6_Cruising Flight Performance.pdf

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


Air Data Computation for

Supersonic Aircraft


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


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


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


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


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)


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)


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

Documents

6_Cruising Flight Performance.pdf