Cruising Flight Performance Robert Stengel, Aircraft Flight Dynamics,

MAE 331, 2012!

Copyright 2012 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"•  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 (Supplement)"

U.S. Standard Atmosphere, 1976"

http://en.wikipedia.org/wiki/U.S._Standard_Atmosphere!

Dynamic Pressure and Mach Number"ρ = airdensity, functionof height= ρsealevele

−βh

a = speed of sound= linear functionof height

Dynamic pressure = q ρV 2 2Mach number =V a

Definitions of Airspeed"

•  Indicated Airspeed (IAS)"

•  Calibrated Airspeed (CAS)*"

•  Airspeed is speed of aircraft measured with respect to air mass"–  Airspeed = Inertial speed if wind speed = 0"

IAS = 2 pstagnation − pambient( ) ρSL =2 ptotal − pstatic( )

ρSL

=2qcρSL

, with qc impact pressure

CAS = IAS corrected for instrument and position errors

=2 qc( )corr−1

ρSL* Kayton & Fried, 1969; NASA TN-D-822, 1961!

Definitions of Airspeed"

•  True Airspeed (TAS)*"

•  Equivalent Airspeed (EAS)*"

•  Airspeed is speed of aircraft measured with respect to air mass"–  Airspeed = Inertial speed if wind speed = 0"

EAS =CAS corrected for compressibility effects =2 qc( )corr−2

ρSL

V TAS = EAS ρSL

ρ(z)= IAScorrected

ρSLρ(z)

•  Mach number"

M =TASa

* Kayton & Fried, 1969; NASA TN-D-822, 1961!

Air Data System"

Air Speed Indicator!Altimeter!Vertical Speed Indicator!

Kayton & Fried, 1969!

•  Subsonic speed: no shock wave ahead of pitot tube"•  Supersonic speed: normal shock wave ahead of pitot tube"

Dynamic and Impact Pressure"

•  Dynamic pressure also can be expressed in terms of Mach number and static (ambient) pressure"

q ρV 2 2 : Dynamic pressureqc = ptotal − pstatic : Impact pressure

pstat z( ) = ρamb z( )RT z( ) [Ideal gas law, R = 287.05 J/kg-°K]

a z( ) = γRT z( ) [Speed of sound, T = absolute temperature, °K, γ = 1.4]

M =V a [Mach number]

q ρamb z( )V 2 2 = γ

2pstat z( )M 2

•  In incompressible flow, dynamic pressure = impact pressure"

Substituting!

•  In subsonic, isentropic compressible flow"

•  Impact pressure is"

ptotal z( )pstatic z( )

= 1+ γ −12

M 2#

$%

&

'(γ γ−1( )

qc ptotal z( )− pstatic z( )"# $%= pstatic z( ) 1+0.2M 2( )3.5 −1"

#&$%'

Compressibility Effects on Impact Pressure"

•  In supersonic, isentropic compressible flow, impact pressure is"

qc = pstatic z( )1+γ2

M 2 γ +1( )2

4γ −2 γ −1( )M 2

#

$

%%%%

&

'

((((

1 γ−1( )

−1

)

*++

,++

-

.++

/++

Flight in the Vertical Plane �

Longitudinal Variables!

Longitudinal Point-Mass Equations of Motion"

V =CT cosα −CD( ) 1

2ρV 2S −mg sinγ

m≈CT −CD( ) 1

2ρV 2S −mg sinγ

m

γ =CT sinα +CL( ) 1

2ρV 2S −mgcosγ

mV≈CL12ρV 2S −mgcosγ

mVh = − z = −vz =V sinγr = x = vx =V cosγ

V = velocityγ = flight path angleh = height (altitude)r = range

•  Assume thrust is aligned with the velocity vector (small-angle approximation for α)"

•  Mass = constant"

Steady, Level Flight"

0 =CT − CD( ) 1

2ρV 2S

m

0 =CL12ρV 2S − mg

mVh = 0r = V

•  Flight path angle = 0"•  Altitude = constant"•  Airspeed = constant"•  Dynamic pressure = constant"

•  Thrust = Drag"

•  Lift = Weight"

Subsonic Lift and Drag Coefficients"

CL = CLo+ CLα

α

CD = CDo+ εCL

2

•  Lift coefficient"

•  Drag coefficient"

•  Subsonic flight, below critical Mach number "

CLo, CLα

, CDo, ε ≈ constant Subsonic!

