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7. Power & Thrust for Cruising Flight

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7. Power & Thrust for Cruising Flight Power and Thrust for Cruising Flight Robert Stengel, Aircraft Flight Dynamics, MAE 331, 2018 Copyright 2018 by Robert Stengel. All rights reserved. For educational use only. http://www.princeton.edu/~stengel/MAE331.html http://www.princeton.edu/~stengel/FlightDynamics.html 1 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 3 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 6 3 Longitudinal Variables 7 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 ( ) 12 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) 14 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 = mV 1,% (% ((τ c −1)+ / −15 1 1 32+$θo − '$θt − ' θoτ c . 62 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 c = (compressor outlet temp. compressor inlet temp.) from Kerrebrock Little change in thrust with airspeed below Mcrit Decrease with increasing altitude 16 8 Airbus E-Fan Electric Propulsion 17 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 19 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 21 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 22 11 Steady, Level Flight 23 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 24 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 25 Thrust Required for Steady, Level Flight 26 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 32 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 33 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⎠⎟ 36 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.
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