Cruising Flight Performance!
!
Robert Stengel, Aircraft Flight Dynamics,
MAE 331, 2014
•!
•!
•!
•!
•!
•!
Learning Objectives!
Definitions of airspeed!
Performance parameters!
Steady cruising flight conditions!
Breguet range equations!
Optimize cruising flight for
minimum thrust and power!
Flight envelope !
Reading:!
Flight Dynamics !
Aerodynamic Coefficients, 118-130!
Airplane Stability and Control!
Chapter 6!
Copyright 2014 by Robert Stengel. All rights reserved. For educational use only.!
http://www.princeton.edu/~stengel/MAE331.html!
http://www.princeton.edu/~stengel/FlightDynamics.html!
1!
U.S. Standard Atmosphere, 1976"
http://en.wikipedia.org/wiki/U.S._Standard_Atmosphere!
2!
Dynamic Pressure and Mach Number"
! = air density, function of height
= ! sealevel e" # h
a = speed of sound
= linear function of height
Dynamic pressure = q ! !V 2 2
Mach number = V a
3!
Definitions of Airspeed"
•! Airspeed is speed of aircraft measured with respect to air mass"
–! Airspeed = Inertial speed if wind speed = 0"
•! Indicated Airspeed (IAS)"
IAS = 2 ( pstagnation ! pambient ) "SL =
=
2 ( ptotal ! pstatic )
"SL
2qc
, with qc ! impact pressure
"SL
•! Calibrated Airspeed (CAS)*"
CAS = IAS corrected for instrument and position errors
=
2 ( qc )corr!1
"SL
* Kayton & Fried, 1969; NASA TN-D-822, 1961!
4!
Definitions of Airspeed"
•! Airspeed is speed of aircraft measured with respect to air mass"
–! Airspeed = Inertial speed if wind speed = 0"
•! Equivalent Airspeed (EAS)*"
EAS = CAS corrected for compressibility effects =
•! True Airspeed (TAS)*"
V ! TAS = EAS
2 ( qc )corr!2
"SL
•! Mach number"
!SL
!SL
= IAScorrected
! (z)
! (z)
* Kayton & Fried, 1969; NASA TN-D-822, 1961!
M=
TAS
a
5!
Flight in the
Vertical Plane!
6!
Longitudinal Variables!
7!
Longitudinal Point-Mass
Equations of Motion"
•! Assume thrust is aligned with the velocity
vector (small-angle approximation for !)"
•! Mass = constant"
1 2
1
#V S " mg sin $ (CT " C D ) #V 2 S " mg sin $
2
2
V! =
%
m
m
1
1
(CT sin ! + CL ) #V 2 S " mg cos $ CL #V 2 S " mg cos $
2
2
$! =
%
mV
mV
h! = "z! = "vz = V sin $
V = velocity = Earth-relative airspeed
(CT cos ! " CD )
r! = x! = vx = V cos $
= True airspeed with zero wind
! = flight path angle
h = height (altitude)
r = range
8!
Conditions for Steady,
Level Flight"
•!
•!
•!
•!
0=
0=
Flight path angle = 0"
Altitude = constant"
Airspeed = constant"
Dynamic pressure = constant"
1
(CT ! CD ) 2 "V 2 S
m
CL
h! = 0
r! = V
1 2
"V S ! mg
2
mV
•! Thrust = Drag"
•! Lift = Weight"
9!
Power and Thrust"
•! Propeller"
Power = P = T ! V = CT
•! Turbojet"
Thrust = T = CT
1 3
"V S # independent of airspeed
2
1 2
!V S " independent of airspeed
2
•! Throttle Effect"
T = Tmax! T = CTmax ! TqS, 0 " ! T " 1
10!
Typical Effects of Altitude and
Velocity on Power and Thrust"
•! Propeller"
•! Turbojet"
11!