Incompressible!

Power and Thrust"•  Propeller"

•  Turbojet"

Power = P = T ×V = CT12ρV 3S ≈ independent of airspeed

Thrust = T = CT12ρV 2S ≈ independent of airspeed

•  Throttle Effect"

T = TmaxδT = CTmaxδTqS, 0 ≤ δT ≤ 1

Typical Effects of Altitude and Velocity on Power and Thrust"

•  Propeller"

•  Turbojet"

Thrust of a Propeller-Driven Aircraft"

T = ηPηI

PengineV

= ηnet

PengineV

•  Efficiencies decrease with airspeed"•  Engine power decreases with altitude"

–  Proportional to air density, w/o supercharger"

•  With constant rpm, variable-pitch propeller"

whereηP = propeller efficiencyηI = ideal propulsive efficiencyηnetmax

≈ 0.85 − 0.9

•  Advance Ratio"

J = VnD

from McCormick!

Propeller Efficiency, ηP, and Advance Ratio, J"

Effect of propeller-blade pitch angle!

whereV = airspeed, m / sn = rotation rate, revolutions / sD = propeller diameter, m

Thrust of a Turbojet Engine"

T = mV θoθo −1#

$%

&

'(

θtθt −1#

$%

&

'( τ c −1( )+ θt

θoτ c

*

+,

-

./

1/2

−1012

32

452

62

•  Little change in thrust with airspeed below Mcrit"•  Decrease with increasing altitude"

wherem = mair + mfuel

θo =pstagpambient

"

#$

%

&'

(γ−1)/γ

; γ = ratio of specific heats ≈ 1.4

θt =turbine inlet temperature

freestream ambient temperature

"

#$

%

&'

τ c =compressor outlet temperaturecompressor inlet temperature

"

#$

%

&'

from Kerrebrock!

Performance Parameters"

•  Lift-to-Drag Ratio"

•  Load Factor"

LD = CL

CD

n = LW = L mg ,"g"s

•  Thrust-to-Weight Ratio" T W = T mg ,"g"s

•  Wing Loading" WS , N m2 or lb ft 2

Steady, Level Flight�

Trimmed CL and α"•  Trimmed lift

coefficient, CL"–  Proportional to

weight"–  Decrease with V2"–  At constant

airspeed, increases with altitude"

•  Trimmed angle of attack, α"–  Constant if dynamic pressure

and weight are constant"–  If dynamic pressure decreases,

angle of attack must increase"

W =CLtrimqS

CLtrim=1qW S( ) = 2

ρV 2 W S( ) = 2 eβh

ρ0V2

#

$%

&

'( W S( )

αtrim =2W ρV 2S −CLo

CLα

=

1qW S( )−CLo

CLα

Thrust Required for Steady, Level Flight"•  Trimmed thrust"

Ttrim = Dcruise =CDo

12ρV 2S

"

#$

%

&'+ε

2W 2

ρV 2S

•  Minimum required thrust conditions"

∂Ttrim∂V

= CDoρVS( ) − 4εW

2

ρV 3S= 0Necessary Condition

= Zero Slope!

Parasitic Drag! Induced Drag!

Necessary and Sufficient Conditions for Minimum

Required Thrust "

∂Ttrim∂V

= CDoρVS( ) − 4εW

2

ρV 3S= 0

Necessary Condition = Zero Slope!

Sufficient Condition for a Minimum = Positive Curvature when slope = 0!

