7. Power & Thrust for Cruising Flight

Home , Blade pitch, Equivalent airspeed, Propeller (aeronautics), Safran Ardiden, Thrust, True airspeed

Power and Thrust for Cruising Flight Robert Stengel, Aircraft Flight Dynamics, MAE 331, 2018

U.S. Standard Atmosphere, 1976

http://en.wikipedia.org/wiki/U.S._Standard_Atmosphere 2

1 Dynamic Pressure and Mach Number ρ = air density, functionof height −βh = ρsealevele a = speed of sound = linear functionof height

Dynamic pressure = q  ρV 2 2 Mach number = V a

Definitions of Airspeed Airspeed is speed of aircraft measured with respect to air mass Airspeed = Inertial speed if wind speed = 0 • Indicated Airspeed (IAS)

2( ptotal − pstatic ) IAS = 2( pstagnation − pambient ) ρSL = ρSL 2q ! c , with q ! impact pressure ρ c SL • Calibrated Airspeed (CAS)* CAS = IAS corrected for instrument and position errors

2(qc ) = corr #1 ρSL 4 * Kayton & Fried, 1969; NASA TN-D-822, 1961

2 Definitions of Airspeed Airspeed is speed of aircraft measured with respect to air mass Airspeed = Inertial speed if wind speed = 0 Equivalent Airspeed (EAS)*

2(qc ) EAS = CAS corrected for compressibility effects = corr # 2 ρSL

True Airspeed (TAS)* Mach number ρ ρ TAS V  TAS = EAS SL = IAS SL M = ρ(z) corrected ρ(z) a

* Kayton & Fried, 1969; NASA TN-D-822, 1961 5

Flight in the Vertical Plane

3 Longitudinal Variables

Longitudinal Point-Mass Equations of Motion • Assume thrust is aligned with the velocity vector (small-angle approximation for α) • Mass = constant

1 2 1 2 (CT cosα − CD ) ρV S − mg sinγ (CT − CD ) ρV S − mg sinγ V! = 2 ≈ 2 m m

1 2 1 2 (CT sinα + CL ) ρV S − mgcosγ CL ρV S − mgcosγ γ! = 2 ≈ 2 mV mV ! h = −z! = −vz = V sinγ V = velocity = Earth-relative airspeed = True airspeed with zero wind r! = x! = vx = V cosγ γ = flight path angle h = height (altitude) 8 r = range

4 Conditions for Steady, Level Flight • Flight path angle = 0 • Altitude = constant • Airspeed = constant • Dynamic pressure = constant

1 2 (CT − CD ) ρV S 0 = 2 • Thrust = Drag m

1 2 CL ρV S − mg 0 = 2 • Lift = Weight mV h = 0

r = V 9

Power and Thrust Propeller 1 Power = P = T ×V = C ρV 3S ≈ independent of airspeed T 2 Turbojet 1 Thrust = T = C ρV 2S ≈ independent of airspeed T 2 Throttle Effect ⎡ ⎤ T = TmaxδT = CT qS δT, 0 ≤ δT ≤ 1 ⎣ max ⎦ 10

5 Typical Effects of Altitude and Velocity on Power and Thrust

• Propeller [Air-breathing engine]

• Turbofan [In between]

• Turbojet

• Battery [Independent of altitude

and airspeed] 11

Models for Altitude Effect on Turbofan Thrust From Flight Dynamics, pp.117-118

1 2 Thrust = CT (V,δT ) ρ(h)V S 2

⎡ n 1 2 ⎤ = ko + k1V ρ(h)V S δT, N ⎢( ) 2 ⎥ ⎣ ⎦

ko = Static thrust coefficient at sea level

k1 = Velocity sensitivity of thrust coefficient n = Exponent of velocity sensitivity [ = −2 for turbojet]

ρ h = ρ e−βh , ρ = 1.225 kg / m3, β = 1/ 9,042 m−1 ( ) SL SL ( )

6 Thrust of a Propeller-Driven Aircraft With constant rpm, variable-pitch propeller P P T = η η engine = η engine P I V net V

ηP = propeller efficiency

ηI = ideal propulsive efficiency

= TV T (V + ΔVinflow ) = V (V + ΔVfreestream 2)

0.85 0.9 ηnetmax ≈ − Efficiencies decrease with airspeed Engine power decreases with altitude Proportional to air density, w/o supercharger 13