Models for Altitude Effect on
Turbofan Thrust"
From Flight Dynamics, pp.117-118"
1
Thrust = CT (V, ! T ) " ( h )V 2 S
2
1
= ( ko + k1V # ) " ( h )V 2 S! T , N
2
ko = Static thrust coefficient at sea level
k1 = Velocity sensitivity of thrust coefficient
! = Exponent of velocity sensitivity
= "2 for turbojet
# T = Throttle setting, ( 0,1)
$ ( h ) = $ SL e" % h , $ SL = 1.225 kg / m 3 , % = (1 / 9,042 ) m "1
12!
Models for Altitude Effect on
Turbofan Thrust"
From AeroModelMach.m in FLIGHT.m, Flight Dynamics,!
http://www.princeton.edu/~stengel/AeroModelMach.m"
[airDens,airPres,temp,soundSpeed] = Atmos(-x(6));"
Thrust = "u(4) * StaticThrust * (airDens / 1.225)^0.7 * (1 - exp((-x(6) – 17000)/2000));"
Atmos(-x(6)) : 1976 U.S. Standard Atmosphere function
-x(6) = h = Altitude, m
airDens = ! = Air density at altitude h, kg/m 3
u(4) = " T = Throttle setting, ( 0,1)
Empirical fit to match known characteristics of powerplant for
generic business jet"
(airDens / 1.225)^0.7 * (1 - exp((-x(6) – 17000)/2000))"
13!
Thrust of a PropellerDriven Aircraft"
•! With constant rpm, variable-pitch propeller"
T = !P!I
Pengine
V
= !net
Pengine
V
where
!P = propeller efficiency
! I = ideal propulsive efficiency
!netmax " 0.85 # 0.9
•! Efficiencies decrease with airspeed"
•! Engine power decreases with altitude"
–! Proportional to air density, w/o supercharger"
14!
Propeller Efficiency, "P,
and Advance Ratio, J"
Effect of propeller-blade pitch angle!
•! Advance Ratio"
J=
V
nD
where
V = airspeed, m / s
n = rotation rate, revolutions / s
D = propeller diameter, m
from McCormick!
15!
Thrust of a
Turbojet
Engine"
1/2
02*#
42
! o &# !t &
!t ! 1,%
T = mV
/ "15
(%
( () c "1) +
! o) c .
23+$ ! o "1 '$ ! t "1 '
26
(
! o = pstag pambient
)
m! = m! air + m! fuel
(" #1)/"
; " = ratio of specific heats $ 1.4
! t = ( turbine inlet temp. freestream ambient temp.)
% c = ( compressor outlet temp. compressor inlet temp.)
from Kerrebrock!
•!
•!
Little change in thrust with airspeed below Mcrit"
Decrease with increasing altitude"
16!
Performance Parameters"
L
•! Lift-to-Drag Ratio"
•! Load Factor"
CL
CD
n = L W = L mg ,"g"s
•! Thrust-to-Weight Ratio"
•! Wing Loading"
D=
T
T
W = mg ,"g"s
W , N m 2 or lb ft 2
S
17!
Steady, Level Flight!
18!
Trimmed Lift Coefficient, CL"
•! Trimmed lift coefficient, CL"
–! Proportional to weight and wing loading factor"
–! Decreases with V2"
–! At constant true airspeed, increases with altitude"
"1
%
W = C Ltrim $ !V 2 ' S = C Ltrim qS
#2
&
C Ltrim
# 2 e" h &
1
2
= (W S ) =
W S) = %
W S)
2 (
2((
q
!V
$ ! 0V '
19!
Trimmed Angle of Attack, !"
•! Trimmed angle of attack, !"
–! Constant if dynamic pressure and weight are
constant"
–! If dynamic pressure decreases, angle of attack
must increase"
! trim
1
W S ) # C Lo
2W "V S # C Lo q (
=
=
C L!
C L!
2
20!
Thrust Required for Steady, Level Flight"
21!
Thrust Required for Steady, Level Flight"
•! Trimmed thrust"
Parasitic Drag!
Induced Drag!
