Category AIRCRAF DESIGN

Figure 10.43 shows the maximum climb SHP and fuel flow (sfc) in nondimensional form for the standard day up to a 30,000-ft altitude for four true air speeds from 50 to 200 kts. Intermediate values may be linearly interpolated. The break in SHP generation up to a 6,000-ft altitude is due to fuel control to keep the EGT low.

Equation 11.15 (see Chapter 11; the turboprop case is not worked out) requires a factor k2 (varies with speed and altitude) to be applied to the SHP. From Figure 10.43a, a value of 0.85 may be used to obtain the initial climb SHP. Initial climb is at an 800-ft altitude. In the example, the uninstalled initial climb power is then 0.85 x 1,075 = 914 SHP. The integrated propeller performance after deducting the installation losses gives the available thrust. Typically, the initial climb starts at a constant EAS of approximately 200 kts, which is approximately Mach 0.3. At a con­stant EAS climb, the Mach number increases with altitude; when it reaches 0.4, it

is held constant. Fuel flow at the initial climb is obtained from Figure 10.43b. The example gives 0.522 x 914 = 477 lb/hr. With varying values of altitude, climb calcu­lations are performed in small increments of altitude, within which the variation is taken as the mean and is kept constant for the increment.

Maximum Cruise Rating

Figure 10.44 shows the maximum cruise SHP and fuel flow in nondimensional form for the standard day from a 5,000- to 30,000-ft altitude for true air speed from 50 to 300 kts. Intermediate values may be linearly interpolated. The graph takes into account the factor k1 (varies with speed and altitude) as indicated in Section 11.3.3, Equation 11.19.

In the example, the design initial maximum cruise speed is given as 300 kts at a 25,000-ft altitude. From Figure 10.44a, the uninstalled power available is 0.525 x 1,075 = 564 SHP. In Figure 10.44b, the corresponding fuel flow is 0.436 x 564 = 246 lb/hr. The integrated propeller performance after deducting the installation losses gives the available installed propeller performance.

Engine performance characteristics vary from type to type. It is cautioned that real engine data are not easy to obtain. The graphs in this section are generic in nature, representing typical current turboprop engines in the class. The graphs include the small amount of jet thrust available at a 70% rating and higher. The jet power is con­verted to SHP and the total ESHP is labeled only “SHP” in the graph. Therefore, in the absence of industry data, readers can continue working with these graphs. In industry, engine manufacturers supply performance data incorporating exact instal­lation losses for accurate computation.

The sizing exercise provides the required thrust at the specified aircraft – performance requirements. Using the propeller performance given in Section 10.10.5, the SHP at the sea-level static condition can be worked. From this informa­tion, engine performance at other ratings can be established for the full flight enve­lope. Takeoff rating maintains constant power but the thrust changes with speed. Figures 10.43 and 10.44 give the typical turboprop thrust and fuel flow (in terms of psfc) at maximum climb and maximum cruise ratings in a nondimensional form.

Typically, the higher the SHP, the lesser is the specific SHP (SHP/ma -SHP/lb/s). There is a similar trend for the specific dry-engine weight (SHPSLS/dry engine weight-lb/lb). Table 10.6 may be used for these computations.

Takeoff Rating

Section 10.10.5 worked out for Tucano class trainer aircraft the uninstalled SHPsls = 1,075 to obtain 4,000 lb installed thrust. Once the SHP at sea-level static condition is established from the sizing exercise, the thrust requirement of an installed four-bladed propeller can be computed.

The psfc of turboprops at takeoff is from 0.475 to 0.6 lb/hr/shp. For an SHP of less than 1,000, use a psfc of 0.6 lb/hr/shp; for more than 1,000, use a psfc of approximately 0.48 lb/hr/shp.

Therefore, fuel flow at SHP^ is 0.5 x 1,075 = 537.5 lb/hr. FromTable 10.6, the intake airmass flow at SHP^ is 0.011 x 1,075 = 11.83 lb/s. The dry-engine weight = 1,075/2.75 = 390 lb.

There are several ways to present piston engine performances. Figure 10.41 shows a Lycoming IO-360 series 180-HP piston engine. Readers may obtain the appropriate engine chart from manufacturers of other engines, or this graph may be scaled for coursework.

