Aircraft Performance General PDF

Getting to Grips with Aircraft Performance GENERAL A. GENERAL 1. THE INTERNATIONAL STANDARD ATMOSPHERE (ISA) 1.1. Standard Atmosphere Modeling The atmosphere is a gaseous envelope surrounding the earth. Its characteristics are diffe...

Getting to Grips with Aircraft Performance GENERAL A. GENERAL 1. THE INTERNATIONAL STANDARD ATMOSPHERE (ISA) 1.1. Standard Atmosphere Modeling The atmosphere is a gaseous envelope surrounding the earth. Its characteristics are different throughout the world. For this reason, it is necessary to adopt an average set of conditions called the International Standard Atmosphere (ISA). 1.1.1. Temperature Modeling The following diagram (Figure A1) illustrates the temperature variations in the standard atmosphere: Altitude (ft) (km) STRATOSPHERE 40000 12 TROPOPAUSE = 36089 ft subsonic jet transport 35000 10 cruise level 30000 25000 8 TROPOSPHERE 20000 6 15000 4 10000 2 5000 -56.5°C 15°C Sea level -60 -40 -20 0 20 40 60 Temperature (°C) Figure A1: ISA temperature The international reference is based on a sea-level temperature of 15°C at a pressure of 1013.25 hPa1. The standard density of the air at sea level is 1.225 kg/m3. 1 1013.25 hPa is equal to 29.92 in Hg, ‘hPa’ meaning hectoPascal and ‘in Hg’ inches of mercury. 11 GENERAL Getting to Grips with Aircraft Performance Temperature decreases with altitude at a constant rate of -6.5°C/1000m or -1.98°C/1000ft up to the tropopause. The standard tropopause altitude is 11,000 m or 36,089 feet. From the tropopause upward, the temperature remains at a constant value of -56.5°C. Therefore, the air which is considered as a perfect gas in the ISA model presents the following characteristics: At Mean Sea Level (MSL): ISA temperature = T0 = +15°C = 288.15 K Above MSL and below the tropopause (36,089 feet): ISA temperature (ºC) = T0 - 1.98 x [Alt(feet)/1000] For a quick determination of the standard temperature at a given altitude, the following approximate formula can be used: ISA temperature (ºC) = 15 - 2 x [Alt(feet)/1000] Above the tropopause (36,089 feet): ISA temperature = -56.5ºC = 216.65 K This ISA model is used as a reference to compare real atmospheric conditions and the corresponding engine/aircraft performance. The atmospheric conditions will therefore be expressed as ISA +/- ∆ISA at a given flight level. Example: Let’s consider a flight in the following conditions: Altitude = 33,000 feet Actual Temperature = -41ºC The standard temperature at 33,000 feet is : ISA = 15 - 2 x 33 = -51ºC, whereas the actual temperature is -41ºC, i.e. 10ºC above the standard. Conclusion: The flight is operated in ISA+10 conditions 12 Getting to Grips with Aircraft Performance GENERAL 1.1.2. Pressure Modeling To calculate the standard pressure P at a given altitude, the following assumptions are made: Temperature is standard, versus altitude. Air is a perfect gas. The altitude obtained from the measurement of the pressure is called pressure altitude (PA), and a standard (ISA) table can be set up (table A1). PAZp PRESSURE ALTITUDE (ft) (km) 40000 12 Zp PA==f(p) f(P) ISA table 10 30000 8 20000 6 4 10000 2 P 200 300 500 850 1013.25 (hPa) Figure A2: Pressure Altitude function of Pressure Pressure (hPa) Pressure altitude (PA) FL= PA/100 (feet) (meters) 200 38661 11784 390 250 34000 10363 340 300 30066 9164 300 500 18287 5574 180 850 4813 1467 50 1013 0 0 0 Table A1: Example of Tabulated Pressure Altitude Values 13 GENERAL Getting to Grips with Aircraft Performance Assuming a volume of air in static equilibrium, the aerostatic equation gives: dP = - ρgdh With ρ = air density at an altitude h g= gravity acceleration (9.80665 m/s2) dh = height of the volume unit dP = pressure variation on dh The perfect gas equation gives: P = RT ρ With R = universal gas constant (287.053 J/kg/K) Consequently: At Mean Sea Level (MSL): P0 = 1013.25 