c- 2: glides, climbs, range, & endurance c- 1: equation of
TRANSCRIPT
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
5.8 Glides5.9 Climbs5.10 Range and Endurance
Unit C-2: List of Subjects
Power-Off Glides
Climbs & Ceilings
Range & Endurance
Page 2 of 18 Unit C-2
Equation of Motion of Power-Off Glide
• Aircraft's flight path angle ( ) is taken as positive downward, and the thrust is zero:
0 sinF ma D W = = = − + => sinD W = (5.27)
2 0 cosF mV r L W ⊥ = = = − => cosL W = (5.28)
Maximum Glide Range
• The aircraft's distance traveled through the air has two components, the vertical altitude lost in the
glide ( )h and the horizontal distance, or range traveled ( )R . For a fixed initial altitude, the range is
maximized when magnitude of the flight path angle ( ) is as small as possible.
• The limit to "how small" can get, while still sustaining steady flight, is set by the force balance in
eq(5.27) and eq(5.28):
1 sintan
cos
D W h
L L D W R
= = = = (5.29)
• The aircraft will achieve its flattest glide angle and its longest glide range when the aircraft is flown
at the speed for ( )max
L D :
L R
D h= (5.30) glide ratio
• The speed for maximum power-off glide range (the best glide ratio) is the speed for minimum thrust
required (means: minimum drag, and induced drag is equal to parasite drag), so that:
0
2
D LC kC= (Condition for the maximum power-off glide range)
• Because L D is a function of LC and because ( )max
L D is achieved for a specific value of LC , the
velocity for maximum glide range increases with weight, but the glide ratio ( )R h does not change
(only true for the simplified drag polar: 0
2
D D LC C kC= + ).
Power-Off Glides
(5.27)
(5.28)
(5.30)
(5.31)
Page 3 of 18 Unit C-2
Minimum Sink Rate
• Of particular interest to those who design, build, and fly sailplane is the speed for minimum sink
rate. Sailplanes are unpowered aircraft that is towed into the air. The most important feature of this
airplane is that minimum sink rate (minimum downward vertical velocity in steady flight) is less
than the upward vertical velocity of the air currents.
• Therefore, although the aircraft is descending through the air, the air is rising faster than the plane's
descending, so airplane's altitude can increase.
• The airplane's sink rate for power-off glide is:
sink rate sin RV D PV
W W
= = = (5.31)
• The speed for minimum sink rate is the speed for minimum power required (induced drag is three
times as great as parasite drag), so that:
0
23 D LC kC= (Condition for the minimum sink rate)
Power-Off Glides (Continued)
Page 4 of 18 Unit C-2
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Solution (5.4)
Example C-2-1(Power-Off Glides)
An aircraft with a drag polar has a wing area of 698 ft2 and weighs 40,000 lb. What is
its and at what speed in ft/s and knots is this achieved in standard sea-level conditions? What if the aircraft weight decreased to 30,000 lb?
Example 5.4
A sailplane’s drag polar is . It has a mass of 500 kg and a wing area of 20 m2. What is its
maximum glide ratio and minimum sink rate at sea level, and at what speeds are these achieved?
Example 5.5
Page 5 of 18 Unit C-2
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Solution (5.5)
Example C-2-1 (Continued)
Page 6 of 18 Unit C-2
Equation of Motion of Climb
• Aircraft's thrust is approximately aligned with the flight path. The flight path angle ( ) is taken as
positive upward:
0 sinF ma T D W = = = − − => sinT D
W
−= (5.32)
2 0 cosF mV r L W ⊥ = = = − => cosL W = (5.33) <= same as eq(5.28)
Best Climb Angle and Maximum Rate of Climb
• The requirement to climb at best angle (maximum height gained for minimum ground distance
traveled) is to determine the maximum sustainable climb angle that will be achieved for conditions
that produce the maximum T D− and minimum aircraft weight.