Readers should note that the power ratings are given in rpm. A Lycoming IO – 360 series takeoff is conducted at a maximum 2,700 rpm, whereas a climb is con­ducted at 2,500 to 2,600 rpm and cruise at 2,100 to 2,400 rpm. A partial throttle descent can be accomplished at 1,800 to 2,000 rpm. Idle is below 1,800 rpm (i. e., around 1,200 to 1,400 rpm; not shown). A fuel-flow graph is shown separately in Figure 10.42.

Piston engine power depends on the amount of airmass inhaled, which is indi­cated by the rpm and manifold pressure, pmanifold, at a particular ambient condition. A throttle valve controls airmass aspiration; when it is closed, there is no power (i. e., pmanifold = 0). When it is fully open and the engine is running at full aspira­tion, suction is created and the pmanifold reads the highest suction values. If there is less propeller load at the same rpm, less power is generated and the valve could be partially closed to inhale less airmass in order to run at equilibrium. At a low rpm, the aspiration level is low and there is a limiting pmanifold line. Therefore, the vari­ables affecting engine power are rpm, pmanifold, altitude, and atmospheric tempera­ture (nonstandard days). If an engine is supercharged, then the graphs indicate the
effect. Figure 10.41 shows the parameters in graphical form; how to use the graphs is explained herein.

Figure 10.41 shows two graphs that must be used together. The left-hand graph provides the starting point for reading conditions at sea level, and the ISA day

Figure 10.42. Lycoming IO-360 series – fuel flow graph

that must be converted to the operating condition at any altitude and atmospheric temperature are shown in the right-hand graph. The given engine condition must be known to obtain the HP at the ambient condition. In the example, the task is to find the HP that the engine is producing at 2,100 rpm, at a 23.75-inch Hg manifold pressure operating at a 2,100-ft altitude with an ambient temperature of 19°F (the ISA day is 511.2 R). The stepwise approach is as follows:

1. In the left-hand graph, locate the point corresponding to 2,100 rpm and at 23.75-inch Hg manifold pressure. Next, the HP at the Y-axis is found to be 114.

2. Transfer the 114-HP point to the Y-axis of the right-hand graph. Then, join that point to the point corresponding to 2,100 rpm and at 23.75-inch Hg manifold pressure in the same graph.

3. From the 2,100-ft altitude at the X-axis, draw a vertical line to the line drawn in Step 2. This gives 119 HP on a standard day.

5. For a nonstandard day, use the expression [BHPact/BHPstd] = j(Tstd/Tact).

6. Figure 10.42 provides information about fuel flow and has two settings – one for best power and one for best economy – which are adjusted by a mixture-ratio lever. For higher power and rpm, the mixture setting is at best power; for cruise, it is at best economy. It is evident that 2,100 rpm results in best economy. For the worked-out example, it is 50 lbs per hour at a power rating of approximately 68%.

This is the highest engine rating that can operate continuously at approximately 90 to 95% of the maximum power. It is more than what is required for climbing at a good fuel economy. Operational demand in this rating arises from specific situa­tions – for example, a very fast climb to altitude (mainly in military use). Typically, a climb is accomplished at around 85% power to reduce stress on the engine and to achieve better fuel economy. For this reason, the maximum continuous rating is not included with some engines and rather is merged with the maximum climb rating.

Maximum Climb Rating

The climb schedule is accomplished at approximately 85 to 90% of the maximum power. A typical climbing time for a civil aircraft is less than 30 to 40 minutes, but it can run continuously.

Maximum Cruise Rating

This rating is approximately 80 to 85% of the maximum power matched to the max­imum cruise-speed capability. Unless there is a need for higher speed, typical cruise is performed at a 70 to 75% power rating, called the cruise rating. This gives the best fuel economy for the LRC. In a holding pattern in an airport vicinity, engines run at still lower power, barely maintaining altitude while waiting for clearance to proceed. The rating depends on the weight of an aircraft; at the end of cruise (lightweight), an approximate 65% rating is sufficient.

Idle Rating

This rating is at approximately 40 to 50% of maximum power and is intended for an engine to run without flameout but also produces practically no thrust. This situation arises at descent, at approach, or on the ground. During a descent, it has been found that better economy can be achieved by descending at partial throttle, at about a 60% power rating. This results in a shallower glide slope to cover more distance and consume less fuel.

