AE317 Aircraft Flight Mechanics & Performance

UNIT C: Performance

ROAD MAP . . .

C-1: Equation of Motion

C-2: Glides, Climbs, Range, & Endurance

C-3: Takeoff, Landing, & Turn

C-4: V-n Diagram & Constraint Analysis

C-5: Performance Analysis Examples

Brandt, et.al., Introduction to Aeronautics: A Design Perspective

Chapter 5: Performance

5.1 Design Motivation5.2 Equations of Motion5.3 Propulsion5.4 Drag and Thrust Required5.5 Lift-to-Drag Ratio5.6 Power Curves5.7 Curve Shifts

Unit C-1: List of Subjects

Equation of Motion

Aircraft Propulsion Systems

Piston Engine/Propeller

Turbojet Engines & Afterburners

Turbofan & Turboprop Engines

Thrust Model Summary

Straight Level Unaccelerated Flight (SLUF)

Lift-to-Drag Ratio & Power Curves

Curve Shifts

Page 2 of 13 Unit C-1

Aircraft performance analysis is the science of predicting what an aircraft can do: how fast and how high

it can fly, how quickly it can turn, how much payload it can carry, how far it can continue flying, and

how short (long) a runway can (needs to) be for takeoff and landing.

Equation of Motion

• The equation of motion for the aircraft can be derived by summing the forces on the aircraft in two

directions: the one in parallel ( ) to the aircraft's velocity vector, and the one in perpendicular ( )⊥

to it.

cos sinTF ma T D W = = − − (5.1) Equation of Motion (in the flight path direction)

2 sin cosTF mV r T L W ⊥ = = + − (5.2) Equation of Motion (in the vertical plane: loop)

: Flight path angle (angle between the horizon and the aircraft's velocity vector, that is opposite

against relative wind)

: Angle of attack (angle between the velocity vector and the aircraft's reference line, often

chosen as the central axis of the fuselage rather than the wing chord line: arbitrary)

T : Thrust angle (angle between the thrust vector and the velocity vector: will not, in general, be

the same as )

r : Radius of turn (if aircraft is turning)

• For many aircraft, it is acceptable to assume that the thrust vector is approximately aligned with the

velocity vector, so that 0T = . This simplifies the equation of motion: cos 1T =

• A very simple but extremely useful condition is that of straight, level, unaccelerated flight, called

SLUF. Under the SLUF condition, 0 = and both components of acceleration are zeros:

T D= and L W= (5.3)

Aircraft Weight

• Aircraft weight will be given as the sum of the aircraft empty weight ( )eW ,the weight of the fuel

( )fW , and the weight of the payload (including pilot and crew) ( )pW :

e f pW W W W= + + (5.4)

• The weight of the payload can be further broken down to permanent payload ( )ppW and expendable

payload ( )xpW : p pp xpW W W= +

Equation of Motion

(5.1)

(5.2)

(5.3)

(5.4)

Page 3 of 13 Unit C-1

Propulsion System Characteristics

• Engine maximum thrust to weight ratio ( )engSLT W : ratio of engine's sea-level output to its own

weight)

• Engine's thrust specific fuel consumption (TSFC): ratio of rate of fuel consumption to thrust

output)

fTSFC W T (5.5) (frequently represented by the symbol: tc )

• If fuel consumption rate has units of "lb/hr" and thrust is in "lb," then TSFC has units of "1/hr."

• An engine that is suitable for a particular flight regime would have a relatively high engSLT W and a

relatively low TSFC.

• Common operating envelopes: ranges of operation altitudes and Mach numbers, and/or true

airspeeds, will be different for each different type of propulsion system.

Aircraft Propulsion Systems

(5.5)

Page 4 of 13 Unit C-1

Piston Engines

• The power produced by a piston engine varies with the size and number of cylinders, the rate at

which the crankshaft rotates, and the density of the air it is using.

• Engine shaft horsepower (SHP) ratings: horsepower (1 hp = 550 ftlb/s) or "kilowatts" in standard

sea-level conditions at a specified maximum rotation rate given in "rev/min" (rpm).