• For non-afterburning turbojets and low-bypass-ratio turbofans, maximum T D− will occur at the
velocity for minD and ( )max
L D because thrust is constant with velocity.
• The climb rate is the vertical component of the aircraft's velocity:
( )Rate of Climb sin A R
V T D P PR C V
W W
− −= = = = (5.34)
• The maximum sustained rate of climb for propeller-driven aircraft is achieved for the lowest possible
aircraft weight and at the airspeed for minimum power required.
• For aircraft powered by other types of propulsion systems, the airspeed for maximum rate of climb
can be found by comparing power-available and power-required curves. The speed where "excess
power" ( )A RP P− is greatest is the speed for maximum rate of climb.
Ceilings
• Design performance requirements for an aircraft can be specified in terms of ceiling.
• At some altitude, thrust available decreases to the point that it just equals the minimum drag.
Maximum angle of climb and maximum rate of climb are zeros, the aircraft can sustain this altitude
only by flying at the minimum drag airspeed: this is aircraft's absolute ceiling.
• Service ceiling: the altitude where a 100 ft/min rate of climb can be sustained.
• Combat ceiling: the altitude where 500 ft/min rate of climb can be sustained.
Climbs & Ceilings
(5.32)
(5.33)
(5.34)
Page 7 of 18 Unit C-2
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Solution (5.6)
Example C-2-2(Climbs & Ceilings)
What are the maximum angle of climb and maximum rate of climb at sea level for the aircraft described in Fig. 5.13 and 5.16 and the speeds at which these occur? Assume the aircraft weighs 6,000 lb.
Example 5.6
Page 8 of 18 Unit C-2
Range and Endurance
• For an aircraft, the ability to fly long distances (range) and/or stay flying long periods of time
(endurance) is among the most important design requirements.
Endurance: Turbojet and Turbofan (Approximation, or "Average Value," Method)
• Because TSFC is modeled as constant with Mach number, the drag (or thrust required) can be
viewed as "fuel-flow-required." For a given thrust required (drag) and a specified weight of fuel
available to be burned ( )fW , the endurance is given by:
f f
f t
W WE
W c D
= = (5.35)
• Maximum endurance for a turbojet or turbofan powered aircraft is achieved for maximum fuel
weight and minimum TSFC, when the aircraft is flying at the speed for minimum thrust required
(drag). However, aircraft's weight (thus its drag) will change as it burns fuel. An approximate
endurance estimate can be made by using the average aircraft weight to calculate an average drag:
avg
f
t
WE
c D
= (5.36) endurance (average value method)
Endurance: Turbojet and Turbofan (the Breguet equation)
• For a more accurate prediction of endurance, the weight change can be expressed in differential
form, with assumption of constant angle of attack (constant LC and L D ) and with lift = weight:
2
1
W
tW
dWE
c D= − (5.37) =>
2 1 1
1 2 2
1 1W W W
L
t t t DW W W
CdW L dW dWE
c D c D W c C W= − = = => 1
2
1lnL
t D
C WE
c C W
=
(5.38)
• This is called the Breguet endurance equation.
• Note that maintaining a constant value of LC at a constant altitude throughout the endurance task
will require the aircraft to fly slower as its weight decreases.
Maximum Endurance for Turbojet and Turbofan Aircraft
• For turbojet and turbofan aircraft: 0
2
D LC kC= (minimum thrust required or minimum drag)
Range & Endurance (1)
(5.35) (5.36)
(5.38)
(5.39)
(5.40) (5.41)
Page 9 of 18 Unit C-2
• Range: Turbojet and Turbofan (Approximation, or "Average Value," Method)
• The range of an aircraft is its endurance multiplied by its velocity:
( )avg avg
f f
t t
W WR E V V
c D c D V
= = = (5.39) range (average value method)
( )avg1 tc D V : Distance (i.e., nautical miles) per fuel pound (similar to a car MPG rating), and
maximizing this parameter will maximize the range
• The ratio avgD V must be minimized to maximize range, the line from the origin that has the lowest
possible slope but still touches the drag curve (a line from the origin "tangent" to the drag curve)
is used to identify the "V for best range."