hPa Above MSL and below the tropopause (36,089 feet): g α αR0 P = P0 (1− h) T0 With P0 = 1013.25 hPa (standard pressure at sea level) T0 = 288.15 K (standard temperature at sea level) α = 0.0065 ºC/m g0 = 9.80665 m/s2 R = 287.053 J/kg/K h = Altitude (m) Note: For low altitudes, a reduction of 1 hPa in the pressure approximately corresponds to a pressure altitude increase of 28 feet. Above the tropopause (36,089 feet): − g 0 ( h −h1 ) RT1 P = P1e With P1 = 226.32 hPa (standard pressure at 11,000 m) T1 = 216.65 K (standard temperature at 11,000 m) 14 Getting to Grips with Aircraft Performance GENERAL h1 = 11,000 m g0 = 9.80665 m/s2 R = 287.053 J/kg/K h = Altitude (m) 1.1.3. Density Modeling To calculate the standard density ρ at a given altitude, the air is assumed to be a perfect gas. Therefore, at a given altitude, the standard density ρ (kg/m3) can be obtained as follows: P ρ= RT with R = universal gas constant (287.053 J/kg/K) P in Pascal T in Kelvin At Mean Sea Level (MSL): ρ0 = 1.225 kg/m3 1.2. International Standard Atmosphere (ISA) Table The International Standard Atmosphere parameters (temperature, pressure, density) can be provided as a function of the altitude under a tabulated form, as shown in Table A2: 15 GENERAL Getting to Grips with Aircraft Performance PRESSURE ALTITUDE TEMP. PRESSURE DENSITY Speed of ALTITUDE (Feet) (°C) RATIO σ = ρ/ρo sound (meters) hPa PSI In.Hg δ = P/Po (kt) 40 000 - 56.5 188 2.72 5.54 0.1851 0.2462 573 12 192 39 000 - 56.5 197 2.58 5.81 0.1942 0.2583 573 11 887 38 000 - 56.5 206 2.99 6.10 0.2038 0.2710 573 11 582 37 000 - 56.5 217 3.14 6.40 0.2138 0.2844 573 11 278 36 000 - 56.3 227 3.30 6.71 0.2243 0.2981 573 10 973 35 000 - 54.3 238 3.46 7.04 0.2353 0.3099 576 10 668 34 000 - 52.4 250 3.63 7.38 0.2467 0.3220 579 10 363 33 000 - 50.4 262 3.80 7.74 0.2586 0.3345 581 10 058 32 000 - 48.4 274 3.98 8.11 0.2709 0.3473 584 9 754 31 000 - 46.4 287 4.17 8.49 0.2837 0.3605 586 9 449 30 000 - 44.4 301 4.36 8.89 0.2970 0.3741 589 9 144 29 000 - 42.5 315 4.57 9.30 0.3107 0.3881 591 8 839 28 000 - 40.5 329 4.78 9.73 0.3250 0.4025 594 8 534 27 000 - 38.5 344 4.99 10.17 0.3398 0.4173 597 8 230 26 000 - 36.5 360 5.22 10.63 0.3552 0.4325 599 7 925 25 000 - 34.5 376 5.45 11.10 0.3711 0.4481 602 7 620 24 000 - 32.5 393 5.70 11.60 0.3876 0.4642 604 7 315 23 000 - 30.6 410 5.95 12.11 0.4046 0.4806 607 7 010 22 000 - 28.6 428 6.21 12.64 0.4223 0.4976 609 6 706 21 000 - 26.6 446 6.47 13.18 0.4406 0.5150 611 6 401 20 000 - 24.6 466 6.75 13.75 0.4595 0.5328 614 6 096 19 000 - 22.6 485 7.04 14.34 0.4791 0.5511 616 5 791 18 000 - 20.7 506 7.34 14.94 0.4994 0.5699 619 5 406 17 000 - 18.7 527 7.65 15.57 0.5203 0.5892 621 5 182 16 000 - 16.7 549 7.97 16.22 0.5420 0.6090 624 4 877 15 000 - 14.7 572 8.29 16.89 0.5643 0.6292 626 4 572 14 000 - 12.7 595 8.63 17.58 0.5875 0.6500 628 4 267 13 000 - 10.8 619 8.99 18.29 0.6113 0.6713 631 3 962 12 000 - 8.8 644 9.35 19.03 0.6360 0.6932 633 3 658 11 000 - 6.8 670 9.72 19.79 0.6614 0.7156 636 3 353 10 000 - 4.8 697 10.10 20.58 0.6877 0.7385 638 3 048 9 000 - 2.8 724 10.51 21.39 0.7148 0.7620 640 2 743 8 000 - 0.8 753 10.92 22.22 0.7428 0.7860 643 2 438 7 000 + 1.1 782 11.34 23.09 0.7716 0.8106 645 2 134 6 000 + 3.1 812 11.78 23.98 0.8014 0.8359 647 1 829 5 000 + 5.1 843 12.23 24.90 0.8320 0.8617 650 1 524 4 000 + 7.1 875 12.69 25.84 0.8637 0.8881 652 1 219 3 000 + 9.1 908 13.17 26.82 0.8962 0.9151 654 914 2 000 + 11.0 942 13.67 27.82 0.9298 0.9428 656 610 1 000 + 13.0 977 14.17 28.86 0.9644 0.9711 659 305 0 + 15.0 1013 14.70 29.92 1.0000 1.0000 661 0 - 1 000 + 17.0 1050 15.23 31.02 1.0366 1.0295 664 - 305 Table A2: International Standard Atmosphere (ISA) 16 Getting to Grips with Aircraft Performance GENERAL 2. ALTIMETRY PRINCIPLES 2.1. General An altimeter (Figure A4) is a manometer, which is calibrated following standard pressure and temperature laws. The ambient