Cruising Flight Envelope Robert Stengel, Aircraft Flight Dynamics, MAE 331, 2018

Transcription

1 Cruising Flight Envelope Robert Stengel, Aircraft Flight Dynamics, MAE 331, 018 Learning Objectives Definitions of airspeed Performance parameters Steady cruising flight conditions Breguet range equations Optimize cruising flight for minimum thrust and power Flight envelope Copyright 018 by Robert Stengel. All rights reserved. For educational use only The Flight Envelope 1

2 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: 00 ft/min Excess thrust provides the ability to accelerate or climb 3 Additional Factors Define the Flight Envelope Piper Dakota Stall Buffet 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 4

3 Boeing 787 Flight Envelope (HW #5, 008) Best Cruise Region 5 Lockheed U- Coffin Corner Stall buffeting and Mach buffeting are limiting factors Narrow corridor for safe flight Climb Schedule 6 3

4 Historical Factoids Air Commerce Act of 196 Airlines formed to carry mail and passengers: Northwest (196) Eastern (197), bankruptcy Pan Am (197), bankruptcy Boeing Air Transport (197), became United (1931) Delta (198), consolidated with Northwest, 010 American (1930) TWA (1930), acquired by American Continental (1934), consolidated with United, 010 Ford Tri-Motor Lockheed Vega Boeing Commercial Aircraft of the 1930s Streamlining, engine cowlings Lockheed 14 Super Electra, Douglas DC-1, DC-, DC-3 Boeing

5 Comfort and Elegance by the End of the Decade Boeing 307, 1 st pressurized cabin (1936), flight engineer, B-17 precursor, large dorsal fin (exterior and interior) Sleeping bunks on transcontinental planes (e.g., DC-3) Full-size dining rooms on flying boats 9 Optimal Cruising Flight 10 5

6 Maximum Lift-to-Drag Ratio CD L D CD ( ) Lift-to-drag ratio + ε Satisfy necessary condition for a maximum 1 + εc ε L + ε ( ) 0 Lift coefficient for maximum L/D and minimum thrust are the same ( ) L / Dmax C D o ε MT 11 Airspeed, Drag Coefficient, and Lift-to-Drag Ratio for L/D max Airspeed V L/Dmax V MT ρ W S ε Drag Coefficient ( C D ) L / Dmax + Maximum L/D ( L / D) max C D o ε 1 ε Maximum L/D depends only on induced drag factor and zero-lift drag coefficient Induced drag factor and zero-lift drag coefficient are functions of Mach number 1 6

7 Cruising Range and Specific Fuel Consumption Thrust Drag Lift Weight 0 ( C T C D ) 1 ρv S m 1 0 ρv S mg mv Level flight!h 0!r V Thrust specific fuel consumption, TSFC c T Fuel mass burned per sec per unit of thrust c T : kg s!m f c T T kn Power specific fuel consumption, PSFC c P Fuel mass burned per sec per unit of power c P : kg s kw!m f c P P 13 Louis Breguet, dr dr dt dm dm dt!r!m Breguet Range Equation for Jet Aircraft Rate of change of range with respect to weight of fuel burned V ( c T T ) V dr L V D c T mg dm Range traveled c T D L D V c T mg Range R R 0 dr W f W i L V D c T g dm m 14 7

8 B-77 Maximum Range of a Jet Aircraft Flying at Constant Altitude At constant altitude and SFC V cruise ( t) W ( t) ρ( h fixed )S Range W f W i C D C D 1 c T g c T g ρs dm m 1 ( ) ρs m 1 1 i m f Range is maximized when C D maximum 15 Breguet Range Equation for Jet Aircraft at Constant Airspeed For constant true airspeed, V V cruise, and SFC MD-83 R L D V cruise c T g ln m L D V cruise c T g ln ( ) m i m i m f m f C V L 1 cruise c T g ln C D m i m f V cruise ( /C D ) as large as possible M -> M crit ρ as small as possible h as high as possible 16 8