Range: Turbojet and Turbofan (the Breguet equation)
• For a more accurate prediction of range, assuming that the angle of attack and L D do not change,
employing the endurance equation in differential form: 2 2 1
1 1 2
1W W W
L
t t t DW W W
V dW V C V dWL dWR
c D c D W c C W
= − − =
• In order to maintain LC at a constant altitude, V must change with changing weight,
( )2 LV W SC = , so that the range equation becomes:
1 1
2 2
1 2
1 2
1 2 2 1W W
L L
t D L t DW W
C CW dW dWR
c C SC W S c C W = = => ( )
1 21 2 1 2
1 2
2 2 L
t D
CR W W
S c C= − (5.40)
• This is called the Breguet Range equation.
• The range is maximized when density is low (high altitude), when TSFC is low (high altitude), when
the weight of fuel available is high, and when 1 2
L DC C is maximum (or 1 2
D LC C is minimum).
For a drag polar of 0
2
D D LC C kC= + , this can be determined by taking the derivative and setting it
equal to zero:
0 0
2
1 2 3 2
1 2 1 2
D L D
D L L
L L
C kC CC C kC
C C
+= = + =>
( )( )0
1 2
1 2
3 2
1 30
2 2
D L D
L
L L
d C C CkC
dC C
= = − +
0
20 3D LC kC= − + => 0
23D LC kC= (5.41)
• Where range is maximized is where parasite drag is three times as great as induced drag.
Maximum Range for Turbojet and Turbofan Aircraft
• For turbojet and turbofan aircraft: 0
23D LC kC= (this is, actually, faster speed than the airspeed for
the maximum endurance).
Range & Endurance (1) (Continued)
Page 10 of 18 Unit C-2
1. At what Mach number should you fly for maximum endurance at this altitude?
2. At what Mach number should you fly for maximum range at this altitude?
3. At what Mach number should you fly for maximum power-off glide range at this altitude?
4. What is your maximum glide ratio for the situation in (3) above?
5. At what Mach number should you fly for minimum power-off sink rate at this altitude?
6. What is your minimum sink rate for the situation in (5) above?
7. At what Mach number should you fly for maximum angle of climb with military thrust (no
afterburner) at this altitude?
8. What is your maximum angle of climb with military thrust for the situation in (7) above?
9. At what Mach number should you fly for maximum rate of climb with military thrust (no
afterburner) at this altitude?
10. What is your maximum rate of climb for the situation in (9) above?
11. Now: you climb T-38 to 20,000 ft and level off (assume standard 20,000 ft atmospheric condition).
The aircraft now weighs 11,000 lb, and you have 2,000 lb of usable fuel. What is your maximum
range at this altitude if you use all of your usable fuel to cruise? T-38 military power (no
afterburner) TSFC at sea level is given: 1.09/h.
12. What is your maximum endurance at this altitude if you use all of your usable fuel to loiter?
Solution (5.7)
(1) At what Mach number should you fly for maximum endurance at this altitude?
• On the T-38 sea-level thrust and drag v.s. Mach number curve, maximum endurance occurs at the
Mach number for minimum drag, which for W = 12,000 lb is: 0.41M = .
(2) At what Mach number should you fly for maximum range at this altitude?
• Maximum range occurs at Mach number where a line drawn from the origin is tangent to the drag
curve, which for W = 12,000 lb is: 0.495M = .
Example C-2-3 (1)(Range & Endurance)
You are flying a T-38 in standard sea-level conditions (over Death Valley in the winter). Your aircraft weighs 12,000 lb. Answer the following questions about this situation:
Example 5.7
Sea-Level Sea-Level
Page 11 of 18 Unit C-2
Solution (5.7) (Continued)
(3) At what Mach number should you fly for maximum power-off glide range at this altitude?
