Cruising Flight Performance Robert Stengel, Aircraft Flight Dynamics, MAE 331, 2016 Copyright 2016 by Robert Stengel. All rights reserved. For educational use only. http://www.princeton.edu/~stengel/MAE331.html http://www.princeton.edu/~stengel/FlightDynamics.html • Definitions of airspeed • Performance parameters • Steady cruising flight conditions • Breguet range equations • Optimize cruising flight for minimum thrust and power • Flight envelope Learning Objectives Reading: Flight Dynamics Aerodynamic Coefficients, 118-130 1 Review Questions ! What is “static margin”? ! Is the airplane’s pitching moment sensitivity to angle of attack linear? ! What factors are most important in defining the airplane’s pitching moment sensitivity to angle of attack ? ! Which is more important: stability or control? ! Is the airplane’s yawing moment sensitivity to sideslip angle linear? ! What effect does the wing dihedral angle have on airplane stability? ! Why would an airplane have a “twin tail”? ! What are “ventral fins”, and why do airplanes have/not have them? 2
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Stall buffeting and Mach buffeting are limiting factors
Narrow corridor for safe flight
Climb Schedule
Air Commerce Act of 1926•! Airlines formed to carry mail and passengers:
–! Northwest (1926)–! Eastern (1927), bankruptcy–! Pan Am (1927), bankruptcy–! Boeing Air Transport (1927), became United (1931)–! Delta (1928), consolidated with Northwest, 2010–! American (1930)–! TWA (1930), acquired by American–! Continental (1934), consolidated with United, 2010
Boeing 40
Ford Tri-Motor Lockheed Vega
http://www.youtube.com/watch?v=3a8G87qnZz4
!!iiss""rriiccaall FFaacc""iiddss
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Commercial Aircraft of the 1930sStreamlining, engine cowlings
Lockheed 14 Super Electra, Boeing 247
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Douglas DC-1, DC-2, DC-3
Comfort and Elegance by the End of the Decade
Boeing 307, 1st pressurized cabin (1936), flight engineer, B-17 pre-cursor, large dorsal fin (exterior and interior)
Sleeping bunks on transcontinental planes (e.g., DC-3)Full-size dining rooms on flying boats
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Seaplanes Became the First TransOceanic Air Transports
•! PanAm led the way–! 1st scheduled TransPacific flights(1935)–! 1st scheduled TransAtlantic flights(1938)–! 1st scheduled non-stop Trans-Atlantic flights (VS-44, 1939)
•! Boeing B-314, Vought-Sikorsky VS-44, Shorts Solent•! Superseded by more efficient landplanes (lighter, less drag)
http://www.youtube.com/watch?v=x8SkeE1h_-A
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Checklist!"!Flight envelope?!
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Optimal Cruising Flight!
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Maximum Lift-to-Drag Ratio
CL( )L /Dmax =CDo
!= CLMT
LD = CL
CD=
CL
CDo+ !CL
2
! CLCD
( )!CL
=1
CDo+ "CL
2 #2"CL
2
CDo+ "CL
2( )2= 0
Satisfy necessary condition for a maximum
Lift-to-drag ratio
Lift coefficient for maximum L/D and minimum thrust are the same
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Airspeed, Drag Coefficient, and Lift-to-Drag Ratio for L/Dmax
VL/Dmax =VMT =2!
WS
"#$
%&'
(CDo
CD( )L /Dmax = CDo+ CDo
= 2CDo
L / D( )max =CDo
!
2CDo
=1
2 !CDo
Maximum L/D depends only on induced drag factor and zero-lift drag coefficient
Induced drag factor and zero-lift drag coefficient are functions of Mach number
Airspeed
Drag Coefficient
Maximum L/D
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Cruising Range and Specific Fuel Consumption
0 = CT !CD( ) 12"V 2S m
0 = CL12"V 2S !mg#
$%&'( mV
•! Thrust = Drag
•! Lift = Weight
•! Thrust specific fuel consumption, TSFC = cT•! Fuel mass burned per sec per unit of thrust
!mf = !cTTcT :kg skN
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!h = 0!r =V
•! Level flight
•! Power specific fuel consumption, PSFC = cP•! Fuel mass burned per sec per unit of power
!mf = !cPPcP :kg skW
Breguet Range Equation for Jet Aircraft
drdm
= dr dtdm dt
=!r!m= V
!cTT( ) = ! VcTD
= ! LD
"#$
%&'
VcTmg
Rate of change of range with respect to weight of fuel burned
Range traveled
Range = R = dr0
R
! = " LD
#$%
&'(
VcT g
#$%
&'(Wi
Wf
!dmm
Louis Breguet, 1880-1955
dr = ! LD
"#$
%&'
VcTmg
dm
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Maximum Range of a Jet Aircraft Flying at
Constant Altitude
Range is maximized when
Range = ! CL
CD
"#$
%&'
1cT g
"#$
%&'
2CL(SWi
Wf
)dmm1 2
=CL
CD
"
#$%
&'2cT g
"#$
%&'
2(S
mi1 2 !mf
1 2( )
Vcruise t( ) =2W t( )
CL! hfixed( )S
At constant altitude and SFC
CL
CD
!
"#$
%&= maximum
B-727
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Breguet Range Equation for Jet Aircraft
For constant true airspeed, V = Vcruise, and SFC
R = ! LD
"#$
%&'VcruisecT g
"#$
%&'ln m( ) mi
m f
= LD
"#$
%&'VcruisecT g
"#$
%&'ln mi
mf
"
#$%
&'
= VcruiseCL
CD
!"#
$%&
1cT g
!"#
$%&ln mi
mf
!
"#$
%&
!! Vcruise(CL/CD) as large as possible!! Respect Mcrit!! ## as small as possible!! h as high as possible
MD-83
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Maximize Jet Aircraft Range Using Optimal Cruise-Climb
Vcruise = 2W CL!S
!R!CL
"! Vcruise
CLCD
( )!CL
=
! VcruiseCL
CDo+ #CL
2( )$
%&&
'
())
!CL
= 0
Assume 2W t( ) ! h( )S = constanti.e., airplane climbs at constant TAS as fuel is burned
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Maximize Jet Aircraft Range Using Optimal Cruise-Climb
a h( ) : Speed of sound; Mcruise-climb : Mach number 59
Step-Climb Approximates Optimal Cruise-Climb
!! Cruise-climb usually violates air traffic control rules!! Constant-altitude cruise does not!! Compromise: Step climb from one allowed altitude
to the next as fuel is burned
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Breguet 890 Mercure
Breguet Atlantique
•! Louis Breguet (1880-1955), aviation pioneer•! Gyroplane (1905), flew vertically in 1907•! Breguet Type 1 (1909), fixed-wing aircraft•! Formed Compagnie des messageries
aériennes (1919), predecessor of Air France•! Breguet Aviation manufactured numerous
military and commericial aircraft until after World War II; teamed with BAC in SEPECAT