"1 2 %
2W 2
Ttrim = Dcruise = C Do $ !V S ' + (
#2
& !V 2 S
•! Minimum required thrust conditions"
Necessary Condition
= Zero Slope!
! Ttrim
4 $W 2
= C Do ( "VS ) #
=0
!V
"V 3S
22!
Necessary and Sufficient
Conditions for Minimum
Required Thrust"
Necessary Condition = Zero Slope!
4 "W 2
C Do ( !VS ) =
!V 3S
Sufficient Condition for a Minimum =
Positive Curvature when slope = 0!
! 2 Ttrim
12#W 2
= C Do ( " S ) +
>0
!V 2
"V 4 S
(+)"
(+)"
23!
Airspeed for
Minimum Thrust in
Steady, Level Flight"
•! Satisfy necessary condition"
# 4! &
2
V =%
W
S
(
)
2(
$ C Do " '
4
•! Fourth-order equation for velocity"
–! Choose the positive root"
VMT
2 "W % (
=
$ '
! # S & C Do
24!
Lift, Drag, and Thrust Coefficients in
Minimum-Thrust Cruising Flight"
Lift coefficient"
C LMT =
=
C Do
(
2
2
!VMT
"W %
$# '&
S
= ( C L )( L/D )
max
Drag and thrust coefficients"
C DMT = C Do + ! C L2MT = C Do + !
= 2C Do " CTMT
C Do
!
25!
Power Required for
Steady, Level Flight"
26!
Power Required for
Steady, Level Flight"
•! Trimmed power"
Induced Drag!
Parasitic Drag!
)
" 1 2 % 2(W 2 ,
Ptrim = TtrimV = DcruiseV = +C Do $ !V S ' +
V
2 .
#2
& !V S *
•! Minimum required power conditions"
! Ptrim
3
2$W 2
2
= C Do ( "V S ) #
=0
2
!V
2
"V S
27!
Airspeed for Minimum
Power in Steady,
Level Flight"
•! Satisfy necessary condition"
•! Fourth-order equation for velocity"
–! Choose the positive root"
•! Corresponding lift and
drag coefficients"
3
2"W 2
2
C Do ( !V S ) =
2
!V 2 S
VMP =
C LMP =
2 "W % (
$ '
! # S & 3C Do
3C Do
!
C DMP = 4C Do
28!
Achievable Airspeeds in ConstantAltitude Flight"
Back Side of the
Thrust Curve"
•! Two equilibrium airspeeds for a given thrust or power setting"
–! Low speed, high CL, high !#
–! High speed, low CL, low !#
•! Achievable airspeeds between minimum and maximum
values with maximum thrust or power#
29!
Achievable Airspeeds for Jet in
Cruising Flight"
Thrust = constant#
Tavail
"1
% 2(W
= C D qS = C Do $ !V 2 S ' +
#2
& !V 2 S
2
2)W
"1
4 %
2
C Do $ !V S ' ( TavailV +
=0
#2
&
!S
2
2
#
W
T
4
V 4 ! avail V 2 +
2 = 0
C Do " S
C Do ( " S )
4th-order algebraic
equation for V#
30!
Achievable Airspeeds for Jet in
Cruising Flight"
Solutions for V2 can be put in quadratic
form and solved easily#
V 2 ! x; V = ± x
Tavail 2
4 #W 2
V !
V +
2 = 0
C Do " S
C Do ( " S )
4
x 2 + bx + c = 0
2
b
" b%
x = ! ± $ ' ! c =V2
# 2&
2
31!
Thrust Required and Thrust
Available for a Typical Bizjet"
•!
•!
•!
Available thrust decreases with altitude, and range of achievable
airspeeds decreases"
Stall limitation at low speed"
Mach number effect on lift and drag increases thrust required at
high speed"
Typical Simplified Jet Thrust Model!
x
x
$ ! ( SL ) e" # h '
Tmax (h) = Tmax (SL) &
= Tmax (SL) $% e" # h '( = Tmax (SL)e" x# h
)
% ! (SL) (
•!