( )available SL SLSHP SHP = (5.6)

Propeller

• The propeller is not 100% efficient at converting engine SHP into thrust. A portion of the engine's

power is used to overcome the aerodynamic drag (profile, induced, and/or wave drags) of the

propeller blades. A constant speed (or "variable-pitch") propeller is designed so that the angle of

attack (or pitch) of the blades can be adjusted to maintain a constant engine rpm. This feature helps

keep the propeller's efficiency high over a wider airspeed range. The variable pitch capability can

also be used to allow the engine to turn at its rated rpm, regardless of the aircraft's airspeed.

• The efficiency of the propeller P is defined as:

P THP SHP = (5.7), where: THP is the thrust horsepower of the engine, THP T V= (5.8)

• Typically, 0.9P for a good constant-speed propeller at the engine rated rpm and aircraft

operating airspeed of below 300 ft/s.

• The available thrust of the engine/propeller combination is then given by:

SL

SL

PAT SHP

V

= (5.9)

Piston Engine/Propeller

(5.6)

(5.7)

(5.8)(5.9)

Page 5 of 13 Unit C-1

Turbojet Engines

• Due to its inefficiency of propellers at high subsonic regime and up, either turbojet or turbofan

engines will be a choice for high subsonic (transonic) and supersonic flight regimes.

• The thrust generated by a turbojet engine is proportional to the rate of change at which the

momentum of the air flowing through the engine:

( )eT m V V= − (5.10) Turbojet Engine Thrust

m : Mass flow rate through the engine

eV : Exhaust gas velocity

• Turbojet thrust varies with altitude and velocity. The maximum thrust ("thrust available") is:

( )SL SLAT T = (5.11)

SLT : Thrust available at sea-level conditions

Afterburners

• The amount of energy that can be added to the gases flowing through a normal turbojet engine is

limited by the temperature that the gases can safely flow through the turbine. Excessive gas

temperatures cause turbine blades to fatigue, deform, or fail.

• However, once the gases have passed through the turbine, it is possible to mix more fuel and burn to

increase the exhaust gas velocity: this is the afterburner.

• Afterburner is not as efficient in converting heat into kinetic energy as the main engine, and so TSFC

increases when afterburner is used. Typically: 50% increase of thrust in full throttle by 200%

increase of fuel flow. Thrust of turbojet with afterburner can be approximated by:

( )( )SL SL 1 0.7AT T M = + (5.12)

Turbojet Engines & Afterburners

(5.10)

(5.11)

(5.12)

Page 6 of 13 Unit C-1

Turbofan Engines

• To reduce the TSFC of a turbojet engine, one of the engine's spools can be connected so that it

drives a larger compressor or fan at the font of the engine. Some of the air drawn in and accelerated

by this fan does not flow through the engine core: this is called, bypass air.

• The ratio of the bypass mass flow rate to the mass flow rate of the air flowing through the engine

core is called, bypass ratio.

• Low-bypass-ratio turbofans behave much like turbojets. On the other hand, high-bypass-ratio

turbofans exhibit a rapid decrease in maximum thrust output with increasing velocity at low

altitudes. Fig. 511 compares low (0.7: with afterburner) and high (5) bypass ratio turbofans.

• For low-bypass-ratio turbofan: eq(5.11) & (5.12) will apply (with/without afterburner)

• For high-bypass-ratio turbofan, thrust can be approximated by:

( ) ( )SL SL0.1AT M T = (5.13)

Supersonic Variable Engine Inlets

• The decrease of maximum thrust in supersonic flight regime will be caused by shock wave

formations at engine inlet. Flow separation caused by shock waves will create loss of total pressure.

• Some inlets have the capability to change shapes so that efficient operation in wider range of flight

regime is possible (i.e., F-15).

Turbofan & Turboprop Engines

(5.13)

(5.14)

Page 7 of 13 Unit C-1

Turboprops

• Turboprop replaces the fans of high-bypass-ratio turbofans with propellers. Operating

characteristics are similar to those of high-bypass-ratio turbofans (typically have lower TSFC).

• Turboprops lose thrust at high speeds more like piston engine/propeller propulsion systems.

• Thrust-to-weight ratio for turboprop is usually higher than piston engine/propeller system, but TSFC

is also usually higher.

• Turboprops are designed so that the high-energy air from the burners expands almost completely to

ambient static pressure in the turbines, so that almost all of the energy is converted to into SHP

(rather than thrust). Any additional thrust (produced by engine exhaust) is included in the sea-level

power rating at the rate of 8 N of thrust per kW (2.5 lb of thrust per horsepower).