atmospheric pressure is the only input parameter used by the altimeter. Zp PA PR ESSU RE ALTITU DE PA = f(P) Zp = f(P) ISA table PA Zp amb amb IA Zi PA Zp set set P 1013.25 (hPa) Pamb Pset Figure A3: Ambient Pressure and Pressure Setting Figure A4: Altimeter Function on PFD Assuming the conditions are standard, the “Indicated Altitude” (IA) is the vertical distance between the following two pressure surfaces (Figure A3): The pressure surface at which the ambient pressure is measured (actual aircraft’s location), and The reference pressure surface, corresponding to the pressure selected by the pilot through the altimeter’s pressure setting knob. IA = f(Pamb) - f(Pset) IA = PAamb - PAset 17 GENERAL Getting to Grips with Aircraft Performance 2.2. Definitions QNH QFE 1013 Radio (AAL) Flight Altitude height Height Level QFE setting QNH setting Standard setting: 1013.25 hPa Figure A5: QNH and Pressure Altitude The pressure setting and the indicated altitude move in the same direction: Any increase in the pressure setting leads to an increase in the corresponding Indicated Altitude (IA). The aim of altimetry is to ensure relevant margins, above ground and between aircraft. For that purpose, different operational pressure settings can be selected through the altimeter’s pressure setting knob (Figure A5): QFE is the pressure at the airport reference point. With the QFE setting, the altimeter indicates the altitude above the airport reference point (if the temperature is standard). Note: The QFE selection is often provided as an option on Airbus aircraft. QNH is the Mean Sea Level pressure. The QNH is calculated through the measurement of the pressure at the airport reference point moved to Mean Sea Level, assuming the standard pressure law. With the QNH setting, the altimeter indicates the altitude above Mean Sea Level (if temperature is standard). Consequently, at the airport level in ISA conditions, the altimeter indicates the topographic altitude of the terrain. Standard corresponds to 1013 hPa. With the standard setting, the altimeter indicates the altitude above the 1013 hPa isobaric surface (if temperature is standard). The aim is to provide a vertical separation between aircraft while getting rid of the local pressure variations throughout 18 Getting to Grips with Aircraft Performance GENERAL the flight. After takeoff, crossing a given altitude referred to as Transition Altitude, the standard setting is selected. The Flight Level corresponds to the Indicated Altitude in feet divided by 100, provided the standard setting is selected. The Transition Altitude is the indicated altitude above which the standard setting must be selected by the crew. The Transition Level is the first available flight level above the transition altitude. The change between the QNH setting and Standard setting occurs at the transition altitude when climbing, and at the transition level when descending (Figure A6). take off approach climb descent 1013 1013 transition transition altitude level QNH sea level QNH QNH 1013 hPa Figure A6: Transition Altitude and Transition Level The transition altitude is generally given on the Standard Instrument Departure (SID) charts, whereas the transition level is usually given by the Air Traffic Control (ATC). 19 GENERAL Getting to Grips with Aircraft Performance 2.3. Effects of Altimeter Setting and Temperature The true altitude of an aircraft is rarely the same as the indicated altitude, when the altimeter setting is 1013 hPa. This is mainly due to the fact that the pressure at sea level is generally different from 1013 hPa, and/or that the temperature is different from ISA. 2.3.1. Altimeter Setting Correction In case of ISA temperature conditions, and a standard altimetric setting, the aircraft true altitude can be obtained from the indicated altitude provided the local QNH is known. True altitude = Indicated altitude + 28 x (QNH [hPa] - 1013) 2.3.2. Temperature Correction Flying at a given indicated altitude, the true altitude increases with the temperature (Figure A7). The relationship between true altitude and indicated altitude can be approximated as follows: T TA = IA TISA TA = True altitude IA = Indicated altitude T = Actual temperature (in Kelvin) TISA = Standard temperature (in Kelvin) An example is provided in Appendix 1 of this manual. 