With empirical correction to force thrust to zero at a given altitude,
hmax. c is a convergence factor."
Tmax (h) = Tmax (SL)e! x" h #$1! e!( h!hmax ) c %&
32!
Thrust Required and Thrust
Available for a Typical Bizjet"
Typical Stall!
Limit!
33!
!is"rical Fac"i#d
•! Aircraft Flight Distance Records"
http://en.wikipedia.org/wiki/Flight_distance_record!
•! Aircraft Flight Endurance Records"
http://en.wikipedia.org/wiki/Flight_endurance_record!
Rutan/Scaled Composites
Voyager!
Rutan/Virgin Atlantic
Global Flyer!
34!
The Flight Envelope!
35!
Flight Envelope Determined by
Available Thrust"
•! All altitudes and airspeeds at which an aircraft can fly "
–! in steady, level flight "
–! at fixed weight"
•! Flight ceiling defined by
available climb rate"
–! Absolute: 0 ft/min"
–! Service: 100 ft/min"
–! Performance: 200 ft/min"
Excess thrust provides the
ability to accelerate or
climb"
36!
Additional Factors Define the
Flight Envelope"
•!
•!
Piper Dakota Stall Buffet"
http://www.youtube.com/watch?v=mCCjGAtbZ4g!
•!
•!
•!
•!
•!
Maximum Mach number"
Maximum allowable
aerodynamic heating"
Maximum thrust"
Maximum dynamic
pressure"
Performance ceiling"
Wing stall"
Flow-separation buffet"
–!
–!
Angle of attack"
Local shock waves"
37!
Boeing 787 Flight
Envelope (HW #5, 2008)"
Best
Cruise
Region"
38!
!is"rical Fac"id#s
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"
Lockheed Vega !
Ford Tri-Motor!
Boeing 40!
http://www.youtube.com/watch?v=3a8G87qnZz4!
•!
•!
39!
Commercial Aircraft of the 1930s"
Streamlining, engine cowlings!
Douglas DC-1, DC-2, DC-3!
•!
Lockheed 14 Super Electra,
Boeing 247, exterior and interior!
40!
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!
41!
•!
•!
•!
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!
42!
Optimal Cruising Flight!
43!
Maximum Lift-to-Drag Ratio"
•! Lift-to-drag ratio"
L
D=
CL
CL
CD = C + !C 2
Do
L
•! Satisfy necessary condition for a maximum"
!
(
CL
CD
! CL
)
=
C Do
1
2" C L2
#
+ " C L2
C Do + " C L2
(
)
2
=0
•! Lift coefficient for maximum L/D
and minimum thrust are the same"
( C L )L / D
max
=
C Do
!
= C LMT
44!
Airspeed, Drag Coefficient, and
Lift-to-Drag Ratio for L/Dmax"
Airspeed!
Drag !
Coefficient!
Maximum !
L/D!
VL/Dmax = VMT =
( C D )L / D
max
2 "W % (
$ '
! # S & C Do
= C Do + C Do = 2C Do
( L / D )max =
C Do !
2C Do
=
1
2 ! C Do
•! Maximum L/D depends only on induced drag factor
and zero-! drag coefficient"
•! Induced drag factor and zero-! drag coefficient are
functions of Mach number"
45!
Cruising Range and
Specific Fuel Consumption"
•! Thrust = Drag" 0 = ( CT ! C D ) 1 "V 2 S m
•! Lift = Weight"
2
1
#
&
0 = % C L "V 2 S ! mg ( mV
$ 2
'
•! Level flight"
h! = 0
r! = V
•! Thrust specific fuel consumption, TSFC = cT"
•! Fuel mass burned per sec per unit of thrust"
cT :
kg s
kN
m! f = !cT T
•! Power specific fuel consumption, PSFC = cP"
•! Fuel mass burned per sec per unit of power"
cP :
kg s
kW
m! f = !cP P
46!