• A turboprop power rating corrected in this way is called, effective shaft horsepower (ESHP):

( )( )SL SLA PT ESHP V = (5.14)

Ramjets

• At very high Mach numbers (supersonic to hypersonic regime), the air that enters a jet engine inlet is

significantly slowed and compressed. The turbomachinery components (compressors and turbines)

are no longer needed. The resulting engine is a little more than an afterburner connected to the inlet

(ramjet engine).

• For ramjets, the air is compressed by ram effect (naturally increased static pressure, when air is

slowed by the inlet). Ramjets cannot function at low speeds (insufficient ram effect). Ramjets need

to be accelerated to the operating speed by some other propulsion system. At Mach numbers above

~ M = 3, ramjets are more efficient than afterburning turbojets.

Rockets

• For extremely high speeds and for space flight, rocket engines are typical choice. Rockets carry

their own oxidizer, so that they do not need to take air at all.

Thrust Model Summary

(5.15)

(5.16)

Page 8 of 13 Unit C-1

Propulsion Type and Thrust Model

• Table 5.1 summarizes the equations that will be used in all performance calculations as models for

the variation of thrust with density and Mach number (or velocity).

TSFC Models

• Thrust specific fuel consumption (TSFC) varies only mildly with Mach number.

• Power specific fuel consumption (PSFC) is the fuel flow required for a given power output (rather

than thrust). Piston and turboprop engine TSFCs vary with Mach number, but PSFCs remains

relatively constant with Mach number and with variations in air temperature.

• PSFC is usually called brake specific fuel consumption (BSFC), because it is measured as the brake

power output for a given fuel flow. Brake power is measured by connecting the engine to a brake

(or dynamometer) that absorbs power and measures the engine's toque and rpm. Because propeller

is not involved in this measurement, propeller efficiency must be included to determine fuel

consumption for a given thrust power output.

• TSFC values for turbine engines generally vary, based on the temperature (or speed of sound):

( )SL SLSL SLt t tc c Temp Temp c a a= = (5.15)/(5.16)

(NOTE) T is "thrust" and Temp is "temperature," in order to avoid confusion.

Installed Thrust and TSFC

• For a variety of reasons, the thrust produced by an engine is frequently less, when it is installed in an

aircraft than when it is tested uninstalled. Some of the sources of this "thrust loss" include viscous

losses in inlets, loss of momentum of cooling air, and power and compressed air bleed requirements

to run engine accessories.

• If installed sea-level thrust and TSFC ratings are supplied by manufacturers, we should use them: if

only uninstalled ratings are available, decrease thrust and increase TSFC by ~ 20% to approximate

the installed values.

Thrust Model Summary (Continued)

Page 9 of 13 Unit C-1

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Solution (5.1)

Example C-1-1(Aircraft Propulsion Systems)

A new afterburning low-bypass-ratio turbofan engine produces 15,000 lb of thrust in military power and 22,000 lb of thrust in afterburner installed in static, sea-level atmospheric condition. Its TSFC for these conditions is 0.8/hr in military power and 2.2/h in afterburner. What are its military and afterburner thrust, as well as TSFC, at h = 20,000 ft and M = 0.8?

Example 5.1

Page 10 of 13 Unit C-1

Drag and Thrust Required

• If the aircraft is in straight level unaccelerated flight (SLUF), forces must be in balance:

LL W C qS= = (5.17) and ( )LC W qS= (5.18)

• Once LC is known, the drag polar can be used to determine DC and drag (D) at that velocity:

0

2

D D LC C kC= + and DD C qS= (5.19)

• Combining eq(5.18) and eq(5.19) yields,

( )0 0 0

2 22

D L D D

W kWD C kC qS C k qS C qS

qS qS

= + = + = +

(5.20)

• Expanding the expression for q yields an equation for drag as a function of velocity:

0

0

2 22

2

2 1

2

D

D

C SkW kWD C qS V

qS S V

= + = + (5.21)

• If the calculation of drag is performed for a range of velocities, and for a fixed aircraft weight and

altitude, a drag curve is generated. The drag is also called thrust required, TR, because it is the

thrust required from the engine to sustain SLUF condition.

• The drag is the sum of parasite drag and induced drag. Parasite drag = induced drag (each making

up half of the total drag) at the point on the curve, this is minimum thrust required.

