Aircraft Flight Dynamics Robert F. Stengel Lecture6 Cruising Flight Performance

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