Representative engine performances of various types at takeoff, maximum climb, and maximum cruise ratings are given in this section for an ISA day. Engine manufacturers also supply performance data for non-ISA days, which is more crit­ical for hot and high-altitude conditions when engines produce considerably less power. To protect engines from heat stress, a fuel-control system is tuned to cut off power generation to a flat-rated value (at an ISA-day engine rating) up to a hot day that can be 20° C above the ISA day. This book does not address non-ISA-day performances; in the industry, they are supplied.

This section describes typical engine outputs. The data are presented in nondimen­sional form. All power plants have prescribed power settings, as discussed herein. Power settings are decided by the engine rpm and/or by the exhaust-pressure ratio

(EPR) at the jet pipe temperature (JPT), which should remain lower than prescribed levels. An engine is identified at its sea-level STD day at takeoff in static conditions. For turbofans, it is denoted as TSLs. Typical power settings (i. e., ratings) of engines are given in the following subsections.

Takeoff Rating

This is the highest rating that produces the maximum power and is rated at 100% (i. e., at static run of turbofans, it is TSLs). At this rating, the engine runs the hottest and therefore has a time limit to ensure a longer life and less maintenance. For civil – aircraft engines, the limit is about 10 minutes; for military aircraft engines, it could extend somewhat longer. A situation may arise in which one engine is inoperative and the operative engine must supply more power at the augmented power rating (APR). Not all engines have an APR, which exceeds 100% power by «5% for a short duration (e. g., «5 minutes).

Using the graphs, a linear extrapolation can be made for two – and five-bladed pro­pellers with a similar AF. Reference [18] discusses the subject in detail with pro­peller charts for other AF.

10.10.1 Propeller Performance at STD Day: Worked-Out Example

In a stepwise manner, thrust from a propeller is worked out as a coursework exer­cise. The method uses the Hamilton Standard charts intended for constant-speed propellers. These charts also can be used for fixed-pitch propellers when the pitch of the propeller should match the best performance at a specific speed: either cruise speed or climb speed. Two forms are shown: (1) from the given thrust, compute the HP (in turboprop case SHP) required; and (2) from the given HP (or SHP), compute the thrust. The starting point is the Cp. Typically, at sea-level takeoff rating at static condition, one SHP produces about 4 pounds on STD day. At the first guesstimate, a factor of 4 is used to obtain SHP to compute the Cp. One iteration may prove sufficient to refine the SHP.

Problem description. Consider a single, 4-bladed, turboprop military trainer air­craft of the class RAF Tucano operating with a constant speed propeller at N = 2,400 rpm giving installed TSLS = 4,000 lbs. The specified aircraft speed is 320 mph at a 20,000-ft altitude (i. e., Mach 0.421). For the aircraft speed, the blade AF is taken as 180. Establish its rated SHP at sea-level static condition and thrusts at various speeds and altitudes. All computations are in STD day.

Case I: Takeoff performance (HP from thrust). This is used to compute the SHP at sea-level takeoff. Guesstimate installed SHP = 4,000/4 = 1,000 SHP.

From Figure 10.39, for a four-bladed propeller, the diameter is taken to be 96 inches, or 8 ft. Figure 10.38 establishes the integrated design Cu; the ratio of the speed of sound at STD day sea level to the altitude, fc = 1.0.

The factor ND x fc = 2,400 x 8 x 1.0 = 19,200. Corresponding to aircraft speed of Mach 0 and the factor NDx fc = 19,200; Figure 10.38 gives the integrated design Cu ^ 0.5.

Equation 10.44 gives:

Cp = (550 x SHP)/(pn3 D5)

or Cp = (550 x 1,000)/(0.00238 x 403 x 85) = 5,50,000/49,91,222 = 0.11

Figure 10.36 (4-blade, AF = 180, CL = 0.5) gives CT/CP = 2.4 corresponding to integrated design Cui = 0.5 and CP = 0.11.

Therefore, installed static thrust, TSLS = (CT/CP)(33,000 x SHP)/ND = (2.4 x 550 x 1,000)/(40 x 8) = 1,320,000/320 = 4,125 lb.

The installed SHP is revised to 970 giving installed thrust TSLS = 4,000 lb. It is close enough to avoid any further iteration.