∂ 2Ttrim∂V 2 = CDo

ρS( ) + 12εW2

ρV 4S> 0

(+)" (+)"

Airspeed for Minimum Thrust in Steady, Level Flight"

•  Fourth-order equation for velocity"–  Choose the positive root"

∂Ttrim∂V

= CDoρVS( ) − 4εW

2

ρV 3S= 0

VMT =2ρWS

"

#$

%

&'

εCDo

•  Satisfy necessary condition" V 4 =

4εCDo

ρ2#

$%%

&

'(( W S( )2

P-51 Mustang Minimum-Thrust Example"

VMT =2ρWS

"

#$

%

&'

εCDo

=2ρ1555.7( ) 0.947

0.0163=76.49ρ

m / s

Wing Span = 37 ft (9.83m)Wing Area = 235 ft 2 (21.83m2 )Loaded Weight = 9,200 lb (3, 465 kg)CDo

= 0.0163ε = 0.0576W / S = 39.3 lb / ft 2 (1555.7 N / m2 )

Altitude, mAir Density, kg/m^3 VMT, m/s

0 1.23 69.112,500 0.96 78.205,000 0.74 89.1510,000 0.41 118.87

Airspeed for minimum thrust!

Lift Coefficient in Minimum-Thrust

Cruising Flight"

VMT =2ρWS

"

#$

%

&'

εCDo

CLMT=

2ρVMT

2WS

"

#$

%

&'=

CDo

ε

•  Airspeed for minimum thrust"

•  Corresponding lift coefficient"

Power Required for Steady, Level Flight"

•  Trimmed power"

Ptrim =TtrimV = DcruiseV = CDo

12ρV 2S

"

#$

%

&'+2εW 2

ρV 2S)

*+

,

-.V

•  Minimum required power conditions"

∂Ptrim∂V

= CDo

32ρV 2S( ) − 2εW

2

ρV 2S= 0

Parasitic Drag! Induced Drag!

Airspeed for Minimum Power in Steady,

Level Flight"

•  Fourth-order equation for velocity"–  Choose the positive root"

VMP =2ρWS

"

#$

%

&'

ε3CDo

•  Satisfy necessary condition"

∂Ptrim∂V

= CDo

32ρV 2S( ) − 2εW

2

ρV 2S= 0

•  Corresponding lift and drag coefficients"

CLMP=

3CDo

εCDMP

= 4CDo

Achievable Airspeeds in Cruising Flight"

•  Two equilibrium airspeeds for a given thrust or power setting"–  Low speed, high CL, high α#–  High speed, low CL, low α#

•  Achievable airspeeds between minimum and maximum values with maximum thrust or power#

Back Side of the Thrust Curve"

Achievable Airspeeds for Jet in Cruising Flight"

Tavail =CDo

12ρV 2S

"

#$

%

&'+2εW 2

ρV 2S

CDo

12ρV 4S

"

#$

%

&'−TavailV

2 +2εW 2

ρS= 0

V 4 −TavailV

2

CDoρS

+4εW 2

CDoρS( )2

= 0

•  Thrust = constant#

•  Solutions for V can be put in quadratic form and solved easily#

€

x ≡V 2; V = ± x

ax 2 + bx + c = 0

x = −b2

±b2$

% & '

( ) 2

− c , a =1

•  4th-order algebraic equation for V#

•  With increasing altitude, available thrust decreases, and range of achievable airspeeds decreases"

•  Stall limitation at low speed"•  Mach number effect on lift and drag increases thrust required at high speed"

Thrust Required and Thrust Available for a Typical Bizjet"

Typical Simplified Jet Thrust Model!

Tmax (h) = Tmax (SL)ρ−nh

ρ(SL), n < 1

= Tmax (SL)ρ−βh

ρ(SL)$

%&

'

()

x

≡ Tmax (SL)σx

where

σ =ρ−βh

ρ(SL), n or x is an empirical constant

Thrust Required and Thrust Available for a Typical Bizjet"

Typical Stall!Limit!

