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Cruising Flight PerformanceRobert Stengel, Aircraft Flight
Dynamics MAE 331, 2010
Copyright 2010 by Robert Stengel. All rights reserved. For educational use only.http://www.princeton.edu/~stengel/MAE331.html
http://www.princeton.edu/~stengel/FlightDynamics.html
U.S. Standard Atmosphere
Air data measurement and computation
Airspeed definitions
Steady, level flight
Simplified power and thrust models
Back side of the power/thrust curve
Performance parameters
Breguet range equation
Jet engine
Propeller-driven
U.S. Standard Atmosphere, 1976
http://en.wikipedia.org/wiki/U.S._Standard_Atmosphere
Dynamic Pressure and Mach Number
! = air density, functionof height
= !sea levele"#h
a = speed of sound, linear functionof height
Dynamic pressure = q =1
2!V 2
Mach number =V
a
Definitions of Airspeed
Definitions of Airspeed
True Airspeed (TAS)*
Indicated Airspeed (IAS)
Calibrated Airspeed (CAS)*
Equivalent Airspeed (EAS)*
Airspeed is speed of aircraft measured withrespect to the air mass Airspeed = Inertial speed if wind speed = 0
IAS = 2 pstag ! pstatic( ) "SL = 2 pt ! ps( ) "SL
CAS = IAS corrected for instrument and position errors
EAS = CAS corrected for compressibility effects
TAS = EAS !SL
!(z)
Mach number
M =TAS
a
* Kayton & Fried, 1969; NASA TN-D-822, 1961
Air Data Measurement andComputation
Air Data System
Air Speed IndicatorAltimeterVertical Speed Indicator
Kayton & Fried, 1969
Air Data ProbesRedundant pitot tubes on F-117
Total and static temperature probe
Total and static pressure portson Concorde
Stagnation/static pressure probe
Redundant pitottubes on FougaMagister
Cessna 172 pitot tube
X-15 Q Ball
Air Data Instruments(Steam Gauges)
Calibrated Airspeed Indicator Altimeter Vertical Speed Indicator
Variometer/Altimeter
True Airspeed Indicator
Machmeter
1 knot = 1 nm / hr
= 1.151 st.mi. / hr = 1.852 km / hr
Air Data Computation for
Subsonic Aircraft
Kayton & Fried, 1969
Air Data Computation for
Supersonic Aircraft
Kayton & Fried, 1969
The Mysterious Disappearance ofAir France Flight 447 (Airbus A330-200)
http://en.wikipedia.org/wiki/AF_447
BEA Interim Reports, 7/2/2009 & 11/30/2009
http://www.bea.aero/en/enquetes/flight.af.447/flight.af.447.php
Suspected Failure of Thales
Heated Pitot Probe
Visual examination showed that the airplane
was not destroyed in flight; it appears to have
struck the surface of the sea in level flight with
high vertical acceleration.
Flight in theVertical Plane
Longitudinal Variables
Longitudinal Point-Mass
Equations of Motion
!V =
CT cos! " CD( )1
2#V 2S " mg sin$
m%
CT " CD( )1
2#V 2S " mg sin$
m
!$ =CT sin! + CL( )
1
2#V 2S " mgcos$
mV%
CL1
2#V 2S " mgcos$
mV
!h = " !z = "vz = V sin$
!r = !x = vx = V cos$ V = velocity
! = flight path angle
h = height (altitude)
r = range
Assume thrust is aligned with the velocityvector (small-angle approximation for !)
Mass = constant
Steady, Level Flight
0 =
CT ! CD( )1
2"V 2S
m
0 =
CL1
2"V 2S ! mg
mV
!h = 0
!r = V
Flight path angle = 0
Altitude = constant
Airspeed = constant
Dynamic pressure = constant
Thrust = Drag
Lift = Weight
Subsonic Lift and DragCoefficients
CL= C
Lo+ C
L!!
