8. Cruising Flight Envelope

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

Learning Objectives • Definitions of airspeed • Performance parameters • Steady cruising flight conditions • Breguet range equations • Optimize cruising flight for minimum thrust and power • Flight envelope

The Flight Envelope

1 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 Excess thrust provides the – Performance: 200 ft/min ability to accelerate or climb

Additional Factors Define the Flight Envelope

§ Maximum Mach number § Maximum allowable Piper Dakota Stall Buffet aerodynamic heating http://www.youtube.com/watch?v=mCCjGAtbZ4g § Maximum thrust § Maximum dynamic pressure § Performance ceiling § Wing stall § Flow-separation buffet § Angle of attack § Local shock waves

2 Boeing 787 Flight Envelope (HW #5, 2008)

Best Cruise Region

Lockheed U-2 “Coffin Corner” Stall buffeting and Mach buffeting are limiting factors Narrow corridor for safe flight

Climb Schedule

3 Historical Factoids 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

Ford Tri-Motor Lockheed Vega

Boeing 40

http://www.youtube.com/watch?v=3a8G87qnZz4 7

Commercial Aircraft of the 1930s Streamlining, engine cowlings Douglas DC-1, DC-2, DC-3 Lockheed 14 Super Electra, Boeing 247

4 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

Optimal Cruising Flight

5 Maximum Lift-to-Drag Ratio Lift-to-drag ratio C L CL L = = 2 D CD C C Do + ε L Satisfy necessary condition for a maximum

CL ∂ 2 ( CD ) 1 2εCL = − 2 = 0 ∂C C + εC 2 2 L Do L C + εC ( Do L )

Lift coefficient for maximum L/D and minimum thrust are the same

C Do (CL ) = = CL L / Dmax ε MT 11

Airspeed, Drag Coefficient, and

Lift-to-Drag Ratio for L/Dmax

2 ⎛ W ⎞ ε Airspeed V = V = L/Dmax MT ⎝⎜ S ⎠⎟ C ρ Do

Drag (CD ) = CD + CD = 2CD Coefficient L / Dmax o o o

Maximum CD ε 1 (L / D) = o = L/D max 2C 2 C Do ε Do 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 12

6 Cruising Range and Specific Fuel Consumption

• Thrust = Drag 1 2 • Level flight 0 = (CT − CD ) ρV S m 2 h! = 0 ⎛ 1 2 ⎞ • Lift = Weight 0 = CL ρV S − mg mV r! = V ⎝⎜ 2 ⎠⎟

• Thrust specific fuel consumption, TSFC = cT • Fuel mass burned per sec per unit of thrust kg s c : m! = −c T T kN f T • Power specific fuel consumption, PSFC = cP • Fuel mass burned per sec per unit of power kg s cP : m! f = −cPP kW 13

Breguet Range Equation

Louis Breguet, for Jet Aircraft 1880-1955 Rate of change of range with respect to weight of fuel burned dr dr dt r! V V ⎛ L ⎞ V = = = = − = −⎜ ⎟ dm dm dt m! −c T c D ⎝ D⎠ c mg ( T ) T T ⎛ L ⎞ V dr = −⎜ ⎟ dm ⎝ D⎠ cT mg Range traveled

R W f ⎛ L ⎞ ⎛ V ⎞ dm Range = R = ∫ dr = − ∫ ⎜ ⎟ ⎜ ⎟ ⎝ D⎠ ⎝ cT g⎠ m 0 Wi 14

7 B-727 Maximum Range of a Jet Aircraft Flying at Constant Altitude At constant altitude and SFC

Vcruise (t) = 2W (t) CL ρ hfixed S ( )

W f ⎛ C ⎞ ⎛ 1 ⎞ 2 dm Range = − L ∫ ⎝⎜ C ⎠⎟ ⎝⎜ c g⎠⎟ C ρS m1 2 Wi D T L

⎛ CL ⎞ ⎛ 2 ⎞ 2 1 2 1 2 = ⎜ ⎟ ⎜ ⎟ (mi − m f ) ⎝ CD ⎠ ⎝ cT g⎠ ρS

⎛ C ⎞ Range is maximized when L ⎜ ⎟ = maximum ⎝ CD ⎠ 15

Breguet Range Equation MD-83 for Jet Aircraft at Constant Airspeed

For constant true airspeed, V = Vcruise, and SFC

⎛ L ⎞ ⎛ Vcruise ⎞ m f R = −⎜ ⎟ ln(m) ⎝ D⎠ ⎜ c g ⎟ mi ⎝ T ⎠ ⎛ CL ⎞ ⎛ 1 ⎞ ⎛ mi ⎞ = ⎜Vcruise ⎟ ⎜ ⎟ ln⎜ ⎟ ⎝ CD ⎠ ⎝ cT g⎠ m f ⎛ L ⎞ ⎛ Vcruise ⎞ ⎛ mi ⎞ ⎝ ⎠ = ⎜ ⎟ ⎜ ⎟ ln⎜ ⎟ ⎝ D⎠ ⎝ cT g ⎠ ⎝ m f ⎠

§ Vcruise(CL/CD) as large as possible § M -> Mcrit § ρ as small as possible § h as high as possible 16