• Maximum glide range occurs at Mach number for minimum drag, which for W = 12,000 lb is:
0.41M = .
(4) What is your maximum glide ratio for the situation in (3) above?
• Maximum glide ratio can be determined by reading off the minimum drag value and then assuming
lift = weight. The maximum glide ratio is equal to the maximum lift-to-drag ratio, and so:
( ) ( ) ( )max max max
12,000 lb 1,000 lb 12R h L D W D= = = = .
(5) At what Mach number should you fly for minimum power-off sink rate at this altitude?
• Minimum sink rate occurs at the speed for minimum power required, which on the T-38 power
required v.s. Mach curve occurs at: 0.34M = .
(6) What is your minimum sink rate for the situation in (5) above?
• The sink rate is equal to the power required divided by weight. From sea-level power-required
curve, the minimum power required for W = 12,000 lb is 450,000 ft lb s , and so the minimum sink
rate is: ( ) ( )450,000 ft lb s 12,000 lb 37.5 ft s 2,250 ft min = = .
Example C-2-3 (2)(Range & Endurance)
20,000 ft
=>
Sea-Level
=>
Sea-Level
=>
Sea-Level
<=
Page 12 of 18 Unit C-2
Solution (5.7) (Continued)
(7) At what Mach number should you fly for maximum angle of climb with military thrust (no
afterburner) at this altitude?
• The climb angle can be found from: ( )sin T D W = − . Maximum angle of climb in military thrust
occurs at the Mach number for minimum drag because military thrust does not vary significantly
with Mach number. For W = 12,000 lb, minimum drag is at: 0.41M = .
(8) What is your maximum angle of climb with military thrust for the situation in (7) above?
• The climb angle can be found from: ( )sin T D W = − . On the T-38 thrust and drag v.s. Mach
number curve, minimum drag is 1,000 lb at M = 0.41, and thrust at this Mach number is 4,250 lb.
For W = 12,000 lb, the maximum climb angle is:
( ) ( )1 1sin sin 4,250 lb 1,000 lb 12,000 lb 0.274 rad 15.7 degT D W − −= − = − = = .
(9) At what Mach number should you fly for maximum rate of climb with military thrust (no
afterburner) at this altitude?
• The climb rate can be found from: ( )A RP P W− . Maximum excess power is at the Mach number
where A RP P− becomes maximum. One technique for quickly finding this point is to draw a
"parallel tangent," a line parallel to the power-available curve tangent to the power-required curve.
The point of tangency identifies where A RP P− is a maximum at: 0.67M = .
(10) What is your maximum rate of climb for the situation in (9) above?
• The climb rate can be found from: ( )A RP P W− . Maximum excess power for W = 12,000 lb occurs
at M = 0.67, where ( )3,500,000 1,350,000 ft lb sA RP P− = − . Maximum climb rate is:
( ) 158.3 ft s 9,500 ft minA RP P W− = = .
(11) What is your maximum range at this altitude if you use all of your usable fuel to cruise?
• You climb T-38 to 20,000 ft and level off (assume standard atmospheric condition). The aircraft
now weighs 11,000 lb, and you have 2,000 lb of usable fuel.
• The speed for maximum range was found in (2), but this time, the weight and altitude are different.