Maximum Lift-to-Drag Ratio"

CL( )L /Dmax =CDo

ε= CLMT

LD = CL

CD=

CL

CDo+ εCL

2

∂ CLCD

( )∂CL

=1

CDo+ εCL

2 −2εCL

2

CDo+ εCL

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"

Airspeed, Drag Coefficient, and Lift-to-Drag Ratio for L/Dmax"

VL /Dmax = VMT =2ρ

WS

"#$

%&'

εCDo

CD( )L /Dmax = CDo+ CDo

= 2CDo

L / D( )max =CDo

ε

2CDo

=1

2 εCDo

•  Maximum L/D depends only on induced drag factor and zero-α drag coefficient"

Airspeed!

Drag !Coefficient!

Maximum !L/D!

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 ε and zero-lift drag"

Note different scales for lift and drag!

P-51 Mustang Maximum L/D

Example"

VL /Dmax = VMT =76.49ρ

m / s

Wing Span = 37 ft (9.83m)Wing Area = 235 ft (21.83m2 )Loaded Weight = 9,200 lb (3, 465 kg)CDo

= 0.0163ε = 0.0576W / S = 1555.7 N / m2

CL( )L /Dmax =CDo

ε= CLMT

= 0.531

CD( )L /Dmax = 2CDo= 0.0326

L / D( )max =1

2 εCDo

= 16.31

Altitude, mAir Density, kg/m^3 VMT, m/s

0 1.23 69.112,500 0.96 78.205,000 0.74 89.1510,000 0.41 118.87

Optimal Cruising Flight�

Cruising Range and Specific Fuel Consumption"

0 =CT − CD( ) 1

2ρV 2S

m

0 =CL12ρV 2S − mg

mVh = 0r = 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[ ]wherewf = fuel weight

€

cP :kg skW

or lb sHP

cT :kg skN

or lb slbf

Breguet Range Equation for Jet Aircraft"

drdw

=dr dtdw dt

=rw=

V−cTT( )

= −VcTD

= −LD"

#$

%

&'VcTW

dr = − LD"

#$

%

&'VcTW

dw

•  Rate of change of range with respect to weight of fuel burned"

•  Range traveled"

Range = R = dr0

R

∫ = −LD#

$%

&

'(VcT

#

$%

&

'(

Wi

Wf

∫ dww

Louis Breguet, 1880-1955!

Breguet Range Equation for Jet Aircraft"

•  For constant true airspeed, V = Vcruise!

R = − LD"

#$

%

&'VcruisecT

"

#$

%

&'ln w( )Wi

Wf

=LD"

#$

%

&'VcruisecT

"

#$

%

&'ln

Wi

Wf

"

#$$

%

&''=

CL

CD

"

#$

%

&'VcruisecT

"

#$

%

&'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

3ε

•  For given initial and final weight, range is maximized when product of V and L/D is maximized"

•  Breguet range equation for constant V = Vcruise"

R = VcruiseCL

CD

!

"#

$

%&1cT

!

"#

$

%&ln

Wi

Wf

!

"##

$

%&&

! Vcruise as fast as possible"! ρ as small as possible"! h as high as possible"

CLMR=

CDo

3ε: Lift Coefficient for Maximum Range

Maximum Range of a Jet Aircraft Flying at Constant True Airspeed"•  Because weight decreases as fuel burns, and V is

assumed constant, altitude must increase to hold CL constant at its best value (�cruise-climb�)"

CLMRq t( )S =W t( )⇒

q t( ) = 12ρ t( )Vcruise2 =

W t( )S

"

#$

%

&'

3εCDo

⇒

ρ(t) = ρoe−βh(t ) =

2Vcruise2

W (t)S

$

%&'

()3εCDo

⇒

h W t( ),Vcruise!" #$

Maximum Range of a Jet Aircraft Flying at

Constant Altitude"

•  Range is maximized when "

Range = − CL

CD

"

#$

%

&'1cT

"

#$

%

&'

2CLρSWi

Wf

∫ dww1 2

=CL

CD

"

#$$

%

&''2cT

"

#$

%

&'

2ρS

Wi1 2 −Wf

1 2( )

Vcruise t( ) =2W t( )

CLρ hfixed( )S•  At constant altitude"

CL

CD

!