8 Maximize Jet Aircraft Range Using Optimal Cruise-Climb

⎡ ⎤ CL CL ∂⎢Vcruise 2 ⎥ ∂ Vcruise C + εC ∂R CD ⎢ ( Do L )⎥ ∝ ( ) = ⎣ ⎦ = 0 ∂CL ∂CL ∂CL

Vcruise = 2W CL ρS

Assume 2W (t) ρ (h)S = constant i.e., airplane climbs at constant TAS as fuel is burned

Maximize Jet Aircraft Range Using Optimal Cruise-Climb

∂⎡V C C + εC 2 ⎤ ∂⎡C1/2 C + εC 2 ⎤ cruise L ( Do L ) 2W L ( Do L ) ⎣ ⎦ = ⎣ ⎦ = 0 ∂CL ρS ∂CL Optimal values: (see Supplemental Material) C C = Do : Lift Coefficient for Maximum Range LMR 3ε C 4 C = C + Do = C DMR Do 3 3 Do

V 2W t C h S a h M cruise−climb = ( ) LMR ρ ( ) = ( ) cruise-climb

a(h) : Speed of sound; M cruise-climb : Mach number 18

9 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

Historical Factoid • 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: built numerous aircraft until after World War II; teamed with BAC in SEPECAT (1966) • Merged with Dassault in 1971

Breguet Atlantique SEPECAT Jaguar

Breguet 14 Breguet 890 Mercure

10 Next Time: Gliding, Climbing, and Turning Flight

Supplemental Material

11 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

Maximize Jet Aircraft Range Using Optimal Cruise-Climb

∂⎡V C C + εC 2 ⎤ ∂⎡C1/2 C + εC 2 ⎤ cruise L ( Do L ) 2w L ( Do L ) ⎣ ⎦ = ⎣ ⎦ = 0 ∂CL ρS ∂CL

2w = Constant; let C1/2 = x, C = x2 ρS L L ⎡ ⎤ C + εx4 − x 4εx3 C − 3εx4 ∂ x ( Do ) ( ) ( Do ) ⎢ ⎥ = 2 = 2 4 4 4 ∂x ⎢ CD + εx ⎥ C + εx C + εx ⎣( o )⎦ ( Do ) ( Do )

Optimal values: C C 4 C Do : C C Do C LMR = DMR = Do + = Do 3ε 3 3 24

12 Breguet Range Equation for

Breguet 890 Mercure Propeller-Driven Aircraft

Rate of change of range with respect to weight of fuel burned dr r V V V " L % 1 = = = − = − = −$ ' dw w −c P c TV c DV # D & c W ( P ) P P P Range traveled R W f # L &# 1 & dw Range = R = dr = − % (% ( ∫ ∫ $ D ' c w 0 Wi $ P '

Breguet Atlantique Breguet Range Equation for Propeller-Driven Aircraft

For constant true airspeed, V = Vcruise

" L %" 1 % W f R = −$ '$ 'ln(w) Wi # D &#cP & " %" % " % CL 1 Wi = $ '$ 'ln$ ' #CD &#cP & #W f &

Range is maximized when

! C $ # L & = maximum = L ( D)max "CD % 26

13 Back Side of the Power Curve Achievable Airspeeds in Propeller-Driven Cruising Flight

Power = constant

Pavail = TavailV P V 4εW 2 V 4 − avail + = 0 C ρS C S 2 Do Do (ρ ) Solutions for V cannot be put in quadratic form; solution is more difficult, e.g., Ferraris method aV 4 + (0)V 3 + (0)V 2 + dV + e = 0

Best bet: roots in MATLAB

P-51 Mustang Minimum-Thrust Example

Wing Span = 37 ft (9.83 m) Wing Area = 235 ft 2 (21.83 m2 ) Loaded Weight = 9,200 lb (3,465 kg) C 0.0163 Do = ε = 0.0576 W / S = 39.3 lb / ft 2 (1555.7 N / m2 )

Airspeed for minimum thrust Air Density,

Altitude, m kg/m^3 VMT, m/s 2 "W % ε 2 0.947 76.49 0 1.23 69.11 VMT = $ ' = (1555.7) = m / s 2,500 0.96 78.20 # S & C 0.0163 5,000 0.74 89.15 ρ Do ρ ρ 10,000 0.41 118.87 28

14 P-51 Mustang Maximum L/D Example

(CD ) = 2CD = 0.0326 L / Dmax o

C Do (CL ) = = CL = 0.531 L / Dmax ε MT Wing Span = 37 ft (9.83 m) 1 2 (L / D) = = 16.31 Wing Area = 235 ft (21.83 m ) max 2 C ε Do Loaded Weight = 9,200 lb (3,465 kg) 76.49 V = V = m / s C = 0.0163 L / Dmax MT Do ρ ε = 0.0576 Air Density, 2 Altitude, m kg/m^3 W / S = 1555.7 N / m 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 29

P-51 Mustang Maximum Range (Internal Tanks only)

! $ ! $ ! $ W = CL qS CL 1 Wi trim R = # & # &ln# & 1 "CD % "cP % W f C W S max " % Ltrim = ( ) q ! 1 $ ! 3,465 + 600 $ βh = (16.31)# &ln# & 2 ⎛ 2 e ⎞ "0.0017 % " 3,465 % = 2 (W S) = 2 (W S) ρV ⎝⎜ ρ V ⎠⎟ 0 =1,530 km ((825 nm)