Taking into account a 7% installation loss at takeoff, the uninstalled TSLS = 4,000/0.93 = 4,300 lb, giving the uninstalled SHP = 1,043. It may now be summa­rized that to obtain 4,000 lb installed thrust, the uninstalled rated power is 1,043 SHP.

Aircraft configuration must ensure ground clearance at a collapsed nose-wheel tire and oleo. A higher number of blades (i. e., higher solidity) could reduce the diameter, at the expense of higher cost. For this aircraft class, it is best to retain the largest propeller diameter permissible, keeping the number of blades to four or five.

If ground clearance is required, then a 1.5-inch radius can be cut off from the tip (i. e., a 3% reduction to a 93-inch diameter), which requires slightly higher

uninstalled power to « 1,043/0.97 = 1,075 SHP to obtain 4,000 lb installed thrust corresponding to installed 1,000 SHP.

Check the diameter with the empirical relation, D = K( P)025 = 18 x (1,050)°-25 = 102.5 inch. The empirical relation is close to the cropped 93-inch diameter computed previously. The smaller diameter is retained for better ground clearance.

Case II: Thrust from HP (worked out with 1,000 SHP installed as maximum takeoff rating at sea-level static condition on STD day). Once the installed SHPsls is known, the thrust for the takeoff rating can be computed. The turboprop fuel control maintains a constant SHP at takeoff rating, keeping it almost invariant. This section computes the thrust available at speeds up to 160 mph to cover liftoff and enter the enroute climb phase. Available thrust is computed at 20, 50, 80,120, and 160 mph, as shown in Table 10.5.

Compute:

J = V/nD = (1.467 x V)/(40 x 8) = 0.00458 x V, where V is in mph

Power coefficient:

Cp = (550 x SHP)/(pn3D5) = (550 x 1,000)/(0.00238 x 403 x 85) = 0.11

For an integrated design, Си = 0.5 and CP = 0.11. The propeller nprop corre­sponding to J and CP is obtained from Figure 10.37 (4-blade, AF = 180, Си = 0.5), as shown in Table 10.5.

Compute thrust: T = (550 nprop x 1,000)/V = 550,000 x nprop/ V, where V is in mph (see Table 10.5). Use Equation 10.46 for the FPS system.

Figure 10.40 plots thrust versus speed at the takeoff rating (see Section 10.11). In a similar manner, thrust at any speed, altitude, and engine rating can be determined from the relevant graphs (Figures 10.41 through 10.44). Section 13.3.4 works out the installed turboprop thrust.

Refer to Section 10.11.2 to obtain the SHP at various engine ratings. For exam­ple, an engine throttles back from the takeoff rating to the maximum climb rating for an enroute climb. Up to about 4,000-ft altitude, it is kept at around 85% of the maximum power and goes down with altitude.

This book does not discuss propeller design. Aircraft designers select propellers offered by the manufacturer, mostly off-the-shelf types, unless they are specially designed in consultation with aircraft designers, such as the rubberized turbofan. This section describes considerations that are necessary and appropriate to aircraft designers in selecting an appropriate propeller to match the sized engine in order to produce thrust for the full flight envelope.

Readers may note that the propeller charts for the number of blades use only three variables: Cp, в, and n (the subscriptp is omitted); they do not specify the pro­peller diameter and rpm. Therefore, similar propellers with the same AF and CLi can use the same chart. Aircraft designers must choose AF or CLi based on the crit­ical phase of operation. The propeller selection requires compromises because opti­mized performance for the full flight envelope is not possible, especially for fixed – pitch propellers.

Recently, certification requirements for noise have affected the issues of com­promise, especially for high-performance propeller designs. A high-tip Mach num­ber is detrimental to noise; to reduce it to n is compromised by reducing the rpm and/or the diameter, thereby increasing J and/or the number of blades. Increasing the number of blades also increases the cost and weight of an aircraft. Propeller curvature is suitable for transonic operation and helps reduce noise.

Figure 10.38. Design CL to avoid compress­ibility loss

Equation 10.22 gives the aerodynamic incidence – that is, the blade angle of attack, a= (в – ф), where у is determined from the aircraft speed and propeller rpm (i. e., function of J = V/nD). It is best to keep a constant along the blade radius to obtain the best Cui (i. e., a is maintained at 6 to 8 deg). The value of 0.7r or 0.75r is used as the reference point – the propeller charts list the reference radius.

