6_Cruising Flight Performance.pdf

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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