!is"rical Fac"i#d
•! 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"
•! Merged with Dassault in 1971"
Breguet Atlantique!
Breguet 14!
SEPECAT Jaguar!
Breguet 890 Mercure!
47!
Louis Breguet,
1880-1955!
Breguet Range Equation
for Jet Aircraft"
Rate of change of range with respect to weight of fuel burned"
dr
dr dt
r!
V
V
" L% V
=
= =
=!
= !$ '
# D & cT mg
dm dm dt m! ( !cT T )
cT D
" L% V
dr = ! $ '
dm
# D & cT mg
Range traveled"
R
Wf
# L & # V & dm
Range = R = ! dr = " ! % ( %
$ D ' $ cT g (' m
0
W
i
48!
Maximum Range of a
Jet Aircraft Flying at
Constant Altitude"
B-727!
•!
At constant altitude"
Vcruise ( t ) =
2W ( t )
(
)
C L ! h fixed S
Wf
" C %" 1 %
2 dm
Range = ! ) $ L ' $
'
C D & # cT g & C L ( S m1 2
Wi #
" CL % " 2 % 2
=$
mi1 2 ! m f 1 2
'
$
'
# C D & # cT g & ( S
(
•!
)
Range is maximized when "
(* ' = minimum
! C $
## L && = maximum and )
*+ h = maximum
" CD %
Breguet Range Equation
for Jet Aircraft"
49!
MD-83!
For constant true airspeed, V = Vcruise!
%
mf
" L%"V
R = ! $ ' $ cruise ' ln ( m ) m
i
# D & # cT g &
% "m %
" L%"V
= $ ' $ cruise ' ln $ i '
# D & # cT g & # m f &
!
C L $ ! 1 $ ! mi $
R = # Vcruise
ln
C D &% #" cT g &% #" m f &%
"
!!
!!
!!
!!
Vcruise as fast as possible"
Respect Mcrit"
$ as small as possible"
h as high as possible"
50!
Maximize Jet Aircraft Range
Using Optimal Cruise-Climb"
!R
"
!C L
(
! Vcruise
CL
!C L
CD
)
$
C
! &Vcruise L
C Do + # C L2
&%
=
!C L
(
)
'
)
)(
=0
Vcruise = 2W C L ! S
Assume 2W ( t ) ! ( h ) S = constant
i.e., airplane climbs at constant TAS as fuel is burned
51!
Maximize Jet Aircraft Range
Using Optimal Cruise-Climb"
(
)
(
)
1/2
2
! #$Vcruise C L C Do + " C L2 %&
2W ! #$C L C Do + " C L %&
=
=0
!C L
!C L
'S
Optimal values: (see Supplemental Material)
C LMR =
C Do
3!
C DMR = C Do +
: Lift Coefficient for Maximum Range
C Do
3
=
4
CD
3 o
Vcruise!climb = 2W ( t ) C LMR " ( h ) S = a ( h ) M cruise-climb
a ( h ) : Speed of sound; M cruise-climb : Mach number
52!
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"
53!
Next Time:!
Gliding, Climbing, and
Turning Flight!
Reading:!
Flight Dynamics !
Aerodynamic Coefficients, 130-141, 147-155!
54!
$upplemental Ma%ria#l
55!
Lift-Drag Polar for a
Typical Bizjet"
•! L/D equals slope of line drawn from the origin"
–! Single maximum for a given polar"
–! Two solutions for lower L/D (high and low airspeed)"
–! Available L/D decreases with Mach number"
•! Intercept for L/Dmax depends only on % and zero-lift drag"
Note different scales
for lift and drag!
56!
Back Side of
the Power
Curve"
•!
Achievable Airspeeds
in Propeller-Driven
Cruising Flight"
Power = constant#
Pavail = TavailV
PavailV
4 #W 2
+
=0
V !
C Do "S C Do ( "S )2
4
•!