The combination of the designed propeller rpm is matched to its diameter to prevent the operation from experiencing a compressibility effect at the maximum speed and specific altitude. A suitable reduction-gear ratio decreases the engine rpm to the preferred propeller rpm. Figure 10.36 is used to obtain the integrated design Cu for the propeller rpm and diameter combination. The factor ND x (ratio of speed of sound at standard day, sea level to the altitude) establishes the integrated design Cu. A spinner at the propeller root is recommended to reduce loss.

The following stepwise observations and information are important to progress the propeller-performance estimation by using the charts in Figures 10.34 through 10.38 (in the figure fc = aait/aSu, where a = speed of sound):

1. Establish the integrated design Cui using Figure 10.38.

2. A typical blade AF is of the following order:

• Low power absorption, 2- to 3-bladed, propellers for homebuilt flying = 80 < AF < 90.

• Medium power absorption, 3- to 4-bladed propellers for piston engines (utility) = 100 < AF < 120.

• High power absorption, 4-bladed and more propellers for turboprops = 140 < AF < 200.

3. Keep the tip Mach number around 0.85 at cruise and ensure that at takeoff, the rpm does not exceed the value at the second segment climb speed.

4. Typically, for a constant-speed, variable-pitch propeller, в is kept low for take­off, gradually increasing at climb speed, reaching an intermediate value at cruise and a high value at the maximum speed. Figure 10.32 shows the benefit of в-control compared to fixed-pitch propellers. Although the figure demonstrates the merit of a constant-speed propeller, its constraints render the governor design and в – control as complex engineering, which requires two modes of operation (not addressed in this book). Design of an automatic blade-control mechanism is specialized engineering.

5. The propeller diameter in inches can be roughly determined by the following empirical relation:

D = K( P)025,

where K = 22 for a 2-blade propeller, 20 for a 3-blade propeller, and 18 for a 4-blade propeller. Power P is the installed power, which is less than the bare engine rating supplied by the engine manufacturer. Figure 10.39 provides the statistics of a typical relationship between engine power and propeller diameter. It is a useful graph for making empirically the initial size of the propeller. If n and J are known in advance, the propeller diameter can be determined using D = 1,056V/(NJ) in the FPS system.

Figure 10.39. Engine power versus diameter

6. Keep at least a 0.5 m (1.6 ft) propeller-tip clearance from the ground; in an extreme demand, this can be reduced slightly. This should prevent the nose – wheel tire from bursting and an oleo collapse.

7. At maximum takeoff static power, the thrust developed by the propellers is about four times the power.

Continue separately (in FPS) with propeller performance for static takeoff and in­flight cruise.

Static Performance (see Figures 10.34 and 10.36)

1. Compute the power coefficient, CP = (550 x SHP)/(pn3D5), where n is in rps.

2. From the propeller chart, find CT/CP.

3. Compute the static thrust, TS = (CT/CP)(33,000 x SHP)/ND, where N is in rpm.

In-Flight Performance (see Figures 10.35 and 10.37)

1. Compute the advance ratio, J = V/(nD).

2. Compute the power coefficient, CP = (550x SHP)/(pn3D5), where n is in rps.

3. From the propeller chart, find efficiency, nP.

4. Compute thrust, T from nP = (TV)/(550 x SHP), where V is in ft/s.

If necessary, off-the-shelf propeller blade tips could be slightly shortened to meet geometrical constraints. Typical penalties are a 1% reduction of diameter affecting 0.65% reduction in thrust; for small changes, linear interpolation may be made.

The practical application of propellers is obtained through blade-element theory, as described herein. A propeller-blade cross-sectional profile has the same functions as that of a wing aerofoil – that is, to operate at the best L/D.

Figure 10.30 shows that a blade-element section, dr, at radius r, is valid for any number of blades at any radius, r. Because blades are rotating elements, their prop­erties vary along the radius.

Figure 10.30 is a velocity diagram showing that an aircraft with a flight speed of V with the propeller rotating at n rps makes the blade element advance in a helical manner. VR is the relative velocity to the blade with an angle of attack a. Here, в is the propeller pitch angle, as defined previously. Strictly speaking, each blade rotates in the wake (i. e., downwash) of the previous blade, but the current treatment ignores this effect and uses propeller charts without appreciable error.

