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!
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