• The thrust and drag curves are not drawn for velocities faster than the speed where thrust available

equal thrust required: the aircraft does not have enough thrust to sustain SLUF (this is maximum

level flight speed: maxV )

• The thrust and drag curves are not drawn in lower speeds, because values of LC that would be

required to maintain level flight at low speed exceed the aircraft's maxLC . The speed where the value

of LC required in order to maintain level flight is just equal to maxLC is called stall speed:

( )maxstall 2 LV W SC= (5.22)

Straight Level Unaccelerated Flight (SLUF)

(5.21)

(5.22)

Page 11 of 13 Unit C-1

Lift-to-Drag Ratio

• The maximum lift-to-drag ratio ( )max

L D describes the maximum efficiency of the aircraft.

• If thrust available v.s. thrust required plot (i.e., fig. 5.13) is available . . . since this is under the

condition of SLUF (lift = weight and weight is constant), if lift is constant, ( )max

L D is achieved

where drag is minimum. This point is identified as "V for minimum thrust required."

• (Important) at the speed for ( )max

L D , parasite drag = induced drag: from eq(5.19).

• Hence some additional relationships can be developed from eq(5.19): 0

2

D LC kC=

0L DC C k= (5.23) and 0 0

2 22 2D D L D LC C kC C kC= + = = (5.24) for ( )max

L D

0

0

2

DL

D D L

C kCL

D C C kC= =

+ => 0

0 0max max

max

1

2 2

DL

D D D

C kCL

D C C kC

= = =

(5.25)

Power Curves for Propeller-Driven Aircraft

• For a propeller-driven aircraft, engine performance is specified in terms of power. With drag

expressed as power required, we need to apply relationship:

R RP T V D V = = (5.26)

• The airspeed where power available equals power required is the aircraft's maxV for that altitude and

aircraft weight. The aircraft where power required is minimum is the speed at which the aircraft can

maintain level flight at that altitude and weight with minimum engine throttle setting.

• For turbojet/turbofan powered aircraft, V for minimum power required occurs at a lower velocity

than V for minimum thrust required. The velocity for minimum thrust required can be determined

by drawing a tangent line from the origin to the power required curve.

• For fig. 5.16, induced drag is three times as great as parasite drag in this "V for minimum power

required" flight condition: 0

23 D LC kC= .

Lift-to-Drag Ratio & Power Curves

(5.25)

(5.26)

Page 12 of 13 Unit C-1

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Solution (5.2)

Solution (5.3)

Example C-1-2(SLUF: Straight Level Unaccelerated Flight)

An aircraft with , k1 = 0.12, and k2 = 0 is flying at h = 30,000 ft and . If the aircraft has a

wing area of 375 ft2 and it weighs 25,000 lb, what is its drag coefficient, and how much drag is it generating? If the aircraft is in SLUF, how much thrust is its engine producing? If its , what is its

stall speed at that altitude?

Example 5.2

What is the power required for the situation in Example 5.2?

Example 5.3

Page 13 of 13 Unit C-1

Curve Shifts for Thrust Required

• Changes in aircraft weight and configuration change the power (and thrust) required curves.

• Generally, changing aircraft configuration involves:

o Extending landing gear

o Extending speed brakes

o Deploying high-lift devices

• All of these increase 0DC without changing k significantly. High-lift devices usually have the largest

effect on 0DC , but they also increase

maxLC (effect on the curve is more complex).

• Extending landing gear and speed breaks will increase parasite drag at all speeds, but induced drag is

unchanged. Because parasite drag is largest at high speeds, the net effect is to shift the drag curve up

and to the left.

• Increasing the aircraft weight changes the induced drag without changing parasite drag. Since

induced drag is higher at low speeds, the net effect is to shift the curve up and to the right.

• Because the true airspeed for a given dynamic pressure increases as density decreases, the effect of

increasing altitude is to shift the curves to the right, without changing their shape (true: as long as the

airspeed does not exceed critical Mach number).

• Above critM , curve shapes change as a result of increased wave drag, as well as shock-induced

separation (pressure) drag.

Curve Shifts for Power Required

• The effects of weight, configuration, and altitude changes on power required curves are very similar

to the curve shifts described in thrust required curves (above).

Curve Shifts