CD= C
Do+ !C
L
2
Lift coefficient
Drag coefficient
Subsonic flight, belowcritical Mach number
CLo, C
L!, C
Do, " # constant
Subsonic
Incompressible
Power and Thrust
Propeller
Turbojet
Power = P = T !V = CT1
2"V 3S # independent of airspeed
Thrust = T = CT1
2!V 2S " independent of airspeed
Throttle Effect
T = Tmax!T = CTmax!TqS, 0 " !T " 1
Typical Effects of Altitude and
Velocity on Power and Thrust
Propeller
Turbojet
Thrust of a Propeller-Driven Aircraft
T = !P!I
Pengine
V= !
net
Pengine
V
Efficiencies decrease with airspeed
Engine power decreases with altitude
Proportional to air density, w/o supercharger
With constant rpm, variable-pitch propeller
where
!P = propeller efficiency
!I = ideal propulsive efficiency
!netmax " 0.85 # 0.9
Advance Ratio
J =V
nD
from McCormick
Propeller Efficiency, "P,and Advance Ratio, J
Effect of propeller-blade pitch angle
where
V = airspeed, m / s
n = rotation rate, revolutions / s
D = propeller diameter, m
Thrust of a
Turbojet
Engine
T = !mV!o
!o"1
#$%
&'(
!t
!t"1
#$%
&'(
)c"1( ) +
!t
!o)c
*
+,
-
./
1/2
"1012
32
452
62
Little change in thrust with airspeed below Mcrit Decrease with increasing altitude
where
!m = !mair + !mfuel
!o =pstag
pambient
"
#$%
&'
(( )1)/(
; ( = ratio of specific heats * 1.4
!t =turbine inlet temperature
freestream ambient temperature
"
#$%
&'
+ c =compressor outlet temperature
compressor inlet temperature
"
#$%
&' from Kerrebrock
Performance Parameters
Lift-to-Drag Ratio
Load Factor
LD=CL
CD
n = LW
= Lmg,"g"s
Thrust-to-Weight Ratio T W= T
mg,"g"s
Wing Loading WS, N m
2or lb ft
2
Steady, Level Flight
Trimmed CL and !
Trimmed liftcoefficient, CL Proportional to
weight
Decrease with V2
At constantairspeed, increaseswith altitude
Trimmed angle of attack, ! Constant if dynamic pressure
and weight are constant
If dynamic pressure decreases,angle of attack must increase
W = CLtrim qS
CLtrim =1
qW S( ) =
2
!V 2W S( ) =
2 e"h
!0V2
#
$%&
'(W S( )
! trim =2W "V 2S # CLo
CL!
=
1
qW S( ) # CLo
CL!
Thrust Required for
Steady, Level Flight
Trimmed thrust
Ttrim
= Dcruise
= CDo
1
2!V 2S"
#$%&'+ (
2W2
!V 2S
Minimum required thrust conditions
!Ttrim
!V= C
Do"VS( ) #
4$W 2
"V 3S= 0Necessary Condition
= Zero Slope
Parasitic Drag Induced Drag
Necessary and Sufficient
Conditions for Minimum
Required Thrust
!Ttrim
!V= C
Do"VS( ) #
4$W 2
"V 3S= 0
Necessary Condition = Zero Slope
Sufficient Condition for a Minimum = Positive Curvature when slope = 0
! 2Ttrim
!V 2= C
Do"S( ) +
12#W 2
"V 4S> 0
(+) (+)
Airspeed for
Minimum Thrust in
Steady, Level Flight
Fourth-order equation for velocity
Choose the positive root
!Ttrim
!V= C
Do"VS( ) #
4$W 2
"V 3S= 0
VMT
=2
!W
S
"#$
%&'
(CDo
Satisfy necessarycondition V
4=
4!CDo"2
#
$%
&
'( W S( )
2
P-51 MustangMinimum-ThrustExample
VMT
=2
!W
S
"#$
%&'
(CDo
=2
!1555.7( )
0.947
0.0163=76.49
!m / s
Wing Span = 37 ft (9.83m)
Wing Area = 235 ft2(21.83m
2)
Loaded Weight = 9,200 lb (3, 465 kg)
CDo = 0.0163
! = 0.0576
W / S = 39.3 lb / ft2(1555.7 N / m
2)
Altitude, m
Air Density,
kg/m^3 VMT, m/s0 1.23 69.112,500 0.96 78.205,000 0.74 89.1510,000 0.41 118.87
Lift Coefficient in
Minimum-Thrust
Cruising Flight
VMT
=2
!W
S
"#$
%&'
(CDo
CLMT
=2
!VMT
2
W
S
"#$
%&'=
CDo
(
Airspeed for minimum thrust
Corresponding lift coefficient
Power Required for
Steady, Level Flight
Trimmed power
Ptrim
= TtrimV = D
cruiseV = C
Do
1
2!V 2S"
#$%&'+2(W 2
!V 2S)
*+
,
-.V
Minimum required power conditions
!Ptrim
!V= C
Do
3
2"V 2S( ) #
2$W 2
"V 2S= 0
Parasitic Drag Induced Drag
Airspeed for Minimum
Power in Steady,
Level Flight
Fourth-order equation for velocity Choose the positive root
VMP
=2
!W
S
"#$
%&'
(3C
Do
Satisfy necessary condition
!Ptrim
!V= C
Do
3
2"V 2S( ) #
2$W 2
"V 2S= 0
Corresponding lift anddrag coefficients
CLMP
=3C
Do
!