Figure 10.30 is the force diagram of the blade element in terms of lift, L, and drag, D, that is normal and parallel, respectively, to VR. Then, the thrust, AT, and force, AF (producing torque), on the blade element can be obtained easily by decomposing lift and drag in the direction of flight and in the plane of the propeller rotation, respectively. Integrating this over the entire blade length (i. e., nondimensionalized as r/R – an advantage applicable to different sizes) gives the thrust, T, and torque – producing force, F, of the blade. The root of the hub (with or without spinner) does not produce thrust, and integration is typically carried out from 0.2 to the tip, 1.0, in terms of r/R. When multiplied by the number of blades, N, this gives the propeller performance.

Therefore, propeller thrust:

1.0

T = Nx ATd(r / R) (10.37)

0.2

and force that produces torque:

1.0

F = Nx AFd(r / R) (10.38)

0.2

By definition, advance ratio: J = V/(nD)

It has been found that from 0.7r (i. e., tapered propeller) to 0.75r (i. e., square propeller), the blades provide the aerodynamic average value that can be applied uniformly over the entire radius to obtain the propeller performance.

It also can be shown that the thrust-to-power ratio is best when the blade ele­ment works at the highest lift-to-drag ratio (L/Dmax). It is clear that a fixed-pitch blade works best at a particular aircraft speed for the given power rating (i. e., rpm) – typically, the climb condition is matched for the compromise. For this rea­son, constant-speed, variable-pitch propellers have better performance over a wider speed range. It is convenient to express thrust and torque in nondimensional form, as follows. From the dimensional analysis (note that the denominator omits the 1/2): Nondimensional thrust,

Tc = Thrust/(p V2 D2)

Thrust coefficient,

Ct = Tc x J = Thrust x [V/(nD)]2/(pV2D2) = Thrust/(pn2D4) (10.39)

In FPS system:

where a = ambient density ratio for altitude performance Nondimensional force (for torque), TF = F/(p V2D2) Force coefficient:

Cf = Tf x J = F x [V/(nD)]2/(pV2D2) = F/(pn2D4) (10.41)

Therefore, torque:

Q = force x distance = Fr = Cf x (pn2 D4) x D/2

Figure 10.34. Static performance: three bladed propeller performance chart – AF100 (for a piston engine)

or torque coefficient:

(10.44)

The wider the blade, the higher the power absorbed to a point when any fur­ther increase would offer diminishing returns in increasing thrust. A nondi­mensional number, defined as the total activity factor (TAF) = N x (105/16) /o’0 (r/R)3(b/D)d(r/R), expresses the integrated capacity of the blade element to absorb power. This indicates that an increase in the outward blade width is more effective than at the hub direction.

A piston engine or a gas turbine drives the propeller. Propulsive efficiency np can be computed by using Equations 10.35,10.39, and 10.44.

Propulsive efficiency,

np = (TV)/[BHP or ESHP]

= [CT x (pn2D4) x V]/[CP x (pn3D5)]

= (Ct/Cp) x [V/(nD)] = (Ct/Cp) x J (10.45)

The theory determines that geometrically similar propellers can be represented in a single nondimensional chart (i. e., propeller graph) combining the nondimen­sional parameters, as shown in Figures 10.34 and 10.35 (for three-bladed propellers) and Figures 10.36 and 10.37 (for four-bladed propellers). Considerable amount of

Figure 10.35. Three-bladed propeller performance chart – AF100 (for a piston engine)

coursework can be conducted using these graphs. These graphs and the procedures to estimate propeller performances are from [16], a courtesy of Hamilton Standard. These graphs are replotted retaining maximum fidelity. The reference provides the full range of graphs for other types of propellers and charts for propellers with a higher activity factor (AF).