20 Getting to Grips with Aircraft Performance GENERAL 1013 ISA+ ∆ISA TA > IA 1013 ISA TA = IA 1013 ISA - ∆ISA TA < IA At a constant Indicated Altitude (IA), the True Altitude (TA) Ê when the Static Air Temperature (SAT) Ê Figure A7: Temperature effect on True Altitude, for a constant Indicated Altitude Conclusion: If the temperature is higher, you fly higher. If the temperature is lower, you fly lower. Temperature correction is important, when flying a departure or arrival procedure in very low temperature conditions. For that purpose, the following table (Table A3) is proposed in the FCOM: 21 GENERAL Getting to Grips with Aircraft Performance Table A3: True Altitude Correction versus Temperature 22 Getting to Grips with Aircraft Performance GENERAL 3. OPERATING SPEEDS Different speed types are used to operate an aircraft. Some of them enable the crew to manage the flight while maintaining some margins from critical areas, whereas others are mainly used for navigational and performance optimization purposes. This is why the following sections propose a review of the different speed types that are used in aeronautics. 3.1. Calibrated Air Speed (CAS) The Calibrated Air Speed (CAS) is obtained from the difference between the total pressure (Pt) and the static pressure (Ps). This difference is called dynamic pressure (q). As the dynamic pressure cannot be measured directly, it is obtained thanks to two probes (Figure A8). q = Pt - Ps Static probes (Stby+ F/O+ Capt.) symmetrical on the other side, to avoid sideslip errors Pitots (Stby + Capt.) F/O on the other side Figure A8: Pitot Tube and Static Probes To obtain the total pressure Pt, airflow is stopped by means of a forward- facing tube, called the pitot tube (Figure A9), which measures the impact pressure. This pressure measurement accounts for the ambient pressure (static aspect) at the given flight altitude plus the aircraft motion (dynamic aspect). The static pressure Ps is measured by means of a series of symmetrical static probes perpendicular to the airflow. This measurement represents the ambient pressure at the given flight altitude (static aspect). CAS = f (Pt-Ps) = f (q) Flying at a constant CAS during a climb phase enables the aerodynamic effect to remain the same as at sea level and, consequently, to eliminate speed variations. 23 GENERAL Getting to Grips with Aircraft Performance Total pressure pick-off: Pt Static port Ps Pi Ps0 Air flow CAS Dynamic: q = Pt - PS Static: PS Figure A9: CAS Determination Process 3.2. Indicated Air Speed (IAS) The Indicated Air Speed (IAS) is the speed indicated by the airspeed indicator. Whatever the flight conditions, if the pressure measurement were accurate, then the IAS should ideally be equal to the CAS. Nevertheless, depending on the aircraft angle of attack, the flaps configuration, the ground proximity (ground effect or not), the wind direction and other influent parameters, some measurement errors are introduced, mainly on the static pressure. This leads to a small difference between the CAS and the IAS values. This difference is called instrumental correction or antenna error (Ki). IAS = CAS + Ki 3.3. True Air Speed (TAS) An aircraft in flight moves in an air mass, which is itself in motion compared to the earth. The True Air Speed (TAS) represents the aircraft speed in a moving reference system linked to this air mass, or simply the aircraft speed in the