Using T-38 thrust and drag curves for h = 20,000 ft and an average cruise weight of 10,000 lb (the
starting weight minus half the fuel weight), the average Mach number for maximum range is:
0.61M = , and the average drag for cruise (the drag at the average weight) is 950 lb. The average
speed for cruise is the average Mach number times the speed of sound at 20,000 ft,
( )0.61 1,036.9 ft s 632.5 ft s 374.3 ktsV Ma= = = = . T-38 military power (no afterburner) TSFC at
sea level is given: 1.09 h , and its TSFC at 20,000 ft is:
sea-level
SL
1,036.9 ft s1.09 1.01
1,116.4 ft st t
ac c h h
a
= = =
The maximum range is: ( )( )
( )avg
2,000 lb374.3 kts 780.2 n.miles
1.01 950 lb
f
t
WR E V V
c D h
= = = =
(12) Using the TSFC calculated for (11) and the minimum drag = 880 lb and W = 10,000 lb, the average
value method yields: ( )( )avg
2,000 lb2.25
1.01 880 lb
f
t
WE h
c D h
= = =
Example C-2-3 (2) (Continued)
Page 13 of 18 Unit C-2
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Solution (5.8)
Example C-2-4(Range & Endurance)
A turbojet-powered trainer aircraft weighs 5,000 lb and is flying at h = 25,000 ft with 1,000 lb of fuel onboard. Its drag polar is
, its wing area is 180 ft2, and the TSFC of its engines is 1.0/h at sea level. What is its maximum range and endurance to tanks dry at this altitude, and at what speed should the pilot initially fly to achieve each?
Example 5.8
Page 14 of 18 Unit C-2
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Solution (5.8) (Continued)
Example C-2-4 (Continued)
Page 15 of 18 Unit C-2
Endurance: Propeller-Driven Aircraft (Approximation, or "Average Value," Method)
• Because fuel consumption for piston engines and turboprops is proportional to power output, the
power curves are the best tools for propeller driven aircraft's endurance & range.
• The average value method prediction for the endurance of propeller-driven aircraft is:
( )( )
( )avgprop prop avg
f f f
f R
W W WE
W c P c D V
= = = (5.42) endurance (average value method)
c : Brake specific fuel consumption (BSFC)
prop : Propeller efficiency
• The speed of maximum endurance of a propeller-driven aircraft is the speed for minimum power
required. Therefore: 0
23 D LC kC= (Condition for the maximum endurance).
Endurance: Propeller-Driven Aircraft (the Breguet equation)
• The more accurate form of the endurance is: 1 1
2 2
prop prop
W W
W W
dW L dWE
c DV c DV W
= = => with constant LC , so that: ( ) ( )2 LV W SC = , then:
( )( )1
2
3 2 3 2prop prop 1 2 1 2
1 23 22
2 2
W
L L
D DW
C CS dW SE W W
c C W c C
− −= = − −
( )3 2
prop 1 2 1 2
2 12L
D
CE S W W
c C
− −= − (5.42) the Breguet endurance equation
• Maximum endurance is achieved for conditions of high propeller efficiency, low BSFC, high density
air (means: low altitude and low temperature), high weight of fuel available, and a maximum value
of the ratio of 3 2
L DC C (or, minimizing 3 2
D LC C ).
• The power required for propeller-driven aircraft is:
R
L D
WP V D V
C C = = , but:
2
L
WV
SC = =>
3
3 2
2 2 1R
L D L L D
W W WP
C C SC S C C = = (5.44)
• For the condition for minimizing 3 2
D LC C :
0 0
2
3 2 1 2
3 2 3 2
D L D
D L L
L L
C kC CC C kC
C C
+= = + =>
( )0
3 2
5 2 1 2
3 10
2 2
D L D
L L L
C C C k
C C C
= = − +
0
20 3 D LC kC= − + => 0
23 D LC kC= (5.45)
Range & Endurance (2)
(5.42) (5.43)
(5.43) (5.45)
(5.46) (5.47)
Page 16 of 18 Unit C-2
slow
er
fast
er
Exactly the same . . .
Range: Propeller-Driven Aircraft (Approximation, or "Average Value," Method)
• The average value method expression for propeller-driven aircraft range is:
( )avg avg
prop prop
f f f
fR
W W WR V E V V V
c cWP D V
= = = =
=>
avg
prop
fWR
cD
= (5.46)
Range: Propeller-Driven Aircraft (the Breguet equation)
• The more accurate form of the range is:
prop 1
2
lnL
D
C WR
c C W
=
(5.47) The Breguet range equation
• Propeller-driven aircraft range is not influenced by air density (altitude), except to the degree that air
density and temperature influence BSFC.