Static computation is problematic when V is zero; then np = 0. Different sets of graphs are required to obtain the values of (Єт/Cp) to compute the takeoff thrust, as shown in Figures 10.34 and 10.36. Finally, Figure 10.38 is intended for selecting the design CL for the propeller to avoid compressibility loss. Thrust for takeoff per­formance can be obtained from the following equations in FPS:

In flight, thrust:

T = (550 x BHP x np)/V, where V is in ft/s = (375 x BHP x np)/V, where V is in mph (10.46)

For static performance (takeoff):

Tto = [(Ct/Cp) x (550 x BHP)]/(nD) (10.47)

Figure 10.36. Four-bladed propeller performance chart – AF180 (for a high – performance turboprop)

Figure 10.37. Four-bladed propeller – performance chart – AF180 (for a high – performance turboprop)

The fundamentals of propeller performance start with the idealized consideration of momentum theory. Its practical application in the industry is based on the subse­quent “blade-element” theory. Both are presented in this section, followed by engi­neering considerations appropriate to aircraft designers. Industrial practices still use a propeller that is supplied by the manufacturer and wind-tunnel-tested generic charts and tables to evaluate its performance. Of the various forms of propeller charts, two are predominant: the NACA method and the Hamilton Standard (i. e., propeller manufacturer) method. This book prefers the Hamilton Standard method used in the industry ([16]). For designing advanced propellers and propfans to oper­ate at speeds greater than Mach 0.6, CFD is important for arriving at the best com­promise, substantiated by wind-tunnel tests. CFD employs more advanced theories (e. g., vortex theory).

Momentum Theory: Actuator Disc

The classical incompressible inviscid momentum theory provides the basis for pro­peller performance ([21]). In this theory, the propeller is represented by a thin actu­ator disc of area, A, placed normal to the free-stream velocity, V0. This captures a stream tube within a CV that has a front surface sufficiently upstream represented by subscript “0” and sufficiently downstream represented by subscript “3” (Fig­ure 10.33). It is assumed that thrust is uniformly distributed over the disc and the tip effects are ignored. Whether or not the disc is rotating is irrelevant because flow through it is taken without any rotation. The station numbers just in front and aft of the disc are designated as 1 and 2.

The impulse given by the disc (i. e., propeller) increases the velocity from the free-stream value of V0, smoothly accelerates to V2 behind the disc, and continues to accelerate to V3 (i. e., Station 3) until the static pressure equals the ambient pres­sure, p0. The pressure and velocity distribution along the stream tube is shown in Figure 10.33. There is a jump in static pressure across the disc (from p1 to p2), but there is no jump in velocity change.

Newton’s law states that the rate of change of momentum is the applied force; in this case, it is the thrust, T. Consider Station 2 of the stream tube immediately behind the disc that produces the thrust. It has a mass flow rate, hi = pAdiscV2, and

Figure 10.33. Control volume showing the stream tube of the actuator disc

the change of velocity is AV = (V3 – V0). This is the reactionary thrust experienced at the disc through the pressure difference multiplied by its area, A.

Thrust produced by the disc T = the rate of the change of momentum = m AV

= P Adisc x (V – Vo)xV>

= pressure across the disc x Adisc = Adisc X (P2 – Pi) (10.23)

Equation 10.23 now can be rewritten as:

P(V3 – V0) x V2 = (p2 – pi) (10.24)

The incompressible flow in Bernoulli’s equation cannot be applied through the disc imparting the energy. Instead, two equations are set up: one for conditions ahead of and the other aft of the disc. Ambient pressure, p0, is the same everywhere.

Ahead of the disc:

P0 + 1/2P V2 = P1 + 1/2P V12 (10.25)

Aft of the disc:

P0 + 1/2P V32 = P2 + 1/2P V22 (10.26)

Subtracting the front relation from the aft relation:

1/2P (V32 – V>2) = (P2 – P1) X 1/2P (V22 – V2) (10.27)

Because there is no jump in velocity across the disc, the last term is omitted. Next, substitute the value of (p2 – p1) from Equation 10.24 in Equation 10.25:

1/2(V32 – V? = (V3 – V0) X V2 or (V3 + V0) = 2V2

Note that (V3 – V0) = AV, when added to Equation 10.26, gives 2V3 = 2V2 + A V, or:

Using conservation of mass, A3V3 = AVi, Equation 10.23 becomes:

T = pAdiscV1 X (V3 – V0) = Adisc(p2 — P1) or (P2 — P1) = p V1 X (V3 — V0) (10.30)

This means that half of the added velocity, AV/2, is ahead of the disc and the remainder, AV/2, is added aft of the disc.

