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 = air density, functionof height = ρ sealevel e − β h a = speed of sound = linear functionof height Dynamic pressure = q  ρ V 2 2 Mach 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 p stagnation − p ambient ( ) ρ SL = 2 p total − p static ( ) ρ SL = 2q c ρ SL , with q c  impact pressure CAS = IAS corrected for instrument and position errors = 2 q c ( ) corr−1 ρ SL * 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 q c ( ) corr−2 ρ SL V  TAS = EAS ρ SL ρ (z) = IAS corrected ρ SL ρ (z) •  Mach number" M = TAS a * 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  ρ V 2 2 : Dynamic pressure q c = p total − p static : Impact pressure p stat z ( ) = ρ amb z ( ) RT z ( ) [Ideal gas law, R = 287.05 J/kg-°K] a z ( ) = γ RT z ( ) [Speed of sound, T = absolute temperature, °K, γ = 1.4] M =V a [Mach number] q  ρ amb z ( ) V 2 2 = γ 2 p stat z ( ) M 2 •  In incompressible flow, dynamic pressure = impact pressure" Substituting! •  In subsonic, isentropic compressible flow" •  Impact pressure is" p total z ( ) p static z ( ) = 1+ γ −1 2 M 2 # $ % & ' ( γ γ −1 ( ) q c  p total z ( ) − p static z ( ) " # $ % = p static z ( ) 1+ 0.2M 2 ( ) 3.5 −1 " # & $ % ' Compressibility Effects on Impact Pressure" •  In supersonic, isentropic compressible flow, impact pressure is" q c = p static z ( ) 1+ γ 2 M 2 γ +1 ( ) 2 4 γ − 2 γ −1 ( ) M 2 # $ % % % % & ' ( ( ( ( 1 γ −1 ( ) −1 ) * + + , + + - . + + / + + Flight in the Vertical Plane Longitudinal Variables! Longitudinal Point-Mass Equations of Motion"  V = C T cos α − C D ( ) 1 2 ρ V 2 S −mg sin γ m ≈ C T − C D ( ) 1 2 ρ V 2 S −mg sin γ m  γ = C T sin α + C L ( ) 1 2 ρ V 2 S −mg cos γ mV ≈ C L 1 2 ρ V 2 S −mg cos γ mV  h = −  z = −v z = V sin γ  r =  x = v x = V cos γ V = velocity γ = flight path angle h = height (altitude) r = range •  Assume thrust is aligned with the velocity vector (small-angle approximation for α )" •  Mass = constant" Steady, Level Flight" 0 = C T − C D ( ) 1 2 ρ V 2 S m 0 = C L 1 2 ρ V 2 S − mg mV  h = 0  r = V •  Flight path angle = 0" •  Altitude = constant" •  Airspeed = constant" •  Dynamic pressure = constant" •  Thrust = Drag" •  Lift = Weight" Subsonic Lift and Drag Coefficients" C L = C L o + C L α α C D = C D o + ε C L 2 •  Lift coefficient" •  Drag coefficient" •  Subsonic flight, below critical Mach number " C L o , C L α , C D o , ε ≈ constant Subsonic! Incompressible! Power and Thrust" •  Propeller" •  Turbojet" Power = P = T × V = C T 1 2 ρ V 3 S ≈ independent of airspeed Thrust = T = C T 1 2 ρ V 2 S ≈ independent of airspeed •  Throttle Effect" T = T max δ T = C T max δ TqS, 0 ≤ δ T ≤ 1 Typical Effects of Altitude and Velocity on Power and Thrust" •  Propeller" •  Turbojet" Thrust of a Propeller- Driven Aircraft" T = η P η I P engine V = η net P engine V •  Efficiencies decrease with airspeed" •  Engine power decreases with altitude" –  Proportional to air density, w/o supercharger" •  With constant rpm, variable-pitch propeller" where η P = propeller efficiency η I = ideal propulsive efficiency η net max ≈ 0.85 − 0.9 •  Advance Ratio" J = V nD from McCormick! Propeller Efficiency, η 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 θ o θ o −1 # $ % & ' ( θ t θ t −1 # $ % & ' ( τ c −1 ( ) + θ t θ o τ c * + , - . / 1/2 −1 0 1 2 3 2 4 5 2 6 2 •  Little change in thrust with airspeed below M crit " •  Decrease with increasing altitude" where  m =  m air +  m fuel θ o = p stag p ambient " # $ % & ' ( γ −1)/ γ ; γ = ratio of specific heats ≈1.4 θ t = turbine inlet temperature freestream ambient temperature " # $ % & ' τ c = compressor outlet temperature compressor inlet temperature " # $ % & ' from Kerrebrock! Performance Parameters" •  Lift-to-Drag Ratio" •  Load Factor" L D = C L C D n = L W = L mg ,"g"s •  Thrust-to-Weight Ratio" T W = T mg ,"g"s •  Wing Loading" W S , N m 2 or lb ft 2 Steady, Level Flight Trimmed C L and α " •  Trimmed lift coefficient, C L " –  Proportional to weight" –  Decrease with V 2 " –  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 = C L trim qS C L trim = 1 q W S ( ) = 2 ρ V 2 W S ( ) = 2 e β h ρ 0 V 2 # $ % & ' ( W S ( ) α trim = 2W ρ V 2 S − C L o C L α = 1 q W S ( ) − C L o C L α Thrust Required for Steady, Level Flight" •  Trimmed thrust" T trim = D cruise = C D o 1 2 ρ V 2 S " # $ % & ' + ε 2W 2 ρ V 2 S •  Minimum required thrust conditions" ∂ T trim ∂ V = C D o ρ VS ( ) − 4 ε W 2 ρ V 3 S = 0 Necessary Condition = Zero Slope! Parasitic Drag! Induced Drag! Necessary and Sufficient Conditions for Minimum Required Thrust " ∂ T trim ∂ V = C D o ρ VS ( ) − 4 ε W 2 ρ V 3 S = 0 Necessary Condition = Zero Slope! Sufficient Condition for a Minimum = Positive Curvature when slope = 0! ∂ 2 T trim ∂ V 2 = C D o ρ S ( ) + 12 ε W 2 ρ V 4 S > 0 (+)" (+)" Airspeed for Minimum Thrust in Steady, Level Flight" •  Fourth-order equation for velocity" –  Choose the positive root" ∂ T trim ∂ V = C D o ρ VS ( ) − 4 ε W 2 ρ V 3 S = 0 V MT = 2 ρ W S " # $ % & ' ε C D o •  Satisfy necessary condition" V 4 = 4 ε C D o ρ 2 # $ % % & ' ( ( W S ( ) 2 P-51 Mustang Minimum-Thrust Example" V MT = 2 ρ W S " # $ % & ' ε C D o = 2 ρ 1555.7 ( ) 0.947 0.0163 = 76.49 ρ m / s 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 D o = 0.0163 ε = 0.0576 W / S = 39.3 lb / ft 2 (1555.7 N / m 2 ) Altitude, m Air Density, kg/m^3 VMT, m/s 0 1.23 69.11 2,500 0.96 78.20 5,000 0.74 89.15 10,000 0.41 118.87 Airspeed for minimum thrust! Lift Coefficient in Minimum-Thrust Cruising Flight" V MT = 2 ρ W S " # $ % & ' ε C D o C L MT = 2 ρ V MT 2 W S " # $ % & ' = C D o ε •  Airspeed for minimum thrust" •  Corresponding lift coefficient" Power Required for Steady, Level Flight" •  Trimmed power" P trim = T trim V = D cruise V = C D o 1 2 ρ V 2 S " # $ % & ' + 2 ε W 2 ρ V 2 S ) * + , - . V •  Minimum required power conditions" ∂ P trim ∂ V = C D o 3 2 ρ V 2 S ( ) − 2 ε W 2 ρ V 2 S = 0 Parasitic Drag! Induced Drag! Airspeed for Minimum Power in Steady, Level Flight" •  Fourth-order equation for velocity" –  Choose the positive root" V MP = 2 ρ W S " # $ % & ' ε 3C D o •  Satisfy necessary condition" ∂ P trim ∂ V = C D o 3 2 ρ V 2 S ( ) − 2 ε W 2 ρ V 2 S = 0 •  Corresponding lift and drag coefficients" C L MP = 3C D o ε C D MP = 4C D o Achievable Airspeeds in Cruising Flight" •  Two equilibrium airspeeds for a given thrust or power setting" –  Low speed, high C L , high α# –  High speed, low C L , 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" T avail = C D o 1 2 ρ V 2 S " # $ % & ' + 2 ε W 2 ρ V 2 S C D o 1 2 ρ V 4 S " # $ % & ' −T avail V 2 + 2 ε W 2 ρ S = 0 V 4 − T avail V 2 C D o ρ S + 4 ε W 2 C D o ρ S ( ) 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 = − b 2 ± b 2 $ % & ' ( ) 2 − c, a = 1 •  4 th -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! T max (h) = T max (SL) ρ −nh ρ (SL) , n < 1 = T max (SL) ρ − β h ρ (SL) $ % & ' ( ) x ≡ T max (SL) σ x where σ = ρ − β h 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" C L ( ) L / D max = C D o ε = C L MT L D = C L C D = C L C D o + ε C L 2 ∂ C L C D ( ) ∂ C L = 1 C D o + ε C L 2 − 2 ε C L 2 C D o + ε C L 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/D max " V L / D max = V MT = 2 ρ W S " # $ % & ' ε C D o C D ( ) L / D max = C D o + C D o = 2C D o L / D ( ) max = C D o ε 2C D o = 1 2 ε C D o •  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/D max depends only on ε and zero-lift drag" Note different scales for lift and drag! P-51 Mustang Maximum L/D Example" V L / D max = V MT = 76.49 ρ m / s Wing Span = 37 ft (9.83 m) Wing Area = 235 ft (21.83 m 2 ) Loaded Weight = 9, 200 lb (3, 465 kg) C D o = 0.0163 ε = 0.0576 W / S = 1555.7 N / m 2 C L ( ) L / D max = C D o ε = C L MT = 0.531 C D ( ) L / D max = 2C D o = 0.0326 L / D ( ) max = 1 2 ε C D o = 16.31 Altitude, m Air Density, kg/m^3 VMT, m/s 0 1.23 69.11 2,500 0.96 78.20 5,000 0.74 89.15 10,000 0.41 118.87 Optimal Cruising Flight Cruising Range and Specific Fuel Consumption" 0 = C T − C D ( ) 1 2 ρ V 2 S m 0 = C L 1 2 ρ V 2 S − mg mV  h = 0  r = V •  Thrust = Drag" •  Lift = Weight" •  Specific fuel consumption, SFC = c P or c T " •  Propeller aircraft" •  Jet aircraft"  w f = −c P P proportional to power [ ]  w f = −c T T proportional to thrust [ ] where w f = fuel weight € c P : kg s kW or lb s HP c T : kg s kN or lb s lbf Breguet Range Equation for Jet Aircraft" dr dw = dr dt dw dt =  r  w = V −c T T ( ) = − V c T D = − L D " # $ % & ' V c T W dr = − L D " # $ % & ' V c T W dw •  Rate of change of range with respect to weight of fuel burned" •  Range traveled" Range = R = dr 0 R ∫ = − L D # $ % & ' ( V c T # $ % & ' ( W i W f ∫ dw w Louis Breguet, 1880-1955! Breguet Range Equation for Jet Aircraft" •  For constant true airspeed, V = V cruise! R = − L D " # $ % & ' V cruise c T " # $ % & ' ln w ( ) W i W f = L D " # $ % & ' V cruise c T " # $ % & ' ln W i W f " # $ $ % & ' ' = C L C D " # $ % & ' V cruise c T " # $ % & ' ln W i W f " # $ $ % & ' ' Dassault ! Etendard IV! [...]... 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 of Air France Flight 447 (Airbus A330-200)" Back Side of the Power Curve " •  Achievable Airspeeds in Propeller-Driven Cruising Flight " Power = constant# Pavail = TavailV V4 − •  PavailV 4 εW 2 + =0... Gliding, Climbing, and Turning Flight Reading Flight Dynamics, 130-141, 147-155 Virtual Textbook, Parts 6,7 Air Data Probes " Stagnation/static pressure probe" Redundant pitot tubes on F-117" Supplemental Material Cessna 172 pitot tube" Total and static temperature probe" Redundant pitot tubes on Fouga Magister" Total and static pressure ports on Concorde" X-15 Q Ball " Flight Testing Instrumentation... Pitot Probe ! BEA Interim Reports, 7/2/2009 & 11/30/2009! http://www.bea.aero/en/enquetes/flight.af.447/flight.af.447.php! •  Best bet: roots in MATLAB# http://en.wikipedia.org/wiki/AF_447! Breguet Range Equation for Propeller-Driven Aircraft " Breguet 890 Mercure! •  Breguet Range Equation for Propeller-Driven Aircraft" •  Rate of change of range with respect to weight of fuel burned" " L %" 1 % Wf R =... the aircraft" # pstagnation ,Tstagnation % pstatic ,Tstatic % z=% αB % % βB $ & # Stagnation pressure and temperature ( % Static pressure and temperature ( % (=% Angle of attack ( % ( % Sideslip angle ' $ Trailing Tail Cones for Accurate Static Pressure Measurement" •  & ( ( ( ( ( ' Air data measurement far from disturbing effects of the aircraft" Air Data Instruments ( Steam Gauges ) " Modern Aircraft. ..Maximum Range of a Jet Aircraft Flying at Constant True Airspeed " Maximum Range of a Jet Aircraft Flying at Constant True Airspeed " •  •  Breguet range equation for constant V = Vcruise" ! C $! 1 $ ! W $ R = #Vcruise L &# & ln # i & C D %" cT % # W f & "... weight, range is maximized when product of V and L/D is maximized" C Do ∂ R ∂ (VC L C D ) = = 0 leading to C LMR = ∂ CL ∂ CL 3ε C LMR = C Do 3ε : Lift Coefficient for Maximum Range Maximum Range of a Jet Aircraft Flying at Constant Altitude " •  At constant altitude " Vcruise ( t ) = 2W ( t ) ( ) C L ρ h fixed S !  Cruise-climb usually violates air traffic control rules" !  Constant-altitude cruise does . Cruising Flight Performance Robert Stengel, Aircraft Flight Dynamics, MAE 331, 2012! Copyright 2012 by Robert Stengel. All rights. kg/m^3 VMT, m/s 0 1.23 69.11 2,500 0.96 78.20 5,000 0.74 89.15 10,000 0.41 118.87 Optimal Cruising Flight Cruising Range and Specific Fuel Consumption" 0 = C T − C D ( ) 1 2 ρ V 2 S m 0. 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 Ma"rial# Air