"##

$

%&&= maximum and ρ =minimum

h = maximum

()*

+*

!  Cruise-climb usually violates air traffic control rules"

!  Constant-altitude cruise does not"!  Compromise: Step climb from

one allowed altitude to the next"

Next Time: �Gliding, Climbing, and

Turning Flight��

Reading �Flight Dynamics, 130-141, 147-155 �

Virtual Textbook, Parts 6,7 �

Supplemental Material

Air Data Probes"Redundant pitot tubes on F-117"

Total and static temperature probe"

Total and static pressure ports on Concorde"

Stagnation/static pressure probe"

Redundant pitot tubes on Fouga Magister"

Cessna 172 pitot tube"

X-15 �Q Ball�"

Flight Testing Instrumentation"•  Air data measurement far from

disturbing effects of the aircraft"

z =

pstagnation ,Tstagnationpstatic ,Tstatic

αB

βB

#

$

%%%%%

&

'

(((((

=

Stagnation pressure and temperatureStatic pressure and temperature

Angle of attackSideslip angle

#

$

%%%%%

&

'

(((((

Trailing Tail Cones for Accurate Static Pressure Measurement"

•  Air data measurement far from disturbing effects of the aircraft"

Air Data Instruments (�Steam Gauges�)"

Altimeter"

1 knot = 1 nm / hr= 1.151 st.mi. / hr = 1.852 km / hr

Calibrated Airspeed Indicator"

Modern Aircraft Cockpit Panels"Cirrus SR-22 Panel" Boeing 777 �Glass Cockpit�"

Air Data Computation for Subsonic Aircraft"

Kayton & Fried, 1969!

Air Data Computation for Supersonic Aircraft"

Kayton & Fried, 1969!

The Mysterious Disappearance of Air 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.�!

Achievable Airspeeds in Propeller-Driven

Cruising Flight"

Pavail = TavailV

V 4 −PavailVCDo

ρS+

4εW 2

CDoρS( )2

= 0

•  Power = constant#

•  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#

Back Side of the Power

Curve"

Breguet Range Equation for Propeller-Driven

Aircraft"

drdw

=rw=

V−cPP( )

= −V

cPTV= −

VcPDV

= −LD"

#$

%

&'1

cPW

•  Rate of change of range with respect to weight of fuel burned"

•  Range traveled"

Range = R = dr0

R

∫ = −LD#

$%

&

'(1cP

#

$%

&

'(

Wi

Wf

∫ dww

Breguet 890 Mercure!

Breguet Range Equation for Propeller-Driven Aircraft"

•  For constant true airspeed, V = Vcruise!

R = − LD"

#$

%

&'1cP

"

#$

%

&'ln w( )Wi

Wf

=CL

CD

"

#$

%

&'1cP

"

#$

%

&'ln

Wi

Wf

"

#$$

%

&''

•  Range is maximized when "

CL

CD

!

"#

$

%&= maximum= L

D( )max

Breguet Atlantique!

P-51 Mustang Maximum Range

(Internal Tanks only)"

W =CLtrimqS

CLtrim=1qW S( ) = 2

ρV 2 W S( ) = 2 eβh

ρ0V2

#

$%

&

'( W S( )

R = CL

CD

!

"#

$

%&max

1cP

!

"#

$

%&ln

Wi

Wf

!

"##

$

%&&

= 16.31( ) 10.0017!

"#

$

%&ln

3,465+6003,465

!

"#

$

%&

=1,530 km (825 nm( )