CDMP
= 4CDo
Achievable Airspeeds in Cruising Flight
Two equilibrium airspeeds for a given thrust or power setting
Low speed, high CL, high !
High speed, low CL, low !
Achievable airspeeds between minimum and maximum valueswith maximum thrust or power
Back Side of the
Thrust Curve
Achievable Airspeeds for Jetin Cruising Flight
Tavail
= CDo
1
2!V 2S"
#$%&'+2(W 2
!V 2S
CDo
1
2!V 4S"
#$%&') T
availV2+2(W 2
!S= 0
V4 )
TavailV2
CDo!S
+4(W 2
CDo
!S( )2= 0
Thrust = constant
Solutions for V can be put in quadratic form and solved easily
x !V 2; V = x
ax2
+ bx + c = 0
x = "b
2
b
2
# $ %
& ' ( 2
" c , a =1
Achievable Airspeeds
in Propeller-Driven
Cruising Flight
Pavail
= TavailV
V4 !
PavailV
CDo"S
+4#W 2
CDo
"S( )2= 0
Power = constant
Solutions for V cannot be put in quadratic form; solution ismore difficult, e.g., Ferrari!s method
aV4+ 0( )V 3 + 0( )V 2 + dV + e = 0
Best bet: roots in MATLAB
Back Side ofthe Power
Curve
With increasing altitude, available thrust decreases, and range ofachievable airspeeds decreases
Stall limitation at low speed
Mach number effect on lift and drag increases thrust required at high speed
Thrust Required and ThrustAvailable for a Typical Bizjet
Typical Simplified Jet Thrust Model
Tmax(h) = T
max(SL)
!"nh
!(SL), n < 1
= Tmax(SL)
!"#h
!(SL)$
%&
'
()
x
* Tmax(SL)+ x
where
! ="#$h
"(SL), n or x is an empirical constant
Thrust Required and ThrustAvailable for a Typical Bizjet
Stall
Limit
Maximum Lift-to-Drag Ratio
CL( )L /Dmax =
CDo
!= C
LMT
LD=CL
CD
=CL
CDo+ !C
L
2
!CL
CD
( )!C
L
=1
CDo+ "C
L
2#
2"CL
2
CDo+ "C
L
2( )2= 0
Satisfy necessary condition for a maximum
Lift-to-drag ratio
Lift coefficient for maximum L/D and minimum thrust arethe same
Airspeed, Drag Coefficient, andLift-to-Drag Ratio for L/Dmax
VL /Dmax
= VMT
=2
!W
S
"#$
%&'
(CDo
CD( )L /Dmax = CDo + CDo = 2CDo
L / D( )max
=CDo
!