Reciprocating-Engine Power and Specific Fuel Consumption (SFC) P(h) ρ (h) = 1.132 − 0.132 PSL ρSL

SFC ∝ Independent of Altitude • Engine power decreases with altitude – Proportional to air density, w/o supercharger – Supercharger increases inlet manifold pressure, increasing power and extending maximum altitude

Anderson (Torenbeek)

7 Propeller Efficiency, ηP, Effect of propeller-blade pitch angle and Advance Ratio, J

Advance Ratio V J = nD where V = airspeed, m / s n = rotation rate, revolutions / s D = propeller diameter, m 15 from McCormick

Thrust of a Turbojet Engine

1/2 0*# &# & - 4 2 θo θt θt 2 T = mV 1,% (% ((τ c −1)+ / −15 2+$θo −1'$θt −1' θoτ c . 2 3 6

m! = m! air + m! fuel (γ −1)/γ θo = ( pstag pambient ) ; γ = ratio of specific heats ≈ 1.4

θt = (turbine inlet temp. freestream ambient temp.) τ = compressor outlet temp. compressor inlet temp. c ( ) from Kerrebrock

Little change in thrust with airspeed below Mcrit Decrease with increasing altitude 16

8 Airbus E-Fan Electric Propulsion

Specific Energy and Energy Density of Fuel and Batteries (typical) • Specific energy = energy/unit mass • Energy density = energy/unit volume Energy Storage Specific Energy, Energy Density, Material MJ/kg MJ/L Lithium-Ion 0.4-0.9 0.9-2.6 Battery Jet Engine Fuel 43 37 (Kerosene) Gasoline 46 34 Methane 56 22 (Liquified) Hydrogen 142 9 (Liquified)

Fuel cell energy conversion efficiency: 40-60%

Solar cell power conversion efficiency: 30-45% Solar irradiance: 1 kW/m2 18

9 Engine/Motor Power, Thrust, and Efficiency (typical)

Engine/Motor Power/Mass, Thermal Propulsive Type kW/kg Efficiency Efficiency Supercharged ~Propeller 1.8 25-50% Radial Engine Efficiency Turboshaft ~Propeller 5 40-60% Engine Efficiency Brushless DC ~Propeller 1-2 - Motor Efficiency Thrust/Weight, - Turbojet Engine 10 40-60% ~1 – |Vexhaust – V|/V Turbofan Engine 4-5 40-60% ~1 – |Vexhaust – V|/V

Zunum ZA-10 Hybrid-Electric Aircraft

• 12-passenger commuter aircraft (2023) • Safran Ardiden 3Z turbine engine, 500kW (~ 650 shp) • Lithium-ion batteries (TBD) • Boeing and Jet Blue funding • Goal: 610-nm (700-sm) range • Turbo Commander test aircraft (2019) 20

10 Performance Parameters

C Lift-to-Drag Ratio L = L D CD

Load Factor L L n = W = mg,"g"s

T T Thrust-to-Weight Ratio W = mg,"g"s

Wing Loading W 2 2 S, N m or lb ft

Historical Factoid • 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 Rutan/Virgin Atlantic Borschberg/Piccard Composites Global Flyer Solar Impulse 2 Voyager

11 Steady, Level Flight

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 2 ⎞ W = C ρV S = C qS Ltrim ⎝⎜ 2 ⎠⎟ Ltrim

1 2 ⎛ 2 eβh ⎞ C = W S = W S = W S Ltrim ( ) 2 ( ) ⎜ 2 ⎟ ( ) q ρV ⎝ ρ0V ⎠

β = 1/ 9,042 m, inverse scale height of air density

12 Trimmed Angle of Attack, α • Trimmed angle of attack, α – Constant if dynamic pressure and weight are constant – If dynamic pressure decreases, angle of attack must increase 1 2 W S C ( ) − Lo 2W ρV S − CL q α = o = trim C C Lα Lα

Thrust Required for Steady, Level Flight

13 Thrust Required for Steady, Level Flight Trimmed thrust Parasitic Drag Induced Drag "1 % 2W 2 T = D = C $ ρV 2S'+ε trim cruise Do #2 & ρV 2S

Minimum required thrust conditions

2 ∂Ttrim 4εW = CD (ρVS) − = 0 ∂V o ρV 3S

Necessary Condition:

Slope = 0 27

Necessary and Sufficient Conditions for Minimum Required Thrust

Necessary Condition = Zero Slope

4εW 2 CD (ρVS) = o ρV 3S

Sufficient Condition for a Minimum = Positive Curvature when slope = 0

2 2 ∂ Ttrim 12εW = CD (ρS) + > 0 ∂V 2 o ρV 4S (+) (+) 28

14 Airspeed for Minimum Thrust in Steady, Level Flight

Satisfy necessary condition

4 ⎛ 4ε ⎞ 2 V = 2 (W S) ⎜ C ⎟ ⎝ Do ρ ⎠ Fourth-order equation for velocity Choose the positive root

2 "W % ε V = $ ' MT # S & C ρ Do 29

Lift, Drag, and Thrust Coefficients in Minimum-Thrust Cruising Flight Lift coefficient 2 ⎛ W ⎞ C = LMT 2 ⎜ ⎟ ρVMT ⎝ S ⎠ C Do = = (CL ) (L/D) ε max Drag and thrust coefficients C C = C + εC 2 = C + ε Do DMT Do LMT Do ε

2C C 30 = Do ≡ TMT

15 Achievable Airspeeds in Constant- Altitude Flight

Back Side of the Thrust Curve

• Two equilibrium airspeeds for a given thrust or power setting

– Low speed, high CL, high α – High speed, low CL, low α • Achievable airspeeds between minimum and maximum values with maximum thrust or power 31

Power Required for Steady, Level Flight

P = T x V

16 Power Required for Steady, Level Flight Trimmed power Parasitic Drag Induced Drag

2 ) "1 2 % 2εW , Ptrim = TtrimV = DcruiseV = +CD $ ρV S'+ .V * o #2 & ρV 2S -

Minimum required power conditions

∂P 3 2εW 2 trim = C ρV 2S − = 0 ∂V Do 2 ( ) ρV 2S

Airspeed for Minimum Power in Steady, Level Flight

• Satisfy necessary condition 3 2εW 2 C ρV 2S = Do 2 ( ) ρV 2S • Fourth-order equation for velocity 2 "W % ε V = $ ' – Choose the positive root MP # S & 3C ρ Do

• Corresponding lift and 3CD drag coefficients C = o LMP ε

C 4C 34 DMP = Do

17 Achievable Airspeeds for Jet in Cruising Flight Thrust = constant

2 ⎛ 1 2 ⎞ 2εW T = C qS = C ρV S + avail D Do ⎜ ⎟ 2 ⎝ 2 ⎠ ρV S

2 ⎛ 1 4 ⎞ 2 2εW C ρV S − T V + = 0 Do ⎜ ⎟ avail ⎝ 2 ⎠ ρS 2T 4εW 2 V 4 − avail V 2 + = 0 C ρS C S 2 Do Do (ρ ) 4th-order algebraic equation for V 35

Achievable Airspeeds for Jet in Cruising Flight Solutions for V2 can be put in quadratic form and solved easily

2 V ! x; V = ± x 2T 4εW 2 V 4 − avail V 2 + = 0 C ρS C S 2 Do Do (ρ ) x2 + bx + c = 0

2 b ⎛ b⎞ 2 x = − ± − c = V 2 ⎝⎜ 2⎠⎟

18 Thrust Required and Thrust Available for a Typical Bizjet

Available thrust decreases with altitude Stall limitation at low speed Mach number effect on lift and drag increases thrust required at high speed

Typical Simplified Jet Thrust Model

x ⎡ ρ(SL)e−βh ⎤ Tmax (h) = Tmax (SL)⎢ ⎥ ⎣ ρ(SL) ⎦ x −βh − xβh = Tmax (SL)⎣⎡e ⎦⎤ = Tmax (SL)e

Empirical correction to force thrust to zero at a given altitude, hmax. c is a convergence factor.

− xβh −(h−hmax ) c Tmax (h) = Tmax (SL)e ⎡1− e ⎤ ⎣ ⎦ 37

Thrust Required and Thrust Available for a Typical Bizjet

Typical Stall Limit

19 Next Time: Cruising Flight Envelope

Supplemental Material

20 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 = ρ = Air density at altitude h, kg/m3 u(4) = δT = 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)) 41

Hybrid-Electric Power System Simulink Design Example

21 NASA Hybrid-Electric V/STOL UAV Concept

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

22 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 ρ Do ρ ρ 5,000 0.74 89.15 10,000 0.41 118.87 45

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 2 Air Density, W / S = 1555.7 N / m Altitude, m 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 46