airflow. It can be obtained from the CAS, using the air density (ρ ρ) and a compressibility correction (K). TAS = ( ρo/ρ ) K CAS 3.4. Ground Speed (GS) The ground speed (GS) represents the aircraft speed in a fixed ground reference system. It is equal to the TAS corrected for the wind component (Figure A10). Ground Speed = True Air Speed + Wind Component 24 Getting to Grips with Aircraft Performance GENERAL Wind TAS GS DA GS = Ground Speed DA = Drift Angle TAS = True Air Speed Figure A10: Ground Speed and Drift Angle 3.5. Mach Number The Mach Number is a comparison between the TAS and the speed of sound. TAS M= a With TAS = True Air Speed a = The speed of sound at the flight altitude The speed of sound in knots is: a(kt) = 39 SAT(K) With SAT = Static Air Temperature (ambient temperature)in Kelvin The speed of sound is solely dependent on temperature. Consequently, the Mach number can be expressed as follows: TAS (kt) M= 39 273 + SAT( ° C) Flying at a given Mach number in the troposphere: When the pressure altitude increases, the SAT decreases and thus the True Air Speed (TAS). Or : higher ⇒ slower 25 GENERAL Getting to Grips with Aircraft Performance Pt and Ps, respectively measured by the aircraft pitot tube and static probes, are also used to compute the Mach number. Therefore,  P − Ps  q M = f  t  = f    Ps   Ps  The TAS indicated on the navigation display of modern aircraft is then obtained from the Mach number: TAS ( Kt ) = 39M 273 + SAT (º C ) 3.6. True Air Speed (TAS) Variations FL 450 400 iso Mach 0.78 tropopause 350 300 Cross-over altitude 250 200 150 100 50 iso CAS 300 TAS (kt) 200 250 300 350 400 450 500 Figure A11: True Air Speed Variations – Climb profile 300 Kt / M0.78 The above graph (Figure A11) illustrates the TAS variations as a function of the pressure altitude for a climb at constant CAS (300 knots) and constant Mach (M0.78). The altitude at which a given CAS is equal to a given Mach number is called the cross-over altitude. 26 Getting to Grips with Aircraft Performance GENERAL 4. FLIGHT MECHANICS For a flight at constant speed in level flight, the drag force must balance the engine thrust. As a general rule, when engine thrust is higher than drag, the aircraft can use this excess thrust to accelerate and/or climb. On the other hand, when the thrust is insufficient to compensate for drag, the aircraft is forced to decelerate and/or descend. In flight, four forces are applied to an aircraft : Thrust, drag, lift and weight. If the aircraft is in steady level flight, the following balance is obtained (Figure A12): The thrust for steady level flight (T) is equal to drag (D = ½ ρ S V2 CD), Weight (mg) is equal to lift (L = ½ ρ S V2 CL). lift thrust drag weight = mg Figure A12: Balance of Forces for Steady Level Flight 4.1.1.1. Standard Lift Equation Weight = mg = ½ ρ S (TAS)2 CL (1) With m = Aircraft mass g = Gravitational acceleration ρ = Air density S = Wing area CL = lift coefficient The lift coefficient, CL, is a function of the angle of attack (α), the Mach number (M), and the aircraft configuration. 27 GENERAL Getting to Grips with Aircraft Performance 4.1.1.2. Standard Drag Equation Thrust = ½ ρ S (TAS)2 CD (2) With CD = Drag coefficient The drag coefficient, CD, is a function of the angle of attack (α), the Mach number (M) and the aircraft configuration. 4.1.1.3. Other Formulas As a function of the Mach number: Lift and drag equations may be expressed with the Mach number M. As a result, the equations are: Weight = 0.7 PS S M2 CL (3) Thrust = 0.7 PS S M2 CD (4) With Ps = Static Pressure As a function of P0: The pressure ratio δ is introduced into the lift and drag equations: Ps δ = (5) P0 With P0 = Pressure at Sea Level Ps = Pressure at Flight Level Therefore, the following equations are independent of pressure altitude: Weight = 0.7 P0 S M 2 C L (6) δ Thrust (7) = 0.7 P0 S M 2 C D δ 28