 C(/rad

Cd = 0.006 + 0.010 (C, – 0.15)2

These are for an 8% thick airfoil. cd, of course, will vary slightly with thickness. With these values for a and Cd, the following quantities can be calculated in the order listed.

Since both induced and rotational losses are included ih the vortex theory, it is more accurate than the momentum-blade element theory. The vortex theory also models the physical propeller more accurately, since the loading vanishes at the blade tip.

The vortex theory was programmed and numerically integrated using a programmable calculator. These calculated points are included in Figures 6.12, 6.13, and 6.14 for a blade angle of 35°. The theory 4s seen to predict the shapes of the thrust and power curves rather well and results in values of CT and Cp that correspond to within Г of the blade angle. In fact, if a blade angle
of 34° is used in the calculations, CV and Cp values are obtained that lie almost exactly on the 35° experimental line over a range of J values from 1.1 to 1.8. Below an advance ratio of 1.1, the section angles of attack become large, so that the blades begin to stall. At this point the prediction of the propeller characteristics becomes very questionable.

The momentum-blade element theory is one means around this difficulty. If we assume a, and the drag-to-lift ratio to be small, then Ve — Vn and

Equation 6.25 can be written approximately for В blades as dT = ^ V2rca(P – ф – a,) cos ф dr

Applying momentum principles to the differential annulus and letting w ~ Vra„ we can also write, for dT,

r

Equating these two expressions for dT/dr gives the following quadratic for a,.

2 , A, <raVr o-aVr + (x+ 8?Vr/ 8?V^ ~Ф)~°

where:

Given the geometry, forward speed, and rotational speed of a propeller, Equation 6.28 can be solved for af. Equation 6.25a and 6.25b can then be numerically integrated. Using equations 6.26 and 6.27 to give the thrust and torque.

The thrust and power of a propeller are normally expressed in coefficient form. These thrust and power coefficients are defined in various ways, depending on what particular reference areas and velocities are used. Test results on propellers almost always define the thrust coefficient, CT, and power coefficient, Cp, as follows.

where n is the rotational speed in revolutions per second and D is the propeller diameter. The thrust, power, p, and D must be in consistent units. For this. convention, one might say that nD is the reference velocity and D2 is the reference area.

One would expect these dimensionless coefficients to be a function only of the flow geometry (excluding scale effects such as Mach number and Reynolds number). From Figure 6.9, the angle of the resultant flow, ф, is seen to be determined by the ratio of V to tor.

ф = tan-1 —

(ОГ

This can be written as

ф = tan ‘ —

TTX

The quantity, J, is called the advance ratio and is defined by

Thus, Ct and Cp are functions of /.

In a dimensionless form, Equations 6.25 and 6.26 can be combined and expressed as

CT = 7Г f (J2+ 7t2x2)(t[Ci cos (ф + a,) — C, j sin (ф + a,)] dx (6.32a) » Jxh

and, since P = (oQ,

Cp = f 7rx(J2 + тт2х2)а[Сі sin (ф + a,) + Q cos (ф + a,)] dx (6.32b) о JXh

xh is the hub station where the blade begins. xh is rather arbitrary, but СУ and Cp are not too sensitive to its value.

To reiterate, one would be given D, V, p, and n. Also, c and /3 would be given as a function of x. At a given station, x, a, is calculated from Equation 6.28. This is followed in order by C, and Cd and, finally, dCTldx and dCpldx. These are then integrated from jch to 1 to give Ct and Cp.

Given J and having calculated CT and Cp, one can now calculate the propeller efficiency. The useful power is defined as TV and P is, of course, the input power. Thus,

TV

т)=Ч – (6.33)

In terms of CT, Cp, and J, this becomes

(6.34)