Solutions for V cannot be put in quadratic form; solution is
more difficult, e.g., Ferrari
s method#
aV 4 + ( 0 )V 3 + ( 0 )V 2 + dV + e = 0
•!
Best bet: roots in MATLAB#
57!
P-51 Mustang
Minimum-Thrust
Example"
Wing Span = 37 ft (9.83 m)
Wing Area = 235 ft 2 (21.83 m 2 )
Loaded Weight = 9, 200 lb (3, 465 kg)
C Do = 0.0163
! = 0.0576
W / S = 39.3 lb / ft 2 (1555.7 N / m 2 )
Airspeed for minimum thrust!
VMT
2 "W % (
2
76.49
0.947
=
=
=
m/s
$ '
(1555.7)
#
&
S
0.0163
!
C Do
!
!
Altitude, m
0
2,500
5,000
10,000
Air Density,
kg/m^3
1.23
0.96
0.74
0.41
VMT, m/s
69.11
78.20
89.15
118.87
58!
P-51 Mustang
Maximum L/D
Example"
( C D )L / D
max
( C L )L / D
max
Wing Span = 37 ft (9.83 m)
=
( L / D )max =
Wing Area = 235 ft (21.83 m 2 )
Loaded Weight = 9, 200 lb (3, 465 kg)
C Do = 0.0163
C Do
!
W / S = 1555.7 N / m 2
Altitude, m
0
2,500
5,000
10,000
= C LMT = 0.531
1
= 16.31
2 ! C Do
VL / Dmax = VMT =
! = 0.0576
76.49
m/s
!
Air Density,
kg/m^3
1.23
0.96
0.74
0.41
VMT, m/s
69.11
78.20
89.15
118.87
59!
Breguet Range Equation
for Propeller-Driven
Aircraft"
Breguet 890 Mercure!
•!
= 2C Do = 0.0326
Rate of change of range with respect to weight of fuel burned"
"L% 1
V
V
V
dr r!
= =
=!
=!
= !$ '
# D & cPW
cPTV
cP DV
dw w! (!cP P )
•!
Range traveled"
Range = R =
Wf
# L &# 1 & dw
dr
=
"
!
! % (% (
0
Wi $ D ' $ cP ' w
R
60!
Breguet Range Equation
for Propeller-Driven
Aircraft"
•!
Breguet Atlantique!
For constant true airspeed, V = Vcruise!
" L %" 1 %
Wf
R = ! $ '$ ' ln ( w ) W
i
# D & # cP &
" C %" 1 % " W
= $ L '$ ' ln $$ i
# C D & # cP & # W f
•!
%
''
&
Range is maximized when "
! CL $
# & = maximum = L D
" CD %
( )
max
61!
P-51 Mustang
Maximum Range
(Internal Tanks only)"
W = C Ltrim qS
C Ltrim =
# 2 e" h &
1
2
W
S
=
W
S
=
( )
( ) % 2 ( (W S )
q
!V 2
$ !0V '
!C $ ! 1 $ !W
R = # L & # & ln ## i
" C D %max " cP % " W f
$
&&
%
! 1 $ ! 3, 465 + 600 $
= (16.31) #
& ln #
&
" 0.0017 % " 3, 465 %
= 1,530 km ((825 nm )
62!
Maximize Jet Aircraft Range
Using Optimal Cruise-Climb"
(
)
(
)
1/2
2
! #$Vcruise C L C Do + " C L2 %&
2w ! #$C L C Do + " C L %&
=
=0
!C L
!C L
'S
2w
= Constant; let C L1/2 = x, C L = x 2
!S
" $
x
&
"x & C Do + # x 4
%
(
)
(
)
(
(
)
' C D + # x 4 * x ( 4 # x 3 ) C D * 3# x 4
o
o
)=
=
2
2
4
)(
C Do + # x
C Do + # x 4
(
)
)
Optimal values:
C LMR =
C Do
3!
: C DMR = C Do +
C Do
3
=
4
CD
3 o
63!