• Propeller-driven aircraft range is maximized by flying in conditions that are characterized by
maximum propeller efficiency, minimum BFSC, maximum weight of fuel available, and minimum
drag (means: maximum L D or L DC C ). recall that maximum L D occurs at the speed where
parasite drag equals induced drag: 0
2
D LC kC= (Condition for maximum cruise range).
Maximum Endurance and Range Conditions: Summary
Power-Off Glide Turbojet/Turbofan Propeller-Driven
Maximum
Endurance
(minimum sink rate)
0
23 D LC kC=
(minimum power required)
0
2
D LC kC=
(minimum thrust required,
or minimum drag)
0
23 D LC kC=
(minimum power required)
Maximum
Range
(best glide ratio)
0
2
D LC kC=
(minimum thrust required, or
minimum drag)
0
23D LC kC=
0
2
D LC kC=
(minimum thrust required,
or minimum drag)
Endurance and Range: What's the Difference?
• For an airplane, the longest endurance is achieved near the airplane's stall speed: the speed at which
it is just able to maintain its altitude. At this speed, its wing just carries its weight so the power
(energy per unit time) that the engine needs to generate to push the aircraft forward is lowest. Since
the aircraft under this condition moves very slowly, its range becomes limited. This is maximum
endurance (but not maximum range).
• Flying an airplane at cruising speed takes more power, since faster speed induces more friction.
However, since the airplane now moves much faster, its range is much longer, even though it cannot
stay up in air, compared against the condition when flying near stall speed. This is maximum range
(but not maximum endurance).
Range & Endurance (2) (Continued)
Cannot fly slower than this (with jet-engine-powered-on) . . .
Page 17 of 18 Unit C-2
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Solution (5.9)
Example C-2-5(Range & Endurance)
An aircraft is being designed to fly on Mars (where the acceleration of gravity is 3.72 m/s2) at an altitude where = 0.01 kg/m3. The aircraft will be powered by a piston engine driving a propeller. The engine has, when tested, burned 50 kg of fuel and 400 kg of oxidizer in 1hwhile producing 104 kW of SHP. The propeller efficiency has been measured in Mars-like conditions at 0.85. The aircraft’s drag polar is
, and its wing area is 50 m2. What will be the aircraft’s maximum range and endurance at this altitude on 500 kg of propellants if its mass with propellants is 1,500 kg?
Example 5.9
Page 18 of 18 Unit C-2
Altitude Variations
• The choice of an appropriate cruising and/or loitering (maximum endurance) altitude is an important
consideration.
• The cruising and loitering altitude choices can be influenced by weather conditions, winds, traffic
congestion, navigation/terrain constraints, training requirements, enemy threat system lethal
envelopes (and warning system coverage), and the cruise speed that can be achieved at an altitude.
• As altitude increases, the decreasing air density causes V for maximum range to increase.
However, this benefit is limited by the ability of the engine(s) to generate sufficient thrust,
increasing wind velocities with increasing altitude, the time and fuel required to climb to higher
altitudes, and Mach number effects.
Best Cruise Mach (BCM) & Best Cruise Altitude (BCA)
• Suppose, if the altitude is where the airspeed for ( )max
L D equals the airspeed corresponding to the
critical Mach number ( )critM . If the aircraft has sufficient thrust to fly this altitude without
afterburner, the altitude where this condition is satisfied results in the absolute maximum range for
that aircraft.
• The altitude and airspeed for this optimum cruise condition are referred to as: the best cruise Mach
(BCM) & the best cruise altitude (BCA).
• Note that, in fig. 5.31, maximum range cruise airspeed at 45,000 ft is about twice the maximum
range cruise airspeed at sea level. Because TSFC is lower at 45,000 ft, the aircraft's range is more
than doubled at this higher altitude!
Range & Endurance (3)