2CDo
=1
2 !CDo
Maximum L/D depends only on induced drag factor andzero-! drag coefficient
Lift-Drag Polar for aTypical 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
P-51 MustangMaximum L/D
Example
VL /Dmax
= VMT
=76.49
!m / s
Wing Span = 37 ft (9.83m)
Wing Area = 235 ft (21.83m2)
Loaded Weight = 9,200 lb (3, 465 kg)
CDo = 0.0163
! = 0.0576
W / S = 1555.7 N / m2
CL( )L /Dmax =
CDo
!= C
LMT= 0.531
CD( )L /Dmax = 2CDo = 0.0326
L / D( )max
=1
2 !CDo
= 16.31
Altitude, m
Air Density,
kg/m^3 VMT, m/s0 1.23 69.112,500 0.96 78.205,000 0.74 89.1510,000 0.41 118.87
Optimal Cruising Flight
Cruising Range andSpecific Fuel Consumption
0 =
CT ! CD( )1
2"V 2S
m
0 =
CL1
2"V 2S ! mg
mV
!h = 0
!r = V
Thrust = Drag
Lift = Weight
Specific fuel consumption, SFC = cP or cT
Propeller aircraft
Jet aircraft
!wf = !cPP proportional to power[ ]
!wf = !cTT proportional to thrust[ ]where
wf = fuel weight
cP :kg s
kWor
lb s
HP
cT :kg s
kNor
lb s
lbf
Breguet Range Equationfor Jet Aircraft
dr
dw=dr dt
dw dt=
!r
!w=
V
!cTT( )
= !V
cTD
= !L
D
"#$
%&'V
cTW
dr = !L
D
"#$
%&'V
cTWdw
Rate of change of range with respect to weight of fuel burned
Range traveled
Range = R = dr0
R
! = "L
D
#$%
&'(
V
cT
#
$%&
'(Wi
Wf
!dw
w
Louis Breguet,1880-1955
Breguet RangeEquation for Jet Aircraft
For constant true airspeed, V = Vcruise
R = !L
D
"#$
%&'Vcruise
cT
"
#$%
&'ln w( )
Wi
Wf
=L
D
"#$
%&'Vcruise
cT
"
#$%
&'ln
Wi
Wf
"
#$
%
&' =
CL
CD
"
#$%
&'Vcruise
cT
"
#$%
&'ln
Wi
Wf
"
#$
%
&'
Dassault
Etendard IV
Maximum Range of a
Jet Aircraft Flying at
Constant True Airspeed
!R
!CL=! VCL CD( )
!CL= 0 leading to CLMR =
CDo
3"
For given initial and final weight, range is maximized whenproduct of V and L/D is maximized
Breguet range equation for constant V = Vcruise
R = VcruiseCL
CD
!
"#$
%&1
cT
!
"#$
%&ln
Wi
Wf
!
"#
$
%&
! Vcruise as fast as possible
! $ as small as possible
! h as high as possible
CLMR
=CDo
3!: Lift Coefficient for Maximum Range
Maximum Range of a Jet Aircraft
Flying at Constant True Airspeed
Because weight decreases as fuel burns, and V isassumed constant, altitude must increase to hold CLconstant at its best value (cruise-climb)
CLMR q t( )S =W t( )!
q t( ) =1
2! t( )Vcruise
2=
W t( )
S
"
#$
%
&'
3(CDo
)
!(t) = !oe"#h(t )
=2
Vcruise
2
W (t)
S
$%&
'()
3*CDo
+
h W t( ),Vcruise
!" #$
Maximum Range of a
Jet Aircraft Flying at
Constant Altitude
Range is maximized when
Range = !CL
CD
"
#$%
&'1
cT
"
#$%
&'2
CL(SWi
Wf
)dw
w1 2
=CL
CD
"
#$
%
&'2
cT
"
#$%
&'2
(SWi
1 2 !Wf1 2( )
Vcruise t( ) =2W t( )
CL! hfixed( )S At constant altitude
CL
CD
!
"#
$
%& = maximum and
' = minimumh = maximum
()*
+*
! Cruise-climb usually violates airtraffic control rules
! Constant-altitude cruise does not
! Compromise: Step climb fromone allowed altitude to the next
Breguet Range Equation
for Propeller-Driven
Aircraft
dr
dw=
!r
!w=
V
!cPP( )
= !V
cPTV
= !V
cPDV
= !L
D
"#$
%&'1
cPW
Rate of change of range with respect to weight of fuel burned
Range traveled
Range = R = dr0
R
! = "L
D
#$%
&'(1
cP
#
$%&
'(Wi
Wf
!dw
w
Breguet 890 Mercure
Breguet Range Equation
for Propeller-Driven
Aircraft For constant true airspeed, V = Vcruise
R = !L
D
"#$
%&'1
cP
"
#$%
&'ln w( )
Wi
Wf
=CL
CD
"
#$%
&'1
cP
"
#$%
&'ln
Wi
Wf
"
#$
%
&'
Range is maximized when
CL
CD
!
"#$
%&= maximum = L
D( )max
Breguet Atlantique
P-51 Mustang
Maximum Range(Internal Tanks only)
Loaded Weight = 9,200 lb (3, 465 kg)
Fuel Weight = 1,320 lb (600 kg)
L / D( )max
= 16.31
cP = 0.0017kg / s
kW
R =CL
CD
!
"#$
%&max
1
cP
!
"#$
%&ln
Wi
Wf
!
"#
$
%&
= 16.31( )1
0.0017
!"#
$%&ln
3,465 + 600
3,465
!"#
$%&
= 1,530 km (825 nm( )
Next Time:Gliding, Climbing, and
Turning Flight
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