9 Maximize Jet Aircraft Range Using Optimal Cruise-Climb C R ( V L cruise CD ) C V L cruise ( + ε ) 0 V cruise W ρs Assume W ( t) ρ ( h)s constant i.e., airplane climbs at constant TAS as fuel is burned 17 Maximize Jet Aircraft Range Using Optimal Cruise-Climb ( ) V cruise + ε W 1/ ρs ( + ε ) 0 Optimal values: (see Supplemental Material) MR C D o : Lift Coefficient for Maximum Range 3ε C DMR + C D o C D o V cruise climb W ( t) MR ρ ( h)s a( h)m cruise-climb a( h) : Speed of sound; M cruise-climb : Mach number 18 9

10 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 19 Historical Factoid Louis Breguet ( ), 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: built numerous aircraft until after World War II; teamed with BAC in SEPECAT (1966) Merged with Dassault in 1971 Breguet Atlantique Breguet 14 SEPECAT Jaguar Breguet 890 Mercure 0 10

11 Next Time: Gliding, Climbing, and Turning Flight 1 Supplemental Material 11

12 Seaplanes Became the First TransOceanic Air Transports PanAm led the way 1 st scheduled TransPacific flights(1935) 1 st scheduled TransAtlantic flights(1938) 1 st 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) 3 Maximize Jet Aircraft Range Using Optimal Cruise-Climb ( ) V cruise + ε w ρs 1/ ( + ε ) 0 x w ρs Constant; let C 1/ L x, x x ( + εx 4 ) C + εx 4 Do + εx 4 ( ) x( 4εx 3 ) ( 3εx 4 ) ( ) ( + εx 4 ) Optimal values: MR C D o 3ε : C D MR + C D o C D o 4 1

13 Breguet 890 Mercure Breguet Range Equation for Propeller-Driven Aircraft Rate of change of range with respect to weight of fuel burned dr dw r w V ( c P P) V Range R c P TV V c P DV " L % 1 $ ' # D & c P W Range traveled R 0 dr W f W i # L &# % ( 1 & dw % ( $ D ' $ c P ' w 5 Breguet Range Equation for Propeller-Driven Aircraft Breguet Atlantique For constant true airspeed, V V cruise " R L %" $ ' 1 % $ 'ln w # D &# c P & " C %" L $ ' 1 % " $ 'ln W % i # C D &# c $ P & # W ' f & W ( ) f Wi Range is maximized when! # " C D $ & maximum L % D ( ) max 6 13

14 Back Side of the Power Curve Achievable Airspeeds in Propeller-Driven Cruising Flight Power constant P avail T avail V V 4 P V avail ρs + 4εW ρs ( ) 0 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 + dv + e 0 Best bet: roots in MATLAB 7 P-51 Mustang Minimum-Thrust Example Wing Span 37 ft (9.83 m) Wing Area 35 ft (1.83 m ) Loaded Weight 9,00 lb (3, 465 kg) ε W / S 39.3 lb / ft ( N / m ) V MT " W % $ ' ρ # S & Airspeed for minimum thrust ε ρ ( ) ρ m / s Altitude, m Air Density, kg/m^3 V MT, m/s , , ,

15 P-51 Mustang Maximum L/D Example ( C D ) L / Dmax Wing Span 37 ft (9.83 m) Wing Area 35 ft (1.83 m ) Loaded Weight 9, 00 lb (3, 465 kg) ε W / S N / m ( ) L / Dmax C D o ε ( L / D) max V L / Dmax V MT ρ MT ε m / s Altitude, m Air Density, kg/m^3 V MT, m/s , , , P-51 Mustang Maximum Range (Internal Tanks only) trim W trim qs 1 ( q W S) ρv W S ( ) eβh ρ 0 V W S ( )! R C $ L # & " % C D max! 1 $! # &ln W $ i " c # P % " W & f %! 1 $! 3, $ ( 16.31) # &ln# & " % " 3,465 % ( ) 1,530 km (85 nm 30 15