Using Equations 10.29 and 10.30, thrust Equation 10.23 can be rewritten as:

T = AdscPV1 x (V3 — V0) = AdscP(V0 + AV/2) x AV (10.31)

Applying this to an aircraft, V0 may be seen as the aircraft velocity, V, by drop­ping the subscript “0”. Then, the useful work rate (power, P) on the aircraft is:

P = TV (10.32)

For the ideal flow without the tip effects, the mechanical work produced in the system is the power, Pideai, generated to drive the propeller force (thrust, T) times velocity, V1, at the disc.

Pideai = T(V + AV/2) (the maximum possible value in an ideal situation) (10.33) Therefore, ideal efficiency:

Пі = P/ Pideai = (TV)/[T(V + AV/2)] = 1/[1 + (AV/2V)] (10.34)

The real effects have viscous, propeller tip effects and other installation effects. In other words, to produce the same thrust, the system must provide more power (for a piston engine, it is seen as the BHP, and for a turboprop, the ESHP), where ESHP is the equivalent SHP that converts the residual thrust at the exhaust nozzle to HP, dividing by an empirical factor of 2.5. The propulsive efficiency as given in Equa­tion 10.4 can be written as:

np = (TV)/[BHP or ESHP] (10.35)

This gives:

np/Пі = {(TV)/[BHP or SHP]}/{1/[1 + (AV/2V)]}

= {(TV)[1 + (A V/2V)]/[BHP or SHP]} = 85 to86% (typically) (10.36)

The industry uses propeller charts that incorporate special terminology. The neces­sary terminology and parameters are defined in this section. Figure 10.30 shows a two-bladed propeller with a blade-element section, dr, at radius r. The propeller has a diameter, D. If ю is the angular velocity, then the blade-element linear velocity at radius r is юг = 2nnr = nnD, where n is the number of revolutions per unit time. An aircraft with a true air speed of V and a propeller angular velocity of ю has the blade element moving in a helical path. At any radius, the relative velocity, VR, has an angle p = tan-1(V/2nnr). At the tip, ptip = tan-1(V/nnD).

D = propeller diameter = 2x r n = revolutions per second (rps) ю = angular velocity N = number of blades

b = propeller-blade width (varies with radius, r)

P = propeller power

Cp = power coefficient (not to be confused with the pressure coefficient) = P/(p n3D5)

T = propeller thrust

Cn = integrated design lift coefficient (CLd = sectional lift coefficient)

CT = propeller thrust coefficient = T/(pn2D4)

в = blade pitch angle subtended by the blade chord and its rotating plane p = propeller pitch = no slip distance covered in one rotation = 2nr tan в (explained previously)

VR = velocity relative to the blade element = V(V2 + ю2г2) (blade Mach num­ber = VR/a)

p = angle subtended by the relative velocity = tan-1(V/2nnr) or tan p = V/пnD (This is the pitch angle of the propeller in flight and is not the same as the blade pitch, which is independent of aircraft speed.) a = angle of attack = (в – p)

J = advance ratio = V/(nD) = ntanp (a nondimensional quantity – analogues to a)

AF = activity factor = (105/16) /010 (r/R)3(b/D)d(r/R)

TAF = total activity factor = Nx AF (it indicates the power absorbed)

However, irrespective of aircraft speed, the inclination of the blade angle from the rotating plane can be seen as a solid-body, screw-thread inclination and is known as the pitch angle, в. The solid-body, screw-like linear advancement through one rota­tion is called pitch, p. The pitch definition is problematic because unlike mechanical screws, the choice of the inclination plane is not standardized. It can be the zero – lift line (which is aerodynamically convenient) or the chord line (which is easy to locate) or the bottom surface – each plane has a different pitch. All of these planes are interrelated by fixed angles. This book uses the chord line for the pitch reference line as shown by the pitch angle, в, in Figure 10.30; this gives pitch, p = 2nr tanв.

Because the blade linear velocity mr varies with the radius, the pitch angle needs to be varied as well to make the best use of the blade-element aerofoil character­istics. When в is varied such that the pitch is not changed along the radius, then the blade has constant pitch. This means that в decreases with increases in r (the variation in в is about 40 deg from root to tip). The blade angle of attack is:

a = (в-ф) = tan-1(p/2nr) – tan-1(V/2nnr) (10.22)

This results in an analog nondimensional parameter, J = advance ratio = V/(nD) = n tan<p.